UFL/COEL96/014
HYDRODYNAMIC MODELING IN SHALLOW WATER
WITH WETTING AND DRYING
by
Justin Ross Davis
Thesis
1996
HYDRODYNAMIC MODELING IN SHALLOW WATER
WITH WETTING AND DRYING
By
JUSTIN ROSS DAVIS
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING
UNIVERSITY OF FLORIDA
1996
ACKNOWLEDGMENTS
First, I would like to thank my advisor, Professor Sheng, for his guidance,
support, and financial assistance throughout my study. In addition, much appreciation is
owed to my committee members, Professor Dean and Professor Thieke, for their review
of this thesis. I would also like to thank Professor Gore, Professor H. G. Wood and Dale
Bass at the University of Virginia whose help with my undergraduate thesis made writing
this one much easier.
I would also like to thank the sponsors of several University of Florida research
projects (Professor Sheng served as the Principal Investigator) for providing funding for
my study and opportunities for me to gain experience in hydrodynamic and water quality
modeling. These projects include the Florida Bay Study funded by the National Park
Service, Everglades National Park and Dry Tortugas National Park; the Sarasota Bay
Field and Modeling Study funded by the Sarasota Bay National Estuary Program and the
United States Geological Survey; and the Indian River Lagoon Hydrodynamic and Water
Quality Study funded by the St. Johns River Water Management District.
I would like to thank DeWitt Smith for providing the Florida Bay data and Ned
Smith for providing the harmonically analyzed data. I would also like to thank the
Florida State University Supercomputing Center for providing some computing
resources.
A world of gratitude is owed to Paul B., Kevin, Mike B., Paul D., Kerry Anne,
Mark G., Matt, Mike K., Wally, Liu, Steve, Hugo, Adam, Eduardo, and Jie, whose help
with classes, research and thesis writing can never be fully appreciated.
Many thanks go to Sandra, John, Lucy, Becky, and Helen for making life easier
and special thanks to Sidney and Subarna who kept the computers running through rain,
snow and gloom of night. Finally, I would like to thank my parents, whose love and
support got me where I am.
TABLE OF CONTENTS
ACKNOW LEDGMENTS ..................... .................. ....... ii
LIST OF TABLES .................................................. .... vi
LIST OF FIGURES ................... ....................... .......... viii
ABSTRACT ................................ .......................... xii
INTRODUCTION ...................................................... 1
1.1 Background .................................................. 1
1.2 Review of Previous Work on Wetting and Drying .................... .4
1.3 Objectives ................................................... 10
1.4 Organization of This Study ..................................... 11
A NUMERICAL HYDRODYNAMICS MODEL ........................... 12
2.1 Governing Equations in Cartesian Coordinates ...................... 12
2.2 Vertical Boundary Conditions in Cartesian Coordinates ............... 13
2.3 Coordinate Transformations ..................................... 16
2.3.1 A Vertically Stretched Grid .............................. 16
2.3.2 Nondimensionalization ................................ 17
2.3.3 VerticallyAveraged Equations in Boundary Fitted Coordinates .. 19
2.4 Finite Difference Equations .................................. 21
2.5 Solution Technique ............................................ 25
2.6 Numerical Stability .......................................... 27
MODEL VERIFICATION ................................................ 30
3.1 Wind Forcing ............................................... 30
3.2 Seiche Test ................................................. 32
3.3 Tidal Forcing ................................................ 35
3.4 Wetting and Drying Test ....................................... 38
3.4.1 Simple Wetting and Drying Test ........................ 38
3.4.2 Wind Forcing in a Closed Basin with Linearly Varying Depth .. 40
3.4.3 Tide Forcing in a Rectangular Basin with Linearly Varying
Depth ........................................... 42
APPLICATION OF MODEL TO FLORIDA BAY ........................... 53
4.1 Overview of Numerical Simulation ............................. 53
4.1.1 Model Domain ....................................... 53
4.1.2 Boundary Fitted Grids ................................. 55
4.1.3 Available Water Level and Wind Data ..................... 59
4.1.4 Forcing Mechanisms and Boundary Conditions ............. 64
4.1.5 Simulation Procedure ..................................65
4.2 M odel Results ................................................ 68
4.2.1 Single Constituent 10Day Simulations ..................... 68
4.2.2 Three constituent 35Day Simulations ...................... 77
CONCLUSIONS AND RECOMMENDATIONS ............................. 99
5.1 Conclusions ................... ............................. 99
5.2 Recommendations .......................................... 100
5.2.1 Improvements to the Model Itself ........................ 100
5.2.2 Improvements to the Boundary Conditions of the Model ...... 101
5.2.3 Improvements in the Model Resolution ................... 101
LIST OF REFERENCES ................................................ 103
BIOGRAPHICAL SKETCH ............................................ 109
LIST OF TABLES
Table p
Table 1.1 A summary of moving boundary models ............................ 10
Table 3.1 Comparison between analytic and model setup surface elevations ........ 31
Table 4.1 Borders and areas, according to three different schemes, of Florida Bay. ... 54
Table 4.2 Locations of the National Park Service inshore stations. ................ 59
Table 4.3 Locations of the offshore tidal stations (Ned Smith, personal communication,
February 10, 1995). ............................................ 62
Table 4.4 Locations of the CMAN wind data stations........................ 62
Table 4.5 Summary of some of the NPS water level records. ..................... 63
Table 4.6 A list of principal tidal constituents for Florida Bay ................... 66
Table 4.7 Major tidal constituents of offshore stations where amplitudes (11) are given in
centimeters and local phase angles (K) are given in degrees (Ned Smith,
personal communication, February 20, 1995). ........................ 67
Table 4.8 The relative importance of individual tidal constituents at each open boundary
station of Florida Bay............................................ 67
Table 4.9 Summary of the singleconstituent runs ............................. 69
Table 4.10 Summary of Florida Bay simulations. Each simulation has M2, K,, and O,
tidal forcing along both the western and southern boundaries ........... 79
Table 4.11 Comparison between simulated and measured tidal amplitudes and phases at
Stations BA, BK, and BN. The best simulations are highlighted ........ 81
Table 4.12
Table 4.13
Table 4.14
Table 4.15
Comparison between simulated and measured tidal amplitudes and phases at
Stations DK, JK, and LM. The best simulations are highlighted. ........ 82
Comparison between simulated and measured tidal amplitudes and phases at
Stations LR, PK, and TC. The best simulations are highlighted. ........ 83
Comparison between measured and simulated tidal amplitudes and phases at
Station WB. The best simulations are highlighted. ................... 84
Rankings of the 35 day simulations. The best simulations are indicated by the
lowest total scores and are highlighted within the table. ............... 85
LIST OF FIGURES
Figure page
Figure 3.1 Surface elevation at three locations within the domain. ................ 31
Figure 3.2 Computational grid (21x5 cells)................................... 33
Figure 3.3 Initial surface elevation for the seiche test. .......................... 34
Figure 3.4 Comparison between simulated (triangles) and theoretical (solid lines) surface
elevation for a seiche oscillation in a closed basin .................... 34
Figure 3.5 The annular section grid (42x5 cells) ............................. 37
Figure 3.6 Comparison between simulated surface elevation and velocity and analytic
solutions for a tidally forced flatbottom annular section .................. 38
Figure 3.7 Simple test diagram. .......................................... 39
Figure 3.8 Simple wetting and drying test results. ......................... ... 39
Figure 3.9 Storm surge diagram for nonwetting and drying simulation ............ 41
Figure 3.10 Storm surge diagram for wetting and drying simulation. .............. 41
Figure 3.11 Comparison of simulated water levels at in a sloping basin, with and
without the wetting and drying model. ............................... 42
Figure 3.12 Wave propagating on a linearly sloping beach diagram ............... 43
Figure 3.13 Computational grid for the wetting and drying test (161x5 cells). ....... 50
Figure 3.14 Nondimensional comparison between wave profiles as predicted by theory
and the numerical model (Time=0 Time=7r/2). ....................... 51
Figure 3.15 Nondimensional comparison between wave profiles as predicted by theory
and the numerical model (Time=27n/3 Time=n). ..................... 52
Figure 4.1 Map of South Florida. ....................................... 54
Figure 4.2 Boundaryfitted "coarse grid" (97x74 cells) used for numerical simulations of
Florida Bay circulation........................................... 56
Figure 4.3 Boundaryfitted "fine grid" (194x148 cells) used for numerical simulations of
Florida Bay circulation........................................... 57
Figure 4.4 Florida Bay bathymetry in the "fine grid" compiled from high resolution (20
m x 20 m) National Park Service data. ............................. 58
Figure 4.5 Location of Florida Bay stations. Interior stations were maintained by the
NPS. The Alligator Reef and Carysfort Reef stations were maintained by the
NOS. The Tennessee Reef station was maintained by the HBOI. ........... 60
Figure 4.6 Locations of CMAN wind data stations. .......................... 61
Figure 4.7 Offshore tidal forcing along the western boundary of Florida Bay. Top panel:
tidal amplitude. Lower panel: tidal phase. ........................ .64
Figure 4.8 Offshore tidal forcing along the southern boundary of Florida Bay. Top panel:
tidal amplitude. Lower panel: tidal phase. ......................... 65
Figure 4.9 Coamplitude chart for the M2 tidal constituent. The chart is constructed from
data measured at 35 study sites (Smith and Pitts 1995a). ............... 70
Figure 4.10 Cophase chart for the M, tidal constituent. The chart is constructed from
data measured at 35 study sites (Smith and Pitts 1995a) ................. 70
Figure 4.11 Coamplitude chart for the K1 tidal constituent. The chart is constructed
from data measured at 35 study sites (Smith and Pitts 1995a) .............. 71
Figure 4.12 Cophase chart for the K, tidal constituent. The chart is constructed from
data measured at 35 study sites (Smith and Pitts 1995a). ................ 71
Figure 4.13 Coamplitude chart for the O1 tidal constituent. The chart is constructed
from data measured at 35 study sites (Smith and Pitts 1995a) .............. 72
Figure 4.14 Cophase chart for the 0, tidal constituent. The chart is constructed from
data measured at 35 study sites (Smith and Pitts 1995a). .............. 72
Figure 4.15 Coamplitude chart of the M,3 simulation. Amplitudes are measured in
centimeters. ................................................73
Figure 4.16 Coamplitude chart of the M,9 simulation. Amplitudes are measured in
centimeters. .................................................73
Figure 4.17 Coamplitude chart of the M210 simulation. Amplitudes are measured in
centimeters. ................................................74
Figure 4.18 Cophase chart of the M,10 simulation ............................ 74
Figure 4.19 Coamplitude chart of the K,10 simulation. Amplitudes are measured in
centimeters. ..................................................75
Figure 4.20 Cophase chart of the K,10 simulation. .......................... 75
Figure 4.21 Coamplitude chart of the 0,10 simulation. Amplitudes are measured in
centimeters. ....................................................76
Figure 4.22 Cophase chart of the O010 simulation. .......................... 76
Figure 4.23 CMAN wind data from September 1 to October 31, 1993. The units of the
xaxis are Julian days. ......................................... 78
Figure 4.24 Comparison of simulated versus measured M2 amplitudes and phases at 10
stations in Florida Bay. ........................................ .86
Figure 4.25 Comparison of simulated versus measured K, amplitudes and phases at 10
stations in Florida Bay. ......................................... 87
Figure 4.26 Comparison of simulated versus measured 0, amplitudes and phases at 10
stations in Florida Bay. .......................................... 88
Figure 4.27 A map of mud banks within Florida Bay (from Enos and Perkins, 1979).. 90
Figure 4.28 Plot of maximum mud bank area for simulation Coarse02 (16.8%). Mud
banks are indicated by the dark regions ............................. 91
Figure 4.29 Plot of maximum mud bank area for simulation Fine02 (25.6%). Mud
banks are indicated by the dark regions. .............................. 92
Figure 4.30 Plot of maximum mud bank area for simulation Fine03 (33.3%). Mud
banks are indicated by the dark regions. .............................. 93
Figure 4.31 Plot of maximum mud bank area for simulation Fine04 (24.3%). Mud
banks are indicated by the dark regions. .............................. 94
Figure 4.32 Plot of maximum mud bank area for simulation Fine05 (26.4%). Mud
banks are indicated by the dark regions. .............................. 95
Figure 4.33 Plot of maximum mud bank area for simulation Fine06 (26.4%). Mud
banks are indicated by the dark regions. .............................. 96
Figure 4.34 Plot of maximum mud bank area for simulation Fine07 (26.2%). Mud
banks are indicated by the dark regions. .............................. 97
Figure 4.35 Plot of maximum mud bank area for simulation Fine08 (26.3%). Mud
banks are indicated by the dark regions. .............................. 98
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering
HYDRODYNAMIC MODELING IN SHALLOW WATER
WITH WETTING AND DRYING
By
Justin Ross Davis
December, 1996
Chairperson: Dr. Y. Peter Sheng
Major Department: Coastal and Oceanographic Engineering
The wetting and drying of shorelines in shallow estuaries, lakes, and coastal
waters takes place routinely and can cause significant physical and ecological
consequences. During hurricanes, storm surges can inundate coastal areas several
kilometers inshore. In Florida Bay, shallow mudbanks are often exposed to the
atmosphere during low tide and dry seasons. A previous study of the tidal circulation in
Florida Bay, which used a curvilineargrid hydrodynamic model, overestimated the tidal
amplitudes in Florida Bay, due to the lack of "wetting and drying" scheme in the model
and the coarse grid resolution. This study reviews several wetting and drying schemes
used in previous numerical modeling studies and based on the model review, a vertically
integrated curvilineargrid model is modified to incorporate a robust wetting and drying
scheme originally developed for a rectangular grid model. Basically, the curvilineargrid
finite difference equations of motion are reformulated such that a Poisson equation of the
water level is first solved using a conjugant gradient method before the velocities are
calculated. The modified verticallyintegrated model is tested with analytical solutions
and then calibrated with realistic tidal data from Florida Bay. The Florida Bay results
showed that the best tidal simulation is obtained when the wetting and drying scheme is
implemented and a fine grid resolution (50 m) is employed.
