Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00084995/00001
## Material Information- Title:
- Hydrodynamic modeling in shallow water with wetting and drying
- Series Title:
- UFLCOEL-96014
- Creator:
- Davis, Justin Ross, 1972-
University of Florida -- Coastal and Oceanographic Engineering Dept - Place of Publication:
- Gainesville Fla
- Publisher:
- Coastal & Oceanographic Engineering Dept., University of Florida
- Publication Date:
- 1996
- Language:
- English
- Physical Description:
- xiii, 109 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Coastal ecology -- Mathematical models -- Florida -- Florida Bay ( lcsh )
Estuarine ecology -- Mathematical models -- Florida -- Florida Bay ( lcsh ) Environmental conditions -- Florida Bay (Fla.) ( lcsh ) Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( lcsh ) Coastal and Oceanographic Engineering thesis, M.E ( lcsh ) - Genre:
- government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt ) theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (M.E.)--University of Florida, 1996.
- Bibliography:
- Includes bibliographical references (leaves 103-108).
- Statement of Responsibility:
- by Justin Ross Davis.
## Record Information- Source Institution:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 37856892 ( OCLC )
## UFDC Membership |

Full Text |

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. ii 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. iii TABLE OF CONTENTS ACKNOWLEDGMENTS............................................. ii LIST OF TABLES .................................................. vi LIST OF FIGURES................................................. viii ABSTRACT ...................................................... xi INTRODUCTION ...................................................1I 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 iv 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 Model 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 v LIST OF TABLES Table pg 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, K1, and 0, 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 Ai Table 4.12 Table 4.13 Table 4.14 Table 4.15 vii 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 viii Figure 3.15 Non-dimensional comparison between wave profiles as predicted by theory and the numerical model (Time=27r/3 Time=Tu)..................... 52 Figure 4.1 Map of South Florida ........................................ 54 Figure 4.2 Boundary-fitted "coarse grid" (97074 cells) used for numerical simulations of Florida Bay circulation ....................................... 56 Figure 4.3 Boundary-fitted "fine grid" (194x 148 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 K, 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 01 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 01 tidal constituent. The chart is constructed from data measured at 35 study sites (Smith and Pitts 1995a) ............... 72 ix 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 M,-40 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 0,410 simulation. Amplitudes are measured in centimeters................................................. 76 Figure 4.22 Co-phase chart of the 01-10 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 M, amplitudes and phases at 10 stations in Florida Bay ........................................ 86 Figure 4.25 Comparison of simulated versus measured K1 amplitudes and phases at 10 stations in Florida Bay......................................... 87 Figure 4.26 Comparison of simulated versus measured 01 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 x 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-OS (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 xi 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 verticallyintegrated curvilinear-grid model is modified to incorporate a robust wetting and drying scheme originally developed for a rectangular grid model. Basically, the curvilinear-grid xii 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. xiii 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 I 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 199 1, 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. 