THREE DIMENSIONAL SIMULATION OF WAVE INDUCED CIRCULATION WITH A DEPTH DEPENDENT RADIATION STRESS

Transcription

1 THREE DIMENSIONAL SIMULATION OF WAVE INDUCED CIRCULATION WITH A DEPTH DEPENDENT RADIATION STRESS By TIANYI 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

2 2010 Tianyi Liu 2

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. Alexandru Sheremet for 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. Finally, I wish to express my most sincere gratitude to my family members. 3

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... 3 LIST OF TABLES... 6 LIST OF FIGURES... 7 ABSTRACT... 9 CHAPTER 1 INTRODUCTION page 1.1 Background Literature Review Objectives Organization METHODOLOGY A Three-Dimensional Circulation Model CH3D Local Wave Model: SWAN The Coupling Process between CH3D and SWAN THE FORMULATIONS OF RADIATION STRESS The Vertically Integrated Radiation Stress The Depth-Dependent Radiation Stress by Xia et al. (2004) The Depth-Dependent Radiation Stress Formulation by Mellor (2008) The Vertical Distribution of the Depth Dependent Radiation Stress TEST SIMULATIONS Wave Set-up The Undertow Test The Wave Set-up on Fringing Reef Simulation of Hurricane Isabel CONCLUSION APPENDIX A TRANSFORMATION OF EQUATIONS OF MOTION FROM CARTESIAN GRID TO THE VERTICALLY-STRETCHED GRID

5 B TRANSFORMATION OF THE RADIATION STRESS TERM FROM CARTESIAN COORDINATE TO CURVILINEAR COORDINATE C BOUNDARY CONDITIONS Wave-Enhanced Surface Stress Wave-Enhanced Bottom Stress Wave-Enhanced Turbulent Mixing D THE TURBULENCE CLOSURE MODEL LIST OF REFERENCES BIOGRAPHICAL SKETCH

6 LIST OF TABLES Table page 3-1 Wave conditions Locations of measurements and water depth The relative RMS error for the simulated current velocities The relative RMS error for the simulated turbulent kinetic energy Studies with different radiation stress formulations

7 LIST OF FIGURES Figure page 2-1 The coupling process between CH3D and SWAN (a) Distribution of depth dependent radiation stress for the wave condition of Case (b) Distribution of depth dependent radiation stress for the wave condition of Case (c) Distribution of depth dependent radiation stress for the wave condition of Case (d) Distribution of depth dependent radiation stress for the wave condition of Case (a) The comparison between analytical and numerical solution for the wave set-up (b) The cross-section of the basin (a) Comparison between the data and numerical results for wave height (b) Comparison between the data and numerical results for wave set-up (c) The cross-section of the basin (a) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using M08 radiation stress (b) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using X04 radiation stress (c) Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using LHS radiation stress (a) Comparison between the simulated and measured current velocity value at Station (b) Comparison between the simulated and measured current velocity value at Station (c) Comparison between the simulated and measured current velocity value at Station (d) Comparison between the simulated and measured current velocity value at Station (e) Comparison between the simulated and measured current velocity value at Station (f) Comparison between the simulated and measured current velocity value at Station (g) Comparison between the simulated and measured current velocity value at Station (a) Wave induced currents simulated by using M08 radiation stress

8 4-5(b) Wave induced currents simulated by using X04 radiation stress (c) Wave induced currents simulated by using LHS radiation stress (a) Comparison of turbulent kinetic energy between model results and data at Station (b) Comparison of turbulent kinetic energy between model results and data at Station (c) Comparison of turbulent kinetic energy between model results and data at Station (d) Comparison of turbulent kinetic energy between model results and data at Station (e) Comparison of turbulent kinetic energy between model results and data at Station (a) Comparison between the data and numerical results for wave height (b) Comparison between the data and numerical results for wave set-up (c) The cross-section of the basin (a) Wave induced currents simulated by using M08 radiation stresses (b) Wave induced currents simulated by using X04 radiation stresses (c) Wave induced currents simulated by using LHS radiation stresses Best track of Hurricane Isabel. (Courtesy of the NHC) 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 Measured and simulated water levels at six stations Measured (a) East-West and (b) North-South currents at Kitty Hawk station Simulated (a) East-West and (b) North-South currents at Kitty Hawk station by using LHS radiation stress formulation Simulated (a) East-West and (b) North-South currents at Kitty Hawk station by using M08 radiation stress formulation Measured and simulated onshore-offshore currents at Kitty Hawk during Hurricane Isabel

