Nonlinear Alfv\'en waves in extended magnetohydrodynamics

TL;DR

This paper derives fully nonlinear Alfvén waves within an extended MHD framework that includes Hall and electron inertia effects, revealing how small-scale effects influence large-scale wave shapes and the interplay of nonlinearity and dispersion.

Contribution

It introduces a new model for nonlinear Alfvén waves that accounts for dispersive effects, showing the interdependence of small and large scales unlike ideal MHD predictions.

Findings

Large-scale wave shapes are constrained by small-scale effects.

The coexistence of scales prevents arbitrary large-scale wave forms.

Nonlinearity and dispersion interplay differs from classical soliton behavior.

Abstract

Large-amplitude Alfv\'en waves are observed in various systems in space and laboratories, demonstrating an interesting property that the wave shapes are stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model predicts that an Alfv\'en wave keeps an arbitrary shape constant when it propagates on a homogeneous ambient magnetic field. However, such arbitrariness is an artifact of the idealized model that omits the dispersive effects. Only special wave forms, consisting of two component sinusoidal functions, can maintain the shape; we derive fully nonlinear Alfv\'en waves by an extended MHD model that includes both the Hall and electron inertia effects. Interestingly, these \small-scale effects" change the picture completely; the large-scale component of the wave cannot be independent of the small scale component, and the coexistence of them forbids the large scale component to have a free wave form. This is a manifestation of the nonlinearity-dispersion interplay, which is somewhat different from that of solitons.