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Understanding properties of extreme ocean wave runup

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Understanding properties of extreme ocean wave runup

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Content Understanding Properties of Extreme Ocean Wave Runup by Luis Montoya A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy (CIVIL ENGINEERING) August 2019 Copyright 2019 Luis Montoya ii Dedication I would like to dedicate this dissertation to God, who has been by my side throughout my life and particularly through my PhD. He has given me the wisdom and He deserves all the credit not me. Proverbs 2:6 says “The Lord giveth wisdom out of His mouth cometh knowledge and understanding”, this was one of the Bible verses that kept me going. I would like to give special thanks to Prof. Lynett, the best advisor that anybody can have. Thank you for your trust, guidance, patience and encouragement. Thanks for “pushing” me to publish and further investigate more on IG waves. I am honored to have been advised these years by you. I also want to thank Prof. Ghanem and Luhar for being a part of my committee, thanks for the valuable support and comments. Also, special thanks to Prof. Synolakis who was often in the Friday meetings to give me feedback and valuable comments, thanks for teaching me to be a great scientist. Additionally, I would like to thank my former lab mates: Nikos, Aykut and Sasan. As well as my current lab mates: Vasilis, Adam, Ezgi and Zili. Thank you, Nikos, for your advice and always being there to help me out with different analysis. Thank you, Adam, for sharing your knowledge and helping me a lot throughout these years. I want to thank my dad Eberth Montoya, mom Luz Cortes for being the best parents anyone can have. Thank you for the support, advice and sacrifice through these years. Thank you, mom, for helping us take care of our daughter. Thank you, dad, for letting my mom come to the USA to help us. I am who I am today because of God and you two. Thanks to my brothers Jose Montoya, Ronald Montoya, Eberth Montoya, Lily Delgado, Stella Montoya and Giovanni Montoya. Finally, I want to thank my beautiful wife, Monica Collazos, my beautiful daughter, Isabel Montoya, and my future son to be, Jonah Montoya. Without your support, love, trust and patience this would not be possible. You guys were my motor and happiness every day as I got home from school. Thank you, Isa, for interrupting daddy in his office just to play for a little bit. You guys are my everything and I love you to the moon. iii Table of Contents DEDICATION ................................................................................................................ II LIST OF TABLES ........................................................................................................... IV LIST OF FIGURES .......................................................................................................... V 1. INTRODUCTION AND BACKGROUND ..................................................................... 1 1.1 OBJECTIVES AND ORGANIZATION ................................................................................ 7 2. MODELING EXTREME INFRA-GRAVITY WAVE RUNUP ........................................... 9 2.1 NUMERICAL MODEL AND SETUP ................................................................................. 9 2.2 OPTIMUM FREQUENCY RESOLUTION .......................................................................... 13 2.3 CONVERGENCE AND COEFFICIENT OF VARIATION .......................................................... 19 2.4 GENERATION MECHANISMS OF EXTREME INFRA-GRAVITY WAVE AND RUNUP .................... 24 2.4.1 Runup Analysis .............................................................................................. 24 2.4.2 Possible Sources of Error ............................................................................... 27 2.4.3 IG Spectral Behavior and Convergence ......................................................... 31 2.4.4 Extreme Runup Occurrence and Timeseries Analysis ................................... 36 2.4.5 Continuous 1-D Wavelet Transform and EOF Analysis .................................. 45 2.4.6 Effects of profile on Runup and Reef vs Planar ............................................. 59 2.4.7 Neural Network Analysis ............................................................................... 63 2.4.8 Synthetic Timeseries Analysis ........................................................................ 65 2.5 CONCLUSIONS ....................................................................................................... 70 3. TSUNAMI RUNUP AND OVERLAND FLOW VELOCITIES ......................................... 73 3.1 FIELD MEASUREMENTS AND OBSERVATIONS ................................................................ 73 3.2 TSUNAMI MODELING ............................................................................................. 74 3.3 RESULTS AND DISCUSSION ....................................................................................... 75 3.3.1 Inter-model Comparison ............................................................................... 75 3.3.2 MOST Model Variability ................................................................................ 83 3.4 CONCLUSIONS ....................................................................................................... 88 4. TSUNAMI VS INFRAGRAVITY SURGE: COMPARISON OF THE PHYSICAL CHARACTER OF EXTREME RUNUP .............................................................................. 89 4.1 METHODOLOGY ..................................................................................................... 89 4.1.1 Field Observations ......................................................................................... 89 4.1.2 Tsunami and Infragravity Modeling .............................................................. 90 4.2 RESULTS AND DISCUSSION ....................................................................................... 92 4.3 CONCLUSIONS ....................................................................................................... 99 5. FUTURE RESEARCH ........................................................................................... 100 REFERENCES ............................................................................................................. 101 iv List of Tables Table 2.1: Linear regression coefficients for several tail measurements within 0.5% error of convergence. Where !=#$+ & since the line is fitted through the origin B is always 0. ................................................................................................... 21 Table 2.2: Tail measurements tested in this study. .......................................................... 22 Table 2.3: List of wave envelope parameters calculated and used in the deterministic analysis. .............................................................................................. 43 Table 2.4: Parameters calculated for the two wave envelopes generating the extreme event. ......................................................................................................... 44 Table 2.5: Profiles tested in this study. Profile 1 is the base profile used in this study. ........................................................................................................................ 60 Table 2.6: Maximum runup around the time of the extreme event ................................ 60 Table 2.7: Dimensionless envelope parameters and their physical representation. ....... 64 Table 3.1: Field data measurements not used in this study. ............................................ 76 Table 3.2: Simulation run times (for 10-hour physical time) and relevant information for each grid resolution. ....................................................................... 84 v List of Figures Figure 2.1: Schematic of numerical setup. Wave maker at the left of the domain, 450 m sponge layer at the left boundary and a wall at the right boundary. Waves take about 185 seconds to get to the shoreline. .......................................... 12 Figure 2.2: Runup time series (runup peaks shown in red dots). All runup peaks are real based on video analysis from the simulations. .................................................. 12 Figure 2.3: (a) Runup cumulative distribution functions (cdf) for configuration #90 were '( = *+ , -./ 01 = +* (23. (b) Averaged runup cdf’s for each frequency resolution. (c) Tails of averaged runup cdf’s. .......................................... 14 Figure 2.4: Scatter plot comparing maximum coarse runup versus maximum fine runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 1.93 degrees, mean error of -0.27 and mean orthogonal distance of 0.23. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. ................................................................................................ 16 Figure 2.5: Scatter plot comparing maximum coarse runup versus maximum finest runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 2.39 degrees, mean error of -0.36 and mean orthogonal distance of 0.22. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. ................................................................................................ 17 Figure 2.6: Scatter plot comparing maximum fine runup versus maximum finest runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 0.29 degrees, mean error of -0.08 and mean orthogonal distance of 0.29. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. ................................................................................................ 17 Figure 2.7: Scatter plot comparing 4+% for all frequency resolutions. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bottom plots shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. ..................... 18 Figure 2.8: Scatter plot comparing 46.8% for all frequency resolutions. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bottom plots shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. .................................................................................................................................. 19 Figure 2.9: Convergence plot of 92% for configuration #90 were ;< =12> and ?@=21 <AB. ............................................................................................................ 20 Figure 2.10: Convergence plot of 46.68% for configuration #100 were '( = *+, and 01 =++ (23. ........................................................................................... 21 Figure 2.11: Dimensionless number (Nc) versus number of waves for convergence within 0.5% error from the mean. ............................................................................ 21 vi Figure 2.12: Coefficient of variations (cv) for all 100 configurations and 24 different tail measurements (Table 2.2). ................................................................................. 23 Figure 2.13: Surf plot for coefficient of variation (cv), wave height ('() and peak period (01 ) for maximum runup. ............................................................................. 23 Figure 2.14: Surf plot for coefficient of variation (cv), wave height ('() and peak period (01 ) for 4+%and 46.8%. ............................................................................ 24 Figure 2.15: Runup, 4+%, measurements from field data versus Stockdon et al (2006) empirical 4+% equation (Terschelling contains 2 data sets). This data set was used for the development of S2006 equation. ............................................ 26 Figure 2.16: COULWAVE predicted 4+% versus Stockdon et al. (2006) empirical equation using the reef slope, 1/13, (black dots), best fit slope, 1/27, (blue dots) and best fit slope using a constant slope on the domain instead of the piecewise slope (red dots). Black dashed line is complete agreement. ................... 26 Figure 2.17: COULWAVE predicted 4+% using a constant slope on the domain instead of the piecewise slope versus Stockdon et al. (2006) (S2006) empirical equation using the best fit slope, 1/27. Red dots represent runup values within the range of the field data used by S2006 (Figure 2.15). Blue dashed line is complete agreement. ..................................................................................... 27 Figure 2.18: Low frequency energy both in the shallow water (h = 0.6 m) (red) and deep water (h = 100 m) (black) for both the (a) reef and (b) planar beach configurations. Solid lines are the mean values from 87 simulations, dashed lines show 95% confidence limits and blue dashed line shows the primary resonant mode from the domain. ............................................................................ 29 Figure 2.19: Frequency spectrum, zoomed in on IG frequencies, of water surface elevation along the one-dimensional reef transect from COULWAVE IG simulations: Plot for simulation using a sponge layer size of (a) 400m and (d) 3500m. ...................................................................................................................... 31 Figure 2.20: Convergence plot of the infragravity (0-0.02 Hz) part of the energy spectrum at 5 different locations on the (a) reef and (b) planar beach transects. For all simulations the wave parameters used for the input energy spectrum were '( =** , and 01 =+* (23. ...................................................... 33 Figure 2.21: Infragravity (0-0.02 Hz) energy spectrum behavior for 5 different locations across the (a) constant depth (with sponge layers on both ends), (b) reef and (c) planar beach transects. Each line is the average of 87 simulations at the specified location. .......................................................................................... 36 Figure 2.22: Frequency spectrum of water surface elevation along the one- dimensional planar beach big domain transect (40 km long) from COULWAVE IG simulation. Depicted in red is the first node (L/4) from each standing wave generated at each frequency. A wall is assumed at x= 3.94*10^4 m since waves start breaking at this location. ....................................................................... 36 Figure 2.23: Runup time series and offshore ($=C86, ) free surface elevations for the configurations that generated an extreme runup event. In red is the time when the extreme runup event occurred and the offshore free surface vii elevations some seconds before and after the extreme event. Event 1 (a and b), event 2 (c and d) and event 3 (e and f). ............................................................... 39 Figure 2.24: Runup time series and offshore ($=C86, ) free surface elevations for the configurations that generated an extreme runup event. In red is the time when the extreme runup event occurred and the offshore free surface elevations some seconds before and after the extreme event. Event 4 (a and b) and event 5 (c and d). ........................................................................................... 40 Figure 2.25: Offshore ($=C86, ) free surface elevations 400 seconds before and 100 seconds after the extreme event occurs for both cases, with a beach and without a beach. Red dashed line is the time when the extreme event occurs and the black dashed line is 185 second mark before the extreme event (since it takes about 185 seconds for a wave to reach the beach). (a) Event 1, (b) event 2, (c) event 3, (d) event 4 and (e) event 5. ...................................................... 41 Figure 2.26: Amplitude spectrum of free surface elevation at several locations in the profile for all the configurations that generated an extreme event. (a) Event 1, (b) event 2, (c) event 3, (d) event 4 and (e) event 5. Magenta line at $=C86, (before the face of the reef), green line at $=*D86, (face of the reef), red line at $=+686, (beginning of reef), ), black line at $= +E86, (middle of reef) and blue line at $=+C66, (end of reef). ..................... 42 Figure 2.27: Wavelet analysis for case 1 (with beach) and case 2 (without a beach). Top two panels correspond to Event 1 and bottom 2 to Event 2. ............................ 46 Figure 2.28: Wavelet analysis for case 1 (with beach) and case 2 (without a beach). Top two panels correspond to Event 3 and bottom 2 to Event 4. ............................ 47 Figure 2.29: Wavelet analysis for case 1 (with beach) and case 2 (without a beach) for Event 5. ............................................................................................................... 48 Figure 2.30: Scalogram from Event # 1 scaled by each frequency level. .......................... 52 Figure 2.31: Scalogram from Event # 1 scaled by each frequency level. .......................... 52 Figure 2.32: Scalogram from Event # 3 scaled by each frequency level. .......................... 53 Figure 2.33: Scalogram from Event # 4 scaled by each frequency level. .......................... 53 Figure 2.34: Scalogram from Event # 5 scaled by each frequency level. .......................... 54 Figure 2.35: Comparison of timeseries of free surface elevation and runup between the reef and beach transect. ..................................................................... 54 Figure 2.36: Comparison of continuous wavelet transform analysis between the (a) reef and (b) planar beach transect. ..................................................................... 55 Figure 2.37: Significant wave height behavior across the (a) reef and the (b) planar beach transects for 4 different period ranges. ......................................................... 56 Figure 2.38: Normalized EOF and theoretical cross-shore modes from event # 1. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. ................................................................................... 57 Figure 2.39: Normalized EOF and theoretical cross-shore modes from event # 2. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. ................................................................................... 58 viii Figure 2.40: Normalized EOF and theoretical cross-shore modes from event # 3. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. ................................................................................... 58 Figure 2.41: Normalized EOF and theoretical cross-shore modes from event # 4. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. ................................................................................... 59 Figure 2.42: Normalized EOF and theoretical cross-shore modes from event # 5. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. ................................................................................... 59 Figure 2.43: Runup timeseries around the time of the extreme event. (a) Event #1, (b) Event #2, and (c) Event #3. Profile #1: Thick Red Dotted Line, Profile #2: Blue dashed line, Profile #3: Red dashed line, Profile #4: Black dashed line, Profile #5: Magenta line, Profile #6: Black line, Profile #7: Thick Pale Green line, Profile #8: Dark Green line, Profile #9: Blue line, and Profile #10: Dark Orange line. .............................................................................................................. 62 Figure 2.44: Runup timeseries around the time of the extreme event. (a) Event #4, and (b) Event #5. Profile #1: Thick Red Dotted Line, Profile #2: Blue dashed line, Profile #3: Red dashed line, Profile #4: Black dashed line, Profile #5: Magenta line, Profile #6: Black line, Profile #7: Thick Pale Green line, Profile #8: Dark Green line, Profile #9: Blue line, and Profile #10: Dark Orange line. .......... 63 Figure 2.45: Correlation coefficient using all possible combinations using 3 parameters. .............................................................................................................. 65 Figure 2.46: Maximum correlation and standard deviation by each parameter sets. .................................................................................................................................. 65 Figure 2.47: Expected runup timeseries from each of the different synthetic timeseries. ................................................................................................................ 