THE INFLUENCE OF VERTICAL CURRENT STRUCTURE ON WIND DRIVEN SURGES IN THE NEAR SHORE REGION By AMANDA TRITINGER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2019
Â© 2019 Amanda Tritinger
To my supportive parents, and my loving teammate .
4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Don T. Resio, who has been an inspirational teacher as he has patiently steered me through my doctoral studies with steadfast support. Without his assistance and superlative knowledge this dissertation would not have been possible. I am also grateful to Dr. Maitane Olabarrieta , who shared her students, lab space, and expertise with me during my time at the University of Florida. I am thankful for my other committee members, Dr. Chris Massey, Dr. Don Slinn, and Dr. Deborah Miller , for providing helpful feedback and guidance during the dissertation process. I would also like to thank Dr. Arnoldo Valle Levinson, Dr. Alex Sheremet, Dr. Cigdem Akan, Dr. W illiam Dally, and Dr. Peter Adams, for sharing their un parallel ed oceanographic insight with me. I gratefully acknowledge the funding provided by the Taylor Engineering Research Institute ( TERI ) for my tuition and would like to thank the Coastal Resilienc e Center ( CRC ) for helping to support my research assistant stipend. In addition, I must thank Mrs. Hollis Klein for her ability to take on and tackle each of m y unexpected technical problems throughout my academic career. I am tha nkful to have had support from my parents, Susan and Dennis Tritinger, my brothe r, Justin, and my partner, Matt Verburg. I have also made many friends during my graduate studies, who are responsible for many great memories, and keeping me moving forward when I needed encouragement ; Dorukhan Ardag, Jackie Brower, Patrick Cooper, Armondo Laurel Castillo, Gisselle Guerra, Nikole Ward, Zhendong Cao, Christian Rojas, Ashley Norton, Gabriel Todaro, Shannon Kay, Abdallah Walid El Safty, Sergio Pena, Collin Reis, Luis Montoya , and Ahmad Yo usif.
5 TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 7 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION, MOTIVATION, BACKGROUND, AND OBJECTIVES ................ 12 1.1 Introduction ................................ ................................ ................................ ....... 12 1.2 Motivation ................................ ................................ ................................ ......... 14 1.3 State of the Art in Storm Surge Modeling ................................ .......................... 15 1.4 Review of Observations of Current 3D Structure ................................ .............. 18 1.5 Review of Theory of 3D Structure of Flow ................................ ........................ 19 1.6 Misspecification of Bottom Stress ................................ ................................ ..... 21 1.7 Objectives ................................ ................................ ................................ ......... 23 2 DEVELOPMENT OF 3D HYDRODYNAMIC MODEL ................................ ............. 26 2.1 3D Hydrodynamic Equations ................................ ................................ ............ 26 2.2 Vertically Resolved Governing Equations ................................ ......................... 26 2.3 Turbulence Closure ................................ ................................ ........................... 29 3 DEVELOPMENT OF 2D HYDRODYNAMIC MODEL ................................ ............. 35 3.1 2D Hydrodynamic Equations ................................ ................................ ............ 35 3.2 Two Dimensional Governing Equations ................................ ............................ 35 3.3 Finite Difference Method ................................ ................................ ................... 36 4 INVESTIGATION INTO VERTICAL CURRENT STRUCTURE IN THE NEARSHORE, AND LIKELY MISREPRESENTATION IN 2DDI MODELS ............. 40 4.1 Potential Misspecification of Bottom Friction ................................ ..................... 40 4.2 Range of Testing ................................ ................................ ............................... 41 4.3 Comparison of Vertically Resolved Behavior in the Nearshore ......................... 41 4.3.1 Test Case 1: Steady State Surface Slope at a Point ............................... 41 4.3.2 Test Case 2: Idealized Water Surface Simulated Along a Transect ........ 43 4.3.3 Test Case 3: Idealized Water Surface Finite Difference Run .................. 44 4.4 Discussion of 2D and 3D Comparison ................................ .............................. 45 4.5 Conclusion from 2D and 3D Comparison ................................ .......................... 48
6 5 ANALYSIS OF VERTICAL STRUCTURE DEPENDENCE ON TIME VARIATION ................................ ................................ ................................ ............ 56 5.1 Analysis of Simulated Vertical Structure ................................ ........................... 56 5.2 Time to Reach Steady State ................................ ................................ ............. 57 5.3 Time Dependence ................................ ................................ ............................. 60 6 IMPLICATIONS TO 2DDI SURGE MODEL APPLICATION ................................ ... 69 6.1 Proposed Multivariate Analysis ................................ ................................ ......... 69 6.2 Multivariate Analysis ................................ ................................ ......................... 71 6.3 Application to 2DDI Storm Surge Models ................................ .......................... 73 7 CONCLUSIONS ................................ ................................ ................................ ..... 81 7.1 Implications from 2DDI vs. VR Bottom Stress ................................ ................... 81 7.2 Future Work ................................ ................................ ................................ ...... 81 7.3 Final Remarks ................................ ................................ ................................ ... 83 APPENDIX A VERTICALLY RESOLVED NUMERICAL SIMULATION ................................ ......... 84 B FORTRAN FINITE DIFFERENCE MODEL SOURCE CODE ................................ . 9 2 C FORTRAN FINITE DIFFERENCE MODEL INPUT TEXT FILE SHALLOW ........... 98 D FORTRAN FINITE DIFFERENCE MODEL INPUT TEXT FILE DEEP ................. 99 E VERTICAL PROFILES FOR VARYING METEOLOGICAL INPUTS ..................... 100 F CROSS SHORE CURRENT TIME VARIATION ANALYSIS ................................ 105 G CROSS SHORE CURRENT MULTIVARIET ANALYSIS (EOF ANALYSIS) ........ 107 H VERTICAL STRUCTURE BOTTOM STRESS ( VSBS) EXAMPLE LOOK UP TABLE MAGNITUDE ................................ ................................ ......................... 119 I VERTICAL STRUCTURE BOTTOM STRESS (VSBS) EXAMPLE LOOK UP TABLE DIRECTION ................................ ................................ ............................ 121 LIST OF REFERENCES ................................ ................................ ............................. 123 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 132
7 LIST OF FIGURES Figure P age 1 1 Acoustic Doppler Current Profiler (ADCP) current profiles off the shore of Melbourne, Florida Burnette and Dally (2017).. ................................ ................ 24 1 2 Wind induced circula tion in the nearshore open coast due to longshore winds, and the response in the current vertical structure. ................................ ... 25 2 1 Fluid element for which momentum is tracked, at each depth throughout the model. ................................ ................................ ................................ ................. 32 2 2 Layers of the fluid element shown in Figure 2 1, at each time it eration during numerical simulation. ................................ ................................ .......................... 33 2 3 Implicit pressure gradient that balances out the slope term at each layer. ......... 34 3 1 Finite difference coordinate and grid system used to conserve energy, momentum and mass in numerical model produced in this chapter. .................. 39 4 1 Test case 1 range of testing to be simulated with highly resolved code developed based on the governing and boundary equations shown in Equations 2 1 through 2 15. ................................ ................................ ............... 51 4 2 24 selected simulations from Test Case 1 ................................ .......................... 52 4 3 Results from one of the simulations from test case 2, showing the surface surge generated by vertically resolved vs. depth averaged numerical simulations.. ................................ ................................ ................................ ........ 53 4 4 Results from Test Case 3.. ................................ ................................ ................. 54 4 5 ( a) Mean bottom currents (at 8.22 meters depth) versus wave height, and (b) Mean bottom currents (at 8.22 meters depth) versus wind speed off the coast of Duck, NC.. ................................ ................................ ................................ ...... 55 5 1 Scaled analysis, the maximum velocity, at the top layer of the water column, is used to non linearize the velocity profile through depth, for wind speeds of 5, 10, 20, and 40 m/s ................................ ................................ .......................... 62 5 2 Relaxation time in both directions, and values are computed to represent the difference between the current time step and the last one.. ......... 63 5 3 Hind Cast wind speed and direction values from ADCIRC model of Hur ricane Matthew.. ................................ ................................ ................................ ............ 64
8 5 4 Hurricane Matthew storm track over WNAT + fine St. Johns Taylor Engineering Inc. grid. ................................ ................................ .......................... 65 5 5 Max elevations (ft.) in North Florida produced during Hind Cast of Hurricane Matthew ADCIRC run. ................................ ................................ ........................ 66 5 6 Simulated cross shore vertical current profiles for 120 minutes of run time, with 20 minute intervals. ................................ ................................ ..................... 67 5 7 Simulated cross shore vertical current profiles for 120 minutes of run time, with 20 minute intervals. The simulations were for (a) wind speeds of 5 m/s with a 5 meter depth. ................................ ................................ .......................... 68 6 1 For a run with 20 m/s onshore winds, at a depth of 5m. ................................ ..... 75 6 2 A 2 hour run with 20 m/s onshore winds, at a depth of 5m ................................ . 76 6 3 The first 15 minutes of a 2 hour run where winds are 20 m/s onshore winds, at a depth of 5m ................................ ................................ ................................ .. 77 6 4 The second 15 minutes of a 2 hour run where winds are 20 m/s onshore winds, at a depth of 5m. ................................ ................................ ..................... 78 6 5 Example bottom stress solution for estimated wind speeds and wind directions by the steady state vertically resolving numerical simulations at a depth of 5 meters. t. ................................ ................................ ............................ 79 6 6 Example bottom stress solution for estimated wind speeds and wind directions by the steady state vertically resolving numerical simulations at a depth of 5 meters. ................................ ................................ ............................... 80
9 LIST OF ABBREVIATIONS 2D 2D DI 2DVR Two Dimensional Two Dimensional Depth Integrated Two Dimensional Vertically Resolved 3D ADCIRC BFC EOF FDM NOAA PCA ROMS VR VS Three Dimensional ADvanced CIRCulation Model Bottom Friction Coefficient Eigen Orthogonal Function Finite Difference Model National Oceanographic and Atmospheric Association Principle Component Analysis Regional Ocean Modeling System Vertically Resolved Vertical Structure VSBS WNAT Vertical Structure Bottom Stress West North Atlantic Tidal
10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE INFLUENCE OF VERTICAL CURRENT STRUCTURE ON WIND DRIVEN SURGES IN THE NEARSHORE REGION By Amanda Sue Tritinger May 2019 Chair: Don ald T. Resio Cochair: Maitane Olabarrieta Lizaso Major: Coastal and Oceanographic Eng ineering 2 Dimensional Depth Integrated ( 2DDI ) models are unable to represent the bottom stress in the nearshore region accurately , mainly because the bottom stress the bottom boundary condition between the ocean floor and the water column is compute d with depth averaged flow . However, the vertical current structure, which often varies significantly in direction and magnitude with respect to the mean current. T his study investigates the general differences in storm surge estimation between two dimens ional ( 2D ) and three dimensional ( 3D ) surge models in open coast areas. A simplified 3D model will be implemented , which is referred to as a two dimensional vertically resolved (2DVR) model, in which it is tacitly assumed that the momentum flux in the vertical is mu ch larger than in the horizontal. A review of previous data and theoretical formulations (Murray, 1975; Meyers, 2008; Fewings et al., 2008 ; Shay, 1989 ) and new analyses ( Burnette & Dally, 2017 ) of long term observations suggest that waves and onshore winds consistently produce significant offshore directed bottom currents near coasts. An investigation of the impacts of using depth averaged currents for open coast surges shows that the neglect of this pervasive phenomenon results in an under prediction of up to 25% in surge levels .
11 A method for calculating momentum distribution through the 2DVR simulation in steady state conditions in the nearshore region , while neglecting wave radiation stress to focus on wind driven momentum fluxes, is developed. In this study, the nearshore region is referred to as an idealized alongshore uniform coast with constant bottom slope, from a few to tens of meters in depth. Though wi nd driven currents are the focus of this work, it can be inferred that similar results would be found if waves are included as an equivalent wind stress . Variation in surge calculation s between the 2DVR and 2DDI methods suggests there is a deficiency asso ciated with depth averaging the wind driven currents. This deficiency must either be included as a level of uncertainty accompanying surge predictions or be identified so that a solution for overcoming it can be developed. The 2DVR simulation method presen ted, with further study, may offer a computationally viable solution for overcoming the deficiencies in 2DDI, 3D, and quasi 3D models.
