Convective instability of self-similar spherical expansion into a vacuum

Abstract

The well-known class of self-similar solutions for an ideal polytropic gas sphere of radius R(t) expanding into a vacuum with velocity $u(r, t) = r\dot{R}/R$ is shown to be convectively unstable. The physical mechanism results from the buoyancy force experienced by anisentropic distributions in the inertial (effective gravitational) field. An equation for the perturbed displacement ξ(r, t), derived from the linearized fluid equations in Lagrangian co-ordinates, is solved by separation of variables. Because the basic state is non-steady, the perturbations do not grow exponentially, but can be expressed in terms of hypergeometric functions. For initial density profiles \[ \rho_0(r)\sim(1-r^2/r^2_0)^{\kappa}, \] modes with angular dependence Y_lm(θ, ϕ) are unstable provided l > 0 and κ < 1/(γ − 1), where γ is the ratio of specific heats. For large l, the characteristic growth time of the perturbations varies as l^−½ and the amplification increases exponentially as a function of l. The radial eigenfunctions are proportional to r^l, and the compressibility and vorticity are both non-zero.