Interaction of turbulence with shock waves

The interaction of turbulence with shock waves is discussed in this dissertation. The heliospherical medium can be highly turbulent. We provide a new mechanism which can explain the unusual long wavelength (0.5-1 AU), low-frequency (∼ 3 x 10-5rm s-1) velocity fluctuation observed by Voyager 2 at 48 AU in the outer heliosphere (Paularena et al., 1996). The velocity oscillations appear to be superimposed on weak linear velocity gradients that are typically increasing. We show that pickup ions can excite low-frequency fluctuations and drive the flow field linearly. The model is compared directly to velocity oscillations observed by the Voyager 2 PLS instrument between 40 and 55AU over the six year period 1993-1998. Since pickup ions (PUIs) play an important role in driving heliospherical instabilities and turbulence, we develop a multi-fluid model of waves with pickup ions and discuss a new pickup ion driven instability. The growth rate of the PUI driven instability is found to be dependant on the ratio of pickup ion number density and electron/proton number density. In Part two of this dissertation, two different methods have been applied to study the interaction of turbulence with shock waves: (1) mean-field theory, and (2) multi-scale reductive perturbation method. Mean-field theory, together with a weak-linearization analysis, is adopted to investigate the statistical properties of the shock structure. The usual Rankine-Hugoniot condition is modified to describe the jump relation between the mean upstream field and the mean downstream field. For a special non-correlated solution, the mean shock position can be calculated from a set of equations, which is possible to be tested with observations. The solutions of turbulent Rankine-Hugoniot (R-H) conditions are compared to the usual steady state R-H conditions. Numerical solutions show that: (1) the downstream Mach number increases with increasing upstream turbulence strength; and (2) real solutions of the turbulent R-H conditions may not exist when strong turbulence interacts with a weak shock, which implies an unstable shock front. A 2D gas-dynamical model describing the interaction of thin shock waves with turbulence is developed by adopting a multi-scale reductive perturbation analysis. This is extended to a 2D MHD model. The interaction is found to be governed by a two-dimensional inviscid Burgers' equation that includes "perturbation terms." Initially prescribed perturbation profiles are explored with numerical simulations to show how the shock front is modified by turbulence. The results indicate that while turbulence can balance the nonlinear steepening of a shock wave at some regions, it can also help to create a larger jump in physical quantities at other regions. The plasma medium in these regions can therefore experience a higher compression, which will result in a downstream state that differs from the usual Rankine-Hugoniot state. A reduced model for parallel MED shocks is then obtained. Numerical simulations for parallel MHD shocks show that magnetic field perturbations play an important role in modifying shock structures.