Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0013880/00001
## Material Information- Title:
- Three-Dimensional Unstructured Finite Difference and Volume Model for Barotropic Coastal and Estuarine Circulation and Application to Hurricane Ivan (2004) and Dennis (2005)
- Creator:
- Lee, Jun
- Publisher:
- University of Florida
- Publication Date:
- 2008
- Language:
- English
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Coastal and Oceanographic Engineering
Civil and Coastal Engineering - Committee Chair:
- Sheng, Y. P.
- Committee Members:
- Hsu, Tian-Jian
Dean, Robert G. Thieke, Robert J. Jaeger, John M. - Graduation Date:
- 5/1/2008
## Subjects- Subjects / Keywords:
- Atmospheric models ( jstor )
Atmospheric pressure ( jstor ) Hertzsprung Russell diagrams ( jstor ) Hurricanes ( jstor ) Modeling ( jstor ) Simulations ( jstor ) Storm surges ( jstor ) Surface water ( jstor ) Velocity ( jstor ) Water depth ( jstor ) dennis, elm, fdm, fvm, hurricane, inundation, ivan, storm, surge, unstructured City of Pensacola ( local ) - Genre:
- Unknown ( sobekcm )
## Notes- Abstract:
- This study focuses on the development and application of three dimensional time-dependant, unstructured grid ocean, coastal, and estuarine numerical circulation model named UFDVM (Unstructured Finite Difference and Volume Model). UFDVM model was developed by combining finite difference and finite volume numerical schemes, taking advantage of the computational efficiency of the finite difference method (FDM), the exact conservation of finite volume method (FVM) and the flexibility of representing complex geometry with an orthogonal unstructured mesh system. The time-explicit Eulerian-Lagrangian Method (ELM) was used to discretize the non-linear advection term, thus removing the CFL condition for stability. To guarantee the conservation of mass, finite volume method was applied to the continuity equation. The propagation term was implemented by a semi-implicit numerical scheme, the so-called ?-method, for numerical stability. UFDVM exploited finite volume numerical scheme in horizontal diffusion term. Because the model used orthogonal unstructured computational grid, much flexibility to resolve complex coastal boundaries was allowed without any transformation of governing equations. The model has ability to treat wetting and drying of land. UFDVM was implemented Z-grid in vertical direction. The equilibrium turbulence model is used to determine the vertical eddy viscosity. After successful comparisons with analytical solutions, UFDVM was applied to the Hurricane-induced storm surge simulations. Hurricane IVAN (2004) and DENNIS (2005) were simulated and compared between simulated results and real data. The model calculated storm surge elevation during the hurricane events using wind and atmospheric pressure fields produced by sophisticated wind models such as WINDGEN, NCEP WNA, NOAA HRD and Holland analytical wind models. Careful comparison with real measured data, combined WNA and HRD wind field with Holland atmospheric pressure produced best results for the simulations. Maximum inundation maps were produced for both hurricanes and compared with measured High Water Marks (HWM). Significant inundation occurred around Pensacola Bay, FL during Hurricane Ivan. The storm surge elevation of I-10 Bridge over Escambia Bay, FL was high enough to inundate the bridge. A number of sensitivity simulations were performed to investigate storm surge level by various model inputs such as different types of simulated wind fields, with and without land reduction, air pressure, tides, bottom frictions, open boundary conditions, and different time step. Sensitivity tests revealed the wind fields and bottom frictions were the most important parameters to reproduce correct surge level and inundation compared to measured data. The land effects on the wind fields were tested. Wind fields with limited land reduction produced best results. Overall, the model UFDVM developed in this study simulated Hurricane IVAN and DENNIS efficiently and accurately. Although UFDVM successfully simulated hurricane induced storm surge, the model does not have wave effect. When compared to storm surge model CH3D-SSMS which has wave effect, the wave component should be included to UFDVM model in order to produce more accurate results.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Jun Lee. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 5/31/2010
- Resource Identifier:
- 689996874 ( OCLC )
## UFDC Membership |

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PAGE 1 1 THREE-DIMENSIONAL UNSTRUCTUR ED FINITE DIFFERENCE AND VOLUME MODEL FOR BAROTROPIC COASTAL AND ESTUARINE CIRCULATION AND APPLICATION TO HURRICANE IVAN (2004) AND DENNIS (2005) By JUN LEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Jun Lee PAGE 3 3 To my Parents and my Family PAGE 4 4 ACKNOWLEDGMENTS I would like to first thank m y advisor, Dr. Y. Peter Sheng, for his guidance and financial assistance throughout my Ph.D study. I would also like to thank the other committee members, Dr. Robert Dean, Dr. Robert J. Thieke, Dr. Tom Hsu, and Dr. John M. Jaeger for their review of my dissertation. In addition, I am greatly indebted to Dr. Ashish Mehta, Dr. Kirk Hatfield in the civil, coastal and oceanographic program at UF, a nd Dr. Wei Shyy at the University of Michigan for their support and encouragement. My research has been supported by research gr ants from a number of agencies including Florida Sea Grant, Florida Atlantic Universi ty, and Southeastern Universities Research Association. Dr. Peter Sheng served as the Prin cipal Investigator of those research grants. I am grateful to my colleagues, Taeyun Kim, Kijin Park, Yeonsik Chang, Yangfeng, Justin, Dave, Jeff, Vadim, Vladimir, Bilge, Ma, Andrew, Detong, Chenxia, Jungwoo, Sangdon, and all students in the coastal program for their help a nd friendship. A big gratitu de is owed to Sidney, Becky, Kim, Helen, Tony, Nancy, Carol, Doretha, a nd some Korean students in our department for making life easier. Most importantly, none of this would have b een possible without love and patience of my family. My father and mother have been a cons tant source of love, conc ern, support and strength all these years. I am grateful to my brothers, sister, and friends for their aide and support throughout this long time journey. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES.......................................................................................................................11 ABSTRACT...................................................................................................................................19 CHAP TER 1 INTRODUCTION..................................................................................................................21 Background and Review of Previous Works..........................................................................21 Structured Grid vs. Unstruct ured Grid or Mesh System......................................................... 21 Structured Grid System................................................................................................... 21 Unstructured Grid System...............................................................................................24 Unstructured Finite Difference, Element vs. Unstructured F inite volume Model................. 25 Finite Difference Models.................................................................................................25 Finite Element Models.................................................................................................... 28 Finite Volume Models..................................................................................................... 28 Objectives and Modeling Strategies....................................................................................... 31 2 GOVERNING EQUATIONS AND BOUN DAR Y CONDITIONS IN THREEDIMENSIONAL CARTESIAN COORDINATES................................................................35 Introduction................................................................................................................... ..........35 Governing Equations in Cartesian Coordinates ...................................................................... 35 Boundary Conditions in Cartesian Coordinates..................................................................... 37 3 THREE-DIMENSIONAL NUMERICAL MODEL WITH ORTHOGONAL UNSTRUCTURED MESH ....................................................................................................41 Introduction................................................................................................................... ..........41 Definition of the Unstructured Orthogonal Mesh and Index .................................................. 42 Finite Volume and Difference Disc retization and Solution A lgorithm.................................. 44 Treatment of Flooding and Drying.................................................................................. 54 Eulerian-Lagrangian Scheme for Advection Term......................................................... 55 Finite Volume Discretization of Horizontal Diffusion Term ..........................................61 Treatment of Coriolis Term............................................................................................. 64 Air Pressure Term............................................................................................................65 Solution of Sparse Matrix: Conjugate Gradient Method ........................................................ 66 Comparison of UFDVM Mode l and UnTRIM Model ........................................................... 68 PAGE 6 6 4 UNSTRUCTURED ORTHOGONAL MESH GENERATION.............................................72 Introduction................................................................................................................... ..........72 Orthogonal Unstructured Triangular Mesh Ge neration with Triangl e Mesh Generator ........72 Mesh Post Processing.............................................................................................................76 5 MODEL VERIFICATION WITH ANALYTICAL SOLUTION.......................................... 83 Introduction................................................................................................................... ..........83 Wind Setup.............................................................................................................................83 Tidal Propagation with Constant Depth................................................................................. 86 Tidal Propagation with N on-Linear Advection ...................................................................... 87 Quarter Annular Tidal Forcing with a Sloping Bottom Test.................................................. 95 Tidal propagation with Coriolis Effect................................................................................. 104 Kelvin Wave Propagation..............................................................................................104 The Governing Equation for Kelvin Waves.................................................................. 104 Comparison between Numerical and Theoretical Solutions ......................................... 107 Tidal Propagation with Bottom Friction Effect.................................................................... 108 Horizontal Diffusion Test..................................................................................................... 113 Wetting and Drying Test over Tidal Flats............................................................................ 118 Atmospheric Pressure Test with Holland Storm Surge Model............................................. 120 Verification of Atmospheric Pressure Term in the Model with Analytical Solution ........... 126 6 HURRICANE IVAN (2004) AND DE NNIS (2 005) SIMULATIONS...............................129 Introduction................................................................................................................... ........129 Storm Surge Model Review.................................................................................................. 130 Wind and Atmospheric Pressure Model............................................................................... 135 NOAA WAVEWATCH III (NWW 3) WNA WI ND Model......................................... 136 WINDGEN Model......................................................................................................... 137 NOAA HRD Surface Wind Analysis System............................................................... 138 HOLLAND =s Analytical Model ...................................................................................138 Wind and Atmospheric Pressure Interpolation..................................................................... 139 Linear Interpolation with Time..................................................................................... 144 Time and spatial Interpolati on (Lagrange Interpolation) ..............................................144 Hurricane IVAN (2004)........................................................................................................147 Computational Domain................................................................................................. 153 Boundary conditions...................................................................................................... 160 Tidal forcing...........................................................................................................160 Wind and atmospheric pressure.............................................................................162 Simulation Results of Hurricane Ivan...................................................................................183 Water Surface Elevation................................................................................................ 183 Flood Level and Maximum Inundation Map.................................................................187 Surge Modeling Sensitivity Tests.........................................................................................197 Base Simulation............................................................................................................. 197 Wind input..............................................................................................................197 Surface wind drag parameterization....................................................................... 199 PAGE 7 7 Bottom friction....................................................................................................... 201 Atmospheric pressure.............................................................................................205 Hurricane Ivan Sensitivity Test.....................................................................................206 Sensitivity to wind input........................................................................................ 207 Sensitivity to bottom friction.................................................................................. 207 Sensitivity to tidal forcing at the tidal open boundary........................................... 210 Sensitivity to atmo spheric pres sure........................................................................ 210 Sensitivity to 2 and 3 dimensionality..................................................................... 211 Sensitivity to coasta l co nfigurations...................................................................... 214 Sensitivity to wave effect on storm surge.............................................................. 215 Sensitivity to time increment.................................................................................. 216 Sensitivity to mesh size.......................................................................................... 218 Summary of sensitivity test.................................................................................... 219 Hurricane Dennis (2005)......................................................................................................221 Computational Domain................................................................................................. 226 Boundary Conditions.....................................................................................................226 Tidal forcing...........................................................................................................226 Wind and pressure.................................................................................................. 226 Simulation Result................................................................................................... 233 Water surface elevation.......................................................................................... 233 7 CONCLUSION AND FUTURE STUDY............................................................................ 237 APPENDIX A SAFFIR-SIMPSON HURRICANE SCALE........................................................................ 244 B FORMULAE TO CALCULATE ERRORS.........................................................................245 C BEST TRACKS FOR HURRICANES IVAN AND DENNIS............................................ 246 D WIND SPEED AND DIRECTION DURING HURRI C ANE IVAN: WINDGEN AND WNA VS. MEASURED......................................................................... 250 E WIND SPEED AND DIRECTION DURIN G HURRIC ANE DENNIS: WNA VS. MEASURED........................................................................................................................253 F COMPARISON OF MEASURED A ND SIMULATED WATER ELEVATION DURING HURRICANE DENNIS ....................................................................................... 254 G DESCRIPTION OF UFDVM SUBROUTINES AND M ODULES.................................... 255 H FLOW CHART OF THE MODEL UFDVM....................................................................... 260 I MANUAL OF UFDVM MODEL........................................................................................ 263 J EXTENSION OF MODEL FROM Z-GR ID TO SIGMA COORDINATE IN VERTICAL DIRECTION....................................................................................................287 PAGE 8 8 LIST OF REFERENCES.............................................................................................................295 BIOGRAPHICAL SKETCH.......................................................................................................305 PAGE 9 9 LIST OF TABLES Table page 1-1 Comparison of grid system................................................................................................ 27 1-2 Comparison of unstructured model.................................................................................... 34 1-3 Features of the present mode............................................................................................. 34 3-1 Tidal potential constants for tidal co nstitu ents and associated effective Earth elasticity factors.................................................................................................................70 3-2 Comparison of numerical met hod for UnTRIM and UFDVM model............................... 71 5-1 Summary of test for analytical solutions and m odel simulations......................................84 5-2 Comparisons between analyt ical and sim ulated wind setup.............................................. 84 5-3 Comparison between anal ytical and num erical soluti on of air pressure term.................127 6-1 Summary of recent develo pm ent of storm surge model.................................................. 136 6-2 Summary of hurricane Ivan features in the US ................................................................ 149 6-3 Selected storm surge elev ations at different locations during hurricane Ivan event........149 6-4 Features of computational domain................................................................................... 155 6-5 Datum in com putational dom ain...................................................................................... 155 6-6 Data stations used for comparison of the m odel during Hurricane Ivan......................... 163 6-7 Tidal constituent parameters for sim ulation Hurricane Ivan based on ADCIRC............163 6-8 Measured data stations in side the computational dom ain................................................ 167 6-9 Peak wind speed, time and peak absolute error at measured station............................... 172 6-10 Comparison of peak air pressure tim e and peak absolute error...................................... 179 6-11 Error of WNA and WINDGEN wind spee d, direction and atmospheric pressure com pared with measured data at stations during hurricane Ivan..................................... 184 6-12 Error of HRD and Holland wind speed, dire ction and atmospheric pressure compared with measured data at stations during hurricane Ivan...................................................... 185 6-13 High water mark comparison for hurricane Ivan simulation........................................... 195 PAGE 10 10 6-14 Hurricane Ivan sensitivity test .........................................................................................199 6-15 NLCD Land cover classificati ons and integrated bottom roughness............................... 202 6-16 Mannings n according to USGS, NLCD land cover classification s............................... 205 6-17 Comparison of computational mesh between coarse and fine m esh............................... 219 6-18 Selected storm surge values at diffe rent location during hurricane Dennis event ...........224 6-19 Summary of hurricane Dennis features in the US........................................................... 224 6-20 Measured data stat ions of Hurricane Dennis ................................................................... 229 A-1 The Saffir-Simpson hurricane scale................................................................................. 244 C-1 Best track for hurricane Ivan, 2 24 September 2004..................................................... 246 C-2 Best track for hurricane Dennis, 4 18 July 2005........................................................... 248 PAGE 11 11 LIST OF FIGURES Figure page 1-1 Simple example of Cartesian rectangular grid ................................................................... 23 1-2 Typical orthogonal curvilinear grid ................................................................................... 24 1-3 Example of non-ort hogonal curvilinear grid...................................................................... 26 1-4 Example of unstructure d triangular m esh or grid.............................................................. 26 1-5 Example of an unstruc tured triangular m esh without local mesh refinement................... 32 1-6 Example of an unstruc tured triangular m esh with local mesh refinement......................... 32 3-1 Orthogonal unstructured mesh........................................................................................... 43 3-2 Location of computational variables.................................................................................. 46 3-3 Definition of velocity components.....................................................................................46 3-4 Numerical stencil of Finite Volume Method..................................................................... 50 3-5 Different dimension of coefficient m atrix A and column vector Uj n+1..............................53 3-6 Determination of wet and dry depth at an elem ent face.................................................... 54 3-7 Schematic diagram for Euleri an grid on Lagrangian fram e...............................................57 3-8 Schematic diagram of backtrac king for the Lagrangian trajectory .................................... 61 3-9 Numerical stencil of finite volum e method for diffusion equation.................................... 63 4-1 Mesh generated by Delaunay triangulation....................................................................... 74 4-2 Mesh generated by a constrained confor ming Delaunay triangulation............................. 75 4-3 Mesh generated by a conforming Delaunay triangulation................................................. 75 4-4 Quality mesh.............................................................................................................. ........77 4-5 Local mesh refinement..................................................................................................... ..77 4-6 Mesh before smoothing and clean up................................................................................80 4-7 Mesh after smoothing and clean up................................................................................... 80 PAGE 12 12 4-8 Mesh that is not renumbered.............................................................................................. 81 4-9 Sparse matrix with 21 half band width corresponding to mesh Figure 4-8....................... 81 4-10 Mesh that is renumbered.................................................................................................. ..82 4-11 Sparse matrix with 5 half band width corresponding to m esh Figure 4-10....................... 82 5-1 2D Comparison between analytical a nd num erical solutions of water surface elevation for wind setup test.............................................................................................. 85 5-2 3D Comparison between analytical and num erical solutions of water surface elevation for wind setup test.............................................................................................. 85 5-3 Computational domain fo r tidal propagation test .............................................................. 88 5-4 Quadrilateral mesh for tidal propagation test..................................................................... 88 5-5 Triangular mesh for tidal propagation test......................................................................... 88 5-6 Comparison of water surface elevation fo r tidal p ropagation test at three different locations with = 1.0, Lines: Analytical solution s, Circles: Numerical solutions............ 89 5-7 Comparison of water surface elevation fo r tidal p ropagation test at three different locations with = 0.501, Lines: Analytical soluti ons, Circles: Numerical solutions........ 90 5-8 Comparison of water velocity for tidal propagation test at thr ee different locations with = 1.0, Lines: Analytical soluti ons, Circles: Num erical solutions........................... 91 5-9 Comparison of water velocity for tidal propagation test at thr ee different locations with = 0.501, Lines: A nalytical soluti ons, Circles: Numerical solutions....................... 92 5-10 Numerical solution of velocity field ( upper) and water surface elevation (lower) of tidal propagation .............................................................................................................. ..93 5-11 Comparison of water elevation for nonlin ear advection at three locations with = 1.0, Line: Analytical Solution, Circle: Numerical Solutions.............................................96 5-12 Comparison of water velocity for nonlin ear advection at th ree locations with = 1.0 Line: Analytical Solution, Circle: Numerical Solutions.................................................... 97 5-13 Annular section, Upper panel: xy plane, Lower panel: x-z plane .................................... 99 5-14 Computational mesh for quarter an nular test. Circles are data stations ........................... 100 5-15 Water depth for quarter annular test ................................................................................ 100 5-16 Comparison between analyt ical and num erical solution for water surface elevation Solid lines: Analytical soluti on, Circles: Numerical solution.......................................... 101 PAGE 13 13 5-17 Comparison between analyt ical and num erical solution for water velocity Solid lines: Analytical solution, Ci rcles: Numerical solution..................................................102 5-18 Water elevation and velocity field at tim e 3.214 days for quarter annular test............... 103 5-19 Water elevation and velocity field at tim e 5.013 days for quarter annular test............... 103 5-20 Comparison of water surface elevation for the Coriolis test at three locatio ns with = 1.0 Lines: Analytical solutions, Circles: Numerical solution..........................................109 5-21 Comparison of water velocity for the Coriolis test at three locations with = 1.0 Lines: Analytical solutions, Circles: Num erical solution................................................ 110 5-22 Velocity field for the Coriolis test at time 72 hours with = 1.0 ....................................111 5-23 Comparison of water surface elevation for bottom friction tes t at three locations with =1.0 Line: Analytical solution, Circles: Numerical solution........................................ 114 5-24 Comparison of water velocity for botto m friction test at three locations with =1.0 Line: Analytical solution, Ci rcles: Numerical solution................................................... 115 5-25 Comparison for the diffusion test at th ree lo cations Lines: Analytical solutions, Symbols: Numerical so lutions Upper panel: X direction, Lower panel: Y direction....... 116 5-26 Numerical solution of temperat ure distribution at tim e = 10 sec.................................... 117 5-27 3D view of numerical solution of temperature distri bution in rectangular plate ............. 117 5-28 Definition of dry and wet element................................................................................... 122 5-29 Wave propagation on a linearly sloping beach................................................................ 122 5-30 Non-dimensional comparison between wa ve profiles as predicted by theory and num erical model of wetting and drying, time = 0 /2.................................................. 123 5-31 Non-dimensional comparison between wa ve profiles as predicted by theory and num erical model of wetting and drying, time = 2 /3 .................................................124 5-32 Numerical solution of water su rface elevation for atmospheric...................................... 127 5-33 Initial atmospheric pressure distribution for analytical test .............................................128 5-34 Numerical solution of water surface elevation after reaching steady state ...................... 128 6-1 NCEP WNA model domain............................................................................................. 140 6-2 WINDGEN model domain.............................................................................................. 140 6-3 Simple grid for Holland model........................................................................................ 140 PAGE 14 14 6-4 NCEP WNA wind field of Hurri cane Ivan at tim e 9/16/04 07:30................................... 141 6-5 WINDGEN pressure field of Hu rricane Ivan at tim e 9/16/04 07:30............................... 141 6-6 WINDGEN wind field of Hurricane Ivan at tim e 9/16/04 07:30.................................... 142 6-7 HRD wind field of Hurricane Ivan.................................................................................. 142 6-8 Holland pressure field of Hurricane Ivan at tim e 9/16/04 07:30..................................... 143 6-9 Holland wind field of Hurrica ne Ivan at tim e 9/16/04 07:30...........................................143 6-10 Time interpolation between the two Hurricane snapshots............................................... 145 6-11 Schematic diagram of shifting of two Hurricane snapshots............................................146 6-12 Time and spatial interpola tion of two Hurricane snapshots ............................................. 146 6-13 Best track of Hurricane Ivan from NOAA NHC............................................................. 150 6-14 Track of Hurricane Ivan with HRD wind field snapshots ...............................................150 6-15 Damage of I-10 Bridge during Hurricane Ivan................................................................ 151 6-16 Hurricane Ivan impact area along Alabama coast........................................................... 152 6-17 Over-washed area in Gulf shores, AL be fore (up per left) and after (lower left), Collapsed buildings before (upper right) and after (lower right) Hurricane Ivan at Orange Beach, AL...........................................................................................................152 6-18 Computational mesh for the hurricane............................................................................. 156 6-19 Water (left) and land (right) elements of the com putational mesh, St. Joseph Island..... 156 6-20 Computational mesh, Escambia bay, FL......................................................................... 157 6-21 Water (left) and Land (right) elements of the computational m esh, Escambia bay........ 157 6-22 Computational mesh, Mobile bay, AL............................................................................. 158 6-23 Water (left) and Land (right) elements of the computational m esh, Mobile bay............ 158 6-24 Computational mesh, LA, MS......................................................................................... 159 6-25 Water (left) and Land (right) elemen ts of the computational m esh, LA, MS.................. 159 6-26 Bathymetry and Topography of the Co m putational Domain (Depth unit: meter)........... 160 6-27 Bathymetry and topography in Mobile Bay..................................................................... 161 PAGE 15 15 6-28 Bathymetry and topogr aphy in Escambia Bay.................................................................161 6-29 Storm surge elevation at 3 differe nt locations obtained from ADCIRC tidal constituent program......................................................................................................... 164 6-30 NOAA predicted tidal el evation at left, center, and right at tidal boundary ....................164 6-31 Measured Data (wind, pressure and water elevation) Stations ....................................... 167 6-32 WINDGEN wind speed (upper) and wind vector............................................................ 168 6-33 WNA wind speed (upper) and wind vector..................................................................... 168 6-34 HRD wind speed (upper) and wind vector...................................................................... 168 6-35 Holland wind speed (upper) and wind vector.................................................................. 169 6-36 Wind fields around Mobile Bay, AL at tim e 6:30, 9/16/04(UTC). Upper left: WINDGEN, Upper right: WNA, Lower left: HRD, Lower right: Holland..................... 170 6-37 Wind fields around Escambia Bay, FL at tim e 6:30, 9/16/04(UTC). Upper left: WINDGEN, Upper right: WNA, Lower left: HRD, Lower right: Holland, Lower: WNA wind and pressure field.......................................................................................... 171 6-38 Comparison between measured and simulated wind speed. Upper left: Grand Isle, LA, Upper right: Dauphin Island, AL Lo wer left: Pensacola, FL, Lower right: Panama City Beach, FL...................................................................................................173 6-39 Comparison of measured and simulated wind direction. Upper left: Grand Isle, LA, Upper right: Dauphin Island, AL, Lower lef t: Pensacola, FL, Lower right: Panam a City Beach, FL................................................................................................................. 174 6-40 Comparison of measured and modeled wind speed and direction at Grand Isle, LA. W INDGEN, WNA, HRD, and Holland wind.................................................................. 175 6-41 Comparison of measured and modeled wind speed and direction at Dauphin Island, AL. W INDGEN, WNA, HRD, and Holland wind.......................................................... 176 6-42 Comparison of measured and modeled wind speed and direction at Pensacola, FL. W INDGEN, WNA, HRD, and Holland wind.................................................................. 177 6-43 Comparison of measured and modeled wind speed and direction at Panam a City Beach, FL, WINDGEN, WNA, HRD, and Holland wind...............................................178 6-44 WINDGEN (upper) and Holland (low er) atm ospheric pressure field............................. 180 6-45 Comparison of measured and simulated atm ospheric pressure. Upper left: Grand Isle, LA, Upper right: Dauphin Island, AL, Lower left: Pensacola, FL, Lower right: Panama City Beach, FL...................................................................................................181 PAGE 16 16 6-46 Air pressure field near Dauphin Isla nd, AL at time 7:30, 9/16/04(UTC) Left: W INDGEN, Right: Holland.............................................................................................182 6-47 Air pressure field near Pensacola, FL at tim e 7:30, 9/16/04( UTC) Left: WINDGEN, Right: Holland..................................................................................................................182 6-48 Root mean square error of simulated wind m odels at measured data stations, Upper: wind speed, Lower: wind direction.................................................................................. 186 6-49 Air pressure Root Mean Square error of si mulated wind models (WINDGEN and Holland) at measured data stations.................................................................................. 186 6-50 Comparison of simulated and measured water surface elevation.................................... 188 6-51 Snapshots of water surface elevation during Hurricane Ivan.......................................... 190 6-52 Maximum inundation map of Hurricane Ivan................................................................. 193 6-53 Inundation map along the coast of Escam bia Bay, FL.................................................... 193 6-54 Before and after Hurricane Ivan imp act areas. A: Breached b arrier island, Pine Beach, AL. B: Over-washed area in Gulf s hore, AL. C: Destroyed Gulf-front houses, Orange Beach, AL, D: Collapsed multi-story building................................................... 194 6-55 Simulated maximum inundation ma p of Hurricane Ivan im pact area............................. 194 6-56 Location of High Water Marks near Escam bia Bay, FL................................................. 195 6-57 Missing spans on the West (left) and East (right) Bound I-10 Bridge (OCA, 2005) during Hurricane Ivan. ..................................................................................................... 196 6-58 Simulated water surface elevation under I-10 Bridge...................................................... 196 6-59 One(left) and ten-minute (right) averaged Hurricane Ivan H RD wind field.................200 6-60 Interpolated 1(left) and 10-m inute (right) averaged Hurricane Ivan HRD wind field on computational domain................................................................................................. 200 6-61 USGS NLCD land cover classifications ..........................................................................203 6-62 Converted land roughness according to USGS NLCD land cover classification s.......... 203 6-63 Hurricane Ivan wind field including land reduction effect ..............................................204 6-64 Mannings n according to USGS, NLCD land cover classification s............................... 204 6-65 Holland atmospheric pressure for Hurricane Ivan........................................................... 206 6-66 Water surface elevation with different wind input ..........................................................208 PAGE 17 17 6-67 Water surface eleva tion with different values of bottom friction.................................... 209 6-68 Comparison of water surface elevation between NOAA pr edicted tide and ADCIRC tidal cons tituent at Panama City Beach, Panama City, and Pensacola............ 212 6-69 Comparison of water surface elevation between NOAA predicte d tide and ADCIRC tidal cons tituent at Biloxi, Dauphin Island and Wave Land............................ 212 6-70 Comparison of water surface elevation be tween run with and without pressure at Florida stations .................................................................................................................213 6-71 Comparison of water surface elevation be tween run with and without pressure at Alabam a, and Mississippi stations................................................................................... 213 6-72 Comparison of 2-D and 3-D simulation. A: Dauphin Island, B: Panam a City, C: Panama City Beach, D: Pensacola................................................................................... 214 6-73 Modification of coastal c onfiguration by rem oving part of barrier island, left: before removing barrier island, right: af ter removing barrier island.......................................... 215 6-74 Simulated water surface elevation with and witho ut the barrier island........................... 216 6-75 Comparison with CH3D with wave result A: Pensacola, FL, B: Panam a City, FL C: Panama City Beach, FL D: Dauphin Island, AL.............................................................217 6-76 Comparison with 5-(red circ les) and 15-(blue line) minute time increment, A: Pensacola, FL, B: Panama City, FL C: Panama City Beach, FL D: Dauphin Island, AL....................................................................................................................................218 6-78 Comparison of coarse (left) and fine m esh...................................................................... 220 6-79 Comparison of water surface elevation with coarse fine mesh........................................ 220 6-80 Best Hurricane Dennis Track from NOAA NHC............................................................ 223 6-81 Track of Hurricane Dennis with HRD wnd field.............................................................223 6-82 Best track of central barometric pres sure and wind speed history for Hurricane Dennis. .............................................................................................................................224 6-83 Hurricane Dennis Impact Area Santa Rosa Island, FL.................................................... 225 6-84 Aerial photos of over-washed Pensac ola, FL (left) and Navarre Beach, FL ................... 225 6-85 Hurricane Ivan and Dennis Pa th, Red: Dennis, Black: Ivan ............................................ 227 6-86 NOAA predicted tide at ope n boundary for Hurricane Dennis ....................................... 