67 Figure 2.48: Wave envelopes preceding the extreme runup from events (a) 1, (b) 2 and (c) 3 ................................................................................................................. 68 Figure 2.49: Wave envelopes preceding the extreme runup from events (a) 4 and (b) 5. .......................................................................................................................... 69 Figure 2.50: Mean runup heights using different number of wave envelopes in the timeseries. ................................................................................................................ 70 Figure 3.1: Maximum tsunami amplitudes (m) predicted by MOST (left panel) and GeoClaw (right panel) in the Sendai plain. ............................................................... 77 Figure 3.2: Comparison between measured east-west distances and model predicted distances in the Sendai plain. ................................................................... 77 Figure 3.3: Comparison of runup height measurements and inundation line between data, MOST and GeoClaw during the 2011 Tohoku event in the Sendai plain. ............................................................................................................. 79 Figure 3.4: Maximum tsunami amplitudes (m) predicted by MOST (left panel) and GeoClaw (right panel) in the Sendai plain. ............................................................... 79 Figure 3.5: Comparison of the runup heights distributions between the interpolated field data, MOST and GeoClaw models. The distribution uses a runup interval spacing of 0.05 m. ............................................................................. 80 ix Figure 3.6: Estimated differences between field data runup heights and the topographic elevations from the numerical grid at the location of the runup measurement. .......................................................................................................... 80 Figure 3.7: Maximum flow velocities predicted by MOST (left panel) and GeoClaw (right panel). ............................................................................................................. 82 Figure 3.8: Comparison of maximum flow velocities at the Sendai plain between Koshimura and Hayashi (2012) measurements (gray triangle), GeoClaw predictions (gray square), MOST predictions using n = 0.025 (black circle), MOST predictions using n = 0.030 (black diamond) and MOST predictions using n = 0.055 (gray upside down triangle). The vertical bars on the model data provide the standard deviation of the predictions in the measurement window. At F2, two measurements were taken. ...................................................... 82 Figure 3.9: (top panel) Comparison between GeoClaw and MOST distributions of maximum shoreline flow velocities and (bottom panel) 1 meter depth maximum flow velocities at the Sendai plain. The distributions use a velocity interval spacing of 0.05 m/s. .................................................................................... 83 Figure 3.10: Comparison of the runup height distributions between the 6 different grid resolutions using MOST. The distributions use a runup interval spacing of 0.05 m. ...................................................................................................................... 85 Figure 3.11: (top panel) Comparison of the maximum shoreline flow velocities distributions and (bottom panel) 1-meter depth maximum flow velocities distributions between the 6 different grid resolutions using MOST. The distributions use a flow velocity interval spacing of 0.05 m/s. ................................. 85 Figure 3.12: Mean flow velocity at different flow depths. The 6-meter flow depth corresponds approximately to the shoreline. The thick black line represents the calculated mean flow velocities using a Froude number of 1. ........................... 87 Figure 3.13: Inland maximum flow velocities across shore in the Sendai plain, (top panel) comparison of the average flow velocities between GeoClaw and the 6 different grid resolutions using MOST (bottom left panel) comparison of peak flow velocities (bottom right panel) comparison of standard deviations. .................................................................................................................................. 87 1 1. Introduction and Background In recent history “sneaker” waves or “king” waves (in Australia), also referred in this study as extreme infragravity (EIG) waves, have been responsible for killing many people throughout the world. Particularly so on the coast of Oregon, where more than 21 people have died since the year 1990 due to this type of phenomenon (The Oregonian, 2016). People that are usually walking by the beach on a “sunny” day are suddenly surprised and washed away by what appears to be a “tsunami-like” wave, which in reality is an EIG wave. Extreme runup events are one of the leading causes of death by drowning in the U.S Pacific Northwest. In 2013 the town of Hernani was destroyed by a Typhoon. Videos from the event reveal that during the storm an EIG wave was generated (producing an extreme runup event) and hit the town destroying some houses in its path (Plan International, 2017). Chien et al. (2002) reported 140 “rogue” waves (better term is sneaker waves since they were reported near the coast) in the coastal zone of Taiwan from 1949-1999. Nikolkina and Didenkulova (2011) reported 39 sneaker wave events at the coast, from 2006-2010, which resulted in 46 fatalities and 79 injuries (in different parts of the world). Many times in the news, EIG waves have been called “tsunami-like” waves due to their similar runup properties. These “rare” and unpredictable events have been recorded and videos can be found online. Comparable to EIG waves, tsunamis also generate extreme inundation in a given area. Landslide and earthquake generated tsunamis, such as those in Sumatra (2004) and Tohoku (2011), have been responsible for killing thousands of people and have generated billions of dollars in damage. These extreme runup events may result in the loss of lives, coastal flooding, beach erosion, damage to coastal structures and damage to marine vessels. There is currently a need to better understand extreme properties of ocean wave runup to assess hazard on coastal areas prone to these events and increase public safety. There has not been significant research on EIG waves and runup, and so they are currently poorly understood (in terms of their generation mechanisms). On the other hand, much research has been done on infragravity (IG) waves; it is expected that to better understand EIG waves one must first understand IG waves and their generation mechanisms. Both are “infragravity”, meaning lower frequency than gravity waves, and the only difference is that EIG waves are an extreme case of IG waves. 2 For many centuries, no one had knowledge of the existence of IG waves. IG waves tend to go unnoticed to human perception as their frequency (20-300s or more) is much lower than that of a wind wave (2-20s), and they can be characterized by slow, gradual motion. It wasn’t until the late 1940s that groundbreaking discoveries in IG waves were made. Munk (1949) and Tucker (1950) were the first to study and observe this phenomenon. Both recorded small amplitude fluctuations with frequency bands in between wind waves (2-20s) and tides (12 and 24hrs). These oscillations were named “surf-beat” by Munk. Since then, many different scientists have endeavored to study this phenomenon and its generation mechanisms. Longuet-Higgins and Stewart (1964) found that IG waves are bound to wave groups which are a result of “radiation stress” variations. This bound long wave is released due to nonlinear interactions between the short waves when wave groups reach the breaking point and waves start to break (Dong et al. 2009a, Herbers et al. 1995b, Gallagher 1971 and Longuet-Higgins and Stewart 1962). Energy is then transferred to lower or higher frequencies through a nonlinear processed called three-wave or triad interactions (Herbers et al. 1994, Abreu et al. 1992, Elgar and Guza 1985). Another mechanism that also involves wave groups and radiation stress was proposed by Symonds el al. (1982). In this mechanism, IG waves are formed due to the varying breaking point of wave groups; larger waves break further offshore than the smaller waves of the wave group. This results in a time variation of the radiations stress which is balanced by a time varying setup therefore generating an IG wave. Several researchers have further studied this mechanism and included it in their models (Bertin et al. 2018, Becker et al. 2016, Schaffer and Svendsen 1988, Lo 1988, List 1992). IG waves dissipate by various dominant mechanisms: bottom friction, breaking, IG-IG interactions, IG-short wave interactions (Inch et al. 2017, Bakker et al 2015, Thomson et al. 2006, Henderson et al 2006, Lowe et al. 2005). IG waves have been found to affect coastal erosion, flooding, oscillation in harbors, and damage on coastal structures (Van Thiel de Vries et al. 2007, Rusell 1993, Okihiro et al. 1993, Holman and Sallenger 1985, Guza and Thornton 1982 and Holman and Bowen 1982). The research discussed above connects properties of IG generation to the nonlinearity of the wave group. Wave nonlinearity is often described in different terms, depending on whether the waves are in deep or shallow water. In deep water, four-wave interactions are predominant (Hasselmann 1962, Hasselman et al 1973, Young and Van Vledder 1993) and result in the exchange of energy between waves. Triad interactions are almost negligible in deep water (Phillips 3 1960) but play a major role in shallow and maybe transitional water depths (Young and Eldeberky 1998). As it was mentioned previously, triad interactions are responsible for transferring energy to lower (IG) or higher frequencies in shallow water. In this nonlinear process energy is transferred to lower frequencies (IG) by the difference interaction (F G =F H −F J ) and to higher frequency by the sum interaction (F G =F H + F J ), where f denotes a wave frequency. Triad interactions only occur in shallow water since in deep water the conditions F H + F J =F G and K H + K J =K G cannot be complied due to the dispersion relation, where k denotes a wavenumber. Triad interactions have been successfully modeled with Boussinesq-type models (Gao et al. 2018, Su and Ma 2018, Norheim and Herbers 1998, Herbers and Burton 1997, Elderberky and Batjes 1996, Madsen and Sorensen 1993, Shaffer et al 1993, Liu et al. 1985, Freilich and Guza 1984). Boussinesq models have also been used to study IG waves in fringing reefs (Roeber and Bricker 2015, Roeber and Cheung 2012, Nwogu and Demiribilek 2010). IG waves have been found important for reef hydrodynamic studies. Coral reefs are known to protect coastal areas from waves since much of the wave energy is dissipated while traveling towards the shore (Lowe et al. 2005). Nevertheless, coastal damage has been reported in areas protected by reefs which has been caused by runup produced by EIG waves and IG oscillations (Roeber and Bricker 2015, Nwogu and Demirbilek 2010). As waves travel from deep to shallow water they break at the face of the reef and release IG waves (Pomeroy et al. 2012, Pequiget et al. 2014). It is very important to study and understand EIG waves and runup in a reef environment to better protect these coastal areas. This study shows the importance of the input energy spectrum frequency resolution when studying IG waves and their extreme events. There has been a great deal of progress made, in the past 20 years, in the study of surf zone hydrodynamics and wave runup. Nevertheless, the generation mechanisms for “rare” or extreme runup events have received considerably less attention. As the EIG wave reaches the shoreline it will generate an extreme runup event. Runup is one of the processes responsible for beach erosion, wave overtopping, inundation and damage to costal structures (Erickson et al. 2007, Stockdon et al 2007, Ruggiero et al. 2001, De Rouck et al 2005, Sallenger 2000). Wave runup is a complex process that depends on water levels (including IG waves), incident wave conditions, surf zone properties, and beach properties. Many field and laboratory studies have shown that runup is directly related to the Irraberen number (function of beach slope, wave height and wavelength) (Batjes, 1974 and Hunt 1959). Several studies have shown that IG waves play a dominant role 4 when analyzing runup (Guza and Thornton 1982). In low sloping beaches for high energy wave conditions, runup will be IG-controlled just as in a reef protected sandy beaches (Bakker et al. 2015, Nwogu and Demirbilek, 2010). Several empirical equations have been developed to calculate runup (9 J% ) based on field and laboratory experiments (Suanez et al. 2015, Paprotny et al 2014, Stockdon et al. 2006, Ruggiero et al. 2004, Ruessink et al. 1998, Holman 1986 and others), but only few are normally used, the most common one being Stockdon et al (2006) (since is is more accurate and it was developed using 10 different data sets including contribution from IG waves). A problem with the 9 J% metric is that it may not correlate with the sneaker (EIG) wave event; a given 9 J% may not indicate a likelihood of EIG events. To better understand extreme runup events, an analysis of the tail of the distribution has to be performed. Few studies have measured and analyzed EIG waves and runup (Roeber and Bricker 2015, Shimozono et al. 2015 and Sheremet et al. 2014). Sheremet et al. (2014) recorded water levels at Banneg Island, France, during a storm event. Measurements collected revealed an EIG wave runup event of more than 2 meters. An 80 second IG wave and a 300 second IG wave were discovered at the time of the runup event. A Boussinesq model was used to understand how these waves were produced. Researchers from the study hypothesize that the 80 second wave (captured by the numerical model) is produced by nonlinear shoaling of the swell and that the 300 second wave (not captured in the numerical model) was produced by the nonlinear interactions between trapped IG waves. Roeber and Bricker (2015) studied an extreme runup event generated by a “tsunami-like” wave (EIG) in Hernani, Philippines. Hernani is a coastal town protected by a fringing reef. This event occurred during Typhoon Haiyan and was successfully modeled using a Boussinesq-type model. Authors suggest that this extreme event was generated by a tsunami-like bore which can result from surf beat. It is also suggested that the reef resonance could have made the event even worse. Shimozono et al. (2015) also studied the runup during typhoon Haiyan using a Boussinesq-type model. The authors suggest that the extreme runup events are due to the coupling of IG waves (generated by waves breaking at the reef) and sea swells. From these studies, it is hypothesized that the EIG wave is produced by nonlinear interactions between waves during high energy wave conditions (usually storm events) or from abrupt breaking of waves at the reef face generating an energetic IG wave. The generation mechanism for EIG events is still unclear since nonlinear interactions and bore formation don’t fully explain extreme runup events. Nonlinear interactions 5 occur constantly and bores were observed multiple times on occasions where there was no extreme event. Also, the importance of spectrum frequency resolution is not mentioned in either study. Recently, Garcia-Medina et al. (2017) studied large runup on gently dissipative beaches. Based on observations on a mildly sloping beach and numerical model results they conclude that bore- bore capture is essential but not sufficient to fully explain large runup events. Bore-bore capture are generated when waves travel at different speed (celerity) and catch up to each other, then nonlinear interactions take place and larger waves are generated. It was found that the phase difference of waves cause a 30% variability in the maximum runup. Finally, the largest offshore waves were not always correlated with large runup events. Again, importance of spectrum frequency resolution is not mentioned and authors do not mention the input frequency resolution used in their models. Frequency and directional spread have also been observed to affect the runup on the shoreline (Guza and Feddersen 2012). To date, several research studies do not report what was used for the IG offshore boundary condition and the input spectrum frequency resolution is rarely reported in the model used (Garcia-Medina et al. 2017, Su et al. 2015, Shimozono et al. 2015, Bakker et al. 2015, Roeber and Bricker 2015, Shimozono et al. 2015, Sheremet et al. 2014, Pomeroy et al. 2012, Bakker et al. 2012, and others). The preliminary research presented in this proposal reveals that input spectrum frequency resolution plays an important role when studying IG waves and runup. Also, it shows that there is an ideal frequency resolution which can make the modeling more accurate and efficient by capturing most of the low frequency energy transfers during the nonlinear wave interactions. Similar to EIG waves, tsunamis are also responsible for extreme runup events. Both EIG waves and tsunamis are long waves (of the order of minutes) but with different generation mechanisms and potentially different hydrodynamic properties. Once the tsunami and EIG wave reach the shoreline their extreme inundation processes are similar. Tsunamis can be generated by earthquakes, landslides, volcanic eruptions or impact from large meteorites. A recent example is the Tohoku earthquake generated tsunami in Japan. This tsunami was generated on March 11, 2011, by a L M =9.0 earthquake 130 km off the coast of Sendai, Japan (Mori et al. 2011). This event is one of the worst in Japan history, killing more than 15,000 people and causing more than $200 billion (USD) in damage. Available data shows that in some areas runup elevations reached 40 m and flow velocities reached more than 14 m/s (Mori et al. 2011, Koshimura and Hayashi 2012). 6 This event raised the safety concerns of many coastal communities. Along the Sendai plain, the tsunami traveled more than 5 km inland with a maximum measured runup of approximately 9.4 m and an average of 2.5 m above Mean Sea Level (MSL) (Mori et al. 2011). The tsunami velocities measured by Koshimura and Hayashi (2012) at different locations on the Sendai plain ranged from 2-8 m/s. Due to the measurements collected during and after the Tohoku event, researchers are provided a great opportunity to model, study and understand the nearshore and onshore hydrodynamics of tsunamis. Numerical model predictions are essential for all Tsunami Hazard Assessments (THA), whether these assessments are probabilistic or deterministic in nature. Generally, methodologies for Probabilistic Tsunami Hazard Assessment (PTHA) use numerical models to predict various products (runup, inundation and flow velocity); this output is processed with a statistical analysis to compute recurrence rates for a given location (Geist and Parsons 2006, Gonzalez et al. 2009, Thio et al. 2012 and Gonzalez et al. 2013). By better understanding model bias and sensitivities, more accurate and reliable THA and PTHA are possible. This will lead to improvement in risk assessment and hazard mitigation in coastal areas susceptible to tsunamis. In past decades, numerical models have been developed that can accurately predict tsunami runup, inundation and flow velocity. Due to the surge of “state of the art” numerical models and their widespread use in this field, there is a need to better understand model predictions and variability for better evacuation and construction planning. In this proposal, The Method of Splitting Tsunami (MOST) (Titov and Synolakis, 1995 and 1998) and GeoClaw (LeVeque et al. 2011, Berger et al. 2011) tsunami models are used to compare runup and flow velocity results to measured field data. Available field survey data and video footage analysis measurements are used to compare model runup and flow velocity predictions. Possible sources of error are analyzed and discussed. This study includes detailed comparisons between observations and numerical simulations in Sendai, focusing on the Sendai plain area. 7 1.1 Objectives and Organization The objective of this study is threefold. Firstly, the effort aims to better understand EIG waves and EIG wave runup during storms. Secondly, it is intended to increase the understanding of the bias and variability in model predictions to locate possible sources of error. Thirdly, present a methodology to physically differentiate between tsunami and IG runup. The following research questions are presented and answered in this study: 1. Does the discretization of frequencies affect the nonlinearity of waves to generate extreme runup? 