
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/UF00075325/00001
Material Information
 Title:
 Winddriven circulation in Lake Okeechobee, Florida the effects of thermal stratification and aquatic vegetation
 Creator:
 Lee, Hye Keun, 1955 ( Dissertant )
Sheng, Y. Peter ( Thesis advisor )
Dean, Robert G. ( Reviewer )
Sheppard, Donald M. ( Reviewer )
Kurzweg, Ulrich H. ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1993
 Copyright Date:
 1993
 Language:
 English
 Physical Description:
 xvi, 202 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Heat flux ( jstor )
Lakes ( jstor ) Modeling ( jstor ) Shengs ( jstor ) Temperature effects ( jstor ) Three dimensional modeling ( jstor ) Turbulence models ( jstor ) Vegetation ( jstor ) Velocity ( jstor ) Water temperature ( jstor ) Coastal and Oceanographic Engineering thesis Ph. D Dissertations, Academic  Coastal and Oceanographic Engineering  UF Lake Okeechobee ( local )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
 Spatial Coverage:
 United States  Florida  Lake Okeechobee
Notes
 Abstract:
 Winddriven circulation in Lake Okeechobee, Florida, is simulated by using a threedimensional curvilineargrid hydrodynamic model and measured field data. Field data show that significant thermal stratification often develops in the vertical water column during daytime in the large and shallow lake. Significant wind mixing due to the lake breeze, however, generally leads to destratification of the water column in the late afternoon and throughout the night. Thus, thermal effect must be considered in the numerical simulation of circulation in shallow lakes. During daytime the lake thermally stratified and wind is relatively weak, momentum transfer is generally limited to the upper layer and hence the bottom currents are much weaker than the surface currents. During the initial phase of significant lake breeze, strong surface currents and opposing bottom currents are developed, followed by oscillatory motions associated with seiche and internal seiche, until they are damped by bottom friction. Lake Okeechobee is covered with submerged and emergent aquatic vegetation over much of the bottom on the western portion of the lake (20% of the surface area). The presence of the vegetation causes damping of the wind, wave and current fields. To provide realistic simulation of winddriven circulation in the presence of vegetation, this study developed a simplified vegetation model which parameterizes the effect of vegetation in terms of added "form drag" terms in the momentum equations. Simulated currents in the open water region in the vicinity of vegetation compare quite well with data. This physical process is successfully modeled by parameterizing the vertical turbulence with a simplified secondorder closure model. Model simulation which assume homogeneous density structure fails to represent the stratification and destratification cycle. On the other hand, simulation which includes thermal effect faithfully reproduced field data.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1993.
 Bibliography:
 Includes bibliographical references (leaves 196201).
 General Note:
 Typescript.
 General Note:
 Vita.
 General Note:
 UFL/COELTR/103
 Funding:
 Technical report (University of Florida. Coastal and Oceanographic Engineering Dept.)
 Statement of Responsibility:
 by Hye Keun Lee.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact Digital Services (UFDC@uflib.ufl.edu) with any additional information they can provide.
 Resource Identifier:
 030333143 ( ALEPH )
30811771 ( OCLC ) AKB0129 ( NOTIS )

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UFL/COELTR/103
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE, FLORIDA: THE EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
by
Hye Keun Lee
Dissertation
1993
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE, FLORIDA: THE EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
By
HYE KEUN LEE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor Dr. Y. Peter Sheng for his continuous guidance, encourgement and financial support throughout my study. I would also like to extend my thanks and appreciation to my doctoral committee members, Dr. Robert G. Dean, Dr. Donald M. Sheppard and Dr. Ulrich H. Kurzweg, for their patience in reviewing this dissertation. My gratitude also extends to Dr. Robert J. Thieke who reviewed my dissertation.
I must thank Dr. Paul W. Chun for reviewing my dissertation and the
great guidance during my stay in Gainesville while he served as a faculty advisor of the Korean Student Association.
Financial support provided by the South Florida Water Management District, West Palm Beach, Florida, through the Lake Okeechobee Phosphorus Dynamics Project is appreciated.
I would like to dedicate this dissertation to my late father and my mother.
Finally, I would like to thank my loving wife, Aesook, for her support and patience, and my beautiful daughter, Mireong, and my smart son, David.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS................................. i
LIST OF FIGURES..................
LIST OF TABLES...................
. vi
... xiii
ABSTRACT..........................................
CHAPTERS
1 INTRODUCTION....................................
2 LITERATURE REVIEW................................
2.1 Numerical Models of Lake Circulation.....................
2.1.1 OneDimensional Model.........................
2.1.2 TwoDimensional Model........................
2.1.3 SteadyState 3D Models........................
2.1.4 TimeDependent 3D Models......................
2.2 Vegetation Models.................................
2.3 Thermal Models...................................
2.4 Turbulence Model..................................
2.4.1 Eddy Viscosity/Diffusivity Concept.................
2.4.2 Constant Eddy Viscosity/ Diffusivity Model.............
2.4.3 MunkAnderson Type Model......................
2.4.4 Reynolds Stress Model..........................
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure
Model ................ .
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model. .. .. .
2.4.7 OneEquation Model (k Model)... .. .. .. .. .. .. .. ..
2.4.8 TwoEquation Model (k f Model).. .. .. .. .. .. ....
2.5 Previous Lake Okeechobee Studies... .. .. .. .. .. .. .. .. ...
2.6 Present Study............ .. .. .. .. .. .. .. .. .. .. ..
3 GOVERNING EQUATIONS.... .. .. .. .. .. ..
3.1 Introduction... .. .. .. .. .. .. .. .. ....
3.2 Dimensional Equations and Boundary Conditions
ordinate System... .. .. .. .. .. .. .. .. ..
3.2.1 Equation of Motion.. .. .. .. .. .. ..
3.2.2 FreeSurface Boundary Condition (z = 77) 3.2.3 Bottom Boundary Condition (z = h) .
3.2.4 Lateral Boundary Condition. .. .. .. .
3.3 Vertical Grid .......................... I
3.4 NonDimensionalization of Equations .. .. .. .
in a Cartesian Co
3.5 Dimensionless Equations in oStretched Cartesian Grid . 27
3.5.1 Vertically Integrated Equations . . 28
3.5.2 Vertical Velocities . . . 29
3.6 Generation of Numerical Grid . . 29
3.6.1 Cartesian Grid . . . 29
3.6.2 Curvilinear Grid . . . 29
3.6.3 Numerical Grid Generation . . 30
3.7 Transformation Rules 31
3.8 Tensor Invariant Governing Equations . . 34
3.9 Dimensionless Equations in BoundaryFitted Grids . 36
3.10 Boundary Conditions and Initial Conditions . 37
3.10.1 Vertical Boundary Conditions . . 37
3.10.2 Lateral Boundary Conditions . . 37
3.10.3 Initial Conditions . . . 37
4 VEGETATION MODEL . . . 39
4.1 Introduction : . 39
4.2 Governing Equations . . . 41
4.2.1 Equations for the Vegetation Layer (Layer 1) . 41
4.2.2 Equations for the VegetationFree Layer (Layer 11) 43
4.2.3 Equations for the Entire Water Column . 44
4.2.4 Dimensionless Equations in Curvilinear Grids . 45
5 HEAT FLUX MODEL . . . 48
5.1 Introduction . . . . 48
5.2 The "Equilibrium Temperature" Method . . 48
5.2.1 ShortWave Solar Radiation . . 49
5.2.2 LongWave Solar Radiation . . 49
5.2.3 Reflected Solar and Atmospheric Radiation . 49
5.2.4 Back Radiation . . . 51
5.2.5 Evaporation . . . 51
5.2.6 Conduction . . . 51
5.2.7 Equilibrium Temperature . . 52
5.2.8 Linear Assumption 52
5.2.9 Procedure for an Estimation of K and T . 53
5.2.10 Modification of the Equilibrium Temperature Method 54
5.3 The "Inverse" Method . . . 54
5.3.1 Governing Equations . . 55
5.3.2 Boundary Conditions . . 55
5.3.3 Finite Difference Equation . . 56
5.3.4 Procedure for an Estimation of Total Heat Flux . 57
6 FINITEDIFFERENCE EQUATIONS . . 58
6.1 Grid System . . . . 58
6.2 External Mode . . . 58
6.3 Internal Mode . . . 61
6.4 Temperature Scheme . . . 62
6.4.1 Advection Terms . . . 63
6.4.2 Horizontal Diffusion Term . . 66
7 MODEL ANALYTICAL TEST...............
7.1 Seiche Test.........................
7.2 Steady State Test.....................
7.3 Effect of Vegetation...................
7.4 Thermal Model Test.... .. .. .. .. .. .. ..
8 MODEL APPLICATION TO LAKE OKEECHOBEE
8.1 Introduction... .. .. .. .. .. .. .. .. ....
8.1.1 Geometry.... .. .. .. .. .. .. .. ...
8.1.2 Temperature.... .. .. .. .. .. .. .. .
8.2 Some Recent Hydrodynamic Data. .. .. .. ..
8.2.1 Wind Data..... .. .. .. .. .. .. .. .
8.2.2 Current Data.... .. .. .. .. .. .. ..
8.2.3 Temperature Data... .. .. .. .. ....
8.2.4 Vegetation Data.. .. .. .. .. .. ....
8.3 Model Setup... .. .. .. .. .. .. .. .. ....
8.3.1 Grid Generation.. .. .. .. .. .. ....
8.3.2 Generation of Bathymetry Array .. .. .
8.4 Model Parameters.... .. .. .. .. .. .. .. ..
8.4.1 Reference Values... .. .. .. .. .. .. .
8.4.2 Turbulence Model and Parameters ....
8.4.3 Bottom Friction Parameters .. .. .. ..
8.4.4 Vegetation Parameters.. .. .. .. .. ..
8.4.5 Wind Stress... .. .. .. .. .. .. .. ..
8.5 Steady State WindDriven Circulation. .. .. .
8.6 WindDriven Circulation without Thermal Stratification. .. .. ..
8.6.1 Tests of Model Performance.... .. .. .. .. .. .. .. ....
8.6.2 Model Results....... .. .. .. .. .. .. .. .. .. .. ....
8.7 WindDriven Circulation with Thermal Stratification: T, Method..
8.8 Simulation of Currents with Thermal Stratification: Inverse Method.
8.8.1 The Diurnal Thermal Cycle.... .. .. .. .. .. .. .. ....
8.9 Sensitivity Tests.... ... .. .. .. .. .. .. .. .. .. .. .. ..
8.9.1 Effect of Bottom Stress... .. .. .. .. .. .. .. .. .. .. .
8.9.2 Effect of Horizontal Diffusion Coefficent.... .. .. .. .. ..
8.9.3 Effect of Different Turbulence Model.... .. .. .. .. .. ..
8.9.4 Effect of Advection Term... .. .. .. .. .. .. .. .. ....
8.10 Spectral Analysis..... .... .. .. .. .. .. .. .. .. .. .. .
9 CONCLUSION..... ... .... .. .. .. .. .. .. .. .. .. .. ..
APPENDIX
A SIMULATED CURRENTS BY IN VERSE METHOD...........
BIBLIOGRAPHY..... .. ... .. .. .. .. .. .. .. .. .. .. .. ..
BIOGRAPHICAL SKETCH.......... .. .. .. .. .. .. .. .. .. .. .
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
76 76 76 76 76 77 81 81 83 83 83
84 88 88 89 91 92 96 98 99 99 99 123 150 150 156 167 168 168 168 169
174
LIST OF FIGURES
3.1 A computational domain and a transformed coordinate system.
4.1 Schematics of flow in vegetation zone... .. .. .. .. .. .. ..
5.1 Meteorological data at Station L006.. .. .. .. .. .. .. ....
6.1 Horizontal and vertical grid system.. .. .. .. .. .. .. ....
7.1 Model results of a seiche test.... .. .. .. .. .. .. .. .. .. .
7.2 Surface elevation contour when the lake is steady state with uniform wind stress of 1 dyne/cm2..... .. .. .. .. .. .. .. .. .
7.3 Effect of vegetation on surface elevation evolution in a winddriven
rectangular lake. Solid line is without vegetation, broken line is with low vegetation density, and dotted line is with high vegetation density..... .. .. .. .. .. .. .. .. .. .. .. .. .. ..
7.4 Time history of wind stress and currents at the center
all five levels. Thermal stratification is not considered. 7.5 Time history of wind stress and currents at the center
all five levels. Thermal stratification is considered.
8.1 Map of Lake Okeechobee.. .. .. .. .. .. .. ....
8.2 Wind rose at Station C... .. .. .. .. .. .. .. ..
8.3 Computation domain of Lake Okeechobee .. .. .. ..
8.4 Curvilinear grid of Lake Okeechobee .. .. .. .. .. .
8.5 Depth contour of Lake Okeechobee when the lake stage
Unit in cm..... .. .. .. .. .. .. .. .. .. .. ..
8.6 Distribution of vegetation height in Lake Okeechobee.
8.7 Distribution of vegetation density in Lake Okeechobee.
of lake at of lake at
. 75
. 78
. 82
. 85
. 86
is 15.5 ft.
. 87
. 95
. 97
Steadystate depthintegrated currents (CM 2s_') in Lake Okeechobee forced by an easterly wind of 1 dyne/cm2 .. .. .. .. ..
S.9 Steadystate surface elevation contour (cm) in Lake Okeechobee
forced by an easterly wind of I dyne/ CM2 . 101
8.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation . 104
8.11 Simulated ((Solid lines) and measured (dotted lines) currents at
Station C Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation . 105
8.12 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation . 106
8.13 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation . 107
8.14 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction). Thermal stratification
was not considered in model simulation . 108
8.15 Simulated ((Solid lines) and measured (dotted lines) currents at
Station C Arm 3: NorthSouth direction) . 109
8.16 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation . 110
8.17 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation' . .
8.18 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation . 112
8.19 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation . 113
8.20 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation . 116
8.21 Simulated solid lines) and measured (dotted fines) currents at
Station E ( rm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation . 117
8.22 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation . 118
8.23 Simulated ((Solid lines) and measured (dotted lines) currents at
Station E Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation . 119
8.24 Stick Diagram of wind stress, measured currents, and sirnualted
currents at Station E . . . 120
8.25 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation . 121
8.26 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation . 122
8.27 Simulated (solid lines) and measured dottedd lines) currents at
Station D (Arm 1: EastWest direction) . 124
8.28 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction) . 125
8.29 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction) . 126
8.30 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) . 127
8.31 Simulated (solid lines) and measured (dotted fines) currents at
Station A (Arm 1: EastWest direction) when thermal effect is
considered 129
8.32 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction) when thermal effect is
considered 130
8.33 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: EastWest direction) when thermal effect is
considered 131
8.34 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction) when thermal effect is
considered I 132
8.35 Simulated solid lines) and measured (dotted lines) currents at
Station B Arm 2: EastWest direction) when thermal effect is
considered 133
8.36 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction) when thermal effect is
considered 134
8.37 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction) when thermal effect is
considered 135
8.38 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction) when thermal effect is
considered 136
8.39 Simulated solid lines) and measured (dotted lines) currents at
Station C Arm 2: EastWest direction) when thermal effect is
considered 137
8.40 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction) when thermal effect is
considered 138
8.41 Simulated (solid lines) and measured (dotted fines) currents at
Station C (Arm 3: EastWest direction) when thermal effect is
considered 139
8.42 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction) when thermal effect is
considered 140
8.43 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction) when thermal effect is
considered 141
8.44 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction) when thermal effect is
considered 142
8.45 Simulated solid lines) and measured (dotted lines) currents at
Station D Arm 2: EastWest direction) when thermal effect is
considered 143
8.46 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) when thermal effect is
considered 144
8.47 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction) when thermal effect is
considered 145
8.48 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction) when thermal effect is
considered 146
8.49 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction) when thermal effect is
considered 147
8.50 Simulated ((Solid lines) and measured (dotted lines) currents at
Station E Arm 2: NorthSouth direction) when thermal effect is
considered 148
8.51 Time history of eddy viscosity at Station C between Julian days
147 and 161 151
8.52 Time history of wind stress and measured currents between Julian
days 150 and 152 152
8.53 Time history of simulated currents *at Station C between Julian
days 150 and 152 153
8.54 Time history of eddy viscosity at Station C between Julian days
150 and 152 154
8.55 Time history of heat fluxes at Station C between Julian days 147
and 161 155
8.56 Temperature contours of data and model at Station C between
Julian days 152 and 155 . . . 157
8.57 Simulated and measured temperatures at Station A . 158
8.58 Simulated and measured temperatures at Station B . 159
8.59 Simulated and measured temperatures at Station C. 160
8.60 Simulated and measured temperatures at Station D . 161
8.61 Simulated and measured temperatures at Station E . 162
8.62 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 'C . . 163
8.63 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 'C . . 164
8.64 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 OC . . 165
8.65 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 'C . . 166
8.66 Spectrum of wind stress and surface elevation . 171
8.67 Spectrum of measured and simulated currents (eastwest direction) at Station C 172
8.68 Spectrum of measured and simulated currents (northsouth direction) at Station C 173
A. 1 Simulated solid lines) and measured (dotted lines) currents at
Station A Arrn 1: EastWest direction). Inverse method was
used for the estimation of heat flux . . 177
A.2 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 178
A.3 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux . . 179
A.4 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 180
A.5 Simulated solid lines) and measured (dotted lines) currents at
Station B Arrn 2: EastWest direction). Inverse method was
used for the estimation of heat flux . . 181
A.6 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 182
A.7 Simulated solid lines) and measured (dotted lines) currents at
Station C Arrn 1: EastWest direction). Inverse method was
used for the estimation of heat flux . . 183
A.8 Simulated (solid lines) and measured (dotted fines) currents at
Station C (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 184
A.9 Simulated solid lines) and measured (dotted lines) currents at
Station C Arrn 2: EastWest direction). Inverse method was
used for the estimation of heat flux . . 185
A.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 186
A.11 Simulated solid lines) and measured (dotted lines) currents at
Station C Arrn 3: EastWest direction). Inverse method was
used for the estimation of heat flux . . 187
A. 12 Simulated (solid lines) and measured (dotted fines) currents at
Station C (Arm 3: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 188
A.13 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux . . 189
A. 14 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 190
A. 15 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux . . 191
A. 16 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 192
A.17 Simulated solid lines) and measured (dotted lines) currents at
Station E Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux . . 193
A. 18 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 194
A. 19 Simulated solid lines) and measured (dotted lines) currents at
Station E Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux . . 195
A.20 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux . . 196
LIST OF TABLES
Selected features of lake models .. .. .. .. ... ... ... ....21
Application features of lake models. .. .. .. .. ... ... ....22
Installation dates and locations of platforms during 1988 and 1989. 79 Instrument mounting, spring deployment. .. .. .. .. ... ....80
Reference values used in the Lake Okeechobee spring 1989 circulation simulation. .. .. .. .. .. ... ... .... ... ... ..89
Vertical turbulence parameters used in the Lake Okeechobee spring 1989 circulation simulation. .. .. .. .. ... ... ... ... ..92
Vegetations in Lake Okeechobee (From Richardson, 1991).. .. ...94 Index of agreement and RMS error at Station C .. .. .. .. ....103
Index of agreement and RMS error at Station B. Index of agreement and RMS error at Station E. Index of agreement and RMS error at Station A. Index of agreement and RMS error at Station D. Index of agreement and RMS error at Station A effect is considered... .. .. .. .. .. .. .. ..
Index of agreement and RMS error at Station B effect is considered... .. .. .. .. .. .. .. ..
............. .... ... ..114
............. ..... ... ..115
............. ..... ... ..123
............. ..... ... ..123
when thermal when thermal
Index of agreement and RMS error at Station C when thermal effect is considered..... .. .. .. .. .. .. .. .. .. .. .. ..
Index of agreement and RMS error at Station D when thermal effect is considered..... .. .. .. .. .. .. .. .. .. .. .. ..
Index of agreement and RMS error at Station E when thermal effect is considered..... .. .. .. .. .. .. .. .. .. .. .. ..
Parameters used in sensitivity tests.. .. .. .. .. .. .. ....
149 149 149 149 149 167
8.7 8.8 8.9
8.10 8.11 8.12 8.13
8.14 8.15 8.16
8.17 Index of agreement and RMS error ..................... 170
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE, FLORIDA: THE EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION By
HYE KEUN LEE
August 1993
Chairman: Dr. Y.P. Sheng
Major Department: Coastal and Oceanographic Engineering
Winddriven circulation in Lake Okeechobee, Florida, is simulated by using a threedimensional curvilineargrid hydrodynamic model and measured field data. Field data show that significant thermal stratification often develops in the vertical water colun during daytime in the large and shallow lake. Significant wind mixing due to the lake breeze, however, generally leads to destratification of the water column in the late afternoon and throughout the night. Thus, thermal effects must be considered in the numerical simulation of circulation in shallow lakes.
During daytime the lake is thermally stratified and wind is relatively weak, momentum transfer is generally limited to the upper layer and hence the bottom currents are much weaker than the surface currents. During the initial phase of significant lake breeze, strong surface currents and opposing bottom currents are developed, followed by oscillatory motions associated with seiche and internal seiche, until they are damped by bottom friction.
Lake Okeechobee is covered with submerged and emergent aquatic vegetation over much of the bottom on the western portion of the lake (20 % of the surface area). The presence of the vegetation causes damping of the wind, wave and current fields. To
provide realistic simulation of winddriven circulation in the presence of vegetation, this study developed a simplified vegetation model which parameterizes the effect of vegetation in terms of added "form drag" terms in the momentum equations. Simulated currents in the open water region in the vicinity of vegetation compare quite well with data. This physical process is successfully modeled by parameterizing the vertical turbulence with a simplified secondorder closure model. Model simulation which assumes homogeneous density structure fails to represent the stratification and destratification cycle. On the other hand, simulation which includes thermal effects faithfully reproduced field data.
CHAPTER 1
INTRODUCTION
Lakes are valuable resource for a variety of human needs: drinking water, agricultural use, navigation, waste water disposal, recreation sites, cooling reservoirs for power plants, etc. Ninetynine percent of Americans live within 50 miles of one of 37,000 lakes (Hanmer, 1984). Lakes can be dangerous to people under certain circumstances such as during flooding and the deterioration of water quality due to the excessive loading of contaminants into the lake.
Water movement in lakes is driven by wind, density gradient, waves, and tributary flow, but primarily is influenced by wind action. During periods of strong wind, severe flooding can be caused by the storm surge. Some examples are the flooding in the Lake Okeechobee area during the 1926 and 1928 hurricanes. The 1926 hurricane caused a storm surge of 7 ft at Moorehaven on the western side of Lake Okeechobee, and about 150 people lost their lives (Helistrom, 1941).
Water quality of lakes is of utmost importance. So long as human activities are limited to a small part of a lake, it may appear that the lake has an unlimited capacity of selfpurification. However, as population and human development increase, a lake may not be able to endure the excessive stresses caused by human actions, and water quality may become deteriorated. Typical evidence of poor water quality includes sudden algal bloom, colored water, fish kill, taste and odor in drinking water, and floating debris of plants. Eutrophication is the process in which excessive loading of nutrients, organic matter, and sediments into lakes results in an increase of primary production. Sources of eutrophication are increased use of fertilizer, waste water discharge, and precipitation of polluted air.
2
Earlier studies on physical processes in lakes concentrated on the observations of periodic up and down motion of water level, i.e., seiche motion. When the wind blows over a certain period, water builds up near the shoreline. After wind ceases, water starts to oscillate as a free long wave. Subsequently, this wave is damped out due to bottom friction. Seiche can be initiated by sudden change of wind speed or direction. the passage of a squall line, an earthquake, or resonance of air and water columns. When the velocity of a squall line is close to the speed of gravity wave, resonance occurs and damages can be more severe. It was reported that severe storm damages in Chicago were caused by a squall line over Lake Michigan on June 26, 1954 (Harris, 1957).
Another important feature in deep temperate lakes is the temperature variation over the depth, which is called thermal stratification. Starting in the early spring, the lake attains a temperature of 4 'C and is more or less isothermal. During the summer season, the surface water starts to become warmer because of increased solar radiation, so that, gradually, a sharp temperature gradient, i.e., thermocline, is formed. The lake remains thermally stratified during summer with a warm surface layer (epilimnion), a thermocline, and a cold layer (hypolimnion). Though strong wind action tends to lower or break the thermocline, the lake generally remains stratified during the entire summer season. During the fall, as the air temperature drops, the net daily heat flux at the water surface becomes negative, i.e., the lake loses heat daily. Hence, water density in the epilimnion often becomes heavier than that in the hypolimnion and causes convective mixing which, in combination with strong wind action, causes the lake to become isothermal again in the winter. This process repeats itself annually. It is important to know the location of thermocline at different times of the year so that water can be withdrawn to a desirable height in deep lakes or reservoirs for various agricultural and municipal uses.
Many processes are influenced by currents and temperature in lakes. For example,
3
the growth rate of all organic matter in lakes is governed by temperature (Goldman, 1979). The growth rate generally increases between some minimum temperature and an optimum temperature, and decreases until it reaches maximum temperature. Cooling water from power plants is mixed with surrounding water by the currents and turbulent mixing which depend on the temperature field as well. Thus, predicting currents and temperature are essential to understanding the transport of various matters and their effects on the ecology.
In the early days, simple analytical models were used to study physical processes in simplified conditions. For example, a setup equation was used to predict the storm surge height (Hellstrom, 1941). Since analytical models could not realistically consider such effects as advection, complex geometry, and topography, they had been applied to limited problems to understand certain basic processes.
Numerical models are valuable tools for simulating and understanding water movement in lakes. Once a rigorously developed model is calibrated with measured data, it can be used to estimate the flow near a manmade structure or to predict the movement of contaminants including oil spill, sediments, etc. During the 1970s and 1980s, vertically averaged twodimensional numerical models, which can compute only the depth average currents and surface elevations, were widely used because they were simple and needed little computer time. However, since they could not give accurate results for cases where the vertical distributions of currents and temperature are required, threedimensional models are needed.
Numerical modeling requires the discretization of the computation domain. Past numerical models which were developed during 1970s generally used a rectangular grid (for example, Sheng, 1975). However, to represent the complex geometry such as the shoreline and the boundary between the vegetation zone and the open water in Lake Okeechobee, a very fine rectangular grid is required. On the other hand, boundaryfitted grids can be and have been used in recent models (for example, Sheng, 1987)
4
to represent the complex geometry with a relatively smaller number of grid points.
In some shallow lakes, aquatic vegetation can grow over large areas. The vegetation can affect the circulation significantly because it introduces additional friction on the flowing water. For example, Lake Okeechobee has vegetation over an area which covers 25% of the total lake surface. Because vegetation consists of stalks with different heights and diameters, a representative diameter and height over each discretized grid cell must be introduced in the model. Additional drag terms must be introduced in the momentum equations to represent the form drag introduced by the vegetation. The consideration of vegetation is necessary to compute the flow and transport of phosphorus between the vegetation area and the open water.
Previously developed numerical models which were applied to deep lakes, e.g., the Great Lakes, cannot be readily applied to shallow lakes such as Lake Okeechobee, since many shallow water processes are not included in these models.
Since 1988, with funding from the South Florida Water Management District and U.S. Environmental Protection Agency, the Coastal and Oceanographic Engineering Department of the University of Florida (under the supervision of Dr. Y. Peter Sheng) has conducted a major study on the hydrodynamics and sediment dynamics and their effects on phosphorus dynamics in Lake Okeechobee. The primary purpose of the study was to quantify the role of hydrodynamics and sediments on the internal loading of phosphorus and the exchange of phosphorus between vegetation zone and open water. As part of the study, field data (wind, air temperature, wave, water current, water temperature, and suspended sediment concentration) were collected over two onemonth periods in 1988 and 1989. Ahn and Sheng (1989) studied the wind waves of Lake Okeechobee. Cook and Sheng (1989) studied the sediment dynamics in Lake Okeechobee. This study focuses on the influences of vegetation and thermal stratification on lake circulation. The objectives of this study are
(1) to obtain a general insight in the winddriven circulation in Lake Okeechobee,
5
(2) to develop a numerical model which can simulate the effect of vegetation on
Lake Okeechobee circulation,
to develop a numerical model which can simulate the thermal stratification and
its effect on circulation, and
(4) to determine the important factors for producing successful simulation of circulation in large shallow lakes.
The literature review will be presented in the Chapter 2, after which the formulation of the threedimensional model will be given in Chapter 3. A vegetation model will be explained in Chapter 4, and a thermal model follows in Chapter 5. Finite difference formulation will be presented in Chapter 6. After the model test in Chapter 7, application to Lake Okeechobee will be described in Chapter 8. Finally, a conclusion will be given in Chapter 9.
CHAPTER 2
LITERATURE REVIEW
2.1 Numerical Models of Lake Circulation Sheng (1986) reviewed numerous models and suggested that models could be classified according to numerical features (dimensionality, horizontal grid, vertical grid, numerical scheme, etc.) and physical features (forcing function, free surface dynamics, spatial scale, turbulence models, etc.). As an example, some models are described in the following.
2.1.1 OneDimensional Model
Onedimensional (1D) models include a single spatial coordinate (longitudinal or vertical). When the lake is elongated and wellmixed in the directions perpendicular to the longitudinal axis, a longitudinal 1D model can be used. A longitudinal 1D system of equations is derived by integrating the continuity and momentum equations over the cross section. Sheng et al. (1990) developed a longitudinal onedimensional model of Indian River Lagoon. Sheng and Chiu (1986) developed a vertical onedimensional model for a location in Atlantic Ocean.
2.1.2 TwoDimensional Model
Twodimensional (2D) models include horizontal 2D models, which assume vertical homogeneity, and vertical 2D models, which assume transverse homogeneity. The equations of motion are obtained by performing integration or averaging in the vertical or transverse directions.
Hsueh and Peng (1973) studied the steadystate verti cally averaged circulation in a rectangular bay by solving a Poisson equation with the method of successive
7
over relaxation (SOR). Their model included the terms of bottom friction, advection, bottom topography, and lateral diffusion, while assuming steady state and homogeniety in density. The specification of the vertical eddy viscosity is not required in the twodimensional model but can give only depthaveraged velocities.
Shanahan and Harleman (1982) developed a transient 2D model which assumed vertical homogeneity. When the lakes are long, deep but relatively narrow, laterally averaged 2D model can be applied (for example, Edinger and Buchak, 1979).
2.1.3 SteadyState 3D Models
An early study on winddriven circulation was conducted by Ekman (1923) who solved momentum equations analytically while neglecting the nonlinear terms. Welander (1957) developed a theory on winddriven currents based on an extension of Ekman's theory. After neglecting inertia terms and horizontal diffusion terms, steadystate momentum equations were combined with the introduction of complex variables. After applying boundary conditions, a solution was obtained in terms of the imposed wind stress and unknown pressure gradient term. By introducing the stream functions for verticallyintegrated flow, the continuity equation could be satisfied unconditionally. The final equation to be solved was reduced to a secondorder partial differential equation for stream function, 0, as follows: V2' = ao + boy + c (2.1)
Ox 'ay
Once 0 is found, the currents can be found by taking the derivatives, 2 and 0 ax ay,
Gedney and Lick (1972) and Sheng and Lick (1972) applied Welander's theory to Lake Erie. The equation for stream function was solved by the successive over relaxation method. The agreement between the field data and model results was good. Eddy viscosity was assumed to be constant but varies with wind speed.
Thomas (1975) used a depthvarying form of vertical eddy viscosity as follows: V= V0(1 + =ku.(h + z) (2.2)
8
where v is eddy viscosity, v0 is eddy viscosity near surface, and k is a constant (0.4).
2.1.4 TimeDependent 3D Models
ModeSplitting
In order to solve the dependent variables with the unsteady threedimensional model, Simons (1974) used a socalled "mode splitting" method for Lake Ontario while Sheng et al. (1978) used a somewhat different method for Lake Erie. Defining the perturbation velocity fi = u U, i = v U where U, U are depthaveraged velocities, and u,v are instantaneous velocities, Sheng and Butler (1982) derived governing equations for i, by subtracting the verticallyaveraged equations from the momentum equations. Therefore the solution procedure consists of an external mode, which includes the surface elevation and U and U, and an internal mode, which includes ii,i3 and temperature.
Time Integration of 2D equations
Time integration is important for improving the efficiency of numerical models. When the explicit method is used, the time step is limited by the Courant condition, which is C 4t < 1. Therefore, explicit method is not desirable for longterm simulations. Leendertse (1967) used the ADI (Alternate Direction Implicit) method to simulate tidal currents in the southern North Sea. All terms in the continuity equation and pressure terms in the momentum equation were treated implicitly, while the other terms were expressed explicitly. After factorization of the finitedifference equations, the resulting unknowns are solved by inversion of tridiagonal matrices in the x sweep and y sweep.
Vertical Grid
Various types of vertical grids are used in numerical models of lake circulation. The earlier models generally used multiple vertical layers of constant fixed thickness (zgrid) which do not change with time ("Eulerian grid") as used by Leendertse (1975). This type of model needs a large number of vertical grid points in order to accurately
9
represent the shallow regions. More recent models use the socalled 0grid which was originally applied in the simulation of atmospheric flow by Phillips (1957). This vertical ustretching uses the same number of vertical grid points in both the shallow and deep regions, with the vertical coordinate defined as follows:
U h(z, y) + C(x, y,t)(23
where h(x, y) is the water depth, and C is the water surface elevation. Governing equations are transformed from the (x, y, z, t) coordinates to (X, y7 0,7 t) coordinates by use of chain rule and become somewhat more complex because of the extra terms introduced by the stretching.
Other types of models (e.g., Simons, 1974) use a "Lagrangian grid" which consists of layers of constant physical property (e.g., density) but timevarying thickness. These models could resolve vertical flow structure with relatively few vertical layers. However, parameterization of the interfacial dynamics is often difficult. Horizontal Grid
One of the challenges in numerical models is the accurate representation of complex geometry. Most models (e.g., Leendertse, 1967) use a rectangular uniform grid to represent the shoreline of a lake or estuary. Thus, a large number of grid points are needed to achieve a fine resolution near the shoreline or islands. Because computational effort is directly related to the number of grid points, grid size should be as small as possible to maintain required resolution near the interest area, so long as the computational effort is not excessive. Therefore, to achieve a balance between resolution and computational efficiency, a nonuniform grid method could be used. Sheng (1975) used smaller grid size used near areas of importance but coarse grid elsewhere.
Use of a boundaryfitted grid is another viable alternative. Johnson (1982) used a boundaryfitted grid to solve depthintegrated equations of motion for rivers. Using chain rules, he transformed the governing equations for a boundaryfitted grid which
10
was generated by using the WESCORA code developed by Thompson (1983). Johnson (1982), however, transformed the equations in terms of the Cartesian velocity components.
The boundaryfitted grid has recently been adapted to threedimensional numerical models. Sheng (1986) applied tensor transformation to derive the threedimensional horizontal equations of motion in boundaryfitted grid in terms of the "contravariant" velocity vectors (a "contravariant" vector consists of components which are perpendicular to the grid line) and the water level. Sigma grid is used in the vertical direction. The resulting equations in the boundaryfitted and sigmastretched grid are rather complex. However, numerous analytical tests were conducted to ensure the accuracy of the model (Sheng, 1986 and Sheng, 1987). The model has been applied to Chesapeake Bay (Sheng et al., 1989a), James River (Sheng et al., 1989b), Lake Okeechobee (Sheng and Lee, 1991a, 1991b), and Tampa/Sarasota Bay (Sheng and Peene, 1992). However, the earlier study on Lake Okeechobee (Sheng and Lee, 1991a) did not consider thermal stratification in their model.
2.2 Vegetation Models
Vegetation can affect the aquatic life and also the water motion in the marsh area. Early studies on the effect of vegetation on flow were conducted in the open channels. Ree (1949) conducted laboratory experiments to produce a set of design curves for vegetated channels. Kouwen et al. (1969) studied the flow retardance in a vegetated channel in the laboratory and proposed the following equation: U = C, + C2An(A (2.4)
U(2.4
where U is average velocity, u* is shear velocity, and C1 and C2 are coefficients. A is a crosssectional area of the channel, and A, is the crosssectional area blocked by the vegetation.
Reid and Whitaker (1976) considered the vegetation effect on flow as an additional term, which is proportional to the quadratic power of the velocity, in the
11
depthintegrated momentum equation. Details of their vegetation models are given in Chapter 4. Their model, however, considered only the linearized equations of motion. In the present study, fully nonlinear equations are considered.
Sheng (1982) developed a comprehensive vegetation model by including the effect of vegetation on mean flow and secondorder correlations in a Reynolds stress model. Although the model was able to faithfully simulate the mean flow and turbulence in the presence of vegetation, it was not used for the present study due to the extra computational effort required when it is combined with a 3D circulation model.
Roig and King (1992) formulated an equivalent continuum model for tidal marsh flows. Neglecting leafiness, flexibility, and vegetation surface roughness, the net resistance force due to vegetation is thought to be related to the following parameters: Tv = f(p,g,p y, u,1.,d,s) (2.5)
where y is the viscosity of water, u is depthaveraged velocity, d is the average diameter of vegetation, 1, is the vegetation height, and s is the spacing between vegetations. Through a dimensional analysis,
= pu f(F, R, ) (2.6)
where F is the Froude Number and R is the Reynold's Number.
To determine the function f, they conducted a simple flume experiment. For each value of s/d, the dimensionless shear parameter I was plotted as a function of R PU28/dwafutinf
and F.
2.3 Thermal Models
Sundaram et al. (1969) used a onedimensional vertical model to demonstrate the formation and maintenance of thermocline in a deep stratified lake. The surface boundary condition was given as follows: aT
q, = pcpKh & = K(T T.) (2.7)
12
where q, is the heat flux, K is heatexchange coefficient,K,, is vertical eddy diffusivity, cp, is specific heat of water, p is density of water, T1, is an equilibrium temperature, and T, is a surface water temperature. They assumed that the annual variation in heat flux can be approximated by the cyclic form of the equilibrium temperature: T, = T, + asin(wt + q0) (2.8)
Following Munk and Anderson (1948), the eddy diffusivity K,, was expressed as the product of the eddy diffusivity under neutral condition and a stability function, which is one under neutral condition but becomes less than one under stable stratification (positive Richardson number).
Price et al. (1986) studied the diurnal thermal cycle in the upper ocean using field data and a vertical 1D thermal model. Their measured data include currents, temperature, and salinity, as well as meteorological data. Field data were collected between April 28, 1980, and May 24, 1980, at about 400 km west of San Diego, California.
Their major findings are the trapping depth of the thermal and velocity response is proportional to r Q112, the thermal response is proportional to Q3/2, and the diurnal jet amplitude is proportional to Q'I2, where Q is the heat flux and r is the wind stress. They also simulated the diurnal thermal cycle using the vertical onedimensional heat equation coupled with the momentum equations.
Gaspar et al. (1990) determined the latent and sensible heat fluxes at the airsea interface using the inverse method. They stated that the total heat flux can be divided into a solar part and a nonsolar part. While the solar radiation data is usually available from direct measurement, the nonsolar part is usually indirectly estimated from the meteorological data. However, this estimation of the nonsolar part involves many empirical formulas and may contain large errors. Gaspar et al. (1990) found that, by using the measured temperature data and solving the vertical onedimensional momentum equation and temperature equation, it is possible to estimate
13
the nonsolar heat flux more accurately. They called this method the "inverse method."
The major advantage of this "inverse" method is that it allows one to use the usual satellite data (wind stress, surface insolation, and sea surface temperature) for the estimation of heat flux at the ocean surface. The disadvantage of the method is that horizontal advection effect is neglected in the analysis.
2.4 Turbulence Model
Flows in natural water bodies are often turbulent, although they can relaminarize during periods of low wind and for tide. Although direct numerical simulation of turbulence can now be performed for simple flow conditions, it is still computationally prohibitive for practical applications in natural water bodies. Thus, "Reynolds averaging", a statistical approach was taken by decomposing the flow variables into a mean and a fluctuating part and averaging the equations over a period of time that is large compared to the turbulent time scale. The resulting equations thus produced are called Reynolds averaged equations. The Reynolds averaged mean flow equations contain terms involving correlations of fluctuating flow variables (i.e., secondorder correlations) that represent fluxes of momentum or scalar quantities caused by turbulent motion. The task of turbulence modeling is to parameterize these unknown correlations in terms of known quantities.
Numerous turbulence models were developed for the parameterization of turbulence. Some models are empirical, while others are based on more rigorous turbulence theory. In this section, some available turbulence models are briefly reviewed with an emphasis on the simplified secondorder closure model.
2.4.1 Eddy Viscosity/Diffusivity Concept
By analogy with molecular transport of momentum, the turbulent stresses are assumed proportional to the meanvelocity gradients. This can be expressed as O Oui
14
where u and are fluctuating velocity components in the xi and xj directions, while ui and u3 are mean velocity components, and vt is the turbulent eddy viscosity.
Similarly, an eddy diffusivity Kt can be defined:
uq= K (2.10)
SO/ xi
where 0' and 0 are the fluctuating and mean temperature or salinity or scalar concentration, and Kt is the turbulent eddy diffusivity.
In order to close the system of mean flow equations for ui, it is necessary to obtain an expression for vt in terms of known mean flow variables. Several options are given in the following:
2.4.2 Constant Eddy Viscosity/Diffusivity Model
Earlier models used constant eddy viscosity/diffusivity models. Although this allows easy determination of analytical solutions for the 1D equation of motion and easy programming, there are many disadvantages. Turbulence is spatially and temporally varying, hence constant eddy viscosity is not realistic. It is difficult to calibrate the constant eddy coefficient model even if extensive field data exists.
2.4.3 MunkAnderson Type Model
Prandtl (1925) assumed that eddy viscosity is proportional to the product of a characteristic fluctuating velocity, V, and a mixing length, L. He suggested that V = lum and A, = 12 u. The only parameter to be specified is length scale, l,, which is assumed to be a linear function of z.
Following Prandtl, we can define the "neutral" vertical eddy viscosity as follows:
AVO=#"y, )2+ (2.11)
where A. is assumed to be a linear function of z increasing with distance above the bottom or below the free surface and with its peak value at middepth, while not exceeding a certain fraction of the local depth. In the presence of strong waves,
15
turbulence mixing in the upper layers may be significantly enhanced. In such case, the length scale A, throughout the upper layers may be assumed to be equal to the maximum value at middepth (Sheng, 1983).
When a lake is stratified, vertical turbulence is affected by buoyancy induced by the vertical nonhomogeneity. In this situation, vertical eddy coefficients should be modified to account for this effect. This is parameterized by introducing the Richardson number:
Ri = P 'a 2 + () 2] (2.12)
Ri is positive when flow is stable (a < 0) and when Ri is negative when flow is unstable (2 > 0). Generally, eddy viscosity and eddy diffusivity are expressed as follows:
A,, = A,,o0i(Ri); K,, = K,,o02(Ri) (2.13)
where q1 and 02 are stability functions and A,,, and Ko are eddy viscosity and eddy diffusivity when there is no stratification. Stability functions have the following forms: 01 = (1 + jRi)m'1; 2 = (1 + 0r2Ri)m2 (2.14)
Based on comparing model results with field data, Munk and Anderson (1948) developed the following formula:
= (1 + lORi)1/2; 2 = (1 + 3.33Ri)3/2 (2.15)
Many similar equations with different coefficients were suggested based on numerous sitespecific studies. These coefficients, however, are not universal, and care must be taken when applying these formulae to a new water body where little data exist.
2.4.4 Reynolds Stress Model
One can obtain an equation for the timeaveraged secondorder correlations by following the procedure: (i) decompose the dependent variables into mean components and fluctuating components, (ii) substitute the decomposition into continuity,
16
momentum equation and heat equation, and (iii) take timeaverage of all equations. For example, the resulting timeaveraged equation for u'u' (e.g., Donaldson, 1973; Sheng, 1982) is
Bu'u) auu _Ou, ,ug up up
+ Uk = iUUk jk + 9j g
at OXk ,k k Po PO
[ '~ (au;'.uj) (2.16)
 2eiktkuuj 2EjkiAZUkUi aXk u. 8p uj 8p 8 u uj cui Bu
+ ' 2v
p 82j p axi aXkaXk Oxk 9Xk
Similar equations for up' and p'p' can be obtained. Unresolved thirdorder correlations and pressure correlations are modeled using the simplest possible forms (Donaldson, 1973).
au'. u bb.q3 avuuu
U + (2.17)
aXk OXk 3A A2
p u u u q r q2
(2xy (u(u) 6" 3i') (2.18)
8(,U, u p Bu p 8qA uiu
 u i u Ou+ pqA u= c(2.19)
+5 axk =v~.
where q is the total fluctuating velocity and A is the turbulence macroscale. The model constants (a, b, and v,) are determined from a wide variety of laboratory data (Lewellen, 1977). Thus, a full Reynolds stress model consists of six equations for velocity fluctuations uju, three equations for the scalar fluxes, uip', and one equation for the variance, p'p'. Considering the required computer storage and CPU time for the turbulence models, it is desirable to use a simplified form of the Reynolds stress model.
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure Model
The complete secondorder closure model is too complicated to be used in a threedimensional model. A simplified secondorder closure model can be developed with
17
the following assumptions: 1) Secondorder correlations have no memory effect. That means correlations at the previous time have no effect on correlations at the next time. Therefore, D = 0. 2) Correlations at a point do not affect the value at another point. Therefore, all the diffusive terms are dropped. These conditions are approximately true if the time scale of turbulence is much less than the mean flow time scale and the turbulence does not vary significantly over the macroscale, i.e., the turbulence is in local equilibrium. Then the remaining equations become as follows (Sheng, 1983): Bu" "u7u p' 7u
0= uju,,i ujp 9j up
0 = u 58k ' u 8uk k 9i Po Po
 2eik ku'u jkeIUtui (2.20)
Sq ,, q q3
A (u u 6; ) 3 612A
, p , ui giP'P'
0 uj s p
'Ox 8 Po
 2ijkjukp' 0.75qU (2.21)
A
 p 0.45qp'p' (2.22
0 = 2unp' + (2.22)
These algebraic equations can be solved with ease, once the mean flow conditions are known. In order to complete the system of equations, q and A need to be solved following the procedure described in Sheng et al., 1989b.
The above "equilibrium closure" model was applied to the Atlantic Ocean (Sheng and Chiu, 1986), Chesapeake Bay (Sheng et al., 1989a) and the James River (Sheng et al., 1989b). More details of the model will be given in Chapter 8.
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model
To introduce some dynamics of turbulence into the simplified secondorder closure model, one can add a dynamic equation for q2(q2 = u'u' + v'v' + w'w'), which is twice the turbulent kinetic energy (Sheng and Villaret, 1989). This TKE closure model has
18
been applied to James River (Choi and Sheng, 1993) and Tampa Bay (Schoellhamer and Sheng, 1993).
Two other turbulence models, which are based on the socalled k c model (Rodi, 1980), are described in the following:
2.4.7 OneEquation Model (k Model)
Using the eddy viscosity/diffusivity concept, the choice of velocity scale can be v/k, where k = (u2 + v2 + w2)/2 is the kinetic energy of the turbulent motion. When this scale is used, the eddy viscosity is expressed as vt =c/L (2.23)
where c' is an empirical constant and L is the length scale. To determine k, an equation is derived from the NavierStokes equation as:
Ok Ok 0 [ u'iu p' ,IOulIOu (.4
"2[Ui( + flgiu v2 ' (2.24)
To obtain a closed set of equations, diffusion term and dissipation term must be modelled. The diffusion flux is often assumed proportional to the gradient k as ui 2 p ak A (2.25)
where Ok is an empirical diffusion constant. The dissipation term c, which is the last term of Eq. 2.24, is usually modelled by the expression k3/2
CD k 3/2 (2.26)
The length scale, L, needs to be specified to complete the turbulence model. Usually L is determined from empirical relations.
2.4.8 TwoEquation Model (k c Model)
To avoid the empirical specification of length scale, another equation for the dissipation c is needed. Then eddy viscosity and eddy diffusivity are expressed as Ct = C Ak (2.27)
f
19
t = (2.28)
at
where cl, is an empirical constant and at is the turbulent Prandtl/Schmidt number.
An equation for e is derived from the NavierStokes equation and is
ae ac a Ovt Oc C 62
 t U = ( cl,k PC3,Gx) cc (2.29)
where P and G are the stress and buoyancy term in Equation 2.24, respectively, and cit, c2. and c3, are empirical model coefficients.
2.5 Previous Lake Okeechobee Studies
Numerous studies on Lake Okeechobee have been performed, but most of them focused on water quality.
Whitaker et al. (1975) studied the storm surges in Lake Okeechobee while considering the vegetation effect in the western marsh area, by using a twodimensional, vertically integrated model. They simulated the seiche in the lake during the 1949 and 1950 hurricanes. In their study, the bottom friction coefficient was parameterized as a function of depth to achieve better agreement of storm surge height.
Schmalz (1986) investigated hurricaneinduced water level fluctuations in Lake Okeechobee. His study consisted of two parts: a hurricane submodel and a hydrodynamic submodel. The hurricane submodel used hurricane parameters such as central pressure depression, radius to maximum winds, maximum wind speed, storm track, storm forward speed, and azimuth of maximum winds, and determined the wind and pressure field that were used as forcing terms for the hydrodynamic submodel.
The hydrodynamic submodel solved the depthaveraged momentum equations and continuity equation. Finitedifference method was used for the numerical solution. For treatment of marsh area, an effective bottom friction which relates the Manning's n to water depth and canopy height was used. The 2D hydrodynamic model can resolve flooding and drying: during strong wind conditions such as a hurricane, a portion of the lake can become dried because of the excessive setdown by wind, while
20
other portion of the lake can become flooded because of excessive setup by wind.
A threedimensional Cartesiangrid hydrodynamic and sediment transport model for Lake Okeechobee was recently developed (Sheng et al., 1991a; Sheng, 1993). In addition, these models were extended to produce a threedimensional phosphorus dynamics model (Sheng, et al., 1991c). These models use the simplified secondorder closure model and the sigma stretched grid, however, did not consider the effects of vegetation and thermal stratification
2.6 Present Study
The present work focuses on the study of effects of vegetation and thermal stratification on winddriven circulation in Lake Okeechobee. As will be shown later, the threedimensional curvilineargrid model (CH3D) will be significantly enhanced to allow accurate simulation of the observed circulation. Model features are compared with model features of some previous lake studies in Tables 2.1 and 2.2. It is apparent that the 3D model developed in this study is more comprehensive than those used in previous lake studies.
Table 2.1: Selected features of lake models.
Author Dimensionality Type of Temporal Turbulence Advection
Model Dynamics
Welander 3D AN T.D. A No
1957
Liggett 3D F.D. T.D. A No
1969
Lee + Liggett 3D F.D. S.S. A No
1970
Liggett + Lee 3D F.D. S.S. A No
1971
Gedney + Lick 3D F.D. T.D. A No
1972
Goldstein + Gedney 3D A.N. B No
1973
Sengupta + Lick 3D F.D. T.D. D Yes
1974
Simons 3D F.D. T.D. B
1974
Sheng 3D F.D. S.S. A No
1975
Thomas 3D F.D. S.S. B No
1975
Whitaker et al. 2D F.D. T.D. Yes
1975
Witten + Thomas 3D F.D. S.S. C No
1976
Lien + Hoopes 3D F.D. S.S. A No
1978
Schmalz 2D F.D. T.D. Yes
1986
Sheng + Lee 3D F.D. T.D. E Yes
1991a
* F.D. : Finite difference
* AN : Analytic
* S.S. : Steady state
* T.D. : Time dependent
* A: Constant
* B : Dependent on wind
* C : Exponential form
* D : MunkAnderson type
* E : Simplified secondorder closure model
Table 2.2: Application features of lake models.
Author Basin Dimen Mean Forcing Vege Grid
tion Depth W H R tation Hor. ver.
Liggett Idealized Yes No No No U
1969 Basin
Lee + Liggett Idealized Yes No No No U
1970 Basin
Liggett + Lee Idealized Yes No No No U
1971 Basin
Gedney + Lick Lake Erie 400 km 20 m Yes No Yes No U
1972 100 km
Sengupta + Lick Squire 1.89 m Yes Yes No No N
1974 Valley
Simons Lake Yes Yes No No U
1974 Ontario
Sheng Lake Erie 400 km 20 m Yes No Yes No N 01
1975 100 km
Thomas Idealized Yes No No No U
1975 Basin
Whitaker et al. Lake 57 km 2.5 m Yes No No Yes U
1975 Okeechobee 60 km
Witten + Thomas Idealized 300 x Max Yes No No No
1976 Basin 87 km 180 m
Lien + Hoopes Lake Yes No No No U
1978 Superior
Schmalz Lake 57 km 2.5 m Yes No No No U
1986 Okeechobee 60 km
Sheng + Lee Lake 57 km 2.5m Yes No No Yes C a
1991a Okeechobee 60 km
* W: Wind
* H : Heating
* R: River
* U : Uniform Cartesian grid
* C : Curvilinear grid
* N : Nonuniform Cartesian grid
*r : Vertically stretched grid
CHAPTER 3
GOVERNING EQUATIONS
3.1 Introduction
This chapter presents the basic equations which govern the water circulation in lakes, reservoirs, and estuaries. Because the details can be found in other references (e.g., Sheng, 1986; Sheng, 1987; Sheng et.al., 1989c), the governing equations are presented here without detailed derivations.
3.2 Dimensional Equations and Boundary Conditions in a Cartesian Coordinate System
The equations which govern the water motion in the water bodies consist of the conservation of mass and momentum, the conservation of heat and salinity, and the equation of state. Because Lake Okeechobee is a fresh water lake, the salinity equation is not considered. The following assumptions are used in the Curvilinear Hydrodynamic Threedimensional Model (C13 D) model.
(1) Reynolds averaging: Three components of velocity, pressure, and temperature are decomposed into mean and fluctuating components and timeaveraged.
(2) Hydrostatic assumption: Vertical length scale in lakes is small compared to the horizontal length scale, and the vertical acceleration is small compared with the gravitational acceleration.
(3) Eddy viscosity concept: After timeaveraging, the secondorder correlation terms in the momentum equation are turbulence stresses, which are related to the product of eddy viscosity and the gradient of mean strain.
(4) Boussinesq approximation: Density variation of water is small, and variable density is considered only in the buoyancy term.
3.2.1 Equation of Motion
With above assumptions, the equations of motion can be written in a righthanded Cartesian coordinate system as follows: Ou Ov Ow
++=0 (3.1)
Ou Ou2 Ouy Ouw lOp 0 AOu
+ + + =fv +T AH
t xy z Po Ox x
+ (AH ) + A, a (3.2)
Ov Ouv 9v2 Ovw f lOp O( Av'
+ + + fu +
Ot Ox Oy 8z Po 9Y Ox ax
+ A. ) + a O (3.3)
ap
z pg (3.4)
OT OuT OvT OwT a (KOT
S+ + H K
Ox D Oy 8z Dz 8
+ KH + T, (3.5)
p = p(T, S) (3.6)
where (u, v, w) are velocities in (x, y, z) directions, f is the Coriolis parameter defined as 2f sine where l is the rotational speed of the earth, q is the latitude, p is density, p is pressure, T is temperature, (AH, KH) are horizontal turbulent eddy coefficients, and (A,, K,) are vertical turbulent eddy viscosities.
For the equation of state, Eqn. 3.6, there are many different formulae that can be used. For the present study, the following equation given by Eckart (1958) is used:
p = (1 + P)/(a + 0.698P)
(3.7)
25
P = 5890 + 38T 0.375T2 + 3S (3.8)
a = 1779.5 + 11.25T 0.0745T2 (3.8 + 0.01T)S (3.9)
where T is water temperature in degree C, S is salinity in ppt, and p is density in g/cm3.
Besides governing equations, boundary conditions should be specified.
3.2.2 FreeSurface Boundary Condition (z = 77)
(1) Kinematic boundary condition:
w = + 7+ v 7 (3.10)
(2) Surface heat flux:
aT
q = If,,T = K(T, T,)3.1 z (3.11)
where T, is the lake surface temperature, T, is the equilibrium temperature, K is a heat transfer coefficient, and q is positive upward. (3) Surface stress: au O
A, = 7,, = v (3.12)
where the wind stresses r., and r, must be specified.
3.2.3 Bottom Boundary Condition (z = h)
(1) Heat flux is specified as zero,i.e., T = 0
(2) Quadratic bottom friction law is used, i.e.,
T. = PCd /U,2 + V12U1,r = PCdQ U2 + v12v1 (3.13)
where ul and v, are velocity components at the first grid point above the bottom.
3.2.4 Lateral Boundary Condition
(1) Heat flux is assumed zero,i.e., T 0
n
(2) No flow through boundaryi.e., u 0 or v = 0
26
3.3 Vertical Grid No natural water bodies have strictly flat bottoms. Therefore to represent the variable bottom topography, a stretching is used by defining a new variable o : z (x, y,t)
Sh(x, y) + ((x, y, t) (314)
The advantage of astretching is that the same vertical model resolution can be maintained in both shallow and deeper parts of a lake. The disadvantage is that it introduces additional terms in the equations. Details of astretching can be found in Sheng and Lick (1980) and Sheng (1983).
3.4 NonDimensionalization of Equations By introducing reference values, the governing equations can be nondimensionalized. The purpose is to make it easier to compare the relative importance of each term. The following relations were used (Sheng, 1986).
(u*, v', w*) = (u,v, wX,/Z,) /U, (x*, y,z*) = (X,y, zXr/Zr) /X, (7;,r;) = (r.r"')/PofZU, t*= tf
q: = To/(Tr To) q/poCfZTo ( = gc/lfUX, = /S, (3.15)
p* = (p po)/(pr Po) T* = (T To)/(Tr To) A*H = AH/AHr
A* = A,,/A,,
KI = KH/KHr
K, = Kv / Kv r
w* = wXI/U,
27
where variables with asterisks are nondimensional variables and variables with r are the reference values.
3.5 Dimensionless Equations in aStretched Cartesian Grid
The transformation relations from a Cartesian coordinate (x, y, z) to a vertically stretched Cartesian coordinate (x,y, o) may be found in Sheng (1983). Using the relation presented in the previous section, the following dimensionless equations are obtained:
ac 8Hu 8Hv Hw
+ a + y + Hat 80y B
l O8Hu ( Ea (A, u
H x + Ay + a
Ro 8(aHuu + 8Huy aHuw
+ E' aH (,x + aAay + H.O.T.
Ro opa 8H 0
H L do + aj pdu + Up
a( E, 8aB
S E a + (A, + au ax H2a k, a]
1 aHv a( E, a ( avv
 at +j ~ A~) u H Ot B y H2 08eRo aHuv aHvv aHvw
H ax ay A
+ EH AH v + y (A ) + H.O.T.
ax ax ay ay
Ro [ Op d aH opd+ )]
Fr2 H J ay +"y J
8( E, 8 Byo
a +,a A, a + By
ay H2aa B ao
1 aHT E, a ( T
Hat PrH, 28a I ,
Ro( aHuT aHvT aHwT H z ax ay r )
(3.16)
(3.17)
(3.18)
(3.19)
28
+ + a) KU Y ) + H.O.T.
PTH ax ax ay ay
p = p(T, S) (3.20)
where H is total depth, / = gZr/Xf' and H.O.T. is higher order terms.
3.5.1 VerticallyIntegrated Equations
The CH3D model can solve the depthintegrated equations and the threedimensional equations. The vertically integrated momentum equations are obtained by integrating the threedimensional equations from bottom to top.
+( + = 0 (3.21)
at ax ay /
au H +r rb + V
t ax
[a 8 UU ) aiUV\
Ro + (3.22)
8L AOU a BUT + EH[ (AH a + aAH a
+ x 8x ay ay
Ro H2 ap
Fr} 2 8x
 Hax
ax
av H +,, U
a T ay
 Ro a (Up)+ (VV) (3.23)
+ EH AH + AH ]
Tax ax y ay
Ro H2 ap
Fr2 2 Oy
 Ha + D
By
3.5.2 Vertical Velocities
(1+ 0 1 (OHu OHv)
= H Ot H ,iO + ay (3.24)
w = Hw+ (l+a) d+ ( Oh _y)
dt+ r~ + v (3.25)
where U = f1 udor, V = f1 vdo,, T*, r.,y are wind stresses at the surface and rb., Tby are bottom stresses.