228 6-87 ADCIRC surge elevation at open boundary for Hurricane Dennis ................................. 228 PAGE 18 18 6-88 Hurricane Denis wind field in comput ational dom ain (upper) and around Escambia Bay (lower), FL at 19:20, July 10th, 2005........................................................................ 229 6-89 Comparison of wind speed and direction for Hurricane Dennis at A: Pens acola, B: Panama City Beach, C: Waveland, D: Dauphin Island...................................................230 6-90 Comparison of simulated and measured wind vector...................................................... 231 6-91 Air Pressure comparison for Hurricane Dennis at A: W aveland, B: Biloxi; C: Pensacola, D: Panama City Beach................................................................................... 232 6-92 Holland analytical atmospheric pressure of Hurricane Dennis........................................232 6-93 Comparison between simulated and m easured water su rface elevation at A: Pensacola, B: Panama City Beach, C: Panama City, D: Dauphin Island........................ 234 6-94 Simulated water surface elevation of Hurrican e Ivan from time 13:40 22:10 07/10/2005.......................................................................................................................235 6-95 Maximum inundation map and created by Hurricane Dennis......................................... 236 6-96 Hurricane Dennis impact areas and ma xim um inundation map around Escambia Bay, FL.....................................................................................................................................236 D-1 Comparison of simulated and measured wind speed....................................................... 250 D-2 Comparison of simulated and measured wind stick........................................................ 251 E-1 Comparison of measured and simulate d water elevation at Grand Isle, LA ................... 252 E-2 Comparison of measured and simulate d water elevation at Pilot Station, L A................ 252 F-1 Comparison of simulated and measured WNA wind speed and direction ...................... 253 I-1 Flow chart of UFDVM model.......................................................................................... 262 K-1 Definition of stretched coordinates system ...................................................................289 K-2 Numerical stencil of finite volum e method..................................................................... 291 PAGE 19 19 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THREE-DIMENSIONAL UNSTRUCTUR ED FINITE DIFFERENCE AND VOLUME MODEL FOR BAROTROPIC COASTAL AND ESTUARINE CIRCULATION AND APPLICATION TO HURRICANE IVAN (2004) AND DENNIS (2005) By Jun Lee May 2008 Chair: Peter Sheng Major: Coastal and Oceanographic Engineering This study focuses on the development and application of three dimensional timedependant, unstructured grid oc ean, coastal, and estuarine nume rical circulation model named UFDVM (Unstructured Finite Difference and Volume Model). UFDVM model was developed by combining finite difference and finite volume numerical schemes, ta king advantage of the computational efficiency of the finite differen ce method (FDM), the exact conservation of finite volume method (FVM) and the flexibility of repr esenting complex geometry with an orthogonal unstructured mesh system. The time-explicit Eule rian Lagrangian Method (ELM) was used to discretize the non-linear advect ion term, thus removing the CFL condition for stability. To guarantee the conservation of mass, finite volume method was applie d to the continuity equation. The propagation term was implemented by a semi-implicit numerical scheme, the so-called method, for numerical stability. UF DVM exploited finite volume nu merical scheme in horizontal diffusion term. Because the model used orthogon al unstructured computational grid, much flexibility to resolve comple x coastal boundaries was allowed without any transformation of governing equations. The model has ability to treat wetting and drying of land. UFDVM was PAGE 20 20 implemented Z-grid in vertical direction. The eq uilibrium turbulence model is used to determine the vertical eddy viscosity. After successful comparisons with analyti cal solutions, UFDVM was applied to the Hurricane-induced storm surge simulations. Hu rricane IVAN (2004) and DENNIS (2005) were simulated and compared between simulated result s and real data. The model calculated storm surge elevation during the hurrican e events using wind and atmosphe ric pressure fields produced by sophisticated wind models such as WI NDGEN, NCEP WNA, NO AA HRD and Holland analytical wind models. Careful comparison with real measured data, combined WNA and HRD wind field with Holland atmos pheric pressure produced best results for the simulations. Maximum inundation maps were produced for both hurricanes and compared with measured High Water Marks (HWM). Significant inundati on occurred around Pensacola Bay, FL during Hurricane Ivan. The storm surge elevation of I-10 Bridge over Escambia Bay, FL was high enough to inundate the bridge. A number of sensitivity simulations were performed to inve stigate storm surge level by various model inputs such as different types of simulated wind fields, with and without land reduction, air pressure, tides, bottom frictions, op en boundary conditions, and different time step. Sensitivity tests revealed the wind fields and bottom frictions were the most important parameters to reproduce correct surge level and inundation compared to measured data. The land effects on the wind fields were tested. Wind fields with limited land reduction produced best results. Overall, the model UFDVM develope d in this study simulated Hurricane IVAN and DENNIS efficiently and accurately. PAGE 21 21 CHAPTER 1 INTRODUCTION Background and Review of Previous Works A large number of num erical models of circula tion in ocean, coastal, estuaries, and lakes have been developed over the last few decades with rectangular structured grids. Structured rectangular grid system is the si mplest grid structure, since it is logically equivalent to a Cartesian coordinate system. C onventional models use only structured Cartesian rectangular grids in the horizontal plane. To allow a more fl exible arrangement and sp atial variation of the grids, one can use an orthogonal curvilinear gird. The transformation of the governing equations will be required. Numerical methods using orthog onal curvilinear co-ordinates are relatively simple because no new terms are introduced by curvilinear transformation of equations. On the other hand, non-orthogonal curvilinea r co-ordinates grids, so cal led boundary-fitted curvilinear grids allow for higher flexibility for fitting comp lex boundaries. This grid is most often used to calculate flows in complex geometries. In this chapter, the advantages and disadva ntages of the structured and unstructured numerical mesh system will be examined. The fi nite difference method (FDM), finite element method (FEM) and finite volume method (FVM) models also are examined by comparing currently used numerical models of circulation in ocean, coastal, estuaries, and lakes. Finally, comparisons will be made between the existing unstructured numerical models and the model developed in this study. Structured Grid vs. Unstruct ured Grid or Mesh System Structured Grid System One pioneering three-dim ensional coastal and estuarine model was developed by Leendertse (1967, 1970). The model uses a time-e xplicit method to solve the three-dimensional PAGE 22 22 hydrostatic equation of motion in Cartesian coordinates with fin ite difference method. Sheng and Butler (1982) developed a three-dimensional mode l of coastal and estu arine circulation in horizontally Cartesian and vertically stretched ( -grid) coordinates. They solved the external mode (water level and ve rtically integrated velocities which govern the surface gravity waves) using a factorized implicit scheme and the intern al mode (vertical struct ure of three-dimensional velocities and temperature and salinity) using a vertically-implicit scheme. These models use only Cartesian rectangular grids in the horizontal plane and have shortcomings for applications where physical boundaries are very complex and thus may require excessively fine grids in order to resolve the complex boundaries in the model. Casulli and Cheng (1992) developed a circulation model named TRIM (Tidal Residua l Inter-tidal Mudflat Model). A Cartesian coordinate system is used in both ho rizontal and vertical directions. The advantages of this rectangular Cartesian grid system are that it is very easy to construct the computational grid system and the matrix of the algebraic equation system has a regular structure, which can be e xploited in developing an efficien t solution technique. However, this grid system has shortcomings for applic ations where physical boundaries are very complex and thus may require excessively fine grids in order to resolve the complex boundaries in the model. The stair steps along th e boundaries are inevitable, al though the finer grids are used. Figure 1-1 shows an example of a simple rectangular grid system As shown Figure 1-1, there are stair steps at the boundary obviously. Structured orthogonal and non-or thogonal curvilinear grid system To allow a more flexible arrangement and spa tial variation of the grids, one can use an orthogonal curvilinear gird (Blumberg and Mellor, 1987). The transformation of the governing equations will be required. Numerical methods us ing orthogonal curvilin ear co-ordinates are PAGE 23 23 relatively simple because no new terms are in troduced by curvilinea r transformation of equations. Figure 1-2 shows an example of an or thogonal curvilinear grid system. The resulting system of algebraic equations is still structured. However, Sheng (1986) indicated orthogonal curvilinear grids are not sufficiently flex ible to fit complex arbitrary geometry. Figure 1-1 Simple example of Cartesian rectangular grid On the other hand, non-orthogonal curvilinear co -ordinates grids, so called boundary-fitted curvilinear grids (Sheng 1986) allow for higher flexibility for fitting complex boundaries. This grid is most often used to calculate flow in complex geometries. The advantage of this grid system is that the grids can be adapted to a ny geometry and easier to achieve boundary fitting than with orthogonal curvilinear grids. Since th e grid lines follow the boundaries, the boundary conditions are more easily implemented than with a step-wise approximation of curved boundaries. The grid can also be adap ted to the flow, i.e., one set of grid lines can be chosen to follow the streamlines which enhance the accuracy and spacing can be made smaller in regions of strong variable variation. Non-orthogonal curvilin ear grids have also several disadvantages, X(km)Y(km)0 5 10 15 20 25 0 5 10 15 PAGE 24 24 such as difficulty in deriving nonlinear and di ffusion terms in equations, more computational efforts due to the new extra terms in the transformed equations and potential numerical dissipation and dispersion problems caused by skew ness and rapid change in grids and larger discretization errors and/or in stability (Oran and Boris 1987). Mo reover, a lot of time consuming and tedious works are required to generate curvilinear grids. Figure 1-3 shows an example of simple boundary-fitted non-orthogon al curvilinear grid system. Figure 1-2 Typical orthogonal curvilinear grid Unstructured Grid System The im portance of boundary fitting has been in vestigated in terms of finite difference methods derived for unstructured triangular grids (Thacker 1977). For very complex geometries, the most flexible type of grid is one which can fit an arbitrary solution domain boundary. Meshes made of triangles or quadrilaterals in 2D, and tetrahedra or hexahe dra in 3D are most often used. In practice, triangular meshes in the horizontal plane are most popular in unstructured mesh system (Ferziger and Peric 2002). Triangular meshes at an arbitrary side can provide an accurate PAGE 25 25 fitting of the irregular complex boundary. In principl e, such grids or meshes could be used with any discretization scheme, but th ey are best adapted to the finite volume method (FVM) and finite element method (FEM). Several finite element algorithms using unstructured triangular elements are currently being studied and app lied (Carey 1995; Walters and Casulli 1998).The elements for FEM or control volumes for FVM may have any shapes of polygons. The advantage of flexibility of unstructured meshes is offset by the disadvantage of the irregularity of the data structure. Node locations and neighbor connectio ns need be specified explicitly. The matrix of the algebraic equation system no longer has regular, diagonal structure; the band width needs to be reduced by reordering of the points. The solv ers for the algebraic equation systems are usually slower than those for regular structured gr id system. Figure 1-4 shows the example of unstructured triangular meshes fo r computation. Table 1-1 is the summary of the grid system Unstructured Finite Difference, Element vs. U nstructured Finite volume Model Finite Difference Models Casulli and Walters (20 00) developed a numerical method for three dimensional free surface flows with orthogonal unstructured grid s called UnTRIM (Unstr uctured Tidal Residual Inter-tidal Mudflat Model). Casu lli and Zanolli (2002) expanded the model method for the three dimensional quasi-hydrostatic, free surface flows. A grid is said to be an unstructured orthogonal grid if within each polygon a center point can be identified in such a way that the segment joining the centers of two adjacent polygons and the side shared by two polygons, have a nonempty intersection and are orthogonal to each other. The governing Reynolds-averaged NavierStokes (RANS) equations are discretized by mean s of finite difference and finite volume algorithms. The momentum equations are disc retized by finite difference method and the PAGE 26 26 continuity equation is discre tized by finite volume method wi th the orthogonal unstructured meshes to guarantee volume conservation. UnTRIM used Eulerian-Lagrangian method (ELM ) for the advection an d horizontal terms. The model has been applied to simulate the tid al flow in the Lagoon of Venice, Italy and the reach of the Big Lost River, ID, USA, with orthogonal triangular meshes. Although the model can resolve the complex geometry well, the grid must be orthogona l; hence to import any unstructured mesh into the model, additional efforts are needed to make the mesh orthogonal. Moreover, the model is not conservative because the non-conservative forms of the differential equations are used. Figure 1-3 Example of non-or thogonal curvilinear grid Figure 1-4 Example of unstructur ed triangular mesh or grid PAGE 27 27 Table 1-1 Comparison of grid system Advantage Disadvantage Rectangular grid (Figure 1-1) Structured grid Easy to construct grid No transformation of equations Well structured algebraic equation Cannot fit boundaries well Stair step boundaries Orthogonal boundary fitted curvilinear grid (Figure 1-2) Structured grid Well structured algebraic equation No extra terms by transformation Transformation of equations Not sufficiently flexible to fit an complex boundaries Need grid generation Non-orthogonal boundary-fitted curvilinear grid (Figure 1-3) Structured grid Well structured algebraic equation Higher flexibility for fitting complex boundaries than orthogonal curvilinear grid Can be adapted to any geometry Transformation of equations Extra terms by transformation More computational efforts Need grid generation Unstructured grid (Figure 1-4) No transformation of equations Any shape of mesh can be used Much higher flexibility for fitting complex boundaries than nonorthogonal curvilinear grid Can be adapted to any geometry Easy local mesh refinement Matrix is unstructured Larger amount of data required for the mesh Solver is slower than structured grid system Based on UnTRIM model, Zhang, et. al. (2004, 2005) developed ELCIRC (Eulerian Lagrangian Circulation) model. The authors included baroclinic terms to the model. The differences between the UnTRIM and ELCIRC is that the calculation of horizontal velocities. UnTRIM converts normal velocities directly to the horizontal velocities at each node. However, ELCIRC calculates both normal and tangential velo cities at the same time both grid faces and grid nodes. Essentially, UnTRIM and ELCIRC models are same. ELCIRC model included the turbulence model in the both salinity and temperature equations. The model applied to Columbia River estuary plume shelf system. PAGE 28 28 Finite Element Models Luettich, Westerink and Sheffner (1991) deve loped an advanced circulation (ADCIRC) model for shelves, coasts and estuaries. The m ode l uses the finite element method (FEM) with unstructured triangular mesh es. Westerink, Luettich and Muccino (1993) developed two dimensional Western North Atla ntic Tidal (WNAT) model based on the ADCIRC. The model was applied to the western North Atlantic, Gulf of Mexico and Caribbean to predict tides. They used local mesh refinement to investigate the influence of the resoluti on of the coastal boundary and found that in near coastal re gions and on the continental shelf, the resolution of the coastal boundary will strongly influence the computationa l results. The general problem of finite element methods is that conservation is not guara nteed without special tr eatment to the scheme. Recently, Zhang, Y.-L. and Baptista, A.M developed a semi-implicit Eulerian-Lagrangian finite-element model (SELFE, 2006). SELFE is an unstructured-grid model designed for the effective simulation of 3D baroclinic circulatio n across river-to-ocean sc ales. It uses a semiimplicit finite-element Eulerian-Lagrangian algorithm to solve the shallow water equations. This model uses hybrid sigma and Z grid in vertical di rection to represent the vertical structures of surface layer flows. The numerical algorithm is high-order, and stable and computationally efficient (but slightly more expensive than ELCIRC). Although not guarant eed in the numerical scheme, the volume conservation is generally good. It also naturally incorporates wetting and drying of tidal flats. While or iginally developed to meet sp ecific modeling challenges for the Columbia River, SELFE has been extensively tested against ocean and coastal benchmarks and applied to a number of bays and estuaries. Finite Volume Models Recently, th e finite volume method has received considerable attention in the numerical computation of fluid dynamics (Dick 1994). The finite volume method (FVM) takes the merits PAGE 29 29 of both finite difference and finite element meth od. In some sense, the finite volume method can be considered as a finite differe nce method applied to the differen tial conservative form of the conservation laws, written in arb itrary coordinate systems. Hence, this method can be applied using any structured grids in the finite differe nce method or unstructured meshes in the finite element method. Generally, the fini te volume method needs less computational effort than finite element method. The finite volume method is ba sed on the integral form of the conservation equation; therefore, the method can ensure that the basic quantities, mass and momentum will remain conserved. Also, a scheme in conservatio n form can be constructed to capture shock waves. Zhao, Tabios, Shen and Lai (1994, 1996) developed a river basin two-dimensional flow model (RBFVM) using the finite volume method to examine river basin flow. The model used unstructured mesh system using a combination of either triangular elem ents or quadrilateral elements. The model was developed without the wind stress, Coriolis and horizontal diffusion terms in governing equations. It has just advection, propagation term in the equations. The model was applied to the flow simulation of the Kissi mmee River, Florida and a levee failure problem in the Mississippi River, Mississippi. Chippada, Dawson, Martinez and Wheeler (1998) have developed a finite volume based model for the numerical solution of the system of shallow water equations. The model is developed in two dimensional with arbitrary triangular meshes and conserves all primary variables such as mass and mo mentum. This model does not ha ve wind stress, Coriolis, and horizontal diffusion terms. The model applied to the tidal waves in the Galveston bay, Texas and compared with the finite element model ADCIRC. PAGE 30 30 Ward (2000) developed an unsteady two dimensional finite volume circulation model to predict sea surface elevation and currents in estuarine and coas tal waters on a system of unstructured triangle meshes. The model is used the semi-implicit solution technique in which the sea surface elevation is solved implicitly using a Gauss-Seidel, point-by-point iterative scheme with successive over-relaxation. Once th e solution for the sea surface elevation is available the velocity are solved explicitly. The model was applied to pred ict tidal circulation in Greenwich Bay, R. The model also used the local mesh refinement. This model does not have wind stress term, Coriolis term, a nd horizontal terms in the model. Chen, et. al. (2003) developed an unstructure d grid, finite-volume, 3D primitive equation, turbulent closure coastal ocean model calle d FVCOM (Finite Volume Community Ocean Model). FVCOM is composed of external and inte rnal modes that are com puted separately using two split steps i.e. external and internal modes to increase computational efficiency. The model uses grid for the vertical direction. FVCOM is solved numerically by the flux calculation in the integral form of primitive equations over non-overlapping, unstructure d triangular grids. The model applied to the Gulf of Maine/Georges Bank region. Valiani, Caleffi, and Zanni (2002, 2003) develope d finite volume model using unstructured quadrilateral meshes to simulate dam breaki ng problem. The algorithm is obtained through the spatial discretization of the shallow water equations by FVM. Particular attention is posed to the numerical treatment on source terms. In order to reduce the numerical instability related to the bottom friction and bottom slope term, a semi-imp licit treatment is used. A mesh refinement technique was used. The model has the ability to handle wetting and drying. The model was applied to dam breaking problem in Malpasse t, France and a flood event in the Toce River PAGE 31 31 Valley, Italy. Table 1-2 shows main features of reviewed finite difference, finite element and finite volume numerical models. Objectives and Modeling Strategies Followed by UnTRIM model of Casulli (2000), th e present work is to develop three dimensional time dependant, triangular and/or qu adrilateral unstructured ocean, coastal, and estuarine numerical circulation model by finite difference and finite volume method to take advantage of merits of the computational effici ency of the finite difference method, conservation of finite volume method and of fl exibility of representing of co mplex geometry of the finite element method. This work should include followings. An Unstructured Orthogonal Triang ular and/or Quadrilateral Model The conventional rectangular gr id system has disadvantages when representing complex geometries. To overcome this, complicated bounda ry fitted curvilinear orthogonal grid system are used. The problem of the curvilinear coordina te system model is th at the governing equation has to be transformed to the curvilinear coor dinate system. The governing equations in the Cartesian coordinate system can be directly discretized to fini te difference and volume equation with any shape of grid or mesh system without any coordinate transformation. The computational mesh will be the unstructured or thogonal triangular or/and quadrilateral mesh system in horizontal directions. Local Mesh Refinement Model Unstructured meshes fit complex geometri es well without resor ting to coordinate transformations. The proposed model will have the ability of the local mesh refinements i.e., the meshes around complex boundaries and the areas wh ere the water depth changes abruptly are fine and the other areas are coar se but the size and shape of tria ngles vary gradually through the PAGE 32 32 flow domain to reduce discre tization error (Ferziger and Pe ric 2002). Figure 1-5 shows an example of an unstructu red triangular mesh without local mesh refinement. Figure 1-6 shows an example of an unstructu red triangular mesh with local mesh refinement when there is deep navigational channel in the middle of computati onal domain. In Figure 1-5, the dotted line is an area of the deep navigational channel. Figure 1-5 Example of an unstructured triangul ar mesh without local mesh refinement Figure 1-6 Example of an unstructured tria ngular mesh with local mesh refinement PAGE 33 33 Handling Wetting and Drying In the shallow coastal water bodies like lagoons, embayments, and estuaries experiencing tidal oscillation of free surface, the extent of areas subjected to alternating wetting and drying (inter tidal flats) can be of the same order of magnitude as the constantly submerged areas. The reproduction of the wetting and drying of the tidal fl ats is one of desirable features of numerical tidal flow models based on the shallow water equa tions. The present model will have the ability to handle the wetting and drying. Inclusion of Wind Stress, Atmospheric Pressure, Coriolis, Horizontal Diffusion, and Bottom friction Terms In most physical situations, the bottom friction terms dominate the lateral diffusion and dispersion. However, there exist situations, especially those involving heavy mixing, where the lateral diffusion terms are important. The Coriolis terms are also important when one simulates the large scale ocean and coastal area. Given the mixing of fresh water and salt water within an estuary, it is possible to establis h baroclinic circula tion (Dyer 1997). In order to present this phenomenon, the baroclinic term should be included in momentum equations. Wind and bottom friction are also important in ci rculation of estuarine system. Th e present model will have the wind stress, Coriolis, horizontal diffusion, bottom friction terms. Comparison with Analytical or CH3D Solutions and Application At the end of each developing stage of the m odel, the present model should be verified by comparing with existing analytical solutions and the result of a state-of -art finite difference numerical model CH3D (Curvilinear Hydr odynamic 3 Dimensional model, Sheng 1986,1987). Then the model will be applied to the storm surge simulations of Hurricane Ivan (2004) and Dennis (2005). PAGE 34 34 Table 1-2 Comparison of unstructured model FDM,FVM FEM FVM Model Name UnTRIM Elcirc Ad Circ SELFE RBFVM FVCOM Authors Casulli and Walters (2000) Y. Zhang Baptista, Myers. (2004, 2005) Luettich, Westerink, Sheffner (1991,1993) Y. Zhang Baptista (2006) Zhao, Tabios, Shen and Lai (1994, 1996) Valiani, Caleffi, and Zanni (2002, 2003) Chippada, Dawson, Martinez (1998) Ward (2000) Chen, et. al. (2002) Dimension 3D 3D 2D/3D 3D 2D 2D 2D 2D 3D Mesh system OT, OQ OT, OQ Triangle T, Q T, Q Q T T T Vertical grid Z Grid Z Grid Z Grid Sigma, Z Sigma grid Equation Differential and integral Differential and integral Integral with shape function Integral with shape function Integral form Integral form Integral Integral Integral (Mode splitting) Application Tidal flow in the Lagoon of Venice, Italy and the reach of the Big Lost River, ID Baroclinic circulation Columbia River, WA Tidal simulation Western North Atlantic Ocean, Gulf of Mexico Baroclinic circulation Columbia River, WA Flow simulation of the Kissimmee River, FL and Mississippi River Dam-Break in Malpasset, France and Flood event, Toce River Valley, Italy Tidal waves in the Galveston bay, TX Tidal circulation Greenwich Bay, RI Tidal simulation to Bohai Sea and Satilla River, GA OT: Orthogonal Triangle Mesh, OQ: Ort hogonal Quadrilateral Mesh, T: Triangle Mesh, Q: Quadrilateral Mesh Table 1-3 Features of the present mode Features of Present model (UFDVM) Grid system Unstructured orthogonal triangular or/and quadrilateral meshes Vertical grid Z grid. Mesh refinement Handles coarse and fine meshes Numerical method Combined finite difference and volume method Conservation Local and global mass Drying and wetting Has the ability to handle wetting and drying Inclusion of terms in equation Include wind stress, atmospheric pressure, Coriolis, horizontal diffusion, bottom friction, and turbulence PAGE 35 35 CHAPTER 2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS IN T HREE-DIMENSIONAL CARTESIAN COORDINATES Introduction In this chapter, the basic three-dim ens ional equations and boundary conditions for estuarine and coastal circ ulation are described with respect to Cartesian (x y z ) coordinate system. Governing Equations in Cartesian Coordinates The four basic assum ptions used for deriving the governing equations are : (1) the flow is incompressible so that the flow is non-divergent, (2) the vertical accelerations are negligible compared to gravity, thus, the vertical pressure distributions satisfies hydrostatic approximation, (3) density is approximated by its mean valu e except when multiplied by gravity, i.e., the Boussinesq approximation is valid, and (4) the ed dy viscosity concept is used with constant turbulent eddy viscosity and diffusivity. Using a right-handed Cartes ian coordinate system ( x y z ) with the origin at the water surface and z measured upward, the equations ar e written as following non-conservative form (Lamb, 1945) : Continuity equation, 0 z w y v x u (2-1) X-momentum equation, fv z u A zy u x u A x p z u w y u v x u u t uv H o 2 2 2 21 (2-2) Y-momentum equation, PAGE 36 36 fu z v A zy v x v A y p z v w y v v x v u t vv H o 2 2 2 21 (2-3) Z-momentum equation, g z p (2-4) Additionally, by using the Leibni tz integration rule, the hydros tatic pressure gradient can be split into the barotropic and the baroclinic components as follows dtyxgtyxtyxgtyxPpz ah),,,(),,(),,,(),,( (2-5) Finally the momentum equations in x and y directions include atmo spheric pressure term are fv z u A z y u x u A P x dz x g x g z u w y u v x u u t uv H o a z 2 2 2 2 0 (2-6) fu z v A z y v x v A P x dz y g y g z v w y v v x v u t vv H o a z o 2 2 2 2 (2-7) Transport equation )() (2 2 2 2z S D z y S x S D z wS y vS x uS t Sv H (2-8) Equation of state, ),( ST (2-9) 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 direction, t is the time, ( x y t ) is the free surface elevation, g is PAGE 37 37 the gravitational acceleration and Ah, Dh and Av, Dv are the horizontal and vertical turbulent eddy coefficients, f is the Coriolis parameter defined as 2 sin where is the rotational speed of the earth and is the latitude, T is temperature, S is scalar transport quantity, o is the density of freshwater, is the density, p is pressure, and Pa is atmospheric pressure at the water surface respectively. Various forms of the equation of state can be used. Eckart (1958) used: ST TT STT P P P )10.08.3(0745.025.115.1779 3375.0385890 )698.0/(2 2 (2-10) where T is in C S is in psu and is in g/ cm3. CH3D uses Equation 2-10, while Blumberg(1978) used: )1( So (2-11) where is a function of temperature, and for a temperature of 15 C and is 7.57E 104. The value of can be derived from t tables (Bowden, 1967). In the present model, Blumberg equation is used because the present mode l does not have a temperature equation. Boundary Conditions in Cartesian Coordinates The boundary conditions at the free surface are specified by the prescribed wind stresses w x and w y : 22 wwwda w x vvuuC z u A (2-12) 22 wwwda w y vvuvC z v A (2-13) where w x and w y are the wind stresses in x and y directions 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 PAGE 38 38 surface and the wind speed at some height above the water surface. For this study, the empirical relationship developed by Garratt (1977) is used. Garratt defined th e drag coefficient as a linear function of wind speed measured at 10 meters above the water surface, Ws : )067.075.0(001.0s daW C (2-14) where Ws is the wind speed in meters per second. At the free surface, the kinematic condition states that ty v x u tDt D w (2-15) Integrating the continuity equation over the dept h and using the kinemati c condition at the free surface leads to the following free surface equation: 0 h hvdz y udz xt (2-16) where h ( x y ) is the water depth measured from the undisturbed water surface and H ( x y t ) is the total water depth, given by H ( x y t ) = h ( x y ) + ( x y t ) The boundary conditions at the sediment-water interface are given by specifyi ng the bottom stress in a form of the ChezyManning formula u z u Ab x v (2-17) v z v Ab y v (2-18) where = g (u2+v2) / Cz 2, u and v are velocities at the bottom layer, and Cz 2 is the Chezy friction coefficient which can be formulated as : n R Cz6 164.4 (2-19) PAGE 39 39 where R is the hydraulic radius given in centimeters and n is Mannings n. In shallow estuaries, the hydraulic radius can be approxi mated by the total depth. In CH3D model, the bottom stresses are calculated differently: ),()(),(11 2/12 1 2 1vuvuCdb b y b x (2-20) where Cda is the bottom friction coefficient, and u1, and v1 represent the velocity components at the first grid point a bove the bottom. Taking z1 as half of the bottom laye r thickness (which starts at the bottom roughness height, zo), Cda, for a hydraulic rough flow, is given by (Sheng, 1983): 2 0 1 2ln z z kCdb (2-21) where k is the von Karman constant. The open boundary condition for the su rface elevation is prescribed by nn n N n nnT t FAtyx 2 cos ),,(1 (2-22) where An, Tn, Fn, and n are the amplitude, peri od, nodal factor and phase angle of the tidal constituents. When open boundary conditions are given in terms of the surface elevation, the normal velocity component is assumed to be of zero slope while th e tangential velocity component may be either (1) zero, or (2) zero slope or (3) computed from the momentum equations. In the present study, it is assume th at the velocity gradients are zero at the open boundary. At the fixed boundary, no flux through the boundary is allowed. Salinity is prescribed as a function of time and depth at the open boundary or salinity is calculated from the one dimensional advection equa tion. In this study, th e salinity at the open boundary is prescribed. At the wa ter surface and the bottom the norma l salinity flux vanishes, i.e. 0 z S Dv (2-23) PAGE 40 40 If the estuary is connected to freshwater river flow then the boundary conditions at the interface are given as S x y z t(,,,) 0 (2-24) uxyzt q hb(,,,) () (2-25) where q is the river flow, usually a constant,(m3/s) and b is width of the corresponding river cells. At the open boundary the sali nity is prescribed as SxyztS(,,,) 0 (2-26) where So is the prescribed salinity value in psu. PAGE 41 41 CHAPTER 3 THREE-DIMENSIONAL NUMERICAL MODEL W ITH ORTHOGONAL UNSTRUCTURED MESH Introduction The governing equations and boundary conditions discussed in Chapter 2 are discretized using finite difference and finite volume method with orthogonal unstructured mesh similar to that described in Casulli a nd Walters (2000) and Casulli and Zanolli (2002, 2005). Before discretizing the gove rning equations, the horizontal (x, y) domain is covered by a set of nonoverlapping convex polygons such as quadrilaterals or triangles. In this study, quadrilateral and/or triangle mesh will be used. A simple stability analysis of the two-dime nsional, vertically averaged shallow water equations showed that the celerity term gh in the equation affect s the stability of the discretized finite difference and volume equati ons. Results of this analysis led to the development of an implicit method (Casulli and Cheng, 1992) and a semi-implicit method (Casulli and Cattani, 1994, Lee, 2000) which has been proven to be unconditionally stable. When the implicit method is used, there is undesirable numerical damping in the numerical solution. To improve the accuracy of the implicit method, an implicitness parameter is introduced such that for = 1, this method is fully implicit and for = 0.5, this method is semi-implicit and remains stable and does not have undesi rable numerical wave damping. For the nonlinear advection term and the Corio lis term, an explicit Eulerian Lagrangian method (ELM) is used (Staniforth and Temp erton, 1986., Casulli, 1990., Nicolaides, 1993., Oliveira and Baptista, 1998, Walters and Casull i, 1998). The Eulerian Lagrangian numerical method is explicit scheme. Moreover, the met hod is unconditionally stable regardless of time step. By using explicit Eulerian Lagrangian me thod, the nonlinear adve ction and the Coriolis PAGE 42 42 terms are computed efficiently. The horizontal diffusion term is discretized by the finite volume method. For stability, the vertical diffusion term wind stress term, and the bottom friction term are discretized by implicit finite difference method. In this chapter, an orthogonal unstructured mesh will be explained in detail and the threedimensional semi-implicit circulation mode l is developed by using the so called scheme or method with finite volume and finite difference method. Tabl e 3-1 shows the summary of numerical discretization method for the equa tions and terms in momentum equation. Table 3-1 Numerical method for the model Equation Term Method Numerical scheme Free surface Propagation Finite Volume method Non-linear Finite Difference Explicit ELM Horizontal Diffusion Finite Volume Explicit cell -centered scheme Vertical Diffusion Finite Difference Implicit central difference Coriolis Finite Difference Explicit ELM Momentum Air pressure Finite Difference method Definition of the Unstructured Orthogonal Mesh and Index A mesh said to be an unstructured orthogonal mesh if within each polygon a point or node can be identified in such a way that the segmen t joining the centers of two polygons and the side shared by two polygons, have a non-empty intersec tion and are orthogonal each other. Figure 3-1 shows the definition of the orthogonal unstructure d mesh for computation. The polygons of solid lines are computational elements and the polygons of dashed lin es are Voronoi regions of the meshes in the figure. In the Figure 3-1, circled dot represents Voronoi cen ter and rectangular dot represents geometric center of triangle. Th e center of the each polygon does not necessarily coincide with their geometrical center. Exam ples of structured orthogonal grids are the rectangular finite difference grid s as will as a grid of uniform equilateral triangles. In these PAGE 43 43 particular cases, the center of each polygon can be identified with its geometrical center. Further examples are the orthogonal curv ilinear grids. Also, an unstruc tured orthogonal mesh is the Delaunay triangulation of a set of points Qi, i = 1, 2, 3, N, provided that the triangulation includes only acute triangles. In this case, the center of the triangles can be identified by the vertices of the corresponding Di richlet tessellati on (Rebay, 1993). Similarly, the Voronoi regions of a Dirichlet tessellation determined by Qi also form an orthogonal mesh with Qi being the centers of the polygons. In Figure 3-1 circled points represent Voronoi centers and square points are the geometrical centers of each triangle. Or thogonality is a requirement for calculation of finite difference approximations of spatial gr adients in unstructured meshes. The accuracy of numerical solutions depends on orthogonality. Ther efore to obtain higher accuracy of numerical solutions, the numerical mesh should be orthogonal. Figure 3-1 Orthogonal unstructured mesh jPi(j,1)Pi(j,2)Pij(i,1) j(i,2) j(i,3) PAGE 44 44 Once the orthogonal meshes are created, the computational mesh system has Np polygons, each having an arbitrar y number of sides Si 3, i = 1, 2, 3, Np. Let Ns be the total number of sides in the meshes, and let j, j = 1, 2, 3, ..., Ns be the length of each side. The sides of the i-th polygon are identified by an index j(i, l), l = 1, 2, 3, ... Si so that 1 j(i, l) Ns, i = 1, 2, 3, ..., Np. The two polygons that share j-th side of the grid ar e identified by the indices i(j,1) and i(j,2) so that 1 j(i, 1) Np, and 1 j(i, 2) Np, j = 1, 2, 3, ... Ns. The distance between the centers of two adjacent polygons that share the j-th side is denoted by j. As shown Figure 3-2, along the vertical dir ection a finite difference discretization, not necessarily uniform, is adopted. The three dimensi onal spatial disc retization consists of prisms whose horizontal faces are the polygons of a given orthogonal mesh and whose height are zk. The discrete velocities and water surface elevation are defined at staggered locations as follows. The water surface elevation n i, the density n i, the salinity Sn i, and the temperature Tn i are located at the center of the i-th polygon. The velocity component normal to each face of a prism is defined at the point of inte rsection between the face and the segment joining the centers of the two prisms that share the face. The water depth hj is defined at the center of the elements. Figure 3-2 shows the location of the com putational variable in a mesh. Finite Volume and Difference Discretization and Solution Algorithm The horizontal gradients of surface elevation in the momentum Equations 2-6 and 2-7, and the velocity gradients in the free surface Equation 2-16 will be discretized by the -scheme, and the vertical mixing terms will be discretized implicitly because the vertical length scale is much smaller than the horizontal one. The Equations 2-1 through 2-8 are invari ant under solid rotation of the x and y axis on the horizontal plane. By usi ng the invariant proper ty of the equations, PAGE 45 45 the momentum Equations 2-6 and 2-7 may be e xpressed as single equation for the velocity component normal to each vertical face of a prism. The equation can be written as u t u u x v u y w u z g x g x dz x P A u x u y z A u z fvo z a o Hv* ** *** *() 2 2 2 2 (3-1) where u* is the velocity component normal to each vertical face of a prism and v* is the tangential velocity component in a right hand coordinate system. The relations of velocity components u, v, u*, and v* are u = u* cos v = u* sin and v* = u* tan where is the angle between u* and x axis (Figure 3-2). For simplicity, is omitted for the equations. As shown in Figures 3-1, 3-2, and 3-3, the spatial mesh consists of triangular or rectangular boxes of face length j and height zk, i.e., a cell-centered grid system is used. zk is the thickness of the k-th layer i.e. the distance between levels k-1 and k. The distance between half-level zk+1/2 is defined as ( zk + zk+1)/2. Each box is numbered at its center with indices i, j, and k. The discrete u velocity is defined at half integer i and integers j and k; velocity v is defined at integers i, k and half integer j; vertical velocity w is defined at integers i, j and half