A modal wave-packet model for the multi-mode Richtmyer–Meshkov instability

Abstract

A model for multimode perturbations subject to the Richtmyer–Meshkov (RM) instability is presented and compared with simulations and experiments for conditions relevant to inertial confinement fusion. The model utilizes the single mode response to the RM impulse whereby its amplitude h(k, t) first grows with an initial velocity V0 ∝ kh(k, 0) that eventually decays in time as 1/kV0t. Both the growth and saturation stages are subject to nonlinearities since they depend explicitly on the initial amplitude. However, rather than using the individual mode amplitude h(k, t), nonlinearity is taken to occur when the root-mean-square amplitude hrms(k, t) of a wave-packet within wavenumbers k ± δk becomes comparable to 1/k. This is done because nearby sidebands can act in unison for an auto-correlation distance 1/δk beyond nonlinearity as observed in the beam-plasma instability. Thus, the nonlinear saturation amplitude for each mode is reduced from the usual 1/k by a phase space factor that depends on the physical dimensionality, as in the Haan model for the Rayleigh–Taylor instability. In addition, for RM, the average value of khrms for the initial spectrum is used to calculate a nonlinear factor FNL that reduces V0, as observed for single modes. For broadband perturbations, the model describes self-similar growth ∝tθ as successively longer wavelength modes reach saturation. The growing and saturated modes must be discerned because only the former promote θ and are enhanced by reshock and spherical convergence. All of these flows are described here by the model in good agreement with simulations and experiments.