2. Will the tail of the runup distributions converge at an “ideal” frequency resolution? 3. What is the cause (generation mechanisms) of EIG waves and runup? 4. How often does this phenomenon occur? The following are the research questions pertaining to the second objective: 5. What is the variability in predictions between the models used? 6. What digital elevation map resolution is ideal for stable products? 7. Does the Froude number change across the inundated area? These are the final questions pertaining to the third objective: 8. Which event, tsunamis or exteme IG, will yield greater flooding flow depths, flow speeds and Froude number at the beach? 9. Can we see the effect of bore-bore capture in the extreme IG event? It is hypothesized that the discretization of frequencies affect the nonlinearity of waves and therefore the generation of extreme runup, and the tail of the runup distribution will “converge” at an “ideal” frequency resolution. It is proposed that the cause of EIG waves is a combination of different factors explained herein, one of the causes being two or more successive sets with specific properties are needed. Based on a statistical analysis a simple calculation is done to determine the predictability of EIG waves. The hypothesis for the second objective are: there is not much variability in the predictions from the numerical models used. These models will yield stable products using an “ideal” grid resolution . Due to the complexity of the overland flow during a 8 tsunami, the Froude number is not stable across the inundated area. The hypothesis for the last objective are: Due to longer tsunami period, maximum overland flow depths and durations of flooding should be larger with tsunami. Due to shorter IG wave period, maximum flow speeds (and thus instantaneous dissipation rates) should be larger with extreme IG waves. Finally, with thinner depths and greater speeds, maximum Froude number will be greater during extreme IG events. This study aims to shed some new light on these subjects and arise the interest of the scientific community to further study it more. The structure of this study is as follows: Chapter 2 describes the methodology used in the EIG wave and runup study. It presents the numerical model employed, experiment setup, statistics used to analyze the data, results, discussion and proposed future research. Chapter 3 focuses on the model bias and variability for the tsunami overland flow study. Tsunami runup (and other products) is compared between two different numerical models and field data. This chapter consists of the methodology, results and a discussion. Finally, Chapter 4 presents a comparison of overland flow characteristics between tsunamis and energetic IG waves. 9 2. Modeling Extreme Infra-Gravity Wave Runup In the field, there are many processes going on at the same time which makes it difficult to study EIG wave runup in a precise way. Therefore, this problem is approached from a numerical perspective since in the field there are many processes going on at the same time which makes it difficult to study this phenomenon (EIG wave runup) in a precise way. From the literature review, it was established that Boussinesq-type models can predict the dynamics and formation of IG waves in fringing reefs and can also predict runup at beaches. The following section describes the numerical model used in this study in more detail. Finally, the EIG wave event that occurred in Hernani during typhoon Haiyan, Philippines, presents an excellent opportunity to test the first part of the hypothesis. Therefore, our experiment setup resembles much of what happened in Hernani. 2.1 Numerical Model and Setup The Cornell University Long Wave (COULWAVE) model is used in this study because of its capabilities in predicting the dynamics and formation of IG waves and runup. COULWAVE is a Boussinesq-based numerical model (Lynett and Liu, 2002) able to simulate wave propagation from fairly deep water to the shoreline ( MNOPQPRSTU VPWTU ≥2) with high accuracy (Wei et al. 1995). COULWAVE has been validated and successfully tested in many studies. It has been used for a wide range of applications such as: wave run-up, propagation, inundation, wave breaking, tsunamis, currents in ports and harbors and hurricane waves (Lynett, 2007, Lovolt et al., 2013, Parsons et al., 2014, Lynett et al., 2014, among others). For the present study, the problem is investigated implementing one-dimensional COULWAVE. This model solves the fully non-linear, weakly-dispersive wave equations given in the one-dimensional conservative form as: ; T + (Z ∝ ;) ] + ^ _ =0 (2.1) 10 (Z ∝ ;) T + (Z ∝ J ;) ] + a;b ] + a;^ ] −Z ∝ ^ _ =0 (2.2) Where ; =b+ ℎ is the total water depth, b is the free surface elevation, h is the water depth, Z ∝ denotes the velocity at a reference elevation d ∝ , ^ ] is the 2 nd order terms of the depth integrated moment equation and averaged velocities, and ^ _ includes the 2 nd order terms of the continuity equation. A spatially constant bottom friction of F =0.035 was used with a quadratic bottom friction law for the model simulations. Manning friction was not used since it over-dissipated wave run-up during the testing of the model. For more details about the equations and numerical schemes employed in the model refer to the references mentioned above. As was mentioned before, the extreme infra-gravity wave that struck the town of Hernani, Philippines, presents an opportunity to test the hypothesis presented in this study. Bathymetry near the coast of Hernani consists of a steep slope (face of the reef) followed by a mild slope, where the reef is located. A schematic for the numerical setups tested in this study are presented in Figure 2.1. The hypothesis from this study are tested using the reef and planar beach configuration. An internal-domain wave-maker is driven by Joint North Sea Wave Project (JONSWAP) input spectrum. A wall is located at the right boundary and a sponge layer at the left boundary. Appropriate length of the sponge layer was found to be at 450 m or wider based on tests to minimize any reflections. A grid resolution of 2 m is used for all the simulations. In this study, we do not impose any special IG boundary conditions. The simulations are forced with an offshore energy spectrum (JONSWAP), which in COULWAVE is broken down into a set of discrete amplitude/frequency/direction sine waves with some specified ∆F and ∆i (when 2D). The offshore wave boundary is then the linear superposition of all of these discrete waves, each with some random phase, created with the internal source generator. IG waves are generated by a nonlinear process. As soon as waves are generated by the wavemaker nonlinearity takes over and IG waves can appear. To further investigate the problem, 100 different configurations of significant wave heights (; j =3 kl 12 m) and peak periods (? W =13 kl 22 sec) are tested (i.e. each ; j value was combined with each ? W value, configuration #1: ; j =3 m and ? W =13 sec, #2: ; j =3 m and 11 ? W =22 sec…#11: ; j =4 m and ? W =13 sec, etc.). To examine the effects of frequency discretization on extreme infra-gravity run-up, three different frequency resolutions are tested in the model: ∆F =0.001 (Blno<A,∆F _ ),0.0001 pFqrA,∆F s t nru 0.00005 (FqrA<k,∆F sT ) Hz. Each individual simulation is run for H ∆s seconds, plus a constant warm-up time. In order to make valid statistical comparisons of the results across the three different frequency resolutions, the same length of time or number of waves must be used. Since the wave pattern repeats itself every H ∆s seconds, each of the three different frequency resolutions will repeat at different times and different number of waves. For ensure the same number of waves are examined across all simulated frequency resolutions, N simulations from the finest need to be compared to 2*N of the fine and to 20*N of the coarse frequency (i.e. configuration 1: ; j =3 m and ? W =13 sec was simulated 100 times using coarse frequency, 10 times using fine frequency and 5 times using the finest frequency; all yield the same number of waves). It is important to note again that each simulation has a unique random phase seed, so that, for example, each of the coarse frequency simulations, while all having the same frequency discretization, yield a different deterministic representation of the input wave energy spectrum. Simulation time for the coarse, fine and finest frequencies was 1900 sec, 10900 sec and 20900 sec, respectively. The first 15 minutes of each simulation are disregarded (not included in the data analysis) to allow the model to reach a quasi-steady state (i.e. a warm-up time). Also, a threshold water depth of 15 cm is used to define the shoreline location. In this study, runup is defined as the vertical distance from the still water level to the point at which the shoreline is located. Only the peaks from the runup time series were used for the data analysis part (red dots in Figure 2.2). A careful review of some videos from the simulations showed that the runup peaks from the time series are real and some of them are localized peaks. The total simulation time for all the runs was about 11,200 wall-clock hours. More than 1 million individual wave crests were simulated. 12 Figure 2.1: Schematic of numerical setup. Wave maker at the left of the domain, 450 m sponge layer at the left boundary and a wall at the right boundary. Waves take about 185 seconds to get to the shoreline. Figure 2.2: Runup time series (runup peaks shown in red dots). All runup peaks are real based on video analysis from the simulations. 13 2.2 Optimum Frequency Resolution Runup is defined as the maximum elevation of the water level on a beach or structure above the still water level. For this section, Cumulative Distribution Functions (CDF) are constructed using only the runup peaks (including localized peaks) from the runup time series. Figure 2.2 shows an example runup timeseries with the runup peaks in read. CDFs of runup elevations were developed for each configuration. Then, CDFs were aggregated for each Hs and Tp combination (i.e. from the fine frequency resolution, ∆F s , 10 different CDFs were averaged for each configuration, 100 for the coarse and 5 for the finest). Figure 2.3a and 2.3b show all the individual runup CDFs and the averaged runup CDFs for a specific configuration (#90) where the significant wave height (; j ) is 12m and the peak period (? W ) is 21 sec. It is very interesting to note that for this particular configuration the mean is the same (2.56m) for all the ∆F′< tested but the ∆F s had the highest maximum (8.74m) compared to the other resolutions. ∆F s had the highest maximum runup for more than half of the configurations (58%) compared to 36% for ∆F sT and 6% for ∆F _ . By changing the ∆F, from ∆F _ to ∆F s , the maximum runup predictions increase by an all- configuration average of 9% (in some configurations the increase is as high as 37% and in others it can decrease as low as 6%). Figure 2.3c shows the tail of the same runup distributions. There is a 10% difference of 9 J% statistics between ∆F _ , ∆F s and ∆F sT . As a particular exceedance runup value approaches the maximum runup the difference between the tested frequencies increases but it remains constants between ∆F s and ∆F sT . 14 Figure 2.3: (a) Runup cumulative distribution functions (cdf) for configuration #90 were ' ( = *+ , -./ 0 1 = +* (23. (b) Averaged runup cdf’s for each frequency resolution. (c) Tails of averaged runup cdf’s. In engineering applications, the 9 J% statistic is often used as a design measure. 9 J% is a statistical measurement that represents the value of runup that will be exceeded by 2% of the individual runup events. It is shown that the problem with the 9 J% statistic is that it is poorly correlated to the sneaker wave (EIG) event which is the main focus of this study. Since the runup produced by the sneaker wave is an extreme event, the tail of the distribution must be considered (Figure 2.3c). The tail of the distribution controls the extreme event. In order to study this type of events a stable “extreme” statistic needs to be used that properly describes the tail of the distribution. Then this statistic must be compared between the different resolutions to determine if it will converge at an ideal resolution or if it will change as ∆F keeps decreasing. In order for the statistical tail measurement to be reliable, it has to be stable, convergent and has to properly represent the tail of the distribution. In this study, to better understand the extreme runup events, we are most concerned with the outliers from the runup dataset, which are sometimes eliminated from the analysis. To establish a reliable ∆F, scatter plots were developed using several different tail measurements from the three ∆F′< mentioned previously. Figure 2.4 and 2.5 show scatter plots of maximum runup predictions from the ∆F _ versus the ∆F s and maximum runup predictions from the ∆F _ versus the ∆F sT , respectively. Both figures show that when using the ∆F _ , the maximum runup is under predicted. Both comparisons yielded a bias greater than 1.5 degrees, mean errors greater than 0.25 m and mean precision greater than 0.2 m. In this study bias is 15 calculated as the absolute difference between the angle from the least square fit line at the origin minus the 45-degree (solid line or perfect agreement) angle. The dashed lines represent the ±20% error. Another statistical measurement used is the mean error between the two measurements (wqn< =x y [9 H ]−9 J , where x y [9 H ] and 9 J are the expected and estimated runup values, respectively). Also, the absolute orthogonal distance from the linear regression line to each data point is used to measure precision or how scattered the data is (uq<(| H ,| J ,(} ~ , Ä )= |(Ç É ÑÇ Ö )] Ü Ñ(] É Ñ] Ö )Ç Ü á] É Ç Ö ÑÇ É ] Ö | à(Ç É ÑÇ Ö ) É á(] É Ñ] Ö ) É , where. | H and | J are the points were the line passes and (} ~ , Ä ) is the point outside the line. Figure 2.6 shows a comparison between the maximum runup predictions from the ∆F s versus the ∆F sT . In this figure, the bias and mean error are closer to 0 (bias of 0.29 degrees and a mean error of 0.08 m) while the precision is 0.29 m. Since the bias from ∆F s versus ∆F sT is closer to zero this means that they are convergent. Henceforth one can trust either of the predictions from the numerical model when using the ∆F s or ∆F sT resolutions; for our purposes, we expect a numerically convergent runup distribution when using ∆F s . It is important to note that identical ∆F s simulations with different random seeds will yield different maximum runup values; however, this is due to a slightly different deterministic realization of the input spectrum, and not the resolution of that realization. Nevertheless, one can see that precision spreads linearly, the higher the runup the more scatter the data is. To see if this convergence between the maximums runup predictions from the two different ∆F′< is physical (since maximums are not a stable statistic and depend on the random seed), several other tail measurements are tested and yield very similar results, some of them shown in Figure 2.7 and 2.8 (9 J% , 9 H% , 9 ~.â% , 9 ~.H% , and several other tail measurement averages see Table 2.2). The primary difference, between the maximum runup and the other tail measurements, was that as the tested tail measurements depart from the maximum (9 äN] ) the data becomes less scattered and bias decreases. Comparisons between ∆F s versus ∆F sT always yield the lowest bias and mean error suggesting that they are convergent and either ∆F can be trusted to better study extreme runup caused by extreme IG waves. Based on this, runup predictions from numerical simulations that use 1000 frequencies in the input energy spectrum are significantly different from those that use 10,000 frequencies because of nonlinearity and runup measurements will be under-predicted. It can be 16 suggested that the coarse ∆F energy spectra does not capture all the exchange in energy between the different frequencies after the waves break at the face of the reef. Figure 2.4: Scatter plot comparing maximum coarse runup versus maximum fine runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 1.93 degrees, mean error of -0.27 and mean orthogonal distance of 0.23. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. 17 Figure 2.5: Scatter plot comparing maximum coarse runup versus maximum finest runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 2.39 degrees, mean error of -0.36 and mean orthogonal distance of 0.22. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. Figure 2.6: Scatter plot comparing maximum fine runup versus maximum finest runup. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bias of 0.29 degrees, mean error of -0.08 and mean orthogonal distance of 0.29. Bottom plot shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. 18 Figure 2.7: Scatter plot comparing 4 +% for all frequency resolutions. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bottom plots shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. 19 Figure 2.8: Scatter plot comparing 4 6.8% for all frequency resolutions. Red dashed line is complete agreement and blue dashed lines are the 20 percent error. Bottom plots shows the absolute orthogonal distance of each data point to the complete agreement line. Red dashed line is the mean distance. 2.3 Convergence and Coefficient of Variation As it was mentioned before, for a tail measurement to be reliable it must be stable, convergent, and be a useful representation the tail of the distribution. Therefore, a convergence test has to be conducted for the tail measurements that were used in the previous section of this study. First, convergence needs to be defined. In this study, convergence is reached when ã(å) ç >Χ, were Χ is a user-defined error level of convergence, A is the cumulative mean of a given tail measurement, N is the highest wave index that exceeds Χ and r is the mean of the tail measurement (see Figure 2.9). In words, convergence is defined in this study as the highest wave index that exceeds a given 20 error level (i.e. 1%). This measurement is taken going from right to left as shown in Figure 2.9. Runup time series were scaled by the peak period in order to analyze the same number of waves for each configuration. Convergence was calculated for every Hs and Tp combination. Figure 2.9 and 2.10 show the convergence for 9 J% and 9 ~.â% for the same configuration, respectively. To present and compare convergence from the different simulation configurations, the dimensionless Nc number (êB = ë U í ì ×A ï ñ ó Ü were h is the water depth, ò Ä is the offshore wavelength and ; j is the significant wave height), was developed. Figure 2.11 shows the number of waves that need to be included in a simulation to achieve convergence of 9 H% at a 0.5% error, for all the ; j and ? W combinations. Table 2.1 shows the coefficient from the 15% error line for the linear regression line fitted at the origin. The linear regression is forced to the origin because Nc goes to zero when ℎ→0, and ? W →∞ which means that ò Ä →∞. From Table 2.1 the number of waves needed to reach convergence at a 0.5% error can be calculated based on the ; j and ? W ranges used in this study. Thus, based on Table 2.1 for a wave condition were ; j =10 m and ? W = 20 secs, the Nc number is 0.40 and to reach convergence for 9 H% at a 0.5% error, 5077 waves need to be simulated at a ∆F s resolution. Table 2.1 can be very useful to quickly calculate the number of waves needed for convergence (stable mean) at a 0.5% error for a wide range of wave conditions. Figure 2.9: Convergence plot of 9 J% for configuration #90 were ; j =12> and ? W =21 <AB. 21 Figure 2.10: Convergence plot of 4 6.68% for configuration #100 were ' ( =*+, and 0 1 = ++ (23. Figure 2.11: Dimensionless number (Nc) versus number of waves for convergence within 0.5% error from the mean. Table 2.1: Linear regression coefficients for several tail measurements within 0.5% error of convergence. Where !