integer k. Water surface elevation is defined at the center of each column i and j. The density of water is defined at the center of each box i, j and k. The water depth h(x, y) is specified at the u and v horizontal points. Using an orthogonal unstructured mesh system, a semi-implicit finite difference discretization for the velocity component normal to each verti cal face of a prism can be derived from the Equation 3-1. Wind stress, Vertical viscosity and bottom fric tion terms are discretized by implicit scheme and the wave propagation term is discretized using a scheme to obtain an efficient numerical algorithm. The advection, horizontal diffusion, Coriolis, and baroclinic terms are discretized by explicit method. PAGE 46 46 Figure 3-2 Location of computational variables Figure 3-3 Definition of velocity components u*x y v* u v k k+1 k-1uj,khjuj,kiiSiTi PAGE 47 47 The finite difference discretization for the ve locity component normal to the each vertical face of a prism with staggered grid system can be derived from Equation 3-1 takes the following form: uFug t g t zz t zjk n jk n j ij n ij n ij n ij n oj jm n ijm n ijm n jk n ijk n ijk n mk N jk njkz,, (,)(,) (,)(,) ,(,),(,),, (,),(,), ,() 1 2 1 1 1 21 21 21 11 2 / ,, ,/ ,/ ,, ,/ 2 1 11 12 12 1 1 1 12v jk n jk n jk nj k v jk n jk n jk nuu z uu z (3-2) where kmmMjjj n,,, 1. In Equation 3-2, un i,j denotes the horizontal velocity component normal to the j-th side of the mesh, at vertical level k and time step n. The positive direction for un i,j has been chosen to be from i(j,1) to i(j,2). F is an explicit finite difference operator, which accounts for the contributions from the discretization of the Cori olis, advection and horizontal friction terms. By using Eulerian Lagrangian scheme, a form for F can be chosen as (Casulli and Waters, 2000). Fu jk n uftv jk t PP t h h u jkjk aij n aij n j,, *, ,(,),(,) 21 (3-3) where u*j,k denotes the horizontal velocity component normal to the j-th side of the mesh and v*j,k is the tangential velocity component in a ri ght-hand coordinate system. Both components are interpolated at time tn at the end of the Lagrangian trajectory based on the values at adjacent mesh points. h is the horizontal Laplacian discretiza tion. In this study, the calculation of Fu is different from Equation 3-3. The nonlinear advec tion term and the Coriolis term are calculated by ELM, the diffusion term is calculated by finite volume method. The calculation of the term Fu will be discussed in following sections. PAGE 48 48 The values of uj,k n+1 above the free surface and below the bottom in Equation 3-2 are eliminated by means of the vertical boundary conditions which are shear stress. At the free surface, wind shear stresses are used. Using the shear stress, the vertical boundary conditions yield following formulae: jM v jM n jM n T n a n jM nu z uu,/ ,/ 12 1 1 12 111 (3-4) and jm v jm n jm n jm n B n jm nuu z u,/ ,, ,/ 12 1 1 1 12 11 (3-5) At the bottom, the log-layer distribution of velo city is used to calculate the bottom stress: B bo bbbzz uuv ln(/)22 (3-6) where is the von Karman constant 0.41, ub and vb are horizontal ve locity components calculated ate a bottom level, Zo = ks / 30 and ks is the bottom roughness. To obtain complete local and global mass c onservation and stabil ity, the free surface Equation 2-16 is discretized by fi nite volume method. Consider a uniform rectangular mesh as shown Figure 3-4. To get a semi-implicit finite volume equation, integrate Equation 2-16 over an area of element Pi, then t xy x udz y vdzxyP hh Pii (3-7) Now, a semi-implicit finite volume discretization for Equation 3-7 gives PAGE 49 49 i n i n ji n ji n ji n ji n km M km M ji n ji n ji n ji n km M km Mt xy zuzu x xy zuzu y x 1 11 1 33 1 22 1 44 1 (,)(,) (,)(,) (,)(,) (,)(,) y zuzu x xy zuzu y xyji n ji n ji n ji n km M km M ji n ji n ji n ji n km M km M ()(,)(,) (,)(,) (,)(,) (,)(,)111 33 22 44 (3-8) where x = j(i,2) = j(i,4), and y = j(i,1) = j(i,3). (3-9) If Equation 3-8 is generalized for any shape of polygons by using relation Equation 3-9, a semi-implicit finite volume discretization for the free surface equation at the center of each polygon is taken to be following form: PPtSzu tSzuii n ii n iljiljilk n km M jilk n l S iljiljilk n km M jilk n l Si i 11 1 11 ,(,)(,),(,), ,(,)(,),(,),() (3-10) where Pi denotes the area of the i-th polygon, i.e. Pi = x y in Figure 3-4 and Si, l is a sign function associated with the orientation of the normal velocity defined on the l-th side of the polygon i. Specifically, Si, l = 1 if a positive velocity on the l-th side corresponds to outflow, Si, l = -1 if positive velocity on the l-th side corresponds to inflow to the i-th water column. Thus, Si, l can be written as: S ijiliijil ijilijilil,(,),(,), (,),(,), 221 21 (3-11) PAGE 50 50 Figure 3-4 Numerical stencil of Finite Volume Method Equations 3-2 and 3-10 constitute a linear system of at most NzNs + Np equations. This system has to be solved at each time step in order to calculate the new field variables. The coefficient matrix of these systems is symmetric and positive definite. Thus the vertical velocity component of the velocity can be readily determ ined by a direct method i.e. Thomas algorithm. Since a linear system of NzNs + Np equations can be large, th e system of Equation 3-2 and 3-10 is first decomposed into a set of Ns independent tri-diagonal systems of Nz equations. Upon multiplication by z, and after including the boundary condi tions of Equations 3-4 and 3-5, Equations 3-2 and 3-10 are first written as matrix form as: PPtSZU tSZUii n ii n il l S jiljil n T jil n il l S jiljil n T jil ni i 1 1 11 1 11 ,(,)(,)(,) ,(,)(,)(,)() (3-12) y x i,2 i,3 i,4 i,1 i PAGE 51 51 After including the boundary c onditions of Equation 3-4 and E quation 3-5, Equations 3-2 and Equation 3-10 are first writt en as matrix form as AUGg t Zj n j n j n j ij n ij n j n 1 2 1 1 1 (,)(,) (3-13) where U u u u Z Z Z Zj n jM n jM n jm n j n jM n jM n jM n 1 1 1 1 1 1 , ,,, G ZFu t gtu ZFu t g ZFu t gj n MjM n j ij n ij n Ta n MjM n j ij n ij n mjm n j ij n ij n ,( ) ( ) ,( ) ( ) ,( ) ( )()() ()() ()( 1 1 121 1 1121 21 ) A zata azaaa azatj n jM n jM n T n jM n jM n jM n jM n jM n jM n jm n jm n jm n B n ,,/ ,/ ,/ ,,/,/,/ ,/ ,,/ 12 1 12 12 11232 32 12 12 10 0 with atzjk n jk n jk n,/,/,//1212 12. PAGE 52 52 Formal substitution of the expression for Uj n1 from Equation 3-9 into Equation 3-10 yields a discrete wave equation for n1 which is given by Pgt S ZAZ PtSZUtSZAGii n iljil jil l S T jil n ijil n ijil n ii n iljil T jil n l S iljil l S T jii ii 122 1 1 1 2 1 1 1 11 11 ,(,) (,) (,) (,),(,), ,(,) (,) ,(,) ((),)l n (3-14) Since matrix Aj n is positive definite, its inverse is also positive definite and therefore [( Z)TA-1 Z]j n is a non-negative number. Hence, Equation 3-14 constitutes a linear sparse system of Np equations for j n+1. This system is strongly diagonally dominant, symmetric and positive definite. Thus it has a unique solution that can be efficiently determined by a preconditioned conjugate gradient method. Once the new free surface location has been computed, Equation 3-12 constitutes a linear, tri-diagonal system for Uj n+1. Each of this tridiagonal systems is independent of the others and is symmetric a nd positive definite. Thus, they can be conveniently solved by a direct tri-diagonal algorithm to determine Uj n+1 throughout the computational domain. Finally, a finite volume discretization of the continuity Equation 2-1 yields the vertical component of the velocity at the new time level. The vertical component of the velocity can be obtained from the incompressibi lity condition, by setting wj n+1 = 0, gives ww P Szu kmmMik n ik n i il l S jiljilk n jilk ni,/,/ ,(,)(,),(,),, ,,, 12 1 12 1 1 11 11 (3-15) Equation 3-15 determines the vertical velocity component on each water column recursively. PAGE 53 53 One very important thing should notice here is treatment of the su rface layer. When the number of vertical layers at time step n and time step n+1 is different, the dimension of coefficient matrix A and new velocity column vector for Uj n+1 in Equation 3-13 is different. Figure 3-5 shows the situation. At time n, the wa ter column has 3 vertical layers, whereas at time n+1, there are 4 vertical layers in z direction because of the changes of water surface elevation with time. In this situation, the matrix and column vector multiplication does not work. Therefore, the velocity at the surface layer is still unknown. Zj n Aj n Uj n+1 Gj n Zj n Zj n+1 Aj n Uj n+1 Gj n Zj n Figure 3-5 Different dimension of coe fficient matrix A and column vector Uj n+1 There are two ways to handle this situation. In Z grid model, the thickness of the surface layer should be big enough to hold water surf ace elevation during the simulation. The surface layer number will not change. In this case, the model does not resolve fine vertical structures of the flow. Another way to avoid th is situation is to use sigma la yer (Sheng, 1989) or mixed sigma and Z grid model (Martin, 1999) in vertical direction. With the use of sigma layer, the number of vertical layers remains same during the simulatio n. In Appendix, the derivation of sigma layer version of UFDVM is explained. However, Z-grid model is used in this study. 3 2 1 4 3 2 1 33 32 31 23 22 21 13 12 11 33 32 31 23 22 21 13 12 11 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 4 3 2 1 PAGE 54 54 Treatment of Flooding and Drying The solution algorithm in Section 3.3 include s the simulation of flooding and drying. As shown Figure 3-6, at each time step, the new total water depth Hj n+1 at the sides of polygon are defined as: Hhhj n jij n jij n 1 1 1 2 10 max,,(,)(,) (3-16) The vertical grid spacing zj n+1 are updated accordingly. Thus, an occurrence of a zero value for the total depth Hj n+1 implies that all the vertical fa ces separating prisms between the water column i (j,1) and i (j,2) are dry and may become wet at a later time when Hj n+1 becomes positive. The height of a dry face and the corresponding normal velocity are taken to be zero. The size and the structure of the linear system given by the discrete wave Equation 3-14 are independent from the vertical resolution. The ver tical resolution affects the assembly of Equation 3-14 and the number of equations for normal velocity Thus, an arbitrarily fi ne vertical resolution can be used with an acceptable increase of the corresponding computing time. Detailed explanation of flooding a nd drying will be discussed in Chapter 5.9. Figure 3-6 Determination of wet a nd dry depth at an element face hji(j,1)i(j,2) PAGE 55 55 Eulerian-Lagrangian Scheme for Advection Term One major challenge in numerical modeling of free surface flows is the accurate and efficient treatment of the advection term in the governing equations. There are numerous numerical schemes for solving the advection term (Lee, 2000) Eulerian-Lagrangian Method (ELM) or semi Lagrangian method is used for this study for the advection term. The Eulerian-Lagrangian Method uses the Lagr angian form of governing equations in an Eulerian computational grid system. The method (Starniforth and Cote, 1991) is generally referred to as semi-Lagrangian method in the comm unity of numerical weat her prediction, but is called Eulerian-Lagrangian Method (ELM) by estu arine and coastal modelers, e.g., Baptista (1987), Casulli and Cheng (1992). The ELM solves th e equation for a variable, e.g., a velocity component or concentration by following a fluid particle along the characteristic backward in time and then interpolating at its foot (departure po int of the particle) to obtain the values of the variables. The ELM was also used by Tang and Adams (1995) with finite difference method and by Baptista (1984), Oliveira and Baptista (1998) and Walters and Casulli (1998) with finite element method. The major advantage of the ELM is that the method is unconditionally stable although it is explicit. Hence, there is no CFL condition constrai nt for numerical stability. In addition, it is transportive and accurate (Casulli, 1994). Despite these advantages, there are several disadvantages of this method. Although the method is accurate, its accuracy is dependent on the interpolation function used. For instance, if a lin ear interpolation function (first order accuracy) is used, there is excessive numerical diffusion in the numerical solution, with the maximum diffusion on the order of x2/(8 t) in the 1D case, although the diffusion can be controlled by reducing the spatial increment or by increasing th e time step size. The Eulerian Lagrangian PAGE 56 56 method is used for solving non-linear term in th e equation in order to take advantage of its simplicity and the enhanced stability and accuracy. Consider the diffusion-free non-conser vative advection equation in 1D c t uc0 (3-17) where the c(x, t) represents any scalar quant ities or velocity vectors, u is velocity field, and is a gradient operator. Most of the fundamental e quations in fluid dynamics can be derived from first principles in either Lagrangian form or Eulerian form. Lagrangian equations describe the evolution of the flow that would be observed fo llowing the motion of an individual parcel of fluid. Eulerian equations describe the evolution that would be observed at a fixed point in space. The one-dimensional Eulerian advection Equation 317 can be expressed in Lagrangian form as Dc Dt0 (3-18) The mathematical equivalence of Equations 3-17 and 3-18 follows fr om the definition of total derivative, D Dtt dx dtx (3-19) and the definition of the velocity, dx dt uxxxi i*(,,)123 (3-20) where i = 1, 2, and 3 and represents a linear interpolant between time step n and n+1. When a Lagrangian numerical treatment is applied to Equation 3-18, the computational grid will be continuously deforming in the general case when u is non-constant. For operational advantage, Equation 3-18 will be discretized on a fixed Eulerian grid system. Lets consider one PAGE 57 57 dimensional time dependant grid in Eulerian gr id shown in Figure 3-7. A finite difference scheme for Equation 3-17 is simply cci n ia n 1 (3-21) where ],)[( tnxaiccn ai and n is time step, i is any mesh or grid point, t is the index at a grid point and a is the CFL number. In general the CFL number is not an integer, therefore ( i-a ) is not the index of a grid poi nt and a proper interpolation fo rmula must be used to definen aic. The stability, numerical diffusion and unphysical oscillati ons of Equation 3-21 depend on the interpolation formula chosen. If a linear interpolation between (i-1) and (i) is used to estimate n aic, one obtains the first order upwind scheme. If a quadratic polynomial fit is used to interpolate between (i-1), (i), and (i+1), one obtains the Leiths method (1971). Figure 3-7 Schematic diagram for Eule rian grid on Lagrangian frame u t tntn+1p x i i-1 i-n i-a PAGE 58 58 The ELM uses a generalization of the interpolation concept of n aic between two or more grid points which do not necessarily include the point (i). Consider that n aic is taken to be a linear interpolation usin g one node upstream and one downstream. For a given a 0 let n be the integer part of a and p the decimal part, then a = n + p, 0 p < 1. In this case Equation 3-21 becomes ccpcci k in k in k in k 1 1() (3-22) or, equivalently cpcpci k in k in k 1 11 () (3-23) Note that if a < 1, then n = 0, p = a, and finite difference equa tions of Equation 3-22 and Equation 3-23 reduce to the firs t order upwind method, since 0 p < 1. The crux of Eulerian Lagrangian method is to determine the Lagrangian trajectory. Since the velocity u is generally non-uniform, the correct value of a can be found from the solution of the ordinary differential Equation 3-20 in three dimension using any backward trajectory computation. The velocity u is known only at time level tn, hence it will be assumed to be constant over one time step. Then, at each grid point ( i ), Equation 3-20 will be integrated numerically from tn to tn+1. Figure 3-8 illustrates the backtracking procedures. The tracking method and associated error analys is are well reviewed by Oliveira and Baptista (1998). In this study the backward Euler method is used for co mputation of the trajectory. To trace the Lagrangian trajectory, 3 dimens ional solution of Equation 3-20 is required. Backtracking for the velocity starts from elements face. To com pute the Lagrangian velocity, the time step t is divided into N equal increments, = t / N, and Equation 3-20 is discretized backward as xxuxxx sNNNssksN i 11221(), ,,,...,, (3-24) PAGE 59 59 where uk(xs) is interpolated with any in terpolation formula. Then, at xi, a can defined by a xx xi 0 The finite difference Equation 3-23 possesses the min-max property (Casulli, 1987), thus this method is entirely free of oscillations. Moreov er the min-max principle also implies stability, hence the numerical approximation of ELM, though explicit, is unc onditionally stable. Rigorous stability analysis can be perform ed using Von Neumann method. Assuming a solution to Equation 3-17 as cAeI k j kijIx() (3-25) where k jA is the amplification factor, I is the grid point index and 1 i. Substituting Equation 3-25 into Equation 3-17 Aaeek ikxiknx11 ()()() (3-26) The magnitude of Equation 3-26 is Aaakxk21211()(cos) (3-27) If the grid CFL number satisfies 0 a 1, the scheme is stable. B ecause the departure point lies within the interpolation interval (i n, i n 1), the magnitude of Equation 3-27 is always less than one, consequently the scheme is unconditi onally stable (Starniforth and Cote, 1991). To examine the artificial di ffusion introduced by Equation 3-22, each term in Equation 322 is expanded into Taylor series to produce the following: Dc Dt x t pp c x HOT 22 22 1 () (3-28) where HOT stands for higher order terms. PAGE 60 60 Since p < 1, the least upper bound for the ar tificial diffusion coefficient is ( x2 /8 t2) which is the same as the upwind method. Howeve r, since the ELM is unconditionally stable and the value of t can be arbitrarily large, the maximum nu merical diffusion can be controlled either by reducing the spatial increment or by increasing the time step. Figure 3-8 illustrates the back tracking of Lagrangian trajectory for an unstructured mesh. The superscript here denotes a variable evaluated at time tn at the end of the Lagrangian trajectory from a computational node. Tracking be gins at a element face of velocity node and Equation 3-20 is used to find the foot of the Lagrangian trajectory. Once the foot of the Lagrangian trajectory is found, the Eulerian velocity is ev aluated by following interpolation function. UU pqUqU pqUqUjilk tt jialkb t jilk t jilk t jilk t jilk t(,), (,), (,), (,), (,), (,), 11 11 11 1 (3-29) where pU tjilk t j(,), (3-30) and qw t zjilk t(,), (3-31) Equations 3-30 and 3-31 are solved the same manner as method to solve Equation 3-24. Finally, the non-linear term in the governing equation is approximated by Equation 3-29 as: NL = Ujialkb t(,), (3-32) PAGE 61 61 Figure 3-8 Schematic diagram of backtracking for the Lagrangian trajectory Finite Volume Discretization of Horizontal Diffusion Term The conservation laws of fl uid motion may be expressed mathematically in either differential form or integral form. When the in tegral form of the equations is utilized, the discretization of the equa tions is finite volume method. To ge neralize the method, consider a two dimensional heat conduction equation: T t A T x T yH 2 2 2 2 (3-33) Equation 3-33 can be written as conservative form; u t A x u xy u yh (3-34) Define F = ( u/ x ) and G = ( u/ y ), then Equation 3-34 is written as T t A F x G yh (3-35) UnUnUnU* PAGE 62 62 Equation 3-34 is integrated over an elements area such as quadrila teral mesh shown in Figure 3-6 or triangular me sh in Figure 3-9, then T t dxdy F x G y dxdyPPii (3-36) Subsequently, Greens Theorem is applied to the right hand side of Equation 3-36. Recall that Greens Theorem converts area integrals to line integrals. T hus, Equation 3-36 is written as u t dxdyAFdyGdxP h Pii (3-37) Using the triangular element in Figure 3-9, for an explicit formulation, the Equation 3-37 can be can be approximated as TTA t P FdyGdxi n i n h i Pi 1 (3-38) and FdyGdxFyFyFyGxGxGxP iabjbckcaiabjbckcai (3-39) where xab = xb xa and yab = yb ya is distance between the two nodes and the rest are the same. The functions F and G at points i, j, and k are calculated by the following relations: F T x T x dxdyA TTyTTyTTyTTyAi i ab ab a nn a n b n bb nn b n a n aab 14 14 111144441 42 / /(3-40) F T x T x dxdyA TTyTTyTTyTTyAj j bc bc b nn b n c n cc nn c n b n bbc 24 24 22224444242 / /(3-41) PAGE 63 63 F T x T x dxdyA TTyTTyTTyTTyAk k ca ca c nn c n a n aa nn a n c n cca 34 34 33334444342 / /(3-42) in Equation 3-40 through 3-42, the x increments are computed by ababyyy Figure 3-9 Numerical stencil of finite volume method for diffusion equation G T y T y dxdyA TTxTTxTTxTTxAi i ab ab a nn a n b n bb nn b n a n aab 14 14 111144441 42 / / (3-43) G T y T x dxdyA TTxTTxTTxTTxAj j bc bc b nn b n c n cc nn c n b n bbc 24 24 22224444242 / / (3-44) G T y T y dxdyA TTxTTxTTxTTxAk k ca ca c nn c n a n aa nn a n c n cca 34 34 333344443 42 / / (3-45) j(i,1) j(i,2) j(i,3)a b c T1 2 3 i j k 4 PAGE 64 64 Again, the x increments are computed by ababxxx where A is the area surrounded by subscript. The corresponding areas are evaluated according to AAA AAA AAAab abcadb bc abcbec ab abccfa14 24 141 3 1 3 1 3 (3-46) Substitution of Equations 3-43 through 3-45 into Equation 3-38 provides the value of Tn+1. When the above method is applied to the diffusion term in the momentum equation, the velocity is defined at the faces of each element. The va lue can be evaluated by interpolating two adjacent elements. Finally the horizontal diffusion term in the Equation 3-1 is approximated as: HD = A t P FdyGdxh i Pi (3-47) Treatment of Coriolis Term Several ways to treat the Coriolis term in the equation can be possible. The explicit discretization of the Co riolis term is unconditionally unsta ble. While the implicit method is unconditionally stable, the computing cost is expens ive. Another way to treat the Coriolis term is to discretize the term explicit in Xmomentum equation, while the term in the Y-momentum equation is discretized implicitly (Sheng, 1983). Although the Coriolis term in the Y-momentum equation is discretized implicitly, the actual co mputation of the term is explicit because the velocity 1 nu is already obtained from the X-momentum equation. From the stability analysis of this method, the time step of the scheme is less than 2/f which is about 6 hours. This is much bigger than the other terms time step constrai nt limits. When the staggered numerical grid is adapted in this method, v is not defined at the u grid point and u is not defined at the v grid point. A simple way to define the undefined velocity at a grid point is to take the average of the PAGE 65 65 velocities at four neighboring points. As discussed in Section 3.3, by the invariant property of the equations, the Coriolis term appears only once in the governing equation 3-1. In this study, the Coriolis term in the equation is treated explicit ELM to take advantage of no time limitation for numerical stability. Consider the Coriolis term in Equation 3-1 u t fv (3-48) By the explicit ELM, Equation 3-48 can be discretized as uutfvijl n ijl n ijl,(,),(,) ,(,) *1 (3-49) where v* is the tangential velocity component in a right hand coordinate system obtained by Eulerian Lagrangian method (see Figure 3-3) This method is unconditionally stable by ELM. Finally the Coriolis term in th e Equation 3-1 is discretized by COR = tfvijl,(,) (3-50) Air Pressure Term The air pressure term in the Equation 3-1 is approximated by semi implicit ( method) in time and forward difference in space. Consider the atmospheric term in the Equation 3-1 u t P xa (3-51) The semi-implicit in time and forward differenci ng in spatial derivative produces a finite difference equation of the form uu t PPjk n jk n a ij n a ij n j,,(,)(,) 1 21 (3-52) Stability analysis indicates that this method is stable when 1. The values of n ap are not calculated but provided by as data each time step. The unit of the atmospheric pressure is mili- PAGE 66 66 bar as input then converted to N/m2 at calculation. The numer ical approximation of the atmospheric term is written as AP PPa ij n a ij n j (,)(,) 21 (3-53) Final numerical discretization of explicit term Fu is written as Fu = NL + HD + COR + AP (3-54) where NL, HD, COR, and AP are defined by Equations 3-29, 3-47, 3-50, and 3-53, respectively. Solution of Sparse Matrix: Conjugate Gradient Method As discussed in Section 3.3, Equation 3-14 constitutes a very large system of equations, which regime a large amount of computing time fo r its solution. Since th e system of Equations 3-14 has symmetric and positive definite sparse ma trix, the equation can be solved efficiently by the conjugate gradient method. Conjugate gradie nt method (Beckman, 1959. Casulli, 1992) is the most popular iterative method for so lving large system of equation, Ax = b, where the matrix A is square, symmetric, and positive definite. Consider a quadratic function of a vector with the form: fxxAxbxcTT()1 2 (3-55) where A is a matrix, x and b a column vector, and c is a scalar constant. If the matrix A is symmetric and positive definite, f(x) is minimized by the solution to Ax = b. The definition of positive definite matrix is that if 0 Tx, for every nonzero vector x, then the matrix A is said to be positive definite matrix. The gradient of a quadratic form (Ahewchuk, 1994) is defined to be PAGE 67 67 fx fx fx fxn'() () () ... () x x x12 (3-56) With Equation 3-56, the gradient of Equation 3-55 is fxAxAxbT'()1 2 1 2 (3-57) If matrix A is symmetric, this equation reduces to fxAxb'() (3-58) Setting the gradient to zero, we obtain Ax = b which we want to solve. Therefore, the solution to Ax = b is a critical point of the function. If A is positive definite as well as symmetric, this solution is a minimum of f(x), so Ax = b can be solved by finding an x that minimizes f(x). It should be noted that if A is not symmetric, the conjugate gr adient method finds the solution to the system (J. S. Ahewchuk, 1994) 1 2 ()AAxbT (3-59) There are a lot of sparse numerical matrix so lver packages (ITPACK, D. R. Kincaid 1982, NSPCG, T.C. Oppe, 1988, and SPARSPAK, A. George, 1981) in public domain. ITPACK matrix solver is used in this study like ELCIRC model. ITPACK is in public domain (http://rene.ma.utexas.edu/CNA/ITPACK/). IT PACK is a collection of seven FORTRAN subroutines for solving large sparse linear sy stems by adaptive accelera ted iterative algorithms. Basic iterative procedures, such as the Jacobi method, the Successive Over-relaxation method, the Symmetric Successive Over-relaxation method, and the RS method for the reduced system PAGE 68 68 are combined, where possible, with acceleration pr ocedures, such as Chebyshev (Semi-Iteration) and Conjugate Gradient, for rapid convergence. Equation 3-14 is solv ed by ITPACK with conjugate gradient method efficiently. The model UFDVM developed in this study is coded with Fortran 90. All the subroutines of UFDVM are explained in Appe ndix F. Flow charts of UFDVM is in Appendix G. A manual of UFDVM is in Appendix H. The manual explains the input files which are mesh file, program control file, and vertical layer files. Comparison of UFDVM Model and UnTRIM Model The UFDVM model developed in this study is based on Casulli and Walters UnTRIM model. Therefore it is worth while to compar e similarities and differences between the two models. Both models use an orthogonal mesh system. UnTRIM uses Cartesian coordinate system. It does not handle longitude and latit ude spherical coordinate system. Unlike UnTRIM, UFDVM model developed in this study is able to handle both Cartesian and longitude and latitude Cylindrical coordinate system via CPP (Carte Para llelo Gramatique Projection) projection method (Westerink, Blain, Luettich 1994). ADCIRC model also uses this CPP method. CPP projection is the e qui-rectangular coordina te system. It is direct scaling of longitude and latitude to distances east and north with the scaling selected such that distortion is minimized along the central latitude pa rallel. The projection is following; xR yRoo ()cos (3-60) where x and y are X and Y component of Cartesian coordinate, R is Earth radius 6356.75 km, is longitude o is center longitude for domain, is latitude, and o is center latitude. Along the conversion of spherical coordinate, -plain approximation (Pedlosky, 1979) is used to minimize the coordinate inconsistency. By -plain approximation, Coriolis parameter is written as: PAGE 69 69 ffyyCC () (3-61) where cf is the Coriolis factor at the mid-latitude of the domain, cy is the Cartesian coordinate at the cf, and is the local derivative of the Coriolis factor at the same latitude. Unlike the UnTRIM model, UFDVM has tidal potential term in the momentum equation. Follow the ADCIRC model, the tidal potential term is included in the momentum equations following manner: U tx (3-62) where is effective Earth el asticity factor which is about 0.69 (Foreman et al., 1993). Follow Reid (1990), the tidal potential is defined as: (,,)()()cos () (), , tCftL tt T jtjn nj jnoj o jn jno 2 (3-63) where is latitude, is longitude, Cj,n is constants characterizing the amplitude of tidal constituent n of species j ( j = 0, declination; j =1, diurnal ; and j = 2, semi-diurnal), to is reference time, fj,n(to) is nodal factors, vj,n is astronomical arguments, Lj() is species specific coefficients (Lo = sin2 L1 = sin(2 ), and L2 = cos2 ) and Tj,n is period of constituent n of species j. Table 3-1 shows tidal potential constants for tidal constituents and associated effective Earth elasticity factors (A.Y. Mukai, J. J. We sterink, R. A. Luettich, Jr., and D. Mark, 2002). ADCIRC model also uses this feat ure (Luettich and Westerink, 2004). UnTRIM uses Eulerian-Lagrangian method fo r non-linear advection term and horizontal diffusion term in the governing equation. The non -linear advection equation is discretized by Eulerian-Lagrangian method in UFDVM. However, the horizontal diffusion term in the governing equation is discretized by finite volume method (Versteeg, 1995, Hoffmann, 1998, Ferziger and Peric, 2002, Leveque, 2002). PAGE 70 70 One of disadvantages of Z grid model is that there will be stair steps in bottom layers. Another difficult is to treat the surface layer as in dicated in finite volume discretization. To avoid those shortcomings, it is common to use transformation such that both the free surface and the bottom become the coordinate surfaces with an equal number of coordinate surfaces in between. This transformation, so called stretching, leads to a smooth representation of the topography with the same order of vertical resolution for the sh allow and deeper parts of the water body. Model UFDVM developed this study ha s the capable of handli ng vertical grid both Z and grid. All the derivation of coordinate is in Appendix K. Table 3-2 is summary of similarities and differences between the UnTR IM and UFDVM model developed in this study. Table 3-1 Tidal potential constants for tidal constituents and associ ated effective Earth elasticity factors Species n Constituent Tj,n (h) Cj,n (m) j,n 1 Mf 13.66079 0.041742 0.693 2 Mm 27.55455 0.022026 0.693 0 3 Ssa 182.6211 0.019446 0.693 1 K1 23.93447 0.141565 0.736 2 O1 25.81934 0.100514 0.695 1 3 Q1 26.86835 0.019256 0.695 1 M2 12.42060 0.242334 0.693 2 S2 12.00000 0.112841 0.693 3 N2 12.65834 0.046398 0.693 2 4 K2 11.96723 0.030704 0.693 PAGE 71 71 Table 3-2 Comparison of numerical method for UnTRIM and UFDVM model Terms UnTRIM UFDVM Horizontal Coordinate System Cartesian Cartesian and Spherical CPP projection method used Horizontal Coordinate System Z grid Z and Sigma grid capable Non-linear advection Eulerian Lagrangian Method (Finite Difference) Eulerian Lagrangian Method (Finite Difference) Horizontal Diffusion Eulerian Lagrangian Method Finite Volume Method Vertical Diffusion Implicit FDM Implicit FDM Propagation Tide Tide and associated Earth tide Surface wave equation Finite Volume Method Finite Volume Method Coriolis Eulerian Lagrangian Method (Finite Difference) Cartesian Coordinates (Constant Coriolis parameter) Eulerian Lagrangian Method (Finite Difference) Beta plain approximation Atmospheric pressure method method PAGE 72 72 CHAPTER 4 UNSTRUCTURED ORTHOGONAL MESH GENERATION Introduction For very complex geometries, the most flexib le type of grid is one which can fit an arbitrary solution domain boundary. The most flex ible grid to fit arbitrary geometry is unstructured mesh. Since the model in this study uses unstructured mesh system, the process of mesh generation is essential. Triangle meshing is by far the most common forms of unstructured mesh generation. Most techniques currently in use to triangulation ar e Delaunay triangulation and advancing front method. The model in this study uses orthogonal unstr uctured triangular or quadrilateral meshes. As discussed in Chapter 3, orthogonality is a requirement for calculation of finite difference approximations of spatial gradients in unstructured meshes. Delaunay triangulation is used in this study because De launay triangulation can generate the orthogonal triangular and/or quadrilateral meshes. There are a lot of unstructured mesh generation packages. Steve Owen1 surveyed bunch of mesh genera tion packages. Among them, ArgusOne2 and Triangle3 mesh generator will be used in this study to generate unstructured mesh. In this chapter, unstructured triangular mesh generation process will be discussed. Orthogonal Unstructured Triangular Mesh Ge neration with Triangle Mesh Generator An orthogonal unstructured triangular mesh is defined as following. Orthogonal triangular mesh is that within each triangul ar element, there exists a point such that the segment joining the center of two adjacent triangles and the side shared by the two triangles have nonempty intersection and are orthogonal to each other. The center of a triangle does not necessarily 1http://www.andrew.cmu.edu/user /sowen/survey/survey.html 2http://www.jeodijital.com/ArgusOne.htm 3http://www-2.cs.cmu.edu/~quake/triangle.html PAGE 73 73 coincide with its geometric center, centroid. Delaunay triangulation is one of algorithms to generate triangular meshes. The Delaunay triangulation is closely related geometrically to the Direchlet tessellation also known as the Voronoi or Theissen tessellations These tessellations split the plane into a number of polygonal regions cal led tiles. Each tile has one sample point in its interior called a generating po int. All other points inside th e polygonal tile are closer to the generating point than to any other. The Delaunay triangulation is created by connecting all generating points which share a common tile edge. Thus formed, the triangle edges are perpendicular bisectors of the tile edges. Although Dela unay triangulation algorithm can generate the triangular meshes, it does not guarantee the center of Voronoi diagram is inside the triangle element. It sometimes is outside tria ngles or has two Voronoi centers inside single triangle. There are several Dela unay triangulation methods. Delauna y triangulation, constrained Delaunay triangulation, conf orming constrained Delaunay triangulation and conforming Delaunay triangulation. A Delaunay triangulation of a ve rtex set is a triangulation of the vertex set with the property that no vertex in the vertex set falls in the interior of the circum-circle (circle that passes through all three vertices) of any triangle in the triangulation. There are three kinds of Delaunay triangulation methods which are constrained Delaunay triangulati on, constrained conforming Delaunay triangulation and confor ming Delaunay triangulation. A constrained Delaunay triangula tion (Figure 4-1) is similar to a Delaunay triangulation, but each boundary segment is present as a sing le edge in the triangulation. A constrained Delaunay triangulation is not trul y a Delaunay triangulati on. Some of its triangles might not be Delaunay, but they are all constrained Delaunay. As seen Figure 4-1, a mesh generated by PAGE 74 74 constrained Delaunay triangulation is not suitable to this study because the Voronoi diagram (dashed line) indicates that the centers of Vo ronoi diagram do not in side mesh elements. A constrained conforming Delaunay triangulati on (Figure 4-2) is a constrained Delaunay triangulation that includes Steiner points. A point is not part of the input set of points which are boundary segments. It usually takes fewer ve rtices to make a good-quality conforming constrained Delaunay triangulation than a goodquality conforming Delaunay triangulation, because the triangles do not need to be Delaunay, although they still must be constrained Delaunay. Although a constrained conforming Delaunay triangulation generates better mesh than generated by a constrained Delaunay triangulation, this mesh is also not suitable to present model. The vertices of Voronoi diagram are out side some triangles. This mesh is generated without any changes of input boundary segments. Figure 4-1 Mesh generated by Delaunay triangulation PAGE 75 75 Figure 4-2 Mesh generated by a constrained conforming Delaunay triangulation Figure 4-3 Mesh generated by a conforming Delaunay triangulation A conforming Delaunay triangulation (Figure 43) is a true Dela unay triangulation in which each boundary segment may have been subdivi ded into several edges by the insertion of additional vertices, called Steine r points. Steiner points are necessa ry to allow the segments to exist in the mesh while maintaining the Delaunay property. Steiner points are also inserted to PAGE 76 76 meet constraints on the minimum angle and ma ximum triangle area. The conforming Delaunay triangulation algorithm generates a true Delaunay me sh and all the vertices of Voronoi diagram are inside all the triangular elements. Thus, a mesh generated by the conforming Delaunay triangulation algorithm will be used in present model. Mesh Post Processing To produce accurate numerical solution, the uns tructured mesh should be well constructed. The mesh generator should have following quantities. (1). The mesh must conform to the boundary of computational region, which may consist of more than one connected component. If the co mputational domain is complex, the boundaries of the generated mesh conform well along the co mplex boundaries. Figure 4-4 shows an example of quality mesh which conforms to boundaries an d resolves complex boundaries. The thicker line represents actual coastal line. The mesh follows coastal boundary well. The mesh also resolves complex area fine enough. If the mesh does not conform to the boundary, then the mesh will be re-meshed so as to conforming the actual boundary. (2) The mesh must be fine enough to produce an acceptable approximation to the original problem. Parts of the domain where the solution is complicated or rapidly changing may require much smaller elements than other parts. The mesh will have the ability of the local mesh refinements i.e., the meshes around complex bounda ries and the areas where the water depth changes abruptly are fine and the other areas are coarse but the size and shape of triangles vary gradually through the flow domain to reduce di scretization error (Ferziger and Peric 2002). Figure 4-5 shows an example of an unstructured triangular mesh with local mesh refinement when there is deep navigational channel in the middle of computational domain. PAGE 77 77 Figure 4-4 Quality mesh. Figure 4-5 Local mesh refinement PAGE 78 78 (3) The individual elements must be well-shaped. There are two important restrictions: No small angles. For some methods, elements with small angle lead to ill-conditioned linear systems that are difficult to solve accura tely (Fried, 1972). Angles close to 180 degree present further problem (Babuska and Aziz, 1976). No obtuse angles. In two dimensions, some methods require the center of circum-circle of each element to lie in the elem ent, so that the perpendicular bisectors of element edges form the planar dual of the mesh. Circum-centers lie within their closed elements if and only if no angle is greater than 90 degree. For circum-cen ters to be will separated from the element boundary, all angles should be bounded away from 90 degree. It is rare that any mesh generation algorithm will be able to define a mesh that is optimal without some form of post-processing to improve the overall quality of the elements. The two main categories of mesh improvement include smoothing and clean-up. Smoothing includes any method that adjusts node locations while main taining the element connectivity. Clean-up generally refers to any process that changes the element connectivity. Figure 4-6 shows mesh area that needs to be smoothed. There are also two disconnected elements in the mesh. The two disconnected elements should be connected to adjacent elements. Figure 4-7 shows the mesh after smoothing and clean-up process. The mesh is smooth and connected to all adjacent elements. The efficiency of some numeri cal simulation depends on the ha lf bandwidth of the matrix they operate on. The bandwidth of the matrix is determined by the numbering of nodes. Each row in the matrix to be solved describes the connectivity of a node (on th e main diagonal) to the other nodes it is connected to. The matrix bandwid th is defined as the difference between the diagonal term and the most distan t non-zero term in the row, ove r all rows. In Figure 4-8, the PAGE 79 79 node number 4 is connected to node 25. Thus, the ha lf bandwidth of the ma trix is equal to 21. Figure 4-9 shows the matrix corres ponding to mesh of Figure 4-8. This matrix is very sparse. The efficiency of matrix solver to Figure 4-9 wi ll be deteriorated by sparseness of the matrix. Renumbering the mesh can result in an impr oved bandwidth by reassigning node numbers such that the numbers of connected nodes are as close as possible. Figure 4-10 shows the mesh after renumbering mesh of Figure 4-8. The resulti ng matrixs half band width equals 5. The renumbered matrix (Figure 411) is much less spar se. The efficiency, thus, of matrix solver will be improved. PAGE 80 80 Figure 4-6 Mesh before smoothing and clean up Figure 4-7 Mesh after smoothing and clean up unsmoothedarea disconnectedelements PAGE 81 81 Figure 4-8 Mesh that is not renumbered Figure 4-9 Sparse matrix with 21 half band width corresponding to mesh Figure 4-8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 PAGE 82 82 Figure 4-10 Mesh that is renumbered Figure 4-11 Sparse matrix with 5 half band width corresponding to mesh Figure 4-10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 PAGE 83 83 CHAPTER 5 MODEL VERIFICATION WITH ANALYTICAL SOLUTION Introduction Before the model is applied to real simulations, it is important to test the correctness of numerical solutions of the model with analytical solutions. In this chapter, results of the threedimensional circulation model will be compared to several analytical solutions. The first analytical