Resonant Interactions of Surface and Internal Waves with Seabed Topography By Louis-Alexandre Couston A dissertation submitted in partial satisfaction of the requirements for the degree of Doctorate of Philosophy in Engineering - Mechanical Engineering in the Graduate Division of the University of California, Berkeley Committee in Charge: Professor Mohammad-Reza Alam, Chair Professor Ronald W. Yeung Professor Philip S. Marcus Professor Per-Olof Persson Spring 2016 Abstract Resonant Interactions of Surface and Internal Waves with Seabed Topography by Louis-Alexandre Couston Doctor of Philosophy in Engineering - Mechanical Engineering University of California, Berkeley Professor Mohammad-Reza Alam, Chair This dissertation provides a fundamental understanding of water-wave transformations over seabed corrugations in the homogeneous as well as in the stratified ocean. Contrary to a flat or mildly sloped seabed, over which water waves can travel long distances undisturbed, a seabed with small periodic variations can strongly affect the propagation of water waves due to resonant wave-seabed interactions{a phenomenon with many potential applications. Here, we investigate theoretically and with direct simulations several new types of wave transformations within the context of inviscid fluid theory, which are different than the classical wave Bragg reflection. Specifically, we show that surface waves traveling over seabed corrugations can become trapped and amplified, or deflected at a large angle (∼ 90◦) relative to the incident direction of propagation. Wave trapping is obtained between two sets of parallel corrugations, and we demonstrate that the amplification mechanism is akin to the Fabry-Perot resonance of light waves in optics. Wave deflection requires three-dimensional and bi-chromatic corrugations and is obtained when the surface and corrugation wavenumber vectors satisfy a newly found class I2 Bragg resonance condition. Internal waves propagating over seabed topography in a stratified fluid can exhibit similar wave trapping and deflection behaviors, but more surprising and intricate internal wave dynamics can also be obtained. Unlike surface waves, internal waves interacting with monochromatic seabed corrugations can simultaneously generate many new waves with different wavenumbers and directions of propagation{a phenomenon which we call chain resonance. Here, we demonstrate that the chain resonance leads to significant energy transfer from long internal waves to short internal waves for almost all angles of the incident waves relative to the orientation of the oblique seabed corrugations. Since short internal waves are prone to breaking, the resonance mechanism, therefore, may have important consequences on the spatial variability of ocean mixing and energy dissipation. Potential applications of the theory of resonant wave-seabed interactions for wave energy extraction and coastal protection are also discussed. 1 A` mes parents, mes fr`eres et ma soeur. Aux penseurs et r^eveursattir´espar les sciences, la d´ecouverte et l'Univers. \La science cherche le mouvement perp´etuel.Elle l'a trouv´e; c'est elle-m^eme.[...] La colossale machine Science ne se repose jamais ; elle n'est jamais satisfaite ; elle est insatiable du mieux, que l'absolu ignore. " Victor Hugo, William Shakespeare (L'Art et la science), 1864 i Acknowledgments It has been an honor and privilege to study at the University of California, Berkeley, alongside professors, researchers, and friends with such impressive and diverse backgrounds, interests, and aspirations. My own research, and the joy I had conducting it, mostly hinges on one person, Professor Mohammad-Reza Alam, who was my research advisor for the past four years and whom I would like to most sincerely thank. Reza is a crafty, sharp, and pragmatic researcher. Adding up the number of hours we spent talking about research, life, and how to write papers, proposals, and news articles would be disconcerting{as it was very many. Reza was relentless in asking for smarter thinking and better delivery, which was frustrating at times, but his persistence paid off as I now feel confident to go out on my own and embrace new, thrilling scientific challenges. I hope that Reza's pragmatism and understanding of what it takes to become a happy and successful researcher in a highly paced, competitive field will be transmitted to all new students of our Theoretical and Applied Fluid Dynamics Lab. My scientific journey at Berkeley was also enriched thanks to a number of professors in the Departments of Mechanical Engineering and Mathematics. I would like to begin by ex- pressing my sincere gratitude to Professor Ronald W. Yeung, who accepted my application to join the Ocean Engineering program at Berkeley, which he oversees. I still vividly remem- ber my excitement before, during, and after our first phone call