12 CHAPTER 1 INTRODUCTION, MOTIVATION, BACKGROUND, AND OBJECTIVES 1.1 Introduction Many major cities are located near the coast . In fact, 67% of cities with a population of more than 10 million, are located in coastal zones and are considered to have a coastal influence (Glasow et al., 2012). Several of these cities are below sea level (Turner et al., 1996) and are thusly mor e vulnerable to storm surge, and sea level rise (Hunt & Watkiss, 2011; Nicholls, 2004). Furthermore, it is estimated that the population exposed to flooding due to storm surge will only increase in the 21 st century (Nicholls et al., 2007) . Storm surge repr esents one of the most critical hazard s , including wind and wave damage, faced by many coastal communities today (Chavas et al., 2012; Malmstadt et al., 2009; Adger et al., 2005 ; Abel et al., 201 1 ) , m aking the need for accurate surge models essential for understanding and quantifying coastal resilience. tropical storm event. A variety of hydrodynamic models exist for c alculating storm surge. However, due to the computer resource constraints, most studies today in both forecasts and hazard studies are conducted with two dimensional depth integrated (2DDI) models rather than fully three dimensional models. This study hypo thesizes that there is a deficiency within 2DDI models , due to the averaging of the vertical current structure over the height of water column, that e ffects its ability to accurately estimat e surges . For simplicity, the analysis presented in this study wil l focus on the near shore region. The near shore region for the purposes of this study, is used in a general context to represent an idealized alongshore uniform coast that has a constant profile,
13 which spans from water depths of a few meters to tens of meters. This idealized nearshore region is a generic representation of many of the coasts encountered in the gulf coast of the United States and along the Atlantic. The majority of surge observed at the coast is generated in nearshore waters (Irish & R esio, 2010), thus this region is the focus of this work. The analysis presented here neglects surface wave radiation stresses, which can be and have been analyzed as an equivalent wind stress (Dally et al., 1985) , but are beyond the scope of this work. Of the many processes present in the nearshore ( Lentz, & Fewings, 2012 ) , m any of the theories that exist for radiation stresses (Longuet Higgins, & Stewart, 1962; Stive, & Wind, 1982; Xia et al., 2004 ), assume that the majority of the momentum flux is leaving the waves in the white caps near the surface, and a b ove the mean water level. Thus, the divergence of the momentum flux from waves occurs on the atmosphere to water surface boundary and can be app lied as a surface stress. Therefore, the wave forcing becomes analogues to the wind forcing. If this analysis were to include wave stress , the divergence of the momentum flux from wave forcing would be implemented as an equivalent wind stress (i.e. the sam e wind stress value would be applied in the same way, with a varying magnitude) . However, t h e equivalent wind stress generated by the radiation stress will not be included in this work so that the relationship between the vertical structure and the wind dr iven currents can be analyzed first. To date, only limited studies have investigated differences between depth integrated and 3 dimensional (3D) models. It is well known that 3D , and quasi 3D models capture a broader range of processes, temporal, and spat ial scales (Zheng et
14 al., 2017; De Vriend & Stive, 1987) than 2DDI models . H owever, most of the studies typically address these differences on site specific , or storm dependent basis . In this paper, a more general case is presented. The investigat ion into differences in surge estimated by a 2D and 3D models is done for idealized open coast cases so that the variation in storm surge calculation will be based on the physically assumptions of the models alone and are not storm or location dependent. Although 3D models are state of the art, 2DDI models tend to represent the status quo in the coastal surge modeling community ; specifically, for risk mapping , i.e., FEMA risk MAP studies (Xian et al., 2015; Brown, 2016), and large scale statistics based surge studi es such as to Cialone et al. (2015) . Often the validation of 2DDI models is based on post calibration techniques, that adjust bottom friction coefficients to help match water level storm observations . This adds to the calibration time demand ed of storm sur ge forecasting and suggests there is a need to adjust the computational methods within 2DDI models to make them universally applicable to all coastal processes. 1.2 Motivation Typically, these areas are low lying regions that can be hig hly susceptible to storm surge. Storm surge has the potential to cause erosion along shorelines, inundation in coastal communities, damage to infrastructure, disruption to industry, and loss of li fe. Though the result of an extreme storm surge event can be disastrous, the physics behind the generation and propagation of storm surge are historically poorly understood (Resio & Westerink, 2008). I mproving the accuracy and reliability of surge forecast ing and hindcasting to quantify the risks resulting from incoming storm events is essential.
15 A strong demand exists for effective two dimensional depth integrated (2DDI) model applications, as they are a valuable resource during storm seasons when forecast ing needs are high and the luxury of time is often not available. Also, the need for accurate 2DDI models in large scale production ensemble runs like that used to Risk MAP Program w ithin the joint probability method (Toro et al., 2010; Irish et al., 2011; Resio et al., 2009; Irish & Resio, 2010; Dietrich et al., 2011; Flowerdew et al., 20 09; Chen et al., 2012) creates serious problems for available computer resources . applications of coastal surge models to critical problems, these models are typically calibrated to obtain reasonable agreement with observed measurements and high water marks for a small set of storms . Frequently, there are only a small set of storm events that impact a given area for which there is measurement data, so this calibration process is limited by available data sets. As 2DDI models are being used in these forecast s and analysis systems , it is necessary to take a step back and objecti vely ask what issues , related to the fundamental physics embodied within these models, may exist and how these issues can be addressed? 1.3 State of the Art in Storm Surge Modeling Storm surge prediction methods began with bathystrophic models and simplif ied empirical models (Freeman, 1953; Reid & Clayton, 1953; Reid, 1955; Reid, 1957 ) . However, storm events have occur red relatively infrequently compared to historical data, and thus source data was sparse. Computational models were then developed based on fundamental hydro dynamical governing equations that represent the surge formation process and physical boundary conditions , such as tidal, air to water surface,
16 water to bottom boundary, and pressure gradient forcings . These models have brought the commu nity of the practice much closer to the prediction of storm surge . H owever, they still rely heavily upon historical data for objective calibration , which creates a dependence on locality and storm intensity. Given computer resource constraints in current risk map applications and other large scale applications, two dimensional and depth integrated (2DDI) models are often chose for storm surge modeling . Two dimensional depth integrated models are an elegant solution to resolve tidal propagation , captur ing amphidromic points, and body forces (pressure gradients and Coriolis acceleration) extremely well. In order to advance 2DDI models beyond their current capabilities for other coastal applications (i.e. the prediction of surge in extreme storm ev ents, transport of contaminants over the surface, naval craft trajectory, etc.) then the fundamental physics being consider must also be able to include forces that are applied only at the top and bottom boundaries . It is well known that the vertical struc ture is not captured within 2DDI models. A variety of hydrodynamic , two dimensional models exist for calculating storm surge (Dube et al., 1986; Luettich et al., 1992; Fan et al., 2016; Li et al., 2014; Kuroiwa et al., 1997; Nwogu, 1993; Wang et al., 2014 ; Xu & Bowen, 1994; Weisberg & Zheng, 2006; Brown et al., 2007; Westerink et al., 2008; Yu et al . , 2017; Dietrich et al.; 2011, Bloemendaal et al., 2018) . Three dimensional (3D), and quasi 3D models also exist that capture a broader range of processes, tem poral, and spatial scales (Haas & Warner, 2009; Zheng et al., 2017; De Vriend & Stive, 1987; Weisberg & Zheng, 2008; Srinivasa et al., 2015; Lakshmi et al., 2017). While these models have been extremely important to our understanding of the impact of storm surge along shorelines, th e research
17 presented in this dissertation suggests that the fundamental bottom stress assumptions in 2DDI models can be improved . 2DDI models implicitly assume that the current is vertically homogeneous; however, decades of theor etical research (Ekman, 1905; Alexander, & Scott, 2008; Meyers, 2008; Svendsen, 1984; Ting, & Kirby, 1995; Svendsen, & Hansen, 1988; Falconer et al., 1991; Svendsen, & Putrevu, 1991; Resio, & Westerink, 2008) and observational studies (Murray, 1975; Shay e t al., 1989; Kuroiwa et al., 199 9 ; Shay, 2010; Jaimes & Shay, 2009; Kocyigit, & Falconer, 2004; Scott et al., 2005; Fewings et al., 2008; Lentz et al., 2008; Uchiyama et al., 2010; Tasnim et al., 2015; Bryant, & Akbar, 2016; Torres et al., 2017) have prove n that the vertical structure of the current in the nearshore region is not well represented as an averaged value, but rather has a process driven , self similar shape that varies with depth. While this mis representation is due to the complex ity of wave ef fects, undertow , and inaccurate depiction of bottom stress , this work will only focus on the latter. Quasi 3D models such as ( Kuroiwa et al., 1999) and multi layer models like (KoÃ§yigit & Falconer, 2004) have been developed to capture the effect the vertical variation of forces has on the distribution of the currents through the water column. Kuroiwa et al., (1999) show how cross shore circulation in the nearshore region is crea ted when the water column begins to encounter the bottom of the ocean floor. KoÃ§yigit et al., (2004) similarly show that when wind blows in the on shore direction, the current at the top of the water column moves on shore and the current at the bottom of t he water moves off shore. Bottom stress concepts in 2DDI models represent situations in which only body forces (i.e. a force that is distributed equally throughout the water
18 column ) exist, multi directional features of the vertical current structure cannot be captured within depth averaged models. When bottom stress values are dependent on depth averaged velocities, they do not correctly estimate the near bottom velocity structure, which is crucial to computed storm surge. 1.4 Review of Observations of Current 3D Structure An example of the nonlinear variation in the vertical structure of the flow in the nearshore is shown in Figure 1 1 , which depicts the mean vertical profile of the cross shore current recorded over ten years in approximately ten m eters of depth off the coast of Melbourne, FL (by Burnette & Dally , 2017). The surface mixed layer is the top layer of the water column where salinity and temperature are vertically uniform and therefore tends to have a current that deviates from the rest of the water column. This profile demonstrates that the surface mixed layer typically does not extend to the sea floor and that the assumption of constant current velocity throughout the water depth does not represent the actual current at this site. Stud ies (Dally et al., 1985; Dally & Osiecki, 1995; Osiecki & Dally, 1997 ) show that energy from breaking waves enters through the air to water surface boundary, similar to the input from wind stress at the same location. The wave driven divergence rate of mom entum through the surface boundary uses the same mechanics as wind driven divergence rate of momentum through the same boundary. This investigation will focus on wind driven surges, assuming that wave driven surges use the same mechanics, and therefore wou ld show similar results. Fewings et al., (2008) performed an observational study on cross shelf flow by cross shelf winds in the continental shelf. This study found that during off shore wind events , cross shore circulation moved offshore in the top of the water column, and on -
19 shore in the bottom, with approximately equivalent net volume transport at the surface as in the bottom layers. They also found that when the winds were in the on shore direction, the result was nearly the reverse scenario. Results fr om Fewings et al., 2008 show vertical velocity profiles that vary with depth for weak wind and wave scenarios. The study contains six years of wind, wave and water velocity data from a water cabled coastal observatory that was in 12 meters of water depth. The cross and along shore current profiles have a mean velocity where if the velocity is moving towards the shore in the top of the water column it is moving off shore in the bottom, and inversely so. Findings from this study, show similar profile shapes f or weak and strong winds, simply varying in magnitude, suggesting self similarity in the profiles that is dependent on wind speed and direction. Observational studies such as this (Fewings et al., 2008; Murray, 1 975 ; Yamashita et al., 199 9 ) , show that the wind driven currents in the cross shore direction vary in magnitude and direction with depth. The length of these studies suggest that this is a persistent and pervasive feature found in the nearshore region. The indication that current ver tical structure self similarity exists pervasivel y in the nearshore suggests that the physics might be captured and reproduced in a simplistic numerical form, and furthermore, introduced into 2DDI models. 1.5 Review of Theory of 3D Structure of Flow Storm surge is the additional surface elevation of water caused by wind and pressure systems such as tropical and extratropical cyclones . The vertical current structure has an effect on storm surge generation and propagation, and is influenced by various coastal driving forces; the pressure gradient (barotropic, baroclinic, and inverse barometric effects ), the Coriolis effect , the force of gravity in the vertical, friction effects
20 at the surface, bottom, and internal (i.e. between mixed layers and the rest of the water column), and the astronomical tide forcing at the boundary layer. These forces can be quantitatively represented with the Reynolds Averaged Navier Stokes Equations. Each force has a varying magnitude of impact on storm surge generation. Analytical solutions of the equations governing acceleration and circulation of the ocean ( Xu & Bowen, 1994; Svendsen et al., 1991 ; Stive & Wind, 1986 ) show there is variation of current velocity through depth. Near a coast the top layer of the water column tends to move in the direction of the wind, and the bottom moves in the off shore direction . Kocyigit & Falconer, ( 2004 ) performed a numerical and analytical study on the prediction of the solution of the steady state vertical current structure for winds at 5 and 1 0 m/s, for a basin that discretizes the water column with uniformly distributed layers . The results show that the top of the water column has the largest velocity at the sea surface boundary, balanced by a distribution of velocities in the opposite directi on at the bottom of the water column. The magnitude of the currents at the bottom of the water column are smaller (and have an opposite sign) than the magnitude at the top . However, they take up a larger layer of the water column tha n those at the top , as is shown in Figure 1 1 . Other existing models show good agreement with observations in the vertical water column (Uchiyama et al., 2010; Kuroiwa et al., 1999) . Kuroiwa et al . 199 9 presents a model that predicts the velocity field in the n ear shore region . T his model considers wave set up, and circulation flow. The findings were like those mentioned from the study of Kocyigit & Falconer, ( 2004 ) . However, this work showed numerical solutions in shallow versus deep water, as well. Findings from this study showe d that
21 the variation with depth becomes more relevant as the water column depth decreased , as expected when the separation between the two surface forces at the top and bottom of the water column diminishes . This highlights the reason why the majority of s torm surge is generated in in the nearshore, because for the same wind stress in shallow water the effect of the wind stress reaches a larger portion of the water column , which is the basis for why this investigation focuses on the nearshore region (Irish and Resio, 2010) . Results from Uchiyama et al., ( 2010 ) also show that the cross shore circulation through depth varies in magnitude and direction. Where the top of the water column moves in the direction of the wind, and the bottom moves in the opposite di rection. This is, of course, inconsistent with assumptions made in 2DDI models. 1.6 Misspecification of Bottom Stress The misspecification of b ottom stress within 2DDI models is the focus in this study . Though many ocean processes are responsible for the propagation and generation of surge, bottom stress has an effect on the vertical structure of the current that is significant on a global scale. While the barotropic pressure gradient creates horizontal distribution of momentum , the momentum transfers due to wind and bottom stress ha ve a vertical distribution. The momentum transferred into the water column from wind stress is dependent on the drag between the air and water layers, and of course the difference in density of materials (water vs. air). Thus, v ertical shears, or gradients develop at the top boundary of the water column. Shear forces are dispersed and dissipate d as momentum is transferred throughout the water column. The implementation of bottom friction in hydrodynamic models has been well stu died for 2DDI models. However, the parameterization and representation of bottom
22 bottom friction coefficient (BFC) that are practically applied to capture the bottom fri ction momentum losses in 2DDI models. In risk mapping and large scale storm surge modeling, the calculated bottom stress is dependent on this objectively chosen BFC and the current velocity (which is averaged over the depth). Thus, the magnitude and direct ion of the bottom stress is often inaccurate, as the magnitude should rely on the velocity at bottom of the water column, and the direction of the bottom stress is dependent on the direction of the flow at the bottom boundary. Intuitively, it is unlikely t hat the magnitude and direction of the averaged velocity are the same as the magnitude and direction of the flow at the bottom of the water column. Figure 1 2 shows the vertical distribution of wind driven currents . This flow pattern is obviously ignored w hen velocities are depth averaged. In a 2DDI model, the velocity for the shallow point (to the right) in Figure 1 2 would be represented with a magnitude less than the maximum velocity in on shore or the off shore direction, and the direction would likely be on shore. However, if the variation of the flow through depth could be captured (as shown in Figure 1 2) and used to calculate bottom stress, then the magnitude of the bottom boundary current would be equal to the arrows shown near the bottom boundary l ayer (not a mean value) and would have an offshore direction. The objective of this work is to investigate the error introduc ed by 2DDI model s, due to their inability to capture the variation in flow throughout depth and its effect on coastal surges . The vertical variation of the flow affects the estimation of the bottom stress, which affects the magnitude of the surge. The surge is mainly a balance between wind stress and bottom stress; if the bottom stress is not correctly characterized, the surge accurately estimated.