CHAPTER 1
INTRODUCTION
1.1 Background
In recent decades, concern over environmental quality has become a major social
and political issue in our society. Communities, once content with the status quo, are
now refocusing on the consequences on the health of human and ecological systems of
such practices as toxic waste dumping, sewage discharge, and the indiscriminate use of
pesticides and fertilizers. One particular area of concern is the health of such water
bodies as estuaries, coastal waters and lakes, which have provided humans with valuable
water resources, leisure activities, and esoteric beauty. In addition, these water bodies
also support numerous wildlife and fishery resources. In short, these water bodies are
very important and their continuing health is of prime societal importance.
To understand the health of water bodies, it is important to understand factors that
affect their wellness. Research has shown that direct exposure to high concentrations of
contaminants is harmful to human and ecological species; however, their pathways
through the environment and the effects of indirect exposure are still not fully understood.
Numerous laboratory and field studies have been, and are being performed in an attempt
to better understand the complex pathways of contaminants. These laboratory methods
2
have proven very useful for quantitative understanding of smallscale and local processes,
but not in quantifying largescale processes. Field studies are more effective in
quantifying largescale and global processes, although the cost is usually very high.
To complement the laboratory and field studies, numerical models can be used.
Through a careful process of model building, calibration and validation, it is possible to
develop models which can simulate the dynamics of various ecosystems.
Modern day fast and reliable computers and better understanding of processes
have helped numerical modeling to gain acceptance throughout the scientific community.
Numerical models can not only be used to study the past and present, but also to predict
the future. There are models capable of simulating chemical reactions on the molecular
scale as well as for simulating ocean circulation patterns on the global scale. There are
models for simulating both manmade and natural disasters which are impossible to
duplicate in laboratory or field studies.
Here at the University of Florida, numerous modeling and field studies have been
conducted in the past ten years to quantitatively understand the various hydrodynamics,
water quality, and ecological processes in estuaries, lakes, and coastal waters. As a result
of these studies, hydrodynamic models have been developed for Chesapeake Bay (Sheng
1989), James River (Sheng et al. 1989a, Choi 1992), Lake Okeechobee (Sheng et al.
1989b, 1990a, Sheng and Lee 1991, Lee and Sheng 1993, Chen 1994), Sarasota Bay and
Tampa Bay (Sheng and Peene 1993, Peene and Sheng 1995, Sheng et al. 1995), Florida
Bay (Sheng 1995), Tampa Bay (Sheng et al. 1994), Lake Apopka (Sheng and Meng 1993)
and Indian River Lagoon (Sheng et al. 1990b, 1993c). Water quality models, which
3
included the modeling of nitrogen cycling and phosphorus cycling, have been developed
for Lake Okeechobee (Sheng et al. 1993a, Chen and Sheng 1995), Roberts Bay (Sheng et
al. 1995a, 1995b), and Tampa Bay (Sheng et al. 1993b, Yassuda 1996). The studies of
Roberts Bay (Sheng et al. 1995a, 1995b) and Tampa Bay (Yassuda 1996) include the
modeling of seagrass in addition to hydrodynamics and water quality.
A robust hydrodynamic model is the prerequisite for successful water quality and
ecological modeling. Sheng (1994) reviewed hydrodynamic models and water quality
models for shallow waters and identified several important model features for shallow
water simulations. One such important feature is the ability of the model to resolve the
wetting and drying of shorelines.
As pointed out by Sheng (1994), most hydrodynamic models for estuaries, lakes,
and coastal waters do not have the ability to simulate wetting and drying of shorelines.
These models generally treat the shoreline as a vertical seawall with a finite depth, instead
of a moving boundary. Thus, a wet grid cell remains wet and a dry grid cell remains dry
all the time. In reality, however, a wet cell may become dry during lower water while a
dry cell may become a wet cell during high water. This feature is essential to successful
simulation of storm surges and tidal circulation in shallow waters, and is the major focus
of this study.
1.2 Review of Previous Work on Wetting and Drying
Many researchers have tackled wetting and drying in their numerical models. A
comprehensive review of the nearly three decades of work into this problem is provided
in this section.
To study storm surges in Galveston Bay, Reid and Bodine (1968) developed a
twodimensional finite difference model based on the vertically integrated equations of
motion. Their model equations included wind stress, rainfall and bottom friction terms,
but no allowance was made for momentum advection, except at wetting and drying
regions of the bay where the effect was included implicitly through the use of empirical
relations. A staggered grid system was used on a Cartesian mesh and each cell had a
uniform depth which provided a stairstep bathymetric approximation. Flow in the
wetted regions was controlled through a series of empirical relations. If the water surface
elevation in a wet cell exceeded the height of a neighboring dry cell, then flow was
permitted into the cell based on a relation for flow over a broad crested barrier. If the
surface elevation did not exceed the dry cell's height, then flow was not permitted. The
model also permitted flow across a submerged barrier. If the surface elevation on both
sides of the barrier exceeded the barrier then the flow rate was calculated using an
empirical relation for flow over a submerged weir. The empirical relations in the model
required the model to be calibrated to determine various coefficients. Two cases were
used to calibrate the model, a spring astronomical tide and Hurricane Carla (1961). The
5
model was then used to predict recorded surge heights during Hurricane Cindy (1963).
The gross features of flooding were predicted and the model's peak surge heights
correlated well when compared to the observed surge. However, the model as well as the
wetting and drying scheme were computational intensive and the empirical coefficients
related to wetting and drying must be determined for each application.
Yeh and Chou (1979) developed a twodimensional model for solving storm surge
problems with moving waterland interfaces. The vertically integrated model used a time
split explicit scheme on a staggered Cartesian grid. The shoreline was treated as a
discrete moving boundary and advanced or retreated according to the rise or fall of the
surge level. Like Reid and Bodine (1968), during a rising surge, a grid point was added
beyond the shoreline if the elevation of a wet cell next to a shoreline was above the height
of the shoreline. During a receding surge, a grid point would be removed if its total depth
decreased below a set value. The authors also designed a procedure to dampen small
wave disturbances caused by the discrete changing of the grid by increasing bottom
friction in shallow waters. Numerical experiments with observed storm surges over the
Gulf of Mexico and the southern coast of Maine showed that their moving boundary
model predicted a considerably lower surge than a fixed grid model, due to a better
representation of the shoreline.
A finite element technique for solving shallow water flow problems with moving
boundaries was developed by Lynch and Gray (1980). Their twodimensional model
accounted for moving boundaries by allowing nodes to move while maintaining their
initial connectivity. Node motion induced extra terms into the standard Galerkin finite
6
element formulation which are easily incorporated into existing fixedgrid programs.
Additionally, if the mesh became unacceptably skew, the domain was rezoned and nodes
were added or deleted as required. The model was tested with two example moving
boundary problems and the results were shown to be quite reasonable.
Benqu6 et al. (1982) developed a vertically integrated moving boundary model
which used the fractional step method in a Cartesian grid. Advection, diffusion, Coriolis,
wind and bottom stress terms were included in the model. The authors solved the
shallow water equations in three steps: advection, diffusion and propagation. A different
numerical scheme was applied at each step with the treatment of the boundary motion
considered in the propagation step and dry land was assumed to be covered with a thin
layer of water. The shallow water equations were first applied to the whole region
including the thin water layer. The flow in the moving boundary region was then
recalculated using equations for bottom friction dominated flow. Good agreement
between numerical results and measured data was presented, based on applications to the
Bay of Saint Brieuc and the River Canche Estuary, France. However, the moving
boundary model was reported to slightly violate the continuity equation.
The moving boundary model developed by Falconer and Owens (1987) solved the
depth integrated shallow water equations in a staggered Cartesian grid. The model
determined the shoreline as a function of the latest computed grid depth. The depths of
the four crosssections of a wetted cell were compared against a critical total depth value.
If the depth of any of the crosssections was less than the critical value, the cell was
removed from the computational field and the cell's Chezy coefficients and velocity
7
components were set to zero. As a cell was allowed to dry, the model assumed the
existence of a thin layer of water which corresponded to the last value before the cell
became dried. A cell became wet again if the average of the four surrounding total depths
exceeded the critical total depth. Along with a critical total depth, the model calculated a
critical surface elevation, below which a grid cell should be removed from the
computational grid. The model was successfully applied to the Humber Estuary, England
with stable and accurate predictions of surface elevation and velocity.
Liu (1988) developed a twodimensional fractional step model based the work of
Benqu6 et al. (1982) which assumed a thin water layer over the dry region. To validate
the moving boundary model, wave propagation onto a linearly sloping beach was studied
and compared to the theoretical solution obtained by Carrier and Greenspan (1958). With
the successful simulation of the sloping beach test, the model was applied to simulate the
wind driven circulation in Lake Okeechobee, Florida. The moving boundary model was
shown to perform better than the fixed boundary model, although mass conservation was
slightly violated.
Akanbi and Katopodes (1989) developed a moving boundary model which was
designed to simulate flood waves propagating on a dry bed. Their twodimensional
model used a dissipative finite element technique in a deforming coordinate system.
Advection, bottom friction and seepage terms were included in their model. Even though
many calculations were avoided by restricting simulations to regions covered by water, a
significant portion of computational time was devoted to grid regeneration and wave front
tracking. The accuracy of the model was tested by experimental measurements from field
8
tests. The field tests corresponded to a step increase in discharge in an initially dry,
permeable rectangular channel. The model appeared to agree well with the field data,
despite the coarse grid used in the computation.
Father and Hubbert (1990) presented a moving boundary study on Morecambe
Bay, a complex estuary on England's west coast. They simulated wetting and drying by
adding and removing cells as the water level rose and fell. To determine whether a cell
was wet or dry, the model checked the total depth at the center and the four sides and also
the surface slope. The model compared depths and surface slopes to critical values.
Once a cell was determined to be dry, the velocities around the cell were set to zero. Wet
cells were calculated from the momentum equations.
Casulli and Cheng (1992) developed a threedimensional semiimplicit finite
difference moving boundary model. The model solved the primitive variable, hydrostatic
equations in a Cartesian grid and included advection, wind stress and bottom friction
terms. The model used an explicit EulerianLagrangian approximation for the advection
terms. At every time step, total water depths at cell edges were checked and set to zero if
the total water depth became negative. The resulting zero depth meant a thin wall barrier
and the flow along the side was identically zero. A cell was considered dry if the total
water depths on all four sides were zero. Thus, the shorelines, defined by the condition of
no mass flux, were automatically determined. The model was applied to San Francisco
Bay, California and the Lagoon of Venice, Italy and reproduced wetting and drying areas
in the regions.
Luo (1993) presented a moving boundary model, tested the model's accuracy
9
against various theoretical solutions, and then applied the model to Lake Okeechobee,
Rupert Bay and Yellow Sea. The threedimensional numerical model solved the shallow
water equations in a Cartesian grid. Shallow region velocities were first calculated with
the standard momentum equations. These velocities were then compared to the velocities
calculated with the modified bottom friction dominated momentum equations and the
small velocity chosen. This bottom friction dominated flow region suppressed wave
disturbance in shallow water area. Dry cells were kept wet continuously through a thin
layer of water, slightly violating mass conservation.
Table 1.1 presents a summary of the works presented herein. Included in the table
is the present model enabling comparison to previous studies. Several different types of
shallow water wetting and drying treatment are shown in the table. Empirical velocity
relations explicitly adjust the flow in shallow waters. Explicitly increasing the bottom
friction in shallow waters is similar to empirical velocity relations with the exception of
only the bottom friction terms being adjusted. Deforming grid systems change the size of
the grid to accommodate the varying wetted waters. A bottom friction dominated flow
system uses a separate set of bottom friction dominated equations in shallow waters. In
the adding and withdrawing of cells scheme, the model only calculates the hydrodynamic
equations in regions which are determined to be wetted. Finally, the last type of wetting
and drying scheme determines the wetting and drying implicitly thru the finite difference
equations.
10
Table 1.1 A summary of moving boundary models.