in February 2011, as well as during my first interactions with him in my hydrodynamics classes. Several professors made themselves available after class to discuss and help me with my research projects. Professors Per-Olof Persson, John Strain, and Jon Wilkening in Mathematics always had their doors open and have been inspiring forces in applied mathematics. In Mechanical Engineering, I found the teaching styles of Professors Tarek I. Zohdi and Philip S. Marcus most interesting, in addition to being very effective; one thing I'll remember for sure is that in numerics, nothing is magic. I would also like to thank Maysamreza Chamanzar and Professor Mir Abbas Jalali who co-authored two papers included in this dissertation. They shared their enthusiasm and expertise about my research, but also about their interests in photonics and planetary sciences. Finally, I wish to thank the professors on my prelims, quals, and disser- tation committees{Ronald W. Yeung, Philip S. Marcus, Omer¨ Sava¸s,Per-Olof Persson, and M.-Reza Alam; thank you for taking the time to assist me, at every step along the way, in completing my PhD degree requirements. It is my intent to stay close to my Bay Area friends, who made this experience delightful and adventurous. Special thanks go to my lab mates Yong Liang and Qiuchen Guo who co- authored two of my papers, as well as to Ahmad Zareei, Joel Tchoufag and Farid Karimpour; to my CMML friends Yichen Jiang, Daniel Grieb, Daewoong Son and Farshad Madhi; and to Anna Lieb and Fran¸coisSoubiran. They have all been very helpful throughout my graduate studies. I will greatly miss my housemates Raja Sutherland, Loni Gray, TJ Sutherland, Jaclyn Leaver, and Evan Pease; what an amazing American family. Fortunately, I will keep joyful memories of shared moments with them, as well as of my friends Soazig Kaam, Christopher Lalau-Keraly, Benjamin Fildier, Karina Cucchi, Sam Kanner, Emma Dobbins, Siva Rama Satyam Bandaru, Katya Cherukumilli, Gabrielle Boisrame, Gopal Penny, and ii Clare Loughran. I am lucky and proud to count all of these amazing people as very dear friends in this world. It is heartbreaking to leave California, where I have had such a wonderful time. Fortu- nately, I owe the highlights of these past five years to being with Caroline Delaire, who I will continue my adventures with. There are no words to describe how important Caroline is to both my personal and professional life{she is driven in her own work, in her relations with other people, and in her fight for equality{and because of her, I rest assured that there is some good in this world, and that the good is strong enough that it will bring happiness to those without it. It is one of my life goals to give her the strength to keep going, for she will change the world for the better. Last but not least, I would like to thank my parents, Alexandra and Vincent Couston; my brothers, Geoffrey and Hubert; my sister Mathilde; and all of my extended family members for their everlasting love and support during this experience far from home. I have always and will continue to crave the moments we get to spend together; they provide me with a warm and comforting feeling that transcends the necessities of life, the difficult times, and my own deepest expectations. Resonant Interactions of Surface and Internal Waves with Seabed Topography c 2016 by Louis-Alexandre Couston iii iv Contents List of Figures vii List of Tables xv 1 Introduction 1 1.1 Introductory remarks . .1 1.2 Theory of water-wave propagation . .2 1.3 Linear wave theory in intermediate water depth . .5 1.4 Resonant wave{seabed interactions . 10 1.5 Dissertation outline . 11 2 Fabry-Perot resonance of water waves 13 2.1 Introduction . 13 2.2 Bragg mirrors for water waves . 14 2.3 Fabry-Perot resonance . 17 2.4 Conclusions . 21 3 Shore protection by oblique seabed bars 25 3.1 Introduction . 25 3.2 Problem formulation . 27 3.3 Class I Bragg deflection for shore protection . 31 3.4 Class I2 Bragg deflection for shore protection . 34 3.4.1 Closed-form solution far from the patch's plane of symmetry . 34 3.4.2 Effect of the patch's plane of symmetry . 37 3.5 Direct simulation . 42 3.5.1 Cross-validation . 42 3.5.2 Effect of detuning and protection efficiency . 43 3.5.3 Perpendicular deflection . 48 3.6 Conclusions . 50 4 Oblique chain resonance of internal waves 55 4.1 Introduction . 55 4.2 Problem formulation . 58 4.2.1 Governing equation and the dispersion relation . 58 4.2.2 Chain resonance conditions . 60 4.3 Three-dimensional multiple-scale analysis . 63 4.3.1 Derivation of the envelope equations . 63 4.3.2 Solution method at the steady state . 67 4.4 Results and discussion . 69 4.4.1 Oblique chain resonance with three perfect resonances . 70 4.4.2 Chain resonance at all corrugation angles . 72 4.4.3 Spatial dynamics of chain resonance . 74 v Contents 4.4.4 Energy flux re-distribution .