On the instability of some upward propagating, exact, nonlinear mountain waves

TL;DR

This paper analyzes the linear stability of upward-propagating mountain waves using the short-wavelength instability method, revealing that instability occurs when wave steepness exceeds a critical value, potentially causing chaotic fluid motion.

Contribution

It applies the short-wavelength instability method to a specific exact mountain wave solution, identifying a critical steepness threshold for instability and potential chaotic behavior.

Findings

Flow becomes unstable when wave steepness exceeds 1/3.

An unstable layer of a few hundred meters beneath the tropopause exists.

Instability may lead to chaotic three-dimensional fluid motion.

Abstract

Using the short-wavelength instability method, we investigate the linear instability of an exact solution describing upward-propagating mountain waves, derived in A. Constantin, \emph{J. Phys. A: Math. Theor.} (2023), under the assumption of a dry adiabatic flow. Within this approach, the stability problem reduces to analysing a system of ordinary differential equations along fluid trajectories. Our results show that the flow becomes unstable when the wave steepness exceeds the critical threshold of $\frac{1}{3}$. Given the representation of the solution in Lagrangian coordinates, the instability analysis will show the existence of an unstable layer of few hundred meters beneath the tropopause where instability may occur, finally leading to a chaotic 3-dimensional fluid motion.