Hydrodynamic modeling in shallow water with wetting and drying

UFL/COEL-96/014

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 Non-dimensionalization ................................ 17
2.3.3 Vertically-Averaged 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 10-Day Simulations ..................... 68
4.2.2 Three constituent 35-Day 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 C-MAN 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 single-constituent 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 flat-bottom 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 non-wetting 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 Non-dimensional comparison between wave profiles as predicted by theory
and the numerical model (Time=0 Time=7r/2). ....................... 51

Figure 3.15 Non-dimensional 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 Boundary-fitted "coarse grid" (97x74 cells) used for numerical simulations of
Florida Bay circulation........................................... 56

Figure 4.3 Boundary-fitted "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 C-MAN 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 Co-amplitude 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 Co-phase 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 Co-amplitude 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 Co-phase 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 Co-amplitude 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 Co-phase 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 Co-amplitude chart of the M,-3 simulation. Amplitudes are measured in
centimeters. ................................................73

Figure 4.16 Co-amplitude chart of the M,-9 simulation. Amplitudes are measured in
centimeters. .................................................73

Figure 4.17 Co-amplitude chart of the M2-10 simulation. Amplitudes are measured in
centimeters. ................................................74

Figure 4.18 Co-phase chart of the M,-10 simulation ............................ 74

Figure 4.19 Co-amplitude chart of the K,-10 simulation. Amplitudes are measured in
centimeters. ..................................................75

Figure 4.20 Co-phase chart of the K,-10 simulation. .......................... 75

Figure 4.21 Co-amplitude chart of the 0,-10 simulation. Amplitudes are measured in
centimeters. ....................................................76

Figure 4.22 Co-phase chart of the O0-10 simulation. .......................... 76

Figure 4.23 C-MAN wind data from September 1 to October 31, 1993. The units of the
x-axis 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 Coarse-02 (16.8%). Mud
banks are indicated by the dark regions ............................. 91

Figure 4.29 Plot of maximum mud bank area for simulation Fine-02 (25.6%). Mud
banks are indicated by the dark regions. .............................. 92

Figure 4.30 Plot of maximum mud bank area for simulation Fine-03 (33.3%). Mud
banks are indicated by the dark regions. .............................. 93

Figure 4.31 Plot of maximum mud bank area for simulation Fine-04 (24.3%). Mud
banks are indicated by the dark regions. .............................. 94

Figure 4.32 Plot of maximum mud bank area for simulation Fine-05 (26.4%). Mud
banks are indicated by the dark regions. .............................. 95

Figure 4.33 Plot of maximum mud bank area for simulation Fine-06 (26.4%). Mud
banks are indicated by the dark regions. .............................. 96

Figure 4.34 Plot of maximum mud bank area for simulation Fine-07 (26.2%). Mud
banks are indicated by the dark regions. .............................. 97

Figure 4.35 Plot of maximum mud bank area for simulation Fine-08 (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 curvilinear-grid 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 curvilinear-grid model is modified to incorporate a robust wetting and drying

scheme originally developed for a rectangular grid model. Basically, the curvilinear-grid

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 vertically-integrated 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 small-scale and local processes,

but not in quantifying large-scale processes. Field studies are more effective in

quantifying large-scale 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 man-made 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

two-dimensional 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 stair-step 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 two-dimensional model for solving storm surge

problems with moving water-land 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 two-dimensional 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 fixed-grid 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 cross-sections of a wetted cell were compared against a critical total depth value.

If the depth of any of the cross-sections 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 two-dimensional 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 two-dimensional

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 three-dimensional semi-implicit 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 Eulerian-Lagrangian 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 three-dimensional 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) 2-D Empirical velocity relations Cartesian Galveston Bay

Yeh and Chou (1979) 2-D Increased bottom friction Cartesian Gulf of Mexico
Add and withdraw cells Southern Coast of Maine

Lynch and Gray (1980) 2-D Bottom friction dominated flow Finite element N/A
Deforming grid

Benque et al. (1982) 2-D Bottom friction dominated flow Cartesian Bay of Saint Brieuc
River Canche Estuary

Falconer and Owens (1987) 2-D Add and withdraw cells Cartesian Humber Estuary

Liu (1988) 2-D Bottom friction dominated flow Cartesian Lake Okeechobee

Akanbi and Katopodes (1989) 2-D Tracked wave front Finite element N/A

Father and Hubbert (1990) 2-D Add and withdraw cells Cartesian Morecambe Bay

Casulli and Cheng (1992) 3-D Implicit Cartesian San Francisco Bay
Lagoon of Venice

Luo (1993) 3-D Bottom friction dominated flow Cartesian Lake Okeechobee
Rupert Bay
Yellow Sea

Present Model (1996) 2-D 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 five-diagonal 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 non-wetting 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 three-dimensional

curvilinear grid model, CH3D, originally developed by Sheng (1987, 1989, and 1994).

