ENERGETICS AND DYNAMICS OF INTERNAL TIDES IN MONTEREY BAY USING NUMERICAL SIMULATIONS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Dujuan Kang November 2010 © 2011 by Kang Dujuan. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/sv691gk5449 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Oliver Fringer, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Stephen Monismith I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Robert Street Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract Mixing processes in the ocean play a key role in controlling the large-scale circulation and energy distribution of the ocean. Internal tide-driven mixing is most important among the processes to mix the ocean interior. In the past decade, significant efforts have been made to understand tidal mixing processes. However, more details and better understanding are still required for some fundamental problems, such as the mechanisms that govern internal tide generation, radiation, and dissipation processes and the associated energy partitioning. This research aims to understand the energetics and dynamics of tidal mixing processes through both theoretical analysis and numerical simulations. The complete form of barotropic and baroclinic energy equations are derived and employed as the theoretical framework for analyzing the tidal energy budget. These equations provide a more accurate and detailed energy analysis because they include the full nonlinear and nonhydrostatic energy flux contributions as well as an improved evaluation of the available potential energy. This approach has been implemented in the hydrodynamic SUNTANS model, which is being employed to study the energetics of barotropic-to- baroclinic tidal conversion over complex bathymetry in the real ocean. Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in the Monterey Bay area are conducted using the SUNTANS model. A de- tailed analysis of the energy budget is performed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy. After that, we then assess how much of this generated baroclinic energy is lost locally versus how much is radiated away and made available for open-ocean mixing. The mechanism of internal tide generation is investigated iv by examining the dependence of barotropic-to-baroclinic energy conversion on three nondimensional parameters, namely the steepness parameter, the tidal excursion pa- rameter, and the Froude number. Finally, a simple parametric model is presented to estimate the barotropic-to-baroclinic energy conversion. v Acknowledgements It is a pleasure to thank everyone who provided me support and made this disser- tation possible. My sincerest gratitude goes to my advisor, Dr. Oliver Fringer, for his excellent guidance, encouragement, and support throughout my PhD study. He taught me numerous modelling skills beyond textbooks and inspired me with his in- novative thinking. Oliver was always available to discuss research and give advice. His patience and encouragement helped me overcome many difficulties and achieve my best. I am truly grateful to have such a wonderful mentor. I am especially grateful to Dr. Robert Street for leading me into the area of ocean sciences and giving me invaluable advice on my studies, research, and future career path. His careful dissertation reading and detailed comments are sincerely appreci- ated. I am very thankful to Dr. Stephen Monismith for helping me develop theoretical background. Many thanks to his insightful comments and useful suggestions to im- prove my dissertation. I would also like to thank Dr. Leif Thomas for serving as one of my oral exam committee members, and for his great ideas to broaden my research horizons. Finally, I thank my defense chair, Dr. Gianluca Iaccarino for his time and kind help. A special thank you goes to Dr. Rocky Geyer for his encouragement and insightful advice on my research and papers. I would also like to thank Dr. Karan Venayag- amoorthy and Dr. Alan Blumberg for providing useful comments and suggestions on my papers. I am very thankful to Dr. Steven Jachec, who has helped me with simulation setup and data analysis. His pioneer work in modelling internal tides in Monterey Bay has been an inspiration to me. I also thank Dr. Jody Klymak, Dr. James