4 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. Flather 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) 2-1) 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 Carteaian Bay of Saint Brieue River Canche Estuary Falconer and Owens (1987) 2-1) Add and withdraw cells Cartesian Humber Estuary Liu (1988) 2-D Bottom friction dominated flow Cartesian Lake Okeechobee Akanhi and Katopodes (1989) 2-D) Tracked wave front Finite element N/A Rather and 1-ubbert (1990) 2-D Add and withdraw cells Cartesian Morecambe Bay Casulli and Cheng (1992) 3-1) Implicit Cartesian San Francisco Bay Lagoon of Venice Luo (1993) 3-1) Bottom friction dominated flow Cartesian Lake Okeechobee Rupert Bay Yellow Sea Present Model (1996) 2-1) 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 CH3D3. " 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). 0 Calibrate and validate the model with field data from Florida Bay. 11 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 verticallyvarying 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): 12 13 au +au 2 + auv auw =- c+f at+ a ay az A-x-g2-+2fa A au(21 "\ ax2 ay 2) aZ\ vazY av+ au+ av2+ avw = 9K fu at ax ay az ay aV aV a a 22 + A + + A4ax2 ay 2) az\ vaz)i au +av + aw =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; ((xy,t) is the free surface elevation; g is the gravitational acceleration and A H and Av are the horizontal and vertical turbulent eddy coefficients, respectively. 2.2 Vertical Boundga Conditions in Cartesian Coordinates The boundary conditions at the free surface are specified by the prescribed wind stresses T.' and T'. au 2 2V pA -rx CdAv~ UW +VW (2.4) 14 pA = = CdVw 2W 2 (2.5) az uwvw where -r' 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: Cd,= 0.001(0.75 + 0.067W) (2.6) where W, is the wind speed in meters per second. At the free surface, the kinematic condition states that W D a + U (+ V .a (2.7) Dt at ax ay t Combining the above equation with Equation (2.3) yields a( + u+av=0 (2.8) at ax ay 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=P~u 2u 2 (2.9) Tb P d~brb +Vb, 2 2 (.0 yb) = P~~dVb Ub + Vb (210 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 d K2 Cd n n2( *Z;) (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 + pA- =z Tx (2.12) pvaV b gVVU22 p~- T= 2 (2.13) az C,1 where C, is the Chezy friction coefficient which can be formulated as: 16 1 C= 4.64 R-6 (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, CH3D3 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-~y +(xy,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, dar but produces a new variable ca =-, which is related to, but different from, the original dt vertical velocity w=- dz. The aY-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 number(s) 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, Ur as the reference horizontal velocity, Wr= -4 Ur as the reference vertical velocity, A Hr as the reference horizontal xr diffusion, Av as the reference vertical diffusion, and Sr as the reference surface elevation, the non-dimensionalization procedure then follows Sheng (1986a) and Sheng et al. (199 1). 18 Dimensionless variables: (u v *w (x *,y *) Uer Ur r W r r Vy I t tr (,rx, Ty (fPOzrUr) AH AH AHr Avr X w ) Ur Dimensionless groups: Vertical Ekman Number: Lateral Ekman Number: Froude Number: Rossby Number: EV Av IZr2 EH AHr JXr FrUr O Ur 0JX '3 zr( Sr Tr (2.16) (2.17) R 0)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 boundaryfitted grid, it is necessary to transform the governing equations from the original (x~y) coordinates to the transformed coordinates ( ,i). 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 - attgk ala {(x~1/.Hid + x~v.HffP) + a(x V/gHai + xjH3)1 (2.19) a x (-Y FgHRR + y/HAgiM) + a y /gHaiP + y,1Fg6HPT)] 20 9 i~21 a + g 22a) g,, 4 a XFgHi " y TirHMo) + xI~gHiiT) + + yv/'HIr 3)]o 1 (2.20) + x/n~gHP3) where ii fudz and P -h depth defined as f vdz are the depth-averaged velocities, H is the total _h H h + and 9= J =vT j is the determinant of the metric tensor g1j, which is defined as x11x + E CX + YVTY 11 9 gi12} 2+ 2 21 922 whose inverse is g 2L(t + 2 -Xx~ + [vI : gi :g} ati (2.21) (2.22) (2.23) (2.24) 911 HR VF9 1-1 921 M V9_I a y Vg_,Hi4i aTI a (X rg'Piif, all 21 Since the model solves for the vertically averaged velocities first, the threedimensional 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 g,~Hii) + (Vg/H3)]=0, (2.25) at F9 l Hg" a(F 0, (2.26) at a a +f + Hg2a F1 0, (2.27) atal 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: 22 ~g i4, U g k ilj.Iv i +j 41+1f V.. 0id.. "I'S1 ~ +~~t+1 0* i~~. (2.28) At(1 -0A3 n n nv~i n~ n+1(i + t Hij'Uii ( 1 ID zPj,u I + AtH~~ og!! (,(n+I n41I =-A HtL fl+1( + tOCI j.I is~ ~ iij*l At(1 Ol)CDP ~ 2.30 j t1,u(l _O)glJv( (n whr F n F4 reeplctfiiedifrnc rersntainftermiigtrs 0 isthedegee o imlictnes ofthepresur