solution is th e steady-state wind-induced setup. The second test is the tidal propagation in a rectangular bay with constant depth. The third solution is the tidal propagation with the nonlinear advection. Next is quart er annular tidal propaga tion test with sloped bottom. This test can verify the nonlinear tidal propagation with sloped bottom. The fifth test is on the Coriolis term. The sixth is a test for the bottom frictional term. Next test is for verification of the diffusion term in the equations. The eighth test is to simulate the wetting and drying over tidal flats. Last is to test atmospheric pressure test. In addition to the wind stress, the air pressure also plays an important role in storm surge simulations. Table 5-1 shows the summary of all the simulations performed in this section for verifying co rrectness of the model presented this work. Wind Setup The steady-state wind-induced setup due to a constant wind stress in a rectangular basin can be written as: ()x gH x Lw 2 (5-1) where is the setup of the water surface, w the applied wind stress, H and L are the depth and length of the basin respectively and the distance from the left boundary is x. The horizontal grid used in the wind setup test is a 21 x 5 cell with a length of 21 km and a width of 5 km. The depth of the computational domain is a constant 5 m and the horizontal grid spacing is 1 km in each PAGE 84 84 direction. A constant wind stress of 1 dyne/cm2 is applied in the posit ive x direction and a time step of 60 seconds is used in the simulation. Ta ble 5-2 shows the calculat ed surface elevation at several locations and Figure 5-1 shows a plot of 2D (x-y) model results and Figure 5-2 is the 3D (5 vertical layers) simulation. Water levels obtained by both the twoand three-dimensional models are exactly the same. Table 5-1 Summary of test for analy tical solutions and model simulations Dimensionality Process AS Dimension Figures Time Purpose 1 Wind setup 1D 2, 3D 5-1, 2 SS To verify wind effect 2 Tidal propagation 1D 2, 3D 5-3, 2, 3, 4, 5, 6, 7 TD To verify linear propagation with flat bottom 3 Non-linear advection effect 1D 2, 3D 5-11, 2 TD To verify non-linear propagation 4 Quarter annular Tidal propagation 1D 2, 3D 5-13, 2, 3, 4, 5 TD To verify non-linear propagation with sloped bottom 5 Coriolis effect 2D 2, 3D 5-20, 2, 3 TD To verify Coriolis effect 6 Bottom friction effect 2D 2, 3D 5-23, 2 TD To verify bottom friction effect 7 Horizontal diffusion test 1D 2D 5-25,2, 3 TD To verify horizontal diffusion effect 8 Wetting and drying 1D 3D 5-28, 2, 3, 4 TD To verify models ability of wetting and drying over tidal flat 9 Atmospheric pressure 2D 2D 5-32, 2, 3 SS To verify atmospheric term in the equations AS: Analytical solution, SS: Steady-state, TD: Time dependent Table 5-2 Comparisons between anal ytical and simulated wind setup xkm() analytical (cm) ModelD2 (cm) ModelD3 (cm) 0.5 -2.04 -2.04 -2.04 10.5 0 0 0 20.5 2.04 2.04 2.04 PAGE 85 85 Figure 5-1 2D Comparison between analytical and numerical solutio ns of water surface elevation for wind setup test Figure 5-2 3D Comparison between analytical and numerical solutio ns of water surface elevation for wind setup test Time(days) WaterSurfaceElevation(cm) 0 0.5 1 1.5 2 -5 -4 -3 -2 -1 0 1 2 3 4 5x=20.5km x=10.5km x=0.5km Time(days) WaterSurfaceElevation(cm) 0 0.5 1 1.5 2 -5 -4 -3 -2 -1 0 1 2 3 4 5x=20.5km x=10.5km x=0.5km PAGE 86 86 Tidal Propagation with Constant Depth Tidal simulation is one of the most important applications of a coastal and estuarine hydrodynamic model. Lynch and Gray (1978) derived the analytical solutions for tidally forced estuaries of various ge ometries and depths. Neglecting the advection terms, diffusion te rms, bottom friction and wind surface stress, the one-dimensional shallow water equa tions in Cartesia n coordinates are U t gh x0 (5-2) t U x0 (5-3) where U = uh is the depth integrated velocity in x-direction. Assuming a closed boundary at x = l and an open boundary at x = 0, the boundary conditions and initial conditions associated with Equations 5-2 and 5-3 are Ulx(,) 0 (5-4) (,)sin 0tato (5-5) (,)xo0 (5-6) U x (,) 00 (5-7) With those initial and boundary conditions, the analytical solutions of the one-dimensional shallow wave equations for water surface elevation and velocity are (Liu 1988): (,) cos() cos sin () sinsinxt aklx kl t a lckk kxtn nn n o 222 0 (5-8) Uxt acklx kl t a lkk kxtn nn n(,) sin() cos cos () coscos 222 0 (5-9) PAGE 87 87 To compare the numerical solution with a lin ear analytical solution, a rectangular basin shown in Figure 5-3 with constant water depth of 10 meters is considered. Input tidal forcing is a periodic tide with 50 centimet ers of amplitude and a period of 12.42 hours along the open boundary. The still water level elevation o is set to be zero. The physical rectangular basin is discretized with 450 quadrilateral and 900 triangular elements as shown in Figure 5-4 and 5-5. A time step of 30 minutes is used. It is noted that when using the implicit numerical scheme, i.e. = 1.0, numerical diffusion is introduced. Thus th e numerical solution should correspond to the first mode of the analytical solution which corr esponding to the first terms of Equations 5-8 and 5-9. If = 0.5, the numerical solution should be comp ared to the complete Equations 5-8 and 5-9 which include the higher mode solution. The test simulations were run with = 1 and 0.501 for twoand three-dimensional cases. When = 0.5, the matrix is best conditioned, a nd the convergence of the conjugate gradient solver is the best th eoretically. However experience suggests that = 0.501 gives similar but more stable results. Figures 56 through 5-9 show the comparis on between various model results vs. analytical solutions. The upper panel in all th e figures is the result at a location near the mouth (x = 10km), the middle panel is that at the middle point of the estuary (x = 30km), and the lower one is near the closed boundary (x = 50km). All the figures show good agreement between the numerical and analytical solutions. Tidal Propagation with Non-Linear Advection If nonlinear advection te rms are included in the two dimensional shallow equations, it is not feasible to obtain analyti cal solution. However, if the equations are reduced to one dimension, we can then obtain the harmonic seri es solution. Neglecting the Coriolis and bottom friction term, the governing equations for one dime nsional non-linear tidal motion can be written as: PAGE 88 88 Figure 5-3 Computational domai n for tidal propagation test Figure 5-4 Quadrilateral mesh for tidal propagation test Figure 5-5 Triangular mesh for tidal propagation test X Y OpenBoundaryClosedBoundaryClosedBoundaryClosedBoundary 60Km 30Km 0 V=0U=0V=0 U X(m) Y (m)0 100002000030000400005000060000 0 10000 20000 30000 X(m) Y(m)0 100002000030000400005000060000 0 10000 20000 30000 PAGE 89 89 Figure 5-6 Comparison of water surface elevation fo r tidal propagation test at three different locations with = 1.0, Lines: Analytical solution s, Circles: Numerical solutions Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 PAGE 90 90 Figure 5-7 Comparison of water surface elevation fo r tidal propagation test at three different locations with = 0.501, Lines: Analytical soluti ons, Circles: Numerical solutions Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150 Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150 Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 150 PAGE 91 91 Figure 5-8 Comparison of water velocity for tidal propagation test at th ree different locations with = 1.0, Lines: Analytical soluti ons, Circles: Numerical solutions Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 PAGE 92 92 Figure 5-9 Comparison of water velocity for tidal propagation test at th ree different locations with = 0.501, Lines: Analytical solu tions, Circles: Numerical solutions Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100 PAGE 93 93 Figure 5-10 Numerical solution of velocity field (upper) and water surface elevati on (lower) of tidal propagation 8 9 10 11 12H(m) 0 10000 20000 30000 40000 50000 60000X ( m ) 0 10000 20000 30000Y ( m ) Y X Z H 13.0 12.6 12.1 11.7 11.3 10.9 10.4 10.0 9.6 9.1 8.7 8.3 7.9 7.4 7.0 TIME=40.00HOURS X(m) Y(m)0100002000030000400005000060000 0 10000 20000 30000 1m/s PAGE 94 94 u t u x gh x0 (5-10) t u x0 (5-11) The boundary conditions are (,)sin() 0tat (5-12) ult(,) 0 (5-13) With the boundary conditions, the analytical soluti on of Equations 5-10 and 5-11 up to first order can be written as (Liu, 1988): (,) cos() cos sin cos sin() cos sin() cos tancos()cosxt aklx kl t ak hkl xklx l kl klx l kl klxlxt 2 28 2 4 2 4 222 (5-14) uxt h acklx kl t ak hkl xklx l k klx l kl klx l kl klklxt(,) sin() cos cos cos cos()sin() cos cos() cos tansin()sin 1 8 2 2 2 4 2 4 2222 2 (5-15) To compare with the analytical solution, nu merical simulations were conducted with the same basin and tidal forcing conditions used. Because the analytical solu tions present up to the first mode solution, the implicit numerical so lution should be compared to the analytical solution. Figures 5-11 and 5-12 show the compar ison of the water surface elevation and water velocity near the mouth, middle of the basin and near the closed bounda ry of the rectangular PAGE 95 95 basin. From the figures, one can see that the numerical and an alytical solutions are in good agreement. Quarter Annular Tidal Forcing with a Sloping Bottom Test This quarter annular te st was developed to assess the pe rformance of the current model (UFDVM) numerical schemes applied to the sh allow-water equations (Lynch and Gray, 1979). The problem contains spatially varying geometry and bathymetry, simultaneously tests the model in both horizontal coordinate dire ctions, is radially symmetric and permits analytical solutions for the 2D and 3D problems (Lynch and Gray, 1978, Lynch and Officer, 1985). Poor numerical schemes will show spurious modes (oscillations) and/or excessive numerical dissipation for this problem. Lynch and Gray (1978) derived the analytical solutions for a tidally forced estuary with a flat bottom and a sloping bottom. A sloping botto m case with the annular section configuration shown in Figure 5-13 was considered for mode l test. Neglecting nonlinear, Coriolis, and horizontal diffusion terms and assume linear bottom friction, one obt ains the following vertically-averaged equations t hv0 (5-5-1) v t gv0 (5-5-2) with the following boundary condi tions: at the closed end, rl, /r = 0; and at the open end, ro, = a cos t, where v is velocity, is water surface elevation, g is gravitational acceleration, is the linear bottom friction coefficient, a is the tidal amplitude, is the tidal frequency, and h is the water depth. Here h is defined as h(r) = Horn, and is defined as 2 = ( 2 i / h)/gHo. Ho is PAGE 96 96 Figure 5-11 Comparison of water elevation for non-linear advection at three locations with = 1.0, Line: Analytical Solution, Circle: Numerical Solutions Time(days) W a t e r s u r f a c e e l e v a t i o n ( c m ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10kmy=15km Time(days) W a t e r s u r f a c e e l e v a t i o n ( c m ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=50kmy=15km Time(days) W a t e r s u r f a c e e l e v a t i o n ( c m ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30kmy=15km PAGE 97 97 Figure 5-12 Comparison of water veloc ity for non-linear advection at three locations with = 1.0 Line: Analytical Solution, Circle: Numerical Solutions Time(days) V e l o c i t y ( c m / s ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10km,y=15km Time(days) V e l o c i t y ( c m / s ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30km,y=15km Time(days) V e l o c i t y ( c m / s ) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=50km,y=15km PAGE 98 98 water depth at the open end. Flow is requi red to be tangent to the solid boundaries at r = r1. A tidal forcing function is specified at r = r2. The analytical solution for a parabolic sloping bottom is (Lynch and Gray, 1978) (,)RertArBressit12 (5-5-3) vrtAsrBsr i H er ss o it(,)Re 1 1 2 1 212 (5-5-4) where A sr srrsrro s ssss 21 2121122 2111, B sr srrsrro s ssss 11 2121121 2112 (5-5-5) ss12 211 (5-5-6) The test geometry is shown in Figure 6.1. Th e computational domain consists of a quarter of an annulus enclosed with land boundaries on three sides and an open ocean boundary on the outer edge. The inner radius has r1 = 60,960 m and the outer radius has ro = 152,400 m. The bathymetric depth along the inner radius is h1 = 3.048 m and drops quadratically (h = h1r2/r2) to the outer radius where h2 = 19.05 m. The computational mesh is shown in Figure 5-14. The mesh has 63 nodes and 96 elements. The discreti zation uses a radial spacing of 15,240 m and an angular spacing of 11.25 degree. The model is started from a state of rest (cold started), a Cartesian coordinate system is specified, and finite amplitude, advection, quadratic bottom friction terms and nonlinearities are included. The elevation boundary is forced with a spatially uniform sinusoidal elevation having a period of 44,712 s (~12.42 hrs, M2 tide), amplitude of 0.3048 m, and phase of 0 deg. The run lasts for 5 days using a time step of 174.656 s. Model water level and ve locity time series are output every time step at 3 different locations in the domain. Figures 5-15 and 5-16 show the comparison between analytical and numerical solu tion for water elevation and velocity at three PAGE 99 99 different locations, respectively. Solid lines are analytical solutions and circles indicate numerical solutions. The numerical and analytical solutions are in very good agreement. To compare the UFDVM to ADCIRC model, exactly same conditions are configured for ADCIRC input file. The results of the two models were exactly same. Figure 517 shows snap shot of water elevation and veloc ity field at day 3.214. Figure 5-13 Annular section, Upper pane l: x-y plane, Lower panel: x-z plane XY050000100000150000 -50000 0 50000 100000 150000r2=152400mr1=60960mo p e n b o u n d a r yS o l i d w a l lSolidwallS o l i d w a l l r1r2B o t t o mO p e n b o u n d a r y S o l i d w a l l PAGE 100 100 Figure 5-14 Computational mesh for quarter annular test. Circles are data stations Figure 5-15 Water depth fo r quarter annular test 0 50000 100000 150000 0 20000 40000 60000 80000 100000 120000 140000 160000Waterdepth( x(m) y(m) 1 2 3 PAGE 101 101 Figure 5-16 Comparison between an alytical and numerical solution for water surface elevation Solid lines: Analytical solution, Circles: Numerical solution Tim e(Days) Waterelevation(m)012345 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1W aterelevationatstation1 Tim e(Days) Waterelevation(m)012345 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1W aterelevationatstation2 Tim e(Days) Waterelevation(m)012345 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1W aterelevationatstation3 PAGE 102 102 Figure 5-17 Comparison between an alytical and numerical solution for water velocity Solid lines: Analytical solution, Circles: Numerical solution Tim e(Days) Velocity(m/s)012345 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1Velocityatstation2 Tim e(Days) Velocitu(m/s)012345 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1Velocityatstation3 Tim e(Days) Velocity(m/s)012345 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Velocityatstation1 PAGE 103 103 Figure 5-18 Water elevation and velocity field at time 3.214 days for quarter annular test Figure 5-19 Water elevation and velocity field at time 5.013 days for quarter annular test XY0 50000100000150000 0 50000 100000 150000ELEVATION 0.50 0.42 0.35 0.27 0.19 0.12 0.04 -0.04 -0.12 -0.19 -0.27 -0.35 -0.42 -0.50 TIME=3.214DAYS (m) XY0 50000100000150000 0 50000 100000 150000 0.5 (m/s) XY0 50000100000150000 0 50000 100000 150000ELEVATION 0.50 0.42 0.35 0.27 0.19 0.12 0.04 -0.04 -0.12 -0.19 -0.27 -0.35 -0.42 -0.50 TIME=5.013DAYS (m) XY0 50000100000150000 0 50000 100000 150000 0.3 (m/s) PAGE 104 104 Tidal propagation with Coriolis Effect In this section, the Coriolis effects on tidal propagation will be examined in detail within the context of Kelvin waves. First the analytical solution of Kelvin wave will be derived then the numerical solution and the theoretical solution of Kelvin wave will be compared to verify the models ability to represent the Coriolis effects. Kelvin Wave Propagation The Kelvin wave is a traveling disturbance that requires the support of a lateral boundary. Therefore, it most often occurs in the ocean wh ere it can travel along coastline. The problem of tidal propagation in estuaries was first consid ered by Taylor (1921). His study was concerned with the reflection of Kelvin waves from the clos ed end of a semi-infinite rectangular channel. He considered two Kelvin waves of equal amplit ude traveling in opposite directions in the canal; the origin of coordinates was chos en such that their phases matched at x = 0. An end-effect term was then added, leaving the position of the cl osed end to be determined. Taylor obtained analytical solutions using the Four ier expansion method. The limitation of this method is that it is difficult to adapt them to the case when Poincar e mode propagation in th e canal is possible. Brown (1973) demonstrated that the solution of Taylors proble m could be obtained by using the method of collocation. The Governing Equation for Kelvin Waves The basic equations for Kelvin waves are obt ained from the full shallow water equations by neglecting the convective terms and frictiona l terms (Rahman, 1982). Thus the governing equations take the form: 0 y V x U t (5-6-1) PAGE 105 105 0 )( fV x hg t U (5-6-2) 0 )( fU y hg t V (5-6-3) where U = u(h + ) and V = v(h + ) represent the flows along x a nd y directions respectively. The initial and boundary conditions associated with Equations 5-6-1, 5-6-2 and 5-6-3 may be written as: Initial conditions: (,,)(,)xyxy00 (5-6-4) UxyUxy(,,)(,) 00 (5-6-5) VxyVxy(,,)(,) 00 (5-6-6) Boundary conditions: (,,) 0ytm (5-6-7) Ulyt(,,)0 (5-6-8) V x t(,,) 00 (5-6-9) Vxbt(,,)0 (5-6-10) where m is the forced water surface eleva tion at the mouth of the basin, l and b are the length and width of the basin as shown in Figure 53. With the boundary condi tions 5-6-3 through 5-610, Rahman (1982) derived the final solutions as following 1 2 1 1 2 1))(exp()(sin)(cos )()(exp )( expn n n n n n n o otixlikybk k fk ybkC tixlikyb fk Rtixliky fk (5-6-11) PAGE 106 106 1 2 1 2 1 1 2 2 2 2))(exp()(sin)(cos )()(exp )( expn n n n n nn o otixlikybk ck f ybkkC c tixlikyb fk Rkc iwtxliky fk kc U (5-6-12) 1 2 1 2 1 2 2 2 2 1 22 2])(exp[)(sin)1(n n n n n nntixlikybk k kf kC f ic V (5-6-13) where o is the Kelvin wave amplitude at the mouth of the basin, R is the reflection coefficient of Kelvin wave, k and are wave number and frequenc y of Kelvin wave, respectively. k1n and k2n are wave numbers of the n-th Poincare mode with respect to the y and x directions. Cn is the amplitude of the n -th Poincare mode. The dispersion relation for Kelvin waves is: 222ck (5-6-14) For Poincare waves, dispersion relation is: 2 22 2 2 2 1c f kknn (5-6-15) From the boundary condition, 0),0,( txV, and b n kn 1 (5-6-16) The unknown coefficients R and Cn are obtained when the boundary condition of U (l y, t) = 0 is satisfied for all y distance and time t. Thus 0)](sin )(cos[1 2 1 2 1 2 1 )]([ 0 )( 0 ybK cK fK ybKKC Rke ken n n n n n n yb fk y fk (5-6-17) To determine the unknown coefficients R and Cn, Equation 5-6-17 should be solved. Equation 5-6-17 presents the proble m of inverting a matrix of infinite order. However if the methods of collocation or poi nt-matching is used, the problem is greatly simplified (Brown PAGE 107 107 1973). Equation 5-6-17 is taken to hold at a finite set of points on [0, y], with the series truncated to a finite number of terms. If only the first N terms of the series are included, there remain N+1 constants to be determined in E quation 5-6-17; thus setting u(0, y) = 0 at N+1 points on [0, b] should provide a solution. The results are N+1 non-homogeneous linear system of equations for the same number of unknowns. Solutions are obtaine d by inverting the matrix of the coefficients. As N is increased, a normally converging sequence of values is found for R and Cn. Comparison between Numerical and Theoretical Solutions Numerical simulation has been made to obtain the tidal propagation in a rectangular basin. Analytical solutions (Rahman 1982) have been obtained from the linearized mathematical model for an estuary. The dimensions of the region of the estuary were taken as 60 30 km with constant depth of 10 m shown in Figure 5-3. The Coriolis paramete r f is chosen to be 0.0001/sec which is a typical value at the middle latitude. Th e period of the forcing tide at the mouth of the basin is 12.42 hours and the amplitude of th e Kelvin wave is 50 cm. Given these basic information, the analytical solution for the tidal propagation inside the basin can be obtained by choosing the real part of Equati ons 5-6-14, 5-6-15 and 5-6-16. For the numerical computation, the time step is chosen to be 30 minutes and the grid system is shown in Figure 5-5. The amplitude of the forcing tide at the open boundary is given by )} exp()](sin)([cos ])([Re) exp(Re{2 1 1 2 1 1 0 0liKybK K fK ybKC iklyb fk xp ikly fkn n n n n n n (5-6-18) which is obtained from Equation 5-6-17 by setting t = 0. The initial values of velocity components U and V are also taken from Equati on 5-6-12 and 5-6-13 by setting t = 0. Figure 517 shows the water surface elevation as a functi on of time at near the open boundary, the middle of the basin and near the closed boundary for 5 days obtained with the two-dimensional model. PAGE 108 108 Figure 5-18 shows the correspondin g water velocities of the tw o-dimensional simulation. The solid lines represent the analytical solution, wh ile the circles are the numerical solutions. The numerical and analytical soluti ons are in good agreements for bot h the water surf ace elevation and velocities at three different locations. Figures 5-20 and 5-21 are the results of threedimensional simulation. The velocities in the three-dimensional case are the depth averaged velocities, which are exactly the same as th e two-dimensional results. Figures 5-22 is a comparison of the velocity fields with Coriolis effects after 60 hours of simulation. One can observe the propagation of the Kelvin waves. Tidal Propagation with Bottom Friction Effect When the bottom friction terms are introduced, the two dimensional shallow water equations can be written as 0 y V x U t (5-7-1) 0 bxx gh t U (5-7-2) 0 byy gh t V (5-7-3) where is the water density, and bx and by are the bottom stresses in x and y direction, respectively. Assume that the water depth h is constant and the bottom stresses can be calculated by linear friction formulae: FUbx (5-7-4) FVby (5-7-5) PAGE 109 109 Figure 5-20 Comparison of water surface elevation fo r the Coriolis test at three locations with = 1.0 Lines: Analytical solutions Circles: Numerical solution Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10kmy=15km Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30kmy=15km Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=50kmy=15km PAGE 110 110 Figure 5-21 Comparison of water velocity for th e Coriolis test at three locations with = 1.0 Lines: Analytical solutions, Circles: Numerical solution Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=50kmy=15km Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30kmy=15km Time(days) Velocity(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10kmy=15km PAGE 111 111 Figure 5-22 Velocity field for the Co riolis test at time 72 hours with = 1.0 where 22 2 max 2 maxhC VUg F (5.7.6) Given these equation, Equation 5-7-1, 57-2, and 5-7-3 can be simplified as: 0y V x U t (5-7-7) 0FU x gh t U (5-7-8) 0FV y gh t V (5-7-9) Rahman (1982) derived the analytical solution to above Equations. The theoretical solutions of the governing equations are )exp( sintan (cos)(cos )exp()sintan(cosRe2 2 2 1 1tixKlKxKybKA tikxklkxAn n n n n n o (5-7-10) X Y 0 10000 20000 30000 40000 50000 60000 0 10000 20000 30000 X Y0100002000030000400005000060000 0 10000 20000 30000 1m/s PAGE 112 112 )}exp()}costan sin )(cos[ )exp()costansin( Re{2 2 2 1 2 1 2 2tixKlKxK ybKKA iF c tikxklkxkA iF c Un n n n n n n o (5-7-11) )}exp()sintan (cos )(cos[ Re{2 2 2 1 1 1 2tixKlKxK ybKKA iF c Vn n n n n n n (5-7-12) where K n bn1 (5-7-13) k c i F2 2 21 () (5-7-14) K c i Fn bn2 2 2 2 21 () (5-7-15) The coefficients nAA .....,,0 can be obtained using the open boundary condition (,,)exp() 0ytitm (5-7-16) at x = 0. They are given as: Adyom b0 (5-7-17) AKbydynm b n0 1cos() (5-7-18) The same basin defined in the tidal forcing te st is used for this test to compare the analytical and numerical solution. The amplitude of the forcing tide is assumed to be constant (50 cm ), which leads to no flow in y-direction, i.e., V = 0 everywhere, and zero values for the coefficients A1 ,..., An.. The period of tidal forcing is 12 hours. The Manning coefficient is 0.02 and the maximum velocity is 50 cm/sec With those parameters, the analytical solutions of PAGE 113 113 Equations 5-7-11 and 5-7-12 can be easily obtain ed. The time step of the numerical model is 30 minutes and the grid system is the same as that shown in Figure 5-5. Comparisons between the numerical and analy tical solutions are s hown in Figures 5-23 and 5-24. Figure 5-23 is the water surface elevati on and velocity in x-direction and Figure 5-24 is the comparison of water velocity. From the figure s, it is clear that the numerical and analytical solutions are in very good agreement for all cases. Horizontal Diffusion Test Neglecting nonlinear terms, Cori olis and the horizontal diffusi on term in y direction, the governing equation can be reduced to a simp le one-dimensional time dependent diffusion equation written as s t K T x x2 2 (5-8-1) for t0, 020 x l cm where T is temperature and Kx is a diffusion coefficient which is 0.86 cm2/s for the test. The boundary and initial conditions are given by st(,) 00 and slt(,)0 for t0 sxCo(,) 0100 for 020 xlcm The analytical solution to E quation 5-8-1 is written as sxt n nK l t n l xn x(,) () exp () sin () 4001 21 21 211 22 2 (5-8-2) Numerical results are obtained by setting l = 20 Cm and diffusion coefficient Kx = Ky.= 0.86 cm2/s .The computational domain is divided into 800 elements for triangular mesh. Figure 5-8 shows the comparison between the anal ytical and numerical solutions at x = 4, 10 and 19 cm in x and y directions for triangular mesh case. PAGE 114 114 Figure 5-23 Comparison of water surface elevation fo r bottom friction test at three locations with =1.0 Line: Analytical solution, Circles: Numerical solution Time(days) Watersurfaceelevation(cm) 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 -100 -75 -50 -25 0 25 50 75 100x=50kmy=15km Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30kmy=15km Time(days) Watersurfaceelevation(cm) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10kmy=15km PAGE 115 115 Figure 5-24 Comparison of water velocity for bottom friction test at three locations with =1.0 Line: Analytical solution, Circles: Numerical solution Figure 5-25 shows the comparison with analytical solution and numerical solution. Figure 5-26 shows the contours of numerical results of the two-dimensional diffusion equation for Time(days) VelocityU(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=30kmy=15km Time(days) VelocityU(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=10kmy=15km Time(days) VelocityU(cm/s) 0 1 2 3 4 5 -100 -75 -50 -25 0 25 50 75 100x=50kmy=15km PAGE 116 116 quadrilateral and triangular mesh es. Figure 5-27 shows 3D view te mperature distribution at time 10 sec. It is seen that the numerical and analytical solutions are in very good agreement. Figure 5-25 Comparison for the diffusion test at three locations Lines: Analytical solutions, Symbols: Numerical solutions Upper panel: X direction, Lower panel: Y direction Time(Sec) Temperature 0 50 100 150 200 0 10 20 30 40 50 60 70 80 90 100 NumericalSol.atY=4Cm NumericalSol.atY=10Cm NumericalSol.atY=19Cm AnalyticalSol.atY=4Cm AnalyticalSol.atY=10Cm AnalyticalSol.atY=19Cm Time(Sec) Temperature 0 50 100 150 200 0 10 20 30 40 50 60 70 80 90 100 NumericalSol.atX=4Cm NumericalSol.atX=10Cm NumericalSol.atX=19Cm AnalyticalSol.atX=4Cm AnalyticalSol.atX=10Cm AnalyticalSol.atX=19Cm PAGE 117 117 Figure 5-26 Numerical solu tion of temperature dist ribution at time = 10 sec Figure 5-27 3D view of numerical solution of temperature dist ribution in rectangular plate 0 20 40 60 80 100T ( D e g r e e C ) 0 5 10 15 20X ( c m ) 0 5 10 15 20Y ( c m ) X Y Z T 100.00 92.86 85.71 78.57 71.43 64.29 57.14 50.00 42.86 35.71 28.57 21.43 14.29 7.14 0.00 TIME=10.00SECONDS X(cm) Y(cm) 0 5 10 15 20 0 5 10 15 20 T 100.00 92.86 85.71 78.57 71.43 64.29 57.14 50.00 42.86 35.71 28.57 21.43 14.29 7.14 0.00 TIME=10.00SECONDS PAGE 118 118 Wetting and Drying Test over Tidal Flats In shallow coastal water bodies like lagoons embayments, and estuaries experiencing tidal oscillation of free surface, the extent of areas subjected to alternating wetting and drying (tidal flats) can be of the same order of magnitude as the cons tantly submerged areas (Reid and Whitaker 1976, Falconer 1986). The reproduction of th e wetting and drying of the tidal flats is a desirable feature of numerical tidal flow models based on shallow water equations. Existing wetting and drying models are very well reviewed by Liu (1988), Davis (1996), Balzano (1999), Lee (2000) and Xie, Pietrafesa and Peng (2003). The major difference between all the models is the way to determine the dr ying and wetting cells and depths. The method to determine drying cells in the pr esent model is based on Casulli and Chengs model (1992, 2002). In this chapter, the method to determine we tting and drying cells in computational cells will be described and the numerical and theoretical results will be compared in terms of the theoretical solution obtained by Ca rrier and Greenspan (1958). The analytical solution was also used by Liu (1998), Luo (1993), Davi s (1996) and Lee (2000) to vali date their numerical models. Determining Flooding and Drying Elements Once the free surface elevation has been co mputed throughout the computational domain, before proceeding to the ne xt time step, some of the vertical grid spacings zjk n,1 have to be updated to account for the new surface location. At each time step, the new total water depth Hj n1at the polygons side s are defined as Hhhj n jij n jij n 1 1 1 2 10 max,,(,)(,) (5-9-1) According to the new total water de pth, the vertical grid spacing zj n+1 is updated. Thus, an occurrence of zero value for the total water depth Hj n+1 implies that all the vertical faces separating prisms between the water column i(j,1) and i(j,2) are dry and may become set at a PAGE 119 119 later time when Hj n+1 becomes positive. The height of a dry face and the corresponding normal velocity at the polygons face are ta ken to be zero. When the total depth is equal to zero, the friction factor at that point will be assumed to be infinity and, accordingly, the corresponding velocity u or v across the side of the cell forced to vanish. The occurren ce of a zero value for the total depth in one side of a cell implies zero velocity or zero mass flux until the total depth becomes positive i.e., the boundary shorelines, wh ich are varying with time, are defined by the condition of no mass flux. This guarantees mass conservation over the co mputational domain. A element is considered a dry cell only if the total water de pths at all side are zero. To reduce computational noise or oscillati on due to very small wet element, some minimum critical dry depth is defined. Normally the critical dry depth is chosen as 0.01 meter in this study. Therefore elements are considered as wet element when the total water depth is greater than the critical depth and dry elements when the total water depth is less than or equal to the critical dry depth. Drying process can take pl ace not only along the coast but also in interior regions such as shoals. For a shallow estuary, even under moderate wind stresses, some interior points over shoals can be dried completely whereas the surrounding elements are still wet. Similarly, when the sea surface elevation at a pr eviously flooded that location, that element returns to dry cell (Figure 5-28) The unwanted numerical oscilla tion due to drying and wetting, such as the presence of a single wet or dry el ement surrounded by dry or wet elements, when the total water depth of a set element drops below the specified threshold depth, drying occur, but an isolated dry element will not be turned into a wet cell until at least one of neighboring elements turns into a wet element as well. PAGE 120 120 Comparison with Analytical Solution To validate the wetting and drying scheme devel oped, 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 wa s also used by Liu (1998), Luo (1993), Davis (1996) and Lee (2000). The deriva tion of the analytical soluti on is well reviewed by Liu and Davis. The side view of rectangular basin is show n Figure 5-29. The computational mesh used in this analytical comparison consists of 100 6 quadrilateral elements with a length of 62 km and a width of 10 km rectangular basin. The bottom slope is 1:2500. The water depth varies from 2 m above mean sea level at x = 0 to 22.8 m below sea level at open boundary. The forcing tide at the open boundary has the form: ()costa T t 2 (5-9-2) where the amplitude, a is 11.24 cm and the period T is 3600 seconds. Figure 5-30 and Figure 5-31 are non-dimensiona l wave profile comparisons between the analytical solution and the numer ical solution. In the figures, non-dimensional variables are defined as X = x* / L t = t* / T and = / ( L ), where superscript is dimensional variables of x distance, time and water surface elevation and is beach slope. From these figures, it is apparent that the analytical and the nu merical solutions are in good agreement. Atmospheric Pressure Test wi th Holland Storm Surge Model One of the most important parts of a numerical model used to simulate storm surge is the storm model itself. A good storm surge model is necessary to determine rapidly changing atmospheric pressure gradient a nd wind stress associated with th e passage of a storm. Planetary boundary layer (PBL) model (Vickery and Twisdale, 1995; Thompson and Cardone, 1996; PAGE 121 121 Vickery et al., 2000) would be the best choice for a storm model. But the PBL model is too complex for the purpose of this study. Instead, a simple storm model a ssuming an exponential decay of pressure from the center of a storm is used (Holland, 1980). The atmospheric pressure field is described as the sum of an axially symm etric part and a large sc ale pressure filed of constant gradient. The symmetric part of the local atmospheric pressure in a storm, P is described in terms of an exponential pressure profile from Holland (1980): PPPPeoo ArB ()/ (5-10-1) where Po is the pressure in the center on the storm, P is the free stream pressure, r is the distance from the center of the storm and A and B are scaling parameters. For the purpose of this study, the simpler local air pressure formulation of Wilson (1957) is used. A is set equal to the radius of maximum wind speed, R and B is set equal to 1. Subtracting P from both sides and rearranging the right hand side yield, PPPPeo Rr ()()/1 (5-10-2) The left hand side of the relative atmospheric pressure, Pa, and Po P is the central pressure drop of the storm, Po. Making these substitutions, the equation can be written as PPeao Rr()/1 (5-10-3) The atmospheric pressure term in the xand y momentum equations consists of derivative of Pa with respect to x and y, respectively. The terms, after negating and dividing by water density are 1 w ao w RrP x P x e / (5-10-4) 1 w ao w RrP y P y e / (5-10-5) PAGE 122 122 Figure 5-28 Definition of dry and wet element Figure 5-29 Wave propagation on a linearly sloping beach PARTIALLY DRYDEPTH DRYDEPTH DRYDEPTH PARTIALLY DRYDEPTH DRYDEPTH PARTIALLY DRYDEPTHDryElementPartiallyDry Element Wetelement th L PAGE 123 123 Figure 5-30 Non-dimensional comparison between wave profiles as predicted by theory and numerical model of wetting and drying, time = 0 /2 X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time=0 Analyticalsolution Numericalsolution Shoreline X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time= X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time= X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time= PAGE 124 124 Figure 5-31 Non-dimensional comparison between wave profiles as predicted by theory and numerical model of wetting and drying, time = 2 /3 X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Analyticalsolution Numericalsolution Shoreline Time= X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time= X 0 10 20 30 40 50 -1.5 -1 -0.5 0 0.5 1 1.5Time= PAGE 125 125 The unstructured finite difference form of th e xand y-momentum atmospheric pressure term used in the model is evaluated at the velocity nodes 121 w ao wj RrRrP x P eeij ij //(,) (,) (5-10-6) The distance, ri( j, 2 ), is measured from the center of the storm to the center of element i ( j ,2 ). The storm also influences the elevation at open boundaries, open = tide + storm. The surface elevation due to the pressure of the storm can be written as storm a w o w RrP g P g e 1/ (5-10-7) The cyclostrophic wind velocity, Uc, is U P R r ec o a Rr/ (5-10-8) The geostrophic wind velocity, Ug, is U P f R r eg o a Rr 2 / (5-10-9) The gradient wind velocity, UG, is UUGC21 (5-10-10) where 1 2V U U Us c c g* (5-10-11) and the resolved part, V* s, of the translational velocity of the storm, Vs is VVss*sin() (5-10-12) where is the angle from the direction of bearing of the storm, to any point inside the storm. The surface wind velocity, Us, in the xand y-direction is then written as PAGE 126 126 UKUsxG cos90 (5-10-13) UKUsyG sin90 (5-10-14) where is an inward rotation angle of 18o and K is the ration of surface wind velocity to gradient wind velocity. Verification of Atmospheric Pressure Term in the Model with Analytical Solution The analytical solution of water surface eleva tion for steady state due to the atmospheric pressure gradient of a storm can be written as P g Ca w (5-10-15) where C is constant value which can be determined from boundary conditions or zero for steady state. To verify the air pressure term, the model is applied to a simple closed rectangular basin with dimension of x = 100 km y = 100 km and uniform depth of 10 m. The domain is discretized 10000 quadrilateral elements and 102 01 nodes. The initial air pre ssure field is setup form Holland analytical storm model. The atmospheric pressure ( Pa) at the center of the domain is 960 mb free stream atmospheric pressure P is set to 1013 mb, and radius to maximum wind, R = 30 km. Figure 5-33 shows the initial atmospheric distribution in the dom ain. The wind field generated from Holland model sets to zero to veri fy air pressure effect only. Figure shows the atmospheric pressure field from the Holland analytical storm m odel with above parameters. The atmospheric pressure fiel d generated by Holland model is minimum of 960.00 mb at the center of the domain and 985.04 mb near boundaries ( x = 90 km y = 50 km ). The analytical solution of water surface elevation is -2.65 cm at x = 90 km y = 50 km and 22.48 cm at the center of the domain. Figure 5-32 shows the numerical solution of the water elevati on after reaching steady state. Table 5-3 shows the comparison between calculated analytical surface elevation and PAGE 127 127 numerical solution at the two lo cations, at the center of domain and near the boundary. Figure 534 shows the numerical solution of atmospheri c pressure after reaching steady state. The analytical and numerical solu tions are in good agreement. Table 5-3 Comparison between analytical and numerical solution of air pressure term x y analytical ( cm ) model ( cm ) 50 km 50 km 22.48 22.48 90 km 50 km -2.65 -2.65 Figure 5-32 Numerical solution of water surface elevation for atmospheric Time(Julianday)Waterelevation(m)257 258 259 260 -0.1 0 0.1 0.2 0.3x=50km,y=50km x=90km,y=50km PAGE 128 128 Figure 5-33 Initial atmospheric pressure distribution for analytical test Figure 5-34 Numerical solution of water su rface elevation after reaching steady state x(m)y ( m )020000400006000080000100000 0 20000 40000 60000 80000 1000000.