Rayleigh-Taylor instability for compressible rotating flows

Abstract

In this paper, we investigate the Rayleigh-Taylor instability problem for two compressible, immiscible, inviscid flows rotating with a 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 [1], we construct normal mode solutions that grow exponentially in time with rate like etc|ξ|−1, where ζ 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 Hk, and leads 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.