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On the stability of viscous Riemann ellipsoids

Joris Labarbe

TL;DR

This work develops a unified linear stability theory for S-type Riemann ellipsoids valid in both the inviscid limit and under weak viscosity. It introduces a generalized Poincaré equation with Cartan-type polynomial solutions to obtain analytic dispersion relations for arbitrary ellipsoidal harmonics, enabling efficient stability analysis beyond low-order modes and without relying on short-wavelength approximations. In the viscous regime, a boundary-layer (Prandtl) approach yields first-order viscous corrections to the inviscid spectrum, revealing viscosity-driven instabilities and avoided crossings near Hamilton–Hopf points, thereby clarifying how dissipation modifies the stability landscape. The results have implications for rotating geophysical and astrophysical flows, offering a tractable framework to study secular evolution and dissipation-driven dynamics in self-gravitating, uniformly strained fluids, with avenues for extending to relativistic and gravitational-wave contexts.

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.

On the stability of viscous Riemann ellipsoids

TL;DR

This work develops a unified linear stability theory for S-type Riemann ellipsoids valid in both the inviscid limit and under weak viscosity. It introduces a generalized Poincaré equation with Cartan-type polynomial solutions to obtain analytic dispersion relations for arbitrary ellipsoidal harmonics, enabling efficient stability analysis beyond low-order modes and without relying on short-wavelength approximations. In the viscous regime, a boundary-layer (Prandtl) approach yields first-order viscous corrections to the inviscid spectrum, revealing viscosity-driven instabilities and avoided crossings near Hamilton–Hopf points, thereby clarifying how dissipation modifies the stability landscape. The results have implications for rotating geophysical and astrophysical flows, offering a tractable framework to study secular evolution and dissipation-driven dynamics in self-gravitating, uniformly strained fluids, with avenues for extending to relativistic and gravitational-wave contexts.

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.
Paper Structure (17 sections, 81 equations, 3 figures)

This paper contains 17 sections, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation of the problem
  3. Equations of motion and equilibrium state
  4. Linear stability theory
  5. Inviscid Riemann ellipsoids
  6. Harmonic expansion of disturbance fields
  7. Normal modes of oscillation
  8. Dispersion relation and stability results
  9. Viscous Riemann ellipsoids
  10. Surface stress conditions
  11. Boundary-layer theory
  12. Analytic solutions and secular instability
  13. Discussion
  14. Description of the ellipsoidal system of coordinates
  15. Geometric relations in curvilinear coordinates
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Sketch of a rigidly rotating and self-gravitating Riemann ellipsoid with uniform strain at its interior. In blue are depicted the elliptic streamlines.
  • Figure 2: Dispersion curves of the $m=n=2$ inviscid Riemann ellipsoid over the planar aspect ratio $\Gamma$. Red (blue) lines denote the frequencies (growth rates) of the system, computed at different values of $f$ according to the legend.
  • Figure 3: Same panels as in Figure \ref{['Fig_inv']} but for the viscous case with $\mathrm{Ek}=10^{-1}$, computed from the first term in expression (\ref{['vis_BC_ell']}).