Rayleigh-Taylor instability for compressible rotating flows

TL;DR

This paper analyzes the Rayleigh-Taylor instability in rotating, compressible fluid flows with a free interface, revealing how rotation affects instability growth and demonstrating ill-posedness of the linearized and nonlinear problems.

Contribution

It introduces a novel method to construct exponentially growing normal mode solutions in rotating compressible flows, showing the ill-posedness of the problem and the impact of rotation on instability.

Findings

Rotation reduces the growth rate of instability.

Solutions can grow arbitrarily quickly in Sobolev spaces.

The linearized problem is ill-posed, leading to ill-posedness of the nonlinear problem.

Abstract

In this paper, we investigate the Rayleigh-Taylor instability problem for two compressible, immiscible, inviscid flows rotating with an constant angular velocity, and evolving with a free interface in the presence of a uniform gravitational field. First we construct the Rayleigh-Taylor steady-state solutions with a denser fluid lying above the free interface with the second fluid, then we turn to an analysis of the equations obtained from linearization around such a steady state. In the presence of uniform rotation, there is no natural variational framework for constructing growing mode solutions to the linearized problem. Using the general method of studying a family of modified variational problems introduced in \cite{Y-I2}, we construct normal mode solutions that grow exponentially in time with rate like $e^{t\sqrt{c|\xi|-1}}$, where $\xi$ is the spatial frequency of the normal mode and the constant $c$ depends on some physical parameters of the two layer fluids. A Fourier synthesis of these normal mode solutions allows us to construct solutions that grow arbitrarily quickly in the Sobolev space $H^k$, and lead to an ill-posedness result for the linearized problem. Moreover, from the analysis we see that rotation diminishes the growth of instability. Using the pathological solutions, we then demonstrate the ill-posedness for the original non-linear problem in some sense.