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 CIRCULATION WITH A DEPTH DEPENDENT RADIATION STRESS Chair: Y. Peter Sheng Major: Coastal and Oceanographic Engineering By Tianyi Liu December 2010 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 dynamically coupled to the model SWAN (Simulating Waves 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 also successfully simulates the wave setup 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 9

10 formulation produces similar storm surge but slightly improved currents, in comparison with data and previous results obtained by depth independent radiation stresses. 10

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 wave induced currents are essential to the bottom sediment transport and shoreline changes. Ocean surface waves are the direct demonstration of unsteady ocean motions. Waves 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 shoaling, refraction, diffraction and dissipation, would occur, and the waves finally break in shallow waters, with the energy dissipated in the surf zone. The gradients of the wave heights and momentum produce the wave-induced nearshore circulation, such as the wave induced set-up, set-down and currents. Longshore variation in wave breaker heights produces longshore currents which cause longshore sediment transport. Wave forcing 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. The term that couples the wave modeling and circulation modeling is the momentum flux produced by waves. The momentum flux, commonly 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 11

12 distributions. To carry out fully 3D simulations by coupling wave models and 3D circulation models, the radiation stress term is necessary to have vertical distributions. Recently, depth dependent radiation stress formulations (Xia et al.., 2004; Mellor, 2003 and 2008) have been developed based on linear wave theory. These depth dependent radiation stresses, when vertically integrated, are consistent with the conventional radiation stress (Longuet-Higgins and Stewart, 1964). In this study, the depth dependent radiation stresses (Xia et al.., 2004; Mellor, 2008) are incorporated into a three-dimensional 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 Waves Nearshore) (Booij et al.., 1999). By comparing simulated vs. observed wave induced circulation, the 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 Review 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. (1984a,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. The model results indicate that the inclusion of the radiation stress improves the accuracy of the computed results slightly by 2%. 12

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 simulated 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 (Dongeren 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 these studies can yield vertically 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. The 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 questioned by Mellor (2003) and Xia et al. (2004). Sheng and Alymov (2002) incorporated the LHS radiation stresses in the CH3D (Sheng, 1986) model and simulated wave setup fields during the 100-year storm event for two study areas in Pinellas County, Florida. The study showed that the grid resolution has some effect on calculated wave setup especially in the areas where bathymetry has steeper gradients. Sun and Sheng (2002) coupled CH3D with the REF/DIF wave model (Kirby and Dalrymple, 1994) and showed significant effects 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- 13

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 account 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-Higgins 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 models and 3D circulation models. Xia et 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 compares well with analytical solution, and the 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. 14

15 Using POM-SWAN and the M04 radiation stress, Xie et al. (2008) 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 waves during Hurricane Hugo, and found that the effect of wave-surge interaction on wave is significant in shallow coastal 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 in 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 process 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 the differences of the depth dependent radiation stress 15

16 distributions are compared and their characteristics are described. In Chapter 4, a few test simulations are conducted by this coupling system, and the model results are compared 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. 16

17 CHAPTER 2 METHODOLOGY 2.1 A Three-Dimensional Circulation Model CH3D CH3D (Curvilinear-grid Hydrodynamics in 3D), a robust three dimensional circulation model originally 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 shoreline and geometry, and a terrain-following σ-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 advective schemes QUICKEST (Leonard, 1979) 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 equation. 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 equation 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: u v w 0 x y z (2.1) 17