The model first solves the vertically-integrated equations of motion, before solving the

equations for the deficit horizontal velocities (the difference between the vertically-

varying horizontal velocities and the vertically-averaged horizontal velocities). The

wetting and drying scheme in this study is implemented on the two-dimensional

vertically-averaged equations. After the solution of the vertically-averaged equations are

obtained, the three-dimensional 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 three-dimensional Cartesian equations describing constant density,

free surface flow can be derived from the Navier-Stokes 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 two-dimensional model are given

using a Chezy formulation:

au b gU 2 g
pAI, x =2 (2.12)
9z C,

av b gvVu-2+
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 a-stretched grid in the vertical directions and a boundary fitted curvilinear

grid in the horizontal direction.

2.3.1 A Vertically Stretched Grid

In three-dimensional modeling of estuaries and lakes, two types of vertical grids

are used. The first grid type, a z-grid, defines constant depth layers along the z-plane.

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 a-grid, 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 a-grid 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 a-transformation 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 a-stretching introduces extra terms, particularly in the
dt
horizontal diffusion terms, to the original equation.

2.3.2 Non-dimensionalization

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 non-dimensionalization procedure then follows Sheng (1986a) and Sheng

et al. (1991).

Dimensionless variables:

(u *, *) =

(x *,y *,z) =

u v Xr W
rw-V
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 Vertically-Averaged Equations in Boundary Fitted Coordinates

In the presence of a complex shoreline, a "boundary-fitted" grid allows

excellent representation of lateral boundaries. Using the elliptic grid generation

technique developed originally by Thompson (1982), a non-orthogonal boundary-fitted

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 two-dimensional

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 + x-H.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 depth-averaged 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(1-0- 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(l-O)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(1-6)f r-- u _+

At(l-6)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 *CJ-l)^
)C (C -I I
lCDl v 1

)lD IiV

(2.35)

( i j )

25

2.5 Solution Technique

Equation (2.31) represents a five-diagonal 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)i-Ij
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+lj-ai, uei- j-aij+l,veij+l-aid,veid-l -

(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,,ij-i

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 semi-implicit 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. Semi-implicit FDEs are the most complicated equations, although, in

principle they have no stability limit and are not numerically diffusive. However, in

practice, semi-implicit schemes tend to be slightly unstable.

29

It can be shown that when non-Cartesian 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 one-dimensional 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 semi-implicitly, 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(m-x)) (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 flat-bottom 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, x2-x1, 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 non-wetting 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- - Non-Wetting 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 one-dimensional 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 non-dimensionalized 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

Gauss-Newton 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 non-dimensional 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
P-0.0 -- ------- 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 Non-dimensional 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 Non-dimensional 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, shallow-water 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 boundary-fitted grid which is non-orthogonal but as orthogonal as possible. Two

grids are generated for this study. A boundary-fitted 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 "fine-grid" 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 Boundary-fitted "coarse grid" (97x74 cells) used for numerical simulations of
Florida Bay circulation.

Kilometers
SHHH 10 1
5 0 5 10 15

Figure 4.3 Boundary-fitted "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), Coastal-Marine Automated Network (C-MAN)

stations (Figure 4.6).

The precise location of the NPS bay water level stations, the NOS offshore water

level stations,, and the C-MAN 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 C-MAN 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 C-MAN 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 tide-driven circulation, and (2) tide- and wind-driven

circulation. For both tide-driven and tide- and wind-driven 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 wind-driven 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 single-constituent simulations, the model is

66

first spun up for 5 days, followed by a 5-day run. For the three-constituent simulations,

the model is first spun up for 5 days, followed by a 30-day 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

Semi-diurnal:
Principal lunar M2 12.42 100

Principal solar S2 12.00 47

Larger lunar elliptic N, 12.66 19

Diurnal:
Luni-solar 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 single-constituent

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 three-constituent 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 10-Day 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 co-amplitude and co-phase 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 co-amplitude 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, co-tidal and co-phase charts appear in Figures 4.9 thru 4.14.