3.6 Generation of Numerical Grid
3.6.1 Cartesian Grid
In order to numerically solve the governing equations, finite difference approximations are introduced to the original governing equations, and solutions are obtained at discrete points within the domain. Therefore, a physical domain of interest must be discretized. When a simple physical domain is considered, cartesian grid can be used and hence grid generation and development of finitedifference equations are relatively easy.
Unfortunately, most physical domains in lakes or estuaries are complex. Previously rectangular grid was widely used. This method has such disadvantages as inaccuracies at boundaries and complications of programming due to unequal grid spacing near boundaries.
3.6.2 Curvilinear Grid
To better resolve the complex geometries in the physical domain, boundaryfitted (curvilinear) grid can be used. In general, a curvilinear grid can be obtained by use of (1) algebraic methods, (2) conformal mapping, and (3) numerical grid generation.
Algebraic grid generation uses an interpolation scheme between the specified boundary points to generate the interior grid points. This is simple and fast computationally, while the smoothness and skewness are hard to control. Conformal
30
mapping method is based on complex variables, so the determination of mapping function is a difficult task. Therefore, many practical applications rely on numerical grid generation techniques.
3.6.3 Numerical Grid Generation
Partial differential equations are solved to obtain the interior grid points with specified boundary points. Thompson (1983) developed an elliptic grid generation code (WESCORA) to generate a twodimensional, boundaryfitted grid in a complex domain.
To help understand the physical reasoning of this method, consider a rectangular domain. When the temperature is specified along the horizontal boundary, then the temperature distribution inside can be obtained by solving the heat equation. Therefore, isothermal lines can be drawn. Also, other isothermal lines can be obtained with the specified temperature in the vertical direction. By superimposing these isothermal lines, intersection points of isothermal lines can be considered as grid points.
WESCORA solves Poisson equations with same idea in a complex domain. Consider the following set of equations (see Figure 3.1):
G. + Gy = P (3.26)
7lxx + 7l'd' = Q (3.27)
with the following boundary conditions:
= (X (xY) onl1and 3
=constant (3.28)
=constant on 2 and 4
'q 7 r(x, Y) (3.29)
31
where the functions P and Q may be chosen to obtain the desired grid resolution and alignment. In practice, one actually solves the following equations which are readily obtained by interchanging the dependent and independent variables in Eqs. 3.26 and
3.27:
ax 20x2, + yx,, + aPxe + 7Qx, = 0 (3.30)
ay 2/y,7, + ty,, + aPye + yQy, = 0 (3.31)
where
a 2 2
/3 = xx7 + YY,
1 2
P = (x7 + y)7 (3.32)
1 2
Q = f(x + y )Q
J = xy, xyy
with the transformed boundary conditions: x = fi(,t 7i) on i = 1 and 3
y = gi(V, qi) (3.33)
x = fi(i, 7) on i = 2 and 4
y = gi(Vi, ) (3.34)
3.7 Transformation Rules Generations of a boundaryfitted grid is an essential step in the development of a boundaryfitted hydrodynamic model. It is, however, only the first step. A more important step is the transformation of governing equations into the boundaryfitted coordinates. A straightforward method is to transform only the independent variables,
C ((.y) y,
or
q* q (,.y) yS
YPZ
PROTOTYPE
T Y R
TRNS FORME
LTRANSFOR MED
Figure 3.1: A computational domain and a transformed coordinate system.
a a(I s X) yjt( It. I a)
3 '
33
i.e., the coordinates, while retaining the Cartesian components of velocities. Johnson (1982) developed such a 2D verticallyintegrated model of estuarine hydrodynamics. The advantage of the method is its simplicity in generating the transformed equations via chain rule. The dimensional forms of the continuity equation and the verticallyintegrated momentum equations are shown by Eqs. (20) and (21) in Appendix A of Sheng (1986). The resulting equations, however, are rather complex. Even when an orthogonal or a conformal grid is used, the equations do not become any simpler. Additional disadvantages are (1) the boundary conditions are quite complicated because the Cartesian velocity components are generally not aligned with the grid lines, (2) the staggered grid cannot be readily used, and (3) numerical instability may develop unless additional variables (e.g., surface elevation or pressure) are solved at additional grid points, (Bernard, 1984, cited in Sheng (1986)).
To alleviate the problems mentioned in the previous paragraph, Sheng (1986) chose to transform the dependent variables as well as the independent variables. Equations in the transformed coordinates ( 77) can be obtained in terms of the contravariant, or covariant, or physical velocity components via tensor transformation (e.g., Sokolnikoff, 1960). As shown in Fig. 8 of Appendix A of Sheng (1986), the contravariant components (u') and physical components u(i) of the velocity vector in the nonCartesian system are locally parallel or orthogonal to the grid lines, while the covariant components (ui) are generally not parallel or orthogonal to the local grid lines. The three components are identical in a Cartesian coordinate system. The following relationships are valid for the three components in a nonCartesian system are
u, = (gi)1/2u(i) (no sum on i) (3.35)
ui = (gii)/2giju(j) (no sum on i) (3.36)
u(i) =_ f(i
(3.37)
34
where g is the diagonal element of the metric tensor gij:
9 xt Xn 6e. (3.38)
which for the twodimensional case of interest is
9i.x + y XCX@ + Y0y7 911 912
9j7XC + Y x 2 Y2 (= 1 922 (3.39)
( + x+~ +yy x+ Y g21 g22
The three components follow different rules for transformation between the prototype and the transformed plane: U = u (3.40)
Oxi
U= u (3.41)
'9xiu
(i) = ( i)1/2 u() (3.42)
where the unbarred quantities represent the components in the prototype system, while the barred quantities represent the components in the transformed system.
3.8 TensorInvariant Governing Equations Before transforming the governing equations, it is essential to first write them in tensorinvariant forms, i.e., equations which are independent of coordinate translation and rotation. For simplicity, unbarred quantities are used to denote the variables in the transformed system unless otherwise indicated.
Following the rules described in the previous paragraph, the following equations are obtained (Sheng, 1986):
(e o xk (V Huk) = 0 (3.43)
1 8Huk
H 8 k ikf ujn
Ro BhukW
n (Huuk), + (3.44)
H I \
E, a auk
+ H2 Oc + EAHUk !m
Ro F 'M.
 H p! da +H!k ( pd +op)
FrD2 L )
where O/OXk is the partial derivative, ge, is the metric tensor while go = J= gy,, x,y is the determinant of the metric tensor, uk is the contravariant velocity, ( ),t represents the covariant spatial derivative, !k represents the contravariant spatial derivative, and ckj is the permutation tensor and 12 1
vo
1
621 = (3.45)
v5o
11 22 = 0
The covariant and contravariant differentiations are defined by
u = U. + Di3 u (3.46)
S!k = gkmS,m (3.47)
where :j represents partial differentiation and Dj represents the Christoffel symbol of the second kind:
D k = gDnik (3.48)
where gi represents the inverse metric tensor, hin, and Dnik is the Christoffel symbol of the first kind:
1
Dijk = (gij:k + gik:i gjk:i) (3.49)
3.9 Dimensionless Equations in BoundaryFitted Grids
Expanding the equation 3.41 and 3.42, the following equations are obtained:
(t +
1 8Hu H at
0V;u
(3.50)
( I l 12 0( g12 g22
9 + g + u +o Vv
 H (Huu) + (Huv) + (2D11 + D 2)Huu
D 2 OHuw
+ (3D2 + D 2)Huv + D22Hvv + 9r
E, 8 ABu H2 Eo (A OU)
Ro Hr2 1 oP
F.,2 8(
(3.51)
+12 do
a77
+ (lH + 12O )pd + p + EHAH(Horizontal Diffusion)
(g 21
Ro
H [
+ g228 +g)
(Huv) + (Hvv) + D 1Huu
8( By
+ (D', + 3D 2)Huv + (D12 + 2D 2)Hvv]
E, a A, v + H a( aa)
H Bov o
Ro H
F,2 I
{21 OH
+ (g +
( 21aH
228H)
22OP do
pdu +up
+ EHAH(Horizontal Diffusion)
where the horizontal diffusion terms are listed in Sheng(1986). The temperature equation can be obtained according to the same procedure as
1 OHT H 8t
_ E, 0 Pr, H2 0cr
K I
F" B
1 OHv
H Ot
(3.52)
g~"
91O U
921
+_V
a( fHv) = 0
84
Ro L [ (HuT) + HvT Ro8HwT
Hv~ (ouT) +g; ( HyT)g
+ EH [gIlT,1 + g12T,1,2 PrH
+ g21T2,1 + g2 T2,2] (3.53)
3.10 Boundary Conditions and Initial Conditions
3.10.1 Vertical Boundary Conditions
The boundary conditions at the free surface (a = 0) are (BuOu H
A"= (,( Ta,,r8=) (3.54)
T H Pr,
= q
5a E, I
The boundary conditions at the bottom (a = 1) are
(au v H
A, = +(rb4, rb7)
U.
= UH,Z,Cd [gru + 2g2uIv1 + g22 ] (UlV1) Avr
9T
= 0 (3.55)
where ul and v, are the contravariant velocity components at the first grid point above the bottom.
3.10.2 Lateral Boundary Conditions
Due to the use of contravariant velocity components, the lateral boundary conditions in the ((, r/, o) grid are similar to those in the (x, y, 0') system. Along the solid boundary, noslip condition dictates that the tangential velocity is zero, while the slip condition requires that the normal velocity is zero. When flow is specified at the open boundary or river boundary, the normal velocity component is prescribed.
3.10.3 Initial Conditions
Initial conditions on vectors, if given in the Cartesian or prototype system, such as the velocity and the surface stress, must be first transformed before being used
38
in the transformed equations. Thus, the surface stress in the transformed coordinate system is given by
1 Q 1 0 2
= a + 2 (3.56)
2 /1 O}2
r T 1
72 = a7i 72 (3.57)
where T1 I2 are the contravariant components of the stress in the transformed system and r1, r 2 are the contravariant components in the Cartesian system. Note that in the Cartesian system, the contravariant, covariant, and physical components of a vector are identical. The contravariant components of the initial velocity vectors can be transformed in the same manner to obtain the proper initial conditions for the transformed momentum equations.
CHAPTER 4
VEGETATION MODEL
4.1 Introduction
The western portion of Lake Okeechobee is covered with an extensive amount of vegetation. The vegetation can affect the circulation in several different ways. First of all, wind stress over the emergent vegetation is reduced below that over the open water. Furthermore, the submerged vegetation introduces drag force to the water column. Because most of the vegetation stalks are elongated cylinders without large leaf areas, the drag force is primarily associated with the profile drag (or form drag) instead of the skin friction drag. The profile drag can reduce the flow and is proportional to the "projected area" of vegetation in the direction of the flow.
The presence of vegetation also can affect the turbulence in the water column. The characteristic sizes of the horizontal and vertical eddies generally are reduced by the vegetation. This usually leads to a reduction of turbulence, although some wake turbulence may be generated on the downstream side of vegetation.
In order to simulate the effects of vegetation, several approaches have been undertaken in previous investigations. For example, Saville (1952) and Sheng et al. (1991b) used an empirical correction factor to simulate the reduction of wind stress over the vegetation area. Sheng et al. (1991b) also adjusted the bottom friction coefficient over the vegetation area. For simplicity, however, Sheng et al. (1991b) did not include the effect of vegetation on mean flow and turbulence in the water column, because the primary focus of that study was the internal loading of nutrients from the bottom sediments in the open water zone. Whitaker et al. (1975) developed a twodimensional, verticallyintegrated model of storm surges in Lake Okeechobee. The profile drag cre
40
ated by the vegetation was included in the linearized verticallyintegrated equations of motion, which did not contain the nonlinear and diffusion terms. Sheng (1982) developed a comprehensive model of turbulent flow over vegetation canopy by considering both the profile drag and the skin friction drag in the momentum equations in addition to the reduction of turbulent eddies and the creation of turbulent wake energy. Detailed vertical structures of mean flow and turbulence stresses were computed by solving the dynamic equations of all the mean flow and turbulent quantities. Model results compared well with available mean flow and turbulence data in a vegetation zone.
For the present study, due to the lack of detailed data on vegetation and mean flow and turbulence in the vegetation zone, a relatively simple vegetation model which is more robust than Whittaker et al.'s model yet simpler than Sheng's 1982 model is developed. Due to the shallow depth in the vegetation zone, it is feasible to treat the water column with no more than two vertical layers. When the height of vegetation is greater than 80% of the total water depth, the flow is considered to be onelayer flow, i.e., the entire water column is considered to contain uniformly distributed vegetation. When the height of vegetation is between 20% and 80% of the total water depth, the flow is considered to be twolayer flow, i.e., the water column consists of a water layer on top of a vegetation layer. The vegetation effect is neglected when the height of vegetation is less than 20% of the total depth. The profile drag introduced by the vegetation can be formulated in the form of a quadratic stress law: Tcanopy = PCdUl I I AN (4.1)
where u is the vertically averaged velocity in the vegetation layer (layer 1), p is the density of water, A is the projected area of vegetation in the direction of the flow, N is the number of stalks per unit horizontal area, and Cd is an empirical drag coefficient. Tickner (1957) performed a laboratory study. Strips of ordinary window screen 0.1 foot in height were placed across a channel to simulate a vegetative canopy. Using
41
Tickner's experimental results, Whitaker et al. (1975) calculated cd(1.77) which was used in this study. Roig and King (1992) showed Cd is a function of Froude Number, Reynolds Number, vegetation height, spacing, and diameter of vegetation. As the water level changes, the flow regime over a vegetation area may change from onelayer to twolayer flow, and vice versa.
4.2 Governing Equations Let us consider an x, y, z coordinate system with the velocity components in the (x, y, z) directions as (u, v, w). The lower layer (layer I) of the water column is covered with vegetation, while the upper layer (layer II) is vegetationfree (Figure 4.1).
Flow in the vegetation layer (Ul, vI) and flow in the vegetationfree layer (u2, v2) both satisfy the equations of motion.
4.2.1 Equations for the Vegetation Layer (Layer I)
Oul Ou Oulvl1 Ou1wl 1 p 0 O Su
+ + + z fv + AH
ax + y 8zp x x Ox
+ [A, + H (4.2)
ay By Oz 80 z
Ov1 Ouivc Ovw Ox 1 1f p 8 I
+ + ful + [AHO
at 8x ay 8z P D y TX ax
+ AH] + [A vl] (4.3)
19y ay 8z 8z
8p
"= pg (4.4)
Integrating Equation (4.4) vertically: p = Pa + Pg(( z) (4.5)
Integrating Equations (4.2) and (4.3) vertically from z = h to z = f:
' + az (U )] y La + gL = f V + (ri 7b F)
at~~ Bax L, aB ,
0, U1 0 [AOU1]
+ I + AH (4.6)
x Ox y ay
Y
S(x,y,t)
Figure 4.1: Schematics of flow in vegetation zone.
 V + 8 ( UxVx
+ Tx LL /
1
gL1(= f U, + (ri, Tby F:y)
p
+O A o [AO 'y
Ox 8x y ay
(4.7)
where L1 = h f and U, and V are the verticallyintegrated velocities within the vegetation layer:
(U1, V1)fh (ul, Vi)dz
(4.8)
7b and 7b, are bottom stresses, 7i, and 7, are interfacial stresses between layer I and layer II, and F, and F, are form drags due to the vegetation canopy.
4.2.2 Equations for the VegetationFree Layer (Layer II)
0u2 Ou2 Ou2v2 at+ + oY
8v2 Ouy, +V2 + 2V2
't ax
y
+ y+
'9y
Bu2w2
Oz
aV2W2 Ozv2w2
8z
lop1 p2 + .u 2 = fv2 + A7 H
O[8 u2 +[ OU2 + y AHy +z Aj
ByLByJ 8z 8z~
1 8p2 0 F8v2l
fu I AHp y2 Oz Ox A
[AHOV] + [ A] + T Y oz 8z
(4.9) (4.10)
Integrating Equation (4.9) and (4.10) vertically from z = t to z = ( and defining L2 = + C:
+
8
U2 V2)
L2
+ gL2(. = f V2 + (r. r)
P
+ AHU' +0 [A,,
Bz ax IDy ay
(4.11)
OV2 + 1 U2V2
at x L2
O (V22\
2 +
1
gL2(y = fU2 + (r, in)
P
S[AHV2 AV2
+ AHI + [AHs
8x ax y ay
(4.12)
where U2 and V2 are verticallyintegrated velocities within the vegetationfree layer: (U2, V2) = (u2,v2)dz (4.13)
aU2
Ot
+Sy (L,+
+ Ux L + 8z \,L2 /
44
4.2.3 Equations for the Entire Water Column
Instead of solving the above equations for the vegetation layer and the vegetationfree layer, it is more convenient to solve the verticallyintegrated equations for the entire water column, which can be readily derived by combining the equations for the two layers. First, the verticallyintegrated velocities over the entire water column in the vegetation zone, U and V, can be defined as
(U, V) (U + U2, V + V2) = (Lii1 + L2ii2, L1ii1 + L202) (4.14) where uz U2, vU and v2 are verticallyaveraged velocities within layer I and layer II, respectively, while L1 and L2 are the thicknesses of layer I and layer II, respectively.
Adding the U1 equation and the U2 equation leads to
au ( 1 U2 Ui U2V2
t + T L, L2 y L, L2)
1
= fV + (r, Trbx F)
P
+ AHU+ AH U] (4.15)
Tx 1AH "] + 19 1
while the summation of the VI1 equation and the V2 equation results in
9t + n + "2 + + + gH(
81t Tz L, L2 y L, L2 )
1
= fU + (,, r8 b F)
P
+ [A V] + AHV] (4.16)
Tz ax y [A5y
All the stress terms are computed as the quadratic power of the flow velocity. For example, r,x and r,y are computed as quadratic functions of the wind speed, rb, and rby are computed as quadratic functions of U and V, and ri, and ri, are quadratic functions of (U2 U1) and (V2 V1). The form drags associated with the vegetation are:
= pcdAN U12 + V() (4.17)
Fx = pcdAxNL (4.17)
JU, + v12 1V1
F, = pcdAvN' L L (4.18)
Ti,= PCai ( 2 + (V )2 (U U (4.19)
L2 L, L2 L, L2 Li1
(U2 _U12 + (V2 V) 2 V
7,= pi( + (4.20)
(L2 L, L2 L, L2 L,
o (4.21)
where Cd and Cdi are empirical drag coefficients.
An additional equation that must be satisfied is the continuity equation: a( au ov
 + 0 (4.22)
4.2.4 Dimensionless Equations in Curvilinear Grids
The above dimensional equations were presented to illustrate the development of the vegetation model. In the curvilineargrid model, however, dimensionless equations in curvilinear grids are solved. These dimensionless equations are presented in the following in terms of the contravariant velocity components in two layers: + Lz[gQ(t + gl2c] 09t
= 912 Ult + U22 + *
+ (Horizontal Diffusion), (UI, U") Ro [Nonlinear Terms(Uz, U")] (4.23)
LL
 + L[g21 + g22 c,1 911 Ul g2 U_2 + r.* * * + (Horizontal Diffusion);, (Uf, U") Ro
+R [Nonlinear Terms(Uf, U')] (4.24)
a U' + L 2[g "(4 + g 12( ] OUt
+ 12 g'1 22 '2 = 92 U2' + 9 U2" +74* 'r, + (Horizontal Diffusion), (U2, U2) +Ro [Nonlinear Terms(U, U ")] (4.25)
+ L2 [g21 g22(n] at
g11 2T( g21 2T 7
.,/ . 2' v 2 + T r Trit1 + (Horizontal Diffusion), (UI, U;') + [Nonlinear Terms (Uf, U ) (4.26)
where
7,*, = r:*. + (* (4.27)
7,*, = q71,Z*. + q77,* (4.28)
S= in frin (4.29)
4ri4 = pfU, I + U rpf UrZ
7* r i + ri (4.30)
" lpfuZL pfU, Z, F,: = ( + Fy (4.31)
P U,. Z, p f U, Z
F* = [,Z] + 77 y z" (4.32)
Defining
U4 = U + (4.33)
U' = U + U (4.34)
we can obtain the following equations for the entire water column:
aU4 +gmlc 1C7 12 U4 +g22 U
at +H [g11+g = U + + T 4 F
v+ (DifVfusion) [U, U.
+ (Diffusion), [U1, U']
+ (Diffusion), [U,, U2] Ro
+ [Nonlinear Terms (Uf, Ur)] Ro
+ 2 [Nonlinear Terms (U2, U2")] (4.35)
aU'7+ H [g2 22( 9 1 U U .
at' r +~2( 22r,=, /o ,7 'r ,7 F17
+ (Diffusion),[Uf, U1'7] + (Diffusion),[U2, U211 Ro
+ [Nonlinear Terms (Uf, U7)]
Ro
+ 2[Nonlinear Terms (Ut, U27)] (4.36)
The dimensionless continuity equation in the curvilinear grids becomes:
Ct + (V Ut) + a(S.U7)] = 0 (4.37)
CHAPTER 5
HEAT FLUX MODEL
5.1 Introduction
All water bodies exchange heat with the atmosphere at airsea interface. To estimate the net heat flux at the airsea interface, it is necessary to consider seven processes: shortwave solar radiation, longwave atmosphere radiation, reflection of solar radiation, reflection of atmospheric radiation, back radiation, evaporation, and conduction.
One way to estimate the net heat flux is to combine the seven complicated processes to an equation in terms of an equilibrium temperature and a heat exchange coefficient (Edinger and Geyer, 1967) as a boundary condition for the temperature equation as follows:
po&vj= q =K(T T) at zi (5.1)
where T,, the equilibrium temperature, is defined as the water surface temperature at which there is no net heat exchange. This method will be called the "Equilibriumn Temperature Method." Method to estimate the heat flux includes the "Inverse Method" developed by Gaspar et al, 1990.
In the following, two methods for defining the boundary condition are described.
5.2 The "Equilibrium Temperature" Method
To determine an equilibrium temperature, following the procedure first developed by Edinger and Geyer (1967), meteorological data and empirical formulas are required. Data from the South Florida Water Management District (SFWMD) and "Climatological Data" published by the National Oceanic Atmospheric Administra
tion (NOAA) were used.
5.2.1 ShortWave Solar Radiation
The amount of shortwave radiation reaching the earth's atmosphere varies with latitude on the earth, time of day, and season of the year. However, the amount of shortwave radiation is reduced as it pass through the atmosphere. Cloud cover, the sun's altitude, and the atmospheric transmission coefficient affect the amount. Empirical formulas are used (Huber and Perez, 1970) to compute the amount of shortwave radiation.
However, the shortwave solar radiation is more easily measured than computed (Edinger and Geyer, 1967). SFWMD measured the solar radiation at the Station L006, which is located at the south of Lake Okeechobee (See figure 5.1). The unit is mLy/min.
5.2.2 LongWave Solar Radiation
The magnitude of the longwave radiation may be estimated by use of empirical formula. Brunt formula (Brunt, 1932) was used.
H, = a(T, + 460)4(C + 0.031Ven) [BTUFtDay1] (5.2)
Ha = Longwave atmospheric radiation,
Ta = Airtemperature in OF measured about six feet
above the water surface,
where ea = Air vapor pressure in mmHg measured about six feet
above the water surface, and
C = A coefficient determined from the air temperature and
ratio of the measured solar radiation to the clear sky
solar radiation.
5.2.3 Reflected Solar and Atmospheric Radiation
The fractions of the solar and atmospheric radiant energy reflected from a water surface are calculated by using the following reflectivity coefficients Rsr = H37 (5.3)
= (5.4)
Ha
50
Solar Radiation 1500
50
100 1.lt.v Humidity
32.5 Air Temperature
30.0
27.5
25.0
2)
22.5
Julian Day
Figure 5.1: Meteorological data at Station L006.
51
The solar reflectivity, R,,, is a function of the sun's altitude and the type and amount of cloud cover. Because there was no cloud data, 0.1 was assumed for R,,. The atmospheric reflectivity, Rar, 0.03, was assumed following the study of Lake Hefner.
5.2.4 Back Radiation
Water sends energy back to the atmosphere in the form of longwave radiation. This can be calculated by the StephanBoltzman fourthpower radiation law (Edinger and Geyer, 1967 and Harleman, et al., 1973): Hbr = yIU(T, + 460)4 (5.5)
Hbr = Rate of back radiation in BTUFt2Day', where YW = Emissivity of water, 0.97,
a = StephanBoltzman constant (4.15 x 1OBTUFt2Day1 R4), and
T, = Water temperature, OF.
5.2.5 Evaporation
Heat is lost from a body of water to the atmosphere through evaporation of the water. Frequently, evaporation is related to meteorological variables (Brutsaert, 1982). The most general form is
H, = (a + bW)(e., eCa) (5.6)
a, b coefficients depending on the evaporation formula employed,
W = Wind speed in miles per hour,
where ea = air vapor pressure in mmHg, and e, and
e, = Saturation vapor pressure of water determined
from the water surface temperature, T,.
e, are related to the temperature of air and relative humidity ( Lowe, 1977). NOAA data include dailyaveraged values of evaporation, wind speed, and air and water temperature. These data are measured at the Belle Glade Station near the southeastern shore of Lake Okeechobee. Those data are averaged daily. By using the least square method, a and b were determined to be 5.663 and 296.36, respectively.
5.2.6 Conduction
Water bodies can gain or lose heat through conduction due to temperature difference between air and water. Heat conduction is related to evaporation by the Bowen
Ratio (Bowen, 1926).
B= H! (5.7)
H
B=C(T T) P (5.8)
(e, ea) 760
where
P = barometric pressure in mmH2,
C' = a coefficient determined from experiments = 0.26.
Thus, conduction is related to the other parameters as follows:
H, = 0.26(a + bW)(T, Ta) [BtuFt2day'] (5.9)
5.2.7 Equilibrium Temperature
Of the seven processes, four processes are independent of surface water temperature: shortwave solar radiation, longwave atmospheric radiation, reflected solar radiation, and reflected atmospheric radiation. The sum of these four fluxes is called absorbed radiation (HR). Thus the net heat flux can be written as follows: AH = HR Hbt H Hc (5.10)
When the net flux AH is zero, HR becomes
HR = y.(T + 460)4 + 0.26(a + bW)(T, Ta) + (a + bW)(e, ea) (5.11)
The net heat flux can be expressed as follows: AH = K(T, T) (5.12)
where K is the heat exchange coefficient.
5.2.8 Linear Assumption
Vapor pressure difference, e. ea, is assumed to have a linear relationship with temperature increment as
e, ea = P(TI Te) (5.13)
53
Also, the fourthpower radiation term can be approximated by a linear term with less than 15% error (Edinger and Geyer, 1967). Therefore, AH become as follows:
AH = 15.7 + (0.26 + /3)(a + bW)(T T,) + 0.051(T T,2) (5.14)
= K(ToT,)
Neglecting the quadratic term, K = 15.7 + (0.26 + 03)(a + bW) (5.15)
Using the above relation, an equation for T, can be derived as follows:
0.051T2 HR 1801 K 15.7 ea cC(3) + 026T
K + K + 02 + 0.26 + (516)
where c(O) is intercept for the temperature and vapor pressure approximation.
5.2.9 Procedure for an Estimation of K and T,
Step 1. Compute HR.
Step 2. Assume T.
Step 3. Find K for given wind and temperature.
Step 4. Compute the right hand side of Eq. 5.16.
Step 5. Compute the left hand side of Eq. 5.16.
Step 6. Compute the difference of Step 4 and Step 5.
Step 7. If error is not within error limit, go to step 2.
54
Step 8. If error is within error limit (0.5 00), K and T, are correct estimated values.
An actual equilibrium temperature file was created using SFWMD data, which were measured at 15mninute intervals. Wind speeds at Station A,B,C,D,E were used for the computation.
5.2.10 Modification of the Equilibrium Temperature Method
By using the equilibrium temperature method, modelpredicted temperature in Lake Okeechobee was found to be unrealistic. Therefore, b of evaporation formula was multiplied by a factor of 0.1. Further, K, the heat exchange coefficient, at Station C was multiplied by 10 to give stronger stratification.
There are many uncertain empirical coefficients in the computation of an equilibrium temperature. First, evaporation data are averaged daily, but model time step is 5 minutes. That means the estimation of evaporation data at a short interval is difficult. Second, wind speed is also averaged daily, and evaporation is correlated with this average wind speed. In actual computation, wind speed at 15minute intervals was used. Surface water temperature data are uncertain. Considering the sharp gradient of water temperature that usually exists near the water surface, the error can be large. Third, all the meteorological data used are from L006 station. Considering the spatial variation of meteorological condition over the lake, error can be large. Most other thermal models simulate the longterm variation of temperature with a time step of one day. Therefore, it seems that a shortterm variation of meteorological data did not create serious problems.
5.3 The "Inverse" Method
When there are insufficient meteorological data, the errors in the estimation of total heat flux at the airsea interface can be large. To better estimate the total flux, the socalled inverse method (Gaspar et al., 1990) was used in this study.
Total flux (qt) can be divided into two parts: solar (q,,,ia,) and nonsolar (qnonsoiar).
55
While incoming solar radiation data are usually available, the nonsolar part is estimated by solving the vertical onedimensional temperature equation coupled with the momentum equation.
5.3.1 Governing Equations
aT 0 DT a = a (Kv T) (5.17)
Ou v a Ou a fv = (A, ) (5.18)
av a Ov
 + fu = (Av ) (5.19)
5.3.2 Boundary Conditions
At the free surface
aT qt
K, (5.20)
8z p
au r
Ao = (5.21) az p
A, v p (5.22)
A z p
where K, is eddy diffusivity, A, is eddy viscosity, qt is the total heat flux, and r, and r, are wind stresses.
At the bottom
BT
z = 0 (5.23)
az
Tbx = pcd I + V1U, (5.24)
r = p 2 + vv (5.25)
Tby = PCd U1 + V1V1 (5.25)
56
where Cd is the drag coefficient and ul and v, are velocities at the lowest grid point above bottom.
Total flux qt cannot be specified a priori because of unknown nonsolar flux, qnonsolar. Therefore, a value of qnonsoiar is first guessed and then corrected until the calculated and measured water temperatures are within an error limit. In this way, the total flux can be determined by summing up the solar and nonsolar parts. This total flux was later used as a boundary condition of temperature equation in the threedimensional simulation.
5.3.3 FiniteDifference Equation
Treating the vertical diffusion term implicitly, Eq. 5.18 is written in the finitedifference form as follows:
At the interior points,
n+1 un+1) +1 n+1
Un+l U! A '+' ~ 2A )
t n AzA
At f. = Az az(5.26)
where At is time increasement and Az is vertical spacing.
Applying free surface boundary condition,
S +un+1l
Un+1 n  Avm1 A
IM A fv =A Az (5.27)
At ~JIM= Az
Where im is the index of surface layer.
Applying bottom boundary condition,
Un+1 uA = 1 (A, + Cd/rl n+1
At fy = Az (5.28)
Similar form for 5.19 is as follows:
At the interior points,
Unfl Un A (VI+IV+I) A' + fu' = "' A, vi (5.29)v
At A Az(5.29)
At Az
Applying free surface boundary condition,
11n+1 _n+1
nm Avim1 IM iml
ni+1 n AVim I + fun = p vm az (5.30)
At t Az
Applying bottom boundary condition,
n+1 o + V1 + fu =
At +
A1+ C+ u + van+ 1 A,, zv v Cd tl 1 1l~
(5.31)
5.17 can be written as follows:
At the interior points,
Tin+1 Tn
At
K (T 1 Ti ) (Tin+ T+11) Az Az
Applying free surface boundary condition, =l 1
T q ,jjM Az At Az
Applying bottom boundary condition, Tn+I Tn K (T+1 Tn+') Az
At Az
5.3.4 Procedure for an Estimation of Total Heat Flux Step 1. Solve for u for given wind stress.
Step 2. Solve for v for given wind stress.
Step 3. Guess nonsolar heat flux and solve for T.
Step 4. compare the computed tempearature to measured temperature.
Step 5. If error is not within error limit (0.5 'C), go to step 3.
Step 6. If error is within error limit (0.5 0C), go to step 1.
(5.32)
(5.33)
(5.34)
CHAPTER 6
FINITEDIFFERENCE EQUATIONS
This chapter describes the finitedifference equations for the governing equations in 77, a coordinate system.
6.1 Grid System
Earlier models used a nonstaggered grid so that all the variables were calculated at the same point (center point). This has a disadvantage. When centered difference scheme is used, onesided difference scheme near the boundary should be used to maintain the same order of accuracy. Therefore, it is inconvenient. CH3D uses a staggered grid as shown Figure 6.1. Surface elevation and temperature are computed at the center of a cell, while the velocities are computed at the face of a cell. A vertical grid is shown in Figure 6.1, and all the variables are computed at the middle of the layer.
6.2 External Mode
The external mode equations consist of the equations for surface displacement and the verticallyintegrated velocities U and V. Treating the wave propagation terms in the finitedifference equations implicitly and factorizing the matrix equation, the following equations for the isweep and 2/sweep are obtained: iSweep Equations
* + TiaAS (VU*) = Cn at(1 T1)8A(V/U') (6.1)
TlVflS (C*) + [I+ t(T2CT
0 U, u a V, v
6 C w. T,
0 0 0
04040
oT o 04040
0 0 04 0A 0 0 A 0
u
dw
o T,p
A
a
0 0
A
0 0
A
s o e
o 0
A
0 0 0 i// 1// 1'"
z+ h +h h
Figure 6.1: Horizontal and vertical grid system.
0
40d 0 0 404
0 o0
60
= Ti47128,((n) + goU" (6.2)
 (1 T1)vf7118("n) (1 T)V7 128,((")
V gU" At(1 C A t(1 T3) + D"
qSweep Equations
(+f + Tla,,( V ) = + Tia,6,(V0V) (6.3)
T1 722n,((n+1) + 2i t (TC7 + n+ (6.4)
T V 7126(*) (1 Tl)Vf712 ((") (1 Ti)47g226()
+ ViV Vn oV At(1 T2)C7 + At( 3 3)2 + Dn
Ti y 128((n+1) + /Un+1 = v/OU* + TiV/712(() (6.5)
where T1, T2, and T3 are all constants between 0 and 1, superscripts indicate the time level, and Cc and C, are the bottom friction terms. For example, the wave propagation terms are treated explicitly if T, = 0, but implicitly if T = 1. Dn and D' are explicit terms in the U and V equations excluding the Coriolis, bottom friction, and wave propagation terms. Additional parameters appearing in the equations are aia
gdA(
1 Atg
ggd .r~7
 A( (6.6)
12 HAtg12
A2
22 HAtg 2
7A=
21 HAtg2
A77
In the Lake Okeechobee application, the external mode is first solved over the entire lake. For the open water region, the above isweep and 4sweep equations
61
are used. For the vegetation zone, however, the (sweep and 9sweep equations are modified by the presence of vegetation and are derived from Equations (4.23) through (4.26). However, it is only necessary to solve the finitedifference equations for the integrated velocities in the entire water column, (U, V), and the velocities in one layer, (U1, V1) or (U2, V2). The velocities in the other layer can be readily obtained by subtracting the onelayer velocities from the total velocities.
6.3 Internal Mode
The internal mode is defined by the equations for the deficit velocity fi and i, (li, i) (u , v ) The equations for ii and i~ are obtained by subtracting the verticallyaveraged equations from the threedimensional equations for u and v:
a g12 22
0Hii = gnHii + gnHi + F3
lift (at I
E, 8 a 8Hi(.7 + H2 (, "'f (r, rb) F2 (6.7)
Sg1 21
Hi = l HiL H) + G3
+ H2a (a A (r, Tb,,) G2 (6.8)
where fi and are the deficit velocities in the (, ) directions, F3 and G3 indicate all the explicit terms in the u and v equations, respectively, while F2 and G2 indicate all the explicit terms in the U and V equations, respectively.
Applying a twotimelevel scheme to the above equations leads to the following finitedifference equations: 12
(Hii)n+x = (Hii)" + AtP H'+lun+x 22
+At z/ fH"t" + At(F3 F2)
+ At E_ A, (H n+lin+l
(Hn)2a 5 A a\/
 (r't Tb)n+ (6.9)
11
(Hi)n+l = (Hfi)" At 9 Hi5n+1 21
At Hu+Ip+1 + At(G3 G2 )n +At E [A, (Ha n+ )]
(fin)2 [v avJ
 (r7,, rb)+1 (6.10)
For the open water zone, the above internal mode equations are solved after the external mode solutions are obtained. For the vegetation zone, no internal mode equations are solved. This is consistent with the assumption that, in the vegetation zone, the velocities are fairly uniform within the vegetation layer and the vegetationfree layer.
6.4 Temperature Scheme This section describes the finite difference equation which is used for solving the temperature equation. Equation 3.53 is written in finite difference form using the forward scheme in time and the centered difference in the vertical diffusion term.
H, +IT+1 ,At. Ro [8H+Tfk H "T At [(HuTVrS) (6.11)
a 1
+ (HvT ) + o (HwT)"
E,(At) 1 De+ (Tn+1 T+l + Hn+ITo. Auk LA+ +' k
, _T
HEH 2At 82T 82T 22 T n
+ TH (9 2 + 2g128 +22 (6.12)
Dividing the above equation by Hn+1 and collecting all the unknown terms in the lefthand side and the known terms in the righthand side, and writing advection terms and diffusion terms separately,
E, At (D, Tn+1 D+ T+1 (6.13)
(Hn+1)2P, Ack ,,k1 + A ,k+
63
E, At D,+ D,
+ 1 + P ,A + T'(HwT)n
= H+1 'Tj,k npl (HuT i + (HvT + io (HwT)"
Hn EH'At 11a2T 1 2g2 a2T 22a2T n
+ 9+l+ 72 + g22 (6.14)
6.4.1 Advection Terms
Many different schemes can be used for the advection terms. When there is a sharp discontinuity, it is difficult to model the convection without numerical diffusion. Leonard (1979) introduces the QUICKEST( Quadratic Upstream Interpolation for Convective Kinematics with Estimated Stream Terms), which gives good results without excessive numerical diffusion.
This QUICKEST scheme treats the advection terms in the idirection as follows:
8 (.,IgHuT)+ (fVgHuT)0(~HuT) = ( (6.15)
where the first term in the right hand side is the flux at the right face of the cell and the second term is at the left face. These two terms are differenced differently depending on the direction of current as follows: At the right face of a cell :
When u is positive :
(voHuT)+ = (v.H)j+,ijui+,ji,k[ (Ti,j,k + Ti+l,j,k)
 (1 (ui+14,jkAt ))(Ti+lj,k 2Tid,k + Ti1,j,k)
1 Ui+1J,k~t T+,
1 U(Ti+1,j,k Tij,k)] (6.16)
When u is negative:
1
(x/HuT)+ = (.\/H)i+,,ui+,,[,k 1 (Ti,j,k + Ti+l,j,k)
1 ui+LIk~t
 (1 ( ui+l jk 2) Ti+2jk 2Ti+l,i,k + Ti,j,k) 6g
1 Ui+li,kAt
I A (Ti+1,j,k Tij,k)] (6.17)
At the left face of a cell :
When u is positive :
(/HuT)_= ( H)iui,j,k (Ti,jk+ Tij,k)
 (1 (Ui kAt ) )(Ti,j,k 2Ti,i,k + Ti2,j,k)
1 Uijk1 UikA t (Ti,j,k i,j, k)] (6.18)
~2 L k
When u is negative :
1
(f/HuT) = (V/ H)i,jui,j,k[ (Tl,j,k + Tid,k)
1~ Uiijkl )2)T
 (1 ui ))(Ti+1,j,k 2Ti,j,k + Ti1,j,k)
1 Ui,j,kAt
2 A (Ti,j,k Ti1,j,k)] (6.19)
The QUICKEST method treats the 77 direction advection term as follows:
8_ ( fogT)+ ( f~ogT)a(V HvT) = ( HvT)+ ( HvT)(6.20)
At the top face of a cell:
When v is positive:
1
(v/oHvT)+ = (gH)ij+lVi,j+1,k[(Ti,j,k + Ti,j+l,j,k)
2 :~
 (1 (Vij+,kt )2 )(Ti,j+,k 2T,j,k + Ti,j1,k)
6 A
1 vj+1,kAt
 2 A (Ti,j+l,k Tij,k)] (6.21)
When v is negative :
(jHvT)+
 (+H)i,j+xVi,j+l,k 1 Ti,j,k Ti,j+l,j,k)
 (1 (vi'j+kA ))(Ti,j+2,j,k 2Ti,j+1,k + Ti,j,k)
 A (Ti,i+1,k Tijk)]
At the bottom face of a cell :
When v is positive :
(\HvT) _
1
(f H) vi,j,k[ (Ti,j1,k + Tij,k)
 (1 ( VikAt)2 )(T,,k 2Ti,l,k + Tij2,k)
1 vi,j,kAt
2 A (T,,k T,1l,k)]
When v is negative :
(f/HvT)_ =
H) 1
(V/H),jvj,k[I (Tjl,k + Tij,k)
1 vi,j,k At (1 ( V ,k ) 2 Ti,j+l,k 2T,, + T ) I vidj,kAtT. Ti,j l)
2 A ,,:k
The QUICKEST method treats the rdirection advection term as follows:
(8w (HwT)+ (HwT)
(HT)
At the top of a cell :
When w is positive :
(HwT)+ = Hi,wi,1,k[ (T,j,k+l + Ti,j,k)
(6.22)
(6.23)
(6.24)
(6.25)
66
 (1 ( I,, 2 )(j,k+ 2Ti,j,k + Ti,j,k1)
6 AOT
SwikAt (Tijk+ Ti1j,k)] (6.26)
When w is negative :
1
(HwT)+ Hiwi,j,k (Ti,j,k+ + Tij,k)
S(1 ( ,kA )2)(Ti,k+2 2Ti,j,k+l + Ti,j,k) SWikAt (Td,k+l Ti,j,k)] (6.27)
2 Aa
At the bottom of a cell:
When w is positive :
1
(HwT) = H,ywi,j,k Il[(Ti,j,k + Ti,j,k1)
11 wi'ikl.At)2)(..
 1 (i l )(Ti,j,k 2Ti,j,k1 + Ti,j,k2) 6AO
1 Wi~ik.1L~
1 i At (Ti,,k Ti,j,k1)] (6.28)
2 au
When w is negative:
(HwT) = Hi,jwi,j,k1 I[(Ti,j,k + Ti,j,k1)
(1 ( i,i,k1l t
S(1 () ))(Tk+1 2Ti,j,k + Ti,j,k1)
1 Woj,k.lAt (Ti,j,k Ti,j,k1)] (6.29)
2 Aot
6.4.2 Horizontal Diffusion Term
The centered difference scheme was used for the horizontal diffusion. For the mixed derivative term, temperature and depth at the corners of a cell are obtained by averaging the values using four neighboring points at the center points of a cell.
11082(HT) 11 (HT)i+l,j,k 2(HT)i,j,k + (HT)i,j,k (6.30)
9 04 2 li,j ( 2 (6.30)
67
1202(HT) 2
S12 = gf,j[(HT)i+1/2,j+/12,k (HT)i+/2,j1/2,k
 (HT)i1/2,j+l/2,k + (HT)i1/2,j1/2,k/Az q (6.31)
222(HT) = (HT)ij+,,k 2(HT)i,j,k + (HT)i,j,l,k (6.32)
9 07 2 A 2 (6.32)
CHAPTER 7
MODEL ANALYTICAL TEST
The purpose of model analytical test is to examine a model's capability to reproduce wellknown physical phenomena for which the model is designed for, by comparing model results with analytical solution.
7.1 Seiche Test
The CH3D model has been tested for winddriven circulation in an idealized enclosed lake which is 11 km long and 11 km wide with a uniform depth of 5 m. A uniform rectangular grid of 1 km grid spacing was used. To perform the seiche test, the initial surface elevation was given as ( = (, cos(27rx/e) where C, is an amplitude and f is a wave length. In the test, (, was set to 5 cm and t was set to 10 km.
Since the lake is of homogeneous density and without bottom friction and diffusion, seiche period can be calculated as T = 21 where f is the basin length and h is the mean depth. For the test basin, the seiche period is 0.87 hours. The simulated surface elevation in the test basin over a 12hour period is shown in Figure 7.1. The result shows that the surface elevation was not damped and the seiche period agrees with analytical seiche period.
7.2 Steady State Test
When a uniform wind blows in the same direction over a rectangular lake with same magnitude over a long period, the lake circulation eventually reaches steady state. Neglecting advection, horizontal diffusion, and bottom friction, the setup equation can be obtained as follows:
g h(7.1)
69
Surface Elevation at North End 6,
4
2
A 4
86 12
Surface levation at west Ee S eface Elevation at Center Surface E
6 6
4 4
2 2 2
a2 2
*4 4
6 6 6 ,
a 6 2 6 12
Tie.eI(Hoursb Te. i~oura i
Surfae. Ele...to.. at South End
2
4
6
* 6 12
Tim. Houwelt
Figure 7.1: Model results of a seiche test.
Tim.e (4uee
70
where p is density of water, 77 is surface elevation, r, is wind stress, and h is water depth. Using the same rectangular grid in the seiche test, a uniform wind stress of 1 dyne/cm2 from east to west was imposed. After 48 hours, a steady state is reached. As shown in Figure 7.2, the surface elevation has a setup in the western part and setdown in the eastern part. The setup across the lake is 1.12cm, which is exactly the same as given by the analytical solution Eqn. 7.1.
7.3 Effect of Vegetation
In order to investigate the ability of vegetation model to represent the effect of vegetation, CH3D was applied to a rectangular lake with a constant depth of 1 m and horizontal dimensions of 4 km by 9 km. At first, vegetation was not considered, and a wind stress of 5 dyne/cm2 was imposed. Then, vegetation canopy with width of 1 cm and density of 500 stalks / m2 was added to the western half of the lake. After that, vegetation density was increased to 5000 stalks / M2. Vegetation height was assumed to be the same as water depth.
With a time step of 5 minutes, the model was run for 24 hours. Time history of surface elevation in the northern end of the vegetation area was plotted in Figure 7.3. As expected, surface elevation rises slowly for the second case and reaches steady state after 5 hours. With highdensity vegetation, surface elevation rises at a slower rate compared to the second case. When wind blows uniformly over long time, vegetation effect disappears and reaches steady state. Additional resistance term due to vegetation becomes smaller because the currents also become smaller at the steady state and wind stress, and pressure gradient and bottom friction are balanced.
7.4 Thermal Model Test
The purpose of this test is to demonstrate how the velocity can be changed with the consideration of thermal stratification. Surface heat flux was idealized using the sine function as follows:
pCj.,,T.7 = K(T T) (7.2)
71
II
oi
* I" I "H
CON'TOU.R FRI3t 1.000 TO I.0000 Q:N TOUR INTERYtR 7a 0.10000 PTIS.3t= 0.61442
Figure 7.2: Surface elevation contour when the lake is steady state with uniform wind stress of1 dyne/cm .
Time history of surface elevation
01
0 5 10 15 20
Hour
Figure 7.3: Effect of vegetation on surface elevation evolution in a winddriven rectangular lake. Solid line is without vegetation, broken line is with low vegetation density, and dotted line is with high vegetation density.
73
Te = 26 + 10 si(27rt (7.3)
where p is density of water, Cp, is specific heat of water, K is heat exchange coefficient, T. is an equilibrium temperature in 0 C, and P is period of 24 hours. Also, wind stress was idealized as shown in Figure 7.4. Time history of currents at all five layers are shown in Figures 7.4 and 7.5. It is apparent that when thermal stratification is considered, currents at the surface layer are much stronger during increasing wind condition because initial momentum is confined to a thinner surface layer.
74
Wind Stress at center
0. 12. 24. 36.
48. 60. 72. 84. 96.
Velocity at center without temperature
0. 12. 2'4. 36. 48.
Hours
60. 72. 84.
Figure 7.4: Time history of wind stress and currents at the center of lake at all five levels. Thermal stratification is not considered.
E
0.0
C
1.5
I I
10.
20.
i i
Wind Stress at center
I i T
I I I
0. 12. 24. 36. 48.
72. 84. 96.
Velocity at center with temperature
I I S I
I I]
K
0. 12. 24. 36. 48. 60. 72. 84. 96.
Hours
Figure 7.5: Time history of wind stress and currents at the center of lake at all five levels. Thermal stratification is considered.
ClJ
E
C
1 .5
10.
IT. "
CHAPTER 8
MODEL APPLICATION TO LAKE OKEECHOBEE
8.1 Introduction
Before the description of the application of CH3D to simulate the winddriven circulation in Lake Okeechobee, it is worth investigating the characteristics of the lake.
8.1.1 Geometry
Lake Okeechobee, located between latitudes 27'12'N and 26'40'N and longitudes 80037'W and 8108'W, is the largest freshwater lake in America, exclusive of the Great Lakes. With an average depth of approximately 3m, and the deepest part less than 5m deep, the basin is shaped like a very shallow saucer. The western part of the lake contains a great deal of emergent and submerged vegetation. According to satellite photos, marsh constitutes 24% of the lake surface area.
8.1.2 Temperature
Due to the location of the lake in subtropical latitude, the annual fluctuations of water temperatures are relatively small. The mean lake temperature based on SFWMD monitoring in the 1970s and 1980s ranges from 15'C to 3400 (Dickinson et al., 1991).
8.2 Some Recent Hlydrodynamic Data
During the fall of 1988 and the spring of 1989, field data were collected by the Coastal and Oceanographic Engineering Department, University of Florida (Sheng et al., 1991a). Details of the field experiments and field data are described by Sheng et al. (1991a). For completeness, some 1989 field data are described briefly in this
section.
Six platforms were set up in Lake Okeechobee. Locations of the six platforms are shown in Figure 8.1. Station A was located in the northern portion of the lake, east of north lake shoal, during the 1988 deployment. This Station A was moved south of the rocky reef area during the 1989 deployment. The other five platforms remained at the same locations during the 1988 and the 1989 deployments. Station B was located near the Indian Prairie Canal. Station C was located in the center of the lake. Station D was located 1.5 krn west of Port Mayaka Lock. Station E was located about 1 mile from the boundary between the vegetation ione and the open water, i.e., the littoral zone. Station F was located within 30 rn from the littoral zone.
Station A was selected to quantify the flow system in the northern zone during the 1988 field survey. Because there was no measurement in the southern zone, it was moved south to quantify the flow during the 1989 field survey. Station B was selected to quantify the flux in the northern littoral zone. Stations C and D were selected to calibrate the model and quantify the flux in the mud zone. Stations E and F were selected to help the computation of phosphorus flux between the vegetation and the open water.
Measured data at these six platforms include wind, current, water temperature, wave, and turbidity. In this study, wind data were used to compute the wind stress field, which is an essential boundary condition for the simulation of the winddriven circulation. Current data were used to calibrate and validate the 3D hydrodynamics model. The installation dates and locations of the platforms during the two deployment periods are shown in Table 8.1.
8.2.1 Wind Data
Wind speed and direction data averaged over 15minute intervals were collected from five stations in Lake Okeechobee during the spring of 1989. Because Station F was close to Station E, wind data were, not collected at Station F. The data collection
[] Emergent Vegetation E] Submerged Vegetation
0 5 10 Miles
i I i
81000'
80045.
Figure 8.1: Map of Lake Okeechobee.
Table 8.1: Installation dates and locations of platforms during 1988 and 1989.
TIME OF LOCATION DATE 1LATIT UDE LONGITUDE DEPTH
YEAR 1(cm)
Site A 092088 27 06.31 80 46.21 396.0
Site B 091788 27 02.78 80 54.31 274.0
FALL Site C 092188 26 54.10 80 47.36 518.0
Site D 092188 26 58.47 80 40.34 457.0
Site E 091888 26 52.81 80 55.96 274.0
______ Site F 091988 26 51.90 80 57.09 183.0
Site A 051689 26 45.67 80 47.83 183.0
Site B 051889 27 02.78 80 54.31 152.0
SPRING Site C 052089 26 54.10 80 47.36 366.0
Site D 052089 26 58.47 80 40.34 335.0
Site E 051989 26 52.81 80 55.96 152.0
______ Site F 051889 26 52.03 80 56.91 91.0
started on Julian Day 136.708. However, the direction of the anemometer was not properly oriented until Julian Day 141.5. The location and height of the anemometer are shown in Table 8.2.
As described in Sheng et al. (1991a), the measured wind over Lake Okeechobee often exhibited significant diurnal variations associated with the lake breeze. During relatively calm periods, significant spatial variation is often found in the wind field.
Water motion in the lake is significantly influenced by the wind. Figure 8.2 shows the wind rose diagram at Station C between Julian days 147 and 161. The number inside the triangle indicates the percentage of wind data in that direction. For example, wind from east to west is 27%. Wind speed between 46 rn/sec is about 45%. The governing wind direction is from east to west due to the location of Lake Okeechobee. The surface area of Lake Okeechobee is big enough to create its own lake breeze. During the daytime, wind blows from lake to land because the air over the land is warmer than that over the lake. Because the Florida peninsula is located between the Atlantic Ocean and the Gulf of Mexico, sea breeze affects the wind direction. As Pielke (1974) indicated, the typical summer wind direction
Table 8.2: Instrument mounting, spring deployment.
SITE ARM CURRENT ELEV AZIM TEMP ELEV OBS ELEV
I I (cm) (deg) (cm) (cm)
A Pressure Sensor 55695, Elev. 96 cm
Wind Sensor 5200, Elev. above MWL 670 cm minus water depth 1 80673 71 342 07 86 0076 86
B Pressure Sensor 55696, Elev. 86 cm
Wind Sensor 5202, Elev. above MWL 518 cm minus water depth 1 80674 25 204 04 43 0078 43
2 80675 114 181 02 132 0075 132
C Pressure Sensor 48228, Elev. 297 cm
Wind Sensor 5203, Elev. above MWL 884 cm minus water depth 1 80679 61 330 01 79 0079 79
2 80681 123 333 09 140 0077 140
3 80680 284 342 06 302 0084 302
D Pressure Sensor 55694, Elev. 254 cm
Wind Sensor 5200, Elev. above MWL 883 cm minus water depth 1 80672 79 270 08 97 0083 97
2 80677 241 255 03 259 0082 259
E Pressure Sensor 55699, Elev. 104 cm
Wind Sensor 5199, Elev. above MWL 518 cm minus water depth 1 80671 36 59 11 53 0081 53
2 80678 116 72 05 135 0080 135
F Pressure Sensor 55697, out of water
No Wind Sensor
1 80676 25 305 10 43 0085 43
81
is from southeast or southwest in south Florida. Those wind systems determine the dominant wind direction in Lake Okeechobee. For the synoptic study of spring 1989, the mean wind speed is about 5.1 rn/see, but it can exceed 11 rn/sec. The wind field is characterized by temporal and spatial nonuniformities. However, during strong wind periods, the wind tends to be more uniform.
8.2.2 Current Data
Current data were collected at 15minute intervals at the six locations. The data collection started on Julian Day 136.708. The number of instrument arms at each station depended upon the water depth and how many vertical levels of data were desired. At station C, which is located in the center of lake, three current meters were installed to measure the vertical variation of currents. Two sensors were installed at Stations B, D and E. One sensor was installed at Stations A and F. The location and height of each of the current meters are shown in Table 8.2.
As discussed in Sheng et al. (1991a), current data showed significant diurnal variations in direct response to the wind. During a period of significant change in wind direction, which usually follows a peak wind period, seiche oscillation generally leads to significant current speed over several seiche periods (multiples of 5 hours).
Magnitudes of currents are very small at all stations. At Station C, mean magnitudes of u and v at arm 3 between Julian day 147 and 156 are 2.09 and 1.52 ern/sec, respectively. Maximum magnitudes of u and v at arm 3 are 11.7 and 6.8 ern/sec, respectively. Considering the accuracy of current meter, 23 ern/sec, currents are very small.
8.2.3 Temperature Data
The lake temperature data showed that a significant vertical temperature gradient can be developed during part of a day when wind is low and atmospheric heating is high. However, over the relatively shallow littoral zone and transition zone, temperature appeared to be well mixed vertically much of the time.
82
LEGEND
24 1
62 .. .. ...
A
WIND ROSE AT STATION C
Figure 8.2: Wind rose at Station C.
83
Lake Okeechobee does not have a strong thermal stratification. But during the daytime, lake becomes stratified. Temperature field data show the water temperature difference between the near top and near bottom at Station C can reach 4 00. However this stratification disappears in a short time due to the strong wind. This stratification can affect the eddy viscosity significantly.
8.2.4 Vegetation Data
The distribution of aquatic vegetation in Lake Okeechobee was determined by the use of recent satellite imageries and ground truth data obtained during field surveys. Satellite imageries from the SPOT satellite were received and processed to create a vegetation map of Lake Okeechobee (J. R. Richardson, personal communication, 1991). This map was overlayed onto a curvilinear grid created by us. The number of pixels with the same color was counted to classify the vegetation type on each grid cell. The vegetation data include the vegetation class, the height of vegetation, the number of stalks per unit area, and the diameter of each class. A total of more than 25 kinds of vegetation were identified. The range of vegetation height is between 0.5 m and 4 m. The density ranges from 10 stalks per m' to 2000 stalks per m' of bottom area, while the diameter of the stalks ranges from 0.25 cm to 15 cm. The most popular type of vegetation was cattail. These vegetation data were provided by John Richardson from the Department of Fisheries and Aquaculture, University of Florida.
8.3 Model Setup
The following describes the procedure for simulation of winddriven circulation in Lake Okeechobee during spring 1989.
8.3.1 Grid Generation
The first step is to select the grid and grid size. Rectangular Cartesian grids were widely used in hydrodynamic models. These grids are easy to generate but have a disadvantage. Because these grids have to represent the boundaries in a stairstepped fashion, they cannot represent the complex geometry accurately unless a large number

Full Text 
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE. FLORIDA: THE
EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
By
HYE KEUN LEE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
66
i (l t..mi 2T..t+Tijti)
1 ujjj'kA t
2 Act
When u) is negative :
.1
(HuT)+ Hi, ju>i (7, j, k+i + Tijtk)
g(! ~ 2Ty.l+1 + Tijj.)
1 ujiijikAt
2 Act
~(Ti,k+1
At the bottom of a cell
When u) is positive :
(HuiT)_ i [2
_ 1(1 TiJj. 2TÂ¡lki + Tijtz)
1 Uj'j'kiAt
2 Act
(Ti,j,k I'iJJcl)]
When u is negative :
rl
(HuT)_ Hi'jUJij'kii^iTij'k + T^_i)
ln ,^i,j,kiAt
~ 6<>"<
Act
)2)(ii,J>+l 27U* + Ti,i,kl)
1
2 Act ^
6.4.2 Horizontal Diffusion Term
(6.26)
(6.27)
(6.28)
(6.29)
The centered difference scheme was used for the horizontal diffusion. For the
mixed derivative term, temperature and depth at the corners of a cell are obtained
by averaging the values using four neighboring points at the center points of a cell.
xld\HT) (.HT)i+hjik 2(HT)i
9 9i,j (AO2
d?
(6.30)
CHAPTER 5
HEAT FLUX MODEL
5.1 Introduction
All water bodies exchange heat with the atmosphere at airsea interface. To
estimate the net heat flux at the airsea interface, it is necessary to consider seven
processes: shortwave solar radiation, longwave atmosphere radiation, reflection of
solar radiation, reflection of atmospheric radiation, back radiation, evaporation, and
conduction.
One way to estimate the net heat flux is to combine the seven complicated pro
cesses to an equation in terms of an equilibrium temperature and a heat exchange
coefficient (Edinger and Geyer, 1967) as a boundary condition for the temperature
equation as follows:
= 9s = K(T Te) at z = t] (5.1)
where Te, the equilibrium temperature, is defined as the water surface temperature
at which there is no net heat exchange. This method will be called the Equilib
rium Temperature Method. Method to estimate the heat flux includes the Inverse
Method developed by Gaspar et al, 1990.
In the following, two methods for defining the boundary condition are described.
5.2 The Equilibrium Temperature Method
To determine an equilibrium temperature, following the procedure first devel
oped by Edinger and Geyer (1967), meteorological data and empirical formulas are
required. Data from the South Florida Water Management District (SFWMD) and
Climatological Data published by the National Oceanic Atmospheric Administra
48
165
V/O TEMP
WITH TEMP
18 HR
19 HR
20 HR
21 HR
U
22 HR
23 HR
U
V
T SCALE
dyne/cm**2
Figure 8.64: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
78
Figure 8.1: Map of Lake Okeechobee.
.00 'LZ ,0MZ
/o
Wind Stress at center
1 .5 : i ] i i t i i t Â¡ i  1 : r
Velocity at center with temperature
Hours
Figure 7.5: Time history of wind stress and currents at the center of lake at all five
levels. Thermal stratification is considered.