23 1. 7 Objective s The objective of this research is to quantify any limitations existing in 2DDI models, specifically, the influence of the vertical current on nearshore storm surge generation . A secondary focus is to develop a strategy to include the vertical structure by parameterizing bottom stress within 2DDI models. This investigation is partitioned into four steps : 1. Implement turbulent closure equations within a 2DVR hydrodynamic model for coastal applications; 2. Implement depth avera ged momentum and mass conservation equations within a 2D hydrodynamic model for coastal applications; 3. Make comparisons of the differences in results from 2D and 2DVR storm surge estimation ; 4. Investigate an effective means of including t he 2DVR vertical mom entum fluxes into the 2D model codes; 5. Suggest application of a simple method for including 2DVR surge estimation solutions in to a 2DDI model .
24 Figure 1 1. Acoustic Doppler Current Profiler (ADCP) current profiles off the shore of Melbourne, Florida Burnette and Dally (2017). Data was collected from 2002 until 2011 and averaged in the cross shore direction. The equipment was placed in a depth of approximate ly 10 meters, with the profiler located in the bottom 2 meters of the water column.
25 Figure 1 2 . Wind induced circulation in the nearshore open coast due to longshore winds, and the response in the current vertical structure.
26 CHAPTER 2 DEVELOP MENT OF 3D HYDRODYNAMIC MODEL 2.1 3D Hydrodynamic Equations 2DDI models assume that b ottom friction and wind input , which are surface stresses , can be treated as body forces that are distributed evenly throughout the water column. In nearshore coastal applications this assumption has previously been recognized as a problem . This analysis attempts to quantify the influence of the return flow on storm surge generation in the nearshore region and to investigate the existence and extent o f this limitation inherent to 2DDI models . To do this, the three dimensional vertical current structure in the nearshore region of an idealized coast was numerically simulated using the 3D equations of motion and compared to 2DDI results for the same condi tions . 2.2 Vertically Resolved Governing Equations The conservation of mass, energy, and momentum of a fluid , found in many publications such as Versteeg & Malalasekera ( 2007), provide the governing equations that will be represented in this numerical investigation. Figure 2 1 shows the coordinate system for which the following explicit numerical model is developed. Figure 2 2 shows the layering used in this numerical experiment, that solves for flow through a given depth for a given time (left), and the element that will be considered (top right) at each layer, with sides equal to , , and . The faces of the element are a distance of , or away from the element point. Figure 2 3 shows an example of this for pressure application. This work assume s horizontal homogeneity in all variables, except pressure, and focuses on the local balance of momentum at one point. Thus, advection terms were neglected.
27 The continuity equation is solved for in this 2DVR method, as well as the 2D method to follow; ( 2 1) where and are horizontal velocity components in the cross shore and along shore directions, over (the depth of a layer in the water column ). T he z axis is in the vertical, as the coordinate system shows in F igure 2 1 . The following general equations , neglecting gradients and atmospheric pressure, represent the conservation of mo mentum of that system (Dube et al . , 1986; Axell, & Liungman, 2001); ( 2 2 ) ( 2 3 ) where is distance in the vertical direction , is the free surface elevation, is the density of water, is time, is the effective diffusivity , is the acceleration of gravity (9.81 m/s 2 ), and is the Coriolis parameter shown below ( s 1 ) where is the rate of the rotation of the earth, and is latitude. ( 2 4 ) It is assumed that the flow is incompressible, that there is hydrostatic balance in the vertical ( , where is pressure), and the system satisfies the Boussinesq approximation , , in which density perturbations are much smaller than the mean density. The upper boun dary conditions (i.e. k = 1 in F igure 2 2 ) for and are
28 ( 2 5 ) ( 2 6 ) ( 2 7 ) ( 2 8 ) where is the wind velocity at 10 meters above the sea surface, is the wind drag coefficient (dimensionless) , is the density of air, and is the wind direction, which is the convention al mathematical basis such that wind along the x axis For simplicity the wind drag coefficient was set to (dimensionless) in these simulations, which is the mean Garrat t (1977) coefficient value for win d intensities of 20 m/s. Garratt (1977) was chosen for this numerical simulation from the various existing coefficient determinations by various authors because it is most widely used and considers sea surfaces with neutra l stability. However, this value may be changed for future simulations, depending on the study purpose. For this study, the purpose is to maintain as a constant in order to analyze only the effect of the vertical flow structure and not the sensitivity to the value of . The lower boundary co nditions (i.e. k = n in F igure 2 2 ) for and are ( 2 9 ) ( 2 10 )
29 where, and are the momentum flux from the bottom of the water column into the ocean floor, represented as such; ( 2 11 ) ( 2 1 2 ) where is a dimensionless coefficient of drag, and horizontal velocity components in the cross shore and along shore directions at the ocean bottom, and . Velocity at the ocean floor is equal to zero, and velocity slightly above the bottom of the water column varies rapidly in time, consequently it is standard practice (i.e. Murray, 1975, Axell, & Liungman, 2001 ; Feddersen et al., 1998 ) to define a given height ) dependent on the depth of the simulation. Bottom stress , , acts in the same di rection as the bottom velocity, . Faria et al. (1998) report stress coefficients in the n ear shore from 0.0006 to 0.012, and in this simulation was set to 0.01. However, if needed, can be adjusted to fit conditions in a given area. 2.3 Turbulence Closure Many turbulent closure schemes have been developed for specific purposes ( Warner et al., 2005; Deigaard et al., 1991; Jacobs, 1984; Murray, 1975; Svendsen & Hansen, 1988; Ting & Kirby, 1995, 1996; Wyrtki, 1961, De Vriend, & Stive, 1987). T he standard model (Axell & Liungman, 2001) was used as a first approximation. Vertica l mixing due to orbital wave motions have not been well studied and are assumed not to influence the vertical distribution of momentum.
30 To solve for the eddy diffusivity value, the estimation from the Kolmogorov, 1942, (the Kolmogorov Prandtl relation ) use d ; ( 2 1 3 ) where is a stability function (set to 0.556), is the turbulent kinetic energy, and is the turbulent length scale calculated using the following algebraic formula, which varies with depth . ( 2 1 4 ) where is the total depth, and is the location in vertical, is the turbulent length scale at level k . The turbulent kinetic energy is solved for by the following transport equation, ( 2 1 5 ) where represents energy losses due to turbulent dissipation. Eddy diffusivity in the horizontal is not considered, as the horizontal momentum fluxes occur at a significantly larger scale than that of the vertical turbulence. Velocity distribution with depth through the wa ter column is governed by the diffusivity of momentum in the vertical. This numerical simulation solves for the momentum flux through finite layers within the water column. Each layer has a height between 0.75 and 0.125 meters, depending on if the initial depth in the simulation ; between 30 and 5 meters. The turbulence closure scheme must be completed by solving for the dissipation term, based on the dimensional argument below ;
31 ( 2 1 6 ) It is assumed that at steady state the slope will react in such a way to conserve energy (i.e. kinetic energy plus potential energy is a constant) . In other words, assuming energy ga ined will become potential energy. Using this numerical approach, various investigat ive scenarios were executed for given depths, wind angles, and other meteorologic parameters. The surface slope is calculated after each iteration and combined with the boundary conditions to allow the solution to a good approximation to a quasi steady state flow to be attained. Idealized scenarios were chosen as case studies in this wo rk. This is a first approximation, however future work will include radiation stress terms as well, to account for the energy created by waves. For now, this study focuses on wind driven momentum only. An example code to be used as a subroutine to solve fo r a given wind speed, direction, and depth is provided in Appendix A .
32 Figure 2 1. Fluid element for which momentum is tracked, at each depth throughout the model.
33 Figure 2 2. Layers of the fluid element shown in Figure 2 1, at each time iteration d uring numerical simulation.
34 Figure 2 3. Implicit pressure gradient th at balances out the slope term at each layer.
35 CHAPTER 3 DEVELOPMENT OF 2D HYDRODYNAMIC MODEL 3.1 2D Hydrodynamic Equations To further explore existing deficiencies within 2DDI models, this investigation will compare the 2DDI method to the vertically resolved solution developed in C hapter two. Two dimensional, depth averaged continuity and momentum equations will be used to calculate the average d velocity in the cross and along shore directions. Bottom and wind stress formulations will be considering the water as moving at a single velocity and direction, averaged over the water column. 3.2 Two Dimensional Governing Equations The depth integrated continuity equation is used to keep track of momentum in the x and y directions in the two dimensional system of equations simulated for this comparison investigation; ( 3 1 ) where and are vertically integrated velocities (i.e. ) over , the total depth . Assuming constant density, neglecting advection, and replacing the tangential stress components, the equations of momentum for the entire water column are given by; ( 3 2) ( 3 3 )
36 These are the basic equations describing depth integrated dynamical processes and used with variations in most 2DDI surge models today . They will be used to compare the and generated by 2DDI simulation versus a vertically resolved (VR) one. 3.3 Finite Difference Method A finite difference method ( FDM ) was developed to investigate deficiencies in 2DDI approaches as well as attempt to capture the three dimensional bottom stress term. The FDM presented here uses the continuity and momentum conservation equations, and is similar to the method presented in Buttolph et al., 2006. These governing equations are solved for and discretized over a staggered grid similar to that shown in F igure 3 1, where each box in the grid has indices or along the or axis (respectively). Velocity values ( and ) are solved for along the edges of each box, while the depth (height plus additional surface elevation ) is solved for at the center of the box. At each iteration the momentum equations are solved for at a given point location within the grid (Figure 3 1). Following this step, the continuity equations are solved for. The x momentum in the finite difference form is explicitly solved by ( 3 4) where is each time step, and and are the points along the rectangular grid along the and axis (respectively) . To see how velocities are oriented along the gridded rectangles see F igure 3 1.
