Stability of stagnation via an expanding accretion shock wave

TL;DR

This paper theoretically analyzes the stability of stagnation via an expanding accretion shock wave in plasma physics, demonstrating that such stagnation is stable with perturbations decaying over time, supported by simulations.

Contribution

It provides the first theoretical stability analysis of stagnation via an expanding shock wave in ideal-gas flows, including explicit eigenvalue formulas and simulation validation.

Findings

Stagnation flow is stable with perturbations decaying over time.

Eigenmodes exhibit power-law, oscillatory, or monotonic decay.

Results support hydro code verification in 2D and 3D simulations.

Abstract

Stagnation of a cold plasma streaming to the center or axis of symmetry via an expanding accretion shock wave is ubiquitous in inertial confinement fusion (ICF) and high-energy-density plasma physics, the examples ranging from plasma flows in x-ray-generating Z pinches [Y. Maron et al., Phys. Rev. Lett. 111, 035001 (2013)] to the experiments in support of the recently suggested concept of impact ignition in ICF [H. Azechi et al., Phys. Rev. Lett. 102, 235002 (2009); M. Murakami et al., Nucl. Fusion 54, 054007 (2014)]. Some experimental evidence indicates that stagnation via an expanding shock wave is stable, but its stability has never been studied theoretically. We present such analysis for the stagnation that does not involve a rarefaction wave behind the expanding shock front and is described by the classic ideal-gas Noh solution in spherical and cylindrical geometry. In either case the stagnated flow has been demonstrated to be stable, initial perturbations exhibiting a power-law, oscillatory or monotonic, decay with time for all the eigenmodes. This conclusion has been supported by our simulations done both on a Cartesian grid and on a curvilinear grid in spherical coordinates. Dispersion equation determining the eigenvalues of the problem and explicit formulas for the eigenfunction profiles corresponding to these eigenvalues are presented, making it possible to use the theory for hydro code verification in two and three dimensions.