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 -0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.12 Waterelevation(m) x(m)y ( m )020000400006000080000100000 0 20000 40000 60000 80000 100000990 988 987 985 984 982 981 979 978 976 975 973 972 970 969 967 966 964 963 961 960 Airpressure(mb) PAGE 129 129 CHAPTER 6 HURRICANE IVAN ( 2004) AND DENNIS (2 005) SIMULATIONS Introduction Hurricanes are the most devastating and da maging hazards impacting the United States. According to NOAA (2005), during the last centu ry, 23 hurricanes had each caused damages in excess of $1 billion dollars. The major damage caused by hurricanes is associated with storm surges and coastal flooding. The surge of high wa ter topped by waves can be devastating. Along the coast, storm surge is the grea test threat to life and property. Accurate storm surge simulations are also essential for producing accurate flood insurance rate maps (FIRMs) for coastal regions. Florida coastal counties alone contribute more than 40% of the total insurance premiums collected by the National Flood Insurance Program (NFIP) administered by the Federal Emergency Mana gement Agency (FEMA). Sheng and Alymov (2002) showed that the FEMA methodology on flood insurance rates in Pinellas county, FL (FEMA, 1988), which is based on the 1-D WHAFIS model, overestimates possible damage that may be caused by the 100-year storm event. The use of a more robust storm surge model will likely result in significant savings in insurance premiums. Hurricanes Ivan made land fall west of Gulf Shore, AL with hurricane scale category 3 September 16th 06:50 (UTC), 2004, and Dennis Santa Rosa Island, FL with category July 10th 19:30 (UTC), 2005. They have similar track and produced similar damages. In this section, several existing storm surg e models used by government agencies (e.g., NOAA and FEMA) and industries are reviewed al ong with some of the models used in the academic world. The main purpose of this review is to define the limitations of the existing models and point out the advantag e of using the storm surge modeling system developed in this study. Next, Hurricane Ivan will be simulated. To examine uncertainties in prime model inputs PAGE 130 130 such as wind, surface wind drag parameterization, bottom friction formulation, and open boundary tidal forcing, a number of tests were performed to produce best results. Finally, Hurricane Dennis will be simulated with UFDVM model. Storm Surge Model Review SPLASH (Special Program to List Amplitudes of Sur ges from Hurricane) Storm Surge modeling started from the 1970's by the National Oceanic and Atmospheric Administrations (NOAA) Special Program to List Amplitudes of Surges from Hurricane (SPLASH) (Jelesnianski, 1972), which was adopt ed by the National Weather Service (NWS). This early stage model is a 2D storm surge model without nonlinear term and fixed boundary on water-land interface. It does not work well in sh allow water domain with complex shoreline and bathymetry and is only suitable for open water. SLOSH (Sea, Lake, and Overland Surges from Hurricanes) Followed by SPLASH model, SLOSH (Overland Surges from Hurricanes) was developed by NOAA (Jelesnianski, et al, 1984 and Jelesnianski, et al, 19 92) and was widely used along Atlantic and Gulf Coasts of the U.S. coast regions to estimate storm surge heights resulting from historical, hypothetical, or pred icted hurricanes. SLOSH model is a 2-D linear barotropic model run by the National Hurricane Center (NHC) to es timate storm surge heights and winds resulting from historical, hypothetical, or pr edicted hurricanes by taking in to account: pressure deficit, size, forward speed, track, and winds. The model does not take into account: tide, precipitation and evaporation, river flow, wind-driven waves. The model uses 2D pol ar coordinates with coarse resolution (0.5 7 km ) and specialized just for surge m odeling. As the primary use of the SLOSH model is to define floodprone areas for evacuation pla nning, the forecasting of storm surge and inundation is designed for a large domain to capture the main feature and is not suitable for a specific hurricane. SLOSH is currently used by NOAA NHS, the U.S. Army PAGE 131 131 Corps of Engineers and FEMA to create flood maps representing the Maximum of the Maximum (MOM) storm surge composite of hypothetical storms. SPH / WIFM SPH (Standard Project Hurricane, Schwerdt et al., 1979, Cialone, 1991) model is a twodimensional, parametric model developed in a stretched Cartesian coordinate system for representing wind and atmospheric pressure fiel ds generated by hurricanes. It is based on the Standard Project Hurricane cr iteria developed by the National Oceanic and Atmospheric Administration (NOAA), and the model's primary outputs are resulting wind velocity and atmospheric pressure fields which can be used in storm surge modeling. The SPH model can be run independently, or it can be invoked from within model WIFM (WES Implicit Flooding Model) dependent, long-wave model for solvi ng the vertically integrated Navier-Stokes equations in a stretched wave hydrodynamics such as tidal circulation and, making use of wind fields produced by SPH, storm surges. WIFM is a simple and outdated model which does not account for wave effect. SPH and WIFN are co mponents of Coastal E ngineering Research Center's Coastal Modeling System used by the U.S. Army Corps of Engineers. HAZUS HAZUS is a software program for estimati ng potential losses from earthquakes, floods, and wind. HAZUS is being developed by FEMA. It has the capability to estimate earthquake losses, and flood and wind models are being de veloped. The Hurricane Loss Estimation Model which is a part of the HAZUS model incorporates sea surface temperature in the boundary layer analysis, and calculates wind speed as a function of central pressure, translation speed, and surface roughness. The model addresses wind pre ssure, wind borne debris, surge, waves, atmospheric pressure change, duration/fatigue, and rain. The Flood Loss Estimation Model is PAGE 132 132 capable of assessing riverine and co astal flooding. It estimates potential damages to all classes of buildings, essential facilities, transportation a nd utility lifelines, and agricultural areas. The model estimates debris, shelter and casualties. Direct losses are estimated based on physical damage to structure, contents, and building interiors. The effects of flood warning and velocity are taken into account. The flood model uses geog raphic information system software to map and display flood hazard data, and the results of damage and loss estimates for building and infrastructure. It also enables users to estimate the effects of flooding on populations. TAOS TAOS (The Arbiter of Storms) model is a 2-D integrated hazards model (Watson, 1995; Watson and Johnson, 1999) that simulates storm surge, wave height, maximum winds, inland flooding, debris and structural da mage. Since tide and wave effect is not included in TAOS, it is similar to SLOSH in the sense of storm surge modeling. POM POM (The Princeton Ocean Model) is a sigma coordinate, free surface, ocean model developed by Blumberg and Mellor. It has be en commonly used for modeling of estuaries, coastal regions, basin and global oceans. An explic it finite difference scheme is used in POM to calculate horizontal differential terms and an implicit scheme is implemented for vertical differentiation. POM saves comput ational time by mode splitting such that the external mode is separated from the internal modes by vertically integrating the govern ing equations. A flooding & drying scheme was recently incorporated in POM (Xie, et al, 2004) and then applied to simulate storm surge and inundation in Charlest on Harbor, South Carolina for Hurricane Hugo and some hypothetical hurricane events (Peng, et al., 2006), as well as in Chesapeake Bay for Hurricane Isabel (Peng, et al., 2006). Moon (2000, 2005) coupled a third-generation wave model PAGE 133 133 WAVEWATCH-II with POM to study wave-curre nt interaction in open water during Typhoon Winnie (1997) in the Yellow and East China Seas. ADCIRC Advanced CIRCulation (ADCIRC, Luettich et al., 1992) is 2 and 3 dimensional finite element model which solves the equations of motion for a fluid on a rotating earth. These equations are based on hydrostatic pressure a nd Boussinesq approximations and have been discretized in space using the finite element method and in time using the finite difference method. Water elevation is obtained from the solu tion of the depth-integrat ed continuity equation in Generalized Wave-Continuity Equation (GWCE) form. Velocity is obtained from the solution of either the 2DDI or 3D mome ntum equations. All nonlinear terms have been retained in these equations. ADCIRC can be run using either a Ca rtesian or a spherical coordinate system. ADCIRC boundary conditions include: specified elevation (harmonic tidal constituents or time series), specified normal flow(harmonic tidal constituents or time series), zero normal slip or no slip conditions for velocity, external barrier ove rflow out of the domain, internal barrier overflow between sections of the domain, surface stress (wind and/or wave radiation stress), atmospheric pressure, outward radiation of waves. ADC IRC can be forced with: elevation boundary conditions, normal flow boundary conditions, surf ace stress boundary conditions, tidal potential, and earth load/self attraction tide. Hurrica ne Betsy (1965), Hugo ( 1989), Floyd (1999), Pam (2004), Ivan (2004), and Katrina (IPET, 2005) were simulated using ADCIRC model. MIKE21 MIKE 21 is a professional engineering so ftware package for 2D free-surface flows developed by Danish Hydraulic Institute (DH I) (DHI, 2002). MIKE 21 is applicable to the simulation of hydraulic and related phenomena in la kes, estuaries, bays, coastal areas, and seas PAGE 134 134 where stratification can be ne glected. The contained comprehensive hydrodynamic module for floodplain modeling in MIKE21 make s it suitable for storm surge simulation in coastal regions when coupling with a storm cyclone wind model. SURGE SURGE is a 3-D hydrodynamic model of ocean ci rculation for coastal areas based on the Princeton Ocean Model (POM). SU RGE simulates and predicts storm surge, flooding, overwash, water recession, and associated horizontal currents. The model makes use of NOAA/NOS bathymetry data and high resolution of US GS/NOAA LIDAR survey data. Hurricane Andrew (1992) and Hurricane Carla (1967) were used for model verification in Louisiana and Lavaca Bay, TX, respectively. CH3D The Curvilinear-grid Hydrodynamics 3D (CH3D) model is a robust state-of-art model first developed by Peter Sheng (1986, 1989). The CH 3D model uses a horizontally non-orthogonal boundary-fitted curvilinear grid and a vertically sigma grid, and he nce is suitable for application to coastal and nearshore waters with complex s horeline and bathymetry. The model also has the ability to handle wetting drying over the tidal flat s. In addition to those features, the model is coupled with wave model SWAN. Hence, CH3D model is one of the best choices to simulate hurricane induced storm surge simulations. CH3D was first used for storm surge simulation for Hurricane Marco (1990) and Floyd (1999). CH3D was updated contin uously to an integrated storm surge modeling system by Peter Sheng and his group at University of Florida, known as CH3D-SSMS (Sheng, et al., 2002; Alymov, 2005). CH3D-SSMS has been applied to several major hurricanes, including Isabel (2003) and Charley and Frances (2004) (Alymov, 2005) and Hurricane Ivan (2004) and Dennis (2005) (Zhang, 2007). PAGE 135 135 UnTRIM and FVCOM UnTRIM (Unstructured Tidal, Residual, Inter-tidal Mudflat) model is 3-D unstructured hydrodynamic model developed by Casulli (Casulli and Walters, 2000) following his previous work (Casulli, V. & R. T. Cheng, 1992; Cheng, R. T., et al., 1993). As discussed in chapter 3, UnTRIM is a semi-implicit finite difference a nd volume model based on the three-dimensional shallow water equations. Comparing to other trad itional models discussed above, it is capable of representing a modeling domain with a very high resolution unstructured grid while maintaining numerical stability and computati on efficiency. This model applied to storm surge simulations of Hurricane Andrew (1992) and Hurri cane Isabel (2003). As discussed in chapter 1, FVCOM model is an unstructured grid, finite-volume, 3D primitive equation, turbulent closure coastal ocean model developed by Chen (2002). FVCOM also used in simulating hurricane induced storm surge. This model was used for the hypothetical storm surge simulation in Tampa Bay, FL (Weisberg and Zheng, 2006). Table 6-2-1 is the summary of recent storm surge model Wind and Atmospheric Pressure Model One of the most important parts of a numerical model used to simulate storm surge is the storm model itself. A good storm surge model is necessary to determine rapidly changing atmospheric pressure gradient and wind stress as sociated with the passage of a storm. Wind and atmospheric pressure gradient are the physical forcing for gene rating storm surge. Wind blowing onshore causes the water mass to pile up agains t a sloping beach and propagate inland. Also, wind generates waves whose effect is twofold. Inaccuracy of wind and pressure fields is generally responsible for errors in storm surge simulations. Four types of wind and atmospheric pressure model data are available. They are NOAA=s WAVEWATCH III (NWW3) WNA WIND MODEL, WINDGEN analysis model, AOM L (Atlantic Oceanographic Meteorological Laboratory) HRD (Hurricane Rese arch Division) surface wind analysis system, and, and PAGE 136 136 HOLLAND analytical model. In th is section, wind and atmosphe ric pressure for storm surge model data will be reviewed. Table 61 shows features of each wind model Table 6-1 Summary of recent development of storm surge model Model Name Authors Dimension Application Domain Case Study Grid Type Wind Data Alymov (2005) 2/3D East coast of Atlantic ocean Isabel (2003) Charley and Frances (2004) Curvilinear structured WINDGEN WNA Holland NOAA Tide HWM CH3D Zhang (2007) 2/3D Northern Gulf of Mexico Ivan(2004) Dennis(2005) Curvilinear structured WNA Holland NOAA Tide HWM Shen (2006a) 2D Chesapeake Bay Isabel (2003) Unstructured Parametric NOAA Tide UnTRIM Shen (2006b) 2D Biscayne Bay Andrew (1992) Unstructured Parametric HWM Shen (2006c) 2D Chesapeake Bay Floyd (1999) Unstructured Parametric NOAA Tide ADCIRC Dietsche (2007) 2D Charleston, SC Hugo (1989) Unstructured Ocean Weather Inc. HWM Mike21 Madsen Jakobsen (2004) 2D Bay of Bengal Cyclone (1991) St ructured Parametric Water level Moon (2003) 3D West coast of Korea Winnie (1997) Structured Holland Water level Peng (2006) 3D Chesapeake Bay Isabel (2003) Charley (2004) Structured Holland Water level Peng (2004) 3D CroatanAlbemarlePamlico Estuarine, NC Emily (1993) Rectangular Holland HWM POM Xie (2004) 3D Hypothetical domain N/A Structured Holland N/A FVCOM Weisberg (2006) 3D Tampa Bay, FL Hypothetical wind Unstructured Holland N/A NOAA WAVEWATCH III (NWW3) WNA WIND Model The Western North Atlantic (WNA) regional wave model was designed to fill the needs of the Eastern and Southern Regions, the Mari ne Prediction Center (M PC) and the Tropical Prediction Center (TPC) which had requested that the area covered by the East Coast and Gulf of Mexico (ECGM) regional wave model be expanded north to 50 N, east to 30 W, south to the Equator, and to include the entire Caribbean Sea. The WNA is based on the NWW3 (Chen, Burroughs, and Tolman 1999) and Tolman (1999a, b, and c). The NWW3 provides the boundary conditions to the WNA. It uses a larger domain (98.25 W 29.75 W by 0.25 S 50.25 N) with PAGE 137 137 275 203 computational grid which spatial resoluti on is 0.25 degrees (28 km). Figure 6-1 shows the computational grid. It covers both ocean and land. Although the WNA model covers a large area, the model provides data for ocean regions only. The WNA model provides 3 hour interval wind field only. It does not provi de atmospheric pressure data. In this study, WNA wind will be used as wind input for Hurricane. However, wind over land will be inter polated from real wind data. Figure6-4 shows an example of wind fiel d of Hurricane Ivan generated by WNA model. WINDGEN Model The WINDGEN is wind and atmospheric filed generation program. The program provides wind and atmospheric pressure output for numerous alternate track models for use by the modelers. The WINDGEN program queries extern al data sources for the acquisition of track, wind, background fields and altern ate track information. The generated wind filed is validated with data from NDBC (National Da ta Buoy Center) in real time. Th ree data sets of wind stress measurements in hurricane force winds were obt ained in the Air-Sea Inte raction Facility using the profile method, the Reynolds stress method, and the momentum budget or surface slope method. The WINDGEN program also uses a larger domain (99 W 30 W by 5 N 53 N) with 346 241 computational grid which spatial resolu tion is 0.2 degree (22 km).Figure 6-2 shows the computational domain used in WINDGEN m odel. The WINDGEN program integrated with the ADCIRC (ADvanced CIRCulation Model for Coasts, Shelves and Estuaries, Luettich, et al., 1992) to produces wind and atmospheric pressure field. The WINDGE N program covers both sea and land. The model provides both wind and at mospheric pressure fiel d. Figures 6-5 and 6-6 show the wind filed and atmospheric field for Hurri cane Ivan. The wind and pressure fields from the WINDGEN model will be used as storm surge model input in this study. PAGE 138 138 NOAA HRD Surface Wind Analysis System Tropical cyclones are monitored globally by spac e -, aircraft -, land and marine based observing systems. Advances in computing and communications have made it possible to obtain these observations in near real-time. In the Atla ntic, Eastern Pacific, and Central Pacific Ocean basins, hurricane wind fields are determined subj ectively based on the specialist's interpretation of flight-level reconnaissance data, satellite observations pressure-wind relationships and available surface data. These fields are represen ted by text portions of the official forecast product as radii (from the storm center) of 34 kt 50 kt, and hurricane force winds in four compass quadrants relative to north. NHC analyzes wind observa tions on a regular 3 or 6 hour schedule consistent with NHC=s warning and forecast cycle. These data are composited relative to the storm over a 4-6 hour peri od. All data are quality controlle d and processed to conform to a common framework for height ( 10 m or 33 feet), exposure (mar ine or open terrain over land), and averaging period (maximum sustained 1 mi nute wind speed). HRD wind analysis system provides just wind fiel d around Hurricane center Figure 6-7 shows HRD wind field of Hurricane Ivan. HRD wind data is useful when Holland wi nd model is used to determine the maximum wind radius and hurricane center. HOLLAND =s Analytical Model HOLLAND=s analytical wind and pressure model is a simple storm model which assumes an exponential decay of pressure fr om the center of a storm (Holland, 1980). The atmospheric pressure field is desc ribed as the sum of an axially symmetric part and a large scale pressure filed of constant gradient. The symmetric part of the local atmospheric pressure in a storm is described in terms of an exponential pressure profile from Holland (1980). Detailed Holland analytical wind model was reviewed Chapter 5.10.This model can be used in any domain as long as the location of the storm, pr essure at the center of the storm and maximum PAGE 139 139 wind point are known. Figure 6-3 shows a simple grid for generating wind and atmospheric pressure field for Hurricane Ivan. Figures 6-8 and 6-9 show wind and atmospheric filed generated by Holland analytical model with Hurricane Ivan. Wind and Atmospheric Pressure Interpolation Time intervals of storm or Hurricane data are one hour for WINDGEN and three hours for WNA and HRD. The time increment in simulation with the current UFDVM is usually less than 30 minutes. When simulation storm surge using WINDGEN, WNA, and HRD data, the data will be interpolated at every time step according to the time increment of simulation. One of two interpolation methods can be applicable for the simulations, linear interp olation with time and time and space interpolation. The assumption for th e interpolation is that the center of Hurricane or storm moves linearly from one data set to another. Table 6-3-1 Summary of wind and atmospheric pressure model Model Domain covered Grid Data Reference Height Land Effect Data Interval Duration of Data NCEP WNA 98.25 29.75 W 0.25 50.25 N Sea area only Figure 6-1 275 203 (28km) Wind only 10 m Open water exposure 3 hours Yearly WINDGEN 99.00 30.00 W 5.00 53.00 N Both sea and land Figure 6-2 346 241 (22km) Wind and pressure Wind: 10m Pressure: Sea level Open water exposure 1 hour Yearly HRD 34 or 50 miles from Hurricane center Figure 6-7 150 200 (5 6 km) Wind only 10m Open Land exposure 3 or 6 Hours Until landfall HOLLAND Any domain Both sea and land Figure 6-8 Any resolution Wind and pressure Wind: 10m Pressure: Sea level Open water exposure Any time Any time PAGE 140 140 Figure 6-1 NCEP WNA model domain Figure 6-2 WINDGEN model domain Figure 6-3 Simple grid for Holland model PAGE 141 141 Figure 6-4 NCEP WNA wind fi eld of Hurricane Ivan at time 9/16/04 07:30(UTC) Figure 6-5 WINDGEN pressure field of Hurricane Ivan at time 9/16/04 07:30(UTC) PAGE 142 142 Figure 6-6 WINDGEN wind fi eld of Hurricane Ivan at time 9/16/04 07:30(UTC) Figure 6-7 HRD wind field of Hurricane Ivan PAGE 143 143 Figure 6-8 Holland pressure field of Hu rricane Ivan at time 9/16/04 07:30(UTC) Figure 6-9 Holland wind field of Hurrica ne Ivan at time 9/16/04 07:30(UTC) PAGE 144 144 Linear Interpolation with Time Consider atmospheric pressure data between the two snapshots which are input data for Hurricane or storm (Figure 6-10). The two sn apshots are come from HRD wind field for Hurricane Ivan at time 9/16/ 04 01:30 and 9/16/04 04:30 UTC. Let the air pressure at ( x, y) in the first snapshot P1 = P ( t = T1, x, y), and P2 = P ( t = T2,x y) in the second snapshot. Then the simple temporal li near interpolation for the pressure at time t is given by P ( t x, y) = (1) P1 ( x, y ) + P2 ( x, y ) (6-3-1) where ( t ) = (t T2) / ( T2 T1) (6-3-2) The middle panel in Figure 6-10 shows the inter polated air pressure data at time t between the two Hurricane data field T1 and T2. As seen from the Figure 6-10, this simple time interpolation method can not capture Hurricane eye. The eye of Hurricane is changed to a elliptic shape not a circle. The maximum wind also decr eased in the middle of the time interpolation process. This is because the interpolation doe s not include the spatial interpolation for the moving Hurricane data field. Hence, this met hod can not be used for storm surge simulation although the method is simple and efficient in computa tional time sense. Time and spatial Interpolation (Lagrange Interpolation) For correct interpolation, spatia l interpolation is required as well as time interpolation. As the same as time interpolation, consider two given snapshots. For this case, spatial shifting of the two snapshots is required. Let=s define the shift distance relativ e to the Hurricane eye. For the first snapshot, x1 = X0 X1, y1 = Y0Y1, and for the second snapshot, x2 = X0 X2, y2 = Y0 Y2 where X0, Y0 are the location of Hurricane eye for the two snapshots and x1 and y1 are shift distance from the first source snapshot, x2 and y2 are shift distance from the second snapshot. Figure 6-11 depicts the spatial shifting. Once th e two snapshots are shifted, inverse distance PAGE 145 145 interpolation for the air pressure and wind speed is performed. The inverse distance algorithm is simple. The value at each data point is weighted by the inverse of the distance between the source data point and the destination data point raised to a power as P ( x, y) = ( Ws P( X0, Y0) ) / Ws (6-3-3) where Ws is the weighting function defined as Ws = D-E. D is the distance between the source and the destination point and E is the exponent valued as betw een 2 and 5. After the spatial interpolation, time interpolation is applied a ccording to the Equation 6-3-1. Figure 6-12 shows the result of time and spatial interpolation. The method captu res the Hurricane eye correctly and the maximum wind is retained in the middle of the interpolation processes. The current UFDVM model uses the time and spatial Lagrange interpolation. Figure 6-10 Time interpolation be tween the two Hurricane snapshots TimeT=T1 WINDSPEED:1014182226303438424650 TimeT=t TimeT=T2 PAGE 146 146 Figure 6-11 Schematic diagram of sh ifting of two Hurricane snapshots Figure 6-12 Time and spatial interp olation of two Hurricane snapshots TimeT=T2 Shiftingdistance TimeT=T1TimeT=tShiftingdistance TimeT=T2 WINDSPEED:1014182226303438424650 TimeT=t TimeT=T1 PAGE 147 147 Hurricane IVAN (2004) Hurricane Ivan was a classical, long-lived hurricane that reached Category 5 (see Appendix A for detailed descrip tion of the scale) strength three times on the Saffir-Simpson Hurricane Scale (SSHS). Figure 6-13 depicts the official "best track" of the Hurricane Ivan's path from NOAA NHC and Figure 6-14 shows track of Hurricane Ivan unt il landfall with NOAA AOML (Atlantic Oceanographic and Meteorological Laboratory) HRD (Hurricane Research Division) wind field. Hurricane Ivan made landfall as a categor y 3 storm on the east side of Mobile Bay west of Gulf Shores Alabama at 06:23 UTC September 16th. The maximum wind observed during Hurricane Ivan in the United Stat es was 40 m/s sustained with a gust to 48 m/s at the Pensacola Naval Air Station, Florida at 06:29 UTC on September 16th. The lowest pressures observed during Ivan's first U. S. landfall were unofficial reports 943.1 mb and 947.9 mb which came from storm chasers based in Fa irhope, Alabama. Although Ivan was weakening as it made its first U.S. landfall, it is estimat ed to have been a category 3 hurricane when it reached the Alabama coast. Rainfall totals generally ranged from 7.6 18 cm along a large swath from Alabama and the Florida panhandle northeastward across the eastern Tennessee Valley and into the New England area. When Ivan was an extra-tropical low pressure system it produced rainfall totals in excess of 18 cm as far north as New Hampshire and as fa r south as the Florida east coast. Even as a weakening tropical depression, Ivan produced rainfall amounts exceeding 18 cm across eastern Texas. Table 6-2 shows features of Hurricane Ivan. Storm surge of 3-4.6 m occurred along the coasts from Destin in the Florida panhandle westward to Mobile Bay/Baldwin County, Alabam a. Storm surge values of 1.8-2.75 m were observed from Destin eastward to St. Marks in the Florida Big Bend region. Lesser va lues of storm surge continued east and southward along the Florida west coast with 1.2 m reported in Hillsborough PAGE 148 148 Bay/Tampa Bay. The highest storm surge elevat ion during Hurricane Ivan in the US was 4.6 m at the Pensacola, FL. There was also a possi ble record observed wave height of 16 m reported by the NOAA Buoy 42040 located in the north central Gulf of Mexico (29.18 N, 88.21 W) south of Alabama. Table 6-3 is summary of storm surge elevation at different locations during Hurricane Ivan. The forces of Ivan were directly responsible for 25 deaths in United States. Ivan caused extensive damage to coastal and inland areas of the United States. Portions of the Interstate 10 bridge system across Pensacola Bay, Florida were severely damaged in several locations as a result of severe wave action on top of the 3 4.6 m storm surge. As much as a quarter-mile of the bridge collapsed into the bay (Figure 6-15). The U.S Highway 90 Causeway across the northern part of the bay was also heavily damaged. Fi gure 6-16 shows Hurricane Ivan impact area from USGS Hurricane and Extreme Storm Impact Studies1. To the south of Pensacola, Florida, Perdido Key bore the brunt of Ivan's fury and wa s essentially leveled. Along the Alabama coast, high surf and wind caused extensive damage to Innerarity Point and Orange Beach. In the Alabama and Florida panhandle areas, widespread over wash occurred along much of the coastal highway system. Figure 6-17 (left) shows over-wa shed island in Gulf Shores, AL, before and after Hurricane Ivan. In additi on, extensive beach erosion caused severe damage to or the destruction of numerous beachfr ont homes, as well as apartment and condominium buildings. Some buildings collapsed due to scouring of th e sand from underneath the foundations caused by the inundating wave action. Figure 6-17 (right) show s the collapsed front of a multistory building before and after Hurricane Ivan at Orange Beach, AL. Ivan was the most destructive hurricane to affect this area in more than 100 years. A tota l of 686,700 claims were filed and the American 1 http://coastal.er.usgs.gov/hurricanes/ivan PAGE 149 149 Insurance Services Group estimates that insured losses in the United States from Hurricane Ivan totaled $7.11 billion, of which more th an $4 billion occurred in Florida alone. Table 6-2 Summary of hurricane Ivan features in the US Value Location Time Maximum wind 40 m/s Pensacola, Florida 06:29 UTC, 9/16/04 Min pressure 943.1 mb Fairhope, Alabama. 06:00 UTC, 9/16/04 Rain fall 18 cm Landfall Mobile Bay west of Gulf Shores, Alabama 06:23 UTC, 9/16/04 Highest storm surge In the U.S 4.6 m Pensacola, Florida Death tall 25 people Economic damage $7.11 billion Table 6-3 Selected storm surge elevations at different locations during hurricane Ivan event State Location Storm surge (m ) Baldwin county 3.1 4.6 Lower Bryant Landing 1.1 Alabama Mobile County 1.8 2.8 Bay County, FL 2.5 3.1 Dixie County 1.2 Escambia County 3.1 4.6 Gulf County 1.2 1.8 Florida Santa Rosa County 3.1 4.6 Grand Isle East Point 0.5 Louisiana SW Pass 0.9 Biloxi Bay 1.0 Mississippi Waveland 1.1 PAGE 150 150 Figure 6-13 Best track of Hu rricane Ivan from NOAA NHC Figure 6-14 Track of Hurricane Ivan with HRD wind field snapshots PAGE 151 151 Figure 6-15 Damage of I-10 Br idge during Hurricane Ivan PAGE 152 152 Figure 6-16 Hurricane Ivan imp act area along Alabama coast Figure 6-17 Over-washed area in Gulf shores, AL before (upper left) and after (lower left), Collapsed buildings before (upper right) and after (lower right) Hurricane Ivan at Orange Beach, AL Orange Beach Gulf Shores Perdido Key PAGE 153 153 Computational Domain Hurricane Ivan affected large areas which include State of Florida, Alabama, Mississippi, and Louisiana, the computational domain includes those 4 States. The do main starts from San Blas Bay, FL and ends Barataria Bay, LA. To include inundation caused by Hurricane, the domain extends land area where the topography is about 50 m above the mean sea level and around 10 30 km from the coast line. Figure 6-18 shows the computational domain for both Hurricane Ivan and Dennis. By making use of advantage of unstructured mesh, the computational mesh was refined around the coasta l line. The computational mesh also resolves MS river, major canals (Including 17th street canal), and I10 bri dge and interstate I10 around Escambia bay. The computational mesh resolves the Hurricane domain very well. Figure 6-19 through 6.5.8 show computational mesh in St. Joseph Island, FL, Escambia Bay, FL, Mobile Bay, AL, and New Orleans area, MS, respectively. The shortest length of mesh is 30 m around coastal line at New Orleans cha nnels and the longest length of the mesh is 22,600 m at the open boundary. The open boundary is away from 200 km from the entrance of Mobil Bay, AL. The computational mesh includes San Blas Bay, FL, West Bay, FL, Choctawhatchee Bay, FL, Escambia Bay, FL, Mobil Bay, AL, Lake Pontch atrain, MS, Lake Maurepas, MS and Barataria Bay, MS. The length from leftmost node to ri ghtmost node is 537 km, top node to bottom node in the domain is 275 km. The computational mesh has total 120,336 nodes and 240,220 elements. Table 6-1 shows the dimension of the computational mesh and domain. For storm surge model evaluation, water level must capture the high water relative to a consistent datum. Since the datum (Mean Sea Le vel) used in ADCIRC si mulation is different from what used UFDVM mesh (North Ameri ca Vertical Datum 88 (NAVD88)), water level simulated by ADCIRC has to be adjusted befo re being imported to UFDVM model mesh. The PAGE 154 154 relationship between NAVD88 and MSL is avai lable at several NOAA stations (NOAA, 2006), which was linearly interpolated into the entire domain. Acco rding to NOAA-NOS (2000), in the US, water level and bathymetric measurements are reference to Mean Lower Low Water (MLLW), the average of all lower water levels during each lunar day over a 19 year Tidal Epoch. Land elevations are referenced to the North American Vertical Datum of 1988 (NAVD88), a geodetic datum determined from a least-squares fit to leveling networks and constrained by mean sea level observed at Fa ther Point/Rimouski, Canada. The relationship between MLLW and NAVD88 may be highly variable in space and time. For example, at the Panama City, Florida, (NOS Station 8729108) Mean Sea Level is 0.203 m above MLLW, NAVD88 is 0.170 m, while at Pe nsacola, FL, (NOS Station 8 729840) Mean Sea Level is 0.188 m above MLLW, NAVD88 is 0.098 m. Table 6-5 shows all the bench mark available in the computational domain (http://tidesandcurrents.no aa.gov). To obtain best numerical solution of the model, the model requires a consistent vert ical datum for water level, bathymetry, and topography.The USGS National El evation Data set (h ttp://seamless.usgs.gov/) was used for calculating topography by spatial in terpolation over the land. Th e data have a resolution of approximately 30 m. The GEODAS bathymetric data set was also used fo r depth interpolation over the water. 90 % of water depth is referenced to mean low lower water. Both data sets were converted to the standard NAVD 88 vertical datum. For storm surge model evaluation, water level must capture the high water relative to a consistent datum. The high resolution USDOT shoreline was utilized to disti nguish between land and water. Fi gure 6-26 shows bathymetry and topography of the computational domain. Figur e 6-27 shows bathymetry and topography at Mobile Bay, LA. Figure 6-28 is those of Escambia Bay, FL. PAGE 155 155 Table 6-4 Features of computational domain Features Dimension Total Number Element 240220 Total Number of Node 120336 Total Horizontal Length 537 km Total Vertical Length 275 km Shortest Length of Element 30 m Longest Length of Element 22.6 km Deepest Water Depth 2,478 m below NAVD Highest Land Elevation 76 m above NAVD Table 6-5 Datum in computational domain Station ID State Location MLLW MSL NAVD 8729108 Panama City FL 30 9.1' N 85 40.0' W 0.000 0.203 0.170 8729840 Pensacola FL 30 24.2' N 87 12.7' W 0.000 0.188 0.098 8729678 Navarre Beach FL 30 22.6' N 86 51.9' W 0.000 0.201 0.144 8735180 Dauphin Island AL 30 15.0' N 88 4.5' W 0.000 0.172 0.070 8737048 Mobil State Docks AL 30 42.5' N 88 2.6' W 0.000 0.241 0.062 8745557 Gulfport Harbor MS 30 21.6' N 89 4.9' W 0.000 0.269 0.130 8747437 Bay Waveland MS 30 19.5' N 89 19.5' W 0.000 0.265 0.099 8747766 Waveland MS 30 16.9' N 89 22.0' W 0.000 0.244 0.066 8761724 Grand Isle LA 29 15.8' N 89 57.4' W 0.000 0.164 -0.166 8761927 USCG New Canal LA 30 1.6' N 90 6.8' W 0.000 0.078 -0.128 PAGE 156 156 Figure 6-18 Computational mesh for the hurricane Figure 6-19 Water (left) and land (right) elements of the computational mesh, St. Joseph Island PAGE 157 157 Figure 6-20 Computational mesh, Escambia bay, FL Figure 6-21 Water (left) and La nd (right) elements of the computational mesh, Escambia bay PAGE 158 158 Figure 6-22 Computational mesh, Mobile bay, AL Figure 6-23 Water (left) and La nd (right) elements of the co mputational mesh, Mobile bay PAGE 159 159 Figure 6-24 Computational mesh, LA, MS Figure 6-25 Water (left) and La nd (right) elements of the computational mesh, MS, LA PAGE 160 160 Figure 6-26 Bathymetry and Topography of the Computational Domain (Depth unit: meter) Boundary conditions A large amount of various data was collected during the passage of Hurricane Ivan. Table 6-6 shows the summary of tide, wi nd, and air pressure data which are used for the application of the model to the Hurricane Ivan simulation. Figure 6-31 indicates the location of the data stations in the computational domain. All the data us ed in this study were obtained from COOPS2 and NDBC3. This section describes the de velopment of the tidal, wind st ress and pressure gradient boundary conditions for simulation of Hurricane Ivan. Tidal forcing Normally when simulation circulation and tr ansport in an estuary exposed to the open ocean, using measured water level as a tidal boundary condition can be properly justified. However when trying to simulate a storm surge, m easured water level cannot be used directly as 2 http://co-ops.nos.noaa.gov 3 http://ndbc.noaa.gov/index.shtml PAGE 161 161 tidal forcing because the measured water level data already contain the storm induced setup which the model is trying to simulate. Figure 6-27 Bathymetry and topography in Mobile Bay Figure 6-28 Bathymetry and topography in Escambia Bay PAGE 162 162 Recently, a tidal data base wa s created using the ADCIRC mode l (Luettich et al., 1992) for the Western North Atlantic, Caribbean and Gulf of Mexico. The tidal database was partially validated by Mukai et, al. except nonlinearly generated constituents M4, M6, and STEADY. M2, S2, N2, K2, O1, K1, Q1, M4, and STEADY tidal constituents are included. In this study, the open tidal boundary conditions are used from ADCIRC tidal base. Seven harmonic tidal constituents are used for simulating Hurricane Ivan. The c onstituents and their corresponding periods and phase lag are listed in Table 6-7. As well as describing tidal elevati on at the open boundaries, storm surge elevations also specified at the op en boundaries. The storm su rge elevations at the boundaries comes from ADCIRC m odel results then, the results are used as boundary storm surge input in present model. Figure 6-29 shows th e storm surge elevation at 3 different locations from ADCIRC model. Boundary 1 is where hur ricane center passed thro ugh, boundary 2 is far from the hurricane center, and boundary 3 indicat es near the hurricane center. The storm elevation is obviously higher wher e hurricane center passed thr ough then far from the hurricane center. NOAA predicted tide also used to provi de tidal boundary elevati on during the Hurricane Ivan as well as tidal constituents. The offshore tid al elevation is interpolated from the inshore tidal stations. Figure 6-30 show s the interpolated tidal elevation from NOAA predicted tide at three different locations, i.e., left, center, and right most tidal boundary elements. Wind and atmospheric pressure Wind is a major force driving a storm surge. Atmospheric pressure gradient is also important role in storm surge modeling. Theref ore when it comes to using wind and atmospheric pressure in a storm surge model, it is very im portant to validate the wind and pressure because their accuracy will be significant fa ctors in the overall accuracy of model's output. Four analysis wind (WINDGEN, WNA, HRD, and Holland) fi elds and two atmospheric pressures (WINDGEN and Holland) were used to simula te Hurricane Ivan. While WINDGEN and Holland PAGE 163 163 models have both wind and pressure data, WN A and HRD have wind da ta only. Those four winds were then converted to wind stress and applied directly to the u and v-momentum equations. Figure 6-31 shows the wind and atmospheric data stations inside the computational domain for comparison with analysis wind a nd pressure and measured data. There are 9 measured data stations inside the computational domain (Table 6-8). Table 6-6 Data stations used for compar ison of the model during Hurricane Ivan Station Location Data State Station Name Lon ( oW) Lat ( oN) Wind Pressure Water elevation Data Source Port Fourchon 90.20 29.11 COOPS Grand Isle 89.96 29.26 COOPS Pilot Station 89.41 28.93 COOPS LA East Bank 90.37 30.05 COOPS Wave Land 89.37 30.28 COOPS MS Biloxi 88.90 30.41 COOPS AL Dauphin Island 88.07 30.25 COOPS Pensacola 87.21 30.40 COOPS Panama City B each 85.88 30.21 COOPS FL Panama city 85.67 30.15 COOPS Table 6-7 Tidal constituent parameters for simulation Hurricane Ivan based on ADCIRC Constituents Period (Hours) Amplifica tion Factor (m) Phase lag (Degree) M2 12.42 0.96943 32.22 S2 12.00 1.00000 0.00 K1 23.70 1.09802 257.94 O1 25.80 1.15884 136.05 K2 11.97 1.26481 335.40 N2 12.70 0.96943 151.19 Q1 26.90 1.15884 255.02 PAGE 164 164 Figure 6-29 Storm surge eleva tion at 3 different locations obtained from ADCIRC tidal constituent program Figure 6-30 NOAA predicted tidal elevation at left, center, and right at tidal boundary TIMES t o r m E l e v a t i o n ( m )258260262264 -0.2 0 0.2 0.4 0.6 0.8 1 Boundary1 Boundary2 Boundary3 PAGE 165 165 Because the grids of analysis wind fields a nd study area are different, the analysis wind fields should be interpolated to computational domain of the study area. As shown Figures 5-1 and 5-7, WINDGEN and Holland wind fields cove r land and water regions. However, WNA wind field does not cover land region (Figure 5-1). HRD wind fields extend only 30 50 miles from hurricane center (Figure 57). While Holland wind fields are directly generated from the formulation, WINDGEN, WNA, and HRD wind fields are interpolated to get wind and pressure fields on the computational grid. Linear interpolation is used for WINDGEN wind field because WINDGEN covers entire computational domai n of the study area. The inverse distance algorithm (AMTEC, 2006) is used for HRD and WN A wind fields. The value at each source data point is weighted by the inverse of the distance between the source data poi nt and the destination point raised to a power as s ss dw w where d and s are the values of the variables at th e destination point a nd the source point, respectively, and sw is the weighting function defined asE sDw D is the distance between the source point and the destinati on point or the minimum distance. E is the exponent and set to 3.5 in this interpolation. The valu es in the land are extrapolated. Figures 6-32, 6-33, 6-34, and 6-35 show interpolated WINDGEN, WNA, HRD, and Holland wind speed (upper panel) and wind vector fields (lower panel) at time 06:30, 9/16/04, UTC after landfall (06:23, 9/16/04, UTC) of hurricane IVAN on the east side of Mobile Bay west of Gulf Shores, AL. In the figures, red line indicates best track of hurricane Ivan by NHC. Black circles present the location where the maximum wind is located. Because there is no air pressure data in WNA and HRD wind data, th e hurricane center (eye) of WNA and HRD was tracked by the location of minimum wind speed. Hurricane center tr acked by minimum air PAGE 166 166 pressure (WINDGEN) and minimu m wind speed (WNA) are slight different from that of NHC. WINDGEN, HRD, and Holland wi nd fields show hurricane ey e clearly, however, WNA wind filed does not show hurricane eye. This is becau se WNA wind filed does not cover land side, it covers water side only. Therefor e, the wind field over land is ex trapolated from wind field over water. The other three wind fields cover both land and water and capture hurricane eye clearly. Hurricane center tracked by mini mum air pressure is exactly same as NHC best track. As indicated before, red line in the figures repr esents the best hurrica ne track by NHC. HRD and Holland wind field follows exactly same path of NHC best track. HRD and Holland wind field was generated based on NHC best track. Howe ver, WINDGEN wind field does not follow NHC best track. The hurricane eye of WINDGEN is s lightly deviated from NHC best track. As indicated before, HRD wind fiel d has land effect over land (Fi gure 6-34). The wind speed over the land is weaker than the other three wind fields (WINDGEN, WNA, and Holland). Figures 636 and 6-37 show detailed wind fields of WINDGEN, WNA, HR D, and Holland around Mobile Bay, AL and Escambia Bay, FL at time 06:30, 9/16/04(UTC). Figure 636 shows atmospheric pressure distribution of WIND GEN (upper panel) and Holland (lower panel) model in the computational domain at time 06:30, 9/16/04 (U TC). WNA and HRD do not have atmospheric pressure data. Figure 6-38 shows comparison be tween the measured and simulated WINDGEN, WNA, HRD, and Holland wind speed at four different locations, Gr and Isle, LA, Dauphin Island, AL, Pensacola, FL, and Panama City B each, FL, respectively. Grand Isle and Dauphin Island are located at left hand si de of hurricane Ivan, whereas Panama City Beach and Pensacola are at right hand side. Data stations Dauphin Island and Pensacola are located within the range of maximum wind radius, hence the winds there peak up to 35 40 m/s. Station Grand Isle and Panama City Beach are located further away from the hurricane center. The wind speeds of the PAGE 167 167 two stations are less than Dauphin Island and Pensacola. Although the distance between the hurricane center and Dauphin Island and that of Panama City Beach is about the same, the wind speed at Pensacola is higher. This is because Pana ma City Beach is located at the right hand side of hurricane eye which has strong wind. Moreover, the wind at Panama City Beach always blow from water to land, thus th ere is no land reduction effect. Figure 6-31 Measured Data (wind, pressure, and water elevation) Stations Table 6-8 Measured data stations inside the computational domain Data station name Location (Degree) Data Panama City, FL 85.67 W 30.15N Wind, Pressure Panama City Beach, FL 85.8 