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Numerical study of the sharp stratification limit towards bilayer models

Théo Fradin

Abstract

In the study of oceanic flows at the geophysical scale, the phenomenon of density stratification plays a central role in the dynamics of the system. Two categories of mathematical models are commonly used to describe the role played by the density stratification: on the one hand, continuously stratified models - such as the stratified Euler equations in a strip, considered in the present article - offer an accurate description of vertical effects, but come with a high level of complexity, both at the theoretical and numerical levels. On the other hand, bilayer models approximate the stratification by a piecewise constant profile. In the latter case, the main point is to study the evolution of the free interface between both layers, which leads to a substantially simplified model. In the present article, we compare both approaches in the framework of the linearized stratified Euler equations around density profiles that are close to piecewise constant profiles, and prove the convergence towards the bilayer Euler equations. However, in the presence of a shear flow, bilayer models have a range of validity limited by the presence of Kelvin-Helmholtz instabilities. In this case, we use a suitable normal modes decomposition to compute numerically the dispersion relation of this linearized model, and provide numerical evidence that the Kelvin-Helmholtz instabilities limit the applicability of two widely used bilayer models, namely the bilayer Euler equations and the bilayer shallow-water equations.

Numerical study of the sharp stratification limit towards bilayer models

Abstract

In the study of oceanic flows at the geophysical scale, the phenomenon of density stratification plays a central role in the dynamics of the system. Two categories of mathematical models are commonly used to describe the role played by the density stratification: on the one hand, continuously stratified models - such as the stratified Euler equations in a strip, considered in the present article - offer an accurate description of vertical effects, but come with a high level of complexity, both at the theoretical and numerical levels. On the other hand, bilayer models approximate the stratification by a piecewise constant profile. In the latter case, the main point is to study the evolution of the free interface between both layers, which leads to a substantially simplified model. In the present article, we compare both approaches in the framework of the linearized stratified Euler equations around density profiles that are close to piecewise constant profiles, and prove the convergence towards the bilayer Euler equations. However, in the presence of a shear flow, bilayer models have a range of validity limited by the presence of Kelvin-Helmholtz instabilities. In this case, we use a suitable normal modes decomposition to compute numerically the dispersion relation of this linearized model, and provide numerical evidence that the Kelvin-Helmholtz instabilities limit the applicability of two widely used bilayer models, namely the bilayer Euler equations and the bilayer shallow-water equations.
Paper Structure (44 sections, 10 theorems, 185 equations, 24 figures)

This paper contains 44 sections, 10 theorems, 185 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. The linear stratified Euler equations, and its normal modes
  4. The stratified Euler equations in isopycnal coordinates
  5. Normal modes
  6. Extension to a non-zero shear flow
  7. The linear bilayer Euler equations
  8. The linear bilayer Euler equations with a shear flow
  9. The irrotational case, with a shear flow
  10. The linear bilayer Euler equations without a shear flow
  11. Numerical scheme
  12. Numerical scheme for the stratified Euler equations
  13. Solving the Sturm-Liouville problem.
  14. Restriction to a finite number of (vertical) modes.
  15. Restriction to a finite number of (horizontal) Fourier modes.
  16. ...and 29 more sections

Key Result

Lemma 3.1

Assume that there exists $c_* > 0$ such that $\rhob \geq c_*$, and such that eqn:stable holds. Let $(f_n)_{n\geq1}, (g_n)_{n\geq 0}, (c_n)_{n\geq1}$ be defined from eqn:SL and eqn:def:gn as above. In particular, $(c_n)_{n\geq1}$ is a decreasing sequence by convention.

Figures (24)

  • Figure 1: Idealized setting. Lines are isopycnals; their color represents the value of the density $\rhob + \rho$ of the fluid, which is a perturbation of the profile $\rhob$. The horizontal velocity is $\Vb + V$, a perturbation of the profile $\Vb$, and the vertical velocity is $w$.
  • Figure 2: The sharp stratification profile $\rhob$ from \ref{['eqn:def:sharp_strat']} and a sharp shear flow $\Vb$, with parameter $\delta > 0$. When $\delta \to 0$, these converge towards piecewise constant functions.
  • Figure 3: The bilayer setting. The unknowns $V_{\pm},w_{\pm}$ are the fluid velocities in the upper and lower layer, as well as the interface $\zeta$ between both layers.
  • Figure 4: Response of the system \ref{['eqn:modes:finite:forcing']} to a localized forcing. Internal waves propagate following Saint-Andrew's cross pattern, in a uniformly stratified fluid. The fluid starts at rest, with an oscillating localized source term at $x=0$ and $r=-0.7$, visible in $(a)$. Colors show the size of the density deviation $\rho$ from the density profile $\rhob^{(1)}$.
  • Figure 5: Two stratification profiles with an upper layer ($r > -0.5+\delta$), a pycnocline ($r \in [-0.5-\delta,-0.5+\delta]$) where the density increases sharply and a lower layer ($r < -0.5-\delta$), with $\delta = 5 . 10^{-2}$ and $\delta = 10^{-3}$. Both the upper and lower layers are stably stratified with a roughly constant Brunt-Väisälä frequency.
  • ...and 19 more figures

Theorems & Definitions (33)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.7
  • Remark 3.8
  • ...and 23 more