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PAGE 1 1 THREE DIMENSIONAL SIMULATION OF WAVE INDUCED CIRCU L ATION WITH A DEPTH DEPENDENT RADIATION STRESS By T IANYI LIU A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 PAGE 2 2 20 10 Tianyi Liu PAGE 3 3 ACKNOWLEDGEMENTS I wish to express my sincere appreciation to my advisor and supervisory committee chairman, Dr. Y. Peter Sheng, for his continuous support and guidance throughout the first two years of my graduate study. I would also like to thank Dr. Alex andru Sheremet fo r reviewing this thesis. I thank Andrew Lapetina, Andrew Condon, Vladimir Paramygin, Bilge Tutak, and Justin Davis for their help and support. Many thanks go to Peng Cheng, Shihfeng Su, Miao Tian, and Yichen Zhang, for their encouragement and assistance. F inally, I wish to express my most sincere gratitude to my family members. PAGE 4 4 TABLE OF CONTENTS p age ACKNOWLEDGEMENTS ................................ ................................ ................................ ............ 3 LIST OF TABLES ................................ ................................ ................................ .......................... 6 LIST OF FIGURES ................................ ................................ ................................ ........................ 7 ABSTRACT ................................ ................................ ................................ ................................ .... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ................. 11 1.1 Background ................................ ................................ ................................ ..................... 11 1.2 Literature R eview ................................ ................................ ................................ ........... 12 1.3 Objectives ................................ ................................ ................................ ....................... 15 1.4 Organization ................................ ................................ ................................ ................... 15 2 METHODOLOGY ................................ ................................ ................................ ................ 17 2.1 A T hree D imensional C irculation M odel CH3D ................................ ...................... 17 2. 2 Local W ave M odel: SWAN ................................ ................................ ............................ 22 2. 3 The C oupling P rocess between CH3D and SWAN ................................ ........................ 24 3 THE FORMULATIONS OF RADIATION STRESS ................................ ........................... 26 3.1 The V ertically I ntegrated R adiation S tress ................................ ................................ ..... 26 3.2 The D epth D ependent R adiation S tress by Xia et al. (2004) ................................ ......... 28 3.3 The D epth D ependent R adiation S tress F ormulation by Mellor (2008) ........................ 29 3.4 The V ertical D istribution of the D epth D ependent R adiation S tress ............................. 29 4 TEST SIMULATIONS ................................ ................................ ................................ .......... 37 4.1 Wave S et up ................................ ................................ ................................ .................... 37 4.2 The U ndertow T est ................................ ................................ ................................ ......... 38 4.3 Th e W ave Set up on F ringing R eef ................................ ................................ ................ 42 4.4 Simulation of Hurricane Isabel ................................ ................................ ....................... 44 5 CONCLUSION ................................ ................................ ................................ ...................... 61 APPENDIX A TRANSFORMATION OF EQUATIONS OF MOTION FROM CARTESIAN GRID TO THE VERTICALLY STRETCHED GRID ................................ ................................ .... 63 PAGE 5 5 B TRANSFORMATION OF THE RADIATION STRESS TERM FROM CARTESIAN COORDINATE TO CURVILINEAR COORDINATE ................................ ........................ 66 C BOUNDARY CONDITIONS ................................ ................................ ............................... 72 Wave Enhanced Surface Stress ................................ ................................ ............................. 74 Wave E nhanced Bottom Stress ................................ ................................ ............................. 75 Wave E nhanced Turbulent M ixing ................................ ................................ ....................... 78 D THE TURBULENCE CLOSURE MODEL ................................ ................................ .......... 80 LIST OF REFERENCES ................................ ................................ ................................ .............. 83 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ........ 87 PAGE 6 6 LIST OF TABLES Table page 3 1 Wave conditions ................................ ................................ ................................ ................ 32 4 1 Locations of measurements and water depth ................................ ................................ .... 46 4 2 The relative RMS error for the simulated current velocities ................................ ............ 46 4 3 The relative RMS error for the simulated turbulent kinetic energy ................................ .. 46 4 4 Studies with different radiation stress formulations ................................ ......................... 47 PAGE 7 7 LIST OF FIGURES Figure page 2 1 The coupling process between CH3D and SWAN ................................ ........................... 25 3 1(a) Distribution of depth dependent radiation stress for the wave condition of Case 1. ........ 33 3 1(b) Distribution of depth dependent radiation stress for the wave condition of Case 2. ........ 34 3 1(c) Distribution of depth dependent radiation stress for the wave condition of Case 3. ........ 35 3 1(d) Distribution of depth dependent radiation stress for the wave condition of Case 4. ........ 36 4 1(a) The comparison between analytical and numeric al solution for the wave set up ............ 48 4 1 ( b ) The cross section of the basin ................................ ................................ ........................... 48 4 2 (a) Comparison between the data and numerical results for wave height .............................. 49 4 2 (b) Comparison between the data and numerical results for wave set up .............................. 49 4 2 (c) T he cross section of the basin. ................................ ................................ .......................... 49 4 3 (a) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using M08 radiation stress ................................ ................................ ........... 50 4 3 (b) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using X04 radiation stress ................................ ................................ ............ 50 4 3 (c) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using LHS radiation stress ................................ ................................ ........... 50 4 4 (a) Comparison between the simulated and measure d current velocity value at Station 1 .... 51 4 4 (b) Comparison between the simulated and measured current velocity value at Station 2 .... 51 4 4 (c) Comparison between the simulated and measured current velocity value at Station 3 .... 51 4 4 (d) Comparison between the simulated and measured current velocity value at Station 4 .... 51 4 4 (e) Comparison between the simulated and measured current velocity value at Station 5 .... 51 4 4 (f) Comparison between the simulated and measured current velocity value at Station 6 .... 51 4 4 (g) Comparison between the simulated and measured current velocity value at Station 7 .... 51 4 5 (a) Wave induced currents simulated by using M08 radiation st ress ................................ ..... 52 PAGE 8 8 4 5 (b) Wave induced currents simulated by using X04 radiation stress ................................ ..... 52 4 5 (c) Wave induced currents simulated by using LHS radiation stress ................................ ..... 52 4 6 (a) Comparison of turbulent kinetic energy between model results and data at Station 3 ..... 53 4 6 (b) Comparison of turbulent kinetic energy between model results and data at Station 4 ..... 53 4 6 (c) Comparison of turbulent kinetic energy between model results and data at Station 5 ..... 53 4 6 (d) Comparison of turbulent kinetic energy between model results and data at Station 6 ..... 53 4 6 (e) Comparison of turbulent kinetic energy between model results and data at Station 7 ..... 53 4 7 (a) Comparison between the data and numerical results for wave height .............................. 54 4 7 (b) Comparison between the data and numerical results for wave set up .............................. 54 4 7 (c) T he cross section of the basin ................................ ................................ ........................... 54 4 8 (a) Wave induced currents simulated by using M08 radiation stresses ................................ 55 4 8 (b) Wave induced currents simulated by using X04 radiation stresses ................................ .. 55 4 8 (c) Wave induced currents simulated by using LHS radiation stresses ................................ 55 4 9 Best track of Hurricane Isabel. (Courtesy of the NHC) ................................ .................... 56 4 10 Isabel track showing locations of measured data and definition of the Chesapeake Bay major axis. Light blue circles represent radiuses of maximum wind at each time .... 