Author(s) Model Type Shallow Water Treatment Grid System Model Applications
Reid and Bodine (1968) 2D Empirical velocity relations Cartesian Galveston Bay
Yeh and Chou (1979) 2D Increased bottom friction Cartesian Gulf of Mexico
Add and withdraw cells Southern Coast of Maine
Lynch and Gray (1980) 2D Bottom friction dominated flow Finite element N/A
Deforming grid
Benque et al. (1982) 2D Bottom friction dominated flow Cartesian Bay of Saint Brieuc
River Canche Estuary
Falconer and Owens (1987) 2D Add and withdraw cells Cartesian Humber Estuary
Liu (1988) 2D Bottom friction dominated flow Cartesian Lake Okeechobee
Akanbi and Katopodes (1989) 2D Tracked wave front Finite element N/A
Father and Hubbert (1990) 2D Add and withdraw cells Cartesian Morecambe Bay
Casulli and Cheng (1992) 3D Implicit Cartesian San Francisco Bay
Lagoon of Venice
Luo (1993) 3D Bottom friction dominated flow Cartesian Lake Okeechobee
Rupert Bay
Yellow Sea
Present Model (1996) 2D Implicit Curvilinear Florida Bay
1.3 Objectives
The objectives of this study include the following:
* Implement the Casulli and Cheng (1992) type wetting and drying scheme in the
3D curvilinear grid model CH3D.
* Test the modified CH3D model with analytical solutions, including the solution
for a wave propagating onto a sloping beach developed by Carrier and Greenspan
(1958).
* Calibrate and validate the model with field data from Florida Bay.
1.4 Organization of This Study
In Chapter 2, the differential momentum and continuity equations are written for a
boundary fitted coordinate system. In a manner similar to Casulli and Cheng (1992), the
differential equations are written in finite difference form and then solved for surface
elevation. The equations are written in vertically averaged form out of which the moving
boundary ability of the model follows directly. A fivediagonal system of equations
results which is solved using a conjugant gradient method. With the new surface
elevation determined the velocities can be backed out using the original finite difference
equations. In Chapter 3, several analytic problems are developed and compared to the
models results. Wind stress, tidal forcing, the moving boundary scheme and curvilinear
grid system are tested. Chapter 4 applies the wetting and drying model to Florida Bay.
Comparisons between the traditional nonwetting and drying model and the model
presented herein are presented and analyzed. Chapter 5 summarizes and concludes the
thesis.
CHAPTER 2
A NUMERICAL HYDRODYNAMICS MODEL
The basic hydrodynamics model used for this study is the threedimensional
curvilinear grid model, CH3D, originally developed by Sheng (1987, 1989, and 1994).
The model first solves the verticallyintegrated equations of motion, before solving the
equations for the deficit horizontal velocities (the difference between the vertically
varying horizontal velocities and the verticallyaveraged horizontal velocities). The
wetting and drying scheme in this study is implemented on the twodimensional
verticallyaveraged equations. After the solution of the verticallyaveraged equations are
obtained, the threedimensional velocities can then be solved similar to the original
CH3D model. In the following, the governing equations of the CH3D model are briefly
described. More detailed equations can be found in Sheng (1987, 1994) and Sheng et al.
(1989).
2.1 Governing Equations in Cartesian Coordinates
The governing threedimensional Cartesian equations describing constant density,
free surface flow can be derived from the NavierStokes equations. After turbulent
averaging, and applying the hydrostatic and Boussinesq approximations, the x and y
momentum and continuity equations have the following form (Sheng 1983):
13
au u2 + av auw a
at ax ay 9z ax.
+ a++a g +fr
2" a y2 a Oy a tZ'
v + vu v2 + avw = fu
at ax ay az ay
(a2v a2v9\ av9 v (22)
+A + +i A I
ax2 ay 2) az Vaz
vu vv vw
au + av 0, (2.3)
ax ay az
where u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) are the velocity components in the horizontal x,
y and vertical z directions; t is the time; ((x,y,t) is the free surface elevation; g is the
gravitational acceleration and AH and Av are the horizontal and vertical turbulent eddy
coefficients, respectively.
2.2 Vertical Boundary Conditions in Cartesian Coordinates
The boundary conditions at the free surface are specified by the prescribed wind
stresses T' and T":
au w 2+
A, = Cdal w+ V, (2.4)
dz
av WV 2
pAv = Ty = CdaVw +V, (2.5)
az
where T" is the wind stress at the free surface; uw and vw are the components of wind
speed measured at some distance above the free surface and Cda is the drag coefficient.
The drag coefficient is normally a function of the roughness of the sea surface and
the wind speed at some height above the water surface. For this study, the empirical
relationship developed by Garratt (1977) is used. Garrat defined the drag coefficient as a
linear function of wind speed measured at 10 meters above the water surface:
C, = 0.001(0.75 + 0.067Ws) (2.6)
where Ws is the wind speed in meters per second.
At the free surface, the kinematic condition states that
w + u + (2.7)
Dt at ax ay at
Combining the above equation with Equation (2.3) yields
( + u + = 0, (2.8)
at ax Oy
where U and V are the vertically integrated velocities.
The bottom boundary conditions for the three dimensional model satisfy the
quadratic stress law such that (Sheng 1983):
15
tb = 2 2+ (2.9)
"r b = P CdUbrUb + ,~Vb,
S2 2 (2.10)
Ty = p PCdVb Ub + Vb, (2.10)
where ub and vb are the near bottom velocities, p is the density of water and Cd is the
drag coefficient. The drag coefficient is defined as
K2
Cd
dln2( s ) (2.11)
where zo is the size of the bottom roughness elements and z1 is the height at which the
velocity is measured. This formulation is appropriate within the constant flux layer above
the bottom (Tennekes and Lumley 1972). In very shallow waters the drag coefficient is
set to a constant value which linearly varies to the value of the above equation at a certain
depth. At the bottom, the boundary conditions for the twodimensional model are given
using a Chezy formulation:
au b gU 2 g
pAI, x =2 (2.12)
9z C,
av b gvVu2+
PA = = 2 (2.13)
a C2,
where Cz is the Chezy friction coefficient which can be formulated as:
1
C 4.64 R (2.14)
n
whereR is the hydraulic radius given in centimeters and n is Manning's n. In shallow
estuaries, the hydraulic radius can be approximated by the total depth.
2.3 Coordinate Transformations
For applications to coastal and estuarine waters with complex geomorphology,
CH3D uses a astretched grid in the vertical directions and a boundary fitted curvilinear
grid in the horizontal direction.
2.3.1 A Vertically Stretched Grid
In threedimensional modeling of estuaries and lakes, two types of vertical grids
are used. The first grid type, a zgrid, defines constant depth layers along the zplane.
This structure represents the physics of the flow with the original simple governing
equations in a (x, y, z) grid. However, to obtain sufficient resolution in shallow waters,
a large number of layers in deep regions are required. Also, if the domain is insufficiently
resolved in the horizontal direction, a stair step representation of a normally smooth
varying bottom topography can occur.
The second grid type, a agrid, defines a constant number of vertical layers. The
17
advantages of this transformation are that the bottom topography is smoothly represented
and the vertical grid resolution is the same throughout the model's domain. The
disadvantages are that additional terms are introduced into the equations of motion and
continuity and, in regions where the bottom topography changes abruptly, errors can be
introduced (Haney 1990). The agrid is defined using
z (x,y,t)
o(x,y,z,t) = (,y,t) (2.15)
h(x,y)+((x,y,t)'
where h is the water depth relative to mean sea level and a is the transformed vertical
coordinate (Phillips 1957). The atransformation retains the original u and v velocities,
da
but produces a new variable o =, which is related to, but different from, the original
dt
vertical velocity w= dz. The astretching introduces extra terms, particularly in the
dt
horizontal diffusion terms, to the original equation.
2.3.2 Nondimensionalization
Dimensionless equations show the relative importance of the terms in the
governing equations. The dimensionless numbers) appear as multipliers in the equation,
thus allowing for easy comparison of all the terms. First, defining Xr and Zr as the
reference lengths in the vertical and horizontal directions, U, as the reference horizontal
Z
velocity, W= Ur as the reference vertical velocity, AHr as the reference horizontal
Xr
diffusion, Avr as the reference vertical diffusion, and Sr as the reference surface
elevation, the nondimensionalization procedure then follows Sheng (1986a) and Sheng
et al. (1991).
Dimensionless variables:
(u *, *) =
(x *,y *,z) =
u v Xr W
rwV
Ur Ur r Wr
xyz
X'XrZ
X Xr Zr
t* = tf
g( _
fr
w w ww
(W* W* (Wvx ,y)
,r( ',y ) P )
(fPoZrUr)
A
A H
Av A
Avr
X
O* = _r
Ur
Dimensionless groups:
Vertical Ekman Number:
Lateral Ekman Number:
Froude Number:
Rossby Number:
A
2
v Zr
AH
EH 2
Ur
F 
S U
fxr
gZr
f2X 2
Sr
WW)
T_ r<,
"r
(2.16)
(2.17)
R 2
r
19
2.3.3 VerticallyAveraged Equations in Boundary Fitted Coordinates
In the presence of a complex shoreline, a "boundaryfitted" grid allows
excellent representation of lateral boundaries. Using the elliptic grid generation
technique developed originally by Thompson (1982), a nonorthogonal boundaryfitted
grid can be generated in the horizontal directions. To solve for flow with a boundary
fitted grid, it is necessary to transform the governing equations from the original (x,y)
coordinates to the transformed coordinates (,rl). During the transformation, the
velocities are also transformed into contravariant velocities. These transformations and
further details can be found in Sheng (1986b, 1987 and 1989).
The crux of the wetting and drying model is implemented for the twodimensional
vertically averaged equations. The simplified equations, neglecting horizontal diffusion,
in the (,ri) grid and in dimensionless form are
__ a 1
a H
+ (L FH) + a(H)= 0, (2.18)
aHii l 1 8( 12 gl a( + 12 922
at 9 a H + 
 (xo( Hau + xH.ffh) + a (xoHiW + x1oHi,)1 (2.19)
Xn[(ysoHR + y oHip) + a(yFoH + YnIFHoT )
__L ll1
S 21a + g 22 )
y (xr
4 a +Xg'Ha
RofrO _
+ y ~Hfig) +
+ xIoHiT3) +
+ y HwI)] (2.20)
+ x/oHPi3)]}
1
where i = udz and 9
H,
h
depth defined as
1
 vdz are the depthaveraged velocities, H is the total
H
H = h + C,
and
g = J= xvy xyv
is the determinant of the metric tensor gij, which is defined as
2 2
x + y
gij =
'Jxx + yVty
xx + y gl g12
2 + 21 g22
x11 + T y9
whose inverse is
x J xt + Y (xx + ) gi g1 12
g (x)
9 P 2 2 21 22
(xex + y t) x + yg
OHt
8t
(2.21)
(2.22)
(2.23)
(2.24)
g11 HR
_ i_
( HyVg
"a (X~rgPiif
all
21
Since the model solves for the vertically averaged velocities first, the three
dimensional velocities are solved in terms of deficit velocities. These equations can be
found in Sheng (1987, 1989) and Yassuda (1996).
2.4 Finite Difference Equations
The vertically averaged differential equations are written in a
simplified form
+ Hi) + ( HP) 0, (2.25)
at F9. all
+ Hg a+F= 0, (2.26)
at a(
H + Hg + F = 0, (2.27)
at arl
where F. and F, are the remaining nonlinear, Coriolis, wind and bottom stress terms.
These simplified equations are written in the following finite difference form:
+, 1 + A tr6B /..C _, I.un _n+l]
,n + g0 u,.,l.i j &ttu )
Atp i .., ,n+l i., _n+l
Atop nii n n
g+ k 0g ,i+l,vvit+l )ijVDij
VoijS At(10 ii.) (2.28)
At(l1)P A n _,n'
H ,iuuij 1 + AtOlCDi1 j, u
+ AtH, .og! (1  t2ju
1n n()
A s$uiJ" 1 At(1 0l)CD2 ) ) (2.29)
AtH, (1 )gjj (ij 
.fijvvij 1 + AtO 1 CDiijv1
+i )' \ (2.30)
+ AtH^jOgJ (z1 C+1J = At
+s/*i* 1  At(1 ol)CDI ,vI
AtH1j,v(lO)g22( (
where F .. and F,4j are explicit finite difference representations of the remaining terms,
0 is the degree of implicitness of the pressure gradient terms and 01 is the degree of
implicitness of the bottom friction terms.
The vertically averaged u and v finite difference equations are substituted into the
continuity equation resulting in
di .(n+l n n+l n n+l n yn+l n .n+l n
Ij 1 Si+lu (+l sij,uilj sj+l ,v +l i1 = qj (2.31)
where
Hn
jat2I AUijugiU +Atocjv (2.32)
s = At2 22 H (2.33)
1 +AtOCDI l.jv
= + i+ u + Si ,u + S + Sij (2.34)
Ij j+ ij i ~ u I j+ ~v ij,vl
and finally
n = n
qij = ij
nn
At(16)f r u _+
At(l6)P r.,v 
10 '~
+ At2O( Fij+l,v uij +
~~1_ A +20( C v igilu
S+At +A(
+ Ajo~i3 F+1j jv
+ At2(1 O6)P( 1
1 +At
+ At2o(1 O)p g iv
1 +A
At2o(1 ) ( ,22
1+At
aud( At(1 9 1)CD ,u
1 +AtOlCDI 1),1 I
g.9.j,u i(1 At(1 O)CDI .juI)\
1 +AtO ICDI 'j,uI
+,v
1 I+AtOlCDEI ,~
t1 At ol )CDjv
Atop(
+ AtOp
,/p. j,
v1,ijl
SAtOep
+ AtOp (
iJ,ui
iJ~v Lj
1 +AtCDI j,
F ni ,v, j
1 +AtelCDIv
t 1CDI +j,uI
toICD iJ+1,v
i.C j *CJl)^
)C (C I I
lCDl v 1
)lD IiV
(2.35)
( i j )
25
2.5 Solution Technique
Equation (2.31) represents a fivediagonal system of linear equations. Large grids
are common in estuarine modeling; therefore, care must be taken when choosing a
method to solve the system of equations. Methods such as Gaussian elimination and LU
decomposition work well; however, they are not well suited to the diagonal sparse matrix
problems common in hydrodynamic modeling. When applied to sparse matrices, these
methods are slow and use a tremendous amount of computer storage.