The co-amplitude charts for the coarse and fine grid cases with a northeast wind, runs M,-

03 and M2-09, 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 M2-10 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 single-constituent 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 Co-amplitude 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 Co-phase 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 Co-amplitude 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 Co-phase 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 Co-amplitude 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 Co-phase 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 Co-amplitude chart of the M2-3 simulation.
Amplitudes are measured in centimeters.

Kilometers

5 0 5 10 15

Figure 4.16 Co-amplitude chart of the M2-9 simulation.
Amplitudes are measured in centimeters.

Kilometers
5 0 5 10 15
5 0 5 10 15

Figure 4.17 Co-amplitude chart of the M2-10 simulation.

Amplitudes are measured in centimeters.

Kilometers

5 0 5 10 15

Figure 4.18 Co-phase chart of the M2-10 simulation.

I~
~ _
-f(~l
r.

4",' ^ "

75

Kilometers

5 0 5 10 15

Figure 4.19 Co-amplitude chart of the K,-10 simulation.
Amplitudes are measured in centimeters.

Kilometers
5 0 10 15
5 0 5 10 15

Figure 4.20 Co-phase chart of the K,-10 simulation.

76

Kilometers
HH- I
5 0 5 10 15

Figure 4.21 Co-amplitude chart of the O,-10 simulation.
Amplitudes are measured in centimeters.

i. ,..

Kilometers

5 0 5 10 15

Figure 4.22 Co-phase chart of the O,-10 simulation.

4.2.2 Three constituent 35-Day Simulations

A series of 35-day 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 real-time 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 C-MAN 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 spin-up 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 35-day runs performed.

Using a 29-day 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 29-day 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 C-MAN wind data from September 1 to October 31, 1993. The units of the
x-axis 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

Coarse-01 Coarse No No 0.03 No

Coarse-02 Coarse Yes No 0.03 No

Fine-01 Fine No No 0.03 No

Fine-02 Fine Yes No 0.03 No

Fine-03 Fine Yes Northeast 0.03 No

Fine-04 Fine Yes Southeast 0.03 No

Fine-05 Fine Yes Real 0.03 No

Fine-06 Fine Yes Real 0.04 No

Fine-07 Fine Yes Real 0.03 Yes

Fine-08 Fine Yes Real 0.04 Yes

In order to identify the "best simulation" from the 10 runs, a non-parametric

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: Fine-04. 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 one-year (1993) data, it is not surprising that the real-time wind cases