8 W 30.21 N Wind, Pressure Pensacola, FL 87.21 W 30.40 N Wind, Pressure Dauphin Island, AL 88.07W 30.25N Wind, Pressure Biloxi, MS 88.90 W 30.41 N Wind, Pressure Waveland, MS 89.37 W 30.28 N Wind, Pressure East Bank, MS 90.37 W 30.05 N Wind, Pressure Grand Isle, LA 89.96 W 29.26 N Wind, Pressure Pilot Station, LA 89.41 W 28.93 N Wind, Pressure PAGE 168 168 Figure 6-32 WINDGEN wind speed ( upper) and wind vector (lower) Figure 6-33 WNA wind speed (upper) and wind vector (lower) Figure 6-34 HRD wind speed (uppe r) and wind vector (lower) PAGE 169 169 Figure 6-35 Holland wind speed (upp er) and wind vector (lower) PAGE 170 170 Figure 6-36 Wind fields around Mobile Bay, AL at time 6:30, 9/16/04(UTC). Upper left: WINDGEN, Upper right: WNA, Lower left: HRD, Lower right: Holland PAGE 171 171 Figure 6-37 Wind fields around Escambia Bay, FL at time 6:30, 9/16/04(UTC). Upper left: WINDGEN, Upper right: WNA, Lower left: HRD, Lower right: Holland, Lower: WNA wind and pressure field PAGE 172 172 However, Grand Isle is exactly opposite situ ation to Panama City Beach. Holland and WNA wind speed over-predicted about 5 m/s than measur ed wind at Grand Isle, LA (upper left panel of Figure 6-38). HRD and WINDGEN model simula ted wind well at the peak. However, HRD wind under-predicted wind after the peak. Holland model over-predicted peak wind speed at Dauphin Island, AL. WINDGEN, WNA, and HRD model simula ted wind speed well. At Pensacola, FL. Holland wind is higher than other three models. There are no measured data after hurricane Ivan landfall at this station. WNA wind over-simulated wind at Panama City Beach, FL. HRD wind under-predicted peak wind about 10 m/s at Panama City Beach, FL. Figure 6.6 11 shows comparison between the measured and simulated WINDGEN, WNA, HRD, and Holland wind direction at the same stations as wind speed. All the four simulated wind directions are in good agreement with measured wind direc tion except Pensacola, FL. This deviation is caused by failure of instrument. Figure 12, 13, 14 and 15 are time series of wind vector at the four different measured stations. Table 6-9 shows the absolute error of peak wind speed and direction at the four stations. In the table, th e peak absolute error at Pensacola station omitted because of uncertainty of instrument errors. Table 6-9 Peak wind speed, time and peak absolute error at measured station Wind \ Stations Dauphin Island Grand Is le Pensacola Panama City Beach Max wind (m/s) 34.8 20.7 26.0 26.5 Measured Time (Julian, UTC) 260.17 259.97 260.21 260.47 Max wind (m/s) 39.06 23.66 40.97 25.20 Time (Julian, UTC) 260.25 260.02 260.33 260.31 Holland Absolute error (%) 10.9 12.5 -5.2 Max wind (m/s) 30.87 19.09 35.75 20.67 Time (Julian, UTC) 260.21 260.0 260.33 260.37 WINDGEN Absolute error (%) -12.7 -8.4 -28.2 Max wind (m/s) 38.13 23.58 43.69 38.13 Time (Julian, UTC) 260.18 260.93 260.31 260.18 HRD Absolute error (%) 8.7 12.2 30.5 Max wind (m/s) 33.51 26.20 31.41 28.70 Time (Julian, UTC) 260.0 260.0 260.37 260.25 WNA Absolute error (%) -3.8 20.9 7.7 PAGE 173 173 Figure 6-38 Comparison between me asured and simulated wind speed. Upper left: Grand Isle, LA, Upper right: Dauphin Island, AL Lower left: Pensacola, FL, Lower right: Panama City Beach, FL PAGE 174 174 Figure 6-39 Comparison of measured and simulated wind direction. Upper left: Grand Isle, LA, Upper right: Dauphin Island, AL, Lower le ft: Pensacola, FL, Lower right: Panama City Beach, FL PAGE 175 175 Figure 6-40 Comparison of measur ed and modeled wind speed and direction at Grand Isle, LA. WINDGEN, WNA, HRD, and Holland wind (from top to bottom) PAGE 176 176 Figure 6-41 Comparison of measured and modele d wind speed and direction at Dauphin Island, AL. WINDGEN, WNA, HRD, and Holla nd wind (from top to bottom) PAGE 177 177 Figure 6-42 Comparison of measur ed and modeled wind speed and direction at Pensacola, FL. WINDGEN, WNA, HRD, and Holland wind (from top to bottom) PAGE 178 178 Figure 6-43 Comparison of measured and modele d wind speed and direction at Panama City Beach, FL, WINDGEN, WNA, HRD, a nd Holland wind (from top to bottom) PAGE 179 179 Figure 6-44 shows atmospheric pressure distribution of WI NDGEN (upper panel) and Holland (lower panel) model in the computa tional domain at time 06:30, 9/16/04 (UTC). WINDGEN and Holland model have atmospheric data. However, WNA a nd HRD do not have atmospheric pressure data. The air pressure of Holland model is calculated by hurricane track, minimum deficit of air pressure and maximum wind radius as discussed Holland model. The standard air pressure was set to 1013 mb The maximum wind radius was calculated from HRD wind field. Figure 6-45 is compar ison of measured and simula ted atmospheric pressure of WINDGEN and Holland model with time at four different stat ions, Grand Isle, LA, Dauphin Island, AL, Pensacola, FL, and Panama City B each, FL. As discussed in Holland model, the model assumes uniform distributi on of air pressure. Hence, the atmospheric pressure near the hurricane center is much close to measured pres sure then those far from hurricane center. As shown Figure 6-44, the air pressure at Dauphin Island and Pensacola is much closer to measured data than those of stations Gr and Isle and Panama City Beach. Table 6-10 shows the absolute error of peak air pressure at the four stati ons. As shown Figure 6-46 and 6-47, Holland model simulates the air pressure lower than WINDGEN air pressure. Table 6-10 Comparison of peak air pressu re, time and peak absolute error Wind Stations Dauphin Island Grand Isle Pensacola Panama City Beach Min pressure(mb) 954 994.3 979.9 996.8 Measured Time (Julian, UTC) 260.29 260.0 260.23 260.33 Min pressure(mb) 961.43 998.07 983.48 1000.77 Time (Julian, UTC) 260.31 260.04 260.33 260.18 Holland Absolute error (%) +0.8 +0.4 +0.4 Min pressure(mb) 968.77 1000.42 988.07 1002.18 Time (Julian, UTC) 260.29 260.0 260.29 260.25 WINDGEN Absolute error (%) +1.5 +0.6 +0.5 PAGE 180 180 Figure 6-44 WINDGEN (upper) and Holland (l ower) atmospheric pressure field PAGE 181 181 Figure 6-45 Comparison of measured and simula ted atmospheric pressu re. Upper left: Grand Isle, LA, Upper right: Dauphi n Island, AL, Lower left: Pensacola, FL, Lower right: Panama City Beach, FL PAGE 182 182 Figure 6-46 Air pressure field near Dauphin Island, AL at tim e 7:30, 9/16/04(UTC) Left: WINDGEN, Right: Holland Figure 6-47 Air pressure field n ear Pensacola, FL at time 7: 30, 9/16/04(UTC) Left: WINDGEN, Right: Holland PAGE 183 183 A thorough error analysis of WNA, WINDGEN, HRD, a nd Holland wind speed, wind direction and atmospheric pressure of Holland and WINDGEN model is shown in Table 6-8. Formula used to calculate th e errors are in Appendix C. For wind speed, Holland model has maximum root mean square error at Dauphin Island (9.28 m/s). WINDGEN model has minimum wind speed RMS error at Grand Isle (2.18 m/s) WNA wind speed RMS error is big compare to other models. This is because WNA model does not cover land. The inte rpolation of WNA wind over land was used wind over water near the stat ions. For wind direction, all the four wind has maximum RMS error at Waveland. Holland mode l has the maximum wind direction RMS error at Waveland (26.5). WINDGEN has minimum wind direction error (6.97) at Dauphin Island. For atmospheric pressure, Holland model has maximum RMS error at Dauphin Island (12.62 mb ). Minimum RMS error for air pressure is 3.17 mb at Waveland of WINDGEN model. Simulation Results of Hurricane Ivan Water Surface Elevation Hurricanes Ivan was simulated with UFDVM mode l developed in this study. According to the wind and pressure analysis discussed prev ious chapter, combined HRD, WNA wind field with limited land exposure and analytical atmosphe ric pressure are used for the simulations. All the other runs with different wind and pressure will be discussed in sensitivity tests. The simulation was performed 5-days from 257 Julian day (09/13/04, 00:00 UTC) to 261 Julian day (09/17/04, 24:00 UTC) which is 2 days before and 2 days after hurrican e Ivans landfall. 300 seconds (5 minutes) time step was used to reso lve finer time series of wind and atmospheric inputs. Time series of water surface elevations were collected at a few NOAA stations along the Gulf coast during Hurricane Ivan. Because wind is strong at right hand side of Hurricane path, large amount of water were pushed into Pensacola and Escambia Bay, FL. PAGE 184 184 Table 6-11 Error of WNA and WINDGEN wind speed, direction and atmospheric pressure compared with measured data at stations during hurricane Ivan WNA WINDGEN Station Error Speed (m/s) Dir () Speed (m/s) Dir () Pressure ( mb ) RMS error 4.70 9.05 3.35 6.97 5.47 Mean absolute error 3.18 4.29 2.62 3.41 4.65 Dauphin Island AL Max absolute error 18.61 33.43 12.86 35.41 14.84 RMS error 6.53 7.94 4.09 9.68 3.43 Mean absolute error 5.34 3.87 3.47 4.92 3.06 East bank LA Max absolute error 16.50 32.21 8.39 35.35 6.85 RMS error 3.25 8.85 2.15 8.65 3.47 Mean absolute error 2.57 4.35 1.80 5.18 3.13 Grand Isle LA Max absolute error 9.25 34.38 5.17 35.37 5.96 RMS error 2.38 6.93 3.86 6.25 3.76 Mean absolute error 1.87 2.73 2.85 3.11 3.17 Pilot Station LA Max absolute error 6.84 35.88 15.86 33.07 11.68 RMS error 7.66 8.82 5.55 7.74 3.17 Mean absolute error 6.29 4.24 4.92 3.93 2.75 West Bank LA Max absolute error 19.88 33.23 13.98 35.46 6.54 RMS error 4.89 20.26 3.37 21.75 4.45 Mean absolute error 3.43 17.62 2.57 19.22 3.91 Waveland MS Max absolute error 15.99 29.93 10.72 35.40 10.25 RMS error 3.89 8.60 3.06 9.20 4.24 Mean absolute error 3.03 5.31 2.36 5.33 3.66 Panama city beach FL Max absolute error 10.23 32.66 9.21 33.50 7.79 RMS error 6.71 15.35 4.05 15.20 4.71 Mean absolute error 5.40 13.07 3.31 13.11 4.43 Pensacola FL Max absolute error 6.23 29.56 11.65 28.04 9.36 PAGE 185 185 Table 6-12 Error of HRD and Holland wind speed, di rection and atmospheric pressure compared with measured data at stations during hurricane Ivan HRD Holland Station Error Speed (m/s) Dir () Speed (m/s) Dir () Pressure ( mb ) RMS error 6.84 10.75 9.28 10.92 12.62 Mean absolute error 4.59 7.38 7.14 7.03 7.95 Dauphin Island AL Max absolute error 25.07 25.22 30.50 24.90 55.73 RMS error 4.39 12.29 4.07 13.57 5.01 Mean absolute error 3.43 8.40 3.15 9.62 4.13 East bank LA Max absolute error 8.97 28.57 8.93 31.55 11.87 RMS error 4.83 12.12 6.20 13.82 6.19 Mean absolute error 3.64 8.50 4.79 10.93 4.95 Grand Isle LA Max absolute error 13.02 29.00 15.08 30.42 15.75 RMS error 2.98 12.41 4.13 12.32 5.01 Mean absolute error 2.65 8.89 3.30 8.60 4.06 West Bank LA Max absolute error 4.97 28.38 10.13 28.40 11.62 RMS error 3.84 23.87 5.12 26.51 7.11 Mean absolute error 2.95 23.64 4.43 26.27 5.59 Waveland MS Max absolute error 11.11 27.60 12.03 28.24 20.60 RMS error 7.12 7.95 6.23 7.14 5.50 Mean absolute error 4.57 5.89 3.79 5.08 4.74 Panama city Beach FL Max absolute error 21.29 18.04 20.53 16.61 12.59 RMS error 3.92 15.27 4.18 15.20 7.06 Mean absolute error 2.68 13.22 2.34 13.11 5.39 Pensacola FL Max absolute error 20.40 27.28 21.56 28.04 26.32 PAGE 186 186 Figure 6-48 Root mean square error of simulated wind models at measured data stations, Upper: wind speed, Lower: wind direction Figure 6-49 Air pressure Root Mean Square error of simulated wind models (WINDGEN and Holland) at measured data stations This strong wind made significant storm surge. The storm surge peaked at the point of landfall between Dauphine Island, AL and Pensacola, FL a nd generated high surge up to 4 meters. Figure 6-50 shows comparisons of measured and simula ted water surface elevations at 6 different measured stations. Unfortunately, the highest water surface elevation at Pensacola, FL was unable to be measured because of instrument fa ilure. Simulated highest water elevation is 2.5 meters. At other stations, the differences between measured and simulated were around 0.3 0.7 0 1 2 3 4 5 6 7 8 9 10Dauph i n I slan d E a s t b a n k Gr an d I s le W est Ba n k W ave la n d P a n a m a city beac h Pe n s a col aWind Speed RMS Error (m/s) WNA WINDGEN HRD Holland 0 5 10 15 20 25 30 Daup h in Island East bank G r a nd Isl e W est Ba n k Wavelan d Panama city beach PensacolaWind Direction RMS Error (Degree ) WNA WINDGEN HRD Holland 0 2 4 6 8 10 12 14Da up hi n Islan d Ea s t b a n k Gr a nd I s l e Wes t Ban k Wave l a n d Panama city bea ch PensacolaAir Pressure RMS Error ( m WINDGEN Holland PAGE 187 187 meter. This is because simulated results do not have wave effect. The wave effect on the surge will be discussed in sensitivity test with comparison with CH3D results. Strong wind will generate high wave run-up. Overall, the measured and simulated water surface elevations are little bit underestimated compared to measured da ta. Again, this is came from the model does not include the wave effects to the surge. In order to see overall storm surge genera ted by Hurricane Ivan in the computational domain, time series snapshot of water surface el evation were drawn. Figure 6-51 shows snapshot of water surface elevation and wind field before a nd after landfall of Ivan every three hours. In the snapshots, the strong wind from the Hurricane Ivan generated a high water surface elevation along the coast of Pensacola and Escambia Bay, FL. As seen the Figure 6-51.A, the highest water surface elevation was occu rred along the coast of Breton S ound, LA before landfall. The wind blew from right to left and the coast of Breton Sound blocked the water flow. As Ivan moves to north, the high water surf ace elevation also moves toward the North. When Ivan made landfall, the highest water elev ation occurred near the Escambia Bay, FL (Figure 6-51.C). However, wind blew from northeast over the Mob ile Bay, AL. This caused lower water surface elevation inside the Mobile Bay. As Hurricane Iv an moves to the North and wind is decreased by land, the water surface elevation is lower down (Figure 6-51.E). Flood Level and Maximum Inundation Map An inundation maps caused by Hurricane Ivan were drawn by robust wetting-and-drying scheme incorporated in the model UFDVM devel oped in this study. During the simulation, the maximum water level over the land was drawn. With this map, inundation area by Hurricane Ivan can be investigated. Figure 6-52 shows maximum simulated inundation level caused by Hurricane Ivan. Figure 6-53 show the maximum simulated inundation map along the coast of Escambia Bay, FL. PAGE 188 188 Figure 6-50 Comparison of simulated and measured water surface elevation PAGE 189 189 Figure 6-51 Snapshots of water surface el evation during Hurricane Ivan, Cont. A B C Breton Sound PAGE 190 190 Figure 6-51 Snapshots of water surf ace elevation during Hurricane Ivan Simulated maximum inundation level before Iv ans landfall occurred near Goose Island, LA, at time 0025 09/16 (UTC) with 4.46 meters. After landfall, the maximum inundation level was 3.45 meter near Milligan Road, FL, at time 1045 09/16 (UTC). There are two ways to investigate storm surge damages. One way is to use aerial photo of hurricane impact area by comparing before and after the hurricane event. Another method is to use so called High Water Marks (HWM). High Water Marks are the primary source of post hurricane high-water level data D E PAGE 191 191 for the evaluation of storm surge models by indi vidual site comparison. The US Army Corps of Engineers and FEMA4 collected some HWMs along the coast region durin g Hurricane Ivan. In order to validate the simu lated inundation, aerial photos5 taken before and after Hurricane Ivan passed over the domain were compared with simulated maximum inundation maps. The photos were taken July 17 2001 and September 17, 2004. Figure 6-54 shows prestorm and post-storm aerial photos of Ivan impact areas. The impacts were breached island at Pine Beach, AL, and over-washed Gulf Shore, AL destroyed houses and multi-story building in Orange Beach, AL. Most of the impacted areas are located the right hand side of Hurricane Ivans trajectory. Those photos were compar ed with simulated maximum inundation map as shown Figure 6-57. As can be seen from the simulated maximum inundation map, those areas were subject to extensive i nundation during storm surge gene rated by Hurricane Ivan. The maximum inundations for those areas were 2 3 meters. This aerial photo validation is rather qualitative but nonetheless important. A more quantitative flood analysis was done based on some evidences obtained from AHurricane Ivan Report@ (Mobil District Engineering Division, 2004) and FEMA. In this report, several locations were examined on presence of high water marks (HWM) left by the flood caused by Hurricane Ivan. Hurricane Ivan caused significant flooding over the northeast Gulf coast, especially around Pensacola Bay and Es cambia Bay, FL. The maximum inundation around the Pensacola Bay and Escambia Bay area is show n in Figure 6-58. According to simulation, a large amount of area east side of the I-10 Bridge and Barrie Is lands were flooded during Ivan. Among the HWMs, the two verified locations are: West Bank of Escambia Bay at the Hwy 90 4 http://chps.sam.usace.army .mil/USHESDATA/Assessments /2004Storms/Ivan/hwms/ 5 http://coastal.er.usgs.gov/hurricanes/ivan/lidar/ PAGE 192 192 Bridge and 1.5 mile from north of I10 Causew ay Bridge. The maximum inundation heights were 3.1 and 3.6 meters at West Bank and north of I1 0 Causeway Bridge, resp ectively. The inundation levels at the other stations are summarized in Table 6-13. During the Hurricane Ivan event, I-10 Bri dge was collapsed by the big storm surge (Figures 6-15). A big storm surge generate d by Hurricane events ma y cause catastrophic damages to coastal structures such as breakwaters jetties, piers, and bridges. As one of big damages during Hurricane Ivan, both west and east ends of I-10 Bridge ov er the Escambia Bay, FL were damaged (Figure 6-57). Big storm surges accompanied by big waves lifted bridge spans. Fortunately, bridge spans in the middle the bridge were not damaged because the surge was not high enough to reach the deck. As storm surge increases the water elevation, waves over the surge hit bridge spans continuously during stor m surge event. The deck height of I-10 Bridge is around 1.05 meter from the sea level at both ends of the br idge. The simulated peak water surface elevation at west end, middle of the Brid ge, and east end was 2.99, 3.02, and 3.05 meters. Water surface elevation at east end is slightly hi gher than west end (Figure 6-58). The peak water surface elevation during Hurricane Ivan under the I-10 Bridge was high enough to inundate the Bridge. PAGE 193 193 Figure 6-52 Maximum inundation map of Hurricane Ivan Figure 6-53 Inundation map along the coast of Escambia Bay, FL PAGE 194 194 Figure 6-54 Before and after Hurricane Ivan im pact areas. A: Breached barrier island, Pine Beach, AL. B: Over-washed ar ea in Gulf shore, AL. C: Destroyed Gulf-front houses, Orange Beach, AL, D: Collapsed multi-story building (lower) Figure 6-55 Simulated maximum inundation map of Hurricane Ivan impact area A C B D PAGE 195 195 Figure 6-56 Location of High Water Marks near Escambia Bay, FL Table 6-13 High water mark comparison for hurricane Ivan simulation Station Name High Water Mark (m) Simulated High Water Level (m) High Water Level Difference (m) Relative High Water Level Difference (%) Perdido Pass Orange Beach 2.6472 3.560 0.918 +34.48 GIWW at Pensacola Gulf Beach 2.9086 3.302 0.39 +13.53 Pensacola Bay At Ft. McRee 2.9147 2.978 0.06 +2.17 Pensacola Bay at Pensacola 3.0649 2.987 -0.07 -2.54 Escambia Bay West Bank at Hwy 90 3.8822 3.631 -0.33 -8.58 Escambia Bay West Bank at North of I-10 3.6418 3.097 -0.544 -14.95 GIWW at Gulf Breeze 3.0 949 2.958 -0 .14 -4.42 PAGE 196 196 Figure 6-57 Missing spans on the We st (left) and East (right ) Bound I-10 Bridge (OCA, 2005) during Hurricane Ivan. Figure 6-58 Simulated water surf ace elevation under I-10 Bridge PAGE 197 197 Surge Modeling Sensitivity Tests To examine uncertainties in prime model inputs such as wind, surface wind drag parameterization, bottom friction formulation, a nd open boundary tidal forcing, a number of tests were performed using the UFDVM model developed in this study. Input wi nds for the test were HRD, WINDGEN, WNA, and Holland analytical wind for Hurricane Ivan. When wind is blowing, the wind field is affected by the type s of land coverage. For the reason, the surface wind drag parameterization is needed. Surface wind drag parameterizations were open water exposure, open land exposure, and limited wind exposure. Among the four types of modeled wind field, only HRD wind fiel d considers the open land expo sure. The other wind models do not consider the land cove rage. In bottom friction sensitivity te st, different values of Mannings n were used from value 0.005 to 0.05. Tidal const ituents and NOAA predicted tidal level at the open boundary were used. Table 6-14 shows the su mmary of the sensitivit y tests performed in this study. Base Simulation All of input parameters used in base simula tion were same as ones used in Hurricane Ivan simulation in previous section. Because WNA wind field does not cover wind over land, combined HRD and WNA wind field was used. For atmospheric pressure input, Holland analytical model was used. As bottom friction pa rameterization, spatially varying USGS NLCD classification was used. In this section, how base input paramete rs were used is explained. Wind input Accurate modeling of storm surge is highly dependent on the accuracy of wind input to the models. Wind speed is the most important fa ctor influencing the st orm surge. Topographic features influence surge devel opment and propagation. Surface wi nd shear stress is the primary PAGE 198 198 forcing for hurricane simulations. Shear stress is non-linearly re lated to wind speed (either a quadratic or cubic dependency depending on the formulation of the wind drag coefficient), so having accurate winds is crucial. Errors in inpu t winds are amplified in a non-linear manner. For the core wind fields in a hurricane, HRD wind fi eld (H*wind) is the most rigorously analyzed. However, the developed wind fields still have inaccuracies and therefore have uncertainty inherent in them, certainly on the order of 5 percent and perhaps more in some situations. Wind fields for the base case, the final H*Wind winds were subjected to a -percent change in magnitude over the entire domain and over the entir e duration of the simulation. There is also uncertainty in the gust factor us ed to convert 1-minute average winds to 30-minute or 10-minute average winds. HRD recommends use of slightly different conversion fa ctors which vary on the order of 5%. Since the air-sea drag laws have been developed assuming 10-minute-averaged winds, a conversion to 10-minute-averaged winds must be implemented. The final HRD wind is 1-minute averaged wind i.e. 1-minute sustained wind speed. A standard pr ocedure to convert 1minute to 10-minute winds is multiplying by 0.8928 (Powell et al., 1996). To convert 1-minute H*wind to 10-minute wind, the conversion factor 0.8928 was applied to the original entire H*wind field. For Hurricane Ivan, there are 28 wind snapshots (07:30, 9/11/04 ~ 07:30, 9/16/04, UTC) in final HRD wind field product. Figure 6-59 shows the original 1-minute averaged HRD wind field for Hurricane IVAN afte r landfall (07:30, 9/16/04, UTC) and the 10-minutes averaged wind filed for the corresponding H*wind filed. Figure 6-60 shows th e interpolated wind speed in the computational domain. Because H*wind provide s the core of Hurricane, the winds outside the core field are extrapolated by inverse distan ce extrapolation, while inside the core field, inverse distance interpolation is used. As wind input sensitivity test, Hurricane Ivan was PAGE 199 199 simulated using 1-minute and 10-minute H*wind fields. Time step used in the simulation is 15 minutes. Surface wind drag parameterization When wind blows over land, the wind fields are affected by the by the types of land cover. The wind will be reduced by different type s of land cover. Any of wind models discussed previous section does not consider the land reduction effect completely. Table 6-14 Hurricane Ivan sensitivity test Wind Open Boundary Bottom Friction Air Pressure HRD WNA WINGEN Analytical Tide Surge L L E O L E N L W L L E O W E L L E O W E L L E PT AC AC uniform 0.005 0.015 0.020 0.050 spatial varying No Holland model Base Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7 Test 8 Test 9 Test 10 3D Test 11 Test 12 Different coastal configuration test Test 13 Different time increment Test 14 Different mesh size test LLC: limited land exposure LC: open land exposure OWC: open water exposure NLW: no land wind PT: Predict Tide, AC: ADCIRC PAGE 200 200 Figure 6-59 One(left) and tenminute (right) averaged Hurricane Ivan HRD wind field (07:30, 9/16/04, UTC) Figure 6-60 Interpolated 1(lef t) and 10-minute (right) averaged Hurricane Ivan HRD wind field on computational domain (07:30, 9/16/04, UTC) HRD wind considers only open land effect. WNA, WINDGEN and Holland wind model are modeled with open water exposure. IPET re port (Vol. 4, 2006) for the Hurricane Katrina study explains the adjustment procedure for the land reduction effects to the modeled wind fields. Following IPET procedures, 10 minute aver aged HRD wind field is converted to limited exposure. The land cover data for the study ar ea is available from the National Land Cover Database (NLCD 2001) on the USGS Seamless Data Distribution web site. The resolution of the PAGE 201 201 land cover data is 1/3 arc second (30 m). Figur e 6-61 shows the Land Cover specifications for the study area. Table 6-15 show s NLCD class. From the land cover data, the bottom roughness for the study area can be obtained. Figure 662 shows converted bottom roughness according to Land Cover Data set. Finally, the HRD wind field can be converted to limited exposure using the integrated bottom roughness. The conversion coe fficient from open water exposure to limited exposure can be written as (Powell. M., 1996 and 1998): f Z Z Z Z Z Zosooso 007061 .lnln where Zos is integrated bottom roughness, Zo is open water bottom roughness, and Z is 10 meters. Table 6-15 shows the integrated bottom roughness for NLCD classifications where NLCD classes was defined by USGS and integrated bottom roughness was defined by FEMA (Federal Emergency Management Association, 2005). Figur e 6-63 shows the final input HRD wind field with limited exposure at Hurricane Ivan land fall. Bottom friction Bottom friction can alter the water elevation during the storm surge event. There are two scenarios in the sensitivity simulations, simula tion with constant Mannings n and spatiallyvarying Mannings n. Various Mannings n va lues according to bottom conditions can be obtained from the USGS land use factors a nd are summarized in Ta ble 6-16. The bottom conditions in the USGS land use factors incl ude herbaceous wetland, woody wetland, swamp, scrubland, orchard, grassland, pasture, crops, recreational grass, fallow, sand, gravel, Cypress forest, deciduous forest, evergreen forest, mixed forest, as well as city conditions including low residential, high residential, and commercial areas. Figure 6-64 shows the spatially-varying Mannings n values interpolated from USGS land use class to computational domain. With PAGE 202 202 constant and spatially-varying Mannings n, it is possible to examine the storm surge elevation with a broad (constant value) description of bo ttom friction versus a deta iled representation of bottom friction. As the constant value, ope n water Mannings n was specified as 0.020. Table 6-15 NLCD land cover classificat ions and integrated bottom roughness NLCD class Description Zos : Integrated bottom roughness 11 Open Water 0.001 12 Ice/Snow 0.012 21 Low Residential 0.330 22 High Residential 0.500 23 Commercial 0.390 31 Bare Rock/Sand 0.090 32 Gravel Pit 0.180 33 Transitional 0.180 41 Deciduous Forest 0.650 42 Evergreen Forest 0.720 43 Mixed Forest 0.710 51 Shrub Land 0.120 61 Orchard/Vineyard 0.270 71 Grassland 0.040 81 Pasture 0.060 82 Row Crops 0.060 83 Small Grains 0.050 84 Fallow 0.050 85 Recreational Grass 0.050 91 Woody Wetland 0.550 92 Herbaceous Wetland 0.110 95 Cypress Forest 0.550 PAGE 203 203 Figure 6-61 USGS NLCD la nd cover classifications Figure 6-62 Converted land roughness according to USGS NLCD land cover classifications PAGE 204 204 Figure 6-63 Hurricane Ivan wind fiel d including land reduction effect Figure 6-64 Mannings n according to USGS, NLCD land cover classifications PAGE 205 205 Table 6-16 Mannings n according to USGS NLCD land cover classifications NLCD class Description Mannings n 11 Open Water 0.020 12 Ice/Snow 0.020 21 Low Residential 0.070 22 High Residential 0.140 23 Commercial 0.050 31 Bare Rock/Sand 0.040 32 Gravel Pit 0.060 33 Transitional 0.100 41 Deciduous Forest 0.120 42 Evergreen Forest 0.150 43 Mixed Forest 0.120 51 Shrub Land 0.050 61 Orchard/Vineyard 0.100 71 Grassland 0.034 81 Pasture 0.030 82 Row Crops 0.035 83 Small Grains 0.035 84 Fallow 0.030 85 Recreational Grass 0.025 91 Woody Wetland 0.100 92 Herbaceous Wetland 0.035 95 Cypress Forest 0.100 Atmospheric pressure The UFDVM model applied in this study requir es wind and pressure fields for the entire modeling domain. The wind fields are discussed pr evious section. HRD, WNA data do not have the atmospheric pressure field. There are two sources to obtain pressure fields for the simulations, WINDGEN and analytical Holland M odel. The pressure field generated for the Hurricane IVAN is from a single exponential pressu re profile (Holland, 1980) which also reflects the central pressure estimates from the NOAA Tropical Prediction Center/National Hurricane Center. In Holland analytical model, the pressure field snapshots, aligned to the storm track, are spatially and temporally interpolated on the id entical fixed computational domain. The track applied represents a linear 3-hr in terpolation of the HRD analysis results for storm position. As a base sensitivity simulation and sensitivity simulations, the analytical Holland atmospheric PAGE 206 206 pressure field inputs were used. Figure 6-65 shows the wind field of Holland model for Hurricane IVAN after landfall (07:30, 9/16/04, UTC). Figure 6-65 Holland atmospheric pr essure for Hurricane Ivan Hurricane Ivan Sensitivity Test Along with the base simulation, 11 set of sensi tivity tests were performed. Seven tests are the simulations for different wind data set. The others are different open boundary tidal elevation, different set of bottom friction conditio ns, and different atmosp heric pressure fields. All the simulations are performed 15 minutes time increment. All the results were compared with measured data at different data stations. Table 6-14 shows the sensitivity tests done by this study. PAGE 207 207 Sensitivity to wind input The first set (base and test 1 test 7) of the sensitivity test for Hurricane Ivan is the storm surge elevation with different wind data set. Th ere are seven test cases in the simulation. The wind fields used in the test are HRD, WNA, WINDGEN, and analytical wind field. The wind fields are three kinds of conditions, limited land exposure (LLE), open land exposure (OLE), and open water exposure (OWE). With these tests, the sensitivity to wind field and land reduction effect can be determined. As shown Table 6-14, the base simulation uses Limited Land Exposure HRD wind field, Holland analyt ical atmospheric pressure, NOAA predicted tide and ADCIRC storm surge elevation at the open boundary, spatially varying bottom roughness according to NLCD. Sensitivity to bottom friction To examine bottom friction effects on storm surge elevation, 4 constant vales of Mannings n and spatially varyi ng Mannings n (Table 6-16) are applied to the model. The constant Mannings n values used in the mode l are 0.001, 0.005, 0.020, and 0.050. In the IPET report, the base Mannings n used in the ADC IRC model was 0.020. In the base simulation, spatially varying Mannings n was used. Figure 667 shows the results of the simulation with 4 different values of constant Mannings n. As expected, surge elevation with small Mannings n 0.001 produced higher water elevation. Larger value of Mannings n 0.05 decreases the surge elevation. However, the difference in surge elevati ons with different Mannings n in water side is small compare to huge changes in values of Ma nnings n. Hence, this test revealed bottom friction with different values of Mannings n changes the storm surge elevation very small. For the inundation simulation, Owing to USGS, NLCD land cover classi fication, spatially varying Mannings n is used in this study. PAGE 208 208 Figure 6-66 Water surface eleva tion with different wind input PAGE 209 209 Figure 6-67 Water surface el evation with different values of bottom friction PAGE 210 210 Sensitivity to tidal forcing at the tidal open boundary In the base simulations, NOAA predicted tidal elevations at the tidal open boundary were used. The Western North Atlantic, Caribbean and Gulf of Mexico Tidal Databases (Makai, 2002) in ADCIRC provides the tidal c onstituents over the region. Wh ile NOAA predicted tide has all the tidal constituents (37 tidal constituents), ADCIRC tidal databases have 9 tidal components. This test is to examine the sensitivity to the tidal forcing at the tidal open boundary. When ADCIRC tidal constituents are imposed along th e open boundary, the water surface elevation is lowered little bit at all stations compared those of NOAA predicted tide case. This means NOAA predicted tide is more accurate than that of ADCIRC. However the difference in water surface elevation between the two tidal forcing is ne gligible. Figure 6-68 shows comparison of water surface elevation at Panama City, FL, Panama City Beach, FL, and Pensacola, FL, when ADCIRC tidal constituents and NOAA predicted tide were used in simulation. The difference in the water surface elevation at Pensacola is about 10 cm. Figure 6-90 shows water surface elevation comparison at Bil oxi, MS, Wave Land, MS, and Dauphin Island, AL. From the comparison sensitivity of storm surge elevation to the tidal forcing can be small. Sensitivity to atmospheric pressure The atmospheric pressure effect on the st orm surge is tested. Holland analytical atmospheric pressure is used fo r the test. As Holland analytical hurricane model, the atmospheric pressure deficit decreases exponentially from th e hurricane center. The atmospheric pressure gradient is account for the varia tions of water surface el evation, therefore its effects on the water surface elevation is limited to ar eas close to the hurricane center. Figures 6-70 and 6-71 show the water surface elevation comparison between with pressure and without pressure. As seen the figures, surge level near the hurri cane center is slightly higher. At stations far from the hurricane center, storm surge level with and without pressu re is almost same. This test reveals that air PAGE 211 211 pressure is less sensitive to the storm surge. Moreover, its effect is only confined to areas near Hurricane center. Sensitivity to 2 and 3 dimensionality Model UFDVM developed in this study is 3-dimensional model. However, all the simulations including sensitivity tests so far we re 2-dimensional simulations. If 2-dimsesional simulation produces good results, it will be a good choice to use 2-dimensional model in computational sense. The two-dimensional resu lts produced reasonably good water level during Hurricane Ivan. As seen in Hurricane Ivan simulations, 2dimsnsional model produced good results. In Hurricane induced storm surge simulati on, time is a crucial factor for forecasting and hind casting of storm surge. Ther efore, if the results of 2 a nd 3-dimensional simulation is comparable, 2-dimensional simulation can be su fficient for storm surge. Although 2-dimensional model produces good storm surge level, if the vert ical structures of fl ow field are needed, 3dimensional model should be used. There are several differences in 2 and 3-dimsnsional model. Velocities in 2-dimensional model are vertica lly averaged ones. The calculation of nonlinear, diffusion and Coriolis terms is based on 2-dimensin al vertically averaged velocity. However, in 3-dimensional model, all of vertical layer veloci ties are used to calculate those terms. This produces more accurate velocity field in the m odel. With 3-dimensional velocity, turbulence model can be used in calculation of vertical e ddy viscosity. 2-dimdnsiona l model uses constant vertical eddy viscosity. The e quilibrium closure turbulence model like CH3D model is applied to calculate eddy viscosity in 3-di mensional model. In 2dimensional model, Chezy friction formulation is used for calculation of the bot tom friction term in the equation. A quadratic bottom friction formulation based on log law is us ed in 3-dimensional model. In 3-dimensional simulation test, 10 layers were used. Simulation time in2-D took 2 and half hours for 5-day run and 7 hours in 3-D. In theory, 3-D simulati on time takes 2-D simulation time multiplied by PAGE 212 212 number of vertical layers. However, in Z-grid sy stem, deeper areas have more vertical layers and shallow areas have just one or two vertical layers. Hence, the simulation time is not increased linearly with number of verti cal layers. Figure 6-72 shows co mparison between 2and 3D simulation. At station Pensacola, the peak of water surface elevation of 3-D result is slightly higher, the other stations, little bit lower. This is because different bottom friction and vertical eddy viscosity are used. Figure 6-68 Comparison of water surface elevation between NOAA predicted tide and ADCIRC tidal constituent at Panama City Beach, Panama City, and Pensacola Figure 6-69 Comparison of water surface eleva tion between NOAA predicted tide and ADCIRC tidal constituent at Bil oxi, Dauphin Island and Wave Land PAGE 213 213 Figure 6-70 Comparison of water surface elevation between run with and without pressure at Florida stations Figure 6-71 Comparison of water surface elevation between run with and without pressure at Alabama, and Mississippi stations PAGE 214 214 Figure 6-72 Comparison of 2-D a nd 3-D simulation. A: Dauphin Is land, B: Panama City, C: Panama City Beach, D: Pensacola Sensitivity to coastal configurations In order to determine what roles do estuaries play in affecting the storm surge and inundation in coastal zones, the barrier island from Gulf Breeze to Pensacola Beach is removed. Figure 6-73 shows the water depth before and afte r removing the part of the barrier island. By removing the barrier island, the entrance of Pens acola Bay is open wide. The barrier island does not protect water input to the bay anymore. Water depth of removing part of the barrier was ser to 10 meters. To compare the sensitivity to surge elevation caused by coastal configuration, A B C D PAGE 215 215 Hurricane Ivan was simulated with and without the barrier island. The numbers 1 to 6 in the Figure 6-73 indicate the surge elev ation stations inside the bay. Figure 6-74 shows the simulation result with and without the barrier island. As seen the figure, there ar e no big differences in storm surge elevations with and without the barr ier island. This reveals that the existence of estuaries or barrier islands do not have significant effect on the storm surge elevation and inundation on coastal configurations. However, th e phase of storm surge is quite different with and without the existence of the barrier island. Because the entrance of Pensacola Bay is widely opened, water pushed by the storm enters easily. Hence, the storm surge moves faster when there is no barrier islands at the entran ce of coastal bays or estuaries. Figure 6-73 Modification of coastal configuration by removing part of barrier island, left: before removing barrier island, right: after removing barrier island Sensitivity to wave effect on storm surge The model UFDVM developed in this study do es not have wave components in the equation. However, as seen in Hurricane Ivan si mulation, the simulated surge level is little bit under-estimated. In order to inve stigate the wave eff ect on storm surge level, simulated surge elevations of CH3D model with and without wave were comp ared to UFDVM model result PAGE 216 216 without wave. As discussed in chapter Storm Surge Model Review, CH3D model has wave component in the model. CH3D model also app lied to storm surge simulation by Hurricane Ivan and Dennis (Zhang, 2007). Figure 6-75 shows comp arison of water surface elevation between CH3D with and without wave and UFDVM without wave at four di fferent measured stations. At Pensacola station, the difference in storm surge level was 0.9 meter. The differences at other stations were about 0.5 meter. Pensacola bay is long, narrow, and shallow bay. This increases wave setup. Hence, the surge elevation in Pens acola bay is higher than the other places. The overall differences in surge elevation with and without ranges 0.3-0.9 meter. Figure 6-74 Simulated water surface eleva tion with and without the barrier island Sensitivity to time increment As discussed in Chapter 3, UFDVM model does not have time limitation. To test the surge elevation of sensitivity to model time incremen t, 5 and 15 minutes time increment were used. The other input parameters are exactly same in two runs. Figure 6-76 shows comparison of water surface elevation for the two runs at four different stations. Red don indicates 5-minute time PAGE 217 217 increment and blue line is 15-minute run. In the Figure, the simulated water surface elevations of 5-minute results are slightly higher than 15-minute results. This is because 5-minute interpolation of wind fields resolves better than that of 15-minutes interpolation. To get better results, small time increment should be recommended if total run time of the model is not long compared to larger time increment. Total run time of 5-minute was 6 hours and 15-minute 4 hours with Intel Pentium 4 3.20GHz machine. Figure 6-75 Comparison with CH3D with wave result A: Pensacola, FL, B: Panama City, FL C: Panama City Beach, FL D: Dauphin Island, AL JulianDayWaterSurfaceElevation(cmNAVD88)258 260 262 -100 -50 0 50 100 150 200 250 300 350 400 Green:Measured Black:UFDVMwithoutwave Red:CH3Dwithoutwave Blue:CH3Dwithwave Orange:PredictedPensacola,FL JulianDayWaterSurfaceElevation(cmNAVD88)258 260 262 -100 -50 0 50 100 150 200 250 Green:Measured Black:UFDVMwithoutwave Red:CH3Dwithoutwave Blue:CH3Dwithwave Orange:PredictedPanamaCity,FL JulianDayWaterSurfaceElevation(cmNAVD88)258 260 262 -100 -50 0 50 100 150 200 250 300 350 400 Green:Measured Black:UFDVMwithoutwave Red:CH3Dwithoutwave Blue:CH3Dwithwave Orange:PredictedPanamaCityBeach,FL JulianDayWaterSurfaceElevation(cmNAVD88)258 260 262 -100 -50 0 50 100 150 200 250 Green:Measured Black:UFDVMwithoutwave Red:CH3Dwithoutwave Blue:CH3Dwithwave Orange:PredictedDauphinIsland,AL A B C D PAGE 218 218 Figure 6-76 Comparison with 5-(r ed circles) and 15-(blue line) minute time increment, A: Pensacola, FL, B: Panama City, FL C: Panama City Beach, FL, D: Dauphin Island, AL Sensitivity to mesh size As final sensitivity test, Hurricane Ivan was simulated with fine and coarse meshes. One of