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On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D

Jingjing Mao, Yan-Lin Wang

Abstract

In this paper, we prove the nonlinear instability of a given vertical shear of velocity between two rigid plane for the 3-D inviscid, non-conducting Boussinesq equations with rotation. When the Rossby number is zero, this rotating inviscid Boussinesq system reduces to the nonlinear geostrophic limit model. For non-zero small Rossby numbers, we establish the nonlinear instability of the shear flow, which is consistent with that of the geostrophic limit model. The proof relies on constructing a precise approximate solution, which comprises a growing profile derived from the nonlinear geostrophic limit model and a higher-order asymptotic expansion with respect to the small Rossby number. Notice that the instabilities (growing modes) are driven by the physical boundaries.

On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D

Abstract

In this paper, we prove the nonlinear instability of a given vertical shear of velocity between two rigid plane for the 3-D inviscid, non-conducting Boussinesq equations with rotation. When the Rossby number is zero, this rotating inviscid Boussinesq system reduces to the nonlinear geostrophic limit model. For non-zero small Rossby numbers, we establish the nonlinear instability of the shear flow, which is consistent with that of the geostrophic limit model. The proof relies on constructing a precise approximate solution, which comprises a growing profile derived from the nonlinear geostrophic limit model and a higher-order asymptotic expansion with respect to the small Rossby number. Notice that the instabilities (growing modes) are driven by the physical boundaries.

Paper Structure

This paper contains 10 sections, 6 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Motivation and the problem
  3. Previous works and main results
  4. Instability for the case $Ro=0$
  5. Linear instability for $Ro=0$
  6. Nonlinear instability for $Ro=0$
  7. Baroclinic instability for the case $Ro\neq 0$
  8. Construction of approximate solution.
  9. Estimates for the error terms
  10. Appendix

Key Result

Theorem 1.1

Assume $Ro>0$ and $0<\delta<1.$ Suppose that $(\textbf{u}^*, \theta^*)(\cdot, t)$ is a strong solution to the perturbed system perturbation_equationsboundary_conditions_w satisfying There exists a positive constant $q_c$ such that if $0<q<q_c,$ then there is a time $T^\delta=O(\log \frac{\eta+C Ro}{\delta})$ so that for some positive constant $C$ and $1\geq \eta>\delta>0.$

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • Remark 3.1
  • Proposition 3.1: cf. TR
  • Lemma 3.2
  • Lemma 3.3