The Vertical Structure of the Wind-driven Circulation

This thesis consists of three loosely related theoretical studies. In chapters 1 3 the physical mechanisms which determine the three dimensional structure of the currents in the Sverdrup interior of a wind-driven gyre are discussed. A variety of simple analytic models suggest that the subsurface geostrophic contours in a wind gyre are closed and so the flow in these regions is not determined by lateral boundary conditions. Instead a turbulent, quasigeostrophic extension of the Batchelor-Prandtl theorem suggests that the potential vorticity is uniform inside these laterally isolated regions. The requirement that the potential vorticity be uniform leads simply and directly to predictions of the shape and extent of the wind gyre and the vertical structure of the currents within it. In chapter 4 the propogation of Rossby wave -trains through slowly varying forced mean flows is examined by solving the linearized potential vorticity equation using the WKB method. If the mean flow is forced the action defined by Bretherton and Garrett (1968) is not conserved. Surprisingly, there is another quadratic wave property which is conserved, the wave enstrophy. In chapter 5 shear dispersion in an oscillatory velocity field, similar to that of an inertial oscillation, is discussed. The goal of this section is to develop intuition about the role of internal waves in horizontal ocean mixing. The problem is examined using a variety of models and techniques. The most important result is (23.2) which is an expression for the effective horizontal diffusivity produced by the interaction of vertical diffusivity and oscillatory verticaT shear. Given an empirical velocity shear spectrum and an estimate of the vertical diffusivity this result could be used to calculate a horizontal eddy diffusivity which parameterizes the horizontal mixing due to the internal wave field. Thesis Supervisor: Dr. P. B. Rhines Title: Senior Scientist, Woods Hole Oceanographic Institution. ACKNOWLEDGEMENTS This is a welcome opportunity to thank my advisor, Peter Rhines. Our partnership has been more exciting and productive than I would have thought possible four brief years ago. His guidance has been inspirational and 'his approach to science has deeply influenced me. The remaining members of my thesis committee, Joe Pedlosky and Glenn Flierl, have done much to improve the readability of this thesis. I am also grateful to them for many enlightening conversations and critical questions which forced me to think mmore clearly. During my studenthood I have enjoyed the company of a remarkably exciting set of office-mates. Their physical proximity encouraged me to expose them to undigested (and often undigestable) ideas. At W.H.O.I. I benefited from the company of Breck Owens, Dan Wright and John Loder. At M.I.T. the denizens of Think-Tank Central, Ellen Brown, Al Campbell, Teri Chereskin and Bill Dewar, provided a resonant audience. That honorary denizen, Glenn Ierley, has the rare distinction of contributing to both my intellectual progress and my physical comfort; my memories of his pecan pies will long endure. I'd also like to thank Mary Ann Lucas who cheerfully and efficiently typed this thesis. Finally, NSF Grant OCE-78-25692 has supported me throughout my stay in the Joint Program. TABLE OF CONTENTS ABSTRACT ACKNOWL EDGEMENTS 3 CHAPTER 1 A DISCUSSION OF THE VERTICAL STRUCTURE OF THE WIND-DRIVEN CIRCULATION. Abstract of Chapter 1. Section 1 The Sverdrup Balance IL Introduction the Sverdrup balance reduces the ii dimensionality of the circulation problem from three to two. An assumption separation of the wind-driven and 11 thermohaline circulations. The quasigeostrophic equations. /6 The nondimensional equations; the significance of is U/sL 2. Scale estimates of i and U in terms of external variables. 17 Section 2 Time Dependent, Dissipationless Circulation Problems zo Introduction the possibility of determining the vertical structure by solving an initial value problem. zo A linear initial value problem. zz Baroclinic flows forced by moving wind-stress patterns 26 -a linear analysis. Nonlinear effects -a two-layer quasigeostrophic model. Section 3 Linear, Steady, Dissipative Circulation Problems 37 Introduction dissipation smooths the singular current distributions suggested by the linear initial value problem. 37 A two layer model with interfacial drag. 37 Continuously stratified models with various dissipative mechanisms.of Chapter 1. Section 1 The Sverdrup Balance IL Introduction the Sverdrup balance reduces the ii dimensionality of the circulation problem from three to two. An assumption separation of the wind-driven and 11 thermohaline circulations. The quasigeostrophic equations. /6 The nondimensional equations; the significance of is U/sL 2. Scale estimates of i and U in terms of external variables. 17 Section 2 Time Dependent, Dissipationless Circulation Problems zo Introduction the possibility of determining the vertical structure by solving an initial value problem. zo A linear initial value problem. zz Baroclinic flows forced by moving wind-stress patterns 26 -a linear analysis. Nonlinear effects -a two-layer quasigeostrophic model. Section 3 Linear, Steady, Dissipative Circulation Problems 37 Introduction dissipation smooths the singular current distributions suggested by the linear initial value problem. 37 A two layer model with interfacial drag. 37 Continuously stratified models with various dissipative mechanisms. The Importance of Closed Geostrophic Contours Introduction the dynamics of closed and blocked geostrophic contours are compared and contrasted. A two layer quasigeostrophic model. A three layer quasigeostrophic model. Uniform potential vorticity in subsurface layers. Some Homogeneous Circulation Problems Introduction Topographically closed geostrophic contours. Sverdrup flow on a broken a-plane. 45 46 50 5 POTENTIAL VORTICITY HOMOGENIZATION Abstract of Chapter 2. An Oceanic Analog of the Batchelor-Prandtl Theorem 66 Introduction removing degeneracy inside closed streamlines using an integral theorem. 66 The Batchelor-Prandtl theorem. 67 Vertical viscosity and a Batchelor-Prandtl theorem for potential vorticity. Some Exegetical Remarks on vi'q' = -/ij q,j 7/ Introduction mesoscale eddies and the removal of degeneracy inside closed geostrophic contours. 7/ The eddy flux of potential vorticity. 72 Potential Vorticity Homogenization a Turbulent Extension of the Batchelor-Prandtl Theorem 7? Introduction the weak eddy assumption. 77 The first proof: use v 'q' = -K q1 explicitly. 77 The second proof: use the enstrophy equation. Section 4of Chapter 2. An Oceanic Analog of the Batchelor-Prandtl Theorem 66 Introduction removing degeneracy inside closed streamlines using an integral theorem. 66 The Batchelor-Prandtl theorem. 67 Vertical viscosity and a Batchelor-Prandtl theorem for potential vorticity. Some Exegetical Remarks on vi'q' = -/ij q,j 7/ Introduction mesoscale eddies and the removal of degeneracy inside closed geostrophic contours. 7/ The eddy flux of potential vorticity. 72 Potential Vorticity Homogenization a Turbulent Extension of the Batchelor-Prandtl Theorem 7? Introduction the weak eddy assumption. 77 The first proof: use v 'q' = -K q1 explicitly. 77 The second proof: use the enstrophy equation. Section 4