The conjugate gradient method is an ideal choice for diagonal sparse matrix
problems because it is a fast and uses a minimal amount of computer memory. There are
two ways the conjugate method can be applied, serially and in parallel. Casulli and
Cheng (1992) present a serial conjugate gradient method which preconditions the matrix
for easier solving. Their method is designed to be applied to five diagonal systems of
linear equations; thus, ideally suited for solving Equation 2.31. The other method, know
as parallel conjugate gradient, is most easily applied on shared memory multiprocessors.
Special care is applied when using these methods to avoid memory conflicts in accessing
data. Wasserbau and Kiiste (1996) and Wang and Hwang (1995) use multicoloring
techniques to avoid this problem. The computational grid is colored such that the data
each processor accesses is never being accessed by another processor at the same time.
Because of its simple nature, the derivation of the conjugate gradient method
presented herein follows Casulli and Cheng (1992). The normalized form of Equation
(2.31) can be written as
n
n+l
(dS 1)iIj
n nnt
(d dd1>)
(
dijd _1
which, by letting
is equivalei
where
n) n+l
eid = 'ij ij
It to
eiJ ai+l,uei+l ai,uei ai+l,veij+1 aei,v ei bij
n
Sij,u
Sa..
Sij,v
aijv 
J,v I I
qij
n
rJ
n
n St~j
nn
Sij~
(
n
q1j
dij~n
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
27
The conjugate gradient algorithm to solve Equations (2.38)(2.41) takes the
following form:
(1) Guess ei(
) () (0) = () (0) (0) (0) (0)
(2) Set = =e ai,l,uei+ljai, uei jaij+l,veij+laid,veidl 
(3) Then for k=0, 1, 2,... and until (r(k),r(k)) < E, calculate
e(ki) e (k) (k)p k),
yi ij ij
ri+1) (k) (k)( p (k))ij,
*ij = id. CC i
p(ki) rik) (k)p (),
ij ij ij'
(2.44)
where p(k) = r+1 +1
(r (k), r (k))
In Equations (2.42) and (2.43), Mp is defined as
(Mp(k) () (k) (k) (k) (k)
(M a,i+  a1i;,vPi. j a,,iji
2.6 Numerical Stability
Consistency, stability and convergence are important when developing a finite
difference scheme. A finite difference equation (FDE) of a partial differential equation
(PDE) is consistent if the FDE reduces to the original PDE as the step sizes approach
where a(k) (r(k)'r (k))
(p (k),Mp (k)
(2.42)
(2.43)
(2.45)
28
zero. A numerical scheme is stable if any error induced in the FDE does not grow with
the solution of the FDE. A finite difference scheme is convergent if the solution of the
FDE approaches that of the PDE as the step sizes approach zero. Consistency and
stability ensure convergence.
Finite difference equations can be written in three different forms which are based
on the point in time that the terms are evaluated. Explicit FDEs evaluate terms at the old
time step, implicit FDEs evaluate terms at the new time step, while semiimplicit FDEs
use both new and old time steps. Explicit FDEs are easy to solve but can result in
stringent stability criteria as in the following example of time step criteria for various
terms:
Advection: At < min
IUmax
Ax mi
Propagation: At < m
S. (2.46)
Bottom Friction: At < 2
2 I maxI
Implicit FDEs lose the stability criteria but the solutions are numerically diffusive. This
formulation is harder to solve because it involves matrix inversion. Also, to produce
accurate results, the timestep is still bounded by the time scales of the physical processes
being simulated. Semiimplicit FDEs are the most complicated equations, although, in
principle they have no stability limit and are not numerically diffusive. However, in
practice, semiimplicit schemes tend to be slightly unstable.
29
It can be shown that when nonCartesian grid systems are used additional criteria
are also necessary. General guidelines are as follows:
Stepping Ratio:
Grid Skewness:
Aspect Ratio:
IAx, Axi1 < 20%
Ax j
kLin > 450
Ay..
Y <_ 10
Axij
is
where (min is the smallest interior angle of the horizontal grid cell.
(2.47)
CHAPTER 3
MODEL VERIFICATION
This chapter verifies the numerical accuracy of the numerical model developed in
the previous chapter though a series of analytical tests. Constant wind stress, seiche, and
tide tests ensure that the forcing mechanisms and boundary conditions are correct.
Additionally, an annular grid and linearly sloped basin tide tests ensure that the boundary
fitted grid, nonlinear terms, and wetting and drying scheme are accurate.
3.1 Wind Forcing
The analytical setup due to a constant wind stress in a rectangular basin can be
written as
C(x) = 1 (3.1)
where C is the setup of the water surface, Tw the applied wind stress, H and L are the
depth and the length of the basin respectively and the distance from the left edge is x.
The grid use in the wind stress test is a 21x5 cell, orthogonal grid with a length of
21 km and a width of 5 km. The depth is a constant 5 m and the grid spacings are fixed at
1 km in each direction. A constant wind stress of 1 dyne/cm2 is applied in the positive x
31
direction and a timestep of 60 s is used in the model. Table 3.1 shows the calculated
surface elevation at several locations and Figure 3.1 shows a plot of model results.
Table 3.1 Comparison between analytic and model setup surface elevations.
x (km) ( .,,, (cm) (., (cm)
0.5 2.04 2.04
10.5 0 0
20.5 2.04 2.04
5
4
3
2
1
0
1
2
3
4
5
x=20.5 km
x=10.5 km
x=0.5 km
0.0 0.5 1.0 1.5
Time (days)
Figure 3.1 Surface elevation at three locations within the domain.
3.2 Seiche Test
The next test assesses the model's ability to simulate a closed basin seiche.
Neglecting diffusion, friction, convective and Coriolis acceleration results in the local
acceleration and surface slope terms balancing. The onedimensional equations of
motion become
au a
S= g, (3.2)
at ax
9u 9C
h ,t (3.3)
ax at
where u is velocity, ( is surface elevation, h is the water depth and g is gravitational
acceleration. Letting 1 represent the basin length, then boundary conditions for this set
of equations are u = 0 at x = 0, I. The lowest mode for the above equations is given by
((x,t) = acos(Gt)cos(kx), (3.4)
u(x,t) = gaksin(t)sin(kx), (3.5)
where a is the wave amplitude, w is the circular frequency and k is the wave number.
The circular frequency is defined as
33
2n7
S= , (3.6)
T
where T is the wave period.
The grid used in the seiche test is a 21x5 cell orthogonal grid with a length, L, of
105 km and a width, W, of 25 km (Figure 3.2). The depth, H, is fixed at 5 m and the
grid spacings, Ax and Ay, are a constant 5 km. For a closed basin the period of the
seiche oscillation is
2L (2)(105,000 m)
T 2L (2)(105,0 m) 30,000 s. (3.7)
Hg H /(9.82 m/S 2)(5 m)
y=W
y=0 __L_ '
x=O x=L
Figure 3.2 Computational grid (21x5 cells).
The initial surface elevation is one half of a wave with a maximum of 10 cm at
x=0, and a minimum of 10 cm at x=L (Figure 3.3).
The model was run semiimplicitly, 0 = 0.51, with a At=60 s, for a time equal to
three seiche periods and a comparison between the analytical and numerical solutions
appears in Figure 3.4. The figure shows the numerical model predicting the theoretical
solution well.
L
15
E
" 10
C
0
0
CO
5
10
0 10 20 30 40 50 60
x (km)
Figure 3.3 Initial surface elevation for the seiche test.
1 5 I I I I 1 i  i I I I I I I I I i
0.0 0.2 0.4 0.6 0.8 1.0
Time (t/T)
Figure 3.4 Comparison between simulated (triangles) and theoretical (solid lines) surface
elevation for a seiche oscillation in a closed basin.
3.3 Tidal Forcing
Tidal simulation is one of the most important applications of an estuarine
hydrodynamic model. Thus, before applying the model to a real estuary, the model
should be compared to analytical tidal forcing problems. Lynch and Gray (1978) derived
analytic solutions for tidally forced estuaries of various geometries and depths.
Neglecting nonlinear, diffusion, friction, and Coriolis terms, the vertically averaged
equations of motion in a Cartesian coordinate system are the same as those given for the
seiche test, Equations (3.2)(3.3).
Again, letting 1 represent basin length, the tidally forced rectangular basin has the
following boundary conditions:
((x,t) = acos(ot), (3.8)
ac =0, (3.9)
where a and o are the tidal amplitude and frequency, respectively.
For a flat bottom, the solution to Equations (3.2)(3.3) is (Lynch and Gray 1978)
((x,t) = Reaeicos(mx)) (3.10)
cos(Pl) '
u(x,t) = Re ie it sin(P(xl) (3.11)
P cos() (3.11)
where Ho is the basin depth and
S= (3.12)
In an annular basin, Lynch and Grey (1978) determined the analytic solution to be
C(r,t) = Re[(AJo(pr) + BYo(pr))eit, (3.13)
u(r,t) = Re[( AJ1(r) BY(Pr)) i e i (3.14)
where Jo, J,, Yo and Y, are Bessel function and
aYl(pr1)
[Jo(pr2)Y1(pr) Yo(Pr2)J(Pr,)]'
(3.15)
aJl(pr,)
B = a (3.16)
[Jo(pr2)Y,(Pr,) Yo(Pr)J,(r)] (
Figure 3.5 is a diagram of the annular section with the numerical grid (42x5 cells)
overlaid. The following values were used in the test:
37
a = 50 cm
27r
CO 
9000 s
= 200
rl = 20 km
r = 83 km
H =lm
0
At = 30 s
The model was run for 10 cycles to attain steady state conditions. Figure 3.6
shows the maximum surface elevation and velocity.
g(e,t)
Figure 3.5 The annular section grid (42x5 cells).
120 30
110
100
E E3 20
'Er W. .
.9 90 E
1) E \
o
S80 \
I, 10
70 
S  Analytic Surface Elevation
60  Analytic Velocity
AP Model Surface Elevation
0 Model Velocity
50 . i ., i i 1 i' 0
0.0 0.2 0.4 0.6 0.8 1.0
Distance from Closed End (r/1)
Figure 3.6 Comparison between simulated surface elevation and velocity and analytic
solutions for a tidally forced flatbottom annular section.
3.4 Wetting and Drying Test
3.4.1 Simple Wetting and Drying Test
A diagram of the simple test is shown in Figure 3.7. Setting the depth of the
basin, H, to be 3 m and the height of the wall, Hwal, to be 1.5 m, and using a timestep of
60 s, the surface elevations of the left and right sides of the wall are shown in Figure 3.8.
This simple wetting and drying test shows qualitatively the model performing properly.
39
77
H
Figure 3.7 Simple test diagram.
300
250
200 \ Left Side
E
I I
150 .. .......... 
100 Right Side
50
0 .I .I .. .1.
0.0 0.2 0.4 0.6 0.8 1.0
Time (days)
Figure 3.8 Simple wetting and drying test results.
3.4.2 Wind Forcing in a Closed Basin with Linearly Varying Depth
Again, this test qualitatively checks the wetting and drying scheme within the
model. A diagram of the test without wetting and drying is shown in Figure 3.9. An
orthogonal grid system is used (50x5 cells). The length of the basin, x2x1, is 50 km and
the width is 5 km. A bottom slope of 1:100000 is used with a depth of 5 cm at the left
most edge and 55 cm at the right side and all the walls are assumed infinitely high.
A diagram of the wetting and drying test us shown in Figure 3.10. An orthogonal
grid (80x5 cells) is used with the same grid spacing and bottom slope as the previous test.
In this test, an additional 30 km of dry cells, from x=0 to x=x1, are included.
Both tests are run for 2 days with a 60 s timestep. The surface elevations at x=x,,
a near shore point where the still water depth is 5 cm, in shown in Figure 3.11. The
results show that models without the a wetting and drying feature significantly over
predict surge height.
41
w
H
X=L, X=L,
Figure 3.9 Storm surge diagram for nonwetting and drying simulation.
X Xw
H
X=O X=L X=L2
Figure 3.10 Storm surge diagram for wetting and drying simulation.
42
50
40
30
c 30 
0
O 20
10
Wetting and Drying Model
S  NonWetting and Drying Model
0, I I I
0.0 0.5 1.0 1.5 2.0
Time (days)
Figure 3.11 Comparison of simulated water levels at x=x1 in a sloping
basin, with and without the wetting and drying model.
3.4.3 Tide Forcing in a Rectangular Basin with Linearly Varying Depth
To validate the wetting and drying scheme developed, a robust analytical test
needs to be developed. Carrier and Greenspan (1958) obtained the theoretical solution to
wave propagation on a linearly sloping beach. Their solution was also used by Liu (1988)
and Luo (1993).
L
Figure 3.12 Wave propagating on a linearly sloping beach diagram.