(Fine-05 thru Fine-08) 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 Coarse-01, Coarse-02, Fine-01, Fine-02, and Fine-04 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
Coarse-01 3.04 -2.44 -44.5%. 30 49 7 70 5 31 2-2 232 59 2.78 1.94 228.6% 58 25
Coarse-02 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 Fine-01 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 Fine-02 1.64 -3.84 -70.1% 12 61 1.51 -0.88 -36.7% 294 121 1.19 0.35 40.8% 85 53
Fine-03 1.97 -3.51 -64.1% 37 86 1.11 -1.28 -53.7% 327 154 0.64 -0.20 -24.0% 110 78
Fine-04 1.74 -3.74 -68.2% 24 73 1.40 -0.99 -41.4% 276 103 0.89 0.04 5.2% 99 67
Fine-05 1.76 -3.72 -67.9% 23 72 1.40 -0.99 -41.5% 294 121 0.80 -0.04 -5.1% 103 71
Fine-06 1.41 -4.07 -74.2% 37 86 1.07 -1.32 -55.3% 311 138 0.49 -0.35 -41.7% 107 74
Fine-07 1.71 -3.77 -68.8% 25 74 1.50 -0.89 -37.2% 299 126 0.84 -0.01 -0.8' 105 72
Fine-08 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
Coarse-01 1.56 0.63 67.7% 326 -17 5.09 3.85 310.8% 315 -133 1.16 0.80 225.5% 10 -88
Coarse-02 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 Fine-01 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 Fine-02 0.96 0.03 3.2% 320 -23 2.13 0.89 71.9% 4 -85 0.75 0.39 109.3% 17 -81
Fine-03 1.08 0.15 16.1% 320 -23 1.63 0.39 31.6% 47 -41 0.45 0.09 26.6% 33 -65
Fine-04 0.91 -0.02 -2.2% 329 -14 2.17 0.94 75.6% 14 -74 0.60 0.25 69.2% 34 -64
Fine-05 1.26 0.33 35.5% 309 -34 1.61 0.38 30.4% 10 -78 0.42 006 17 41 262 164
Fine-06 0.97 0.04 4.0% 312 -31 1.27 0.03 2.2% 28 -60 043 0312 3-1 241 143
Fine-07 1.25 0.32 34.1% 306 -37 1.59 0.35 28.1% 15 -74 0.49 0.14 38.1% 256 158
Fine-08 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
Coarse-01 1.35 0.13 10.7% 227 -125 3.87 2.46 173.5% 242 167 1.12 0.84 304.3% 270 171
Coarse-02 1.12 -010 -8.2, 211 -1411 2.31 0.90 63.3% 268 -168 0.91 0.63 227.8% 245 147
0 Fine-01 I 37 i 12 3-. 242 -110 3.59 2.17 153.7% 247 171 1.34 1.07 386.0% 282 -176
1 Fine-02 0.86 -0.36 -29.5% 227 -125 1.64 0.23 16.1% 283 -152 0.70 0.43 154.4% 271 173
Fine-03 1.10 -0.12 -9.8% 218 -134 1.29 -0.13 -9.1% 330 -105 0.49 0.21 76.61. 284 -174
Fine-04 0.90 -0.32 -26.2% 227 -125 1.70 0.28 19.9% 294 -142 0.61 0.34 121.3% 278 -180
Fine-05 0.93 -0.29 -23.8% 227 -125 1.58 0.16 11.4% 295 -140 1.03 0.76 273.4% 275 176
Fine-06 0.72 -0.50 -41.2% 235 -117 1.15 -0.26 -18.7% 312 -124 0.70 0.43 154.4% 283 -175
Fine-07 0.90 -0.32 -26.2% 228 -124 1.45 003 2.20 295 -141 0.95 0.68 245.1% 269 171
Fine-08 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
Coarse-01 3.10 2.33 301.4% 72 25 17.30 -2.48 -12.6% 196 101 2.56 1.65 180.4% 62 18
Coarse-02 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 Fine-01 2.25 1.48 191.5% 55 9 17 3 u -12 -1: 197 102 1.93 1.01 110.8% 43 -1
2 Fine-02 1.23 0.45 58.4% 9I V 20.10 0.31 1.6% 197 102 1.23 0.32 34.6% 85 41
Fine-03 0.59 -0.19 -24.3% 181 135 20.71 0.93 4.7% 198 103 0.70 -0.21 -22.9% 121 77
Fine-04 0.92 0.15 19.5% 124 77 19.39 -0.39 -2.0% 198 103 0.87 -0.05 -5.3% 100 56
Fine-05 0.86 0.09 11.5. 128 82 20.20 0.42 2.1% 197 102 0.94 0.02 2.30 106 62
Fine-06 0.53 -0.25 -31.9% 145 98 19.43 -0.36 -1.8% 199 105 0.61 -0.30 -33.2% 126 82
Fine-07 0.96 0.19 24.7% 128 82 20.17 0.38 1.9% 197 102 0.94 0.03 2.7% 110 66
Fine-08 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
Coarse-01 1.29 0.64 99.1% 21 -60 8 61 3 .. c7. 8- .73 1.27 0.09 7.3% 16 -14
Coarse-02 061 -0.04 -5.5 13 -68 8 70 3 56 i. 286 -71 069 0 0 -22' 5: 8 -22
K Fine-01 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 Fine-02 0.60 -0.04 -6.6% 23 -58 8.93 3.78 73.5% 282 -75 0.75 -0.43 -36.4% 16 -13
Fine-03 0.52 -0.13 -20.2% 71 -10 9.11 3.97 77.1% 281 -76 0.43 -0.75 -63.4% 34 4
Fine-04 0.60 -0.04 -6.6% 47 -34 8.78 3.64 70.7% 285 -72 0.57 -0.62 -52.2% 33 3
Fine-05 0.79 0.14 22.2% 168 87 8.89 3.74 72.8% 281 -75 0.58 -0.60 -51.0% 332 -58
Fine-06 0.92 0.27 41.7% 170 89 8.68 3.54 68.8% 284 -73 0.54 -0.65 -54.7% 341 -48
Fine-07 0.77 0.12 19.3% 170 89 8.90 3.76 73.0% 281 -75 0.76 -0.42 -35.4% 335 -55
Fine-08 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
Coarse-01 1.17 0.53 82.4% 272 167 5.78 1.48 34.3% 220 -139 1.12 0.79 241.4% 262 -150
