Nonlinear Alfvén waves in a finite-beta plasma

Abstract

The DNLS equation for parallel nonlinear and weakly dispersive MHD waves is extended to finite beta values as well as to three spatial dimensions, by means of the reductive perturbation method. Kinetic effects are included by means of the hybrid fluid and kinetic guiding-centre model of Grad (1961). The resulting equation contains a nonlinear and non-local term representing the effect of resonant particles.

Fluid models for kinetic effects on coherent nonlinear Alfvén waves. I. Fundamental theory

“…A fully kinetic calculation was undertaken by Rogister [5] for the case of a high-beta plasma. For parallel propagation, his results counside with equations obtained later by Mjølhus and Wyller [12,6] and Spangler [13,14]. The effect of Landau damping appears in the DNLS via an additional cubic term which is an integral operator over space.…”

“…We first use one-fluid MHD equations with parallel dissipation coefficients µ and χ which are constant and independent of mode frequency and wave-number. Later we replace constant χ with the integral operator representation of Hammet and Perkins [17], and obtain a modifield DNLS similar to that obtained by Mjølhus and Wyller [12,6] and Spangler [13,14]. However, our approach yields expressions for coefficients of the nonlinear terms which are much simpler than theirs and allows clear, unambiguous physical interpretation of the results.…”

“…Previous studies have shown [4][5][6] that nonlinear Alfvén and fast magnetosonic waves are described by the derivative nonlinear Schrödinger equation (DNLS). In this model, a parallel ponderomotive force of a highamplitude wave perturbs the plasma density.…”

“…Thus, with the notable exceptions of Ref. [6,[12][13][14], the theoretical 'lore' of nonlinear Alfvén waves in a collisionless, β ∼ 1 plasma is built upon a conceptual paradigm constructed for a collisional, β < 1 system. This state of affairs is due, in part, to the intractability and unwieldiness of straightforward kinetic analysis of nonlinear Alfvén waves.…”

“…The other attempts to include Landau damping in the nonlinear Alfvén wave evolution used kinetics to calculate second-order density (or pressure) perturbation, but the DNLS-like equation (for b) was obtained from MHD equations. Mjølhus and Wyller [12,6] used a guiding center formalism and Spangler [13,14] used the full Vlasov equation. Both methods predict a nonlocal integral term, and the coefficients of the nonlinear terms coinside with those of Rogister [5].…”