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Chaotic advection, mixing, and property exchange in three-dimensional ocean eddies and gyres by Genevieve Jay Brett Submitted to the Joint Program in Applied Ocean Science & Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY and the WOODS HOLE OCEANOGRAPHIC INSTITUTION June 2018 ©2018 Genevieve Brett. All rights reserved. The author hereby grants to MIT and WHOI permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author................................................................................... Joint Program in Physical Oceanography Massachusetts Institute of Technology & Woods Hole Oceanographic Institution May 11, 2018 Certified by. Larry Pratt Senior Scientist Woods Hole Oceanographic Institution Thesis Supervisor Certified by. Irina Rypina Associate Scientist Woods Hole Oceanographic Institution Thesis Supervisor Accepted by.............................................................................. Larry Pratt Chairman, Joint Committee for Physical Oceanography Woods Hole Oceanographic Institution 2 Chaotic advection, mixing, and property exchange in three-dimensional ocean eddies and gyres by Genevieve Jay Brett Submitted to the Joint Program in Physical Oceanography Massachusetts Institute of Technology & Woods Hole Oceanographic Institution on May 11, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This work investigates how a Lagrangian perspective applies to models of two oceanographic flows: an overturning submesoscale eddy and the Western Alboran Gyre. In the first case, I focus on the importance of diffusion as compared to chaotic advection for tracers in this system. Three methods are used to quantify the relative contributions: scaling arguments including a Lagrangian Batchelor scale, statistical analysis of ensembles of trajectories, and Nakamura effective diffusivity from numerical simulations of dye release. Through these complementary methods, I find that chaotic advection dominates over turbulent diffusion in the widest chaotic regions, which always occur near the center and outer rim of the cylinder and sometimes occur in interior regions for Ekman numbers near 0.01. In thin chaotic regions, diffusion is at least as important as chaotic advection. From this analysis, it is clear that identified Lagrangian coherent structures will be barriers to transport for long times if they are much larger than the Batchelor scale. The second case is a model of the Western Alboran Gyre with realistic forcing and bathymetry. I examine its transport properties from both an Eulerian and Lagrangian perspective. I find that advection is most often the dominant term in Eulerian budgets for volume, salt, and heat in the gyre, with diffusion and surface fluxes playing a smaller role. In the vorticity budget, advection is aslarge the effects of wind and viscous diffusion, but not dominant. For the Lagrangian analysis, Iconstruct a moving gyre boundary from segments of the stable and unstable manifolds emanating from two persistent hyperbolic trajectories on the coast at the eastern and western extent of the gyre. These manifolds are computed on several isopycnals and stacked vertically to construct a three-dimensional Lagrangian gyre boundary. The regions these manifolds cover is the stirring region, where there is a path for water to reach the gyre. On timescales of days to weeks, water from the Atlantic Jet and the northern coast can enter the outer parts of the gyre, but there is a core region in the interior that is separate. Using a gate, I calculate the continuous advective transport across the Lagrangian boundary in three dimensions for the first time. A Lagrangian volume budget is calculated, and challenges in its closure are described. Lagrangian and Eulerian advective transports are found to be of similar magnitudes. 