37 The y momentum in the finite difference form is explicitly solved by ( 3 5 ) The continuity equation is solved by the finite difference method using each rectangular grid as a control difference (left f igure in F igure 3 1). The mass flux is calculated at the face of each rectangle, and and values from previous iteration s are entered as the spatial derivations. The FDM approximation of the continuity equation is calculated as ( 3 6 ) where the change in is given by ( 3 7 ) The array system depends on an or cell row and column indexing system that is looped through during every iteration of the simulation. It should be noted, if this model is used, the Courant condition (Courant number, ) must be met to retain stability (Richtmyer & Morton, 1967) ; ( 3 8 ) with stability maintained when the Courant number is less than or equal to one and not maintained elsewise . The FDM was coded using Fortran and can be found in Appendix B of this paper. The FDM stores its values in an array instead of using a formal matrix. The initial depths of this array, as well as initial values such as stress coefficients, are read in as a text file
38 D (C has a shallow grid and D has a deep grid) of this paper.
39 Figure 3 1. Finite difference coordinate and grid system used to conserve energy, momentum and mass in numerical model produced in this chapter.
40 CHAPTER 4 INVESTIGATION INTO VERTICAL CURRENT STRUCTURE IN THE NEARSHORE, AND LIKELY MISREPRESENTATION IN 2DDI MODELS 4.1 Potential Misspecification of Bottom Friction Three test cases were run for this investigation. The first simulation was executed for a range of depths (from shallow to deep water ) on an idealized coastline with constant offshore slope. The vertical structures produced by the simulation s were analyzed to provide insight into its characteristic form and to create a data base of highly resolved, qua si steady state solutions for time varying scenarios. Second, a simulation was executed along transects (i.e. from deep to shallow water), where the additional surge generated at each depth for the vertically resolved (VR) and a depth averaged (2DDI) model were accumulated into shallow water and the difference compared. This simulation was done for varying meteorological scenarios to show how much additional surge was created by the VR numerical simulations. Finally, a simplistic depth averaging Finite Diff erence Model ( FDM ) was developed. The 2D FDM was implemented for idealized nearshore storm simulations with varying meteorological forces. The same meteorological inputs were used in a second set of simulations; however, this set used the bottom stress val ue estimated by the VR steady state numerical simulation and applied it to each time step at each point within the FDM . The results from the depth averaged bottom stress FDM executions and the executions using the vertically resolved bottom stress value we re compared and analyzed. This was done using a Vertical Structure Bottom Stress (VSBS) value from the vertically resolved model. Surge levels from these three tests were then analyzed to quantify the extent to which depth averaging differs from model resu lts
41 which include vertical variation in the currents. The following were the objectives of these test cases; Investigate the potential impact of the vertical structure of bottom friction on in idealized coastal areas, and the effect of these impacts on surge calibration . Capture the physics of the multilayer vertical structure using well established, simplified numerical model. 4.2 Range of Testing 2DDI models provide reasonable water level and current values for many storm scenarios. However, often the results need to be calibrated by adjusting the bottom friction values within the simulation. This localized bottom friction calibrating may validate the water levels at specific points of interest but may also introduce uncertainty and error when used for different wind forcing conditions . It is important to investigate the extent of the error being introduced on a storm by storm basis , in order to understand the uncertainty provided by water level and current values predicted for storm events yet to occur . Every calibration is site and storm specific, therefore the following three test cases were made to provide insight into what is happening in the nearshore during various storm conditions. To ensure symmetry in the solutions, for validation purposes, the latitude was set to zero in these numerical simulations. 4.3 Comparison of Vertically Resolved Behavior in the Near s hore 4.3.1 Test Case 1: Steady State Surface Slope at a Point The first simulation was run for given depths of 5 and 30 meters (shallow and deep water) on an idealized surface slope shoreline. The wind magnitude forcing fields were not ramped, so that further investigation could be done into variations in time, and the run time needed to reach steady state. The surface is constrained so that t he force balance is zero, which requires that the slope force balance the total momentum flux
42 into the water column, minus the bottom stress. Initial currents were set to zero. The vertical distribution of horizontal velocities with depth produced by the s imulations was analyzed to provide insight into what this structure theoretically should look like, as well as used to create a data base of highly resolved, quasi steady state solutions for varying scenarios. Scope of t esting . Vertically Resolved (VR) numerical simulations were executed using the methodology developed above. The range of testing included ; seven wind angles ( 90, 45, 22.5,0,22.5,45,90), four wind speeds (5 m/s, 10 m/s, 20 m/s, 40 m/s), five bottom slopes (1/10,1/ 100,1/500,1/1000,1/10000), and two depths (5 and 30 meters). The model ran for 3 hours of simulated time for each test case with a time step of 0.5 seconds (shown in Figure 4 1). Results . The depth varying cross shore and along shore VR velocities of test case one provided basic understanding into the shape of the vertical current structure for various meteorologic forcings. Figure 4 2 shows the results from twenty four simulation s where blue lines which are darker represent higher wind speeds. The top of the water column (in the cross shore direction) moves in the direction of the wind, and the bottom moves in the opposite direction . The magnitude of the maximum current velocities (bot h in the direction of the wind and in the opposite direction) varies wi th wind speed. Overall, the model behaves as would be expected for these simplified cases. The results shown here are for a constant wind forcing, this is the reason why the model reaches steady state over three hours of simulated run time. As is shown in Figure 4 2, at steady state, there is a consistent shape to the vertical structure from each simulation which depends mostly on the wind speed and
43 wind direction, and not as much on water depth . Further investigation is presented in the C hapter 5 , in which an assessment of the relative contribution of wind and water depth is presented . Similar results from the other simulated meteorologic inputs tests modeled for varying depths can be found in the Appendix E. These figures show that there is a similar shape on the vertical structure that is dependent on wind speed and direction. The magnitude of that shape appears to be dependent on depth. 4.3.2 Test Case 2: Steady State Surface Slope Along a Transect Secon d , a simulation was executed along transects (i.e. from deep to shallow water), where the additional surge computed for each previous cell was used as a boundary condition for the next one, for the vertically resolved and a depth averaged model was accumul ated into shallow water and the difference compared. Simulations were uncoupled, in that the additional surface elevation at each depth starting at 30 meters, was calculated and added to the following depth (i.e. 30 meters to 29, and 29 to 28) initial surface elevation . In other words, the increase in height is driven by the increase in volume into that area (conservation of mass ) and balanced by the surface slope needed to make potential energy equal kineti c energy . This provides a reasonable approximation of additional surge accumulated along the transect and offers an estimate of th e inherent difference between the surge at the coast using the method and that produced by depth averaged models. Scope of t e sting . This simulation was done for varying meteorological scenarios, similar to the range in test case 1, to show how much additional surge is created by the vertically resolved numerical simulations. Calculation of surge started in deep water (30 meters) and moved to shallow water (5 meters).
44 Results . While case 1 does a good job providing insight to what the vertical structure is doing in the nearshore of an idealized inner shelf profile in both shallow and deep water, test case 2 provides a means to est imate the surge at the coast. The tests here investigate the differences between the vertically resolve surges and the 2DDI surges . Figure 4 3 shows the results of a transect running from 30 meters to 5 meters water depth and compares the amount of surge g enerated by the highly resolved vertical simulation versus a depth averaged calculation. This is run for a 30 m/s wind blowing on shore for 3 hours of simulated run time. Over those three hours, an additional 0.1 meters of surge is neglected by the depth a veraged calculation. That is approximately 25% of the surge generated by the depth averaged model. Th e result s illustrate that the error is not small when the vertical structures is averaged over and when the return flow is considered in the vertical prof ile, the surge could be almost 25% higher than calculated by a purely depth averaged model. 4.3.3 Test Case 3: Idealized Finite Difference Method Run s Finally, a simplistic depth averaging Finite Difference Model ( FDM ) was developed using metho dology s imilar to (Buttolph et al., 2006). The same meteorological inputs were used in a second simulation, however, this time the bottom stress value was estimated by the VR steady state numerical simulation and applied to each time step at each point within the FDM . This is why the run time was so short, as the entire vertical structure was well resolved at each point and time. The results were compared and analyzed between the purely depth averaged FDM and the runs where the VSBS value was used. Scope of t esting . The FDM was implemented for idealized nearshore shorelines with an onshore wind speed of 30 m/s and run for a simulation time of 360 seconds with
45 0.5 second time steps. This was for shallow (5 meters) and deep (30 meters) 9 by 4 grids with 100 mete rs simulated spacing (grid available in appendix C and D). NX is equal to 10 because an ocean boundary is applied at the first row. Results . While test case 2 answers the question of does it matter if the vertical structure is neglected in a depth average model, test case 3 attempts to capture this structure within the depth averaged model . M ost importantly case 3 focuses on the implications that the vertical structure has on the bottom stress which influences the total momentum within the water column. Th rough this simplistic model, the bottom stress value created by the highly resolved simulation replaced the bottom stress value in a depth averaged FDM . Figure 4 4 shows the results from this test case. Where the depth averaged results are on the left two pa nel s and the simulation where the highly resolved vertical structure bottom stress value is on the right. Results show the differences in purely depth averaged (2DDI) model and quasi vertically resolved (2DVR) model. The difference between the two in sh allow water, after only 360 seconds of simulated run time is approximately 0. 005 m additional meters of surface elevation, which is 0.53 times the amount of surge generated by the depth averaged model over this time. It seems like when the bottom stress va lue is dependent on the vertical structure , this has an effect on surge estimation. It can be assumed, likewise, that the tuning of the drag force coefficient during model calibration has a significant impact on surge estimation of the model. 4. 4 Discus sio n of 2D and 3D Comparison The representation of bottom friction in many of the 2DDI models used today is an analogue to frictional effects in stream flow, where the energy lost in the bottom
46 stress is found to balance the gain in energy in the downslope fl ow. In this case, the force balance is between the barometric pressure gradient and the bottom stress acting at the bottom of the flow, which is treated empirically as a loss over the entire water column. In open channel flows, such a balance has been foun d to be reasonably well represented by Chezy and Manning proposed laws, which are based on experimental findings from friction slopes in laboratory flumes with idealized flows (Machiels et al. 2009) . However, this is not equivalent to the conventional quad ratic drag law used in most physics based applications. As shown in Test Cases 1 3, this is probably good assumption for tidally driven flows and for post storm drainage from a flooded area, this approximation cannot provide a good representation for wind and wave driven flows, particularly near coasts. Since most surge models are calibrated for applications in (or Chezy) values to match observations in specific areas; h owever, it is not clear that calibration for a specific storm or small set of storms can actually provide consistent behavior over a significant range of storms or locations. Although there have been a small number of careful near bottom measurements of cu rrents near a coast, most have focused on longshore currents rather than cross shore currents. Data collected off the coast of Duck, NC at the Field Research Facility have recently been processed. Long term measurements provide a substantial climatological perspective of the nearshore cross shore and longshore currents near the bottom of the water column. A clear conclusion from this work is that, when relatively strong winds blow in the onshore direction, near bottom currents consistently move in the offsh ore direction.