=#$+ & since the line is fitted through the origin B is always 0. Coefficient R 2% R 1.5% R 1% R 0.5% R 0.3% A 12496 12564 12586 12610 12637 B 0 0 0 0 0 Another statistical parameter that is very useful is the coefficient of variation (cv). The cv shows the extent of the variability with respect to the mean. The cv is calculated in the following 22 manner: Bõ = ú ù where û is the standard deviation and µ is the mean. For the 100 different configurations, using a ∆F s resolution, the cv is calculated for 24 different tail measurements (see Table 2.2) and is presented in Figure 2.12. It is very interesting to observe that the calculated cv’s are bounded, between 0.14-0.25, for a wide range of cases (configurations). These results are very useful when analyzing a deterministic case. From that one case, based on the ; j and ? W , the variation of the standard deviation and the average is known. From Table 2.1 the number of waves needed to come up with a convergent statistic of the tail can be calculated. By running one case, based on Figure 2.12, one can know the range the cv is going to fall into which is very good when performing a deterministic analysis. Finally, Figure 2.13 and 2.14 shows a surf plot to correlate the cv with ; j and ? W for different R-values. It can be observed that the cv is higher for ; j ≤6> and often for lower period waves. For large ; j and long period waves, variation is lower. It is suggested that for large waves with long periods, rare runup is strongly controlled by the IG waves and therefore there will be less runup variability about the mean. Runup variability increases when wave energy decreases and IG waves decrease. Table 2.2: Tail measurements tested in this study. Average Tail Measurement Tail Measurement R H.â% −R H% R J% R H.â% −R ~.¢% R H.â% R H.â% −R ~.J% R H% R H% −R ~.£% R ~.§% R H% −R ~.¢% R ~.•% R H% −R ~.â% R ~.£% R H% −R ~.J% R ~.¢% R H% −R ~.H% R ~.â% R ~.¢% −R ~.J% R ~.¶% R ~.£% −R ~.â% R ~.G% R ~.£% −R ~.H% R ~.J% R ~.â% −R ~.H% R ~.H% 23 Figure 2.12: Coefficient of variations (cv) for all 100 configurations and 24 different tail measurements (Table 2.2). Figure 2.13: Surf plot for coefficient of variation (cv), wave height (' ( ) and peak period (0 1 ) for maximum runup. 24 Figure 2.14: Surf plot for coefficient of variation (cv), wave height (' ( ) and peak period (0 1 ) for 4 +% and 4 6.8% . 2.4 Generation Mechanisms of Extreme Infra-Gravity Wave and Runup 2.4.1 Runup Analysis Extreme wave runup events are important for the prediction of beach erosion, wave overtopping, inundation, and damage to coastal structures. The generation mechanisms of EIG waves and runup are still poorly understood. Before proceeding with further analysis on extreme runup events, runup predictions from the numerical model need to be compared to runup predictions from a well stablished empirical equation. Several empirical equations have been developed to calculate runup (9 J% ) based on field and laboratory experiments (Suanez et al. 2015, Paprotny et al 2014, Stockdon et al. 2006, Ruggiero et al. 2004, Ruessink et al. 1998, Holman 1986 and others). Only few of these equations are used in practice, with the most common and comprehensive one being Stockdon et al (2006) (S2006). S2006 is accurate and it was developed using 10 different data sets including the contribution from IG waves. Figure 2.15 shows a scatter plot of the 9 J% statistic from the field data used in S2006 versus the predictions using the empirical equation developed in their study. It is interesting to note that there is some scatter in the plot and 25 the largest runup observed in these experiments was around 3.5 m. It is suggested that for runup predictions greater than 3.5 m, the empirical equation from S2006 may not be applicable. Figure 2.16 shows a comparison of runup predictions from COULWAVE versus predictions using S2006 empirical equation. The reader is reminded that the reef configuration is composed of three piece-wise linear slopes, a steep 1/13 at the reef break, a mild 1/250 slope on the reef, and the dry beach slope of 1/70 (see Figure 2.1). When using S2006, a single characteristic profile slope must be used. Poor agreement is found when the reef slope (1/13) is used when calculating runup using the S2006 empirical equation. Although waves break at this slope when approaching the reef, it is not appropriate to use this slope when calculating S2006 9 J% . The slope that produced the best agreement between the datasets was 1/27 which lies between the reef slope and the beach slope. Better agreement is found using this value but with some discrepancies at the lower and upper limits. These discrepancies are due to the three piece-wise linear slopes present in the profile. In order to ensure that the 9 J% predicted by COULWAVE are accurate, 6 simulation runs (since it takes about 4300 waves for 9 J% to reach convergence) were modeled using a uniform and constant slope of 1/27. Figure 2.17 shows a comparison between the predicted 9 J% from COULWAVE and the 9 J% from S2006. Good agreement is found between the two data sets. Highlighted in red are the data points within the limits (0-3.5 m) that S2006 developed their empirical equation (Figure 2.16). Based on the scatter of Fig. 2.15 it is very reasonable for COULWAVE to under predict the small 9 J% predictions. In general, COULWAVE does a resonable job predicting 9 J% . Large runup values seem to be over predicted but due to the fact that S2006 analyzed runup heights less than 3.5 m it is not expected to be as accurate on runup values greater than that. 26 Figure 2.15: Runup, 4 +% , measurements from field data versus Stockdon et al (2006) empirical 4 +% equation (Terschelling contains 2 data sets). This data set was used for the development of S2006 equation. Figure 2.16: COULWAVE predicted 4 +% versus Stockdon et al. (2006) empirical equation using the reef slope, 1/13, (black dots), best fit slope, 1/27, (blue dots) and best fit slope using a constant slope on the domain instead of the piecewise slope (red dots). Black dashed line is complete agreement. 27 Figure 2.17: COULWAVE predicted 4 +% using a constant slope on the domain instead of the piecewise slope versus Stockdon et al. (2006) (S2006) empirical equation using the best fit slope, 1/27. Red dots represent runup values within the range of the field data used by S2006 (Figure 2.15). Blue dashed line is complete agreement. 2.4.2 Possible Sources of Error When modeling IG waves there are two main sources of domain configuration error that have to be considered: the basin resonance modes and the performance of the sponge layers. Both numerical and lab experiments are prone to be contaminated by basin resonance which can be attributed to poor sponge layer performance in the numerical simulations or in the lab not waiting sufficient time for water to come to complete rest in between runs. As an initial check, the basin resonance performance of the sponge layers for this problem is tested using the two different transects (Figure 2.1). In COULWAVE the sponge layers damp both momentum and mass. The IG generation is essentially a transient problem for each pulse, and without mass damping, poor sponge absorption can lead to wave reflections. To calculate the primary resonance mode from both numerical domains (Figure 2.1), a single simulation is run with a wall at the left boundary of the domain. With this setup, there is no energy absorption at the boundaries. A Fast Fourier Transform (FFT) is run on the free surface elevation (fse) time series across both profile configurations and yield a primary basin resonance mode of 0.00231 Hz for the reef and of 0.00252 Hz for the planar beach. If there is resonance contamination 28 in a numerical simulation significant a spectra energy peak at the basin’s primary frequency mode will be present at the ends of the domain. This would imply that the long period motions are not being absorbed by the sponge layers at the boundary. Figure 2.18 shows the energy spectrum averaged over ~90 phase realizations for both the reef and planar beach configurations. Each simulation had a duration of 10,900 sec using an energetic input wave spectrum (; j =11 > nru ? W =21 <AB). If basin resonance is present in the simulations there should be a large peak around 0.00231 Hz for the reef and around 0.00252 Hz for the planar beach at the two end locations (700m and 4200m), with less energy at the offshore locations. Here, little energy at the basin resonance frequency is observed at the offshore spectra for both configurations suggesting that there is no basin resonance contamination. These long period motions are being absorbed by the sponge layer. Figure 2.18a shows that in the reef configuration most of the IG energy comes from the reef primary resonant mode (~0.00151 Hz). Some energy can also be observed between 0.005 and 0.008 Hz which may correspond to the other reef’s resonant modes. Figure 2.18b shows that the majority of the energy for the planar beach configuration is contained within 0.003 and 0.007 Hz. The planar beach IG spectrum appears broader than the reef’s spectrum since the reef IG energy is dominated by the reef resonant modes. The next section explains more in detail the IG behavior in both configurations. 29 Figure 2.18: Low frequency energy both in the shallow water (h = 0.6 m) (red) and deep water (h = 100 m) (black) for both the (a) reef and (b) planar beach configurations. Solid lines are the mean values from 87 simulations, dashed lines show 95% confidence limits and blue dashed line shows the primary resonant mode from the domain. To visualize the behavior of the low frequency energy two animations were developed of the low-pass filtered (in time) wave surface, with two different filter cutoffs (150 s and 300 s). The videos can be found here: https://drive.google.com/open?id=1mrKak4Udou4Eo3AOiHTwJ2_1YQcaae8T. It is evident that 30 the free surface elevation near x=0->400m, where the sponge exists, stays at zero. Thus, there is no energy making it through the sponge, bouncing off the back wall, and going back into the domain. Also standing long waves are not visible in the basin. Energy from x=500m to ~ 1800m is mostly IG energy leaking off the shelf and some "locked" energy in the incident wave signal. Energy at x>1800m is from nonlinear interactions within the wave spectrum which excites the reef at the resonant modes. To check if the size of the sponge layer would affect the results, two different sizes of sponge layer, 400m and 3,500m, were tested in the reef and planar beach (not shown) configuration. Figure 2.19a and b shows the energy across the reef transect for both the small and large sponge layer, respectively. It can be observed that the energy behaves the same in both cases therefore the length of the sponge layer doesn’t affect the IG energy behavior in our simulations. Spectral results are identical regardless of the size of the sponge layer, but an initial calculation has to be done, based on the modeled wavelengths, to establish the minimum size of the sponge layer. Simulations were run with domains of constant depth = 100m with the same input wave generation, sponge layers at both boundaries, and no long period energy was present. The constant depth simulations are described in more detail in the next section. 31 Figure 2.19: Frequency spectrum, zoomed in on IG frequencies, of water surface elevation along the one-dimensional reef transect from COULWAVE IG simulations: Plot for simulation using a sponge layer size of (a) 400m and (d) 3500m. 2.4.3 IG Spectral Behavior and Convergence It is important to properly model and capture IG wave generation and energy transfers in order to predict reliable runup heights at the beach. As it was previously stated, for our work, we do not impose any special IG boundary conditions. In COULWAVE the simulations are forced with an offshore energy spectrum (JONSWAP), which in the model is broken into a set of discrete 32 amplitude/frequency/direction sine waves with some specified ∆F and ∆i. The offshore boundary is the linear superposition of all these discrete waves, each with a random phase, created with an internal source generator. After the waves are generated, IG energy is subsequently amplified by nonlinear wave-wave interactions within the modeling domain. It is well known that a single model simulation yields a very noisy un-averaged frequency spectrum since it is just a single-phase realization. To properly capture the statistical properties of the spectral realizations, the phase-resolved wave field must be modeled for a duration of 1/∆F, as was done in this study. With a different random phase seed, the spectral pattern is likely to also look noisy but noisy in a different way, with peaks at different nearby frequencies. One of the reasons to model multiple phase realizations is to average the spectra which will properly represent the spectral properties, reduce the noise and reveal any relevant modes. In general, from a synthetic energy spectrum, theoretically, there can be an infinite number of free surface elevation (fse) time series each with a unique set of phases. The variability in the fse time series (wave envelopes, wave sequencing, max. wave height, envelope duration etc..) will affect the runup and the IG energy in the spectra. Phase angles affect the timing of the interactions of the individual waves with each other and with the bathymetry. With different timing of interactions, the spectral pattern for each realization will change, as will the the maximum runup. The main objective of this section is to establish how many runs it takes to get a reasonably converged IG (0-0.02 Hz) spectrum. To address IG convergence, a total of 87 simulation are run and the averages calculated for both the reef and planar beach configuration, each simulation was run for 10,000 sec (∆F = 0.0001 Hz) and with the wave maker located at x=500m (Figure 1a). The wave parameters for the input wave spectrum are the same as in the previous sections (; j = 11 > nru ? W =21 <AB). The root means square error (rmse), in percent, is calculated from 0-0.02 Hz at a 0.0001 Hz resolution. For this study rmse (à∑(} ä −} V ) J ∑(} V ) J ⁄ , where } ä is the model prediction and } V is the data prediction) is used to establish IG convergence. Figure 2.20 shows that rmse reduces exponentially from 5% to near 0% for both the reef and planar beach configurations. The reef configuration shows more variability in the rmse than the planar beach across the entire transect. If an IG rmse of 1% is desired at any location across the transect, an approximately 30 simulations have to be run for both transects. Both configurations show that after 30 different phase simulations, IG rmse reduces slowly. The exponential decrease for the first 10 simulations reveals that 87 runs is appropriate number to properly define convergence. 33 Convergence was only tested using the reef and planar beach configuration shown in Figure 2.1 and using a very energetic input spectrum, ; j =11 > nru ? W =21 <AB. Figure 2.20: Convergence plot of the infragravity (0-0.02 Hz) part of the energy spectrum at 5 different locations on the (a) reef and (b) planar beach transects. For all simulations the wave parameters used for the input energy spectrum were ' ( =** , and 0 1 =+* (23. Three different animations were created that show the IG (0-0.02) behavior, for the average of 87 simulations, across the constant depth (with sponge layers on both ends), reef and planar beach transects. The videos can be found here: https://drive.google.com/open?id=1YfTizv1T0CaWfFwzeuX_uQ9_lcZEAFOG. Figure 2.21 just shows the IG behavior at 5 different locations across the 3 configurations. It is important to 34 reiterate that in the previous section is was established that there is no basin resonance contamination in the domain during the simulations. Figure 2.21a reveals that it takes ~10 wavelengths (of the peak period) of propagation to reach this quasi-steady IG spectrum, for this high ka condition. The IG energy we see for this constant depth case is at least 1/50th the energy we see in the reef and planar cases at the offshore gage. Figure 2.21b shows that the reef resonance is the dominant IG energy. The IG energy at the deep- water wave gage is offshore directed energy, leaked from the reef resonance. It can also be observed that when the waves start breaking (x~1800m) significant energy starts appearing at frequencies lower than 0.004 Hz. In the middle of the reef (purple line) more energy is transferred to the reef resonance mode and a secondary energy peak appears at 0.004 Hz which corresponds to a secondary mode. The closer to the beach the more energy is observed at the reef resonance frequency (0.00156 Hz). Figure 2.21c shows the planar beach IG spectrum. The IG energy at the deep-water wave gage is offshore directed energy, similar to the reef configuration. Two humps of energy can be observed at the deep-water gage. As the waves start to feel the slope, shoal and then break the two energy humps become more distinguishable. Finally, when the waves approach the shoreline only one energetic hump can be observed centered at 0.006 Hz. This IG energy comes from wave-wave interactions (including IG waves), free waves being released (bound waves), and reflected energy. It would be enlightening to quantify the amount of energy transferred from each mechanism since this can help establish which process is predominant at each section of the transect. Another area that requires further investigation is the origin of the two humps in the IG spectrum. These areas need further investigation and are out of the scope of this study. An interesting phenomenon occurs for the planar beach configuration. Very low frequency energy (f<0.002 Hz) starts to appear around x=3700 m, h=15 m and grows steadily until the swash. This energy does not appear at depths less than 15 m. If it was wave it would reflect off the beach, and there would be a signal offshore, which there is not. Figure 2.22 shows the frequency spectrum across the planar beach transect using a 40 Km long domain. It also shows (in red) the first node (L/4) of the standing waves generated at each frequency. It can be observed, that the nodes from a standing wave become very long for these low frequency waves. Therefore, it can be concluded that the low frequency energy is not present in the smaller domain because the offshore area before 35 the low frequency energy appears (x<3700 m) lies on a standing wave node. But, when the size of the domain increases energy starts appearing further offshore. 36 Figure 2.21: Infragravity (0-0.02 Hz) energy spectrum behavior for 5 different locations across the (a) constant depth (with sponge layers on both ends), (b) reef and (c) planar beach transects. Each line is the average of 87 simulations at the specified location. Figure 2.22: Frequency spectrum of water surface elevation along the one-dimensional planar beach big domain transect (40 km long) from COULWAVE IG simulation. Depicted in red is the first node (L/4) from each standing wave generated at each frequency. A wall is assumed at x= 3.94*10^4 m since waves start breaking at this location. 