Girton, vi Dr. Eric Kunze and Samantha Brody for kindly providing field data in support of this dissertation. Many thanks to Dr. Edward Gross and the UCLA ROMS group for their great help with my research work on the ROMS-SUNTANS coupling. I would like to thank Dr. Jeffrey Koseff, who helped me adapt to the new study environment when I first came to Stanford and gave me great support when I was searching for future career opportunities. I greatly appreciate Dr. Dale Haidvogel and Dr. Enrique Curchitser for their support and understanding, which allow me to focus on completing this dissertation. My sincere thanks also go to Dr. Andreas Thurnherr and Dr. Sonya Legg for useful discussions and suggestions on my research. The Environmental Fluid Mechanics Laboratory (EFML) is a comfortable place to work in. I feel very fortunate to have been part of it and would like to thank all the faculty, researchers, post-docs, students, and staffs for creating such a big warm family. A special thank you goes to our research group, including the current mem- bers: Bing Wang, Yi-Ju Chou, Subbayya Sankaranarayanan, Vivien Chua, Phillip Wolfram, Sean Vitousek, Goncalo Gil, Sergey Koltakov, and the former: Zhonghua Zhang, Steven Jachec, Karan Venayagamoorthy, Jan Wang, Gang Zhao, Sheng Chen, Mike Barad, and Jun Lee. I also thank Jill Nomura and Sandra Wetzel for the kind help. I have been very lucky to have many good friends during my study at Stanford. I really enjoyed knowing all of them. I am particularly thankful to Yifang Chen for her support and care when I was writing this dissertation. My deepest gratitude is to my parents, whose endless love and constant encouragement have always been the strongest power supporting me. Finally, I give my unique thanks to my husband, Wei Li. This PhD dissertation is dedicated to him for his love, care, patience, and support. Thank you. My research and dissertation were funded by ONR Grant N00014-05-1-0294 (Sci- entific officers: Dr. C. Linwood Vincent, Dr. Terri Paluszkiewicz and Dr. Scott Harper). This support is gratefully acknowledged. vii Dedication To my husband, Wei Li, for his love, care, patience, and support. viii Contents Abstract iv Acknowledgements vi Dedication viii 1 Introduction 1 1.1 Motivation and Background . 1 1.2 Project Objectives . 4 1.3 Dissertation Layout . 5 2 Literature Review 6 2.1 Linear Internal Wave Theory . 6 2.2 Internal Tide Generation Mechanisms . 10 2.3 Numerical Modelling of Internal Tides . 15 2.4 Internal Tides in Monterey Bay . 16 2.4.1 Field Observations . 17 2.4.2 Numerical Simulations . 17 2.5 Summary . 20 3 Numerical Methodology 22 3.1 SUNTANS . 22 3.2 Nonhydrostatic Pressure Solver: Time Accuracy . 24 3.2.1 Governing Equations . 25 ix 3.2.2 Numerical Discretization . 26 3.2.3 Time Accuracy of Pressure Methods . 27 3.2.4 Temporal Convergence Test . 29 3.2.5 Conclusions . 34 3.3 Nonhydrostatic Pressure Solver: Computational Efficiency . 34 3.3.1 Pressure-Poisson Equation . 35 3.3.2 Preconditioned Conjugate Gradient Method . 37 3.3.3 Convergence Test of CG and PCG . 38 3.3.4 Conclusions . 41 3.4 One-way ROMS-SUNTANS Coupling . 41 3.4.1 ROMS vs. SUNTANS . 41 3.4.2 One- and Two-way Coupling . 42 3.4.3 Interpolation Algorithm . 45 3.4.4 Implementation of One-way Nesting . 49 3.5 Summary . 50 4 Calculation of Available Potential Energy in Internal Wave Fields 51 4.1 Introduction . 52 4.2 Interpretation of APE . 53 4.3 Energy Conservation Laws . 55 4.4 Energetics of Progressive Internal Waves . 57 4.4.1 Numerical Setup . 57 4.4.2 Evolution of First-mode Internal Waves . 58 4.4.3 Energetics . 60 4.5 Conclusions . 63 5 Theoretical Framework: Barotropic and Baroclinic Energy Equa- tions 64 5.1 Governing Equations . 64 5.1.1 Boundary Conditions . 66 5.1.2 Definitions and Assumptions . 68 5.2 Energy Equations . 72 x 5.2.1 Kinetic Energy Equation . 73 5.2.2 Perturbation Potential Energy Equation . 74 5.2.3 Available Potential Energy Equation . 74 5.2.4 Total Energy Equation . 75 5.3 Barotropic Energy Equations . 77 5.4 Baroclinic Energy Equations . 79 6 Numerical Simulations: Energetics and Dynamics of Internal Tides in the Monterey Bay Area 84 6.1 Introduction . 85 6.2 Theoretical Framework . 88 6.3 Simulation Setup . 92 6.3.1 The SUNTANS Model . 92 6.3.2 Domain and Grid .