On the stability of viscous Riemann ellipsoids

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.

💡 Research Summary

The paper presents a unified linear stability theory for Riemann ellipsoids, covering both the inviscid limit and the weakly viscous regime. Riemann ellipsoids are homogeneous, incompressible, self‑gravitating fluid bodies whose velocity field is a linear function of position, combining a solid‑body rotation Ω with a uniform internal vorticity ζ that is aligned with the rotation axis (the S‑type family). The authors first reformulate the classical Poincaré equation for small oscillations by incorporating the uniform strain associated with the internal vorticity. This yields a “generalised Poincaré equation” that remains valid for any harmonic degree ℓ. By extending Cartan’s polynomial solutions, they construct an explicit family of ellipsoidal harmonics that satisfy the equation for arbitrary ℓ, and from these obtain an analytic dispersion relation ω(ℓ, f, Γ, Ξ), where f = ζ/Ω measures the ratio of internal vorticity to rotation, and Γ, Ξ are the semi‑axis ratios of the ellipsoid. The dispersion relation is algebraic, computationally inexpensive, and provides direct spectral access to all modes, in contrast to the traditional virial‑tensor approach (which becomes cumbersome at high ℓ) or short‑wavelength WKB approximations (which are limited to asymptotically large ℓ).

In the viscous extension the authors employ a Prandtl‑type boundary‑layer analysis. Because the free surface is stress‑free in the inviscid case, the presence of a small Ekman number (Ek ≪ 1) necessitates a thin viscous sublayer adjacent to the ellipsoid’s surface. Within this layer the Navier–Stokes equations retain the leading viscous term, while the interior flow remains governed by the inviscid equations. Matching the interior and boundary‑layer solutions yields first‑order viscous corrections to the eigenvalues: ω = ω₀ + δ − iγ, where ω₀ is the inviscid frequency, γ > 0 represents viscous damping, and δ is a real shift that can be either stabilising or destabilising. Crucially, for certain combinations of f, Γ, and Ξ the viscous correction drives a pair of eigenvalues across the imaginary axis, producing a dissipation‑induced (or secular) instability. This mechanism mirrors the classic instability identified by Thompson & Tait for Maclaurin spheroids, but here it appears in the richer context of Riemann ellipsoids where rotation, strain, and vorticity coexist.

The authors map the stability landscape over the three‑dimensional parameter space (f, Γ, Ξ). Their diagrams reveal several key trends: (i) when f ≈ 0 (pure rotation) viscosity mainly damps the inviscid elliptic instability, shrinking the unstable region; (ii) for large |f| (strain‑dominated configurations) high‑order harmonics (ℓ ≥ 3) become increasingly susceptible to viscous destabilisation, illustrating a classic elliptic instability amplified by diffusion; (iii) intermediate values of f exhibit a delicate balance between gyroscopic stabilisation (due to rotation) and strain‑driven destabilisation, leading to narrow bands where even a tiny viscosity triggers growth. The analysis also recovers known limits: the Maclaurin spheroid (Γ = 1, ζ = 0) and the Dedekind ellipsoid (|f| → ∞) appear as special cases of the general framework.

From a physical perspective, the results have direct relevance to geophysical and astrophysical flows. In planetary interiors or stellar radiative zones, differential rotation and internal shear often coexist; the present theory predicts when such configurations will be linearly unstable, potentially seeding turbulence or enhancing angular‑momentum transport. In compact objects, where viscosity may be anomalously low but still finite, the identified dissipation‑induced modes could affect the excitation of oscillation spectra and, consequently, gravitational‑wave emission mechanisms such as the Chandrasekhar–Friedman–Schutz (CFS) instability.

Methodologically, the paper’s contribution lies in (a) providing an analytically tractable, high‑order harmonic solution to the generalized Poincaré problem, and (b) integrating a systematic boundary‑layer correction that yields explicit viscous eigenvalue shifts. This combination offers a powerful tool for future studies of rotating, strained fluid bodies with free surfaces, extending beyond ellipsoidal geometry to more general configurations.

In summary, the work delivers a comprehensive, mathematically rigorous, and physically insightful treatment of the linear stability of viscous Riemann ellipsoids, bridging a long‑standing gap in the literature and opening avenues for applications in both Earth‑science fluid dynamics and relativistic astrophysics.