142
Velocity at Platform D* 3D. Intarpo 1 a ted(Arm 1) W/ Temp
JulIan Day
Figure 8.44: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction) when thermal effect is considered.
95
Vegetation Height
Unit i m
77
section.
Six platforms were set up in Lake Okeechobee. Locations of the six platforms are
shown in Figure 8.1. Station A was located in the northern portion of the lake, east
of north lake shoal, during the 1988 deployment. This Station A was moved south of
the rocky reef area during the 1989 deployment. The other five platforms remained at
the same locations during the 1988 and the 1989 deployments. Station B was located
near the Indian Prairie Canal. Station C was located in the center of the lake. Station
D was located 1.5 km west of Port Mayaka Lock. Station E was located about 1 mile
from the boundary between the vegetation zone and the open water, i.e., the littoral
zone. Station F was located within 30 m from the littoral zone.
Station A was selected to quantify the flow system in the northern zone during
the 1988 field survey. Because there was no measurement in the southern zone, it was
moved south to quantify the flow during the 1989 field survey. Station B was selected
to quantify the flux in the northern littoral zone. Stations C and D were selected to
calibrate the model and quantify the flux in the mud zone. Stations E and F were
selected to help the computation of phosphorus flux between the vegetation and the
open water.
Measured data at these six platforms include wind, current, water temperature,
wave, and turbidity. In this study, wind data were used to compute the wind stress
field, which is an essential boundary condition for the simulation of the winddriven
circulation. Current data were used to calibrate and validate the 3D hydrodynamics
model. The installation dates and locations of the platforms during the two deploy
ment periods are shown in Table 8.1.
8.2.1 Wind Data
Wind speed and direction data averaged over 15minute intervals were collected
from five stations in Lake Okeechobee during the spring of 1989. Because Station F
was close to Station E, wind data were not collected at Station F. The data collection
184
Velocity at Platform C 3D, Interpolated(Arm 1) Inverse
JulIan Day
Figure A.8: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
51
The solar reflectivity, R,r, is a function of the suns altitude and the type and amount
of cloud cover. Because there was no cloud data, 0.1 was assumed for Rsr. The
atmospheric reflectivity, Rar, 0.03, was assumed following the study of Lake Hefner.
5.2.4 Back Radiation
Water sends energy back to the atmosphere in the form of longwave radiation.
This can be calculated by the StephanBoltzman fourthpower radiation law (Edinger
and Geyer, 1967 and Harleman, et al., 1973):
Hbr = 7w
(5.5)
where
5.2.5
Hbr = Rate of back radiation in BTUFt~2Day1,
7u, = Emissivity of water, 0.97,
cr = StephanBoltzman constant (4.15 x 108BTUFt~2Day1 R~4), and
Ts = Water temperature, F.
Evaporation
Heat is lost from a body of water to the atmosphere through evaporation of
the water. Frequently, evaporation is related to meteorological variables (Brutsaert,
1982). The most general form is
He = (a + bW)(e3 ea)
(5.6)
a,b =
W =
where ea =
ej =
ea are related to
coefficients depending on the evaporation formula employed,
Wind speed in miles per hour,
air vapor pressure in mmHg, and e3 and
Saturation vapor pressure of water determined
from the water surface temperature, Ts.
the temperature of air and relative humidity ( Lowe, 1977). NOAA
data include dailyaveraged values of evaporation, wind speed, and air and water tem
perature. These data are measured at the Belle Glade Station near the southeastern
shore of Lake Okeechobee. Those data are averaged daily. By using the least square
method, a and b were determined to be 5.663 and 296.36, respectively.
5.2.6 Conduction
Water bodies can gain or lose heat through conduction due to temperature differ
ence between air and water. Heat conduction is related to evaporation by the Bowen
{degree)
162
Temperature at Platform E i MODEL
Figure 8.61: Simulated and measured temperatures at Station E.
168
are generally small while eastwest currents are stronger. Therefore, magnitude of
bottom stress in the northsouth direction is small and affects currents little.
Drag coefficient can be estimated from the turbulent theory in the bottom bound
ary lyer. Roughness heights of 0.5 and 0.01 were used (cases 9 and 10). The index
of agreement was not changed much. Therefore, the choice of 0.01 in the base run
(case 1) seems to be appropriate.
A different method to estimate the drag coefficient in the smooth bottom bound
ary was used, and the index of agreement was dropped (case 11).
8.9.2 Effect of Horizontal Diffusion Coefficent
Horizontal diffusion coefficients of 104 and 103 were used (cases 4 and 5). The
change of index of agreement is hardly noticeable.
8.9.3 Effect of Different Turbulence Model
Three different turbulence models were used (cases 1, 7, and 8). Constant eddy
viscosity of 20cm2/sec was used in case 8. Simulated currents did not agree well
compared to case 1. Turbulence can not be appropriately resolved by use of constant
eddy viscosity because turbulence motion varies spatially and temporally.
The MunkAnderson type was used for the parameterization of turbulence (case
7). The effect of stratification due to temperature and/or salinity is considered by
multiplying the stability function. However, as discussed by Sheng (1983), there
are many formulas, and lots of field data are required to get bestfit. The index of
agreement was not improved compared to case 1, which used a simplified secondorder
closure model.
8.9.4 Effect of Advection Term
To investigate the importance of the advective term, CH3D was run with the
advection term (case 11). The Rossby number, which is an indicator of the importance
of the advective term relative to the Coriolis term, is defined as U/ fL. Taking
the charcteristic values (U = 10cm/sec, f = 6.62e~5, and L = 37km), the Rossby
49
tion (NOAA) were used.
5.2.1 ShortWave Solar Radiation
The amount of shortwave radiation reaching the earths atmosphere varies with
latitude on the earth, time of day, and season of the year. However, the amount
of shortwave radiation is reduced as it pass through the atmosphere. Cloud cover,
the suns altitude, and the atmospheric transmission coefficient affect the amount.
Empirical formulas are used (Huber and Perez, 1970) to compute the amount of
shortwave radiation.
However, the shortwave solar radiation is more easily measured than computed
(Edinger and Geyer, 1967). SFWMD measured the solar radiation at the Station
L006, which is located at the south of Lake Okeechobee (See figure 5.1). The unit is
mLyjmin.
5.2.2 LongWave Solar Radiation
The magnitude of the longwave radiation may be estimated by use of empirical
formula. Brunt formula (Brunt, 1932) was used.
Ha =
where
5.2.3
H0 = Longwave atmospheric radiation,
Ta = Airtemperature in F measured about six feet
above the water surface,
ea = Air vapor pressure in mmHg measured about six feet
above the water surface, and
C = A coefficient determined from the air temperature and
ratio of the measured solar radiation to the clear sky
solar radiation.
Reflected Solar and Atmospheric Radiation
(5.2)
The fractions of the solar and atmospheric radiant energy reflected from a water
surface are calculated by using the following reflectivity coefficients :
H,
jy _
" ~ H,
(5.3)
Rar =
Ha
Ha
(5.4)
dyne/cm**2
144
Velocity at Platform D 3D. InterpoI ated(Arm 2) W/ Temp
JulIan Day
Figure 8.46: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction) when thermal effect is considered.
ulcm/secl dyne/cm**2
74
Wind Stress at center
Velocity at center without temperature
Hours
Figure 7.4: Time history of wind stress and currents at the center of lake at all five
levels. Thermal stratification is not considered.
cm**2.sec dyne**2/cm**4.sec dyne**2/cm**4.sec
171
Figure 8.66: Spectrum of wind stress and surface elevation
11
depthintegrated momentum equation. Details of their vegetation models are given
in Chapter 4. Their model, however, considered only the linearized equations of
motion. In the present study, fully nonlinear equations are considered.
Sheng (1982) developed a comprehensive vegetation model by including the effect
of vegetation on mean flow and secondorder correlations in a Reynolds stress model.
Although the model was able to faithfully simulate the mean flow and turbulence in
the presence of vegetation, it was not used for the present study due to the extra
computational effort required when it is combined with a 3D circulation model.
Roig and King (1992) formulated an equivalent continuum model for tidal marsh
flows. Neglecting leafiness, flexibility, and vegetation surface roughness, the net re
sistance force due to vegetation is thought to be related to the following parameters:
tv = f(p,9,l*,u,l3,d,s) (2.5)
where p is the viscosity of water, u is depthaveraged velocity, d is the average diameter
of vegetation, l3 is the vegetation height, and s is the spacing between vegetations.
Through a dimensional analysis,
T=pu*jf(F,R,i) (2.6)
where F is the Froude Number and R is the Reynolds Number.
To determine the function /, they conducted a simple flume experiment. For each
value of s/d, the dimensionless shear parameter pvaj was plotted as a function of R
and F.
2.3 Thermal Models
Sundaram et ah (1969) used a onedimensional vertical model to demonstrate
the formation and maintenance of thermocline in a deep stratified lake. The surface
boundary condition was given as follows:
?. = pKh = K(T,~ T.)
(2.7)
96
whereupon the equivalent vegetation density is
N = Sf(A* Hcanopy d
canopy ) (823)
where dcanopy is assumed to be 1 cm. However, if we assume the height of each
vegetation type is the same as average water depth in the cell, h, is replaced by D in
Equation (8.21) and Hcanopy is replaced by D in Equation (8.23). The distribution
of N in Lake Okeechobee is shown in Figure 8.7.
The value of the profile drag coefficient for a cylinder, c, was set to be 1.77 in
this study. Whitaker et al. (1975) estimated the interfacial stress coefficient, c,
by calibrating the onedimensional surge model with the observed quasi steadystate
surface profile in Lake Okeechobee.
8.4.5 Wind Stress
In Lake Okeechobee, as in most shallow lakes, momentum is imparted to the water
primarily by the action of surface wind stress. Wind over Lake Okeechobee varies on
a time scale less than one hour, and a spatial scale of about 10 km. Therefore, a
time and spacevarying wind stress field is required. Wind speed and direction data
averaged over 15minute intervals were collected from five stations in Lake Okeechobee
during the spring of 1989. As for the bottom stress, the wind stress at the surface
was computed from the quadratic form:
riX = PaCdaWxyJW2 + Wy2 (8.24)
TSV = PaCdaWyy/W* + WÂ¡ (8.25)
where rsx and t,v are the surface wind stresses in the x and y directions, respectively;
pa = 127 x 103gcm~3 is the density of air; Wx and Wy are the wind speeds in
the x and y directions, respectively; and Ca is the drag coefficient of air. The drag
coefficient of air was computed from Garratts formula:
Cda = 0.001 (o.75 + 0.00067^^2 + Wy2)
(8.26)
dyne/cm**2
121
Wind Stress at Platform Ai tau x (MODEL)
Velocity at Platform A* 3D. InterpoI ated(Arm 1) W/0 Temp
Julian Day
Figure 8.25: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction). Thermal stratification was not considered in model '
simulation.
16
momentum equation and heat equation, and (iii) take timeaverage of all equations.
For example, the resulting timeaveraged equation for uu' (e.g., Donaldson, 1973;
Sheng, 1982) is
du;u
t a 1 (
ut OXk
rrduj
'dxk
}rdui Ujp Uip
 uiukx ^ Si + 9j
dxk p0 p0
 2eiktkutuj 2ejkÂ£ltukui
(dukUjUj)
dxk
(2.16)
u\ dp u'j dp d2uiu': du'i du]
3 + v\r22u 1 3
p dxj p dxi dxkdxk
dxk dxk
Similar equations for u{p and p p can be obtained. Unresolved thirdorder cor
relations and pressure correlations are modeled using the simplest possible forms
(Donaldson, 1973).
du{ du'j bSijq3 avu\u
Vdxkdxk 3A A2
(2.17)
p du'i du'j,
r ( i "J \ H f I T r 9 \
p dx, + dxj A u,Uj 6x1 3 1
(2.18)
d du'ip t du'jp dqA tqu'
gXj + dx. ~ 'c dxt dxt
(2.19)
where q is the total fluctuating velocity and A is the turbulence macroscale. The
model constants (a, 6, and vc) are determined from a wide variety of laboratory data
(Lewellen, 1977). Thus, a full Reynolds stress model consists of six equations for
velocity fluctuations three equations for the scalar fluxes, u\p', and one equation
for the variance, p'p'. Considering the required computer storage and CPU time for
the turbulence models, it is desirable to use a simplified form of the Reynolds stress
model.
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure Model
The complete secondorder closure model is too complicated to be used in a three
dimensional model. A simplified secondorder closure model can be developed with
A. 14 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 190
A.15 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 191
A. 16 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 192
A.17 Simulated isolid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 193
A.18 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 194
A. 19 Simulated isolid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 195
A.20 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 196
xii
8
where v is eddy viscosity, vQ is eddy viscosity near surface, and & is a constant (0.4).
2.1.4 TimeDependent 3D Models
ModeSplitting
In order to solve the dependent variables with the unsteady threedimensional
model, Simons (1974) used a socalled mode splitting method for Lake Ontario
while Sheng et al. (1978) used a somewhat different method for Lake Erie. Defining
the perturbation velocity = u TZ, v = v v where u,v are depthaveraged veloci
ties, and u,v are instantaneous velocities, Sheng and Butler (1982) derived governing
equations for u, v by subtracting the verticallyaveraged equations from the momen
tum equations. Therefore the solution procedure consists of an external mode, which
includes the surface elevation and u and , and an internal mode, which includes ,v
and temperature.
Time Integration of 2D equations
Time integration is important for improving the efficiency of numerical models.
When the explicit method is used, the time step is limited by the Courant condition,
which is Cj^; < 1. Therefore, explicit method is not desirable for longterm simu
lations. Leendertse (1967) used the ADI (Alternate Direction Implicit) method to
simulate tidal currents in the southern North Sea. All terms in the continuity equa
tion and pressure terms in the momentum equation were treated implicitly, while
the other terms were expressed explicitly. After factorization of the finitedifference
equations, the resulting unknowns are solved by inversion of tridiagonal matrices in
the x sweep and y sweep.
Vertical Grid
Various types of vertical grids are used in numerical models of lake circulation.
The earlier models generally used multiple vertical layers of constant fixed thickness
(zgrid) which do not change with time (Eulerian grid) as used by Leendertse (1975).
This type of model needs a large number of vertical grid points in order to accurately
71
Figure 7.2: Surface elevation contour when the lake is steady state with uniform wind
stress of 1 dyne/cm2.
42
Figure 4.1: Schematics of flow in vegetation zone.
181
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B> 3D, InterpoIatedI Arm 21 Inverse
Julian Day
Figure A.5: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
LIST OF FIGURES
3.1 A computational domain and a transformed coordinate system. 32
4.1 Schematics of flow in vegetation zone 42
5.1 Meteorological data at Station L006 50
6.1 Horizontal and vertical grid system 59
7.1 Model results of a seiche test 69
7.2 Surface elevation contour when the lake is steady state with uni
form wind stress of 1 dyne/cm2 71
7.3 Effect of vegetation on surface elevation evolution in a winddriven
rectangular lake. Solid line is without vegetation, broken line is
with low vegetation density, and dotted line is with high vegetation
density. 72
7.4 Time history of wind stress and currents at the center of lake at
all five levels. Thermal stratification is not considered 74
7.5 Time history of wind stress and currents at the center of lake at
all five levels. Thermal stratification is considered 75
8.1 Map of Lake Okeechobee 78
8.2 Wind rose at Station C 82
8.3 Computation domain of Lake Okeechobee 85
8.4 Curvilinear grid of Lake Okeechobee 86
8.5 Depth contour of Lake Okeechobee when the lake stage is 15.5 ft.
Unit in cm 87
8.6 Distribution of vegetation height in Lake Okeechobee 95
8.7 Distribution of vegetation density in Lake Okeechobee 97
8.8 Steadystate depthintegrated currents (cm2s~1) in Lake Okee
chobee forced by an easterly wind of 1 dyne/cm2 100
vi
169
number is 0.04. Compared with case 1, the index of agreement was not changed
much. Therefore the advection term is negligible.
8.10 Spectral Analysis
Spectral analysis can reveal the important frequencies which are related to the
physical processes. Surface elevation and currents at Station C were used for spectral
analysis. Data on Julian day 147 were excluded because CH3D started from zero
velocity. Also, data after Julian day 158 were excluded because measured data had
bad measurements. Dominant periods of wind are 5.8 day, 23 hour and 11.6 hour as
shown Figure 8.66. A component of 5.8 day period is related to the longterm trend
of wind. Components of 23 hour and 11.6 hour are related to the diurnal variation of
wind.
As shown in Figure 8.66, the spectrum of surface elevation shows the frequencies
of diurnal and semidiurnal components. Magnitude of the spectrum corresponding
to the first mode seiche period (about 4 hours) is relatively small compared with the
diurnal component because Station C is located near the nodal point. The spectrum
of simulated currents agrees well with the spectrum of measured currents as shown
in Figure 8.68. Not only period but also magnitudes of the spectrums agree well.
38
in the transformed equations. Thus, the surface stress in the transformed coordinate
system is given by
r1 = r + r!
ox ay
r* = dvr 1,^2
dx ^ dy
(3.56)
(3.57)
where r1, r2 are the contravariant components of the stress in the transformed system
and r1, r2 are the contravariant components in the Cartesian system. Note that in
the Cartesian system, the contravariant, covariant, and physical components of a
vector are identical. The contravariant components of the initial velocity vectors can
be transformed in the same manner to obtain the proper initial conditions for the
transformed momentum equations.
189
Wind Stress at Platform D tau x (MODELJ
Velocity at Platform D 3D, Interpolated(Arm 1) Inverse
Ju l Ian Day
Figure A.13: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
132
Velocity at Platform B 3D, Interpo1 ated(Arm 1) W/ Temp
Julian Day
Figure 8.34: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: NorthSouth direction) when thermal effect is considered.
187
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform C. 3D, InterpolatedI Arm 3) Inverse
Julian Day
Figure A.ll: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform Ci 3D, Interpo1ated1Arm 3) W/ Temp
Julian Day
Figure 8.41: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction) when thermal effect is considered.
22
Table 2.2: Application features of lake models.
Author
Basin
tion
Dimen
Mean
Depth
Forcing
Vege
tation
Grid
W
H
R
Hor.
ver.
Liggett
1969
Idealized
Basin
Yes
No
No
No
U
Lee + Liggett
1970
Idealized
Basin
Yes
No
No
No
U
Liggett + Lee
1971
Idealized
Basin
Yes
No
No
No
u
Gedney f Lick
1972
Lake Erie
400 km
100 km
20 m
Yes
No
Yes
No
u
Sengupta + Lick
1974
Squire
Valley
1.89 m
Yes
Yes
No
No
N
Simons
1974
Lake
Ontario
Yes
Yes
No
No
u
Sheng
1975
Lake Erie
400 km
100 km
20 m
Yes
No
Yes
No
N
cr
Thomas
1975
Idealized
Basin
Yes
No
No
No
U
Whitaker et al.
1975
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
Yes
u
Witten + Thomas
1976
Idealized
Basin
300 x
87 km
Max
180 m
Yes
No
No
No
Lien + Hoopes
1978
Lake
Superior
Yes
No
No
No
u
Schmalz
1986
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
No
u
Sheng + Lee
1991a
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
Yes
c
a
* W : Wind
* H : Heating
* R : River
* U : Uniform Cartesian grid
* C : Curvilinear grid
* N : Nonuniform Cartesian grid
* a : Vertically stretched grid
Watt/M* *2
155
1500
1000
500
0
500
1000
1500
Figure 8.55: Time history of heat fluxes at Station C between Julian davs 147 and
161.
Heat Flux
148
150
152
154
Julian Day
20
other portion of the lake can become flooded because of excessive setup by wind.
A threedimensional Cartesiangrid hydrodynamic and sediment transport model
for Lake Okeechobee was recently developed (Sheng et al., 1991a; Sheng, 1993). In
addition, these models were extended to produce a threedimensional phosphorus
dynamics model (Sheng, et al., 1991c). These models use the simplified secondorder
closure model and the sigma stretched grid, however, did not consider the effects of
vegetation and thermal stratification
2.6 Present Study
The present work focuses on the study of effects of vegetation and thermal strat
ification on winddriven circulation in Lake Okeechobee. As will be shown later, the
threedimensional curvilineargrid model (CH3D) will be significantly enhanced to
allow accurate simulation of the observed circulation. Model features are compared
with model features of some previous lake studies in Tables 2.1 and 2.2. It is apparent
that the 3D model developed in this study is more comprehensive than those used
in previous lake studies.
59
o u, u
a v, v
A
B
w. T,
a
A
A
A
B
A
A
A
A
0
A
A
A
A
a
A
B
O A
a
o A
a
o A
o A
O A
B
u
A w
O T, p
A Z
o
A
o
A
o
A
o
A
o
irrrfTrr
__ Z'C z
C h
Figure 6.1: Horizontal and vertical grid system.
87
CONTOUR FROM 20.>00 TO 420.00 CONTOUR INTERVAL OF 20.000
Figure 8.5: Depth contour of Lake Okeechobee when the lake stage is 15.5 ft. Unit
m cm
37
Tj^\k^HuT)+i^HvT)
I
PrH
,21 r
RodHuT
H da
+ ^Iur.u+,T
+ 9*'Tu + gT,2,2
3.10 Boundary Conditions and Initial Conditions
3.10.1Vertical Boundary Conditions
The boundary conditions at the free surface (a = 0) are
4,
fdu du\
jfo'lfo)
dT
da
H_
Ev
HPrv
E
Tsrt)
The boundary conditions at the bottom (a = 1) are
(du dv\ H .
Av Idada) ~ E}T^T^)
HrZrCd + 25ri2iiiWi +^22^1 (ui,ui)
Avr 1
dT
da
= 0
(3.53)
(3.54)
(3.55)
where ui and Ui are the contravariant velocity components at the first grid point
above the bottom.
3.10.2Lateral Boundary Conditions
Due to the use of contravariant velocity components, the lateral boundary condi
tions in the (Â£,rÂ¡,a) grid are similar to those in the (x,y,a) system. Along the solid
boundary, noslip condition dictates that the tangential velocity is zero, while the slip
condition requires that the normal velocity is zero. When flow is specified at the open
boundary or river boundary, the normal velocity component is prescribed.
3.10.3Initial Conditions
Initial conditions on vectors, if given in the Cartesian or prototype system, such
as the velocity and the surface stress, must be first transformed before being used
29
3.5.2 Vertical Velocities
u>
(1 +cr)dC 1 r dHu dHv\
/3H dt H J1 { dx + dy )
(1 + a) d( ( dh dh\
(3.24)
(3.25)
where U = f\ uda, V = vda, r3X, r3y are wind stresses at the surface and r^,
are bottom stresses.
3.6 Generation of Numerical Grid
3.6.1 Cartesian Grid
In order to numerically solve the governing equations, finite difference approxima
tions are introduced to the original governing equations, and solutions are obtained
at discrete points within the domain. Therefore, a physical domain of interest must
be discretized. When a simple physical domain is considered, cartesian grid can be
used and hence grid generation and development of finitedifference equations are
relatively easy.
Unfortunately, most physical domains in lakes or estuaries are complex. Pre
viously rectangular grid was widely used. This method has such disadvantages as
inaccuracies at boundaries and complications of programming due to unequal grid
spacing near boundaries.
3.6.2 Curvilinear Grid
To better resolve the complex geometries in the physical domain, boundaryfitted
(curvilinear) grid can be used. In general, a curvilinear grid can be obtained by use
of (1) algebraic methods, (2) conformal mapping, and (3) numerical grid generation.
Algebraic grid generation uses an interpolation scheme between the specified
boundary points to generate the interior grid points. This is simple and fast com
putationally, while the smoothness and skewness are hard to control. Conformal
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE, FLORIDA: THE
EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
By
HYE KEUN LEE
August 1993
Chairman: Dr. Y.P. Sheng
Major Department: Coastal and Oceanographic Engineering
Winddriven circulation in Lake Okeechobee, Florida, is simulated by using a
threedimensional curvilineargrid hydrodynamic model and measured field data. Field
data show that significant thermal stratification often develops in the vertical water
column during daytime in the large and shallow lake. Significant wind mixing due to
the lake breeze, however, generally leads to destratification of the water column in the
late afternoon and throughout the night. Thus, thermal effects must be considered
in the numerical simulation of circulation in shallow lakes.
During daytime the lake is thermally stratified and wind is relatively weak, mo
mentum transfer is generally limited to the upper layer and hence the bottom currents
are much weaker than the surface currents. During the initial phase of significant
lake breeze, strong surface currents and opposing bottom currents are developed, fol
lowed by oscillatory motions associated with seiche and internal seiche, until they are
damped by bottom friction.
Lake Okeechobee is covered with submerged and emergent aquatic vegetation over
much of the bottom on the western portion of the lake (20 % of the surface area). The
presence of the vegetation causes damping of the wind, wave and current fields. To
xv
dyne/cm**2
141
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D. 3D, Interpo1atedlArm 1) W/ Temp
Julian Day
Figure 8.43: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction) when thermal effect is considered.
159
Temperature at Platform B MODEL
Temperature at Platform B DATA
*147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.58: Simulated and measured temperatures at Station B.
uI cm/sec) dyne/cm**2
135
Wind Stress at Platform C tau x (MODEL)
Velocity at Platform C* 3D. Interpolated(Arm 1) W/ Temp
Jullan Day
Figure 8.37: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction) when thermal effect is considered.
175
5. In order to consider thermal effects, at first, heat fluxes at the water surface
were estimated by using the meteorological data and expressed as timevarying equi
librium temperatures. However, due to the insufficient meteorological data, simulated
temperatures did not agree well with field data although currents results were much
improved.
6. Another estimation of heat fluxes was tried with the socalled inverse method.
Assuming the total heat flux consists of a solar part and a nonsolar part, the nonsolar
part was estimated by solving the vertical onedimensional temperature equation.
Total heat fluxes were used as a boundary condition at the water surface for the
temperature equation of CH3D. Results showed that the predicted temperature agrees
well with field data. Therefore, when there are insufficient meteorological data, the
inverse method can be a good method to estimate the heat flux with given wind data
and measured surface temperature.
cm Vs cmVs Watl/m!
151
HMt Flux Estimated by Equifibrium Tampsratura
Eddy Viacoaiy at C (temperatura considerad)
JufisnDsy
Figure 8.51: Time history of eddy viscosity at Station C between Julian days 147 and
161.
dyne/cm**2
183
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform Ci 3D. InterpolatedI Arm 1) Inverse
Julian Day
Figure A.7: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
164
6HR
7HR
8HR
9HR
10HR
11HR
W/0 TEMP
WITH TEMP
U
V
U
V T SCALE
Figure 8.63: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor Dr. Y. Peter Sheng for
his continuous guidance, encourgement and financial support throughout my study.
I would also like to extend my thanks and appreciation to my doctoral committee
members, Dr. Robert G. Dean, Dr. Donald M. Sheppard and Dr. Ulrich H. Kurzweg,
for their patience in reviewing this dissertation. My gratitude also extends to Dr.
Robert J. Thieke who reviewed my dissertation.
I must thank Dr. Paul W. Chun for reviewing my dissertation and the
great guidance during my stay in Gainesville while he served as a faculty advisor
of the Korean Student Association.
Financial support provided by the South Florida Water Management District,
West Palm Beach, Florida, through the Lake Okeechobee Phosphorus Dynamics
Project is appreciated.
I would like to dedicate this dissertation to my late father and my mother.
Finally, I would like to thank my loving wife, Aesook, for her support and patience,
and my beautiful daughter, Mireong, and my smart son, David.
li
CHAPTER 9
CONCLUSION
A threedimensional hydrodynamic model (CH3D) was significantly enhanced to
study the winddriven circulation in Lake Okeechobee considering the effects of veg
etation and thermal stratification. Space and timevarying wind stresses were used
to drive the model. The effect of vegetation was parameterized as increased pro
file drag on the flow. Vertical turbulence was parameterized by a simplified second
orderclosure model.
CH3D was used to study the winddriven circulation during the period of May
27, 1989, to June 10, 1989, in Lake Okeechobee to simulate the currents driven by
winds. Followings are major conclusions from this study.
1. Both the simulated and measured currents in the vicinity of the vegetation
zone were found to be primarily in the direction parallel to the vegetation boundary,
thus suggesting relatively little transport across the vegetation boundary.
2. For Lake Okeechobee, it was determined that thermal stratification effects
were critical to the successful simulation of circulation under increasing winds.
3. Without considering thermal effects, the longterm trend of simulated currents
followed well that of field data. But the simulated currents did not show the peaks
well, which were quite obvious in the field data. Therefore, thermal effect in shallow
lakes was considered by solving the temperature equation which was coupled with the
momentum equations.
4. With thermal effect, simulated currents not only revealed the peaks well but
also followed field data on shortterm trends quite well, indicating the proper param
eterization of turbulence.
174
dyne/cm**2
148
Wind Stress at Platform E> tau y (MODEL)
Julian Day
Figure 8.50: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: NorthSouth direction) when thermal effect is considered.
26
3.3 Vertical Grid
No natural water bodies have strictly flat bottoms. Therefore to represent the
variable bottom topography, a stretching is used by defining a new variable a :
a = I7i^M (3.U)
h(x,y) + ((x,y,t)
The advantage of crstretching is that the same vertical model resolution can be main
tained in both shallow and deeper parts of a lake. The disadvantage is that it in
troduces additional terms in the equations. Details of crstretching can be found in
Sheng and Lick (1980) and Sheng (1983).
3.4 NonDimensionalization of Equations
By introducing reference values, the governing equations can be nondimensionalized.
The purpose is to make it easier to compare the relative importance of each term.
The following relations were used (Sheng, 1986).
(u*,u*,u>*)
(u,V,wXr/Zr) /Ur
(x*,yn,z*)
=
(x,y,zXr/Zr) /Xr
(Tx>Ty)
=
W,T?)/PofZTUr
t*
=
tf
Â£
=
T0/(Tr T0) q/poCpfZrT0
c
=
gC/fUrXr = C/Sr
p*
=
(P~ Po)/{Pr ~ Po)
T*
=
(T T0)/(Tr T0)
A*h
=
Ah/Aht
K
=
Av/ Avt
K'h
=
Kh/ KHt
K
=
Kv/Kvr
J*
=
uXT/UT
(3.15)

dyne/cm**2
116
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E* 3D, Interpolatad(Arm 1) W/0 Temp
Ju\Ian Day
Figure 8.20: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation.
83
Lake Okeechobee does not have a strong thermal stratification. But during the
daytime, lake becomes stratified. Temperature field data show the water temperature
difference between the near top and near bottom at Station C can reach 4 C. However
this stratification disappears in a short time due to the strong wind. This stratification
can affect the eddy viscosity significantly.
8.2.4 Vegetation Data
The distribution of aquatic vegetation in Lake Okeechobee was determined by the
use of recent satellite imageries and ground truth data obtained during field surveys.
Satellite imageries from the SPOT satellite were received and processed to create a
vegetation map of Lake Okeechobee (J.R.Richardson, personal communication, 1991).
This map was overlayed onto a curvilinear grid created by us. The number of pixels
with the same color was counted to classify the vegetation type on each grid cell. The
vegetation data include the vegetation class, the height of vegetation, the number of
stalks per unit area, and the diameter of each class. A total of more than 25 kinds of
vegetation were identified. The range of vegetation height is between 0.5 m and 4 m.
The density ranges from 10 stalks per m2 to 2000 stalks per m2 of bottom area, while
the diameter of the stalks ranges from 0.25 cm to 15 cm. The most popular type
of vegetation was cattail. These vegetation data were provided by John Richardson
from the Department of Fisheries and Aquaculture, University of Florida.
8.3 Model Setup
The following describes the procedure for simulation of winddriven circulation in
Lake Okeechobee during spring 1989.
8.3.1 Grid Generation
The first step is to select the grid and grid size. Rectangular Cartesian grids were
widely used in hydrodynamic models. These grids are easy to generate but have a
disadvantage. Because these grids have to represent the boundaries in a stairstepped
fashion, they cannot represent the complex geometry accurately unless a large number
90
braic equation may be obtained from Equations (2.18), (2.19) and (2.20).
0 = 3A2b2sQ4 + A[(bs + 3b + 7b2s)Ri Abs(l 26)]Q2
+b(s + 3 + Abs)Ri2 + (bs A)(l 2b) Ri
(8.4)
where
Q =
V(fe)2+(&);
(8.5)
and
Ri =
Po 9z
(8.6)
(Si) +(fi)
The total rms turbulent velocity q can be obtained from the above equations after
the mean flow variables are determined at each time step. The vertical eddy viscosity
is then computed from
where Sm is defined as
Av SniAq
c A + uj ww
~A
A to q 2
(8.7)
(8.8)
and where u = Ri/(AQ2), u> = 1 u)/bs, and
126
ww =
(8.9)
3(1 2lJ)
The length scale A is assumed to be a linear function of vertical distance immediately
above the bottom or below the free surface. In addition, the length scale A must
satisfy the following relationships (Sheng and Chiu, 1986):
dA
dz
< 0.65
A < fz,H
(8.10)
(8.11)
5
(2) to develop a numerical model which can simulate the effect of vegetation on
Lake Okeechobee circulation,
( 3) to develop a numerical model which can simulate the thermal stratification and
its effect on circulation, and
(4) to determine the important factors for producing successful simulation of circu
lation in large shallow lakes.
The literature review will be presented in the Chapter 2, after which the for
mulation of the threedimensional model will be given in Chapter 3. A vegetation
model will be explained in Chapter 4, and a thermal model follows in Chapter 5.
Finitedifference formulation will be presented in Chapter 6. After the model test in
Chapter 7, application to Lake Okeechobee will be described in Chapter 8. Finally,
a conclusion will be given in Chapter 9.
provide realistic simulation of winddriven circulation in the presence of vegetation,
this study developed a simplified vegetation model which parameterizes the effect
of vegetation in terms of added form drag terms in the momentum equations.
Simulated currents in the open water region in the vicinity of vegetation compare
quite well with data. This physical process is successfully modeled by parameterizing
the vertical turbulence with a simplified secondorder closure model. Model simulation
which assumes homogeneous density structure fails to represent the stratification and
destratification cycle. On the other hand, simulation which includes thermal effects
faithfully reproduced field data.
xvi
27
where variables with asterisks are nondimensional variables and variables with r are
the reference values.
3.5 Dimensionless Equations in aStretched Cartesian Grid
The transformation relations from a Cartesian coordinate (x, y, z) to a vertically
stretched Cartesian coordinate (x,y,a) may be found in Sheng (1983). Using the
relation presented in the previous section, the following dimensionless equations are
obtained:
d( adHu BHv TTdu A
Â¥ + ^ + ^ + ^ = 0
1 dHu
H dt
+
dÂ£ E^d_ f du
dx + H2 da \ vda,
Ro dHuu dHuv
H \ dx + dy
 + v
dHuu\
+ ~dT)
ac,_]kJL(A
dx + H2 da V v da)
+ Bx
(3.16)
(3.17)
1 dHv
H dt
dC Ev d
+
7T Mv
dv
dy H2 da \ v da
u
Ro ( dHuv dHvv dHvu'
H \ dx dy + da /
+ E" +
Ro
Fd
d<
H
f
dx
dP
d_
dy
dH
dv'
~ d
dy dy
dyJ
(/>+ ap
+ H.O.T.
dv
dy H2 da V da
+ Bv
(3.18)
1 dHT Ev d ( dT\
H dt ~ Prv H2da ^ da)
Ro idHuT dHvT dHuT\
H \ dx + dy + da )
(3.19)
dyne/cm**2
182
Velocity at Platform B* 3D, Intarpo 1 ated(Arm 2) Inverse
Julian Day
Figure A.6: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 2: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
35
+
Ro
Ev
d
H2 da
Ro
(Hueu% +
dhuku
da
(3.44)
+ EHAu*mr
where d/dxk is the partial derivative, gn is the metric tensor while g0 = J =
x^yv xvyz is the determinant of the metric tensor, uk is the contravariant veloc
ity, ( )/ represents the covariant spatial derivative, \k represents the contravariant
spatial derivative, and is the permutation tensor and
,12
,21
,n
1
V90
1
V90
e22 = 0
(3.45)
The covariant and contravariant differentiations are defined by
UJ = uj + D*iua
(3.46)
S\k = gkmS,m (3.47)
where :j represents partial differentiation and D'aj represents the Christoffel symbol
of the second kind:
D)k = rDjk (3.48)
where g'n represents the inverse metric tensor, h,, and Dnjk is the Christoffel symbol
of the first kind:
1, .
t'ijk 2 \9ijk T gik.j ~ gjk.i)
(3.49)
69
Surfaea Elevation at Mart* End
6
4
 2
a
I
M
2
4
6
6 *2
Tas IHourel
Srfc* Elatalltn at Mast tna
t f
Surface Elevation at Canto*
Surface Elevation at Eaet End
Surface Elevation at South End
6 ,
6 L >
6 12
Tlee (Houra I
Figure 7.1: Model results of a seiche test
CHAPTER 1
INTRODUCTION
Lakes are valuable resource for a variety of human needs: drinking water, agri
cultural use, navigation, waste water disposal, recreation sites, cooling reservoirs for
power plants, etc. Ninetynine percent of Americans live within 50 miles of one of
37,000 lakes (Hanmer, 1984). Lakes can be dangerous to people under certain cir
cumstances such as during flooding and the deterioration of water quality due to the
excessive loading of contaminants into the lake.
Water movement in lakes is driven by wind, density gradient, waves, and tributary
flow, but primarily is influenced by wind action. During periods of strong wind, severe
flooding can be caused by the storm surge. Some examples are the flooding in the
Lake Okeechobee area during the 1926 and 1928 hurricanes. The 1926 hurricane
caused a storm surge of 7 ft at Moorehaven on the western side of Lake Okeechobee,
and about 150 people lost their lives (Hellstrom, 1941).
Water quality of lakes is of utmost importance. So long as human activities are
limited to a small part of a lake, it may appear that the lake has an unlimited capacity
of selfpurification. However, as population and human development increase, a lake
may not be able to endure the excessive stresses caused by human actions, and water
quality may become deteriorated. Typical evidence of poor water quality includes
sudden algal bloom, colored water, fish kill, taste and odor in drinking water, and
floating debris of plants. Eutrophication is the process in which excessive loading of
nutrients, organic matter, and sediments into lakes results in an increase of primary
production. Sources of eutrophication are increased use of fertilizer, waste water
discharge, and precipitation of polluted air.
1
dyne/cm**2
124
Wind Stress at Platform Di tau x (MODEL)
Velocity at Platform Di 3D. InterpoI atedIArm 1) W/0 Temp
Julian Day
Figure 8.27: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction).
32
* C<*.y) r.*,<*,t*
(o) or (e>)
^ n(*,y) y* y,( ,.**
. 3*
*> y*
TRANSFORMED
Figure 3.1: A computational domain and a transformed coordinate system.
dyne/cm**2
112
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B. 3D, Interpo1 ated(Arm 2) W/0 Temp
Julian Day
Figure 8.18: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation.
85
Figure 8.3: Computation domain of Lake Okeechobee.
156
effect. Until 8:00 a.m, the lake is thermally homogeneous. Vertical profiles of currents
are very similar independent of thermal effect. However, after sunrise the lake starts
to be stratified due to heating at the water surface.
Starting at 9:00 a.m., the vertical profiles of temperature show that the stratifica
tion and vertical profiles of simulated currents with thermal effect start differing from
those without thermal effect. Generally, currents with thermal effect are stronger.
This is because the momentum transfer is limited to the upper layer, and magnitudes
of currents become larger. Also, the currents in the lower layer increase due to return
flow.
Afternoon lake breeze starts at 5:00 p.m., and the reduced solar heating causes the
destratification. At 7:00 p.m., the thermocline breaks and the lake becomes thermally
homogeneous. Therefore, thermal effect becomes insignificant throughout the night.
This stratification and destratification cycle is repeated daily. Therefore, the
simulated currents without the thermal effect are generally smaller than the measured
currents. As will be explained in the sensitivity test, the adjustments of bottom drag
coefficient and roughness height did not improve the model results. In general, if a
numerical model underestimates the current magnitude but reveals the general trend
of measured data, the usual approach to improve the results is often the reduction
of bottom stress. However, this can fail because of neglecting of the diurnal thermal
cycle in shallow lakes.
8.9 Sensitivity Tests
The successful run presented in the previous section was produced with a partic
ular set of model parameters. In order to ensure that the particular choice of model
parameters is not arbitrary, it is essential to conduct a sensitivity analysis on the
models response to changes in parameters. This sensitivity analysis is also necessary
because the measured data always contain some measurement error.
ulcm/secl dyne/cm**2
185
Wind Stress at Platform C< tau x (MODEL)
Velocity at Platform Ct 3D. Intarpo I ated(Arm 2) Inverse
Julian Day
Figure A.9: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
ulcm/sec) dyne/cm**2
147
Wind Stress at Platform E> tau x (MODEL)
Velocity at Platform E> 3D. Interpo1ated1 Arm 2) W/ Temp
Figure 8.49: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction) when thermal effect is considered.
64
1 (1 (^Aty)(TÂ¡+2jt 2TÂ¡+Â¡jt + TÂ¡,k)
1 A
2 A
(Ti+i'k T{[,j,A:)]
At the left face of a cell
When u is positive :
(y/f'HuT). (y/g^H) j u, j, [ (T, i, fc +
_ 1 (* (i^)2)(3n. ., 27',.,,,* + Ti.2,j,k)
1 ut,j,fcAt
2 AÂ£
(Ti,j,k Tiij'k)]
When u is negative
(6.17)
(6.18)
(y/glHuT)_ + T.j^)
 (i (!ii^)2)(rt.+liii, 22U* + Tiij,*:)
1 u,j,fcAi
(^U* T<U,fc)]
2 AÂ£
The QUICKEST method treats the rj direction advection term as follows:
a (k/slHvT)+ (Jg'HvT).
a~^HvT) Aj
At the top face of a cell :
When v is positive :
(6.19)
(6.20)
(\/
 j(l (!~1f(TÂ¡jW 2+ Ty.u)
1 Vij+tjA t
2 AÂ£
(7v,j+1,A: ^ij'.fc)]
(6.21)
When v is negative :
53
Also, the fourthpower radiation term can be approximated by a linear term with less
than 15% error (Edinger and Geyer, 1967). Therefore, AH become as follows:
AH = 15.7 + (0.26 + /?)(a + bW)(Ts Te) + 0.051(TS2 Te2) (5.14)
= K(Ta Te)
Neglecting the quadratic term,
K = 15.7f (0.26 + P){a + bW) (5.15)
Using the above relation, an equation for Te can be derived as follows:
0.0517J Hr 1801 K 15.7,e0 c(/3) 0.26T ,
T+ ~ = k + K~''[o3eTJ + mT1 (5'16)
where c(/3) is intercept for the temperature and vapor pressure approximation.
5.2.9 Procedure for an Estimation of K and Te
Step 1. Compute Hr.
Step 2. Assume Te.
Step 3. Find K for given wind and temperature.
Step 4. Compute the right hand side of Eq. 5.16.
Step 5. Compute the left hand side of Eq. 5.16.
Step 6. Compute the difference of Step 4 and Step 5.
Step 7. If error is not within error limit, go to step 2.
dyne/cm**2
133
Wind Stress at Platform B< tau x (MODEL)
Velocity at Platform B> 3D, Interpo1 ated(Arm 2) W/ Temp
Julian Day
Figure 8.35: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction) when thermal effect is considered.
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INGEST IEID EWLAW3ADI_ZJZ1JZ INGEST_TIME 20170714T21:42:03Z PACKAGE UF00075325_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
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31
where the functions P and Q may be chosen to obtain the desired grid resolution and
alignment. In practice, one actually solves the following equations which are readily
obtained by interchanging the dependent and independent variables in Eqs. 3.26 and
3.27:
ctx^Â£ 2b b otPx^ b 'yQxq
al/ ~ + 7yrm + aPyt + lQyv
where
a = xr, + yn
P = xiXr, + ytyv
7 = + y
P = ^
Q = jM + vl
J = XtVr, XM
with the transformed boundary conditions:
x = on i = 1 and 3
V = 9i((,Vi)
x = fi(Zi,rÂ¡) on i = 2 and 4
V = 9i(M (334)
3.7 Transformation Rules
Generations of a boundaryfitted grid is an essential step in the development of
a boundaryfitted hydrodynamic model. It is, however, only the first step. A more
important step is the transformation of governing equations into the boundaryfitted
coordinates. A straightforward method is to transform only the independent variables,
(3.32)
(3.33)
= 0 (3.30)
= 0 (3.31)
33
i.e., the coordinates, while retaining the Cartesian components of velocities. Johnson
(1982) developed such a 2D verticallyintegrated model of estuarine hydrodynamics.
The advantage of the method is its simplicity in generating the transformed equations
via chain rule. The dimensional forms of the continuity equation and the vertically
integrated momentum equations are shown by Eqs. (20) and (21) in Appendix A of
Sheng (1986). The resulting equations, however, are rather complex. Even when an
orthogonal or a conformal grid is used, the equations do not become any simpler. Ad
ditional disadvantages are (1) the boundary conditions are quite complicated because
the Cartesian velocity components are generally not aligned with the grid lines, (2)
the staggered grid cannot be readily used, and (3) numerical instability may develop
unless additional variables (e.g., surface elevation or pressure) are solved at additional
grid points, (Bernard, 1984, cited in Sheng (1986)).
To alleviate the problems mentioned in the previous paragraph, Sheng (1986)
chose to transform the dependent variables as well as the independent variables.
Equations in the transformed coordinates (Â£, 77) can be obtained in terms of the con
travariant, or covariant, or physical velocity components via tensor transformation
(e.g., SokolnikofF, 1960). As shown in Fig. 8 of Appendix A of Sheng (1986), the
contravariant components (tt) and physical components u(i) of the velocity vector
in the nonCartesian system are locally parallel or orthogonal to the grid lines, while
the covariant components (it,) are generally not parallel or orthogonal to the local
grid lines. The three components are identical in a Cartesian coordinate system. The
following relationships are valid for the three components in a nonCartesian system
are
' = (gu) 1/2u(i)
(no sum on i)
(3.35)
i = (9ii)~1/29iMj)
(no sum on i)
(3.36)
u(i) = i
(3.37)
14
where u\ and n' are fluctuating velocity components in the xt and x2 directions, while
u; and Uj are mean velocity components, and vt is the turbulent eddy viscosity.
Similarly, an eddy diffusivity Kt can be defined:
<210>
where ' and are the fluctuating and mean temperature or salinity or scalar con
centration, and Kt is the turbulent eddy diffusivity.
In order to close the system of mean flow equations for it,, it is necessary to obtain
an expression for vt in terms of known mean flow variables. Several options are given
in the following:
2.4.2 Constant Eddy Viscosity/Diffusivity Model
Earlier models used constant eddy viscosity/diffusivity models. Although this
allows easy determination of analytical solutions for the 1D equation of motion and
easy programming, there are many disadvantages. Turbulence is spatially and tempo
rally varying, hence constant eddy viscosity is not realistic. It is difficult to calibrate
the constant eddy coefficient model even if extensive field data exists.
2.4.3 MunkAnderson Type Model
Prandtl (1925) assumed that eddy viscosity is proportional to the product of a
t A A
characteristic fluctuating velocity, V, and a mixing length, L. He suggested that
V = /mÂ§^ and Av = The only parameter to be specified is length scale, lm,
which is assumed to be a linear function of z.
Following Prandtl, we can define the neutral vertical eddy viscosity as follows:
A
 H
' du
dv'
1 0.5
^ + 77
dz
dz.
(2.11)
where A0 is assumed to be a linear function of z increasing with distance above the
bottom or below the free surface and with its peak value at middepth, while not
exceeding a certain fraction of the local depth. In the presence of strong waves,
dyne/cm#*2
105
Velocity at Platform Ci 3D, InterpolatedtArm 1) W/0 Temp
JulIan Day
Figure 8.11: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 1: NorthSouth, direction). Thermal stratification was not considered in model
simulation
44
4.2.3 Equations for the Entire Water Column
Instead of solving the above equations for the vegetation layer and the vegetation
free layer, it is more convenient to solve the verticallyintegrated equations for the
entire water column, which can be readily derived by combining the equations for the
two layers. First, the verticallyintegrated velocities over the entire water column in
the vegetation zone, U and V, can be defined as
{U, V) = (I/i + U2, Vx 4 V2) (L\Ui f Liu2, L\V\ + L2v2)
(4.14)
where Hi u2, T>i and v2 are verticallyaveraged velocities within layer I and layer II,
respectively, while L\ and L2 are the thicknesses of layer I and layer II, respectively.
Adding the U\ equation and the U2 equation leads to
dt dx l L\ L2
a (UiVj U2V2\
+ ^{LT+~Lr)+9H
fV H (tsx TÂ¡>r F:x)
P
d
a au'
d
'a dU
dx
+ dy
Ah~k~
l y\
+
while the summation of the V\ equation and the V2 equation results in
(4.15)
dV d {U\V\ U2V2]
dt + dx\ Lx + L2 )
d (V? V22\
+ %(lT+l7J+/c
fU 4" ~{Tay Tby &'ey)
d
\ 9V'
d
\ dV
dx
dx
+ dy
Ahz
L dy
(4.16)
All the stress terms are computed as the quadratic power of the flow velocity. For
example, riX and r,y are computed as quadratic functions of the wind speed, tÂ¡,x and
T^y are computed as quadratic functions of U and V, and rtx and r,y are quadratic
functions of (U2 U\) and (V2 V\). The form drags associated with the vegetation
are:
Fcx = pcdAxN3
M+fUt
L1
(Â£)
(4.17)
dyne/cm**2
145
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E* 3D. Intarpolatad(Arm 1) W/ Temp
JulIan Day
Figure 8.47: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction) when thermal effect is considered.
CHAPTER 2
LITERATURE REVIEW
2.1Numerical Models of Lake Circulation
Sheng (1986) reviewed numerous models and suggested that models could be
classified according to numerical features (dimensionality, horizontal grid, vertical
grid, numerical scheme, etc.) and physical features (forcing function, free surface
dynamics, spatial scale, turbulence models, etc.). As an example, some models are
described in the following.
2.1.1 OneDimensional Model
Onedimensional (1D) models include a single spatial coordinate (longitudinal or
vertical). When the lake is elongated and wellmixed in the directions perpendicular
to the longitudinal axis, a longitudinal 1D model can be used. A longitudinal 1D
system of equations is derived by integrating the continuity and momentum equations
over the cross section. Sheng et al. (1990) developed a longitudinal onedimensional
model of Indian River Lagoon. Sheng and Chiu (1986) developed a vertical one
dimensional model for a location in Atlantic Ocean.
2.1.2 TwoDimensional Model
Twodimensional (2D) models include horizontal 2D models, which assume ver
tical homogeneity, and vertical 2D models, which assume transverse homogeneity.
The equations of motion are obtained by performing integration or averaging in the
vertical or transverse directions.
Hsueh and Peng (1973) studied the steadystate verticallyaveraged circulation
in a rectangular bay by solving a Poisson equation with the method of successive
6
84
of grid points are used. Complex boundaries can be represented more accurately by
using the curvilinear grid for a general 2D region with boundaries of arbitrary shape
and with boundary intrusions and internal obstacles, such as islands.
To generate a curvilinear grid, first design a computational domain as shown
Figure 8.3. Next step is to digitize the boundary points of Lake Okeechobee using a
detailed map. After that, WESCORA code is used for the grid generation. As shown
in Figure 8.4, a horizontal grid consisting of 23 by 28 points was generated for this
study. During the spring of 1989, the lake stage was dropped to 12.5 ft. Therefore,
the most western part became dry land, and the cells in this area was excluded in the
model simulations.
8.3.2 Generation of Bathymetry Array
The accurate representation of bathymetry is an important factor for the simula
tion. The map used is a nautical chart which is published by National Oceanographic
and Atmosphere Administration (Chart Number 11428, February 8, 1986). From the
bathymetric data on the map, digitized depths at a total of 983 points were obtained.
Based on the digitized depths, the depths at the grid points are evaluated using a
threepoint interpolation scheme as follows:
3 WkDk
where Dk is the digitized depths at the three nearest points, and
(8.1)
Wk =
(xi,j ~ xk)2 + (y. j yk)2
(8.2)
Wo = ^2 Wk
k=l
(8.3)
Depth contour is in Figure 8.5. maximum depth is about 4 m and minimum
depth is 20 cm.
199
Lewellen, W. S., 1977: Use of invariant modeling. Handbook of Turbulence, 1,
(W. Frost, ed.), Plenum Publishing Corp., New York, NY., pp. 237280.
Lien, S. L. and J. A. Hoopes, 1978: Winddriven, steady flows in Lake Superior.
Limnology and Oceanography, 23, pp. 91103.
Liggett, J. A., 1969: Unsteady circulation in shallow, homogeneous lakes. Journal
of the Hydraulics Division, 95, pp. 12731288.
Liggett, J. A. and K. K. Lee, 1971: Properties of circulation in stratified lakes.
Journal of the Hydraulics Division, 97, pp. 1529.
Lowe, P. R., 1977: An approximating polynomial for the computation of saturation
vapor pressure. J.Appl.Meteorol, 16, pp. 100103.
Munk, W. H. and E. R. Anderson, 1948: Notes on a theory of the thermocline.
Journal of Marine Research, 7, pp. 276295.
Orlob, G. T., 1959: Eddy diffusion in homogeneous turbulence. J. Hyd. Div., Proc.
A.S.C.E., 85, pp. 75101.
Philips, N. A., 1957: A coordinate system having some special advantages for
numerical forcasting. Journal of Meteorology, If, pp. 184185.
Pielke, R. A., 1974: A threedimensional numerical model of the sea breeze over
south Florida. Monthly Weather Review, 102, pp. 115139.
Prandtl, L., 1925: Uber die ausgebildete Turbulenz. ZAMM, 5, 136.
Price, J. F., R. A.. Weller and R. Pinkel, 1986: Diurnal cycling: Observations
and models of the upper ocean response to diurnal heating, cooling, and wind
mixing. Journal of Geophysical Research, 91, pp. 84118427.
Ree, W. O., 1949: Hydraulic characteristics of vegetation for vegetated waterways.
Agricultural Engineering, pp.184 189.
Reid, R. 0. and R. E. Whitaker, 1976: Winddriven flow water influenced by a
canopy. Journal of Waterways, Harbors and Coastal Engineering Division, pp.
6177.
Rodi, W., 1980: Turbulence models and their application in hydraulics, Mono
graph, International Association for Hydraulic Research, Delft, The Nether
lands.
Roig, L. C. and I. P. King, 1992: Continuum model for flows in emergent marsh
vegetation. Estuarine and Coastal Modeling, (M.L. Spaulding, ed.), A.S.C.E.,
pp. 268279.
Saville, T., 1952: Wind setup and waves in shallow water. Technical Memoran
dum No. 27, Beach Erosion Board, Office of the Chief of Engineers, Corps of
Engineers.
dyne/cm**2
108
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform Ci 3D, I nterpoI atedIArm 3) W/0 Temp
147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.14: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction). Thermal stratification was not considered in model
simulation
u(cm/s) u(cm/s)
153
Currents at C (without thermal stratification)
Currents at C (with thermal stratification)
Figure 8.53: Time history of simulated currents at Station C between Julian days 150
and 152.
19
rt =
(2.28)
where is an empirical constant and crt is the turbulent Prandtl/Schmidt number.
An equation for e is derived from the NavierStokes equation and is
dt dt d dut dt t e
dt dx{ dxi ^ dat dxi
) + C1 e^(Pc3eG) C2 c~
(2.29)
where P and G are the stress and buoyancy term in Equation 2.24, respectively, and
Cie, c2e and c3c are empirical model coefficients.
2.5 Previous Lake Okeechobee Studies
Numerous studies on Lake Okeechobee have been performed, but most of them
focused on water quality.
Whitaker et al. (1975) studied the storm surges in Lake Okeechobee while con
sidering the vegetation effect in the western marsh area, by using a twodimensional,
vertically integrated model. They simulated the seiche in the lake during the 1949
and 1950 hurricanes. In their study, the bottom friction coefficient was parameterized
as a function of depth to achieve better agreement of storm surge height.
Schmalz (1986) investigated hurricaneinduced water level fluctuations in Lake
Okeechobee. His study consisted of two parts: a hurricane submodel and a hydrody
namic submodel. The hurricane submodel used hurricane parameters such as central
pressure depression, radius to maximum winds, maximum wind speed, storm track,
storm forward speed, and azimuth of maximum winds, and determined the wind and
pressure field that were used as forcing terms for the hydrodynamic submodel.
The hydrodynamic submodel solved the depthaveraged momentum equations
and continuity equation. Finitedifference method was used for the numerical solution.
For treatment of marsh area, an effective bottom friction which relates the Mannings
n to water depth and canopy height was used. The 2D hydrodynamic model can
resolve flooding and drying: during strong wind conditions such as a hurricane, a
portion of the lake can become dried because of the excessive setdown by wind, while
dyne/cm**2
118
Wind Stress at Platform Ei tau x (MODEL)
Velocity at Platform E< 3D, Interpo1 ated1 Arm 2) W/0 Temp
147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.22: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation.
the nonsolar heat flux more accurately. They called this method the inverse method.
The major advantage of this inverse method is that it allows one to use the
usual satellite data (wind stress, surface insolation, and sea surface temperature) for
the estimation of heat flux at the ocean surface. The disadvantage of the method is
that horizontal advection effect is neglected in the analysis.
2.4 Turbulence Model
Flows in natural water bodies are often turbulent, although they can relaminar
ize during periods of low wind and for tide. Although direct numerical simulation of
turbulence can now be performed for simple flow conditions, it is still computation
ally prohibitive for practical applications in natural water bodies. Thus, Reynolds
averaging, a statistical approach was taken by decomposing the flow variables into
a mean and a fluctuating part and averaging the equations over a period of time that
is large compared to the turbulent time scale. The resulting equations thus produced
are called Reynolds averaged equations. The Reynolds averaged mean flow equations
contain terms involving correlations of fluctuating flow variables (i.e., secondorder
correlations) that represent fluxes of momentum or scalar quantities caused by tur
bulent motion. The task of turbulence modeling is to parameterize these unknown
correlations in terms of known quantities.
Numerous turbulence models were developed for the parameterization of turbu
lence. Some models are empirical, while others are based on more rigorous turbulence
theory. In this section, some available turbulence models are briefly reviewed with an
emphasis on the simplified secondorder closure model.
2.4.1 Eddy Viscosity/Diffusivity Concept
By analogy with molecular transport of momentum, the turbulent stresses are
assumed proportional to the meanvelocity gradients. This can be expressed as
dyne/cm**2
178
Velocity at Platform A* 3D. Interpolated(Arm 1) Inverse
Julian Day
Figure A.2: Simulated (solid lines) and measured (dotted lines) currents at Station A
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
192
Velocity at Platform D. 3D, InterpoIated(Arm 2) Inverse
Julian Day
Figure A.16: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
194
Wind Stress at Platform Ei tau y (MODEL)
Velocity at Platform E 3D, InterpoI ated(Arm 1) Inverse
Julian Day
Figure A.18: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
L
160
Temperature at Platform C MODEL
Temperature at Platform C DATA
*147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.59: Simulated and measured temperatures at Station C.
198
Gaspar, P., J. C. Andre, and J. M Lefevre, 1990: The determination of the latent
and sensible heat fluxes at the sea surface viewed as an inverse problem. Journal
of Geophysical Research, 95, pp. 1616916178.
Gedney, R. T. and W. Lick, 1972: Winddriven currents in Lake Erie. Journal of
Geophysical Research, 77, pp. 27142723.
Goldman, J. C., 1979: Temperature effects on steadystate growth, phosphorus
uptake, and the chemical composition of a marine phytoplankton. Microbial.
Ecol., 5, pp. 153166.
Hanmer, R. W., 1984: EPAs emerging nonpoint source role. EPA 440/5184001.
Harleman, D. R. F, D. N. Brocard, and T. 0. Najarian, 1973: A predictive model for
transient temperature distributions in unsteady flows. Report No. 175, Ralph M.
Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA.
Harris, D. L., 1957: The effect of a moving pressure disturbance on the water level
in a lake. Meteorological Monographs, Vol. 2, No. 10., American Meteorological
Society.
Hellstrom, 1941: Wind effect on lakes and rivers. Bulletin No. 41 of the Institution
of Hydraulics at the Royal Institute of Technology, Stockholm, Sweden. 191 pp.
