Numerical studies of waves and particle acceleration in shocks

Abstract

Aspects of the self consistent acceleration and transport of cosmic rays in astrophysical fluid flows and associated numerical methods are studied. Problems investigated are: (i) magnetohydrodynamic (MHD) wave interactions and instabilities in two-fluid models of cosmic ray modified shocks and flows; (ii) two dimensional, self consistent models of cosmic ray acceleration by the first order Fermi mechanism in supernova remnant shocks; (iii) new Riemann solver for the two-dimensional Euler equations and adaptive mesh refinement scheme for the coupled MHD and cosmic ray transport equations. The interaction of short wavelength MHD waves and instabilities in cosmic ray modified flows are investigated using asymptotic analysis and numerical simulations, with application to cosmic ray driven squeezing instabilities in supernova remnant shocks. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow; cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic-rays. Numerical solutions of the fully nonlinear two-fluid cosmic ray MHD equations are compared with solutions of the wave mixing equations for oblique, cosmic ray modified shocks. A two-dimensional, self-consistent, adaptive mesh refinement numerical algorithm is developed for the solution of the ideal magnetohydrodynamic equations coupled to the kinetic transport equation for energetic charged particles. The method is used to simulate the evolution of the momentum distribution function of the cosmic rays accelerated at supernova remnant shocks. The numerical methods were tested on a variety of fluid dynamics and MHD problems, and previous models of cosmic ray modified supernova remnant shocks. A Riemann solver based on two-dimensional multi-state Riemann problems was developed. The scheme generalizes the traditional one-dimensional flux calculation to include contributions to the flux through the cell edges of the waves originating at cell corners. The multidimensional flux corrections increase the accuracy and stability of the scheme. An adaptive mesh refinement technique was used to study the Von Neumann paradox associated with the formation of three shocks, when a low Mach number, supersonic flow impinges on a thin wedge. For the first time, the region near the triple point has been resolved in a numerical solution of the Euler equations.