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

TL;DR

This paper proves the nonlinear instability of shear flows in 3-D inviscid, non-conducting Boussinesq equations with rotation, highlighting boundary-driven instabilities and their relation to geostrophic limit models.

Contribution

It establishes the nonlinear instability of shear flows in the 3-D inviscid Boussinesq system with rotation, extending understanding of boundary-driven instabilities in geophysical flows.

Findings

Nonlinear instability of shear flows proven for small Rossby numbers.

Instabilities are driven by physical boundaries.

Results connect to the geostrophic limit model.

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.