Hsueh, Y. and C. Y. Peng, 1973: A numerical study of the steady circulation in an
open bay. Journal of Physical Oceanography, 3, pp. 220225.
Huber, W. C. and A. I. Perez, 1970: Prediction of Solar and Atmospheric Radiation
for Energy Budget Studies of Lakes and Streams, Publication No. 10, Water
Resources Research Center, University of Florida, Gainesville, FL.
Johnson, B. J., 1982: Numerical modeling of estuarine hydrodynamics on a
boundaryfitted coordinate system. Numerical Grid Generation
(J. F. Thompson, ed.), Elsevier Publishing Company, New York, NY, pp. 409
436.
Kouwen, N., T. E. Unny, and H. M. Hill, 1969: Flow retardance in vegetated
channels. J. Irrigation and Drainage Div., Proc. A.S.C.E., ( IR 2), pp. 329342.
Lee, K. K. and J. A. Liggett, 1970: Computation for circulation in stratified lakes.
Journal of the Hydraulics Division, 96, pp. 20892115.
Leendertse, J. J., 1967: Aspects of a computational model for long period water
wave propagation. Rand Corporation, Santa Monica, CA, Rep. RM5294PR,
165 pp.
Leendertse, J. J. and S.K. Liu, 1975: A threedimensional model for estuaries
and coastal seas: Volume II, Aspects of computation. Rand Corporation, Santa
Monica, CA, Rep. R1764OWRT, 123 pp.
Leonard, B. P., 1979: Adjusted quadratic upstream algorithms for transient incom
pressible convection. AIAA Journal, pp. 226233.
BIBLIOGRAPHY
Ahn, K. M. and Y. P. Sheng, 1989: Wind wave hindcasting and estimation of
bottom shear stress in Lake Okeechobee. UFL/COEL Report 89 027, Coastal
and Oceanographic Engineering Department, University of Florida, Gainesville,
FL.
Bowen, I. S., 1926: The ratio of heat losses by conduction and by evaporation from
any water surface. Physical Review, 27, No. 2. pp.779787.
Brunt, D., 1932: Notes on radiation in the atmosphere. Quart.J.Roy.Meteorol.Soc.,
58, pp. 389420.
Brutsaert, W, 1982: Evaporation into the atmosphere, D.Reidel Publishing Com
pany, Dordrecht, Holland, p.289.
Choi, J. K. and Y. P. Sheng, 1993: Threedimensional curvilineargrid modeling
of baroclinic circulation and mixing in a partiallymixed estuary. UFL/COEL
TR /094, Coastal and Oceanographic Engineering Department, University of
Florida, Gainesville, FL.
Cook, V. J. and Y. P. Sheng, 1989: Vertical mixing and resuspension of fine sed
iments in Lake Okeechobee. UFL/COEL 89/025, Coastal and Oceanographic
Engineering Department, University of Florida, Gainesville, FL.
Dickinson, R. E., W. C. Huber and C. D. Pollman, 1991: Modeling of phospho
rus dynamics of Lake Okeechobee. Final Report to the South Florida Water
Management District, Envoronmental Engineering Department, University of
Florida, Gainesville, FL.
Donaldson, C. duP, 1973: Atmospheric turbulence and the dispersal of atmospheric
pollutants. Proceedings of Workshop on Meteorology, American Meteorological
Society, (ed. D.A. Haugen), Science Press, pp. 313390.
Eckart, C., 1958: The equation of state of water and sea water at low temperature
and pressure. American J. of Science, 256, pp. 240250.
Edinger, J. E. and E. M. Buchak, 1979: Reservoir longitudinal and vertical implicit
hydrodynamics. Environmental Effects of Hydraulic Engineering Works, E.E.
Driver and W.O. Wunderlich, Eds., Tennessee Valley Authority, Knoxville, TN.
Edinger, J. E. and J. C. Geyer, 1967: Heat Exchange in the environment Pu.
No. 65902, Edison Electric Institute, New York, NY.
Ekman, V. W., 1923: Veber horizontal cirkulation bei winderzeuogten Meeres Stro
mungen. Ark. Mat. Astron. Fys., 17, p. 26.
197
79
Table 8.1: Installation dates and locations of platforms during 1988 and 1989.
TIME OF
YEAR
LOCATION
DATE
LATITUDE
LONGITUDE
DEPTH
(cm)
FALL
Site A
092088
27 06.31
80 46.21
396.0
Site B
091788
27 02.78
80 54.31
274.0
Site C
092188
26 54.10
80 47.36
518.0
Site D
092188
26 58.47
80 40.34
457.0
Site E
091888
26 52.81
80 55.96
274.0
Site F
091988
26 51.90
80 57.09
183.0
SPRING
Site A
051689
26 45.67
80 47.83
183.0
Site B
051889
27 02.78
80 54.31
152.0
Site C
052089
26 54.10
80 47.36
366.0
Site D
052089
26 58.47
80 40.34
335.0
Site E
051989
26 52.81
80 55.96
152.0
Site F
051889
26 52.03
80 56.91
91.0
started on Julian Day 136.708. However, the direction of the anemometer was not
properly oriented until Julian Day 141.5. The location and height of the anemometer
are shown in Table 8.2.
As described in Sheng et al. (1991a), the measured wind over Lake Okeechobee
often exhibited significant diurnal variations associated with the lake breeze. During
relatively calm periods, significant spatial variation is often found in the wind field.
Water motion in the lake is significantly influenced by the wind. Figure 8.2
shows the wind rose diagram at Station C between Julian days 147 and 161. The
number inside the triangle indicats the percentage of wind data in that direction.
For example, wind from east to west is 27%. Wind speed between 46 m/sec is
about 45%. The governing wind direction is from east to west due to the location
of Lake Okeechobee. The surface area of Lake Okeechobee is big enough to create
its own lake breeze. During the daytime, wind blows from lake to land because the
air over the land is warmer than that over the lake. Because the Florida peninsula
is located between the Atlantic Ocean and the Gulf of Mexico, sea breeze affects
the wind direction. As Pielke (1974) indicated, the typical summer wind direction
CHAPTER 7
MODEL ANALYTICAL TEST
The purpose of model analytical test is to examine a models capability to re
produce wellknown physical phenomena for which the model is designed for, by
comparing model results with analytical solution.
7.1 Seiche Test
The CH3D model has been tested for winddriven circulation in an idealized
enclosed lake which is 11 km long and 11 km wide with a uniform depth of 5 m. A
uniform rectangular grid of 1 km grid spacing was used. To perform the seiche test,
the initial surface elevation was given as Â£ = (0cos(2irx/) where (0 is an amplitude
and Â£ is a wave length. In the test, (0 was set to 5 cm and t was set to 10 km.
Since the lake is of homogeneous density and without bottom friction and diffu
sion, seiche period can be calculated as T 7S where l is the basin length and h
ygh
is the mean depth. For the test basin, the seiche period is 0.87 hours. The simulated
surface elevation in the test basin over a 12hour period is shown in Figure 7.1. The
result shows that the surface elevation was not damped and the seiche period agrees
with analytical seiche period.
7.2 Steady State Test
When a uniform wind blows in the same direction over a rectangular lake with
same magnitude over a long period, the lake circulation eventually reaches steady
state. Neglecting advection, horizontal diffusion, and bottom friction, the setup equa
tion can be obtained as follows:
dr] tw
99 dx ~ h
(7.1)
68
UFL/COELTR/103
WINDDRIVEN CIRCULATION IN LAKE
OKEECHOBEE, FLORIDA: THE EFFECTS OF
THERMAL STRATIFICATION AND AQUATIC
VEGETATION
by
Hye Keun Lee
Dissertation
1993
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE. FLORIDA: THE
EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
By
HYE KEUN LEE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1993
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor Dr. Y. Peter Sheng for
his continuous guidance, encourgement and financial support throughout my study.
I would also like to extend my thanks and appreciation to my doctoral committee
members, Dr. Robert G. Dean, Dr. Donald M. Sheppard and Dr. Ulrich H. Kurzweg,
for their patience in reviewing this dissertation. My gratitude also extends to Dr.
Robert J. Thieke who reviewed my dissertation.
I must thank Dr. Paul W. Chun for reviewing my dissertation and the
great guidance during my stay in Gainesville while he served as a faculty advisor
of the Korean Student Association.
Financial support provided by the South Florida Water Management District,
West Palm Beach, Florida, through the Lake Okeechobee Phosphorus Dynamics
Project is appreciated.
I would like to dedicate this dissertation to my late father and my mother.
Finally, I would like to thank my loving wife, Aesook, for her support and patience,
and my beautiful daughter, Mireong, and my smart son, David.
li
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES vi
LIST OF TABLES xiii
ABSTRACT XV
CHAPTERS
1 INTRODUCTION 1
2 LITERATURE REVIEW 6
2.1 Numerical Models of Lake Circulation 6
2.1.1 OneDimensional Model 6
2.1.2 TwoDimensional Model 6
2.1.3 SteadyState 3D Models 7
2.1.4 TimeDependent 3D Models 8
2.2 Vegetation Models 10
2.3 Thermal Models 11
2.4 Turbulence Model 13
2.4.1 Eddy Viscosity/Diffusivity Concept 13
2.4.2 Constant Eddy Viscosity/Diffusivity Model 14
2.4.3 MunkAnderson Type Model 14
2.4.4 Reynolds Stress Model 15
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure
Model 16
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model 17
2.4.7 OneEquation Model (k Model) 18
2.4.8 TwoEquation Model (k t Model) 18
2.5 Previous Lake Okeechobee Studies 19
2.6 Present Study 20
3 GOVERNING EQUATIONS 23
3.1 Introduction 23
3.2 Dimensional Equations and Boundary Conditions in a Cartesian Co
ordinate System 23
3.2.1 Equation of Motion 24
3.2.2 FreeSurface Boundary Condition (z rj) 25
3.2.3 Bottom Boundary Condition (z = h) 25
3.2.4 Lateral Boundary Condition 25
3.3 Vertical Grid 26
3.4 NonDimensionalization of Equations 26
iii
3.5 Dimensionless Equations in crStretched Cartesian Grid 27
3.5.1 VerticallyIntegrated Equations 28
3.5.2 Vertical Velocities 29
3.6 Generation of Numerical Grid 29
3.6.1 Cartesian Grid 29
3.6.2 Curvilinear Grid 29
3.6.3 Numerical Grid Generation 30
3.7 Transformation Rules 31
3.8 TensorInvariant Governing Equations 34
3.9 Dimensionless Equations in BoundaryFitted Grids 36
3.10 Boundary Conditions and Initial Conditions 37
3.10.1 Vertical Boundary Conditions 37
3.10.2 Lateral Boundary Conditions 37
3.10.3 Initial Conditions 37
4 VEGETATION MODEL 39
4.1 Introduction 39
4.2 Governing Equations 41
4.2.1 Equations for the Vegetation Layer (Layer I) 41
4.2.2 Equations for the VegetationFree Layer (Layer II) 43
4.2.3 Equations for the Entire Water Column 44
4.2.4 Dimensionless Equations in Curvilinear Grids 45
5 HEAT FLUX MODEL 48
5.1 Introduction 48
5.2 The Equilibrium Temperature Method 48
5.2.1 ShortWave Solar Radiation 49
5.2.2 LongWave Solar Radiation 49
5.2.3 Reflected Solar and Atmospheric Radiation 49
5.2.4 Back Radiation 51
5.2.5 Evaporation 51
5.2.6 Conduction 51
5.2.7 Equilibrium Temperature 52
5.2.8 Linear Assumption 52
5.2.9 Procedure for an Estimation of K and Te 53
5.2.10 Modification of the Equilibrium Temperature Method 54
5.3 The Inverse Method 54
5.3.1 Governing Equations 55
5.3.2 Boundary Conditions 55
5.3.3 FiniteDifference Equation 56
5.3.4 Procedure for an Estimation of Total Heat Flux 57
6 FINITEDIFFERENCE EQUATIONS 58
6.1 Grid System 58
6.2 External Mode 58
6.3 Internal Mode 61
6.4 Temperature Scheme 62
6.4.1 Advection Terms 63
6.4.2 Horizontal Diffusion Term 66
IV
7 MODEL ANALYTICAL TEST 68
7.1 Seiche Test 68
7.2 Steady State Test 68
7.3 Effect of Vegetation 70
7.4 Thermal Model Test 70
8 MODEL APPLICATION TO LAKE OKEECHOBEE 76
8.1 Introduction 76
8.1.1 Geometry 76
8.1.2 Temperature 76
8.2 Some Recent Hydrodynamic Data 76
8.2.1 Wind Data 77
8.2.2 Current Data 81
8.2.3 Temperature Data 81
8.2.4 Vegetation Data 83
8.3 Model Setup 83
8.3.1 Grid Generation 83
8.3.2 Generation of Bathymetry Array 84
8.4 Model Parameters 88
8.4.1 Reference Values 88
8.4.2 Turbulence Model and Parameters 89
8.4.3 Bottom Friction Parameters 91
8.4.4 Vegetation Parameters 92
8.4.5 Wind Stress 96
8.5 Steady State WindDriven Circulation 98
8.6 WindDriven Circulation without Thermal Stratification 99
8.6.1 Tests of Model Performance 99
8.6.2 Model Results 99
8.7 WindDriven Circulation with Thermal Stratification: Te Method . 123
8.8 Simulation of Currents with Thermal Stratification: Inverse Method 150
8.8.1 The Diurnal Thermal Cycle 150
8.9 Sensitivity Tests 156
8.9.1 Effect of Bottom Stress 167
8.9.2 Effect of Horizontal Diffusion Coefficent 168
8.9.3 Effect of Different Turbulence Model 168
8.9.4 Effect of Advection Term 168
8.10 Spectral Analysis 169
9 CONCLUSION 174
APPENDIX
A SIMULATED CURRENTS BY INVERSE METHOD 176
BIBLIOGRAPHY 197
BIOGRAPHICAL SKETCH 203
LIST OF FIGURES
3.1 A computational domain and a transformed coordinate system. 32
4.1 Schematics of flow in vegetation zone 42
5.1 Meteorological data at Station L006 50
6.1 Horizontal and vertical grid system 59
7.1 Model results of a seiche test 69
7.2 Surface elevation contour when the lake is steady state with uni
form wind stress of 1 dyne/cm2 71
7.3 Effect of vegetation on surface elevation evolution in a winddriven
rectangular lake. Solid line is without vegetation, broken line is
with low vegetation density, and dotted line is with high vegetation
density. 72
7.4 Time history of wind stress and currents at the center of lake at
all five levels. Thermal stratification is not considered 74
7.5 Time history of wind stress and currents at the center of lake at
all five levels. Thermal stratification is considered 75
8.1 Map of Lake Okeechobee 78
8.2 Wind rose at Station C 82
8.3 Computation domain of Lake Okeechobee 85
8.4 Curvilinear grid of Lake Okeechobee 86
8.5 Depth contour of Lake Okeechobee when the lake stage is 15.5 ft.
Unit in cm 87
8.6 Distribution of vegetation height in Lake Okeechobee 95
8.7 Distribution of vegetation density in Lake Okeechobee 97
8.8 Steadystate depthintegrated currents (cm2s~1) in Lake Okee
chobee forced by an easterly wind of 1 dyne/cm2 100
vi
5.9 Steadystate surface elevation contour (cm) in Lake Okeechobee
forced by an easterly wind of 1 dyne/cm2 101
8.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 104
8.11 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 105
8.12 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 106
8.13 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 107
8.14 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction). Thermal stratification
was not considered in model simulation 108
8.15 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction) 109
8.16 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 110
8.17 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation Ill
8.18 Simulated (solid fines) and measured (dotted fines) currents at
Station B (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 112
8.19 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 113
8.20 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 116
8.21 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 117
8.22 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 118
vii
8.23 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 119
8.24 Stick Diagram of wind stress, measured currents, and simualted
currents at Station E 120
8.25 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 121
8.26 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 122
8.27 Simulated (solid lines) and measured (dtted lines) currents at
Station D (Arm 1: EastWest direction) 124
8.28 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction) 125
8.29 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction) 126
8.30 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) 127
8.31 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction) when thermal effect is
considered 129
8.32 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction) when thermal effect is
considered 130
8.33 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: EastWest direction) when thermal effect is
considered 131
8.34 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction) when thermal effect is
considered 132
8.35 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: EastWest direction) when thermal effect is
considered 133
8.36 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction) when thermal effect is
considered 134
Vlll
8.37 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction) when thermal effect is
considered 135
8.38 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction) when thermal effect is
considered 136
8.39 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction) when thermal effect is
considered 137
8.40 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction) when thermal effect is
considered 138
8.41 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction) when thermal effect is
considered 139
8.42 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction) when thermal effect is
considered 140
8.43 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction) when thermal effect is
considered 141
8.44 Simulated (solid lines) and measured (dotted lines) currents at'
Station D (Arm 1: NorthSouth direction) when thermal effect is
considered 142
8.45 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction) when thermal effect is
considered 143
8.46 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) when thermal effect is
considered 144
8.47 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction) when thermal effect is
considered 145
8.48 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction) when thermal effect is
considered 146
8.49 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction) when thermal effect is
considered 147
IX
8.50 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction) when thermal effect is
considered 148
8.51 Time history of eddy viscosity at Station C between Julian days
147 and 161 151
8.52 > Time history of wind stress and measured currents between Julian
days 150 and 152 152
8.53 Time history of simulated currents at Station C between Julian
days 150 and 152 153
8.54 Time history of eddy viscosity at Station C between Julian days
150 and 152 154
8.55 Time history of heat fluxes at Station C between Julian days 147
and 161 155
8.56 Temperature contours of data and model at Station C between
Julian days 152 and 155 157
8.57 Simulated and measured temperatures at Station A 158
8.58 Simulated and measured temperatures at Station B 159
8.59 Simulated and measured temperatures at Station C 160
8.60 Simulated and measured temperatures at Station D 161
8.61 Simulated and measured temperatures at Station E 162
8.62 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 163
8.63 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 164
8.64 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 165
8.65 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 166
8.66 Spectrum of wind stress and surface elevation 171
8.67 Spectrum of measured and simulated currents (eastwest direc
tion) at Station C 172
8.68 Spectrum of measured and simulated currents (northsouth direc
tion) at Station C 173
x
A.l Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 177
A.2 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 178
A.3 Simulated isolid lines) and measured (dotted lines) currents at
Station B (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 179
A.4 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 180
A.5 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 181
A.6 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 182
A.7 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 183
A.8 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 184
A.9 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 185
A.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 186
A. 11 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction). Inverse method was
used for the estimation of heat flux 187
A. 12 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction). Inverse method was
used for the estimation of heat flux 188
A.13 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 189
xi
A. 14 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 190
A.15 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 191
A. 16 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 192
A.17 Simulated isolid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 193
A.18 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 194
A. 19 Simulated isolid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 195
A.20 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 196
xii
LIST OF TABLES
2.1 Selected features of lake models 21
2.2 Application features of lake models 22
8.1 Installation dates and locations of platforms during 1988 and 1989. 79
8.2 Instrument mounting, spring deployment 80
8.3 Reference values used in the Lake Okeechobee spring 1989 circu
lation simulation 89
8.4 Vertical turbulence parameters used in the Lake Okeechobee spring
1989 circulation simulation 92
8.5 Vegetations in Lake Okeechobee (From Richardson, 1991) 94
8.6 Index of agreement and RMS error at Station C 103
8.7 Index of agreement and RMS error at Station B 114
8.8 Index of agreement and RMS error at Station E 115
8.9 Index of agreement and RMS error at Station A 123
8.10 Index of agreement and RMS error at Station D 123
8.11 Index of agreement and RMS error at Station A when thermal
effect is considered 149
8.12 Index of agreement and RMS error at Station B when thermal
effect is considered 149
8.13 Index of agreement and RMS error at Station C when thermal
effect is considered 149
8.14 Index of agreement and RMS error at Station D when thermal
effect is considered 149
8.15 Index of agreement and RMS error at Station E when thermal
effect is considered 149
8.16 Parameters used in sensitivity tests 167
xiii
Index of agreement and RMS error
xiv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
WINDDRIVEN CIRCULATION IN LAKE OKEECHOBEE, FLORIDA: THE
EFFECTS OF THERMAL STRATIFICATION AND AQUATIC VEGETATION
By
HYE KEUN LEE
August 1993
Chairman: Dr. Y.P. Sheng
Major Department: Coastal and Oceanographic Engineering
Winddriven circulation in Lake Okeechobee, Florida, is simulated by using a
threedimensional curvilineargrid hydrodynamic model and measured field data. Field
data show that significant thermal stratification often develops in the vertical water
column during daytime in the large and shallow lake. Significant wind mixing due to
the lake breeze, however, generally leads to destratification of the water column in the
late afternoon and throughout the night. Thus, thermal effects must be considered
in the numerical simulation of circulation in shallow lakes.
During daytime the lake is thermally stratified and wind is relatively weak, mo
mentum transfer is generally limited to the upper layer and hence the bottom currents
are much weaker than the surface currents. During the initial phase of significant
lake breeze, strong surface currents and opposing bottom currents are developed, fol
lowed by oscillatory motions associated with seiche and internal seiche, until they are
damped by bottom friction.
Lake Okeechobee is covered with submerged and emergent aquatic vegetation over
much of the bottom on the western portion of the lake (20 % of the surface area). The
presence of the vegetation causes damping of the wind, wave and current fields. To
xv
provide realistic simulation of winddriven circulation in the presence of vegetation,
this study developed a simplified vegetation model which parameterizes the effect
of vegetation in terms of added form drag terms in the momentum equations.
Simulated currents in the open water region in the vicinity of vegetation compare
quite well with data. This physical process is successfully modeled by parameterizing
the vertical turbulence with a simplified secondorder closure model. Model simulation
which assumes homogeneous density structure fails to represent the stratification and
destratification cycle. On the other hand, simulation which includes thermal effects
faithfully reproduced field data.
xvi
CHAPTER 1
INTRODUCTION
Lakes are valuable resource for a variety of human needs: drinking water, agri
cultural use, navigation, waste water disposal, recreation sites, cooling reservoirs for
power plants, etc. Ninetynine percent of Americans live within 50 miles of one of
37,000 lakes (Hanmer, 1984). Lakes can be dangerous to people under certain cir
cumstances such as during flooding and the deterioration of water quality due to the
excessive loading of contaminants into the lake.
Water movement in lakes is driven by wind, density gradient, waves, and tributary
flow, but primarily is influenced by wind action. During periods of strong wind, severe
flooding can be caused by the storm surge. Some examples are the flooding in the
Lake Okeechobee area during the 1926 and 1928 hurricanes. The 1926 hurricane
caused a storm surge of 7 ft at Moorehaven on the western side of Lake Okeechobee,
and about 150 people lost their lives (Hellstrom, 1941).
Water quality of lakes is of utmost importance. So long as human activities are
limited to a small part of a lake, it may appear that the lake has an unlimited capacity
of selfpurification. However, as population and human development increase, a lake
may not be able to endure the excessive stresses caused by human actions, and water
quality may become deteriorated. Typical evidence of poor water quality includes
sudden algal bloom, colored water, fish kill, taste and odor in drinking water, and
floating debris of plants. Eutrophication is the process in which excessive loading of
nutrients, organic matter, and sediments into lakes results in an increase of primary
production. Sources of eutrophication are increased use of fertilizer, waste water
discharge, and precipitation of polluted air.
1
2
Earlier studies on physical processes in lakes concentrated on the observations of
periodic up and down motion of water level, i.e., seiche motion. When the wind blows
over a certain period, water builds up near the shoreline. After wind ceases, water
starts to oscillate as a free long wave. Subsequently, this wave is damped out due to
bottom friction. Seiche can be initiated by sudden change of wind speed or direction,
the passage of a squall line, an earthquake, or resonance of air and water columns.
When the velocity of a squall line is close to the speed of gravity wave, resonance
occurs and damages can be more severe. It was reported that severe storm damages
in Chicago were caused by a squall line over Lake Michigan on June 26, 1954 (Harris,
1957).
Another important feature in deep temperate lakes is the temperature variation
over the depth, which is called thermal stratification. Starting in the early spring, the
lake attains a temperature of 4 C and is more or less isothermal. During the summer
season, the surface water starts to become warmer because of increased solar radiation,
so that, gradually, a sharp temperature gradient, i.e., thermocline, is formed. The lake
remains thermally stratified during summer with a warm surface layer (epilimnion),
a thermocline, and a cold layer (hypolimnion). Though strong wind action tends to
lower or break the thermocline, the lake generally remains stratified during the entire
summer season. During the fall, as the air temperature drops, the net daily heat flux
at the water surface becomes negative, i.e., the lake loses heat daily. Hence, water
density in the epilimnion often becomes heavier than that in the hypolimnion and
causes convective mixing which, in combination with strong wind action, causes the
lake to become isothermal again in the winter. This process repeats itself annually. It
is important to know the location of thermocline at different times of the year so that
water can be withdrawn to a desirable height in deep lakes or reservoirs for various
agricultural and municipal uses.
Many processes are influenced by currents and temperature in lakes. For example,
3
the growth rate of all organic matter in lakes is governed by temperature (Goldman,
1979). The growth rate generally increases between some minimum temperature
and an optimum temperature, and decreases until it reaches maximum temperature.
Cooling water from power plants is mixed with surrounding water by the currents and
turbulent mixing which depend on the temperature field as well. Thus, predicting
currents and temperature are essential to understanding the transport of various
matters and their effects on the ecology.
In the early days, simple analytical models were used to study physical processes
in simplified conditions. For example, a setup equation was used to predict the
storm surge height (Hellstrom, 1941). Since analytical models could not realistically
consider such effects as advection, complex geometry, and topography, they had been
applied to limited problems to understand certain basic processes.
Numerical models are valuable tools for simulating and understanding water
movement in lakes. Once a rigorously developed model is calibrated with measured
data, it can be used to estimate the flow near a manmade structure or to predict
the movement of contaminants including oil spill, sediments, etc. During the 1970s
and 1980s, vertically averaged twodimensional numerical models, which can compute
only the depth average currents and surface elevations, were widely used because they
were simple and needed little computer time. However, since they could not give ac
curate results for cases where the vertical distributions of currents and temperature
are required, threedimensional models are needed.
Numerical modeling requires the discretization of the computation domain. Past
numerical models which were developed during 1970s generally used a rectangular grid
(for example, Sheng, 1975). However, to represent the complex geometry such as the
shoreline and the boundary between the vegetation zone and the open water in Lake
Okeechobee, a very fine rectangular grid is required. On the other hand, boundary
fitted grids can be and have been used in recent models (for example, Sheng, 1987)
4
to represent the complex geometry with a relatively smaller number of grid points.
In some shallow lakes, aquatic vegetation can grow over large areas. The vegeta
tion can affect the circulation significantly because it introduces additional friction on
the flowing water. For example, Lake Okeechobee has vegetation over an area which
covers 25% of the total lake surface. Because vegetation consists of stalks with differ
ent heights and diameters, a representative diameter and height over each discretized
grid cell must be introduced in the model. Additional drag terms must be introduced
in the momentum equations to represent the form drag introduced by the vegetation.
The consideration of vegetation is necessary to compute the flow and transport of
phosphorus between the vegetation area and the open water.
Previously developed numerical models which were applied to deep lakes, e.g.,
the Great Lakes, cannot be readily applied to shallow lakes such as Lake Okeechobee,
since many shallow water processes are not included in these models.
Since 1988, with funding from the South Florida Water Management District and
U.S. Environmental Protection Agency, the Coastal and Oceanographic Engineering
Department of the University of Florida (under the supervision of Dr. Y. Peter
Sheng) has conducted a major study on the hydrodynamics and sediment dynamics
and their effects on phosphorus dynamics in Lake Okeechobee. The primary purpose
of the study was to quantify the role of hydrodynamics and sediments on the internal
loading of phosphorus and the exchange of phosphorus between vegetation zone and
open water. As part of the study, field data (wind, air temperature, wave, water
current, water temperature, and suspended sediment concentration) were collected
over two onemonth periods in 1988 and 1989. Ahn and Sheng (1989) studied the wind
waves of Lake Okeechobee. Cook and Sheng (1989) studied the sediment dynamics
in Lake Okeechobee. This study focuses on the influences of vegetation and thermal
stratification on lake circulation. The objectives of this study are
(1) to obtain a general insight in the winddriven circulation in Lake Okeechobee,
5
(2) to develop a numerical model which can simulate the effect of vegetation on
Lake Okeechobee circulation,
( 3) to develop a numerical model which can simulate the thermal stratification and
its effect on circulation, and
(4) to determine the important factors for producing successful simulation of circu
lation in large shallow lakes.
The literature review will be presented in the Chapter 2, after which the for
mulation of the threedimensional model will be given in Chapter 3. A vegetation
model will be explained in Chapter 4, and a thermal model follows in Chapter 5.
Finitedifference formulation will be presented in Chapter 6. After the model test in
Chapter 7, application to Lake Okeechobee will be described in Chapter 8. Finally,
a conclusion will be given in Chapter 9.
CHAPTER 2
LITERATURE REVIEW
2.1Numerical Models of Lake Circulation
Sheng (1986) reviewed numerous models and suggested that models could be
classified according to numerical features (dimensionality, horizontal grid, vertical
grid, numerical scheme, etc.) and physical features (forcing function, free surface
dynamics, spatial scale, turbulence models, etc.). As an example, some models are
described in the following.
2.1.1 OneDimensional Model
Onedimensional (1D) models include a single spatial coordinate (longitudinal or
vertical). When the lake is elongated and wellmixed in the directions perpendicular
to the longitudinal axis, a longitudinal 1D model can be used. A longitudinal 1D
system of equations is derived by integrating the continuity and momentum equations
over the cross section. Sheng et al. (1990) developed a longitudinal onedimensional
model of Indian River Lagoon. Sheng and Chiu (1986) developed a vertical one
dimensional model for a location in Atlantic Ocean.
2.1.2 TwoDimensional Model
Twodimensional (2D) models include horizontal 2D models, which assume ver
tical homogeneity, and vertical 2D models, which assume transverse homogeneity.
The equations of motion are obtained by performing integration or averaging in the
vertical or transverse directions.
Hsueh and Peng (1973) studied the steadystate verticallyaveraged circulation
in a rectangular bay by solving a Poisson equation with the method of successive
6
7
over relaxation (SOR). Their model included the terms of bottom friction, advection,
bottom topography, and lateral diffusion, while assuming steady state and homoge
niety in density. The specification of the vertical eddy viscosity is not required in the
twodimensional model but can give only depthaveraged velocities.
Shanahan and Harleman (1982) developed a transient 2D model which assumed
vertical homogeneity. When the lakes are long, deep but relatively narrow, laterally
averaged 2D model can be applied (for example, Edinger and Buchak, 1979).
2.1.3 SteadyState 3D Models
An early study on winddriven circulation was conducted by Ekman (1923) who
solved momentum equations analytically while neglecting the nonlinear terms. We
lander (1957) developed a theory on winddriven currents based on an extension
of Ekmans theory. After neglecting inertia terms and horizontal diffusion terms,
steadystate momentum equations were combined with the introduction of complex
variables. After applying boundary conditions, a solution was obtained in terms of
the imposed wind stress and unknown pressure gradient term. By introducing the
stream functions for verticallyintegrated flow, the continuity equation could be sat
isfied unconditionally. The final equation to be solved was reduced to a secondorder
partial differential equation for stream function, t/>, as follows:
V^ = ^+6W+C (21)
Once is found, the currents can be found by taking the derivatives, and ^.
Gedney and Lick (1972) and Sheng and Lick (1972) applied Welanders theory
to Lake Erie. The equation for stream function was solved by the successive over
relaxation method. The agreement between the field data and model results was
good. Eddy viscosity was assumed to be constant but varies with wind speed.
Thomas (1975) used a depthvarying form of vertical eddy viscosity as follows:
%
V = i/0(l + t) ku,(h + z)
(2.2)
8
where v is eddy viscosity, vQ is eddy viscosity near surface, and & is a constant (0.4).
2.1.4 TimeDependent 3D Models
ModeSplitting
In order to solve the dependent variables with the unsteady threedimensional
model, Simons (1974) used a socalled mode splitting method for Lake Ontario
while Sheng et al. (1978) used a somewhat different method for Lake Erie. Defining
the perturbation velocity = u TZ, v = v v where u,v are depthaveraged veloci
ties, and u,v are instantaneous velocities, Sheng and Butler (1982) derived governing
equations for u, v by subtracting the verticallyaveraged equations from the momen
tum equations. Therefore the solution procedure consists of an external mode, which
includes the surface elevation and u and , and an internal mode, which includes ,v
and temperature.
Time Integration of 2D equations
Time integration is important for improving the efficiency of numerical models.
When the explicit method is used, the time step is limited by the Courant condition,
which is Cj^; < 1. Therefore, explicit method is not desirable for longterm simu
lations. Leendertse (1967) used the ADI (Alternate Direction Implicit) method to
simulate tidal currents in the southern North Sea. All terms in the continuity equa
tion and pressure terms in the momentum equation were treated implicitly, while
the other terms were expressed explicitly. After factorization of the finitedifference
equations, the resulting unknowns are solved by inversion of tridiagonal matrices in
the x sweep and y sweep.
Vertical Grid
Various types of vertical grids are used in numerical models of lake circulation.
The earlier models generally used multiple vertical layers of constant fixed thickness
(zgrid) which do not change with time (Eulerian grid) as used by Leendertse (1975).
This type of model needs a large number of vertical grid points in order to accurately
9
represent the shallow regions. More recent models use the socalled crgrid which
was originally applied in the simulation of atmospheric flow by Phillips (1957). This
vertical cstretching uses the same number of vertical grid points in both the shallow
and deep regions, with the vertical coordinate defined as follows:
a
Hx,y) + ((x,y,t)
(2.3)
where h(x, y) is the water depth, and ( is the water surface elevation. Governing
equations are transformed from the (x,y,z,t) coordinates to (x,t/,cr, t) coordinates
by use of chain rule and become somewhat more complex because of the extra terms
introduced by the stretching.
Other types of models (e.g., Simons, 1974) use a Lagrangian grid which consists
of layers of constant physical property (e.g., density) but timevarying thickness.
These models could resolve vertical flow structure with relatively few vertical layers.
However, parameterization of the interfacial dynamics is often difficult.
Horizontal Grid
One of the challenges in numerical models is the accurate representation of com
plex geometry. Most models (e.g., Leendertse, 1967) use a rectangular uniform grid
to represent the shoreline of a lake or estuary. Thus, a large number of grid points
are needed to achieve a fine resolution near the shoreline or islands. Because com
putational effort is directly related to the number of grid points, grid size should be
as small as possible to maintain required resolution near the interest area, so long
as the computational effort is not excessive. Therefore, to achieve a balance between
resolution and computational efficiency, a nonuniform grid method could be used.
Sheng (1975) used smaller grid size used near areas of importance but coarse grid
elsewhere.
Use of a boundaryfitted grid is another viable alternative. Johnson (1982) used
a boundaryfitted grid to solve depthintegrated equations of motion for rivers. Using
chain rules, he transformed the governing equations for a boundaryfitted grid which
10
was generated by using the WESCORA code developed by Thompson (1983). John
son (1982), however, transformed the equations in terms of the Cartesian velocity
components.
The boundaryfitted grid has recently been adapted to threedimensional nu
merical models. Sheng (1986) applied tensor transformation to derive the three
dimensional horizontal equations of motion in boundaryfitted grid in terms of the
contravariant velocity vectors (a contravariant vector consists of components
which are perpendicular to the grid line) and the water level. Sigma grid is used
in the vertical direction. The resulting equations in the boundaryfitted and sigma
stretched grid are rather complex. However, numerous analytical tests were conducted
to ensure the accuracy of the model (Sheng, 1986 and Sheng, 1987). The model has
been applied to Chesapeake Bay (Sheng et al., 1989a), James River (Sheng et al.,
1989b), Lake Okeechobee (Sheng and Lee, 1991a, 1991b), and Tampa/Sarasota Bay
(Sheng and Peene, 1992). However, the earlier study on Lake Okeechobee (Sheng and
Lee, 1991a) did not consider thermal stratification in their model.
2.2 Vegetation Models
Vegetation can affect the aquatic life and also the water motion in the marsh area.
Early studies on the effect of vegetation on flow were conducted in the open channels.
Ree (1949) conducted laboratory experiments to produce a set of design curves for
vegetated channels. Kouwen et al. (1969) studied the flow retardance in a vegetated
channel in the laboratory and proposed the following equation:
Â£ = C, + CM A) (2.4)
where U is average velocity, u* is shear velocity, and C\ and Ci are coefficients. A
is a crosssectional area of the channel, and Av is the crosssectional area blocked by
the vegetation.
Reid and Whitaker (1976) considered the vegetation effect on flow as an ad
ditional term, which is proportional to the quadratic power of the velocity, in the
11
depthintegrated momentum equation. Details of their vegetation models are given
in Chapter 4. Their model, however, considered only the linearized equations of
motion. In the present study, fully nonlinear equations are considered.
Sheng (1982) developed a comprehensive vegetation model by including the effect
of vegetation on mean flow and secondorder correlations in a Reynolds stress model.
Although the model was able to faithfully simulate the mean flow and turbulence in
the presence of vegetation, it was not used for the present study due to the extra
computational effort required when it is combined with a 3D circulation model.
Roig and King (1992) formulated an equivalent continuum model for tidal marsh
flows. Neglecting leafiness, flexibility, and vegetation surface roughness, the net re
sistance force due to vegetation is thought to be related to the following parameters:
tv = f(p,9,l*,u,l3,d,s) (2.5)
where p is the viscosity of water, u is depthaveraged velocity, d is the average diameter
of vegetation, l3 is the vegetation height, and s is the spacing between vegetations.
Through a dimensional analysis,
T=pu*jf(F,R,i) (2.6)
where F is the Froude Number and R is the Reynolds Number.
To determine the function /, they conducted a simple flume experiment. For each
value of s/d, the dimensionless shear parameter pvaj was plotted as a function of R
and F.
2.3 Thermal Models
Sundaram et ah (1969) used a onedimensional vertical model to demonstrate
the formation and maintenance of thermocline in a deep stratified lake. The surface
boundary condition was given as follows:
?. = pKh = K(T,~ T.)
(2.7)
12
where q, is the heat flux, K is heatexchange coefficient,is vertical eddy diffusivity,
cp is specific heat of water, p is density of water, Te is an equilibrium temperature,
and Ta is a surface water temperature. They assumed that the annual variation in
heat flux can be approximated by the cyclic form of the equilibrium temperature:
T' = Te + asin(ut + ) (2.8)
Following Munk and Anderson (1948), the eddy diffusivity Kh was expressed as the
product of the eddy diffusivity under neutral condition and a stability function, which
is one under neutral condition but becomes less than one under stable stratification
(positive Richardson number).
Price et al. (1986) studied the diurnal thermal cycle in the upper ocean using
field data and a vertical 1D thermal model. Their measured data include currents,
temperature, and salinity, as well as meteorological data. Field data were collected
between April 28, 1980, and May 24, 1980, at about 400 km west of San Diego,
California.
Their major findings are the trapping depth of the thermal and velocity response
is proportional to r Q1/2, the thermal response is proportional to Q3/2, and the diurnal
jet amplitude is proportional to Q1^2, where Q is the heat flux and r is the wind stress.
They also simulated the diurnal thermal cycle using the vertical onedimensional heat
equation coupled with the momentum equations.
Gaspar et al. (1990) determined the latent and sensible heat fluxes at the air
sea interface using the inverse method. They stated that the total heat flux can
be divided into a solar part and a nonsolar part. While the solar radiation data
is usually available from direct measurement, the nonsolar part is usually indirectly
estimated from the meteorological data. However, this estimation of the nonsolar part
involves many empirical formulas and may contain large errors. Gaspar et al. (1990)
found that, by using the measured temperature data and solving the vertical one
dimensional momentum equation and temperature equation, it is possible to estimate
the nonsolar heat flux more accurately. They called this method the inverse method.
The major advantage of this inverse method is that it allows one to use the
usual satellite data (wind stress, surface insolation, and sea surface temperature) for
the estimation of heat flux at the ocean surface. The disadvantage of the method is
that horizontal advection effect is neglected in the analysis.
2.4 Turbulence Model
Flows in natural water bodies are often turbulent, although they can relaminar
ize during periods of low wind and for tide. Although direct numerical simulation of
turbulence can now be performed for simple flow conditions, it is still computation
ally prohibitive for practical applications in natural water bodies. Thus, Reynolds
averaging, a statistical approach was taken by decomposing the flow variables into
a mean and a fluctuating part and averaging the equations over a period of time that
is large compared to the turbulent time scale. The resulting equations thus produced
are called Reynolds averaged equations. The Reynolds averaged mean flow equations
contain terms involving correlations of fluctuating flow variables (i.e., secondorder
correlations) that represent fluxes of momentum or scalar quantities caused by tur
bulent motion. The task of turbulence modeling is to parameterize these unknown
correlations in terms of known quantities.
Numerous turbulence models were developed for the parameterization of turbu
lence. Some models are empirical, while others are based on more rigorous turbulence
theory. In this section, some available turbulence models are briefly reviewed with an
emphasis on the simplified secondorder closure model.
2.4.1 Eddy Viscosity/Diffusivity Concept
By analogy with molecular transport of momentum, the turbulent stresses are
assumed proportional to the meanvelocity gradients. This can be expressed as
14
where u\ and n' are fluctuating velocity components in the xt and x2 directions, while
u; and Uj are mean velocity components, and vt is the turbulent eddy viscosity.
Similarly, an eddy diffusivity Kt can be defined:
<210>
where ' and are the fluctuating and mean temperature or salinity or scalar con
centration, and Kt is the turbulent eddy diffusivity.
In order to close the system of mean flow equations for it,, it is necessary to obtain
an expression for vt in terms of known mean flow variables. Several options are given
in the following:
2.4.2 Constant Eddy Viscosity/Diffusivity Model
Earlier models used constant eddy viscosity/diffusivity models. Although this
allows easy determination of analytical solutions for the 1D equation of motion and
easy programming, there are many disadvantages. Turbulence is spatially and tempo
rally varying, hence constant eddy viscosity is not realistic. It is difficult to calibrate
the constant eddy coefficient model even if extensive field data exists.
2.4.3 MunkAnderson Type Model
Prandtl (1925) assumed that eddy viscosity is proportional to the product of a
t A A
characteristic fluctuating velocity, V, and a mixing length, L. He suggested that
V = /mÂ§^ and Av = The only parameter to be specified is length scale, lm,
which is assumed to be a linear function of z.
Following Prandtl, we can define the neutral vertical eddy viscosity as follows:
A
 H
' du
dv'
1 0.5
^ + 77
dz
dz.
(2.11)
where A0 is assumed to be a linear function of z increasing with distance above the
bottom or below the free surface and with its peak value at middepth, while not
exceeding a certain fraction of the local depth. In the presence of strong waves,
15
turbulence mixing in the upper layers may be significantly enhanced. In such case,
the length scale A0 throughout the upper layers may be assumed to be equal to the
maximum value at middepth (Sheng, 1983).
When a lake is stratified, vertical turbulence is affected by buoyancy induced
by the vertical nonhomogeneity. In this situation, vertical eddy coefficients should
be modified to account for this effect. This is parameterized by introducing the
Richardson number:
Ri =
p dz
(2.12)
Ri is positive when flow is stable (ff < 0) and when Ri is negative when flow is
unstable (^ > 0). Generally, eddy viscosity and eddy diffusivity are expressed as
follows:
Av = AV02(Ri) (2.13)
where 2 are stability functions and Avo and Kvo are eddy viscosity and eddy
diffusivity when there is no stratification. Stability functions have the following forms:
l = {l + 2 = (1 + a2Ri)mi (2.14)
Based on comparing model results with field data, Munk and Anderson (1948) devel
oped the following formula:
i = (1 + 10ifc)"1/2; 2 = (1 + 3.33 Ri)'3'2 (2.15)
Many similar equations with different coefficients were suggested based on numerous
sitespecific studies. These coefficients, however, are not universal, and care must be
taken when applying these formlete to a new water body where little data exist.
2.4.4 Reynolds Stress Model
One can obtain an equation for the timeaveraged secondorder correlations by
following the procedure: (i) decompose the dependent variables into mean compo
nents and fluctuating components, (ii) substitute the decomposition into continuity,
16
momentum equation and heat equation, and (iii) take timeaverage of all equations.
For example, the resulting timeaveraged equation for uu' (e.g., Donaldson, 1973;
Sheng, 1982) is
du;u
t a 1 (
ut OXk
rrduj
'dxk
}rdui Ujp Uip
 uiukx ^ Si + 9j
dxk p0 p0
 2eiktkutuj 2ejkÂ£ltukui
(dukUjUj)
dxk
(2.16)
u\ dp u'j dp d2uiu': du'i du]
3 + v\r22u 1 3
p dxj p dxi dxkdxk
dxk dxk
Similar equations for u{p and p p can be obtained. Unresolved thirdorder cor
relations and pressure correlations are modeled using the simplest possible forms
(Donaldson, 1973).
du{ du'j bSijq3 avu\u
Vdxkdxk 3A A2
(2.17)
p du'i du'j,
r ( i "J \ H f I T r 9 \
p dx, + dxj A u,Uj 6x1 3 1
(2.18)
d du'ip t du'jp dqA tqu'
gXj + dx. ~ 'c dxt dxt
(2.19)
where q is the total fluctuating velocity and A is the turbulence macroscale. The
model constants (a, 6, and vc) are determined from a wide variety of laboratory data
(Lewellen, 1977). Thus, a full Reynolds stress model consists of six equations for
velocity fluctuations three equations for the scalar fluxes, u\p', and one equation
for the variance, p'p'. Considering the required computer storage and CPU time for
the turbulence models, it is desirable to use a simplified form of the Reynolds stress
model.
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure Model
The complete secondorder closure model is too complicated to be used in a three
dimensional model. A simplified secondorder closure model can be developed with
17
the following assumptions: 1) Secondorder correlations have no memory effect. That
means correlations at the previous time have no effect on correlations at the next time.
Therefore, ^ = 0. 2) Correlations at a point do not affect the value at another point.
Therefore, all the diffusive terms are dropped. These conditions are approximately
true if the time scale of turbulence is much less than the mean flow time scale and
the turbulence does not vary significantly over the macroscale, i.e., the turbulence is
in local equilibrium. Then the remaining equations become as follows (Sheng, 1983):
0 = U;U
duj
dxk
~ UjUk
duj
dxk
9i
Ujp
9j
UiP
26 kUU j Â£jk(QUkUÂ¡
H / t f C V \ C
j{uiUj Sijj) 6,
,J 12A
(2.20)
0 =
/ / &P it Ollx
9iPP
*
Po
2eijkSljUkp 0.75
t *
UiP
(2.21)
n cCT7dP 45qpp
0 = 2up d7Â¡ + a
(2.22)
These algebraic equations can be solved with ease, once the mean flow conditions are
known. In order to complete the system of equations, q and A need to be solved
following the procedure described in Sheng et al., 1989b.
The above equilibrium closure model was applied to the Atlantic Ocean (Sheng
and Chiu, 1986), Chesapeake Bay (Sheng et al., 1989a) and the James River (Sheng
et al., 1989b). More details of the model will be given in Chapter 8.
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model
To introduce some dynamics of turbulence into the simplified secondorder closure
model, one can add a dynamic equation for q2(q2 = u'u' + v'v' + ww1), which is twice
the turbulent kinetic energy (Sheng and Villaret, 1989). This TKE closure model has
18
been applied to James River (Choi and Sheng, 1993) and Tampa Bay (Schoellhamer
and Sheng, 1993).
Two other turbulence models, which are based on the socalled k e model (Rodi,
1980), are described in the following:
2.4.7 OneEquation Model (k Model)
Using the eddy viscosity/diffusivity concept, the choice of velocity scale can be
y/k, where k = (u2 + v2 + u>2)/2 is the kinetic energy of the turbulent motion. When
this scale is used, the eddy viscosity is expressed as
vt = c'^VkL (2.23)
where is an empirical constant and L is the length scale. To determine k, an
equation is derived from the NavierStokes equation as:
dk dk d ,Xu' P x, rrdu'i ttj du'i du\
m + uaTi = &71"'( V + t)1 ^ "
(2.24)
2 p/J ~'~3 dxj dxjdxj
To obtain a closed set of equations, diffusion term and dissipation term must be
modelled. The diffusion flux is often assumed proportional to the gradient k as
ut dk
<(UiU3 P \
Ui(~ + 71 = ^ ax.
(2.25)
where crfc is an empirical diffusion constant. The dissipation term e, which is the last
term of Eq. 2.24, is usually modelled by the expression
A:3/2
e = Cjy
(2.26)
The length scale, L, needs to be specified to complete the turbulence model. Usually
L is determined from empirical relations.
2.4.8 TwoEquation Model (k e Model)
To avoid the empirical specification of length scale, another equation for the
dissipation e is needed. Then eddy viscosity and eddy diffusivity are expressed as
k2
i>t cM
(2.27)
19
rt =
(2.28)
where is an empirical constant and crt is the turbulent Prandtl/Schmidt number.
An equation for e is derived from the NavierStokes equation and is
dt dt d dut dt t e
dt dx{ dxi ^ dat dxi
) + C1 e^(Pc3eG) C2 c~
(2.29)
where P and G are the stress and buoyancy term in Equation 2.24, respectively, and
Cie, c2e and c3c are empirical model coefficients.
2.5 Previous Lake Okeechobee Studies
Numerous studies on Lake Okeechobee have been performed, but most of them
focused on water quality.
Whitaker et al. (1975) studied the storm surges in Lake Okeechobee while con
sidering the vegetation effect in the western marsh area, by using a twodimensional,
vertically integrated model. They simulated the seiche in the lake during the 1949
and 1950 hurricanes. In their study, the bottom friction coefficient was parameterized
as a function of depth to achieve better agreement of storm surge height.
Schmalz (1986) investigated hurricaneinduced water level fluctuations in Lake
Okeechobee. His study consisted of two parts: a hurricane submodel and a hydrody
namic submodel. The hurricane submodel used hurricane parameters such as central
pressure depression, radius to maximum winds, maximum wind speed, storm track,
storm forward speed, and azimuth of maximum winds, and determined the wind and
pressure field that were used as forcing terms for the hydrodynamic submodel.
The hydrodynamic submodel solved the depthaveraged momentum equations
and continuity equation. Finitedifference method was used for the numerical solution.
For treatment of marsh area, an effective bottom friction which relates the Mannings
n to water depth and canopy height was used. The 2D hydrodynamic model can
resolve flooding and drying: during strong wind conditions such as a hurricane, a
portion of the lake can become dried because of the excessive setdown by wind, while
20
other portion of the lake can become flooded because of excessive setup by wind.
A threedimensional Cartesiangrid hydrodynamic and sediment transport model
for Lake Okeechobee was recently developed (Sheng et al., 1991a; Sheng, 1993). In
addition, these models were extended to produce a threedimensional phosphorus
dynamics model (Sheng, et al., 1991c). These models use the simplified secondorder
closure model and the sigma stretched grid, however, did not consider the effects of
vegetation and thermal stratification
2.6 Present Study
The present work focuses on the study of effects of vegetation and thermal strat
ification on winddriven circulation in Lake Okeechobee. As will be shown later, the
threedimensional curvilineargrid model (CH3D) will be significantly enhanced to
allow accurate simulation of the observed circulation. Model features are compared
with model features of some previous lake studies in Tables 2.1 and 2.2. It is apparent
that the 3D model developed in this study is more comprehensive than those used
in previous lake studies.
21
Table 2.1: Selected features of lake models.
Author
Dimensionality
Type of
Model
Temporal
Dynamics
Turbulence
Advection
Welander
1957
3D
AN
T.D.
A
No
Liggett
1969
3D
F.D.
T.D.
A
No
Lee + Liggett
1970
3D
F.D.
S.S.
A
No
Liggett + Lee
1971
3D
F.D.
S.S.
A
No
Gedney + Lick
1972
3D
F.D.
T.D.
A
No
Goldstein + Gedney
1973
3D
A.N.
B
No
Sengupta + Lick
1974
3D
F.D.
T.D.
D
Yes
Simons
1974
3D
F.D.
T.D.
B
Sheng
1975
3D
F.D.
S.S.
A
No
Thomas
1975
3D
F.D.
S.S.
B
No
Whitaker et al.
1975
2D
F.D.
T.D.
Yes
Witten 4 Thomas
1976
3D
F.D.
S.S.
C
No
Lien + Hoopes
1978
3D
F.D.
S.S.
A
No
Schmalz
1986
2D
F.D.
T.D.
Yes
Sheng + Lee
1991a
3D
F.D.
T.D.
E
Yes
* F.D. : Finite difference
* AN : Analytic
* S.S. : Steady state
* T.D. : Time dependent
* A : Constant
* B : Dependent on wind
* C : Exponential form
* D : MunkAnderson type
* E : Simplified secondorder closure model
22
Table 2.2: Application features of lake models.
Author
Basin
tion
Dimen
Mean
Depth
Forcing
Vege
tation
Grid
W
H
R
Hor.
ver.
Liggett
1969
Idealized
Basin
Yes
No
No
No
U
Lee + Liggett
1970
Idealized
Basin
Yes
No
No
No
U
Liggett + Lee
1971
Idealized
Basin
Yes
No
No
No
u
Gedney f Lick
1972
Lake Erie
400 km
100 km
20 m
Yes
No
Yes
No
u
Sengupta + Lick
1974
Squire
Valley
1.89 m
Yes
Yes
No
No
N
Simons
1974
Lake
Ontario
Yes
Yes
No
No
u
Sheng
1975
Lake Erie
400 km
100 km
20 m
Yes
No
Yes
No
N
cr
Thomas
1975
Idealized
Basin
Yes
No
No
No
U
Whitaker et al.
1975
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
Yes
u
Witten + Thomas
1976
Idealized
Basin
300 x
87 km
Max
180 m
Yes
No
No
No
Lien + Hoopes
1978
Lake
Superior
Yes
No
No
No
u
Schmalz
1986
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
No
u
Sheng + Lee
1991a
Lake
Okeechobee
57 km
60 km
2.5 m
Yes
No
No
Yes
c
a
* W : Wind
* H : Heating
* R : River
* U : Uniform Cartesian grid
* C : Curvilinear grid
* N : Nonuniform Cartesian grid
* a : Vertically stretched grid
CHAPTER 3
GOVERNING EQUATIONS
3.1 Introduction
This chapter presents the basic equations which govern the water circulation in
lakes, reservoirs, and estuaries. Because the details can be found in other references
(e.g., Sheng, 1986; Sheng, 1987; Sheng et.al., 1989c), the governing equations are
presented here without detailed derivations.
3.2 Dimensional Equations and Boundary Conditions in a Cartesian Coordinate System
The equations which govern the water motion in the water bodies consist of
the conservation of mass and momentum, the conservation of heat and salinity, and
the equation of state. Because Lake Okeechobee is a fresh water lake, the salinity
equation is not considered. The following assumptions are used in the Curvilinear
Hydrodynamic Threedimensional Model (CH3D) model.
(1) Reynolds averaging: Three components of velocity, pressure, and temperature
are decomposed into mean and fluctuating components and timeaveraged.
(2) Hydrostatic assumption: Vertical length scale in lakes is small compared to
the horizontal length scale, and the vertical acceleration is small compared with the
gravitational acceleration.
(3) Eddy viscosity concept: After timeaveraging, the secondorder correlation
terms in the momentum equation are turbulence stresses, which are related to the
product of eddy viscosity and the gradient of mean strain.
(4) Boussinesq approximation: Density variation of water is small, and variable
density is considered only in the buoyancy term.
23
24
3.2.1 Equation of Motion
With above assumptions, the equations of motion can be written in a righthanded
Cartesian coordinate system as follows:
du dv dw
dx dy dz
(3.1)
du du2
dt ^ dx
+
duv duw 1 dp d f du'
dy + dz V pQ dx A dx \ H dx /
d (A du
ay \ H dyt
a (A du'
+ dz [AvdzJ
(3.2)
dv duv
dt A dx
dv2 dvw 1 dp d f dv'
dy + dz U p0dyJrdx\Hdx/
d( dv\ d( dv\
+ dH{A"a7j) +& (/'&)
(3.3)
dp
dz
= pg
(3.4)
dT duT dvT dwT d_ ( dT'
dt "** dx + dy + dz dx\HdxJ
, d / dT\ d / dT\
dy \H dy) + dz V v dz)
P = P(T,S)
(3.5)
(3.6)
where (u,v,w) are velocities in (x, y, z) directions, f is the Coriolis parameter defined
as 20 sm. where 1 is the rotational speed of the earth, is the latitude, p is density,
p is pressure, T is temperature, (Ah,Kh) are horizontal turbulent eddy coefficients,
and (riv, Kv) are vertical turbulent eddy viscosities.
For the equation of state, Eqn. 3.6, there are many different formulae that can be
used. For the present study, the following equation given by Eckart (1958) is used:
p = (1 + P)/(a + 0.698P)
(3.7)
25
P = 5890 + 387 0.37572 + 3S
(3.8)
a = 1779.5 + 11.257 0.074572 (3.8 + 0.017)5 (3.9)
where T is water temperature in degree C, S is salinity in ppt, and p is density in
g/crn3.
Besides governing equations, boundary conditions should be specified.
3.2.2FreeSurface Boundary Condition (z = tj)
(1) Kinematic boundary condition:
dt] dq dn dn
W = f u h V ~
dt dx dy dt
(3.10)
(2) Surface heat flux:
q = Kv^ = K{TsTe) (3.11)
where Ta is the lake surface temperature, Te is the equilibrium temperature, K is a
heat transfer coefficient, and q is positive upward. (3) Surface stress:
Av
du
~d~z
(3.12)
where the wind stresses tx and ry must be specified.
3.2.3Bottom Boundary Condition (z h)
(1) Heat flux is specified as zero,i.e., = 0
(2) Quadratic bottom friction law is used, i.e.,
= pCdyJua2 + vi2uury
= pCdy/ui2 + v i2ui
(3.13)
where u\ and tq are velocity components at the first grid point above the bottom.
3.2.4Lateral Boundary Condition
(1) Heat flux is assumed zero,i.e., = 0
(2) No flow through boundary,i.e., u 0 or v = 0
26
3.3 Vertical Grid
No natural water bodies have strictly flat bottoms. Therefore to represent the
variable bottom topography, a stretching is used by defining a new variable a :
a = I7i^M (3.U)
h(x,y) + ((x,y,t)
The advantage of crstretching is that the same vertical model resolution can be main
tained in both shallow and deeper parts of a lake. The disadvantage is that it in
troduces additional terms in the equations. Details of crstretching can be found in
Sheng and Lick (1980) and Sheng (1983).
3.4 NonDimensionalization of Equations
By introducing reference values, the governing equations can be nondimensionalized.
The purpose is to make it easier to compare the relative importance of each term.
The following relations were used (Sheng, 1986).
(u*,u*,u>*)
(u,V,wXr/Zr) /Ur
(x*,yn,z*)
=
(x,y,zXr/Zr) /Xr
(Tx>Ty)
=
W,T?)/PofZTUr
t*
=
tf
Â£
=
T0/(Tr T0) q/poCpfZrT0
c
=
gC/fUrXr = C/Sr
p*
=
(P~ Po)/{Pr ~ Po)
T*
=
(T T0)/(Tr T0)
A*h
=
Ah/Aht
K
=
Av/ Avt
K'h
=
Kh/ KHt
K
=
Kv/Kvr
J*
=
uXT/UT
(3.15)

27
where variables with asterisks are nondimensional variables and variables with r are
the reference values.
3.5 Dimensionless Equations in aStretched Cartesian Grid
The transformation relations from a Cartesian coordinate (x, y, z) to a vertically
stretched Cartesian coordinate (x,y,a) may be found in Sheng (1983). Using the
relation presented in the previous section, the following dimensionless equations are
obtained:
d( adHu BHv TTdu A
Â¥ + ^ + ^ + ^ = 0
1 dHu
H dt
+
dÂ£ E^d_ f du
dx + H2 da \ vda,
Ro dHuu dHuv
H \ dx + dy
 + v
dHuu\
+ ~dT)
ac,_]kJL(A
dx + H2 da V v da)
+ Bx
(3.16)
(3.17)
1 dHv
H dt
dC Ev d
+
7T Mv
dv
dy H2 da \ v da
u
Ro ( dHuv dHvv dHvu'
H \ dx dy + da /
+ E" +
Ro
Fd
d<
H
f
dx
dP
d_
dy
dH
dv'
~ d
dy dy
dyJ
(/>+ ap
+ H.O.T.
dv
dy H2 da V da
+ Bv
(3.18)
1 dHT Ev d ( dT\
H dt ~ Prv H2da ^ da)
Ro idHuT dHvT dHuT\
H \ dx + dy + da )
(3.19)
28
+
1z\JL(k ^\a.
