Nonlinear evolution of waves and shocks in the solar wind

The nonlinear evolution of finite amplitude low-frequency waves is studied by means of analytic theory and numerical simulation, with special emphasis on their relation to shock waves. The steepening rate for finite amplitude magnetohydrodynamic (MHD) waves is defined. It is argued analytically and proved numerically that a wave will steepen to form a shock only when the steepening rate is greater than the collisionless damping rate. This simple criterion suggests that, whereas the MHD fast model should ordinarily steepen, steepened slow waves may occur rarely near 1 AU, in agreement with spacecraft observations. Numerical simulations using a hybrid code show that slow mode waves are subject to strong Landau damping. Nonlinearly steepened waves are also detected in the earth's foreshock, in association with beam ions backstreaming from the bowshock, as a possible free energy source for wave growth. However, linear instability analysis shows that, regardless of the beam ion types, the maximum growth occurs for parallel propagation for which the waves are purely electromagnetic and noncompressive, and cannot steepen.