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Nonlinear long wave amplification in the shadow zone of offshore islands

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Nonlinear long wave amplification in the shadow zone of offshore islands

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Content NONLINEAR LONG WAVE AMPLIFICATION IN THE SHADOW ZONE OF OFFSHORE ISLANDS by Vassilios Skanavis A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CIVIL ENGINEERING) December 2020 Copyright 2020 Vassilios Skanavis Acknowledgments My PhD journey has been a life-changing experience, made possible because of the support and guidance that I received from several wonderful people. It is the norm to have a supervisor who supports and encourages you during your studies, but my supervisor is one of a kind! Professor Costas Synolakis went, in every situation, the extra mile, to make me reach my goal successfully! He believed in me, he believed that I could make it in harsh competitions, that I could excel in difficult scenarios – in the classrooms/laboratories/field work, that I could solve complex experiments and simulations. Professor Costas Synolakis managed to make me test my skills and abilities to the max and he was also great in encouraging me, to always be calm and patient! Our long discussions, expeditions, even the simple talks and diners filled me with new experiences and ideas! No time with Professor Costas Synolakis is ever boring! The truth, though, remains that, I am a very lucky person having walked next to him during my doctoral studies. Professor, thank you for everything you have done for me! Special “thanks” to Patrick Lynett, Professor of Civil and Environmental Engi- neering at USC; to Mitul Luhar, Assistant Professor of Aerospace and Mechanical ii EngineeringandCivilandEnvironmentalEngineeringatUSC;EmileOkal, Profes- sor Emeritus Department of Earth and Planetary Sciences Northwestern Univer- sity; Felipe De Barros, Associate Professor of Civil and Environmental Engineer- ing at USC; Thanasis Fokas, Professor of Applied Mathematics and Theoretical Physics University of Cambridge. Every single one of the above mentioned Profes- sors has marked this period of mine with excellent guidance, support, feedback and fulfillment of my pertinent needs and mostly memorable moments of collaboration that assisted me in making it during my doctoral studies training period. I gratefully acknowledge the funding received towards my PhD from • the Viterbi fellowships offered by the University of Southern California. • Myronis fellowships offered by the University of Southern California. • Gerondelis Foundation fellowship. • Academy of Athens fellowship offered by Academy of Athens. • The work presented in this thesis was funded by the National Science Foun- dation NEES Research program, with award number 1538624. A heartfelt "thank you" to my great friend and colleague at USC lab and everyday life experience, Dr. Nikos Kalligeris from the Institute of Geodynam- ics, National Observatory of Athens. I am grateful for his endless hours of working in numerous ways during the various stages of my PhD. Niko, you helped me enormously, especially with the mammoth task of doing the final touch for my dissertation. I am grateful for his encouragement in spouts and challenging me to exceed myself. iii To my friends, “Thank to you” for understanding my ups and downs. This an exciting and stressful period of my life, and I am happy you were there. Emelia, thanks for “not complaining” and your support! Last but with great value, thank you Brother Constantinos. You are the best brother I could have asked for! To my Father Leonidas and my Mother Con- stantina: You were the ones who insisted that I embark on this journey in the first place, Mom and Dad, who, I love you. ToyouGrandmotherMaria, alwaysbelievedinmeandencouragedmetofollow my dreams, I dedicate this dissertation, as a small token for all your unconditional love and care! iv Contents Acknowledgments ii List of Tables vii List of Figures viii Abstract xx 1 Introduction 1 1.1 The specific motivation for this study - the 2010 Mentawai tsunami 8 1.2 Motivation and plan of the present study . . . . . . . . . . . . . . . 15 2 Laboratory Setup 20 2.1 Cameras and light . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 One–island configuration . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Camera field of view (FOV) . . . . . . . . . . . . . . . . . . 23 2.2.2 Boundary conditions and water level . . . . . . . . . . . . . 24 2.3 Two–island configuration . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Camera field of view . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2 Boundary conditions and water level . . . . . . . . . . . . . 27 2.4 Wave gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Post-Processing of Laboratory Data 31 3.1 Image pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Particle identification . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Particle Tracking Velocimetry (PTV) . . . . . . . . . . . . . . . . . 34 3.4 Camera parametrization . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.1 Intrinsic parametrization . . . . . . . . . . . . . . . . . . . . 35 3.4.2 Extrinsic parametrization . . . . . . . . . . . . . . . . . . . 36 3.5 3D surface velocity extraction, with one island in place. . . . . . . . 41 3.6 3D surface elevation extraction one island . . . . . . . . . . . . . . 43 3.7 Edge detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7.1 The coastline around the island . . . . . . . . . . . . . . . . 46 v 3.7.2 Analytical solutions for runup on a conical island . . . . . . 48 3.7.3 The coastline on the sloping beach . . . . . . . . . . . . . . 50 3.8 3D surface elevation extraction for the two island case . . . . . . . . 54 4 Numerical Models at Laboratory Scales 58 4.1 MOST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1.1 The history of MOST . . . . . . . . . . . . . . . . . . . . . . 58 4.1.2 Simulation of the one–island laboratory experiment using MOST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Modeling the laboratory experiments with COULWAVE . . . . . . 67 4.2.1 The particular version of the Boussinesq equations used in COULWAVE . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.2 Comparison of laboratory measurements with COULWAVE results – One island . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.3 Comparison of laboratory measurements with COULWAVE results – Two islands . . . . . . . . . . . . . . . . . . . . . . 76 5 Numerical Models at Geophysical Scales 81 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Validation of MOST . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Numerical evaluation of the runup around a conical island and on the sloping beach behind it . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.1 Free surface flow kinematics . . . . . . . . . . . . . . . . . . 99 6 Conclusions 111 A Labratory data for Runup 114 Reference List 123 vi List of Tables 5.1 Physical parameter ranges used for 2+1 D numerical experiments used for validation of MOST . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Parameter ranges for computational experiments in this section. . . 91 A.1 Laboratory data for solitary waves runup non–breaking and breaking114 vii List of Figures 1.1 Map of Babi island location. . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Pictures of Babi island. Credit: Dr. Costas Synolakis . . . . . . . . 4 1.3 Longwaverunupontheleeofaconicalisland. Experimentsinspired by the 1992 Flores tsunami, see Yeh et al. (1994b). Numerical sim- ulations by Titov & Synolakis (1995) -left subfigure- pictures from the laboratory experiments of Briggs et al. (1995), on the right. The arrows indicate the direction of the wave attack. This classic set of experiments is still widely used for code validation. During these experiments, runup was not measured in the beach the islands shad- ows, because it was not know at the time that the coast was also effected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Simulationofthemaximumwaveheightofthe10/25/2010Mentawais tsunami. Significant runup was modeled and observed on coastal areas behind small offshore islands. (Hill et al., 2012) . . . . . . . . 9 1.5 SchematicofthegeometryoftheexperimentalsetupfromStefanakis etal.(2014). Theparameter tanθ i variedfrom 0.05−0.2, tanθ b from 0.05− 0.2, d from 0− 5 km, h from 100− 1000 m and the effective "frequency" of the incoming solitary–like wave from 0.01− 0.1 rad/s. 10 viii 1.6 Run-up amplification (RA) as a function of the wavelength to the island radius (at its base) ratio. J = tanθ b / q H 0 /L 0 Different col- ors indicate values of the surf similarity parameter (a.k.a. Iribarren number) computed with the beach slope and multiplied with the square root of wave height over wave length at deep water. (Ste- fanakis et al., 2014). . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Snapshots of the evolution of the free surface elevation measured in meters as the wave passes the island and runs up the beach behind it. The island focuses the amplified wave as it propagates towards the beach. The color bar on the right is in logarithmic scale, for visualization. Here, the max run-up amplification behind the island is 1.59. (Stefanakis et al., 2014). . . . . . . . . . . . . . . . . . . . . 14 2.1 Schematic of OSU basin . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The figure on the left represents the schematic of the mounted cam- eras in relation to the water surface. The image on the right illus- trates the experimental set-up using the light diffusers. . . . . . . . 22 2.3 The experimental set–up for the one island configuration. The lines are contours of every 0.1 m of elevation. The red dashed box is the study area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Camerapositionsforthe3Dstereomatchingforthefirstconfiguration. 24 2.5 Wavemaker boundary conditions. Top plot shows the piston dis- placement time history, starting from the fully retreated position and reaching maximum stroke. The bottom plot shows the pre- dicted (shallow water theory) surface elevation at the face of the wavemaker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ix 2.6 The experimental set-up for the two island configuration. The lines are contours at every 0.1 m of elevation. Red dashed line box is the study area for the 3D PVT analysis. . . . . . . . . . . . . . . . . . 26 2.7 Shows the study area for the experimental set-up of the 3D stereo matching for the second configuration. . . . . . . . . . . . . . . . . 27 2.8 The wavemaket motion for generating the leading depression N– wave used in the subsequent experiments. The top plot shows the piston displacement time history, starting from the fully retreated position and reaching maximum stroke. The bottom plot shows the predicted (shallow water theory) surface elevation at the face of the wavemaker. The water depth, these trajectories were used, was 30 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 On the top image one island configuration, on the bottom image two islands configuration (Credit: Dr. Costas Synolakis). . . . . . . 30 3.1 On the top left figure is the original frame, on the bottom right is the LED light, on the right top figure is the transformation to a gray scale and on the bottom right is the gray scale flipping the colors. . 33 3.2 An example of particle identification: yellow circles denote the iden- tified particles and the red dots the clumped particles that were rejected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Intersection of two rays associated with the left and right camera viewpoints (Douxchamps et al., 2005). (X,Y ) in the figure corre- spond to the image coordinates (u,v) in the notation used here. . . 39 3.4 Table with control points to determine the camera extrinsic param- eters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 x 3.5 Figure on the top shows the scattered data – Figure on the bottom shows interpolation from the scattered data at the same time, which is 10.51 s from the start of the wavemaker. . . . . . . . . . . . . . . 42 3.6 Figure shows velocities u,v in different position around the island (see Figure 3.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 3D view of free surface elevation as the solitary wave goes past the side of the island at t = 10.5 s from wavemaker start. . . . . . . . . 44 3.8 Free surface elevation time-series at different locations around the conical island. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.9 An example of shoreline extraction using edge detection for a frame in position 1. The yellow line corresponds to the water-surface inter- face detected through the step in image intensity. . . . . . . . . . . 47 3.10 Runup velocity of at the center of the island. On the abscissa the dimensionless number x ∗ /d is used, where d is the depth and x∗ is the dimensionless length scale. The time in the ordinate is non- dimensionalized using the constant depth d. . . . . . . . . . . . . . 48 3.11 Maximum runup extracted from edge detection compared to the analytical prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.12 An example of shoreline extraction using edge detection for the slop- ingbeach. Theyellowlinecorrespondstothewater-surfaceinterface detected through the step in image intensity. . . . . . . . . . . . . . 51 3.13 Runupvelocityofatthecenteroftheslopingbeach. Ontheabscissa the dimensionless number x ∗ /d is used, where d is the depth and x∗ is the dimensionless length scale. The time in the ordinate is non-dimensionalized using the constant depth d. . . . . . . . . . . . 51 xi 3.14 Maximum run-up of a solitary wave as a function of slope angle for non-breaking and breaking waves for several incident relative wave heights (Li & Raichlen, 2002). . . . . . . . . . . . . . . . . . . . . . 52 3.15 Shows the maximum run-up of the sloping beach with the black edge detection method in comparison to the red crosses depicting the total station points in comparison to the analytical model (with a constant slope without an island) shown by the blue dashed line. Black dashed line is the shoreline constant depth 30 cm. . . . . . . 53 3.16 Free surface elevation time-series at different locations around the two conical island. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.17 Shows the maximum run-up of the sloping beach. Laboratory data recorded using total station. . . . . . . . . . . . . . . . . . . . . . . 56 3.18 Two island configurations (Credit: Dr. Pedro Lomonaco). . . . . . . 57 4.1 Snapshots of the top views of the flow evolution of a 0.4 solitary wave interacting with a conical island and a sloping beach behind it, for the geometrical parameters in section 2.2. Computations using MOST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 An≈ 0.4Solitarywavetravellingoverconstantdepth, usingMOST, but with zero initial velocity. The initial wave is a 0.8 wave splits into two waves, each with≈ 0.4 wave height, which as expected in 1+1D propagation, per D’Alembert’s solution. The water depth is 30 cm. The wave breaks almost immediately after the motion starts. 64 4.3 Comparison of the PTV-extracted free surface elevation with the MOST predictions. Black line corresponds to the laboratory data, and the red dashed line to MOST. See Figure 3.8 for the locations. 65 xii 4.4 Comparison of the PTV-extracted u− and v− velocities with the MOST predictions. Black line corresponds to the laboratory data, and the red dashed line to MOST. See Figure 3.8 for the locations. 66 4.5 Comparison of the PTV-extracted free surface elevation with the COULWAVE predictions. Black line corresponds to the experimen- tal data, and the red dashed line to COULWAVE. See Figure 3.8 for the locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.6 Comparison of the PTV-extracted u− and v−velocities with the COULWAVE predictions. Black line corresponds to the experimen- tal data, and the red dashed line to COULWAVE. See Figure 3.8 for the locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Top view of the conical island for an H/d = 0.4 solitary wave and compares analytical (green line) , numerical (violet line) predictions with the laboratory measurements, obtained using edge detection, as described in section 3.7. Concentric solid elevation lines are every 10cm, so the undisturbed water depth is 30cm. Surprisingly, the analytical predictions of Kânoğlu & Synolakis (1998) predict the maximum runup about as well as COULWAVE, even for this highly nonlinear wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 xiii 4.8 Comparison of the maximum runup along the beach behind the island inferred from the laboratory measurements (edge detection), measurements using a total station, predictions from COULWAVE and the breaking-solitary wave model of Li & Raichlen (2002) which is based on Synolakis (1987). On the top. there is the shoreline pro- file along the cross–shore coordinate (a top view) while the bottom figure is the maximum runup along the cross-shore coordinate ( Fig- ure 4.9 for visualizing the location). . . . . . . . . . . . . . . . . . . 73 4.9 Top view of the maximum wave height over all time steps, for a 0.4 solitary wave interacting with the topography with parameters shown in section 2.2. Computations using COULWAVE. The black line is the runup from the laboratory data, red line is the predic- tions from COULWAVE and with the blue dashed is the breaking – solitary wave model (Li & Raichlen, 2002). . . . . . . . . . . . . . . 74 4.10 Snapshots of the top views of the flow evolution of a 0.4 solitary wave interacting with a conical island and a sloping beach behind it, for the geometrical parameters insec 2.2. Computations using COULWAVE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.11 Snapshots of the top views of the flow evolution of a N–wave inter- acting with two conical islands and a sloping beach behind it, for the geometrical parameters in section 2.3. Computations using COUL- WAVE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.12 Comparison of the PTV-extracted free surface elevation with the COULWAVE predictions. Black line corresponds to the labratory data, and the red dashed line to COULWAVE. See Figure 3.16 for the locations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 xiv 4.13 Black line corresponds to the COULWAVE predictions for the sur- fave velocities u and v. See Figure 3.16 for the locations. . . . . . . 79 4.14 Top view of the maximum wave height over all time steps, for a N–wave interacting with the topography with parameters shown in section 2.3 (Computing using COULWAVE). The black dashed line (circle and vertical) are the initial positions of the shoreline on the island and the sloping beach, respectively. Comparison of the maximum runup along the beach behind the island inferred from the laboratory measurements using a total station (black line) and predictions from COULWAVE (red dashed line). . . . . . . . . . . . 80 5.1 Schematic of the geometry . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 a: The figure on the left shows the runup amplification as predicted by the runup law versus the prediction of the runup amplification fromMOST,forthe125computationsofsolitary–likewaves. b: The figure on the right shows the dimensional runup prediction from the runup law versus the numerical prediction of MOST. . . . . . . . . 85 5.3 The runup amplification as a function of the runup law. Compari- son of the integral evaluation (analytical), its asymptotic expansion (analytical–asymptotic expansion) and results from MOST. . . . . . 86 5.4 Therunupamplificationasafunctionoftherunuplaw. Comparison of the integral evaluation (blue line), its asymptotic expansion (red line) and laboratory data from Li & Raichlen (2001), as shown in Table A.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 xv 5.5 The runup amplification as a function of the runup law. Compar- ison of the integral evaluation (analytical), its asymptotic expan- sion (analytical–asymptotic expansion) and the universal analytical using Equation 5.19. . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.6 Schematicofthegeometryofasingle–humpsolitary–likewaveapproach- ing a conical island of angle θ i distant d from the toe of a sloping beach of angle θ b . The offshore depth is h. . . . . . . . . . . . . . . 92 5.7 Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.05, d = 0 m, h = 325 m, ω = 0.0175 rad/s, with n-wave coefficients a =−0.0005235, b =−1140, c = 0.0001613, d = 1875. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8 Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.05, d = 0 m, h = 100 m, ω = 0.005 rad/s, with n- wave coefficients a =−0.0001276, b = 7558, c = 0.0001101, d = 7749. 94 5.9 Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.0875, d = 0 m, h = 100 m, ω = 0.005 rad/s, with n-wave coefficients a =−0.0003248, b = 3440, c = 0.0003002, d = 1279. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.10 Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.0875, d = 1250 m, h = 775 m, ω = 0.0112 rad/s, with n-wave coefficients a =−0.0001926,b = 558.7,c = 8.05e− 05, d = 5144. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.11 Plots of the variation of the parameter a in the N-wave type fit of Equation (5.21) of the maximal shoreline motion. Variations with respect to the wavelength, the island radius, the distance between the toe of the island to the toe of the beach and with the water depth. 96 xvi 5.12 whereL is the wavelength,X 0 =h/ tanθ b ,h is the water depth and r is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.0006718x− 0.0006188. . . . . . . . . 97 5.13 where λ is the wavelength and r is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.808 ln(0.3651x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.14 where λ is the wavelength and r is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.563csch(0.075x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.15 whereλ is the wavelength andr is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) =−9.34x 4 + 22.2x 3 − 17.63x 2 + 4.45x + 0.34. . . . . . . . . . . . . . . . . . . . . 98 5.16 Runupamplificationasafunctionoftheratioofthewavelengthover the island radius using VOLNA and MOST. The original Stefanakis et al. (2014) results are shown in circles, while crosses within these circlesidentifythoseresultswhichareforwaveswhicharedispersive, i.e., with λ/d< 20. The black circles identify results obtained with MOST for the same parameters used by Stefanakis et al. (2014) with VOLNA. Clearly, most differences are observed for dispersive waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.17 Runup amplification as a function of the ratio of the wavelength over the island radius using MOST. The black x-s show results with MOST, for the entire parameter range (see Table 5.2), with red circle is the maximum and minimum runup amplification. . . . . . . 101 xvii 5.18 A definition sketch. The solid red is the coastline, the dashed red is the toe of the sloping beach and the dashed black line shows the semi-circle along which the wave height is shown in subsequent figures.102 5.19 Snapshotsoftheflowelevationaroundtheisland, inincreasingtime, from left to right, for the case shown in Figure 5.23b. The color bar is dimensional, and the numbers are in meters. The initial wave height is 1.5 m. RA = 0.67, λ/r = 2. . . . . . . . . . . . . . . . . . 104 5.20 Snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island. The blue line is with the island in place, the red dashed line is without the island. Two humps appear in the top snapshots, in both cases, because the profiles are along a semi-circle. This is the same case as in Figure 5.19, with H− 1.5 m. RA = 0.67 and λ/r = 2, i.e., the case with the lowest recorded runnup amplification. . . . . . . . . . . . . . . . . . . . . . 105 5.21 Maps of the surface elevation at different times corresponding to the case of maximum runup amplification RA=1.37. . . . . . . . . . . . 108 5.22 Snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island. The blue line is with the island in place, the red dashed line is without the island. This is as in Figure 5.19, with H = 1.5 m. RA = 1.37 and λ/r = 10, i.e., the case with the highest computed RA. . . . . . . . . . . . . . . . . . 109 xviii 5.23 On the upper figure, the maximum surface elevation for the highest runup amplification, as calculated with MOST and as seen in Fig- ure 5.17. RA = 1.37 and λ/r = 10, tanθ i = 0.1625, tanθ b = 0.2, h = 100 m, d = 0 m, and ω = 0.03 rad/s. On the lower figure, the evolution of the maximum surface elevation, for the case with lowest runup amplification, as seen in Figure 5.17. RA = 0.67, λ/r = 2, tanθ i = 0.05, tanθ b = 0.05, h = 100 m, d = 1250 m, and ω = 0.03 rad/s. In both figures, the dotted green line is the prediction of Synolakis (1991), the blue lines show the evolution of the maxi- mum along an axis which bisects the island. The red lines show the equivalent evolution of the maximum elevation along the bisector, but without the island in place. The two points corresponding to the two cases in these two subfigures are also marked in Figure 5.17. 110 A.1 The normalized maximum runup of solitary waves versus the nor- malized wave height. All runup laboratory data available in lit- erature for breaking and non–breaking solitary waves on a sloping beach of constant slope (see Table A.1). . . . . . . . . . . . . . . . 