57 4 11 Measured and simulated water levels at six stations ................................ ........................ 58 4 12 Measured (a) East West and (b) North South currents at Kitty Hawk station ................. 59 4 13 Simulated (a) East West and (b) North South currents at Kitty Hawk station by using LHS radiation stress formulation ................................ ................................ ...................... 59 4 14 Simulated (a) East West and (b) North South currents at Kitty Hawk station by using M08 radiation stress formulation ................................ ................................ ...................... 60 4 15 Measured and simulated onshore offshore currents at Kitty Hawk during Hurricane Isabel ................................ ................................ ................................ ................................ 60 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science THREE DIMENSIONAL SIMULATION OF WAVE INDUCED CIRCU L ATION WITH A DEPTH DEPENDENT RADIATION STRESS By Tianyi Liu December 20 10 Chair: Y. Peter Sheng Major: Coastal and Oceanographic Engineering In this study, a three dimensional current wave modeling system, CH3D SWAN, has been enhanced with a depth dependent radiation stress formulation. The three dimensional modeling system consists of a three dimensional hydrodynamic model CH3D (Curvilinear Hydrodynamics in 3D) which is dynami cally coupled to the model SWAN (Simulating W a ves Nearshore). This study considers two depth dependent and one depth independent radiation stresses, and compares the performances of three different formulations. Results of the coupled CH3D SWAN compare well with analytical solution and observation of wave set up. Wave induced currents and turbulence observed during a laboratory experiment on undertow are also successfully simulated. The coupled modeling system als o successfully simulates the wave set up over a laboratory model of a fringing reef. While the simulated wave set up is in agreement with the laboratory data, wave induced currents over the fringing reef are also calculated by the coupled modeling system. Using the CH3D SSMS and a depth dependent radiation stress formulation, storm surge and currents during Hurricane Isabel are also simulated. The depth dependent radiation stress PAGE 10 10 formulation produces similar storm surge but slightly improved currents, in c omparison with data and previous results obtained by depth independent radiation stresses. PAGE 11 11 CHAPTER 1 INTRODUCTION 1.1 Background The coastal area is a dynamic region of the ocean where complex processes of wave transformation and circulation occur. The wave circulation interaction is of great importance for the nearshore hydrodynamics, and has significant impacts on human activities. Wave induced set up produces a rise of the water elevation above the still water level, and the wa ve induced current s are essential to the bottom sediment transport and shoreline changes. Ocean surface waves are the direct demonstration of unsteady ocean motions W aves and wave induced circulation are two of the most important mechanisms in nearshore area As ocean surface waves propagate over the beach wave transformations, such as sh oaling, refraction, diffraction and dissipation would occur and the waves finally break in shallow water s with the energy dissipated in the surf zone. The gradients o f the wave heights and momentum produce the wave induced nearshore circulation such as th e wave induced set up, set down and currents Longshore variation in wave breaker heights produce s longshore currents which cause longshore sediment transport. Wave f orc ing on circulation in the surf zone is a major cause of sediment transport and beach morphology evolution The understanding and prediction of wave induced circulation are essential for the study of nearshore dynamics The numerical modeling of the water waves and ocean circulation has been advanced tremendously in the recent few decades. T h e term that couples the wave modeling and circulation modeling is the momentum flux produced by waves. T h e momentum flux, common ly referred to as radiation stress was firstly introduced by Longuet Higgins and Stewart (1962, 1964 hereafter referred as LHS ) as a vertically integrated tensor which does not have vertical PAGE 12 12 distributions To carry out fully 3D simulations by coupling wave model s and 3D circulation model s the radiation stress term is necessary to have vertical distribution s Recently, d epth dependent radiation stress formulations (Xia et al.. 2004; Mellor, 2003 and 2008) have been developed based on linear wave theory T hese depth dependent radiation s tresses, when vertically integrated, are consistent with the conventional radiation stress (Longuet Higgins and Stewart, 1964) In this study, the depth dependent radiation stress es (Xia et al.. 2004 ; Mellor 2008) are incorporated into a three dimensiona l current wave modeling system, CH3D SSMS ( Sheng et al.. 2010) which consists of a three dimensional hydrodynamic model CH3D (Curvilinear Hydrodynamics in 3D) (Sheng, 1986) and a wave model SWAN (Simulating W a ves Nearshore) (Booij et al.. 1999). By comparing simulated vs. observed wave induced circulation, t he performances of the two depth dependent radiation stress formulations (Xia et al. 2004 ; Mellor 2008) and the depth independent LHS radiation stress are assessed. 1.2 Literature R eview The development of the radiation stress concept by Longuet Higgins and Stewart (1964) led to a greater understanding of wave induced circulation such as the longshore current study by Longuet Higgins (1970), the cross shore current study by Svendsen et al. (1 984a,b), Stive and Wind (1986), Okayasu et al. (1988). Some studies applied 2DH (two dimensional horizontal) circulation models with the LHS radiation stress to examine the effects of wave on storm surge simulations. Zhang and Li (1996) implemented the LHS radiation stress formulation into two dimensional ocean circulation equations and studied the importance of radiation stress in calculation of storm surge. T he model results indicate that the inclusion of the radiation stress improves the accuracy of t he computed results slightly by 2%. PAGE 13 13 Roland et al. ( 2009) developed a 2D coupled wave current model with unstructured grids and the coupling is based on the LHS radiation stress. Including wave effects in circulation simulation, the si mulated surge levels are improved in comparison with observations Using the 2DH circulation model and the LHS radiation stress, vertical variations in horizontal velocities are introduced by splitting the velocity into a wave component and a current component (Do ngeren et al. 1994; Haas et al. 2003; Wang et al. 2008). Their models are often referred to as quasi 3 D models. Recent studies have used 3D circulation model and LHS radiation stress to study wave induced circulation and t hese studies can yield vertic ally varying currents and eddy coefficients. Xie et al. (2001) studied the effects of surface waves on ocean currents in the coastal waters using a coupled wave current modeling system. T he coupling system employs the LHS radiation stress which is averaged over depth for fully 3D simulations The assumption of a uniform distribution of radiation stress is question ed by Mellor (2003) and Xia et al. (2004) Sheng and Alymov (2002) incorporated the LHS radiation s tresses in the CH3D (Sheng, 198 6 ) model and simulated wave setup fi elds during the 100 year storm event for two study areas in Pinellas County, Florida. T he study showed that the grid resolution has some eff ect on calculated wave setup especially in the areas where b athymetry has steeper gradients. Sun and Sheng (2002) coupled CH3D with the REF/DIF wave model (Kirby and Dalry mple 1994 ) and showed signi fi cant e ff ects of waves on water level and coastal currents. The model results were compared with analytical solution and laboratory measurements on the wave induced currents. Sheng et al. (2010) studied the importance of waves on storm surge, currents and inundation during Hurricane Isabel by using an integrated storm surge modeling system, CH3D PAGE 14 14 SSMS. It is found that the wave induced surface stress (inside and outside the estuary) and radiation stresses (outside the estuary) are more important than the wave induced bottom stress in affecting the water level. Inside the Chesapeake Bay, the wave effects can a ccount for 5~10% for the peak surge elevation, and outside the Chesapeake Bay, the wave effects can account for 20% of the peak surge level. The radiation stress term is applied by averaging the vertically integrated radiation stress formulation (Longuet H iggins and Stewart, 1964) over depth to carry out the three dimensional simulation for simplicity. The advancement of the depth dependent radiation stress led to the fully 3D current wave simulations by coupling wave model s and 3D circulation models. Xia e t al. (2004 hereafter referred to as M04 ) extended the vertically integrated radiation stress to vertically varying based on linear wave theory. After applying this radiation stress to a 3D circulation model, a laboratory experiment was simulated in Xia et al. (2004) The simulated wave set up compare s well with analytical solution, and t he simulated wave induced currents show a two gyres flow pattern over the slope. Mellor (2003) derived the formulation for depth dependent radiation stress based on linear wave theory and sigma coordinate. But using this formulation, Ardhuin et al. (2008 ) found an unreasonable phenomenon that the mean currents were produced in deep water with