18 u uu uv uw 1 Sxx 1 Sxy 1 Pa g t x y z x y x x w w w 2 2 g u u u dz fv A z H A 2 2 V w x x y z z ( ) ( ) (2.2) v vu vv vw 1 Syx 1 Syy 1 Pa g t x y z x y y y w w w 2 2 g v v v dz fu A z H A 2 2 V w y x y z z ( ) ( ) (2.3) The hydrostatic assumption states: p g z (2.4) where u( x, y, z, t ), v( x, y, z, t) and w( x, y, z, t ) are the velocity vector components [LT -1 ] in x-, y-, and z-coordinate directions, respectively; t is time [T]; ζ(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 boundary 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 sigmastretching coordinate, which allows for 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: t g 0 H ( g Hu g Hv 0 ) ( 0 ) 0 (2.5) 18

19 1 Hu P P g g H t g g ( g g ) ( g g ) ( u v) 0 0 R 0 c c c c x ( y g 0 S xx y g 0 S xy ) ( y g 0 S xy y g 0 S yy ) g 0 c c c c y ( x g0sxx x g0 Sxy ) ( x g0sxy x g0 S yy ) R g H S S xx 2 xy Syy 2 y 2 x y x H H S S xx xy Syy y y ( x y x y ) x x R x ( y g Huu y g Huv) ( y g Huv y g Hvv) gh 0 y ( x g0 Huu x g0 Huv) ( x g0 Huv x g0 Hvv) g Ev u ( A 2 v ) EH AH (Horizontal Diffusion of u) H R H 12 H H ( g g ) d ( g g )( d p) 2 F r 0 Hu (2.6) 19

20 1 Hv P P g g H t g g ( g g ) ( g g ) ( u v) 0 0 R 0 c c c c x ( y g 0 S yx y g 0 S yy ) ( y g 0 S yx y g 0 S yy ) g 0 c c c c y ( x g0sxx x g0 S yx ) ( x g0s yx x g0 S yy ) R g H S S xx xy Syy y y ( x y x y ) x x H H S S S xx 2 xy yy 2 y 2 x y x R x ( y g Huv y g Hvv) ( y g Huv y g Hvv) gh 0 y ( x g Huu x g Huv) ( x g Huv x g Hvv) g Ev v ( A 2 v ) EH AH (Horizontal Diffusion of v) H R H 22 H H ( g g ) d ( g g )( d p) 2 F r Hv (2.7) where, and are the transformed coordinates; z ( x, y, t) h( x, y) ( x, y, t) u, v, w are non-dimensional contra-variant velocities in curvilinear grid (,, ). g 0 is the Jacobian of horizontal transformation; g 11, g 22, g, g, g are the metric coefficients of coordinate transformations; is non-dimensional parameter; is water level; 20

21 The dimensionless variables are: * * * ( x, y, z ) ( x, y, zx r / Zr ) / X r * * * ( u, v, w ) ( u, v, wx r / Zr ) / Ur * * * * w w ( x, y ) ( x, y ) / ( 0 fzru r ) ( x, y ) / r * t tf * g / ( fu X ) r r S S / ( U ) * 2 ij ij w r * 0 r 0 ( ) / ( ) A A / A * H H Hr A A / A * V V Vr * Xr / Ur The dimensionless groups are: Vertical Ekman Number: Lateral Ekman Number: E A fz 2 V vr / ( r ) E A fx 2 H Hr / ( r ) Vertical Prandtl Number: P A / K v vr vr Lateral Prandtl Number: P A / K H Hr Hr Vertical Schmidt Number: S A / D v vr vr Later Schmidt Number: S A / D H Hr Hr Froude Number: F U / ( gz ) r r r 1/2 Rossby Number: R U / ( fx ) o r r 21

22 Densimetric Froude Number: F F / When wave effects are considered in the simulation, the velocities are divided into a current component and Stokes drift after phase averaging (Mellor, 2008): u uˆ u, w wˆ (2.8a,b) S Where α denotes the horizontal coordinates, ( uˆ, ˆ w) is the current component of the velocity, us is the Stokes drift. The boundary conditions and wave implementations in CH3D are shown in Appendix C. 2.2 Local Wave Model: SWAN The wave field information for the nearshore circulation is provided by the wave model SWAN. The SWAN (Simulating WAves Nearshore) model (Booij et al., 1999) is a thirdgeneration 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 represents 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 nonlinear 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- 22 rd r