47 Figure 4 5 shows near bottom cross shore current measurements for all cases in September 2000 in which the wind direction fell with an angle band of 47.5 to 92.5 degrees relative to the shore normal at Duck, NC. The y axis shows the mean nea r bottom cross shore current in a depth of 8 meters, where negative values depict currents moving off shore. This data shows that as wind speeds and wave heights increase, so do the off shore near bottom current velocities at this site. Similar results can be found in observational studies ( Hansen & Svendsen, 1985; De Vriend & Stive, 1987; Dube et al. , 1986; Pechon & Teisson, 1994; Lentz et al., 2001 ; Shay et al., 1989), which also show persistent vertical structure in storm events with highly sheared currents and surface and bottom currents moving in opposing directions. Computational simulations (Kuroiwa et al. , 2001 ; Svendsen & Putrevu, 1991 ) provi de additional support for the vertical distribution of the return flow nearshore and resulting distribution of velocity magnitude and direction with depths that are consistent with the results of simulations shown here. Studies comparing the diffe rence between a 2D and 3D simulations (Zheng et al., 2013) show that depth integrated models cannot reproduce the results from three dimensional models without making significant adjustments to coefficients within the 2DDI model. Results from additional o bservations and previous studies are consistent with results found here. The idealized case examined here simplifies comparisons between depth integrated and highly resolved vertical layer modeling paradigms. In this case, the inherent differences are stra ightforward to identify and quantify, which allows a general perspective for the potential significance of the vertical structure on surges at the coast. The inclusion of the vertical structure into the shallow water equations produces
48 important difference s in surges, and also in motions within the top and bottom layers of the water column. The majority of storm surge is generated in shallow water, so this investigation only focused on nearshore idealized open coast scenarios. The nearshore region is also where bottom friction has the largest impact on wind driven flow. The simulations run here were also very simplistic by design, so that the difference between the highly resolved runs to depth averaged ones could be attributed to the lack or addition of th e current vertical structure component. There is a need for accurate prediction of near bottom and near surface currents for decisions related to fish/oyster migration, pollution/oil spill mitigation, and sediment transfer. Although traditional 3D model s capture some of the vertical structure within their flow representations, the type of resolution required for accurate estimates of near bottom and near surface current structure is typically much higher than what is used in these models. Additional inve stigation must be done into the incorporation of the VSBS into 2DDI models. 4. 5 Conclusion from 2D and 3D Comparison The simulations run in this experiment were based on a simplistic k scaling turbulence closure schemed. Although it could be argued that a more refined turbulent closure scheme could produce somewhat different results, the results obtained here were shown to be consistent with extensive observations at the USACE Field Research . The simulations run in this experiment were based on a simplistic k scaling turbulence closure scheme. Although it could be argued that a more refined turbulent closure scheme could produce somewhat different results, the results obtained here were shown to be consistent with extensive observations at the USACE Field Re search
49 Facility in Duck North Carolina. General results for a set of 24 of runs (Figure 4 2 ) show that the pattern of deviation from the 2DDI assumption is a function of both wind speed and directions. For example, an onshore wind with a wind speed of 20 m /s produces an offshore current of approximately 0.15 m/s consistent with measurements from the FRF at, Duck, NC. (Ward, 2018) and a surge deviation of 0.06 m, while a wind speed of 20 m/s at a direction of 45 degrees produce a sur ge deviation of 0.25 m (F igure 4 5 ). Figure 4 5 shows a combination of wind and wave forcings that reach current speeds similar to that found in Test Case 1. As mentioned before, wave and wind stresses are both surface stress , and Test Case 1 considers both if wave stress is inclu ded as an equivalent wind stress). Although locally calibrated 2DDI models can provide a reasonable fit to observations in a specific storm or a set of similar storms, the dependence of these deviations on wind direction and speed and on wave conditions makes it extremely problematic for the 2DDI paradigm to match results obtained when vertical current structure is considered. This work demonstrates the need for a model that can resolve the vertical structure in currents. The next step of this work will b e the parameterization of bottom drag and therefore return flow in the nearshore in 2DDI models. Example application of this method are decreasing the bottom stress calibration tuning time needed to validate a hind casted storm event, increased accuracy in prediction of surge in an incoming storm due to the lack of tuning needed, more accurate surge p rediction for simulated storms that are developed for statistics purposes, and the capability to capture return flow in the nearshore open coast, which would help models capture contamination transport, or the drogue passage.
50 In C hapter 6 of this work , a preliminary way to include the highly resolved results from the simulation used here to investigate misestimating of bottom stress in 2DDI models will be develop ed. Friction only needs to be considered in this detail in the nearshore, where the current fee ls the bottom, so additional computational need will not be great, however accuracy will increase. Capturing the bottom boundary physics is a more efficient way to predict and plan for inundation due to storm surge rather than locally tuning friction coeff icients to specific storm events. Test case 1 provided insight into the vertical current structure in the nearshore. Test case 2 shows that when this vertical structure is not captured in 2DDI models, there is a significant difference in storm surge genera tion. Test case 3 further suggests a difference exists when the vertical structure is or is not included in a 2DDI simulation, and investigated the application of including the structure through adaptive, time dependent, bottom stress parameterization . This work will investigate the limitations and time dependence of capturing the vertical structure using the code developed for this study, and the possible application to existing 2DDI models.
51 Figure 4 1. Test case 1 range of testing to b e simulated with highly resolved code developed based on the governing and boundary equations shown in Equations 2 1 through 2 15 .
52 Figure 4 2. 24 selected simulations from T est C ase 1 for wind speeds of 5, 10, 20, and 40 m/s at the following angles and for the following depths; (a) a 45 degree angle onshore at 5 meters of depth , (b) a 45 degree angle onshore at 30 meters of depth , (c) a 0 degree angle onshore at 5 meters of depth , (d) a 0 degree angle onshore at 30 meters of dept h, (e) a 45 degree angle onshore at 5 meters of depth , and (f) a 45 degree angle onshore at 30 meters of depth.
53 Figure 4 3 . Results from one of the simulations from test case 2, showing the surface surge generated by vertically resolved vs. depth averaged numerical simulations. This specific run is of an idealized nearshore bottom, with a slope of 1 over 1000 (m/m) at 1 meter i ncrements, with winds blowing on shore at 30 meters per second.
54 Figure 4 4. Results from T est C ase 3. (a) Depth averaged FDM at 5 meters of depth, (b) d epth averaged FDM with vertically resolved bottom stress value pulled at each point and time at 5 meters of depth, (c) d epth averaged FDM at 30 meters of depth, and (d) d epth averaged FDM with vertically resolved bottom stress value pulled at each point and time at 30 meters of depth.
55 Figure 4 5. (a ) Mean bottom currents (at 8.22 meters depth) versus wave height, and (b ) Mean bottom currents (at 8.22 meters depth) versus wind speed off the coast of Duck, NC. Only on shore and shore normal winds were considered (i.e. 35 to 75 degrees). (a) ( b )
56 CHAPTER 5 ANALYSI S OF VERTICAL STRUCTURE DEPENDENCE ON TIME VARIATION 5.1 Analysis of Simulated Vertical Structure Results from test case 2 that a significant error exists when the vertical structure is not considered in a 2DDI storm surge model and test case 3 provides insight into the possibility of including the vertical structure through a representative bottom stress value and potentially near surface current. The focus in this dissertation is on storm effects on storm surge, so related effects on near surface curren t in coastal areas are not addressed here. Before additional investigations into methods for the inclusion of the Vertical Structure Bottom Stress (VSBS) inclusion into 2DDI models are explored, results from the numerical simulations developed in this stud y must be analyzed for dependence on time variation. Results from test case 1 in Chapter 4 it was shown the vertical structure that develops over three hours of run time. Figure 4 2 shows the end result at the last time steep in the simulation. Recall, wi nds were not ramped for these runs, and initial velocity values through depth were set to 0 m/s. The inclusion of the VSBS will be based on steady state values. Thus, relaxation time, or the time needed for the model to reach steady state, must be examined . Figure 5 1 shows velocity profiles produced in test case one for depths of 5, 10, 15, 20, 25, and 30 meters with wind speeds of 5, 10, 20, and 40 m/s at a 45 degree angle, scaled through depth by the maximum velocity produced at the top of the water column . The result is a self similar shape that reaches approximately between and at . It is likely the variation between and is due to the selection of the bottom
57 boundary reference height (where the roughness is applied in a logarithmic fashion). If this point were permitted to vary with time and current velocity, then momentum would be removed through the bottom boundary depe ndent on the speed of the height from the bottom to the top of the bottom boundary layer, and it likely we would see identical scaled velocity profile for every depth, speed, and height. Similar results were seen with simulations angles other than 45 degre es, which also produced varying self similar shape s . 5.2 Time to Reach Steady State The last vertical current profile developed by the vertically resolving numerical simulation will be considered the final and fully solved solution. To calculate relaxation time, values are computed in each direction. The criterion for relaxation is defined in terms of the time at which the changes over the entire profile become less than some threshold value. These values represent the difference between the velocities at each depth each time step from that of the previous time step, converted into a standard deviation, and , i.e. ( 5 1) ( 5 2 ) where is the number of layers in the water column. The solution will be considered to have attained a reasonable approximation to a steady state when and are less than 1 cm/s. The time at which this occurs is taken as the relaxation times, . Figure 5 1 shows the time required for the
58 vertically resolved (VR) numerical simulations to reach steady state for various wind speeds, di rections, and water depths. Results show that less time is needed to reach steady state for winds that are directly on shore, than for winds that are blowing perpendicular to the shoreline ( 90 and 90). The relaxation time is also larger for higher wind sp eeds. The smallest relaxation time is around 1 minute, and the largest values are approximately 70 minutes. The larger relaxation times are present during along shore winds with high wind speeds, and in the deepest water tested. Winds are typically input i nto surge models at 15 minute intervals that are sometimes even interpolated from hourly wind data. Depending on which interval one considers, the most extreme relaxation times will take one two four wind input intervals worth of data to reach steady state . Considering and less than 1 cm/s is an extremely stringent value, this suggests that on a 15 minute interval, steady state could be reached if the entire VR simulation model was integrated into a 2DDI system, especially for wind speeds less than 20 m/s and in shallow depths. It is unlikely that wind input values will vary as much as zero to these maximum wind speeds in a given interval change. In this way, the maximum relaxation values shown here are purely academic. As an example, Figure 5 2 shows observed and modeled wind sp eed and direction from Hurricane Matthew, a hurricane that hit North Florida in early October, 2016. The observed winds are collected every 6 minutes at the Fernandina and Trident Point National Oceanic and Atmospheric Association (NOAA) tide gauge station s ( https://tidesandcurrents.noaa.gov/ ). The winds and pressure fields that drove circulation within this model were read in every 15 minutes, and output wind
59 speed and direction from the model was printed every 6 minutes to make a comparison to NOAA wind o bservations. T he 2DDI finite element numerical model ADvanced CIRCulation or ADCIRC ( Luettich et al ., 1992) was used to model the storm surge that occurred during Matthew. The West North Atlantic Tidal model (WNAT) mesh depicted were provided by Taylor En gineering, Inc., and contain the Gulf of Mexico, the Atlantic Ocean, and the Caribbean Sea (all west of the 60 degree west meridian), which was used to capture the tidal forcing througho ut the domain. The mesh is bounded to the south, north, and west by So uth America, North America, and Central America, respectively. Overall the ers approximately 8.4 million square kilometers . The WNAT mesh other tributarie s throughout the east coast of Florida. The mesh resolution varies from 50 kilometers to 5 meters, with nodal attributes assigned to each node in the unstructured mesh, prescribing bottom friction, eddy viscosity, sea surface height above geoid, and the su rface effective roughness length. The wind and pressure fields obtained from Oceanweather, Inc. were used at a resolution of both 15 kilometers and 28 kilometers. A ramp up value of 5 days and a time step of second were utilized throughout the model run, O ctober 1, 2016 00:00 UT C to October 10, 2016 00:00 UTC, at 15 minute time steps. Figure 5 3 shows Hurricane F igure 5 4 shows the resultant maximum elevation over the 10 day simulation. The take away, shown in Figure 5 2, is that even for extreme storm events, wind speed may increase by up to 10 m/s over an entire day, whereas the VR numerical
60 simulations capture a 0 to 10 (and at most 0 to 40) m/s instantaneous increase in wind speed. Thus, in application, the extreme relaxation times shown in Figure 5 1 will likely never occur. Even if they were, however, a 70 minute relaxation time is easily captured within one to four wind input time step intervals. Therefore, it can be concluded that time dependence is finite. 5.3 Time Dependence Further investigation was done into time dependence, and factors influencing relaxation time and steady state current structure. Fro m test case 1 , it is shown that the direction cross shore current varies with depth, whereas the along cause for a larger introduction of error when the velocity is averaged over depth, therefore, cross shore circulation will be the focus of the following investigation. Figure 5 5 shows the results of four VR numerical runs, each at a depth of 20 meters, with a wind direction of 0 degrees (on shore). The si mulations were calculated for wind speed s of 5, 10, 15 and 20 m/s. The simulate d run time was 120 minutes, and the figures show the vertical profile that was produced every 20 of those minutes. Of the four speeds tested, the 5 m/s wind speed run is the only one that required more than 20 minutes to reach its steady state vertical pro file shape. Note, the x axis is scaled to the maximum negative and positive value reached in the vertical profile, as the focus is not a magnitude of the velocities produced, but the shape of the profile. Figure 5 6 shows results from similar VR numerical simulations, these runs vary wind speed linearly with depth (i.e. the first run is at a depth of 5 meters and has a wind speed of 5 m/s, 10 meters and 10 m/s, and so forth). Again, the only shape that needs over 20 minutes to reach its shape is the first ( a) simulation of 5 m/s and 5 meters. Whereas the others make the final VR current profile shape by 20 minutes. The 20
61 meters and 20 m/s (d) run overshoots the last vertical profile at first, but reaches the shape by the first 20 minute mark. The 15 m/s and 15 meter (c) simulation undershoots it, but reaches the shape of the profile. The 10 m/s and 10 meter (b) simulation attains a reasonable approximation to make the steady state vertical structure profile by 20 minutes. There is a similarity from top to bottom in all of the results in F igure 5 5 and Figure 5 6, which means there is organization in the vertical current structure. The existence of such organization suggests that there is a relationship between inputs to the model and the shape of the VR str ucture. From Figure 5 6 and Figure 5 7 it appears that the variation in time dependence of the vertical current structure produced by this model is dependent on the wind speed. The relationship of this dependence, and magnitude of variation must be investi gated in order to determine implications for including the vertical current structure in a 2DDI model through VSBS values .