gaient tem and0 is h dgeeo imliines ofte oto rctotrs The vrical avrae u an vt fiit difrec eqaiosar usttte nth cniuteqan resulting inI tn 23 d ij2 l S j+ lju~ j~lj Si ,( - S j i v i = q1v Ig. j (2 .3 1 ) where Hn =t. At2 2 AgOU1, ,U I +AtO cj K (2.32) n Sn At2~ 22 H j 1 +AtOlCDI I7 (2.33) ng n fl Sn +j +1s + sij+ + + + i v (2.34) and finally 24 n2 = i At(1 0O) (n01+1iJ + At2Op ( F ,i+ijur i~i 19j, '+AtelCDI''+1j,uI +AgeOI( F t g +1" +At2O(1 O)p(g' 1++ At~o(11 1 +At( +At2o(1 O) 3 ( 22 1 +A At2o(1 -~( 22 9_ i+~ At(1 O1)CD11+j~l 1 +At'ICDI <+1j,uI 1 I)CD~I VI VF._it_+~lh 1,v +tlD tO 6dC IVUIl -Atop-( + Atop( -Atop( + Atop( telCD I Vi'+Iju p ij (n iju((n 3 1 CD V 4,u I ((n ij+l tO I CD ij, Iv ,n i,,gn (n ij ij 0) g0 H'1 n ijU -gT i IJVH Ln 1 +Atol CD u7 1 +Atel CDII ) (2.35) )lCD~ ~i.VI ) - Atu OICDI ~ 1 +AtOlCDI'P'j+,vI jg _,~gj,,nj l- At(1 0,)CDj~'jv) 1 +AtO ICD I V,, I I (ij) ) )ICDI' jyl 25 2.5 Solution Technique Equation (2.3 1) 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 2.3 1. 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 (2.3 1) can be written as (ddi d-1) n nnt (ddjjd 1>) which, by letting is equivalei where it to n aij~ Sj~u n Sij~v nn Id~ 26 n n St~j nn Sij~ (d, n q1j (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: (0) (1) Guess e1, (2) Set j -a~jil ajiI-aj,~ej,- ~ij)-b~ (3) Then for k=-O, 1, 2.... and until (r k)r (k)) < E, calculate (kid') = (k) -x(k)p(k), yi =e1 pj where a (k) -(rk)r () 1k+) (k a(k)(Mp (k))i.. ( k ) = .( k + ) k p k) p.. -i p i)jk (2.44) where (rk r ______(r (k)r (k)) In Equations (2.42) and (2.43), Mp is defined as (M(k))i =(k) ail~p(k) (k (k) (k) (MpL, pIJ aiP- aiiYjl jp-* 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 (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 EDE. 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 < AXmin I Umax I Propagation: At <- xn Bottom Friction: At < 2 2 maxI Implicit EDEs 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 EDEs 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: IAxi, A-xi+1j 20% ktin > 450 < 10 Ax.. where ( mj. 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, -c, 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 I1km in each direction. A constant wind stress of 1 dyne/cm2 is applied in the positive x 30 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 (kin) ~Ttl(cm) (U,1 (cm) 0.5 -2.04 -2.04 10.5 0 0 20.5 2.04 2.04 5 4 3 x=20.5 km -2 0.0 0.5 1.0 1.52. Tie(as Fiue31SraeeeainaChrelctoswtiOh oan 32 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_ 9 a(- (3.2) at ax h aua (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, 1. The lowest mode for the above equations is given by =~~t gak iwtsnk)(35 where a is the wave amplitude, w is the circular frequency and k is the wave number. The circular frequency is defined as 33 = 2~r(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 kmn and a width, W, of 25 kmn (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 T =2L -(2)(105,000 m) -30,000 s. (3.7) 9H (9.82 MI/S 2)(5 mn) Y=W 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 cmn at x=0, and a minimum of -10 cm at x=L (Figure 3.3). The model was run semi-implicitly, 0 = 0.5 1, 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 L 34 15 ,2, 10 C 0 W0) CO) -~5 -10 0 10 20 3 0 5 0 7 0 9 0 x (kin) Figure 3.3 Initial surface elevation for the seiche test. 15 10 x=02 01 00020.4 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. 35 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)[ X = acos((,t), (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) (Yxt) = Reae jot cos(P(x-xi))1 (3.10) L cos(Pl) 1 36 u(x,t) = Re[ iw fe iwtsin(P(X-xi))(311 PO cos(1) (.1 where H is the basin depth and P H (3.12) In an annular basin, Lynch and Grey (1978) determined the analytic solution to be C(r,t) = Re[(AJO(Pr) + BY(13r))e II (3.13) u(r,t) = Re[(- AJ1(Pr) BY(r)i)e iGWt] (3.14) where JO, JI, Y0 and Y, are Bessel function and A aY(13rl) [J0(Pr2)YI(Pr 1) YO(Pr2)Jj(Prj1)] (3.15) B = aJ1(Prl) (.6 [Jo,(Pr2)Yi(Pr,) YO(Pr2)J1(3r,1](.6 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 2nr 9000 S =200 =r 20 kmn r2 8 3 kmn H Ilm 0 A t =30 s The model was run for 10 cycles to attain steady state conditions. Figure 3.6 shows the maximum surface elevation and velocity. r2 Figure 3.5 The annular section grid (42x5 cells). 38 120 30 110 100 0 E E 20 .2 90 -E C)800 10 70' [2/ Analytic Surface Elevation 60 -- -- Analytic Velocity A Model Surface Elevation 0l Model Velocity 00020.4 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 mn and the height of the wall, H wall' 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 Figure 3.7 Simple test diagram. 