Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-convergence of the rotating stratified flows toward the quasi-geostrophic dynamics

Min Jun Jo, Junha Kim, Jihoon Lee

Abstract

The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.

Non-convergence of the rotating stratified flows toward the quasi-geostrophic dynamics

Abstract

The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.
Paper Structure (30 sections, 17 theorems, 158 equations)

This paper contains 30 sections, 17 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Quasi-geostrophy of rotating stratified fluids
  3. Goal of this paper
  4. Summary
  5. Background
  6. Three-scale singular limit
  7. Previous works for the QG system
  8. Main results
  9. Non-convergence for $\mu=1$
  10. Fast convergence vs. slow non-convergence for $\mu\to 1$
  11. $\mu$-universal non-convergence in $L_t^{\infty} H_x^{s}$
  12. $\mu$-universal convergence in $H^s$
  13. Generalized non-convergence theorem
  14. Preliminaries
  15. Local well-posedness
  16. ...and 15 more sections

Key Result

Theorem 2.1

Fix any $m\in \mathbb{N}$ with $m\geq 6$. For any $u_0=(v_0,\theta_0)\in H^m(\mathbb R^3)$ with $\nabla \cdot v_0=0$ and $u_0-P_1 u_0 \neq 0$ where $P_{1}$ is a projection operator defined in Pmu with $\mu=1$, there exists a constant $A>0$ depending only on $u_0$ such that the unique classical local for some sufficiently small $t_0\in(0,T_{\ast}),$ where $T_{\ast}$ is the maximal existence time of

Theorems & Definitions (48)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Remark 2.8
  • Theorem 2.9: A variant of the result in Grenier
  • Remark 2.10
  • ...and 38 more