big advantages of unstructured grid model is that it is easy to refine mesh finer. To test model sensitivity to mesh size, coarse mesh was genera ted based on finer mesh. The fine mesh is about 4 times finer than coarse mesh (Figure 6-78). Table 6-17 shows comparison of fine and coarse meshes. Figure 6-79 shows the comparison of wate r surface elevation with coarse and fine mesh. As seen in the Figure, at stations Pensacola, Panama City and Panama City Beach, the surge A B C D PAGE 219 219 heights were under estimated about 0.3 0.5 mete r. However, at station Biloxi, MS, the surge level was over estimated about 0.2 meter. This te st reveals mesh size does not effect the surge elevation much. However, finer mesh produces surge elevation well. Table 6-17 Comparison of computational mesh between coarse and fine mesh Features Coarse Fine Total Number Element 64324 240220 Total Number of Node 32520 120336 Total number of Tidal Element 67 72 Total number of Land Boundary Nodes 648 337 Summary of sensitivity test To investigate sensitivity to surge elevati on according to paramete r input, several tests were performed compared to a base test. As expected, wind input was one of the most important parameters to effect storm surge elevation. The surge elevation depends on wind field. If wind is strong, surge level is higher, vice versa. In addition to various wind input, bottom friction parameterization is the second most important fa ct to influence the surge level. As bottom friction coefficient (Mannings n here) is increased the surge level is decrease. Phase lag of the storm surge is dependant on tidal open boundary inputs. NOAA predicted input produces more accurate phase and surge elevation than ADCIRC tidal constituents. Because Hurricane storm surge is short time event, the difference is negligible. Atmospheric was effect on surge level. However, the difference was confined to near the hu rricane center where pre ssure gradient is big. The dimensionality of the model was not effect the surge elevation significantly. If vertical structures of flow are needed, 3-D simulati on should be used. Coastal configuration of computational domain does not effect on the surg e level. However, coastal configuration has effect on the phase of storm surge. Waves had si gnificant effect on the surge level. Comparison with CH3D model with wave e ffect showed about 0.5 0.9 meter differences in surge level PAGE 220 220 without waves. The effects of time increment and mesh size of the model were small. Overall, wind parameterization, and wa ves were effect on the storm surge level significantly. Figure 6-78 Comparison of coarse (left) and fine mesh (right) Figure 6-79 Comparison of water surface elevation with coarse fine mesh PAGE 221 221 Hurricane Dennis (2005) Dennis formed from a tropical wave that moved westward from the coast of Africa on June 29th. Figure 6-80 depicts the official "best track" of the tropical cyclone's path from NOAA NHC and Figure 6-81 shows track of Hurricane De nnis until landfall with NOAA AOML (Atlantic Oceanographic and Meteorological Laboratory) HRD (Hurricane Re search Division) wind field. The track of Hurricane Dennis is very similar to Hurricane Ivan (2004) Hurricane Ivan passed through east of Mobil Bay, AL, while Dennis in the middle of Escambia Bay, FL. Dennis reached hurricane strength early on July 7th, then intensified into a Category 4. Dennis weakened to a Category 3 hurricane while passing across southeastern Caribbean Sea. Dennis gradually intensified for the next 6-12 hours over the Gulf of Mexico. Dennis made landfall on Santa Rosa Island, Florida, between Navarre Beach and Gulf Breeze, 19:30 UTC July 10th. The Florida Coastal Monitoring Program (FCMP) tower at th e Pensacola Airport measured a pressure of 956.3 mb at 19:43 UTC 10 July, while the FCMP towe r in Navarre measure a pressure of 965.2 mb at 19:09 UTC that day. A storm chaser in Pace, Florida, measur ed an unofficial pressure of 945 mb at 19:10 UTC 10 July as the eye passed over. Dennis continued north-northwestward after landfall, with the center moving across th e western Florida Panhand le into southwestern Alabama before it weakened into a tropical st orm. In the United States, Dennis produced widespread heavy rainfall along the track from the western Florida Panhandle to the Ohio Valley. Total rainfall at Camden, Alabama, was 32.5 cm while Monticello, Florida, reported 17.65 cm An instrumented tower run by the FCMP at Navarre measured 1-min average winds (5-m elevation) of 44.24 m/s at 19:21 UTC on July 10th. Another FCMP tower at the Pensacola Airport measured 1-min average winds (10-m elevation) of 37 m/s just west of the eye at 19:46 UTC. Coastal Marine Automated Station (C-M AN) at Sand Key, Florida, reported 10-min PAGE 222 222 average winds (13.1-m elevation) of 27.78 m/s at 08:20 UTC on July 9th, while the C-MAN station at Sombrero Key, Florida, reported 2-min average winds (48.5-m elevation) of 33 m/s at 08:00 UTC on July 9th. A National Ocean Service station at Panama City Beach, Florida, reported 6-min average winds (6.1-m elevation) of 26.24 m/s at 18:00 UTC on July 10th. Table 618 shows the features of Hurricane Dennis. Fi gure 6-82 shows the best track of central barometric pressure and wind speed history for Hurricane Dennis by NHC. Dennis produced a storm surge of 1.8 2.1 m above normal tide levels on Santa Rosa Island near where the center made landfall. This surge over-washed Santa Rosa Island near and west of Navarre Beach. Figur e 6-83 shows Santa Rosa Island impacted by Hurricane Dennis from USGS Hurricane and Extreme Storm Impact Studies6. Left panel of Figure 6-83 shows over-washed Pensacola Beach, FL, pre and post Hurricane Dennis. Over-wash during Hurricane Dennis extends landward of the Gulf-front ro ad. Right panel of Figure 6-84 shows building damage and over-wash at Navarre Beach, FL. Ov er-wash deposits resulting from Hurricane Dennis cover the main road with sand and extend landward into the bay. Overwash in those areas will be compared numerical results. A storm surge of 1.8 2.8 m above normal tide levels occurred in Apalachee Bay, Florida, which inunda ted parts of the town of St. Marks and other nearby areas. This surge was likely triggered by an oceanic trapped shelf wave that propagated northward along the Florida west coast. Table 6-19 shows storm surge values at several locations during Hurricane Dennis. Dennis is directly resp onsible for 3 deaths in the United States. The American Insurance Services Group estimates the insured property damage in the United States at $1.12 billion. Based on a doubling of this figur e to account for uninsured property damage, the total U. S. damage estimate for Dennis is $2.23 billion. 6 http://coastal.er.usgs.gov/hurricanes PAGE 223 223 .. Figure 6-80 Best Hurricane De nnis Track from NOAA NHC Figure 6-81 Track of Hurricane Dennis with HRD wnd field PAGE 224 224 Figure 6-82 Best track of central barometric pressure and wind speed history for Hurricane Dennis (NHC, 2005). Table 6-18 Selected storm surge values at different location during hurricane Dennis event State Location Storm surge ( m ) Time(UTC) Panama City Beach 1.74 7/10 18:00 Pensacola 1.27 7/10 19:00 Florida Apalachicola 2.12 7/10 17:00 Louisiana SW Pass 0.39 7/10 23:06 Biloxi 0.67 7/10 19:23 Mississippi Waveland 0.51 7/10 22:54 Table 6-19 Summary of hurricane Dennis features in the US Value Location Time(UTC) Maximum wind 44 m/s Navarre, FL 19:21 7/10/05 Min pressure 956.3 mb Pensacola, FL 19:43 7/10/05 Rain fall 33 cm Camden, AL Landfall Santa Rosa Island, FL 19:30 7/10/05 Highest storm surge 2.12 m Apalachicola, FL Death tall 3 people Economic damage $2.23 billion Date(July2005,UTC) B arome t r i c P ressure ( m b ) WindSpeed(kt) 5 6 7 8 9 10 11 12 13 14 880 900 920 940 960 980 1000 1020 1040 0 20 40 60 80 100 120 140 160 180 BarometricPressure WindSpeed Landfall PAGE 225 225 Figure 6-83 Hurricane Dennis Impact Area Santa Rosa Island, FL (Map form USGS Hurricane Impact Study) Figure 6-84 Aerial photos of over-washed Pensacola, FL (left) and Navarre Beach, FL (Photos from USGS Hu rricane Impact Study) PAGE 226 226 Computational Domain The path of Hurricane Dennis was similar to Hurricane Ivan. The two Hurricanes made landfall with category 3. As seen Figure 6-85, Hurricane Ivan traveled just right side of Hurricane Ivans path. Their impacts on coastal areas were similar. Because of their similarity, computational domain for Hurrica ne Dennis was used for simu lation of Hurricane Dennis. Boundary Conditions Like Hurricane Ivan simulation, a large amount of various data was collected for simulation of Hurricane Dennis. WNA, HRD wind data were used for wind fields. NOAA predicted tidal input and ADCIRC surge elev ation at the open boundary were used. For comparison of simulation results, water elevation da ta were collected at measured station. Table 6-20 shows measured data stations. Tidal forcing As seen in sensitivity tests, when NOAA pred icted tide was used as tidal input, the model produced more accurate surge elevations. Hence, NOAA predicted tide data was used as tidal forcing at the tidal open boundary. In order to get storm surge elevation at the open boundary, ADCIRC was used. Figure 6-86 and Figure 6-87 show NOAA predicted tide and ADCIRC surge elevation at three different open boundary elemen ts for Hurricane Dennis, respectively. Because Hurricane Dennis approached to right side of th e computational domain, the surge elevation at right side was higher than eleva tion at left side (Figure 6-87) Wind and pressure Similar to simulations of Hurricane Ivan, combined WNA and HRD wi nd fields were used. Analytical pressure was used for atmospheri c pressure input. Unfo rtunately, WINDGEN wind fields and pressure are not unavailable. Because Hurricane Dennis is very similar to Hurricane Ivan, same grid, topography and bathymetry were used for the simulations. As discussed in PAGE 227 227 Hurricane Ivan simulation, combined HRD, WNA wind fields and analytical pressure were used as inputs of the model. Table 6-20 shows measured data stations. In order to get best simulati on results, wind fields were modified including limited land exposure according NLCD land cover specificati on (Figure 6-65). Figure 6-88 shows combined WNA and HRD wind field at tim e 7/10/05 19:20 (UTC). Figure 6-89 shows wind field near Escambia Bay, FL at same time. The simulate d wind and pressure were compared measured data. Figure 6-89 and Figure 6-90 show the co mparison with simulated and measured data. Unlike Hurricane Ivan, the maximum wind radius of Hurricane Dennis is compact. The error of simulated wind is less than 10 %. Figure 6-85 Hurricane Ivan and Dennis Path, Red: Dennis, Black: Ivan PAGE 228 228 Figure 6-86 NOAA predicted tide at open boundary for Hurricane Dennis Figure 6-87 ADCIRC surge elevation at open boundary for Hurricane Dennis PAGE 229 229 Table 6-20 Measured data st ations of hurricane Dennis Station Location Data State Station Name Lon ( oW) Lat ( oN) Wind Pressure Water elevation Data Source LA Port Fourchon 90.20 29.11 COOPS Waveland 89.37 30.28 COOPS MS Biloxi 88.90 30.41 COOPS AL Dauphin Island 88.07 30.25 COOPS Pensacola 87.21 30.40 COOPS Panama City B each 85.88 30.21 COOPS FL Panama city 85.67 30.15 COOPS Figure 6-88 Hurricane Denis wind field in com putational domain (upper) and around Escambia Bay (lower), FL at 19:20, July 10th, 2005 (UTC) PAGE 230 230 Figure 6-89 Comparison of wind speed and direction for Hurricane Dennis at A: Pensacola, FL, B: Panama City Beach, FL, C: Waveland, D: Dauphin Island Air pressure for Dennis is calculated from th e analytical Holland Model solution based on official hurricane track, maximum wind radius and deficit air pressure. Figure 6-91 shows comparison between simulated and measured air pressure time series for Hurricane Dennis. Because the spatial pressure variation in the domain is totally based on the Holland model input and assumes uniform, the pressure variation far away from the Hurricane center was not fully developed. Therefore, the simula ted pressure at stat ions close to the hurricane center shows A B D C PAGE 231 231 better comparison than stations that far away fr om the hurricane center. For example, simulated air pressure at station Waveland is not good compar e to other stations. However, as seen in the governing equations, the pressure do es not effect on the surface elev ation. The spatial gradient of pressure modifies water surface el evation. The trend of pressure difference is similar. Therefore this absolute pressure differences between simu lated and measured data have much less effects on simulated water. Figure 6-92 shows Holland an alytical pressure distribution of Hurricane Dennis when it made landfall at time 7/10/05 19:30 (UTC). Figure 6-90 Comparison of simulated and measured wind vector TIME(JULIANDAY)8 189 190 191 192 193 194 1 Measured WindstickatPensacola,FL 5m/s WNA Measured Simulated TIME(JULIANDAY)189 190 191 192 193 194 Measured WindstickatDauphinIsland,AL 5m/s WNA Measured Simulated TIME(JULIANDAY)189 190 191 192 193 194 Measured WindstickatBiloxi,MS 5m/s WNA Measured Simulated PAGE 232 232 Figure 6-91 Air Pressure comparison for Hurrica ne Dennis at A: Waveland, B: Biloxi; C: Pensacola, D: Panama City Beach. Figure 6-92 Holland analytical atmosphe ric pressure of Hurricane Dennis. A B C D PAGE 233 233 Simulation Result Water surface elevation Hurricanes Dennis was simulated with comb ined HRD and WNA winds and Holland air pressure. NLCD bottom friction cl assification was used for fric tion parameters. Simulation was performed 5-days from 188 Julian day (07/08/04, 00:00 UTC) to 194 Julian day (07/13/04, 24:00 UTC) which is 2 days before and 2 days after hurricane Ivans landf all day. 300 seconds (5 minutes) time step was used. Simulated water elevations were compared with measured water elevation at four different meas ured stations in Figure 6-93. All the other simulated versus measured water elevation results during Hurrica ne Dennis are shown in Appendix G. As shown in the figures, simulated water surface elevations with Hurricane Dennis wind field were slightly underestimated at all four locations. In Fi gure 6-93 A is comparison with simulated and measured water surface elevation at Pensacola, FL. Among the 4 measured stations, simulated water surface elevation at Panama City Beach (Figure 6-92 B) was most underestimated by 0.6 m. The simulated peak elevation was before the measured peak. Overall, simulated water surface elevation was lower at all stations. This is because the model UFDVM does not have wave induced setup. Figure 6-94 show s water elevation entire computational domain during the Hurricane Dennis event. As expected, water surface elevation is always hi gher at right hand side of Dennis track. Flood level and maximum inundation map For Hurricane Dennis, there is no High Water Mark available. Figure 6-95 A shows the simulated maximum inundation map created by Hurricane Dennis. Figure 6-95 B shows the maximum inundation map around Escambia Bay, FL. Since Hurricane Dennis was weaker than Hurricane Ivan, the extension of maximum inundation is less than that of Hurricane Ivan. The maximum inundation occurred under the I-10 Bridge around 2.63 meters at time 19:30 7/10/2005 PAGE 234 234 (UTC). Figure 6-96 shows the Hurricane Dennis impact area from USGS Hurricane Dennis Impact Study (2005). The maximum inundation height of the imp act area was around 0.5 to 2.5 meters. In contrast Hurricane Ivan, Hurricane Dennis is slight weaker than Hurricane Ivan. The Hurricane Dennis impacted less area than Hurricane Ivan. Inundated elevation is less than 3 meters by the present model. As seen water elevation comparison with measured data, the present model underestimated the water surface elevation by 20 50 cm by absence of wave model. Figure 6-93 Comparison between simulated and measured water surface elevation at A: Pensacola, B: Panama City Beach, C: Panama City, D: Dauphin Island A B C D PAGE 235 235 Figure 6-94 Simulated water surface elevation of Hurricane Ivan from time 13:40 22:10 07/10/2005 (UTC) PAGE 236 236 Figure 6-95 Maximum inundation map and created by Hurricane Dennis Figure 6-96 Hurricane Dennis impact areas a nd maximum inundation map around Escambia Bay, FL A B PAGE 237 237 CHAPTER 7 CONCLUSION AND FUTURE STUDY The goal of this study was to develop a 3-di m ensional numerical model with orthogonal unstructured mesh for coastal and estuarine ci rculation named UFDVM (Unstructured Finite Difference and Volume Method). The model was im plemented with combined finite difference and finite volume scheme. To eliminate time step constraint for non-linear advection terms in the governing equations, the ELM (Eul erian-Lagrangian Method) nume rical scheme was used. Wave propagation term was implemented using so-called -scheme which is stable in value between 0.5 and 1.0. If value is 0.5 then the scheme is semi -implicit method. In this case the higher order terms of wave can be captured. If value 1.0 is used, the scheme is fully implicit. By implementing with ELM for advection term, -scheme for propagation term, the model can be used any time increment. In addition to elimin ation of time constraint conservation is also important. To take advantage of exact numeri cal conservation, the c ontinuity equation was implemented by finite volume method. By adva ntage of unstructured mesh system, complex coastal area was well resolved. Na tural coastal boundaries are very complex. While other models such as CH3D, POM use transformation of equa tion for fit complex boundari es in the horizontal directions, the model developed in this study uses unstructured orthogonal mesh system without any transformation of governing equations. With triangles and quadrilater als, the computational mesh follows complex boundaries well. Moreover, mesh refinement creates higher resolution of computational grid where the areas finer resolutions are needed. Unlike non-orthogonal unstructured mesh generation, orthogonal mesh ge neration requires some additional efforts. In this study, Voronoi triangulation technique was used to create orthogonal computational meshes. In the vertical directi on, Z-grid system which is not necessa rily uniform was used in the model UFDVM. With Z-grid system, the model saved computational time in 3-dimensional simulation PAGE 238 238 because the number of vertical grid cells are less in shallow wa ter, vice-versa. However, the model requires one big surface layer. This reduces 3-D to 2D in shallow water or inundation areas. The present model also has the ability to treat wetting and drying processes over the tidal flat or inundation areas. Before applying model to real simulations, an alytical test is essential to verify correctness of the model. Numerical solutions of the model UFDVM developed in this study were tested and compared with analytical solutions by term by term. Extensive analytical test showed the models numerical solutions were very good agreem ent with analytical solutions. After successfully performed a series of analytical tests, the model applied to one of the most devastating coastal phenomena, Hurricane-induced storm surge i nundation of seawater. The two major hurricanes, Hurricane Ivan (2004) and Dennis (2005) were simulated by UFDVM developed in this study. In hurricane-induced storm surge simulations, the most important factor in parameters is hurricane wind field. A thorough analysis of the WNA, WINDGEN, HRD and Holland wind fields were performed to simulate Hurrica ne Ivan and Dennis. While WINDGEN and Holland analytical model have wind and atmospheric pressure, WNA and HRD model have wind field only. WINDGEN and Holland model cover both s ea and land, WNA covers only sea side. HRD wind fields cover around Hurri cane eye. Usually, wind stre ss conversion uses 10-minute averaged wind. All of the wind models are 1-mi nute averaged data. All of the 1-minute wind fields were converted 10-minut e ones. HRD model has land redu ction effects, others not. According to USGS, NLCD classification, wind fields were modi fied to produce best results. The analysis revealed that results from both at mospheric models compared well with measured wind speed and direction. While WINDGEN mode l is commercial WNA and HRD wind fields are maintained by NOAA. Considering all th e above factors, WNA and HRD winds are PAGE 239 239 combined to get best wind model. Wind field are provided with large time intervals (WINDGEN: 1 hour, WNA: 3 hours, HRD: 3-6 hours). However, time increment of numerical model is much smaller than those. Consequently interpolation of wind field with time and space is another important factor. The time and space Lagrange interpolation technique was used to improve the accuracy between wind snapshots, especially around hurricane centers. For simulation of Hurricane Ivan and Dennis, combined WNA and HRD wind field with Lagrange interpolation. The analytical Holland model is used to generate atmospheric pressure based on NHC best track, maximum wind radius and pressure deficit. For storm surge simulation, coarse and fine meshes were generated based on GEODAS coastal line data. The fine mesh resolved complex coastal areas well. For topography a nd bathymetry in the computational domain, GEODAS, SEAMLESS, and USGS Coastal Relief data base were used. Bathymetry and topography data were converted to NAVD. After calibrating wind field, atmospheric pr essure, bottom friction, UFDVM applied to storm surge simulation with Hurricane Ivan (2004) and Dennis (2005). As open boundary tidal forcing, NOAA predicted tide data was used. ADCIRC model was used to get surge elevations at the open boundary. Simulated water surface elev ations by the model were compared with measured data stations. In addition to water surface elevations, maximum inundation map was drawn thanks to drying wetting feat ure of the model. Hurricane Iv an was a classical, long-lived hurricane that reached Category 5 and made landfall as a categor y 3 storm on the east side of Mobile Bay west of Gulf Shores. Ivan caused significant flooding and damage at the coastal region. One of damage was collapse of I-10 Br idge over the Escambia Bay, FL. The highest surge of around 4 m was observed at the north end of Escambia Bay and was simulated by UFDVM. Overall comparison of surge el evation was successful although the model PAGE 240 240 underestimated the level little bit. The maximum surge errors are around 10 15% at most stations. The maximum inundation map was created and compared to High Water Mark (HWM). The most flooded domains were found to the east of Pensacola Bay and to the east of Black Water Bay. Model UFDVM simulated storm surg e by Hurricane Ivan well with good sets of bathymetry and topography, accurate wind fi eld, and open boundary data. According to procedures of Hurricane simulati ons, Hurricane Dennis (2005) also simulated. The features of Hurricane Dennis are very similar to those of Hurricane Ivan. Although the two hurricanes are same category 3 at landfall, Dennis has smaller sc ale than Ivan. Hence, the wave had on effects on surge small. Simulated water elevations were compared with measured data better than those of Ivan. To scrutinize the differences between simulate d and measured data and sensitivity to surge elevation by different parameterizations, a bunch of sensitivity test was performed. Wind field was most sensitive to surge elevation among the pa rameters. Four different of wind data sets were considered including HR D wind, WNA wind, WINDGEN wi nd, and analytical Holland wind without any modification of the wind fields. Different surge levels were obtained from each of the four wind fields; sometimes the differences were quite big at seve ral stations. However, the trend of differences was sa me for the different data sets WINDGEN produced the highest surge elevation. This is because WINDGEN wind filed does not have any land reduction to eh wind field and covers both sea and land sides. Land reduces wind by diff erent land types. Land reduction effect on wind field is important for domains where wind blow from inland. After modifying wind with limited land reduction, the simu lated surge elevations reduced little bit. The atmospheric pressure also had on effect on the su rge elevation. The differences were occurred around where hurricane eye was locate d. This is because pressure gradient are quite big around PAGE 241 241 Hurricane eye. Sensitivity of tidal forcing at the open boundary was investigated. There are two kinds of tidal forcing data. One is ADCIRC tidal constituents and NOAA predicted tide. NOAAs predicted tide produced better results than ADCIRC tidal constituents. This is because ADCIRC uses 8 components of tidal constituen t; NOAA predicted tide uses 32 constituents including nonlinear ones. However the differences between the two were not big. The dimensionality of the model was not effect th e surge elevation significantly. The biggest difference in 2and 3-D model is the use of different formula of bottom friction. While 2-D uses Manningn, 3-D model log layer bottom friction. Because bottom friction of 2and 3-D is very small the results of simulated su rge elevation are also small. Although the differences in surge elevation were small, if vertical structures of flow are needed, 3-D simulation should be used. To test how coastal configuration of computational domain changes st orm surge level, some part of barrier islands were eliminated from the domain. Sensitivity test showed coastal configuration of computational domain does not effect on the surge level. The second most sensitivity factor to storm surge elevation was the existence of wave in the model. Because UFDVM model does not have wave components in the governing equati ons, the simulated surge elevations were compared with the results of CH3D model with wave effects. Wa ve sensitivity test revealed waves had significant effect on the surge level. Comparison with CH3D m odel with wave effect showed about 0.5 0.9 meter differences in surge level without waves. Time is one of critical factors for storm surge simula tions. UFDVM model ran with two different time increment, 5 minutes and 15 minutes. The effects of different tim e increment were not discernable. Therefore, bigger time increment can be used. The total run time with 5-minute was 6 hours for 5-day simulation, 15-minute 4 hours. As final sensitivity test, two different mesh size of computational domain were tested. The differences in simulated water surface elevation between the two PAGE 242 242 meshes were not significant. However, the i nundation maps produced by the two meshes were quite different. Because coarse mesh cannot resolve the complex coastal line, the coastal lines were exaggerated. This modi fies topography of the computational domain over the inundation lands. Hence, the model produced different inun dation results between the coarse and fine meshes. From the simulations of the two major Hu rricane Ivan and Dennis, the model UFDVM developed in this study can be used to simulate hurricane-induced stor m simulations. The model simulates wetting and drying of water over th e land good. Although the simulated water surface elevation was underestimated, the overall comparison with measur ed data was good within error range of 10 15 %. Maximum inundation map produced by UFDVM, the results can be used to maintain the flood protection plan and desi gn human-protective coastal structures. According to the results of Hurricane Ivan and Dennis, the model UFDVM developed in this study needs some more work to simula te storm surge model well in the future. The inclusion of wave setup process is obvious. Both Hurricane Ivan and Dennis simulation, the model underestimated the peak of storm surge. This came from the absence of wave model in the present mode l. Coupling of SWAN wave model will be one of choices. CH3D has wave model with SWAN coupling. UFDVM is 3D model. However, the surface layer should keep all the surface elevation changes in a one level as disc ussed in finite difference a nd volume discretization. This reduces 3-D model into 2-D model in shallow water areas. To resolve shallow water fine, or S-Z grid in vertical direc tion should be used. Extension of or S-Z grid in vertical direction of UFDVM model is required to resolve vertical structures of shallow water areas. PAGE 243 243 UFDVM model uses orthogonal unstructured meshes. Ge neration of orthogonal unstructured mesh requires some extra work. One of options of this extra work should improve algorithm itself. Fully finite volum e method does not require orthogonality of computational mesh. However, the will be some more computational cost in computations. To simulate transport process, the trans port equation should be implemented in the model. Although the present model does not have time step constrai nt, parallel implementation of the model will speed up simulation. PAGE 244 244 APPENDIX A SAFFIR-SIMPSON HURRICANE SCALE The Saffir-Sim pson Hurricane Scale is a 1-5 ratin g based on the hurricanes present intensity. This is used to give an estimate of the poten tial property damage and flooding expected along the coast from a hurricane landfall. The scale was fo rmulated in 1969 by Herbert Saffir, a consulting engineer, and Dr. Bob Simpson, director of the National Hurricane Center. Table A-1 The Saffir-Simpson hurricane scale Category Wind speed (mph) Description Examples 1 74-95 No real damage to buildings. Damage to unanchored mobile homes. Some damage to poorly constructed signs. Also, some coastal flooding and minor pier damage. Irene 1999 Allison 1995 2 96-110 Some damage to building roofs, doors and windows. Considerable damage to mobile homes. Flooding damages piers and small craft in unprotected moorings may break their moorings. Some trees blown down. Bonnie 1998, Georges 1998 Gloria 1985 3 111-130 Some structural damage to small residences and utility buildings. Large trees blown down. Mobile homes and poorly built signs destroyed. Flooding near the coast destroys smaller structures with larger structures damaged by floating debris. Terrain may be flooded well inland. Keith 2000, Fran 1996, Opal 1995, Alicia 1983 Betsy 1965 4 131-155 More extensive curtain wall failures with some complete roof structure failure on small residences. Major erosion of beach areas. Terrain may be flooded well inland. Hugo 1989 Donna 1960 5 156 and up Complete roof failure on many residences and industrial buildings. Some complete building failures with small utility buildings blown over or away. Flooding causes major damage to lower floors of all structures near the shoreline. Massive evacuation of residential areas may be required. Andrew 1992, Camille 1969 Labor Day 1935 PAGE 245 245 APPENDIX B FORMULAE TO CALCULATE ERRORS 2 1 1 1RMSError MeanAbsoluteError MaximumAbsoluteErrormax ErroratPeakN ii i N ii i ii iN peakpeakSM N SM N SM MS (B. 1) where S and M are simulated and measured values, respectively. PAGE 246 246 APPENDIX C BEST TRACKS FOR HURRICANES IVAN AND DENNIS The following tables show best tracks for Hurricanes IVAN and DENNI S. The information was obtained from NOAA and contains time, lati tude/longitude position, pressure in the middle of the storm, maximum wind sp eed, and storm stage according to the Saffir-Simpson Hurricane Scale. Table C-1 Best track for hurricane Ivan, 2 24 September 2004 Position Date/Time (UTC) Lat ( N) Lon ( W) Pressure (mb) Wind Speed (kt) Stage 02 / 1800 03 / 0000 03 / 0600 04 / 0600 04 / 1200 05 / 0600 05 / 1200 06 / 0600 06 / 1200 07 / 0600 07 / 1200 08 / 0600 08 / 1200 09 / 0600 09 / 1200 10 / 0600 10 / 1200 11 / 0600 11 / 1200 12 / 0600 12 / 1200 13 / 0600 13 / 1200 14 / 0600 14 / 1200 15 / 0600 15 / 1200 16 / 0000 16 / 0600 16 / 1200 16 / 1800 17 / 0000 17 / 0600 17 / 1200 9.7 9.7 9.7 8.9 8.9 9.5 9.8 10.8 11.0 11.3 11.6 12.3 12.6 13.7 14.2 15.7 16.2 17.4 17.7 18.4 18.8 19.9 20.4 22.4 23.0 25.6 26.7 28.9 30.0 31.4 32.5 33.8 34.7 35.4 27.6 28.7 30.3 36.5 38.2 43.4 45.1 50.5 52.5 57.8 59.4 64.1 65.5 69.5 70.8 73.8 74.7 77.6 78.4 80.4 81.2 83.5 84.1 85.6 86.0 87.4 87.9 88.2 87.9 87.7 87.4 86.5 85.7 84.0 1009 1007 1005 997 997 987 977 950 955 965 963 946 955 925 919 930 934 923 925 915 919 920 915 924 930 93 5 939 931 943 965 975 986 991 994 25 30 35 50 50 65 85 110 110 95 100 120 120 140 140 125 125 130 125 135 135 140 140 140 125 120 115 110 105 70 50 30 25 20 tropical depression tropical storm hurricane tropical storm tropical depression Continue on next page PAGE 247 247 17 / 1800 18 / 0600 18 / 1800 19 / 0000 19 / 0600 19 / 1200 19 / 1800 20 / 0000 20 / 0600 20 / 1200 20 / 1800 21 / 0000 21 / 0600 21 / 1200 21 / 1800 22 / 0000 22 / 0600 22 / 1200 22 / 1800 23 / 0000 23 / 0600 23 / 1200 23 / 1800 24 / 0000 24 / 0600 24 / 1200 12 / 0000 13 / 2100 16 / 0650 24 / 0200 21 / 0600 21 / 1200 21 / 1800 22 / 0000 22 / 0600 22 / 1200 22 / 1800 23 / 0000 23 / 0600 23 / 1200 23 / 1800 24 / 0000 24 / 0600 24 / 1200 12 / 0000 13 / 2100 16 / 0650 24 / 0200 36.2 37.7 38.0 37.5 36.0 34.5 32.8 31.0 29.0 27.5 26.4 26.1 5.9 25.8 25.2 24.8 25.1 26.0 26.5 27.1 27.9 28.9 29.2 29.6 30.1 18.2 21.2 30.2 29.8 80.6 81.7 82.8 84.1 86.1 87.3 88.6 89.5 91.0 92.2 92.7 93.2 94.2 79.6 84.8 87.9 83.6 82.3 78.5 75.5 74.0 74.0 74.5 75.8 77.5 78.5 78.7 79.1 79.7 80.6 81.7 82.8 84.1 86.1 87.3 88.6 89.5 91.0 92.2 92.7 93.2 94.2 79.6 84.8 87.9 83.6 80.6 81.7 82.8 84.1 86.1 87.3 88.6 89.5 91.0 92.2 92.7 93.2 94.2 79.6 84.8 87.9 83 .6 996 998 1002 1003 1005 1008 1008 1008 1008 1009 1009 1009 1009 1009 1010 1010 1010 1010 1008 1007 1007 998 1003 1003 1009 910 910 946 1004 1009 1009 1010 1010 1010 1010 1008 1007 1007 998 1003 1003 1009 910 910 946 1004 20 15 25 35 35 35 35 35 35 30 25 25 25 25 25 25 25 25 30 35 35 50 40 30 25 145 140 105 30 25 25 25 25 25 25 30 35 35 50 40 30 25 145 140 105 30 extratropical low tropical depression tropical storm tropical depression dissipated inland minimum pressure minimum pressure 1st U.S. landfall, Pine Beach, AL, 2nd U.S. landfall,Holly Beach, LA" low tropical depression tropical storm tropical depression di ssipated inland minimum pressure minimum pressure 1st U.S. landfall, Pine Beach, AL, 2nd U.S. landfall,Holly Beach, LA PAGE 248 248 Table C-2 Best track for hurricane Dennis, 4 18 July 2005 Position Date/Time (UTC) Lat ( N) Lon ( W) Pressure (mb) Wind Speed (kt) Stage 04 / 1800 05 / 0000 05 / 0600 05 / 1200 05 / 1800 06 / 0000 06 / 0600 06 / 1200 06 / 1800 07 / 0000 07 / 0600 07 / 1200 07 / 1800 08 / 0000 08 / 0600 08 / 1200 08 / 1800 09 / 0000 09 / 0600 09 / 1200 09 / 1800 10 / 0000 10 / 0600 10 / 1200 10 / 1800 11 / 0000 11 / 0600 11 / 1200 11 / 1800 12 / 0000 12 / 0600 12 / 1200 12 / 1800 13 / 0000 13 / 0600 13 / 1200 13 / 1800 14 / 0000 14 / 0600 14 / 1200 14 / 1800 15 / 0000 15 / 0600 15 / 1200 15 / 1800 16 / 0000 12.0 12.2 12.5 13.0 13.6 14.3 14.7 15.1 15.6 16.2 16.7 17.6 18.5 19.4 20.3 20.9 22.0 22.7 23.4 24.3 25.2 26.1 27.2 28.5 29.9 31.5 32.6 33.9 35.3 36.4 37.1 37.7 38.1 38.5 38.9 39.2 39.2 39.2 39.0 38.7 38.4 38.1 37.9 38.1 38.4 38.6 60.8 62.5 64.2 65.9 67.3 68.5 69.7 70.9 71.9 73.0 74.1 74.9 76.1 77.1 78.4 79.5 80.6 81.6 82.5 83.4 84.2 85.0 85.8 86.3 86.9 87.7 88.5 88.8 89.1 89.2 89.0 88.7 88.3 87.8 87.2 86.5 85.8 85.7 85.6 85.6 85.6 85.9 86 .2 86.4 86.6 86.8 1010 1009 1008 1007 1005 1000 995 991 989 982 972 967 957 951 953 938 941 960 973 967 962 942 935 930 942 970 991 997 1002 1003 1005 1007 1008 1009 1010 1010 1010 1009 1009 1010 1010 1009 1010 1012 1012 1011 25 30 30 35 40 45 50 55 60 70 80 90 100 120 110 130 120 100 75 80 90 110 125 120 110 45 30 25 20 20 15 15 15 15 15 15 15 10 10 10 10 10 10 10 10 10 tropical depression tropical storm " hurricane " " " tropical storm tropical depression " " remnant low " " C ontinued on next page PAGE 249 249 16 / 0600 16 / 1200 16 / 1800 17 / 0000 17 / 0600 17 / 1200 17 / 1800 18 / 0000 18 / 0600 18 / 1200 04 / 2100 08 / 0245 08 / 1845 10 / 1930 10 / 1200 08 / 1200 39.4 40.2 40.8 41.3 42.2 43.1 43.9 44.6 45.8 12.1 19.9 22.1 30.4 28.5 20.9 86.5 86.2 85.2 84.1 83.2 82.3 81.4 80.5 79.8 61.6 77.6 80.7 87.1 86.3 79.5 1013 1014 1014 1013 1013 1013 1012 1010 1009 1009 956 941 946 930 938 10 10 10 10 10 10 10 10 10 30 120 120 105 120 130 continued from previous page " " absorbed by large landfall, Grenada landfall, Punta del Ingles, Cuba landfall, Punta Mangles Altos, Cuba landfall, Santa Rosa Island, Florida minimum pressure maximum wind PAGE 250 250 APPENDIX D WIND SPEED AND DIRECTION DURING HURRI C ANE IVAN: WINDGEN AND WNA VS. MEASURED Figure D-1 Comparison of simula ted and measured wind speed WindSpeedatEastbank,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME(JULIANDAY)W I N D S P E E D ( m / s )258 260 262 264 0 5 10 15 20 25 30 35 40 WindSpeedatEastbank,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME(JULIANDAY)W I N D S P E E D ( m / s )258 260 262 264 0 5 10 15 20 25 30 35 40 WindSpeedatPilotstation,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME ( JULI A ND A Y ) W I N D S P E E D ( m / s )258 260 262 264 0 5 10 15 20 25 30 35 40 WindSpeedatPilotstation,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME ( JULI A ND A Y ) W I N D S P E E D ( m / s )258 260 262 264 0 5 10 15 20 25 30 35 40 WindSpeedatWaveland,MSSolidline:Measuredwindspeed Dashedline:WINDGENwindspeed TIME(JULIANDAY)W I N D S P E E D ( m / s )258 260 262 264 0 5 10 15 20 25 30 35 40 WindSpeedatWaveland,MSSolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME(JULIANDAY)W I N D S P E E D ( m / s )258 260 262 26 4 0 5 10 15 20 25 30 35 40 PAGE 251 251 Figure D-2 Comparison of simula ted and measured wind stick TIM E(JULIA NDA Y) 258 260 262 264 15m /s W indspeedanddirectionatEastBank,LA Solidline:m easuredw ind Dottedlind:W INDGENw ind W indspeedanddirectionatEastbank, A LSolidline:m easuredw ind Dashedlind:W NAw ind TIM E(JULIA NDA Y)258 260 262 264 15m /s W indspeedanddirectionatPilot_Station,LA Solidline:m easuredw ind Dottedlind:W INDGENw ind TIME(JULIA NDA Y) 258 260 262 264 15m /s W indspeedanddirectionatPilotstation,L A Solidline:m easuredw ind Dashedlind:W NAw ind TIME(JULIA NDA Y)258 260 262 264 15m /s W indspeedanddirectionatW aveLand,M S Solidline:m easuredw ind Dottedlind:W INDG ENw ind TIM E(JULIA NDA Y) 258 260 262 264 15m /s W indspeedanddirectionatWaveland,M SSolidline:m easuredw ind Dashedlind:W NAw ind TIM E(JULIA NDA Y)258 260 262 264 15m /s PAGE 252 252 APPENDIX E COMPARISON OF MEASURED AND SI MULATED WATER ELEVATION DURING HURRICANE IVAN Figure E-1 Comparison of measured and simulated water elevation at Grand Isle, LA Figure E-2 Comparison of measured and simula ted water elevation at Pilot Station, LA PAGE 253 253 APPENDIX F WIND SPEED AND DIRECTION DURING HURRI C ANE DENNIS: WNA VS. MEASURED Figure F-1 Comparison of simulated and measured WNA wind speed and direction HurricaneDennis WindSpeedatEastbank,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed T I M E ( J U L I A N D A Y ) W I N D S P E E D ( m / s )188 190 192 194 0 5 10 15 20 25 30 WindspeedanddirectionatEastbank,LA Solidline:Measuredwind Dashedline:WNAwind TIME(JULIANDAY)188 190 192 194 15m /sHurricaneDennis HurricaneDennis WindSpeedatPilotstation,LASolidline:Measuredwindspeed Dashedline:WNAwindspeed TIME(JULIANDAY)W I N D S P E E D ( m / s )188 190 192 194 0 5 10 15 20 25 30 WindspeedanddirectionatPilotstation,LA Solidline:Measuredwind Dashedline:WNAwind TIME(JULIANDAY)188 190 192 194 15m /sHurricaneDennis PAGE 254 254 APPENDIX G COMPARISON OF MEASURED AND SI MULATED WATER ELEVATION DURING HUR RICANE DENNIS Figure G-1 Comparison of measured and simulated water elevation TIME(JULIANDAY)W a t e r e l e v a t i o n ( m N A V D )188 190 192 194 -1 -0.5 0 0.5 1 1.5 2 2.5 3HurricaneDennis WaterElevationatDauphin_Island,MS Redline:Measured Blueline:Simulated T IME ( JULI A ND A Y ) W a t e r e l e v a t i o n ( m N A V D )190 192 194 -1 -0.5 0 0.5 1 1.5 2 2.5 3HurricaneDennis WaterElevationatWaveLand,MS Blueline:Simulated Redline:Measured T IME ( JULI A ND A Y ) W a t e r e l e v a t i o n ( m N A V D )188 190 192 194 -1 -0.5 0 0.5 1 1.5 2 2.5 3HurricaneDennis WaterElevationatEastBank,LA Redline:Measured Blueline:Simulated TIME(JULIANDAY)W a t e r e l e v a t i o n ( m N A V D )188 190 192 194 -1 -0.5 0 0.5 1 1.5 2 2.5 3HurricaneDennis WaterElevationatGrandIsle,LA Blueline:Simulated Redline:Measured PAGE 255 255 APPENDIX H DESCRIPTION OF UFDVM SUBROUTINES AND MODULES The m odel UFDVM developed in this study has subroutines. In this appendix explains the subroutines 1. UFDVM_MAIN.f90 This is main program of the model UFDVM. 2. UFDVM_TIME_STAMP.f90 This subroutine calculates total program execution time, cpu time. 3. UFDVM_INITIALIZE_VARIABLES.f90 All the variables includi ng arrays are initialized in this subroutine. 4. UFDVM_READ_VERTICAL_LAYER.f90 Read vertical layer data from fort.300 and set vertical layers. 5. UFDVM_READ_PARA_AND_MESH.f90 In this subroutine, mesh depth file fort.100 and program control file, fort.200 are read. After reading mesh data, mesh geometry (element area, face number, adjacent element number, face length, boundary nodes and element, water depth at face, etc.) is determined. 6. UFDVM_CALCULATE_CORIOLIS_FACTOR.f90 Coriolis parameters are calculated in this subroutine 7. UFDVM_SETUP_COLD_START.f90 This subroutine sets up cold start according to program control parameter. If simulation is hot start, this subroutine r ead all the hot start va riables from fort.1000. 8. UFDVM_SETUP_HOT_START.f90 This subroutine sets up hot start according to program control parameter. If simulation is hot start, this subroutine read all th e hot start variables from fort.1000. PAGE 256 256 9. UFDVM_UPDATE_VERTICAL_LAYERS.f90 After calculating a new total water depth, new vertical layers are updated according to total water depth and vertical layer depth. It usually determines the surface layer number and bottom layer number with wetting and drying 10. UFDVM_CALCULATE_VELOCITIY_AT_NODE.f90 UFDVM program calculate water velocity at element faces. According to water velocity at the element face, this subroutine interpol ates and calculates water velocity at nodes. 11. UFDVM_MOVIE_FILE_OPEN_ASCII.f90 This subroutine opens output field data file with ASCII data format according to control parameter. If mesh dimension is big, the size of output file will be huge. In this case, BINARY file format will be good choice. 12. UFDVM_MOVIE_FILE_OPEN_BINARY.f90 Output file format is BINARY. Sa me as UFDVM_MOVIE_FILE_OPEN_ASCII.f90 13. UFDVM_TIME_SERIES_FILE_OPEN.f90 This subroutine opens time series output file in Tecplot format. 14. UFDVM_JULIAN_TO_LOCALTIME.f90 In this subroutine Julian day is converted in local time (UTC) 15. UFDVM_JULIAN.f90 In this subroutine local time is converted to Julian day 16. UFDVM_READ_WIND_AND_PRESSURE_DATA.f90 This subroutine reads wind and atmospheric pressure data from fort.22 according to time step. The atmospheric pressure da ta are converted to milibar to N/m2. PAGE 257 257 17. UFDVM_SETUP_SPINUP.f90 If spin up flag is on, this subroutine turn s on the spin up procedures the spin up by ramp function. 18. UFDVM_CALCULATE_BOTTOM_FRICTION.f90 This subroutine calculates bottom friction 19. UFDVM_AIR_PRESSURE.f90 The atmospheric pressure term is calculated. The atmospheric pressure is read several data stations. 