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A linear model of separation for western boundary currents with bathymetry

Anne-Laure Dalibard, Corentin Gentil

TL;DR

The paper develops a linear, highly structured model for rotating, stratified flows on a $eta$-plane with topography to capture western boundary current separation. It derives a 3D quasi-geostrophic core coupled to two boundary-layer types: a quasi-geostrophic Munk layer and a non-geostrophic Ekman layer, and constructs an approximate solution to arbitrary order in an $ ablaeta$-scaled expansion, with rigorous justification of validity. A key novelty is the detailed treatment of Ekman layers in the presence of strong topography and stratification, yielding new dynamical features and a mechanism for boundary-current detachment. The work provides both periodic and Cauchy-problem stability results and supports the theoretical construction with numerical illustrations of separation, suggesting a pathway to parameterizations of topography-stratification interactions in coarse ocean models.

Abstract

This paper is devoted to the asymptotic analysis of strongly rotating and stratified fluids, under a $β$-plane approximation, and within a three-dimensional spatial domain with strong topography. Our purpose is to propose a linear idealized model, which is able to capture one of the key features of western boundary currents, in spite of its simplicity: the separation of the currents from the coast. Our simplified framework allows us to perform explicit computations, and to highlight the intricate links between rotation, stratification and bathymetry. In fact, we are able to construct approximate solutions at any order for our system, and to justify their validity. Each term in the asymptotic expansion is the sum of an interior part and of two boundary layer parts: a ``Munk'' type boundary layer, which is quasi-geostrophic, and an ``Ekman part'', which is not. Even though the Munk part of the approximation bears some similarity with previously studied 2D models, the analysis of the Ekman part is completely new, and several of its properties differ strongly from the ones of classical Ekman layers. Our theoretical analysis is supplemented with numerical illustrations, which exhibit the desired separation behavior.

A linear model of separation for western boundary currents with bathymetry

TL;DR

The paper develops a linear, highly structured model for rotating, stratified flows on a -plane with topography to capture western boundary current separation. It derives a 3D quasi-geostrophic core coupled to two boundary-layer types: a quasi-geostrophic Munk layer and a non-geostrophic Ekman layer, and constructs an approximate solution to arbitrary order in an -scaled expansion, with rigorous justification of validity. A key novelty is the detailed treatment of Ekman layers in the presence of strong topography and stratification, yielding new dynamical features and a mechanism for boundary-current detachment. The work provides both periodic and Cauchy-problem stability results and supports the theoretical construction with numerical illustrations of separation, suggesting a pathway to parameterizations of topography-stratification interactions in coarse ocean models.

Abstract

This paper is devoted to the asymptotic analysis of strongly rotating and stratified fluids, under a -plane approximation, and within a three-dimensional spatial domain with strong topography. Our purpose is to propose a linear idealized model, which is able to capture one of the key features of western boundary currents, in spite of its simplicity: the separation of the currents from the coast. Our simplified framework allows us to perform explicit computations, and to highlight the intricate links between rotation, stratification and bathymetry. In fact, we are able to construct approximate solutions at any order for our system, and to justify their validity. Each term in the asymptotic expansion is the sum of an interior part and of two boundary layer parts: a ``Munk'' type boundary layer, which is quasi-geostrophic, and an ``Ekman part'', which is not. Even though the Munk part of the approximation bears some similarity with previously studied 2D models, the analysis of the Ekman part is completely new, and several of its properties differ strongly from the ones of classical Ekman layers. Our theoretical analysis is supplemented with numerical illustrations, which exhibit the desired separation behavior.
Paper Structure (33 sections, 15 theorems, 176 equations, 2 figures)

This paper contains 33 sections, 15 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation from physical oceanography
  3. Results and numerical illustrations
  4. Assumptions on the parameters.
  5. Assmptions on the source term.
  6. Numerical experiments.
  7. Strategy of proof and comparison with previous works
  8. Plan of the paper
  9. Notation
  10. Strategy for building an approximate solution
  11. Derivation of the quasi-geostrophic equation for the stream function
  12. Determination of the first corrective term
  13. Determination and role of Ekman boundary layers
  14. Analysis of the quasi-geostrophic equation
  15. Proof of Lemma on characteristic roots
  16. ...and 18 more sections

Key Result

Proposition 1.1

Let $N\geq 0$, $m\geq 0$ be arbitrary. Assume that assumptions (H0)-(H4) are satisfied, with sufficiently large exponents $q,Q$ depending on $a,b,d, e$, $m$ and $N$. Assume furthermore that $\sin \alpha <0$ (western boundary). Then there exists an approximate solution $(u^{\mathrm{app}},\rho^\mathrm Furthermore, $(u^{\mathrm{app}},\rho^\mathrm{app})$ can be constructed explicitly in terms of the s

Figures (2)

  • Figure 1: Local and global coordinate systems in the case $\sin \alpha<0$ (western boundary).
  • Figure 2: Illustration of the role of topography in the behaviour of the stream function, in the $(x_1,x_2)$ plane, at $x_3$ fixed.

Theorems & Definitions (36)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6: Possible extensions
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2: Green function
  • ...and 26 more