300 250 200 Left Side E 1) 150 . . . . . 100 Right Side 50 0 .. .1 0.0 0.2 0.4 0.6 0.8 1.0 Time (days) Figure 3.8 Simple wetting and drying test results. 40 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-x,, 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 kmn 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 xl 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 IH X-L2 X=L, Figure 3.9 Storm surge diagram for non-wetting and drying simulation. tw H x=o X-L1 X=L2 Figure 3.10 Storm surge diagram for wetting and drying simulation. 42 50 40 0 .0 CO 20 10 Wetting and Drying Model Non-Wetting and Drying Model 0~ I I 0.0 0.5 1.0 1.5 2.0 Time (days) Figure 3.11 Comparison of simulated water levels at x=xl 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). 43 W() U Figure 3.12 Wave propagating on a linearly sloping beach diagram. The one-dimensional nonlinear shallow water equations can be written as a + a[(11 + h*)* 0 at* ax *[ I 0 a*+ U~u + -aI0, at* ax* (3.18) (3.19) where asterisks denote dimensional quantities, il 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) F4 g U() = gL, (3.21) where ( is the beach angle. The equations are then non-dimensionalized using the following relations: x L t * T '1 T (3.22) h h U U 0 Defining h* 2 h 1 h + il x +TI, (3.23) Equations (3.18) and (3.19) then become T+ [(n + Xx = 0, (3.24) 45 u t+ U U+ 1. 0 (3.25) Rewriting Equations (3.24) and (3.25) in terms of u and c gives 2t+ 2ucx + cu.,= 0, ut + uu + 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 along Defining the characteristic variables and as u + 2c t, =u 2c -t. Equation (3.28) becomes C= constant along dx = U + C alongdt =const along dx= -C dt Assuming x and t are functions of ( and then for ( = constant or = constant we dx =~ ~ (3.28) (3.29) (3.30) (3.31) (3.32) 46 get dx =ala dt a a dx =ala dt a(t a( if C=constant, if constant. From these two equations, we get X = tj(u + c), XC=tc(u c). From Equations (3.29) and (3.30), we can obtain U+c (3C +t 4 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( ) (t2) tC(( 3E) + t2) 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) 47 2(( + t + 3(tC + Q~ = 0, (3.41) a linear partial differential equation. It is convenient to introduce new variables a and X defined as X = C= 2(t (3.42 (3.43) Equation (3.41) then becomes + 3ta G Since t x + u from Equation (3.42), u must also satisfy Equation (3.44) 2 UIX= UG +3u, Introducing a "potential", T(u,)X), defined as (P0 U =- (3.44) (3.45) (3.46) then Equation (3.45) becomes =P; (00, + C ar (3.47) Equation (3.47) is a single partial differential equation whose boundary condition at the a = + =4c. 48 shoreline is (Y = 0, (3.48) which corresponds to the condition c = 0, 1. e., the total water depth at the shoreline must equal zero at all times. In terms of the variables a, X, and the potential pXCarrier and Greenspan proposed the following expressions for t x, il, u, and c: t X + U= 2 2 (7 (3.49) U 2 2 (PxL 1 (pX Cy x= -+ C + -=+ I 2 4 24 16' T= 2-X= y2 -X (I G2 ~1=16y~ 4 16' C= 4 (3.50) (3.51) (3.52) (3.53) If (p(a,A) is given, then Equations (3.49)-(3.53) give t x, TI, u, and c parametrically in terms of a; and X. In general, it is difficult to obtain direct function relationships for 11 and u in terms of x and t. 49 Carrier and Greenspan pointed out a solution to Equation (3.47) T(a,X) =-8A0J0( ~)sin( ),(3.54) where AO is an arbitrary amplitude parameter and J0 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 p(u,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 (Y and X, are determined using a Gauss-Newton method so that ri(x,t) and u(x,t) are easily obtained from Equations (3.5 1) 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, ax, 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 kmn to 15 kin, 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 kin, the grid spacing is fixed at 1 km. The model is forced atx=L with a periodic forcing function of the form ~()=acos( T3 t), (3.55) where the amplitude, a, is 11.24 cm and the period, T, is 3600 s. 50 y=W !II;(t)IIII I y=O H x=O x=L Figure 3.13 Computational grid for the wetting and drying test (16 1x5 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 1.0 0.5 -0.5 -1.0 10 to0 30 40 1.5 1.0. 0.5 -0.5 -1.0 -.'010 20 x30 40 1.5 1.0. 0.57 -0.5 -1.0 -1.5 1.5 1.0 0.5 P_ 0.0 0 10 20 x30 4b Time=ir/2 Theoretical Solution -0.5 Numerical Solution -1.0 --------Mean Water Level Shoreline 01. 