120 A.2 The normalized maximum runup of solitary waves climbing up dif- ferent beaches versus the normalized wave height (see Table A.1). . 121 A.3 The normalized maximum runup of solitary waves climbing up dif- ferentbeachesversussurfparameterξ −1 s . Where (ξ −1 s =s(H/h) −9/10 ), H is the wave height and h is the constant depth. All runup labo- ratory data available in literature for breaking solitary waves on a sloping beach of constant slope (see Table A.1). . . . . . . . . . . . 122 xix Abstract In effective crisis management, prevention, preparedness, response and recovery are of crucial importance to improve resilience. It is inevitable that humanity will often be be faced with natural disasters and other threats, and survival hinges on knowledge and experience at all levels of risk management at local, national and global levels. In this regard, being able to identify previously unrecognized threats is essential. Thisisastudyoftherunupamplificationintheshadowzoneofoffshoreislands. Runup amplification (RA) is the ratio of the runup along the beach shadowed by the island to the runup that would be observed if there was no island present. Field survey reports from recent tsunamis suggest that local residents in mainland areas shadowed by nearby islands maybe under the impression that these islands protect them from tsunamis. Recent numerical results using machine learning and examining only non–breaking waves have generated substantial attention in world media, because they suggest that, in most cases, islands amplify tsunamis in the shadow zones behind them. Thestudyusestheidealizedbathymetryofacircularislandfrontingauniformly slopingbeach, asinthestudyofStefanakisetal.(2014). Thebulkofthestudyused MOST, a NSW solver, while several simulations were run with the Boussinesq– type equation solver COULWAVE, The study used measurements from laboratory xx experimentstoassesstheveracityofbothcodesatsmallscales,thenexactsolutions of the SW to check the veracity of MOST at geophysical scales. Two sets of experiments are presented, with one and two 1 : 2 slope islands fronting a 1 : 10 uniformly sloping beach. The particle tracking algorithm imple- mented by Kalligeris (2017) was used to track the particles between successive video frames. For a given inter–particle spacing, the tracking became more diffi- cult as the flow speed increases. In other words, for faster flows, one needs more frames per second. The detection threshold turned out as 1 m/sec. The laboratory measurements were compared with both analytical results and numerical predictions. It was found that the analytical formulation of Kânoğlu & Synolakis (1998) proved capable of capturing the maximum runup for the one islandconfiguration, whereitisvalid, thereisnoanalyticalsolutionfortwoislands. Also, itappearsthatbothMOSTandCOULWAVEreproducethemaximumrunup very well around the island and along the sloping beach shadowed by the island. COULWAVE also reproduces the runup velocities very well. Surprisingly, for very long waves on very steep beaches, the classic analytical expression for a solitary wave climbing up a sloping beach and known as Synolakis’ runup law does not work, instead, it was determined that the runup amplification R/H asymptotes to RA= 2. The results suggest that both runup ampplification (RA>1) and shadowing (RA<1) occur, depending primarily on whether the initial wave breaks of not on the front face of the island. Shorter solitary–like initial waves break on islands with mild faces and the islands do not shelter the coastlines behind them. Longer waves climbing up islands with steeper front faces evolve in the beginning as predicted by Synolakis (1991), then continue to climb up the sloping beach and produce high runup amplification. This is the case more likely to occur in nature. xxi The counter intuitive phenomena described here suggest public education is urgently needed to help people identify locales at higher risk than others, as is the provision of high–resolution inundation maps which account for such phenomena, for evacuation planning. Successfulresponsesinnaturaldisasters,includeunderstandingtherisk,report- ing the related disasters and formulating a communication code with all involved. Communication tailored to the needs of citizens, civil servants, political institu- tions and the media, will play a significant role in reducing significantly human cost. xxii Chapter 1 Introduction Tsunamis may now be ubiquitously known around the world, and used as metaphorssuchasAlanGreenspan’sfinancialtsunamitodescribethe2008crisisin the world’s economies, but this was not always the case. Before the middle 1990s, tsunamis were believed quite rare, at worst occurring once every decade somewhere around the world. Back then, simulation tools for 2D propagation and inundation were still under development, and the then possible rudimentary tsunami warn- ings were based solely on earthquake detection, epicenter and size determination, and historic reports as to whether earthquakes in adjacent locations had generated tsunamis in the past. A brief history of the evolution of tsunami science in the past twenty five years maybe found in Kânoğlu et al. (2015). We now know (Synolakis & Okal, 2005) that tsunamis with runup of the order of a few meters occur about once a year, on average, and tsunamis with runup of the order of 10 m occur every five years. Since the 1992 Nicaraguan tsunami, there have been comprehensive field surveys documenting the flow impacts along coast- lines that were struck by tsunamis and fairly consistent measurements of tsunami inundation were made, in the sense that there was a realization of differences in tsunami overland flow depth, maximum tsunami runup and tsunami height at the initial shoreline. Earlier reports of field measurements did not always differentiate between them. Eyewitness accounts or fortuitous videos which captured the evolu- tion of tsunamis have revealed rich flow patterns that tsunamis may follow, which sometimes defy intuition, see (Synolakis & Bernard, 2006). It is the objective of 1 this thesis to investigate one such unexpected flow phenomenon, identified only recently. OnDecember12th1992,alargeearthquakeM = 5.1×10 27 dyn-cm(Dziewonski et al., 1994) struck off Flores, Indonesia. Flores is 354 km long and about 50 km wide and is located in Nusa Tengara, east of Java and west of the island of Guinea. According to Synolakis & Okal (2005), the seisme was likely caused by subduction of what was then proposed as a block named Banda below the Australian plate, under a complex regime of back–arc compression. This event resulted in over 2500 deaths and triggered a large tsunami with runup reaching up to 30 m, only three months after the 1992 Nicaraguan tsunami. The latter is widely considered the event which ushered the modern era of tsunami field measurements and modeling. The field results on inundation that differed substantially from the then state-of- the art computations, which only included propagation calculations and no runup. AnInternationalTsunamiSurveyTeamwasdeployedinFlorestodocumentthe effects of the tsunami. There was extensive shaking damage in the island’s capital city Maumere. The about 1800 deaths reported there couldn’t be differentiated as shaking or tsunami victims. About 90,000 inhabitants were left homeless and it has been reported by some that, as many as 90% of the structures suffered damage. Tsunami runup ranged up to 6 locally, and in the neighborhood of Wurhing there was overland flow with the wave attacking a tiny densely populated peninsula from two sides. The worst damage was in the small 2.4 km diameter volcanic island Babi, off Flores, where 700 died because of the tsunami. 2 122˚20' 122˚20' 122˚30' 122˚30' 122˚40' 122˚40' 122˚50' 122˚50' −8˚30' −8˚30' −8˚20' −8˚20' −8˚10' −8˚10' −8˚00' −8˚00' Babi island Kodia Flores M7.8 Riangpuho Figure 1.1: Map of Babi island location. As per Synolakis [personal communication] the Indonesian authorities at first were quite reluctant to facilitate access to Babi, claiming there was nothing to find. Professors Synolakis and Kawata splintered off the main survey group, hired a small outrigger and anyway went to Babi, and took the first measurements. Two survivors who had returned to collect belongings and look for bodies of family members described gruesome scenes with human remains impaled on trees. One survivor had to swim across the 3.4 km channel to Flores to bring the news of the catastrophe to the Synolakis and Kawata saw that the wave devastated two fishing villages on its lee side, normally protected from wind waves and swell. Once back, they notified the entire tsunami survey team, and the Indonesian authorities, and a military helicopter transfer was arranged. 3 Figure 1.2: Pictures of Babi island. Credit: Dr. Costas Synolakis While maximum runup values in Babi reached approximately 7 m, there was a location in Riangrioko in North Flores where runup reached 26 m. This was later attributed to a local submarine landslide, itself triggered by the earthquake. Overall, in 1992, there was little understanding of how to differentiate between maximum runup from overland flow depths. It almost took a decade and the reanalysis of the measurements from the 1998 landslide tsunami of Papua New Guinea Lynett et al. (2002), for the tsunami community, to understand these differences. The Flores event triggered renewed interest in the understanding of long wave phenomena. Byserendipity, NSF hadthen justfundeda seriesof laboratoryexper- iments funded at what was then called the Coastal Engineering Research Center at the Waterways Experiment Station, at Vicksburgh, Mississipi; it is now called 4 Coastal and Hydraulic laboratory. According to Synolakis [personal communica- tion], the joint proposal written by Professor Philip Liu, Harry Yeh and himself in 1991 had included experiments with the simplified geometry of a solitary wave climbing up a conical island. The objective then was to have high quality measure- ments from an idealized topography to help benchmark tsunami numerical codes. The subsequent experiments run by Professor Synolakis and reported in Briggs et al. (1995), led to numerical computations by Liu et al. (1995), and analytical models by Kânoğlu & Synolakis (1998). They all have demonstrated that long waves can amplify wave runup on the lee side of conical islands in comparison to the runup on the front side of the island. Figure1.3: Longwaverunupontheleeofaconicalisland. Experimentsinspiredby the 1992 Flores tsunami, see Yeh et al. (1994b). Numerical simulations by Titov & Synolakis(1995)-leftsubfigure-picturesfromthelaboratory experiments ofBriggs et al. (1995), on the right. The arrows indicate the direction of the wave attack. This classic set of experiments is still widely used for code validation. During these experiments, runup was not measured in the beach the islands shadows, because it was not know at the time that the coast was also effected. 5 On December 26th 2004, the infamous Boxing Day tsunami (Synolakis & Bernard(2006) andmany others)struckoff BandaAcehin Sumatraand embedded the term tsunami in most world languages. To the date of this writing, it remains the deadliest natural disaster of Sweden, given the number of Swedish citizens who perished while visiting Thailand as tourists. The event greatly raised public awareness of the effect of tsunamis and the perils of improper evacuation plan- ning and inadequate warning systems. Since then, significant advances have been made worldwide in operational warnings due to the occurrence of over eighteen additional metric and decimetre tsunamis. With the possible exception of the Mediterranean, most high–risk coastlines in the world’s oceans are now able to receive a warning message with details about the hazard and in some instances, specifically identifying locales at risk. Ideally, a warning should be distributed minutes after a suspect earthquake and a specific forecast made as to its actual coastal flooding. In this regard, the March 11th 2011 GreatTohokutsunami(Fujiietal.(2011);Ideetal.(2011);Morietal.(2011);Fritz et al. (2012)), possibly the most measured and studied tsunami this far, illustrated that preparation of risk communities remains the only effective countermeasure to save lives. In Banda Aceh, it has been estimated that about 90% of the population in inundated areas died, in Japan only 6 years later, only 10% did so. The 2011 Japan tsunami is also the most widely filmed tsunami in history, as it occurred during the day, in a highly developed country where news media routinely use helicopters for documenting news, not to mention the ubiquitous availability of cellphone cameras. Unfortunately, these videos highlight the fact that in some cases the population of high-risk communities waited too long to evacuate with disastrous outcomes. While such behavior may be understandable for unsuspecting residents in nations with little tsunami education and awareness. 6 Japan has been at the forefront of evacuation planning and was considered to be the most tsunami ready in the world Synolakis & Kânoğlu (2015). Evidently, significant social–science work is required to improve the understanding of the human responses and to better guide outreach efforts. Another aspect which needs not only social–science work, but also basic phys- ical science advances is the understanding of what are otherwise counter intu- itive flow phenomena. Synolakis & Bernard (2006), Synolakis & Kânoğlu (2015), Kânoğlu et al. (2015) have all presented one explanation why sometimes people wait too long to evacuate and some had been filmed posing in front of an advanc- ing tsunami. They had described based on Synolakis (1987) work how a wavefront slows down as it approaches the initial shoreline, but that the shoreline accelerates rapidly once the wave hits it. They claim that victims confuse shoreline motion with wavefront motion. In other words, observers initially see a wave slowing down as it shoals into increasingly shallower water and are led to believe that they can outrun what appears a rapidly decelerating wave. The sudden acceleration of the shoreline and the wave front once it reaches the initial shoreline astonishes even trained scientists. The significance of this finding was not fully comprehended until the 2004 Boxing Day tsunami. Folklore about natural hazards abounds among coastal residents worldwide is fanciful and more based on myth than on science or hard facts. In surveys following the 2009 Samoan tsunami and also the 2011 Japan tsunami in Guam, many eyewitnesses commented that fringing reefs had protected them and proudly showed them off to the scientists surveying the impact. In fact, it is doubtful that reefs do offer substantial protection, and, to this date, the conditions under which they do so are not well understood. On the contrary, reef openings often funnel long wave energy and increase impact on the shoreline across the opening, which 7 is usually where villages are located for easier through navigation of fishermen. This was observed in a follow-up survey in 1995 of the 1992 Nicaraguan event, where there were substantial differences in runup between beaches only distant 2 km from each other; there was an opening in the fringing reef where most of the devastation occurred. The understanding of the effects of reefs was so poor, even in the scientific community, that several studies had been published trying to match the measurements in a locale in front of the reef opening, but without including the reef in the bathymetric grids, basically because its presence was not known before 1995. Another counter intuitive flow pattern is described and studied in this thesis. One would normally expect small offshore islands to protect the shoreline behind them. As it turns out, they sometimes don’t, and in fact they may amplify the tsunamiinundation, atleastascomparedtoadjacentlocationswithouttheoffshore islands. 1.1 The specific motivation for this study - the 2010 Mentawai tsunami On October25th 2010, a tsunamioccurred offthe coastof theMentawai islands of Sumatra in Indonesia spreading death, destruction and huge economic loss at the affected sites. Villages located in the high-risk coastlines, were formally built in the belief that the offshore islands protected them from wind waves, creating a false assurance of safety from destructive long waves. However, due to the large differences in wavelengths, tsunamis behave differently than wind waves and gener- ate substantial annihilation and wreckage. In reality, villages were built in some of 8 the most vulnerable spots along the coastline. The above tsunami provided a criti- cal moment for tsunami research as it increased public understanding of tsunamis, and raised awareness and the need for preparation for communities at-risk. Fur- thermore, the destruction, collected measurements and eyewitness accounts by the International Tsunami Survey Team after this event, prompted increased research and data analysis on the effects of long waves and their associated currents, run-up and devastation in the coastal zones shadowed by offshore islands. Figure 1.4: Simulation of the maximum wave height of the 10/25/2010 Mentawais tsunami. Significant runup was modeled and observed on coastal areas behind small offshore islands. (Hill et al., 2012) Specifically, Stefanakis et al. (2014) investigated whether what was observed in the Mentawais in 2010 is more general and not a one-of-a-kind occurrence that sometimes confounds geoscientists. The experimenters used the nonlinear shallow- water wave (NLSW) solver VOLNA Dutykh et al. (2011a) and assumed idealized 9 topography consisting of a conical island, in a constant depth region, in front of a uniformly sloping beach, as in Figure 1.5. The incoming solitary-like wave had the same crest length as the beach, and without the islands it was first confirmed that the runup did not vary in the alongshore direction. The runup amplification was calculated, i.e., the ratio of the runup on the plane beach in the shadow zone of the island to that at a neighboring location, not shadowed by the island, was computed. Their main objective was to find the maximum runup amplification with the least number of simulations. In their active learning experiment, they varied the island slope, the beach slope, the water depth, the distance between the island and the plane beach and the incoming wavelength, over a wide range of realistic values. Had they used a straightforward approach with each variable taking, for example, one of ten values sequentially, and had they varied each while holding the others constant, they would have needed 100,000 simulations, which is compu- tationally prohibitively expensive at the fine resolution of 2 m used for the runup computations. Figure 1.5: Schematic of the geometry of the experimental setup from Stefanakis et al. (2014). The parameter tanθ i varied from 0.05− 0.2, tanθ b from 0.05− 0.2,d from 0− 5 km,h from 100− 1000 m and the effective "frequency" of the incoming solitary–like wave from 0.01− 0.1 rad/s. 10 As argued by Stefanakis et al. (2014), machine learning allows numerical simu- lations to be of low computations cost, which is particular important as the com- plexity of the model. For example, a typical "blind" grid search over a model space of dimensionn each variable being allowedm values leads tom×n simulations to determine an approximate functional variation. Stefanakis et al. (2014) discussed sampling techniques developed to reduce the number of numerical experiments by finding a representative sample in the input space. These techniques are usually referred to as "experimental design" - e.g. Sacks et al. (1989) - and are static, meaning that the design (sampling) is made initially, before the execution of the experiments, and the selection of the future query points is not guided by the prior numerical results. While all such designed points are queried, at computational cost, static design has been a substantial advancement, compared to the usual mn approach. Yet, static design may fail when a phenomenon is counter intuitive or when there are unexpected ranges of values where "special effects" occur. Adaptive design was used by Santner et al. (2003) and machine–learning algo- rithms have been developed for ”active experimental design”, where already com- puted results are used as a guide to select future query points Gramacy & Lee (2009), and the process then repeated until a set goal is reached. In their proto- col, a statistical model was built and updated as new results arrived. Using the predictions of this statistical model (emulator), future query points are selected according to the objective of the experiment, until it is achieved. Stefanakis et al. (2014) asserted that such dynamic approaches further reduce the computational costs and allow the investigation of larger parameter ranges. Building such emulators has additional advantages, since they can be used instead of the actual simulators, being substantially less computationally demanding to 11 evaluate. If this approach is indeed efficacious, it could possibly offer substan- tial advantages whenever a quick forecast is needed, by cycling through ranges of seismic parameters that are poorly constrained in seismic inversions, although any operational applications would need to be tested extensively and compared to current methods. Depending on the emulator, it is possible to perform sensitivity analyses of the model output to input parameters. Sarri et al. (2012) emulator was apparently the first in the context of tsunami research; they studied landslide- tsunamis on a plane beach, based on the theoretical model of Sammarco & Renzi (2008). Stefanakis et al. (2014) focused on a small solitary island close to shore. To reduce the numbers of simulations, they built an emulator based on Gaussian processes to guide the selection of the query points in a five physical-variable space. Using the newly developed method for Active Experimental Design of Con- tal et al. (2013), their active learning approach reduced substantially the compu- tations required to determine the maximum runup amplification. Stefanakis et al. (2014) also presented metrics for comparison of the performance of different learning strategies. Inalloftheirnumericalexperiments, asguidedbytheactivelearningalgorithm, small islands produced amplification in their shadow zones which was greater than one. In most cases, the islands amplified the run-up in the shadow zone behind them compared to adjacent unshadowed locales, and in the remaining cases the islands made no difference. Therefore, the islands, never provided any additional protection negating the earlier belief of high-risk residents. 12 Figure1.6: Run-upamplification(RA)asafunctionofthewavelengthtotheisland radius (at its base) ratio. J = tanθ b / q H 0 /L 0 Different colors indicate values of the surf similarity parameter (a.k.a. Iribarren number) computed with the beach slope and multiplied with the square root of wave height over wave length at deep water. (Stefanakis et al., 2014). 13 Figure 1.7: Snapshots of the evolution of the free surface elevation measured in meters as the wave passes the island and runs up the beach behind it. The island focusestheamplifiedwaveasitpropagatestowardsthebeach. Thecolorbaronthe right is in logarithmic scale, for visualization. Here, the max run-up amplification behind the island is 1.59. (Stefanakis et al., 2014). Figure 1.7 shows one example of the tsunami evolution around the island. Figure 1.6 presents the computed results for the runup amplification (RA) in terms 14 of the ratio of a measure of the wavelength of the tsunami to the island radius, grouped by color by the so called Irribaren number. It thus appears that, contrary to popular belief and intuition, small islands can act as tsunami lenses focusing energy behind them. Clearly, the entire phe- nomenologyoftsunamiamplificationisevenricherthanenvisionedbyBerry(2007) and Kânoğlu et al. (2014). But is this effect as general as this preliminary study suggested ? 1.2 Motivation and plan of the present study Formally, the most utilized quantitative indicator for tsunami impact is the run-up. Run-up is defined as the elevation of the maximum wave uprush on a surface above still water level. It is a great indicator for comparison purposes, for example when asking the question if offshore islands amplify the runup on the beach behind them, or whether the runup on the front side of the island is greater than the runup behind it. The term runup amplification is used to compare runup on a beach to some other measure of the incoming wave height, for example, it was used for decades to relate the runup measured to the slip Okal & Synolakis (2004). The study discussed in the previous section suggests that long waves amplify wave runup in the shadow zone of islands fronting sloping beaches, instead of reducing it. An earlier laboratory study by Keen & Lynett (2019) described the flow details and included a runup measurements for five different waves and four island geometries, of two offshore islands fronting a sloping beach. Interestingly, the runup profiles measured did not suggest runup amplification, but no such conclusions were drawn, possibly because of the small number of experiments. 