62 Figure 5 1 . Scaled analysis, the maximum velocity, at the top layer of the water column, is used to non linearize the velocity profile through depth, for wind speeds of 5, 10, 20, and 40 m/s (where darker lines represent higher speeds), and a wind direction of 45 degr ees. The depths simulated vary, (a) shows a simulated 5 meters of depth, (b) shows a simulated 10 meters of depth, (c) shows a simulated 15 meters of depth, (d) shows a simulated 20 meters of depth, (e) shows a simulated 25 meters of depth, and (f) shows a simulated 30 meters of depth . (a) ( b ) ( c ) ( d ) ( e ) ( f )
63 Figure 5 2 . Relaxation time in both directions, and values are computed to represent the difference between the current time step and the last one. When and are less than 1 cm/s that time is called the relaxation time. This figure shows that time for various wind directions, speeds, and water depths, where (a) is an onshore (0 degree) wind, (b) is a 90 degree wind, (c) is a 90 degree wind, (d) is a 45 degree wind, (e) is a 45 degree wind, (f) is a 22.5 degree wind, and (g) is a 22.5 degree wind.
64 Figure 5 3 . Hind Cast wind speed and direction values from ADCIRC model of Hurricane Matthew. (a) Wind speeds from the observation and modeled at Trident station, (b) wind directions from the observation and modeled at Trident station , (c) wind speeds from the observa tion and modeled at Fernandina Station, and (d) wind directions from the observation and modeled at Fernandina Station. ( a ) ( b ) ( c ) ( d )
65 Figure 5 4 . Hurricane Matthew storm track over W N A T + fine St. Johns Taylor Engineering Inc. grid.
66 Figure 5 5 . Max elevations ( ft. ) in North Florida produced during Hind Cast of Hurricane Matthew ADCIRC run . Longitude Latitude
67 Figure 5 6 . Simulated cross shore vertical current profiles for 120 minutes of run time, with 20 minute intervals. The simulations were for (a) wind speeds of 5 m/s with a 20 meter depth, (b) wind speeds of 10 m/s with a 20 meter depth, (c) wind speeds of 15 m/s with a 20 meter depth, and (d) wind speeds of 20 m/s with a 2 0 meter depth. ( a ) ( b ) ( c ) ( d )
68 Figure 5 7 . Simulated cross shore vertical current profiles for 120 minutes of run time, with 20 minute intervals. The simulations were for (a) wind speeds of 5 m/s with a 5 meter depth, (b) wind speeds of 10 m/s with a 10 meter depth, (c) wind speeds of 15 m/s with a 15 meter depth, and (d) wind speeds of 20 m/s with a 20 meter depth. ( a ) ( b ) ( c ) ( d )
69 CHAPTER 6 IMP LICATIONS TO 2DDI SURGE MODEL APPLICATION 6.1 Proposed Multivariate Analysis O rganiza tion within the multi dimensional vertical current structure was analyzed prior to identifying methods for 2DDI VR applications. Such analyses can be helpful for selecting a coordinate system in which best obj ectively captures organization. S pecifically, axes may be chosen that represent the maximum variance, while maintaining orthogonality. This method reduces the dimensions needed to represent the variance among the various vertical current structures. Principle Component Analysis (PCA) is t he method used in this study to select such a coordinate system and reduce the dimensions needed to represent the vertical current structure for varying meteorologic inputs, and water depths. The data produced by VR numerical simulation executions (velocit y through depth over time) were put into a data matrices , with the mean value subtracted; producing covariance matrices . To solve for the eigenfunctions of this covariance matrix, , was then transformed (rotated) into a diagonal f orm, where the diagonal elements, are the only non zero values in the transformed matrix. The solution for a given set of homogeneous linear equations with unknowns, can be represented by this algebraic relationship ( 6 1) w here is a matrix, is the diagonal matrix, and is a set of is the orthogonal column vectors.
70 In the fields of meteorology and oceanography, eigenvectors are often referred to as Empirical Orthogonal Functions (EOFs). If the equation above is r ewritten the following is produced, ( 6 2) Or ( 6 3) where is the identity matrix, or the solution if the determinant is singular. The values in the diagonal matrix, , are called eigenvalues, and the set of n orthogonal column vectors, , are termed eigenvec tors. The eigenvalues represent the amount of variance, within the covariance matrix, that correspond s with each eigenvector. The amount of variance in a given dimension, for example the 2 nd dimension, is defined by the variance along the second eigenvector, in this case , equals the associated eigenvalue, in this case , which equals the total variance of the data along the axis. An example of this method in coastal application can be found in Resio et al., 1973. Note that these sets of eigenvalues and eigenvectors have the ability to reconstruct the original data set matrix by multiplying the matr ix of eigen vectors by a vector that contains the eigen values. The PCA was used on simulated cross shore vertical structures developed from a simulated VR numerical execution for an onshore (0 degrees) wind, with a wind speed of 20 m/s, at a depth of 5 meters. Figure 6 1 shows the vertical current str ucture as profiles (a) that are printed every 20 simulated minutes (for a run of 2 hours with a time step of 0.5 seconds), as well as the variation through time of the velocities produced through depth (b).
71 As mentioned in Chapter 5 , the structure approac hes its shape within the first 20 minutes (a), and in fact is 0.1 m/s away from the final solution by the 1000 th time step, or approximately 8 minutes into the simulation. While this is the result for one simulation, appendix F shows results for similar ru ns of varying depths, speeds, and wind directions to give a fuller understanding of the variation of the velocities with time and depth. The PCA technique was applied to each of these runs to find the covariance matrix, eigenvectors and eigenvalues that r epresent the variance within the vertical structure through simulated time. Figure 6 2 through 6 4 shows the results from the simulation shown in F igure 6 1 for three different time intervals (full simulated time, first 15 minutes, and second 15 minutes), and appendix G shows the solutions from the simulations in appendix F for those same time intervals. 6.2 Multivariate Analysis Figure 6 2 shows the Empirical Orthogonal Function (EOF) results for the entire simulated two hours of the VR run (where winds were not ramped). The covariance matrix (a) shows the variance by depth in the cross shore current. The top three dimensions, or modes represent 100% of the run. The EOF or eigenvector for each mode is shown in (b). At a glance, the first mode imitates the vertical current structure produced by the VR runs. This EOF accounts for 91% of the variance. The second and third modes makes up less than 10% of the variance, and from the shape of the EOF through depth, have less of a physical relationship to the vertical current structure. (c) shows the EOFs for each mode through time. The second mode oscillates between negative and positive v alues through time. This suggests this mode represents an oscillation related to the discontinuous mode in which the simulation is initiated (no
72 ramp) and diminishes only on the same time scale at the total solution relaxation time. When the three modes ar e reconstructed, the result is very similar to the vertical structure shown in (a) from F igure 6 1; positive, larger values at the top of the water column, and negative values near the bottom with a lesser magnitude that take up a larger volume. This is th see from Fi gure 6 1, steady state seems to be reached within the first 8 minutes. Figure 6 3 shows a similar PCA performed for the first 15 minutes of the simulation. The covariance matrix (a ) seems to be similar to that in 6 2, however the maximum variance, at the top of the water column, is 5 times as large, which is expected since the relaxation rate has a somewhat exponential decay form. The EOF make up 93%, 4% and 3% of the variance for t he first, second, and third mode/dimension (respectively). This is an increase for the first mode from the total run by 2%. The variance through time (c) for the first mode looks somewhat similar to that from the full simulation, however the second mode sh ows a clearer oscillation that focuses around the 2 meter mark. This suggests even further that this is a computer model made variance, in that the middle of the water column is where the two varying boundary conditions equilibrate slowest, since they are the farthest from the boundaries. The reconstruction of these three modes makes up 99%, which suggests that the relaxation time likely happens around 15 minutes of run time. Figure 6 4 represents the PCA performed for the second (15 30) 15 minute interval of run time. Again, the covariance matrix (a) shows the maximum variance at the top layer of the water column. However, variance elsewhere is limited. The first
73 mode (b) accounts for essentially100% of the variance through depth. The reconstruction of mod es 1, 2, and 3 (c) seems less accurate than the first mode alone. Which again, suggests that mode 1 contains the physical properties that define the variation of velocity through depth, more than mode 2 or 3. Figure 6 2 through 6 3 shown the PCA for the si mulation in F igure 6 1, however the results are similar to what was developed through the PCA method for simulation shown in appendix F, and this can be seen in appendix G. This suggests that the first mode represents physical variance through depth of the cross shore velocity in the nearshore, and that steady state is reached within the first 15 to 30 minutes of run time, with the alongshore relaxation being the slowest. 6.3 Application to 2DDI Storm Surge Models So far, this study has shown there is a pro blem in the application of the bottom friction term when the velocities are averaged over vertical water column compared with the VR numerical simulations that produced the vertical current structures for various physical and meteorological components. The results from each of these simulations were into a highly resolved multi dimensional data base. It is suggested that this data base be used as a look up table for bottom stress within a 2DDI storm surge model. This creates a time based, directionally vari ant bottom stress coefficient value, based on the VR data base findings of this study, which could significantly improve the bottom stress estimation within 2DDI models. An example look up tables were generated, and contains the VSBS values (steady state) that were solved for input values with wind speeds of 0 to 40 m/s (at 2 m/s intervals), wind directions of 0 to 365 degrees (at 5 degree intervals), and water depths of 3 to 30 meters (at 1 meter in tervals). Appendix H contains the values for 5
74 meters of depth, which are shown as a contour plot in Figure 6 5. If a value is in between these intervals , say 1.2 m/s winds, an interpolated value between 0 and 2 m/s can be calculated or the value could be round up to 2 m/s could be used. For simplicity , the latter method is used here. Figure 6 6 shows examples for the bottom stress solution from the 2DVR simulations for varying wind directions and wind speeds. The top (a) shows the along shore bottom stress solutions, and the bottom (b) shows th e cross shore bottom stress solutions. Note, the latitude, which can be adjusted within the 2DVR model, was set to 0 degrees of latitude. The bottom stress solutions meet expectations for their response to varying wind angles, and have a uniformity that su ggests, with further work, there may be a simple function that calls the bottom stress solution into a 2DDI model via an empirical relationship instead of needed to implement a look up table.
75 Figure 6 1 . For a run with 20 m/s onshore winds, at a depth of 5m; (a ) Cross shore vertical profile plotted at 20 min intervals for a 2 hour run, (b ) Time variation in cross shore current through depth and time, comparison is made using v alues at the last timestep. Th e run time was 0 to 120 simulated minutes (14400 timesteps, at dt = 0.5 seconds). (a) ( b )
76 Figure 6 2 . A 2 hour run with 20 m/s onshore winds, at a depth of 5m; (a ) Covariance with depth throughout the entire run, Bottom (b ) Top three normal modes through depth, (c ) Mode 1, Mode 2, Mode 3, and Modes 1 3 combined, variation through depth and time. (a) ( b ) ( c )
77 Figure 6 3 . The first 15 minutes of a 2 hour run where winds are 20 m/s onshore winds, at a depth of 5m; (a ) Covariance with depth through time , (b ) Top three normal modes through depth, (c ) Mode 1, Mode 2, Mode 3, and Modes 1 3 combined, variation through depth and time. (a) ( b ) ( c )
78 Figure 6 4 . The second 15 minutes of a 2 hour run where winds are 20 m/s onshore winds, at a depth of 5m; (a ) Covariance with depth through time , (b ) Top three normal modes through depth, (c ) Mode 1, Mode 2, Mode 3, and Modes 1 3 combined, variation through depth and time . (a) ( b ) ( c )
79 Figure 6 5 . Example bottom stress solution for estimated wind speeds and wind directions by the steady state vertically resolving numerical simulations at a depth of 5 meters. Where (a) is the magnitude of the stress component and (b) is the direct ion of the stress component. (a) ( b )
80 Figure 6 6 . Example bottom stress solution for estimated wind speeds and wind directions by the steady state vertically resolving numerical simulations at a depth of 5 meters. Whe re (a) is bottom stress in the along shore direction and (b) is bottom stress in the cross shore direction (a) ( b )
81 CHAPTER 7 CONCLUSIONS 7 .1 Implications from 2D DI vs. VR Bottom Stress N ear boundary flows driven by wind forcing are not well represented by a mean depth averaged flow. 2DDI models r equire bottom friction coefficient calibration to hindcast extreme storm events, which introduces uncertainty, in that the bottom stress doe s not accurately represen t the physics of momentum exchange processes at t he bottom boundary. The need to adjust bottom friction on a storm by storm and location ed results for a different storm and/or a different location are likely inaccurate. An idealized set of tests alon g a straight open coast were conducted to investigate potential deviations in surge heights that might arise from the misrepresentation of bottom stress in 2DDI models. Results show that when the current structure is neglected then the bottom stresses wil l be misrepresented and cause an estimation error in storm surge calculations which can be very significant for both real time predictions and probabilistic estimates of inundation levels. 7. 2 Future Work The representation and parameterization of bottom s tress through the use of the depth averaged current velocities instead of the return flow velocities in 2DDI models introduces uncertainty into the model that must be adjusted by calibrations. This value is often attuned to make modeled storm surge match a set of observed values which are often spatially sparse and may include their own uncertainties. The highly resolved, multidimensional data base developed in this study can be applied to 2DDI models, by including the wind speed, and direction dependent ve rtical structure bottom stress
82 (VSBS) through a subroutine or look up table fashion. This can reduce the need for re calibration 2DDI models, using a storm by storm and location by location basis to represent a more comprehensive, physics based first appro ximation to bottom stress. The numerical model can be adjusted to include radiation stress, as it is well known that waves and wave br e aking also create an increase in surface elevation. Radiation stress can be included through the divergence of the moment um flux as it approaches the shore. The additional momentum is released into the water from waves breaking at the surface and thus can be introduced as an equivalent wind stress. If the inclusion of the VSBS values is found to produce more accurate storm s urge predictions (with or without post simulation calibration), then further work could be done to include a numerical relationship between wind speed, and direction to the VSBS. Once this relationship is developed, a simple function or subroutine can be a dded to 2DDI models instead of needing to include a look up table, if found advantageous. Though this study focuses on 2DDI models, 3D and quasi 3D models may also benefit from including the 2DVR based bottom stress values. 3D and quasi 3D ocean circulati on models often use the same depth averaging technique from 2DDI models at each layer of the 3D model. For example, the regional oceanic modeling system (ROMS) divides the water column into layers , that can vary in thickness, and can be evenly distributed or have separate settings for top and bottom boundary thickness. The thickness of the layers is limited by computational power and the availability of computation time. If this were adapted , h owev er, the 3D density gradie nt would need to be considered when calculating the transfer of momentum in order to account for highly stratified zones.