2.4.4 Extreme Runup Occurrence and Timeseries Analysis An extreme runup event is defined in this study as a runup elevation prediction that is 5 standard deviations from the mean. It is important to point out that by definition of the normal distribution, a five-sigma event should yield a 1 in 3.5 million chance. For this study, an extreme event is yielded approximately 1 in 375,000 chance. This is because the limits of the runup distribution are not “normal”. Out of the 1,875,000 simulated waves a total of 5 extreme events were observed from the fine frequency input spectrum simulations, 0 events from the finest and 0 37 from the coarse. With a simple calculation, for every 69 days of energetic wave conditions at a single location, we expect one extreme runup event to occur. The wave parameters used for the development of the input energy spectrum for the cases that generated an extreme event are: • Event 1 ; j =12 > nru ? W =20 <AB • Event 2 ; j =11 > nru ? W =21 <AB • Event 3 ; j =11 > nru ? W =22 <AB • Event 4 ; j =12 > nru ? W =22 <AB • Event 5 ; j =12 > nru ? W =22 <AB Figure 2.23 and 2.24 show the time series of runup and free surface elevation (fse) (at } = 750 >, before the waves reach the face of the reef) for the simulations that produced an extreme event. It can be observed that large offshore waves are not always correlated to large runup events, supporting observations from Garcia-Medina et al (2017). To better understand the incident wave conditions that produced the extreme runup events, additional simulations were run in COULWAVE in a domain without beach and with sponge layers on both boundaries (refer hereafter as case 2 and case 1 is the domain composed of three piece-wise slopes). The purpose of these “case 2” simulations is to shed light on the properties of the incoming wave train, with all beach reflections eliminated. To correlate the fse time series from both cases, input energy spectrum and phase from case 1 is used for case 2. Fse is recorded at the same location in both cases, at } =750>.. Figure 2.25 shows the time series of free surface elevations at 400 seconds before the maximum runup occurs through 100 seconds after, for case 1 and 2. Energy being reflected from the face of the reef and shoreline doesn’t affect much the evolution of the fse but might change the properties of the incident wave. There is something unique about these waves that cause these extreme runup events since the other millions of waves did not produced such an event. After analyzing the ht envelopes within the time series and videos from the simulations several characteristics can be observed from these unique wave envelopes. First, 2 pulses or more of energy (wave envelopes) are needed to produce the extreme runup event. The mean zero up-crossing period of the wave envelopes is in between 16-22 seconds. The mean wave height of the first envelope is from 7-15m and the second is from 10.4-13.2 m. The total period of the two wave packages producing the event 38 is more than 210 seconds and the total number of waves from both packets is in between 10-17 waves. By comparing the parameters between the envelopes from all the timeseries, including the ones that did not produce an extreme event, it was determined that the ranges of Hzmean, Tzmean, Ur mean, H/L mean, and cumulative wave power are unique for an extreme event. The wave envelope parameters used in the deterministic analysis are listed in Table 2.3 and 2.4. The extreme event depends on the arrival time of the pulses, nonlinearity, the amplitudes and periods of the waves in each packet, the duration of each packet, and the geometric properties of the shelf. Figures 2.26 show the IG (low) frequency spectrum at 5 different locations in the profile for all the EIG cases with the reef and beach. It can be noted that in all the events significant energy is present in the IG frequencies. Most of this energy is found between 0.001-0.002 Hz. For these configurations, the IG waves contribute time-averaged wave amplitudes of 0.5 m to almost 1m near the shoreline. IG energy at the offshore wave gauge is due to the reflected IG from the beach as there was no IG energy found at this location in the case tested with no beach. For a better understanding of the generation mechanisms of the EIG wave the reef resonance properties must be calculated to determine its contribution to the extreme event. Also, it has to be established if IG waves (100-250 sec) are locked in the carrier signal or if they are being reflected by the beach as mentioned above. Wavelet analysis is used to further investigate local properties of the wave signal that produced an extreme since it provides the evolution of period and phase of the signal across time. The resonant period of a reef can be calculated using the following equation: ? çPjÄRNR_P = ¶í (JRáH)àSU r=0,1,2… Where L is the cross-shore length of the reef, h is the mean water depth over the reef, g is the gravity and n is the oscillation mode. From the profile used in this study ò = 750 >,ℎ=3 > nru r=0. Yielding a reef resonant period of 553 secs (0.0018 Hz). Nevertheless, during energetic wave conditions the wave setup increases therefore extending the shoreline position landward, and increasing L. This leads to an increase of the resonant period. In all the events, there was an average shoreline increase of 25% (~180 m). Yielding a resonant period of 679 secs (0.00147 Hz). This explains the various peaks in the frequency spectra between 0.001- 0.002 Hz in Figure 2.26. 39 Figure 2.23: Runup time series and offshore ($=C86, ) free surface elevations for the configurations that generated an extreme runup event. In red is the time when the extreme runup event occurred and the offshore free surface elevations some seconds before and after the extreme event. Event 1 (a and b), event 2 (c and d) and event 3 (e and f). 40 Figure 2.24: Runup time series and offshore ($=C86, ) free surface elevations for the configurations that generated an extreme runup event. In red is the time when the extreme runup event occurred and the offshore free surface elevations some seconds before and after the extreme event. Event 4 (a and b) and event 5 (c and d). 41 Figure 2.25: Offshore ($=C86, ) free surface elevations 400 seconds before and 100 seconds after the extreme event occurs for both cases, with a beach and without a beach. Red dashed line is the time when the extreme event occurs and the black dashed line is 185 second mark before the extreme event (since it takes about 185 seconds for a wave to reach the beach). (a) Event 1, (b) event 2, (c) event 3, (d) event 4 and (e) event 5. 42 Figure 2.26: Amplitude spectrum of free surface elevation at several locations in the profile for all the configurations that generated an extreme event. (a) Event 1, (b) event 2, (c) event 3, (d) event 4 and (e) event 5. Magenta line at $=C86, (before the face of the reef), green line at $= *D86, (face of the reef), red line at $=+686, (beginning of reef), ), black line at $=+E86, (middle of reef) and blue line at $=+C66, (end of reef). 43 Table 2.3: List of wave envelope parameters calculated and used in the deterministic analysis. Wave Parameter 1 Number of Waves 2 mean, max, min(Hz) 3 mean, max, min(Tz) 4 Envelope Duration 5 Cum(Power) 6 Cum(Energy) 7 mean, max, min(H/L) 8 mean, max, min(Ursell) 44 Table 2.4: Parameters calculated for the two wave envelopes generating the extreme event. Mean Hz (m) Mean Tz (sec) Duration (sec) Number of Waves Mean H/L Mean Ursell Cumulative Power (kW/m) Event 1 Envelope #1 7.4 14.2 142.0 10 0.025 1.1 4585 Envelope #2 13.2 18.6 130.0 7 0.023 5.5 15846 Event 2 Envelope #1 15.0 22.0 110.2 5 0.020 8.4 12932 Envelope #2 12.7 19.9 99.3 5 0.020 5.5 9577 Event 3 Envelope #1 6.7 19.5 156.1 8 0.013 2.7 4096 Envelope #2 13.3 20.8 228.4 11 0.020 6.4 23306 Event 4 Envelope #1 14.7 22.4 112.0 5 0.019 9.1 12591 Envelope #2 11.2 21.4 128.2 6 0.016 5.9 8543 Event 5 Envelope #1 11.9 21.1 126.4 6 0.017 6.5 10598 Envelope #2 10.4 17.0 101.9 6 0.022 4.0 7893 45 2.4.5 Continuous 1-D Wavelet Transform and EOF Analysis Figures 2.27-2.29 show the results from the wavelet analysis for the cases with beach and without beach (at x=750m). Some of the similarities from the cases with beach are: distinct resonance period energy (continuous throughout the time series) at around 600 secs, discontinuous resonance period at around 180 secs (clearer when color axis is increased), and “plumes” or leakage of energy from low periods to higher periods suggesting triad interactions are taking place. The plumes indicate that the wave train is nonlinear and that energy is being transferred to higher harmonics. Resonance modes should appear continuous but the constant shift in water level due to IG surging is likely to be shifting the resonance modes as well. To determine if energy from low frequencies is being transferred after the breaking or if it comes locked in the wave signal a closer look has to be given to “case 2” simulations since the exhibit “plumes” throughout the time series. Events 1-3 show a distinct plume, with more energy in the time series and leaking to IG frequencies than the other plumes, right before the extreme events occurs (event 1: around 1000 sec, event 2: around 10,000 sec and event 3 around 8000 sec). Events 4-5 also show leakage of energy right before the extreme event but with less energy than the other plumes from the time series. But for event 5, the leakage of energy before the extreme event reaches to higher periods than the other plumes in the time series. Perhaps a plume analysis could shed more light on the generation mechanisms of these extreme events. This might indicate that some IG energy is locked in the carrier signal. Finally, it is worth mentioning that bores and bore-bore capture were not analyzed in depth in this study to date. Bores were present at the time of the extreme runup events but they were also observed multiple times when there was no extreme event. Bores play a role in the generation of extreme runup events but do not fully explain it. 46 Figure 2.27: Wavelet analysis for case 1 (with beach) and case 2 (without a beach). Top two panels correspond to Event 1 and bottom 2 to Event 2. 47 Figure 2.28: Wavelet analysis for case 1 (with beach) and case 2 (without a beach). Top two panels correspond to Event 3 and bottom 2 to Event 4. 48 Figure 2.29: Wavelet analysis for case 1 (with beach) and case 2 (without a beach) for Event 5. Videos showing the continuous wavelet analysis, across the reef and planar beach transect, were developed for the 5 extreme events to better understand nonlinear interactions and energy transfer across the profile. These videos are very rich in information, showing how and when the energy is being transferred to the lower frequencies (the resonance modes). The videos can be found in: https://drive.google.com/open?id=1YfTizv1T0CaWfFwzeuX_uQ9_lcZEAFOG. The following was observed, for the reef transect, at the time of the extreme events throughout the profile: • x = ~750m: a weak plume is observed from the incident wave signal more energetic than other plumes present throughout the time series. Significant amount of energy is observed around the reef resonant period (>600 sec). The weak plume makes a clear connection between the incident wave signal and IG periods. 49 • x = ~1250m (reef face): Significant energy is being transferred to ~150 sec forming an “eddy” like shape. • x = ~1500m (reef face): Plume coming out from the incident wave signal gets more energetic and connects with an “eddy” at ~150 sec. Another plume can be observed from ~150 sec to >650 sec. • x = ~1700m (reef face): Waves start breaking and significant energy dissipates from the incident wave signal. Many plumes start appearing from the incident wave signal. • x = ~1800m (reef face): More plumes appear throughout the incident wave signal. Less energy is observed at periods >600 sec than before. Significant energy is transferred to periods ~150 sec. • x = ~1900m-2000m (end of reef face): Significant amount of energy is transferred to periods >600 sec. Plume, from incident wave signal, is more energetic at the time of the extreme event than in other times. • x > ~2000m (reef): More energy transferred to periods > 600 sec. Eddy at ~600 sec is more energetic at the reef than other locations. Significant energy can be observed around 250 sec and 400 sec. The following was observed, for the planar beach transect, at the time of the extreme events throughout the profile: • x = ~750m: a weak plume is observed from the incident wave signal more energetic than other plumes present throughout the time series. Significant amount of energy is observed around 200 sec were an “eddie” shape is formed. The weak plume makes a clear connection between the incident wave signal and IG periods. The majority of IG energy at the deepwater wave gage is offshore directed energy • 750<x<3500m: More plumes are formed and a gradual transfer of energy around 200 sec due to wave shoaling and wave-wave interactions. • X>3500 m: This area is were waves start shoaling and breaking. Energy from the incident wave signal starts to dissipate. and energy going to low frequencies (T>200sec) the IG is generated along the beach during the shoaling. Waves start breaking and an eddie starts to form at around 344 sec, free wave released once the 50 waves break. The eddies remain connected all the time to plumes coming from the incident wave signal showing that energy transfers are still happening. To analyze how energetic the “plumes” are throughout the timeseries (CWT) we scaled the results by the maximum energy at each frequency level. The continuous wavelet transform is performed to the timeseries located at x = 750m. Figures 2.30-2.34 show that at the time of the extreme runup the plumes coming out from the carrier signal are more active or energetic than at any other time in the time series. Nonlinear interactions are at the “peak” just before the extreme occurs, the sets of waves that are generating the extreme, are highly nonlinear and are transferring energy into the lower frequencies at higher quantities than any other time in the time series, suggesting that the reef’s resonant modes are more energetic at this instant in time. At other instances in the timeseries the plumes appear energetic, but they do not reach low frequencies as at the time of the extreme. Similar behavior was observed in the other 4 extreme events. Not only the reef’s 0th resonance mode is important the other resonant modes (1 st , 2 nd , 3 rd ) need to be excited by the incident wave group for the extreme runup event to occur. Another comparison was made between a planar beach profile and a reef profile (base case). The goal of this comparison is to show the IG behavior in a simple profile (planar) vs a more complex one (reef). A slope of 1/35 is used for the planar beach since it represents the average slope of the entire reef configuration. Figure 2.35 presents a comparison timeseries of fse and runup between the reef and planar beach configuration. Fse is almost identical between both configurations but there are major differences in the runup height timeseries. The average runup for the reef is 2.27 m and 2.79 for the planar beach. Runup standard deviation is 1.22 m and 2.2 for the reef and planar beach respectively. The largest difference was found in the predicted maximum runup, the maximum predicted runup for the reef is 8.1m and 11.8m for the planar beach. The planar beach configuration predicted a 45% greater maximum runup than the reef. Similar results were found with the other extreme events (not shown). Also, the timeseries from the other events (#2, #4 and #5) revealed that the maximum runup from the planar beach configuration didn’t occur at the same time as in the reef configuration because the planar beach case is more sensitive to individual big waves and not the wave envelope as with the reef configuration. To better compare and observe the behavior of IG energy in both configurations a CWT analysis was performed. It can be observed, in Figure 2.36, that the plumes reach higher periods 51 during the reef configuration due to the reef dynamics, as it was mentioned previously. By analyzing videos of cwt across the transect, eddies of energy are formed around 150 sec in both configurations. For the planar beach case, as waves approach breaking and after breaking energy leaks into higher periods (~250 sec) and for the reef case, energy keeps leaking to even higher periods (>600 sec). For both configurations, the incident wave signal has locked energy in the IG periods that is released as free waves when the big waves start breaking. Figure 2.37 shows the ! " across the profile from 4 period ranges and reveals very interesting insight into the behavior of IG in both configurations. ! " from wind waves and swell (5s<T<30s) in both configurations behave similar: slight increase, due to shoaling, until waves break and then it tends to decrease for the rest of the transect. ! " from IG (0.5min<T<3min) also have a similar behavior with one notable difference: steady increase is observed as the waves shoal and the maximum IG ! " is observed soon after waves break, sum interactions occur at this part of the transect which explain the increase in ! " . The notable difference is the jump in ! " near the entrance of the reef which might be explained by self-self interactions taking place. Maybe the change in bathymetry might influence the intensity in sum interactions. It is very interesting to note that the ! " from IG (3min<T<8min) has very similar behavior in both configurations, in the reef and at the beach for the reef case. A slight increase in ! " is observed for a couple hundred meters followed by a noticeable decrease and ending with an increase. The increase in energy at these periods can be attributed to difference frequency interactions. When the decrease is happening, this can be due to this energy going to higher or lower period waves, since energy at these both periods is increasing. ! " from IG (8min<T<17min) has a very different behavior in both configurations due to the dynamics of the reef. In the reef configuration, a steady linear amplification is observed with the largest amplitude at the shore. This increase in energy is due to the reef resonant modes and interactions of the incoming IG waves with the reef. For the planar beach configuration, there is only a small amount of energy at the IG periods just before reaching the shoreline. It is not clear where this energy might come from, maybe beach resonance or very long waves being generated. A more in-depth comparison of overland flow characteristics between the two transects can be found in Chapter 4. 52 Figure 2.30: Scalogram from Event # 1 scaled by each frequency level. Figure 2.31: Scalogram from Event # 1 scaled by each frequency level. 53 Figure 2.32: Scalogram from Event # 3 scaled by each frequency level. Figure 2.33: Scalogram from Event # 4 scaled by each frequency level. 54 Figure 2.34: Scalogram from Event # 5 scaled by each frequency level. Figure 2.35: Comparison of timeseries of free surface elevation and runup between the reef and beach transect. 55 Figure 2.36: Comparison of continuous wavelet transform analysis between the (a) reef and (b) planar beach transect. 56 Figure 2.37: Significant wave height behavior across the (a) reef and the (b) planar beach transects for 4 different period ranges. The Empirical Orthogonal Function (EOF) analysis is used to distinguish the most energetic resonant modes inside the reef. EOF has been widely used to study the variability of dynamical processes at the nearshore (Winant et al 1975). EOF is used in this study to analyze dominant cross-shore temporal and spatial patterns of IG motions. By using EOF analysis Peguinet et al 2009 and Peguinet et al 2014; discovered cross-shoe standing modes on the reef during tropical storm ManYi and large wave events. Similar methodology to Peguinet et al 2009 is used in this study. Becker and Yoon 2016 also used EOFs to analyze cross-shore IG motions on reefs. 