15 Early numerical results showed that under no conditions (among those studied) do the offshore islands provide protection from tsunamis to the background shorelines. Similar results will be discussed pertaining to nonlinear long wave amplification in the shadow zone of islands. So, the matter as to whether indeed islands always amplify waves in the beaches they shadow, remains open, in the sense that there hasn’t been a massive effort to check whether the phenomenon persists over the entire parameter ranges of geophysical interest. The previous laboratory experiments of Keen & Lynett (2019) at the National Tsunami Basin at Oregon State University focused on high–end flow visualization and determining time–histories of surface elevation at specific locales. Since this was anyway a follow-on study from that of Keen & Lynett (2019), the focus was on detailed optical measurements (using particle tracking velocimetry - PTV) of the surface elevation and surface velocities for two different geometric combinations of islands. It also includes computations with two numerical codes and comparisons with the PTV measurements with the numerical predictions, so as to gain confi- dence in using the codes to produce estimated of flow variables in a wider range of parameters than those tried in the laboratory. Repeating, the objective was to confirm the machine learning results of Stefanakis et al. (2014), which predicted that under no conditions do the islands provide protection from long waves. This would be the first time that results from machine learning will confirm labora- tory studies pertaining to water waves, where it has been previously challenging to define the nonlinear results. Even the slightest of changes of wave breaking and large differences in coastal amplification could trigger phenomena reminiscent of the butterfly analysis. Normally, laboratory experiments with waves or other free surface water flows, involve measuring time histories of surface elevations with instruments referred to 16 as wave gauges; changes in depth result in capacitance or resistance changes which are recorded. In a complex 2D geometry as utilized in laboratory experiments, and with a finite number of wave gages (for example, Keen & Lynett (2019) used 24), it would be necessary to repeat the experiment so that the gages can be moved to a different location, so that eventually there is sufficient coverage of the entire flow field. Moreover, wave gauges require frequent re-calibration due to electronic drifts. Utilizing optical measurements, the surface was seeded with tracers and an array of cameras which monitor the surface. The tracers provided targets so that particle tracking analysis could occur. If the image analysis was strictly 2D, the imaging processing problems would be tractable. Instead using 3D analy- sis, allowed an experimenter the ability to track the tracers movement in all three spatial directions more reminiscent of how a wave moves into the field of view. Using separate cameras for stereo matching to extract the elevation of the surface, along with the velocities, the technique has been tested to achieve the necessary accuracy of comparing time series of surface elevation from the image tracking with measurements from conventional wave gauges. The advantage is that once the PTV analysis is done, one has the complete flow field for all locations and relevant times. Earlierresults byKalligeris (2017)havesuggestedthat the3D imageprocessing developed is capable of simplifying the measurements and it is used here exten- sively. However, there are limitations, for example when the particles move faster than the intrinsic capabilities of the optical system, which relied on the framing rate of the camera. All of this is discussed in chapters 2 and 3. The former describes the laboratory set up, with two different geometries with one and two 17 offshore islands. Chapter 3 discusses the methodology of the optical measurements - i.e. the PTV methodology. Chapter 3 also compares the analytical results of Kânoğlu & Synolakis (1998) with the edge–detection algorithm developed here, for the runup on the conical island itself. The objective was to validate the edge–detection methodology, which was then used to determine the runup in the two–island configuration. Chapter 4 compares laboratory measurements with numerical predictions from two codes. One is the nonlinear shallow–water wave equation solver MOST (Titov, 1988) while the other is COULWAVE (Kim & Lynett, 2011) a Boussinesq–type equation solver. The codes are tested with a highly nonlinear 0.4 solitary wave interacting with the island–sloping beach geometry. The objective is to identify ranges of parameters where MOST and COULWAVE may differ from the labora- tory measurements - this is a big question, given the vastly different computational requirements of the two codes. If MOST proved adequate for estimating runup amplification, the computational times could be reduced by a factor of ten. Chapter 5 discusses the application of MOST at geophysical scales and includes afurtherbenchmarkingofMOSTwiththeanalyticalasymptoticresultofSynolakis (1987) known as runup law, a simple algebraic expression which relates runup with water depth, beach slope and incoming wave height, for classic solitary waves interacting with a plane beach. The chapter discusses the extension of this result over a wider range of parameters, then proceeds to use MOST to answer the basic question of this thesis, i.e., is runup amplification omnipresent for all reasonable combinationsofgeophysicalparameters, oraretheregeometriesforwhichanisland just offshore a sloping beach provides sheltering. Oneshouldnotforgettheoverarchingobjectiveofallresearchintsunamihydro- dynamics, which is is to help improve tsunami models, and help identify regions 18 where existing models may need more resolution, or they may not produce satisfac- tory projections. Even a decade ago, smaller islands were removed from numerical grids to help stabilize the analysis, under the pretext that the final results for on-land inundation would have been more conservative, as the islands would have protected the coastlines. If this thesis has managed to raise a red flag to avoid such simplistic arguments when human lives are at risk, then this entire five year effort has been worthwhile. 19 Chapter 2 Laboratory Setup The experiments took place at Oregon State University’s O.H. Hinsdale Wave Research Laboratory. Their wave basin is 44 m long, 25.5 m wide with a of 2.1 m. The basin has its own standard coordinate system, which is used for reference in the hundreds of physical simulations, that take place at OSU. In this set of experiments, oneortwoislandsandauniformlyslopingbeacharefeatured, located opposite to the wave generator. The origin of the coordinate system is at the center of wavemaker, (see Figure 2.1). wavemaker 44 m 26.5 m x y Bridge Figure 2.1: Schematic of OSU basin 20 The slope is located across of the wavemaker on the positive side of the x–axis and begins 21.6 m away from the origin. From this point, the basin of the beach (the tow) has a constant slope of 1:10 until the end of the basin at the 44 m mark. The slope is built from metallic plates and the bottom and surrounding walls of the basin are concrete. The instrumentation bridge spans over the width of the basin and is sitting on rails allowing it to easily slide along the basin’s x–axis. 2.1 Cameras and light A substantial part of this study involved optical measurements of surface eleva- tion and water currents in around the islands and on the sloping beach, including maximum runup. To make them possible, a 3D particle tracking technique (PTV) was used as described in Kalligeris (2017), here summarized in chapter 3. The flow was seeded with 8.5 mm hydrophobic neutrally buoyant particles. Stereo photography was necessary to allow the inference of the surface elevation. TwoPanasoniccameras(model: AW-HE60S)wereusedforthestereomatching. Theycanrecordhighdefinition(HD)videoandhavepan,tiltandzoomcapabilities to fine–tune the field of view (FOV). For all runs and positions, video was recorded with HD resolution (1280x720p) at a 59.94 frames per second rate. The two cameras were mounted on the basin’s bridge (Figure 2.2). The FOV overlap was maximized for the 3D PTV experiments to achieve a stereo camera configuration. Thestereoimagepairsallowfortheextractionofspatialcoordinates in all three axes. The cameras were positioned 3.3 m above the water surface level, achieving a detailed resolution at the water surface (4 pixel/cm). Particles with diameter of 8.5 mm were used to track them in time and extract velocities from the water surface. The cameras were moved to different positions to cover the area 21 around the island by either moving the bridge (in the x-direction) or the camera mounting (in the y-direction). Light sources were utilized to increase the shutter speed of the camera and avoid over-exposure of the tracer paths. Six flood lights were used around the FOV, mounted below the bridge (Figure 1.5). Custom-made light diffusers were needed to soften the light intensity and diffuse the light across the image (Figure 2.2). Figure 2.2: The figure on the left represents the schematic of the mounted cameras inrelationtothewatersurface. Theimageontherightillustratestheexperimental set-up using the light diffusers. 2.2 One–island configuration The first configuration involved a single conical island, located near the center of the basin at (x = 19.62 m, y =−0.03 m), Figure 2.1. The island has a base– diameter of 4 m, and a height of 1 m. It was constructed of steel plates creating a cone-shape with a constant slope of 1 : 2, and was screwed to the floor of the basin. The toe of the beach was touching the toe of the island (see Figure 2.3). 22 wavemaker 44 m 26.5 m x y WG1 WG2 WG3 x y Figure 2.3: The experimental set–up for the one island configuration. The lines are contours of every 0.1 m of elevation. The red dashed box is the study area. Figure 2.3 shows a schematic top–view of the basin. The blue circles contour the island and the vertical lines are the contours of the beach slope. The island was painted white for improve the contrast with the surface tracers and∼ 100 ground control points were marked around the island to create ground control points for the camera calibration. 2.2.1 Camera field of view (FOV) Five different stereo matching positions were used with FOV dimensions of 3 m along x–axis and 2 m along y–axis. These positions were scattered around the island(seeFigure2.5). Itwasnecessarytousemultiplepositionstocovertheentire study area in higher resolution. For every position, at least four experimental trials were run, with surface tracers uniformly distributed within the FOV. 23 Figure 2.4: Camera positions for the 3D stereo matching for the first configuration. With the camera positions above, it was also possible to extract the dynamic coastline around the island. Two additional camera positions were used to measure the runup on the sloping beach. 2.2.2 Boundary conditions and water level The boundary conditions in thewave basinconsisted of solid walls on two of the basin’ssides, ametallicbeachslopeandawavemakerontheotherside(Figure2.3). The piston-type wavemaker consists of 29 individual boards with 2 m maximum stroke and 2 m/s maximum velocity. It is capable of creating monochromatic, polychormatic and multi–directional waves, with periods ranging between 0.5 to 10 s (OSU website). The wavemaker motion was optimized to generate a non– breaking solitary wave with an amplitude of 0.14 m and a period of∼ 1.5 s at a water depth of 0.3 m. The wavemaker displacement time-history and the resulting free surface elevation recorded by a wave gauge are shown in Figure 2.5. The wavemaker face stayed in the forward position for the duration of data collection and subsequently returned to its starting position to generate the solitary wave for 24 the next experimental trial. The same boundary conditions were used for all the one-island configuration experiments. 0 0.5 1 1.5 2 2.5 3 3.5 4 -1.5 -1 -0.5 0 Wavemaker Displacement [m] 0 0.5 1 1.5 2 2.5 3 3.5 4 time [s] 0 0.06 0.12 Elevation at Wavemaker [m] Figure 2.5: Wavemaker boundary conditions. Top plot shows the piston displace- ment time history, starting from the fully retreated position and reaching maxi- mum stroke. The bottom plot shows the predicted (shallow water theory) surface elevation at the face of the wavemaker. 2.3 Two–island configuration The second configuration used the cone–shaped island discussed and another frustum–of–a–cone shaped island, (Figure 2.6). The upper island in the figure had a flat top (this was the cone–frustum island), a diameter of 4 m, height of 0.5 m (with a constant slope of 1 : 2), and its center was located at basin coordinates (19.62 m,4.97 m). This island was also firmly mounted on the basin floor with 25 screws. The islands were painted white, with 50 ground control points created along its perimeter for camera calibration. The bottom island is the same as in the one–island configuration. The gap between the two islands was 1 m (at the base of the cones). Both islands have their bases tangent to the toe of the sloping beach. wavemaker 44 m 26.5 m x y WG1 WG2 WG3 x y Figure 2.6: The experimental set-up for the two island configuration. The lines are contours at every 0.1 m of elevation. Red dashed line box is the study area for the 3D PVT analysis. 2.3.1 Camera field of view For the two-island configuration, the stereo cameras were moved in thirteen different positions around the two islands to cover the whole study area (Figure 2.7). The cameras were placed in two additional positions to capture the moving beach shoreline. The FOV of each camera measured 3 m along the x-axis and 2 26 m along the y-axis. Video was recorded for at least four different trials in each camera position. Figure 2.7: Shows the study area for the experimental set-up of the 3D stereo matching for the second configuration. 2.3.2 Boundary conditions and water level In the two-island configuration, instead of forcing a solitary wave through the wavemaker, an N–wave was created (Figure 2.8). This was similar to the N– wave of Keen & Lynett (2019), but their experiments were at 0.5 m depth. The piston displacement time-history consisted of a slow backwards motion and a push forward. This wavemaker displacement created a single asymmetric N-wave pulse with a 20 s period at a 0.3 m water depth. The same boundary conditions were 27 used for all the two–island experimental trials. The wavemaker face stayed in the final forward position for the duration of data collection. 20 25 30 35 40 45 -1 -0.5 0 0.5 1 Wavemaker Displacement [m] 20 25 30 35 40 45 time [s] -0.1 -0.05 0 0.05 0.1 Elevation at Wavemaker [m] Figure 2.8: The wavemaket motion for generating the leading depression N–wave used in the subsequent experiments. The top plot shows the piston displacement time history, starting from the fully retreated position and reaching maximum stroke. The bottom plot shows the predicted (shallow water theory) surface eleva- tion at the face of the wavemaker. The water depth, these trajectories were used, was 30 cm. 2.4 Wave gauges Four wave gauges were used to examine the repeatability of the experiment. One wave gauge was placed on the wavemaker and the other three were located mid way between the island and the wavemaker at basin coordinates (9.5, -0.02), 28 (8.3, 2.5) and (9.5, 4.8) (Figure 2.3 & Figure 2.6). The gauges were mounted on the basin floor and were connected to the DAQ system via wires. The gauges were used in all experimental trials. The wave gauges used were of resistant–type. Electric current is applied through wires forming a resistor, the load of which depends on the level of immer- sion in the conducting fluid, in a classic Wheatstone bridge configuration. Prior to the experimental set-up, a simple calibration was performed to provide the relationship between voltage and immersion level. Voltage measurements were collected for 1 min at a 50 Hz sampling frequency, at each calibration elevation. 29 Figure 2.9: On the top image one island configuration, on the bottom image two islands configuration (Credit: Dr. Costas Synolakis). 30 Chapter 3 Post-Processing of Laboratory Data In this chapter, the post–processing steps and the laboratory results for the one-island and two island configurations are discussed. Also, for reference, the measurements are compared with the analytical results of Kânoğlu & Synolakis (1998) for the runup on a conical island. As a preamble, the measurements involved interpretation of stereo–matched pairs of videos with multiple surface tracers, to infer water surface elevation and velocities. 3.1 Image pre-processing To identify the surface tracers in the recorded videos, it was necessary to first extract the still frames. The still images were subsequently post–processed to mask irrelevant objects out (such as parts of the instrument carriage or a careless inves- tigator’s hand) and enhance the appearance of the particles. The final step was to synchronize the stereo videos and reference the starting time to the wavemaker motion. For the second step, a mask was applied for the objects that were not needed inside the frame (e.g. still objects, or the dry part of an island). Finally, the image was transformed from color to gray scale (Figure 3.1). 31 Every different stereo position needed to be fine tuned differently. This was necessary due to variations in the light and the varying positions, therefore the gray–scale intensity varied. The gray–scale intensity was needed for the identi- fication of the particles inside the frame of the image. The particles were black and the island background was white. Therefore, using a gray–scale image easily separated the background color from the particles. Two different methods were applied to reference the video recording time to the wavemaker time–history. The time at which video recording started was saved in the video meta–data files. The wavemaker displacement time–history has its own meta–data file, making it easy to match the recording time with the wavemaker displacementtime–history. Thesecond(andmoreaccurate)methodtosynchronize the frames, this utilized an LED light that was facing the stereo cameras (Figure 3.1), and the LED was blinking with a known frequency. Matching the blinking sequence extracted from the video to the one prescribed allowed to reference the video recording time to the DAQ time. Referencing the video recording time to the DAQ time also served to synchronize the two cameras. The two cameras were recording in parallel, but started at different times. One of the two videos was used as a time reference, and the frame numbers of the other were adjusted accordingly, so that the pairs of individual video frames were properly matched. 32 Figure 3.1: On the top left figure is the original frame, on the bottom right is the LED light, on the right top figure is the transformation to a gray scale and on the bottom right is the gray scale flipping the colors. 3.2 Particle identification This section describes the methodology to identify the tracer centers in each frame. The method of Crocker & Grier (1996) employed here is an automated par- ticle identification technique and was also used at OSU in the context of studying large–scale coherent structures evolving around a breakwater by Kalligeris (2017). The user needs to specify the particle diameter and its geometrical characteristics so as to distinguish the tracer from background noise. Among the other geometri- cal characteristics, the radius of gyration is a useful parameter to detect clumping of particles which were rejected from the detection process. Figure 3.2 shows with outlined yellow circles the accepted particles and with red dots the rejected (clumped) particles. 33 This method works very well when the flow speed is less than 1 m/s. During higher flow speeds, the tracers captured in the still frames transformed from a circular shape towards an ellipsoidal shape. When the shape was not uniformly circular, themethodwasunabletoidentifytheparticle. Thisphenomenonhappens because the camera shutter speed is not high enough to account for the high flow speed. In the one-island configuration experiments, the flow was slow enough for the automated algorithm to be able to identify the particles. Figure 3.2: An example of particle identification: yellow circles denote the identi- fied particles and the red dots the clumped particles that were rejected. 3.3 Particle Tracking Velocimetry (PTV) The particle tracking algorithm of Crocker & Grier (1996) as implemented by Kalligeris (2017) was used to track the particles between successive video frames. To speed-up the processes, inter–frame matching is sought within a search radius from the last known tracer position, which represents the maximum distance a tracer is expected to travel between successive frames. For this to work, the mean 34 inter–particle distance in the frame needs to be large compared to the inter–frame displacement. For a given inter–particle spacing, the tracking becomes more diffi- cult as the flow speed increases. In other words, for faster flows, one needs more frames per second. For the flow speed considered in the one-island experiment, the algorithm was largely able to track the particles, except where the wave was breaking on the side of the island. The video–frame rate turned out to be a severe limitation in the two island case, particularly in the high–speed region between the islands. In each experimental trial, the particles were identified and traced in both cameras. The tracer paths referenced in image coordinates were stored for further processing. To convert the tracer paths from image to world coordinates, and thus extract surface velocities in physical units, the imaging geometry needs to be parameterized. This is the subject of the next section. 3.4 Camera parametrization This section describes the steps to extract the camera intrinsic and the extrinsic parameters. The extrinsic parameters define the location and the field of view of the camera, whereas the intrinsic correspond to the properties of the lens and the camera. 3.4.1 Intrinsic parametrization In order to extract quantitative data from the video frames, it was necessary to take into count and remove the lens distortion - most camera lenses create a distor- tion to the image. The intrinsic parameterization procedure involves photograph- ing a flat checkerboard of known dimensions from various angles and positions. 35 The Bouguet (2015) solver uses these checkerboard images to obtain both radial and tangential distortion coefficients, allowing to remove the distortion from the still frames. This process was repeated for all zoom levels used in the experiments. 3.4.2 Extrinsic parametrization This section discusses imaging geometry, the use of the calibration table, direct linear transformation coefficients and ray intersection and matching algorithms. Image-to-world coordinate transformation Using the intrinsic parameterization from section 3.4.1, it is possible to remove the distortion from the still frames, and thus obtain image coordinates in the so- called pinhole camera model. Objects with world coordinates (x,y,z), relative to an arbitrary 3D Cartesian system, can be transformed to image coordinates (u,v) using the so–called Direct Linear Transformation (DLT) equations: u = L 1 x +L 2 y +L 3 z +L 4 L 9 x +L 10 y +L 11 z + 1 v = L 5 x +L 6 y +L 7 z +L 8 L 9 x +L 10 y +L 11 z + 1 (3.1) whereL 1 ,L 2 ,...,L 11 are the DLT coefficients. The coefficients can be determined from ground control points (GCPs for which both the image and world coordinates are known) using the least-square method. The GCPs were defined on a mobile calibration table which is described in the next section. Alternatively, image coordinates (u,v) can be transformed to world coordinates using the 3D ray parametric equation (Douxchamps et al., 2005) α        u v 1        =A        x y z        +b, (3.2) 36 where matrix A and vector b contain constants similar to the DLT coefficients which can also be estimated using a least-square fit and utilizing GCPs. The dif- ference between the DTL Equation (3.1) and the 3D parametric Equation (3.2) is the free parameter α, which allows to define the image-to-world coordinate trans- formation along an optical ray beaming through the camera lens. Both systems of Equations (3.1) and (3.2) are underdetermined for the inverse problem of solving for the basin coordinates (x,y,z) (two equations with three unknowns). The inverse transformation is only possible if the elevationz is known (i.e. the object is positioned on a known surface), or two cameras are used in stereo configuration. For two cameras, Equation 3.1 becomes (Holland et al., 1997).            (L L 1 −L L 2 u L ) (L L 2 −L L 10 u L ) (L L 3 −L L 1 u L ) (L L 5 −L L 9 v L ) (L L 6 −L L 10 v L ) (L L 7 −L L 11 v L ) (L R 1 −L R 2 u R ) (L R 2 −L R 10 u R ) (L R 3 −L R 1 v R ) (L R 5 −L R 9 v R ) (L R 6 −L R 10 v R ) (L R 7 −L R 11 v R )                   x y z        =            (u L −L L 4 ) (v L −L L 8 ) (u R −L R 4 ) (v R −L R 8 )            , (3.3) where superscripts L,R correspond to the two cameras (L = left, R = right). In order to use this system of equations for the current application, the image coordinates of a surface tracer need to be matched between the two cameras. This is achieved through the ray intersection and particle matching method proposed by Douxchamps et al. (2005), which is explained in the next section. Ray intersection and matching Every field of view has a set of two cameras, a camera on the left (V =L) and a camera on the right(V =R). R (V ) = (u (V ) ,v (V ) ) is associated with each camera view point. Transformation can be modeled as a perspective projection according 37 to Faugeras et al. (2001). Each viewpoint consists of a matrix A (V ) and vector b (V ) defined as r (V ) (α) =p (V ) +αq (V ) , (3.4) wherep (V ) andq (V ) are the position of the projection center and the ray direction, respectively, defined as p (V ) =−(A (V ) ) −1 b (V ) q (V ) =−(A (V ) ) −1        u (V ) v (V ) 1        . (3.5) R (L) i andR (R) j aretheimagecoordinatesfortwocandidatesofastereoscopicmatch. Equation 3.4 is a parametric equation that gives the corresponding rays (passing through the camera lens and the candidate surface tracers): r (L) i (α) =p (L) i +αq (L) i r (R) j (β) =p (R) j +βq (R) j , (3.6) where β is the free ray free parameter for the second camera. With reference to Figure 3.3, minimizing the distance between the two camera rays translates to solving for the free parameters α and β through the system of equations given by    q (L) i ·q (L) i −q (L) i ·q (R) j q (R) j ·q (L) i −q (R) j ·q (R) j       α β    =    q (L) i · (p (R) j −p (L) i ) q (R) j · (p (R) j −p (L) i )   . (3.7) 38 Figure 3.3: Intersection of two rays associated with the left and right camera viewpoints (Douxchamps et al., 2005). (X,Y ) in the figure correspond to the image coordinates (u,v) in the notation used here. The midpoint r ij and distance d ij encountered between the two points of min- imum are given by: r ij = 1 2 r (L) i (α) +r (R) j (β) d ij = r (R) i (β) +r (L) j (α) . (3.8) To find the optimum match between the surface tracers identified in each cam- era still frame, the goal becomes to minimize the global penalty P i d ij . A method to minimize the global penalty is Vogel’s approximation, which assigns the highest order of matching to the particles that are more likely to match. Once the match- ing particle between the two cameras is obtained, the system of Equations (3.3) was used to obtain the basin coordinates of the surface tracers detected from the cameras. 39 Theaccuracyoftheimage-to-basincoordinatetransformationforthetwostereo camera configuration can be found by computing the mean distance between the real GCP coordinates (x,y,z) and the predicted (x p ,y p ,z p ), i.e. r = 1 N N X i=1 q (x p (i)−x(i)) 2 + (y p (i)−y(i)) 2 + (z p (i)−z(i)) 2 (3.9) where N is the total number of GCPs that is used for the extrinsic parameteriza- tion. The mean error from all the runs for the first configuration was found to be r−xy = 4 mm for the horizontal and r−z = 1.5 mm for vertical. Calibration Table The calibration table featured 15x7 (105) GCPs at two different elevations (Figure 3.4). The first elevation was on the water level in the form of a square– celled grid painted on the board, and the upper GCPs were∼ 10 cm above the still water level marked on 4x4’s. Each time the cameras were re–positioned, the calibration table was placed in the basin, centered in the camera FOVs, and its four corner coordinates were measured using a total station. The four corner reference coordinates were utilized to obtain the absolute basin coordinates of all the GCPs using a rigid body coordinate transformation. If the island was physically located too close to the calibration table and compromised the FOV, GCPs from the island were used to fill the gap in the FOV. 40 Figure3.4: Tablewithcontrolpointstodeterminethecameraextrinsicparameters. 3.5 3D surface velocity extraction, with one island in place. The paths of the tracers (from 2D PTV, see Section 3.3) that were matched through the ray intersection method were used to extract 3D surface velocities. Since the tracer matching is now known, the image coordinates of the two cam- era paths for each surface tracer was converted to world coordinates using Equa- tion (3.3). The velocities in all three directions were then computed using the backward finite difference scheme. As an example, Figure 3.5 shows the surface velocity vectors extracted at t = 10.5 s. 41 Figure 3.5: Figure on the top shows the scattered data – Figure on the bottom shows interpolation from the scattered data at the same time, which is 10.51 s from the start of the wavemaker. Figure 3.6 shows velocities (u,v) along x and y extracted at the locations around the island shown in Figure 3.5. The continuous curves are a result of linear spatial interpolation of the scattered PTV velocity vectors. With the exception of location #5, and the first peak of the (u,v) velocities in location #4, spatial interpolation yields robust results. Location #5 is right behind the island, where the wave arrives from the two sides and the fronts collide at the location of the numerical gauge. Thus, the noise in the velocity time-series can be attributed to the high spatial variability of the velocity vectors in the particular region. 42 5 101520 -0.5 0 0.5 1 1 u m/s v m/s 5 101520 -0.5 0 0.5 1 2 u m/s v m/s 5 101520 -0.5 0 0.5 1 velocity [m/s] 3 u m/s v m/s 5 101520 -0.5 0 0.5 1 4 u m/s v m/s 5 101520 time [sec] -0.5 0 0.5 1 5 u m/s v m/s 5 101520 time [sec] -0.5 0 0.5 1 6 u m/s v m/s Figure 3.6: Figure shows velocitiesu,v in different position around the island (see Figure 3.5). 3.6 3D surface elevation extraction one island The stereoscopically-matched tracers were used to extract surface elevations through Equation (3.3), it is important to note that surface elevation extraction doesn’t require tracking the particles between successive frames. As an example, Figure 3.7 shows a 3D surface elevation map for the one-island configuration at t = 10.5 s. The stereoscopic particle matching algorithm was mostly successful 43 exceptinsomeareasofhigh-speedflowwhereitwasdifficulttoidentifytheparticle centers. Figure 3.7: 3D view of free surface elevation as the solitary wave goes past the side of the island at t = 10.5 s from wavemaker start. Figure 3.8 shows time-series of free surface elevation at different locations around the island. The curves were created by linear interpolation of the scat- tered tracer elevations. Locations #1 and #2 are placed on the wavemaker side of the island, and thus pick up the wave front before it interacts with the island. Location #3 is located on the side of the island (along the center-line of the island with respect to the basin y-axis), and locations #4, #5 and #6 are behind the island. The amplitude of the leading wave in the free surface elevation time-series 5-6 is higher compared to the time-series in locations 1-3, partly due to the shal- lower water depth, but also due to the convergence of the fronts of the waves wrapping around the island. 44 010 20 -0.1 0 0.1 0.2 1 010 20 -0.1 0 0.1 0.2 2 010 20 -0.1 0 0.1 0.2 [m] 3 010 20 -0.1 0 0.1 0.2 4 010 20 time [sec] -0.1 0 0.1 0.2 5 010 20 time [sec] -0.1 0 0.1 0.2 6 Figure 3.8: Free surface elevation time-series at different locations around the conical island. 45 3.7 Edge detection The shorelines on the conical island and on the sloping beach behind it were extracted from the video still frames using the edge detection image processing methodology. This technique identifies the sharp edge in image intensity at the interface between the edge of the water and the island or beach. Since the interface islocatedonaknownsurface,itispossibletoextractthetwohorizontalcoordinates of the shoreline from a single camera. The shoreline data are subsequently used to find the maximum runup and compute the shoreline velocity. 3.7.1 The coastline around the island To convert the island shoreline extracted through edge detection to world coor- dinates, the image coordinates from the single camera have to be mapped to the surface. For this particular surface geometry, it is convenient to switch to a polar coordinate system (r,θ) x =r cos(θ), y =r sin(θ), z = (2−r)/2, (3.10) where r is the radial distance from the center of the island, and θ is the azimuth around the cone. The island is symmetric with a radius of 2 m and a height of 1 m and the slope of the island is constant. Cartesian coordinates (x,y,z) in the co–linearity Equation (3.1) are substituted with (r,θ) through Equation (3.10), which results in a system of two equations with two unknowns u = L 1 r cos(θ) +L 2 r sin(θ) +L 3 (2−r)/2 +L 4 L 9 r cos(θ) +L 10 r sin(θ) +L 11 (2−r)/2 + 1 , v = L 5 r cos(θ) +L 6 r sin(θ) +L 7 (2−r)/2 +L 8 L 9 r cos(θ) +L 10 r sin(θ) +L 11 (2−r)/2 + 1 . (3.11) 46 The system of equations does not give an exact correspondence between image and world coordinates, but can be solved using a least-square method. The limitation of this methodology is that the mapping surface needs to be perfectly symmetric to be described in that fashion. Figure 3.9: An example of shoreline extraction using edge detection for a frame in position 1. The yellow line corresponds to the water-surface interface detected through the step in image intensity. This technique was applied to all positions using a single camera. It is impor- tant to capture the run-up since the particles did not reach the shoreline. The run-up velocity profile in the center of the island is shown in Figure 3.10. For the run–down is was not possible to extract the shoreline using edge detection, because a very thin layer of water stays on the island slope leaving no contrast with the underlying surface. 47 -2 -1.5 -1 -0.5 0 x*/d 0 1 2 3 4 5 6 Figure 3.10: Runup velocity of at the center of the island. On the abscissa the dimensionlessnumberx ∗ /disused, wheredisthedepthandx∗isthedimensionless length scale. The time in the ordinate is non-dimensionalized using the constant depth d. 3.7.2 Analytical solutions for runup on a conical island The only existing analytical solution for runup on a conical island is that of Kânoğlu & Synolakis (1998). They considered first waves propagating over con- stant depth h 1 in polar coordinates and derived the linear theory solution, as per below. η(r,θ,t) = +∞ X n=−∞ n A n J n (kr) +B n Y n (kr) o e i(nθ−ωt) , (3.12) whereJ n is the first kind of Bessel functions, Y n is the second kind of Bessel func- tions, k =ω/h 1/2 1 , h 1 = is the constant depth over which waves are propagating. They then considered a pillbox (sill) of radius R whose upper surface was at depthh 2 . Theythenmatchedthefreesurfaceelevationandthesurfaceslopeatthe edge of the pillbox, and they obtained the solution for axisymmetric surface waves 48 propagating over the topography of a pillbox sitting on top of a constant depth region. They repeated this procedure to find the solution of an approximation of the conical island which looked like a wedding cake. As they wrote, given that the solutionisknownforevolutionovereachsill, solvingtheentireproblemformultiple co-axial sills of decreasing radius involves matching solutions at the interface of the sills. They eventually derived this solution for the shoreline motion fro the run–up around the conical island, as follows R(t) = (4/3) π √ ab Z ∞ −∞ cosech(αω) ∞ X m=−∞ e iΘ H (1)0 m (ωb) dω, (3.13) where, b is the radius of the base of the island, a is the radius of the island at the height of the water level, α = π/2 q 3H/4, with H the wave height, Θ = mθ+ω(x s −t)+(−m+1)π/2, andH (1) m is the Hankel functions of the first kind, and of course,H (1)0 m is its derivative. Kânoğlu & Synolakis (1998) evaluated this integral by first deriving its Laurent expansion, and then using asymptotic approximations for the Hankel functions. Here, a numerical evaluation of the integral was done, to avoid asymptotics, which while they worked well for the cases discussed in the 1998 paper, they may not work equally well for the parameter ranges used here. d dω H (1) m (ω) = mH (1) m (ωb) ω −H (1) m+1 (ωb). (3.14) 49 Figure 3.11: Maximum runup extracted from edge detection compared to the analytical prediction. Figure 3.11 shows a comparison of the the analytical solution results and with the measurements derived using edge detection. However, edge detection shows higher runup on the back side of the island than the linear theory predicts. This is possibly due to the collision of the two waves which split as the solitary wave starts its interaction with the island. 3.7.3 The coastline on the sloping beach Using a single camera, the coastline was extracted from the edge detection. The LiDAR data of the beach slope (x,y,z) was used and converted to the image coordinates (u,v) using the Equation 3.1. So utilizing this method, the image was rectified in basin coordinates and the run-up of the sloping beach was extracted for different time and positions. 50 Figure3.12: Anexampleofshorelineextractionusingedgedetectionforthesloping beach. The yellow line corresponds to the water-surface interface detected through the step in image intensity. -12 -10 -8 -6 -4 -2 0 x*/d 0 10 20 Figure 3.13: Runup velocity of at the center of the sloping beach. On the abscissa the dimensionless numberx ∗ /d is used, whered is the depth andx∗ is the dimen- sionless length scale. The time in the ordinate is non-dimensionalized using the constant depth d. Comparison with analytical model Figure 3.15 shows the maximum runup on the beach slope. The black line is the edge detection while the points are extrapolated from the total station points. The comparison shows good result of the to points of maximum runup. Using 51 Figure 3.14 maximum run-up of a solitary wave as a function of slope angle for breaking waves for H/h o = 0.4 showed with blue dashed line. Figure 3.14: Maximum run-up of a solitary wave as a function of slope angle for non-breaking and breaking waves for several incident relative wave heights (Li & Raichlen, 2002). 52 Figure 3.15: Shows the maximum run-up of the sloping beach with the black edge detectionmethodincomparisontotheredcrossesdepictingthetotalstationpoints in comparison to the analytical model (with a constant slope without an island) shown by the blue dashed line. Black dashed line is the shoreline constant depth 30 cm. 53 3.8 3D surface elevation extraction for the two island case Similar with the one island configuration were used stereoscopically–matched tracers to extract surface elevations through Equation (3.3). Figure 3.16 shows time-series of free surface elevation at different locations around the island. The curves were created by linear interpolation of the scattered tracer elevations. Locations #1, #2, #3, #7 are placed on the wave maker side of the island, and thus pick up the wave front before it interacts with the island. Location #4 and #8 are located on the sides of the island (along the center-line of the island with respect to the basin y-axis), and locations #5, #6 and #9 are behind the island. 54 Figure 3.16: Free surface elevation time-series at different locations around the two conical island. 55 Figure 3.17: Shows the maximum run-up of the sloping beach. Laboratory data recorded using total station. We were unable to extract water particle velocities for the particular combi- nation of parameters, because the detection threshold is∼ 1 m/s. When the experiments were planned, there was no benefit from numerical results, and the large water particle velocities observed between the islands were not expected. Also, with hindsight, there was undue expectation that the algorithms that had worked so well for the single island would work equally well for the two islands. 56 12 34 56 Figure 3.18: Two island configurations (Credit: Dr. Pedro Lomonaco). 57 Chapter 4 Numerical Models at Laboratory Scales 4.1 MOST 4.1.1 The history of MOST MOST is a suite of codes solving the 2+1D shallow–water equations. Its evo- lution and history is discussed in Titov et al. (2016). It is based on the splitting method, also known as the method of fractional steps, (Godunov, 1973). Dimen- sional splitting produces two 1+1D hyperbolic systems, one in each propagation direction. Each 1+1D problem is then solved with the method of characteristics, andtheearlier1+2DrealizationofMOSTisdescribedinTitov&Synolakis(1995). The 2+1D code was developed at USC where inundation computations onto dry land were added to what was a rudimentary 1+1D propagation code. Per the description in papers too numerous to individually cite, but highlighted first by Synolakis (1987) and then by Liu et al. (1991), this calculating runup is a vexing problem with moving and deforming boundaries, where specifying adequate mathematical conditions is tricky. Earlier computations of the climb of a bore on a beach by Hibberd & Peregrine (1979) used ad hoc algorithms, as discussed by Synolakis (1986). 58 The resulting shoreline algorithm of Titov & Synolakis (1995) has the ability to model the run-up of mildly breaking waves. These are simulated in MOST as bore–like fronts, only using the inherent dissipation of energy that finite-difference schemesprovide; themereprocessofnumericaldifferentiationofthedifferentterms in the equations of motion appears to dissipate energy. This energy dissipation seems not to affect the preservation of momentum and mass of the NSW. Titov & Synolakis (1995) commented that “why these simple equations [referring to the depth-averaged shallow water-wave equations] can even model some details of breaking is still quite puzzling.” These observations were recently rediscovered by Couston et al. (2015). Titov&Synolakis(1997)usedanearlierversionofMOST(referredtoasVTCS- 3) to model a large-scale 2+1D laboratory experiment with solitary waves evolving around a conical island. The basin was 30 m wide and 25 m long, and the island diameteratitsbasewas 7.2m, withaslopeof 1 : 4. Theexperimentswererunat 32 cm and 42 cm depths. According to Costas Synolakis (personal communication) the experiments had been planned in an NSF proposal submitted in 1991, but they were completed right after the 1992 Flores tsunami (Yeh et al., 1994a). This tsunamihitFloresinNusaTengara, Indonesiaandkilledanestimated2200persons and left 40,000 homeless (Wikipedia). The event is best known for the high run-up on the back side of Babi, a small island off Flores Synolakis & Okal (2005). In the laboratoryexperiments, oncethesolitarywaveshitthefrontsideoftheisland, they split into two fronts, moving around with their crests nearly in radial directions before finally colliding behind it, creating spectacular interference patterns and counter intuitive high run-up, which was similar to that observed in the field (Liu et al. (1995); Briggs et al. (1995); Kânoğlu & Synolakis (1998)). As Titov et al. (2016) argued, comparisons of VTSC–3 results with the laboratory data not only 59 showedgoodagreementinfrontoftheislandbutalsobehindit, wherethetwowave fronts collided, for both time histories of surface elevations and for the maximum run-up. The 1992 island laboratory experiment did not feature a sloping beach behind the island, instead there was horse–hair to absorb the waves passing the island. Hence, the opportunity to observe the enhanced runup on the beach shadowed by the island was lost. Such enhanced runup was observed during the 2010 Mentawai tsunami(Hilletal.,2012). ThelattertriggeredthestudyofStefanakisetal.(2014) who used active learning to understand whether there were any conditions under which the island provided sheltering on the beach behind it, instead of enhancing the runup. More details are explained in the introduction section 1.1 and in Figure 1.5. 4.1.2 Simulation of the one–island laboratory experiment using MOST This subsection discusses the laboratory experiment with one–island as described in chapter 2, which is of course different than the 1992 experiment described in the previous section. For starters, refer to Figure 4.1, which shows six aerial views of an initial 0.8 solitary wave propagating towards the conical island. The wave splits into two waves, as seen in the subfigure with t = 9.94 sec, long before it interacts with the island, which is approximately at this time. This splitting of the initial condition in one dimensional propagation is a stan- dard feature of 1D classic wave equations, and can be explained by D’Alembert’s solution – see Butkov (1973), p. 594. In D’Alembert’s solution, an initial condition of the form η(x,t = 0) = f(x) with ∂η/∂t(x,t = 0) = 0 for a wave equation over constant depth in terms of η(x,t) results into the solution 60 η(x,t) = 1/2f(x−ct) + 1/2f(x +ct). So, to obtain a travelling solitary wave with a given H/d, one has to assign a solitary wave with 2 H/d, then wait for the wave to split up. This can be seen in Figure 4.2, which shows profiles along the centerline of the numerical basin, for different times. The 0.4 wave breaks, hence the sequence of crests seen fort> 10 sec. (Note again that in, this simulation, the initial velocity is zero. In the particular figure, one sees the wave moving to the left, that is towards the wave generator, while the wave moving towards the right interacts with the island, hence this kind of comparison of profiles would be less useful. This interaction is described below. Returning to Figure 4.1, the wave evolves, and byt = 11.19 sec the wave crests are joining again, and by timet = 12.44 sec, a singe–crest wave is seen to approach the island, with a stem just behind it. Subsequent sub figures corresponding to later times show the enhanced runup, which is more visible in the online PDF than on the printed version. Figure 4.3 shows comparisons of free surface elevations between PTV–extracted free-surface elevations as described in chapter 3, with the locations of the gauges shown in 3.8. Locations 1 through 3 are on the front, locations 4 through 6 behind the island, and locations 5 and 6 are along the symmetry axis in the along stream direction. MOST underpredicts the laboratory realization of the experiment, likely because, in MOST, the wave impinging on the island is not a 0.4 solitary wave, but a breaking front, as shown in Figure 4.2. The comparisons between the PTV– extracted laboratory measurements for the u and v horizontal particle velocities are shown in Figure 4.4. Again, MOST with zero initial velocity is seen to under- predict the actual velocities in locations 4 through 6, behind the island. The other comparisons are adequate. 61 While it would have been useful to try MOST with nonzero initial velocity. here, a higher order model is needed to better model the measurements, and this will be explored in the next section. 62 Figure 4.1: Snapshots of the top views of the flow evolution of a 0.4 solitary wave interacting with a conical island and a sloping beach behind it, for the geometrical parameters in section 2.2. Computations using MOST. 63 8 10121416182022 0 10 20 time = 8.71 sec 8 10121416182022 0 10 20 time = 9.46 sec 8 10121416182022 0 10 20 [cm] time = 10.21 sec 8 10121416182022 0 10 20 time = 10.96 sec 8 10121416182022 x [m] 0 10 20 time = 11.71 sec Figure 4.2: An≈ 0.4 Solitary wave travelling over constant depth, using MOST, but with zero initial velocity. The initial wave is a 0.8 wave splits into two waves, each with≈ 0.4 wave height, which as expected in 1+1D propagation, per D’Alembert’s solution. The water depth is 30 cm. The wave breaks almost immediately after the motion starts. 64 010 20 -0.1 0 0.1 0.2 1 010 20 -0.1 0 0.1 0.2 2 010 20 -0.1 0 0.1 0.2 [m] 3 010 20 -0.1 0 0.1 0.2 4 010 20 time [sec] -0.1 0 0.1 0.2 5 010 20 time [sec] -0.1 0 0.1 0.2 6 Figure 4.3: Comparison of the PTV-extracted free surface elevation with the MOST predictions. Black line corresponds to the laboratory data, and the red dashed line to MOST. See Figure 3.8 for the locations. 65 0 5 10 15 20 25 -1 0 1 1 0 5 10 15 20 25 -1 0 1 2 0 5 10 15 20 25 -1 0 1 velocity u [m/s] 3 0 5 10 15 20 25 -1 0 1 4 0 5 10 15 20 25 -1 0 1 5 0 5 10 15 20 25 -1 0 1 6 0 5 10 15 20 25 -0.2 0 0.2 1 0 5 10 15 20 25 -0.2 0 0.2 2 0 5 10 15 20 25 -0.2 0 0.2 velocity v [m/s] 3 0 5 10 15 20 25 -0.2 0 0.2 4 0 5 10 15 20 25 time [sec] -0.2 0 0.2 5 0 5 10 15 20 25 time [sec] -0.2 0 0.2 6 Figure 4.4: Comparison of the PTV-extracted u− and v− velocities with the MOST predictions. Black line corresponds to the laboratory data, and the red dashed line to MOST. See Figure 3.8 for the locations. 66 4.2 Modeling the laboratory experiments with COULWAVE 4.2.1 The particular version of the Boussinesq equations used in COULWAVE As discussed, a good question is whether the less than optimal fits of the laboratory measurements with the MOST predictions are due to the fact that the initial wave is specified with no initial velocity, or whether a code solving a higher– order approximation of the Navier–Stokes equations is needed. The discussion in Synolakis & Kânoğlu (2015) not withstanding, Boussinesq–type models are the most accessible alternative to NSW equation solvers. In this section, the laboratory measurements are compared with predic- tions from the Boussinesq-type numerical model COULWAVE (Lynett & Liu, 2004). The model solves the fully-nonlinear, weakly dispersive, weakly rotational Boussinesq-type equations through a finite volume solver on a regular grid, of Kim et al. (2009) and Kim & Lynett (2011), which in conservative form are given by (Lynett et al., 2012) ∂η ∂t +∇· [H~ u] +E DISP +E VISC = 0, (4.1) ∂~ u ∂t +~ u·∇~ u +g∇η + ~ F DISP + ~ F VISC + ~ M HOR + ~ M VERT + τ ρH = 0. (4.2) Equation 4.1 is the conservation of mass, where η is the surface elevation, and the E DISP and E VISC terms represent 2nd–order corrections to the basic shallow 67 water model for dispersion and viscous diffusion, respectively. Equation 4.2 is the momentum equation in vector form, where ~ u is the reference horizontal velocity vector,H is water depth,g is gravitational acceleration,∇ is the horizontal gradi- ent operator, t is time, τ is the bottom stress and ρ is the fluid density. Equation 4.2 has 2nd order forcing ( ~ F) and mixing ( ~ M) terms; subscript "VERT" stands for vertical and subscript "HOR" for horizontal. Wave energy is dissipated through bottom friction and through wave breaking, which is modeled using a transport- based breaking model. 