New Global Instabilities of the Riemann Ellipsoids

Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability

“…If we consider the tidal synchronisation problem, the pair of fastest growing modes typically have frequencies in the fluid (Ω) frame approximately equal to half the frequency of tidal forcing, so that temporal resonance occurs if the frequencies in the fluid frame ωΩ,1 ∼ ωΩ,2 ∼ γ (more generally we require ωΩ,1 ± ωΩ,2 = 2γ). Spatial resonance requires the modes to have azimuthal wavenumbers m1 ± m2 = 2 (since the tidal deformation has m = 2), and also ℓ1 = ℓ2 (Kerswell 1994;Lebovitz & Lifschitz 1996a). Modes with different ℓ do not couple, as has been proved by Kerswell (1993) and Lebovitz & Lifschitz (1996a).…”

“…This also has the significant advantage that much hard work in devising the formalism has already been carried out. The Lagrangian theory is elucidated in Lebovitz (1989a,b) and has been applied to the stability of Riemann S-type ellipsoids (with no tidal potential) by Lebovitz & Lifschitz (1996a). The equations describing linearised Lagrangian perturbations (working in the bulge frame) are…”

“…Spatial resonance requires the modes to have azimuthal wavenumbers m1 ± m2 = 2 (since the tidal deformation has m = 2), and also ℓ1 = ℓ2 (Kerswell 1994;Lebovitz & Lifschitz 1996a). Modes with different ℓ do not couple, as has been proved by Kerswell (1993) and Lebovitz & Lifschitz (1996a). In the limit of small A, the elliptical instability is only possible if n/Ω ∈ [−1, 3], since IMs are restricted to ωΩ ∈ [−2Ω, 2Ω], but as we will show below (and further demonstrate using a local WKB analysis in Appendix D), this can be violated when A is sufficiently large.…”

“…A similar construction can be carried out for ℓ > 2, for which we refer the reader to Lebovitz & Lifschitz (1996a) for further details.…”

An Approximate Solver for Riemann and Riemann‐like Ellipsoidal Configurations

“…Previous linear stability analyses (Chandrasekhar 1969;Lifschitz & Lebovitz 1993;Lebovitz & Lifschitz 1996;Lebovitz & Saldanha 1999) have revealed a number of hydrodynamic instabilities that might arise in incompressible Riemann ellipsoids, but no corresponding numerical work using state-of-the-art three-dimensional hydrodynamic techniques has been carried out. In particular, Riemann S-type ellipsoids have been found to be subject to a hydrodynamic strain instability associated with elliptical stream lines ( Lebovitz & Lifschitz 1996), which raises concerns about the stability of certain types of geophysical flows and leads to suspicions about the evolutionary path of stars that are driven by gravitational-radiation-reaction (GRR) forces toward the Dedekind sequence ( Lai & Shapiro 1995).…”