Coarse-02 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 Fine-01 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 Fine-02 0.60 -0.04 -5.8% 273 169 6.13 1.83 42.5% 220 -138 0.71 0.38 115.7% 271 -140
Fine-03 O) .0 10 .i; : 321 -143 6.28 1.98 46.0% 220 -139 0.46 0.13 40.7% 285 -126
Fine-04 066 0.02 3.5'. ;97 .1-7 6.00 1.70 39.4% 225 -134 0.57 0.24 73.5% 279 -133
Fine-05 1.08 0.44 69.1% 266 162 6.14 1.84 42.8% 221 -138 0.79 0.46 140.6% 271 -140
Fine-06 0.75 0.12 18.1% 271 167 6.00 1.70 39.5% 223 -135 0.46 0.13 40.60o 280 -132
Fine-07 1.07 0.43 67.7% 265 161 6.16 1.86 43.1% 221 -138 0.76 0.43 131.7% 267 -145
Fine-08 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
Coarse-01 3.11 0.53 20.7% 219 107 524 -2.25 -30.1". 335 30 309 2.36 325.0% 70 17
Coarse-02 67 -1 ? -739 217 105 1 7 -. 71 -.-.. 4: ? 2 9 2.21 305.0% 93 39
M Fine-01 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 Fine-02 0.90 -1.68 -65.2% 321 -151 2.83 -4.66 -62.2% 334 29 1.28 0.56 76.8% 92 39
Fine-03 0.64 -1.93 -75.1% 323 -149 2.86 -4.63 -61.8% 347 42 0.09 -0.63 -87.4% 197 144
Fine-04 0.51 -2.07 -80.2% 332 -140 2.78 -4.71 -62.9% 342 37 0.92 0.19 26.7% 127 73
Fine-05 0.89 -1.69 -65.5% 322 -151 2.86 -4.63 -61.8% 340 35 0.92 0.19 26.8% 135 81
Fine-06 0.65 -1.93 -75.0% 330 -142 2.58 -4.91 -65.6% 350 45 0.62 -0.10 -14.4% 153 99
Fine-07 0.82 -1.76 -68.2% 329 -143 2.88 -4.61 -61.5% 341 36 1.02 0.29 40.5% 136 83
Fine-08 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
Coarse-01 4.86 3.18 188.8% 288 -82 1.98 0.13 7.3% 285 -35 1.24 0.26 27.1% 18 -103
Coarse-02 3.66 1.97 117.3% 286 -85 1.88 0.04 2.1% 271 -49 0 .0 -0 37 -.330': 13 -108
K Fine-01 4.26 2.58 153.2% 293 -77 2 0 37 2 295 -26 1.70 0.73 74.6% 22 -98
1 Fine-02 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
Fine-03 3.09 1.41 83.8% 303 -68 1.98 0.14 7.7% 288 -32 0.33 -0.65 -66.6% 174 53
Fine-04 2.46 0.77 46.0% 303 -68 1.75 -0.09 -4.7% 286 -34 0.56 -0.42 -42.9% 47 -74
Fine-05 3.04 1.36 81.0% 305 -66 2.15 0.31 16.9% 290 -30 0.48 -0.49 -50.3% 175 54
Fine-06 2.65 0.97 57.5% 311 -60 1.90 0.06 3.1% 295 -26 0.57 -0.40 -41.4% 153 32
Fine-07 3.07 1.39 82.4% 304 -67 2.35 0.51 27.5% 283 -37 0.38 -0.60 -61.4% 173 52
Fine-08 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
Coarse-01 3.16 1.39 78.4% 214 -166 209 0.02 0.8. 177 -153 1.13 0.47 69.8% 272 163
Coarse-02 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 Fine-01 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 Fine-02 1.74 -0.04 -2.1% 215 -165 1.57 -0.51 -24.5% 188 -142 0.65 -0.02 -2.3% 274 165
Fine-03 2.18 0.41 23.1% 216 -164 1.89 -0.19 -9.1% 186 -144 0.33 -0.34 -51.2% 80 -29
Fine-04 1.77 0.00 0 0o 217 -162 1.73 -0.35 -16.6% 183 -147 0.57 -0.10 -15.4% 290 -179
Fine-05 1.67 -0.10 -5.8% 214 -166 1.67 -0.41 -19.6% 189 -141 0.87 0.20 30.2% 261 152
Fine-06 1.53 -0.24 -13.5% 223 -157 1.50 -0.58 -27.9% 190 -140 066 0.00 -0.5% 268 159
Fine-07 1.65 -0.12 -6.9% 217 -163 1.68 -0.40 -19.1% 185 -145 0.90 0.23 34.4% 258 149
Fine-08 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
Coarse-01 1.70 1.17 217.4% 295 81
Coarse-02 1.60 1.07 199.4% 31 178
M Fine-01 3.03 2.50 466.1% 271 57
2 Fine-02 0.93 0.39 73.0% 349 135
Fine-03 0.29 -0.24 -45.4% 65 -149
Fine-04 0.60 0.06 11.4% 345 131
Fine-05 0.69 0.16 29.5% 1 147
Fine-06 0.39 -0.15 -27.3% 15 161
Fine-07 0.72 0.19 34.7% 3 149
Fine-08 0.42 -0.12 -21.8% 16 162
Measured 1.12 0.00 0.0% 69 0
Coarse-01 2.80 1.68 150.9% 325 -104
Coarse-02 1.45 0.34 30.1% 333 -96
K Fine-01 3.33 2.21 198.3% 327 -102
1 Fine-02 1.51 0.39 35.3% 359 -70
Fine-03 1.02 -0.09 -8.5% 48 -21
Fine-04 1.75 0.63 56.8% 13 -56
Fine-05 0.80 -0.32 -28.5% 8 -61
Fine-06 0.41 -0.71 -63.4% 50 -19
Fine-07 0.80 -0.31 -27.9% 5 -64
Fine-08 0.56 -0.55 -49.6% 45 -24
Measured 1.37 0.00 0.0% 71 0
Coarse-01 2.14 0.76 55.8% 244 173
Coarse-02 1.35 -0.02 -1.5% 242 172
0 Fine-01 2.44 1.07 78.0% 249 178
1 Fine-02 1.11 -0.26 -19.3% 271 -159
Fine-03 0.75 -0.63 -45.6% 323 -108
Fine-04 1.32 -0.05 -4.0% 288 -142
Fine-05 1.33 -0.05 -3.3% 277 -154
Fine-06 0.88 -0.49 -36.0% 287 -143
Fine-07 1.26 -0.11 -8.1% 278 -153
Fine-08 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)