4.2.2 Comparison of laboratory measurements with COULWAVE results – One island For the particular realization of COULWAVE, the free surface elevation time– series recorded in the laboratory at (x = 8.27,y = 2.45) m at 30 cm depth was used as surface forcing, alongx = 8.27 m inside the model domain to initialize the code. Reflective and absorbing boundary conditions were used for the side walls and along the wavemaker face, respectively, while wave runup was simulated on the beach and conical island. Figure4.5showsthecomparisonbetweentheexperimentalandmodel-predicted free– surface elevation time–series at six different locations, identified in Figure 3.8. The comparisons are excellent, even for this highly nonlinear 0.4 solitary wave. Figure 4.6 show the comparison between predictions and measurements for the u− and v−velocities. The data match well for the leading elevation wave, providing confidence in the combined numerical scheme/laboratory experiments methodology. The time-series deviate after the leading wave at locations 4 to 6; recall that location 5 is at the intersection of the centerline axis of the island and the toe of the sloping beach, 4 is located further alongshore the toe, while 68 6 is further up along the centerline, about 1/2 up the beach. The complexity of the flow just behind the island would likely require the use of a depth-resolving numerical model to better capture the flow details. However, overall, COULWAVE produces better–than–just–satisfactory predictions for the velocities. Figure 4.7 shows a top view of the conical island for an H/d = 0.4 solitary wave and compares analytical (green line), numerical (violet line) predictions with the laboratory measurements, obtained using edge detection, as described in sec- tion 3.7. Surprisingly, the analytical predictions of Kânoğlu & Synolakis (1998) –described in section 3.7.1 predict the maximum runup about as well as COUL- WAVE. Recall that the analytical prediction is based on an idealized topographic model where the island is represented by four concentric cylinders, of decreasing diameter, i.e, it is modeled as a wedding cake. On the back of the island, there is less runup than on the front, because this is a case where the waves breaks, which makes the fit with the linear theory even more surprising indeed. Figure 4.8 compares the maximum runup along the beach behind the island inferredfromthelaboratorymeasurements(edgedetection), measurementsusinga total station, predictions from COULWAVE and the breaking-solitary wave model of Li & Raichlen (2002) which is based on Synolakis (1987). Figure on the top is the runup profile along x and figure on the bottom is the runup along z. Snapshots of the top views of the flow evolution for the same solitary wave with amplitude of 0.12 m interacting with a conical island and a sloping beach behind it, for the geometrical parameters in section 2.2, these snapshots are shown in figure 4.10. The results were derived using COOLWAVE. The snapshots are provided every 1.5 sec and show how the initial wave-front splits into two waves (t = 14.59 sec) and then climbs up on the beach. In this particular case, there is 69 runup amplification, as the maximum runup just behind the island is higher than the runup alongshore in regions not sheltered by the island. 010 20 -0.1 0 0.1 0.2 1 010 20 -0.1 0 0.1 0.2 2 010 20 -0.1 0 0.1 0.2 [m] 3 010 20 -0.1 0 0.1 0.2 4 010 20 time [sec] -0.1 0 0.1 0.2 5 010 20 time [sec] -0.1 0 0.1 0.2 6 Figure 4.5: Comparison of the PTV-extracted free surface elevation with the COULWAVE predictions. Black line corresponds to the experimental data, and the red dashed line to COULWAVE. See Figure 3.8 for the locations. 70 0 5 10 15 20 25 -1 0 1 1 0 5 10 15 20 25 -1 0 1 2 0 5 10 15 20 25 -1 0 1 velocity u [m/s] 3 0 5 10 15 20 25 -1 0 1 4 0 5 10 15 20 25 -1 0 1 5 0 5 10 15 20 25 -1 0 1 6 0 5 10 15 20 25 -0.2 0 0.2 1 0 5 10 15 20 25 -0.2 0 0.2 2 0 5 10 15 20 25 -0.2 0 0.2 velocity v [m/s] 3 0 5 10 15 20 25 -0.2 0 0.2 4 0 5 10 15 20 25 time [sec] -0.2 0 0.2 5 0 5 10 15 20 25 time [sec] -0.2 0 0.2 6 Figure 4.6: Comparison of the PTV-extracted u− and v−velocities with the COULWAVE predictions. Black line corresponds to the experimental data, and the red dashed line to COULWAVE. See Figure 3.8 for the locations. 71 Figure 4.7: Top view of the conical island for an H/d = 0.4 solitary wave and compares analytical (green line) , numerical (violet line) predictions with the lab- oratory measurements, obtained using edge detection, as described in section 3.7. Concentric solid elevation lines are every 10cm, so the undisturbed water depth is 30cm. Surprisingly, the analytical predictions of Kânoğlu & Synolakis (1998) predict the maximum runup about as well as COULWAVE, even for this highly nonlinear wave. 72 -10 -5 0 5 10 26 28 30 32 x [m] Total Station Edge Detection Analytical Breaking COULWAVE -10 -5 0 5 10 y [m] 0 0.2 0.4 0.6 runup [m] Figure 4.8: Comparison of the maximum runup along the beach behind the island inferredfromthelaboratorymeasurements(edgedetection), measurementsusinga total station, predictions from COULWAVE and the breaking-solitary wave model of Li & Raichlen (2002) which is based on Synolakis (1987). On the top. there is the shoreline profile along the cross–shore coordinate (a top view) while the bottom figure is the maximum runup along the cross-shore coordinate ( Figure 4.9 for visualizing the location). 73 Figure 4.9: Top view of the maximum wave height over all time steps, for a 0.4 solitary wave interacting with the topography with parameters shown in section 2.2. Computations using COULWAVE. The black line is the runup from the labo- ratory data, red line is the predictions from COULWAVE and with the blue dashed is the breaking – solitary wave model (Li & Raichlen, 2002). 74 Figure 4.10: Snapshots of the top views of the flow evolution of a 0.4 solitary wave interacting with a conical island and a sloping beach behind it, for the geometrical parameters insec 2.2. Computations using COULWAVE. 75 4.2.3 Comparison of laboratory measurements with COULWAVE results – Two islands One interesting problems is the flow around two islands, as described in chapter 2. For the record, this is the same configuration referred to as C in Keen & Lynett (2019). As a reminder, the one island is conical and is shown in the lower portion of the figures to follow, the other island is a frustum of a cone, an island which is identical in shape to the one (but not size) with the island used in the 1992 experiments. The initial wave is an N–wave as discussed in section 2, as also used by Keen & Lynett (2019). This particular N-wave bares no resemblance to the waves in Tadepalli & Synolakis (1994), because of limitations in the trajectory options being available. OSU has only one option for a leading-depression N–wave and it was not possible to reprogram the wavemaker. Figure 4.11 shows snapshots of the top views of the flow evolution of a N–wave interacting with two conical islands and a sloping beach behind it, at different times, for the geometrical parameters in section 2.3. Figure 4.14 shows a map of the maximum wave height observed in the entire numerical experiment, as modeled by COULWAVE. Note that no runup is shown in the upper island, because it is the frustum, and the wave runs over it. Note the flow between the two islands. One would have expected a jet like flow causing higher runup along the sloping beach behind them. Instead, one observes that the maximum runup is less in this region than in the sections of the sloping beach the islands shadow. Referring to Keen & Lynett (2019), similar behavior is observed for solitary waves, see their Figure 8. 76 Figure 4.11: Snapshots of the top views of the flow evolution of a N–wave inter- acting with two conical islands and a sloping beach behind it, for the geometrical parameters in section 2.3. Computations using COULWAVE. 77 40 50 60 -0.1 0 0.1 0.2 1 40 50 60 -0.1 0 0.1 0.2 2 40 50 60 -0.1 0 0.1 0.2 3 40 50 60 -0.1 0 0.1 0.2 [m] 4 40 50 60 -0.1 0 0.1 0.2 5 40 50 60 -0.1 0 0.1 0.2 6 40 50 60 -0.1 0 0.1 0.2 7 40 50 60 time [sec] -0.1 0 0.1 0.2 8 40 50 60 -0.1 0 0.1 0.2 9 Figure 4.12: Comparison of the PTV-extracted free surface elevation with the COULWAVE predictions. Black line corresponds to the labratory data, and the red dashed line to COULWAVE. See Figure 3.16 for the locations. 78 30 40 50 -0.5 0 0.5 1 1 30 40 50 -0.5 0 0.5 1 2 30 40 50 -0.5 0 0.5 1 3 30 40 50 -0.5 0 0.5 1 velocity u [m/s] 4 30 40 50 -0.5 0 0.5 1 5 30 40 50 -0.5 0 0.5 1 6 30 40 50 -0.5 0 0.5 1 7 30 40 50 -0.5 0 0.5 1 8 30 40 50 -0.5 0 0.5 1 9 30 40 50 -0.2 0 0.2 0.4 1 30 40 50 -0.2 0 0.2 0.4 2 30 40 50 -0.2 0 0.2 0.4 3 30 40 50 -0.2 0 0.2 0.4 velocity v [m/s] 4 30 40 50 -0.2 0 0.2 0.4 5 30 40 50 -0.2 0 0.2 0.4 6 30 40 50 -0.2 0 0.2 0.4 7 30 40 50 time [sec] -0.2 0 0.2 0.4 8 30 40 50 -0.2 0 0.2 0.4 9 Figure 4.13: Black line corresponds to the COULWAVE predictions for the surfave velocities u and v. See Figure 3.16 for the locations. 79 Figure 4.14: Top view of the maximum wave height over all time steps, for a N–wave interacting with the topography with parameters shown in section 2.3 (ComputingusingCOULWAVE).Theblackdashedline(circleandvertical)arethe initial positions of the shoreline on the island and the sloping beach, respectively. Comparisonofthemaximumrunupalongthebeachbehindtheislandinferredfrom thelaboratorymeasurementsusingatotalstation(blackline)andpredictionsfrom COULWAVE (red dashed line). 80 Chapter 5 Numerical Models at Geophysical Scales 5.1 Introduction Thebasicpremiseofthisthesisistoexaminetheunusualtsunamiamplification phenomena on plane beaches in the shadow zone of islands, as observed in 2010 in the Mentawais, and to check the results of Stefanakis et al. (2014) who found that there was no reduction of the runup behind the islands under none of the conditions they tested. This premise is examined in this section. The experiments described in the previous section served the purpose to bench- mark both MOST and COULWAVE with one and two island geometries, but also to understand the flow dynamics behind the islands. It is yet unclear how well the codes perform when calculating the runup over a wider range of parameters (beach slope, island slope, depth). Hence, before proceeding with examining the flow with the island shadowing a plane beach, MOST will be tested by examin- ing the runup of a solitary–like wave on a plane beach, in parameter ranges of geophysical relevance. 81 5.2 Validation of MOST As Synolakis et al. (2008) argued, "tsunami models have evolved in the last two decades through careful and explicit validation/verification by comparing their predictions with benchmark analytical solutions, laboratory experiments, and field measurements. While there is in principle no assurance that a numerical code that has performed well in all benchmark tests will always produce realistic inunda- tion predictions, validated/verified codes largely reduce the level of uncertainty in their results to the uncertainty in the geophysical initial conditions. Furthermore, when coupled with real-time free-field tsunami measurements from tsunameters, validated/verified codes are the only choice for realistic forecasting of inundation." MOST is a code which has been benchmarked repeatedly Titov et al. (2016). In the present context, it is important to check how well the specific version of MOST used predicts the available analytical results for solitary waves, and in particular, Synolakis (1987) runup law. This is one of the standard benchmark tests, used by the tsunami community, (Synolakis & Bernard, 2006). Such a validation would show that MOST solves the shallow water wave equations and produces results which are as close to the analytical solution as grid resolutions allow. Such a check would also allow for examining the effect of the inherent numerical dissipation, omnipresent in numerical schemes, but of course absent from analytical solutions. Here, the predictions of MOST will be compared with Synolakis’ runup law and to numerical evaluations of the MacLaurin series. 82 Figure 5.1: Schematic of the geometry Parameter Range tanθ b 0.05-0.2 h 100-1000 [m] ω 0.005-0.03 [rad s −1 ] Table 5.1: Physical parameter ranges used for 2+1 D numerical experiments used for validation of MOST The standard form of the 2+1 dimensional nonlinear shallow water wave equa- tions (NSW) equations are, h t + (uh) x + (vh) y = 0 (5.1) u t =uu x +vu y +gh x =gd x (5.2) v t +uv x +vv y +gh y =gd y (5.3) where h = η(x,y,t) +d(x,y,t), η(x,y,t) is the wave amplitude, d(x,y,t) is the water depth, u(x,y,t) and v(x,y,t) are the depth-averaged velocities; these are the equations solved by Titov & Synolakis (1998). The linearized shallow-water equations can be obtained as follows - the particular derivation is due to Segur (2009): 83 ∂η ∂t + ∂(uh) ∂x + ∂(vh) ∂y = 0, (5.4) ∂u ∂t +g ∂η ∂x = 0, (5.5) ∂v ∂t +g ∂η ∂y = 0, (5.6) where the advective terms have been eliminated from the LSW. Multiplying Equa- tion 5.4 by √ g, and both Equation 5.5 and Equation 5.6 by √ h, one obtains, ∂ ∂t (η √ g) + ∂ ∂x (u √ h q gh) + ∂ ∂y (v √ h q gh) = 0, (5.7) ∂ ∂t (u √ h + q gh ∂ ∂x (η √ g) = 0, (5.8) ∂ ∂t (v √ h) + q gh ∂ ∂y (η √ g) = 0. (5.9) One can readily eliminateu √ h andv √ h by taking the time derivative of Equation (5.7) and the space derivatives of Equation (5.8) and Equation (5.9), and then ∂ 2 ∂t 2 (η √ g) =∇· [gh·∇(η √ g)]. (5.10) In dimensionless variables, ∂ 2 η ∂t 2 = ∂ ∂x h ∂η ∂x + ∂ ∂y h ∂η ∂y , (5.11) Synolakis (1986, 1987) solved the 1+1D version of the NSW equations. This is applicable in this case, since there is no long–shore variation, i.e., when a solitary 84 wave of infinite crest length evolves over constant depth and then on the sloping beach. In this case, Synolakis (1987) runup law requires that R d = 2.931 q cotθ b H d 5/4 . (5.12) If instead of the classic Boussinesq style solitary wave with η(x,t = 0) =Hsech 2 γ(X 0 −ct) , one allows γ to vary, then one obtains that 02468 10 0 1 2 3 4 5 6 7 8 9 10 5m grid C cell size 1:1 0 5 10 15 0 5 10 15 1:1 Figure 5.2: a: The figure on the left shows the runup amplification as predicted by the runup law versus the prediction of the runup amplification from MOST, for the 125 computations of solitary–like waves. b: The figure on the right shows the dimensional runup prediction from the runup law versus the numerical prediction of MOST. Figures 5.3 shows the runup normalised by the initial wave height R/H as calculated analytically using the full integral evaluation, instead of the analytical 85 runup law, as a function of the R/H as calculated numerically using MOST. A perfect representation of the SW solution would require that all points collapse along the one to one line, shown in the figure. In fact, as expected, the MOST predictions are slight lower than the results from the analytical solution, due to the numerical dissipation inherent in MOST, as in all other numerical codes, whether externally added or not. Figure 5.2b shows the same results, but dimensional. Both figures only include 125 (5 3 ) results with solitary-like wave and slope from table 5.2. In other words, five values for each of the the three parameters, beach slope, omega and water depth were used. 02468 0 1 2 3 4 5 6 7 8 Analytical Analytical-asymptotic expansion MOST Figure 5.3: The runup amplification as a function of the runup law. Compari- son of the integral evaluation (analytical), its asymptotic expansion (analytical– asymptotic expansion) and results from MOST. 86 Figure5.2showstherunupamplificationfortheresultsincludedinFigures5.2.a and 5.2.b as a function of the prediction of the runup law. On close examination, it appears that in subfigure b there is an region where MOST runup prediction are almost exactly 3.0 m (recall that the input wave is 1.5 m high), regardless of the other parameters, yet the analytical runup varies (based on the runup law). This needs to be further investigated. Another way to view the same results is to plot the runup amplification values against those predicted by the runup law, Figure 5.3. The figure compares the results from the analytical solution without Synolakis’s asymptotic approximation (numerical integration, blue line), the results using the runup law (asymptotic result, 1 : 1 red line), against the MOST predictions. It is clear that there is an asymptote atR/H = 2, which corresponds to the really long waves in the sample, waves which are solitary wave-like, but with longer wavelengths. Itishelpfultoinvestigatethis, henceconsiderthederivationofSynolakis’runup law. Synolakis (1986, 1987) used contour integration and derived the Laurent expansion of the solution for a solitary wave propagating over constant depth and approaching a sloping beach, as shown in Figure 5.1, R(t) d = 8 H d ∞ X n=1 (−1) n+1 n exp[−2γ(X 1 −X 0 −ct)n] I 0 (4γX 0 n) +I 1 (4γX 0 n) . (5.13) Synolakis proceeded using the asymptotic expansion of the Bessel functions for √ cotβ( H d ) 1 4 for large values, and simplified the resulting Laurent expansion. However, for really small waves or very steep beaches, √ cotβ( H d ) 1 4 is not small, and presumably the asymptotic expansion is not valid, hence one needs to use a different method to sum the series. Staring with Equation 5.14 and following N. Kalligeris (Personal Communica- tion), 87 R(t) d = 8 H d ∞ X n=1 (−1) n+1 n exp[−2γ(θ−ct)n] I 0 (4γX 0 n) +I 1 (4γX 0 n) , (5.14) correcting Synolakis’ typo of (−1) n which should be (−1) n+1 , otherwise R can came out negative, one can use the asymptotic expansion of the modified Bessel function for small arguments 4γX 0 n→ 0 (this is the limit for the smallest R H ) I 0 (4γX 0 n)→ 1 and I 1 (4γX 0 n)→ 2γX 0 n. Then, Equation 5.14 becomes R(t) d ≈ 8 ∞ X n=1 (−1) n+1 n exp[2γ(θ +ct)n] 1−i(2γX 0 n) . (5.15) Multiplying the nominator and denominator with the complex conjugate of the denominator, one obtains, R(t) d ≈ 8 ∞ X n=1 (−1) n+1 n exp[2γ(θ +ct)n] 1 + (2γX 0 n) 2 + 2γX 0 i n 2 (−1) n+1 exp[2γ(θ +ct)n] 1 +i(2γX 0 n) 2 . (5.16) If one keeps only the real part, then R(t) d ≈ 8 ∞ X n=1 (−1) n+1 n exp[2γ(θ +ct)n] 1 + (2γX 0 n) 2 , (5.17) when γX 0 n is very small, that is for very long solitary-like waves and very steep beaches with cotθ b =X 0 ≈O 1 , then R(t) d ≈ 8 ∞ X n=1 (−1) n+1 n exp[2γ(θ +ct)n]. (5.18) 88 Following Synolakis’ methodology, the series is of the form P ∞ n=1 (−1)n(−1) n+1 χ n , its maximum values is 1/4 and occurs at χ = 0.981 = exp[−0.0192]. So at the lowest limit, max runup is R H = 2. Clearly, the runup amplification asymptotes to 2 for very long waves climbing on very steep beaches, as if running up a vertical wall. Again, as expected, very small solitary waves have very long wavelengths, and when climbing up on steep beaches, they do not “feel” shoaling, instead interact with the sloping beach, as they would with a vertical wall. 0123456 0 1 2 3 4 5 6 Analytical Analytical-asymptotic expansion Li & Raichlen 2001 Figure 5.4: The runup amplification as a function of the runup law. Compari- son of the integral evaluation (blue line), its asymptotic expansion (red line) and laboratory data from Li & Raichlen (2001), as shown in Table A.1. A universal formula for non-breaking can be derived based on the results from the analytical solution without Synolakis’s asymptotic approximation shown in 89 Figure 5.3. The full analytical solution (blue line) approaches R H = 2 for x-axis values <∼ 1, and the 1:1 line (asymptotic expansion) for x-axis values >∼ 4. Thus, a hyperbola of the form R/H = (x a + 2 a ) 1/a can be fitted based on the two asymptotes. The optimal factor is a = 4, then a universal analytical solution for non–breaking solitary waves is given below and in Figure 5.5 with the green dashed line. R H =   3387.1 cotβ 0 T ! 2 d g + 16   1/4 (5.19) 02468 0 1 2 3 4 5 6 7 8 Analytical Analytical-asymptotic expansion Universal analytical Figure 5.5: The runup amplification as a function of the runup law. Compari- son of the integral evaluation (analytical), its asymptotic expansion (analytical– asymptotic expansion) and the universal analytical using Equation 5.19. In summary, in this subsection MOST was not only validated, but also a pre- viously unrecognized asymoptote to the analytical runup results was discovered. 90 5.3 Numerical evaluation of the runup around a conical island and on the sloping beach behind it In this section, MOST will be used to evaluate the flow field around the topog- raphy of a conical island fronting a sloping beach. As discussed earlier, this is the same computational experiment as performed by Stefanakis et al. (2014) using the numerical code VOLNA (Dutykh et al., 2011b). Here, the veracity of those results will be checked, i.e., whether runup amplification exists over a wider range of "periods" (0.005 < ω < 0.03 rad/s) than as used by Stefanakis et al. (2014), whose range is (0.01<ω< 0.1 rad/s). In essence, the present experiments exam- ine longer waves than Stefanakis’ analysis. These ranges of parameters are believed to be of geophysical relevance. The initial wave profile given by Equation 5.20. η(t) = 1.5sech 2 (ωt− 2.6) (5.20) Parameter Range tanθ i 0.05-0.2 tanθ b 0.05-0.2 d 0-5000 [m] h 100-1000 [m] ω 0.005-0.03 [rad s −1 ] Table 5.2: Parameter ranges for computational experiments in this section. As reminder, the initial condition for the computations is a single–hump wave Equation 5.20, and the topography is shown in Figure 5.6. This initial profile is 91 used instead of the Boussinesq solitary wave - as in Synolakis (1987)- because it allows for testing shorter or longer single-hump waves which may not be solitary waves, in the usual sense, i.e., they are not of constant form when propagating over constant depth. Figure 5.6: Schematic of the geometry of a single–hump solitary–like wave approaching a conical island of angle θ i distant d from the toe of a sloping beach of angle θ b . The offshore depth is h. A total of 21, 875 runs were made with MOST version III, seven for each of the 3, 125 unique combinations of parameters in Table 5.2. Five values for each of the five parameters in Table 5.2 were chosen, resulting into 5 5 combinations. (Recall, that in the earlier section, where the 2+1 D on the runup on a plane beach was computed, there were 5 3 cases, again because five values were assigned to each of the three parameters of Table 5.1. Each of the 3, 125 combinations was run seven different times, horizontally moving the fine grid where runup is computed to different locations, to capture the entire runup flow dynamics. This was necessitated because a resolution of 5 m was required. Note that the computations were made with 100− 1, 000 m offshore depths and 1000− 22, 000 m island diameters. Version III can only have one grid C, later versions can have multiple grids C, (Titov et al., 2016). To repeat, runup 92 amplification is the ratio of the runup at the point of the beach at the intersection of the cross–shore island axis and the "free field" runup at a point along the beach not shadowed by the island. A single grid C would not have allowed calculation of the entire runup in this region. Examples of the cross–shore runup distribution along the plane beach behind the island are shown in Figures (5.7, 5.8, 5.9, 5.10). Each figure shows the along- shore variation of the maximum vertical runup, which is symmetric along an axis which runs from the center of the island and is perpendicular to the initial position of the shoreline. Figure 5.9 shows the various modes of maximal shoreline profiles. The runup amplification is also shown for each of the profiles shown. Each of the figures shows the results from the computations with the initial wave interacting with the topography with the island fronting the beach and with- out. For clarity, the computed maximum runup with the wave striking the beach without an island has been subtracted, so that is the axis y = 0 shows the maxi- mum runup of the island bathymetry with respect to the profile that would have resulted had the same initial wave reached the sloping beach, without the island in place. Far from the island centerline axis (indicated with y = 0 in each figure), the relative runup is zero, because far from the centerline axis the presence of the island is not felt. These runup profiles look conspicuously like the generalized N–waves of Tade- palli & Synolakis (1994), so an attempt was made to fit the N-wave profile below, η =a(x−b)sech 2 c(x−d) (5.21) with the observed maximum runup profiles with the island in place. a,b,c,d are free parameters, and one would expect that they correlate with the five parameters 93 in Table 5.2. If there is such strong correlation, this would seem to imply that one of the five parameters is less important in the maximum runup distribution. 