Elliptical instability of compressible flow in ellipsoids

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 ΩP 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 ΩP/ΩF, with ΩF 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