Buoyancy instability of homologous implosions

Abstract

I consider the hydrodynamic stability of imploding ideal gases as an idealized model for inertial confinement fusion capsules, sonoluminescent bubbles and the gravitational collapse of astrophysical gases. For oblate modes (short-wavelength incompressive modes elongated in the direction of the mean flow), a second-order ordinary differential equation is derived that can be used to assess the stability of any time-dependent flow with planar, cylindrical or spherical symmetry. Upon further restricting the analysis to homologous flows, it is shown that a monatomic gas is governed by the Schwarzschild criterion for buoyant stability. Under buoyantly unstable conditions, both entropy and vorticity fluctuations experience power-law growth in time, with a growth rate that depends upon mean flow gradients and, in the absence of dissipative effects, is independent of mode number. If the flow accelerates throughout the implosion, oblate modes amplify by a factor$(2C)^{|N_{0}|t_{i}}$, where$C$is the convergence ratio of the implosion,$N_{0}$is the initial buoyancy frequency and$t_{i}$is the implosion time scale. If, instead, the implosion consists of a coasting phase followed by stagnation, oblate modes amplify by a factor$\exp ({\rm\pi}|N_{0}|t_{s})$, where$N_{0}$is the buoyancy frequency at stagnation and$t_{s}$is the stagnation time scale. Even under stable conditions, vorticity fluctuations grow due to the conservation of angular momentum as the gas is compressed. For non-monatomic gases, this additional growth due to compression results in weak oscillatory growth under conditions that would otherwise be buoyantly stable; this over-stability is consistent with the conservation of wave action in the fluid frame. The above analytical results are verified by evolving the complete set of linear equations as an initial value problem, and it is demonstrated that oblate modes are the fastest-growing modes and that high mode numbers are required to reach this limit (Legendre mode$\ell \gtrsim 100$for spherical flows). Finally, comparisons are made with a Lagrangian hydrodynamics code, and it is found that a numerical resolution of${\sim}30$zones per wavelength is required to capture these solutions accurately. This translates to an angular resolution of${\sim}(12/\ell )^{\circ }$, or${\lesssim}0.1^{\circ }$to resolve the fastest-growing modes.