10 20 x V Figure 3.14 Non-dimensional comparison between wave profiles as predicted by theory and the numerical model (Time=O Time=Tc/2). 52 1.5 r 1.0 0.5 97 0.0 -0.5 -1.0 -1.5 1.5 1.0 0.5 P'0.0 -0.5 -1.0 -1.5 0 10 20 x30 40 U -'U 4U 1.5 Theoretical Solution 1.0 Tm iuA Numerical Solution ----- Mean Water Level 0.5 Shoreline -0.5 -1.0 10l 2'0 30 sot Figure 3.15 Non-dimensional comparison between wave profiles as predicted by theory and the numerical model (Time=27r/3 Time=r). 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 53 54 of the keys, to accommodate the use of measured water level data along the reef tract. -Gulf Atlantic Ocean of Mexico Florida ~ : 82 81 Longitude (W) 80 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 81 *05'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 1 995a) Cape Sable to Matecumbe Keys Keys 1550 km2 (Edge of Shallow Mud Banks) (Physical) (Scholl 1966) 26.0 z 25.5 CD -a 25.0 24.5 G 55 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. 56 Kilometers 5 0 5 10 15 Figure 4.2 Boundary-fitted "coarse grid" (97074 cells) used for numerical simulations of Florida Bay circulation. 57 Kilometers 5 0 5 10 15 Figure 4.3 Boundary-fitted "fine grid" (194x 148 cells) used for numerical simulations of Florida Bay circulation. 58 4 4, Depth (in) 5 0 5 1011 Figue 44 Forid Ba bahymery n te "ine rid copile frm hgh rsoltio (2 m x 0 m)NatonalParkSerice ata 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 25 007.3 80'50.0' BN -Butternut Key 25 0O5.1V 80'3 1.1V DK -Ducky Key 25 0O1.8' 80'29.4' JK -Johnson Key 250*03.1V 80'54.2' LM Little Madeira Bay 25010.1' 80'37.9' LR Little Rabbit Key 24058.9' 80'49.6' PK -Peterson Key 24055.1' 80044.8' TC Trout Cove 25 012.7 80032.0' WB Whipray Basin 25 004.7 80'43.7' 60 0 Carysfort Reef g %WB 0 o LRA 41 p "V. Tennessee Reef 0 *Alligator Reef Kilometers 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 11130. 'AC -le 61 p Molasses ,ng Keyv(LON Fl)* Reef (MLRF1) Kilometers 5 0 5 10 15 0Sombrero Reef (SMKFI) Figure 4.6 Locations of C-MAN wind data stations. Lc ,A:9 62 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' 24 044.1l' 24 037.6 24 027.2! Longitude (W) 80c'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 25 000.0! 80024.0' LONF I 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. 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 0 1, 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 May26, 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 Oct 09, 1990 Jul 16, 1992 Oct 02, 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 64 4.1.4 Forcing Mechanisms and Boundar 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 30 o20 E 1 0-- - - - 0 350- 300 -- - -- - M, 250 -------- K1 0 0 0 LogKy(oth lrd ailn Nrh Fiur 150fsoetdlfocn ln h etenbudr fFord a.Tppnl tia mltd.Lwr ae:tdlpae 65 50 40 -2 30 o~20 E 10 0 350 300 250 - - - - 00) Cv 200 2 (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, K1, 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 N2 12.66 19 Diurnal: Luni-solar diurnal K, 23.93 58 Principal lunar diurnal 01 25.82 42 Principal solar diurnal P1 24.07 19 67 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). Tidal constituent Station M, S2 N, K, 0 P, Carysfort Reef il 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 il 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 il 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 rq 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 il 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 58% 54% 41% 49% 41% 48% 7% 13% 10% 12% 12% 11% Tidal constituent N, K, 19% 8% 12% 10% 10% 15% 9% 14% 8% 19% 12% 13% Total 8% 11% 19% 16% 20% 15% (cm) 56.69 51.05 66.52 46.51 42.46 52.65 0% 0% 5% 0% 0% 1% 10% 100% 100% 100% 100% 100% 68 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 (I 995a) 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 M,-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,-l10 (Figure 4.17), shows the benefit of both the finer grid and the wetting and drying mode. The cophase chart for simulation M,40O is shown in Figure 4.18. Results from the most accurate K, and 01 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 70 M2 Amplitude (cm 5 15,~ C? 0 20km 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). M2 Phase Angle (0)....: 150 / % I, 1i 1801, 60O.,, /1'-., k306/ ~270 0 20km I I 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). 71 K1 Amplitude (cm) -:6- ;,-f 0 20km 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 Angle () ...... 30 / 330-. 