57 The theoretical cross-shore structure of the resonant mode can be calculated by the following equation # $ (&)= cos,(2. +1) 1 22 &3 . = 0,1,2… where x is the cross-shore distance from the shoreline and L is the width of the shelf. The first 4 fundamental modes of the reef range from 700s to 100s. A total of 104 gauges in the reef are used for the EOF analysis which allow several modes to be resolved. The EOF analysis was performed for the 104 timeseries across the reef. EOFs are computed as eigenfunctions of the covariance matrix. A band pass filter with a cut off frequency of 80 seconds is used to examine the energy in the first 5 resonant frequencies, periods from 80 to more than 700s. Figure 2.38-2.41 show that the cross-shore structure of the EOF modes match the structure of the theoretical resonant modes for the 5 extreme events in this study. The four EOF leading modes explain 98% of the IG variance on the reef. The variance is strongly dominated by the first mode since its eigen value explains 70% of the IG variance on the reef. The eigen value from the second mode explains about 20% of the IG variance. It can be suggested that the first 2 resonant modes are are the most important since they explain almost 90% of the variance. Energy from the big incoming waves and IG waves are interacting with these modes and pushing this water into the beach. Figure 2.38: Normalized EOF and theoretical cross-shore modes from event # 1. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. 58 Figure 2.39: Normalized EOF and theoretical cross-shore modes from event # 2. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. Figure 2.40: Normalized EOF and theoretical cross-shore modes from event # 3. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. 59 Figure 2.41: Normalized EOF and theoretical cross-shore modes from event # 4. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. Figure 2.42: Normalized EOF and theoretical cross-shore modes from event # 5. Top 3 panels correspond to first three modes, 0, 1, and 2. Bottom 2 panels correspond to mode 3 and 4. 2.4.6 Effects of profile on Runup and Reef vs Planar For this section only, the reef transect is used to better understand IG driven runup. Figure 2.1a shows the profile used, refereed as the “base” case in this section. To better understand the effects 60 of the bathymetry on the maximum runup, several profiles were tested with different reef face slope, reef slope, reef initial depth, and beach slope (see Table 2.5). A total of 9 different profiles were tested using the same incident wave conditions that generated the extreme events. Figure 2.43 and 2.44 and Table 2.6 show the maximum runup heights with each profile. It can be observed that profiles #5-7 generated larger runup in almost all the events. In particular for all events, predictions from profile #6 and 7 were an average of 10% more than the base case. Predictions from profiles #1-3 were an average of 30-25% lower than the base case predicted runup, profiles #8 and 9 predicted an average of 20-25% below the base case, and profiles #5 and 10 predicted an average of <10% below the base case max runup prediction. The biggest increase was observed using profile # 7, from event #1 (Table 2.6), predicting 24% above the base case maximum runup. The lowest decrease was observed using profile #4, event #5, predicting 47% below the base case maximum runup. These results show that the beach slope has a greater influence on maximum runup height predictions than the reef face slope and the reef slope, the steeper the beach the larger the runup height. Table 2.5: Profiles tested in this study. Profile 1 is the base profile used in this study. Profiles Tested 1 Base Case: Reef Face 1/13, Reef 1/250 (3m depth), Beach 1/71 2 Reef Face 1/25 3 Reef Face 1/35 4 Reef Face 1/55 5 Beach: 1/60 6 Beach: 1/35 7 Beach: 1/25 8 Reef: 1/333 9 Reef: 1/416 10 Reef Depth: 5m and Beach: 1/35 Table 2.6: Maximum runup around the time of the extreme event Maximum Runup (m) Profile # Event #1 Event #2 Event #3 Event #4 Event #5 1 8.3 8.1 7.8 8.7 8.5 61 2 7.7 6.3 6.1 5.6 5.0 3 6.9 5.9 5.9 6.3 4.8 4 6.7 5.7 5.8 5.1 4.4 5 8.7 7.3 7.3 7.8 9.3 6 10.1 9.7 7.9 9.5 8.8 7 10.3 8.4 8.4 10.1 8.8 8 6.7 7.0 7.1 6.2 6.3 9 6.9 9.5 8.1 5.1 6.6 10 7.9 8.5 8.2 7.3 7.0 62 Figure 2.43: Runup timeseries around the time of the extreme event. (a) Event #1, (b) Event #2, and (c) Event #3. Profile #1: Thick Red Dotted Line, Profile #2: Blue dashed line, Profile #3: Red dashed line, Profile #4: Black dashed line, Profile #5: Magenta line, Profile #6: Black line, Profile #7: Thick Pale Green line, Profile #8: Dark Green line, Profile #9: Blue line, and Profile #10: Dark Orange line. 63 Figure 2.44: Runup timeseries around the time of the extreme event. (a) Event #4, and (b) Event #5. Profile #1: Thick Red Dotted Line, Profile #2: Blue dashed line, Profile #3: Red dashed line, Profile #4: Black dashed line, Profile #5: Magenta line, Profile #6: Black line, Profile #7: Thick Pale Green line, Profile #8: Dark Green line, Profile #9: Blue line, and Profile #10: Dark Orange line. 2.4.7 Neural Network Analysis Artificial Neural Networks (ANN) are widely used in various different fields of industry, business and science (Widrow et al. 1994). The ANN approach is simply a nonlinear blackbox model and has many advantages over empirical models since it can be continuously retrained and adapted to new data. ANN have been used to predict tides (Chaudhari 1998, Mandal 2001), predict significant wave heights and peak periods (Tsai et al 2002, Kalra et al. 2005a, Rao and Mandal 2005). Table 2.7 shows the list of dimensionless envelope parameters used for the estimation of R/Hs during the ANN application. The sample size was increased by running more simulations using the same Hs and Tp combinations that generated an extreme. More than 20,00 envelopes 64 were analyzed. Figure 2.45 shows the variability of the correlation coefficient when testing different combinations using 3 dimensionless parameters to predict R/Hs. The single parameter that yielded the best correlation for predicting runup is the maximum wave crest height with a testing correlation of about 0.6 (Figure 2.46). When using two parameters, the highest correlation was around 0.7 by using the max wave crest height and number of waves in the envelope. For 3 parameters, the shoaling effect, mean Ursell and number of waves in the envelope yielded the best correlation. A reason why max wave crest height didn’t appear in the 3 parameters is because in Figure 2.45 there were other combinations that are close to the max. When using 4 or more parameters it gets more complicated to track down influential parameters. Overall, mean max. crest height and number of waves are the ones that influence the correlation coefficient the most. Stefanakis 2011, found that highly nonlinear waves generate larger runup which may be also true for a reef setup. Further investigation is suggested to better comprehend the effects of the various parameters. A total of 7 parameters are needed to reach a maximum possible correlation of 0.8. This means that information is needed from previous envelopes to better explain the extreme event. Maybe by incorporating the parameters from 3 or more previous envelopes a better correlation can be achieved. Table 2.7: Dimensionless envelope parameters and their physical representation. Dimensionless Envelope Parameter Physical Representation 1 mean(h/L) Shoaling effect 2 mean(H/h) Wave height stability 3 mean(Iribarren) Breaking parameter 4 mean(H/L) Wave steepness 5 mean(Ursell) Nonlinearity 6 Num. of Waves Number of waves in group 7 mean(Dur/Tmax) Dimensionless duration 8 mean(a+)/h Effect of crest 9 mean(a-)/h Effect of trough 10 max(a+)/h Effect of max. crest 11 min(a-)/h Effect of min. trough 12 mean(g*T*Dur/(2pi*H)) Envelope steepness 65 Figure 2.45: Correlation coefficient using all possible combinations using 3 parameters. Figure 2.46: Maximum correlation and standard deviation by each parameter sets. 2.4.8 Synthetic Timeseries Analysis 66 Some of the questions that remain unanswered are: Is the extreme IG wave event localized or globalized? How many previous envelopes are conditioning the shoreline for the event to occur? Synthetic timeseries with the characteristics of the envelopes producing the extremes were created and tested. A statistical analysis is performed to better study the impact of the envelopes preceding the extreme events . The goal of this section is to determine if the extreme event is localized or global, meaning if 1 envelope is generating the extreme event or various envelopes. A total of 30 timeseries, each with a different random phase, are used to test the hypothesis. Each timeseries was generated from an input spectrum with \a ∆9 = 0.0001 Hz a Hs=8m and Tp=21 sec. The envelope that generated the extreme runup and up to 3 prior envelopes are tested from the 5 events. Case 1 consist of the envelope generating the extreme event. Case 2 includes case 1 plus 1 more envelope. Case 3 includes case 2 plus one more envelope adding to a total of 3 envelopes. Case 4 consists of case 3 plus 1 more envelope, adding to a total of 4 envelopes. Figure 2.48 and 2.49 shows the cases from each event included in the timeseries. These envelopes were added in 3 different sections/times in each of the timeseries, having a total sample size of 90 runup events for each of the different tested cases. Mean and standard deviation are calculated and discussed below. The working hypothesis for this section is that the envelopes (case #2-4) prior to the envelope generating the extreme runup (case 1) are important since they are conditioning the reef and shoreline for the extreme event to happen. The variance of the maximum runup will be better explained by adding the prior envelopes. These envelopes are causing the reef and shoreline to resonate so that the last envelope, which has the bigger waves, can push this water towards the beach. Figure 2.47 shows a good example of what the author is expecting to happen for all the cases from each extreme event. 67 Figure 2.47: Expected runup timeseries from each of the different synthetic timeseries. 68 Figure 2.48: Wave envelopes preceding the extreme runup from events (a) 1, (b) 2 and (c) 3 69 Figure 2.49: Wave envelopes preceding the extreme runup from events (a) 4 and (b) 5. Figure 2.50 shows the runup height generated using a different number of envelopes for each event. These results support the hypothesis postulated. The more envelopes are included into the signal the more the variance of the runup is explained. Previous envelopes are important because they are conditioning the reef and shoreline for the maximum runup to occur. The only event that does not show an increase in runup with more envelopes is event # 2. The last envelope from event 2 is relatively long compared to the other events and might condition the reef and shoreline without the need of previous envelopes. It is still unclear to the author why by adding 3 and 4 envelopes to the signal of event 2, the maximum runup slightly decreases. It is suggested that if the number of samples is increased, the runup height will tend to be constant for event 2. Finally, it is interesting to note that the standard deviation decreases when the number of envelopes increase which also supports the postulated hypothesis. 70 Figure 2.50: Mean runup heights using different number of wave envelopes in the timeseries. 2.5 Conclusions COULWAVE, a Boussinesq-type model, is used in this study to analyze extreme properties of ocean wave runup and generation mechanisms of EIG waves. The experiment setup resembles much of what happened in Hernani during Typhoon Haiyan in 2013 where an EIG wave struck the town and destroyed several houses in its path. For high energy wave conditions, with beaches that have IG-dominated runup, in order to get numerically convergent extreme tail measurement (: ;% -: =.?% ) values at the 0.5% error level convergence, a ∆9 of 0.0001-0.00005 Hz is needed. This is more than 100 times smaller than what is typically used in these Boussinesq / coastal phase-resolving models. The reason for such a small ∆9 is because the integrated low frequency energy transfer is sensitive to a fine resolution of the interacting frequencies. It is strongly suggested to the scientific community to indicate the input frequency resolution or IG boundary conditions used in their study. When comparing the three ∆9’s used in this study, it is established that better predictions are obtained when using the ∆9 @ or ∆9 @A resolutions, 0.0001 and 0.00005 respectively. The bias, from Figure 2.6, is closer to zero which means that convergence is achieved at either ∆9’s. Therefore, runup predictions from numerical simulations that use 10,000 frequencies in the input energy spectrum are going to be more accurate than using less frequencies. Coarser ∆9 energy 71 spectra will not capture all the exchange in energy between the different frequencies when nonlinearity takes over. The Nc number (dimensionless) was developed to better present and compare convergence of several tail measurements from the different configurations. Table 2.1 was created to quickly calculate the number of waves needed for convergence (stable mean) at a 0.5% error for a wide range of wave conditions. The coefficient of variation (cv) is calculated for 24 different tail measurements (see Table 2.2) and was found to be bounded between 0.14-0.25, for a wide range of configurations (combination of significant wave heights (! " ) and peak periods (B C )). This is very useful when analyzing a deterministic case because from that one case, based on the ! " and B C , the variation of the standard deviation and the average is going to be known. In general, COULWAVE does a very good job predicting : ;% when compared to the S2006 : ;% empirical equation. Both were compared using a constant slope of 1/27. Higher runup values were over predicted due to the fact that S2006 developed their empirical equation analyzing runup heights lower than 3.5 m so accuracy decreases when dealing with large runup predictions. Higher runup values predictions (>3.5 m) from S2006 are not expected to be as accurate since their equation was developed with runup height of <3.5 G. There are at least two sources of error in a numerical domain that have to be considered and tested: contamination from basin resonance modes and poor performance of the sponge layer. The sponge layers, in COULWAVE, had an excellent performance since no energy makes it through the sponge, bouncing off the back wall, and going back into the domain. All the energy is being damped by the sponge layer. Also, simulations from both configurations showed that not much energy, at the basin resonance frequency, is observed at the offshore spectra suggesting that there is no basin resonance contamination. Additionally, two different sizes of sponge layers are tested and revealed that energy behaved the same across the transect therefore, the length of the sponge layer doesn’t affect the IG energy behavior in the simulations. To get a reasonably converged IG (0-0.02 Hz) spectrum, with a rmse of 1% at any location across the transect, an approximate of 30 simulations have to be run for both the reef and planar beach configurations. A total of 5 extreme runup events were identified out of 1,875,000 simulated waves (~16 sec waves). Based on this we can expect an extreme event happening every 69 days of energetic wave conditions the envelope analysis of the offshore free surface elevations, revealed that extreme 72 events depend on the arrival time of the pulses, the amplitudes and periods of the waves in each packet, the duration of each packet, and the geometric properties of the shelf. The wavelet analysis revealed how reef resonance (553-679 sec) contributes to the extreme events. Also, in the cases where no beach was present, 3 out of the 5 events show an energetic nonlinear transfer plumes right before the event. These plumes indicate that there is leakage of energy from low periods to higher periods suggesting triad interactions are taking place. Events 4 and 5 show a nonlinear energy transfer reaching higher periods (~600 secs) and a “n” shape feature at the time of the extreme. The cases with a beach showed various plumes, that were less energetic, throughout the time series when other runup peaks occurred. This suggests that for an extreme event to happen there should be significant amount of energy leaked or transferred to low frequencies and in resonance with the reef. Finally, the synthetic timeseries analysis revealed that most of the times more than 2 envelopes are needed to condition the shoreline, excite the reef and generate an extreme runup event. For all the events, except # 2, when more envelopes where included into the signal the variance of the runup was better explained. Event # 2 behaved different because the envelope preceding the extreme event has far greater duration than from the other events. 73 3. Tsunami Runup and Overland Flow Velocities 3.1 Field Measurements and Observations The field survey data published in Mori et al. (2011) and the flow velocity measurements from Koshimura and Hayashi (2012) are used to compare accuracy and reliability of numerical model predictions. More than 5300 measurements were recorded by a large group of scientists and researchers. A total of 63 universities and 297 people were involved in this project covering 2000 km of the Japanese coast. In Sendai the maximum measured runup elevation was 9.4 m (The 2011 Tohoku Earthquake Tsunami Joint Survey Group, 2011). Only 10 percent of the runup measurements were greater than 5 m. For this study, we will be focusing on the Sendai plain (particularly from 38.10° N to 38.28° N). The wave front at the Sendai plain reached more than 5 km inland from the shoreline with the average being 4.2 km. Flow velocities estimates measured by Koshimura and Hayashi (2012) were obtained from a 2- D projective transformation video analysis. One of the two locations where measurement estimates were made was the Sendai plain. The video used in the analysis was taken by a Japanese broadcasting company. Flow velocity estimates were made at four different locations within the Sendai plain. These locations are at a distance of 1000-3000 m from the coastline. The maximum measured flow speed was 8.0 m/s. The grids used in this study are from the M7000 digital contoured bathymetric data and the GSI 10-m digital elevation models (http://fgd.gsi.go.jp/download/). Five nested grids were used in the numerical models. The propagation grid, or grid A, was the coarsest grid at 3 arc-minutes. Four additional nested grids (1 arc-min, 20 arc-sec, 4 arc-sec and 1 arc-sec) were used covering the area of interest. Also, five additional grids were created (3 arc-sec, 2arc-sec, 0.67 arc-sec, 0.50 arc-sec and 0.33 arc-sec) by interpolating the 4 arc-sec and 1 arc-sec grid. These grids were used to analyze convergence and variability within the MOST model predictions. All the grids are referenced to Mean Sea Level (MSL) vertical datum and to the World Geodetic System of 1984 (WGS 84) horizontal datum. 