## Elliptical instability of compressible flow in ellipsoids

Institute of Geophysics, University of Göttingen,
Friedrich-Hund-Platz 1,
37077
Göttingen,
Germany

e-mail: niels.clausen@geo.physik.uni-goettingen.de; andreas.tilgner@geo.physik.uni-goettingen.de

Received:
8
October
2013

Accepted:
10
December
2013

*Context. *Elliptical instability is due to a parametric resonance of two
inertial modes in a fluid velocity field with elliptical streamlines. This flow is a
simple model of the motion in a tidally deformed, rotating body. Elliptical instability
typically leads to three-dimensional turbulence. The associated turbulent dissipation
together with the dissipation of the large scale mode may be important for the
synchronization process in stellar and planetary binary systems.

*Aims. *In order to determine the influence of the compressibility on the
stability limits of tidal flows in stars or planets, we calculate the growth rates of
perturbations in flows with elliptical streamlines within ellipsoidal boundaries of small
ellipticity. In addition, the influence of the orbiting frequency of the tidal perturber
Ω_{P} and the viscosity of the fluid are taken into account.

*Methods. *We studied the linear stability of the flow to determine the
growth rates. We solved the Euler equation and the continuity equation. The viscosity was
introduced heuristically in our calculations. We assumed a power law for the radial
dependence of the background density. Together with the use of the anelastic
approximation, this enabled us to use semi-analytical methods to solve the equations.

*Results. *It is found that the growth rate of a certain mode combination
depends on the compressibility. However, the influence of the compressibility is
negligible for the growth rate maximized over all possible modes if viscous bulk damping
effects can be neglected. The growth rate maximized over all possible modes determines the
stability of the flow. The stability limit for the compressible fluid confined to an
ellipsoid is the same as for incompressible fluid in an unbounded domain. Depending on the
ratio Ω_{P}/Ω_{F}, with Ω_{F} the spin rate of
the central object in the frame of the rotating tidal perturber, certain pairs of modes
resonate with each other. The size of the bulk damping term depends on the modes that
resonate with each other. Therefore the growth rate of the viscous flow depends on the
compressibility. Estimates for the stability limit in viscous fluids are given.

Key words: hydrodynamics / instabilities / waves

*© ESO, 2014*