KINETIC EFFECTS ON ALFVEN-WAVE NONLINEARITY .2. THE MODIFIED NONLINEAR-WAVE EQUATION

The study of kinetic effects on Alfvén wave nonlinearity is continued. Previously obtained expressions for the perturbed (by an Alfvén wave) ion and electron distribution functions are used to obtain a nonlinear wave equation for parallel-propagating, circularly polarized waves. The results are cast in the form of a modified version of the familiar derivative nonlinear Schrödinger equation. The approach in obtaining this equation is a hybrid one; fluid theory is used to the greatest extent possible, and kinetic theory is introduced where the correction is believed to be most important. Fluid theory at two levels of sophistication is employed. The first uses a simple scalar pressure term. This approach yields physical insight and illuminates the field-aligned fluid flow and the associated plasma density perturbation as a major contributor to Alfvén wave nonlinearity. The second approach employs a tensor pressure term that in general will be necessary. The results indicate that kinetic effects in general produce a nonlinear wave equation that is of a different functional form than the derivative nonlinear Schrödinger equation, as previously reported by Mjølhus and Wyller [Phys. Scr. 33, 442 (1986); J. Plasma Phys. 40, 229 (1988)]. The coefficient of the derivative cubic term depends on the plasma beta in a way which, in general, is quite different from the fluid expression. In addition, a functionally novel term appears in the modified equation. The magnitude of this term, named the "nonlocal term" by Mjølhus and Wyller, can be large when the plasma beta is comparable to unity. The susceptibility of the modified equation to modulational instability is studied. Kinetic effects cause modulational instability of wave packets, even when fluid theory would predict modulational stability. This modulational instability occurs for both right- and left-hand polarized waves. This latter result might explain a property of waves near the Earth's bow shock that was enigmatic within the context of fluid theory. © 1990 American Institute of Physics.