On the stability of viscous Riemann ellipsoids

TL;DR

This paper analyzes the linear stability of Riemann ellipsoids considering both inviscid and weakly viscous effects, providing analytic dispersion relations and stability diagrams relevant to geophysical and astrophysical flows.

Contribution

It derives a generalized Poincare equation for small oscillations and incorporates viscous effects via boundary-layer analysis, extending classical results with efficient computational methods.

Findings

Analytic dispersion relation for 3D ellipsoidal disturbances

First-order viscous corrections to the inviscid spectrum

Stability diagrams illustrating effects of rotation, strain, and viscosity

Abstract

The present study investigates the linear stability of Riemann ellipsoids in both the inviscid limit and in the presence of weak viscosity. In the inviscid regime, we derive a generalised Poincare equation governing small fluid oscillations and construct a family of polynomial solutions that extends the classical results of Cartan to flows with a uniform strain field. This formulation provides an analytic dispersion relation for three-dimensional ellipsoidal disturbances and remains computationally efficient at arbitrary harmonic degree, in contrast to the virial tensor method or to short-wavelength (WKB) approximations. The viscous effects are incorporated through a boundary-layer analysis based on Prandtls theory, leading to first-order viscous corrections to the inviscid spectrum and allowing a systematic investigation of viscosity-driven instabilities. Stability diagrams are presented over the space of admissible Riemann ellipsoids, illustrating the roles of rotation, internal strain, and diffusion, with implications for rotating shear flows in geophysical and astrophysical contexts.