Pth dx \ H dx J dy
+ H.O.T.
P = p{T, S)
(3.20)
where H is total depth, /? = gZr/X2f2 and H.O.T. is higher order terms.
3.5.1 VerticallyIntegrated Equations
The CH3D model can solve the depthintegrated equations and the threedimensional
equations. The vertically integrated momentum equations are obtained by integrating
the threedimensional equations from bottom to top.
(Â£+Â£)
V OX dy J
= 0
(3.21)
dU
dt
H + Tsx Tbx + V
dx
Ro
d_ fUU^ m d_ fUV}
dx{ H ) + dy{ H )
+ E
d_
dx
Ro H2 dp
Fr 2 dx
Hir + Dx
dx
H
mr
dx
+
d ( dU'
dy V H dy,
dV_ rrd(M TT
dt ~ Hdy+Tay Thy U
Ro
d i
dx'
, H ) + dy V H )
+ Eh
r d
< dv\ d (
dV M
dx
{Ah dx ) + dy dy)
Ro H2 dp
Frl 2 dy
= H^ + Dy
dy
(3.22)
(3.23)
29
3.5.2 Vertical Velocities
u>
(1 +cr)dC 1 r dHu dHv\
/3H dt H J1 { dx + dy )
(1 + a) d( ( dh dh\
(3.24)
(3.25)
where U = f\ uda, V = vda, r3X, r3y are wind stresses at the surface and r^,
are bottom stresses.
3.6 Generation of Numerical Grid
3.6.1 Cartesian Grid
In order to numerically solve the governing equations, finite difference approxima
tions are introduced to the original governing equations, and solutions are obtained
at discrete points within the domain. Therefore, a physical domain of interest must
be discretized. When a simple physical domain is considered, cartesian grid can be
used and hence grid generation and development of finitedifference equations are
relatively easy.
Unfortunately, most physical domains in lakes or estuaries are complex. Pre
viously rectangular grid was widely used. This method has such disadvantages as
inaccuracies at boundaries and complications of programming due to unequal grid
spacing near boundaries.
3.6.2 Curvilinear Grid
To better resolve the complex geometries in the physical domain, boundaryfitted
(curvilinear) grid can be used. In general, a curvilinear grid can be obtained by use
of (1) algebraic methods, (2) conformal mapping, and (3) numerical grid generation.
Algebraic grid generation uses an interpolation scheme between the specified
boundary points to generate the interior grid points. This is simple and fast com
putationally, while the smoothness and skewness are hard to control. Conformal
30
mapping method is based on complex variables, so the determination of mapping
function is a difficult task. Therefore, many practical applications rely on numerical
grid generation techniques.
3.6.3 Numerical Grid Generation
Partial differential equations are solved to obtain the interior grid points with
specified boundary points. Thompson (1983) developed an elliptic grid generation
code (WESCORA) to generate a twodimensional, boundaryfitted grid in a complex
domain.
To help understand the physical reasoning of this method, consider a rectangular
domain. When the temperature is specified along the horizontal boundary, then
the temperature distribution inside can be obtained by solving the heat equation.
Therefore, isothermal lines can be drawn. Also, other isothermal lines can be obtained
with the specified temperature in the vertical direction. By superimposing these
isothermal fines, intersection points of isothermal fines can be considered as grid
points.
WESCORA solves Poisson equations with same idea in a complex domain. Con
sider the following set of equations (see Figure 3.1):
U + {yy = P (3.26)
T]xx d* Vyy = Q (3.27)
with the following boundary conditions:
Â£ = Â£(x,y) on 1 and 3
7] = constant (3.28)
( = constant
T) = r](x,y)
on 2 and 4
(3.29)
31
where the functions P and Q may be chosen to obtain the desired grid resolution and
alignment. In practice, one actually solves the following equations which are readily
obtained by interchanging the dependent and independent variables in Eqs. 3.26 and
3.27:
ctx^Â£ 2b b otPx^ b 'yQxq
al/ ~ + 7yrm + aPyt + lQyv
where
a = xr, + yn
P = xiXr, + ytyv
7 = + y
P = ^
Q = jM + vl
J = XtVr, XM
with the transformed boundary conditions:
x = on i = 1 and 3
V = 9i((,Vi)
x = fi(Zi,rÂ¡) on i = 2 and 4
V = 9i(M (334)
3.7 Transformation Rules
Generations of a boundaryfitted grid is an essential step in the development of
a boundaryfitted hydrodynamic model. It is, however, only the first step. A more
important step is the transformation of governing equations into the boundaryfitted
coordinates. A straightforward method is to transform only the independent variables,
(3.32)
(3.33)
= 0 (3.30)
= 0 (3.31)
32
* C<*.y) r.*,<*,t*
(o) or (e>)
^ n(*,y) y* y,( ,.**
. 3*
*> y*
TRANSFORMED
Figure 3.1: A computational domain and a transformed coordinate system.
33
i.e., the coordinates, while retaining the Cartesian components of velocities. Johnson
(1982) developed such a 2D verticallyintegrated model of estuarine hydrodynamics.
The advantage of the method is its simplicity in generating the transformed equations
via chain rule. The dimensional forms of the continuity equation and the vertically
integrated momentum equations are shown by Eqs. (20) and (21) in Appendix A of
Sheng (1986). The resulting equations, however, are rather complex. Even when an
orthogonal or a conformal grid is used, the equations do not become any simpler. Ad
ditional disadvantages are (1) the boundary conditions are quite complicated because
the Cartesian velocity components are generally not aligned with the grid lines, (2)
the staggered grid cannot be readily used, and (3) numerical instability may develop
unless additional variables (e.g., surface elevation or pressure) are solved at additional
grid points, (Bernard, 1984, cited in Sheng (1986)).
To alleviate the problems mentioned in the previous paragraph, Sheng (1986)
chose to transform the dependent variables as well as the independent variables.
Equations in the transformed coordinates (Â£, 77) can be obtained in terms of the con
travariant, or covariant, or physical velocity components via tensor transformation
(e.g., SokolnikofF, 1960). As shown in Fig. 8 of Appendix A of Sheng (1986), the
contravariant components (tt) and physical components u(i) of the velocity vector
in the nonCartesian system are locally parallel or orthogonal to the grid lines, while
the covariant components (it,) are generally not parallel or orthogonal to the local
grid lines. The three components are identical in a Cartesian coordinate system. The
following relationships are valid for the three components in a nonCartesian system
are
' = (gu) 1/2u(i)
(no sum on i)
(3.35)
i = (9ii)~1/29iMj)
(no sum on i)
(3.36)
u(i) = i
(3.37)
34
where g is the diagonal element of the metric tensor gtJ:
dxldxn c
9iJ = WWn
(3.38)
which for the twodimensional case of interest is
_ f x\ + V\ x(xri + y(Vv\ ( 9n 912 \
\ xnxt + yvyt x2n + yl ) ^ g21 g22 )
(3.39)
The three components follow different rules for transformation between the prototype
and the transformed plane:
 d?_ j
u dx>u
(3.40)
(3.41)
(>) = (3.42)
where the unbarred quantities represent the components in the prototype system,
while the barred quantities represent the components in the transformed system.
3.8 TensorInvariant Governing Equations
Before transforming the governing equations, it is essential to first write them in
tensorinvariant forms, i.e., equations which are independent of coordinate translation
and rotation. For simplicity, unbarred quantities are used to denote the variables in
the transformed system unless otherwise indicated.
Following the rules described in the previous paragraph, the following equations
are obtained (Sheng, 1986):
+ =0 <343>
1 dHuk
H dt
Okgnjekiu
35
+
Ro
Ev
d
H2 da
Ro
(Hueu% +
dhuku
da
(3.44)
+ EHAu*mr
where d/dxk is the partial derivative, gn is the metric tensor while g0 = J =
x^yv xvyz is the determinant of the metric tensor, uk is the contravariant veloc
ity, ( )/ represents the covariant spatial derivative, \k represents the contravariant
spatial derivative, and is the permutation tensor and
,12
,21
,n
1
V90
1
V90
e22 = 0
(3.45)
The covariant and contravariant differentiations are defined by
UJ = uj + D*iua
(3.46)
S\k = gkmS,m (3.47)
where :j represents partial differentiation and D'aj represents the Christoffel symbol
of the second kind:
D)k = rDjk (3.48)
where g'n represents the inverse metric tensor, h,, and Dnjk is the Christoffel symbol
of the first kind:
1, .
t'ijk 2 \9ijk T gik.j ~ gjk.i)
(3.49)
36
3.9 Dimensionless Equations in BoundaryFitted Grids
Expanding the equation 3.41 and 3.42, the following equations are obtained:
Ct +
^(sfcHu) + FjglHv)
dii
= 0
(3.50)
1 dHu
H~dT
(9lI+s2) + i^+^
Ro
~H
y/9o y/90
O Q
(Huu) + Huv) + (2 D\x + D\2)Huu
+ (3.Di2 + D\2)Huv 4 D\2Hvv +
dHuu>
da
+
K d_
H2 da
Ro
F
' du'
V r\
. da j
 Â£[/;(*%>'%)*
4 Â£//A//(Horizontal Diffusion)
(3.51)
1 dHv
H dt
(
#11 9 21
:U 4 V
,V9~o y/9o
Ro
~H
o C\
(Huv) + (Hvv) 4 D^Huu
4 (Dh 4 iD\2)Huv 4 (D\2 4 2D\2)Hvv
4
E^d_
H2 da
Ro
F,
1 [ft
21 22 dp
9 8(+9 toÂ¡,
da
+ (921^ + 922^)(lpda + aP
4 '//^//(Horizontal Diffusion)
(3.52)
where the horizontal diffusion terms are listed in Sheng(1986). The temperature
equation can be obtained according to the same procedure as
1 dHT Ev d ( dT\
H dt ~ Prv H2da \Kv da)
37
Tj^\k^HuT)+i^HvT)
I
PrH
,21 r
RodHuT
H da
+ ^Iur.u+,T
+ 9*'Tu + gT,2,2
3.10 Boundary Conditions and Initial Conditions
3.10.1Vertical Boundary Conditions
The boundary conditions at the free surface (a = 0) are
4,
fdu du\
jfo'lfo)
dT
da
H_
Ev
HPrv
E
Tsrt)
The boundary conditions at the bottom (a = 1) are
(du dv\ H .
Av Idada) ~ E}T^T^)
HrZrCd + 25ri2iiiWi +^22^1 (ui,ui)
Avr 1
dT
da
= 0
(3.53)
(3.54)
(3.55)
where ui and Ui are the contravariant velocity components at the first grid point
above the bottom.
3.10.2Lateral Boundary Conditions
Due to the use of contravariant velocity components, the lateral boundary condi
tions in the (Â£,rÂ¡,a) grid are similar to those in the (x,y,a) system. Along the solid
boundary, noslip condition dictates that the tangential velocity is zero, while the slip
condition requires that the normal velocity is zero. When flow is specified at the open
boundary or river boundary, the normal velocity component is prescribed.
3.10.3Initial Conditions
Initial conditions on vectors, if given in the Cartesian or prototype system, such
as the velocity and the surface stress, must be first transformed before being used
38
in the transformed equations. Thus, the surface stress in the transformed coordinate
system is given by
r1 = r + r!
ox ay
r* = dvr 1,^2
dx ^ dy
(3.56)
(3.57)
where r1, r2 are the contravariant components of the stress in the transformed system
and r1, r2 are the contravariant components in the Cartesian system. Note that in
the Cartesian system, the contravariant, covariant, and physical components of a
vector are identical. The contravariant components of the initial velocity vectors can
be transformed in the same manner to obtain the proper initial conditions for the
transformed momentum equations.
CHAPTER 4
VEGETATION MODEL
4.1 Introduction
The western portion of Lake Okeechobee is covered with an extensive amount
of vegetation. The vegetation can affect the circulation in several different ways.
First of all, wind stress over the emergent vegetation is reduced below that over the
open water. Furthermore, the submerged vegetation introduces drag force to the
water column. Because most of the vegetation stalks are elongated cylinders without
large leaf areas, the drag force is primarily associated with the profile drag (or form
drag) instead of the skin friction drag. The profile drag can reduce the flow and is
proportional to the projected area of vegetation in the direction of the flow.
The presence of vegetation also can affect the turbulence in the water column.
The characteristic sizes of the horizontal and vertical eddies generally are reduced by
the vegetation. This usually leads to a reduction of turbulence, although some wake
turbulence may be generated on the downstream side of vegetation.
In order to simulate the effects of vegetation, several approaches have been under
taken in previous investigations. For example, Saville (1952) and Sheng et al. (1991b)
used an empirical correction factor to simulate the reduction of wind stress over the
vegetation area. Sheng et al. (1991b) also adjusted the bottom friction coefficient
over the vegetation area. For simplicity, however, Sheng et al. (1991b) did not include
the effect of vegetation on mean flow and turbulence in the water column, because the
primary focus of that study was the internal loading of nutrients from the bottom sed
iments in the open water zone. Whitaker et al. (1975) developed a twodimensional,
verticallyintegrated model of storm surges in Lake Okeechobee. The profile drag cre
39
40
ated by the vegetation was included in the linearized verticallyintegrated equations of
motion, which did not contain the nonlinear and diffusion terms. Sheng (1982) devel
oped a comprehensive model of turbulent flow over vegetation canopy by considering
both the profile drag and the skin friction drag in the momentum equations in addi
tion to the reduction of turbulent eddies and the creation of turbulent wake energy.
Detailed vertical structures of mean flow and turbulence stresses were computed by
solving the dynamic equations of all the mean flow and turbulent quantities. Model
results compared well with available mean flow and turbulence data in a vegetation
zone.
For the present study, due to the lack of detailed data on vegetation and mean
flow and turbulence in the vegetation zone, a relatively simple vegetation model which
is more robust than Whittaker et a/.s model yet simpler than Shengs 1982 model is
developed. Due to the shallow depth in the vegetation zone, it is feasible to treat the
water column with no more than two vertical layers. When the height of vegetation is
greater than 80% of the total water depth, the flow is considered to be onelayer flow,
i.e., the entire water column is considered to contain uniformly distributed vegetation.
When the height of vegetation is between 20% and 80% of the total water depth, the
flow is considered to be twolayer flow, i.e., the water column consists of a water layer
on top of a vegetation layer. The vegetation effect is neglected when the height of
vegetation is less than 20% of the total depth. The profile drag introduced by the
vegetation can be formulated in the form of a quadratic stress law:
^canopy = pCU,\\u\\AN (4*1)
where u is the vertically averaged velocity in the vegetation layer (layer 1), p is the
density of water, A is the projected area of vegetation in the direction of the flow, N is
the number of stalks per unit horizontal area, and c is an empirical drag coefficient.
Tickner (1957) performed a laboratory study. Strips of ordinary window screen 0.1
foot in height were placed across a channel to simulate a vegetative canopy. Using
41
Tickners experimental results, Whitaker et al. (1975) calculated c( 1.77) which was
used in this study. Roig and King (1992) showed c< is a function of Froude Number,
Reynolds Number, vegetation height, spacing, and diameter of vegetation. As the
water level changes, the flow regime over a vegetation area may change from one
layer to twolayer flow, and vice versa.
4.2 Governing Equations
Let us consider an x, y, z coordinate system with the velocity components in the
(x, y, z) directions as (u, v, w). The lower layer (layer I) of the water column is covered
with vegetation, while the upper layer (layer II) is vegetationfree (Figure 4.1).
Flow in the vegetation layer (ui, ux) and flow in the vegetationfree layer (u2, v2)
both satisfy the equations of motion.
4.2.1 Equations for the Vegetation Layer (Layer I)
du i du\ duivi duiWi
dt dx dy dz
dv\ du\V\ dv\ dv\w\
dt dx ^ dy ^ dz
_ 1 dpi d
JV1 + TT"
p dx dx
+
d_
dy
a dlLl
Ah~h~
dy
+
dz
1 dpi d
~ ~fu' ~ J aT + di
iff
du\
Ah
dx
dui
dz
dv x
dx
d
. dv i
d
. dvi
+ dy
Ahtt
[ v J
1 ^
co
+
1 N
CO
s
T
dp
Tz
pa
Integrating Equation (4.4) vertically:
P = Pa+ pg(C z)
Integrating Equations (4.2) and ( 4.3) vertically from z = h to z =
dUi
dt
+ h (zr)+5? (tt) + >L,(*=fVl+(tw ~ Fa)
+
d_
dx
Ah
dUi
dx
+
d_
dy
Ah
dUi
dy
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
42
Figure 4.1: Schematics of flow in vegetation zone.
43
W + (tt) + ) (Â£) + si.C, = M + n Fw)
d
r. dv;l
d
\ dVi
dx
1
Â¡5
ca
H
1
+ dy
Ah a 
L dy J
(4.7)
where L\ = h i and U\ and V\ are the verticallyintegrated velocities within the
vegetation layer:
(U1,vl)= f (u,i,Vi)dz (4.8)
J h
Tf,r and Tby are bottom stresses, rtx and r)y are interfacial stresses between layer I and
layer II, and and are form drags due to the vegetation canopy.
4.2.2 Equations for the VegetationFree Layer (Layer II)
du2 du\ du2v2 du2w2 1 dp2 d
dt dx dy dz V2 pdx+dx
Ah
du2
dx
d_
+ dy
A
du2
d
dy dz
a du2
Avz
az
(4.9)
dv2 du2v2 dv\ dv2w2 1 dp2 d
+ ~aT~ + a7 + ~5T =
A dV2
A~F~
dx
d
a dv2
d
a dv2
+ dy
AH~~
[ dy J
+ d~z
Av~d7
(4.10)
Integrating Equation (4.9) and (4.10) vertically from 2 = to z = ( and defining
L2 i +
("Tjx Tjx)
P
+
d
\a9U2 1
d
4
\a9UA
dx
OX
dy
1 ^ Cl
L dy
(4.11)
dV2
dt
d (U2V2\ d (V?\ T rrr K
+ di \~lt) + Ty (17) + +(rTiy)
d
\a dV2]
d
\ t dV2
dx
Ahw1
dx
+ dy
Ah p
L dy
(4.12)
where U2 and V2 are verticallyintegrated velocities within the vegetationfree layer:
(U2,V2) = J ^(u2,v2)dz (4.13)
44
4.2.3 Equations for the Entire Water Column
Instead of solving the above equations for the vegetation layer and the vegetation
free layer, it is more convenient to solve the verticallyintegrated equations for the
entire water column, which can be readily derived by combining the equations for the
two layers. First, the verticallyintegrated velocities over the entire water column in
the vegetation zone, U and V, can be defined as
{U, V) = (I/i + U2, Vx 4 V2) (L\Ui f Liu2, L\V\ + L2v2)
(4.14)
where Hi u2, T>i and v2 are verticallyaveraged velocities within layer I and layer II,
respectively, while L\ and L2 are the thicknesses of layer I and layer II, respectively.
Adding the U\ equation and the U2 equation leads to
dt dx l L\ L2
a (UiVj U2V2\
+ ^{LT+~Lr)+9H
fV H (tsx TÂ¡>r F:x)
P
d
a au'
d
'a dU
dx
+ dy
Ah~k~
l y\
+
while the summation of the V\ equation and the V2 equation results in
(4.15)
dV d {U\V\ U2V2]
dt + dx\ Lx + L2 )
d (V? V22\
+ %(lT+l7J+/c
fU 4" ~{Tay Tby &'ey)
d
\ 9V'
d
\ dV
dx
dx
+ dy
Ahz
L dy
(4.16)
All the stress terms are computed as the quadratic power of the flow velocity. For
example, riX and r,y are computed as quadratic functions of the wind speed, tÂ¡,x and
T^y are computed as quadratic functions of U and V, and rtx and r,y are quadratic
functions of (U2 U\) and (V2 V\). The form drags associated with the vegetation
are:
Fcx = pcdAxN3
M+fUt
L1
(Â£)
(4.17)
45
(4.18)
(4.19)
where c and c are empirical drag coefficients.
An additional equation that must be satisfied is the continuity equation:
d( dU dV
4 1 = (
dt dx dy
4.2.4 Dimensionless Equations in Curvilinear Grids
(4.22)
The above dimensional equations were presented to illustrate the development of
the vegetation model. In the curvilineargrid model, however, dimensionless equations
in curvilinear grids are solved. These dimensionless equations are presented in the
following in terms of the contravariant velocity components in two layers:
dU[
dt
+ Â£i[s,11Ci + s^C?]
912
+ (Horizontal Diffusion)^ (f/f, U^)
+ [Nonlinear Terms(Â£/f, )]
(4.23)
dU?
dt
3ll Tji 321 JJD * P*
'U1 7=Ui + Tirt ~ Tbrt *
Vd~o~l '9,?
+ (Horizontal Diffusion)^ (Â£/f, U^)
Ro
IT
ctf
+
[Nonlinear Terms (t/f, i/^J
(4.24)
46
where
Defining
jji
Q[ + L2\gU
912 Tjt 4. 922 TJV i T* r*
v J + vr2+,<
+ (Horizontal Diffusion)^ {U2,U2)
+ y [Nonlinear Terms (/ U2)]
dU2
dt
= i2LÂ£Â£ + r* _r.*
2 V9~o 2 sv
+ (Horizontal Diffusion)^ (U2, U2)
+ [Nonlinear Terms (U2,U2)^
r.i =
r =
T.i =
T.'r, =
ctj
Â£*r + yTÂ¡y
+ Vy^sy
'T**
T]x
f.
0*
T\x
+ 4
Tiy
[pfUrZr\
.PfUrZr\
7~ix
+ 9y
Ty
[pfUrZr
[pfurzr\
Fax
+ tv
Fey
[pfUrZr
,pfurzr J
Fax
+ Vy
f Fcv 1
[pfUrZr_
[pfUrZr\
u( = u{ +
IP = Ui + Ui
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
we can obtain the following equations for the entire water column:
au*
dt
+ H[gnQ+9l%] =
<7l2
9 22
_l/i + ^c/, + T.T._Fi
+ (Diffusion^ [Ui,Ui]
47
+ (Diffusion)^ [U^U^]
+ ~~ [Nonlinear Terms (U^, Ui)]
Ro
+ [Nonlinear Terms (C/, /2)] (4.35)
Lo
dU*
dt
+ ffbaiC< + *MC,] =
+
9 n
U*~
921
\/9o \/9o
(Diffusion) [C/*,Ui]
U + t; tL f;
677
crt
+ (Diffusion)J?[7> U2]
+ ^[Nonlinear Terms (t/f, U*)\
L1
+ [Nonlinear Terms (C/, C/2)] (4.36)
L 2
The dimensionless continuity equation in the curvilinear grids becomes:
= 0
(4.37)
CHAPTER 5
HEAT FLUX MODEL
5.1 Introduction
All water bodies exchange heat with the atmosphere at airsea interface. To
estimate the net heat flux at the airsea interface, it is necessary to consider seven
processes: shortwave solar radiation, longwave atmosphere radiation, reflection of
solar radiation, reflection of atmospheric radiation, back radiation, evaporation, and
conduction.
One way to estimate the net heat flux is to combine the seven complicated pro
cesses to an equation in terms of an equilibrium temperature and a heat exchange
coefficient (Edinger and Geyer, 1967) as a boundary condition for the temperature
equation as follows:
= 9s = K(T Te) at z = t] (5.1)
where Te, the equilibrium temperature, is defined as the water surface temperature
at which there is no net heat exchange. This method will be called the Equilib
rium Temperature Method. Method to estimate the heat flux includes the Inverse
Method developed by Gaspar et al, 1990.
In the following, two methods for defining the boundary condition are described.
5.2 The Equilibrium Temperature Method
To determine an equilibrium temperature, following the procedure first devel
oped by Edinger and Geyer (1967), meteorological data and empirical formulas are
required. Data from the South Florida Water Management District (SFWMD) and
Climatological Data published by the National Oceanic Atmospheric Administra
48
49
tion (NOAA) were used.
5.2.1 ShortWave Solar Radiation
The amount of shortwave radiation reaching the earths atmosphere varies with
latitude on the earth, time of day, and season of the year. However, the amount
of shortwave radiation is reduced as it pass through the atmosphere. Cloud cover,
the suns altitude, and the atmospheric transmission coefficient affect the amount.
Empirical formulas are used (Huber and Perez, 1970) to compute the amount of
shortwave radiation.
However, the shortwave solar radiation is more easily measured than computed
(Edinger and Geyer, 1967). SFWMD measured the solar radiation at the Station
L006, which is located at the south of Lake Okeechobee (See figure 5.1). The unit is
mLyjmin.
5.2.2 LongWave Solar Radiation
The magnitude of the longwave radiation may be estimated by use of empirical
formula. Brunt formula (Brunt, 1932) was used.
Ha =
where
5.2.3
H0 = Longwave atmospheric radiation,
Ta = Airtemperature in F measured about six feet
above the water surface,
ea = Air vapor pressure in mmHg measured about six feet
above the water surface, and
C = A coefficient determined from the air temperature and
ratio of the measured solar radiation to the clear sky
solar radiation.
Reflected Solar and Atmospheric Radiation
(5.2)
The fractions of the solar and atmospheric radiant energy reflected from a water
surface are calculated by using the following reflectivity coefficients :
H,
jy _
" ~ H,
(5.3)
Rar =
Ha
Ha
(5.4)
Degrees Celsius mLangioy/min
50
Solar Radiation
Julian Day
Figure 5.1: Meteorological data at Station L006.
51
The solar reflectivity, R,r, is a function of the suns altitude and the type and amount
of cloud cover. Because there was no cloud data, 0.1 was assumed for Rsr. The
atmospheric reflectivity, Rar, 0.03, was assumed following the study of Lake Hefner.
5.2.4 Back Radiation
Water sends energy back to the atmosphere in the form of longwave radiation.
This can be calculated by the StephanBoltzman fourthpower radiation law (Edinger
and Geyer, 1967 and Harleman, et al., 1973):
Hbr = 7w
(5.5)
where
5.2.5
Hbr = Rate of back radiation in BTUFt~2Day1,
7u, = Emissivity of water, 0.97,
cr = StephanBoltzman constant (4.15 x 108BTUFt~2Day1 R~4), and
Ts = Water temperature, F.
Evaporation
Heat is lost from a body of water to the atmosphere through evaporation of
the water. Frequently, evaporation is related to meteorological variables (Brutsaert,
1982). The most general form is
He = (a + bW)(e3 ea)
(5.6)
a,b =
W =
where ea =
ej =
ea are related to
coefficients depending on the evaporation formula employed,
Wind speed in miles per hour,
air vapor pressure in mmHg, and e3 and
Saturation vapor pressure of water determined
from the water surface temperature, Ts.
the temperature of air and relative humidity ( Lowe, 1977). NOAA
data include dailyaveraged values of evaporation, wind speed, and air and water tem
perature. These data are measured at the Belle Glade Station near the southeastern
shore of Lake Okeechobee. Those data are averaged daily. By using the least square
method, a and b were determined to be 5.663 and 296.36, respectively.
5.2.6 Conduction
Water bodies can gain or lose heat through conduction due to temperature differ
ence between air and water. Heat conduction is related to evaporation by the Bowen
Ratio (Bowen, 1926).
52
11 e
D_C,(T,Ta) P
(e, ea) 760
where
P = barometric pressure in mmHg,
C7 a coefficient determined from experiments = 0.26.
Thus, conduction is related to the other parameters as follows:
Hc = 0.26(a + bW)(T3 Ta) [BtuFt^day1] (5.9)
5.2.7 Equilibrium Temperature
Of the seven processes, four processes are independent of surface water temper
ature: shortwave solar radiation, longwave atmospheric radiation, reflected solar
radiation, and reflected atmospheric radiation. The sum of these four fluxes is called
absorbed radiation (Hr). Thus the net heat flux can be written as follows:
AH Hr Hbr HeHc (5.10)
When the net flux AH is zero, Hr becomes
Hr = ~fwcr(Te + 460)4 + 0.26(a + bW)(Te Ta) + (a + bW)(ee ea) (5.11)
The net heat flux can be expressed as follows:
AH = K(T, Te) (5.12)
where K is the heat exchange coefficient.
5.2.8 Linear Assumption
Vapor pressure difference, e3 ea, is assumed to have a linear relationship with
temperature increment as
e. e, = i9(T, T,j
(5.7)
(5.8)
(5.13)
53
Also, the fourthpower radiation term can be approximated by a linear term with less
than 15% error (Edinger and Geyer, 1967). Therefore, AH become as follows:
AH = 15.7 + (0.26 + /?)(a + bW)(Ts Te) + 0.051(TS2 Te2) (5.14)
= K(Ta Te)
Neglecting the quadratic term,
K = 15.7f (0.26 + P){a + bW) (5.15)
Using the above relation, an equation for Te can be derived as follows:
0.0517J Hr 1801 K 15.7,e0 c(/3) 0.26T ,
T+ ~ = k + K~''[o3eTJ + mT1 (5'16)
where c(/3) is intercept for the temperature and vapor pressure approximation.
5.2.9 Procedure for an Estimation of K and Te
Step 1. Compute Hr.
Step 2. Assume Te.
Step 3. Find K for given wind and temperature.
Step 4. Compute the right hand side of Eq. 5.16.
Step 5. Compute the left hand side of Eq. 5.16.
Step 6. Compute the difference of Step 4 and Step 5.
Step 7. If error is not within error limit, go to step 2.
54
Step 8. If error is within error limit (0.5 C), K and Te are correct estimated
values.
An actual equilibrium temperature file was created using SFWMD data, which
were measured at 15minute intervals. Wind speeds at Station A,B,C,D,E were used
for the computation.
5.2.10 Modification of the Equilibrium Temperature Method
By using the equilibrium temperature method, modelpredicted temperature in
Lake Okeechobee was found to be unrealistic. Therefore, b of evaporation formula was
multiplied by a factor of 0.1. Further, K, the heat exchange coefficient, at Station C
was multiplied by 10 to give stronger stratification.
There are many uncertain empirical coefficients in the computation of an equilib
rium temperature. First, evaporation data are averaged daily, but model time step
is 5 minutes. That means the estimation of evaporation data at a short interval is
difficult. Second, wind speed is also averaged daily, and evaporation is correlated with
this average wind speed. In actual computation, wind speed at 15minute intervals
was used. Surface water temperature data are uncertain. Considering the sharp gra
dient of water temperature that usually exists near the water surface, the error can be
large. Third, all the meteorological data used are from L006 station. Considering the
spatial variation of meteorological condition over the lake, error can be large. Most
other thermal models simulate the longterm variation of temperature with a time
step of one day. Therefore, it seems that a shortterm variation of meteorological
data did not create serious problems.
5.3 The Inverse Method
When there are insufficient meteorological data, the errors in the estimation of
total heat flux at the airsea interface can be large. To better estimate the total flux,
the socalled inverse method (Gaspar et al., 1990) was used in this study.
Total flux (qt) can be divided into two parts: solar (q,0Â¡aT) and nonsolar {qnonsoiar)
55
While incoming solar radiation data are usually available, the nonsolar part is esti
mated by solving the vertical onedimensional temperature equation coupled with the
momentum equation.
5.3.1 Governing Equations
5.3.2 Boundary Conditions
At the free surface
?L (K
dt dzK v dz]
(5.17)
du d du
m/v = Tz(A"a;)
(5.18)
dv d dv.
Tt+fu = Tz(Am)
(5.19)
oz p
(5.20)
A
r\
oz p
(5.21)
A
OZ P
(5.22)
where Kv is eddy diffusivity, Av is eddy viscosity, qt is the total heat flux, and rx and
Ty are wind stresses.
At the bottom
dT_
dz
= 0
(5.23)
Tbx = PCd\Ju\ + V^Ui
(5.24)
Tby = PCd\Ju\ + VÂ¡Vi
(5.25)
56
where c is the drag coefficient and Ui and v\ are velocities at the lowest grid point
above bottom.
Total flux qt cannot be specified a priori because of unknown nonsolar flux,
Qncmaolar Therefore, a value of qnonsoiar is first guessed and then corrected until the
calculated and measured water temperatures are within an error limit. In this way,
the total flux can be determined by summing up the solar and nonsolar parts. This
total flux was later used as a boundary condition of temperature equation in the
threedimensional simulation.
5.3.3 FiniteDifference Equation
Treating the vertical diffusion term implicitly, Eq. 5.18 is written in the finite
difference form as follows:
At the interior points,
u?+1 u?
'n + 1 Am A,,!
~M =
Az
A z
At J'' Az
where At is time increasement and Az is vertical spacing.
Applying free surface boundary condition,
u
n+l
U;
At
~ Mm =
A (ti U .
A 2 tm Â¡m1
Az
Where im is the index of surface layer.
Applying bottom boundary condition,
u
At JVl Az
+i
Similar form for 5.19 is as follows:
At the interior points,
vi vi f,.n Az Az
At +JUÂ¡ A Z
Applying free surface boundary condition,
u"+1 _
m tm
At
Za. A (^Irn1 1 )
r n Â£ Az
+1 im Az
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
57
Applying bottom boundary condition,
r+1 r.,. 4. c^u\+
At + /Ul Az
5.17 can be written as follows:
At the interior points,
//Tin+l 1 \
rrn+1 T
ij T AV,_1 X7
At
Az
Applying free surface boundary condition,
rpn + l rpn 1s (TÂ¡X )
im im 1 Az
Ai Az
Applying bottom boundary condition,
jin+1 j^n
At Az
(Tn+i_Tr+l)
1Xvl A j.
5.3.4 Procedure for an Estimation of Total Heat Flux
Step 1. Solve for u for given wind stress.
(5.31)
(5.32)
(5.33)
(5.34)
Step 2. Solve for v for given wind stress.
Step 3. Guess nonsolar heat flux and solve for T.
Step 4. compare the computed tempearature to measured temperature.
Step 5. If error is not within error limit (0.5 C), go to step 3.
Step 6. If error is within error limit (0.5 C), go to step 1.
CHAPTER 6
FINITEDIFFERENCE EQUATIONS
This chapter describes the finitedifference equations for the governing equations
in Â£, r?, <7 coordinate system.
6.1 Grid System
Earlier models used a nonstaggered grid so that all the variables were calculated
at the same point (center point). This has a disadvantage. When centered difference
scheme is used, onesided difference scheme near the boundary should be used to
maintain the same order of accuracy. Therefore, it is inconvenient. CH3D uses a
staggered grid as shown Figure 6.1. Surface elevation and temperature are computed
at the center of a cell, while the velocities are computed at the face of a cell. A
vertical grid is shown in Figure 6.1, and all the variables are computed at the middle
of the layer.
6.2 External Mode
The external mode equations consist of the equations for surface displacement
( and the verticallyintegrated velocities U and V. Treating the wave propagation
terms in the finitedifference equations implicitly and factorizing the matrix equation,
the following equations for the Â£sweep and ^sweep are obtained:
Â£Sweep Equations
C + W(v^*) = C ^ TxMV9~oUn) (6.1)
av6n(V&yn)
riv^7u^(n +
1 + At
V9~oU*
58
59
o u, u
a v, v
A
B
w. T,
a
A
A
A
B
A
A
A
A
0
A
A
A
A
a
A
B
O A
a
o A
a
o A
o A
O A
B
u
A w
O T, p
A Z
o
A
o
A
o
A
o
A
o
irrrfTrr
__ Z'C z
C h
Figure 6.1: Horizontal and vertical grid system.
60
= Tly/gÂ¡'r12Sfl(C) + V9^n
~ (1Tx)v^7n^(n (1 r1)v^7%(Cw)
(6.2)
V9~oUn
At{lT2)CrtAt(lT3)
912
y/g;.
+ Dfn
rjSweep Equations
Cn+1 + r, aMValv**1) = r + r,Q(v/sW)
(6.3)
ri^7,(C+1) +
1 + Ai ( T2CTrÂ¡ + T3
912
VWJ
n+1
(6.4)
= ^^^(narov^y^niri)^22^)
+ V9~oVn^Â¡Vn
At(l T2)Crv A(l T3)
912
y/9o,
+ D"
i^71%"+) + ^"+1 = vS^ + rlv^712,(C")
(6.5)
where 7\, T2, and 73 are all constants between 0 and 1, superscripts indicate the
time level, and CT$ and CTrÂ¡ are the bottom friction terms. For example, the wave
propagation terms are treated explicitly if T\ = 0, but implicitly if 7\ = 1. D'^ and Â£>'n
are explicit terms in the U and V equations excluding the Coriolis, bottom friction,
and wave propagation terms. Additional parameters appearing in the equations are
<1
ai
9dA(
0At
av
9dAr)
9d
=
y/To
,11
HAtg11
7
AC
712
=
HAtg12
AC
22
HAtg22
7
At]
.,21
HAtg12
7
Ag
In the Lake Okeechobee application, the external mode is first solved over the
entire lake. For the open water region, the above Â£sweep and 77sweep equations
61
are used. For the vegetation zone, however, the Â£sweep and 7/sweep equations are
modified by the presence of vegetation and are derived from Equations (4.23) through
(4.26). However, it is only necessary to solve the finitedifference equations for the
integrated velocities in the entire water column, (U, V), and the velocities in one
layer, (Ui,Vi) or (U2, Vj). The velocities in the other layer can be readily obtained
by subtracting the onelayer velocities from the total velocities.
6.3 Internal Mode
The internal mode is defined by the equations for the deficit velocity and v,
(,v) = (u ~ jÂ¡i v jj) The equations for and v are obtained by subtracting the
verticallyaveraged equations from the threedimensional equations for u and v:
dtHu ^
g12 g22
y H+ + F3
+
EzJL
H2 da
y/Qo
 (u Tbt) f2
(6.7)
d_
dt
Hv
+
,11
21
H iHv + G3
y/So y/do
Evd( dHv\
H2 da da )
(Tsti 7"6tj) G2
(6.8)
where and v are the deficit velocities in the (Â£,77) directions, F3 and G3 indicate all
the explicit terms in the u and v equations, respectively, while F2 and G2 indicate all
the explicit terms in the U and V equations, respectively.
Applying a twotimelevel scheme to the above equations leads to the following
finitedifference equations:
A2
(f)n+1 = {H)n + At^z Hn+ln+1
\/9o
22
+ At^Hnn + At(F3 F2)n
J A t
y/Ho
Ev d
(Hn)2 da
(T*i n()n+1
Av(Hn+1un+1)
(6.9)
62
,ii
(Hv)n+1 = (Hv)n At? Hnvn+1
\ZlTo
a21
 At^Hn+1vn+1 + At(G3 G2)n
y/fo
4* A t
ev d
(.HnY da
('T*v ~ n)B+1
Av^(Hn+1vn+1)
(6.10)
For the open water zone, the above internal mode equations are solved after the
external mode solutions are obtained. For the vegetation zone, no internal mode
equations are solved. This is consistent with the assumption that, in the vegetation
zone, the velocities are fairly uniform within the vegetation layer and the vegetation
free layer.
6.4 Temperature Scheme
This section describes the finite difference equation which is used for solving the
temperature equation. Equation 3.53 is written in finite difference form using the
forward scheme in time and the centered difference in the vertical diffusion term.
jyn+lyn+l jjnrpn
t)jyk ,j tk
A tR0
y/9o
HuT,/Â£)
(6.11)
+
+ A{HvT^ +
Ev(At) 1
#B+1rc. Aak
Dv
D
+
A cr_
HEjj 2At
Aa+
(rfc'v7T+1)
,11
d2T
+ 2g
12
d2T
+ 9
22
d2T'
(6.12)
 CH
\ dÂ£2 dÂ£drj drÂ¡2 /
Dividing the above equation by Hn+1 and collecting all the unknown terms in the
lefthand side and the known terms in the righthand side, and writing advection
terms and diffusion terms separately,
EvAt
Ey rpn+l A+_mn+l
vA
(H*+')2Prm Aak
(6.13)
63
+
1 +
Hn
Ev At ( Dv+ Dv_
(Hn+'yPrvAak VAcr+ + A
'Tn+l
1 >jA
TJ1,
At R0
Hn+1 ij* ~ Hn+l Jg~o
Hn Eh At f n d2T
Hn+1 prH v dt2
6.4.1 Advection Terms
HvTy" + ^A{HwTr
+
+ 2gn
d2T
d^drj
+9
d2T\n
dr,2)
(6.14)
Many different schemes can be used for the advection terms. When there is a
sharp discontinuity, it is difficult to model the convection without numerical diffu
sion. Leonard (1979) introduces the QUICKEST( Quadratic Upstream Interpolation
for Convective Kinematics with Estimated Stream Terms), which gives good results
without excessive numerical diffusion.
This QUICKEST scheme treats the advection terms in the ^direction as follows:
( /7THnT\ (V^HuT)+ IV9~qHuT)_
(6.15)
where the first term in the right hand side is the flux at the right face of the cell
and the second term is at the left face. These two terms are differenced differently
depending on the direction of current as follows:
At the right face of a cell :
When u is positive :
(yg~0HuT)+ (y/gÂ¡H)i+i,jUi+itjik[(Tiijk + 7i+i,,fc)
1(1 _iy){T.+ut _2T..t +
1 Ui+i,j,kAt
AÂ£
(6.16)
When u is negative
(y/9oHuT)+ (\/^i7){+i (^',j,fc + TJ+ij^)
64
1 (1 (^Aty)(TÂ¡+2jt 2TÂ¡+Â¡jt + TÂ¡,k)
1 A
2 A
(Ti+i'k T{[,j,A:)]
At the left face of a cell
When u is positive :
(y/f'HuT). (y/g^H) j u, j, [ (T, i, fc +
_ 1 (* (i^)2)(3n. ., 27',.,,,* + Ti.2,j,k)
1 ut,j,fcAt
2 AÂ£
(Ti,j,k Tiij'k)]
When u is negative
(6.17)
(6.18)
(y/glHuT)_ + T.j^)
 (i (!ii^)2)(rt.+liii, 22U* + Tiij,*:)
1 u,j,fcAi
(^U* T<U,fc)]
2 AÂ£
The QUICKEST method treats the rj direction advection term as follows:
a (k/slHvT)+ (Jg'HvT).
a~^HvT) Aj
At the top face of a cell :
When v is positive :
(6.19)
(6.20)
(\/
 j(l (!~1f(TÂ¡jW 2+ Ty.u)
1 Vij+tjA t
2 AÂ£
(7v,j+1,A: ^ij'.fc)]
(6.21)
When v is negative :
65
{y/9lHvT)+ (\/{hH)itj+lViij+itk[(Tijtk + Tij+ij'it)
1
,Vitj+hkAt
6^ AÂ£
) )(Ti,j+2,j,k + Tij'k)
1 Vj,j+i,kAt
2 AÂ£
At the bottom face of a cell :
When v is positive :
(Tij+1.*
(6.22)
(y/g0HvT). (y/%H)ijVitjtk[(Titj_ltk +
~ i(l C^nTij* ~ TTijij, + Tij2t)
1 VijtkA t
2 AÂ£
(Tijt Tiji,*)]
(6.23)
When v is negative
(VglHvT)_ {y/9lH)ijV{jt[(Tijxjk + Tij'k)
1 ri (Vi,j,kAt
6l 1 AÂ£
)2)(Ti,j+i,k 2 Tijtk + Tijitk)
1 VitjikA t
(Tij'k Tij_itk)}
2 AÂ£
The QUICKEST method treats the trdirection advection term as follows:
(6.24)
d_
da
(.HuT) =
(HuT)+ (HloT)_
A a
(6.25)
At the top of a cell :
When uj is positive :
(HuT)+ + Tijtk)
66
i (l t..mi 2T..t+Tijti)
1 ujjj'kA t
2 Act
When u) is negative :
.1
(HuT)+ Hi, ju>i (7, j, k+i + Tijtk)
g(! ~ 2Ty.l+1 + Tijj.)
1 ujiijikAt
2 Act
~(Ti,k+1
At the bottom of a cell
When u) is positive :
(HuiT)_ i [2
_ 1(1 TiJj. 2TÂ¡lki + Tijtz)
1 Uj'j'kiAt
2 Act
(Ti,j,k I'iJJcl)]
When u is negative :
rl
(HuT)_ Hi'jUJij'kii^iTij'k + T^_i)
ln ,^i,j,kiAt
~ 6<>"<
Act
)2)(ii,J>+l 27U* + Ti,i,kl)
1
2 Act ^
6.4.2 Horizontal Diffusion Term
(6.26)
(6.27)
(6.28)
(6.29)
The centered difference scheme was used for the horizontal diffusion. For the
mixed derivative term, temperature and depth at the corners of a cell are obtained
by averaging the values using four neighboring points at the center points of a cell.
xld\HT) (.HT)i+hjik 2(HT)i
9 9i,j (AO2
d?
(6.30)
67
i2 d\HT)
9 dCdrj
9ij[{HT)i+i/2,j+i/2,k (HT)i+i/2i/2,k
(HT)i_il2J+i/2,k + (HT)i_1/2Ji/2,k]/AÂ£AT1
,22
d2{HT)
drj2
a22
(HT)iJ+lJt 2(HT)iJtk + mju
(Arj)2
(6.31)
(6.32)
CHAPTER 7
MODEL ANALYTICAL TEST
The purpose of model analytical test is to examine a models capability to re
produce wellknown physical phenomena for which the model is designed for, by
comparing model results with analytical solution.
7.1 Seiche Test
The CH3D model has been tested for winddriven circulation in an idealized
enclosed lake which is 11 km long and 11 km wide with a uniform depth of 5 m. A
uniform rectangular grid of 1 km grid spacing was used. To perform the seiche test,
the initial surface elevation was given as Â£ = (0cos(2irx/) where (0 is an amplitude
and Â£ is a wave length. In the test, (0 was set to 5 cm and t was set to 10 km.
Since the lake is of homogeneous density and without bottom friction and diffu
sion, seiche period can be calculated as T 7S where l is the basin length and h
ygh
is the mean depth. For the test basin, the seiche period is 0.87 hours. The simulated
surface elevation in the test basin over a 12hour period is shown in Figure 7.1. The
result shows that the surface elevation was not damped and the seiche period agrees
with analytical seiche period.
7.2 Steady State Test
When a uniform wind blows in the same direction over a rectangular lake with
same magnitude over a long period, the lake circulation eventually reaches steady
state. Neglecting advection, horizontal diffusion, and bottom friction, the setup equa
tion can be obtained as follows:
dr] tw
99 dx ~ h
(7.1)
68
69
Surfaea Elevation at Mart* End
6
4
 2
a
I
M
2
4
6
6 *2
Tas IHourel
Srfc* Elatalltn at Mast tna
t f
Surface Elevation at Canto*
Surface Elevation at Eaet End
Surface Elevation at South End
6 ,
6 L >
6 12
Tlee (Houra I
Figure 7.1: Model results of a seiche test
70
where p is density of water, y is surface elevation, tw is wind stress, and h is water
depth. Using the same rectangular grid in the seiche test, a uniform wind stress of
1 dyne/cm2 from east to west was imposed. After 48 hours, a steady state is reached.
As shown in Figure 7.2, the surface elevation has a setup in the western part and
setdown in the eastern part. The setup across the lake is 1.12cm, which is exactly
the same as given by the analytical solution Eqn. 7.1.
7.3 Effect of Vegetation
In order to investigate the ability of vegetation model to represent the effect of
vegetation, CH3D was applied to a rectangular lake with a constant depth of 1 m and
horizontal dimensions of 4 km by 9 km. At first, vegetation was not considered, and
a wind stress of 5 dyne/cm2 was imposed. Then, vegetation canopy with width of 1
cm and density of 500 stalks / m2 was added to the western half of the lake. After
that, vegetation density was increased to 5000 stalks / m2. Vegetation height was
assumed to be the same as water depth.
With a time step of 5 minutes, the model was run for 24 hours. Time history
of surface elevation in the northern end of the vegetation area was plotted in Figure
7.3. As expected, surface elevation rises slowly for the second case and reaches steady
state after 5 hours. With highdensity vegetation, surface elevation rises at a slower
rate compared to the second case. When wind blows uniformly over long time, vege
tation effect disappears and reaches steady state. Additional resistance term due to
vegetation becomes smaller because the currents also become smaller at the steady
state and wind stress, and pressure gradient and bottom friction are balanced.
7.4 Thermal Model Test
The purpose of this test is to demonstrate how the velocity can be changed with
the consideration of thermal stratification. Surface heat flux was idealized using the
sine function as follows:
pCrK~ = K(T r.)
(7.2)
71
Figure 7.2: Surface elevation contour when the lake is steady state with uniform wind
stress of 1 dyne/cm2.
Surface lcvation (cm)
72
Time history of surface elevation
Figure 7.3: Effect of vegetation on surface elevation evolution in a winddriven rectan
gular lake. Solid line is without vegetation, broken line is with low vegetation density,
and dotted line is with high vegetation density.
73
Te = 26 + 10 sin() (7.3)
where p is density of water, Cp is specific heat of water, K is heat exchange coefficient,
Te is an equilibrium temperature in 0 C, and P is period of 24 hours. Also, wind stress
was idealized as shown in Figure 7.4. Time history of currents at all five layers are
shown in Figures 7.4 and 7.5. It is apparent that when thermal stratification is
considered, currents at the surface layer are much stronger during increasing wind
condition because initial momentum is confined to a thinner surface layer.
ulcm/secl dyne/cm**2
74
Wind Stress at center
Velocity at center without temperature
Hours
Figure 7.4: Time history of wind stress and currents at the center of lake at all five
levels. Thermal stratification is not considered.
/o
Wind Stress at center
1 .5 : i ] i i t i i t Â¡ i  1 : r
Velocity at center with temperature
Hours
Figure 7.5: Time history of wind stress and currents at the center of lake at all five
levels. Thermal stratification is considered.
CHAPTER 8
MODEL APPLICATION TO LAKE OKEECHOBEE
8.1Introduction
Before the description of the application of CH3D to simulate the winddriven
circulation in Lake Okeechobee, it is worth investigating the characteristics of the
lake.
8.1.1 Geometry
Lake Okeechobee, located between latitudes 2712'N and 2640iV and longitudes
8037 Hr and 8108 VK, is the largest freshwater lake in America, exclusive of the
Great Lakes. With an average depth of approximately 3m, and the deepest part less
than 5m deep, the basin is shaped like a very shallow saucer. The western part of
the lake contains a great deal of emergent and submerged vegetation. According to
satellite photos, marsh constitutes 24% of the lake surface area.
8.1.2 Temperature
Due to the location of the lake in subtropical latitude, the annual fluctuations
of water temperatures are relatively small. The mean lake temperature based on
SFWMD monitoring in the 1970s and 1980s ranges from 15C to 34C' (Dickinson et
al., 1991).
8.2Some Recent Hydrodynamic Data
During the fall of 1988 and the spring of 1989, field data were collected by the
Coastal and Oceanographic Engineering Department, University of Florida (Sheng et
al., 1991a). Details of the field experiments and field data are described by Sheng
et al. (1991a). For completeness, some 1989 field data are described briefly in this
76
77
section.
Six platforms were set up in Lake Okeechobee. Locations of the six platforms are
shown in Figure 8.1. Station A was located in the northern portion of the lake, east
of north lake shoal, during the 1988 deployment. This Station A was moved south of
the rocky reef area during the 1989 deployment. The other five platforms remained at
the same locations during the 1988 and the 1989 deployments. Station B was located
near the Indian Prairie Canal. Station C was located in the center of the lake. Station
D was located 1.5 km west of Port Mayaka Lock. Station E was located about 1 mile
from the boundary between the vegetation zone and the open water, i.e., the littoral
zone. Station F was located within 30 m from the littoral zone.
Station A was selected to quantify the flow system in the northern zone during
the 1988 field survey. Because there was no measurement in the southern zone, it was
moved south to quantify the flow during the 1989 field survey. Station B was selected
to quantify the flux in the northern littoral zone. Stations C and D were selected to
calibrate the model and quantify the flux in the mud zone. Stations E and F were
selected to help the computation of phosphorus flux between the vegetation and the
open water.
Measured data at these six platforms include wind, current, water temperature,
wave, and turbidity. In this study, wind data were used to compute the wind stress
field, which is an essential boundary condition for the simulation of the winddriven
circulation. Current data were used to calibrate and validate the 3D hydrodynamics
model. The installation dates and locations of the platforms during the two deploy
ment periods are shown in Table 8.1.
8.2.1 Wind Data
Wind speed and direction data averaged over 15minute intervals were collected
from five stations in Lake Okeechobee during the spring of 1989. Because Station F
was close to Station E, wind data were not collected at Station F. The data collection
78
Figure 8.1: Map of Lake Okeechobee.
.00 'LZ ,0MZ
79
Table 8.1: Installation dates and locations of platforms during 1988 and 1989.
TIME OF
YEAR
LOCATION
DATE
LATITUDE
LONGITUDE
DEPTH
(cm)
FALL
Site A
092088
27 06.31
80 46.21
396.0
Site B
091788
27 02.78
80 54.31
274.0
Site C
092188
26 54.10
80 47.36
518.0
Site D
092188
26 58.47
80 40.34
457.0
Site E
091888
26 52.81
80 55.96
274.0
Site F
091988
26 51.90
80 57.09
183.0
SPRING
Site A
051689
26 45.67
80 47.83
183.0
Site B
051889
27 02.78
80 54.31
152.0
Site C
052089
26 54.10
80 47.36
366.0
Site D
052089
26 58.47
80 40.34
335.0
Site E
051989
26 52.81
80 55.96
152.0
Site F
051889
26 52.03
80 56.91
91.0
started on Julian Day 136.708. However, the direction of the anemometer was not
properly oriented until Julian Day 141.5. The location and height of the anemometer
are shown in Table 8.2.
As described in Sheng et al. (1991a), the measured wind over Lake Okeechobee
often exhibited significant diurnal variations associated with the lake breeze. During
relatively calm periods, significant spatial variation is often found in the wind field.
Water motion in the lake is significantly influenced by the wind. Figure 8.2
shows the wind rose diagram at Station C between Julian days 147 and 161. The
number inside the triangle indicats the percentage of wind data in that direction.
For example, wind from east to west is 27%. Wind speed between 46 m/sec is
about 45%. The governing wind direction is from east to west due to the location
of Lake Okeechobee. The surface area of Lake Okeechobee is big enough to create
its own lake breeze. During the daytime, wind blows from lake to land because the
air over the land is warmer than that over the lake. Because the Florida peninsula
is located between the Atlantic Ocean and the Gulf of Mexico, sea breeze affects
the wind direction. As Pielke (1974) indicated, the typical summer wind direction
80
Table 8.2: Instrument mounting, spring deployment.
SITE
ARM
CURRENT
ELEV
(cm)
AZIM
(deg)
TEMP
ELEV
(cm)
OBS
ELEV
(cm)
A
Pressure Sensor 556
)5, Elev.
96 cm
Wind Sensor 5200, Elev. above MWL 670 cm minus water depth
1
80673
71
342
07
86
0076
86
B
Pressure Sensor 55696, Elev. 86 cm
Wind Sensor 5202, Elev. above MWL 518 cm minus water depth
1
80674
25
204
04
43
0078
43
2
80675
114
181
02
132
0075
132
Pressure Sensor 48228, Elev. 297 cm
Wind Sensor 5203, Elev. above MWL 884 cm minus water depth
1
80679
61
330
01
79
0079
79
2
80681
123
333
09
140
0077
140
3
80680
284
342
06
302
0084
302
D
Pressure Sensor 55694, Elev. 254 cm
Wind Sensor 5200, Elev. above MWL 883 cm minus water depth
1
80672
79
270
08
97
0083
97
2
80677
241
255
03
259
0082
259
Pressure Sensor 55699, Elev.
04 cm
Wind Sensor 5199, Elev. above MWL 518 cm minus water depth
1
80671
36
59
11
53
0081
53
2
80678
116
72
05
135
0080
135
Pressure Sensor 55697, out of water
No Wind Sensor
80676
25
305
10
43
0085
81
is from southeast or southwest in south Florida. Those wind systems determine the
dominant wind direction in Lake Okeechobee. For the synoptic study of spring 1989,
the mean wind speed is about 5.1 m/sec, but it can exceed 11 m/sec. The wind field
is characterized by temporal and spatial nonuniformities. However, during strong
wind periods, the wind tends to be more uniform.
8.2.2 Current Data
Current data were collected at 15minute intervals at the six locations. The data
collection started on Julian Day 136.708. The number of instrument arms at each
station depended upon the water depth and how many vertical levels of data were
desired. At station C, which is located in the center of lake, three current meters were
installed to measure the vertical variation of currents. Two sensors were installed at
Stations B, D and E. One sensor was installed at Stations A and F. The location and
height of each of the current meters are shown in Table 8.2.
As discussed in Sheng et al. (1991a), current data showed significant diurnal
variations in direct response to the wind. During a period of significant change in
wind direction, which usually follows a peak wind period, seiche oscillation generally
leads to significant current speed over several seiche periods (multiples of 5 hours).
Magnitudes of currents are very small at all stations. At Station C, mean magni
tudes of u and v at arm 3 between Julian day 147 and 156 are 2.09 and 1.52 cm/sec,
respectively. Maximum magnitudes of u and v at arm 3 are 11.7 and 6.8 cm/sec,
respectively. Considering the accuracy of current meter, 23 cm Â¡sec, currents are
very small.
8.2.3 Temperature Data
The lake temperature data showed that a significant vertical temperature gradi
ent can be developed during part of a day when wind is low and atmospheric heating
is high. However, over the relatively shallow littoral zone and transition zone, tem
perature appeared to be well mixed vertically much of the time.
82
LEGEND
1
m/%
m/%
m/%
m/%
m/%
V
WIND ROSE AT STATION C
Figure 8.2: Wind rose at Station C
83
Lake Okeechobee does not have a strong thermal stratification. But during the
daytime, lake becomes stratified. Temperature field data show the water temperature
difference between the near top and near bottom at Station C can reach 4 C. However
this stratification disappears in a short time due to the strong wind. This stratification
can affect the eddy viscosity significantly.
8.2.4 Vegetation Data
The distribution of aquatic vegetation in Lake Okeechobee was determined by the
use of recent satellite imageries and ground truth data obtained during field surveys.
Satellite imageries from the SPOT satellite were received and processed to create a
vegetation map of Lake Okeechobee (J.R.Richardson, personal communication, 1991).
This map was overlayed onto a curvilinear grid created by us. The number of pixels
with the same color was counted to classify the vegetation type on each grid cell. The
vegetation data include the vegetation class, the height of vegetation, the number of
stalks per unit area, and the diameter of each class. A total of more than 25 kinds of
vegetation were identified. The range of vegetation height is between 0.5 m and 4 m.
The density ranges from 10 stalks per m2 to 2000 stalks per m2 of bottom area, while
the diameter of the stalks ranges from 0.25 cm to 15 cm. The most popular type
of vegetation was cattail. These vegetation data were provided by John Richardson
from the Department of Fisheries and Aquaculture, University of Florida.
8.3 Model Setup
The following describes the procedure for simulation of winddriven circulation in
Lake Okeechobee during spring 1989.
8.3.1 Grid Generation
The first step is to select the grid and grid size. Rectangular Cartesian grids were
widely used in hydrodynamic models. These grids are easy to generate but have a
disadvantage. Because these grids have to represent the boundaries in a stairstepped
fashion, they cannot represent the complex geometry accurately unless a large number
84
of grid points are used. Complex boundaries can be represented more accurately by
using the curvilinear grid for a general 2D region with boundaries of arbitrary shape
and with boundary intrusions and internal obstacles, such as islands.
To generate a curvilinear grid, first design a computational domain as shown
Figure 8.3. Next step is to digitize the boundary points of Lake Okeechobee using a
detailed map. After that, WESCORA code is used for the grid generation. As shown
in Figure 8.4, a horizontal grid consisting of 23 by 28 points was generated for this
study. During the spring of 1989, the lake stage was dropped to 12.5 ft. Therefore,
the most western part became dry land, and the cells in this area was excluded in the
model simulations.
8.3.2 Generation of Bathymetry Array
The accurate representation of bathymetry is an important factor for the simula
tion. The map used is a nautical chart which is published by National Oceanographic
and Atmosphere Administration (Chart Number 11428, February 8, 1986). From the
bathymetric data on the map, digitized depths at a total of 983 points were obtained.