The onedimensional nonlinear shallow water equations can be written as
_* + O (1 + h*)u* = 0, (3.18)
at* ax *
u ,*u* rl'*
+* + g 0, (3.19)
at x ax*
where asterisks denote dimensional quantities, ir is the water surface elevation above the
mean water level, h is the still water depth which varies linearly with x, u is the velocity
44
in the x direction. Letting L be the characteristic length scale of the wave. Then we can
define time and velocity scales as
T = L (3.20)
= vgL, (3.21)
where ( is the beach angle. The equations are then nondimensionalized using the
following relations:
X
x
L
t_ t*
T
11 (3.22)
4L
h *
U
U
u 
u
Defining
2 h* +11"*
2 + h + l = x + T, (3.23)
4L
Equations (3.18) and (3.19) then become
it + [(n + x)u =0, (3.24)
L Jx
ut + uux + r1x = 0.
Rewriting Equations (3.24) and (3.25) in terms of u and c gives
2c, + 2ucx + cux = 0,
ut + uux + 2ccx = 1.
(3.26)
(3.27)
Carrier and Greenspan transformed Equations (3.26) and (3.27) into a problem with only
one linear equation through a series of elegant transformations. A brief derivation will be
presented here.
Adding and subtracting Equations (3.26) and (3.27) gives
d
(u 2c t) = 0
dt
along
Defining the characteristic variables ( and as
S= u + 2c t,
( = u 2c t.
Equation (3.28) becomes
C = constant
dx
along = u + c,
dt
dx
 = const along u c.
dt
Assuming x and t are functions of ( and , then for ( = constant or = constant we
dx
dt u c.
dt
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
(3.25)
dx = ax /at
dt 9^ 88
dx = x it
dt a( C(
if C = constant,
if ( = constant.
From these two equations, we get
x = tj(u + c),
x = tt(u c).
From Equations (3.29) and (3.30), we can obtain
(3C ()
u + c + t,
4
(( 3E)
u  c 3 + t.
4
Substituting Equations (3.37) and (3.38) into Equations (3.35) and (3.36) yields
the following transform relationship between (x,t) and (C,):
tj(3( ) t2)
4 2
tC(( 3) (t2
x4 = 3
4 2
Eliminating x from Equations (3.39) and (3.40) results in
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
(3.38)
(3.39)
(3.40)
2(( + 0)tt + 3(tC + t) = 0,
a linear partial differential equation. It is convenient to introduce new variables a and X
defined as
XA = C = 2(t u),
(3.42)
(3.43)
Equation (3.41) then becomes
3t,
t = too
a
Since t = + u from Equation (3.42), u must also satisfy Equation (3.44)
2
3u,
ux = Uoo + 
Introducing a "potential", q(o,X), defined as
(P0
U=
a
(3.44)
(3.45)
(3.46)
then Equation (3.45) becomes
(P,
(PI;L= P0,, +
a
(3.47)
Equation (3.47) is a single partial differential equation whose boundary condition at the
a = + = 4c.
(3.41)
shoreline is
a = 0,
(3.48)
which corresponds to the condition c = 0, I. e., the total water depth at the shoreline
must equal zero at all times.
In terms of the variables a, X, and the potential p(o,X), Carrier and Greenspan
proposed the following expressions for t, x, rl, u, and c:
t = +U =
2 2 o
(3.49)
u 2 P 1 (pX2 y2
X + C + + + 
2 4 2 4 16'
2 o2 P o2
C .
4
(3.50)
(3.51)
(3.52)
(3.53)
If (p(o,X) is given, then Equations (3.49)(3.53) give t, x, Tr, u, and c parametrically in
terms of a and X. In general, it is difficult to obtain direct function relationships for r1
and u in terms of x and t.
49
Carrier and Greenspan pointed out a solution to Equation (3.47)
((oa,) = 8AoJo( sin( ), (3.54)
where Ao is an arbitrary amplitude parameter and Jo is a zeroth order Bessel function of
the first kind. This potential represents a standing wave solution resulting from the
perfect reflection of a unit frequency wave. With qp(o,X) given, Equations (3.49)(3.53)
will implicitly give the solution of the standing wave.
To evaluate ri(x,t) and u(x,t) for a given x and t, Equations (3.49)(3.53) must
be solved numerically. For specific values of x and t, a and X, are determined using a
GaussNewton method so that ri(x,t) and u(x,t) are easily obtained from Equations (3.51)
and (3.52), respectively.
The grid used in this analytical comparison is a 161x5 cell orthogonal grid with a
length, L, of 62 km and a width, W, of 10 km (Figure 3.13). The bottom slope, a, is
1:2500. The depth, h, varies from 2 m above mean sea level at x=0, to 22.8 m below sea
level at x=L. The Ay is a fixed value of 2 km while the Ax is variable. From 0 to 10.5
km the grid spacing is fixed at 100 m, from 10.5 km to 15 km, the grid spacing starts at
100 m and adds an additional 100 m each cell to a maximum of 1 km at the 15 km point.
From 15 km to 62 km, the grid spacing is fixed at 1 km. The model is forced atx=L with
a periodic forcing function of the form
(t) = acos( t), (3.55)
where the amplitude, a, is 11.24 cm and the period, T, is 3600 s.
y=W I II(t)
y=O
x=O x=L
Figure 3.13 Computational grid for the wetting and drying test (161x5 cells).
Figures 3.14 and Figure 3.15 are nondimensional wave profile comparisons
between the analytical solution and the model. From these figures, it is evident that the
model agrees well with the theory and the results are as good as those presented in the
previous wetting and drying studies of Liu (1988) and Luo (1993).
51
1.5
Time=0
1.0
0.5
0.0   
0.5
1.0
15 10 20 x 30 40
1.5
1.5 Time=7n/6
1.0
0.5
P0.0   A  A A
0.5
1.0
1.5 0 10 20 x 30 40
1.5
1h 5 Time=2/3
1.0
0.5
P '0.0  
0.5
1.0
0 10 20 X 30 40
1.5 Time=n/2
1.0
0.5
P'0.0
Theoretical Solution
0.5 Numerical Solution
1.0 . Mean Water Level
Shoreline
1.50 10 20 30 40
Figure 3.14 Nondimensional comparison between wave profiles as predicted by theory
and the numerical model (Time=O Time=Tc/2).
1.5
Time=2n/3
1.0
0.5
g '0.0  
0.5
1.0
15 10 20 X 30 40
1.5
Time=5x7/6
1.0
0.5
P'0.0  
0.5
1.0
1.5 b 10 20 3040
1.5 Theoretical Solution
1.0 Time= A Numerical Solution
.... Mean Water Level
0.5 Shoreline
P,0.0       
0.5
1.0
15 10 20 X 30 40
Figure 3.15 Nondimensional comparison between wave profiles as predicted by theory
and the numerical model (Time=27T/3 Time=in).
CHAPTER 4
APPLICATION OF MODEL TO FLORIDA BAY
In this chapter, a numerical experiment is presented to demonstrate an application
of the wetting and drying model to the prediction of surface elevations in Florida Bay.
Florida Bay is a very shallow estuary with significant wetting and drying of mud banks.
4.1 Overview of Numerical Simulation
In this section, the following are presented: model domain, model grid, model
bathymetry, data stations, forcing mechanisms, model boundary conditions, and
simulation procedure.
4.1.1 Model Domain
Florida Bay is the triangular, shallowwater estuary located directly south of the
Florida peninsula (Figure 4.1). The bay is bordered to the southeast by the Florida Keys
and to the west by the Gulf of Mexico. There is no well defined border between the gulf
and the bay and three differing definitions of the domain exist (Table 4.1). For the
purposes of this report, the western boundary is defined by a line from Cape Sable to
Matecumbe Keys and the southern boundary is defined by the Keys and the surface area
of 1550 km2 (Scholl 1966) will be used. The domain of the model extends further south
54
of the keys, to accommodate the use of measured water level data along the reef tract.
26.0 
25.5 Gulf of Mexico Atlantic
Ocean
25.0 Florida B
24.5
82 81 80
Longitude (W)
Figure 4.1 Map of South Florida.
Table 4.1 Borders and areas, according to three different schemes, of Florida Bay.
Western Boundary Southeastern Boundary Surface Area
Longitude 81005'W Keys 2140 km2
(Arbitrary) (Physical) (Smith and Pitts 1995a)
East Cape to Fiesta Key Intracoastal Waterway 1645 km2
(Everglades National Park Domain) (Everglades National Park Domain) (Smith and Pitts 1995a)
Cape Sable to Matecumbe Keys Keys 1550 km2
(Edge of Shallow Mud Banks) (Physical) (Scholl 1966)
4.1.2 Boundary Fitted Grids
The southern boundary of the model grid is positioned along the reef tract where
some NOS tide monitoring stations were located, while the western edge is placed along
the authorized boundary of the Everglades National Park. The coastline is then fitted
with a boundaryfitted grid which is nonorthogonal but as orthogonal as possible. Two
grids are generated for this study. A boundaryfitted grid for Florida Bay is shown in
Figure 4.2. This grid has a minimum spacing of about 100 m and will be referred to as
the "coarse grid" from hereon. A fine grid, which is generated by equally dividing each
cell of the coarse grid into four cells, is shown in Figure 4.3. Using the high resolution
(20 m x 20 m) bathymetry of the National Park Service, the "finegrid" bathymetry is
developed and shown in Figure 4.4.
I I
~3f
Kilometers
5 0 5 10 15
5 0 5 10 15
Figure 4.2 Boundaryfitted "coarse grid" (97x74 cells) used for numerical simulations of
Florida Bay circulation.
Kilometers
SHHH 10 1
5 0 5 10 15
Figure 4.3 Boundaryfitted "fine grid" (194x148 cells) used for numerical simulations of
Florida Bay circulation.
"C "
Vr ,
I .g' a'."
a   
r."
'' Q o " : .
I, ,
4t:. "' ,,' '%;." ': '"!) ..
1 .... *" & "y. '
., 
1; 
Kilometers
5 0 5 10 15
Depth (m)
10
9
8
7
6
5
4
3
2
1
Figure 4.4 Florida Bay bathymetry in the "fine grid" compiled from high resolution (20
m x 20 m) National Park Service data.
59
4.1.3 Available Water Level and Wind Data
Hydrodynamic monitoring of Florida Bay was conducted by a variety of
organizations including the National Park Service (NPS), the National Ocean Service
(NOS) and the Harbor Branch Oceanographic Institution (HBOI). Historical water level
data are available stations shown in Figure 4.5. In addition to water level data which can
be used for forcing and calibration of the model, wind data are also available at several
National Data Buoy Center's (NDBC), CoastalMarine Automated Network (CMAN)
stations (Figure 4.6).
The precise location of the NPS bay water level stations, the NOS offshore water
level stations,, and the CMAN wind stations are shown in Tables 4.3, 4.3, and 4.4,
respectively.
Table 4.2 Locations of the National Park Service inshore stations.
Name Latitude (N) Longitude (W)
BA Bob Allen Key 25001.6' 80040.9'
BK Buoy Key 25007.3' 80050.0'
BN Butternut Key 25 05.1' 80031.1'
DK Ducky Key 2501.8' 8029.4'
JK Johnson Key 25003.1' 80054.2'
LM Little Madeira Bay 25010.1' 80037.9'
LR Little Rabbit Key 24058.9' 80049.6'
PK Peterson Key 24055.1' 80044.8'
TC Trout Cove 25 12.7' 80032.0'
WB Whipray Basin 25004.7' 80043.7'
*
Carysfort Reef
S WB .WB
SBA
LR
PK
* Alligator Reef
Tennessee Reef
*0
Kilometers
HH 0 5 1 I
5 0 5 10 15
Figure 4.5 Location of Florida Bay stations. Interior stations were maintained by the
NPS. The Alligator Reef and Carysfort Reef stations were maintained by the NOS. The
Tennessee Reef station was maintained by the HBOI.
D
Molasses Reef (MLRF1)
Long Key (LONF1) * ,
Kilometers
5 0 5 10 15
5 0 5 10 15
* Sombrero Reef (SMKF1)
Figure 4.6 Locations of CMAN wind data stations.
Table 4.3 Locations of the offshore tidal stations (Ned Smith, personal communication,
February 10, 1995).
Station
Carysfort Reef
Alligator Reef
Tennessee Reef
Sombrero Key
Sand Key Light
Latitude (N)
25013.3'
24051.1'
24 44.1'
24037.6'
24027.2'
Longitude (W)
80 12.7'
80037.1'
80046.6'
81006.8'
81052.7'
Table 4.4 Locations of the CMAN wind data stations..
Station Latitude (N) Longitude (W)
MLRF1 Molasses Reef 25000.0' 80024.0'
LONF1 Long Key 24050.6' 80051.7'
SMKF1 Sombrero Reef 24036.0' 81006.0'
A summary of the NPS water level records for the inshore stations appears in
Table 4.5. Based on the water level data, harmonic analysis is conducted to produce the
amplitudes and phases of the major tidal constituents. The constituent data near the open
boundaries are used to produce the constituent data along the entire open boundary.
Based on the wind data, a wind stress field over the model grid can be produced.
I I,
63
Table 4.5 Summary of some of the NPS water level records.