23 induced wave breaking. In deep water the evolution of the spectrum is dominated by the wavewave 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 dissipated 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 significant 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 bottom 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, shoaling 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: generation 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. 23

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(σ,θ) equal to the energy density divided by the relative frequency: N(σ,θ)=E(σ,θ)/σ. The evolution of the wave spectrum is described by the spectral action balance equation: S N cxn cyn cn cn t x y (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 term represents shifting of the relative frequency due to variations in depths and currents (with propagation velocity c σ in σ- space). The fifth term represents depth-induced and current-induced refraction (with propagation velocity c θ in θ-space). The expressions for these propagation speeds are taken from linear wave theory. The term S (=S(σ,θ)) at the right hand side of the action balance equation is the source term in terms of energy density representing the effects of generation, dissipation and nonlinear wave-wave interactions, etc The Coupling Process between CH3D and SWAN The circulation and wave models, CH3D and SWAN, are dynamically coupled in the way that SWAN provides the wave field information like wave height, period, and directions at each grid cell, and the 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 24

25 propagation 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 25

26 CHAPTER 3 THE FORMULATIONS OF RADIATION STRESS 3.1 The Vertically Integrated Radiation Stress 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 setdown 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 waves by the mean flux in the absence of the waves, which are: xx 0 2 ( ) h h 0 (3.1) S p u dz p dz yy 0 2 ( ) h h 0 (3.2) S p v dz p dz S xy uvdz (3.3) h S yx vudz (3.4) h Where h is the water depth, ζ is the water elevation of the free surface, ρ 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 expressions for radiation stress in Equations 3.1~3.4 can be simplified as: Sxx E[ n(cos 1) ] (3.5) Syy E[ n(sin 1) ] (3.6) E Sxy Syx nsin 2 (3.7) 2 26

27 Where E is the wave energy, θ is the angle wave propagating to the onshore direction and n is the ratio of group velocity to wave celerity: n=(1+2kh/sinh2kh)/2. The radiation stress can be divided into a momentum part S m and a pressure part S p according to Haas and Svendsen (2000), and rewritten as: S e S S m P (3.8) 2 cos w cos w sin w e 2 sinw cos w sin w (3.9) 1 (1 ) ; S 16 2 Sm gh G p gh G (3.10) Where α β is the wave angle to the x- coordinate, G=2kh/sinh2kh; k is the wave number, h is the water depth. The wave induced mass transport is given in Svendsen (1984b) as: Q gh c k k 2 w B0 (3.11) Where, α 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 (1984b) 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 represents an increase in radiation stress which according to Svendsen (1984b) can be written as: S h 0.9 gh L (3.12) ' 2 m b The wave induced mass transport inside the surf zone is: 2 2 gh c A h k Q w ( B0 ) 2 c gh H L k (3.13) 27

28 Where, L is the wave length and A is the area for the surface roller of breaking waves. A is given as: A=0.06HL (Okayasu et al.., 1988). 3.2 The Depth-Dependent Radiation Stress 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 integration of his formulation is consistent with the formulation by Longuet-Higgins and Stewart (1964), Phillips (1977). From the vertically integrated LHS radiation stress, Xia et al. (2004) developed the new formulation in vertically stretching σ coordinate, with σ=(z-ζ)/(h+ζ), and assumed σ=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: k k S z E k z h E k z h sinh 2kh sinh 2kh Ez k( z h)sinh k( z h) E cosh k( z h) E [1 ] 2 xx ( ) [cosh 2 ( ) 1]cos [cosh 2 ( ) 1] 2 D hcosh kh D cosh kh k k S z E k z h E k z h sinh 2kh sinh 2kh Ez k( z h)sinh k( z h) E cosh k( z h) E [1 ] 2 yy ( ) [cosh 2 ( ) 1]sin [cosh 2 ( ) 1] 2 D hcosh kh D cosh kh (3.14) (3.15) k Sxy ( z) E [cosh 2 k( z h) 1]sin cos (3.16) sinh 2kh S ( z) S ( z) (3.17) yx xy Where, E is the wave energy, k is the wave number, h is the water depth. θ is the wave angle to the x- coordinate. 28