Based on the digitized depths, the depths at the grid points are evaluated using a
threepoint interpolation scheme as follows:
3 WkDk
where Dk is the digitized depths at the three nearest points, and
(8.1)
Wk =
(xi,j ~ xk)2 + (y. j yk)2
(8.2)
Wo = ^2 Wk
k=l
(8.3)
Depth contour is in Figure 8.5. maximum depth is about 4 m and minimum
depth is 20 cm.
85
Figure 8.3: Computation domain of Lake Okeechobee.
r
86
Figure 8.4: Curvilinear grid of Lake Okeechobee.
87
CONTOUR FROM 20.>00 TO 420.00 CONTOUR INTERVAL OF 20.000
Figure 8.5: Depth contour of Lake Okeechobee when the lake stage is 15.5 ft. Unit
m cm
88
8.4 Model Parameters
Because of the great flexibility of CH3D, many parameters must be specified for
its proper use. A convenient way to describe the input parameters is to classify the
parameters as reference values, parameters associated with turbulence, parameters as
sociated with bottom friction, parameters associated with vegetation, and parameters
associated with wind stress.
8.4.1 Reference Values
Reference values are values which are characteristic scales for the processes of
interest. The reference values shown in Table 8.3 were used to simulate the Lake
Okeechobee Spring 1989 circulation. The reference value for the horizontal scale (Xr)
for Lake Okeechobee is the average length of grid, while the reference value for the
vertical scale (Zr) is the characteristic depth of the lake.
The reference velocity (Ur) shown in Table 8.3 is a typical speed for windforced
currents in Lake Okeechobee. In the formulation of the dimensionless equations, time
is scaled by the Coriolis parameter, / = 2f2sin<Â£, where fi is the rotational speed
of the earth and <}> is the latitude. The value for / that is shown in Table 8.3 was
obtained by using fl = 7.29 x 105 s1 and the average latitude of Lake Okeechobee
= 27N. The two physical constants in Table 8.3, which are used as reference
values, are the gravitational acceleration (g) and the density of fresh water (p0). The
reference value shown in Table 8.3 for the horizontal eddy viscosity (Aht) is set to a
scale length dependent value based upon the 4/3 Law suggested by Stommel (1949),
Ah = eÂ£4/3, where e is an empirical constant and is a scale length. Using the value
presented by Orlob (1959) for the empirical constant, e = 4.53 x 10~4 m2/3 s"1, it can
be shown that the value for Aht shown in Table 8.3 is characteristic of processes with
horizontal scale lengths of i = 1.81km, which is just slightly smaller that the grid
interval Ax = 2 km. Thus, the value used for Aur is reasonable for representing the
effect of subgrid scale motion on horizontal mixing. The reference value for vertical
89
Table 8.3: Reference values used in the Lake Okeechobee spring 1989 circulation
simulation.
Parameter
Units
Value
Xr
cm
3.11 x 105
ZT
cm
150
UT
cm s1
10
f
cm s1
6.62 x 105
9
cm s2
981
Po
g cm3
1
AHr
cm2 s1
105
AVr
cm2 s1
1
eddy viscosity (AVr) which is shown in Table 8.3 is characteristic of calm conditions
in Lake Okeechobee. During periods of strong wind mixing, vertical eddy viscosities
can exceed 100 cm2 s1.
8.4.2 Turbulence Model and Parameters
The vertical turbulence was parameterized by using a simplified secondorder
turbulence closure model developed by Sheng et al. (1986b). As described by Sheng
et al. (1989b), the simplified secondorder closure model is obtained by neglecting the
evolution and diffusion terms from the full Reynolds stress equations in the complete
secondorder closure turbulence model, and then replacing the mass flux equations
with temperature and salinity flux equations which are applicable to water bodies.
The tensor equations for the Reynolds stress (u(u'), mass flux (u[p'), and variance
(p'p1) are then, respectively, as shown in Eqs. (2.18), (2.19) and (2.20). Although
details of the turbulence model were given elsewhere ( Sheng and Chiu, 1986; Sheng
et al., 1989b), the equations are given in the following for completeness.
By considering only vertical gradients of the mean variables, the following alge
90
braic equation may be obtained from Equations (2.18), (2.19) and (2.20).
0 = 3A2b2sQ4 + A[(bs + 3b + 7b2s)Ri Abs(l 26)]Q2
+b(s + 3 + Abs)Ri2 + (bs A)(l 2b) Ri
(8.4)
where
Q =
V(fe)2+(&);
(8.5)
and
Ri =
Po 9z
(8.6)
(Si) +(fi)
The total rms turbulent velocity q can be obtained from the above equations after
the mean flow variables are determined at each time step. The vertical eddy viscosity
is then computed from
where Sm is defined as
Av SniAq
c A + uj ww
~A
A to q 2
(8.7)
(8.8)
and where u = Ri/(AQ2), u> = 1 u)/bs, and
126
ww =
(8.9)
3(1 2lJ)
The length scale A is assumed to be a linear function of vertical distance immediately
above the bottom or below the free surface. In addition, the length scale A must
satisfy the following relationships (Sheng and Chiu, 1986):
dA
dz
< 0.65
A < fz,H
(8.10)
(8.11)
91
A
<
/ Hv
(8.12)
A
<
Qcut &q2
(8.13)
A
<
<7
N
(8.14)
where fza is usually within the range of 0.1 to 0.25, H is the total depth, Hp is the
depth of the pycnocline, Qcut is a coefficient between 0.1 and 0.25, 6q2 is the spread
of turbulence determined from the q2 profile, and N is the BruntVaisala frequency
defined as
(8.15)
Some of the parameterization of turbulence is controlled by userinput in the
CH3D code. The model constants A, b, and s, however, are invariant. Besides the
vertical turbulence reference value AVr described above in Table 8.3, the vertical turbu
lence parameters which are useradjustable are Amtn (minimum turbulent macroscale),
Qcut (fraction of turbulence spread), fZ3 (fraction of depth), AVmin (minimum eddy
viscosity), and AVmaz (maximum eddy viscosity). The values of the vertical turbu
lence parameters which were used to simulate the circulation in Lake Okeechobee
during spring 1989 circulation (see Table 8.4) are those which gave the best results
from an independent onedimensional vertical model of flow and sediment for Lake
Okeechobee at Station C during the same time period.
8.4.3 Bottom Friction Parameters
Momentum imparted at the water surface by wind stress is dissipated from the
water column by bottom friction. Bottom friction in CH3D is in the form of a
quadratic bottom stress law:
Ti* = pcduiyjgnu\ + 2gi2uivi + g22vj
nr, = pCd.V\\jgnu\ + 2guiiiVi + g22vl (8.16)
where are contravariant velocities at the first grid point above the bottom, and
cj is an empirical drag coefficient. Depending on the value of the flag for bottom
92
Table 8.4: Vertical turbulence parameters used in the Lake Okeechobee spring 1989
circulation simulation.
Parameter
Value
Amin
0.5
Qcut
0.2
fzs
0.2
A
0.5
100
friction, the drag coefficient is either set to a constant specified by the input value
CTB, or a variable according to the law of the wall. For the simulation of the Lake
Okeechobee spring 1989 circulation, a variable drag coefficient was used according to
the law of the wall as
where k = 0.4 is von Karmans constant, z\ is the height of the first grid point
above the bottom, and z0 is the roughness height. In CH3D, the roughness height
is specified as the input parameter BZ1. For the Lake Okeechobee simulation, the
roughness height was taken to be z0 = 0.1 cm.
8.4.4 Vegetation Parameters
There are more than 25 kinds of vegetation in the littoral zone. Each type of
vegetation has a different diameter and height. For simplicity, in this study it was as
sumed that all types of vegetation have cylindrical stalks. Table 5 shows the various
types of vegetation found in Lake Okeechobee (J.R. Richardson, personal commu
nication, 1991). To represent the vegetation distribution accurately, the concept of
equivalent vegetation density is introduced. The equivalent vegetation density is
defined as the equivalent number of stalks with 1 cm diameter per unit horizontal
93
area which will give the same total projected area. If, however, the effect of skin
friction drag due to vegetation is more important, then the equivalent vegetation
density has to be defined in terms of the total wetted area rather than the total
projected area. The procedure for incorporating the vegetation information into
our model is described as follows.
Let
di = the diameter of the ith vegetation type [cm],
Pi = the number of stalks of the ith vegetation type per unit
area,
hi = the height of the ith vegetation type in the grid cell [m],
D = the average water depth in the grid cell [m],
A = the total area of the grid cell [m2], and
Pi = the percent area occupied by the ith vegetation type
within the total area of the grid cell (%).
The sum of the heights of all the plants in the grid cell, H, is
H = ^ pi hi Pi/100 A (8.18)
i
and the total number of stalks of the plants in the grid cell, T, is
T = ^2pi* Pi/100 A (8.19)
whereupon the average height of vegetation in the grid cell, Hcanopy, is
He* nopy = H/T (8.20)
The distribution of Lfcanopy in Lake Okeechobee determined by this method is
shown in Figure 8.6. The projected area occupied by the ith vegetation type in the
grid cell is
o;, = pi di hi Pi/100 A (8.21)
and the total projected area occupied by plants in the cell is
S ~ 2 Qi
(8.22)
94
Table 8.5: Vegetations in Lake Okeechobee (From Richardson, 1991).
Value
Points
Acres
%
Height
[m]
Density
[#/m2]
Diameter
[cm]
Description
0
41631
38228.64
0
1
36
33.058
0.01
2.5
10
2
Buttonbush
2
2737
2513.315
0.57
4
30
8
Melaleuca
3
5699
5233.241
1.18
2.5
10
5
Willow
4
43
39.486
0.01
1
2000
0.25
Spartina
5
43433
39883.37
9.03
2.5
50
7
Cattail
6
3975
3650.138
0.83
1.5
40
15
Sawgrass
7
1195
1097.337
0.25
5
10
2
Mixed Upland
8
4712
4326.905
0.98
1
500
0.25
Rhynchospora
9
36
33.058
0.01
0.5
10
1
Sagitaria/pontedaria
10
11458
10521.57
2.38
1
2000
0.25
Mixed grasses
11
1316
1208.448 .
0.27
0
7
1
Nymphae
12
6836
6277.318
1.42
1
60
0.25
Eleocharis
13
36
33.058
0.01
5
10
2
Guava
14
44
40.404
0.01
0
100
5
Hyacinth
15
4029
3699.724
0.84
2.5
50
2
Scirpus
16
370750
340449.9
77.08
0
0
0
Open Water
17
196
179.982
0.04
0
0
0
Periphyton
18
5344
4907.254
1.11
1
100
0.25
Eleocharis/periphyton mix
19
2365
2171.717
0.49
1
100
0.3
Nymphae/eleocharis mix
20
2759
2533.517
0.57
0.5
15
3.5
Lotus
21
2238
2055.096
0.47
0
*
*
Submerged
22
2724
2501.377
0.57
2
12
5
Cattail/nymphae mix
23
714
655.647
0.15
2.5
50
5
Phragmites
24
278
255.28
0.06
1
50
0.3
Maidencane
25
3952
3629.017
0.82
**
**
**
Excluded
26
873
801.653
0.18
1
2000
0.25
Spartina/panicum mix
27
30
27.548
0.01
0
0
0
Airboat trails
28
3181
2921.028
0.66
1
100
1
Successional disturbed
1 provides no wind resistance but considerable cross section resistance to flow
2 ** area diked preventing surface water flow
95
Vegetation Height
Unit i m
96
whereupon the equivalent vegetation density is
N = Sf(A* Hcanopy d
canopy ) (823)
where dcanopy is assumed to be 1 cm. However, if we assume the height of each
vegetation type is the same as average water depth in the cell, h, is replaced by D in
Equation (8.21) and Hcanopy is replaced by D in Equation (8.23). The distribution
of N in Lake Okeechobee is shown in Figure 8.7.
The value of the profile drag coefficient for a cylinder, c, was set to be 1.77 in
this study. Whitaker et al. (1975) estimated the interfacial stress coefficient, c,
by calibrating the onedimensional surge model with the observed quasi steadystate
surface profile in Lake Okeechobee.
8.4.5 Wind Stress
In Lake Okeechobee, as in most shallow lakes, momentum is imparted to the water
primarily by the action of surface wind stress. Wind over Lake Okeechobee varies on
a time scale less than one hour, and a spatial scale of about 10 km. Therefore, a
time and spacevarying wind stress field is required. Wind speed and direction data
averaged over 15minute intervals were collected from five stations in Lake Okeechobee
during the spring of 1989. As for the bottom stress, the wind stress at the surface
was computed from the quadratic form:
riX = PaCdaWxyJW2 + Wy2 (8.24)
TSV = PaCdaWyy/W* + WÂ¡ (8.25)
where rsx and t,v are the surface wind stresses in the x and y directions, respectively;
pa = 127 x 103gcm~3 is the density of air; Wx and Wy are the wind speeds in
the x and y directions, respectively; and Ca is the drag coefficient of air. The drag
coefficient of air was computed from Garratts formula:
Cda = 0.001 (o.75 + 0.00067^^2 + Wy2)
(8.26)
Vegetation Distribution
Number of sta1ks/Cm**2*10000.
Figure 8.7: Distribution of vegetation density in Lake Okeechobee.
98
where the wind speeds are in cgs units (cms1). A maximum value of 0.003 was
imposed upon Cda. Wind stress from the five stations was interpolated onto the
curvilinear grid by inversedistancesquared weighting. The inversedistancesquared
interpolation is done as follows. The distance from the corner point of each grid cell
to each of the wind stations is first calculated. The wind stress at each corner point
is then computed by a summation of the wind stress from the five wind stations,
weighted according to the inverse square of the distance from the corner point to each
wind station. This interpolation process can be summarized as
N
r3)nR0/Rn (8.27)
n=1
where N = 5 wind stations, and
5 O.'J *n)2 + (Vi,j ~ yn)2
and
Rn = (xitj Xn) + (Vi,j lIn)
(8.28)
(8.29)
The surface wind stresses Ts^,Tsr) at the u,v points, respectively, are then obtained
from the values at the corner points of the grid cells by transformation relation.
8.5 Steady State WindDriven Circulation
To investigate the steadystate circulation, an easterly wind of 1 dyne/cm,2 was
imposed at the free surface. As shown in Figure 8.8, the depth integrated field
shows two circulation gyres were formed: one clockwise gyre in the southern part
and another counterclockwise gyre in the northern part. Currents follow the wind
direction in the shallow northern and southern part and return flow in the central
deep area to satisfy the continuity.
Figure 8.9 shows the surface elevation contour when the lake reaches steady state.
Though the bottom is not flat and geometry is complex, approximate setup height
can be estimated by use of setup equation. Neglecting bottom stress, setup equation
99
is fj = With Ax = 19km and h = 2.5m, setup is 7.8 cm. This is close to the
maximum setup in Figure 8.9.
8.6 WindDriven Circulation without Thermal Stratification
8.6.1 Tests of Model Performance
Comparing the simulated results with measured data is not a simple task. Two
methods are used for the comparison. The first method is to plot simulated results
and measured data. While this method can give a quick intuition, this can mislead
the readers to the evaluation of the models predictive ability because this method
can show the models ability qualitatively.
Therefore quantitative measurements of the models performance are necessary.
Typical parameters are mean values of model and data, and the correlation coefficient
between the model and data. A new parameter, suggested by Willmott (1981), is the
index of agreement used for the comparison as follows:
,, EfaiCfi Qif
where Ot is the observed value, P, is the predicted value, O is the average of O,, P is
the average of P, P{ = P, O, and 0\ = 0,0. When the index of agreement is
unity, predicted values perfectly agree with observed values in magnitude and sign.
8.6.2 Model Results
Currents At Station C
Time series of wind stress at Station C show that the wind field is temporarily
varying. Therefore, spatially and temporarily varying wind stress field was used for
the simulation. Time series of the wind stress field shows that there is a significant
diurnal variation of wind speed (Figure 8.10). Wind is calm early in the morning
and strong in the afternoon and becomes calm again in the late night.
The comparison of peak wind stress and peak current of field data shows that there
is a time lag between the current and the wind, for example, Julian Day 150.8 in Figure
100
Figure 8.8: Steadystate depthintegrated currents (cm2s1) in Lake Okeechobee
forced by an easterly wind of 1 dyne/cm2.
101
CONTOUR FROM 6.3000 TO 8.1000 CONTOUR INTERVAL OF 0.90000
Figure 8.9: Steadystate surface elevation contour (cm) in Lake Okeechobee forced
by an easterly wind of 1 dyne/cm2.
102
8.14. When the wind blows in the same direction over a considerable time period,
water is piled up to create a setup of surface elevation. Earlier studies indicated
that it takes approximately one hour (Saville, 1952) for the current to respond to the
wind. This is also manifested in the field data. The peak times of model currents
compare quite well with the peak times of measured currents. When the wind changes
direction or dies out, the wind setup is released and the seiche starts to travel back
and forth in the lake. Seiche period is computed by T = 2L/\/gK. When an average
depth of 2.5 m and a length of 37 km, which is the length of lake excluding the marsh
area in eastwest direction, is used, the seiche period is about four hours. The model
currents clearly show this period.
Currents at Station C are measured at Arms 1, 2 and 3, which are at the height
of 17%, 33%, and 78% of the total depth above the bottom, respectively. To find
out the corresponding model currents is not easy because the model uses cr stretching
vertically. Therefore, the nearest layers are determined according to the percentage
of total depth at the instrument heights. After that, linear interpolation between
two layers is used for the model currents to be compared with field data. As shown
in Figures 8.10 8.15, the longterm trend of time series of model current agrees
well with that of field data. However, magnitudes of model currents (particularly the
peak currents everyday) are generally smaller than those of field data. This can be
explained with two possible reasons. First of all, the model currents are forced by 15
minute averaged wind data, while the measured currents are forced by the raw wind,
which contains many spikes which are stronger than the 15minute averaged wind.
Thus, the simulated currents are expected to be less than the measured currents.
Another possibility is due to some numerical dissipation which is always present in
realistic simulations. The numerical dissipation can be reduced by using a smaller
time step.
Measured currents have another trend which the model could not simulate well.
103
When the wind is light, the measured currents are usually small. As the wind increases
in the afternoon, the surface currents follow the wind direction. However, when
the wind speed reduces, the setup is released and the currents change the direction.
Therefore, general time history of measured currents have two peaks: a positive peak
when the flow is from west to east during increasing easterly wind and a negative
peak when the flow is from east to west during decreasing easterly wind.
While the model could simulate the positive peaks well, it missed many negative
peaks as shown in Figure 8.14. As it will be shown later, the reason for this is that
the model did not consider the thermal effect, and hence the eddy viscosity was not
calculated correctly. A reduction of the time step did not lead to improved model
results.
The index of agreement of this model simulation is shown in Table 8.6. The
average index is 0.75 and 0.66 for the eastwest currents and northsouth currents,
respectively.
Table 8.6: Index of agreement and RMS error at Station C.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.78
0.69
1.98
1.46
2
0.74
0.67
2.47
1.69
3
0.73
0.62
2.12
2.01
Currents At Station B
Wind direction is mainly toward the west during the two weeks. Measured cur
rents at both arms show that the direction is toward the southwest. Arm 1 was at
16% of the total depth above the bottom. The average current at arm 1 is 3 cm/sec.
A comparison of the model results with field data at arm 1 shows that the directions
agree well. As shown in Figures 8.16 and 8.17, the model could simulate the time
of peak currents well. There is a discrepancy of approximately 1 cm/sec between
the simulated and measured average currents. Considering the accuracy of current
dyne/cm**2
104
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform C> 3D, In t er po 1 a i ed.l Ar m 1) W/0 Temp
Julian Day
Figure 8.10: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation
dyne/cm#*2
105
Velocity at Platform Ci 3D, InterpolatedtArm 1) W/0 Temp
JulIan Day
Figure 8.11: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 1: NorthSouth, direction). Thermal stratification was not considered in model
simulation
dyne/cm**2
106
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform C* 3D. Interpolated(Arm 2) W/0 Temp
Julian Day
Figure 8.12: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation
dyne/cm**2
107
Velocity at Platform C 3D, Interpolated(Arm 2) W/Q Temp
JulIan Day
Figure 8.13: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation
dyne/cm**2
108
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform Ci 3D, I nterpoI atedIArm 3) W/0 Temp
147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.14: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction). Thermal stratification was not considered in model
simulation
vlcm/sec) dyne/cm**2
109
Velocity at Platform C 3D, Interpolated(Arm 3) W/0 Temp
Julian Day
Figure 8.15: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction).
dyne/cm**2
no
Wind Stress at Platform B tau x (MODEL)
Velocity at Platform B* 3D. Interpolated(Arm 1) W/0 Temp
Jul I an Day
Figure 8.16: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation.
dyne/cm*#2
111
Velocity at Platform B* 3D. Interpo1 ated(Arm 11 W/0 Temp
Julian Day
Figure 8.17: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
112
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B. 3D, Interpo1 ated(Arm 2) W/0 Temp
Julian Day
Figure 8.18: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
113
Velocity at Platform B 3D, Interpo1 ated(Arm 2) W/0 Temp
JulIan Day
Figure 8.19: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation.
114
meters, this difference is considered negligible. However, the peak values of model
currents are somewhat smaller than the field data, as it was found for Station C.
Arm 2 wras at a height of 75% of the total depth. A comparison of the model
current with field data at arm 2 shows generally good agreement in current direction
over two weeks (Figures 8.18 and 8.19). Model currents are somewhat stronger
than the field data. The agreement between the model currents and field data in
the northsouth direction becomes worse after Julian Day 154. Considering that the
wind becomes strong in the northsouth direction, the field data remains from north
to south. This seems to indicate that data have drifting problems.
Both the measured and simulated currents at Station B indicate that flow is gen
erally in the southwestnortheast direction, which is parallel to the boundary between
the vegetation and the open water.
Table 8.7: Index of agreement and RMS error at Station B.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.76
0.68
1.86
1.70
2
0.75
0.65
2.50
3.07
Currents At Station E
The depth at this station was 152 cm. As shown in Figure 8.20, the dominant
wind direction was toward the west. However, the anemometer did not work properly
after Day 157. Thus, the wind stress after Day 157 was computed with the wind data
obtained at the other stations. The average wind stress was about 0.5 dyne/cm2.
Measured current at arm 2, which was 116 cm above the bottom, shows that u
has a negative trend and v has a positive trend over the twoweek period (Figures
8.22 and 8.23). Therefore flow direction is parallel to the boundary between the
vegetation area and the open water. Measured current at arm 1, which was at 36
cm above the bottom, also shows the same trend (Figures 8.20 and 8.21). The flow
115
direction was almost the same at both arms.
Average current speeds at arm 2 were 4 cm/sec in the x direction and 7 cm/sec in
the y direction. Average speeds at arm 1 were 3 cm/sec and 6 cm/sec, respectively.
On Julian Day 157, there was a short period of strong wind which created a strong
seiche. Measured data at both arms clearly showed this seiche.
Arm 2 was at a height of 76% of the total depth. As shown in Figure 8.22, time
series of the model results indicated that u has a negative trend and v has a positive
trend. Therefore, it appears that the model could simulate the longterm trend of
flow direction correctly.
Arm 1 was at a height of 24% of the total depth. As shown in Figures 8.20 and
8.21, model could simulate the currents in terms of direction and magnitude.
Figure 8.24 shows the stick diagram of wind stress, the model results, and the
field data. It clearly shows flow is generally from the southeast to northwest, which
is parallel to the boundary between the vegetation and the open water. Model could
simulate this trend very well.
Table 8.8: Index of agreement and RMS error at Station E.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.38
0.40
2.86
5.66
2
0.28
0.29
2.66
4.46
Currents At Station A
One arm was installed at a height of 71 cm above the bottom. Measured data
show that currents respond well to the change in wind stress. Comparing the model
results with field data at arm 1 shows that the directions agree quite well while the
measured current speeds are underestimated by the model, as shown in Figures 8.25
and 8.26.
dyne/cm**2
116
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E* 3D, Interpolatad(Arm 1) W/0 Temp
Ju\Ian Day
Figure 8.20: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
117
Wind Stress at Platform E< tau y (MODEL)
f
Figure 8.21: Simulated (solid lines) and measured (dotted lines) currents at Station E
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
118
Wind Stress at Platform Ei tau x (MODEL)
Velocity at Platform E< 3D, Interpo1 ated1 Arm 2) W/0 Temp
147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.22: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
119
Wind Stress at Platform Ei tau y (MODEL)
Velocity at Platform E' 3D. Interpo1 atad(Arm 2) W/0 Temp
Julian Day
Figure 8.23: Simulated (solid lines) and measured (dotted lines) currents at Station E
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation.
120
Wind Stress
1 dyne/cm2
Station E
Figure 8.24: Stick Diagram of wind stress, measured currents, and simualted currents
at Station E.
dyne/cm**2
121
Wind Stress at Platform Ai tau x (MODEL)
Velocity at Platform A* 3D. InterpoI ated(Arm 1) W/0 Temp
Julian Day
Figure 8.25: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction). Thermal stratification was not considered in model '
simulation.
dyne/cm**2
122
Velocity at Platform A* 3D, Interpo1ated(Arm 1) W/0 Temp
JulIan Day
Figure 8.26: Simulated (solid lines) and measured (dotted lines) currents at Station A
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
123
Table 8.9: Index of agreement and RMS error at Station A.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.39
0.37
3.73
2.07
Currents At Station D
Arm 1 was at a height of 20% of the total depth. The model results agree well
with field data at arm 1 in terms of the flow direction while the model underestimates
the peak currents (Figure 8.27 and 8.28). The model results at arm 2 shown in Figures
8.29 and 8.30 agree well with field data in terms of the flow direction.
Table 8.10: Index of agreement and RMS error at Station D.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.42
0.68
2.22
2.16
2
0.44
0.62
2.11
3.13
8.7 WindDriven Circulation with Thermal Stratification: Te Method
To improve the model performance, vertical thermal stratification is considered.
At first, the equilibrium temperature method is used and equilibrium temperatures
and heat exchange coefficients were estimated by the method described in Chapter 5.
These estimated equilibrium temperatures at Station C are then assumed to be the
same at all other grid points.
LongTerm Dynamics
Figures 8.31 through 8.50 show the simulated currents and field data. Tables
8.11 through 8.15 show the index of agreements.
As shown in Table 8.13, the model results with thermal effect agree well with field
data. During the daytime, the lake becomes thermally stratified. The temperature
difference between Arm 1 and Arm 3 at Station C can sometimes becomes 3C. This
small temperature difference causes the density of water to vary vertically. Although
dyne/cm**2
124
Wind Stress at Platform Di tau x (MODEL)
Velocity at Platform Di 3D. InterpoI atedIArm 1) W/0 Temp
Julian Day
Figure 8.27: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction).
dyne/cm**2
125
Velocity at Platform D 3 Interpolated(Arm 1) W/0 Temp
Ju1 an Day
Figure 8.28: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction).
dyne/cm**2
126
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D 3D. Interpolated(Arm 2) W/0 Temp
Jullan Day
Figure 8.29: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction).
v(cm/secl dyne/cm*#2
127
Velocity at Platform D* 3D, InterpolatedlArm 2) W/0 Temp
Julian Day
Figure 8.30: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction).
128
the density difference is small, the resulting buoyancy and eddy viscosity can become
much different from that of homogeneous water. Figure 8.51 shows the time history
of eddy viscosity at Station C in homogeneous and thermally stratified cases. In the
thermally stratified case, the peak values of eddy viscosity become larger than that
of a homogeneous lake. Also, during the daytime when the lake is stratified and the
wind is weak, eddy viscosity becomes small as expected.
ShortTerm Dynamics
To understand the physics of a lake during the typical daytime, the time histories
of wind stress, heat flux, and measured currents at Station C during Julian day 150
and 152 were plotted in Figure 8.52. Because wind direction is from east to west,
northsouth components of wind stress and currents are small. Therefore eastwest
direction components are plotted. As shown in Figure 8.52, currents and wind stress
follow typical diurnal variations. During the daytime, currents at Arm 3 are from
east to west, while currents at Arm 1 are from west to east. Therefore, the lake
behaves as a twolayer system. Figure 8.53 shows the model results. Model simulated
currents with thermal effect agree well with field data. Time history of eddy viscosity
at Station C is shown in Figure 8.54. Around 7 a.m. of Julian day 150, wind is
mild and wind stress is less than 0.3 dyne/cm2. Because forcing is small at the water
surface, the eddy viscosity should be negligible. At around 2:30 p.m (Julian day
150.6), wind stress starts to increase while the lake is thermally stratified. Therefore,
momentum transfer is limited to the upper layer. As a consequence, eddy viscosity
above the thermocline increases, but eddy viscosity at the bottom layer remains small.
About three hours later, after the strong wind has resulted in the overturning of the
thermocline eddy viscosity at the bottom layer also increases. Around 9 p.m., the
lake is affected by the seiche which is caused by the reduced wind stress.. During the
dyne/cm**2
129
Wind Stress at Platform Ai tau x (MODEL)
Velocity at Platform A 3D, Intarpolated(Arm 1) W/ Temp
Julian Day
Figure 8.31: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction) when thermal effect is considered.
v{cm/sec) dyne/cm**2
130
Velocity at Platform A 3D. Interpo 1 ated(Arm 1) W/ Temp
Ju1 Ian Day
Figure 8.32: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: NorthSouth direction) when thermal effect is considered.
dyne/cm**2
131
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B> 3D, Interpo1ated(Arm 1) W/ Temp
Julian Day
Figure 8.33: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction) when thermal effect is considered.
132
Velocity at Platform B 3D, Interpo1 ated(Arm 1) W/ Temp
Julian Day
Figure 8.34: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: NorthSouth direction) when thermal effect is considered.
dyne/cm**2
133
Wind Stress at Platform B< tau x (MODEL)
Velocity at Platform B> 3D, Interpo1 ated(Arm 2) W/ Temp
Julian Day
Figure 8.35: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction) when thermal effect is considered.
vlcm/sec) dyne/cm##2
134
Velocity at Platform B* 3D, InterpolatedlArm 2) W/ Temp
Julian Day
Figure 8.36: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: NorthSouth direction) when thermal effect is considered.
uI cm/sec) dyne/cm**2
135
Wind Stress at Platform C tau x (MODEL)
Velocity at Platform C* 3D. Interpolated(Arm 1) W/ Temp
Jullan Day
Figure 8.37: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction) when thermal effect is considered.
dyne/cm**2
136
Velocity at Platform C* 3D. Intarpo 1ated(Arm 1) W/ Temp
JulIan Day
Figure 8.38: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: NorthSouth direction) when thermal effect is considered.
Wind Stress at Platform C* tau x (MODEL)
Velocity at Platform C 3D, InteroolatedlArm 2) W/ Temp
Jutlan Day
Figure 8.39: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction) when thermal effect is considered.
dyne/cm**2
138
Velocity at Platform C 3D, Interpolated(Arm 2) W/ Temp
Jul!an Day
Figure 8.40: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: NorthSouth direction) when thermal effect is considered.
dyne/cm**2
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform Ci 3D, Interpo1ated1Arm 3) W/ Temp
Julian Day
Figure 8.41: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction) when thermal effect is considered.
140
Julian Day
Figure 8.42: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction) when thermal effect is considered.
L
dyne/cm**2
141
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D. 3D, Interpo1atedlArm 1) W/ Temp
Julian Day
Figure 8.43: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction) when thermal effect is considered.
142
Velocity at Platform D* 3D. Intarpo 1 a ted(Arm 1) W/ Temp
JulIan Day
Figure 8.44: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction) when thermal effect is considered.
143
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D* 3D, Intarpolated(Arm 2) W/ Temp
Figure 8.45: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction) when thermal effect is considered.
dyne/cm**2
144
Velocity at Platform D 3D. InterpoI ated(Arm 2) W/ Temp
JulIan Day
Figure 8.46: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction) when thermal effect is considered.
dyne/cm**2
145
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E* 3D. Intarpolatad(Arm 1) W/ Temp
JulIan Day
Figure 8.47: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction) when thermal effect is considered.
v(cm/secl dyne/cm**2
146
Wind Stress at Platform Ei tau y (MODEL)
Figure 8.48: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: NorthSouth direction) when thermal effect is considered.
ulcm/sec) dyne/cm**2
147
Wind Stress at Platform E> tau x (MODEL)
Velocity at Platform E> 3D. Interpo1ated1 Arm 2) W/ Temp
Figure 8.49: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction) when thermal effect is considered.
dyne/cm**2
148
Wind Stress at Platform E> tau y (MODEL)
Julian Day
Figure 8.50: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: NorthSouth direction) when thermal effect is considered.
149
Table 8.11: Index of agreement and RMS error at Station A when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.35
0.45
4.23
2.07
Table 8.12: Index of agreement and RMS error at Station B when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.75
0.69
1.89
1.65
2
0.77
0.67
2.53
3.10
Table 8.13: Index of agreement and RMS error at Station C when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.86
0.81
1.86
1.38
2
0.75
0.77
2.87
1.64
3
0.78
0.71
2.23
2.05
Table 8.14: Index of agreement and RMS error at Station D when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.82
0.76
1.74
1.94
2
0.58
0.61
2.60
3.37
Table 8.15: Index of agreement and RMS error at Station E when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.37
0.39
3.15
5.86
2
0.32
0.39
2.62
4.09
150
seiche, currents at all the three arms have same directions. This diurnal variation is
repeated during the next day.
8.8 WindDriven Circulation with Thermal Stratification: Inverse Method
To improve the temperature prediction and currents, the inverse method was used
to estimate the heat flux at the water surface. Assuming the negligible advection ef
fect, total fluxes at five stations were obtained by solving the vertical onedimensional
temperature equation coupled with momentum equations.
As shown in Figure 8.55, nonsolar heat flux fluctuates while the solar part is
regular. Using total fluxes, CH3D was run. Trends and magnitudes of currents were
similar to those with equilibrium temperature (figures showing simulated currents
and data are included in the Appendix).
However, temperature prediction was much improved. Figure 8.56 shows the
temperature contour with time between Julian day 152 and 155. Generally, the lake
is vertically homogeneous in the night. As sun rises, the lake becomes thermally
stratified. When wind blows strong in the afternoon, the lake is destratified due to
windinduced mixing. Field data show the daily thermal stratification and destratifi
cation. The contours of simulated temperature indicate clearly this process. Figures
8.57 through 8.61 show the time histories of temperature data and the model. In
general, the model can simulate the stratification well at all stations.
8.8.1 The Diurnal Thermal Cycle
From the time history of temperature, diurnal variations of stratification and de
stratification are obvious. In spite of the maximum temperature difference between
Arm 1 and Arm 3 at Station C during a day, thermal effect on the currents is signif
icant as shown in Figures 8.62 through 8.65.
Typical vertical profiles of currents and temperature at Station C on Julian Day
155 are shown sequentially, starting at midnight. The first two columns are simulated
current profiles without thermal effect, and the other three profiles are with thermal
cm Vs cmVs Watl/m!
151
HMt Flux Estimated by Equifibrium Tampsratura
Eddy Viacoaiy at C (temperatura considerad)
JufisnDsy
Figure 8.51: Time history of eddy viscosity at Station C between Julian days 147 and
161.
u(cm/s) dyne/cm**2
152
Wind Stress (EastWest) at Station C
Julian Day
Measured Currents at Station C
Julian Day
Figure 8.52: Time history of wind stress and measured currents between Julian days
150 and 152.
u(cm/s) u(cm/s)
153
Currents at C (without thermal stratification)
Currents at C (with thermal stratification)
Figure 8.53: Time history of simulated currents at Station C between Julian days 150
and 152.
cmÂ¡/s cmVs
154
Eddy Viscosity at C (temperatura considered)
Figure 8.54: Time history of eddy viscosity at Station C between Julian days 150 and
152.
Watt/M* *2
155
1500
1000
500
0
500
1000
1500
Figure 8.55: Time history of heat fluxes at Station C between Julian davs 147 and
161.
Heat Flux
148
150
152
154
Julian Day
156
effect. Until 8:00 a.m, the lake is thermally homogeneous. Vertical profiles of currents
are very similar independent of thermal effect. However, after sunrise the lake starts
to be stratified due to heating at the water surface.
Starting at 9:00 a.m., the vertical profiles of temperature show that the stratifica
tion and vertical profiles of simulated currents with thermal effect start differing from
those without thermal effect. Generally, currents with thermal effect are stronger.
This is because the momentum transfer is limited to the upper layer, and magnitudes
of currents become larger. Also, the currents in the lower layer increase due to return
flow.
Afternoon lake breeze starts at 5:00 p.m., and the reduced solar heating causes the
destratification. At 7:00 p.m., the thermocline breaks and the lake becomes thermally
homogeneous. Therefore, thermal effect becomes insignificant throughout the night.
This stratification and destratification cycle is repeated daily. Therefore, the
simulated currents without the thermal effect are generally smaller than the measured
currents. As will be explained in the sensitivity test, the adjustments of bottom drag
coefficient and roughness height did not improve the model results. In general, if a
numerical model underestimates the current magnitude but reveals the general trend
of measured data, the usual approach to improve the results is often the reduction
of bottom stress. However, this can fail because of neglecting of the diurnal thermal
cycle in shallow lakes.
8.9 Sensitivity Tests
The successful run presented in the previous section was produced with a partic
ular set of model parameters. In order to ensure that the particular choice of model
parameters is not arbitrary, it is essential to conduct a sensitivity analysis on the
models response to changes in parameters. This sensitivity analysis is also necessary
because the measured data always contain some measurement error.
157
Temperature at C (Model)
Julian Day
Fiffure 8.56: Temperature contours of data and model at Station C between Julian
days 152 and 155.
(degree)
158
Temperature at Platform A i MODEL
147. 149. 151. 153. 155. 157. 159. 161
Julian Day
Figure 8.57: Simulated and measured temperatures at Station A.
159
Temperature at Platform B MODEL
Temperature at Platform B DATA
*147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.58: Simulated and measured temperatures at Station B.
160
Temperature at Platform C MODEL
Temperature at Platform C DATA
*147. 149. 151. 153. 155. 157. 159. 161.
Julian Day
Figure 8.59: Simulated and measured temperatures at Station C.
161
Temperature at Platform D i MODEL
Temperature at Platform D DATA
Julian Day
Figure 8.60: Simulated and measured temperatures at Station D.
{degree)
162
Temperature at Platform E i MODEL
Figure 8.61: Simulated and measured temperatures at Station E.
163
W/O TEMP
WITH TEMP
U
V
u
V T
0 HR
]
\
1 HR
[
7
2 HR
I
!
3 HR
I
'
4 HR
I
Â¡
1
5 HR
f
SCALE
10 cm/sec
5 C
1 dyne/cm**2
Figure 8.62: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
164
6HR
7HR
8HR
9HR
10HR
11HR
W/0 TEMP
WITH TEMP
U
V
U
V T SCALE
Figure 8.63: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
165
V/O TEMP
WITH TEMP
18 HR
19 HR
20 HR
21 HR
U
22 HR
23 HR
U
V
T SCALE
dyne/cm**2
Figure 8.64: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
166
10 HR
19 HR
20 HR
21 HR
22 HR
23 HR
W/0 TEMP
WITH TEMP
Figure 8.65: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
167
It is thus important to quantify the sensitivity/uncertainty of model results while
comparing them with field data. Sensitivity analysis may reveal those parameters to
which the model is particularly sensitive. If a slight variation in a particular model
parameter produces a significantly different result, then that particular parameter
should be determined precisely. In contrast, those parameters to which the model is
less sensitive need not be estimated as accurately. As shown in Tables 8.16 and 8.17,
Table 8.16: Parameters used in sensitivity tests.
Case
Law of
the
wall
Zq
cm
Drag
Coeff.
Ah
cm2/s
Second
order
closure
Munk
Anderson
Av
cm,2/s
Smooth
bottom
Advec
tion
1
Yes
0.1
No
10s
Yes
No
No
No
No
2
No
No
0.001
105
Yes
No
No
No
No
3
No
No
0.004
105
Yes
No
No
No
No
4
Yes
0.1
No
104
Yes
No
No
No
No
5
Yes
0.1
No
103
Yes
No
No
No
No
6
No
No
0.01
105
Yes
No
No
No
No
7
Yes
0.1
No
105
No
Yes
No
No
No
8
Yes
0.1
No
105
No
No
20
No
No
9
Yes
0.5
No
105
Yes
No
No
No
No
10
Yes
0.01
No
105
Yes
No
No
No
No
11
Yes
0.1
No
10s
Yes
No
No
Yes
No
12
Yes
0.1
No
105
Yes
No
No
No
Yes
12 test runs were performed. Model performance was measured by use of an index
of agreement between simulated currents and measured currents at three different
heights at Station C.
8.9.1 Effect of Bottom Stress
Five tests were performed by using the different formula and coefficients. Bottom
stress was expressed as a quadratic law. Different constant drag coefficients (0.001,
0.004, and 0.01) were used (cases 2, 3, and 6, respectively). Compared with case 1,
cases 2 and 6 were not very sensitive, but the index of agreement dropped substantially
with use of Cd = 0.004. This is only in the eastwest direction. Northsouth currents
168
are generally small while eastwest currents are stronger. Therefore, magnitude of
bottom stress in the northsouth direction is small and affects currents little.
Drag coefficient can be estimated from the turbulent theory in the bottom bound
ary lyer. Roughness heights of 0.5 and 0.01 were used (cases 9 and 10). The index
of agreement was not changed much. Therefore, the choice of 0.01 in the base run
(case 1) seems to be appropriate.
A different method to estimate the drag coefficient in the smooth bottom bound
ary was used, and the index of agreement was dropped (case 11).
8.9.2 Effect of Horizontal Diffusion Coefficent
Horizontal diffusion coefficients of 104 and 103 were used (cases 4 and 5). The
change of index of agreement is hardly noticeable.
8.9.3 Effect of Different Turbulence Model
Three different turbulence models were used (cases 1, 7, and 8). Constant eddy
viscosity of 20cm2/sec was used in case 8. Simulated currents did not agree well
compared to case 1. Turbulence can not be appropriately resolved by use of constant
eddy viscosity because turbulence motion varies spatially and temporally.
The MunkAnderson type was used for the parameterization of turbulence (case
7). The effect of stratification due to temperature and/or salinity is considered by
multiplying the stability function. However, as discussed by Sheng (1983), there
are many formulas, and lots of field data are required to get bestfit. The index of
agreement was not improved compared to case 1, which used a simplified secondorder
closure model.
8.9.4 Effect of Advection Term
To investigate the importance of the advective term, CH3D was run with the
advection term (case 11). The Rossby number, which is an indicator of the importance
of the advective term relative to the Coriolis term, is defined as U/ fL. Taking
the charcteristic values (U = 10cm/sec, f = 6.62e~5, and L = 37km), the Rossby
169
number is 0.04. Compared with case 1, the index of agreement was not changed
much. Therefore the advection term is negligible.
8.10 Spectral Analysis
Spectral analysis can reveal the important frequencies which are related to the
physical processes. Surface elevation and currents at Station C were used for spectral
analysis. Data on Julian day 147 were excluded because CH3D started from zero
velocity. Also, data after Julian day 158 were excluded because measured data had
bad measurements. Dominant periods of wind are 5.8 day, 23 hour and 11.6 hour as
shown Figure 8.66. A component of 5.8 day period is related to the longterm trend
of wind. Components of 23 hour and 11.6 hour are related to the diurnal variation of
wind.
As shown in Figure 8.66, the spectrum of surface elevation shows the frequencies
of diurnal and semidiurnal components. Magnitude of the spectrum corresponding
to the first mode seiche period (about 4 hours) is relatively small compared with the
diurnal component because Station C is located near the nodal point. The spectrum
of simulated currents agrees well with the spectrum of measured currents as shown
in Figure 8.68. Not only period but also magnitudes of the spectrums agree well.
1
2
3
4
5
6
7
8
9
10
11
12
170
Table 8.17: Index of agreement and RMS error.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.86
0.85
1.79
1.17
2
0.78
0.80
2.55
1.48
3
0.78
0.73
2.13
1.81
1
0.86
0.85
1.87
1.20
2
0.78
0.81
2.66
1.49
3
0.79
0.72
2.16
1.88
1
0.70
0.84
3.64
1.43
2
0.65
0.79
4.37
1.70
3
0.65
0.70
3.44
2.12
1
0.86
0.83
1.79
1.25
2
0.78
0.79
2.56
1.51
3
0.78
0.71
2.15
1.88
1
0.86
0.83
1.82
1.25
2
0.78
0.79
2.61
1.51
3
0.77
0.71
2.20
1.87
1
0.86
0.82
1.70
1.21
2
0.79
0.78
2.35
1.49
3
0.76
0.73
2.19
1.77
1
0.75
0.67
2.04
1.54
2
0.71
0.64
2.45
1.84
3
0.73
0.64
2.12
1.90
1
0.77
0.68
1.99
1.51
2
0.73
0.65
2.39
1.77
3
0.70
0.62
2.13
1.93
1
0.86
0.82
1.69
1.21
2
0.79
0.77
2.34
1.50
3
0.75
0.73
2.24
1.76
1
0.84
0.86
2.15
1.21
2
0.76
0.81
2.93
1.52
3
0.77
0.72
2.27
1.92
1
0.79
0.85
2.56
1.28
2
0.72
0.80
3.38
1.59
3
0.75
0.72
2.49
1.98
1
0.87
0.84
1.73
1.19
2
0.80
0.80
2.45
1.47
3
0.78
0.72
2.14
1.86
cm**2.sec dyne**2/cm**4.sec dyne**2/cm**4.sec
171
Figure 8.66: Spectrum of wind stress and surface elevation
cm**2/sec 001**2/560 cm**2/seo
172
Figure 8.67: Spectrum of measured and simulated currents (eastwest direction) at
Station C
cm*+2/sec cm**2/sec cm**2/sec
173
Figure 8.68: Spectrum of measured and simulated currents (northsouth direction) at
Station C
CHAPTER 9
CONCLUSION
A threedimensional hydrodynamic model (CH3D) was significantly enhanced to
study the winddriven circulation in Lake Okeechobee considering the effects of veg
etation and thermal stratification. Space and timevarying wind stresses were used
to drive the model. The effect of vegetation was parameterized as increased pro
file drag on the flow. Vertical turbulence was parameterized by a simplified second
orderclosure model.
CH3D was used to study the winddriven circulation during the period of May
27, 1989, to June 10, 1989, in Lake Okeechobee to simulate the currents driven by
winds. Followings are major conclusions from this study.
1. Both the simulated and measured currents in the vicinity of the vegetation
zone were found to be primarily in the direction parallel to the vegetation boundary,
thus suggesting relatively little transport across the vegetation boundary.
2. For Lake Okeechobee, it was determined that thermal stratification effects
were critical to the successful simulation of circulation under increasing winds.
3. Without considering thermal effects, the longterm trend of simulated currents
followed well that of field data. But the simulated currents did not show the peaks
well, which were quite obvious in the field data. Therefore, thermal effect in shallow
lakes was considered by solving the temperature equation which was coupled with the
momentum equations.
4. With thermal effect, simulated currents not only revealed the peaks well but
also followed field data on shortterm trends quite well, indicating the proper param
eterization of turbulence.
174
175
5. In order to consider thermal effects, at first, heat fluxes at the water surface
were estimated by using the meteorological data and expressed as timevarying equi
librium temperatures. However, due to the insufficient meteorological data, simulated
temperatures did not agree well with field data although currents results were much
improved.
6. Another estimation of heat fluxes was tried with the socalled inverse method.
Assuming the total heat flux consists of a solar part and a nonsolar part, the nonsolar
part was estimated by solving the vertical onedimensional temperature equation.
Total heat fluxes were used as a boundary condition at the water surface for the
temperature equation of CH3D. Results showed that the predicted temperature agrees
well with field data. Therefore, when there are insufficient meteorological data, the
inverse method can be a good method to estimate the heat flux with given wind data
and measured surface temperature.
APPENDIX A
SIMULATED CURRENTS BY INVERSE METHOD
176
dyne/cm**2
177
Wind Stress at Platform A tau x (MODEL)
Velocity at Platform A* 3D, Interpolated(Arm 1) Inverse
Jullan Day
Figure A.l: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
178
Velocity at Platform A* 3D. Interpolated(Arm 1) Inverse
Julian Day
Figure A.2: Simulated (solid lines) and measured (dotted lines) currents at Station A
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
179
Wind Stress at Platform tau x (MODEL)
Velocity at Platform B* 3D, Interpolated(Arm 1 ) Inverse
Julian Day
Figure A.3: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
180
Velocity at Platform B 3D, InterpoI ated(Arm 1) Inverse
Julian Day
Figure A.4: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
181
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B> 3D, InterpoIatedI Arm 21 Inverse
Julian Day
Figure A.5: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
182
Velocity at Platform B* 3D, Intarpo 1 ated(Arm 2) Inverse
Julian Day
Figure A.6: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 2: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
183
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform Ci 3D. InterpolatedI Arm 1) Inverse
Julian Day
Figure A.7: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
184
Velocity at Platform C 3D, Interpolated(Arm 1) Inverse
JulIan Day
Figure A.8: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
ulcm/secl dyne/cm**2
185
Wind Stress at Platform C< tau x (MODEL)
Velocity at Platform Ct 3D. Intarpo I ated(Arm 2) Inverse
Julian Day
Figure A.9: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm*#2
186
Velocity at Platform Ct 3D. Interpolated(Arm 2) Inverse
Jullan Day
Figure A. 10: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
187
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform C. 3D, InterpolatedI Arm 3) Inverse
Julian Day
Figure A.ll: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
188
Velocity at Platform Ct 3D, Interpo1 ated(Arm 3) Inverse
JulIan Day
Figure A.12: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
189
Wind Stress at Platform D tau x (MODELJ
Velocity at Platform D 3D, Interpolated(Arm 1) Inverse
Ju l Ian Day
Figure A.13: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
190
Velocity at Platform D* 3D, Interpolated(Arm 11 Inverse
JulIan Day
Figure A.14: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
dyne/cm*#2
191
Wind Stress at Platform D> tau x (MODEL)
Velocity at Platform D. 3D. Interpo1 atsd(Arm 2) Inverse
Julian Day
Figure A.15: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
192
Velocity at Platform D. 3D, InterpoIated(Arm 2) Inverse
Julian Day
Figure A.16: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
dyne/cm**2
193
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E 3D, Interpolated(Arm 1) Inverse
147. 149. 151. 153. 155. 157. 159. 161.
JulI an Day
Figure A. 17: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
194
Wind Stress at Platform Ei tau y (MODEL)
Velocity at Platform E 3D, InterpoI ated(Arm 1) Inverse
Julian Day
Figure A.18: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
L
195
Wind Stress at Platform Ei tau x (MODEL)
Velocity at Platform E> 3D, InterpoIatedI Arm 2) Inverse
Julian Day
Figure A.19: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
L
dyne/cm**2
196
Wind Stress at Platform E tau y (MODEL!
Velocity at Platform E* 3D, Intarpolated(Arm 2) Inverse
JulIan Day
Figure A.20: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
BIBLIOGRAPHY
Ahn, K. M. and Y. P. Sheng, 1989: Wind wave hindcasting and estimation of
bottom shear stress in Lake Okeechobee. UFL/COEL Report 89 027, Coastal
and Oceanographic Engineering Department, University of Florida, Gainesville,
FL.
Bowen, I. S., 1926: The ratio of heat losses by conduction and by evaporation from
any water surface. Physical Review, 27, No. 2. pp.779787.
Brunt, D., 1932: Notes on radiation in the atmosphere. Quart.J.Roy.Meteorol.Soc.,
58, pp. 389420.
Brutsaert, W, 1982: Evaporation into the atmosphere, D.Reidel Publishing Com
pany, Dordrecht, Holland, p.289.
Choi, J. K. and Y. P. Sheng, 1993: Threedimensional curvilineargrid modeling
of baroclinic circulation and mixing in a partiallymixed estuary. UFL/COEL
TR /094, Coastal and Oceanographic Engineering Department, University of
Florida, Gainesville, FL.
Cook, V. J. and Y. P. Sheng, 1989: Vertical mixing and resuspension of fine sed
iments in Lake Okeechobee. UFL/COEL 89/025, Coastal and Oceanographic
Engineering Department, University of Florida, Gainesville, FL.
Dickinson, R. E., W. C. Huber and C. D. Pollman, 1991: Modeling of phospho
rus dynamics of Lake Okeechobee. Final Report to the South Florida Water
Management District, Envoronmental Engineering Department, University of
Florida, Gainesville, FL.
Donaldson, C. duP, 1973: Atmospheric turbulence and the dispersal of atmospheric
pollutants. Proceedings of Workshop on Meteorology, American Meteorological
Society, (ed. D.A. Haugen), Science Press, pp. 313390.
Eckart, C., 1958: The equation of state of water and sea water at low temperature
and pressure. American J. of Science, 256, pp. 240250.
Edinger, J. E. and E. M. Buchak, 1979: Reservoir longitudinal and vertical implicit
hydrodynamics. Environmental Effects of Hydraulic Engineering Works, E.E.
Driver and W.O. Wunderlich, Eds., Tennessee Valley Authority, Knoxville, TN.
Edinger, J. E. and J. C. Geyer, 1967: Heat Exchange in the environment Pu.
No. 65902, Edison Electric Institute, New York, NY.
Ekman, V. W., 1923: Veber horizontal cirkulation bei winderzeuogten Meeres Stro
mungen. Ark. Mat. Astron. Fys., 17, p. 26.
197
198
Gaspar, P., J. C. Andre, and J. M Lefevre, 1990: The determination of the latent
and sensible heat fluxes at the sea surface viewed as an inverse problem. Journal
of Geophysical Research, 95, pp. 1616916178.
Gedney, R. T. and W. Lick, 1972: Winddriven currents in Lake Erie. Journal of
Geophysical Research, 77, pp. 27142723.
Goldman, J. C., 1979: Temperature effects on steadystate growth, phosphorus
uptake, and the chemical composition of a marine phytoplankton. Microbial.
Ecol., 5, pp. 153166.
Hanmer, R. W., 1984: EPAs emerging nonpoint source role. EPA 440/5184001.
Harleman, D. R. F, D. N. Brocard, and T. 0. Najarian, 1973: A predictive model for
transient temperature distributions in unsteady flows. Report No. 175, Ralph M.
Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA.
Harris, D. L., 1957: The effect of a moving pressure disturbance on the water level
in a lake. Meteorological Monographs, Vol. 2, No. 10., American Meteorological
Society.
Hellstrom, 1941: Wind effect on lakes and rivers. Bulletin No. 41 of the Institution
of Hydraulics at the Royal Institute of Technology, Stockholm, Sweden. 191 pp.
Hsueh, Y. and C. Y. Peng, 1973: A numerical study of the steady circulation in an
open bay. Journal of Physical Oceanography, 3, pp. 220225.
Huber, W. C. and A. I. Perez, 1970: Prediction of Solar and Atmospheric Radiation
for Energy Budget Studies of Lakes and Streams, Publication No. 10, Water
Resources Research Center, University of Florida, Gainesville, FL.
Johnson, B. J., 1982: Numerical modeling of estuarine hydrodynamics on a
boundaryfitted coordinate system. Numerical Grid Generation
(J. F. Thompson, ed.), Elsevier Publishing Company, New York, NY, pp. 409
436.
Kouwen, N., T. E. Unny, and H. M. Hill, 1969: Flow retardance in vegetated
channels. J. Irrigation and Drainage Div., Proc. A.S.C.E., ( IR 2), pp. 329342.
Lee, K. K. and J. A. Liggett, 1970: Computation for circulation in stratified lakes.
Journal of the Hydraulics Division, 96, pp. 20892115.
Leendertse, J. J., 1967: Aspects of a computational model for long period water
wave propagation. Rand Corporation, Santa Monica, CA, Rep. RM5294PR,
165 pp.
Leendertse, J. J. and S.K. Liu, 1975: A threedimensional model for estuaries
and coastal seas: Volume II, Aspects of computation. Rand Corporation, Santa
Monica, CA, Rep. R1764OWRT, 123 pp.
Leonard, B. P., 1979: Adjusted quadratic upstream algorithms for transient incom
pressible convection. AIAA Journal, pp. 226233.
199
Lewellen, W. S., 1977: Use of invariant modeling. Handbook of Turbulence, 1,
(W. Frost, ed.), Plenum Publishing Corp., New York, NY., pp. 237280.
Lien, S. L. and J. A. Hoopes, 1978: Winddriven, steady flows in Lake Superior.
Limnology and Oceanography, 23, pp. 91103.
Liggett, J. A., 1969: Unsteady circulation in shallow, homogeneous lakes. Journal
of the Hydraulics Division, 95, pp. 12731288.
Liggett, J. A. and K. K. Lee, 1971: Properties of circulation in stratified lakes.
Journal of the Hydraulics Division, 97, pp. 1529.
Lowe, P. R., 1977: An approximating polynomial for the computation of saturation
vapor pressure. J.Appl.Meteorol, 16, pp. 100103.
Munk, W. H. and E. R. Anderson, 1948: Notes on a theory of the thermocline.
Journal of Marine Research, 7, pp. 276295.
Orlob, G. T., 1959: Eddy diffusion in homogeneous turbulence. J. Hyd. Div., Proc.
A.S.C.E., 85, pp. 75101.
Philips, N. A., 1957: A coordinate system having some special advantages for
numerical forcasting. Journal of Meteorology, If, pp. 184185.
Pielke, R. A., 1974: A threedimensional numerical model of the sea breeze over
south Florida. Monthly Weather Review, 102, pp. 115139.
Prandtl, L., 1925: Uber die ausgebildete Turbulenz. ZAMM, 5, 136.
Price, J. F., R. A.. Weller and R. Pinkel, 1986: Diurnal cycling: Observations
and models of the upper ocean response to diurnal heating, cooling, and wind
mixing. Journal of Geophysical Research, 91, pp. 84118427.
Ree, W. O., 1949: Hydraulic characteristics of vegetation for vegetated waterways.
Agricultural Engineering, pp.184 189.
Reid, R. 0. and R. E. Whitaker, 1976: Winddriven flow water influenced by a
canopy. Journal of Waterways, Harbors and Coastal Engineering Division, pp.
6177.
Rodi, W., 1980: Turbulence models and their application in hydraulics, Mono
graph, International Association for Hydraulic Research, Delft, The Nether
lands.
Roig, L. C. and I. P. King, 1992: Continuum model for flows in emergent marsh
vegetation. Estuarine and Coastal Modeling, (M.L. Spaulding, ed.), A.S.C.E.,
pp. 268279.
Saville, T., 1952: Wind setup and waves in shallow water. Technical Memoran
dum No. 27, Beach Erosion Board, Office of the Chief of Engineers, Corps of
Engineers.
200
Schmalz, R. A., 1986: A numerical investigation of hurricaneinduced water level
fluctuations in Lake Okeechobee. Miscellaneous Report CERC8612, U.S. Army
Corps of Engineers.
Schoellhamer, D. H. and Y. P. Sheng, 1993: Simulation and analysis of sediment re
suspension observed in Old Tampa Bay, Florida. UFL/COEL TR /091, Coastal
and Oceanographic Engineering Department, University of Florida, Gainesville,
FL.
Sengupta, S. and W. J. Lick, 1974: A numerical model for wind driven circula
tion and temperature fields in lakes and ponds. FTAS/TR7499, Case Western
Reserve University, Cleveland, OH.
Shanahan, P. and D. R. F. Harleman, 1982: Linked hydrodynamic and biogeo
chemical models of water quality in shallow lakes. Report No. 268, Ralph M.
Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA.
Sheng, Y. P., 1975: Winddriven currents and contaminants dispersion in the
nearshore of large lakes. Technical Report H751, U.S. Army Engineer Wa
terways Experiment Station, Vicksburg, MS.
Sheng, Y. P., 1982: Hydraulic applications of a secondorder closure model of tur
bulent transport, pp. 106119. In (P. Smith, ed.) Proceedings 1982 ASCE Hy
draulic Division Specialty Conference on Applying Research to Hydraulic Prac
tice, American Society of Civil Engineers, Jackson, MS.
Sheng, Y. P., 1983: Mathematical modeling of threedimensional coastal currents
and sediment dispersion: model development and application. Technical Report
CERC832, U.S. Army Eng. Waterways Experiment Station, Vicksburg, MS.
Sheng, Y. P., 1984: A turbulent transport model of coastal processes. Proceed
ings 19th International Conference on Coastal Engineering, American Society
of Civil Engineers, Houston, TX, pp. 23802396.
Sheng, Y. P., 1986: CH3D: A threedimensional numerical model of coastal and
estuarine circulation and transport in generalized curvilinear grids. Technical
Report No. 587, Aeronautical Research Associates of Princeton, Princeton, NJ.