bottom variations Mellor (2008 hereafter referred to as M08 ) improved the previous derivation (Mellor 2003), and produced a new formulation of the depth dependent radiation stress The wave energy equation is also modified due to the effects of currents This formulation solved the problem stated by Ardhuin et al. (2008 ), although it has not been verified PAGE 15 15 Using POM SWAN and the M04 radiation stress Xie et al. (200 8 ) studied the effects of wave current interaction on circulation during Hurricane Hugo ; Liu and Xie (2009) studied the effects of wave current surge interaction on wave s during Hurricane Hugo and found that the effect of wave surge interaction on wave is significant in shallow co astal waters, but relatively small in deep water, and the influence of wave current interaction on wave propagation is relatively insignificant. However, the validity of the depth dependent radiation stresses of Xia et al. (2004) and Mellor (2008) have not been quantitatively confirmed by comparing simulated vs. observed wave induced circulation (e.g., wave set up, wave induced currents) and i n this study performances of the two depth dependent radiation stress formulations (X04 ; M08) and the depth independent LHS radiation stress would be assessed 1.3 Objectives The objectives of this study are as follows: Incorporate the depth dependent radiation stresses into the three dimensional current wave modeling system, CH3D SSMS Test the coupling system and assess the performances of depth dependent and depth independent radiation stresses by comparing model results with analytical solutions and laboratory observations Evaluate the significance of the depth dependent radiation stress in storm surge simulation. 1.4 Organization The organization of the paper is as follows: in Chapter 2 the governing equations for the three dimensional circulation model CH3D and the two dimensional wave model SWAN are presented. The coupling pr ocess between the two models is demonstrated. Chapter 3 states the development of the radiation stress concept, from the conventional vertically integrated tensor to depth dependent formulations and t he differences of the depth dependent radiation stress PAGE 16 16 distributions are compared and their characteristics are described In Chapter 4, a few test simulations are conducted by this coupling system and t he model results are co mpared with analytical solution as well as observations in laboratory and Hurricane Isabel The performances of the depth dependent and depth independent radiation stresses were compared and assessed Chapter 5 summarizes and concludes the thesis. PAGE 17 17 CHAPTER 2 METHODOLOGY 2.1 A Three Dimensional Circulation M odel CH3D CH3D (Curvilinear grid Hydrod ynamics in 3D) a robust three dimensional circulation model originall y developed by Sheng (1986 ), has been successfully applied to simulate the estuarine, coastal and riverine circulation driven by tide, wind and density gradients. The model uses a boundary fitted non orthogonal curvilinear grid in the horizontal directions to resolve the complex sh oreline and geometry, and a terrain following grid in the vertical direction. The model uses a Smagorinski type horizontal diffusion coefficient, a robust turbulence closure model (Sheng and Villaret, 1989) for the vertical mixing, and highly accurate ad vective schemes QUICKEST (Leonard, 19 79 ) and Ultimate QUICKEST (Leonard, 1991). The governing equations are simplified Navier Stokes equations based on four assumptions. First, the water is assumed to be incompressible, which simplifies the continuity equa tion. The second one is the hydrostatic assumption that the characteristic vertical length scale is much smaller than the horizontal length scale and the vertical velocity and small and the vertical acceleration could be neglected, thereby, the momentum eq uation in the vertical direction is simplified into hydrostatic pressure relation. Third, with the Boussinesq approximation, an average density can be used in the equations except in the buoyancy term. Fourth, the application of the eddy viscosity concept, which assumes that he turbulent Reynolds stresses are the product of the mean velocity gradients and eddy viscosities. The equations of motion for CH3D are: (2.1) PAGE 18 18 (2.2) (2.3) The hydrostatic assumption states: (2.4) w here and are the velocity vector components [LT 1 ] in x y and z coordinate directions, respectively; t is time [T]; ( x,y,t ) is the free surface elevation [L]; g is the acceleration of gravity [L 2 T ] ; A H and A v are the horizontal and vertical turbulent eddy coefficients, respectively [L 2 T 1 ]; and f is the Coriolis component [T 1 ]. S xx S xy S yx and S yy are radiation stress terms. P a is atmospheric pressure. The model employs the non orthogonal curvilinear and bo undary fitted grid system for the continuity and momentum equations, in order to resolve complex boundaries and geometries in the coastal or riverine area. In addition, in the vertical direction, the model uses a sigma stretching coordinate, which allows f or dealing with the bathymetry variations at the bottom of the basin. The non dimensional form of governing equations in curvilinear non orthogonal boundary fitted grid system is as follows: ( 2.5 ) PAGE 19 19 ( 2.6 ) PAGE 20 20 ( 2.7 ) where and are the transformed coordinates; are non dimensional contra variant veloc ities in curvilinear grid ( ). is the Jacobian of horizontal transformation; are the metric coefficients of coordinate transformations; is non dimensional parameter; is water level; PAGE 21 21 The dimensionless variables are: The dimensionless groups are: Vertical Ekman Number: Lateral Ekman Number: Vertical Prandtl Number: Lateral Prandtl Number: Vertical Schmidt Number: Later Schmidt Number: Froude Number: Rossby Num ber: PAGE 22 22 Densimetric Froude Number: When wave effects are considered in the simulation, the velocities are divided into a current component and Stokes drift after phase averaging (Mellor 2008): (2. 8 a,b) W here denotes the horizontal coordinates, is the current component of the velocity, is the Stokes drift. The boundary conditions and wave implementations in CH3D are shown in Appendix C. 2. 2 Local Wave M odel: SWAN The wave field information for the nearshore circulation is provided by the wave model SWAN. The SWAN (Simulating W Aves Nearshore) model (Booij et al. 1999) is a third generation wave model which computes random, short crested waves in coastal regions and inland waters. It accounts for wave propagation in time and space, shoaling, refraction due to current and depth, frequency shifting due to currents and non stationary depth, wave generation by wind, bottom friction, depth induced breaking, and transmission through and reflection from obstacles. SWAN represent s waves using a two dimensional wave action density energy spectrum and the evolution of the spectrum is described by the spectral action balance equation in which a local rate of change of action density in time is related to the propagation of action in geographical space, shifting of relative frequency due to currents and depths, depth induced and current induced refraction all balance by the source term in terms of energy density representing the effects of energy generation, energy dissipation and nonl inear wave wave interactions. Generation of waves due to wind in SWAN is described as a sum of linear and exponential growth. The dissipation of wave energy consists of whitecapping, bottom friction and depth PAGE 23 23 induced wave breaking. In deep water the evolut ion of the spectrum is dominated by the wave wave quadruplet interactions which transfer wave energy from the peak of the spectrum. In very shallow water, triad wave wave interactions transfer energy from lower to higher frequencies where the energy is dis sipated by whitecapping. SWAN is stationary and optionally non stationary and can use a boundary fitted curvilinear grid which is irregular, quadrangular, and not necessarily orthogonal spherical or unstructured grid It calculates various important wave and wave related parameters such as signi fi cant wave height, swell wave height, mean wave direction, peak wave direction, direction of energy transport, mean absolute wave period, mean relative wave period, current velocity, energy dissipation due to bott om friction, wave breaking and whitecapping, fraction of breaking waves due to depth induced breaking, transport of energy, wave induced force, the RMS value of the maxima of the orbital velocity near the bottom, the RMS value of the orbital velocity near the bottom, average wavelength, average wave steepness, wave spectrum, etc. The following wave propagation processes are represented in SWAN: propagation through geographic space, refraction due to spatial variations in bottom and current, diffraction, sh oaling due to spatial variations in bottom and current, blocking and reflections by opposing currents and transmission through, blockage by or reflection against obstacles. The following wave generation and dissipation processes are represented in SWAN: ge neration by wind, dissipation by whitecapping, dissipation by depth induced wave breaking, dissipation by bottom friction and wave wave interactions in both deep and shallow water. PAGE 24 24 The SWAN model predicts a 2 D wave field on the grid points, and the waves are described with the tow dimensional wave action density spectrum N ( ) equal to the energy density divided by the relative frequency: N ( )= E ( )/ The evolution of the wave spectrum is described by the spectral action balance equation: (2. 