29 3.3 The Depth-Dependent Radiation Stress Formulation by Mellor (2008) Mellor (2008) developed the depth-dependent radiation stress term by deriving the three dimensional continuity and momentum equations when waves 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 (1977). 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: kk S ke( F ) 2 CS FCC FSC FSS ED (3.18) k F F F F SS CS SC CC sinh k( z h) (3.19a) sinh kd cosh k( z h) (3.19b) sinh kd sinh k( z h) (3.19c) cosh kd cosh k( z h) (3.19d) cosh kd Where, α, β 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: E 0 if z D and E E h Ddz (3.20) The Vertical Distribution of the Depth Dependent Radiation Stress The X04 and M08 depth dependent radiation stress formulations, when vertically integrated, are in agreement with previous studies (Longuet-Higgins and Stewart, 1964; Phillips, 1977). However, their vertical distributions are different in deep or shallow waters. The wave conditions for checking the distributions are listed in Table

30 In Table 3-1, k is 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 shown 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 (σ=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 discontinuous 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 M08 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 M08 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, when waves travel normally to the coast in Figure 3-1(c), S xx predicted by M08 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 30

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 using X04 formulation, S xx and S yy increases almost linearly from the surface to the bottom, which may indicate stronger wave effects 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 M08 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 distributions of S xy and S yx predicted by the two formulations have the same style, which decreases exponentially from the surface to the bottom. The comparison of the two formulations in Figure 3-1 indicates that the two radiation stresses 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. 31

32 Table 3-1. Wave conditions Wave height (m) Wave period (s) Wave angle ( ) Depth (m) Case Deep water Case Deep water Case Shallow water Case Shallow water kd 32

33 Figure23-1(a). Distribution of depth dependent radiation stress for the wave condition of Case 1. 33

34 Figure 3-1(b). Distribution of depth dependent radiation stress for the wave condition of Case 2. 34

35 Figure43-1(c). Distribution of depth dependent radiation stress for the wave condition of Case 3. 35

36 Figure53-1(d). Distribution of depth dependent radiation stress for the wave condition of Case 4. 36

37 CHAPTER 4 TEST SIMULATIONS 4.1 Wave Set-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 water 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: dsxx dx d gh ( ) 0 (4.1) dx The analytical solutions of wave set-up inside the surf zone and set-down outside the surf zone based on linear wave theory are: 2 1 ak (4.2) 2 sinh 2kh 2 3 / / 8 ( x) [ h h( x)] b b (4.3) where ζ 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, κ is the breaking index. 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 shown in Figure 4-1(b). The incident wave height is 0.6 m; the wave period is 5 s. 37

38 Three different methods of applying the radiation stress term are employed for comparisons. The 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 κ=0.73. The simulated wave set-up and analytical solution is shown in Figure 4-1(a), which suggests that the numerical results from the three methods do not have much difference, and all match well with the analytical solution. 4.2 The Undertow Test 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 Wind, 1986; Okayasu et al., 1988; Ting and Kirby, 1994). While some (e.g. Svendsen, 1984b; 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 measurements. Christensen (2006) used a large eddy simulation to study the turbulence and the undertow induced by spilling and plunging breakers; the 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 induced by waves over a sloped bottom; the simulated undertow compares well for some of the stations, however, the mass conservation in this study is 38

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 system and the performances of depth dependent radiation stresses, a laboratory experiment of undertow by Ting and Kirby (1994) is simulated. The experiment was conducted in a two dimensional wave tank, which is 40 m long, 0.6 m wide and 1.0 m deep. The experimental 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 height 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 friction 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 39

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: relative RMS error 1 N x i model max( x ) i data i x data 2 (4.4) Where 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 for 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. 40

41 The flow direction by using the LHS 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 match 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 characteristics, 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 reflections or equipment disturbance, and the uncertainties in the laboratory experiment may influence the accuracy of the data. 41