Station Name
BA Bob Allen Key
BK Buoy Key
BN Butternut Key
DK Duck Key
JK Johnson Key
LM Little Madeira Bay
LR Little Rabbit Key
PK Peterson Key
TC Trout Cove
WB Whipray Basin
Beginning Date
Jan 03, 1990
Apr 17,1992
Jun 25, 1993
Jan 03, 1990
Apr 08, 1991
Apr 01, 1992
Oct 13, 1992
Jun 25, 1993
Jan 03, 1990
Nov 21, 1991
Apr 13, 1992
Mar 12, 1993
Jun 15, 1993
Nov 20, 1993
Jan 05, 1990
Nov 19, 1991
Sep 15, 1992
Aug 06, 1993
Nov 28, 1989
Mar 26, 1990
Sep 08, 1990
Oct 23,1991
Feb 10, 1993
Mar 24, 1993
Sep 09, 1993
Jan 04, 1990
Feb 16, 1993
Sep 07, 1993
Jan 03, 1990
Mar 24, 1993
Jun 25, 1993
Dec 27, 1993
Jan 05, 1990
Nov 18, 1991
Aug 14, 1992
Nov 27, 1992
Jun 25, 1993
Jan 04, 1990
Nov 08, 1990
Sep 03, 1992
Nov 23, 1992
Feb 12, 1993
Apr 23, 1993
Jun 14, 1993
Nov 10, 1993
Jan 05, 1990
Mar 19, 1993
Jun 25, 1993
Ending Date
Mar 19, 1992
May 26, 1993
Jan 25, 1994
Mar 15,1991
Mar 19, 1992
Oct 08, 1992
May 21, 1993
Jan 20, 1994
Oct 17,1991
Mar 30, 1992
Mar 11, 1993
May 27, 1993
Nov 11, 1993
Jan 24, 1994
Apr30, 1990
Aug 17, 1992
May 18, 1993
Jan 12, 1994
Feb 09, 1990
Aug 04, 1990
Sep 17, 1991
May 13, 1992
Mar 13, 1993
May 26, 1993
Oct 25, 1993
Dec 31, 1992
May 19, 1993
Jan 13, 1994
Mar 12, 1993
May 26, 1993
Nov 12, 1993
Jan 25, 1994
Nov 14, 1991
Jul 17, 1992
Nov 25, 1992
May 26, 1993
Jan 25, 1994
Oct09, 1990
Jul 16, 1992
Oct02, 1992
Dec 30, 1992
Mar 11, 1993
May 17,1993
Oct 21, 1993
Jan 12, 1994
Mar 12, 1993
May 19, 1993
Jan 25, 1994
I
64
4.1.4 Forcing Mechanisms and Boundary Conditions
In reality, Florida Bay circulation is driven by tide, wind, and density gradient. In
this study, the focus is on (1) purely tidedriven circulation, and (2) tide and winddriven
circulation. For both tidedriven and tide and winddriven circulation, tidal constituents
along the open boundaries are determined from the available water level data. The
amplitudes and phases of major constituents along the water boundaries are shown in
Figure 4.7 while those along the southern boundary are shown in Figure 4.8.
50
40
E
 30
CD
_. 20
E
10 
0
350 
300 ..... 
M2
250 . .. K1
250 . K,
v 200 .. O,
S150
Az
100
50
0
0 '= *=='=== ....=' 
Long Key (South) Florida Mainland (North)
Figure 4.7 Offshore tidal forcing along the western boundary of Florida Bay. Top panel:
tidal amplitude. Lower panel: tidal phase.
65
50
40
S30
20
E
10
0
350
300
250 .. . ..
v 200
C 150
M,
100
50  K1
0
Tennessee Reef (southwest) Alligator Reef Carysfort Reef (northeast)
Figure 4.8 Offshore tidal forcing along the southern boundary of Florida Bay. Top panel:
tidal amplitude. Lower panel: tidal phase.
For the tide and winddriven circulation, the wind stress field is applied over the
model domain, with the same tidal constituents along the open boundaries.
4.1.5 Simulation Procedure
Model simulations are conducted for each major constituent, and then with a
combination of three constituents. For the singleconstituent simulations, the model is
66
first spun up for 5 days, followed by a 5day run. For the threeconstituent simulations,
the model is first spun up for 5 days, followed by a 30day run to allow for the harmonic
analysis which uses 29 days of data. For each run, tidal forcing is imposed along both the
western and southern boundaries according to the curves shown in Figures 4.7 and 4.8.
Principal tidal constituents considered for this study are shown in Table 4.6. Tidal
constituents at the open boundary stations are shown in Table 4.7 (DeWitt Smith,
personal communication, 1994). In order to determine the relative importance of the
individual constituents, the percentage of total tidal amplitude was calculated for each
tidal constituent at each offshore station (Table 4.8). The three major constituents at the
offshore stations are the M2, K,, 0, tides which contribute three quarters of the total tidal
signal. These three constituents are then used to develop the tidal forcing of the model.
Table 4.6 A list of principal tidal constituents for Florida Bay.
Species and name Symbol Period Relative
(hours) Size
Semidiurnal:
Principal lunar M2 12.42 100
Principal solar S2 12.00 47
Larger lunar elliptic N, 12.66 19
Diurnal:
Lunisolar diurnal K, 23.93 58
Principal lunar diurnal 0, 25.82 42
Principal solar diurnal P, 24.07 19
Table 4.7 Major tidal constituents of offshore stations where amplitudes (rl) are given in
centimeters and local phase angles (K) are given in degrees (Ned Smith, personal
communication, February 20, 1995).
Tidal constituent
Station M, S, N, K, O, P,
Carysfort Reef ri 32.80 3.84 11.03 4.45 4.57 N/A
K 242.7 282.4 235.1 211.1 215.2 N/A
Alligator Reef i1 27.68 6.58 5.94 5.00 5.85 N/A
K 232.4 294.9 210.8 241.4 247.7 N/A
Tennessee Reef rl 27.55 6.68 6.27 10.03 12.65 3.34
K 217.8 264.9 187.4 138.7 262.3 138.7
Sombrero Key rI 22.62 5.39 4.48 6.64 7.38 N/A
K 244.0 260.4 225.1 262.4 262.6 N/A
Sand Key Light ri 17.28 4.97 3.57 8.11 8.53 N/A
K 246.2 267.6 229.0 272.9 269.7 N/A
Table 4.8 The relative importance of individual tidal constituents at each open boundary
station of Florida Bay.
Station
Carysfort Reef
Alligator Reef
Tennessee Reef
Sombrero Key
Sand Key Light
Average
M,
58%
54%
41%
49%
41%
48%
S,
7%
13%
10%
12%
12%
11%
Tidal constituent
N, K,
19% 8%
12% 10%
10% 15%
9% 14%
8% 19%
12% 13%
Total
0,
8%
11%
19%
16%
20%
15%
(cm)
56.69
51.05
66.52
46.51
42.46
52.65
(%)
100%
100%
100%
100%
100%
100%
I
I

4.2 Model Results
Two types of model simulations are performed to validate the wetting and drying
model. The first type of model simulation is for 10 days and contains singleconstituent
tidal forcing. Simulated amplitudes and phases for the constituent are plotted over the
entire model domain and then compared to those determined from measured data. The
second type of model simulation is 35 days and contains threeconstituent tidal forcing.
For several stations within the bay, a harmonic analysis is performed on the simulated
data and compared to the harmonically analyzed measured data. Additionally, simulated
mud banks are plotted and compared against those reported in the literature.
4.2.1 Single Constituent 10Day Simulations
Simulations are performed for both the coarse and fine grids along with various
combinations of wind forcing and are summarized in Table 4.9. Simulations are run with
a 60 s timestep for 10 days. The coamplitude and cophase charts are produced using
only the results during the last two tidal cycles of the 10 day run. Also, any setup
incurred during the model run is subtracted from the coamplitude charts. Bottom friction
is calculated implicitly with a Manning's n of 0.025. Smith and Pitts (1995a) present
plots of tidal amplitude and phase for several tidal constituents in the interior of Florida
Bay. Their M2, K, and 0, cotidal and cophase charts appear in Figures 4.9 thru 4.14.
The coamplitude charts for the coarse and fine grid cases with a northeast wind, runs M,
03 and M209, are shown in Figures 4.15 and 4.16, respectively. Comparing these cases
69
with the fine grid, wetting and drying simulation with a northeast wind, run M,10 (Figure
4.17), shows the benefit of both the finer grid and the wetting and drying mode. The co
phase chart for simulation M210 is shown in Figure 4.18. Results from the most accurate
K, and 0, model simulations are shown in Figures 4.19 thru 4.22.
Table 4.9 Summary of the singleconstituent runs.
Run Number Grid Wetting and drying Wind
01 Coarse No No
02 Coarse Yes No
03 Coarse No Northeast
04 Coarse Yes Northeast
05 Coarse No Southeast
06 Coarse Yes Southeast
07 Fine No No
08 Fine Yes No
09 Fine No Northeast
10 Fine Yes Northeast
11 Fine No Southeast
12 Fine Yes Southeast
M2 Amplitude (cm)
S 5 '
i _o" or
Figure 4.9 Coamplitude chart for the M2 tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
30
I
I I
i^"ss
0 20km
I I
Figure 4.10 Cophase chart for the M2 tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
K1 Amplitude (cm)
" *0 20km
I I
Figure 4.11 Coamplitude chart for the K, tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
K1 Phase
0 20km
t I
Figure 4.12 Cophase chart for the K, tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
0 0 20km
I 1
Figure 4.13 Coamplitude chart for the 0, tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
01 Phase Angle (
0 20km
! !__
Figure 4.14 Cophase chart for the 0, tidal constituent.
The chart is constructed from data measured at 35 study
sites (Smith and Pitts 1995a).
Kilometers
5 0 5 10 15
5 0 5 10 15
Figure 4.15 Coamplitude chart of the M23 simulation.
Amplitudes are measured in centimeters.
Kilometers
5 0 5 10 15
Figure 4.16 Coamplitude chart of the M29 simulation.
Amplitudes are measured in centimeters.
Kilometers
5 0 5 10 15
5 0 5 10 15
Figure 4.17 Coamplitude chart of the M210 simulation.
Amplitudes are measured in centimeters.
Kilometers
5 0 5 10 15
Figure 4.18 Cophase chart of the M210 simulation.
I~
~ _
f(~l
r.
4",' ^ "
75
Kilometers
5 0 5 10 15
Figure 4.19 Coamplitude chart of the K,10 simulation.
Amplitudes are measured in centimeters.
Kilometers
5 0 10 15
5 0 5 10 15
Figure 4.20 Cophase chart of the K,10 simulation.
76
Kilometers
HH I
5 0 5 10 15
Figure 4.21 Coamplitude chart of the O,10 simulation.
Amplitudes are measured in centimeters.
i. ,..
Kilometers
5 0 5 10 15
Figure 4.22 Cophase chart of the O,10 simulation.
4.2.2 Three constituent 35Day Simulations
A series of 35day model simulations of Florida Bay circulation has been
performed to examine the effects of grid (coarse and fine grids), wetting and drying,
wind, bottom friction and Coriolis acceleration on the tidal circulation. For those runs
which include wind forcing, both realtime and seasonally predominant wind data are
used. According to Smith and Pitts (1995b), a southeast wind is predominant from
February thru September and a northeast wind is predominant from October thru January.
Vector plots of wind speed measured at the three CMAN stations from October to
September, 1993, are shown in Figure 4.23. The model is run from Julian Day 255 to
290. Five days are allotted for a hydrodynamic spinup of the system, after which 29 days
are used for harmonic analysis with one extra day left over. Table 4.10 presents a
summary of the ten 35day runs performed.
Using a 29day Fourier analysis program for tidal prediction, the tidal
constituents, M2, K,, and O, were calculated at 10 stations in the interior of Florida Bay.
With a 29day time series of hourly water level along with the longitude of the station as
input, the program calculates the amplitude and local phase as output. These values are
then compared to the measured values from Julian day 260 (September 17) through
Julian day 289 (October 16), 1993. Relative and percent errors of simulated amplitudes
and phases versus the measured ones are summarized in Tables 4.11 thru 4.14.
LONF1
MLRF1
SMKF1
5 m/sec
5 m/sec
5 m/sec
Figure 4.23 CMAN wind data from September 1 to October 31, 1993. The units of the
xaxis are Julian days.
Table 4.10 Summary of Florida Bay simulations. Each simulation has M2, K1, and 0,
tidal forcing along both the western and southern boundaries.
Name Grid Wetting and drying Wind Manning's n Coriolis
Coarse01 Coarse No No 0.03 No
Coarse02 Coarse Yes No 0.03 No
Fine01 Fine No No 0.03 No
Fine02 Fine Yes No 0.03 No
Fine03 Fine Yes Northeast 0.03 No
Fine04 Fine Yes Southeast 0.03 No
Fine05 Fine Yes Real 0.03 No
Fine06 Fine Yes Real 0.04 No
Fine07 Fine Yes Real 0.03 Yes
Fine08 Fine Yes Real 0.04 Yes
In order to identify the "best simulation" from the 10 runs, a nonparametric
ranking system is used. Each of the 10 runs is ranked in terms of both amplitude and
phase angle, from 1 to 10, 1 being the closest to the measured value and 10 being the
furthest from the measured value. Table 4.15 presents the rankings for each of the
individual forcing constituents as well as rankings for the diurnal constituents and all of
the constituents combined. Combining the amplitude ranking with the phase ranking
produces a combined ranking which, when applied to the combined constituent case,
highlights the best simulation: Fine04. This run uses the fine grid and the wetting and
80
drying scheme with a southeast wind. Since the tidal forcing is based on constituents
determined from oneyear (1993) data, it is not surprising that the realtime wind cases
(Fine05 thru Fine08) are not the best simulations, while the southeast wind case (Fine
04) is the best simulation. A comparison of simulated versus measured M,, K,, and 0,
constituents for runs Coarse01, Coarse02, Fine01, Fine02, and Fine04 are shown in
Figures 4.24, 4.25, and 4.26, respectively. These figures show graphically the
improvement provided by the coarse grid and the wetting and drying model in simulating
constituents.