30 ~0 20km 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 1995 a). 72 01 Amplitude (cm) 0 20kmn Figure 4.13 Co-amplitude chart for the 01 tidal constituent. The chart is constructed from data measured at 35 study sites (Smith and Pitts 1995a). 01 Phase Ang'le (0) ~ * 3QO? 36 0 20kmn 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 1995 a). 73 Kilometers 5 0 5 10 1s Figure 4. 15 Co-amplitude chart of the MK-3 simulation. Amplitudes are measured in centimeters. Kilometers 5 0 5 10 15 Figure 4. 16 Co-amplitude chart of the M,-9 simulation. Amplitudes are measured in centimeters. r 74 -~ -,'--,. 4 'So Kilomter 5 0 5 10 is Figure 4.17 Co-amplitude chart of the M,-I10 simulation. Amplitudes are measured in centimeters. 300-- Kilometers 5 0 5 10 15 Figure 4.18 Co-phase chart of the M,-O simulation. 75 Figure 4.19 Co-amplitude chart of the K,-10 simulation. Amplitudes are measured in centimeters. Kilome.ter 5 0 5 10 15 Figure 4.20 Co-phase chart of the K,-10 simulation. 76 Kilometers 5 0 5 10 15 Figure 4.21 Co-amplitude chart of the 0,-10 simulation. Amplitudes are measured in centimeters. 330 Kilometers 5 0 5 10 15 Figure 4.22 Co-phase chart of the Q-10 simulation. 77 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 thai. September and a northeast wind is predominant from October thai 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, K1, and 01, 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. 78 5 rn/sec 250 260 MLRF1 270 280 290 300 I' 5 rn/sec 250 260 270 280 290 300 5 rn/sec Figure 4.23 C-MAN wind data from September I to October 31, 1993. The units of the x-axis are Julian days. SMK 250 260 270 280 290 300 LON 79 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-Ol Coarse No No 0.03 No Coarse-02 Coarse Yes No 0.03 No Fine-Ol 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 (Fine04) is the best simulation. A comparison of simulated versus measured M2, K1, and 0, constituents for runs Coarse-Ol, Coarse-02, Fine-Ol, 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. ReL Err. Mag. Rel. Err. Per. Err. Mag. Rel. Err. Measured 5.48 0.0 0.0%- 311 U 2.39 0.00 0.0% 173 0 0.85 0.00 0.0% 33 0 Coarse-0l ~.0 .O A -2.44 44.5' 360 49 7.70 5.31 222.2% 232 $ 2.78 1.94 228.6% 58 25 Caarse-O2 2e.99 -2.49 -45.4%/ 20 69 2-3 -00 .v, 263 89 2.47 1.63 191.9% 79 47 M Fine-Ol 2.34 -3.14 -57.3% j3-12.. 31 7.38 4.98 208.5% 238 65 2.09 1.24 146.4% 48 13 2 Fine-O2 1.64 -3.84 -70.1% 12 61 1.51 -0.88 -36.7% 294 121 1.19 0.35 40.8% 85 530 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-O4 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-OS 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-OS 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> ;90s.8 105 72 Fine-OS 1.46 -4.02 -73.4% 37 86 1.06 -1.33 -55.7% 313 140 0.58 0'.26 -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-al 1.56 0 .3 67.7%/ 326 -17 5.09 3.85 310.8% 315 -133 1.16 0.80 225.5% 10 -88 Coarse-O2 2 -0 1' 1 312 -31 2.95 1.72 138.6% 346 -102 0.63 0.28 78.1% 17 -81 K Fine-Ol 1.84 0.91 97.8% 333 -10 4.63 3.39 273.8% 321 -128 1.65 1.30 364.3% 12 -86 1 Fine-O2 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-O3 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 6 Fine-05 1.26 0.33 35.5% 309 -34 1.61 0.38 30.4% 10 -78 [0.42~ 0. 17.4', 262 164 Fine-OS 0.97 0.04 4.0% 312 -31 127 0.03 2.9 28 -60 0.48 0.12 34.9%-/ 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-OS 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-al 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 T -010 -8.2' 211 -141 2.31 0.90 63.3% 268 -168 0.91 0.63 227.8% 245 147~ 0 Fine-Ol 1.37 0.15 12.3% 1242K -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 .2 76.6A 284 -174 Fine-04 0.90 -0.32 -26.2% 227 -125 1.70 0.28 19.9% 294 -142 0.61 0.34 12 1.3%-/ 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-OS 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 .4 0.03 212%1, 295 -141 0.95 0.68 245.1% 269 171 Fine-OS 0.75 -0.47 -38.3% 233 -119 1.12 -0.29 -20.8%/ 311 -124 0.68 0.41 147.7% 279 -179 00 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-Ol 3.10 2.33 301.4% 72 25 17.30 -2.48 -12.6% 19 101l 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-0l 2.25 1.48 191.5% 55<' 9 ~ 17.34 -2.44 -12.4% 197 102 1.93 1.01 110.8% K:43 .' 1 2 Fine-02 1.23 0.45 58.4% 91l 45 ~20.10 0.1 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 0.8 88 .09g 1l1.5%1, 128 82 20.20 0.42 2.1% 197 102 0.4 00 .' 