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 closure 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 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.34 for calculating wave enhanced eddy viscosity simplifies the process of wave induced turbulence, and may also bring errors. 4.3 The Wave Set-up on Fringing Reef 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. The study of wave setup over fringing reefs requires an accurate estimation 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 42

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 fringing 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. The cross-section profile of this reef-beach system is shown in Figure 4-7(c). One case of experiments 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 and variable bottom friction with bottom roughness z 0 =0.01, as well as wave enhanced eddy viscosity and bottom friction, are activated in the simulation. The 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 measurements 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 flow 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 43

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 the observations of undertow. The studies with different radiation stress formulations are summarized in Table 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 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 The coastal domain as shown in this figure has 548,240 grid cells. The 2-D vertically-averaged version of ADCIRC (Luettich et al , IPET ) 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 ). 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 open boundaries of 44

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 stress 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. Without 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 currents 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 The M08 radiation stress gives a different current field, in comparison with previous results, and the comparison in Figure 4-15 suggests slightly 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 using the M08 radiation stress may not dominate and make significant difference in the hurricane environment. 45

46 Table 4-1.2Locations of measurements and water depth Stations x (m) d (m) h (m) Table 4-2.3The relative RMS error for the simulated current velocities Mellor (2008) (%) Xia (2004) (%) LHS (1964) (%) Station Station Station Station Station Station Station Overall Table 4-3. The relative RMS error for the simulated turbulent kinetic energy Mellor (2008) (%) Xia (2004) (%) LHS (1964) (%) Station Station Station Station Station Overall

47 Table 4-4.5Studies with different radiation stress formulations Radiation Circ. Bottom Study stress model friction Zhang and Li (1995) Dongeren, et al.. (1994) Xie, et al.. (2001) Xie, et al.. (2008) LHS, 1964 LHS, 1964 LHS, 1964 Xia et al., DH Quasi- 3D 3D 3D Chezy- Manning formulation Chezy- Manning formulation Turbulent bottom boundary layer Turbulent bottom boundary layer Turbulent mixing Strength Weakness N/A Calculated from Coffey and Nielsen (1984) and Battjes (1975) Second-order turbulence closure scheme of Mellor and Yamada (1982) Second-order turbulence closure scheme of Mellor and Yamada (1982) Waves are considered in circulation model Provides vertical structure of currents Fully 3D simulation Fully 3D simulation Radiation stress has vertical distribution Simulation is 2D Vertical distribution of currents is not available Model is basically 2D Assuming vertically uniform distribution of radiation stress No wave induced flow outside the surf zone 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, D Turbulent bottom boundary layer Equilibrium 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 47

48 Figure64-1. (a) The comparison between analytical and numerical solution for the wave set-up; (b) The cross-section of the basin. 48

49 Figure74-2. Comparison between the data and numerical results for (a) wave height; (b) wave set-up. (c) the cross-section of the basin. 49

50 Figure84-3. Comparison between the simulated (black arrow) and measured (red arrow) current velocities by using (a) M08; (b) X04; (c) LHS radiation stresses 50

51 Figure94-4. 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. 51

52 Figure 4-5. Wave induced currents simulated by using (a) M08; (b) X04; (c) LHS radiation stresses. Figure 10 52

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 53

54 Figure 4-7. Comparison between the data and numerical results for (a) wave height; (b) wave setup. (c) the cross-section of the basin. Figure 12 54

55 Figure 4-8. Wave induced currents simulated by using (a) M08; (b) X04; (c) LHS radiation stresses. Figure 13 55

56 Figure 4-9. Best track of Hurricane Isabel. (Courtesy of the NHC)Figure 14 56

57 Figure 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 57

58 Figure Measured and simulated water levels at six stations. Figure 16 58

59 Figure Measured (a) East-West and (b) North-South currents at Kitty Hawk station. Figure 17 Figure Simulated (a) East-West and (b) North-South currents at Kitty Hawk station by using LHS radiation stress formulation. Figure 18 59

60 Figure Simulated (a) East-West and (b) North-South currents at Kitty Hawk station by using M08 radiation stress formulation. Figure 19 Figure Measured and simulated onshore-offshore currents at Kitty Hawk during Hurricane Isabel. Figure 20 60