Table 4.11 Comparison between simulated and measured tidal amplitudes and phases at Stations BA, BK, and BN. The best
simulations are highlighted.
BA BK BN
Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg)
_Mag. Rel. Err. Per. Err. Mag. Rel. Err. Mag. Rel. Err. Per. Err. Mag. Rl. Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err.
Measured 5.48 0.00 0.0% 311 0 2.39 0.00 0.0% 173 0 0.85 0.00 0.0% 33 0
Coarse01 3.04 2.44 44.5%. 30 49 7 70 5 31 22 232 59 2.78 1.94 228.6% 58 25
Coarse02 2.99 2.49 45.4% 20 3 2.33 006 2.6'5 2 .3 .9 2.47 1.63 191.9% 79 47
M Fine01 2.34 3.14 57.3% 342 31 7 39 4 8 0i 5:: 2?3 6 2.09 1.24 146.4% 46 13
2 Fine02 1.64 3.84 70.1% 12 61 1.51 0.88 36.7% 294 121 1.19 0.35 40.8% 85 53
Fine03 1.97 3.51 64.1% 37 86 1.11 1.28 53.7% 327 154 0.64 0.20 24.0% 110 78
Fine04 1.74 3.74 68.2% 24 73 1.40 0.99 41.4% 276 103 0.89 0.04 5.2% 99 67
Fine05 1.76 3.72 67.9% 23 72 1.40 0.99 41.5% 294 121 0.80 0.04 5.1% 103 71
Fine06 1.41 4.07 74.2% 37 86 1.07 1.32 55.3% 311 138 0.49 0.35 41.7% 107 74
Fine07 1.71 3.77 68.8% 25 74 1.50 0.89 37.2% 299 126 0.84 0.01 0.8' 105 72
Fine08 1.46 4.02 73.4% 37 86 1.06 1.33 55.7% 313 140 0 5.A .0 .:. .31 0 111 78
Measured 0.93 0.00 0.0% 343 0 1.24 0.00 0.0% 88 0 0.36 0.00 0.0% 98 0
Coarse01 1.56 0.63 67.7% 326 17 5.09 3.85 310.8% 315 133 1.16 0.80 225.5% 10 88
Coarse02 0.92 0.01 1.11 312 .31 2.95 1.72 138.6% 346 102 0.63 0.28 78.1% 17 81
K Fine01 I r4 0 91 97 8 333 10 4.63 3.39 273.8% 321 128 1.65 1.30 364.3% 12 86
1 Fine02 0.96 0.03 3.2% 320 23 2.13 0.89 71.9% 4 85 0.75 0.39 109.3% 17 81
Fine03 1.08 0.15 16.1% 320 23 1.63 0.39 31.6% 47 41 0.45 0.09 26.6% 33 65
Fine04 0.91 0.02 2.2% 329 14 2.17 0.94 75.6% 14 74 0.60 0.25 69.2% 34 64
Fine05 1.26 0.33 35.5% 309 34 1.61 0.38 30.4% 10 78 0.42 006 17 41 262 164
Fine06 0.97 0.04 4.0% 312 31 1.27 0.03 2.2% 28 60 043 0312 31 241 143
Fine07 1.25 0.32 34.1% 306 37 1.59 0.35 28.1% 15 74 0.49 0.14 38.1% 256 158
Fine08 0.96 0.03 2.8% 311 32 1.20 0.04 3.4% 29 60 0.43 0.07 20.2% 234 136
Measured 1.22 0.00 0.0% 352 0 1.41 0.00 0.0% 76 0 0.28 0.00 0.0% 98 0
Coarse01 1.35 0.13 10.7% 227 125 3.87 2.46 173.5% 242 167 1.12 0.84 304.3% 270 171
Coarse02 1.12 010 8.2, 211 1411 2.31 0.90 63.3% 268 168 0.91 0.63 227.8% 245 147
0 Fine01 I 37 i 12 3. 242 110 3.59 2.17 153.7% 247 171 1.34 1.07 386.0% 282 176
1 Fine02 0.86 0.36 29.5% 227 125 1.64 0.23 16.1% 283 152 0.70 0.43 154.4% 271 173
Fine03 1.10 0.12 9.8% 218 134 1.29 0.13 9.1% 330 105 0.49 0.21 76.61. 284 174
Fine04 0.90 0.32 26.2% 227 125 1.70 0.28 19.9% 294 142 0.61 0.34 121.3% 278 180
Fine05 0.93 0.29 23.8% 227 125 1.58 0.16 11.4% 295 140 1.03 0.76 273.4% 275 176
Fine06 0.72 0.50 41.2% 235 117 1.15 0.26 18.7% 312 124 0.70 0.43 154.4% 283 175
Fine07 0.90 0.32 26.2% 228 124 1.45 003 2.20 295 141 0.95 0.68 245.1% 269 171
Fine08 0.75 0.47 38.3% 233 119 1.12 0.29 20.8% 311 124 0.68 0.41 147.7% 279 179
Table 4.12 Comparison between simulated and measured tidal amplitudes and phases at Stations DK, JK, and LM. The best
simulations are highlighted.
DK JK LM
Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg)
Mag. Rel. Err. Per. Err. Mag. Rel. Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err.
Measured 0.77 0.00 0.0% 46 0 19.78 0.00 0.0% 95 0 0.91 0.00 0.0% 44 0
Coarse01 3.10 2.33 301.4% 72 25 17.30 2.48 12.6% 196 101 2.56 1.65 180.4% 62 18
Coarse02 2.95 2.18 281.7% 93 47 17.81 1.97 10.0% 198 103 2.57 1.65 180.5% 84 40
M Fine01 2.25 1.48 191.5% 55 9 17 3 u 12 1: 197 102 1.93 1.01 110.8% 43 1
2 Fine02 1.23 0.45 58.4% 9I V 20.10 0.31 1.6% 197 102 1.23 0.32 34.6% 85 41
Fine03 0.59 0.19 24.3% 181 135 20.71 0.93 4.7% 198 103 0.70 0.21 22.9% 121 77
Fine04 0.92 0.15 19.5% 124 77 19.39 0.39 2.0% 198 103 0.87 0.05 5.3% 100 56
Fine05 0.86 0.09 11.5. 128 82 20.20 0.42 2.1% 197 102 0.94 0.02 2.30 106 62
Fine06 0.53 0.25 31.9% 145 98 19.43 0.36 1.8% 199 105 0.61 0.30 33.2% 126 82
Fine07 0.96 0.19 24.7% 128 82 20.17 0.38 1.9% 197 102 0.94 0.03 2.7% 110 66
Fine08 0.60 0.17 21.8% 151 104 19.43 0.35 1.8% 199 105 0.62 0.29 31.8% 123 79
Measured 0.65 0.00 0.0% 81 0 5.14 0.00 0.0% 357 0 1.18 0.00 0.0% 30 0
Coarse01 1.29 0.64 99.1% 21 60 8 61 3 .. c7. 8 .73 1.27 0.09 7.3% 16 14
Coarse02 061 0.04 5.5 13 68 8 70 3 56 i. 286 71 069 0 0 22' 5: 8 22
K Fine01 1.75 1.11 171.5% 19 63 8.26 3.12 60.60 2: .72 1 15 0 37 31 .1 13 17
1 Fine02 0.60 0.04 6.6% 23 58 8.93 3.78 73.5% 282 75 0.75 0.43 36.4% 16 13
Fine03 0.52 0.13 20.2% 71 10 9.11 3.97 77.1% 281 76 0.43 0.75 63.4% 34 4
Fine04 0.60 0.04 6.6% 47 34 8.78 3.64 70.7% 285 72 0.57 0.62 52.2% 33 3
Fine05 0.79 0.14 22.2% 168 87 8.89 3.74 72.8% 281 75 0.58 0.60 51.0% 332 58
Fine06 0.92 0.27 41.7% 170 89 8.68 3.54 68.8% 284 73 0.54 0.65 54.7% 341 48
Fine07 0.77 0.12 19.3% 170 89 8.90 3.76 73.0% 281 75 0.76 0.42 35.4% 335 55
Fine08 0.83 0.18 28.2% 174 93 8.64 3.49 67.9% 284 73 0.54 0.64 54.1% 340 49
Measured 0.64 0.00 0.0% 104 0 4.30 0.00 0.0% 359 0 0.33 0.00 0.0% 52 0
Coarse01 1.17 0.53 82.4% 272 167 5.78 1.48 34.3% 220 139 1.12 0.79 241.4% 262 150
Coarse02 0.91 0.27 42.8% 255 151 5.84 1.54 35.8% 224 135 0.92 0.59 179.4% 253 159
0 Fine01 1.34 0.70 109.8% 283 179 5.60 1.30 30.3. 221 138 1.22 0.90 272.3% 278 134
1 Fine02 0.60 0.04 5.8% 273 169 6.13 1.83 42.5% 220 138 0.71 0.38 115.7% 271 140
Fine03 O) .0 10 .i; : 321 143 6.28 1.98 46.0% 220 139 0.46 0.13 40.7% 285 126
Fine04 066 0.02 3.5'. ;97 .17 6.00 1.70 39.4% 225 134 0.57 0.24 73.5% 279 133
Fine05 1.08 0.44 69.1% 266 162 6.14 1.84 42.8% 221 138 0.79 0.46 140.6% 271 140
Fine06 0.75 0.12 18.1% 271 167 6.00 1.70 39.5% 223 135 0.46 0.13 40.60o 280 132
Fine07 1.07 0.43 67.7% 265 161 6.16 1.86 43.1% 221 138 0.76 0.43 131.7% 267 145
Fine08 0.80 0.16 25.1% 266 162 6.00 1.70 39.4% 223 135 0.55 0.22 66.4% 271 140
Table 4.13 Comparison between simulated and measured tidal amplitudes and phases at Stations LR, PK, and TC. The best
simulations are highlighted.
LR PK TC
Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg) Amplitude (cm) Phase (deg)
_Mag. Rel. Err. Per. Err. Mag. Rel. Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err.
Measured 2 i IJI 0 Oi: 11 0 701 i)I) 0 3015 0 0 73 0.00 0.0% 53 0
Coarse01 3.11 0.53 20.7% 219 107 524 2.25 30.1". 335 30 309 2.36 325.0% 70 17
Coarse02 67 1 ? 739 217 105 1 7 . 71 ... 4: ? 2 9 2.21 305.0% 93 39
M Fine01 3.14 0.56 21.8% 231 119 3.80 3.69 49.3% 319 14 2.23 1.50 207.2% 54 1
2 Fine02 0.90 1.68 65.2% 321 151 2.83 4.66 62.2% 334 29 1.28 0.56 76.8% 92 39
Fine03 0.64 1.93 75.1% 323 149 2.86 4.63 61.8% 347 42 0.09 0.63 87.4% 197 144
Fine04 0.51 2.07 80.2% 332 140 2.78 4.71 62.9% 342 37 0.92 0.19 26.7% 127 73
Fine05 0.89 1.69 65.5% 322 151 2.86 4.63 61.8% 340 35 0.92 0.19 26.8% 135 81
Fine06 0.65 1.93 75.0% 330 142 2.58 4.91 65.6% 350 45 0.62 0.10 14.4% 153 99
Fine07 0.82 1.76 68.2% 329 143 2.88 4.61 61.5% 341 36 1.02 0.29 40.5% 136 83
Fine08 0.50 2.08 80.6% 343 129 2.57 4.92 65.7% 351 46 0.62 0.11 15.3% 155 102
Measured 1.68 0.00 0.0% 11 0 1.84 0.00 0.0% 320 0 0.97 0.00 0.0% 121 0
Coarse01 4.86 3.18 188.8% 288 82 1.98 0.13 7.3% 285 35 1.24 0.26 27.1% 18 103
Coarse02 3.66 1.97 117.3% 286 85 1.88 0.04 2.1% 271 49 0 .0 0 37 .330': 13 108
K Fine01 4.26 2.58 153.2% 293 77 2 0 37 2 295 26 1.70 0.73 74.6% 22 98
1 Fine02 2.49 0.81 48.0% 299 71 1.86 0.02 1.1. ..8 4 .2f. 0.67 0.30 31.1% 23 98
Fine03 3.09 1.41 83.8% 303 68 1.98 0.14 7.7% 288 32 0.33 0.65 66.6% 174 53
Fine04 2.46 0.77 46.0% 303 68 1.75 0.09 4.7% 286 34 0.56 0.42 42.9% 47 74
Fine05 3.04 1.36 81.0% 305 66 2.15 0.31 16.9% 290 30 0.48 0.49 50.3% 175 54
Fine06 2.65 0.97 57.5% 311 60 1.90 0.06 3.1% 295 26 0.57 0.40 41.4% 153 32
Fine07 3.07 1.39 82.4% 304 67 2.35 0.51 27.5% 283 37 0.38 0.60 61.4% 173 52
Fine08 26, 0 5E 9:: 310 60 1 ,1 7 4:. ?'I ; :1 4,6 .J'. ':: 1 .6 ?!