106 62 Fine-06 0.53 -0.25 -3.9 145 98 19.43 -0.36 -1.8% 199 105 0.6 1 -600 -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 0.0 0.0%n 30 0 Coarse-Ol 1.29 0.64 99.1%/ 21 -60 8.61 3.46 67.3% 284 -73 1.7 7 T'% 16 -14 Coarse-02 0 2T 0.0 5.5 13 -68 ,8.70 3.56 69.2% 28 7 0.68 -0.50 -42.5% 8 -22 K Fine-Ol 1.75 1.11 171.5% 19 -63 8.26, 3.12 60.6% 284 -72 1.55 0.37 31.4% 13 -17 1 Fine-02 0.60 -0.04 -6.6% 23 -58 8.93i 3.78 73.5% 282 -75 0.75 -0.43 -36.4% 16 -13 Fine-03 0.52 -0.13 -20.2% ~7 -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% 2~'3 3" Fine-OS 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-OB 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-0l 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-0l 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 6j.13 1.83 42.5% 220 -138 0.71 0.38 115.7% 271 -140 Fine-03 0.54 -0.10 15.4 % 321 -143 6.28 1.98 46.0% 220 -139 0.46 0.13 40.7% A85 -12262 Fine-04 0.66 0.02 3.5%7 287 -177 6.00 1.70 39.4% ~225 -134 0.57 0.24 73.5% 279 -133 Fine-OS 1.08 0.44 69.1% 266 162 6.14 1.84 42.8% 221 -138 0.79 0.46 140.6%l 271 -140 Fine-06 0.75 0.12 18.1% 271 167 6.00 1.70 39.5% 223 -135 0.4 0.13'~' 406- 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 00 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 TO 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 2.58 0.00 0.0%4 112 0 7.49 0.00 0.0% 305 0 0.73 0.00 0.0% 53 0 Coarse-Ol 3.11 0.5 20.7' 21q 107 5.24 -225 -30.1' 335 30 3.09 2.36 325.0% 70 17 Core0 ~7 -1.90 73j. 9%/ 217 105~ 4.78 -2:.7i1 -3.% 343 38 2.94 2.21 305.0% 93 39 M Fine-al 3.14 0.56 21.8% 231 119 3.80 -3.69 -49.3% 319< 14 2.23 1.50 207.2% 541 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-OS 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 -14.4o,> 153 99 Fine-07 0.82 -1.76 -68.2% 329 -143 2.88 -4.61 -61.5% 341 36 1.02 0.29J 40.5%/ 136 83 Fine-OS 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 00% 121 0 Coarse-al 4.86 3.18 188.8% 288 -82 1.98 0.13 7.3% 285 -35 1.24 0 ~.26 2.19 18 -103 Coarse-O2 3.66 1.97 117.3% 286 -85 1.88 0.04 2.1% 271 -49 0.60 -0.37 -38.0%/ 13 -108 KFine-al 4.26 2.58 153.2% 293 -77 2.21 0.37 20.2c 295 -26 1.70 0.73 74.6% 22 -98 1 Fine-O2 2.49 0.81 48.0% 299 -71 1.867i 0.02 ~ 281 e4 -6 0.67 -0.30 -31.1% 23 -98 Fine-03 3 .9 1.41 83.8% 303 -68 1.98 0.14 7.7%1 288 -32 0.33 -0.65 -66.6% 174 53 Fine-O4 2.6 0,7 .O% 303 -68 1.75 -0.09 -4.7% 286 -34 0.56 -0.42 -42.9% 47 -74 Fine-OS 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-OS 2.65 0.97 57.5% 311 -60 1.90 0.06 3.1% 29 2 0.57 -0.40 -41.4% 153 32 Fine-07 3.07 1.39 82.4% 304 -67 2.35 0.51 27.5% 283l -37 0.38 -0.60 -61.4% 173 52Fine-OS 2.62 0.94 55.9% 310 -6 1.98 0.14 7.4% 291 -29 0.51 -0.46 -47.7% 156 35 Measured 1.77 0.00 0.0% 20 0 2.08 0.00 0.0% 330 0 0.67 0.00 0.0% 110 0 Coarse-al 3.16 1.39 78.4% 214 -166 2.09 0,02 0.81% 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-al 2.81 1.04 58.7% 218 -161 1.75 -0.32 -15.6% <1961 -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-O3 2.18 0.41 23.1% 216 -164 1.89 -0.19 -9.1% 186 -144 0.33 -0.34 -51.2% so~ 29 Fine-04 .1.77 0.00 0.0% 217 -162 1.73 -0.35 -16.6% 183 -147 0.57 -0.10 -15.4% 290 -179 Fine-O5 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-O6 1.53 -0.24 -13.5% 223 -157 1.50 -0.58 -27.9% 190 -140 0 .6 -0S% 268 159 Fine-O7 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 00 84 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-Ol 1.70 1.17 217.4% 295 81 Coarse-02 1.60 1.07 199.4% 31 178 M Fine-Ol 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~ 11.40/ 345 131 Fine-OS 0.69 0.16 29.5% 1 147 Fine-OG 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-Ol 2.80 1.68 150.9% 325 -104 Coarse-02 1.45 0.34 30.1% 333 -96 K Fine-Ol 3.33 2.21 198.3% 327 -102 1Fine-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-OS 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-OB 0.56 -0.55 -49.6% 45 -24 Measured 1.37 0.00 0.0% 71 0 Coarse-Ol 2.14 0.76 55.8% 244 173 Coarse-02 1.35 -0.2 -14.5% 242 172 0 Fine-Ol 2.44 1.07 78.0% 249 178 1Fine-02 1.11 -0.26 -19.3% 271 -159 Fine-03 0.75 -0.63 -45.6% 3f_23 -10 Fine-04 1.32 -0.05 -4.0% 288 -142 Fine-OS 1.33 -0.05 -3.3% 277 -154 Fine-OG 0.88 -0.49 -36.0% 287 -143 Fine-07 1.26 -0.11 -8.1% 278 -153 Fine-OB 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 PaeCombined Amplitude Rankings Total Phase Rankings Total Combined Rankings Total _________BA BK RN DK JK LM IS PK TCWF Score BA BK BN DK JK LM IR PK TC WB Scare BA BK RN OK JK LM Lii PK TC WB Score CearseOi 1 10 10 10 1 0 9 1 1 10 B 71 2 1 2 2 1 2 2 3 2 2 19 3 11 12 12 11 11 3 4 12 i 90 Coarse2 2 1 B 9 B 10 6 2 B B 64 4 3 3A 4 B 3A 1 7 4 10 47 B 4 1 2 