0 102030405060 Alongshore [km] 3.5 4 4.5 5 5.5 6 6.5 Runup [m] 1234567 Computed Smoothed Fitted Grid C Figure 5.7: Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.05,d = 0 m,h = 325 m,ω = 0.0175 rad/s, with n-wave coefficients a =−0.0005235, b =−1140, c = 0.0001613, d = 1875. 0 102030405060 Alongshore [km] 3 3.5 4 4.5 5 5.5 Runup [m] Computed Smoothed Fitted Figure 5.8: Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.05,d = 0 m,h = 100 m,ω = 0.005 rad/s, with n-wave coefficients a =−0.0001276, b = 7558, c = 0.0001101, d = 7749. 94 0 5 10 15 20 25 Alongshore [km] 2.5 3 3.5 4 4.5 Runup [m] Computed Smoothed Fitted Figure 5.9: Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.0875, d = 0 m, h = 100 m, ω = 0.005 rad/s, with n-wave coefficients a =−0.0003248, b = 3440, c = 0.0003002, d = 1279. 0 20406080 100 120 Alongshore [km] 3 3.5 4 4.5 5 5.5 6 Runup [m] Computed Smoothed Fitted Figure 5.10: Runup profile, parameters for computational experiments: tanθ i = 0.05, tanθ b = 0.0875, d = 1250 m, h = 775 m, ω = 0.0112 rad/s, with n-wave coefficients a =−0.0001926, b = 558.7, c = 8.05e− 05, d = 5144. 95 Figure 5.11 shows plots of the variation of the parameter a in the N-wave type fit of Equation (5.21) of the maximal shoreline motion. The subfigures show the variations with respect to the wavelength, the island radius, the distance between the toe of the island to the toe of the beach and with the water depth. The objective here is to examine if there are consistent trends, that would help identify how the parameter a varies, in lieu of results from dimensional analysis. 0 5 10 15 wavelength [m] 10 4 -4 -3 -2 -1 0 fit parameter a (m) 10 -3 012 Island radius [m] 10 4 -4 -3 -2 -1 0 fit parameter a (m) 10 -3 024 total distance from the toe [m] 10 4 -4 -3 -2 -1 0 fit parameter a [m] 10 -3 0 500 1000 water depth [m] -4 -3 -2 -1 0 fit parameter a (m) 10 -3 Figure 5.11: Plots of the variation of the parameter a in the N-wave type fit of Equation (5.21) of the maximal shoreline motion. Variations with respect to the wavelength, the island radius, the distance between the toe of the island to the toe of the beach and with the water depth. Figure 5.12 shows the variation of the parameter a with respect to a dimension- less combination of parameters, inferred from examining the four subplots. There is substantial scatter. 96 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 -2.5 -2 -1.5 -1 -0.5 0 10 -3 Figure 5.12: where L is the wavelength, X 0 =h/ tanθ b , h is the water depth and r is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.0006718x− 0.0006188. 0 5 10 15 20 25 30 35 40 -6 -4 -2 0 2 4 6 Figure 5.13: whereλ is the wavelength andr is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.808 ln(0.3651x). 97 10 1 10 -1 10 0 Figure 5.14: whereλ is the wavelength andr is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) = 0.563csch(0.075x). 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1.5 2 Figure 5.15: where λ is the wavelength and r is the radius of the island on the sea floor. The red solid line is the fitted curve with f(x) =−9.34x 4 + 22.2x 3 − 17.63x 2 + 4.45x + 0.34. 98 5.3.1 Free surface flow kinematics Recall that the objective has been to check whether runup amplification on the sloping beach fronted by the conical island persists for all physically realistic parameter ranges. One of the main conclusions of Stefanakis et al. (2014) has been that there were no cases identified by machine learning where the island offered protection in the shoreline behind it. This analysis can now turn and examine the runup amplification as a function of the wavelength over the island radius, for the 3, 125 runs with parameters in the ranges shown in Table 5.1. The computed results are shown in Figure 5.17. The same plot includes the results of Stefanakis et al. (2014). The trends are an entirely different trend than what was observed by Stefanakis et al. (2014), who run about 200 cases Figure 1.6, selected from machine learning. Remember that the latter used VOLNA (Dutykh et al., 2011b) another shallow–water equation solver. Note that the larger differences are observed in the shorter waves, raising the questions if this could be that VOLNA and MOST produce different results for dispersive waves? Figure 5.17 groups computational results in four categories. The original Ste- fanakis et al. (2014) results are shown in circles, while crosses within these circles identify those results which are for waves which are dispersive, i.e., withλ/d< 20. The red x-s identify results obtained with MOST for the same parameters used by Stefanakis et al. (2014) with VOLNA. The black x-s show results with MOST, for the entire parameter range. Clearly, most differences are observed for dispersive waves. We selected five values for each of the five parameters. For the record, the maximum runup amplification in the present results is 1.37, while the minimum is 0.67. 99 02468 10 12 0 / r 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 RA VOLNA vs MOST all VOLNA results VOLNA results with no dispersion all MOST results MOST results with no dispersion Figure 5.16: Runup amplification as a function of the ratio of the wavelength over the island radius using VOLNA and MOST. The original Stefanakis et al. (2014) results are shown in circles, while crosses within these circles identify those results which are for waves which are dispersive, i.e., with λ/d < 20. The black circles identify results obtained with MOST for the same parameters used by Stefanakis et al. (2014) with VOLNA. Clearly, most differences are observed for dispersive waves. 100 0 5 10 15 20 25 30 35 0.5 1 1.5 Figure 5.17: Runup amplification as a function of the ratio of the wavelength over the island radius using MOST. The black x-s show results with MOST, for the entire parameter range (see Table 5.2), with red circle is the maximum and minimum runup amplification. 101 The question arises as to what causes the observed differences in the runup amplification RA, as shown in Figure 5.17. One likely explanation is the splitting of the incoming wave front into two fronts, each propagating on either side of the island. Figure 5.18: A definition sketch. The solid red is the coastline, the dashed red is the toe of the sloping beach and the dashed black line shows the semi-circle along which the wave height is shown in subsequent figures. Figure 5.19 shows snapshots of the surface elevation around the island at differ- ent times, for λ/r = 2 resulting to RA = 0.67, that corresponds to the minimum RA from the Figure 5.17. Figure 5.20 shows snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island, as shown in the schematic Figure 5.18. The particular conditions are with a sea floor radius of 4000m, tanθ i = 0.05, tanθ b = 0.05, d = 0 m, h = 100 m and ω = 0.03 rad/s. For example, the lower of the 5 lines is eta (r = 4000 m, t), i.e. the profile at dimen- sionless timet ∗ = 459.78, while the upper line is at dimensionless timet ∗ = 585.06. 102 It is important to note that in this case the island reduces the runup on the beach (RA<1), compared to the no–island case. t ∗ = (t−t 0 ) r g h , t 0 = x island −x solitary √ gh , (5.22) where h is the constant depth, x island is the center of the island in the x-axis and x solitary is the center of the initial solitary–like wave in the x–axis. i.e., the position of its crest. 103 Figure 5.19: Snapshots of the flow elevation around the island, in increasing time, from left to right, for the case shown in Figure 5.23b. The color bar is dimensional, and the numbers are in meters. The initial wave height is 1.5 m. RA = 0.67, λ/r = 2. 104 0 1 2 3 t * = 459.78 0 1 2 3 t * = 491.10 0 1 2 3 [m] t * = 522.42 0 1 2 3 t * = 553.74 -80 -60 -40 -20 0 20 40 60 80 deg ° 0 1 2 3 t * = 585.06 Figure 5.20: Snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island. The blue line is with the island in place, the red dashed line is without the island. Two humps appear in the top snapshots, in both cases, because the profiles are along a semi-circle. This is the same case as in Figure 5.19, with H− 1.5 m. RA = 0.67 and λ/r = 2, i.e., the case with the lowest recorded runnup amplification. 105 Figure 5.21 shows snapshots of the surface elevation around the island at dif- ferent times, forλ/r = 10 resulting toRA = 1.37, corresponding to the maximum RA from Figure 5.17. Figure 5.22 shows snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island, as shown in the schematic Figure 5.18. In this particular geometry, the island’s seafloor radius is 1231m, tanθ i = 0.1625, tanθ b = 0.2, d = 0 m, h = 100 m and ω = 0.03 rad/s. For example, the lower of the 5 lines is eta (r = 2481 m, t), i.e. the profile at time t ∗ = 331.17, while the upper line is at time t ∗ = 205.89. It is important to note that these are not images of shoreline motion. Instead, they correspond to a combination of parameters which produce maximum runup amplification. The maximum surface elevation behind the island is 3.06 m for an offshore wave of 1.5 m, indicating > 2 x amplification. All times shown in the figure are normalized. Figure 5.23 shows the evolution of the maximum amplitude of the incoming solitary–like wave for two extreme cases, i.e., the one with the lowest and the one with the highest runup amplification as observed in Figure 5.17, where the two extreme values are indicated in red. Both subfigures include results obtained with MOST. Both cases involve the same incoming wave, i.e, the two cases have the same initial condition but different geometric characteristics. The upper figure 5.23 corresponds to the higher amplification values of RA = 1.37 and λ/r = 10, tanθ i = 0.1625, tanθ b = 0.2, h = 100 m, d = 0 m, and ω = 0.03 rad/s. The toe of the sloping beach starts at x = 15.6. The blue line shows the evolution of the maximum along an axis which bisects the island. The red line shows the equivalent evolution of the maximum elevation along the bisector, but without the island in place. As the incoming wave evolves, it seems to gradually shoal, per Synolakis & Skjelbreia (1993), then continue on with the shoaling up the slopind beach, it is as if the island were not there. The resulting runup is higher. 106 The lower Figure in 5.23 shows the the evolution of the maximum surface elevation, for the case with lowest runup amplification, as depicted in Figure 5.17, and corresponds to RA = 0.67, λ/r = 2, tanθ i = 0.05, tanθ b = 0.05, h = 100 m, d = 1250 m, and ω = 0.03 rad/s. In other words, in this case, the island provides shadowing. As in the upper figure, the blue line shows the evolution of the maximum along an axis which bisects the island. The red line shows the equivalent evolution of the maximum elevation along the bisector, but without the island in place. As the incoming wave evolves, it seems to gradually shoal, then shoal rapidly, again as per Synolakis & Skjelbreia (1993). On the lee side of the island, the wave maximum wave height diminishes rapidly in a reverse Green’s law effect, see Synolakis (1991) for the derivation which is solitary wave specific. In this regard, Synolakis (ibid) has argued that Green’s law was derived for periodic waves and did not include reflection off a beach. Given that the reflection process introduces frequency–specific phase shifts, it was not apri ori clear that a complex wave wave form such a s solitary wave would necessarily evolve per the classic Green’s law for periodic waves. Hence Synolakis (ibid) provided a proof based on asymptotics of the linear theory solution for a solitary wave evolving up a sloping beach. Examining the figures, it becomes clear that for the wave shoaling over the milder sloped island (lower figure), the reflection appears to affect the maximum height of the island near the front toe of the island. For the same wave climbing up the steeper island, the reflection appears to affect the maximum height earlier, which implies that there is interaction between the incoming and outgoing wave. This is as expected, typically the milder the slope, the later the reflection process starts. Synolakis (1991) analysis appears valid, almost to breaking. 107 Figure 5.21: Maps of the surface elevation at different times corresponding to the case of maximum runup amplification RA=1.37. 108 0 1 2 3 t * = 205.89 0 1 2 3 t * = 237.21 0 1 2 3 [m] t * = 268.53 0 1 2 3 t * = 299.85 -80 -60 -40 -20 0 20 40 60 80 deg ° 0 1 2 3 t * = 331.17 Figure 5.22: Snapshots of the distribution of wave heights along a semi–circle centered in the centre of the island. The blue line is with the island in place, the red dashed line is without the island. This is as in Figure 5.19, with H = 1.5 m. RA = 1.37 and λ/r = 10, i.e., the case with the highest computed RA. Comparing the upper and lower figures 5.23, it becomes more clear when there is runup amplification (RA>1) and when there is reduction (RA<1). When the wave breaks, as indicated in the zone of rapid shoaling in the lower figure, the wave front does not recover, and it climbs up the beach with its wave height reduced. On the other hand, then the wave doesn’t break (as in the upper subfigure), the 109 wave continues to climb, over what appears to it as a much longer beach, which is the combo of the front end of the island and the sloping beach behind it. 10 11 12 13 14 15 16 x [km] -1 0 1 2 3 4 max [m] 10 15 20 25 x [km] -1 0 1 2 3 4 max [m] Figure 5.23: On the upper figure, the maximum surface elevation for the highest runup amplification, as calculated with MOST and as seen in Figure 5.17. RA = 1.37 andλ/r = 10, tanθ i = 0.1625, tanθ b = 0.2,h = 100 m,d = 0 m, andω = 0.03 rad/s. On the lower figure, the evolution of the maximum surface elevation, for the case with lowest runup amplification, as seen in Figure 5.17. RA = 0.67, λ/r = 2, tanθ i = 0.05, tanθ b = 0.05, h = 100 m, d = 1250 m, and ω = 0.03 rad/s. In both figures, the dotted green line is the prediction of Synolakis (1991), the blue lines show the evolution of the maximum along an axis which bisects the island. The red lines show the equivalent evolution of the maximum elevation along the bisector, but without the island in place. The two points corresponding to the two cases in these two subfigures are also marked in Figure 5.17. 110 Chapter 6 Conclusions This was a study of unusual tsunami amplification phenomena on sloping beaches fronted by circular islands. The study was motivated by field observations Hill et al. (2012) and numerical modeling results by Jose Borrero (personal com- munication) on the 25 October 2010 tsunami in the Mentawai islands in Indonesia. The study used the idealized bathymetry of a circular island fronting a uni- formly sloping beach, as in the study of Stefanakis et al. (2014). The bulk of the study used MOST, a NSW solver, while several simulations were run with the Boussinesq–type equation solver COULWAVE. The study used measurement from laboratory experiments to assess the veracity of both codes at small scales, then exact solutions of the SW to check the veracity of MOST at geophysical scales. In terms of the laboratory experiments, there were two sets of experiments, with one and two islands of slope 1 : 2, fronting a 1 : 10 uniformly sloping beach. The direct measurements involved surface–piercing resistance–type wave gauges monitoring elevation and acoustic Doppler velocimeter (ADVs) monitoring the two horizontal velocities. The indirect measurements involved matching pairs of videos recording the flows seeded with 0.85 mm diameter particles. The particle tracking algorithm of Crocker & Grier (1996) as implemented by Kalligeris (2017) was used to track the particles between successive video frames. For a given inter–particle spacing, the tracking becomes more difficult as the flow speed increases. In other words, for faster flows, one needs more frames per second. Thedetectionthresholdturnedouttobe 1m/sec. Hence,thealgorithmcouldtrack 111 the flows in the one-island experiment, except where the wave was breaking on the side of the island. The video–frame rate turned out to be a severe limitation in the two island case, particularly in the high–speed region between the islands. The laboratory measurements were compared with both analytical results and numerical predictions. The specific findings follow: 1. The analytical formulation of Kânoğlu & Synolakis (1998) proved capable to capturing the maximum runup for the one island configuration; there is no analytical solution for two islands. 2. Both MOST and COULWAVE reproduce the maximum runup very well around the island and along the sloping beach shadowed by the island. 3. COULWAVE reproduces well the maximum runup behind the two islands, particularly in the region directly shadowed by the islands. It also reproduces well the flow velocities in the one island case. 4. For very long waves on very steep beaches, the classic analytical expression for a solitary wave climbing up a sloping beach known as Synolakis’ runup law does not work, instead, it was determined that the runup amplification R/H = 2. 5. The shoreline shape at maximum runup when a solitary–like wave climbs up a sloping beach fronted by a conical island is described by an N–wave with four parameters. An attempt was made to determine how these parameters depend on the geometric configuration. This study started with the objective of determining whether there is indeed runupamplificationalongthebeachshadowedbyaconicalisland, andthusconfirm the results of Stefanakis et al. (2014). Recall that the latter argued that under 112 no conditions the islands provide shadowing, that is the runup behind the island being less than on parts of the sloping beach not fronted by the island. Their conclusion was based on about 200 numerical experiments whose param- eters were selected through machine learning from 100,000 possible combinations of parameters. The results herein suggest that for small values of the wavelength over the radius (λ 0 /r), the islands indeed provide protection, for other values they don’t. It appears, that if the wave crest splits into two crests – which is more prominent for smaller wavelengths or for islands with mild slopes, there is no runup amplification, i.e., RA<1. If the incoming wave is very long, there is less shoaling on the sides of the island, the wave appears to "pass through", and induce high runup just behind the island. In both high (RA=1.37) and low (RA=0.67) runup amplification cases detailed here, Synolakis (1991) result appears to hold for the initial climb of the solitary– like initial waves up the front side of the island. It also holds for the rundown behind the island, in a reverse–Green’s law manner, as the depth increases, the height decreases. As in many basic–science investigations, this study raised more questions than it answered. While the matter of the runup amplification appeared closed, it was opened here again and closed herewith, undoubtedly to be opened again by future investigators. In the meantime, nearshore counter–intuitive tsunami flow phenomena will continue to kill people, who often are less able to timely react even when intuition works as hoped. Clearly public education to help people identify locales at higher risk than others is urgently needed, as is the provision of high– resolution inundation maps which account for such phenomena, for evacuation planning. 113 Appendix A Labratory data for Runup Table A.1: Laboratory data for solitary waves runup non–breaking and breaking H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.250 0.506 19.85 Synolakis (1987) 0.442 1.205 9.35 Baldock et al. (2012) 0.072 0.233 19.85 Synolakis (1987) 0.466 1.247 9.35 Baldock et al. (2012) 0.448 0.723 19.85 Synolakis (1987) 0.479 1.282 9.35 Baldock et al. (2012) 0.078 0.251 19.85 Synolakis (1987) 0.597 1.372 9.35 Baldock et al. (2012) 0.384 0.621 19.85 Synolakis (1987) 0.581 1.466 9.35 Baldock et al. (2012) 0.097 0.274 19.85 Synolakis (1987) 0.635 1.513 9.35 Baldock et al. (2012) 0.462 0.659 19.85 Synolakis (1987) 0.642 1.513 9.35 Baldock et al. (2012) 0.236 0.467 19.85 Synolakis (1987) 0.005 0.008 10.00 Lo et al. (2013) 0.294 0.542 19.85 Synolakis (1987) 0.007 0.015 10.00 Lo et al. (2013) 0.610 0.780 19.85 Synolakis (1987) 0.010 0.028 10.00 Lo et al. (2013) 0.591 0.790 19.85 Synolakis (1987) 0.015 0.045 10.00 Lo et al. (2013) 0.607 0.805 19.85 Synolakis (1987) 0.019 0.064 10.00 Lo et al. (2013) 0.607 0.780 19.85 Synolakis (1987) 0.024 0.079 10.00 Lo et al. (2013) 0.601 0.801 19.85 Synolakis (1987) 0.029 0.099 10.00 Lo et al. (2013) 0.090 0.270 19.85 Synolakis (1987) 0.034 0.118 10.00 Lo et al. (2013) 0.259 0.519 19.85 Synolakis (1987) 0.048 0.162 10.00 Lo et al. (2013) 0.590 0.810 19.85 Synolakis (1987) 0.038 0.140 10.00 Lo et al. (2013) 0.298 0.551 19.85 Synolakis (1987) 0.043 0.164 10.00 Lo et al. (2013) 0.322 0.591 19.85 Synolakis (1987) 0.048 0.184 10.00 Lo et al. (2013) 0.170 0.407 19.85 Synolakis (1987) 0.075 0.294 10.00 Lo et al. (2013) 0.273 0.487 19.85 Synolakis (1987) 0.103 0.374 10.00 Lo et al. (2013) 0.276 0.495 19.85 Synolakis (1987) 0.132 0.477 10.00 Lo et al. (2013) 0.633 0.842 19.85 Synolakis (1987) 0.139 0.499 10.00 Lo et al. (2013) 0.625 0.825 19.85 Synolakis (1987) 0.350 0.872 10.00 Lo et al. (2013) 0.626 0.862 19.85 Synolakis (1987) 0.005 0.011 20.00 Lo et al. (2013) 0.283 0.527 19.85 Synolakis (1987) 0.008 0.020 20.00 Lo et al. (2013) 0.286 0.513 19.85 Synolakis (1987) 0.011 0.027 20.00 Lo et al. (2013) 0.323 0.555 19.85 Synolakis (1987) 0.016 0.044 20.00 Lo et al. (2013) 0.036 0.124 19.85 Synolakis (1987) 0.020 0.060 20.00 Lo et al. (2013) 0.188 0.409 19.85 Synolakis (1987) 0.025 0.075 20.00 Lo et al. (2013) 0.271 0.513 19.85 Synolakis (1987) 0.030 0.093 20.00 Lo et al. (2013) 0.416 0.686 19.85 Synolakis (1987) 0.040 0.126 20.00 Lo et al. (2013) 0.159 0.384 19.85 Synolakis (1987) 0.049 0.162 20.00 Lo et al. (2013) 0.160 0.384 19.85 Synolakis (1987) 0.060 0.181 20.00 Lo et al. (2013) 0.143 0.366 19.85 Synolakis (1987) 0.080 0.225 20.00 Lo et al. (2013) 0.036 0.121 19.85 Synolakis (1987) 0.100 0.267 20.00 Lo et al. (2013) 0.394 0.641 19.85 Synolakis (1987) 0.094 0.238 20.00 Lo et al. (2013) 0.048 0.182 19.85 Synolakis (1987) 0.101 0.250 20.00 Lo et al. (2013) 0.267 0.507 19.85 Synolakis (1987) 0.140 0.306 20.00 Lo et al. (2013) 0.039 0.152 19.85 Synolakis (1987) 0.150 0.307 20.00 Lo et al. (2013) 0.040 0.156 19.85 Synolakis (1987) 0.187 0.369 20.00 Lo et al. (2013) 0.021 0.076 19.85 Synolakis (1987) 0.202 0.376 20.00 Lo et al. (2013) 0.014 0.049 19.85 Synolakis (1987) 0.233 0.431 20.00 Lo et al. (2013) 0.051 0.191 19.85 Synolakis (1987) 0.252 0.448 20.00 Lo et al. (2013) 0.075 0.258 19.85 Synolakis (1987) 0.306 0.487 20.00 Lo et al. (2013) 0.073 0.248 19.85 Synolakis (1987) 0.326 0.526 20.00 Lo et al. (2013) 0.065 0.228 19.85 Synolakis (1987) 0.372 0.569 20.00 Lo et al. (2013) 0.055 0.207 19.85 Synolakis (1987) 0.417 0.616 20.00 Lo et al. (2013) 0.056 0.207 19.85 Synolakis (1987) 0.052 0.199 20.00 Wu et al. (2018) 0.034 0.144 19.85 Synolakis (1987) 0.053 0.195 20.00 Wu et al. (2018) 0.018 0.074 19.85 Synolakis (1987) 0.051 0.195 20.00 Wu et al. (2018) 0.009 0.036 19.85 Synolakis (1987) 0.062 0.214 20.00 Wu et al. (2018) 0.018 0.075 19.85 Synolakis (1987) 0.055 0.209 20.00 Wu et al. (2018) 0.027 0.108 19.85 Synolakis (1987) 0.055 0.214 20.00 Wu et al. (2018) 0.038 0.146 19.85 Synolakis (1987) 0.059 0.213 20.00 Wu et al. (2018) 0.047 0.195 19.85 Synolakis (1987) 0.053 0.214 20.00 Wu et al. (2018) 0.047 0.195 19.85 Synolakis (1987) 0.061 0.231 20.00 Wu et al. (2018) 0.188 0.425 19.85 Synolakis (1987) 0.059 0.230 20.00 Wu et al. (2018) Continued on next page 114 Table A.1 – continued from previous page H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.019 0.078 19.85 Synolakis (1987) 0.058 0.227 20.00 Wu et al. (2018) 0.019 0.076 19.85 Synolakis (1987) 0.059 0.230 20.00 Wu et al. (2018) 0.084 0.288 19.85 Synolakis (1987) 0.058 0.230 20.00 Wu et al. (2018) 0.009 0.041 19.85 Synolakis (1987) 0.093 0.320 20.00 Wu et al. (2018) 0.005 0.019 19.85 Synolakis (1987) 0.098 0.326 20.00 Wu et al. (2018) 0.006 0.022 19.85 Synolakis (1987) 0.103 0.330 20.00 Wu et al. (2018) 0.007 0.026 19.85 Synolakis (1987) 0.106 0.330 20.00 Wu et al. (2018) 0.028 0.123 19.85 Synolakis (1987) 0.149 0.376 20.00 Wu et al. (2018) 0.008 0.029 19.85 Synolakis (1987) 0.160 0.383 20.00 Wu et al. (2018) 0.023 0.087 19.85 Synolakis (1987) 0.161 0.383 20.00 Wu et al. (2018) 0.017 0.063 19.85 Synolakis (1987) 0.150 0.370 20.00 Wu et al. (2018) 0.034 0.088 19.85 Synolakis (1987) 0.148 0.364 20.00 Wu et al. (2018) 0.012 0.048 19.85 Synolakis (1987) 0.172 0.389 20.00 Wu et al. (2018) 0.014 0.052 19.85 Synolakis (1987) 0.175 0.392 20.00 Wu et al. (2018) 0.009 0.036 19.85 Synolakis (1987) 0.176 0.392 20.00 Wu et al. (2018) 0.193 0.426 19.85 Synolakis (1987) 0.204 0.426 20.00 Wu et al. (2018) 0.044 0.182 19.85 Synolakis (1987) 0.203 0.423 20.00 Wu et al. (2018) 0.022 0.098 19.85 Synolakis (1987) 0.203 0.423 20.00 Wu et al. (2018) 0.039 0.162 19.85 Synolakis (1987) 0.247 0.479 20.00 Wu et al. (2018) 0.040 0.168 17.00 Sriram et al. (2016) 0.248 0.483 20.00 Wu et al. (2018) 0.040 0.164 16.00 Sriram et al. (2016) 0.243 0.479 20.00 Wu et al. (2018) 0.040 0.156 15.00 Sriram et al. (2016) 0.296 0.545 20.00 Wu et al. (2018) 0.040 0.152 14.00 Sriram et al. (2016) 0.303 0.548 20.00 Wu et al. (2018) 0.040 0.148 13.00 Sriram et al. (2016) 0.303 0.545 20.00 Wu et al. (2018) 0.040 0.144 12.00 Sriram et al. (2016) 0.242 0.476 20.00 Wu et al. (2018) 0.040 0.136 11.00 Sriram et al. (2016) 0.244 0.473 20.00 Wu et al. (2018) 0.040 0.128 10.00 Sriram et al. (2016) 0.245 