Buoyancy instability of homologous implosions

Hot spot turbulence is a potential contributor to yield degradation in inertial confinement fusion (ICF) capsules, although its origin, if present, remains unclear. In this work, a perturbation analysis is performed of an analytical homologous solution that mimics the hot spot and surrounding cold fuel during the late stages of an ICF implosion. It is shown that the flow is governed by the Schwarzschild criterion for buoyant stability, and that during stagnation, short wavelength entropy and vorticity fluctuations amplify by a factor $\exp\left(\pi \left|N_0\right| t_s\right)$, where $N_0$ is the buoyancy frequency at stagnation and $t_s$ is the stagnation time scale. This amplification factor is exponentially sensitive to mean flow gradients and varies from $10^3$--$10^7$ for realistic gradients. 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 the evolution of vorticity accurately. This translates to an angular resolution of ${\sim}(12/\ell)^\circ$, or ${\sim}0.1^\circ$ to resolve the fastest growing modes (Legendre mode $\ell > 100$).