3 Thesis Supervisor: Larry Pratt Title: Senior Scientist Woods Hole Oceanographic Institution Thesis Supervisor: Irina Rypina Title: Associate Scientist Woods Hole Oceanographic Institution 4 Biography Genevieve Brett, called Jay by her friends and colleagues, was raised in Connecticut. She attended Hopkins School in New Haven before studying for her B.A. in Mathematics and Physics at Skidmore College in Saratoga Springs, NY. Her first major research was done with Jim Gunton and his group at Lehigh University in the summer of 2011. She was then introduced to physical oceanography by Greg Gerbi, who joined the Skidmore physics faculty in 2011. Encouraged by Gerbi, she was a Winter Fellow at WHOI in 2012, working with Young-Oh Kwon, Magdalena Andres, and then- student Derya Akkaynak. During that fellowship, she became interested in Larry Pratt’s work, which led to her joining the MIT-WHOI Joint Program later that year. After graduation, Jay will begin a postdoctoral position with Kelvin Richards of the University of Hawai’i M¯anoa. Acknowledgments Funding for this work was provided by Woods Hole Oceanographic Institution and the Office of Naval Research. The ONR support was through the MURI on Dynamical Systems Theory and Lagrangian Data Assimilation in 4D and the CALYPSO project. I would like to thank the scientists, admins, and peers who made the Joint Program a welcoming place for me. Everyone who has asked me about my work and listened to me talk through an issue with it has contributed— thank you. Of course, I would not have gotten to this point without my advisors, who were with me at every turn, or my committee, who shared their insights. Thanks to Greg Gerbi, Magdalena Andres, and Young-Oh Kwon for my introduction to oceanography and my excellent experience as a Winter Fellow at WHOI. I also thank Mike Neubert and the math ecology group for reminding me that math is fun. My family, both biological and chosen, and especially my partner, has supported me throughout my studies. I will be forever thankful to you all, and I hope I am supporting you well in turn. 5 6 Contents 1 Introduction 25 2 Competition between Chaotic Advection and Diffusion: Stirring and Mixing in a 3D Eddy Model 35 2.1 Introduction . 36 2.2 Kinematic Model . 38 2.2.1 Steady Symmetry-Breaking Perturbation . 40 2.2.2 Comparison to Dynamic Model . 42 2.3 Observing Chaotic Advection: Can it overcome diffusion? . 45 2.3.1 Scaling Derivations . 47 2.3.2 Scaling Results . 53 2.3.3 Spreading of Trajectory Ensembles . 55 2.3.4 Tracer Release Simulations . 62 2.4 Discussion . 72 2.5 Conclusion . 74 2.6 References . 77 3 The Western Alboran Gyre: an Eulerian Analysis of its Properties and their Budgets 81 3.1 Introduction and Background . 83 3.2 Model Description . 85 3.2.1 Model Setup . 85 7 3.2.2 Model Validation . 87 3.2.3 Model Output Description . 88 3.3 Results . 96 3.3.1 Salt and Heat . 102 3.3.2 Dynamics of the WAG: Vorticity . 113 3.4 Discussion and Conclusions . 120 3.5 References . 122 4 Chaotic Advection in the Alboran Sea, I: Lagrangian Geometrical Analysis of the Western Alboran Gyre 125 4.1 Introduction and Background . 127 4.2 Methods . 129 4.2.1 Model . 129 4.2.2 Velocity Field Analysis . 130 4.3 Results . 134 4.3.1 WAG core and stirring regions . 134 4.3.2 Advective Transport by Lobes . 135 4.4 Discussion and Conclusions . 139 4.5 References . 143 5 Chaotic Advection in the Alboran Sea, II: Lagrangian Analysis of Property Budgets of the Western Alboran Gyre 145 5.1 Introduction . 147 5.2 Methods . 147 5.3 Results . 151 5.3.1 Gate Transport . 151 5.3.2 Budgets using the gate method . 152 5.4 Discussion . 166 5.5 References . 168 6 General Discussion and Conclusions 171 8 A Rotating Cylinder Appendix 177 A.1 Bifurcation Analysis . 177 A.2 Gaussian Tracer in Linear Strain . 180 A.3 Nakamura Effective Diffusivity . 183 A.3.1 Derivation . 183 A.3.2 Long-time Limit . 187 B Eulerian Analysis Appendix 189 B.1 MITgcm Output Comparison to Climatology . 189 B.1.1 5m depth . 189 B.1.2 100m depth . 198 B.1.3 400m depth . 206 B.1.4 1000m depth . 214 B.2 MITgcm Eulerian Budget Methods . 214 B.2.1 Overview . 214 B.2.2 Diagnostics . 214 B.2.3 Budget Terms . 225 B.3 Western Alboran Gyre Budget Examples . 231 B.3.1 Overview . 231 B.3.2 Full WAG salt budget . 231 B.3.3 Momentum Budgets . 233 C Biological-Physical Interactions in the Alboran Sea 239 C.1 Phytoplankton in the Alboran Sea: Physical Interactions and Functional Types . 239 C.1.1 Introduction and Background . 240 C.1.2 Physics . 240 C.1.3 Biology . 241 C.1.4 Proposed Work . 243 C.1.5 Goal and Questions . 243 C.1.6 Approach . ..