Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Christian Puntini

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.

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

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 . 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.

Paper Structure

This paper contains 4 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of the exact solution in Constantin et al. (2023)
  3. Short-wavelength instability analysis
  4. Discussion and conclusions

Figures (4)

  • Figure 1: Sketch of the two main types of mountain waves: upward propagating and trapped lee waves. Image generated (partially) with the use of u:ai of the University of Vienna.
  • Figure 2: Particle path of \ref{['solution']} with $a=s=0$ and $b=-0.1$ (black), $b=-0.3$ (blue), $b=-1$ (red) as time evolves, with $t\in [0,10]$. Moreover, $U=0.1$, $W=1$ and $k=20/3$. At time $t=0$, the particle initial position is in the bottom-left corner, and at final time $t=10$ its position is in the top right corner. Due to the exponential decrease of the wave amplitude with $b$, for $b=-1$ the oscillations are basically not visible.
  • Figure 3: Trochoidal particle path of \ref{['solution']} at fixed time $t=0$, with $a \in [0,3]$, $s=0$ and $b=-0.1$ (black), $b=-0.3$ (blue), with $k=20/3$. For $b=-1$ the oscillations are not visible, so we decided to not plot it.
  • Figure 4: Sketch of upward-propagating mountain waves and their instability. Orographic lift along the slope is relatively uniform, while the wave travels through a laminar flow layer. Turbulence typically forms below due to overturning eddies (rotors) on the lee side, and above from instability in the upper layer just beneath the tropopause. Image generated (partially) with the use of u:ai of the University of Vienna.