83 7. 3 Final Remarks 2DDI surge estimation models do not appear to reproduce the correct behavior with respect to bottom stress. Specifically demonstrated in this dissertation the differences include; 1. Variation of the alongshore and cross shore currents with depth, generating a flow in the opposite from the flow at the surface as numerically determined in test case 1; 2. A dditional storm surge generated by the inclusion of the vertical structure as shown in test case 2; 3. A nd additional surface elevation numerically calculated by the 2DDI Finite Difference Method ( FDM ) when the Vertical Structure Bottom Stress (VSBS) value was considered, shown in test case 3. When the VSBS is included into 2DDI storm surge models, it will be expected that the physics being modeled are in better agreement with observation, particularl y when the near bottom flow is in the opposite direction from the depth averaged flow. It is expect that the bottom stress should be more accurately modeled. Improved physics in models typically results in less need for frequent retuning of model on a sit e by site and storm by storm basis.
84 APPENDIX A VERTICALLY RESOLVED NUMERICAL SIMULATION CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C DIVERGENCE OF MOMENTUM CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C UNIVER SITY OF NORTH FLORIDA C ADVISOR DR. DON RESIO C AMANDA TRITINGER CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC subroutine DIVMOMENTUM (FCORR,DEP,WIND0,WANG,DX, & ETA,IDEP,SLOPEINU3D, DMOMDT,lunv,lunu) real WANG, D EP, CD, CB, C0, g, ANGLE, DIR, b real SMALL, NSMALL, OMEGA, FCORR, ETA, FVBAR, SLOPE real GRAdPdx, GRAdPdy, DT, UTOP, VTOP, z0, DZ, AREA0 real TWOPI, RADC, RHORA, UW, SLOPEINU3D, NWINTER real ZLTINSQ, ZLBINSQ, CBR, UBAR, VBAR, UWSQ real WIND0, TAUW3D, DX, UB3D, UB3DX, UB3DY real TB2DU, TB2DV, TAUB3DU, TAUB3DV, DUDT, DVDT, TA2DU, TA2DV integer TIME, NWTIME, NTIMES, NZ, NW, WTIME, NMOD, NWMOD, REFPT integer LL, IDEP, lunv, lunu, cc, nwend, nwi parameter (N PT=30000) dimension U(NPT), V(NPT), DUTOP(NPT) dimension TK(NPT), ZL(NPT), VISC(NPT), DUBOT(NPT), DVBOT(NPT) dimension DELU(NPT), DELV(NPT), SHEAR(NPT), EPS(NPT), DVTOP(NPT) dimension FLUXXTOP(NPT), FLUXYTOP(NPT), FLUXXBOT(NPT) dimension FLUXYBOT(NPT), FU(NPT), FV(NPT), DETADX(NPT) dimension VELNW(NPT), DIRNW(NPT) character filename CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC TWOPI = 8.0*atan(1.0) RADC = TWOPI/360.0 OMEGA = 7.27*10**( 5 .0) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Print Values C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC print*, ' | ', 'Speed (m/s)', WIND0, ' |' print*, ' | ', 'Angle (deg)', WANG/RADC, &' |' print*, ' | ', 'Depth (m) ', DEP, ' |' print*, ' | ', 'Eta (m)', ETA,' |' DEP = DEP + ETA print*, ' | ', 'Depth + Eta ', DEP, ' |'
85 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Starting Values C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC TIME = 180 WTIME = 1 DT = 0.5 NW = 1 NZ = 40 DZ = DEP/NZ DX = 100 DY = 100 REFPT = 5 if(IDEP.lt.15) REFPT = REFPT*2 RHORA = 0.0012 AREA0 = 0.0 SLOPE = 0.0 GRAdPdx = 0.0 GRAdPdy = 0.0 cc = 0 c SET NW to 0 for no ramping c SET NW to 1 for ramping c SET NW to 2 to read in WINDS c SET NW to 3 to ramp AND read in WINDS CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C NW = 1 ( RAMPING ) C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c if NW = 1, below set the time you want to ramp for: NWTIME = 7200 if(NW.eq.1) TIME = TIME + NWTIME CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C NW = 2 or 3 ( read w inds in w/o RAMPING ) C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c if NW = 2 or 3, user must set wind input interval in minutes (i.e. 15 min) c multiply by 60 to get to seconds (in sec) if(NW.eq.2 .or. NW.eq.3) then NWINTER = 6*60 NWMOD = (NWINTER/DT+0.5) open(21,file="vel.txt",status='old') open(22,file="dir.txt",status='old') nwend = ((TIME 1)/NWINTER) do 200 nwi = 1,nwend read(21,*) VELNW(nwi)
86 read(22,*) DIRNW(nwi) 200 continue close(21) close(22) endif print*, ' | ', 'Run Time (sec) ', TIME, ' |' print*, ' | ', 'Write Time(sec)', WTIME,' |' CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCC CC C Set Constants C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CD = 0.0022 CB = 0.01 C0 = 0.5562 g = 9.81 SMALL = 1.0E 8 NSMALL = 1.00*(SMALL) NTIMES = TIME/DT+0.5 NMOD = WTIME/DT+0.5 UW = WIND0 DIR = WANG c Parameterization of bottom drag z0 = (REFPT*DZ)*exp( 0.4/(sqrt(CB))) c Because we are using the reference point bin from the bottom, to apply c t he drag, we're taking the mid point of that bin minus the mid c point from the bottom ((REFPT+.5) .5) = Z > ln(z/z0) CBR = abs(alog((REFPT+.5)*z0)/alog(0.5*z0)) print*, '************************************************' pr int*, 'Beginning Simulation' print*, '************************************************' CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c initialize values CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC UBAR = 0.0 VBAR = 0.0 c 2D Inputs UB3DX = 0.0 UB3DY = 0.0 DUDT = 0.0 DVDT = 0.0 TA2DU = 0.0 TA2DV = 0.0 TB2DU = 0.0
87 TB2DV = 0.0 TAUB3DU = 0.0 TAUB3DV = 0.0 do 20 J = 1,NZ+2 U(J) = 0.0 V(J) = 0.0 TK(J) = 0.0 EPS(J) = 0.0 SHEAR(J) = 0.0 DELV(J) = 0.0 DELU(J) = 0.0 DUTOP(J) = 0.0 DVTOP(J) = 0.0 DUBOT(J) = 0.0 DVBOT(J) = 0.0 c Solve for the turbulent length scale: a parabolic profile c See L.B. Axell and O.Liungman, 2001 (equ 24) ZLTINSQ=1.0/(0.4*DZ*(J 0.5))**2 ZLBINSQ=1.0/(0.4*DZ*(NZ+0.5 J))**2 ZL(J)=sqrt(1./(ZLTINSQ+ZLBINSQ)) if (ZL(J).lt.0.) ZL(J)=sqrt((1.0/(0.4*DZ))**2) 20 continue CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C time steps: CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC do 30 IT = 1,NTIMES CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C NW = 1 ( RAMPING ) C CCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC if(NW.eq.1) then if(IT.le.NWTIME/DT) UW = WIND0*IT/(NWTIME/DT) endif CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C NW = 2 or 3 ( read w inds in w/o RAMPING ) C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC if(NW.eq.2 .or. NW.eq.3) then if(mod(IT,(NWMOD/1)).eq.1) then cc = cc + 1
88 UW = VELNW(cc) WANG = DIRNW(cc) en dif if(IT.gt.(NTIMES NWINTER)) then UW = VELNW(cc 1) WANG = DIRNW(cc 1) else c Interpolate Between Times UW = (VELNW(CC+1) VELNW(CC))*(mod(IT,(NWMOD/1))/NWMOD)+ & VELNW(CC) WANG = (DIRNW(CC+1) DIRNW(CC))*(mod(IT,(NWMOD/1))/NWMOD)+ & DIRNW(CC) endif WANG = 270 WANG if(WANG.lt.0) WANG = WANG + 360 DIR = WANG * RADC endif UTOP = 1.0*CD*(RHORA)*((UW abs(U(1)))**2)* DT*cos(DIR)/DZ VTOP = 1.0*CD*(RHORA)*((UW abs(V(1)))**2)*DT*sin(DIR)/DZ U(1) = U(1)+UTOP V(1) = V(1)+VTOP CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C Momentum Flux through Top and Bottom of each Level CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC do 50 IZ = 1, NZ+1 c Solve for eddy viscosity, using stability constant, VISC(IZ) = C0*sqrt(TK(IZ))*ZL(IZ) if (IZ.eq.1) then DUTOP(IZ) = 0.0 DVTOP(IZ) = 0.0 else DUTOP(IZ) = U(IZ 1) U(IZ) DVTOP(IZ) = V(IZ 1) V(IZ) endif DUBOT(IZ) = U(IZ) U(IZ+1) DVBOT(IZ) = V(IZ) V(IZ+1) FLUXXTOP(IZ) = VISC(IZ)*DUTOP(IZ) FLUXYTOP(IZ) = VISC(IZ)*DVTOP(IZ) FV(IZ) = FCORR* V(IZ)
89 FU(IZ) = FCORR*U(IZ) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C because no effect at top if(iz.eq.1) then FLUXXTOP(IZ) = 0.0 FLUXYTOP(IZ) = 0.0 FU(IZ) = 0.0 FV(IZ) = 0.0 E ndif CCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC FLUXXBOT(IZ) = VISC(IZ+1)*DUBOT(IZ) FLUXYBOT(IZ) = VISC(IZ+1)*DVBOT(IZ) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C because friction with bottom if(IZ.eq.NZ) then FLUXXBOT(IZ) = CB*U(NZ REFPT)*abs(U(NZ REFPT)) FLUXYBOT(IZ) = CB*V(NZ REFPT)*abs(V(NZ REFPT)) E ndif CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC DELU(IZ)= (((FLUXXTOP(IZ) FLUXXBOT(IZ))/DZ)+FV(IZ) GRAdPdx)*DT DELV(IZ)= (((FLUXYTOP(IZ) FLUXYBOT(IZ))/DZ) FU(IZ) GRAdPdy)*DT SHEAR(IZ)= 0.5*(DUTOP(IZ)**2+DUBOT(IZ)**2+SMALL)*DT & +0.5*(DVTOP(IZ)**2+DVBOT(IZ)**2+SMALL)*DT 50 continue CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C do 60 IZ = 1, NZ U(IZ) = U(IZ)+DELU(IZ) V(IZ) = V(IZ)+DELV(IZ) UBAR = UBAR+U(IZ) VBAR = VBAR+V(IZ) c Close the Turbulence argument by solving for Eps (eqution 19) EPS(IZ) = DT*((C0**3)*(abs(TK(IZ))**1.5)/ZL(IZ)) c Where tk = (shear (ps) eps (e) plus change ( Axell and Liungman)
90 c Assuming constant density, the buoyance frequency is turned off. TK(IZ) = TK(IZ)+SHEAR(IZ) EPS(IZ) if(TK(IZ).lt.0.0) TK(IZ)=SMALL 60 continue UBAR = UBAR/(NZ) VBAR = VBAR/(NZ) U(NZ+1) = 0.0 V(NZ+1) = 0.0 c using the reference bin from the bottom as the point that will be referenced c for CD coefficient of drag to be applied (u(*)^2 = Cd*u(ref)^2 U(NZ) = U(NZ REFPT)*(CBR) V(NZ) = V(NZ REFPT)*(CBR) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C RECORD CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC if(mod(IT,(NMOD/1)).eq.1)then do 70 IZ = 1, NZ C write (lunu,*) U(IZ), (DZ*IZ)+DZ/2.0 C write(lunv,*) V(IZ), (DZ*IZ)+DZ/2.0 70 continue C write(lunu,*) U(NZ+1), DZ*(NZ) C write(lunv,*) V(NZ+1), DZ*(NZ) endif CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC if(abs(U(NZ REFPT)).lt.SMALL) then ANGLE = 0.0 UB3D = SMALL else ANGLE = atan2(V(NZ REFPT),U(NZ REFPT)) UB3D = sqrt((U(NZ REFPT)**2)+(V(NZ REFPT)**2)) endif UB3DX = UB3D*cos(ANGLE) UB3DY = UB3D* sin(ANGLE) TAUB3DU = U(NZ REFPT)*abs(U(NZ REFPT))*CBR