Sheng, Y. P., 1987: Onmodeling threedimensional estuarine hydrodynamics.
ThreeDimensional Models of Marine and Estuarine Hydrodynamics, (J.Nihoul,
ed.), Elsevier Oceanography Series, Elsevier, pp. 3554.
Sheng, Y. P., 1990: Evolution of a threedimensional curvilineargrid hydrodynamic
model for estuaries, lakes and coastal waters: CH3D. Estuarine and Coastal
Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 4049.
Sheng, Y. P., 1993: Hydrodynamics, sediment transport and their effects on phos
phorus dynamics in Lake Okeechobee. Nearshore and Estuarine Cohesive Sedi
ment Transport, (A.J. Mehta, ed.), American Geophysical Union, pp. 558571.
Sheng, Y. P. and H. L. Butler, 1982: Modeling coastal currents and sediment
transport. Proceedings 18th International Conference on Coastal Engineering,
American Society of Civil Engineers, Cape Town, Republic of South Africa, pp.
11271148.
201
Sheng, Y. P. and S. S. Chiu, 1986: Tropical cyclone generated currents. Proceedings
of the 20th International Conference on Coastal Engineering, American Society
of Civil Engineers, Taipei, Taiwan, pp. 737751.
Sheng, Y. P. and H. K. Lee, 1991a: The effect of aquatic vegetation on winddriven
circulation in Lake Okeechobee. Report No. UFL/COEL91019, Coastal and
Oceanographic Engineering Department, University of Florida, Gainesville, FL.
Sheng, Y. P., and H. K. Lee, 1991b: Computation of phosphorus flux between
the vegetation area and the open water in Lake Okeechobee. Report No.
UFL/COEL91022, Coastal and Oceanographic Engineering Department, Uni
versity of Florida, Gainesville, FL.
Sheng, Y. P. and W. Lick, 1972: Winddriven currents in a partially icecovered
lake. Proceedings of the 16th Conference on Great Lake Research, pp. 10011008.
Sheng, Y. P. and W. Lick, 1980: A two mode freesurface numerical model for the
threedimensional timedependent currents in large lakes. U.S. Environmental
Protection Agency Report EPA 600/380047.
Sheng, Y. P. and S. J. Peene, 1992: Circulation and its effect on water quality in
Sarasota Bay. In Framework for Action, Sarasota Bay, Sarasota Bay National
Estuary Program, Sarasota, FL.
Sheng, Y. P. and C. Villaret, 1989: Modeling the effect of suspended sediment
stratification on bottom exchange processes. Journal of Geophysical Research,
94, pp. 1442914444.
Sheng, Y. P., J. K. Choi and A. Y. Kuo, 1989b: Threedimensional numerical
modeling of tidal circulation and salinity transport in James River Estuary.
Estuarine and Coastal Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 209218.
Sheng, Y. P., J. K. Choi and P. F. Wang, 1989c: A threedimensional numer
ical model of the James River and Hampton Road estuarine system. Report
No. UFL/COEL89020, Coastal and Oceanographic Engineering Department,
University of Florida, Gainesville, Florida.
Sheng, Y. P., D. E. Eliason and X. J. Chen, 1993: Modeling threedimensional
circulation and sediment transport in lakes and estuaries Estuarine and Coastal
Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 105115.
Sheng, Y. P., H. K. Lee and K. H.Wang, 1989a: On numerical strategies of estu
arine and coastal modeling. Estuarine and Coastal Modeling, (M.L. Spaulding,
ed.), A.S.C.E., pp. 291301.
Sheng, Y. P., S. Peene and Y. M. Liu, 1990: Numerical modeling of tidal hydro
dynamics and salinity transport in the Indian River Lagoon. Florida Scientist,
53, pp. 147168.
Sheng, Y. P., D. E. Eliason, X. J. Chen and J. K. Choi, 1991a: A three
dimensional numerical model of hydrodynamics and sediment transport in lakes
and estuaries: Theory, model development and documentation. Final Report to
U.S. Environmental Protection Agency, Coastal and Oceanographic Engineer
ing Department, University of Florida, Gainesville, FL.
202
Sheng, Y. P., D. E. Eliason, J. K. Choi, X. J. Chen and H. K. Lee, 1991b:
Numerical simulation of threedimensional winddriven circulation and sediment
transport in Lake Okeechobee during spring 1989. Final Report to the South
Florida Water Management District, Coastal and Oceanographic Engineering
Department, University of Florida, Gainesville, FL, 91p.
Sheng, Y. P., D. E. Eliason, R. E. Dickinson and J. K. Choi, 1991c: A threemonth
simulation of winddriven circulation, sediment transport and phosphorus trans
port in Lake Okeechobee. Report No UFL/COEL91023, Coastal and Oceano
graphic Engineering Department, University of Florida, Gainesville, Florida
Sheng, Y. P., W. Lick, R. T. Gedney and F. B. Molls, 1978: Numerical simulation
of threedimensional circulation in Lake Erie: A comparison of a freesurface
model and a rigidlid model. Journal of Physical Oceanography, 8, pp. 713727.
Simons, T. J., 1974: Verifications of numerical models of Lake Ontario, Part I:
Circulation in spring and early summer. Journal of Physical Oceanography, 4i
pp. 507523.
Sokolnikoff, I. S., 1960: Tensor Analysis, John Wiley and Sons, New York.
Stommel, H., 1949: Horizontal diffusion due to oceanic turbulence. J. Mar. Res, 8,
pp. 199.
Sundaram, T.R., Easterbrook, C. C., Piech, K. R. and Rudinger, G., 1969: An
investigation of the physical effects of thermal discharges into Cayuga Lake.
Report VT261602, Nov. 1969. Cornell Aeronautical Laboratory, Buffalo, NY.
Thomas, J. H., 1975: A theory of steady winddriven currents in shallow water
with variable eddy viscosity. Journal of Physical Oceanography, 5, pp. 136142.
Thompson, J. F., 1983: A boundaryfitted coordinate code for general two
dimensional regions with obstacles and boundary intrusions. Technical Report
E838, U.S. Army Eng. Waterways Experiment Station, Vicksburg, MS.
Tickner, E. G., 1957: Effects of bottom roughness on wind tide in shallow water.
Technical Memorandum No. 95, Beach Erosion Board, Office of the Chief of
Engineers, Corps of Engineers.
Welander, P., 1957: Wind action on a shallow sea: Some generalizations of Ekmans
theory. Tellus, 9, pp. 4552.
Whitaker, R. E., R. 0. Reid and A. C. Vastano, 1975: An analysis of drag coefficient
at hurricane windspeeds from a numerical simulation of dynamical water level
changes in Lake Okeechobee, Florida. Technical Memorandum No. 56, Coastal
Engineering Research Center, Corps of Engineers.
Willmott, C. J, 1981: On the validation of models. Physical Geography, 1981, pp.
184195.
Witten, A. J. and J. H. Thomas, 1976: Steady winddriven currents in a large lake
with depthdependent eddy viscosity. Journal of Physical Oceanography, 6, pp.
8592.
vlcm/sec) dyne/cm**2
109
Velocity at Platform C 3D, Interpolated(Arm 3) W/0 Temp
Julian Day
Figure 8.15: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction).
47
+ (Diffusion)^ [U^U^]
+ ~~ [Nonlinear Terms (U^, Ui)]
Ro
+ [Nonlinear Terms (C/, /2)] (4.35)
Lo
dU*
dt
+ ffbaiC< + *MC,] =
+
9 n
U*~
921
\/9o \/9o
(Diffusion) [C/*,Ui]
U + t; tL f;
677
crt
+ (Diffusion)J?[7> U2]
+ ^[Nonlinear Terms (t/f, U*)\
L1
+ [Nonlinear Terms (C/, C/2)] (4.36)
L 2
The dimensionless continuity equation in the curvilinear grids becomes:
= 0
(4.37)
dyne/cm**2
119
Wind Stress at Platform Ei tau y (MODEL)
Velocity at Platform E' 3D. Interpo1 atad(Arm 2) W/0 Temp
Julian Day
Figure 8.23: Simulated (solid lines) and measured (dotted lines) currents at Station E
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation.
88
8.4 Model Parameters
Because of the great flexibility of CH3D, many parameters must be specified for
its proper use. A convenient way to describe the input parameters is to classify the
parameters as reference values, parameters associated with turbulence, parameters as
sociated with bottom friction, parameters associated with vegetation, and parameters
associated with wind stress.
8.4.1 Reference Values
Reference values are values which are characteristic scales for the processes of
interest. The reference values shown in Table 8.3 were used to simulate the Lake
Okeechobee Spring 1989 circulation. The reference value for the horizontal scale (Xr)
for Lake Okeechobee is the average length of grid, while the reference value for the
vertical scale (Zr) is the characteristic depth of the lake.
The reference velocity (Ur) shown in Table 8.3 is a typical speed for windforced
currents in Lake Okeechobee. In the formulation of the dimensionless equations, time
is scaled by the Coriolis parameter, / = 2f2sin<Â£, where fi is the rotational speed
of the earth and <}> is the latitude. The value for / that is shown in Table 8.3 was
obtained by using fl = 7.29 x 105 s1 and the average latitude of Lake Okeechobee
= 27N. The two physical constants in Table 8.3, which are used as reference
values, are the gravitational acceleration (g) and the density of fresh water (p0). The
reference value shown in Table 8.3 for the horizontal eddy viscosity (Aht) is set to a
scale length dependent value based upon the 4/3 Law suggested by Stommel (1949),
Ah = eÂ£4/3, where e is an empirical constant and is a scale length. Using the value
presented by Orlob (1959) for the empirical constant, e = 4.53 x 10~4 m2/3 s"1, it can
be shown that the value for Aht shown in Table 8.3 is characteristic of processes with
horizontal scale lengths of i = 1.81km, which is just slightly smaller that the grid
interval Ax = 2 km. Thus, the value used for Aur is reasonable for representing the
effect of subgrid scale motion on horizontal mixing. The reference value for vertical
dyne/cm**2
104
Wind Stress at Platform C> tau x (MODEL)
Velocity at Platform C> 3D, In t er po 1 a i ed.l Ar m 1) W/0 Temp
Julian Day
Figure 8.10: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation
120
Wind Stress
1 dyne/cm2
Station E
Figure 8.24: Stick Diagram of wind stress, measured currents, and simualted currents
at Station E.
82
LEGEND
1
m/%
m/%
m/%
m/%
m/%
V
WIND ROSE AT STATION C
Figure 8.2: Wind rose at Station C
45
(4.18)
(4.19)
where c and c are empirical drag coefficients.
An additional equation that must be satisfied is the continuity equation:
d( dU dV
4 1 = (
dt dx dy
4.2.4 Dimensionless Equations in Curvilinear Grids
(4.22)
The above dimensional equations were presented to illustrate the development of
the vegetation model. In the curvilineargrid model, however, dimensionless equations
in curvilinear grids are solved. These dimensionless equations are presented in the
following in terms of the contravariant velocity components in two layers:
dU[
dt
+ Â£i[s,11Ci + s^C?]
912
+ (Horizontal Diffusion)^ (f/f, U^)
+ [Nonlinear Terms(Â£/f, )]
(4.23)
dU?
dt
3ll Tji 321 JJD * P*
'U1 7=Ui + Tirt ~ Tbrt *
Vd~o~l '9,?
+ (Horizontal Diffusion)^ (Â£/f, U^)
Ro
IT
ctf
+
[Nonlinear Terms (t/f, i/^J
(4.24)
163
W/O TEMP
WITH TEMP
U
V
u
V T
0 HR
]
\
1 HR
[
7
2 HR
I
!
3 HR
I
'
4 HR
I
Â¡
1
5 HR
f
SCALE
10 cm/sec
5 C
1 dyne/cm**2
Figure 8.62: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
201
Sheng, Y. P. and S. S. Chiu, 1986: Tropical cyclone generated currents. Proceedings
of the 20th International Conference on Coastal Engineering, American Society
of Civil Engineers, Taipei, Taiwan, pp. 737751.
Sheng, Y. P. and H. K. Lee, 1991a: The effect of aquatic vegetation on winddriven
circulation in Lake Okeechobee. Report No. UFL/COEL91019, Coastal and
Oceanographic Engineering Department, University of Florida, Gainesville, FL.
Sheng, Y. P., and H. K. Lee, 1991b: Computation of phosphorus flux between
the vegetation area and the open water in Lake Okeechobee. Report No.
UFL/COEL91022, Coastal and Oceanographic Engineering Department, Uni
versity of Florida, Gainesville, FL.
Sheng, Y. P. and W. Lick, 1972: Winddriven currents in a partially icecovered
lake. Proceedings of the 16th Conference on Great Lake Research, pp. 10011008.
Sheng, Y. P. and W. Lick, 1980: A two mode freesurface numerical model for the
threedimensional timedependent currents in large lakes. U.S. Environmental
Protection Agency Report EPA 600/380047.
Sheng, Y. P. and S. J. Peene, 1992: Circulation and its effect on water quality in
Sarasota Bay. In Framework for Action, Sarasota Bay, Sarasota Bay National
Estuary Program, Sarasota, FL.
Sheng, Y. P. and C. Villaret, 1989: Modeling the effect of suspended sediment
stratification on bottom exchange processes. Journal of Geophysical Research,
94, pp. 1442914444.
Sheng, Y. P., J. K. Choi and A. Y. Kuo, 1989b: Threedimensional numerical
modeling of tidal circulation and salinity transport in James River Estuary.
Estuarine and Coastal Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 209218.
Sheng, Y. P., J. K. Choi and P. F. Wang, 1989c: A threedimensional numer
ical model of the James River and Hampton Road estuarine system. Report
No. UFL/COEL89020, Coastal and Oceanographic Engineering Department,
University of Florida, Gainesville, Florida.
Sheng, Y. P., D. E. Eliason and X. J. Chen, 1993: Modeling threedimensional
circulation and sediment transport in lakes and estuaries Estuarine and Coastal
Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 105115.
Sheng, Y. P., H. K. Lee and K. H.Wang, 1989a: On numerical strategies of estu
arine and coastal modeling. Estuarine and Coastal Modeling, (M.L. Spaulding,
ed.), A.S.C.E., pp. 291301.
Sheng, Y. P., S. Peene and Y. M. Liu, 1990: Numerical modeling of tidal hydro
dynamics and salinity transport in the Indian River Lagoon. Florida Scientist,
53, pp. 147168.
Sheng, Y. P., D. E. Eliason, X. J. Chen and J. K. Choi, 1991a: A three
dimensional numerical model of hydrodynamics and sediment transport in lakes
and estuaries: Theory, model development and documentation. Final Report to
U.S. Environmental Protection Agency, Coastal and Oceanographic Engineer
ing Department, University of Florida, Gainesville, FL.
dyne/cm**2
177
Wind Stress at Platform A tau x (MODEL)
Velocity at Platform A* 3D, Interpolated(Arm 1) Inverse
Jullan Day
Figure A.l: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
54
Step 8. If error is within error limit (0.5 C), K and Te are correct estimated
values.
An actual equilibrium temperature file was created using SFWMD data, which
were measured at 15minute intervals. Wind speeds at Station A,B,C,D,E were used
for the computation.
5.2.10 Modification of the Equilibrium Temperature Method
By using the equilibrium temperature method, modelpredicted temperature in
Lake Okeechobee was found to be unrealistic. Therefore, b of evaporation formula was
multiplied by a factor of 0.1. Further, K, the heat exchange coefficient, at Station C
was multiplied by 10 to give stronger stratification.
There are many uncertain empirical coefficients in the computation of an equilib
rium temperature. First, evaporation data are averaged daily, but model time step
is 5 minutes. That means the estimation of evaporation data at a short interval is
difficult. Second, wind speed is also averaged daily, and evaporation is correlated with
this average wind speed. In actual computation, wind speed at 15minute intervals
was used. Surface water temperature data are uncertain. Considering the sharp gra
dient of water temperature that usually exists near the water surface, the error can be
large. Third, all the meteorological data used are from L006 station. Considering the
spatial variation of meteorological condition over the lake, error can be large. Most
other thermal models simulate the longterm variation of temperature with a time
step of one day. Therefore, it seems that a shortterm variation of meteorological
data did not create serious problems.
5.3 The Inverse Method
When there are insufficient meteorological data, the errors in the estimation of
total heat flux at the airsea interface can be large. To better estimate the total flux,
the socalled inverse method (Gaspar et al., 1990) was used in this study.
Total flux (qt) can be divided into two parts: solar (q,0Â¡aT) and nonsolar {qnonsoiar)
CHAPTER 8
MODEL APPLICATION TO LAKE OKEECHOBEE
8.1Introduction
Before the description of the application of CH3D to simulate the winddriven
circulation in Lake Okeechobee, it is worth investigating the characteristics of the
lake.
8.1.1 Geometry
Lake Okeechobee, located between latitudes 2712'N and 2640iV and longitudes
8037 Hr and 8108 VK, is the largest freshwater lake in America, exclusive of the
Great Lakes. With an average depth of approximately 3m, and the deepest part less
than 5m deep, the basin is shaped like a very shallow saucer. The western part of
the lake contains a great deal of emergent and submerged vegetation. According to
satellite photos, marsh constitutes 24% of the lake surface area.
8.1.2 Temperature
Due to the location of the lake in subtropical latitude, the annual fluctuations
of water temperatures are relatively small. The mean lake temperature based on
SFWMD monitoring in the 1970s and 1980s ranges from 15C to 34C' (Dickinson et
al., 1991).
8.2Some Recent Hydrodynamic Data
During the fall of 1988 and the spring of 1989, field data were collected by the
Coastal and Oceanographic Engineering Department, University of Florida (Sheng et
al., 1991a). Details of the field experiments and field data are described by Sheng
et al. (1991a). For completeness, some 1989 field data are described briefly in this
76
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INGEST IEID E5U04JEEI_IWEN7S INGEST_TIME 20171128T19:40:15Z PACKAGE UF00075325_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
dyne/cm**2
190
Velocity at Platform D* 3D, Interpolated(Arm 11 Inverse
JulIan Day
Figure A.14: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
65
{y/9lHvT)+ (\/{hH)itj+lViij+itk[(Tijtk + Tij+ij'it)
1
,Vitj+hkAt
6^ AÂ£
) )(Ti,j+2,j,k + Tij'k)
1 Vj,j+i,kAt
2 AÂ£
At the bottom face of a cell :
When v is positive :
(Tij+1.*
(6.22)
(y/g0HvT). (y/%H)ijVitjtk[(Titj_ltk +
~ i(l C^nTij* ~ TTijij, + Tij2t)
1 VijtkA t
2 AÂ£
(Tijt Tiji,*)]
(6.23)
When v is negative
(VglHvT)_ {y/9lH)ijV{jt[(Tijxjk + Tij'k)
1 ri (Vi,j,kAt
6l 1 AÂ£
)2)(Ti,j+i,k 2 Tijtk + Tijitk)
1 VitjikA t
(Tij'k Tij_itk)}
2 AÂ£
The QUICKEST method treats the trdirection advection term as follows:
(6.24)
d_
da
(.HuT) =
(HuT)+ (HloT)_
A a
(6.25)
At the top of a cell :
When uj is positive :
(HuT)+ + Tijtk)
73
Te = 26 + 10 sin() (7.3)
where p is density of water, Cp is specific heat of water, K is heat exchange coefficient,
Te is an equilibrium temperature in 0 C, and P is period of 24 hours. Also, wind stress
was idealized as shown in Figure 7.4. Time history of currents at all five layers are
shown in Figures 7.4 and 7.5. It is apparent that when thermal stratification is
considered, currents at the surface layer are much stronger during increasing wind
condition because initial momentum is confined to a thinner surface layer.
dyne/cm**2
122
Velocity at Platform A* 3D, Interpo1ated(Arm 1) W/0 Temp
JulIan Day
Figure 8.26: Simulated (solid lines) and measured (dotted lines) currents at Station A
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
55
While incoming solar radiation data are usually available, the nonsolar part is esti
mated by solving the vertical onedimensional temperature equation coupled with the
momentum equation.
5.3.1 Governing Equations
5.3.2 Boundary Conditions
At the free surface
?L (K
dt dzK v dz]
(5.17)
du d du
m/v = Tz(A"a;)
(5.18)
dv d dv.
Tt+fu = Tz(Am)
(5.19)
oz p
(5.20)
A
r\
oz p
(5.21)
A
OZ P
(5.22)
where Kv is eddy diffusivity, Av is eddy viscosity, qt is the total heat flux, and rx and
Ty are wind stresses.
At the bottom
dT_
dz
= 0
(5.23)
Tbx = PCd\Ju\ + V^Ui
(5.24)
Tby = PCd\Ju\ + VÂ¡Vi
(5.25)
28
+
1z\JL(k ^\a.
Pth dx \ H dx J dy
+ H.O.T.
P = p{T, S)
(3.20)
where H is total depth, /? = gZr/X2f2 and H.O.T. is higher order terms.
3.5.1 VerticallyIntegrated Equations
The CH3D model can solve the depthintegrated equations and the threedimensional
equations. The vertically integrated momentum equations are obtained by integrating
the threedimensional equations from bottom to top.
(Â£+Â£)
V OX dy J
= 0
(3.21)
dU
dt
H + Tsx Tbx + V
dx
Ro
d_ fUU^ m d_ fUV}
dx{ H ) + dy{ H )
+ E
d_
dx
Ro H2 dp
Fr 2 dx
Hir + Dx
dx
H
mr
dx
+
d ( dU'
dy V H dy,
dV_ rrd(M TT
dt ~ Hdy+Tay Thy U
Ro
d i
dx'
, H ) + dy V H )
+ Eh
r d
< dv\ d (
dV M
dx
{Ah dx ) + dy dy)
Ro H2 dp
Frl 2 dy
= H^ + Dy
dy
(3.22)
(3.23)
200
Schmalz, R. A., 1986: A numerical investigation of hurricaneinduced water level
fluctuations in Lake Okeechobee. Miscellaneous Report CERC8612, U.S. Army
Corps of Engineers.
Schoellhamer, D. H. and Y. P. Sheng, 1993: Simulation and analysis of sediment re
suspension observed in Old Tampa Bay, Florida. UFL/COEL TR /091, Coastal
and Oceanographic Engineering Department, University of Florida, Gainesville,
FL.
Sengupta, S. and W. J. Lick, 1974: A numerical model for wind driven circula
tion and temperature fields in lakes and ponds. FTAS/TR7499, Case Western
Reserve University, Cleveland, OH.
Shanahan, P. and D. R. F. Harleman, 1982: Linked hydrodynamic and biogeo
chemical models of water quality in shallow lakes. Report No. 268, Ralph M.
Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA.
Sheng, Y. P., 1975: Winddriven currents and contaminants dispersion in the
nearshore of large lakes. Technical Report H751, U.S. Army Engineer Wa
terways Experiment Station, Vicksburg, MS.
Sheng, Y. P., 1982: Hydraulic applications of a secondorder closure model of tur
bulent transport, pp. 106119. In (P. Smith, ed.) Proceedings 1982 ASCE Hy
draulic Division Specialty Conference on Applying Research to Hydraulic Prac
tice, American Society of Civil Engineers, Jackson, MS.
Sheng, Y. P., 1983: Mathematical modeling of threedimensional coastal currents
and sediment dispersion: model development and application. Technical Report
CERC832, U.S. Army Eng. Waterways Experiment Station, Vicksburg, MS.
Sheng, Y. P., 1984: A turbulent transport model of coastal processes. Proceed
ings 19th International Conference on Coastal Engineering, American Society
of Civil Engineers, Houston, TX, pp. 23802396.
Sheng, Y. P., 1986: CH3D: A threedimensional numerical model of coastal and
estuarine circulation and transport in generalized curvilinear grids. Technical
Report No. 587, Aeronautical Research Associates of Princeton, Princeton, NJ.
Sheng, Y. P., 1987: Onmodeling threedimensional estuarine hydrodynamics.
ThreeDimensional Models of Marine and Estuarine Hydrodynamics, (J.Nihoul,
ed.), Elsevier Oceanography Series, Elsevier, pp. 3554.
Sheng, Y. P., 1990: Evolution of a threedimensional curvilineargrid hydrodynamic
model for estuaries, lakes and coastal waters: CH3D. Estuarine and Coastal
Modeling, (M.L. Spaulding, ed.), A.S.C.E., pp. 4049.
Sheng, Y. P., 1993: Hydrodynamics, sediment transport and their effects on phos
phorus dynamics in Lake Okeechobee. Nearshore and Estuarine Cohesive Sedi
ment Transport, (A.J. Mehta, ed.), American Geophysical Union, pp. 558571.
Sheng, Y. P. and H. L. Butler, 1982: Modeling coastal currents and sediment
transport. Proceedings 18th International Conference on Coastal Engineering,
American Society of Civil Engineers, Cape Town, Republic of South Africa, pp.
11271148.
70
where p is density of water, y is surface elevation, tw is wind stress, and h is water
depth. Using the same rectangular grid in the seiche test, a uniform wind stress of
1 dyne/cm2 from east to west was imposed. After 48 hours, a steady state is reached.
As shown in Figure 7.2, the surface elevation has a setup in the western part and
setdown in the eastern part. The setup across the lake is 1.12cm, which is exactly
the same as given by the analytical solution Eqn. 7.1.
7.3 Effect of Vegetation
In order to investigate the ability of vegetation model to represent the effect of
vegetation, CH3D was applied to a rectangular lake with a constant depth of 1 m and
horizontal dimensions of 4 km by 9 km. At first, vegetation was not considered, and
a wind stress of 5 dyne/cm2 was imposed. Then, vegetation canopy with width of 1
cm and density of 500 stalks / m2 was added to the western half of the lake. After
that, vegetation density was increased to 5000 stalks / m2. Vegetation height was
assumed to be the same as water depth.
With a time step of 5 minutes, the model was run for 24 hours. Time history
of surface elevation in the northern end of the vegetation area was plotted in Figure
7.3. As expected, surface elevation rises slowly for the second case and reaches steady
state after 5 hours. With highdensity vegetation, surface elevation rises at a slower
rate compared to the second case. When wind blows uniformly over long time, vege
tation effect disappears and reaches steady state. Additional resistance term due to
vegetation becomes smaller because the currents also become smaller at the steady
state and wind stress, and pressure gradient and bottom friction are balanced.
7.4 Thermal Model Test
The purpose of this test is to demonstrate how the velocity can be changed with
the consideration of thermal stratification. Surface heat flux was idealized using the
sine function as follows:
pCrK~ = K(T r.)
(7.2)
67
i2 d\HT)
9 dCdrj
9ij[{HT)i+i/2,j+i/2,k (HT)i+i/2i/2,k
(HT)i_il2J+i/2,k + (HT)i_1/2Ji/2,k]/AÂ£AT1
,22
d2{HT)
drj2
a22
(HT)iJ+lJt 2(HT)iJtk + mju
(Arj)2
(6.31)
(6.32)
12
where q, is the heat flux, K is heatexchange coefficient,is vertical eddy diffusivity,
cp is specific heat of water, p is density of water, Te is an equilibrium temperature,
and Ta is a surface water temperature. They assumed that the annual variation in
heat flux can be approximated by the cyclic form of the equilibrium temperature:
T' = Te + asin(ut + ) (2.8)
Following Munk and Anderson (1948), the eddy diffusivity Kh was expressed as the
product of the eddy diffusivity under neutral condition and a stability function, which
is one under neutral condition but becomes less than one under stable stratification
(positive Richardson number).
Price et al. (1986) studied the diurnal thermal cycle in the upper ocean using
field data and a vertical 1D thermal model. Their measured data include currents,
temperature, and salinity, as well as meteorological data. Field data were collected
between April 28, 1980, and May 24, 1980, at about 400 km west of San Diego,
California.
Their major findings are the trapping depth of the thermal and velocity response
is proportional to r Q1/2, the thermal response is proportional to Q3/2, and the diurnal
jet amplitude is proportional to Q1^2, where Q is the heat flux and r is the wind stress.
They also simulated the diurnal thermal cycle using the vertical onedimensional heat
equation coupled with the momentum equations.
Gaspar et al. (1990) determined the latent and sensible heat fluxes at the air
sea interface using the inverse method. They stated that the total heat flux can
be divided into a solar part and a nonsolar part. While the solar radiation data
is usually available from direct measurement, the nonsolar part is usually indirectly
estimated from the meteorological data. However, this estimation of the nonsolar part
involves many empirical formulas and may contain large errors. Gaspar et al. (1990)
found that, by using the measured temperature data and solving the vertical one
dimensional momentum equation and temperature equation, it is possible to estimate
Wind Stress at Platform C* tau x (MODEL)
Velocity at Platform C 3D, InteroolatedlArm 2) W/ Temp
Jutlan Day
Figure 8.39: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction) when thermal effect is considered.
dyne/cm**2
117
Wind Stress at Platform E< tau y (MODEL)
f
Figure 8.21: Simulated (solid lines) and measured (dotted lines) currents at Station E
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
(degree)
158
Temperature at Platform A i MODEL
147. 149. 151. 153. 155. 157. 159. 161
Julian Day
Figure 8.57: Simulated and measured temperatures at Station A.
dyne/cm**2
180
Velocity at Platform B 3D, InterpoI ated(Arm 1) Inverse
Julian Day
Figure A.4: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 1: NorthSouth direction). Inverse method was used for the estimation of heat
flux.
91
A
<
/ Hv
(8.12)
A
<
Qcut &q2
(8.13)
A
<
<7
N
(8.14)
where fza is usually within the range of 0.1 to 0.25, H is the total depth, Hp is the
depth of the pycnocline, Qcut is a coefficient between 0.1 and 0.25, 6q2 is the spread
of turbulence determined from the q2 profile, and N is the BruntVaisala frequency
defined as
(8.15)
Some of the parameterization of turbulence is controlled by userinput in the
CH3D code. The model constants A, b, and s, however, are invariant. Besides the
vertical turbulence reference value AVr described above in Table 8.3, the vertical turbu
lence parameters which are useradjustable are Amtn (minimum turbulent macroscale),
Qcut (fraction of turbulence spread), fZ3 (fraction of depth), AVmin (minimum eddy
viscosity), and AVmaz (maximum eddy viscosity). The values of the vertical turbu
lence parameters which were used to simulate the circulation in Lake Okeechobee
during spring 1989 circulation (see Table 8.4) are those which gave the best results
from an independent onedimensional vertical model of flow and sediment for Lake
Okeechobee at Station C during the same time period.
8.4.3 Bottom Friction Parameters
Momentum imparted at the water surface by wind stress is dissipated from the
water column by bottom friction. Bottom friction in CH3D is in the form of a
quadratic bottom stress law:
Ti* = pcduiyjgnu\ + 2gi2uivi + g22vj
nr, = pCd.V\\jgnu\ + 2guiiiVi + g22vl (8.16)
where are contravariant velocities at the first grid point above the bottom, and
cj is an empirical drag coefficient. Depending on the value of the flag for bottom
15
turbulence mixing in the upper layers may be significantly enhanced. In such case,
the length scale A0 throughout the upper layers may be assumed to be equal to the
maximum value at middepth (Sheng, 1983).
When a lake is stratified, vertical turbulence is affected by buoyancy induced
by the vertical nonhomogeneity. In this situation, vertical eddy coefficients should
be modified to account for this effect. This is parameterized by introducing the
Richardson number:
Ri =
p dz
(2.12)
Ri is positive when flow is stable (ff < 0) and when Ri is negative when flow is
unstable (^ > 0). Generally, eddy viscosity and eddy diffusivity are expressed as
follows:
Av = AV02(Ri) (2.13)
where 2 are stability functions and Avo and Kvo are eddy viscosity and eddy
diffusivity when there is no stratification. Stability functions have the following forms:
l = {l + 2 = (1 + a2Ri)mi (2.14)
Based on comparing model results with field data, Munk and Anderson (1948) devel
oped the following formula:
i = (1 + 10ifc)"1/2; 2 = (1 + 3.33 Ri)'3'2 (2.15)
Many similar equations with different coefficients were suggested based on numerous
sitespecific studies. These coefficients, however, are not universal, and care must be
taken when applying these formlete to a new water body where little data exist.
2.4.4 Reynolds Stress Model
One can obtain an equation for the timeaveraged secondorder correlations by
following the procedure: (i) decompose the dependent variables into mean compo
nents and fluctuating components, (ii) substitute the decomposition into continuity,
92
Table 8.4: Vertical turbulence parameters used in the Lake Okeechobee spring 1989
circulation simulation.
Parameter
Value
Amin
0.5
Qcut
0.2
fzs
0.2
A
0.5
100
friction, the drag coefficient is either set to a constant specified by the input value
CTB, or a variable according to the law of the wall. For the simulation of the Lake
Okeechobee spring 1989 circulation, a variable drag coefficient was used according to
the law of the wall as
where k = 0.4 is von Karmans constant, z\ is the height of the first grid point
above the bottom, and z0 is the roughness height. In CH3D, the roughness height
is specified as the input parameter BZ1. For the Lake Okeechobee simulation, the
roughness height was taken to be z0 = 0.1 cm.
8.4.4 Vegetation Parameters
There are more than 25 kinds of vegetation in the littoral zone. Each type of
vegetation has a different diameter and height. For simplicity, in this study it was as
sumed that all types of vegetation have cylindrical stalks. Table 5 shows the various
types of vegetation found in Lake Okeechobee (J.R. Richardson, personal commu
nication, 1991). To represent the vegetation distribution accurately, the concept of
equivalent vegetation density is introduced. The equivalent vegetation density is
defined as the equivalent number of stalks with 1 cm diameter per unit horizontal
43
W + (tt) + ) (Â£) + si.C, = M + n Fw)
d
r. dv;l
d
\ dVi
dx
1
Â¡5
ca
H
1
+ dy
Ah a 
L dy J
(4.7)
where L\ = h i and U\ and V\ are the verticallyintegrated velocities within the
vegetation layer:
(U1,vl)= f (u,i,Vi)dz (4.8)
J h
Tf,r and Tby are bottom stresses, rtx and r)y are interfacial stresses between layer I and
layer II, and and are form drags due to the vegetation canopy.
4.2.2 Equations for the VegetationFree Layer (Layer II)
du2 du\ du2v2 du2w2 1 dp2 d
dt dx dy dz V2 pdx+dx
Ah
du2
dx
d_
+ dy
A
du2
d
dy dz
a du2
Avz
az
(4.9)
dv2 du2v2 dv\ dv2w2 1 dp2 d
+ ~aT~ + a7 + ~5T =
A dV2
A~F~
dx
d
a dv2
d
a dv2
+ dy
AH~~
[ dy J
+ d~z
Av~d7
(4.10)
Integrating Equation (4.9) and (4.10) vertically from 2 = to z = ( and defining
L2 i +
("Tjx Tjx)
P
+
d
\a9U2 1
d
4
\a9UA
dx
OX
dy
1 ^ Cl
L dy
(4.11)
dV2
dt
d (U2V2\ d (V?\ T rrr K
+ di \~lt) + Ty (17) + +(rTiy)
d
\a dV2]
d
\ t dV2
dx
Ahw1
dx
+ dy
Ah p
L dy
(4.12)
where U2 and V2 are verticallyintegrated velocities within the vegetationfree layer:
(U2,V2) = J ^(u2,v2)dz (4.13)
25
P = 5890 + 387 0.37572 + 3S
(3.8)
a = 1779.5 + 11.257 0.074572 (3.8 + 0.017)5 (3.9)
where T is water temperature in degree C, S is salinity in ppt, and p is density in
g/crn3.
Besides governing equations, boundary conditions should be specified.
3.2.2FreeSurface Boundary Condition (z = tj)
(1) Kinematic boundary condition:
dt] dq dn dn
W = f u h V ~
dt dx dy dt
(3.10)
(2) Surface heat flux:
q = Kv^ = K{TsTe) (3.11)
where Ta is the lake surface temperature, Te is the equilibrium temperature, K is a
heat transfer coefficient, and q is positive upward. (3) Surface stress:
Av
du
~d~z
(3.12)
where the wind stresses tx and ry must be specified.
3.2.3Bottom Boundary Condition (z h)
(1) Heat flux is specified as zero,i.e., = 0
(2) Quadratic bottom friction law is used, i.e.,
= pCdyJua2 + vi2uury
= pCdy/ui2 + v i2ui
(3.13)
where u\ and tq are velocity components at the first grid point above the bottom.
3.2.4Lateral Boundary Condition
(1) Heat flux is assumed zero,i.e., = 0
(2) No flow through boundary,i.e., u 0 or v = 0
CHAPTER 3
GOVERNING EQUATIONS
3.1 Introduction
This chapter presents the basic equations which govern the water circulation in
lakes, reservoirs, and estuaries. Because the details can be found in other references
(e.g., Sheng, 1986; Sheng, 1987; Sheng et.al., 1989c), the governing equations are
presented here without detailed derivations.
3.2 Dimensional Equations and Boundary Conditions in a Cartesian Coordinate System
The equations which govern the water motion in the water bodies consist of
the conservation of mass and momentum, the conservation of heat and salinity, and
the equation of state. Because Lake Okeechobee is a fresh water lake, the salinity
equation is not considered. The following assumptions are used in the Curvilinear
Hydrodynamic Threedimensional Model (CH3D) model.
(1) Reynolds averaging: Three components of velocity, pressure, and temperature
are decomposed into mean and fluctuating components and timeaveraged.
(2) Hydrostatic assumption: Vertical length scale in lakes is small compared to
the horizontal length scale, and the vertical acceleration is small compared with the
gravitational acceleration.
(3) Eddy viscosity concept: After timeaveraging, the secondorder correlation
terms in the momentum equation are turbulence stresses, which are related to the
product of eddy viscosity and the gradient of mean strain.
(4) Boussinesq approximation: Density variation of water is small, and variable
density is considered only in the buoyancy term.
23
195
Wind Stress at Platform Ei tau x (MODEL)
Velocity at Platform E> 3D, InterpoIatedI Arm 2) Inverse
Julian Day
Figure A.19: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
L
63
+
1 +
Hn
Ev At ( Dv+ Dv_
(Hn+'yPrvAak VAcr+ + A
'Tn+l
1 >jA
TJ1,
At R0
Hn+1 ij* ~ Hn+l Jg~o
Hn Eh At f n d2T
Hn+1 prH v dt2
6.4.1 Advection Terms
HvTy" + ^A{HwTr
+
+ 2gn
d2T
d^drj
+9
d2T\n
dr,2)
(6.14)
Many different schemes can be used for the advection terms. When there is a
sharp discontinuity, it is difficult to model the convection without numerical diffu
sion. Leonard (1979) introduces the QUICKEST( Quadratic Upstream Interpolation
for Convective Kinematics with Estimated Stream Terms), which gives good results
without excessive numerical diffusion.
This QUICKEST scheme treats the advection terms in the ^direction as follows:
( /7THnT\ (V^HuT)+ IV9~qHuT)_
(6.15)
where the first term in the right hand side is the flux at the right face of the cell
and the second term is at the left face. These two terms are differenced differently
depending on the direction of current as follows:
At the right face of a cell :
When u is positive :
(yg~0HuT)+ (y/gÂ¡H)i+i,jUi+itjik[(Tiijk + 7i+i,,fc)
1(1 _iy){T.+ut _2T..t +
1 Ui+i,j,kAt
AÂ£
(6.16)
When u is negative
(y/9oHuT)+ (\/^i7){+i (^',j,fc + TJ+ij^)
dyne/cm**2
126
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D 3D. Interpolated(Arm 2) W/0 Temp
Jullan Day
Figure 8.29: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction).
v(cm/secl dyne/cm**2
146
Wind Stress at Platform Ei tau y (MODEL)
Figure 8.48: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: NorthSouth direction) when thermal effect is considered.
24
3.2.1 Equation of Motion
With above assumptions, the equations of motion can be written in a righthanded
Cartesian coordinate system as follows:
du dv dw
dx dy dz
(3.1)
du du2
dt ^ dx
+
duv duw 1 dp d f du'
dy + dz V pQ dx A dx \ H dx /
d (A du
ay \ H dyt
a (A du'
+ dz [AvdzJ
(3.2)
dv duv
dt A dx
dv2 dvw 1 dp d f dv'
dy + dz U p0dyJrdx\Hdx/
d( dv\ d( dv\
+ dH{A"a7j) +& (/'&)
(3.3)
dp
dz
= pg
(3.4)
dT duT dvT dwT d_ ( dT'
dt "** dx + dy + dz dx\HdxJ
, d / dT\ d / dT\
dy \H dy) + dz V v dz)
P = P(T,S)
(3.5)
(3.6)
where (u,v,w) are velocities in (x, y, z) directions, f is the Coriolis parameter defined
as 20 sm. where 1 is the rotational speed of the earth, is the latitude, p is density,
p is pressure, T is temperature, (Ah,Kh) are horizontal turbulent eddy coefficients,
and (riv, Kv) are vertical turbulent eddy viscosities.
For the equation of state, Eqn. 3.6, there are many different formulae that can be
used. For the present study, the following equation given by Eckart (1958) is used:
p = (1 + P)/(a + 0.698P)
(3.7)
167
It is thus important to quantify the sensitivity/uncertainty of model results while
comparing them with field data. Sensitivity analysis may reveal those parameters to
which the model is particularly sensitive. If a slight variation in a particular model
parameter produces a significantly different result, then that particular parameter
should be determined precisely. In contrast, those parameters to which the model is
less sensitive need not be estimated as accurately. As shown in Tables 8.16 and 8.17,
Table 8.16: Parameters used in sensitivity tests.
Case
Law of
the
wall
Zq
cm
Drag
Coeff.
Ah
cm2/s
Second
order
closure
Munk
Anderson
Av
cm,2/s
Smooth
bottom
Advec
tion
1
Yes
0.1
No
10s
Yes
No
No
No
No
2
No
No
0.001
105
Yes
No
No
No
No
3
No
No
0.004
105
Yes
No
No
No
No
4
Yes
0.1
No
104
Yes
No
No
No
No
5
Yes
0.1
No
103
Yes
No
No
No
No
6
No
No
0.01
105
Yes
No
No
No
No
7
Yes
0.1
No
105
No
Yes
No
No
No
8
Yes
0.1
No
105
No
No
20
No
No
9
Yes
0.5
No
105
Yes
No
No
No
No
10
Yes
0.01
No
105
Yes
No
No
No
No
11
Yes
0.1
No
10s
Yes
No
No
Yes
No
12
Yes
0.1
No
105
Yes
No
No
No
Yes
12 test runs were performed. Model performance was measured by use of an index
of agreement between simulated currents and measured currents at three different
heights at Station C.
8.9.1 Effect of Bottom Stress
Five tests were performed by using the different formula and coefficients. Bottom
stress was expressed as a quadratic law. Different constant drag coefficients (0.001,
0.004, and 0.01) were used (cases 2, 3, and 6, respectively). Compared with case 1,
cases 2 and 6 were not very sensitive, but the index of agreement dropped substantially
with use of Cd = 0.004. This is only in the eastwest direction. Northsouth currents
115
direction was almost the same at both arms.
Average current speeds at arm 2 were 4 cm/sec in the x direction and 7 cm/sec in
the y direction. Average speeds at arm 1 were 3 cm/sec and 6 cm/sec, respectively.
On Julian Day 157, there was a short period of strong wind which created a strong
seiche. Measured data at both arms clearly showed this seiche.
Arm 2 was at a height of 76% of the total depth. As shown in Figure 8.22, time
series of the model results indicated that u has a negative trend and v has a positive
trend. Therefore, it appears that the model could simulate the longterm trend of
flow direction correctly.
Arm 1 was at a height of 24% of the total depth. As shown in Figures 8.20 and
8.21, model could simulate the currents in terms of direction and magnitude.
Figure 8.24 shows the stick diagram of wind stress, the model results, and the
field data. It clearly shows flow is generally from the southeast to northwest, which
is parallel to the boundary between the vegetation and the open water. Model could
simulate this trend very well.
Table 8.8: Index of agreement and RMS error at Station E.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.38
0.40
2.86
5.66
2
0.28
0.29
2.66
4.46
Currents At Station A
One arm was installed at a height of 71 cm above the bottom. Measured data
show that currents respond well to the change in wind stress. Comparing the model
results with field data at arm 1 shows that the directions agree quite well while the
measured current speeds are underestimated by the model, as shown in Figures 8.25
and 8.26.
Index of agreement and RMS error
xiv
dyne/cm**2
196
Wind Stress at Platform E tau y (MODEL!
Velocity at Platform E* 3D, Intarpolated(Arm 2) Inverse
JulIan Day
Figure A.20: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
vlcm/sec) dyne/cm##2
134
Velocity at Platform B* 3D, InterpolatedlArm 2) W/ Temp
Julian Day
Figure 8.36: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 2: NorthSouth direction) when thermal effect is considered.
62
,ii
(Hv)n+1 = (Hv)n At? Hnvn+1
\ZlTo
a21
 At^Hn+1vn+1 + At(G3 G2)n
y/fo
4* A t
ev d
(.HnY da
('T*v ~ n)B+1
Av^(Hn+1vn+1)
(6.10)
For the open water zone, the above internal mode equations are solved after the
external mode solutions are obtained. For the vegetation zone, no internal mode
equations are solved. This is consistent with the assumption that, in the vegetation
zone, the velocities are fairly uniform within the vegetation layer and the vegetation
free layer.
6.4 Temperature Scheme
This section describes the finite difference equation which is used for solving the
temperature equation. Equation 3.53 is written in finite difference form using the
forward scheme in time and the centered difference in the vertical diffusion term.
jyn+lyn+l jjnrpn
t)jyk ,j tk
A tR0
y/9o
HuT,/Â£)
(6.11)
+
+ A{HvT^ +
Ev(At) 1
#B+1rc. Aak
Dv
D
+
A cr_
HEjj 2At
Aa+
(rfc'v7T+1)
,11
d2T
+ 2g
12
d2T
+ 9
22
d2T'
(6.12)
 CH
\ dÂ£2 dÂ£drj drÂ¡2 /
Dividing the above equation by Hn+1 and collecting all the unknown terms in the
lefthand side and the known terms in the righthand side, and writing advection
terms and diffusion terms separately,
EvAt
Ey rpn+l A+_mn+l
vA
(H*+')2Prm Aak
(6.13)
v{cm/sec) dyne/cm**2
130
Velocity at Platform A 3D. Interpo 1 ated(Arm 1) W/ Temp
Ju1 Ian Day
Figure 8.32: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: NorthSouth direction) when thermal effect is considered.
5.9 Steadystate surface elevation contour (cm) in Lake Okeechobee
forced by an easterly wind of 1 dyne/cm2 101
8.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 104
8.11 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 105
8.12 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 106
8.13 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 107
8.14 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction). Thermal stratification
was not considered in model simulation 108
8.15 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction) 109
8.16 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 110
8.17 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation Ill
8.18 Simulated (solid fines) and measured (dotted fines) currents at
Station B (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 112
8.19 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 113
8.20 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 116
8.21 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 117
8.22 Simulated (solid fines) and measured (dotted fines) currents at
Station E (Arm 2: EastWest direction). Thermal stratification
was not considered in model simulation 118
vii
103
When the wind is light, the measured currents are usually small. As the wind increases
in the afternoon, the surface currents follow the wind direction. However, when
the wind speed reduces, the setup is released and the currents change the direction.
Therefore, general time history of measured currents have two peaks: a positive peak
when the flow is from west to east during increasing easterly wind and a negative
peak when the flow is from east to west during decreasing easterly wind.
While the model could simulate the positive peaks well, it missed many negative
peaks as shown in Figure 8.14. As it will be shown later, the reason for this is that
the model did not consider the thermal effect, and hence the eddy viscosity was not
calculated correctly. A reduction of the time step did not lead to improved model
results.
The index of agreement of this model simulation is shown in Table 8.6. The
average index is 0.75 and 0.66 for the eastwest currents and northsouth currents,
respectively.
Table 8.6: Index of agreement and RMS error at Station C.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.78
0.69
1.98
1.46
2
0.74
0.67
2.47
1.69
3
0.73
0.62
2.12
2.01
Currents At Station B
Wind direction is mainly toward the west during the two weeks. Measured cur
rents at both arms show that the direction is toward the southwest. Arm 1 was at
16% of the total depth above the bottom. The average current at arm 1 is 3 cm/sec.
A comparison of the model results with field data at arm 1 shows that the directions
agree well. As shown in Figures 8.16 and 8.17, the model could simulate the time
of peak currents well. There is a discrepancy of approximately 1 cm/sec between
the simulated and measured average currents. Considering the accuracy of current
166
10 HR
19 HR
20 HR
21 HR
22 HR
23 HR
W/0 TEMP
WITH TEMP
Figure 8.65: Vertical profiles of currents and temperature during a typical day. Base
temperature is 25 C.
u(cm/s) dyne/cm**2
152
Wind Stress (EastWest) at Station C
Julian Day
Measured Currents at Station C
Julian Day
Figure 8.52: Time history of wind stress and measured currents between Julian days
150 and 152.
123
Table 8.9: Index of agreement and RMS error at Station A.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.39
0.37
3.73
2.07
Currents At Station D
Arm 1 was at a height of 20% of the total depth. The model results agree well
with field data at arm 1 in terms of the flow direction while the model underestimates
the peak currents (Figure 8.27 and 8.28). The model results at arm 2 shown in Figures
8.29 and 8.30 agree well with field data in terms of the flow direction.
Table 8.10: Index of agreement and RMS error at Station D.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.42
0.68
2.22
2.16
2
0.44
0.62
2.11
3.13
8.7 WindDriven Circulation with Thermal Stratification: Te Method
To improve the model performance, vertical thermal stratification is considered.
At first, the equilibrium temperature method is used and equilibrium temperatures
and heat exchange coefficients were estimated by the method described in Chapter 5.
These estimated equilibrium temperatures at Station C are then assumed to be the
same at all other grid points.
LongTerm Dynamics
Figures 8.31 through 8.50 show the simulated currents and field data. Tables
8.11 through 8.15 show the index of agreements.
As shown in Table 8.13, the model results with thermal effect agree well with field
data. During the daytime, the lake becomes thermally stratified. The temperature
difference between Arm 1 and Arm 3 at Station C can sometimes becomes 3C. This
small temperature difference causes the density of water to vary vertically. Although
56
where c is the drag coefficient and Ui and v\ are velocities at the lowest grid point
above bottom.
Total flux qt cannot be specified a priori because of unknown nonsolar flux,
Qncmaolar Therefore, a value of qnonsoiar is first guessed and then corrected until the
calculated and measured water temperatures are within an error limit. In this way,
the total flux can be determined by summing up the solar and nonsolar parts. This
total flux was later used as a boundary condition of temperature equation in the
threedimensional simulation.
5.3.3 FiniteDifference Equation
Treating the vertical diffusion term implicitly, Eq. 5.18 is written in the finite
difference form as follows:
At the interior points,
u?+1 u?
'n + 1 Am A,,!
~M =
Az
A z
At J'' Az
where At is time increasement and Az is vertical spacing.
Applying free surface boundary condition,
u
n+l
U;
At
~ Mm =
A (ti U .
A 2 tm Â¡m1
Az
Where im is the index of surface layer.
Applying bottom boundary condition,
u
At JVl Az
+i
Similar form for 5.19 is as follows:
At the interior points,
vi vi f,.n Az Az
At +JUÂ¡ A Z
Applying free surface boundary condition,
u"+1 _
m tm
At
Za. A (^Irn1 1 )
r n Â£ Az
+1 im Az
(5.26)
(5.27)
(5.28)
(5.29)
(5.30)
Degrees Celsius mLangioy/min
50
Solar Radiation
Julian Day
Figure 5.1: Meteorological data at Station L006.
202
Sheng, Y. P., D. E. Eliason, J. K. Choi, X. J. Chen and H. K. Lee, 1991b:
Numerical simulation of threedimensional winddriven circulation and sediment
transport in Lake Okeechobee during spring 1989. Final Report to the South
Florida Water Management District, Coastal and Oceanographic Engineering
Department, University of Florida, Gainesville, FL, 91p.
Sheng, Y. P., D. E. Eliason, R. E. Dickinson and J. K. Choi, 1991c: A threemonth
simulation of winddriven circulation, sediment transport and phosphorus trans
port in Lake Okeechobee. Report No UFL/COEL91023, Coastal and Oceano
graphic Engineering Department, University of Florida, Gainesville, Florida
Sheng, Y. P., W. Lick, R. T. Gedney and F. B. Molls, 1978: Numerical simulation
of threedimensional circulation in Lake Erie: A comparison of a freesurface
model and a rigidlid model. Journal of Physical Oceanography, 8, pp. 713727.
Simons, T. J., 1974: Verifications of numerical models of Lake Ontario, Part I:
Circulation in spring and early summer. Journal of Physical Oceanography, 4i
pp. 507523.
Sokolnikoff, I. S., 1960: Tensor Analysis, John Wiley and Sons, New York.
Stommel, H., 1949: Horizontal diffusion due to oceanic turbulence. J. Mar. Res, 8,
pp. 199.
Sundaram, T.R., Easterbrook, C. C., Piech, K. R. and Rudinger, G., 1969: An
investigation of the physical effects of thermal discharges into Cayuga Lake.
Report VT261602, Nov. 1969. Cornell Aeronautical Laboratory, Buffalo, NY.
Thomas, J. H., 1975: A theory of steady winddriven currents in shallow water
with variable eddy viscosity. Journal of Physical Oceanography, 5, pp. 136142.
Thompson, J. F., 1983: A boundaryfitted coordinate code for general two
dimensional regions with obstacles and boundary intrusions. Technical Report
E838, U.S. Army Eng. Waterways Experiment Station, Vicksburg, MS.
Tickner, E. G., 1957: Effects of bottom roughness on wind tide in shallow water.
Technical Memorandum No. 95, Beach Erosion Board, Office of the Chief of
Engineers, Corps of Engineers.
Welander, P., 1957: Wind action on a shallow sea: Some generalizations of Ekmans
theory. Tellus, 9, pp. 4552.
Whitaker, R. E., R. 0. Reid and A. C. Vastano, 1975: An analysis of drag coefficient
at hurricane windspeeds from a numerical simulation of dynamical water level
changes in Lake Okeechobee, Florida. Technical Memorandum No. 56, Coastal
Engineering Research Center, Corps of Engineers.
Willmott, C. J, 1981: On the validation of models. Physical Geography, 1981, pp.
184195.
Witten, A. J. and J. H. Thomas, 1976: Steady winddriven currents in a large lake
with depthdependent eddy viscosity. Journal of Physical Oceanography, 6, pp.
8592.
41
Tickners experimental results, Whitaker et al. (1975) calculated c( 1.77) which was
used in this study. Roig and King (1992) showed c< is a function of Froude Number,
Reynolds Number, vegetation height, spacing, and diameter of vegetation. As the
water level changes, the flow regime over a vegetation area may change from one
layer to twolayer flow, and vice versa.
4.2 Governing Equations
Let us consider an x, y, z coordinate system with the velocity components in the
(x, y, z) directions as (u, v, w). The lower layer (layer I) of the water column is covered
with vegetation, while the upper layer (layer II) is vegetationfree (Figure 4.1).
Flow in the vegetation layer (ui, ux) and flow in the vegetationfree layer (u2, v2)
both satisfy the equations of motion.
4.2.1 Equations for the Vegetation Layer (Layer I)
du i du\ duivi duiWi
dt dx dy dz
dv\ du\V\ dv\ dv\w\
dt dx ^ dy ^ dz
_ 1 dpi d
JV1 + TT"
p dx dx
+
d_
dy
a dlLl
Ah~h~
dy
+
dz
1 dpi d
~ ~fu' ~ J aT + di
iff
du\
Ah
dx
dui
dz
dv x
dx
d
. dv i
d
. dvi
+ dy
Ahtt
[ v J
1 ^
co
+
1 N
CO
s
T
dp
Tz
pa
Integrating Equation (4.4) vertically:
P = Pa+ pg(C z)
Integrating Equations (4.2) and ( 4.3) vertically from z = h to z =
dUi
dt
+ h (zr)+5? (tt) + >L,(*=fVl+(tw ~ Fa)
+
d_
dx
Ah
dUi
dx
+
d_
dy
Ah
dUi
dy
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
3
the growth rate of all organic matter in lakes is governed by temperature (Goldman,
1979). The growth rate generally increases between some minimum temperature
and an optimum temperature, and decreases until it reaches maximum temperature.
Cooling water from power plants is mixed with surrounding water by the currents and
turbulent mixing which depend on the temperature field as well. Thus, predicting
currents and temperature are essential to understanding the transport of various
matters and their effects on the ecology.
In the early days, simple analytical models were used to study physical processes
in simplified conditions. For example, a setup equation was used to predict the
storm surge height (Hellstrom, 1941). Since analytical models could not realistically
consider such effects as advection, complex geometry, and topography, they had been
applied to limited problems to understand certain basic processes.
Numerical models are valuable tools for simulating and understanding water
movement in lakes. Once a rigorously developed model is calibrated with measured
data, it can be used to estimate the flow near a manmade structure or to predict
the movement of contaminants including oil spill, sediments, etc. During the 1970s
and 1980s, vertically averaged twodimensional numerical models, which can compute
only the depth average currents and surface elevations, were widely used because they
were simple and needed little computer time. However, since they could not give ac
curate results for cases where the vertical distributions of currents and temperature
are required, threedimensional models are needed.
Numerical modeling requires the discretization of the computation domain. Past
numerical models which were developed during 1970s generally used a rectangular grid
(for example, Sheng, 1975). However, to represent the complex geometry such as the
shoreline and the boundary between the vegetation zone and the open water in Lake
Okeechobee, a very fine rectangular grid is required. On the other hand, boundary
fitted grids can be and have been used in recent models (for example, Sheng, 1987)
CHAPTER 4
VEGETATION MODEL
4.1 Introduction
The western portion of Lake Okeechobee is covered with an extensive amount
of vegetation. The vegetation can affect the circulation in several different ways.
First of all, wind stress over the emergent vegetation is reduced below that over the
open water. Furthermore, the submerged vegetation introduces drag force to the
water column. Because most of the vegetation stalks are elongated cylinders without
large leaf areas, the drag force is primarily associated with the profile drag (or form
drag) instead of the skin friction drag. The profile drag can reduce the flow and is
proportional to the projected area of vegetation in the direction of the flow.
The presence of vegetation also can affect the turbulence in the water column.
The characteristic sizes of the horizontal and vertical eddies generally are reduced by
the vegetation. This usually leads to a reduction of turbulence, although some wake
turbulence may be generated on the downstream side of vegetation.
In order to simulate the effects of vegetation, several approaches have been under
taken in previous investigations. For example, Saville (1952) and Sheng et al. (1991b)
used an empirical correction factor to simulate the reduction of wind stress over the
vegetation area. Sheng et al. (1991b) also adjusted the bottom friction coefficient
over the vegetation area. For simplicity, however, Sheng et al. (1991b) did not include
the effect of vegetation on mean flow and turbulence in the water column, because the
primary focus of that study was the internal loading of nutrients from the bottom sed
iments in the open water zone. Whitaker et al. (1975) developed a twodimensional,
verticallyintegrated model of storm surges in Lake Okeechobee. The profile drag cre
39
LIST OF TABLES
2.1 Selected features of lake models 21
2.2 Application features of lake models 22
8.1 Installation dates and locations of platforms during 1988 and 1989. 79
8.2 Instrument mounting, spring deployment 80
8.3 Reference values used in the Lake Okeechobee spring 1989 circu
lation simulation 89
8.4 Vertical turbulence parameters used in the Lake Okeechobee spring
1989 circulation simulation 92
8.5 Vegetations in Lake Okeechobee (From Richardson, 1991) 94
8.6 Index of agreement and RMS error at Station C 103
8.7 Index of agreement and RMS error at Station B 114
8.8 Index of agreement and RMS error at Station E 115
8.9 Index of agreement and RMS error at Station A 123
8.10 Index of agreement and RMS error at Station D 123
8.11 Index of agreement and RMS error at Station A when thermal
effect is considered 149
8.12 Index of agreement and RMS error at Station B when thermal
effect is considered 149
8.13 Index of agreement and RMS error at Station C when thermal
effect is considered 149
8.14 Index of agreement and RMS error at Station D when thermal
effect is considered 149
8.15 Index of agreement and RMS error at Station E when thermal
effect is considered 149
8.16 Parameters used in sensitivity tests 167
xiii
21
Table 2.1: Selected features of lake models.
Author
Dimensionality
Type of
Model
Temporal
Dynamics
Turbulence
Advection
Welander
1957
3D
AN
T.D.
A
No
Liggett
1969
3D
F.D.
T.D.
A
No
Lee + Liggett
1970
3D
F.D.
