Multiple–scales perturbation methods are used to derive equations describing wave–wave interactions in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension. Two problems are considered: (a) the interaction between short-wavelength waves in a non-uniform large-scale background flow, and (b) wave interactions between long-wavelength waves propagating through a uniform background medium. The short-wavelength wave equations describe the interaction between the backward and forward thermal gas sound waves and the contact discontinuity eigenmode via ‘wave mixing’, in which the waves are reflected by gradients in the large scale background flow. The equations also contain quadratic wave interaction terms describing (i) the Burgers self-wave interaction term, (ii) mean–wave-field interaction terms in which a specific wave interacts with the mean field of the other waves, and (iii) three-wave resonant interactions that describe how a sound wave resonantly interacts with the contact discontinuity to generate a reverse sound wave. In the limit of no interaction between the cosmic rays and thermal gas, and for a uniform background state, the equations reduce to two coupled, integro-differential Burgers equations derived by Majda and Rosales for resonantly interacting waves in adiabatic gasdynamics. The short-wavelength equations also contain squeezing instability terms associated with the large-scale cosmic-ray pressure gradient, which were first investigated by Drury and Dorfi. A generalized wave-action equation, or canonical wave-energy equation, variational principles, and WKB analyses of the linearized equations are used to investigate the modification of the cosmic-ray squeezing instability by wave mixing. Coupled Burgers equations are also derived in the long-wavelength regime that describe resonant wave interactions for weak, diffusively smoothed cosmic-ray-modified shocks. In the latter equations, the cosmic-ray-modified sound waves can resonantly interact with either the contact discontinuity or the cosmic-ray pressure-balance mode to generate a reverse sound wave.
Abstract.The interaction of the solar wind termination shock with disturbances incident from the upstream solar wind is considered using a one-dimensional MHD (both transverse and oblique) model. This paper extends the work of Story and Zank [1995] (Paper 1) to include the effects of the interplanetary magnetic field. A decay law is derived to describe the damping of compound rarefaction/shock structures.The inclusion of a parallel magnetic field component leads to the production of slow-mode rarefactions and shocks. Therefore, depending on parameters, the interaction of the termination shock with interplanetary disturbances may serve to generate both slow-and fast-mode magnetosonic waves that propagate through the heliosheath. Furthermore, it follows from the simulations presented here that slow-mode waves and rotational discontinuities can be expected to occur, at least to some extent, at any region of the solar system where collisions between oblique MHD shocks occur. Some discussion regarding the relative local orientation of the heliospheric magnetic field, with respect to the interstellar magnetic field, is presented to address the possibility of a global rotational discontinuity(s) which may be necessary to connect the interplanetary magnetic field to the interstellar field. Some further results concerning the interaction of waves and shocks reflected from heliosheath structure with the termination shock are also presented.
A one‐dimensional gasdynamic model is used to analyze the interaction of the solar wind termination shock (TS) with various interplanetary disturbances. These include density enhancements and depletions, Gaussian “shock” pulses, and forward‐reverse shock pairs. Our simulations suggest that the termination shock is unlikely to be stationary. Both the interaction state and the postinteraction state of the termination shock are discussed. The very complicated state and structure of the TS during interaction with interplanetary disturbances is examined carefully. It is found that the TS is not readily identifiable from the other simultaneously present structures (reverse shock, contact discontinuities, forward shocks and so on.). This may complicate the identification of the TS by the Voyager or Pioneer spacecraft significantly. It is also found that a number of interesting structures propagate and evolve downstream after a collision. These include sharp spikes in the density, which advect outward at the postshock flow speed, and secondary damped shock pulses produced in the collision, which propagate supersonically. Some implications of both the density enhancements and the production of damped shock waves in the heliosheath and at the heliopause are discussed.
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