20. UFDVM_AIR_PRESSURE_ALL_NODES.f90 Same as subroutine UFDVM_AIR_PRESSURE.f90. The data used in this subroutine are fort.22 21. UFDVM_SETUP_WIND.f90 Setup wind related parameters. 22. UFDVM_CALCULATE_WIND_STRESS.f90 The input wind u and v velocities at 10 meters above the water level are converted to wind stress according Garett formula. 23. UFDVM_COMPUTE_BO UNDARY_VELO.f90 This subroutine calculates the velocity component at open and closed boundaries. 24. UFDVM_SOLVE_NONLINEAR_ADVECTION.f90 Non linear advection term in the governing equation is solved by Eulerian-Larangian numerical method. 25. UFDVM_SOLVE_MOMENTUM_EQUATION.f90 PAGE 258 258 The momentum equation is solved in this subroutine 26. UFDVM_COMPUTE_ETA.f90 The new water elevations at the Voronoi cen ter of the elements are computed by finite volume method and updated. The finite volu me methods ensure volume conservation. 27. UFDVM_CALCULATE_VELOCITY_AT_FACE.f90 After calculating the new water elevation, wate r velocity at element faces is determined. 28. UFDVM_CHECK_CONSERVATION.F90 This subroutine check conservation of the model 29. UFDVM_MAKE_MOVIE_FILE_ASCII.f90 After all the calculations, this subroutine outpu ts all the field data with ASCII file for Tecplot 30. UFDVM_MAKE_MOVIE_FILE_BINARY.f90 Same as UFDVM_MAKE_MOVIE_FILE_ASCII.f90 except write BINARY file format. 31. UFDVM_MAKE_TIME_SERIES_FILE.f90 Write individual time series output of time series variables, for examples, 2 and 3D velocity, water elevation. 32. UFDVM_TRIDIAGONAL_SOLVER.F90 The tri-diagonal matrix come from the mome ntum equation is solv ed by Tri-Diagonal Matrix Algorithm (TDMA or Thomas algorithm) 33. UFDVM_LINEAR_INTERPOLATION.f90 This subroutine interpolates variables by inve rse distance interpolation algorithms. This subroutine used for interpolated wind and atmospheric pressure interpolation with time and space. PAGE 259 259 34. UFDVM_HOLLAND_STORM_SURGE.f90 In this subroutine, Holland storm surge anal ytical model is calculated for storm surge simulations. Modules There are two modules in the program fo r defining variables and function interface 1. UFDVM_MODULE_VARIABLES.F90 This module defines all the variable s and arrays needed in the program 2. UFDVM_FUNCTION_LIBRARY.f90 This is interface for defining some functions for example; calculate area of elements, converts longitude and latitude to meter scale. PAGE 260 260 APPENDIX I FLOW CHART OF THE MODEL UFDVM The m odel developed in this study named UFDVM consists of three parts of computational procedures. The first part sets up coefficients, initializes all th e arrays, and reads a mesh depth file and open boundary conditions a nd wind data. The second part is to compute the numerical solutions of the present model. The last part is to write the computational results. After all the coefficients and arrays are set up, the program checks if a stor m surge simulation flag is on. According to storm surge simulation type, the program setups wind and atmospheric pressure. The model first solves the advection equations using Eulerian-Lagrangian scheme. After solving advection terms, then the horiz ontal diffusion terms are calculate d by the finite volume method. The bottom friction terms are calculated by explicit numerical method. All the above explicit terms are combined and set up the tri-diagona l matrix and solved by tri-diagonal matrix algorithm (TDMA or Thomas algorithm). After solving momentum equation, a system of sparse matrix is established to calculate new water elevati on. Then the system of sparse matrix is solved by preconditioned conjugate gradient method. On ce new water elevation is obtained, three dimensional water velocities at element faces are calculated. Af ter obtaining water elevation and water velocity, all the variables should be updated and dry and wet element are determined by total water depth. Following is the flowchart of mode l UFDVM developed in this study. PAGE 261 261 Figure I-1 Flow chart of UFDVM model (continue) BEGIN PROGRAMUFDVM_MAIN Set up program : read mesh and setup simulation environment UFDVM_TIME_STAMPUFDVM_INITIALIZE__VARIABLESURDVM_READ_MESH_AND_PARAUFDVM_SETUP_VERTICAL_LAYERS If wind and atmospheric pressure terms are included turned on UFDVM_SETUP_WIND Which storm model is used1 = Wind and pressure data are read from fort.222 = Holland analytical storm model UFDVM_READ_WIND_AND_PRESSURE_DATA UFDVM_HOLLADN_STORM_SURGE Start simulation UFDVM_CALCULATE_WIND_STRESS UFDVM_CALCULATE_BOTTOM_FRICTION PAGE 262 262 Figure I-1 Flow chart of UFDVM model UFDVM_CALCULATE_BOUNDARY_VELO Calculate nonlinear advection term UFDVM_SOLVE_NONLINEAR_ADVECTION UFDVM_SLOVE_MOMENTUM_EQUATION UFDVM_COMPUTE_ETA UFDVM_CALCULATE_VELOCITY_AT_FACE UFDVM_UPDATE_VERTICAL_LAYERS Output resultsUFDVM_WRITE_MOVIE_FILEUFDVM_WRITE_TIME_SERIES_FILES End simulation UFDVM_TIME_STAMPEnd simulation NO YES YESNO PAGE 263 263 APPENDIX F MANUAL OF UFDVM MODEL This document is the m anual for the UFDVM model developed in th is study. There are three mandatory input files to run the model, fort.100, 200, and 300. This manual explains the three main input files for the model. 1. Mesh file: fort.100 Polygons as well as face of a mesh stored in a file named fort.100 must be stored in an appropriate way. This is required to reasons of efficiency and correctness. The mesh will be generated by mesh generator and must be orthogonal. The following orderings must be fulfilled. Mesh nodes: 1. For each polygon (triangle or/and quadrilateral) the nodes must be given in counter-clockwise orientation (program will stop if this is not the case). 2. No sorting is required fo r the global mesh node numbers. Polygons (triangle or / and quadrilateral): 1. For reasons of efficiency and accuracy, a red-black-ordering of sparse matrix of the polygons is recommended: (a) The red polygons must become the first [1, Npr] ones out of the Np polygons ; (b) the black polygons must be numbered [ Np + 1, Np] if no red-black ordering is used, Npr = Np must hold true. 2. If there are Np polygons along the open boundary where th e water level shal l be prescribed, they must be assigned the lowermost numbers [1, Np]. The mesh file is sub-divided into th e mesh node section, polygon section, boundary section which includes all the land, island, wa ll, and open boundary sections. Final line is the modification of water depth section Mesh node section: PAGE 264 264 1. The first three lines: mesh file description or mesh title 2. The fourth line: Mesh coordinate system 1 = Cartesian coordinate 2 = Longitude and Latitude spherical coordinate 3. The fifth and sixth line: Mesh dimension Total number elements in mesh, Total number of nodes 4. The seventh to tenth lines: Comments lines 5. Next line is mesh nodes data 1st column: node number 2nd and 3rd columns: X and Y coordinate of nodes Unit: meter for Cartesian and degree for spherical 4th column: water depth at the node Unit: meter After the mesh node section, next is polygon of mesh element section Polygon section 1st column: Element number 2nd column: shape of polygons ; if 3 = Triangles 4 = Quadrilaterals The rest columns: Node number of each mesh If shape of polygon is triangle, three node numbers are needed If shape of polygon is quadrilateral, four node numbers should be specified Boundary section PAGE 265 265 Boundaries include land, island, wall, river, and open boundaries. Open boundary includes tidal and river boundaries. Land, island, and wall are classi fied as closed land boundaries. The format of the boundary section is following 1st line of boundary section: Total nu mber of open boundary segment along the boundary of the computational domain. 2nd line: Total number of open boundary nodes along the open boundary segments. 3rd line: Total number of open boundary nodes at each open boundary segment. From the next line, the 1st column is the open boundary node number and 2nd column is serial number of open boundary for justification Next line is the land, is land, and wall boundaries. 1st line of land boundary section: Total number of land boundary segment along the boundary of the computational domain. 2nd line: Total number of land boundary nodes along the land boundary segments. 3rd line: Total number of open boundary nodes at each land boundary segment. From the next line, the 1st column is the land boundary node number and 2nd column is serial number of land boundary for justification Modification of water depth section If users want to modify water depth at el ement faces, users must specify the face number and water depth at the face in a file fort.150. This is for the case of river banks and thin walls. Note that this is not water depth at nodes but water depth at element faces. If 1 then modify the water depth at faces 0 then no modificatio n of water depth at face PAGE 266 266 The following is an example of fort.150 file Format of fort.150 Line 1: header of fort.150 Line 2 -: 1st column: face number of mesh (integer) 2nd column: modified water depth (real) FACE_NO WATER_DEPTH_AT_FACE 780 -15.0 781 -15.2 783 -15.5 784 -15.2 881 -10.5 883 -10.6 884 -12.8 885 -12.3 An example of Input mesh file fort.100: The following is an example of com putational mesh node, polygons, and boundary. Mesh file description MESH TITLE: IVAN MESH FOR UFDVM, ADAPTED FROM CH3D QUAD GRIDS Mesh Coordinate system 1: Ca rtesian, 2: Lon, Lat system 1 NO_OF_ELEMENTS NO_OF_NODES 68729 69466 END OF TOTAL NUMBER OF ELEMENTS AND MODE NODAL DATA NODE_NUMBER X(I) Y(I) WATER DEPTH(M) 1 0.2269141E+06 0.3172216E+07 0.8158200E+02 2 0.2287896E+06 0.3172415E+07 0.8230000E+02 3 0.2306545E+06 0.3172615E+07 0.8246700E+02 4 0.2325079E+06 0.3172812E+07 0.8244200E+02 5 0.2343496E+06 0.3173009E+07 0.8272100E+02 6 0.2361821E+06 0.3173204E+07 0.8349900E+02 7 0.2380031E+06 0.3173399E+07 0.8374800E+02 8 0.2398109E+06 0.3173592E+07 0.8454100E+02 continued PAGE 267 267 69458 0.5090578E+06 0.3396395E+07 -0.5647600E+01 69459 0.5095996E+06 0.3396554E+07 -0.5299600E+01 69460 0.5101411E+06 0.3396712E+07 -0.5626700E+01 69461 0.5106829E+06 0.3396871E+07 -0.6154100E+01 69462 0.5112248E+06 0.3397030E+07 -0.6776400E+01 69463 0.5117663E+06 0.3397188E+07 -0.3295000E+02 69464 0.5123082E+06 0.3397347E+07 -0.4230100E+02 69465 0.5128498E+06 0.3397506E+07 -0.4262400E+02 69466 0.5136624E+06 0.3397744E+07 -0.4262400E+02 END OF NODAL DATA ELEMENT DATA ELE_NO SHAPE FACE1 FACE2 FACE3 FACE4 1 4 1 2 445 444 2 4 2 3 446 445 3 4 3 4 447 446 4 4 4 5 448 447 5 4 5 6 449 448 6 4 6 7 450 449 7 4 7 8 451 450 continued 68718 4 69324 69325 69455 69454 68719 4 69325 69326 69456 69455 68720 4 69326 69327 69457 69456 68721 4 69327 69328 69458 69457 68722 4 69328 69329 69459 69458 68723 4 69329 69330 69460 69459 68724 4 69330 69331 69461 69460 68725 4 69331 69332 69462 69461 68726 4 69332 69333 69463 69462 68727 4 69333 69334 69464 69463 68728 4 69334 69335 69465 69464 68729 4 69335 69336 69466 69465 END OF ELEMENT DATA TIDAL, RIVER AND ISLAND S BOUNDARY SECTION 1 NO_OF_OPEN_BND_SEGMENT 1 TOTAL NUMBER OF TIDAL OPEN BOUNDARY SEGMENT(S) 67 NO_OF_OPEN_BND_NODE 67 NO_OB_NODE_AT_EACH_SEGMENT(K=1) 100 1 101 2 102 3 106 4 108 5 PAGE 268 268 continued 164 65 165 66 167 67 END OF TIDAL, RIVER BOUNDARYT DATA DATA: LAND BOUNDARY INFORMATION 1 : TOTAL_ NO_OF_LAND_BND_SEGMENT 1472 : TOTAL_NO_OF_LAND_BND_NODE 1472 : NO_LAND_BND_AT_EACH_SEGMENT(K) = 1 50284 1 51975 2 52666 3 53357 4 54048 5 55739 6 56430 7 continued 63357 1469 64048 1470 65739 1471 66430 1472 I_WATER_DEPTH_MODIFICATION_FLAG 1 2. Program Control Input Data File: fort.200 The input data file is a file to control all the input parameters and options for the program run. Line 1: Program description Line 2: Model name: UFDVM Line 3, 4, and 5: Start time of the simulation PAGE 269 269 Column 1: Simulation start year (integer) Column 2: Simulation start month (integer) Column 3: Simulation start year (integer) Column 4: Simulation start day (integer) Column 5: Simulation start hour (integer) Column 6: Simulation start minute (integer) Column 3: Simulation start second (real) Line 6 and 7: Column 1: Hot or cold start: if 0 = cold start, 1: hot start Column 2: Output to the screen file flag: if 0 = do nothing 1 = output simulati on process to the screen Column 3: Output to a file flag: if 0 = do nothing 1 = output simulation pr ocess to a file named fort.500 Column 4: Output to a file flag: if 0 = do nothing 1 = output simulation pr ocess to a file named fort.500 Line 8 and 9: Column 1: screen output interv al(integer): screen output step Column 2: screen output flag: if 0 = do nothing 1 = output simulati on process to the screen PAGE 270 270 Line 10 and 11: Column 1: Mesh input file coordinate system flag: (integer) if 0 = Spherical coordinate system 1 = Cartesian coordinate system Column 2: Mean longitude of computational domain (real) Column 3: Mean latitude of computational domain (real) Line 12 and 13: Column 1: Total number of simulation day(s) (real) Column 2: Time increment (delta T) in second (real) Line 14 and 15 Column 1: Spin-up flag for tide (integer) if 0 = Do nothing 1 = Spin-up time for tide Column 2: Spin-up period for tide in days (real) Line 16 and 17: Implicitness of theta method (real) The value of theta should be between 0.5 and 1 0.5 = semi-implicit method, 1.0 = implicit method Line 18 and 19: Transport model flag (integer) if 0 = Turn off the transport equation 1 = Turn transport equation ( Salinity simulation) Line 20 and 21: Baroclinic a nd barotropic flag (integer) Column 1: Baroclinic flag if 0 = Baroclinic term is off PAGE 271 271 1 = Baroclinic term is on Column 2: Barotropic flag if 0 = Barotropic term is off 1 = Barotropic term is on Line 22 and 23: Parameters for barotropic simulation Column 1: Mean temperature (real) Column 2: Maximum temperature (real) Column 3: minimum salinity (real) Column 4: Maximum salinity (real) Line 24 and 25: Advection term flag (integer) if 0 = Advection term is off 1 = advection term is on and ELM scheme applied to nonlinear term Line 26 and 27: ELM back-tracking flag and number of sub-iteration of ELM Column 1: ELM tracking flag (integer) if 0 = No ELM 1 = ELM backtracking is on Column 2: Number of subiteration for ELM (integer) Column 3: Number of mini mum iteration for ELM (integer) Column 4: Number of maxi mum iteration for ELM (integer) Line 28 and 29: Critical dry depth for wetting and drying (real, unit: meter) Line 30 and 31: Bottom friction flag Column 1: Bottom fricti on flag option (integer) if 0 = Constant bottom friction will be used for the simulation PAGE 272 272 1 = Logarithmic bottom friction is applied If variable bottom friction is used, user s need a file specified bottom friction coefficient (fort.33) Line 32 and 33: Coriolis term flag Column 1: Coriolis term option (integer) If -1 = Constant Coriolis parameter will be sued 0 = Variable Coriolis parameter will be sued Column 2: Latitude for defining Coriolis parameter (Degree, real) Column 3: Coriolis parameter for constant Coriolis term (real) The next section is for wind and atmospheric pre ssure flag for simulations. This flags include storm surge simulation based on various wind and pressure data sets. Line 34 and 35: wind and pressure term flag Column 1: Wind and pressu re input flag for storm surge simulation (integer) If 0 = No wind and pressure term is considered in simulation 1 = Wind and pressure term will be considered. Wind and pressure data will be read from. Following is format of wind and pressure data. The 1st column is node number, 2nd and 3rd columns are u and v components of wind velocity with m/s and 4th column is atmospheric pressure unit of milibar. 1 17.06876755 -19.23584938 1013.00000000 2 17.21857452 -19.35668755 1013.00000000 3 17.36643600 -19.48005295 1013.00000000 4 17.51576424 -19.60027313 1013.00000000 5 17.66893959 -19.71701813 1013.00000000 6 17.81451416 -19.84219170 1013.00000000 7 17.96371269 -19.96085930 1013.00000000 8 18.11673546 -20.07708931 1013.00000000 PAGE 273 273 Column 2: wind and pressure data input time interval in second (real) The wind and pressure data are read w ith this time interval. Intermediate wind and pressure data are then interpolated in the program with time and space. Line 36 and 37: Julian day of wind and pressure input Column 1: Start Julian day of calcula tion of wind and pressure term (real). Column 2: End Julian day of calculati on of wind and pressure term (real). Wind and pressure terms are considered between the start and end Julian days Line 38 and 39: Storm el evation input at open bo undary flag (integer) If 1: Storm elevation data is read from fort.61. fort.61 file is obtained from ADCIRC model. Strom data (meter) should be read at every time step of UFDVM run. If 0: No storm elevation data are read at the open boundaries Following is format of fort. 61 storm elevation data file. Line 1, 2, and 3: ADCIRC header From Line 4: 1st column: serial number of open boundary mesh number 2nd column: strom elevation data at open boundary Example of fort.61 data file eastcoast_95d_ll_select.grd 4320 67 0.3000000E+03 5 1 86700.0000000000 2890 1 5.484932900000000E-002 2 3.951800100000000E-003 3 5.442489200000000E-002 4 4.934874500000000E-002 5 4.856100100000000E-002 PAGE 274 274 continue 60 3.004638600000000E-002 61 3.883277400000000E-002 62 4.609906900000000E-002 63 5.022676100000000E-002 64 5.659150600000000E-002 65 5.263936400000000E-002 66 5.959315200000000E-002 67 6.108589300000000E-002 Line 40 and 41: Storm elevati on input interval and total num ber of storm elevation mesh 1st column: Storm elevation data input interval with second unit (real). This should be exactly same as time increment of the UFDVM model. 2nd column: Total number of storm elevation mesh number (integer) Line 42 and 43: Total number of stor m elevation boundary segment (integer) Line 44 and 45: Storm elevation boundary segment number (integer) along the boundary Line 46 and 47: Total number of storm el evation mesh at each segment (integer) Line 48 and 49: Wind only simulation flag This flag is used when wind data are provided only limited wind stations. Wind data are interpolated with time and space. 1st column: wind only simulation flag (integer) if 0: no wind simulation 1: Constant wind field is applied to the whole domain 2: Time and space varying wind data are applied to whole doamin 3: Constant wind is applied to some wind stations 4: Time and space varying wind data are applied to some wind stations 5: Time and space varyi ng wind data are read from fort.22 2nd column: Wind data read time interval (real) PAGE 275 275 3rd column: Total number of wind stations (integer) Line 50 and 51: X, and Y C oordinate of wind stations 1st and 2nd column: X, and Y Coordinate of wind stations (real) Line 52 and 53: Julian day of wind data input Column 1: Start Julian day of calculation of wind (real). Column 2: End Julian day of calculation of wind (real). Line 54 and 55: Spin-up flag for wind simulation Column 1: Spin-up flag for wind simulation (integer) if 0: No spin-up is applied to wind 1: Spin-up time is applied for wind Column 2: Spin-up period for wind simulation in day(s) (real) Line 56 and 57: Atmospheric pressure simulation flag This flag is used when atmospheric pressure is provided only limite d pressure stations like wind simulation. 1st column: Atmospheric pressure simulation flag (integer) if 0: no atmospheric simulation 1: Read atmospheric pressure data from some data stations. Atmospheric data will be read from data file fort.640 2: Time and space varying atmospheric pressure data are applied to whole domain. Data will be read from fort.640 2nd column: Number of atmospheric pressure stations Line 58 and 59: X, and Y Coordinate of atmospheric pressure data stations 1st and 2nd column: X, and Y Coordinate of atmospheric pressure data stations (real) PAGE 276 276 Line 60 and 61: Julian day of atmospheric pressure data input Column 1: Start Julian day of calculation of pressure (real). Column 2: End Julian day of calculation of pressure (real). Line 62 and 63: Holland storm su rge simulation flag (integer) if 0: No Holland storm surge simulation 1: Holland storm surge model will be applied to model. In this case fort.650 data file is needed to supply storm or Hurricane data. Data format of fort.650 data file 1st line: Data header 1st Column: Year (integer) 2nd Column: Month (integer) 3rd Column: Day (integer) 4th Column: Hour (integer) 5th Column: Minute (integer) 6th and 7th Column: X, Y coordinate of Hurricane or storm center 8th Column: Latitude of Hurricane or storm center 9th Column: Pressure drop in Hg Pressure drop is the differen ce between standard atmospheric pressure and pressure at the center of storm or Hurricane 10th Column: Maximum wind radius in miles Maximum wind radius is the distance from storm center to the location where maximum wind is located. PAGE 277 277 Example: fort.650 YEAR MM DD HH MM X Y LAT DP(in hg) MWR(nmi) 2004 09 13 00 00 941021.60 2161556.00 19.50 -2.86 86.95 2004 09 13 03 00 898474.80 2182743.00 19.70 -2.81 85.78 2004 09 13 06 00 866525.30 2204229.00 19.90 -2.75 85.22 2004 09 13 09 00 823985.90 2236642.00 20.20 -2.81 87.26 2004 09 13 12 00 802680.10 2258418.00 20.40 -2.89 91.83 2004 09 13 15 00 770998.80 2280044.00 20.60 -2.95 100.52 continue 2004 09 17 09 00 691876.30 3863966.00 34.90 -.59 83.06 2004 09 17 12 00 772457.40 3921537.00 35.40 -.56 58.48 Line 64 and 65: Julian day of Holla nd storm surge simulation data input Column 1: Start Julian day of Holland model (real). Column 2: End Julian day of Holland model (real). Line 66, 67, and 68: Matrix solver option and parameters Column 1: Matrix solver (integer) If 1: Conjugate gradient (CG) method 2: Successive over-relaxation (SOR) method Column 2: Maximum allowed number of iteration (integer) Column 3: Temporary flag, not used (integer) set zero Column 4: Zeta value (real) set to 5.0E-6 Column 5: Tolerance for c onvergence (real) set to 1.0E-13 Next section is for Turbulence option. Curre ntly constant eddy viscosity is used. Line 69 and 70: Turbulence option (integer) Set to 0: constant eddy viscosity Line 71 and 72: Values of eddy viscosity PAGE 278 278 Column 1: constant vertical eddy viscosity Column 2: Constant vertical eddy diffusivity Line 73 and 74: Horizontal eddy viscosity Column 1: Constant eddy viscosity Column 2: Smagorinsky parameter Next section is for tidal open boundary condition input. Line 75 and 76: Earth tide parameters Column 1: Total number of earth tidal potential species Column 2: Cut-off depth for earth tide (meter) If water depth is less than the cutoff depth for earth tide, earth tide is not considered Line 77 and 78: Earth tide species data Column 1: Earth tidal species (integer) Column 2: Amplitude of earth tide (meter, real) Column 3: Frequency of the earth tide (real) Column 4: Nodal factor of the earth tide (real) Column 5: Astronomical phase of the earth tide (degree, real) Next section is tidal open boundary forcing parameters Line 79 and 80: Total number of tidal constituent considered at the tidal open boundary (integer). Line 81 and 82: Periods of tidal constituents Column 1: Period of M2 tide (12.40) Column 2: Period of S2 tide (12.00) Column 3: Period of K1 tide (23.70) PAGE 279 279 Column 4: Period of O1 tide (25.80) Column 5: Period of K2 tide (11.97) Column 6: Period of N2 tide (12.70) Column 7: Period of Q1 tide (26.90) Column 8: Period of P1 tide (24.10) Line 83, 84, and 85: Tidal am plitude of the constituents Column 1: Amplitude of M2 tide Column 2: Amplitude of S2 tide Column 3: Amplitude of K1 tide Column 4: Amplitude of O1 tide Column 5: Amplitude of K2 tide Column 6: Amplitude of N2 tide Column 7: Amplitude of Q1 tide Column 8: Amplitude of P1 tide Line 86, 87, and 88: Tidal Phase la g of the constituents (degree) Column 1: Phase lag of M2 tide Column 2: Phase lag of S2 tide Column 3: Phase lag of K1 tide Column 4: Phase lag of O1 tide Column 5: Phase lag of K2 tide Column 6: Phase lag of N2 tide Column 7: Phase lag of Q1 tide Column 8: Phase lag of P1 tide PAGE 280 280 Line 89 and 90: Total number of open boundary segment. This is just for checking the correctness of the number of open boundary segments and should be same number as defined in fort.100 mesh depth file (integer). Next section is the data for open bou ndary types including tide and river Line 91 and 92: Types of open boundary Column 1: Total number of open boundary element (integer) Column 2: Type of water elevation forcing (integer) If 1: this boundary is forced by constant water elevation 2: This boundary of forced by tidal constituent Column 3: Boundary flow type (integer) If 1: This boundary is forced by constant river discharge 2: This boundary is for ced by time varying river discharge Line 93 and 94: Tidal constituents name used in simulation (1 byte character) M2 S2 K1 O1 K2 N2 Q1 P1 Line 95 and 96: Tidal amplitudes and phase lag for constituents used in simulation Column 1: Amplitude of M2 tide if used in simulation Column 2: Amplitude of S2 tide if used in simulation Column 3: Amplitude of K1 tide if used in simulation Column 4: Amplitude of O1 tide if used in simulation Column 5: Amplitude of K2 tide if used in simulation Column 6: Amplitude of N2 tide if used in simulation Column 7: Amplitude of Q1 tide if used in simulation Column 8: Amplitude of P1 tide if used in simulation PAGE 281 281 Column 9: Phase lag of M2 tide if used in simulation Column 10: Phase lag of S2 tide if used in simulation Column 11: Phase lag of K1 tide if used in simulation Column 12: Phase lag of O1 tide if used in simulation Column 13: Phase lag of K2 tide if used in simulation Column 14: Phase lag of N2 tide if used in simulation Column 15: Phase lag of Q1 tide if used in simulation Column 16: Phase lag of P1 tide if used in simulation Column 17: Serial number of tidal open boundary element number (integer) Column 18: Actual number of element number (integer) Nest section describes for output Line 97 and 98: Output parameters Column 1: ASCII movie flag if 0: Do nothing 1: output file is stored by ASCII format Column 2: Binary movie flag (integer) If 0: Do nothing 1: Output file is stored by Bi nary file. This saves a lot of storage Line 99 and 100: Ou tput frequency Column 1: Start iteration number for movie file (integer) Column 2: End iteration numb er for movie file (integer) Column 3: Option for movie file if integer number: movie file stored at this number of iteration PAGE 282 282 Column 4: Option for constant movie file if integer number: Movie file stored at every this number of iteration Line 101 and 102: Individual da ta station number for output Column 1: Total number of data station for output (integer) Column 2: Frequency of data output (integer) Line 103 and 104: Time series individual data station mesh number (integer) Line 105 and 106: Name of data station ( character set). Must be character string Line 107 and 108: Total number of wind and atmospheric pressu re data station for output Line 109 and 110: Wind and pressure data station node number Next section is output for 3 di mensional velocity time series Line 111 and 112: Vertical layer number for output time series of 3 D velocity Column 1: Surface layer nu mber for output (integer) Column 2: Middle layer number for output (integer) Column 3: Bottom layer number for output (integer) Example of fort.200 input file: HURRICANE IVAN SIMULATION WITH WNA WIND ONLY WITHOUT PRESSURE UFDVM_CIRCULATION MODEL START_TIME_OF_SIMULATION SIMULATION START YEAR MONTH DAY HOUR MINUTE SECOND 2004 9 13 0 0 0.0 I_HOT_OR_COLD_START, I_OUTPUT_FILE_FLAG, I_WRITE_TO_FILE_FLAG 0 0 0 0 I_SCREEN_OUTPUT_INTERVAL, I_SCREEN_OUTPUT_FLAG 1 0 I_COORD_FRAME_FLAG,ANGLE_MEAN_L ONGITUDE_OF_DOMAIN,ANGLE_MEAN_ LATITUDE_OF_DOMAIN 1 0.0 0.0 SIMULATION_DAY DT(IN SECONDS) 7.0 300.0 PAGE 283 283 I_SPINUP_FLAG_FOR_TIDE, SPINUP _PERIOD_FOR_TIDE_IN_DAY 0 0.1 THETA FOR IMPLICITNESS OF PROPAGATION TERM 1.0 I_ADVECTION_FLAG 1 I_ELM_BACKTRACE_FLAG,NUM_SUB_IT ERATION,NUM_MIN_ELM_ITER,NUM_MA X_ELM_ITER 0 1 1 100 DRY_DEPTH 0.01 I_BOTTOM_FRICTION_OPTION, CONST_BOTM_FRIC_COEF 0 0.0 I_CORIOLIS_FLAG, CORIOLIS_LATIT UDE_DEGREE, CORIOLIS_COEFFICIENT -1 29.5 6.732294781D-5 I_WIND_PRES_FLAG, WIND_PRES_INPUT_TIME_SEC 1 10800.0 WIND_PRES_START_JDAY, WIND_PRES_END_JDAY 257.0 264.0 I_STORM_ELEVATION_AT_BOUNDARY_FL AG,STORM_ELEVATION_INPUT_TIME_S EC,NO_WIND_PRES_STORM_OB_ELEMENT 1 300.0 67 I_WIND_FLAG, WIND_INTERVAL_SECOND, NO_OF_WIND_STATION 0 3600.0 0 WIND_STATION_COORD_X2(I), WIND_STATION_COORD_Y2(I= NO_OF_WIND_STATION) WIND_START_JDAY, WIND_END_JDAY I_SPINUP_FLAG_FOR_WIND, SPINUP_PERIOD_FOR_WIND_IN_DAY !(IF I_WIND_FLAG > 0 THEN SPECITY VALUES FOLLOWING LINE) I_AIR_PRESSURE_FLAG, NO_OF_AIR_P_STATION 0 9 AIR_P_STATION_COORD_X(I), AIR_P_STATION_COORD_Y(I=NO_OF_AIR_P_STATION) AIR_P_START_JDAY, AIR_P_END_JDAY I_HOLLAND_STORM_SURGE_FLAG 0 HOLLAND_STORM_START_JDAY, HOLLAND_STORM_END_JDAY I_HEAT_MODEL_FLAG, I_SALT_MODEL_FLAG 0 0 INITIAL_TEMPERATURE_FIELD_FLAG, INITIAL_SALINITY_FIELD_FLAG 1 1 I_CHECK_CONSERVATION_FLAG IHCHECK 0 0 NUM_SUBCYCLING_STEP 1 I_DENSITY_FLAG PAGE 284 284 0 MATRIX SOLVER DATA MATRIX_SOLVER MAX_USER_DEFINED _ITERATION IREMOVE ZETA TOLERANCE_FOR_CONVERGE 1 1000 0 5.e-6 1.e-13 I_TURBULENCE_FLAG 0 CONSTANT_VERTICAL_EDDY_VISCOSITY, CONSTANT_VERTICAL_EDDY_DIFFUSIVITY 0.e-2 1.e-4 I_HORIZONTAL_EDDY_VISCOSITY_FLAG, SMAGORINSKY_PARAMETER 0 0.0 I_SPONGE_LAYER_FLAG 0 NO_EARTH_TIDE_POTENTIAL_SPECIES, EARTH_TIDE_CUTOFF_DEPTH 0 40. EARTH_TIDE_SPECIES_J(I),EARTH_TIDE_AMPLITUDE(I),EARTH_TIDE_FREQUENCY (I),EARTH_TIDE_NODAL_FACTOR(I),EARTH_TIDE_ASTRO_ARG_DEGREE(I) NO_OF_TIDAL_CONSTITUENT_AT_OB 7 M2 S2 K1 O1 K2 N2 Q1 P1 12.40 12.00 23.70 25.80 11.97 12.70 26.90 24.10 TIDE_OB_INPUT_AMPLIFICATION_FACTOR M2 S2 K1 O1 K2 N2 Q1 P1 0.96943 1.00000 1.09802 1.15884 1.26481 0.96943 1.15884 1.00000 GREENWICH PHASE LAG IN DEGREE AT TIDAL OPEN BOUNDARY FOR TIDALCONSTITUENT M2 S2 K1 O1 K2 N2 Q1 P1 32.22 0.00 257.94 136.05 335.40 151.19 255.02 97.67 NO_OF_OPEN_BND_SEGMENT_FOR_CHECK 1 NO_OB_ELEMENT ELE_TYPE FLOW _TYPE TEMP_TYPE SAL_TYPE 67 3 0 3 3 TIDAL CONSTITUENT(S) NAME USED M2 S2 K1 O1 K2 N2 Q1 P1 TIDAL AMPLITUDE TIDAL PHASE LAG M2 S2 K1 O1 K2 N2 Q1 M2 S2 K1 O1 K2 N2 Q1 I_SERIAL_ELEMENT_NO_LEFT_TO_RIGHT I_OB_ELEMENT_NUMBER I_OB_INDEX_FOR_COUNT 0.01140 0.00779 0.15078 0.14858 0.00333 0.00386 0.03064 145.587 121.570 16.133 19.248 95.361 150.146 3.345 1 360 1 0.02491 0.01385 0.14864 0.14687 0.00434 0.00518 0.03047 94.239 89.732 17.209 20.207 87.473 102.089 4.326 44 362 2 0.01126 0.00775 0.15074 0.14859 0.00323 0.00379 0.03064 145.097 120.710 16.236 19.354 93.871 149.651 3.361 2 363 3 0.01816 0.00041 0.14171 0.13745 0.00157 0.00566 0.02822 207.697 213.010 18.871 20.906 107.915 167.675 5.044 65 393 4 0.01802 0.00061 0.14175 0.13750 0.00167 0.00570 0.02823 204.430 160.222 18.639 20.718 105.330 165.779 4.828 64 407 5 0.01781 0.00304 0.14395 0.13968 0.00233 0.00594 0.02868 185.181 125.812 17.857 20.074 102.538 156.035 4.138 62 413 6 0.01003 0.00707 0.15087 0.14896 0.00319 0.00337 0.03071 138.992 115.805 16.740 19.735 100.001 147.943 3.720 10 579 7 0.01003 0.00707 0.15087 0.14896 0.00319 0.00337 0.03071 138.992 115.805 16.740 19.735 100.001 147.943 3.720 10 579 8 0.01080 0.00749 0.15034 0.14828 0.00328 0.00362 0.03057 142.914 119.464 16.376 19.456 96.532 149.219 3.521 5 581 9 0.01097 0.00760 0.15034 0.14825 0.00328 0.00367 0.03054 143.172 119.856 16.252 19.350 95.195 149.027 3.433 4 582 10 0.01067 0.00740 0.15046 0.14844 0.00325 0.00359 0.03059 142.723 118.833 16.435 19.499 97.985 149.114 3.537 6 585 11 PAGE 285 285 Continue 0.02911 0.01538 0.14974 0.14721 0.00502 0.00648 0.03050 100.688 92.948 16.577 19.603 89.927 105.523 3.923 50 713 54 0.02873 0.01508 0.14999 0.14729 0.00503 0.00658 0.03050 103.430 94.294 16.563 19.560 90.977 107.828 3.892 51 716 55 0.02810 0.01460 0.15016 0.14731 0.00502 0.00664 0.03049 106.859 96.099 16.571 19.531 92.404 110.631 3.873 52 719 56 0.02698 0.01371 0.15026 0.14721 0.00498 0.00671 0.03044 112.954 99.575 16.621 19.520 95.089 115.349 3.876 53 722 57 0.02534 0.01239 0.15015 0.14685 0.00479 0.00672 0.03033 121.339 103.812 16.715 19.520 97.561 121.323 3.836 54 724 58 0.02333 0.01094 0.14968 0.14617 0.00452 0.00660 0.03015 129.423 106.927 16.793 19.477 98.855 126.788 3.728 55 729 59 0.02197 0.00940 0.14900 0.14526 0.00409 0.00656 0.02994 140.131 111.579 16.905 19.480 99.997 133.014 3.627 56 731 60 0.02110 0.00799 0.14819 0.14426 0.00373 0.00653 0.02969 150.567 116.782 17.060 19.545 101.487 138.739 3.641 57 733 61 0.01991 0.00571 0.14672 0.14251 0.00306 0.00638 0.02929 167.537 124.873 17.350 19.697 102.661 147.595 3.762 59 740 62 0.01948 0.00465 0.14551 0.14126 0.00277 0.00628 0.02901 175.111 129.128 17.574 19.872 103.928 151.420 3.928 60 742 63 0.01759 0.00188 0.14276 0.13852 0.00201 0.00580 0.02843 194.000 129.803 18.240 20.391 104.086 160.433 4.493 63 744 64 0.01818 0.00352 0.14429 0.14004 0.00247 0.00602 0.02875 182.045 127.013 17.778 20.026 102.821 154.654 4.085 61 745 65 0.01863 0.00064 0.14164 0.13736 0.00154 0.00570 0.02820 209.854 242.241 19.062 21.070 110.089 169.130 5.215 66 761 66 0.01949 0.00113 0.14148 0.13714 0.00144 0.00577 0.02817 212.837 256.548 19.211 21.196 111.535 171.186 5.338 67 765 67 *************** END OF DATA FOR OPEN BOUNDARY SEGMENT 1 ******** ASCII_MOVIE_FLAG BINARY_MOVIE_FLAG 0 0 I_START_MOVIE_FREQUENCY_IT I_END_MOVIE_FREQUENCY_IT I_VARYING_MOVIE_FREQUENCY_IN_ITERATION I_CONST_MOVIE_FREQUENCY_IN_ITERATION 846 1152 1 36 TOTAL TIME SERIES STATION I_TI ME_SERIES_FREQUENCY_IN_ITERATION 10 1 TIME SERIES STATION NUMBER(ELEMENT NUMBER) 26163 27812 31335 27029 36800 49528 60429 25004 14459 12756 STATION_NAME(I) 'Port Fourchon,LA' 'Grand Isle,LA' 'East Bank,LA' 'Pilot Station,LA' 'Wave Land,MS' 'Biloxi,MS' 'Dauphin Island,MS' 'Pensacola,FL' 'Panamacity Beach,FL' 'Panamacity,FL' VERTICAL LAYER NUMBER FOR 3D LAYER VELOCITY FIELD SURFACE MIDDLE BOTTOM 5 3 1 3. Vertical layer data input file: fort.300 fort.300 file is for vertical layer data file for UFDVM input It specifies total number of vertical layers and la yer depth (Delta Z). Line 1 and 2: Column 1: Total number of vertical layers (integer) PAGE 286 286 Column 2: Mean sea level (real) Line 3 and 4: Column 1: Layer number (integer) Column 2: Delta Z of each layer (real) Column 3: Z level from the bottom (real) Example of fort.300 file: NO_OF_Z_LAYER, Z_MEAN_SEA_LEVEL 1 2480.0 LAYER_NO_DUMMY, DELTA_Z(I), Z_LEVEL(I) 1 2560.00 2560.00 PAGE 287 287 APPENDIX K EXTENSION OF MODEL FROM Z-GRID TO SIGMA COORDINATE IN VERTICAL DIRECTION To simulate circulation in coastal and es tuarine water bodies with gradual bathymetric variation, it is common to use -transformation such that both the free surface and the bottom become the coordinate surfaces with an equal numb er of coordinate surfaces in between. This transformation, so called -stretching, leads to a smooth repr esentation of the topography with the same order of vertical resolution for the shallow and deeper parts of the water body. The transformation introduces extra terns to the orig inal Cartesian governing equations of motion. However, most of these extra te rms appear in the horizontal diffu sion terms, which are generally not very important compared to the other terms. The -grid model is suitable for simulating flows in regions of gradual bathymetric and t opographic variations and gives a more accurate estimation of bottom stress than Z-grid model, which resolves the depth with stair-step grids. As discussed in Chapter 3, there are two shortcom ings in Z-grid model developed in this study. Those are stair-step in vertical grid and diffi culty of treatment of surface layer. To overcome those weakness in Z-grid model, the use of Sigm a or combined Sigma-Z grid model. In this appendix, the extension of Z-grid to Sigma-grid model is explained. The transformation from the Cartesian grid to vertically-stretched sigma grid is defined as H z h z (K-1) By transformation, continuity and mo mentum equation can be written as: tx udz y vdzhh0 (K-2) PAGE 288 288 u t u u x v u yh u g x A u x u y hz A h u fvh v 2 2 2 21 (K-3) u t u v x v v yh v g x A v x v y hz A h v fuh v 2 2 2 21 (K-4) where is vertical velocity in sigma coordinate defined as: dt d h All the other variables are same as defined in Z-grid system. The new vertical velocity in sigma coordinate is related to w by the following relationship: wu h xx v h yy h tt (K-5) The finite difference discretization for the ve locity component normal to the each vertical face of a prism with staggered grid system can be derived from the Equations K-1 takes the following form: uFug t g t hh t hjk n jk n j ij n ij n ij n ij n oj jm n ijm n ijm n jk n ijk n ijk n mk N jk njz,, (,)(,) (,)(,) ,(,),(,),, (,),(,), ,() 1 2 1 1 1 21 21 211 2 k v jk n jk n jk nj k v jk n jk n jk n oj a ij n a ij nuu h uu h t PP 12 1 11 12 12 1 1 1 12 21 / ,, ,/ ,/ ,, ,/ (,)(,) (K-6) where kmmMjjj n,,, 1. PAGE 289 289 Figure K-1 Definition of stretched coordinates system By using Eulerian Lagrangian scheme, a form for F can be chosen as Fu jk n uftv jk t PP t h h u jkjk aij n aij n j,, *, ,(,),(,) 21 (K-7) The values of uj,k n+1 above the free surface and below the bot tom in Equation K-6 are eliminated by means of the vertical boundary conditions whic h are shear stress. At the free surface, wind shear stresses are used. Using the shear stress, the vertical boundary conditions yield following formulae: jM v jM n jM nT n a n jM nu h uu,/ ,/ 12 1 1 12 111 (K-8) u*x y v* u v 1 H=h + y z 0 PAGE 290 290 jm v jm n jm n jm nB n jm nuu h u,/ ,, ,/ 12 1 1 1 12 11 (K-9) To obtain complete local and global mass conser vation and stability, th e free surface Equation K-1 is discretized by finite vol ume method. Consider a uniform rectangular mesh as shown Figure K-1. To get a semi-implicit finite volume equation, integrate Equa tion K-2 over an area of element Pi, then t xy x udz y vdzxyP hh Pii (K-10) Now, a semi-implicit finite volume di scretization for Equation K-10 gives: yx y uh uh yx x uh uh yx y uh uh yx x uh uh yx tM mk M mk n ij n ij n ij n ij M mk M mk n ij n ij n ij n ij M mk M mk n ij n ij n ij n ij M mk M mk n ij n ij n ij n ij n i n i )4,()4,( )2,()2,( )3,()3,( )1,()1,( 1 )4,()4,( 1 )2,()2,( 1 )3,()3,( 1 )1,()1,( 1)1( (K-11) where x = j(i,2) = j(i,4), and y = j(i,1) = j(i,3) If the equation K-10 is generalized for a ny shape of polygons by using relation K-11, a semi-implicit finite volume discretization for the free surface equation at the center of each polygon is taken to be following form: PPtShu tShuii n ii n iljiljilk n km M jilk n l S iljiljilk n km M jilk n l Si i 11 1 11 ,(,)(,),(,), ,(,) (,),(,),() (K-12) PAGE 291 291 where Pi denotes the area of the i -th polygon, i.e. Pi = x y in Figure K-2 and Si, l is a sign function associated with the orientation of the normal velocity defined on the l -th side of the polygon i Specifically, Si, l = 1 if a positive velocity on the l -th side corresponds to outflow, Si, l = -1 if positive velocity on the l -th side corresponds to inflow to the i -th water column. Thus, Si, l can be written as S ijiliijil ijilijilil,(,),(,), (,),(,), 221 21 (K-13) Equations K-6 and K-12 constitute a linear system of at most NzNs + Np equations. This system has to be solved at each time step in order to calculate the new field variables. The coefficient matrix of these systems is symm etric and positive definite. Thus, the vertical velocity component of the velocity can be readily determined by a direct method i.e. Thomas algorithm. Figure K-2 Numerical stenci l of finite volume method Since a linear system of NzNs + Np equations can be large, th e system of equation K-6 and K-12 is first decomposed into a set of Ns independent tri-diagonal systems of Nz equations. Upon y x i,2 i,3 i,4 i,1 i PAGE 292 292 multiplication by H and after including the boundary c onditions, equations K-6 and K-12 are first written as matrix form as: PPtShU tShUii n ii n il l S jiljil n T jil n il l S jiljil n T jil ni i 1 1 11 1 11 ,(,)(,)(,) ,(,)(,)(,)() (K-14) AUGg t hj n j n j n j ij n ij n j n 1 2 1 1 1 (,)(,) (K-15) where U u u u K K Kj n jM n jM n jm n 1 1 1 1 11 1 1, ,, / / / G HFu t gt u HFu t g HFu t gj n j n jM n j ij n ij n Ta n j n jM n j ij n ij n j n jm n j ij n ij ,( ) ( ) ,( ) ( ) ,( ) (()() ()() ()( 1 1 121 1 12 1 2,))1n A Hata aHaaa aHatj n j n jM n T n jM n jM n j n jM n jM n jM n jm n j n jm n B n ,/ ,/ ,/ ,/,/,/ ,/ ,/ 12 1 12 12 1232 32 12 12 10 0 PAGE 293 293 with at Hjk n jk n jk n,/,/,//1212 12. Formal substitution of the expression for Uj n1 from equation K-14 into K-15 yields a discrete wave equation for n1 which is given by Pgt S hAZ PtShUtSii n iljil jil l S n T jil n ijil n ijil n ii n iljil n T jil n l S iljii i 122 1 1 1 2 1 1 1 12 21 ,(,) (,) (,) (,),(,), ,(,) (,) ,(,()l l S ij nT T jil nihAG)/, (,) 1 12 12 (K-16) Since matrix Aj n is positive definite, its inverse is also positive definite and therefore [(H )TA-1 ]j n is a non-negative number. Hence, equation K-16 constitutes a linear sparse system of Np equations for j n+1. This system is strongly diagonally dominant, symmetric and positive definite. Thus it has a unique solution that can be efficiently determined by a preconditioned conjugate gradient method. Once the new free surface location has been computed, equation K-15 constitutes a linear, tri-diagonal system for Uj n+1. Each of this tridiagonal systems is independent of the others and is symmetric a nd positive definite. Thus, they can be conveniently solved by a direct tri-diagonal algorithm to determine Uj n+1 throughout the computational domain. Finally, a finite volume discretization of the continuity equation K-2 yields the vertical component of the velocity at the new time level. The vertical component of the velocity can be obtained from the incompressibi lity condition, by setting wj n+1 = 0, gives 1 ),,(),( 1 1 1 ,1n klijlij S l li i n j n j n kj n kjuS Pt HHi (K-17) PAGE 294 294 t t HH H uS wn i n i k n i n i k j n i k n klijlij S l li j n i n ki n kii 1 1 1 1 ),,(),( 1 1 1 1 (K-18) Equation K-17 determines the vertical velocity component on each water column recursively. PAGE 295 295 LIST OF REFERENCES 1. Alymov, V. V. Integrated Modeling of Stor m Surges during Hurricanes Isabel, Charley, and Franses, Dissertation, Univ. Of Florida, 2005 2. 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Journal of Hydraulic Engineering Vol. 122, No. 12, 1996 PAGE 305 305 BIOGRAPHICAL SKETCH Jun Lee was born Buan, Korea. Buan is loca ted in Beonsan penins ula conn ected to the Yellow Sea. He was interested in the rise and fa ll of the Yellow Sea level. He attended the Korea Maritime University in Pusan, Korea. He joined military for 3 years. He obtained a Bachelor of Engineering in Ocean Engineeri ng from Ocean Engineering Department in the spring of 1992. He continued to study his Masters degree and earn ed a Master of Engineering from the same Department in Korea Maritime University in the spring of 1994. He remained the University as a assistant researcher in the Un iversity. He joined the Coasta l and Oceanographic Engineering Department at the University of Florida in th e fall of 1996. He earned a Master of Science degree and Ph. D from the Civil an d Coastal Engineering Department. |