13q 16 13 7 A 13 lB 111Il M Fie__1 39 88 98 23 810 8 1<211 1VI'1 1 1 17 4 11 6 .969 4 j11 66 2 Fine 02 B 2 6 7 1 7 3 7 6 7 64 3 6 4 3 24 B923 4 39 11 7 10 10 2 It 12 B B It 9)3 Fine_03 4 6 4 4 7 4 B 6 7 6 66 B 10 B 10 7 B B B 10 7 B5 12 16 13 14 14 12 16 14 17 13 141 FinneB4 A64 3263BB3A11 44 6 4 B 6656 6663 60 126 B 7 11814 14B8 4 14 Fin_9 o-0 5 ~ dt~~ 2 6 44 2 6 6 6 4 6 10 4 6 6 66 10 11 B 7 10 7 14 9 10 B 06 Fine__66 10 7 7 6 3 6 7 B 1 3 69 10 8 B B 10 10 6 B B 8 60 20 16 10 14 13 16 13 lB 9 11 144 Fine_07 7 3 1 6 4 2 6 4 6 6 41 7 7 7 7 3 7 7 5 7 6 63 14 1 0 B 1 2 7 6 1 2 9 12 11 1 04 Fine 08 9 B 6 3 2 6 10 16 2 2 66 9 B 10 B B B 4 10 6 B 87 lB 17 16 12 11 14 14 20 11 11 143 Cee:rse:00 B 10 B 6 2 1 10 6 1 9 65 3 10 6 4 4 4 9 7 6 10 66 12 20 15 13 6 6 19 12 10 19 131 C.enel 1 B 7 1 5 5 8 2 2 4 44 6 B 2 6 1 6 10 10 10 8 67 7 16 9 7 6 11 lB 12 13 12 ill K Fine_01 106910 101 2 A B 10 10 BO 1 B 6 5 2 6 B 2 7 9 63 11 16 15 15 3 7 17 11 17 16 133 1 F n"_2 "t r 8 a q-421 S 4 7 3 3 9 3 7 B 8 7 69 6 13 11 6 16 7 B 6 10 12 103 Fine03 6 6 3 5 10 10 7 3 6 1 4 1 10 2 6 6 4 2 40 11 6 7 6 20 12 13 12 13 3 10.3 Fine 04 2 7 6 2 0 7 1 4 6 7 47 1 2C 1 tt ~1, C'6#3 4t 245i1 1,7 9 8 1P 711 1 8 FineS0 B 4 1 6 7 6 6 B 7 3 60 9 6 10771 4 5 5 66 17 1 11 13, 14 16 B12 12 B 121 Fin&AB 6 1 4 8 4 B 4 3 4 B 50 2 B B 6 7 2 1 1 1 44 12 3 12 17 10 16 6 4 5 9 14 Fine-67 7 3 6 4 B 3 6 10 6 2 6 10 4 B B B 9 4 B 3 6 70 1 7 7 1 4 1 2 1 6 1 2 10 19 11 8 126 Fine08 3 22 7 3 83 6 66 40 6 3 710 6681 3 23 60116 5 917B8-164 9B 8 0 6 Cea:rseOl 3 10 B B 2 0 10 1 9 9 71 5082 7 9 6 B 8 7 0 74 B 16 11 16 11 16 109 10 16 lB 140 Cnerse2 1 8 6 6 3 B B 2 7 B 67 10 6 1 2 2 10 10 10 2 B 64 11 1 7 7 8 05l 16 12 9 1 6 1 21 0 Finol0 4 B 10 10 1 10 B 4 10 10 77 1 10 B 10 5 4 3 1 9 10 61 5 I9 lB 20 6 14 12 6 11 20 136 1 Fine 02 B 4 4 2 7 6 2 B 2 7 41 B 7 4 B 6 6 7 6 B 7 66 16 11 6 10 13 11 9 13 10 14 110 Fine03 221 1 02In9736 3 Q 1 616110 16 6 11 41 fi 1 6 420 3139t 9 2 6so Fine__04 7% T7~ C 2 ~ 1 t1f, 7 610 6 1 3 4 8 10 2 60 14 12 12 10 6 7 6 13 14 41 66 Flne 06 6b BB73766 60 6 4 7 0776 4 4 6 6 1 1 7 105 13 I5 14 11 II 16 12 116 Fine 06 106 6 4 6 1 5 1 3 40 F 2~~~ 2 2 rJ,At t 1 2 7 11 10 10 3 7 11 7 6 04 Finer 6 1 7 7 9 6 4 6 6 0 67 4 6 3 3 B B 7 3 0 10 6 10 10 17 14 6 13 B 10 106 Fine 06 B 7 3 6 6 3 6 10 3 4 66_ 3 3 6 4 3 5 1 3 5 4 40 12 10 12 6 6 8 7 13 B 6 95 Ceemne01 12 20 lB 16 4 10 20 6 10 18 136 B 16 B 11 13 13 16 16 16 16 140 20 38 26 26 17 23 38 22 26 37 276 Cas:e 2 16 13 7 8 13 16 4 10 12 101 16 1 7 3 B 3 1 6 20 20 1 2 1 6 1 31 lB 33 16 16 11 26 30 24 22 28 232 KOJ Fine_0l 14 16 20 20 2 12 16 13 20 20 167 2 11 13 16 7 6 11 3 16 lB 114 16 37 33 36 6 21 26 16 36 30 271 1 Fine 62 12 10 12 6 16 B 4 B 4 12 63 12 14 7 11 16 9 14 13 16 14 126 24 24 16 16 31 16 18 22 20 20 21B Fie 03 B 7 4 6 20 12 14 10 17 2 10 14 2 6 2 20 3 12 11 5 3 61 1 6 1 10 40 10 26 21 22 6 1 Fine_04 13 R 2 1 ii11 2 83 9 11 11 11 4 4 6 14 16 6 66 is 19 4* 4 4 111 '23t25i15 178 Fine 06 13 7 14 16. 13 B 16i 12 110 109 10, 17 12 11 14 1 l B B 1 I N 2 20 17 26 26 20i 30 16 23 21 20 231 FinoB0 166 6 91 2100 9 1 2 6 9 Il 66 t~~ 1b2 4 T 7 4 7 24 1 0 23 27 20 161 3 15 12 16 178 Fine 07 1 3 4 12 11 17 9 10 16 14 7 113 1 4 6 12 11 16 17 6 16 6 11 121 27 13 24 22 33 20 lB 32 20 18 234 Fine 08 12 6 6 12 6 11 6 1 6 9 10 1 01 I1 6 1 6 1 4 8 1 3 2 6 7 7_ 60 23 16 21 26 16 24 11 22 16 17 11 Coerse 0l 13 30 28 28 14 16 21 7 20 27 207 10 10 10 13 14 10 20 19 lB 21 159 23 49 38 41 20 34 41 26 36 48 366 Caerse602 4 17 22 16 10 23 22 6 16 20 166 30 20 6 121 11 16 21 27 16; 26 176 24 37 2B 26 27 42 43 33 36 46 343 M O Fine-01 17 27 26 26 11 20 20 16 26 36 220 221 6 4 IG1 1 4 1~ 6 Kis 20 46 43 44 23 30 34 20 46 60 36 2 I 1 Fine02 20 12 16 12 17 16 7 16 10 19 147 15, 16 I11, 14 17 13,2 10 1610 164 35 31 26 26 34 26 30 31 26 37 311 Fine_03 12 13 12 2 16 22 16 24 B 168 3 12l 12 27 112 1 16 10 166 34 25r 26 24 04 27 43 36 36 lB 324 Fine__04 1t r 1 5 51 1 2 15 15 16 16 10 9 14 20 21 6 145 30 1' 27t21 2325 7p3 1 72 Fine 05 lB1-11 11 22 61 6 201 10 23 18 16 23 21 12 16 16 162 38 26 34 -33 3i3 33 32 31 29 334 Fine r0 26 13 16 16 13 16 16 21 6 14 158 16 12 22 23 20 16 10 12 10 12 164 44 26 38 41 33 36 26 33 21 26 322 Finn 07 20 7 13 16 21 11 16 20 16 12 104 21 16 19 18 lB 24 16 21 13 17 164 41 23 32 34 40 36 31 41 32 26 338 Fine08 21 17 10 10 10 16 lB 26 11 12 167 20 16 26 23 17 22 6 16 16 16 177 41 32 36 38 27 36 25 42 27 26 334 86 00 JK9 J ,, k. BA K1II1~-Q K LegendP 3$ Figure 4.24 Comparison of simulated versus measured M2 amplitudes and phases at 10 stations in Florida Bay. |