Measured 1.77 ,.,,j 0 .: 0 3 : 2?,) ,0 ,j) r; I 10 0
Coarse01 3.16 1.39 78.4% 214 166 209 0.02 0.8. 177 153 1.13 0.47 69.8% 272 163
Coarse02 2.68 0.91 51.2% 207 173 2.12 0.04 2.1% 172 158 0.91 0.24 35.6% 255 146
0 Fine01 2.81 1.04 58.7% 218 161 1.75 0.32 15.6% 196 134 1.31 0.64 95.8% 281 171
1 Fine02 1.74 0.04 2.1% 215 165 1.57 0.51 24.5% 188 142 0.65 0.02 2.3% 274 165
Fine03 2.18 0.41 23.1% 216 164 1.89 0.19 9.1% 186 144 0.33 0.34 51.2% 80 29
Fine04 1.77 0.00 0 0o 217 162 1.73 0.35 16.6% 183 147 0.57 0.10 15.4% 290 179
Fine05 1.67 0.10 5.8% 214 166 1.67 0.41 19.6% 189 141 0.87 0.20 30.2% 261 152
Fine06 1.53 0.24 13.5% 223 157 1.50 0.58 27.9% 190 140 066 0.00 0.5% 268 159
Fine07 1.65 0.12 6.9% 217 163 1.68 0.40 19.1% 185 145 0.90 0.23 34.4% 258 149
Fine08 1.46 0.31 17.5% 227 153 1.49 0.58 28.1% 190 140 0.69 0.02 3.2% 264 155
Table 4.14 Comparison between measured and simulated tidal amplitudes and phases
at Station WB. The best simulations are highlighted.
WB
Amplitude (cm) Phase (deg)
Mag. Rel. Err. Per. Err. Mag. Rel. Err.
Measured 0.54 0.00 0.0% 214 0
Coarse01 1.70 1.17 217.4% 295 81
Coarse02 1.60 1.07 199.4% 31 178
M Fine01 3.03 2.50 466.1% 271 57
2 Fine02 0.93 0.39 73.0% 349 135
Fine03 0.29 0.24 45.4% 65 149
Fine04 0.60 0.06 11.4% 345 131
Fine05 0.69 0.16 29.5% 1 147
Fine06 0.39 0.15 27.3% 15 161
Fine07 0.72 0.19 34.7% 3 149
Fine08 0.42 0.12 21.8% 16 162
Measured 1.12 0.00 0.0% 69 0
Coarse01 2.80 1.68 150.9% 325 104
Coarse02 1.45 0.34 30.1% 333 96
K Fine01 3.33 2.21 198.3% 327 102
1 Fine02 1.51 0.39 35.3% 359 70
Fine03 1.02 0.09 8.5% 48 21
Fine04 1.75 0.63 56.8% 13 56
Fine05 0.80 0.32 28.5% 8 61
Fine06 0.41 0.71 63.4% 50 19
Fine07 0.80 0.31 27.9% 5 64
Fine08 0.56 0.55 49.6% 45 24
Measured 1.37 0.00 0.0% 71 0
Coarse01 2.14 0.76 55.8% 244 173
Coarse02 1.35 0.02 1.5% 242 172
0 Fine01 2.44 1.07 78.0% 249 178
1 Fine02 1.11 0.26 19.3% 271 159
Fine03 0.75 0.63 45.6% 323 108
Fine04 1.32 0.05 4.0% 288 142
Fine05 1.33 0.05 3.3% 277 154
Fine06 0.88 0.49 36.0% 287 143
Fine07 1.26 0.11 8.1% 278 153
Fine08 0.79 0.58 42.4% 288 143
85
Table 4.15 Rankings of the 35 day simulations. The best simulations are indicated by the
lowest total scores and are highlighted within the table.
Amplitude Phase Combined
Amplitude Rankings Total Phase Rankings Total Combined Rankings Total
BA BK BN DK JK LM LR PK TCW Score BA BK BN DK JK LM LR PK TC WB Score BA BK BN DK JK LM LR PK TC WB Score
CoarseOl 1 10 10 10 10 9 1 1 10 9 71 2 1 2 2 1 2 2 3 2 2 19 3 11 12 12 11 11 3 4 12 11 90
Coarse02 2 1 9 9 8 10 6 2 9 8 64 4 3 3 4 8 3 1 7 4 10 47 6 4 12 13 16 13 7 9 13 18 111
M Fine_01 3 9 8 8 9 8 2 3 8 10 68 1 2 1 1 5 1 3 1 1 1 17 I1 11 9 9 1 9 4 9 11 85
2 Fine 02 8 2 6 7 1 7 3 7 6 7 54 i J : i 3. I, :11 I. . 3 II
Fine_03 4 6 4 4 7 4 8 6 7 6 56 8 10 9 10 7 8 8 8 10 7 85 12 16 13 14 14 12 16 14 17 13 141
Fine 04 6 4 3 2 5 3 9 8 3 1 44 6 4 5 5 6 5 5 6 5 3 50 12 8 8 7 11 8 14 14 8 4 94
Fine 05 & 5 2 1 8 1 4 5 14 2 37 5 6 6 6 4 6 10 4 6 5 58 10 11 8 7 10 7 14 9 10 9 95
Fine 06 1** .1 ; 1 I : 10 8 8 8 10 10 6 9 8 8 85 20 15 15 14 13 16 13 18 9 11 144
Fine_07 7 3 1 5 4 2 5 4 5 5 41 7 7 7 7 3 7 7 5 7 6 63 14 10 8 12 7 9 12 9 12 11 104
Fine 08 9 8 5 3 2 5 10 10 2 2 56 9 9 10 9 9 9 4 10 9 9 87 18 17 15 12 11 14 14 20 11 11 143
CoarseOl 9 10 9 9 2 1 10 5 1 9 65 3 10 6 4 4 4 9 7 9 10 66 12 20 15 13 6 5 19 12 10 19 131
Coarse02 1 8 7 1 5 5 8 2 3 4 44 6 8 2 6 1 6 1010 10 8 67 7 16 9 7 6 11 18 12 13 12 111
K Fine_01 10 9 10 10 1 2 9 9 10 10 80 1 9 5 5 2 5 8 2 7 9 53 11 18 15 15 3 7 17 11 17 19 133
1 Fine_02 4 6 8 3 9 4 2 1 2 5 41 4 7 3 3 9 3 7 8 8 7 59 8 13 11 6 18 7 9 9 10 12 103
Fine 03 i,  i ; 5 1 4 1 10 2 6 5 4 2 40 11 6 7 6 20 12 13 12 13 3 103
Fine 04 2 7 6 2 6 7 1 4 5 7 47 5 2 1 1 5 1 E 4 35 4 12 7 1 a a 1 I I 11
Fine 05 8 4 1 6 7 6 5 8 7 3 55 r I,. i~ : I . I i 1I1 i I i. i: 1. 1
Fine 06 5 1 4 8 4 9 4 3 4 8 50 7 2 8 9 6 7 2 1 1 1 44 12 3 12 17 10 16 6 4 5 9 94
Fine 07 7 3 5 4 8 3 6 10 8 2 56 10 4 9 8 8 9 4 9 3 6 70 17 7 14 12 16 12 10 19 11 8 126
Fine 08 3 2 2 7 3 8 3 6 6 6 46 8 3 7 10 5 8 1 3 2 3 50 11 5 9 17 8 16 4 9 8 9 96
Coarse01 3 10 9 9 2 9 10 1 9 9 71 5 8 2 7 9 9 9 9 7 9 74 8 18 11 16 11 18 19 10 16 18 145
Coarse02 1 8 6 6 3 8 8 2 7 8 57 10 9 1 2 2 10 10 10 2 8 64 11 17 7 8 5 18 18 12 9 16 121
0 Fine 01 4 9 10 10 1 10 9 4 10 10 77 1 10 8 10 5 4 3 1 9 10 61 5 19 18 20 6 14 12 5 19 20 138
1 Fine 02 8 4 4 2 7 5 2 8 2 7 49 8 7 4 8 6 6 7 5 8 7 66 16 11 8 10 13 11 9 13 10 14 115
Fine 03 2 2 1 3 10 2 7 3 8 1 39 9 1 5 1 10 1 6 6 1 1 41 11 3 6 4 24) 3 13 9 9 2 60
Fine_04 7 6 2 I J 4 1 5 J 2 36 7 6 10 9 1 3 4 8 10 2 60 i! I: I I. r i I l
Fine 05 5 3 8 8 8 7 3 7 5 6 60 6 4 7 5 7 7 8 4 4 6 58 11 7 15 13 15 14 11 11 9 12 118
Fine 06 10 5 5 4 6 1 5 9 1 3 49 2 2 6 4 2 22 6 3 5 1.' 7 11 10 10 3 7 11 7 6 84
Fine 07 6 1 7 7 9 6 4 6 6 5 57 4 5 3 3 8 8 5 7 3 5 51 10 6 10 10 17 14 9 13 9 10 108
Fine 08 9 7 3 5 5 3 6 10 3 4 55 3 3 9 4 3 5 1 3 5 4 40 12 10 12 9 8 8 7 13 8 8 95
CoarseOl 12 20 18 18 4 10 20 6 10 18 136 8 18 8 11 13 13 18 16 16 19 140 20 38 26 29 17 23 38 22 26 37 276
Coarse02 2 16 13 7 8 13 16 4 10 12 101 16 17 3 8 3 16 20 20 12 16 131 18 33 16 15 11 29 36 24 22 28 232
KO Fine_01 14 18 20 20 2 12 18 13 20 20 157 2 19 13 15 7 9 11 3 16 19 114 16 37 33 35 9 21 29 16 36 39 271
1 1 Fine02 12 10 12 5 16 9 4 9 4 12 93 12 14 7 11 15 9 14 13 16 14 125 24 24 19 16 31 18 18 22 20 26 218
Fine 03 8 7 4 8 20 12 14 10 17 2 102 14 2 9 2 20 3 12 11 5 3 81 22 9 13 10 40 15 26 21 22 5 183
Fine 04 9 13 8 3 i 11 2 899 983 11 11 11 4 4 9 14 16 6 95 18 22 19 I 1 15 11 2 5 15 1B7
Fine 05 1 I I n i I ." I.: t" p' lI I I 11 1 1 : .1 I:
Fine06 15 6 9 12 10 10 9 12 5 11 99 0 3 1i 15 10 9 4 3 7 4 79 i 10 23 27 20 19 13 15 12 15 178
Fine 07 13 4 12 11 17 9 10 16 14 7 113 .1 i. 11 I' I *i iI 1.1 :' 13 24 22 33 26 19 32 20 18 234
Fine 08 12 9 5 12 8 11 9 16 9 10 101 11 6 16 14 8 13 2 6 7 7 90 23 15 21 26 16 24 11 22 16 17 191
CoarseOl 13 30 28 28 14 19 21 7 20 27 207 10 19 10 13 14 15 20 19 18 21 159 23 49 38 41 28 34 41 26 38 48 366
Coarse02 4 17 22 16 16 23 22 6 19 20 165 20 20 6 12 11 19 21 27 16 26 178 24 37 28 28 27 42 43 33 35 46 343
MKO Fine 01 17 27 28 28 11 20 20 16 28 30 225 3 21 1.6 16 110 14 17 20 111 .1 48 42 44 23 30 34 20 45 50 356
2 1 1 Fine 02 20 121812 17 16 7 16 10 19 147 1 1. I 1 I 1 I' Ir :. I i. I .' I 31 29 26 34 29 30 31 29 37 311
Fine_03 12 13 8 12 27 16 22 16 24 8 158 22 12 18 12 27 11 20 19 15 10 166 34 25 26 24 54 27 42 35 39 18 324
Fine 04 15 17 11 5 15 l 11 17 12 1 127 15 15 1 16 160 9 14 20 21 9 145 TO 32 27 21 25 ? 25 17 33 t1 27?
Fine 0 I I I I' : I I ... I l 20 16 23 18 18 23 21 12 15 16 182 38 28 34 33 39 37 33 32 31 29 334
Fine 06 25 13 16 18 13 16 16 21 6 14 158 19 12 22 23 20 19 10 12 15 12 164 44 25 38 41 33 35 26 33 21 26 322
Fine 07 20 7 13 16 21 11 15 20 19 12 154 21 16 19 18 19 24 16 21 13 17 184 41 23 32 34 40 35 31 41 32 29 338
Fine 08 21 17 10 15 10 16 19 26 11 12 157 20 15 26 23 17 22 6 16 166 1 177 41 32 36 38 27 38 25 42 27 28 334
 II *II II IIII[LY
WB* 1
JKO
jIjj 0 l. BA
*LR
*. *PK PK
0
*
I __
Legend
'I r ,U J,
II)
7r 1.,
'.
Figure 4.24 Comparison of simulated versus measured M2 amplitudes and phases at 10
stations in Florida Bay.
'Ac 
