General Temporal Instability Criteria For Stably Stratified Inviscid Flow

TL;DR

This paper derives new criteria for the temporal instability of stably stratified inviscid flows using the Taylor-Goldstein equation, revealing conditions under which flow becomes unstable or stable, and extending classical instability criteria.

Contribution

It extends classical instability criteria to stratified inviscid flows and challenges the applicability of the Miles-Howard theorem in this context.

Findings

Flow instability occurs when total Froude number squared $Fr_t^2 \\leq 1$.

Flow stability is associated with $Fr_t^2 > 1$ and specific velocity profiles.

Unstable perturbations are long-wave scale and local instability relates to $Ri > 1/4".

Abstract

The temporal instability of stably stratified flow was investigated by analyzing the Taylor-Goldstein equation theoretically. According to this analysis, the stable stratification $N^2\geq0$ has a destabilization mechanism, and the flow instability is due to the competition of the kinetic energy with the potential energy, which is dominated by the total Froude number $Fr_t^2$. Globally, $Fr_t^2 \leq 1$ implies that the total kinetic energy is smaller than the total potential energy. So the potential energy might transfer to the kinetic energy after being disturbed, and the flow becomes unstable. On the other hand, when the potential energy is smaller than the kinetic energy ($Fr_t^2>1$), the flow is stable because no potential energy could transfer to the kinetic energy. The flow is more stable with the velocity profile $U'/U'''>0$ than that with $U'/U'''<0$. Besides, the unstable perturbation must be long-wave scale. Locally, the flow is unstable as the gradient Richardson number $Ri>1/4$. These results extend the Rayleigh's, Fj{\o}rtoft's, Sun's and Arnol'd's criteria for the inviscid homogenous fluid, but they contradict the well-known Miles-Howard theorem. It is argued here that the transform $F=\phi/(U-c)^n$ is not suitable for temporal stability problem, and that it will lead to contradictions with the results derived from the Taylor-Goldstein equation. However, such transform might be useful for the study of the Orr-Sommerfeld equation in viscous flows.