74 3.2 Tsunami Modeling The Method of Splitting Tsunami (MOST) model was developed as part of the Early Detection and Forecast of Tsunami (EDFT) project and introduced by Titov and Synolakis (1995 and 1998). This model is currently used by NOAA for propagation and inundation forecasting (Titov, 2009). MOST has been validated and successfully tested in various studies (Synolakis et al., 2007; Titov and Synolakis, 1998; Titov and Gonzalez 1997). Wei et al. (2011) modeled the 2011 Tohoku tsunami with MOST and presented a detailed analysis of runup height and inundation along the Japanese coast. MOST solves the 2+1 nonlinear shallow water equations: ℎ A +(Iℎ) J +(Kℎ) L = 0 (3.1) I A +II J +KI L +Mℎ J = MN J −PI (3.2) K A +IK J +KK L +Mℎ L = MN L −PK (3.3) Where Q(&,R,S) = wave amplitude, d is the water depth, ℎ(&,R,S)= Q(&,R,S)+N(&,R,S) , I(&,R,S) and K(&,R,S) are the depth-averaged velocities, and P(ℎ,I,K) is the drag coefficient computed by equation 4: P(ℎ,I,K)= . ; Mℎ T U V W XI ; +K ; (3.4) Runup and inundation were only performed in the higher resolution grids (3 arc-sec, 2 arc-sec, 1 arc-sec, 0.67 arc-sec, 0.5 arc-sec and 0.4 arc-sec, or approximately 90m, 60m, 30m, 20m, 15m and 10m, respectively). To evaluate the MOST model sensitivity to the Manning’s coefficient three different values were used for the simulations, n = 0.025, 0.030 and 0.035. For a detailed description of MOST refer to Titov and Synolakis (1995 and 1998). GeoClaw, developed by LeVeque (1997, 2002), is an open source tsunami model approved by the United States National Tsunami Hazard Mitigation Program (NTHMP). It has been validated by comparing real and artificial data (runup, inundation and flow velocity) with model results (Arcos and LeVeque 2015, Gonzalez et al. 2011, LeVeque et al. 2011, Berger et al. 2011, George 2008, and LeVeque and George 2008). GeoClaw uses finite volume methods to solve the two dimensional nonlinear shallow water equations in conservative form: 75 ℎ A +(Iℎ) J +(Kℎ) L = 0 (3.5) (ℎI) A +YℎI ; + 1 2 Mℎ ; Z J +(ℎIK) L = −Mℎ[ J −PℎI (3.6) (ℎK) A +(ℎIK) J +YℎK ; + 1 2 Mℎ ; Z L = −Mℎ[ L −PℎK (3.7) Where ℎ(&,R,S) the fluid depth, I(&,R,S) and K(&,R,S) are the depth-averaged velocities, [(&,R,S) is the topography or bathymetry and P(ℎ,I,K) is the drag coefficient computed by equation 4 with the Manning coefficient, n=0.025 constant throughout the grid. For a detailed description of GeoClaw refer to LeVeque (2011) and Berger et al. (2011). For the present study two modifications were done to the GeoClaw code to perform the tsunami simulation. Firstly, the code was modified to use fixed grids instead Adaptive Mesh Refinement (AMR). Lastly, the modified code uses the Generic Mapping Tool Network Common Data Form (GMT NetCDF) files as inputs and outputs. Most of the parameters are the same as those used in MacInnes et al. (2013) except for the number of auxiliary variables in the aux array (clawdata.num_aux) which was set to 4, minimum allowable water depth (geo_data.dry_tolerance) set to 0.01m, and maximum depth at which bottom friction is included in the calculations (geo_data.friction_depth) set to 100 m. The initial condition used in both models is an initial sea-surface deformation based on Yokota et al. (2011). This source model was created by carrying out a quadruple joint inversion of the strong motion, teleseismic, geodetic and tsunami datasets. The resulting model has a maximum coeseismic slip of approximately 35 m and a seismic moment of 4.2*10 22 Nm, which yields Mw = 9.0. 3.3 Results and Discussion 3.3.1 Inter-model Comparison A 30m resolution grid was used for the inter-model comparison analysis. Also as it was previously mentioned, three different Manning n values (n = 0.025, 0.030 and 0.035) were used in 76 MOST to evaluate its sensitivity to predict runup elevations and overland flow velocities. Figure 3.1 shows the maximum free surface elevation during the Tohoku tsunami event for MOST and GeoClaw. Both models predict that higher free surface elevations occur at the central part of the Sendai plain, around 38.2° N 140.975° E, with maximum wave amplitudes ranging from 8-12m. Both models agree relatively well with each other when predicting sea surface elevation near the shoreline, with MOST yielding slightly higher predictions. For consistency purposes, 7 runup measurements from the field data were removed from the analysis, given in Table 3.1. These runup measurements were located very close to the shoreline and led to an irregular inundation line when combined with the other runup points. A total of 46 runup measurements were used from the Sendai plain in this analysis. Figure 3.2 presents the east- west distances measured vs the east-west distances predicted by both models. It shows that the model predictions are in general 77% and 63% of the measured values, for MOST and GeoClaw respectively. The estimated correlation for Geoclaw is 0.57 and 0.58 for MOST. In general, both models over-predict the east-west distances in the Sendai plain. Figure 3.3 (left panel) shows the field data runup measurements and the predicted runup by both models. Average runup from the 46 field data measurements is 1.89 m with a standard deviation of 0.70 m while the average runup calculated by MOST and GeoClaw is 3.01 m and 3.34 m respectively. Thus, much of the model runup results lay approximately 2 to 3 standard deviations away from the mean of the field data. The runup standard deviation for both models is 0.16 m and 0.33 m for MOST and GeoClaw respectively. Figure 3.3 (right panel) shows the inundation line predicted by both models and the field data runup height measurements at the Sendai plain (38.10° N to 38.28° N). Both models provide a reasonably accurate prediction of the inundation line. This would seem to indicate an inconsistency, in that the inundation line is well predicted, but the runup elevation is not; this will be addressed later in this section. Table 3.1: Field data measurements not used in this study. Lat. (ºN) Lon. (ºE) 38.1725 140.9538 38.1822 140.9583 38.2394 140.9533 38.2718 140.9981 38.2724 140.9980 77 38.2799 141.0506 38.2799 141.0484 Figure 3.1: Maximum tsunami amplitudes (m) predicted by MOST (left panel) and GeoClaw (right panel) in the Sendai plain. Figure 3.2: Comparison between measured east-west distances and model predicted distances in the Sendai plain. Figure 3.4 presents the variability of runup and inundation line predictions using different n- values. It can be seen that runup decrease with higher n-values therefore reducing the models error. The Sendia-average runup calculated by MOST is 2.59 m and 2.51 using n-values of 0.030 and 0.035 respectively. It is very interesting to note that there is a much higher difference in the runup and inundation line predictions when increasing the n-value from 0.025 to 0.030 than from 0.030 78 to 0.035. This would indicate that accurate runup elevation predictions require both high precision and accuracy in bottom friction in this area. Figure 3.5 shows the distributions of the runup height from the field data and the models. This normalized histogram and all histograms to be presented in this paper, are generated using all relevant field or modeled data between 38.10° N to 38.28° N along the Sendia plain. The distribution uses a runup interval spacing (histogram bin width) of 0.05 m. The model distributions have a similar shape with means within 10% of each other, but it can be clearly noted that they overestimate the observed runup. By using different higher n-values the shape of the distributions remains similar but the peaks tend to move left decreasing the error of the MOST model predictions. To further analyze the runup predicted by both models, we compared the field data runup height measurements with the elevation from the numerical topographic grid at the location of the runup measurements. The Sendai 30m resolution topography was used in this analysis. Figure 3.6 presents a histogram of the estimated differences. It shows that most of the differences range between (-3m) to (-1m). This indicates that there is some error in the topography, one that cannot be simply attributed to a datum inconsistency, due to the spread of the histogram. The differences between runup elevations and topographic grid elevations would also indicate that it should not be possible for a model to agree with both the inundation line and the runup elevation when using this GSI topography data. This inconsistency is due to the fact that runup measurements include small-scale local topography while the grid doesn’t. Such errors are particularly significant for flat coastal areas such as the Sendai Plain. The outliers in the histogram correspond to mountains or low elevation points that are small to resolve by a 30m grid. 79 Figure 3.3: Comparison of runup height measurements and inundation line between data, MOST and GeoClaw during the 2011 Tohoku event in the Sendai plain. Figure 3.4: Maximum tsunami amplitudes (m) predicted by MOST (left panel) and GeoClaw (right panel) in the Sendai plain. 80 Figure 3.5: Comparison of the runup heights distributions between the interpolated field data, MOST and GeoClaw models. The distribution uses a runup interval spacing of 0.05 m. Figure 3.6: Estimated differences between field data runup heights and the topographic elevations from the numerical grid at the location of the runup measurement. In addition to flow depths, tsunami flow velocities have to be analyzed to understand the tsunami hazard at a particular location. For example, Synolakis (2004) stated that currents are more destructive than wave height amplitudes during many tsunami events. Figure 3.7 presents the maximum flow velocities predicted by MOST and GeoClaw, respectively, in the Sendai plain. Both models agree on their predictions and locations of high flow velocities. They also both show a rather complex profile of overland flow velocity, with a number of local maxima. These local maxima are due to topographic features and properties of the incident wave form. 81 Koshimura and Hayashi (2012) measured the tsunami flow velocities at 4 different locations in the Sendai plain. Figure 3.8 shows the modeled tsunami flow velocities and the field measurements at these locations. When using n value of 0.025, both models under-predict the flow velocity at F1 which is close to the coastline and over predict at F2, Y1 and K3 which are further inland. Figure 3.8 also shows the modeled flow velocities predicted by MOST using different n-values. It can be seen that the error increases near the shoreline (less than 1500 m) when a higher value of n is used. Farther inland the error decreases and the predicted velocities move closer to the measured velocity but still over-predicting at Y1 and under-predicting at K3. It can be suggested to use higher values of Manning coefficients (n = 0.030 or 0.035) when analyzing overland flow properties far from the coastline (>1500m) to decrease error in the predictions. Both models use uniform Manning’s n coefficients throughout the grid and this might be a reason to why the models over-predict and under-predict farther inland. This observation is an indication that spatially variable bottom roughness is likely necessary to capture local flow speeds; such an implementation is certainty viable but would require spatial maps of ground properties, which could be used to construct friction factor maps. Furthermore, the ground properties will change in time following interaction with the tsunami (i.e. erosion, flattening of vegetation), leading to a substantial modeling challenge. Other reasons that may cause a numerical model to overestimate or underestimate flow velocity measurements are: complex and unresolved bathymetry/topography, improper friction coefficients, no inclusion of tides, and numerical dispersion and dissipation errors. Figure 3.9 shows a comparison of the distributions of modeled maximum shoreline flow velocity and flow velocity at the 1 meter inland flow depth. The distributions use a flow velocity interval spacing of 0.05 m/s. These two locations are meant to represent limits of the overland flow area; one comparison at the shoreline and another near the inundation limit, but still at a significant flow depth. Since there is no available data at these locations it is very difficult to assess the model accuracy. Many of the model velocity predictions at the shoreline are between 5-9 m/s, with means of 7.30 and 7.34 m/s for GeoClaw and MOST respectively. The shapes of the shoreline flow velocity distributions tend to agree well with small differences in their means. Figure 3.9 (bottom panel) shows that both models agree well when predicting the maximum flow velocities at the 1 meter flow depth. The peak of the GeoClaw distribution is located around 1.63 m/s and with an average of 1.60 m/s while the peak for MOST is located around 1.20 m/s with an average of 1.63 m/s. 82 Figure 3.7: Maximum flow velocities predicted by MOST (left panel) and GeoClaw (right panel). Figure 3.8: Comparison of maximum flow velocities at the Sendai plain between Koshimura and Hayashi (2012) measurements (gray triangle), GeoClaw predictions (gray square), MOST predictions using n = 0.025 (black circle), MOST predictions using n = 0.030 (black diamond) and MOST predictions using n = 0.055 (gray upside down triangle). The vertical bars on the model data provide the standard deviation of the predictions in the measurement window. At F2, two measurements were taken. 83 Figure 3.9: (top panel) Comparison between GeoClaw and MOST distributions of maximum shoreline flow velocities and (bottom panel) 1 meter depth maximum flow velocities at the Sendai plain. The distributions use a velocity interval spacing of 0.05 m/s. 3.3.2 MOST Model Variability Unquantified variabilities within a model can lead to unknown errors in a THA. In this section, we use MOST to further explore and understand the possible sources of error within a model. For this part of the analysis, 6 inundation grids (3 arc-sec, 2 arc-sec, 1 arc-sec, 0.67 arc-sec, 0.50 arc- sec and 0.33 arc-sec) were used to compare inundation, runup and velocity predictions made by the MOST model. As was previously mentioned, finer grids were created by interpolating the 4 arc-sec and 1 arc-sec topography data. Overall run-times for each resolution are presented in Table 3.2. All the simulations were performed for a physical time of 10 hours. The CPU used was an AMD Opteron 6140 running at 2.6 GHz. Run times are almost increased by a factor of four when the grid resolution is increased by a factor of two (i.e. from 30m to 15m). Figure 3.10 shows a distribution of the runup height calculations from the different grids. Both runup and inundation line predictions numerically converge within the tested grid sizes. There are small deviations in the inundation line and runup calculations when using different grid resolutions (30m-10m). In this case, it would seem reasonable to conclude that there is no need to use inundation grids finer than 30m when calculating runup and inundation lines. 84 Figure 3.11 shows a comparison of maximum shoreline flow velocity distributions and 1 meter flow depth maximum velocity distributions for the different grid resolutions. Averages for shoreline velocities are around 7.50 m/s and for 1 meter flow depth, velocities are around 1.70 m/s. While there appears to be numerical convergence between the 30m, 20 m, and 15 m resolution simulations at the shoreline, the 10m resolution grid diverges, with an average maximum velocity of 8.03 m/s. While this divergence is relatively small with a change of 7% in mean values between the 15 m and 10 m results, it is a difference that is not easy to reconcile. From inspection of the results, this variance between the 15 m and 10 m results appears to be driven by a difference in the prediction of the steep front of the incoming bore, and with the understanding that breaking in this model is controlled through numerical dissipation, it is difficult to assess whether this variance is physical (better resolution of the process) or numerical (different numerical errors). Stable numerical results were not achievable for grid sizes less than 10 m. Such a divergence with finer resolutions is not found at the inland location. Table 3.2: Simulation run times (for 10-hour physical time) and relevant information for each grid resolution. Grid Resolution (m) Run Time (min) Threads used nx ny Time step (sec) 90 20 8 415 355 0.5 60 41 8 623 533 0.5 30 190 8 1244 1064 0.5 20 431 8 1607 1595 0.5 15 750 8 2142 2126 0.5 10 1713 8 3729 3189 0.25 85 Figure 3.10: Comparison of the runup height distributions between the 6 different grid resolutions using MOST. The distributions use a runup interval spacing of 0.05 m. Figure 3.11: (top panel) Comparison of the maximum shoreline flow velocities distributions and (bottom panel) 1-meter depth maximum flow velocities distributions between the 6 different grid resolutions using MOST. The distributions use a flow velocity interval spacing of 0.05 m/s. Figure 3.12 shows the calculated mean flow velocity at 6 different flow depths (where 6m corresponds to approximately the shoreline). It is very interesting to note that the mean speeds, with the exception of the 90 and 60 m resolution results, converge at a flow depth of 1 m. Also, 86 the greatest increase in flow velocity between flow depths was found to be from 2 to 3 meters with an average increase of 2.69 m/s for the tested grids. The Froude number (Fr), \ XMℎ ⁄ , where V is the magnitude of the overland flow velocity and h is the overland flow depth, is commonly used to constrain flow depth and speed for tsunamis when tracing deposits. Figure 3.12 presents the variability of the Fr at the Sendai plain showing an irregular Fr profile. The Fr is very near one at the shoreline, greater than one (supercritical flow) at flow depths of 3-4 meters and less than one (subcritical flow) at flow depths of 1-2 meters. It is reasonable to expect a steady decrease in velocity, if using the assumption of a simple beach, as the wave front makes its way inland; however, the simulation results provided in Figure 3.13 (top panel) show otherwise. Between 400m-1200m inland, maximum velocities (again the mean of the maximum velocities across the studied Sendia Plain) appear to be constant. After analyzing numerical output, we observe that small bathymetry/topography features can cause large changes in predicted flow velocities, producing secondary peaks. Further inland, results from MOST show a steady decline in velocity from 1200m-2800m with a negative slope of about -0.002 for all grid resolutions. Peak flow velocities fluctuate from 6-17 m/s and standard deviations from 0.9-1.7 m/s. Finally, it is interesting to note that Figure 3.13 (bottom left panel) shows abrupt changes in the peak flow velocities near the shoreline. These results agree with the fact that when a breaking bore hits the shoreline the front suddenly accelerates; the first data point shows this sudden acceleration. This phenomenon was investigated by Synolakis (1987). 87 Figure 3.12: Mean flow velocity at different flow depths. The 6-meter flow depth corresponds approximately to the shoreline. The thick black line represents the calculated mean flow velocities using a Froude number of 1. Figure 3.13: Inland maximum flow velocities across shore in the Sendai plain, (top panel) comparison of the average flow velocities between GeoClaw and the 6 different grid resolutions using MOST (bottom left panel) comparison of peak flow velocities (bottom right panel) comparison of standard deviations. 88 3.4 Conclusions This study presents a comparison between field data and model predictions of the 2011 Tohoku tsunami. Runup elevation was, in general, over predicted by the models. On the other hand, the inundation line predicted by the modeling was in good agreement with the observed, and this inconsistency can be attributed to errors in the topographical data. By comparing observed runup elevation measurements to DEM elevations at the same location, the topography bias ranged from (-3m) to (-1m). This bias is very significant for this particular location since the Sendai Plain is, of course, relatively flat. Thus, with this topography it should not be possible to get both the runup elevation and inundation line correct. For inter-model agreement, both numerical models, MOST and GeoClaw, agree relatively well with each other when predicting maximum sea surface elevation and maximum velocities, with MOST yielding slightly higher predictions for both. Also, it is important to note that the two models predict similar maximum velocity at the shoreline (Figure 9), which is typically the most important place to do so. The MOST sensitivity analysis to the Manning coefficient revealed that higher n-values yielded more precise runup elevation. The error for overland flow velocity increased at distances less than 1500m and greatly decreased at larger distances. When predicting runup heights and inundation lines with MOST, numerical convergence was achieved using the 30m resolution inundation grid. It can be suggested that grids finer than 30m resolution are not necessary when calculating these products at this particular location. Generally, when trying to simulate velocities, it is recommended to use higher resolution topography as small local changes in bathymetry/topography can cause similarly large local changes in the speed. Predictions of overland flow showed that the Froude number varies at different flow depths. At flow depths of 3-4m the flow is considered supercritical, which indicates that speed would increase in the presence of buildings or structures. It was expected that the maximum flow velocity would decrease as the wave makes its way inland, but this pattern was not obvious between the 400m and 1200m contour lines, as complexities in the topography and flooding waves obscure this idealized expectation. Finally, it is suggested that an uncertainty analysis, such as a probabilistic description of input parameters leading to probabilistic output, would provide a more complete understanding of the distribution of possible model predictions. 