0.490 20.00 Wu et al. (2018) 0.040 0.124 9.00 Sriram et al. (2016) 0.243 0.479 20.00 Wu et al. (2018) 0.040 0.120 8.00 Sriram et al. (2016) 0.243 0.483 20.00 Wu et al. (2018) 0.040 0.112 7.00 Sriram et al. (2016) 0.245 0.473 20.00 Wu et al. (2018) 0.040 0.100 6.00 Sriram et al. (2016) 0.244 0.468 20.00 Wu et al. (2018) 0.060 0.240 11.00 Sriram et al. (2016) 0.243 0.479 20.00 Wu et al. (2018) 0.060 0.228 10.00 Sriram et al. (2016) 0.243 0.473 20.00 Wu et al. (2018) 0.060 0.222 9.00 Sriram et al. (2016) 0.244 0.470 20.00 Wu et al. (2018) 0.060 0.216 8.00 Sriram et al. (2016) 0.163 0.407 20.00 Wu et al. (2018) 0.060 0.204 7.00 Sriram et al. (2016) 0.174 0.414 20.00 Wu et al. (2018) 0.060 0.192 6.00 Sriram et al. (2016) 0.172 0.406 20.00 Wu et al. (2018) 0.060 0.180 5.00 Sriram et al. (2016) 0.176 0.414 20.00 Wu et al. (2018) 0.040 0.168 85.00 Sriram et al. (2016) 0.178 0.409 20.00 Wu et al. (2018) 0.040 0.148 66.00 Sriram et al. (2016) 0.175 0.411 20.00 Wu et al. (2018) 0.040 0.124 46.00 Sriram et al. (2016) 0.170 0.407 20.00 Wu et al. (2018) 0.040 0.092 25.00 Sriram et al. (2016) 0.170 0.415 20.00 Wu et al. (2018) 0.040 0.048 6.00 Sriram et al. (2016) 0.172 0.412 20.00 Wu et al. (2018) 0.024 0.089 60.00 Hsiao et al. (2008) 0.172 0.409 20.00 Wu et al. (2018) 0.035 0.097 60.00 Hsiao et al. (2008) 0.173 0.415 20.00 Wu et al. (2018) 0.048 0.114 60.00 Hsiao et al. (2008) 0.062 0.220 20.00 Wu et al. (2018) 0.057 0.124 60.00 Hsiao et al. (2008) 0.065 0.221 20.00 Wu et al. (2018) 0.077 0.139 60.00 Hsiao et al. (2008) 0.059 0.201 20.00 Wu et al. (2018) 0.090 0.146 60.00 Hsiao et al. (2008) 0.067 0.221 20.00 Wu et al. (2018) 0.096 0.153 60.00 Hsiao et al. (2008) 0.069 0.221 20.00 Wu et al. (2018) 0.129 0.168 60.00 Hsiao et al. (2008) 0.065 0.221 20.00 Wu et al. (2018) 0.156 0.181 60.00 Hsiao et al. (2008) 0.066 0.221 20.00 Wu et al. (2018) 0.165 0.189 60.00 Hsiao et al. (2008) 0.065 0.220 20.00 Wu et al. (2018) 0.183 0.199 60.00 Hsiao et al. (2008) 0.060 0.209 20.00 Wu et al. (2018) 0.208 0.208 60.00 Hsiao et al. (2008) 0.061 0.207 20.00 Wu et al. (2018) 0.213 0.215 60.00 Hsiao et al. (2008) 0.062 0.207 20.00 Wu et al. (2018) 0.227 0.226 60.00 Hsiao et al. (2008) 0.060 0.207 20.00 Wu et al. (2018) 0.248 0.232 60.00 Hsiao et al. (2008) 0.061 0.205 20.00 Wu et al. (2018) 0.264 0.236 60.00 Hsiao et al. (2008) 0.095 0.298 20.00 Wu et al. (2018) 0.288 0.249 60.00 Hsiao et al. (2008) 0.100 0.305 20.00 Wu et al. (2018) 0.322 0.256 60.00 Hsiao et al. (2008) 0.101 0.307 20.00 Wu et al. (2018) 0.338 0.261 60.00 Hsiao et al. (2008) 0.101 0.304 20.00 Wu et al. (2018) 0.011 0.046 60.00 Hsiao et al. (2008) 0.140 0.366 20.00 Wu et al. (2018) 0.020 0.076 60.00 Hsiao et al. (2008) 0.148 0.385 20.00 Wu et al. (2018) 0.030 0.093 60.00 Hsiao et al. (2008) 0.150 0.379 20.00 Wu et al. (2018) 0.041 0.099 60.00 Hsiao et al. (2008) 0.148 0.377 20.00 Wu et al. (2018) 0.052 0.119 60.00 Hsiao et al. (2008) 0.149 0.384 20.00 Wu et al. (2018) 0.054 0.111 60.00 Hsiao et al. (2008) 0.150 0.384 20.00 Wu et al. (2018) 0.053 0.113 60.00 Hsiao et al. (2008) 0.189 0.438 20.00 Wu et al. (2018) 0.055 0.120 60.00 Hsiao et al. (2008) 0.199 0.445 20.00 Wu et al. (2018) 0.053 0.108 60.00 Hsiao et al. (2008) 0.198 0.448 20.00 Wu et al. (2018) 0.053 0.113 60.00 Hsiao et al. (2008) 0.197 0.440 20.00 Wu et al. (2018) 0.052 0.114 60.00 Hsiao et al. (2008) 0.197 0.443 20.00 Wu et al. (2018) 0.070 0.139 60.00 Hsiao et al. (2008) 0.290 0.547 20.00 Wu et al. (2018) Continued on next page 115 Table A.1 – continued from previous page H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.086 0.146 60.00 Hsiao et al. (2008) 0.303 0.566 20.00 Wu et al. (2018) 0.100 0.157 60.00 Hsiao et al. (2008) 0.303 0.558 20.00 Wu et al. (2018) 0.118 0.170 60.00 Hsiao et al. (2008) 0.301 0.569 20.00 Wu et al. (2018) 0.120 0.165 60.00 Hsiao et al. (2008) 0.298 0.564 20.00 Wu et al. (2018) 0.120 0.167 60.00 Hsiao et al. (2008) 0.234 0.463 20.00 Wu et al. (2018) 0.119 0.168 60.00 Hsiao et al. (2008) 0.246 0.489 20.00 Wu et al. (2018) 0.120 0.162 60.00 Hsiao et al. (2008) 0.220 0.488 20.00 Wu et al. (2018) 0.120 0.164 60.00 Hsiao et al. (2008) 0.224 0.491 20.00 Wu et al. (2018) 0.120 0.165 60.00 Hsiao et al. (2008) 0.250 0.483 20.00 Wu et al. (2018) 0.137 0.177 60.00 Hsiao et al. (2008) 0.320 0.563 20.00 Wu et al. (2018) 0.152 0.188 60.00 Hsiao et al. (2008) 0.313 0.558 20.00 Wu et al. (2018) 0.152 0.180 60.00 Hsiao et al. (2008) 0.300 0.529 20.00 Wu et al. (2018) 0.152 0.187 60.00 Hsiao et al. (2008) 0.299 0.528 20.00 Wu et al. (2018) 0.152 0.184 60.00 Hsiao et al. (2008) 0.205 0.432 20.00 Wu et al. (2018) 0.152 0.177 60.00 Hsiao et al. (2008) 0.206 0.436 20.00 Wu et al. (2018) 0.153 0.182 60.00 Hsiao et al. (2008) 0.207 0.434 20.00 Wu et al. (2018) 0.152 0.183 60.00 Hsiao et al. (2008) 0.167 0.387 20.00 Wu et al. (2018) 0.020 0.083 60.00 Hsiao et al. (2008) 0.172 0.398 20.00 Wu et al. (2018) 0.036 0.093 60.00 Hsiao et al. (2008) 0.173 0.403 20.00 Wu et al. (2018) 0.045 0.110 60.00 Hsiao et al. (2008) 0.171 0.407 20.00 Wu et al. (2018) 0.054 0.123 60.00 Hsiao et al. (2008) 0.151 0.393 20.00 Wu et al. (2018) 0.070 0.138 60.00 Hsiao et al. (2008) 0.154 0.401 20.00 Wu et al. (2018) 0.086 0.153 60.00 Hsiao et al. (2008) 0.153 0.393 20.00 Wu et al. (2018) 0.054 0.200 20.00 Chang et al. (2009) 0.107 0.331 20.00 Wu et al. (2018) 0.094 0.317 20.00 Chang et al. (2009) 0.100 0.321 20.00 Wu et al. (2018) 0.164 0.400 20.00 Chang et al. (2009) 0.101 0.326 20.00 Wu et al. (2018) 0.173 0.423 20.00 Chang et al. (2009) 0.101 0.326 20.00 Wu et al. (2018) 0.220 0.460 20.00 Chang et al. (2009) 0.065 0.258 20.00 Wu et al. (2018) 0.235 0.491 20.00 Chang et al. (2009) 0.065 0.251 20.00 Wu et al. (2018) 0.027 0.049 2.08 Li & Raichlen (2001) 0.055 0.233 20.00 Wu et al. (2018) 0.064 0.141 2.08 Li & Raichlen (2001) 0.060 0.246 20.00 Wu et al. (2018) 0.072 0.164 2.08 Li & Raichlen (2001) 0.060 0.248 20.00 Wu et al. (2018) 0.089 0.211 2.08 Li & Raichlen (2001) 0.298 0.553 20.00 Wu et al. (2018) 0.109 0.262 2.08 Li & Raichlen (2001) 0.304 0.534 20.00 Wu et al. (2018) 0.114 0.296 2.08 Li & Raichlen (2001) 0.304 0.555 20.00 Wu et al. (2018) 0.136 0.354 2.08 Li & Raichlen (2001) 0.303 0.550 20.00 Wu et al. (2018) 0.147 0.406 2.08 Li & Raichlen (2001) 0.247 0.480 20.00 Wu et al. (2018) 0.165 0.434 2.08 Li & Raichlen (2001) 0.247 0.494 20.00 Wu et al. (2018) 0.174 0.490 2.08 Li & Raichlen (2001) 0.205 0.428 20.00 Wu et al. (2018) 0.198 0.558 2.08 Li & Raichlen (2001) 0.204 0.423 20.00 Wu et al. (2018) 0.202 0.567 2.08 Li & Raichlen (2001) 0.202 0.432 20.00 Wu et al. (2018) 0.230 0.681 2.08 Li & Raichlen (2001) 0.175 0.394 20.00 Wu et al. (2018) 0.236 0.684 2.08 Li & Raichlen (2001) 0.178 0.396 20.00 Wu et al. (2018) 0.258 0.765 2.08 Li & Raichlen (2001) 0.175 0.401 20.00 Wu et al. (2018) 0.270 0.838 2.08 Li & Raichlen (2001) 0.155 0.373 20.00 Wu et al. (2018) 0.281 0.880 2.08 Li & Raichlen (2001) 0.150 0.371 20.00 Wu et al. (2018) 0.287 0.879 2.08 Li & Raichlen (2001) 0.150 0.366 20.00 Wu et al. (2018) 0.307 0.963 2.08 Li & Raichlen (2001) 0.102 0.312 20.00 Wu et al. (2018) 0.316 1.025 2.08 Li & Raichlen (2001) 0.103 0.317 20.00 Wu et al. (2018) 0.322 1.067 2.08 Li & Raichlen (2001) 0.102 0.317 20.00 Wu et al. (2018) 0.339 1.122 2.08 Li & Raichlen (2001) 0.063 0.239 20.00 Wu et al. (2018) 0.050 0.147 5.67 Li & Raichlen (2002) 0.064 0.233 20.00 Wu et al. (2018) 0.055 0.152 5.67 Li & Raichlen (2002) 0.060 0.228 20.00 Wu et al. (2018) 0.064 0.182 5.67 Li & Raichlen (2002) 0.444 0.667 20.00 Wu et al. (2018) 0.068 0.197 5.67 Li & Raichlen (2002) 0.448 0.653 20.00 Wu et al. (2018) 0.076 0.211 5.67 Li & Raichlen (2002) 0.449 0.656 20.00 Wu et al. (2018) 0.073 0.226 5.67 Li & Raichlen (2002) 0.447 0.660 20.00 Wu et al. (2018) 0.082 0.257 5.67 Li & Raichlen (2002) 0.445 0.653 20.00 Wu et al. (2018) 0.086 0.250 5.67 Li & Raichlen (2002) 0.445 0.656 20.00 Wu et al. (2018) 0.091 0.247 5.67 Li & Raichlen (2002) 0.316 0.521 20.00 Wu et al. (2018) 0.088 0.302 5.67 Li & Raichlen (2002) 0.319 0.521 20.00 Wu et al. (2018) 0.095 0.297 5.67 Li & Raichlen (2002) 0.317 0.524 20.00 Wu et al. (2018) 0.098 0.287 5.67 Li & Raichlen (2002) 0.315 0.542 20.00 Wu et al. (2018) 0.104 0.291 5.67 Li & Raichlen (2002) 0.315 0.528 20.00 Wu et al. (2018) 0.108 0.317 5.67 Li & Raichlen (2002) 0.316 0.517 20.00 Wu et al. (2018) 0.109 0.330 5.67 Li & Raichlen (2002) 0.170 0.382 20.00 Wu et al. (2018) 0.121 0.347 5.67 Li & Raichlen (2002) 0.170 0.375 20.00 Wu et al. (2018) 0.114 0.377 5.67 Li & Raichlen (2002) 0.170 0.364 20.00 Wu et al. (2018) 0.110 0.445 5.67 Li & Raichlen (2002) 0.172 0.364 20.00 Wu et al. (2018) 0.133 0.404 5.67 Li & Raichlen (2002) 0.170 0.364 20.00 Wu et al. (2018) 0.128 0.428 5.67 Li & Raichlen (2002) 0.169 0.367 20.00 Wu et al. (2018) 0.141 0.435 5.67 Li & Raichlen (2002) 0.116 0.293 20.00 Wu et al. (2018) 0.137 0.471 5.67 Li & Raichlen (2002) 0.114 0.296 20.00 Wu et al. (2018) 0.143 0.483 5.67 Li & Raichlen (2002) 0.114 0.293 20.00 Wu et al. (2018) Continued on next page 116 Table A.1 – continued from previous page H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.152 0.540 5.67 Li & Raichlen (2002) 0.115 0.293 20.00 Wu et al. (2018) 0.162 0.511 5.67 Li & Raichlen (2002) 0.116 0.300 20.00 Wu et al. (2018) 0.163 0.564 5.67 Li & Raichlen (2002) 0.116 0.293 20.00 Wu et al. (2018) 0.168 0.551 5.67 Li & Raichlen (2002) 0.089 0.250 20.00 Wu et al. (2018) 0.169 0.589 5.67 Li & Raichlen (2002) 0.101 0.250 20.00 Wu et al. (2018) 0.179 0.587 5.67 Li & Raichlen (2002) 0.089 0.250 20.00 Wu et al. (2018) 0.211 0.666 5.67 Li & Raichlen (2002) 0.090 0.250 20.00 Wu et al. (2018) 0.244 0.779 5.67 Li & Raichlen (2002) 0.088 0.250 20.00 Wu et al. (2018) 0.253 0.806 5.67 Li & Raichlen (2002) 0.090 0.253 20.00 Wu et al. (2018) 0.258 0.834 5.67 Li & Raichlen (2002) 0.370 0.130 100.00 Wu et al. (2018) 0.297 0.960 5.67 Li & Raichlen (2002) 0.054 0.074 100.00 Wu et al. (2018) 0.425 1.248 5.67 Li & Raichlen (2002) 0.056 0.075 100.00 Wu et al. (2018) 0.066 0.229 19.85 Li & Raichlen (2002) 0.058 0.078 100.00 Wu et al. (2018) 0.073 0.233 19.85 Li & Raichlen (2002) 0.056 0.071 100.00 Wu et al. (2018) 0.074 0.249 19.85 Li & Raichlen (2002) 0.058 0.071 100.00 Wu et al. (2018) 0.076 0.258 19.85 Li & Raichlen (2002) 0.112 0.089 100.00 Wu et al. (2018) 0.079 0.251 19.85 Li & Raichlen (2002) 0.110 0.087 100.00 Wu et al. (2018) 0.091 0.268 19.85 Li & Raichlen (2002) 0.115 0.095 100.00 Wu et al. (2018) 0.095 0.288 19.85 Li & Raichlen (2002) 0.115 0.094 100.00 Wu et al. (2018) 0.098 0.275 19.85 Li & Raichlen (2002) 0.113 0.089 100.00 Wu et al. (2018) 0.144 0.366 19.85 Li & Raichlen (2002) 0.165 0.099 100.00 Wu et al. (2018) 0.161 0.384 19.85 Li & Raichlen (2002) 0.168 0.097 100.00 Wu et al. (2018) 0.172 0.407 19.85 Li & Raichlen (2002) 0.168 0.100 100.00 Wu et al. (2018) 0.190 0.409 19.85 Li & Raichlen (2002) 0.167 0.099 100.00 Wu et al. (2018) 0.190 0.426 19.85 Li & Raichlen (2002) 0.168 0.096 100.00 Wu et al. (2018) 0.195 0.427 19.85 Li & Raichlen (2002) 0.189 0.101 100.00 Wu et al. (2018) 0.238 0.467 19.85 Li & Raichlen (2002) 0.190 0.101 100.00 Wu et al. (2018) 0.252 0.505 19.85 Li & Raichlen (2002) 0.191 0.104 100.00 Wu et al. (2018) 0.261 0.517 19.85 Li & Raichlen (2002) 0.189 0.104 100.00 Wu et al. (2018) 0.269 0.506 19.85 Li & Raichlen (2002) 0.193 0.105 100.00 Wu et al. (2018) 0.275 0.487 19.85 Li & Raichlen (2002) 0.212 0.111 100.00 Wu et al. (2018) 0.273 0.510 19.85 Li & Raichlen (2002) 0.205 0.111 100.00 Wu et al. (2018) 0.278 0.495 19.85 Li & Raichlen (2002) 0.215 0.114 100.00 Wu et al. (2018) 0.285 0.525 19.85 Li & Raichlen (2002) 0.204 0.113 100.00 Wu et al. (2018) 0.288 0.512 19.85 Li & Raichlen (2002) 0.207 0.113 100.00 Wu et al. (2018) 0.296 0.539 19.85 Li & Raichlen (2002) 0.142 0.098 100.00 Wu et al. (2018) 0.299 0.548 19.85 Li & Raichlen (2002) 0.142 0.099 100.00 Wu et al. (2018) 0.325 0.554 19.85 Li & Raichlen (2002) 0.083 0.089 100.00 Wu et al. (2018) 0.324 0.589 19.85 Li & Raichlen (2002) 0.235 0.108 100.00 Wu et al. (2018) 0.386 0.620 19.85 Li & Raichlen (2002) 0.235 0.114 100.00 Wu et al. (2018) 0.396 0.638 19.85 Li & Raichlen (2002) 0.232 0.107 100.00 Wu et al. (2018) 0.409 0.656 19.85 Li & Raichlen (2002) 0.228 0.108 100.00 Wu et al. (2018) 0.419 0.683 19.85 Li & Raichlen (2002) 0.232 0.107 100.00 Wu et al. (2018) 0.450 0.722 19.85 Li & Raichlen (2002) 0.251 0.113 100.00 Wu et al. (2018) 0.464 0.658 19.85 Li & Raichlen (2002) 0.265 0.117 100.00 Wu et al. (2018) 0.007 0.015 30.00 Briggs et al. (1995) 0.265 0.118 100.00 Wu et al. (2018) 0.016 0.040 30.00 Briggs et al. (1995) 0.264 0.121 100.00 Wu et al. (2018) 0.024 0.067 30.00 Briggs et al. (1995) 0.262 0.118 100.00 Wu et al. (2018) 0.039 0.108 30.00 Briggs et al. (1995) 0.195 0.106 100.00 Wu et al. (2018) 0.058 0.141 30.00 Briggs et al. (1995) 0.198 0.115 100.00 Wu et al. (2018) 0.082 0.177 30.00 Briggs et al. (1995) 0.198 0.101 100.00 Wu et al. (2018) 0.125 0.231 30.00 Briggs et al. (1995) 0.198 0.114 100.00 Wu et al. (2018) 0.171 0.272 30.00 Briggs et al. (1995) 0.198 0.115 100.00 Wu et al. (2018) 0.005 0.030 30.00 Briggs et al. (1995) 0.256 0.114 100.00 Wu et al. (2018) 0.014 0.063 30.00 Briggs et al. (1995) 0.245 0.122 100.00 Wu et al. (2018) 0.024 0.095 30.00 Briggs et al. (1995) 0.248 0.115 100.00 Wu et al. (2018) 0.040 0.133 30.00 Briggs et al. (1995) 0.239 0.125 100.00 Wu et al. (2018) 0.057 0.164 30.00 Briggs et al. (1995) 0.247 0.119 100.00 Wu et al. (2018) 0.080 0.207 30.00 Briggs et al. (1995) 0.258 0.125 100.00 Wu et al. (2018) 0.132 0.250 30.00 Briggs et al. (1995) 0.265 0.125 100.00 Wu et al. (2018) 0.179 0.286 30.00 Briggs et al. (1995) 0.340 0.140 100.00 Wu et al. (2018) 0.274 0.343 30.00 Briggs et al. (1995) 0.346 0.135 100.00 Wu et al. (2018) 0.351 0.373 30.00 Briggs et al. (1995) 0.347 0.140 100.00 Wu et al. (2018) 0.099 0.377 11.20 Smith et al. (2017) 0.349 0.144 100.00 Wu et al. (2018) 0.118 0.456 11.20 Smith et al. (2017) 0.346 0.144 100.00 Wu et al. (2018) 0.198 0.637 11.20 Smith et al. (2017) 0.267 0.106 100.00 Wu et al. (2018) 0.296 0.829 11.20 Smith et al. (2017) 0.268 0.110 100.00 Wu et al. (2018) 0.393 0.983 11.20 Smith et al. (2017) 0.269 0.109 100.00 Wu et al. (2018) 0.486 1.153 11.20 Smith et al. (2017) 0.281 0.112 100.00 Wu et al. (2018) 0.120 0.444 4.50 Jensen et al. (2003) 0.280 0.118 100.00 Wu et al. (2018) 0.530 1.749 4.50 Jensen et al. (2003) 0.276 0.115 100.00 Wu et al. (2018) 0.335 1.173 4.50 Jensen et al. (2003) 0.344 0.127 100.00 Wu et al. (2018) 0.340 1.054 4.50 Jensen et al. (2003) 0.286 0.120 100.00 Wu et al. (2018) 0.620 1.922 4.50 Jensen et al. (2003) 0.286 0.122 100.00 Wu et al. (2018) Continued on next page 117 Table A.1 – continued from previous page H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.655 2.031 4.50 Jensen et al. (2003) 0.290 0.125 100.00 Wu et al. (2018) 0.201 0.158 57.29 Hafsteinsson et al. (2017) 0.152 0.098 100.00 Wu et al. (2018) 0.300 0.190 57.29 Hafsteinsson et al. (2017) 0.150 0.093 100.00 Wu et al. (2018) 0.401 0.216 57.29 Hafsteinsson et al. (2017) 0.149 0.088 100.00 Wu et al. (2018) 0.501 0.242 57.29 Hafsteinsson et al. (2017) 0.295 0.123 100.00 Wu et al. (2018) 0.601 0.269 57.29 Hafsteinsson et al. (2017) 0.293 0.123 100.00 Wu et al. (2018) 0.703 0.290 57.29 Hafsteinsson et al. (2017) 0.294 0.128 100.00 Wu et al. (2018) 0.201 0.440 19.08 Hafsteinsson et al. (2017) 0.303 0.122 100.00 Wu et al. (2018) 0.301 0.559 19.08 Hafsteinsson et al. (2017) 0.302 0.124 100.00 Wu et al. (2018) 0.401 0.614 19.08 Hafsteinsson et al. (2017) 0.303 0.123 100.00 Wu et al. (2018) 0.500 0.701 19.08 Hafsteinsson et al. (2017) 0.337 0.134 100.00 Wu et al. (2018) 0.602 0.801 19.08 Hafsteinsson et al. (2017) 0.337 0.133 100.00 Wu et al. (2018) 0.702 0.888 19.08 Hafsteinsson et al. (2017) 0.343 0.133 100.00 Wu et al. (2018) 0.201 0.609 9.51 Hafsteinsson et al. (2017) 0.346 0.138 100.00 Wu et al. (2018) 0.200 0.653 9.51 Hafsteinsson et al. (2017) 0.202 0.116 100.00 Wu et al. (2018) 0.300 0.814 9.51 Hafsteinsson et al. (2017) 0.316 0.134 100.00 Wu et al. (2018) 0.300 0.841 9.51 Hafsteinsson et al. (2017) 0.320 0.132 100.00 Wu et al. (2018) 0.300 0.870 9.51 Hafsteinsson et al. (2017) 0.312 0.126 100.00 Wu et al. (2018) 0.400 1.001 9.51 Hafsteinsson et al. (2017) 0.323 0.128 100.00 Wu et al. (2018) 0.401 1.046 9.51 Hafsteinsson et al. (2017) 0.322 0.134 100.00 Wu et al. (2018) 0.500 1.152 9.51 Hafsteinsson et al. (2017) 0.326 0.131 100.00 Wu et al. (2018) 0.500 1.188 9.51 Hafsteinsson et al. (2017) 0.151 0.094 100.00 Wu et al. (2018) 0.500 1.257 9.51 Hafsteinsson et al. (2017) 0.274 0.118 100.00 Wu et al. (2018) 0.601 1.310 9.51 Hafsteinsson et al. (2017) 0.282 0.119 100.00 Wu et al. (2018) 0.600 1.375 9.51 Hafsteinsson et al. (2017) 0.275 0.118 100.00 Wu et al. (2018) 0.600 1.449 9.51 Hafsteinsson et al. (2017) 0.299 0.125 100.00 Wu et al. (2018) 0.702 1.486 9.51 Hafsteinsson et al. (2017) 0.294 0.125 100.00 Wu et al. (2018) 0.701 1.518 9.51 Hafsteinsson et al. (2017) 0.296 0.123 100.00 Wu et al. (2018) 0.700 1.549 9.51 Hafsteinsson et al. (2017) 0.264 0.117 100.00 Wu et al. (2018) 0.042 0.120 5.00 Gedik et al. (2005) 0.264 0.116 100.00 Wu et al. (2018) 0.046 0.143 5.00 Gedik et al. (2005) 0.269 0.116 100.00 Wu et al. (2018) 0.050 0.150 5.00 Gedik et al. (2005) 0.248 0.122 100.00 Wu et al. (2018) 0.054 0.162 5.00 Gedik et al. (2005) 0.244 0.120 100.00 Wu et al. (2018) 0.059 0.177 5.00 Gedik et al. (2005) 0.249 0.115 100.00 Wu et al. (2018) 0.065 0.190 5.00 Gedik et al. (2005) 0.313 0.124 100.00 Wu et al. (2018) 0.067 0.192 5.00 Gedik et al. (2005) 0.312 0.131 100.00 Wu et al. (2018) 0.070 0.196 5.00 Gedik et al. (2005) 0.318 0.120 100.00 Wu et al. (2018) 0.071 0.199 5.00 Gedik et al. (2005) 0.148 0.090 100.00 Wu et al. (2018) 0.073 0.207 5.00 Gedik et al. (2005) 0.255 0.114 100.00 Wu et al. (2018) 0.075 0.220 5.00 Gedik et al. (2005) 0.257 0.118 100.00 Wu et al. (2018) 0.085 0.246 5.00 Gedik et al. (2005) 0.258 0.114 100.00 Wu et al. (2018) 0.087 0.248 5.00 Gedik et al. (2005) 0.324 0.124 100.00 Wu et al. (2018) 0.090 0.252 5.00 Gedik et al. (2005) 0.324 0.125 100.00 Wu et al. (2018) 0.114 0.320 5.00 Gedik et al. (2005) 0.327 0.127 100.00 Wu et al. (2018) 0.115 0.322 5.00 Gedik et al. (2005) 0.151 0.092 100.00 Wu et al. (2018) 0.116 0.327 5.00 Gedik et al. (2005) 0.199 0.108 100.00 Wu et al. (2018) 0.118 0.329 5.00 Gedik et al. (2005) 0.197 0.107 100.00 Wu et al. (2018) 0.120 0.333 5.00 Gedik et al. (2005) 0.328 0.129 100.00 Wu et al. (2018) 0.122 0.337 5.00 Gedik et al. (2005) 0.334 0.132 100.00 Wu et al. (2018) 0.124 0.339 5.00 Gedik et al. (2005) 0.335 0.131 100.00 Wu et al. (2018) 0.125 0.341 5.00 Gedik et al. (2005) 0.344 0.132 100.00 Wu et al. (2018) 0.127 0.346 5.00 Gedik et al. (2005) 0.344 0.134 100.00 Wu et al. (2018) 0.138 0.382 5.00 Gedik et al. (2005) 0.347 0.134 100.00 Wu et al. (2018) 0.150 0.412 5.00 Gedik et al. (2005) 0.203 0.107 100.00 Wu et al. (2018) 0.153 0.412 5.00 Gedik et al. (2005) 0.305 0.127 100.00 Wu et al. (2018) 0.161 0.438 5.00 Gedik et al. (2005) 0.306 0.126 100.00 Wu et al. (2018) 0.176 0.472 5.00 Gedik et al. (2005) 0.305 0.128 100.00 Wu et al. (2018) 0.179 0.480 5.00 Gedik et al. (2005) 0.184 0.099 100.00 Wu et al. (2018) 0.185 0.493 5.00 Gedik et al. (2005) 0.190 0.098 100.00 Wu et al. (2018) 0.190 0.510 5.00 Gedik et al. (2005) 0.363 0.128 100.00 Wu et al. (2018) 0.041 0.101 5.00 Gedik et al. (2005) 0.191 0.093 100.00 Wu et al. (2018) 0.048 0.106 5.00 Gedik et al. (2005) 0.193 0.096 100.00 Wu et al. (2018) 0.053 0.128 5.00 Gedik et al. (2005) 0.240 0.110 100.00 Wu et al. (2018) 0.060 0.130 5.00 Gedik et al. (2005) 0.254 0.109 100.00 Wu et al. (2018) 0.050 0.208 5.00 Gedik et al. (2005) 0.369 0.129 100.00 Wu et al. (2018) 0.065 0.174 5.00 Gedik et al. (2005) 0.366 0.126 100.00 Wu et al. (2018) 0.067 0.237 5.00 Gedik et al. (2005) 0.366 0.128 100.00 Wu et al. (2018) 0.074 0.205 5.00 Gedik et al. (2005) 0.188 0.093 100.00 Wu et al. (2018) 0.084 0.165 5.00 Gedik et al. (2005) 0.188 0.093 100.00 Wu et al. (2018) 0.086 0.190 5.00 Gedik et al. (2005) 0.304 0.116 100.00 Wu et al. (2018) 0.084 0.224 5.00 Gedik et al. (2005) 0.307 0.118 100.00 Wu et al. (2018) 0.088 0.210 5.00 Gedik et al. (2005) 0.311 0.123 100.00 Wu et al. (2018) 0.113 0.291 5.00 Gedik et al. (2005) 0.342 0.125 100.00 Wu et al. (2018) 0.137 0.286 5.00 Gedik et al. (2005) 0.349 0.125 100.00 Wu et al. (2018) Continued on next page 118 Table A.1 – continued from previous page H/d [-] R/d [-] Slope [ o ] Source H/d [-] R/d [-] Slope [ o ] Source 0.118 0.473 5.00 Gedik et al. (2005) 0.342 0.126 100.00 Wu et al. (2018) 0.123 0.501 5.00 Gedik et al. (2005) 0.306 0.111 100.00 Wu et al. (2018) 0.151 0.465 5.00 Gedik et al. (2005) 0.193 0.106 100.00 Wu et al. (2018) 0.150 0.482 5.00 Gedik et al. (2005) 0.189 0.099 100.00 Wu et al. (2018) 0.152 0.508 5.00 Gedik et al. (2005) 0.194 0.103 100.00 Wu et al. (2018) 0.161 0.494 5.00 Gedik et al. (2005) 0.376 0.140 100.00 Wu et al. (2018) 0.161 0.566 5.00 Gedik et al. (2005) 0.201 0.108 100.00 Wu et al. (2018) 0.175 0.512 5.00 Gedik et al. (2005) 0.196 0.098 100.00 Wu et al. (2018) 0.179 0.530 5.00 Gedik et al. (2005) 0.361 0.136 100.00 Wu et al. (2018) 0.185 0.380 5.00 Gedik et al. (2005) 0.374 0.144 100.00 Wu et al. (2018) 0.191 0.337 5.00 Gedik et al. (2005) 0.364 0.129 100.00 Wu et al. (2018) 0.205 0.613 5.00 Gedik et al. (2005) 0.377 0.150 100.00 Wu et al. (2018) 0.274 0.435 5.00 Gedik et al. (2005) 0.374 0.133 100.00 Wu et al. (2018) 0.277 0.629 5.00 Gedik et al. (2005) 0.385 0.119 100.00 Wu et al. (2018) 0.277 0.680 5.00 Gedik et al. (2005) 0.258 0.128 100.00 Wu et al. (2018) 0.277 0.731 5.00 Gedik et al. (2005) 0.255 0.110 100.00 Wu et al. (2018) 0.291 0.732 5.00 Gedik et al. (2005) 0.354 0.123 100.00 Wu et al. (2018) 0.295 0.681 5.00 Gedik et al. (2005) 0.341 0.144 100.00 Wu et al. (2018) 0.338 0.627 5.00 Gedik et al. (2005) 0.347 0.131 100.00 Wu et al. (2018) 0.183 0.429 3.50 Gedik et al. (2005) 0.387 0.138 100.00 Wu et al. (2018) 0.186 0.373 3.50 Gedik et al. (2005) 0.396 0.125 100.00 Wu et al. (2018) 0.208 0.437 3.50 Gedik et al. (2005) 0.385 0.119 100.00 Wu et al. (2018) 0.216 0.525 3.50 Gedik et al. (2005) 0.250 0.099 100.00 Wu et al. (2018) 0.272 0.625 3.50 Gedik et al. (2005) 0.257 0.110 100.00 Wu et al. (2018) 0.329 0.609 3.50 Gedik et al. (2005) 0.343 0.119 100.00 Wu et al. (2018) 0.319 0.663 3.50 Gedik et al. (2005) 0.347 0.126 100.00 Wu et al. (2018) 0.330 0.686 3.50 Gedik et al. (2005) 0.400 0.134 100.00 Wu et al. (2018) 0.335 0.732 3.50 Gedik et al. (2005) 0.375 0.123 100.00 Wu et al. (2018) 0.157 0.334 2.50 Gedik et al. (2005) 0.384 0.123 100.00 Wu et al. (2018) 0.167 0.325 2.50 Gedik et al. (2005) 0.405 0.119 100.00 Wu et al. (2018) 0.227 0.482 2.50 Gedik et al. (2005) 0.325 0.130 100.00 Wu et al. (2018) 0.249 0.504 2.50 Gedik et al. (2005) 0.324 0.113 100.00 Wu et al. (2018) 0.288 0.562 2.50 Gedik et al. (2005) 0.318 0.109 100.00 Wu et al. (2018) 0.317 0.612 2.50 Gedik et al. (2005) 0.394 0.131 100.00 Wu et al. (2018) 0.330 0.578 2.50 Gedik et al. (2005) 0.403 0.125 100.00 Wu et al. (2018) 0.333 0.614 2.50 Gedik et al. (2005) 0.394 0.126 100.00 Wu et al. (2018) 0.137 0.460 9.35 Baldock et al. (2012) 0.398 0.126 100.00 Wu et al. (2018) 0.124 0.643 9.35 Baldock et al. (2012) 0.256 0.105 100.00 Wu et al. (2018) 0.157 0.545 9.35 Baldock et al. (2012) 0.307 0.118 100.00 Wu et al. (2018) 0.165 0.591 9.35 Baldock et al. (2012) 0.317 0.121 100.00 Wu et al. (2018) 0.175 0.589 9.35 Baldock et al. (2012) 0.388 0.126 100.00 Wu et al. (2018) 0.207 0.613 9.35 Baldock et al. (2012) 0.392 0.129 100.00 Wu et al. (2018) 0.216 0.786 9.35 Baldock et al. (2012) 0.391 0.129 100.00 Wu et al. (2018) 0.234 0.763 9.35 Baldock et al. (2012) 0.408 0.131 100.00 Wu et al. (2018) 0.267 0.805 9.35 Baldock et al. (2012) 0.404 0.129 100.00 Wu et al. (2018) 0.275 0.885 9.35 Baldock et al. (2012) 0.405 0.119 100.00 Wu et al. (2018) 0.319 0.913 9.35 Baldock et al. (2012) 0.260 0.118 100.00 Wu et al. (2018) 0.329 1.054 9.35 Baldock et al. (2012) 0.395 0.134 100.00 Wu et al. (2018) 0.363 1.019 9.35 Baldock et al. (2012) 0.399 0.129 100.00 Wu et al. (2018) 0.406 1.087 9.35 Baldock et al. (2012) 0.412 0.131 100.00 Wu et al. (2018) 119 10 -2 10 -1 10 0 10 -2 10 -1 10 0 Synolakis (1987) Sriram et al. (2016) Hsiao et al. (2008) Chang et al. (2009) Li & Raichken (2001) Li & Raichken (2002) Briggs et al. (1995) Smith et al (2017) Jensen et al. (2003) Hafsteimsson et al. (2017) Gedik et al. (2005) Baldock et al. (2012) Lo et al. (2013) Wu et al. (2018) Figure A.1: The normalized maximum runup of solitary waves versus the normalized wave height. All runup laboratory data available in literature for breaking and non–breaking solitary waves on a sloping beach of constant slope (see Table A.1). 120 10 -2 10 -1 10 0 10 1 10 -2 10 -1 10 0 Synolakis (1987) Sriram et al. (2016) Hsiao et al. (2008) Chang et al. (2009) Li & Raichken (2001) Li & Raichken (2002) Briggs et al. (1995) Smith et al (2017) Jensen et al. (2003) Hafsteimsson et al. (2017) Gedik et al. (2005) Baldock et al. (2012) Lo et al. (2013) Wu et al. (2018) Figure A.2: The normalized maximum runup of solitary waves climbing up different beaches versus the normalized wave height (see Table A.1). 121 10 0 10 1 0 1 2 3 4 5 6 Synolakis (1987) Sriram et al. (2016) Hsiao et al. (2008) Chang et al. (2009) Li & Raichken (2001) Li & Raichken (2002) Briggs et al. (1995) Smith et al (2017) Jensen et al. (2003) Hafsteimsson et al. (2017) Gedik et al. (2005) Baldock et al. (2012) Lo et al. (2013) Wu et al. (2018) Equation Wu et al. 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Abstract (if available)