9 ) The first term in the LHS represents the local rate of change of action density in time, the second and third term represent propagation of action in geographical space (with propagation velocities c x and c y ) in x and y space, respectively). The fourth t erm represents shifting of the relative frequency due to variations in depths and currents (with propagation velocity c in space). The fifth term represents depth induced and current induced refraction (with propagation velocity c in space). The expressions for these propagation speeds are taken from linear wave theory. The term S (= S ( )) at the right hand side of the action balance equation is the source term in terms of energy density representing the effects of generation, dissip ation and nonlinear wave wave interactions, etc.. 2. 3 The Coupling P rocess between CH3D and SWAN The circulation and wave models, CH3D and SWAN are dynamically coupled in the way that SWAN provide s the wave field information like wave height, period, and dir ections at each grid cell and t he wave parameters are used to estimate the radiation stress as the forcing in the water momentum equation in CH3D The wave enhanced bottom friction and eddy viscosity are also calculated by the wave field information from SWAN. The wave set up and wave induced currents are computed in the simulation by CH3D, and the wave set up may also change the wave propagation process by altering total water depth; the currents would affect the wave PAGE 25 25 propagat ion by inducing wave refraction. Therefore, the wave and current interact with each other. The coupling process of CH3D and SWAN is shown in Figure 2 1. Figure 2 1 The coupling process between CH3D and SWAN PAGE 26 26 CHAPTER 3 THE FORMULATIONS OF RADIATION STRESS 3.1 The Vertically Integrated Radiation S tress The radiation stress concept was developed by Longuet Higgins and Stewart (1964) and Phillips (1977) as a vertically integrated tensor. This concept explains the wave set up and set down inside and outside the surf zone, rip current and wave current interactions. The radiation stress is derived by subtracting the total flux of horizontal momentum due to w aves by the mean flux in the absence of the waves, which are: (3.1) (3.2) (3.3) (3.4) W here h is the water depth, is the water elevation of the free surface, is the density of the fluid, p is the total pressure and p 0 is the hydrostatic pressure in the absence of waves. Based on linear wave theory, the expre ssions for radiation stress in E quations 3.1~3.4 can be simplified as: (3.5) (3.6) (3.7) PAGE 27 27 W here E is the wave energy, is the angle wave propagating to the onshore direction and n is the ratio of group velocity to wave celerity: n =(1+2 kh /sinh2 kh )/2. The radiation stress can be divided into a momentum part S m and a pressure part S p according to Ha a s and Svendsen (2000), and rewritten as: ( 3.8 ) ( 3.9 ) ; ( 3.10 ) Where is the wave angle to the x coordinate, G =2 kh /sinh2 kh ; k is the wave number, h is the water depth. The wave induced mass transport is given in Svendsen (1984b) as: ( 3.11 ) Where, is the horizontal coordinate, H is the wave height, c is the wave phase speed, k is the wave number, B 0 is wave shape factor which is defined as: B 0 =(1+ G )/16. The surface roller term developed by Svendsen (1984 b ) plays an important part in mass, momentum and energy balance in the surf zone and is the primary driving mechanism for the undertow. The roller r epresents an increase in radiation stress which according to Svendsen (1984 b ) can be written as: ( 3 .1 2 ) The wave induced mass transport inside the surf zone is: ( 3 .1 3 ) PAGE 28 28 Where, L is the wave length and A is the area for the surface roller of breaking waves. A is given as: A =0.06 HL (Okayasu et al. ., 1988). 3 .2 The Depth Dependent Radiation S tress by Xia et al. (2004) Xia et al. (2004) extended the vertically integrated concept of radiation stress to vertically varying based on linear wave theory, and the integr ation of his formulation is consistent with the formulation by Longuet Higgins and Stewart (1964), Phillips (197 7 ). From the vertically integrated LHS radiation stress, Xia et al. (2004) developed the new formulation in vertically stretching coordinate with = ( z )/( h + ) and assumed = z / h in some of the derivations so some terms are neglected and part of radiation stress between wave crest and mean water level is ignored The small amplitude assumption ( a << L) which works well in deep water is also applied and all terms of higher order than ka are neglected. The formulation of depth dependent radiation stress by Xia et al. is: ( 3 .1 4 ) ( 3 .1 5 ) ( 3 .1 6 ) ( 3 .1 7 ) Where, E is the wave energy, k is the wave number, h is the water depth. is the wave angle to the x coordinate. PAGE 29 29 3 .3 The Depth Dependent Radiation Stress F ormulation by Mellor (2008) Mellor (2008) developed the depth dependent radiation stress term by deriving the three dimensional c ontinuity and momentum equations when wave s are included and the integration of this formulation is also consistent with the vertically integrated radiation stress by Longuet Higgins and Stewart (1964) and Phillips (197 7 ). The derivation is also based on linear wave theory which works well for deep water, and assumes small ka values and small bottom slope. The formulation of the depth dependent radiation stress by Mellor (2008) is: ( 3 .1 8 ) (3.19a) (3.19b) (3.19c) (3.19d) W here, are horizontal coordinates, k is the wave number, E is wave energy, E D is a modified Dirac delta function, and is defined according to Mellor (2008) as: if and ( 3 .2 0 ) 3 .4 T h e Vertical Distribution of the Depth Dependent Radiation S tress The X04 and M 08 depth dependent radiation stress formulations, when vertically integrated, are in agreement with previous studies (Longuet Higgins and Stewart, 1964; Phillips, 197 7 ). However, their vertical distributions are different in deep or shallow waters The wave conditions for checking th e distributions are listed in Table 3 1 PAGE 30 30 In Table 3 1, k i s the wave number; D is the total water depth. The shallow or deep water waves are estimated by Dean and Dalrymple (1991). The distributions of the depth dependent radiation stress determined by the two formulations (Xia et al. 2004, Mellor 2008) under the wave conditions sh own in Table 3 1 are shown in Figure 3 1. In shallow water, when waves travel normal toward the shore, the distribution of the M08 radiation stress in Figure 3 1(a) indicates that S xx distributes uniformly over the depth, except the discontinuous point at the surface ( =0) because of the Dirac delta function in Equation 3.20 representing the radiation stress between the wave crest and the wave level. S xy and S yx are zero in the water column due to zero wave angle. The absolute value of S yy decades exponentially from the surface to the bottom, and reaches zero at the bottom; the disco ntinuous point due to Equation 3.20 exists at the surface. The distribution of S xx by X04 formulation gives an opposite direction to that by M 08 formulation. It drops from the surface to the bottom. S xy and S yx are also zero due to zero wave angle; S yy varies over the depth, and reaches zero at the bottom. With a 30 wave angle toward the shore, Figure 3 1(b) shows that the distributions of S yy predicted by both formulations are similar as that with a zero wave angle. When waves propagate obliquely toward the shore, the S xx term by M 08 formulation is not uniform over the depth, but decades exponentially from the surface to the bottom. S xy and S yx are not zero since the wave angle is not zero, and the two formulations give the same pattern of distribution for S xy and S yx which decreases exponentially from the surface to the bottom, over the water column. In deep water, w hen waves travel normally t o the coast in Figure 3 1(c) S xx predicted by M 08 formulation distributes uniformly over the water column, the discontinuous point at the surface indicate the radiation stress between the wave crest and the water level. S xy and S yx are PAGE 31 31 both zero due to zero wave angle. S yy drops exponentially from the surface to the bottom and reaches zero at the bottom. By u sing X04 formulation, S xx and S yy increases almost linearly from the surface to the bottom, which may indicate stronger wave effect s to the current at the bottom than the surface. S xy and S yx are both zero due to zero wave angle. When waves travel obliquely toward the shore in deep water, Figure 3 1(d) shows that S xx and S yy predicted by M 08 formulation varies over the depth and the absolute value decreases from the surface to the bottom, while the S xy and S yx predicted by X04 formulation increases almost linearly from the surface to the bottom. The distribution s of S xy and S yx predicted by t he two formulations have the same style, which decreases exponentially from the surface to the bottom. T he comparison of th e two formulations in Figure 3 1 indicates that the two radiation stress es are quite different except the S xy and S yx terms. Test simulations by the two formulations are presented in Chapter 4, and the performances of the two formulations are evaluated. PAGE 32 32 Table 3 1 Wave conditions Wave height (m) W ave period (s) W ave angle ( ) D epth (m) kD Case 1 0.5 1 0 2 8.0570 D eep water Case 2 0.5 1 30 2 8.0570 D eep water Case 3 0.5 10 0 2 0.2877 S hallow water Case 4 0.5 10 30 2 0.2877 S hallow water PAGE 33 33 Figure 2 3 1(a). Distribution of depth dependent radiation stress for the wave condition of Case 1. PAGE 34 34 Figure 3 1(b). Distribution of depth dependent radiation stress for the wave condition of C ase 2. PAGE 35 35 Figure 4 3 1(c). Distribution of depth dependent radiation stress for the wave condition of Case 3. PAGE 36 36 Figure 5 3 1(d). Distribution of depth dependent radiation stress for the wave condition of Case 4. PAGE 37 37 CHAPTER 4 TEST SIMULATIONS 4 .1 Wave S et up Wave set up generally occurs in the surf zone. As the waves shoal and break on a beach, they produce excess momentum flux in the shoreward direction. At steady state, the shoreward decrease of radiation stress is balanced by a shoreward increase in the wat er level. This raises the water surface elevation within the surf zone to be higher than the still water level and produce setup. It also pushes the water level outside the surf zone to be lower than the still water level and produce set down. The momentum balance according to Longuet Higgins and Stewart (1964) is: ( 4 .1) T he analytical solutions of wave set up inside the surf zone and set down outside the surf zone based on linear wave theory are: ( 4 .2) ( 4 .3) w here is the water elevation, a is the wave amplitude, k is the wave number, h is the water depth, h b is the water depth at the breaker line, is the breaking i ndex. To test the model with the analytical solution, a simple test case was used: the basin is 150 m by 150 m; the slope of the bottom is 1:40, with 2.1 m depth at the flat bottom part, and 0.1 m at the shallowest part. The cross section of the basin is s hown in Figure 4 1 (b). The incident wave height is 0.6 m; the wave period is 5 s. PAGE 38 38 Three different methods of applying the radiation stress term are employed for comparisons. T h e first method applies the depth dependent M08 radiation stress; the second one uses the vertical varying X04 radiation stress; the third one uses the LHS radiation stress, as well as a surface roller term described in Chapter 3. The grid resolution is 10 m by 10 m in the horizontal and 16 layers in the vertical; the bottom roughness is z 0 =0.4; the breaking index is selected as =0.73. The simulated wave set up and analytical solution is shown in Figure 4 1 (a) which suggests that the numerical results fro m the three methods do not have much difference, and all match well with the analytical solution. 4 .2 The Undertow T est The undertow which is a near bottom compensating flow for mass transport and surface roller drift in the surf zone, was firstly studied by Bagnold (1940). Laboratory experiments have also been done to measure the undertow over sloped bottom (Hansen and Svendsen, 1984; Stive and W i nd, 1986; Okayasu et al. 1988; Ting and Kirby, 1994). While some (e.g. Svendsen, 1984 b ; Stive and Wind, 1986 ; Putrevu and Svendsen, 1993) used theoretical methods to predict the undertow, others developed numerical models. Svendsen et al. (2003) used two wave models and the quasi 3D model SHORECIRC to simulate the wave induced currents, and compared with the mea surements. Christensen (2006) used a large eddy simulation to study the turbulence and the undertow induced by spilling and plunging breakers ; t he undertow is successfully simulated, but the simulated turbulence still has some difference with the observations. Wang (2008) derived the new expressions of radiation stress and volume flux based on nonlinear wave theories, and used the Boussinesq type nonlinear wave model COULWAVE to simulate the currents induce d by waves over a sloped bottom; t he simul ated undertow compares well for some of the stations, however, the mass conservation in this study is PAGE 39 39 questionable, because the vertical integration of the simulated currents do not seem to be zero and the mean inflow does not balance the outflow. With the recently developed depth dependent radiation stress a fully 3D simulation of undertow is possible by coupling a 3D circulation model and a wave model, and the vertical variations of the currents can be simulated. To test the CH3D SWAN modeling sy stem and the performances of depth dependent radiation stresses a laboratory experiment of undertow by Ting and Kirby (1994) is simulated The experiment w as conducted in a two dimensional wave tank which is 40 m long, 0.6 m wide and 1.0 m deep. The expe rimental arrangement is shown in Figure 4 2 (c). The bottom slope is 1:35, and the water depth at the horizontal region is 0.4 m The test of experiments with a wave he ight 0.128 m at the horizontal region and 5 s wave period was simulated The locations of measurements are shown in Table 4 1 : d is the total depth h is the water depth. In the experiment, it is noted that the wave breaks at the location x =7.795 m, which is Station 2, so Station1 is outside the surf zone, and Station 3~7 are inside the surf zone. The grid spacing in the horizontal direction is 35cm and the number of layers in the vertical direction is 16 The simulated period is 15 minutes with 0.005s time step until the simulation reaches steady state The flooding and drying is activated in CH3D which allows water to occupy land cells A variable eddy viscosity and variable bottom f riction with bottom roughness z 0 =0. 4 are applied. The wave enhanced eddy viscosity and bottom friction are also activated in the simulation The wave field is significant for simulating the wave induced currents and set up. The comparison of the wave height calculated by SWAN and the measurement is shown in Figure 4 2 (a) which indicates that the simulated wave height agrees well with the data. The comparison PAGE 40 40 of the simulated wave set up and observations is shown in Figure 4 2 (b) which suggests that they are fit well The fully 3D simulation of the wave induced currents with a depth dependent radiation stress produces a vertical structure of currents, as is shown in Figure 4 3 ~4 5. Figure 4 5 demonstrates different mean flow patterns of the wave induced currents over the whole basin by using three different radiation stresses. The relative RMS error for the simulated results compared with the observation is shown in Table 4 2. The relative RMS error is calculated as follows: ( 4.4 ) W here x model and x data are simulated and observed results respectively. Simulations with different radiation stresses produce different flow patterns, as is shown in Figure 4 3~4 5. The relative RMS error calculated by Equation 4.4 for the simulated currents vs. observations in Table 4 2 indicates using the M08 radiation stress gives more accuracy than the other two. With the X04 radiation stress, the simulated currents give a two gyres flow which corresponds to the model results fo r the experiment by Bijker et al. (1974) in Xia et al. (2004). However, the simulated currents are in a direction opposite to the observations in the surf zone, and the simulated downwelling at the breaker line contradicts the measurements at Station 2. The seaward undertow near the bottom is not successfully simulated by using this method. The reversed flow pattern in the surf zone alters the direction of the bottom friction, and then affects the force balance between bottom friction, pressure gradient and radiation stress. PAGE 41 41 T h e flow direction by using the LH S radiation stress is consistent with the observation in the surf zone, however, outside the surf zone, the simulated currents are almost zero, which is against the measurements at Station 1, and the simulated upwelling at the breaker line also does not ma tch with the observation at Station 2. Using the M08 radiation stress, the direction of the simulated currents is consistent with the observations. However, some divergences still exist between simulated currents and observations. The possible reasons are: (1) The M08 radiation stress is developed based on linear wave theory, and SWAN is a linear wave model. In the nearshore areas, especially in the surf zone, the nonlinear process could play an important role. (2) The viscous effects of the boundary layer could be significant for the dynamics in the surf zone, besides the radiation stress, wave enhanced bottom friction and eddy viscosity. (3) As indicated by Ting and Kirby (1994), different wave breakers may generate mean flow with different characteristic s, however, in this coupling system, the parameter accounting for different wave breakers has not been considered yet. (4) Turbulence generated by breaking waves is important in determining the mean flow in the surf zone; however, the simulated turbulent kinetic energy shown in Figure 4 6 has errors in comparison with observations. (5) The measurement in the laboratory can also be affected by various factors such as wave reflect ions or equipment disturbance, and the uncertainties in the laboratory experiment may influence the accuracy of the data. PAGE 42 42 The simulated turbulent kinetic energy (TKE) is compared to observations in Figure 4 6. The turbulent kinetic energy is calculated by the equilibrium turbulent clos ure model developed by Sheng and Villaret (1989). The wave enhanced turbulence contributed by the wave energy dissipation D b and the roller energy dissipation D r described in Chapter 2 are both activated in the simulation. The wave breaking affects the vertical eddy viscosity A v as is shown in Equation C 34 where the parameter M is selected to be 0.025 according to Vriend and Stive (1987). The relative RMS error of the simulated TKE is shown in Table 4 3 and this indicates that the depth dependent radiation stress (M08, X04) gives more accurate results of TKE than the LHS radiation stress. The differences between the numerical results and observations may come from the error of the simulated currents, since the turbulence is mainly produced by the vertical shear in the water column. Moreover, the Equation C.3 4 for calculating wave enhanced eddy viscosity simplifies the process of wave induced turbulence, and may also bring errors. 4 .3 The W ave S et up on Fringing R eef Corel reefs ring a large number of the islands at the coast. The fringing reefs have a wide, shallow and flat bottom near the coastline, and drop into the deep water with a large slope. Wave often breaks at the edge of the reef while propagating toward the coast, causing set up over the flat part. During hurricanes or typhoons, high waves and storm surge may cause damage and inundation over the relatively shallow area of the fringing reef. T he study of wave setup over fringing reefs requires an accurate es timation of the wave field over the reef by the well developed wave models, as well as the calculation of the setup and inundation by circulation models. Using the CH3D SWAN current wave coupling system, a PAGE 43 43 fully 3D simulation of the wave induced set up and currents over fringing reefs is possible and the accuracy of the model results can be evaluated by comparing with the observations. The laboratory experiment selected to test the model is from Z Demirbilek et al. (2007). A physical model of a 2 D fringi ng reef was built in a 35 m long by 0.7 m wide wind wave flume at the University of Michigan, and a series of experiments have been conducted in this wave flume. T h e cross section profile of this reef beach system is shown in Figure 4 7(c). One case of exp eriments was selected, with incident wave height 0.075m and wave period is 1.5 s. The computational grid is 0.25 m by 0.25 m in the horizontal direction, and the water column is divided into 8 layers in the vertical direction. A variable eddy viscosity an d variable bottom f riction with bottom roughness z 0 =0. 01, as well as wave enhanced eddy viscosity and bottom friction, are activated in the simulation. T he time step is 0.05 s, and the simulated time is 30 min until the simulation reaches steady state. The comparison of the wave height calculated by SWAN and the observations over the reef is shown in Figure 4 7(a) and the simulated wave height agrees well with the measurement. The comparison of the simulated wave set up and measure ments in Figure 4 7(b) suggests that the numerical results are fit well for the observations The simulated currents are shown in Figure 4 8. Unfortunately, the measurements of the mean currents are not available for comparison and validation of the model results. The three flo w patterns in Figure 4 8 all present a rather weak flow over the flat part of the reef. However, the mean flow patterns over the slope part are different: the currents from the M08 radiation stress show a shoreward flow at the surface and undertow near the bottom; the X04 radiation stress produces a two gyre flow, while the LHS radiation stress gives no currents PAGE 44 44 outside the surf zone. As discussed in Section 4.2, the current field simulated by using X04 or LHS radiation stress could be inconsistent with th e observations of undertow. The studies with different radiation stress formulations are summarized in Table 4 4 4.4 Simulation of Hurricane Isabel This section is based on the work of Sheng et al. (2010). In this study, the vertically uniform LHS radiation stress is replaced by the depth dependent M08 radiation stress in the CH3D SSMS, and the model results by using the M08 radiation stress are compared with data and previous results. Hurricane Isabel is a tropical cyclone which strikes portions of northeastern North Carolina and east central Virginia in 2003 The track of the hurricane is shown in Figure 4 9 Hurricane Isabel has been studied by Alymov (2005) and Sheng et al. (2010), and it was found that the wave effects during Isabel are significant to affect the surge level currents and inundation The Isabel track and locations of all data stations are shown in Figure 4 10. T he coastal domain as shown in this figure has 548,240 grid cells. The 2 D vertically averaged version of ADCIRC (Luettich et al. 1992, IPET 192 2006) is used to simulate the regional/basin scale surge over the entire Gulf of Mexico and western North Atlantic represented by the EC95d 194 (ADCIRC Tidal Database, version ec_95d) grid with 31,435 nodes, and to provide water elevation along the open boundaries of the coastal surge model CH3D. A high resolution grid is used by CH3D and ADCIRC employs the coarse offshore grid. Tides along the CH3D open boundaries are provided by the ADCIRC tidal constituents (Mukai et al. 2002). The third generation wave model SWAN is used for wave simulation in the CH3D domain. The stationary SWAN was applied. In the deep water, the model results of WAVEWATCH III (WW3) (Tolman 1999 ) are used to provide the wave conditions along the ope n boundaries of PAGE 45 45 the CH3D/SWAN domain. The domain of the WW3 model is similar to the ADCIRC domain. WW3 uses the WNA wind, which is based on the GFDL hurricane wind model. In this study, several wind models used in Sheng et al. (2010) were applied: the WNA wind provided by NCEP, the WINDGEN (WGN) wind provided through the University of Miami. The resolution of the WGN wind is 0.2 degrees and the resolution of the WNA wind is 0.25 degrees The simulations used WGN or WNA wind, as well as the M08 radiation st ress formulation to account for wave effects. The simulated water levels with WGN wind are compared with previous results and observations at six stations in Figure 4 11, which shows that the wave effects contribute significantly for the surge level. Witho ut considering the wave effects, the surge level would be highly underestimated. However, the M08 radiation stress does not produce significant difference in comparison to the application of LHS radiation stress, and the surge level is only 1~3 cm higher than previous results with LHS radiation stress. The observations and simulated c urrents with WNA wind at Kitty Hawk are shown in Figure 4 12~4 14, and the comparisons at several times at the peak of the storm surge are shown in Figure 4 15. The M08 radiation stress gives a different current field, in comparison with previous results, and the comparison in Figure 4 15 suggests slight ly improved results of simulated currents by using the M08 radiation stress. The currents during hurricanes are contributed by various processes (e.g., wind, tide, waves), and the wave induced currents by us ing the M08 radiation stress may not dominate and make significant difference in the hurricane environment. PAGE 46 46 Table 4 1 2 Locations of measurements and water depth Stations 1 2 3 4 5 6 7 x (m) 7.295 7.795 8.345 8.795 9.295 9.795 10.395 d (m) 0.169 0.156 0.142 0.128 0.113 0.096 0.079 h (m) 0.166 0.154 0.143 0.132 0.119 0.104 0.090 Table 4 2 3 The relative RMS error for the simulated current velocities Mellor (2008) (%) Xia (2004) (%) LHS (1964) (%) Station 1 37.1 61.4 56.6 Station 2 39.2 74.6 44.4 Station 3 52.5 107.8 55.4 Station 4 73.6 142.7 95.4 Station 5 44.4 111.7 60.9 Station 6 48.8 130.6 72.6 Station 7 50.5 104.9 72.4 Overall 38.5 82.6 51.5 Table 4 3 The relative RMS error for the simulated turbulent kinetic energy Mellor (2008) (%) Xia (2004) (%) LHS (1964) (%) Station 3 37.5 6.8 24.2 Station 4 24.9 32.3 41.8 Station 5 19.7 24.1 27.4 Station 6 15.7 18.5 28.5 Station 7 13.0 21.7 30.5 Overall 16.5 19.9 26.8 PAGE 47 47 Table 4 4. 5 Studies with different radiation stress formulations Study Radiat ion stress Circ. model B ottom friction Turbulent mixing S trength Weakness Zhang and Li (1995) LHS, 1964 2DH Chezy Manning formula tion N/A Waves are considered in circulation model Simulation is 2D Vertical distribution of currents is not available Dongeren, et al. (1994) LHS, 1964 Quasi 3D Chezy Manning formula tion Calculated from Coffey and Nielsen (1984) and Battjes ( 1975 ) Provides vertical structure of currents Model is basically 2D Xie, et al. (2001) LHS, 1964 3D Turbulent bottom boundary layer S econd order turbulence closure scheme of Mellor and Yamada ( 1982 ) F ully 3D simulation Assuming vertically uniform distribution of radiation stress N o wave induced flow outside the surf zone Xie, et al. (2008) Xia et al. 2004 3D Turbulent bottom boundary layer S econd order turbulence closure scheme of Mellor and Yamada ( 1982 ) Fully 3D simulation Radiation stress has vertical distribution Wave induced currents inside the surf zone have inaccurate direction Radiation stress formulation is based on linear wave theory Viscous effects of the boundary layer are neglected This study Mellor, 2008 3D Turbulent bottom boundary layer E quilibrium turbulence closure model of Sheng and Villaret (1989) & turbulence induced by breaking waves (Battjes, 1975) Fully 3D simulation Radiation stress has vertical distribution Wave induced currents have correct direction Radiation stress formulation is based on linear wave theory Viscous effects of the boundary layer are neglected PAGE 48 48 Figure 6 4 1. (a) The comparison between analytical and numerical solution for the wave set up; (b) The cross section of the basin. PAGE 49 49 Figure 7 4 2. Comparison between the data and numerical results for (a) wave height; (b) wave set up. (c) the cross section of the basin. PAGE 50 50 Figure 8 4 3. Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using (a) M08 ; (b) X04 ; (c) LHS radiation stresses PAGE 51 51 Figure 9 4 4. Comparison between the simulated and measured current velocity value at (a) Station 1; (b) Station 2; (c) Station 3; (d) Station 4; (e) Station 5; (f) Station 6; (g) Station 7. PAGE 52 52 Figure 4 5. Wave induced currents simulated by using (a) M08 ; (b) X04 ; (c) LHS radiation stresses Figure 10 PAGE 53 53 Figure 4 6 Comparison of turbulent kinetic energy between model results and data at (a) Station 3; (b) Station 4; (c) Station 5; (d) Station 6; (e) Station 7; Figure 11 PAGE 54 54 Figure 4 7. Comparison between the data and numerical results for (a) wave height; (b) wave set up. (c) the cross section of the basin. Figure 12 PAGE 55 55 Figure 4 8. Wave induced currents simulated by using (a) M08 ; (b) X04 ; (c) LHS radiation stresses Figure 13 PAGE 56 56 Figure 4 9. Best track of Hurricane Isabel. (Courtesy of the NHC) Figure 14 PAGE 57 57 Figure 4 10. Isabel track showing locations of measured data and definition of the Chesapeake Bay major axis. Light blue circles represent radiuses of maximum wind at each time. Figure 15 PAGE 58 58 Figure 4 11 Measured and simulated water levels at six stations. Figure 16 PAGE 59 59 Figure 4 1 2 Measured (a) East West and (b) North South currents at Kitty Hawk station. Figure 17 Figure 4 1 3 Simulated (a) East West and (b) North South currents at Kitty Hawk station by using LHS radiation stress formulation. Figure 18 PAGE 60 60 Figure 4 1 4 Simulated (a) East West and (b) North South currents at Kitty Hawk station by using M08 radiation stress formulation Figure 19 Figure 4 1 5 Measured and simulated onshore offshore currents at Kitty Hawk during Hurricane Isabel. Figure 20 PAGE 61 61 CHAPTER 5 CONCLUSION In this study, a three dimensional current wave modeling system, CH3D SWAN, has been enhanced with depth dependent radiation stresses. This modeling system consists of a three dimensional hydrodynamic model CH3D which is dynamically coupled to the model SWAN which provides wave field infor mation ( e.g., wave height, period, direction or wave spectrum). The wave effects considered in the circulation simulation are radiation stress, wave induced mass flux, wave enhanced bottom stress, wave enhanced surface stress and wave in d u ce d mixing. The water level change and wave induced currents from the circulation simulation are also reflected to the wave field simulation in SWAN during the simulation. Two depth dependent (X04, M08) and one depth independent (LHS) radiation stresses are considere d in this study, and their performances are compared. Results of the coupled CH3D SWAN compare well with analytical solution and laboratory observation of wave set up, for which the three radiation stresses produce very slight difference. Wave induced cu rrents and turbulence observed during a laboratory experiment on undertow are also simulated, and the three different radiation stresses produce different mean flow patterns and turbulent kinetic energy. In the analysis of relative RMS error of the numeric al results with observations in the undertow experiment the M08 radiation stress gives the most accurate mean currents and turbulence, and the simulated currents have the correct direction inside and outside the surf zone. Using the X04 radiation stress, the simulated currents inside the surf zone have a reversed flow pattern in comparison with the observations. The LHS radiation stress produces very weak flow outside the surf zone, with the velocities almost zero, which is inconsistent with the measuremen ts of undertow. However, using the M08 radiation stress to simulate undertow is limited by several aspects such as the linear wave theory which the M08 PAGE 62 62 radiation stress and the wave model are based on, the viscous effects of the boundary layer inside the s urf zone, and different breaker types. Further study should be done to more accurately predict the wave induced currents in the nearshore area. The coupled modeling system also successfully simulates the wave height and wave set up over a laboratory model of a fringing reef. While the simulated wave set up is in agreement with the laboratory data, wave induced currents over the fringing reef are also calculated by the coupled modeling system, and three different radiation stresses generate different mean fl ow patterns. Using the CH3D SSMS and the M08 depth dependent radiation stress, storm surge and currents during Hurricane Isabel are also simulated. Waves contribute significantly for the surge level. The M08 depth dependent radiation stress formulation produces similar storm surge but slightly impro ved the currents which are contributed by various processes (e.g., wind, tide and waves), in comparison with data and previous results obtained by depth independent radiation stresses. PAGE 63 63 A PPENDIX A TRANSFORMATION OF EQUATIONS O F MOTION FROM CARTESIAN GRID TO THE VERTICALLY STRETCHED GRID To simulate the circulation in water bodies with gradual bathymetric variation, it is common to use transformation such that both the free surface and the bottom b ecome the coordinate surfaces with an equal number of coordinate surfaces in between. This transformation, the so called stretching, leads to a smooth representation of the topography with the same order of vertical resolution for the shallow and deeper parts of the water body. The grid model is suitable for simulating flow and salinity transport in regions of gradual bathymetric variation and gives a more accurate estimation of bottom stress than the z grid model, which resolves the depth with stair s tep grids. The transformation from Cartesian grid ( x y z t ) to the vertically stretched grid is defined as: (A.1) W here h is the water depth relative to mean sea level, and is the free surface elevation relative to mean sea level, and is the transformed vertical coordinate such that =0 at the free surface and =1 at the bottom. Using the chain rule, the following partial diff erential expression can be derived: (A.2) w here The first order differential operators in the coordinate system are related to those in the coordinate system as: (A.3) PAGE 64 64 (A.4) (A.5) (A.6) w here After a pplying the transformation rules and non dimensionalizing the variables t he dimensionless equations in vertically stretched grid are: (A.7) (A.8) (A.9) PAGE 65 65 w here the vertical velocity, is: (A.10) w hich is a result of the coordinate transformation. PAGE 66 66 A PPENDIX B TRANSFORMATION OF THE RADIATION STRESS TERM FROM CARTESIAN COORDINATE TO CURVILINEAR COORDINATE Cartesian coordinates are sometimes employed to simulate the process in relatively simple domain, for example, a rectangular basin. However, in reality, the geo metry of the coastal area ( shoreline, estuaries or rivers ) is rather complicated and the Cartesian coordinates cannot represent the boundary accurately. Therefore, for very complex shaped domains, curvilinear coordinates are often used to get the grids bou ndary fitted. When this approach is used, we need to modify not just the variables, but also our governing equations. One of the widely used coordinate systems is the curvilinear coordinates for these types of transformation. Cylindrical, spherical or pola r coordinates are some of the special cases of the curvilinear coordinates. For example, if you are modeling a sphere, using a spherical coordinate system would be advantageous as compared to the Cartesian coordinate system. The equations of motion in non orthogonal curvilinear co ordinate are shown in Equations 2.5~2.7 Here we will discuss how to transform the depth dependent radiation stress term from Cartesian coordinate to curvilinear coordinate. Th e transformation requires the transformation of variabl es, and partial derivatives of the variables. The Cartesian forms of the variables are given with and the curvilinear coordinates are designated with the variables The transformation between these variables can be done using the transformation matrix and inverse transformation matrix. The transformation matrix, which transforms is given as : PAGE 67 67 ( B .1) While the inverse transformation matrix, which transforms is give as; ( B .2) Perpendicular vectors in curvilinear coordinates are defined by the contravariant vector components. Each one of the contravariant velocity components (at each face) can be defined as a transformation of Cartesian vector components. The basic vectors for transforming from Car tesian to curvilinear coordinate are: The contravariant length vector: ( B .3) ( B .4) The covariant length vector: ( B .5) ( B .6) T he radiation stress tensor is: ( B .7) With the contravariant and covariant vectors, t he contravariant radiation stress tensor can be obtained by: PAGE 68 68 ( B .8) ( B .9) ( B .10) The governing equation s for CH3D in Cartesian coordinate are shown in Equations 2. 1 ~2.3 The radiation stress terms in vertically stretching grid, as shown in Appendix C, are: ( B .11) ( B .12) a) The transformation of the radiation stress term into curvilinear coordinate The momentum equation with the local acceleration and radiation stress terms in Cartesian can be written in vector form: ( B .13) Applying th e tensor calculation rules, the equation s of motion in vector form can be written as: ( B .14) w here Therefore, the vector form of momentum equation in curvilinear coordinate is: PAGE 69 69 ( B .15) The equation of motion in curvilinear coordinate after non dimensionalization is: ( B .16) ( B .17) b) The transformation of the radiation stress term into curvilinear coordinate The momentum equation with local acceleration and radiation stress terms is : ( B .18) ( B .19) In the transformation from ( x y ) to the form of the would not change : PAGE 70 70 ( B .20) So the equations can be written as: ( B .21) ( B .22) Rewrite the above equations into vector form: ( B .23) ( B .24 ) ( B .25 ) ( B .26) Transform the equations into curvilinear coordinate: ( B .27) PAGE 71 71 ( B .28) Therefore, the radiation stress terms in vertical stretching and non orthogonal curvilinear coordinate are: ( B .29) ( B .30) PAGE 72 72 APPENDIX C BOUNDARY CONDITIONS Boundary conditions are required to compute the variables at the boundary. If the closed boundary is fixed in space, the component of the fluid velocity normal to it should be zero ( where is the unit vector normal to the boundary). I f the fluid is viscous, the no slip condition ( where is the unit vector tangential to the boundary) is applied. The closed and no slip conditions are made so that the fluid at the boundary having zero velocity with respect to the boundary. At the closed boundary, no normal flux of momentum, salt or energy e xists since the velocity is zero. T h e kinematic and dynamic boundary conditions without considering the wave effects at the free surface are specified as: ( C 1 ) (C.2) (C.3) w here is water elevation, and wx w y are wind shear stresses at the free surface, and can be computed from the quadratic law: ( C 4 ) ( C 5 ) where and are wind speed components. is the drag coefficient of air, which can be calculated using Garratt (1977) formulation: PAGE 73 73 ( C 6 ) The kinematic and dynamic boundary conditions without considering the wave effects at the bottom are: ( C 7 ) ( C 8 ) ( C 9 ) w here b x by are the bottom shear stresses, which can be given by the quadratic law: ( C 10 ) ( C 11 ) where and are bottom velocities and is the drag coefficient which is defined usin g the formulation by Sheng (198 6 ): ( C 12 ) where is the von Karman constant. The formulation states that the coefficient is a function of the size of the bottom roughness, and the height at which is measured, is within the constant flux layer above the bottom. The size of the bottom roughness can be expressed in terms of the Nikuradse equivalent sand grain size, using the relation In the two dimensional mode, the bottom boundary conditions are given using the Chezy Manning formulation : PAGE 74 74 ( C 13 ) ( C 14 ) where is the Chezy friction coefficient defined as ( C 15 ) where is the hydraulic radius which can be approximated by the total depth given in centimeters, and is Manning's n. Wave Enhanced Surface Stress Wave enhanced surface roughness, and drag coefficient, developed by Donelan et al. (1993), are used to calculate w ind stress at the free surface B oth the surface roughness and the drag coefficient are functions of wave age and roughness increases when the waves are young and makes the wind stress higher compared to when the waves are fully developed. ( C.1 6 ) where is the wind speed at 10 m above air sea interface. Follow ing the relation between and yields the wave enhanced drag coefficient ( C. 1 7 ) PAGE 75 75 where is wave phase speed and represents the inverse wave age. Wave E nhanced Bottom Stress Two methods have been implemented in CH3D for the wave enhanced bottom stress. T h e first one applies a simplified formulation by Signell et al. (1990) based on the Gran t and Madsen (1979) theory for a wave averaged bottom boundary layer. T he second method resolves a turbulent wave current boundary level by using a comprehensive look up table for wave current bottom stress developed with a turbulent closure model of Sheng and Villaret (1989). The Grant and Madsen (1979) formulation is given by the quadratic law, but is the wave enhanced drag coefficient. ( C. 18 ) ( C. 1 9 ) The main assumption use d in the formulation is that for a co linear flow, the maximum bottom shear stress is defined as ( C. 2 0 ) where is the bottom stress due to current and is the maximum stress due to waves which can be determined from ( C. 2 1 ) where is the near bottom wave orbital velocity ; is the wave friction factor which depends on the bottom roughness, The final expression for the wave enhanced drag coefficient at the reference height is PAGE 76 76 ( C. 2 2 ) Follow ing Signell et al. (1990), the reference height was specified as 20 cm and cm 3 at one meter above the bed in the absence of waves. T he effective drag coefficient is used to compute bottom stress as defined by E quations ( C 1 8) and ( C 1 9). The second formulation uses a turbulent closure model (Sheng and Villaret, 1989) to calculate the wave current bottom shear stress inside a turbulent wave current bottom boundary layer The wave resolving governing equations for the combined wave current bottom boundary layer are : ( C. 2 3 ) ( C. 2 4 ) with the following bottom b oundary conditions: ( C. 2 5 ) ( C. 2 6 ) where are velocity components at the lowest grid point near the bottom and is computed by: ( C. 2 7 ) where is the bottom roughness and is the von Karman constant. PAGE 77 77 Boundary conditions at the top of the bottom boundary layer are: ( C. 28 ) ( C. 29 ) To drive a wave induced oscillatory motion inside the boundary layer a pressure gradient from the linear wave theory is applied: ( C. 30 ) ( C. 3 1 ) where g is gravitational acceleration, k is wave number, is wave height, is wave direction, and is angular wave frequency. To drive a current inside the boundary layer a constant pressure gradient is applied in the y direction: ( C. 3 2 ) The vertical turbulent eddy viscosity i nside the turbulent wave current boundary layer i s determined using a TKE closure model developed by Sheng and Villaret (1989) T he water depth is defined as half of the vertical grid spacing subtracted by the roughness length, and the wave height corresponded to the point is determined according to the linear wave theory: ( C. 3 3 ) PAGE 78 78 where is local water depth, is water surface elevation, and is wave height at the surface. Wave E nhanced T urbulent M ixing Turbulence is of great significance to the physical process of the ocean. In the surf zone, t h e role of turbulence is more apparent. Wave breaking on a sloping breach cause a large amount of energy to release and turn into turbulence. The generation of turbulence by breaki ng wave starts at the toe of the breaking wave front and takes place mainly inside and in the surface roller. Turbulence is then spread downwards by convection. As pointed out by Svendsen (1987), the intensity of turbulence is fairly constant below the wav e trough level. However, many important characteristics of turbulence in the surf zone are still not well understood. In this study, a constant horizontal eddy viscosity coefficient is used. For the vertical eddy viscosity, the equilibrium turbulence closu re scheme developed by Sheng and Villaret (1989) was modified to account for wave effects. Following Battjes (1975) and Vriend and Stive (1987), the wave enhanced vertical eddy viscosity is as follows: ( C.3 4 ) Where A zc is the eddy viscosity related to the mean currents as computed by the equilibrium closure model implemented in the CH3D model. D b is the wave energy dissipation resulted from wav e breaking and bottom friction, h is the water depth and M is constant. The wave energy dissipation, D b was calculated according to Battjes and Janssen (1978) as follows: ( C.3 5 ) PAGE 79 79 W here Q b is the fraction of breaking wave ; H m is the maximum wave height that can exist at this depth, and T is the wave period. The fraction of breaking waves was calculated from the following implicit relation: ( C.3 6 ) Where E tot is the total wave energy. Besides the wave energy dissipation D b due to wave breaking and bottom friction, the roller dissipation D r plays an important role in affecting the turbulent process in the surf zone (Mocke 2001). The rate of energy dissipation in the wave roller ( D r ), according to Narin et al. (1990) is calculated as follows: (C. 3 7 ) W here E r is the roller energy, is a measure of the mean slope of the roller on the front face of the wave, g is the acceleration of gravity and c is the phase speed of the wave. PAGE 80 80 APPENDIX D THE TURBULENCE CLOSURE MODEL The equations for the equilibrium closure model (Sheng and Villaret 1989) are: ( D .1) ( D .2) ( D .3) is the turbulence length scale, earth rotation T he coefficients contained in the above equations were determined from laboratory data, and remain constant in the application of the equilibrium closure model. q 2 can be determined from the following equation when the mean flow are known (Sheng and Villaret 1989): ( D .4) Where A b and s are constants and have the values of 0.75, 0.125 and 1.8 respectively, and: ( D .5) ( D .6) PAGE 81 81 The total root mean square turbulent velocity q can then be obtained from ( D .7) Where is the von Karman constant, is the shear velocity of wave and current boundary layer. After the mean flow variables are determined. The vertical eddy viscosity coefficients can then be computed from: ( D .8) ( D .9) ( D .10) ( D .11) ( D .12) The turbulent macro scale is often assumed to satisfy a number of integral equations. First of all, it is assumed to be a linear function of the vertical distance immediately above the bottom or below the free surface. In addition, it must satisfy the following relationships: ( D .13) ( D .14) ( D .15) ( D .16) PAGE 82 82 ( D .17) Where C 1 is usually between 0.1 and 0.25, H is the total depth, H p is then depth of pynocline; C 2 ranging from 0.1 to 0.25, is the fractional cut off limitation of turbulent macro scale b ased on the spread of the turbulence determined from the turbulent kinetic energy profile, and N is the Brunt Vaisala frequency, defined as: ( D .18) PAGE 83 83 LIST OF REFERENCES Alymov, V. V., 2005. Integrated modeling of storm surge during Hurricanes Isabel, Charley and Frances. 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A numerical study on the effect of wave s c urrent interactions on the storm surge and inundation associated with Hurricane Hugo 1989. Ocean Modelling 20, 252 269. Zhang, M. Y. Li, Y. S. 1996 The synchronous coupling of a third generation wave model and a two dimensional storm surge model. Ocean Eng. 23:533 543. PAGE 87 87 BIO GRAPHICAL SKETCH Tianyi Liu was born in 1986 in Liaocheng, Shandong Province, China. He started college in the Department of Naval Architecture and Ocean Engineering in Huazhong University of Science and Technology in 2004, and received a Bachelor of Engineering degree in 2008. In th e fall of 2 008, he entered the Coastal and Oceanographic E ngineering program in University of Florida for his graduate study. 