89 4. Tsunami vs Infragravity Surge: Comparison of the Physical Character of Extreme Runup Recent observations of energetic infragravity (IG) flooding events, such as those in the Philippines during Typhoon Haiyan, suggest that IG surges may approach the coast as breaking bores with periods of minutes; a very tsunami-like characteristic. Energetic IG waves have been observed in various locations around the world and have led to loss of lives and damages to property. In this study, a comparison of overland flow characteristics between tsunamis and energetic IG wave events is presented. In general, whenever the tsunamis and energetic IG waves have similar runup, tsunamis tend to generate greater flow depths and longer flood durations than IG. However, flow velocities and Froude number are larger for IG primarily due to bore-bore capture. This study provides a statistical and physical discriminant between tsunami and IG, such that in areas exposed to both, a proper interpretation of overland transport, deposition, and damage is possible. 4.1 Methodology 4.1.1 Field Observations On November 8, 2013 Typhoon Haiyan generated large surge and waves, and devastated, among many others, the coastal town of Hernani in the Philippines (Nobuoka et al. 2014). Typhoon Haiyan killed more than 6300 people and injured more than 28,689 in the Philippines (Lagmay et al. 2014). In Hernani, deposits of sand were found ~300 m inland after the storm (Soria et al. 2018). Since Hernani is protected by an extensive fringing reef, a hazard reduction from the storm was expected (Ferrario 2014). During the typhoon, several extreme IG waves or “tsunami-like” waves were generated and struck the coastal town (Roeber and Bricker, 2015). Video recording of one of the extreme IG waves shows that in a matter of seconds a house, located about 3 m above mean sea level (Roeber and Bricker, 2015), is washed away. This documented event has been very useful in the study of these types of phenomena and is used in the present study to test the hypothesis postulated. 90 Another IG event took place near Half Moon Bay, California, USA, during a surf competition on February 13, 2010. Spectators were enjoying a sunny day when suddenly an extreme IG wave washed away several people causing injuries to 13 of them (The Times, 2011). In Oregon’s past storm seasons, more than 21 people have died since 1990 due to “sneaker” (IG) waves (The Oregonian, 2016). Recently several people have died in Cabo San Lucas, Mexico, when they were dragged out to sea by an extreme IG wave during a sunny day walking at the beach (Chicago Tribune, 2017). From these events it has become clear that extreme IG waves often occur during stormy conditions, but not always. 4.1.2 Tsunami and Infragravity Modeling The Cornell University Long Wave (COULWAVE) model is used in this study because of its capabilities in predicting the dynamics and formation of IG waves and runup. COULWAVE is a Boussinesq-based numerical model (Lynett and Liu 2002, Kim et al. 2009) able to simulate wave propagation from deep water ( ^_`aba$cAd eaCAd ≥2) to the shoreline with high accuracy (Wei et al. 1995). COULWAVE has been validated and it has been used for a wide range of applications such as: wave run-up, propagation, inundation, wave breaking, tsunamis, currents in ports and harbors and hurricane waves (Lynett, 2007, Lovolt et al., 2013, Parsons et al., 2014, Lynett et al., 2014, among others). This model solves the fully non-linearly, weakly-dispersive wave equations given in the one-dimensional conservative form as: ! A +(g ∝ !) J + P i = 0 (4.1) (g ∝ !) A +(g ∝ ; !) J +M!j J +M!P J −g ∝ P i = 0 (4.2) Where ! = j+ℎ is the total water depth, g ∝ denotes the velocity at a reference elevation k ∝ , P J is the 2 nd order terms of the depth integrated momentum equation and averaged velocities, and 91 P i includes the 2 nd order terms of the continuity equation. A spatially constant bottom friction of 9 = 0.0035 was used with a quadratic bottom friction law for the model simulations. For the generation of energetic IG waves an internal-domain wave-maker is used in COULWAVE, driven by a Joint North Sea Wave Project (JONSWAP) input spectrum. A wall is located at the right boundary and a sponge layer at the left boundary. A grid resolution of 2 m is used for all the simulations. The tsunamis are modeled as a single symmetric amplitude dipole. A grid resolution of 20 m is used for all tsunami simulations. The rest of the parameters from COULWAVE were kept the same as in the IG wave simulations. In this study, we do not impose any special IG boundary conditions. The IG simulations are forced with an offshore JONSWAP spectrum, which in COULWAVE is broken down into a set of discrete amplitude/frequency sine waves with some specified ∆9. The offshore wave boundary is then the linear superposition of these discrete waves, each with a random phase, created with the internal source generator. IG energy is subsequently amplified by nonlinear wave-wave interactions within the modeling domain. For the IG simulations, the incident short-wave condition is discretized with a very small frequency resolution (10 -5 Hz). Different configurations of significant wave heights (! " = 3 Sl 12 m) and peak periods (B C = 13 Sl 22 sec) were tested (total of 100 configurations). Each configuration was simulated 10 times and had a unique random phase seed yielding a different deterministic representation of the input wave energy spectrum. After simulating more than a million waves, only 5 extreme IG runup events were detected (generated when ! " > 11 m and B m > 20 sec). An extreme or very rare IG runup is defined here as a runup elevation that is 5 standard deviations above the mean, or a 5-sigma event. Two different profiles (planar beach and reef case) are used to study the overland flow dynamics of both tsunamis and IG waves (Figure 1). As was previously mentioned, tsunamis are modeled as a single symmetric amplitude dipole with both a leading and following depression. The period of the tsunamis tested in this study, defined as two times the duration between the peak of the crest and minimum of the trough, range from 6 min to 22 min. To make a valid comparison between the tsunami and extreme IG event two different control criteria were used: maximum runup elevation and maximum offshore crest elevation. For the first criterion, where the maximum runup elevation for both the tsunami and IG are equal, different offshore tsunami amplitudes had to be tested until we found a tsunami with the same runup as the extreme IG event. Alternatively, 92 when matching maximum offshore crest elevation, tsunami simulations were initialized with an offshore (at 100 m depth) crest elevation equal to the offshore wind wave amplitude that produced the extreme runup in the IG simulation. Figure 4.1. Transects used for COULWAVE tsunami and IG simulations, Reef case (a) and planar beach case (b). A sponge layer and a wall are used at the left and right boundary, respectively. 4.2 Results and Discussion Figure 2a shows the 100-m depth ocean surface elevation time series (& = 750G) for the IG reef case simulation. Only 1 out of the 5 extreme events is used in this analysis as all have similar runup elevations and overland flow behavior. The wave parameters for the input spectrum that generated this extreme are: ! " = 11 G and B C = 21 opq. The maximum runup elevation generated for the reef bathymetry is 8.05 m with a dominant runup period of about 650 sec (Figure 2b). Dominant runup period is defined in this study as the approximate time in between runup peaks. A spectrum of low frequency runup height oscillations can also be observed in Figure 2b. The maximum wave crest height of the incident wave train that generated the maximum runup is 11.37 m. For the planar beach case, a slope of 1/35 is used since it represents the average slope of the entire reef configuration. For the planar beach, the maximum runup is 11.76 m and a dominant runup period of about 200 sec (not shown). To match the maximum tsunami runup elevation with 93 the IG event runup, offshore tsunami crest heights (at 100 m depth) ranging from 4.2-5 and from 3.2-4 m are used for the reef configuration and the planar beach, accordingly. The tsunami height changes depending on the tsunami period and whether there is a leading or following trough. Figure 2c presents a frequency spectrum across the reef for the IG wind-wave simulation. It shows that the offshore incident waves break just before the reef at & = 2,000 G. On the reef, energy in the gravity wave periods is very low but significant energy can be observed at the IG frequencies (9 <0.02 !k). It can be inferred that when the waves start shoaling, before breaking, energy is transferred to lower frequencies by three wave interactions since the magnitude of the triad interactions increase (Gallagher 1971, Young and Elderberky 1998). From Figure 2d local reef resonance of IG can be observed at around 9 = 0.0017 !k. The energy at lower frequencies than the resonance period is generated by the static setup in the mean water level. Finally, small amounts of energy can be observed at IG frequencies at the reef slope and offshore due to IG reflection as free waves, from the nearshore to the offshore. 94 Figure 2. (a) Offshore (& = 750G) free surface elevations and (b) runup time series for the IG configuration that generated an extreme runup event (reef case). A maximum runup elevation of 8.05 m and a dominant runup period of about 650 sec was predicted by COULWAVE. Frequency spectrum of water surface elevation along the one-dimensional reef transect from COULWAVE IG simulations (reef case): (c) Plot for a broad frequency range and (d) zoomed in on IG frequencies. 95 To better understand the overland flow dynamics between tsunamis and IG waves this study presents comparisons of flood duration, flow depth, maximum overland flow velocity and Froude number (Fr). Maximum velocities are calculated using a 0.3 m depth threshold in order to reduce the relevance of very thin, but fast moving flows. Measurements are compared at four different profile elevations for the reef configuration (0 m, 2 m, 4 m and 6 m) and six elevations for the planar beach configuration (0 m, 2 m, 4 m, 6 m, 8 m, and 10 m). Figure 3a shows that for the reef configuration, the ground was usually flooded longer during the tsunami event for the tsunami periods tested, as compared to the IG event flood duration. However, flood durations are very similar for both mechanisms when the tsunami has a period of 6 min. For the planar beach, Figure 3b shows that flood duration is greater for tsunami waves than IG for all tsunami periods. Also, comparable to the reef, both events tend to yield similar flood durations for small tsunami periods (<6min). There is little difference in the results when there is a leading depression first in the tsunami. Overall, tsunamis tend to flood the entire profile for a longer time in both model configurations. For the reef, Figure 3c shows that tsunami and energetic IG flow depths are very similar for tsunami periods comparable to the IG dominant runup period (~11 min). When the tsunami period is greater than the IG dominant period, the tsunami flow depths tend to be larger. Also, tsunami flow depths are found to be greater than IG flow depths at the upper beach (6 m elevation) for all tsunami periods. Figure 3d shows that for the planar beach, tsunami flow depths are usually greater than IG flow depths. When the tsunami period is 6 min, flow depths from both events are almost the same. Generally, tsunamis tend to have a greater flow depth throughout the entire profile in both configurations; however, when the IG dominant runup period and tsunami period are similar their flow depths are similar. 96 Figure 3. Comparison of flood duration and flow depths between IG and tsunami for the reef case (a and c) and planar beach case (b and d). Leading crest and leading depression tsunami predictions, for a period of 6 min, are presented with a circle and a black cross, respectively. Tsunami period of 11 min are presented with a square (leading crest) and a red cross (leading depression). Tsunami period of 22 min are presented with a diamond (leading crest) and a blue cross (leading depression). Figure 4a shows that for the reef, IG maximum flow speeds are higher than tsunami flow speeds throughout the beach profile, except at the shoreline when tsunami period is 6 min. A similar behavior is observed for the Fr (Figure 4c). Away from the immediate shoreline, the maximum Fr is always greater for the energetic IG event. Finally, for the planar beach, IG flow velocities and Fr were much larger than tsunami flow speeds as shown in Figure 4b and 4d. 97 Regardless of the tsunami period the Fr for IG is always higher, and for some locations approaches a value of 5. As the waves travel to the beach they have a Fr of less than 1, but as they begin to break near the beach the Fr gets close to 1 and in some instances greater than 1 after breaking. In order for the Fr to be greater than 1 after the waves break there has to be an increase in the flow at low depths. This happens with bore-bore capture, i.e. as a bore collapses on the beach the transferred momentum increases the velocities at low depths (García-Medina et al. 2017). Therefore, bore renewal with bore-bore capture contributes to these larger flow speeds and increases the Fr. Energetic IG show greater speeds and Fr with smaller depths in both configurations tested. Figure 4. Comparison of flow speeds and Froude number between IG and tsunami for the reef case (a and c) and planar beach case (b and d). Leading crest and leading depression tsunami 98 predictions, for a period of 6 min, are presented with a circle and a black cross, respectively. Tsunami period of 11 min are presented with a square (leading crest) and a red cross (leading trough). Tsunami period of 22 min are presented with a diamond (leading crest) and a blue cross (leading trough). To further investigate the overland flow behavior of these two flooding mechanism, a second control criterion is used in which the same maximum offshore wave crest height from the IG simulations is used for the tsunami simulations. An offshore wave crest height of 11.37 m for the tsunami is adopted, as previously mentioned. It is important to note that the tsunami runup is much larger than the IG runup (~20 m vs ~9 m). For this comparison, only results from the reef are presented as very similar results are obtained for the planar beach. Figure 5a shows that for tsunami periods less than the dominant run-up period, tsunami velocities are larger than IG velocities. As tsunami period increases, the velocities decrease throughout the beach profile. Fr is greater across the profile for the tsunamis, except at 2-4 m elevation, when the tsunami period is less than the IG runup dominant period as shown in Figure 5b. When the tsunami period increases, Fr is greater for IG everywhere along the profile except at the shoreline. It is worthwhile to reiterate that for this offshore-amplitude-match scenario, where the maximum tsunami runup elevation is much greater than the IG runup, there are still locations along the beach profile where the extreme IG flooding leads to larger maximum fluid speeds and larger maximum Froude numbers. 99 Figure 5: Comparison of flow speeds (a) and Fr (b) between IG and tsunami for the reef. Leading crest and leading depression tsunami predictions, for a period of 6 min, are presented with a circle and a black cross, respectively. Tsunami period of 11 min are presented with a square (leading crest) and a red cross (leading trough). Tsunami period of 22 min are presented with a diamond (leading crest) and a blue cross (leading trough). 4.3 Conclusions Extreme IG-driven runup events can appear “tsunami-like” and, for the wave conditions prescribed herein, can lead to runup elevations in excess of 10m above storm water level. In general, whenever the tsunamis and energetic IG waves have similar runup it is determined that tsunami flow depths and flood durations are greater. On the other hand, the maximum IG flow speeds are greater in most of the cases, particularly when tsunamis have a long period (>6 min). Also, IG maximum Fr is greater than the tsunami maximum Fr in the majority of cases. For the IG simulations, the large Fr appear to be generated by bore-bore capture. Furthermore, when tsunami runup is much larger than IG runup (with matching maximum incident offshore crest heights) the IG speeds can be greater when tsunami periods are greater than the dominant IG period. Tsunamis and energetic IG of similar amplitude/period lead to differences in flow properties, and it is consequently expected that these differences influence the resulting sedimentology. There are several challenges that exist in order to discriminate tsunami events from extreme IG events. First is the lack of identified extreme IG deposits. Furthermore, for coastal areas that are prone to both large tsunamis and large wind wave events, the local statistics of extreme IG/wind wave runup needs to be analyzed and understood. The periods at which the local beach excite, or trap IG energy needs to be investigated since the longer the IG period, the further onshore the water can travel. Also, the return periods of tsunamis and large wind wave runup needs to be compared at hazard levels of interest. Areas where the return periods are not comparable could be used to identify where deposits, boulders and design hazards can be more reliably attributed. 100 5. Future Research A model can be developed to predict extreme events based on offshore real-time data. It was revealed that the envelopes characteristics are very crucial for the prediction of extreme events. By analyzing these characteristics from a real-time offshore buoy it will be possible to predict when an extreme event can occur. A methodology can be developed to characterize nearshore IG spectrum based on the offshore IG spectrum. An approximate of 30 phase realizations are needed to have a converged IG spectrum (1% error). By simulating only 30 runs for each possible (realistic) wave parameters combination one can have converged spectrums that can be used to characterize the nearshore IG spectrum based on the offshore IG spectrum. Also, the IG part can be characterized based on the initial input spectrum. Finally, direction discretization is going to be analyzed by using 2D COULWAVE. 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Abstract (if available)

Abstract In this study a methodology is established to model infragravity wave and their runup using a Boussinesq-type model. The importance of input energy frequency resolution is shown by comparing runup height predictions from three different frequency resolutions. Several extreme infragravity wave runup generation mechanisms are presented and discussed. Furthermore, runup, inundation and overland flow velocity predictions are are compared using field data, MOST and GeoClaw. Finally, a comparison of overland flow characteristics is presented between tsunamis and energetic infragravity waves. The purpose of this study is to better understand the properties of extreme wave runup.

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Creator Montoya, Luis Humberto (author)

Core Title Understanding properties of extreme ocean wave runup

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 06/10/2019

Defense Date 04/29/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Boussinesq,extreme runup,infragravity,OAI-PMH Harvest,overland flow,tsunami

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lynett, Patrick (committee chair), Ghanem, Roger (committee member), Luhar, Mitul (committee member)

Creator Email lhmontoy@usc.edu,luishmontoyacortes@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-172219

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