91 TAUB3DV = V(NZ REFPT)*abs(V(NZ REFPT))*CBR ccccccccccccccc Balance additional ass cccccccccccccccccccccccccc AREA0=AREA0+UBAR*DEP*DT b = 2.0*AREA0/(DEP*DX) SLOPE = b /(DEP*DY) GRAdPdx = 0.5*g*SLOPE CCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C 2D Bottom Stress Calc C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC TB2DU = CB*(DUDT*abs(DUDT))/DEP TB2DV = CB*(DVDT*abs(DVDT))/DEP TA2DU = (CD*RHORA*UW**2)*cos(DIR)/DEP TA2DV = (CD*RHORA*UW**2)*sin(DIR)/DEP DUDT = DUDT + (( FCORR*DVDT) TB2DU + TA2DU)*DT DVDT = DVDT + ((FCORR*DUDT) TB2DV + TA2DV)*DT 30 continue CCCC CCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C TIME LOOP END C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC print*, 'UTOP, VTOP, ANGLE', UTOP, VTOP, DIR/RADC print*, 'TAUB3DU, TAUB 3DV, TB2DU, TB2DV, U and V' print*, TAUB3DU, TAUB3DV, TB2DU, TB2DV, U(NZ REFPT), V(NZ REFPT) end subroutine
92 APPENDIX B FORTRAN FINITE DIFFERENCE MODEL SOURCE CODE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C AMANDA TRITINGER C Storm Surge Finite Difference Model C UNF Spring 2018 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC program FDM real CD, HR, CC, CB, MN, WIND, WDIR, FCORR, LAT real RHORA, RHOair, RHOwat, TIME, WTIME, SMALL, OMEGA real GRA, TWOPI, RADC, FU, FV, dx, dy, dt, dh real UTOT, TAUbu, TAUbv, UTOP, VTOP, UBOT, VBOT real DELU, DELV, DELH, UMID, VMID, ETAMID, DHY, DHX real NETA, OETA, TAUB3D integer NX, NY, ix, iy, it, NMOD, NTIMES, jx, area integer WAV, ADV, UorV parameter (NYP=1000, NXP=1000) dimension UOLD(NXP,NYP), VOLD(NXP,NYP), HOLD(NXP,NYP) dimension UNEW(NXP,NYP), VNEW(NXP,NYP), HNEW(NXP,NYP) dimensio n OETA(NXP,NYP), NETA(NXP,NYP), IDEP(NXP,NYP) open(14,file='SUB 2D_Shall_HsurgeOUT.txt',status='unknown') C open(14,file='SUB 2D_Deep_HsurgeOUT.txt',status='unknown') open(15,file='UsurgeOUT.txt',status='unknown') open(16,f ile='VsurgeOUT.txt',status='unknown') open(12,file='surgeOUT.txt',status='unknown') open(11,file='surgeIN.txt',status='old') CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C READ in the INPUT values (coef, time, etc.) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC read(11,*), CD, HR, WIND, WDIR, LAT, TIME, WTIME read(11,*), WAV read(11,*), ADV read(11,*), NX, NY CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C constants... CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC dx = 100.0 dy = 100.0
93 area = dx*dy dt = 1.0 NTIMES = TIME/dt+0.5 NMOD = WTIME/dt+0.5 GRA = 9.81 TWOPI = 8.0*atan(1.0) RADC = TWOPI/360 WDIR = RADC*WDIR LAT = RADC*LAT RHOair = 1.20 RHOwat = 999.00 RHORA = RHOair/RHOwat SMALL = 1.0E 9 OMEGA = 7.27*10**( 5.0) FCORR = 2.0*OMEGA*sin(LAT) MN = 0.002 CC = HR**(1/6)/MN CB = GRA/(CC**2) CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C READ in the INITIAL grid values (H) at CENTER CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC do 5 iy = 1, NY read(11,*), (HOLD (jx,iy), jx = 1, NX) do 15 ix = 1, NX IDEP(jx,iy) = HOLD(ix,iy) UOLD(ix,iy) = 0.0 VOLD(ix,iy) = 0.0 OETA(ix,iy) = 0.0 UNEW(ix,iy) = 0.0 VNEW(ix,iy) = 0.0 NETA(ix,iy) = 0.0 TAUB3D = 0.0 15 continue 5 continue c NOTE: All velocities are at the edges of difference squares
94 c while the H values are at the center of these squares. do 10 iy = 2, NY 1 write(14,*) (OETA(jx,iy), jx = 2, NX 1) write(15,*) (UOLD(jx,iy), jx = 2, NX 1) write(16,*) (VOLD(jx,iy), jx = 2, NX 1) 10 continue do 55 it = 1,NTIMES if(mod(it,NMOD).eq.1) print*, 'Time = ', it C Calculate the change in U @ (ix+1,iy) because you need C H @ (ix,iy) and (ix+1,iy) to center H. C Set Boundary so that N = N 1, 1 = 2 do 45 ix = 1, NX+1 UOLD(ix,1) = UOLD(ix,2) HOLD(ix,1) = HOLD(ix,2) UOLD(ix,NY+1) = UOLD(ix,NY) HOLD(ix,NY+1) = HOLD(ix,NY) 45 continue do 145 iy = 1, NY+1 UOLD(1,iy) = UOLD(2,iy) HOLD(1,iy) = HOLD(2,iy) UOLD(NX+1,iy) = UOLD(NX,iy) HOLD(NX+1,iy) = HOLD(NX,iy) 145 continue UorV = 0 do 25 ix = 3, NX 1 do 35 iy = 2, NY 1 ETAMIDMID = (HOLD(ix 1,iy) + HOLD(ix,iy))/2. VMID = (VOLD(ix 1,iy)+VOLD(ix,iy)+VOLD(ix,iy+1)+ & VOLD(ix 1,iy+1))/4. FV = FCORR * VMID DHX = GRA*(OETA(ix 1,iy) OETA(ix,iy))/dx TAUbu = CB*(UOLD(ix,iy)*abs(UOLD(ix,iy)))/ ETAMIDMID TAUwu = CD*RHORA*(WIND**2)*cos(WDIR)/ETAMIDMID C Un Comment to run 2DVR subroutine C call DIVMOMENTUM (LAT,ETAMIDMID,WIND,WDIR,dx,IDEP(jx,iy),DHX, C & TAUB3D, UorV)
95 C TAUbu = TAUB3D DELU = ( FV + TAUwu TAUbu + DH X ) *dt UNEW(ix,iy) = UOLD(ix,iy) + DELU 35 continue 25 continue C Calculate the change in V @ (ix,iy+1) because you need C H @ (ix,iy) and (ix+1,iy) to center H. UorV = 1 do 125 ix = 2, NX 1 do 135 iy = 3, NY 1 ETAMIDMID = (HOLD(ix,iy 1) + HOLD(ix,iy))/2. UMID = (UOLD(ix,iy 1)+UOLD(ix+1,iy 1)+UOLD(ix+1,iy)+ & UOLD(ix,iy))/4. write(12,*) 'HMID =', ETAMIDMID , 'UMID =', UMID FU = FCORR * UMID DHY = GRA*((OETA(ix,iy 1) OETA(ix,iy)))/dy C Un Comment to run 2DVR subroutine C call DIVMOMENTUM (LAT,ETAMIDMID,WIND,WDIR,dx,IDEP(jx,iy),DHX, C & TAUB3D, UorV) C TAUbv = TAUB3D TAUbv = CB*(VOLD(ix,iy)*abs(VOLD(ix,iy)))/ETAMIDMID TAUwv = CD*RHORA*(WIND**2)*sin(WDIR)/ETAMIDMID DELV = (FU + TAUwv TAUbv + DHY) *dt VNEW(ix,iy) = VOLD(ix,iy) + DELV write(12,*) 'DELV = ', DELV, 'TAUbv = ', TAUbv, 'TAUwv = ', TAUwv write(12,*) 'FU = ', FU, 'DHY = ', DHY 135 continue 125 continue C Calculate new H value @ (ix,iy) based on the C continuity equation. do 175 ix = 2, NX 1 do 185 iy = 2, NY 1
96 C Find U and V values at center of square, and at time C equal to 1/2 (i.e. in between time steps). UTOP = (UOLD(ix+1,iy) + UNEW(ix+1,iy))/2. UBOT = (UOLD(ix,iy) + UNEW(ix,iy))/2. VTOP = (VOLD(ix,iy+1) + VNEW(ix,iy+1))/2. VBOT = (VOLD(ix,iy) + VNEW(ix,iy))/2. DELH = ((UTOP UBOT)/dx + ( VTOP VBOT)/dy)*dt NETA(ix,iy) = OETA(ix,iy) DELH HNEW(ix,iy) = HOLD(ix,iy) + NETA(ix,iy) 185 continue 175 continue CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C WRITE CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCC do 20 iy = 2, NY 1 if (ISNAN(HNEW(NX,iy))) print*, 'ERRRORRR!! at time= ', IT if (ISNAN(HNEW(NX,iy))) goto 9999 write(14,*) (NETA(jx,iy), jx = 2, NX 1) write(15,*) (UNEW(jx,iy), jx = 2, NX 1) write(16,*) (VNEW(jx,iy), jx = 2, NX 1) 20 continue do 75 ix = 1, NX do 85 iy = 1, NY UOLD(ix,iy) = UNEW(ix,iy) VOLD(ix,iy) = VNEW(ix,iy) OETA(ix,iy) = NETA(ix,iy) 85 continue 75 continue CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 55 continue
97 9999 print*, 'Program Ended' end program FDM
98 APPENDIX C FORTRAN FINITE DIFFERENCE MODEL INPUT TEXT FILE SHALLOW CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C AMANDA TRITINGER C SurgeIN.txt CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 0.0022 2 20 0 0 14400 60 !CD, HR, W(m/s), Wdir, LAT, TIME, WTIME 0 !Waves on or off (1 or 0) 0 !Advection on or off (1 or 0) 10 4 !NX, by NY 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 0 0 5 5 5 5 5 5 5 5 5 0 0 5 5 5 5 5 5 5 5 5 0 0 0 0 0 0 0 0 0 0 0 0
99 APPENDIX D FORTRAN FINITE DIFFERENCE MODEL INPUT TEXT FILE DEEP CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C AMANDA TRITINGER C SurgeIN.txt CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 0.0022 2 20 0 0 14400 60 !CD, HR, W(m/s), Wdir, LAT, TIME, WTIME 0 !Waves on or off (1 or 0) 0 !Advection on or off (1 or 0) 10 4 !NX, by NY 0 0 0 0 0 0 0 0 0 0 0 0 30 30 30 30 30 30 30 30 30 0 0 30 30 30 30 30 30 30 30 30 0 0 30 30 30 30 30 30 30 30 30 0 0 0 0 0 0 0 0 0 0 0 0
100 APPENDIX E VERTICAL PROFILES FOR VARYING METEOLOGICAL INPUTS
105 APPENDIX F CROSS SHORE CURRENT TIME VARIATION ANALYSIS
107 APPENDIX G CROSS SHORE CURRENT MULTIVARIET ANALYSIS (EOF ANALYSIS)
119 APPENDIX H VERTICAL STRUCTURE BOTTOM STRESS (VSBS) EXAMPLE LOOK UP TABLE MAGNITUDE VSBS solution for 5 meters ;
121 APPENDIX I VERTICAL STRUCTURE BOTTOM STRESS (VSBS) EXAMPLE LOOK UP TABLE DIRECTION VSBS solution for 5 meters;
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132 BIOGRAPHICAL SKETCH Amanda Tritinger earned her Bachelor of Science degree in environmental e ngineering from the University of Central Florida (UCF) in May 2013. At UCF she was an undergraduate research assistant in the Coastal Hydroscience Analysis, Modeling, and Predictive Simulations Laboratory (CHAMPS lab) , under the leadership of Dr. Scott Hagen. In May 2015 she received a ful ly funded Master of Science in civil e ngineering with a focus in coastal modeling degree from the University of North Florida under the guidance of Dr. Peter Bacopolous . She spent four years as a graduate research assistant in the fully funded joint doctoral program between the University of Florida (UF) and the University of North Florida (UNF). Amanda studied under advisors from both schools; Dr. Do n Resio (UNF) and Dr. Maitane Olabarrieta (UF). She earned her Doctor of Philosophy in c oastal and oceanographic e ngineering from UF in 2019.