Rayleigh-Taylor instability for compressible rotating flows

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.

💡 Research Summary

The paper addresses the Rayleigh‑Taylor instability (RTI) in a two‑layer compressible, inviscid, immiscible fluid system that rotates uniformly about a fixed axis while being subjected to a constant gravitational field. The authors first construct a family of steady‑state configurations in which the denser fluid lies above the lighter one, separated by a free interface. In the rotating frame the equilibrium balances gravity, pressure gradients, and the centrifugal force; the density and pressure profiles are obtained explicitly from the compressible equation of state (p=A\rho^\gamma) ((\gamma>1)).

Linearizing the Euler equations about this equilibrium yields a system for the perturbation of the interface, velocity, and pressure. After Fourier transforming in the horizontal directions, the perturbations are represented as normal modes (\exp(i\xi\cdot x + \sigma t)) with spatial frequency (\xi\in\mathbb{R}^2) and growth rate (\sigma). The Coriolis term appears only in the phase of the mode and does not directly affect (\sigma). However, the presence of rotation destroys the usual variational structure that underlies classical RTI analysis; there is no coercive energy functional whose critical points give the growing modes.

To overcome this obstacle the authors adopt the “modified variational” framework introduced in their earlier work (referenced as \cite{Y‑I2}). They introduce an auxiliary parameter (\lambda) and consider a family of (\lambda)-dependent variational problems whose Euler‑Lagrange equations reproduce the linearized normal‑mode equations. By carefully tuning (\lambda) they obtain a dispersion relation of the form
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