7-r 1.,
'.

Figure 4.24 Comparison of simulated versus measured M2 amplitudes and phases at 10
stations in Florida Bay.

'Ac --

MILLISECOND	CLASS.METHOD	MESSAGE
0	sobekcm_page_globals.constructor
0	sobekcm_page_globals.constructor	Application State validated or built
0	sobekcm_database.verify_item_lookup_object
0	sobekcm_page_globals.constructor	Navigation Object created from URI query string
0	sobekcm_database.verify_item_lookup_object
0	sobekcm_page_globals.display_item	Retrieving item or group information
0	sobekcm_page_globals.get_entire_collection_hierarchy	Retrieving hierarchy information
0	sobekcm_assistant.get_entire_collection_hierarchy
0	cached_data_manager.retrieve_item_aggregation
0	cached_data_manager.retrieve_item_aggregation	Found item aggregation on local cache
0	item_aggregation_builder.get_item_aggregation	Found 'all' item aggregation in cache
0	system.web.ui.page.page_load (ufdc.page_load)
0	sobekcm_page_globals.constructor.on_page_load
0	html_echo_mainwriter.add_style_references	Adding style references to HTML
0	html_echo_mainwriter.add_text_to_page	Reading the text from the file and echoing back to the output stream
72	html_echo_mainwriter.add_text_to_page	Finished reading and writing the file