S.S.
A
No
Liggett + Lee
1971
3D
F.D.
S.S.
A
No
Gedney + Lick
1972
3D
F.D.
T.D.
A
No
Goldstein + Gedney
1973
3D
A.N.
B
No
Sengupta + Lick
1974
3D
F.D.
T.D.
D
Yes
Simons
1974
3D
F.D.
T.D.
B
Sheng
1975
3D
F.D.
S.S.
A
No
Thomas
1975
3D
F.D.
S.S.
B
No
Whitaker et al.
1975
2D
F.D.
T.D.
Yes
Witten 4 Thomas
1976
3D
F.D.
S.S.
C
No
Lien + Hoopes
1978
3D
F.D.
S.S.
A
No
Schmalz
1986
2D
F.D.
T.D.
Yes
Sheng + Lee
1991a
3D
F.D.
T.D.
E
Yes
* F.D. : Finite difference
* AN : Analytic
* S.S. : Steady state
* T.D. : Time dependent
* A : Constant
* B : Dependent on wind
* C : Exponential form
* D : MunkAnderson type
* E : Simplified secondorder closure model
157
Temperature at C (Model)
Julian Day
Fiffure 8.56: Temperature contours of data and model at Station C between Julian
days 152 and 155.
36
3.9 Dimensionless Equations in BoundaryFitted Grids
Expanding the equation 3.41 and 3.42, the following equations are obtained:
Ct +
^(sfcHu) + FjglHv)
dii
= 0
(3.50)
1 dHu
H~dT
(9lI+s2) + i^+^
Ro
~H
y/9o y/90
O Q
(Huu) + Huv) + (2 D\x + D\2)Huu
+ (3.Di2 + D\2)Huv 4 D\2Hvv +
dHuu>
da
+
K d_
H2 da
Ro
F
' du'
V r\
. da j
 Â£[/;(*%>'%)*
4 Â£//A//(Horizontal Diffusion)
(3.51)
1 dHv
H dt
(
#11 9 21
:U 4 V
,V9~o y/9o
Ro
~H
o C\
(Huv) + (Hvv) 4 D^Huu
4 (Dh 4 iD\2)Huv 4 (D\2 4 2D\2)Hvv
4
E^d_
H2 da
Ro
F,
1 [ft
21 22 dp
9 8(+9 toÂ¡,
da
+ (921^ + 922^)(lpda + aP
4 '//^//(Horizontal Diffusion)
(3.52)
where the horizontal diffusion terms are listed in Sheng(1986). The temperature
equation can be obtained according to the same procedure as
1 dHT Ev d ( dT\
H dt ~ Prv H2da \Kv da)
dyne/cm**2
188
Velocity at Platform Ct 3D, Interpo1 ated(Arm 3) Inverse
JulIan Day
Figure A.12: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
18
been applied to James River (Choi and Sheng, 1993) and Tampa Bay (Schoellhamer
and Sheng, 1993).
Two other turbulence models, which are based on the socalled k e model (Rodi,
1980), are described in the following:
2.4.7 OneEquation Model (k Model)
Using the eddy viscosity/diffusivity concept, the choice of velocity scale can be
y/k, where k = (u2 + v2 + u>2)/2 is the kinetic energy of the turbulent motion. When
this scale is used, the eddy viscosity is expressed as
vt = c'^VkL (2.23)
where is an empirical constant and L is the length scale. To determine k, an
equation is derived from the NavierStokes equation as:
dk dk d ,Xu' P x, rrdu'i ttj du'i du\
m + uaTi = &71"'( V + t)1 ^ "
(2.24)
2 p/J ~'~3 dxj dxjdxj
To obtain a closed set of equations, diffusion term and dissipation term must be
modelled. The diffusion flux is often assumed proportional to the gradient k as
ut dk
<(UiU3 P \
Ui(~ + 71 = ^ ax.
(2.25)
where crfc is an empirical diffusion constant. The dissipation term e, which is the last
term of Eq. 2.24, is usually modelled by the expression
A:3/2
e = Cjy
(2.26)
The length scale, L, needs to be specified to complete the turbulence model. Usually
L is determined from empirical relations.
2.4.8 TwoEquation Model (k e Model)
To avoid the empirical specification of length scale, another equation for the
dissipation e is needed. Then eddy viscosity and eddy diffusivity are expressed as
k2
i>t cM
(2.27)
114
meters, this difference is considered negligible. However, the peak values of model
currents are somewhat smaller than the field data, as it was found for Station C.
Arm 2 wras at a height of 75% of the total depth. A comparison of the model
current with field data at arm 2 shows generally good agreement in current direction
over two weeks (Figures 8.18 and 8.19). Model currents are somewhat stronger
than the field data. The agreement between the model currents and field data in
the northsouth direction becomes worse after Julian Day 154. Considering that the
wind becomes strong in the northsouth direction, the field data remains from north
to south. This seems to indicate that data have drifting problems.
Both the measured and simulated currents at Station B indicate that flow is gen
erally in the southwestnortheast direction, which is parallel to the boundary between
the vegetation and the open water.
Table 8.7: Index of agreement and RMS error at Station B.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.76
0.68
1.86
1.70
2
0.75
0.65
2.50
3.07
Currents At Station E
The depth at this station was 152 cm. As shown in Figure 8.20, the dominant
wind direction was toward the west. However, the anemometer did not work properly
after Day 157. Thus, the wind stress after Day 157 was computed with the wind data
obtained at the other stations. The average wind stress was about 0.5 dyne/cm2.
Measured current at arm 2, which was 116 cm above the bottom, shows that u
has a negative trend and v has a positive trend over the twoweek period (Figures
8.22 and 8.23). Therefore flow direction is parallel to the boundary between the
vegetation area and the open water. Measured current at arm 1, which was at 36
cm above the bottom, also shows the same trend (Figures 8.20 and 8.21). The flow
60
= Tly/gÂ¡'r12Sfl(C) + V9^n
~ (1Tx)v^7n^(n (1 r1)v^7%(Cw)
(6.2)
V9~oUn
At{lT2)CrtAt(lT3)
912
y/g;.
+ Dfn
rjSweep Equations
Cn+1 + r, aMValv**1) = r + r,Q(v/sW)
(6.3)
ri^7,(C+1) +
1 + Ai ( T2CTrÂ¡ + T3
912
VWJ
n+1
(6.4)
= ^^^(narov^y^niri)^22^)
+ V9~oVn^Â¡Vn
At(l T2)Crv A(l T3)
912
y/9o,
+ D"
i^71%"+) + ^"+1 = vS^ + rlv^712,(C")
(6.5)
where 7\, T2, and 73 are all constants between 0 and 1, superscripts indicate the
time level, and CT$ and CTrÂ¡ are the bottom friction terms. For example, the wave
propagation terms are treated explicitly if T\ = 0, but implicitly if 7\ = 1. D'^ and Â£>'n
are explicit terms in the U and V equations excluding the Coriolis, bottom friction,
and wave propagation terms. Additional parameters appearing in the equations are
<1
ai
9dA(
0At
av
9dAr)
9d
=
y/To
,11
HAtg11
7
AC
712
=
HAtg12
AC
22
HAtg22
7
At]
.,21
HAtg12
7
Ag
In the Lake Okeechobee application, the external mode is first solved over the
entire lake. For the open water region, the above Â£sweep and 77sweep equations
89
Table 8.3: Reference values used in the Lake Okeechobee spring 1989 circulation
simulation.
Parameter
Units
Value
Xr
cm
3.11 x 105
ZT
cm
150
UT
cm s1
10
f
cm s1
6.62 x 105
9
cm s2
981
Po
g cm3
1
AHr
cm2 s1
105
AVr
cm2 s1
1
eddy viscosity (AVr) which is shown in Table 8.3 is characteristic of calm conditions
in Lake Okeechobee. During periods of strong wind mixing, vertical eddy viscosities
can exceed 100 cm2 s1.
8.4.2 Turbulence Model and Parameters
The vertical turbulence was parameterized by using a simplified secondorder
turbulence closure model developed by Sheng et al. (1986b). As described by Sheng
et al. (1989b), the simplified secondorder closure model is obtained by neglecting the
evolution and diffusion terms from the full Reynolds stress equations in the complete
secondorder closure turbulence model, and then replacing the mass flux equations
with temperature and salinity flux equations which are applicable to water bodies.
The tensor equations for the Reynolds stress (u(u'), mass flux (u[p'), and variance
(p'p1) are then, respectively, as shown in Eqs. (2.18), (2.19) and (2.20). Although
details of the turbulence model were given elsewhere ( Sheng and Chiu, 1986; Sheng
et al., 1989b), the equations are given in the following for completeness.
By considering only vertical gradients of the mean variables, the following alge
dyne/cm**2
136
Velocity at Platform C* 3D. Intarpo 1ated(Arm 1) W/ Temp
JulIan Day
Figure 8.38: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 1: NorthSouth direction) when thermal effect is considered.
cm**2/sec 001**2/560 cm**2/seo
172
Figure 8.67: Spectrum of measured and simulated currents (eastwest direction) at
Station C
dyne/cm**2
179
Wind Stress at Platform tau x (MODEL)
Velocity at Platform B* 3D, Interpolated(Arm 1 ) Inverse
Julian Day
Figure A.3: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
dyne/cm**2
113
Velocity at Platform B 3D, Interpo1 ated(Arm 2) W/0 Temp
JulIan Day
Figure 8.19: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation.
3.5 Dimensionless Equations in crStretched Cartesian Grid 27
3.5.1 VerticallyIntegrated Equations 28
3.5.2 Vertical Velocities 29
3.6 Generation of Numerical Grid 29
3.6.1 Cartesian Grid 29
3.6.2 Curvilinear Grid 29
3.6.3 Numerical Grid Generation 30
3.7 Transformation Rules 31
3.8 TensorInvariant Governing Equations 34
3.9 Dimensionless Equations in BoundaryFitted Grids 36
3.10 Boundary Conditions and Initial Conditions 37
3.10.1 Vertical Boundary Conditions 37
3.10.2 Lateral Boundary Conditions 37
3.10.3 Initial Conditions 37
4 VEGETATION MODEL 39
4.1 Introduction 39
4.2 Governing Equations 41
4.2.1 Equations for the Vegetation Layer (Layer I) 41
4.2.2 Equations for the VegetationFree Layer (Layer II) 43
4.2.3 Equations for the Entire Water Column 44
4.2.4 Dimensionless Equations in Curvilinear Grids 45
5 HEAT FLUX MODEL 48
5.1 Introduction 48
5.2 The Equilibrium Temperature Method 48
5.2.1 ShortWave Solar Radiation 49
5.2.2 LongWave Solar Radiation 49
5.2.3 Reflected Solar and Atmospheric Radiation 49
5.2.4 Back Radiation 51
5.2.5 Evaporation 51
5.2.6 Conduction 51
5.2.7 Equilibrium Temperature 52
5.2.8 Linear Assumption 52
5.2.9 Procedure for an Estimation of K and Te 53
5.2.10 Modification of the Equilibrium Temperature Method 54
5.3 The Inverse Method 54
5.3.1 Governing Equations 55
5.3.2 Boundary Conditions 55
5.3.3 FiniteDifference Equation 56
5.3.4 Procedure for an Estimation of Total Heat Flux 57
6 FINITEDIFFERENCE EQUATIONS 58
6.1 Grid System 58
6.2 External Mode 58
6.3 Internal Mode 61
6.4 Temperature Scheme 62
6.4.1 Advection Terms 63
6.4.2 Horizontal Diffusion Term 66
IV
dyne/cm**2
107
Velocity at Platform C 3D, Interpolated(Arm 2) W/Q Temp
JulIan Day
Figure 8.13: Simulated (solid lines) and measured (dotted lines) currents at Station C
(Arm 2: NorthSouth direction). Thermal stratification was not considered in model
simulation
dyne/cm**2
129
Wind Stress at Platform Ai tau x (MODEL)
Velocity at Platform A 3D, Intarpolated(Arm 1) W/ Temp
Julian Day
Figure 8.31: Simulated (solid lines) and measured (dotted lines) currents at Station
A (Arm 1: EastWest direction) when thermal effect is considered.
Surface lcvation (cm)
72
Time history of surface elevation
Figure 7.3: Effect of vegetation on surface elevation evolution in a winddriven rectan
gular lake. Solid line is without vegetation, broken line is with low vegetation density,
and dotted line is with high vegetation density.
v(cm/secl dyne/cm*#2
127
Velocity at Platform D* 3D, InterpolatedlArm 2) W/0 Temp
Julian Day
Figure 8.30: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: NorthSouth direction).
81
is from southeast or southwest in south Florida. Those wind systems determine the
dominant wind direction in Lake Okeechobee. For the synoptic study of spring 1989,
the mean wind speed is about 5.1 m/sec, but it can exceed 11 m/sec. The wind field
is characterized by temporal and spatial nonuniformities. However, during strong
wind periods, the wind tends to be more uniform.
8.2.2 Current Data
Current data were collected at 15minute intervals at the six locations. The data
collection started on Julian Day 136.708. The number of instrument arms at each
station depended upon the water depth and how many vertical levels of data were
desired. At station C, which is located in the center of lake, three current meters were
installed to measure the vertical variation of currents. Two sensors were installed at
Stations B, D and E. One sensor was installed at Stations A and F. The location and
height of each of the current meters are shown in Table 8.2.
As discussed in Sheng et al. (1991a), current data showed significant diurnal
variations in direct response to the wind. During a period of significant change in
wind direction, which usually follows a peak wind period, seiche oscillation generally
leads to significant current speed over several seiche periods (multiples of 5 hours).
Magnitudes of currents are very small at all stations. At Station C, mean magni
tudes of u and v at arm 3 between Julian day 147 and 156 are 2.09 and 1.52 cm/sec,
respectively. Maximum magnitudes of u and v at arm 3 are 11.7 and 6.8 cm/sec,
respectively. Considering the accuracy of current meter, 23 cm Â¡sec, currents are
very small.
8.2.3 Temperature Data
The lake temperature data showed that a significant vertical temperature gradi
ent can be developed during part of a day when wind is low and atmospheric heating
is high. However, over the relatively shallow littoral zone and transition zone, tem
perature appeared to be well mixed vertically much of the time.
149
Table 8.11: Index of agreement and RMS error at Station A when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.35
0.45
4.23
2.07
Table 8.12: Index of agreement and RMS error at Station B when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.75
0.69
1.89
1.65
2
0.77
0.67
2.53
3.10
Table 8.13: Index of agreement and RMS error at Station C when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.86
0.81
1.86
1.38
2
0.75
0.77
2.87
1.64
3
0.78
0.71
2.23
2.05
Table 8.14: Index of agreement and RMS error at Station D when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.82
0.76
1.74
1.94
2
0.58
0.61
2.60
3.37
Table 8.15: Index of agreement and RMS error at Station E when thermal effect is
considered.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.37
0.39
3.15
5.86
2
0.32
0.39
2.62
4.09
102
8.14. When the wind blows in the same direction over a considerable time period,
water is piled up to create a setup of surface elevation. Earlier studies indicated
that it takes approximately one hour (Saville, 1952) for the current to respond to the
wind. This is also manifested in the field data. The peak times of model currents
compare quite well with the peak times of measured currents. When the wind changes
direction or dies out, the wind setup is released and the seiche starts to travel back
and forth in the lake. Seiche period is computed by T = 2L/\/gK. When an average
depth of 2.5 m and a length of 37 km, which is the length of lake excluding the marsh
area in eastwest direction, is used, the seiche period is about four hours. The model
currents clearly show this period.
Currents at Station C are measured at Arms 1, 2 and 3, which are at the height
of 17%, 33%, and 78% of the total depth above the bottom, respectively. To find
out the corresponding model currents is not easy because the model uses cr stretching
vertically. Therefore, the nearest layers are determined according to the percentage
of total depth at the instrument heights. After that, linear interpolation between
two layers is used for the model currents to be compared with field data. As shown
in Figures 8.10 8.15, the longterm trend of time series of model current agrees
well with that of field data. However, magnitudes of model currents (particularly the
peak currents everyday) are generally smaller than those of field data. This can be
explained with two possible reasons. First of all, the model currents are forced by 15
minute averaged wind data, while the measured currents are forced by the raw wind,
which contains many spikes which are stronger than the 15minute averaged wind.
Thus, the simulated currents are expected to be less than the measured currents.
Another possibility is due to some numerical dissipation which is always present in
realistic simulations. The numerical dissipation can be reduced by using a smaller
time step.
Measured currents have another trend which the model could not simulate well.
dyne/cm*#2
191
Wind Stress at Platform D> tau x (MODEL)
Velocity at Platform D. 3D. Interpo1 atsd(Arm 2) Inverse
Julian Day
Figure A.15: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction). Inverse method was used for the estimation of heat
flux.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF FIGURES vi
LIST OF TABLES xiii
ABSTRACT XV
CHAPTERS
1 INTRODUCTION 1
2 LITERATURE REVIEW 6
2.1 Numerical Models of Lake Circulation 6
2.1.1 OneDimensional Model 6
2.1.2 TwoDimensional Model 6
2.1.3 SteadyState 3D Models 7
2.1.4 TimeDependent 3D Models 8
2.2 Vegetation Models 10
2.3 Thermal Models 11
2.4 Turbulence Model 13
2.4.1 Eddy Viscosity/Diffusivity Concept 13
2.4.2 Constant Eddy Viscosity/Diffusivity Model 14
2.4.3 MunkAnderson Type Model 14
2.4.4 Reynolds Stress Model 15
2.4.5 A Simplified SecondOrder Closure Model: Equilibrium Closure
Model 16
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model 17
2.4.7 OneEquation Model (k Model) 18
2.4.8 TwoEquation Model (k t Model) 18
2.5 Previous Lake Okeechobee Studies 19
2.6 Present Study 20
3 GOVERNING EQUATIONS 23
3.1 Introduction 23
3.2 Dimensional Equations and Boundary Conditions in a Cartesian Co
ordinate System 23
3.2.1 Equation of Motion 24
3.2.2 FreeSurface Boundary Condition (z rj) 25
3.2.3 Bottom Boundary Condition (z = h) 25
3.2.4 Lateral Boundary Condition 25
3.3 Vertical Grid 26
3.4 NonDimensionalization of Equations 26
iii
dyne/cm*#2
111
Velocity at Platform B* 3D. Interpo1 ated(Arm 11 W/0 Temp
Julian Day
Figure 8.17: Simulated (solid lines) and measured (dotted lines) currents at Station B
(Arm 1: NorthSouth direction). Thermal stratification was not considered in model
simulation.
8.23 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction). Thermal stratification
was not considered in model simulation 119
8.24 Stick Diagram of wind stress, measured currents, and simualted
currents at Station E 120
8.25 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction). Thermal stratification
was not considered in model simulation 121
8.26 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Thermal stratification
was not considered in model simulation 122
8.27 Simulated (solid lines) and measured (dtted lines) currents at
Station D (Arm 1: EastWest direction) 124
8.28 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: NorthSouth direction) 125
8.29 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction) 126
8.30 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) 127
8.31 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction) when thermal effect is
considered 129
8.32 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction) when thermal effect is
considered 130
8.33 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: EastWest direction) when thermal effect is
considered 131
8.34 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 1: NorthSouth direction) when thermal effect is
considered 132
8.35 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: EastWest direction) when thermal effect is
considered 133
8.36 Simulated (solid lines) and measured (dotted fines) currents at
Station B (Arm 2: NorthSouth direction) when thermal effect is
considered 134
Vlll
128
the density difference is small, the resulting buoyancy and eddy viscosity can become
much different from that of homogeneous water. Figure 8.51 shows the time history
of eddy viscosity at Station C in homogeneous and thermally stratified cases. In the
thermally stratified case, the peak values of eddy viscosity become larger than that
of a homogeneous lake. Also, during the daytime when the lake is stratified and the
wind is weak, eddy viscosity becomes small as expected.
ShortTerm Dynamics
To understand the physics of a lake during the typical daytime, the time histories
of wind stress, heat flux, and measured currents at Station C during Julian day 150
and 152 were plotted in Figure 8.52. Because wind direction is from east to west,
northsouth components of wind stress and currents are small. Therefore eastwest
direction components are plotted. As shown in Figure 8.52, currents and wind stress
follow typical diurnal variations. During the daytime, currents at Arm 3 are from
east to west, while currents at Arm 1 are from west to east. Therefore, the lake
behaves as a twolayer system. Figure 8.53 shows the model results. Model simulated
currents with thermal effect agree well with field data. Time history of eddy viscosity
at Station C is shown in Figure 8.54. Around 7 a.m. of Julian day 150, wind is
mild and wind stress is less than 0.3 dyne/cm2. Because forcing is small at the water
surface, the eddy viscosity should be negligible. At around 2:30 p.m (Julian day
150.6), wind stress starts to increase while the lake is thermally stratified. Therefore,
momentum transfer is limited to the upper layer. As a consequence, eddy viscosity
above the thermocline increases, but eddy viscosity at the bottom layer remains small.
About three hours later, after the strong wind has resulted in the overturning of the
thermocline eddy viscosity at the bottom layer also increases. Around 9 p.m., the
lake is affected by the seiche which is caused by the reduced wind stress.. During the
dyne/cm**2
106
Wind Stress at Platform Ci tau x (MODEL)
Velocity at Platform C* 3D. Interpolated(Arm 2) W/0 Temp
Julian Day
Figure 8.12: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: EastWest direction). Thermal stratification was not considered in model
simulation
80
Table 8.2: Instrument mounting, spring deployment.
SITE
ARM
CURRENT
ELEV
(cm)
AZIM
(deg)
TEMP
ELEV
(cm)
OBS
ELEV
(cm)
A
Pressure Sensor 556
)5, Elev.
96 cm
Wind Sensor 5200, Elev. above MWL 670 cm minus water depth
1
80673
71
342
07
86
0076
86
B
Pressure Sensor 55696, Elev. 86 cm
Wind Sensor 5202, Elev. above MWL 518 cm minus water depth
1
80674
25
204
04
43
0078
43
2
80675
114
181
02
132
0075
132
Pressure Sensor 48228, Elev. 297 cm
Wind Sensor 5203, Elev. above MWL 884 cm minus water depth
1
80679
61
330
01
79
0079
79
2
80681
123
333
09
140
0077
140
3
80680
284
342
06
302
0084
302
D
Pressure Sensor 55694, Elev. 254 cm
Wind Sensor 5200, Elev. above MWL 883 cm minus water depth
1
80672
79
270
08
97
0083
97
2
80677
241
255
03
259
0082
259
Pressure Sensor 55699, Elev.
04 cm
Wind Sensor 5199, Elev. above MWL 518 cm minus water depth
1
80671
36
59
11
53
0081
53
2
80678
116
72
05
135
0080
135
Pressure Sensor 55697, out of water
No Wind Sensor
80676
25
305
10
43
0085
8.37 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction) when thermal effect is
considered 135
8.38 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction) when thermal effect is
considered 136
8.39 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction) when thermal effect is
considered 137
8.40 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction) when thermal effect is
considered 138
8.41 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction) when thermal effect is
considered 139
8.42 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction) when thermal effect is
considered 140
8.43 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction) when thermal effect is
considered 141
8.44 Simulated (solid lines) and measured (dotted lines) currents at'
Station D (Arm 1: NorthSouth direction) when thermal effect is
considered 142
8.45 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: EastWest direction) when thermal effect is
considered 143
8.46 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 2: NorthSouth direction) when thermal effect is
considered 144
8.47 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: EastWest direction) when thermal effect is
considered 145
8.48 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 1: NorthSouth direction) when thermal effect is
considered 146
8.49 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: EastWest direction) when thermal effect is
considered 147
IX
1
2
3
4
5
6
7
8
9
10
11
12
170
Table 8.17: Index of agreement and RMS error.
Arm
Index of Agreement
RMS Error
u
V
u(cm/sec)
v(cm/sec)
1
0.86
0.85
1.79
1.17
2
0.78
0.80
2.55
1.48
3
0.78
0.73
2.13
1.81
1
0.86
0.85
1.87
1.20
2
0.78
0.81
2.66
1.49
3
0.79
0.72
2.16
1.88
1
0.70
0.84
3.64
1.43
2
0.65
0.79
4.37
1.70
3
0.65
0.70
3.44
2.12
1
0.86
0.83
1.79
1.25
2
0.78
0.79
2.56
1.51
3
0.78
0.71
2.15
1.88
1
0.86
0.83
1.82
1.25
2
0.78
0.79
2.61
1.51
3
0.77
0.71
2.20
1.87
1
0.86
0.82
1.70
1.21
2
0.79
0.78
2.35
1.49
3
0.76
0.73
2.19
1.77
1
0.75
0.67
2.04
1.54
2
0.71
0.64
2.45
1.84
3
0.73
0.64
2.12
1.90
1
0.77
0.68
1.99
1.51
2
0.73
0.65
2.39
1.77
3
0.70
0.62
2.13
1.93
1
0.86
0.82
1.69
1.21
2
0.79
0.77
2.34
1.50
3
0.75
0.73
2.24
1.76
1
0.84
0.86
2.15
1.21
2
0.76
0.81
2.93
1.52
3
0.77
0.72
2.27
1.92
1
0.79
0.85
2.56
1.28
2
0.72
0.80
3.38
1.59
3
0.75
0.72
2.49
1.98
1
0.87
0.84
1.73
1.19
2
0.80
0.80
2.45
1.47
3
0.78
0.72
2.14
1.86
10
was generated by using the WESCORA code developed by Thompson (1983). John
son (1982), however, transformed the equations in terms of the Cartesian velocity
components.
The boundaryfitted grid has recently been adapted to threedimensional nu
merical models. Sheng (1986) applied tensor transformation to derive the three
dimensional horizontal equations of motion in boundaryfitted grid in terms of the
contravariant velocity vectors (a contravariant vector consists of components
which are perpendicular to the grid line) and the water level. Sigma grid is used
in the vertical direction. The resulting equations in the boundaryfitted and sigma
stretched grid are rather complex. However, numerous analytical tests were conducted
to ensure the accuracy of the model (Sheng, 1986 and Sheng, 1987). The model has
been applied to Chesapeake Bay (Sheng et al., 1989a), James River (Sheng et al.,
1989b), Lake Okeechobee (Sheng and Lee, 1991a, 1991b), and Tampa/Sarasota Bay
(Sheng and Peene, 1992). However, the earlier study on Lake Okeechobee (Sheng and
Lee, 1991a) did not consider thermal stratification in their model.
2.2 Vegetation Models
Vegetation can affect the aquatic life and also the water motion in the marsh area.
Early studies on the effect of vegetation on flow were conducted in the open channels.
Ree (1949) conducted laboratory experiments to produce a set of design curves for
vegetated channels. Kouwen et al. (1969) studied the flow retardance in a vegetated
channel in the laboratory and proposed the following equation:
Â£ = C, + CM A) (2.4)
where U is average velocity, u* is shear velocity, and C\ and Ci are coefficients. A
is a crosssectional area of the channel, and Av is the crosssectional area blocked by
the vegetation.
Reid and Whitaker (1976) considered the vegetation effect on flow as an ad
ditional term, which is proportional to the quadratic power of the velocity, in the
dyne/cm*#2
186
Velocity at Platform Ct 3D. Interpolated(Arm 2) Inverse
Jullan Day
Figure A. 10: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: NorthSouth direction). Inverse method was used for the estimation of
heat flux.
dyne/cm**2
138
Velocity at Platform C 3D, Interpolated(Arm 2) W/ Temp
Jul!an Day
Figure 8.40: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 2: NorthSouth direction) when thermal effect is considered.
dyne/cm**2
193
Wind Stress at Platform E tau x (MODEL)
Velocity at Platform E 3D, Interpolated(Arm 1) Inverse
147. 149. 151. 153. 155. 157. 159. 161.
JulI an Day
Figure A. 17: Simulated (solid lines) and measured (dotted lines) currents at Station
E (Arm 1: EastWest direction). Inverse method was used for the estimation of heat
flux.
46
where
Defining
jji
Q[ + L2\gU
912 Tjt 4. 922 TJV i T* r*
v J + vr2+,<
+ (Horizontal Diffusion)^ {U2,U2)
+ y [Nonlinear Terms (/ U2)]
dU2
dt
= i2LÂ£Â£ + r* _r.*
2 V9~o 2 sv
+ (Horizontal Diffusion)^ (U2, U2)
+ [Nonlinear Terms (U2,U2)^
r.i =
r =
T.i =
T.'r, =
ctj
Â£*r + yTÂ¡y
+ Vy^sy
'T**
T]x
f.
0*
T\x
+ 4
Tiy
[pfUrZr\
.PfUrZr\
7~ix
+ 9y
Ty
[pfUrZr
[pfurzr\
Fax
+ tv
Fey
[pfUrZr
,pfurzr J
Fax
+ Vy
f Fcv 1
[pfUrZr_
[pfUrZr\
u( = u{ +
IP = Ui + Ui
(4.25)
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
we can obtain the following equations for the entire water column:
au*
dt
+ H[gnQ+9l%] =
<7l2
9 22
_l/i + ^c/, + T.T._Fi
+ (Diffusion^ [Ui,Ui]
Vegetation Distribution
Number of sta1ks/Cm**2*10000.
Figure 8.7: Distribution of vegetation density in Lake Okeechobee.
dyne/cm**2
no
Wind Stress at Platform B tau x (MODEL)
Velocity at Platform B* 3D. Interpolated(Arm 1) W/0 Temp
Jul I an Day
Figure 8.16: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction). Thermal stratification was not considered in model
simulation.
dyne/cm**2
131
Wind Stress at Platform Bi tau x (MODEL)
Velocity at Platform B> 3D, Interpo1ated(Arm 1) W/ Temp
Julian Day
Figure 8.33: Simulated (solid lines) and measured (dotted lines) currents at Station
B (Arm 1: EastWest direction) when thermal effect is considered.
99
is fj = With Ax = 19km and h = 2.5m, setup is 7.8 cm. This is close to the
maximum setup in Figure 8.9.
8.6 WindDriven Circulation without Thermal Stratification
8.6.1 Tests of Model Performance
Comparing the simulated results with measured data is not a simple task. Two
methods are used for the comparison. The first method is to plot simulated results
and measured data. While this method can give a quick intuition, this can mislead
the readers to the evaluation of the models predictive ability because this method
can show the models ability qualitatively.
Therefore quantitative measurements of the models performance are necessary.
Typical parameters are mean values of model and data, and the correlation coefficient
between the model and data. A new parameter, suggested by Willmott (1981), is the
index of agreement used for the comparison as follows:
,, EfaiCfi Qif
where Ot is the observed value, P, is the predicted value, O is the average of O,, P is
the average of P, P{ = P, O, and 0\ = 0,0. When the index of agreement is
unity, predicted values perfectly agree with observed values in magnitude and sign.
8.6.2 Model Results
Currents At Station C
Time series of wind stress at Station C show that the wind field is temporarily
varying. Therefore, spatially and temporarily varying wind stress field was used for
the simulation. Time series of the wind stress field shows that there is a significant
diurnal variation of wind speed (Figure 8.10). Wind is calm early in the morning
and strong in the afternoon and becomes calm again in the late night.
The comparison of peak wind stress and peak current of field data shows that there
is a time lag between the current and the wind, for example, Julian Day 150.8 in Figure
CHAPTER 6
FINITEDIFFERENCE EQUATIONS
This chapter describes the finitedifference equations for the governing equations
in Â£, r?, <7 coordinate system.
6.1 Grid System
Earlier models used a nonstaggered grid so that all the variables were calculated
at the same point (center point). This has a disadvantage. When centered difference
scheme is used, onesided difference scheme near the boundary should be used to
maintain the same order of accuracy. Therefore, it is inconvenient. CH3D uses a
staggered grid as shown Figure 6.1. Surface elevation and temperature are computed
at the center of a cell, while the velocities are computed at the face of a cell. A
vertical grid is shown in Figure 6.1, and all the variables are computed at the middle
of the layer.
6.2 External Mode
The external mode equations consist of the equations for surface displacement
( and the verticallyintegrated velocities U and V. Treating the wave propagation
terms in the finitedifference equations implicitly and factorizing the matrix equation,
the following equations for the Â£sweep and ^sweep are obtained:
Â£Sweep Equations
C + W(v^*) = C ^ TxMV9~oUn) (6.1)
av6n(V&yn)
riv^7u^(n +
1 + At
V9~oU*
58
4
to represent the complex geometry with a relatively smaller number of grid points.
In some shallow lakes, aquatic vegetation can grow over large areas. The vegeta
tion can affect the circulation significantly because it introduces additional friction on
the flowing water. For example, Lake Okeechobee has vegetation over an area which
covers 25% of the total lake surface. Because vegetation consists of stalks with differ
ent heights and diameters, a representative diameter and height over each discretized
grid cell must be introduced in the model. Additional drag terms must be introduced
in the momentum equations to represent the form drag introduced by the vegetation.
The consideration of vegetation is necessary to compute the flow and transport of
phosphorus between the vegetation area and the open water.
Previously developed numerical models which were applied to deep lakes, e.g.,
the Great Lakes, cannot be readily applied to shallow lakes such as Lake Okeechobee,
since many shallow water processes are not included in these models.
Since 1988, with funding from the South Florida Water Management District and
U.S. Environmental Protection Agency, the Coastal and Oceanographic Engineering
Department of the University of Florida (under the supervision of Dr. Y. Peter
Sheng) has conducted a major study on the hydrodynamics and sediment dynamics
and their effects on phosphorus dynamics in Lake Okeechobee. The primary purpose
of the study was to quantify the role of hydrodynamics and sediments on the internal
loading of phosphorus and the exchange of phosphorus between vegetation zone and
open water. As part of the study, field data (wind, air temperature, wave, water
current, water temperature, and suspended sediment concentration) were collected
over two onemonth periods in 1988 and 1989. Ahn and Sheng (1989) studied the wind
waves of Lake Okeechobee. Cook and Sheng (1989) studied the sediment dynamics
in Lake Okeechobee. This study focuses on the influences of vegetation and thermal
stratification on lake circulation. The objectives of this study are
(1) to obtain a general insight in the winddriven circulation in Lake Okeechobee,
61
are used. For the vegetation zone, however, the Â£sweep and 7/sweep equations are
modified by the presence of vegetation and are derived from Equations (4.23) through
(4.26). However, it is only necessary to solve the finitedifference equations for the
integrated velocities in the entire water column, (U, V), and the velocities in one
layer, (Ui,Vi) or (U2, Vj). The velocities in the other layer can be readily obtained
by subtracting the onelayer velocities from the total velocities.
6.3 Internal Mode
The internal mode is defined by the equations for the deficit velocity and v,
(,v) = (u ~ jÂ¡i v jj) The equations for and v are obtained by subtracting the
verticallyaveraged equations from the threedimensional equations for u and v:
dtHu ^
g12 g22
y H+ + F3
+
EzJL
H2 da
y/Qo
 (u Tbt) f2
(6.7)
d_
dt
Hv
+
,11
21
H iHv + G3
y/So y/do
Evd( dHv\
H2 da da )
(Tsti 7"6tj) G2
(6.8)
where and v are the deficit velocities in the (Â£,77) directions, F3 and G3 indicate all
the explicit terms in the u and v equations, respectively, while F2 and G2 indicate all
the explicit terms in the U and V equations, respectively.
Applying a twotimelevel scheme to the above equations leads to the following
finitedifference equations:
A2
(f)n+1 = {H)n + At^z Hn+ln+1
\/9o
22
+ At^Hnn + At(F3 F2)n
J A t
y/Ho
Ev d
(Hn)2 da
(T*i n()n+1
Av(Hn+1un+1)
(6.9)
A.l Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 177
A.2 Simulated (solid lines) and measured (dotted lines) currents at
Station A (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 178
A.3 Simulated isolid lines) and measured (dotted lines) currents at
Station B (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 179
A.4 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 180
A.5 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 181
A.6 Simulated (solid lines) and measured (dotted lines) currents at
Station B (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 182
A.7 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 183
A.8 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 1: NorthSouth direction). Inverse method was
used for the estimation of heat flux 184
A.9 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: EastWest direction). Inverse method was
used for the estimation of heat flux 185
A.10 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 2: NorthSouth direction). Inverse method was
used for the estimation of heat flux 186
A. 11 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: EastWest direction). Inverse method was
used for the estimation of heat flux 187
A. 12 Simulated (solid lines) and measured (dotted lines) currents at
Station C (Arm 3: NorthSouth direction). Inverse method was
used for the estimation of heat flux 188
A.13 Simulated (solid lines) and measured (dotted lines) currents at
Station D (Arm 1: EastWest direction). Inverse method was
used for the estimation of heat flux 189
xi
143
Wind Stress at Platform D tau x (MODEL)
Velocity at Platform D* 3D, Intarpolated(Arm 2) W/ Temp
Figure 8.45: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 2: EastWest direction) when thermal effect is considered.
2
Earlier studies on physical processes in lakes concentrated on the observations of
periodic up and down motion of water level, i.e., seiche motion. When the wind blows
over a certain period, water builds up near the shoreline. After wind ceases, water
starts to oscillate as a free long wave. Subsequently, this wave is damped out due to
bottom friction. Seiche can be initiated by sudden change of wind speed or direction,
the passage of a squall line, an earthquake, or resonance of air and water columns.
When the velocity of a squall line is close to the speed of gravity wave, resonance
occurs and damages can be more severe. It was reported that severe storm damages
in Chicago were caused by a squall line over Lake Michigan on June 26, 1954 (Harris,
1957).
Another important feature in deep temperate lakes is the temperature variation
over the depth, which is called thermal stratification. Starting in the early spring, the
lake attains a temperature of 4 C and is more or less isothermal. During the summer
season, the surface water starts to become warmer because of increased solar radiation,
so that, gradually, a sharp temperature gradient, i.e., thermocline, is formed. The lake
remains thermally stratified during summer with a warm surface layer (epilimnion),
a thermocline, and a cold layer (hypolimnion). Though strong wind action tends to
lower or break the thermocline, the lake generally remains stratified during the entire
summer season. During the fall, as the air temperature drops, the net daily heat flux
at the water surface becomes negative, i.e., the lake loses heat daily. Hence, water
density in the epilimnion often becomes heavier than that in the hypolimnion and
causes convective mixing which, in combination with strong wind action, causes the
lake to become isothermal again in the winter. This process repeats itself annually. It
is important to know the location of thermocline at different times of the year so that
water can be withdrawn to a desirable height in deep lakes or reservoirs for various
agricultural and municipal uses.
Many processes are influenced by currents and temperature in lakes. For example,
57
Applying bottom boundary condition,
r+1 r.,. 4. c^u\+
At + /Ul Az
5.17 can be written as follows:
At the interior points,
//Tin+l 1 \
rrn+1 T
ij T AV,_1 X7
At
Az
Applying free surface boundary condition,
rpn + l rpn 1s (TÂ¡X )
im im 1 Az
Ai Az
Applying bottom boundary condition,
jin+1 j^n
At Az
(Tn+i_Tr+l)
1Xvl A j.
5.3.4 Procedure for an Estimation of Total Heat Flux
Step 1. Solve for u for given wind stress.
(5.31)
(5.32)
(5.33)
(5.34)
Step 2. Solve for v for given wind stress.
Step 3. Guess nonsolar heat flux and solve for T.
Step 4. compare the computed tempearature to measured temperature.
Step 5. If error is not within error limit (0.5 C), go to step 3.
Step 6. If error is within error limit (0.5 C), go to step 1.
cm*+2/sec cm**2/sec cm**2/sec
173
Figure 8.68: Spectrum of measured and simulated currents (northsouth direction) at
Station C
9
represent the shallow regions. More recent models use the socalled crgrid which
was originally applied in the simulation of atmospheric flow by Phillips (1957). This
vertical cstretching uses the same number of vertical grid points in both the shallow
and deep regions, with the vertical coordinate defined as follows:
a
Hx,y) + ((x,y,t)
(2.3)
where h(x, y) is the water depth, and ( is the water surface elevation. Governing
equations are transformed from the (x,y,z,t) coordinates to (x,t/,cr, t) coordinates
by use of chain rule and become somewhat more complex because of the extra terms
introduced by the stretching.
Other types of models (e.g., Simons, 1974) use a Lagrangian grid which consists
of layers of constant physical property (e.g., density) but timevarying thickness.
These models could resolve vertical flow structure with relatively few vertical layers.
However, parameterization of the interfacial dynamics is often difficult.
Horizontal Grid
One of the challenges in numerical models is the accurate representation of com
plex geometry. Most models (e.g., Leendertse, 1967) use a rectangular uniform grid
to represent the shoreline of a lake or estuary. Thus, a large number of grid points
are needed to achieve a fine resolution near the shoreline or islands. Because com
putational effort is directly related to the number of grid points, grid size should be
as small as possible to maintain required resolution near the interest area, so long
as the computational effort is not excessive. Therefore, to achieve a balance between
resolution and computational efficiency, a nonuniform grid method could be used.
Sheng (1975) used smaller grid size used near areas of importance but coarse grid
elsewhere.
Use of a boundaryfitted grid is another viable alternative. Johnson (1982) used
a boundaryfitted grid to solve depthintegrated equations of motion for rivers. Using
chain rules, he transformed the governing equations for a boundaryfitted grid which
98
where the wind speeds are in cgs units (cms1). A maximum value of 0.003 was
imposed upon Cda. Wind stress from the five stations was interpolated onto the
curvilinear grid by inversedistancesquared weighting. The inversedistancesquared
interpolation is done as follows. The distance from the corner point of each grid cell
to each of the wind stations is first calculated. The wind stress at each corner point
is then computed by a summation of the wind stress from the five wind stations,
weighted according to the inverse square of the distance from the corner point to each
wind station. This interpolation process can be summarized as
N
r3)nR0/Rn (8.27)
n=1
where N = 5 wind stations, and
5 O.'J *n)2 + (Vi,j ~ yn)2
and
Rn = (xitj Xn) + (Vi,j lIn)
(8.28)
(8.29)
The surface wind stresses Ts^,Tsr) at the u,v points, respectively, are then obtained
from the values at the corner points of the grid cells by transformation relation.
8.5 Steady State WindDriven Circulation
To investigate the steadystate circulation, an easterly wind of 1 dyne/cm,2 was
imposed at the free surface. As shown in Figure 8.8, the depth integrated field
shows two circulation gyres were formed: one clockwise gyre in the southern part
and another counterclockwise gyre in the northern part. Currents follow the wind
direction in the shallow northern and southern part and return flow in the central
deep area to satisfy the continuity.
Figure 8.9 shows the surface elevation contour when the lake reaches steady state.
Though the bottom is not flat and geometry is complex, approximate setup height
can be estimated by use of setup equation. Neglecting bottom stress, setup equation
34
where g is the diagonal element of the metric tensor gtJ:
dxldxn c
9iJ = WWn
(3.38)
which for the twodimensional case of interest is
_ f x\ + V\ x(xri + y(Vv\ ( 9n 912 \
\ xnxt + yvyt x2n + yl ) ^ g21 g22 )
(3.39)
The three components follow different rules for transformation between the prototype
and the transformed plane:
 d?_ j
u dx>u
(3.40)
(3.41)
(>) = (3.42)
where the unbarred quantities represent the components in the prototype system,
while the barred quantities represent the components in the transformed system.
3.8 TensorInvariant Governing Equations
Before transforming the governing equations, it is essential to first write them in
tensorinvariant forms, i.e., equations which are independent of coordinate translation
and rotation. For simplicity, unbarred quantities are used to denote the variables in
the transformed system unless otherwise indicated.
Following the rules described in the previous paragraph, the following equations
are obtained (Sheng, 1986):
+ =0 <343>
1 dHuk
H dt
Okgnjekiu
150
seiche, currents at all the three arms have same directions. This diurnal variation is
repeated during the next day.
8.8 WindDriven Circulation with Thermal Stratification: Inverse Method
To improve the temperature prediction and currents, the inverse method was used
to estimate the heat flux at the water surface. Assuming the negligible advection ef
fect, total fluxes at five stations were obtained by solving the vertical onedimensional
temperature equation coupled with momentum equations.
As shown in Figure 8.55, nonsolar heat flux fluctuates while the solar part is
regular. Using total fluxes, CH3D was run. Trends and magnitudes of currents were
similar to those with equilibrium temperature (figures showing simulated currents
and data are included in the Appendix).
However, temperature prediction was much improved. Figure 8.56 shows the
temperature contour with time between Julian day 152 and 155. Generally, the lake
is vertically homogeneous in the night. As sun rises, the lake becomes thermally
stratified. When wind blows strong in the afternoon, the lake is destratified due to
windinduced mixing. Field data show the daily thermal stratification and destratifi
cation. The contours of simulated temperature indicate clearly this process. Figures
8.57 through 8.61 show the time histories of temperature data and the model. In
general, the model can simulate the stratification well at all stations.
8.8.1 The Diurnal Thermal Cycle
From the time history of temperature, diurnal variations of stratification and de
stratification are obvious. In spite of the maximum temperature difference between
Arm 1 and Arm 3 at Station C during a day, thermal effect on the currents is signif
icant as shown in Figures 8.62 through 8.65.
Typical vertical profiles of currents and temperature at Station C on Julian Day
155 are shown sequentially, starting at midnight. The first two columns are simulated
current profiles without thermal effect, and the other three profiles are with thermal
30
mapping method is based on complex variables, so the determination of mapping
function is a difficult task. Therefore, many practical applications rely on numerical
grid generation techniques.
3.6.3 Numerical Grid Generation
Partial differential equations are solved to obtain the interior grid points with
specified boundary points. Thompson (1983) developed an elliptic grid generation
code (WESCORA) to generate a twodimensional, boundaryfitted grid in a complex
domain.
To help understand the physical reasoning of this method, consider a rectangular
domain. When the temperature is specified along the horizontal boundary, then
the temperature distribution inside can be obtained by solving the heat equation.
Therefore, isothermal lines can be drawn. Also, other isothermal lines can be obtained
with the specified temperature in the vertical direction. By superimposing these
isothermal fines, intersection points of isothermal fines can be considered as grid
points.
WESCORA solves Poisson equations with same idea in a complex domain. Con
sider the following set of equations (see Figure 3.1):
U + {yy = P (3.26)
T]xx d* Vyy = Q (3.27)
with the following boundary conditions:
Â£ = Â£(x,y) on 1 and 3
7] = constant (3.28)
( = constant
T) = r](x,y)
on 2 and 4
(3.29)
APPENDIX A
SIMULATED CURRENTS BY INVERSE METHOD
176
cmÂ¡/s cmVs
154
Eddy Viscosity at C (temperatura considered)
Figure 8.54: Time history of eddy viscosity at Station C between Julian days 150 and
152.
Ratio (Bowen, 1926).
52
11 e
D_C,(T,Ta) P
(e, ea) 760
where
P = barometric pressure in mmHg,
C7 a coefficient determined from experiments = 0.26.
Thus, conduction is related to the other parameters as follows:
Hc = 0.26(a + bW)(T3 Ta) [BtuFt^day1] (5.9)
5.2.7 Equilibrium Temperature
Of the seven processes, four processes are independent of surface water temper
ature: shortwave solar radiation, longwave atmospheric radiation, reflected solar
radiation, and reflected atmospheric radiation. The sum of these four fluxes is called
absorbed radiation (Hr). Thus the net heat flux can be written as follows:
AH Hr Hbr HeHc (5.10)
When the net flux AH is zero, Hr becomes
Hr = ~fwcr(Te + 460)4 + 0.26(a + bW)(Te Ta) + (a + bW)(ee ea) (5.11)
The net heat flux can be expressed as follows:
AH = K(T, Te) (5.12)
where K is the heat exchange coefficient.
5.2.8 Linear Assumption
Vapor pressure difference, e3 ea, is assumed to have a linear relationship with
temperature increment as
e. e, = i9(T, T,j
(5.7)
(5.8)
(5.13)
7 MODEL ANALYTICAL TEST 68
7.1 Seiche Test 68
7.2 Steady State Test 68
7.3 Effect of Vegetation 70
7.4 Thermal Model Test 70
8 MODEL APPLICATION TO LAKE OKEECHOBEE 76
8.1 Introduction 76
8.1.1 Geometry 76
8.1.2 Temperature 76
8.2 Some Recent Hydrodynamic Data 76
8.2.1 Wind Data 77
8.2.2 Current Data 81
8.2.3 Temperature Data 81
8.2.4 Vegetation Data 83
8.3 Model Setup 83
8.3.1 Grid Generation 83
8.3.2 Generation of Bathymetry Array 84
8.4 Model Parameters 88
8.4.1 Reference Values 88
8.4.2 Turbulence Model and Parameters 89
8.4.3 Bottom Friction Parameters 91
8.4.4 Vegetation Parameters 92
8.4.5 Wind Stress 96
8.5 Steady State WindDriven Circulation 98
8.6 WindDriven Circulation without Thermal Stratification 99
8.6.1 Tests of Model Performance 99
8.6.2 Model Results 99
8.7 WindDriven Circulation with Thermal Stratification: Te Method . 123
8.8 Simulation of Currents with Thermal Stratification: Inverse Method 150
8.8.1 The Diurnal Thermal Cycle 150
8.9 Sensitivity Tests 156
8.9.1 Effect of Bottom Stress 167
8.9.2 Effect of Horizontal Diffusion Coefficent 168
8.9.3 Effect of Different Turbulence Model 168
8.9.4 Effect of Advection Term 168
8.10 Spectral Analysis 169
9 CONCLUSION 174
APPENDIX
A SIMULATED CURRENTS BY INVERSE METHOD 176
BIBLIOGRAPHY 197
BIOGRAPHICAL SKETCH 203
140
Julian Day
Figure 8.42: Simulated (solid lines) and measured (dotted lines) currents at Station
C (Arm 3: NorthSouth direction) when thermal effect is considered.
L
dyne/cm**2
125
Velocity at Platform D 3 Interpolated(Arm 1) W/0 Temp
Ju1 an Day
Figure 8.28: Simulated (solid lines) and measured (dotted lines) currents at Station
D (Arm 1: NorthSouth direction).
100
Figure 8.8: Steadystate depthintegrated currents (cm2s1) in Lake Okeechobee
forced by an easterly wind of 1 dyne/cm2.
17
the following assumptions: 1) Secondorder correlations have no memory effect. That
means correlations at the previous time have no effect on correlations at the next time.
Therefore, ^ = 0. 2) Correlations at a point do not affect the value at another point.
Therefore, all the diffusive terms are dropped. These conditions are approximately
true if the time scale of turbulence is much less than the mean flow time scale and
the turbulence does not vary significantly over the macroscale, i.e., the turbulence is
in local equilibrium. Then the remaining equations become as follows (Sheng, 1983):
0 = U;U
duj
dxk
~ UjUk
duj
dxk
9i
Ujp
9j
UiP
26 kUU j Â£jk(QUkUÂ¡
H / t f C V \ C
j{uiUj Sijj) 6,
,J 12A
(2.20)
0 =
/ / &P it Ollx
9iPP
*
Po
2eijkSljUkp 0.75
t *
UiP
(2.21)
n cCT7dP 45qpp
0 = 2up d7Â¡ + a
(2.22)
These algebraic equations can be solved with ease, once the mean flow conditions are
known. In order to complete the system of equations, q and A need to be solved
following the procedure described in Sheng et al., 1989b.
The above equilibrium closure model was applied to the Atlantic Ocean (Sheng
and Chiu, 1986), Chesapeake Bay (Sheng et al., 1989a) and the James River (Sheng
et al., 1989b). More details of the model will be given in Chapter 8.
2.4.6 A Turbulent Kinetic Energy (TKE) Closure Model
To introduce some dynamics of turbulence into the simplified secondorder closure
model, one can add a dynamic equation for q2(q2 = u'u' + v'v' + ww1), which is twice
the turbulent kinetic energy (Sheng and Villaret, 1989). This TKE closure model has
UFL/COELTR/103
WINDDRIVEN CIRCULATION IN LAKE
OKEECHOBEE, FLORIDA: THE EFFECTS OF
THERMAL STRATIFICATION AND AQUATIC
VEGETATION
by
Hye Keun Lee
Dissertation
1993
8.50 Simulated (solid lines) and measured (dotted lines) currents at
Station E (Arm 2: NorthSouth direction) when thermal effect is
considered 148
8.51 Time history of eddy viscosity at Station C between Julian days
147 and 161 151
8.52 > Time history of wind stress and measured currents between Julian
days 150 and 152 152
8.53 Time history of simulated currents at Station C between Julian
days 150 and 152 153
8.54 Time history of eddy viscosity at Station C between Julian days
150 and 152 154
8.55 Time history of heat fluxes at Station C between Julian days 147
and 161 155
8.56 Temperature contours of data and model at Station C between
Julian days 152 and 155 157
8.57 Simulated and measured temperatures at Station A 158
8.58 Simulated and measured temperatures at Station B 159
8.59 Simulated and measured temperatures at Station C 160
8.60 Simulated and measured temperatures at Station D 161
8.61 Simulated and measured temperatures at Station E 162
8.62 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 163
8.63 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 164
8.64 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 165
8.65 Vertical profiles of currents and temperature during a typical day.
Base temperature is 25 C 166
8.66 Spectrum of wind stress and surface elevation 171
8.67 Spectrum of measured and simulated currents (eastwest direc
tion) at Station C 172
8.68 Spectrum of measured and simulated currents (northsouth direc
tion) at Station C 173
x
94
Table 8.5: Vegetations in Lake Okeechobee (From Richardson, 1991).
Value
Points
Acres
%
Height
[m]
Density
[#/m2]
Diameter
[cm]
Description
0
41631
38228.64
0
1
36
33.058
0.01
2.5
10
2
Buttonbush
2
2737
2513.315
0.57
4
30
8
Melaleuca
3
5699
5233.241
1.18
2.5
10
5
Willow
4
43
39.486
0.01
1
2000
0.25
Spartina
5
43433
39883.37
9.03
2.5
50
7
Cattail
6
3975
3650.138
0.83
1.5
40
15
Sawgrass
7
1195
1097.337
0.25
5
10
2
Mixed Upland
8
4712
4326.905
0.98
1
500
0.25
Rhynchospora
9
36
33.058
0.01
0.5
10
1
Sagitaria/pontedaria
10
11458
10521.57
2.38
1
2000
0.25
Mixed grasses
11
1316
1208.448 .
0.27
0
7
1
Nymphae
12
6836
6277.318
1.42
1
60
0.25
Eleocharis
13
36
33.058
0.01
5
10
2
Guava
14
44
40.404
0.01
0
100
5
Hyacinth
15
4029
3699.724
0.84
2.5
50
2
Scirpus
16
370750
340449.9
77.08
0
0
0
Open Water
17
196
179.982
0.04
0
0
0
Periphyton
18
5344
4907.254
1.11
1
100
0.25
Eleocharis/periphyton mix
19
2365
2171.717
0.49
1
100
0.3
Nymphae/eleocharis mix
20
2759
2533.517
0.57
0.5
15
3.5
Lotus
21
2238
2055.096
0.47
0
*
*
Submerged
22
2724
2501.377
0.57
2
12
5
Cattail/nymphae mix
23
714
655.647
0.15
2.5
50
5
Phragmites
24
278
255.28
0.06
1
50
0.3
Maidencane
25
3952
3629.017
0.82
**
**
**
Excluded
26
873
801.653
0.18
1
2000
0.25
Spartina/panicum mix
27
30
27.548
0.01
0
0
0
Airboat trails
28
3181
2921.028
0.66
1
100
1
Successional disturbed
1 provides no wind resistance but considerable cross section resistance to flow
2 ** area diked preventing surface water flow
40
ated by the vegetation was included in the linearized verticallyintegrated equations of
motion, which did not contain the nonlinear and diffusion terms. Sheng (1982) devel
oped a comprehensive model of turbulent flow over vegetation canopy by considering
both the profile drag and the skin friction drag in the momentum equations in addi
tion to the reduction of turbulent eddies and the creation of turbulent wake energy.
Detailed vertical structures of mean flow and turbulence stresses were computed by
solving the dynamic equations of all the mean flow and turbulent quantities. Model
results compared well with available mean flow and turbulence data in a vegetation
zone.
For the present study, due to the lack of detailed data on vegetation and mean
flow and turbulence in the vegetation zone, a relatively simple vegetation model which
is more robust than Whittaker et a/.s model yet simpler than Shengs 1982 model is
developed. Due to the shallow depth in the vegetation zone, it is feasible to treat the
water column with no more than two vertical layers. When the height of vegetation is
greater than 80% of the total water depth, the flow is considered to be onelayer flow,
i.e., the entire water column is considered to contain uniformly distributed vegetation.
When the height of vegetation is between 20% and 80% of the total water depth, the
flow is considered to be twolayer flow, i.e., the water column consists of a water layer
on top of a vegetation layer. The vegetation effect is neglected when the height of
vegetation is less than 20% of the total depth. The profile drag introduced by the
vegetation can be formulated in the form of a quadratic stress law:
^canopy = pCU,\\u\\AN (4*1)
where u is the vertically averaged velocity in the vegetation layer (layer 1), p is the
density of water, A is the projected area of vegetation in the direction of the flow, N is
the number of stalks per unit horizontal area, and c is an empirical drag coefficient.
Tickner (1957) performed a laboratory study. Strips of ordinary window screen 0.1
foot in height were placed across a channel to simulate a vegetative canopy. Using
r
86
Figure 8.4: Curvilinear grid of Lake Okeechobee.
7
over relaxation (SOR). Their model included the terms of bottom friction, advection,
bottom topography, and lateral diffusion, while assuming steady state and homoge
niety in density. The specification of the vertical eddy viscosity is not required in the
twodimensional model but can give only depthaveraged velocities.
Shanahan and Harleman (1982) developed a transient 2D model which assumed
vertical homogeneity. When the lakes are long, deep but relatively narrow, laterally
averaged 2D model can be applied (for example, Edinger and Buchak, 1979).
2.1.3 SteadyState 3D Models
An early study on winddriven circulation was conducted by Ekman (1923) who
solved momentum equations analytically while neglecting the nonlinear terms. We
lander (1957) developed a theory on winddriven currents based on an extension
of Ekmans theory. After neglecting inertia terms and horizontal diffusion terms,
steadystate momentum equations were combined with the introduction of complex
variables. After applying boundary conditions, a solution was obtained in terms of
the imposed wind stress and unknown pressure gradient term. By introducing the
stream functions for verticallyintegrated flow, the continuity equation could be sat
isfied unconditionally. The final equation to be solved was reduced to a secondorder
partial differential equation for stream function, t/>, as follows:
V^ = ^+6W+C (21)
Once is found, the currents can be found by taking the derivatives, and ^.
Gedney and Lick (1972) and Sheng and Lick (1972) applied Welanders theory
to Lake Erie. The equation for stream function was solved by the successive over
relaxation method. The agreement between the field data and model results was
good. Eddy viscosity was assumed to be constant but varies with wind speed.
Thomas (1975) used a depthvarying form of vertical eddy viscosity as follows:
%
V = i/0(l + t) ku,(h + z)
(2.2)
101
CONTOUR FROM 6.3000 TO 8.1000 CONTOUR INTERVAL OF 0.90000
Figure 8.9: Steadystate surface elevation contour (cm) in Lake Okeechobee forced
by an easterly wind of 1 dyne/cm2.
93
area which will give the same total projected area. If, however, the effect of skin
friction drag due to vegetation is more important, then the equivalent vegetation
density has to be defined in terms of the total wetted area rather than the total
projected area. The procedure for incorporating the vegetation information into
our model is described as follows.
Let
di = the diameter of the ith vegetation type [cm],
Pi = the number of stalks of the ith vegetation type per unit
area,
hi = the height of the ith vegetation type in the grid cell [m],
D = the average water depth in the grid cell [m],
A = the total area of the grid cell [m2], and
Pi = the percent area occupied by the ith vegetation type
within the total area of the grid cell (%).
The sum of the heights of all the plants in the grid cell, H, is
H = ^ pi hi Pi/100 A (8.18)
i
and the total number of stalks of the plants in the grid cell, T, is
T = ^2pi* Pi/100 A (8.19)
whereupon the average height of vegetation in the grid cell, Hcanopy, is
He* nopy = H/T (8.20)
The distribution of Lfcanopy in Lake Okeechobee determined by this method is
shown in Figure 8.6. The projected area occupied by the ith vegetation type in the
grid cell is
o;, = pi di hi Pi/100 A (8.21)
and the total projected area occupied by plants in the cell is
S ~ 2 Qi
(8.22)
161
Temperature at Platform D i MODEL
Temperature at Platform D DATA
Julian Day
Figure 8.60: Simulated and measured temperatures at Station D.