Abstract In effective crisis management, prevention, preparedness, response and recovery are of crucial importance to improve resilience. It is inevitable that humanity will often be be faced with natural disasters and other threats, and survival hinges on knowledge and experience at all levels of risk management at local, national and global levels. In this regard, being able to identify previously unrecognized threats is essential. ❧ This is a study of the runup amplification in the shadow zone of offshore islands. Runup amplification (RA) is the ratio of the runup along the beach shadowed by the island to the runup that would be observed if there was no island present. Field survey reports from recent tsunamis suggest that local residents in mainland areas shadowed by nearby islands maybe under the impression that these islands protect them from tsunamis. Recent numerical results using machine learning and examining only non-breaking waves have generated substantial attention in world media, because they suggest that, in most cases, islands amplify tsunamis in the shadow zones behind them. ❧ The study uses the idealized bathymetry of a circular island fronting a uniformly sloping beach, as in the study of Stefanakis et. al. (2014). The bulk of the study used MOST, a NSW solver, while several simulations were run with the Boussinesq-type equation solver COULWAVE, The study used measurements from laboratory experiments to assess the veracity of both codes at small scales, then exact solutions of the SW to check the veracity of MOST at geophysical scales. ❧ Two sets of experiments are presented, with one and two 1:2 slope islands fronting a 1:10 uniformly sloping beach. The particle tracking algorithm implemented by Kalligeris et. al. (2017) was used to track the particles between successive video frames. For a given inter-particle spacing, the tracking became more difficult as the flow speed increases. In other words, for faster flows, one needs more frames per second. The detection threshold turned out as 1 m/sec. ❧ The laboratory measurements were compared with both analytical results and numerical predictions. It was found that the analytical formulation of Kânoğlu & Synolakis (1998) proved capable of capturing the maximum runup for the one island configuration, where it is valid, there is no analytical solution for two islands. Also, it appears that both MOST and COULWAVE reproduce the maximum runup very well around the island and along the sloping beach shadowed by the island. COULWAVE also reproduces the runup velocities very well. Surprisingly, for very long waves on very steep beaches, the classic analytical expression for a solitary wave climbing up a sloping beach and known as Synolakis' runup law does not work, instead, it was determined that the runup amplification R/H asymptotes to RA=2. ❧ The results suggest that both runup ampplification (RA>1) and shadowing (RA<1) occur, depending primarily on whether the initial wave breaks of not on the front face of the island. Shorter solitary-like initial waves break on islands with mild faces and the islands do not shelter the coastlines behind them. Longer waves climbing up islands with steeper front faces evolve in the beginning as predicted by Synolakis (1991), then continue to climb up the sloping beach and produce high runup amplification. This is the case more likely to occur in nature. ❧ The counter intuitive phenomena described here suggest public education is urgently needed to help people identify locales at higher risk than others, as is the provision of high-resolution inundation maps which account for such phenomena, for evacuation planning. ❧ Successful responses in natural disasters, include understanding the risk, reporting the related disasters and formulating a communication code with all involved. Communication tailored to the needs of citizens, civil servants, political institutions and the media, will play a significant role in reducing significantly human cost.

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Asset Metadata

Creator Skanavis, Vassilios (author)

Core Title Nonlinear long wave amplification in the shadow zone of offshore islands

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 11/30/2020

Defense Date 12/13/2020

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

Tag image analysis,laboratory data,long wave,OAI-PMH Harvest,runup,tsunami

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Synolakis, Costas (committee chair)

Creator Email billskan@gmail.com,skanavis@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-400778

Unique identifier UC11668212

Identifier etd-SkanavisVa-9168.pdf (filename),usctheses-c89-400778 (legacy record id)

Legacy Identifier etd-SkanavisVa-9168.pdf

Dmrecord 400778

Document Type Dissertation

Rights Skanavis, Vassilios

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA