Nonlinear dynamics of small-scale Alfvén waves

Abstract

We study the nonlinear evolution of very oblique small-scale Alfvén waves with k ⊥ d i > ∼ 1. At these scales, the waves become significantly compressive, unlike in MHD, due to the Hall term in the equations. We demonstrate that when frequencies are small compared to the ion gyrofrequency and amplitudes small compared to unity, no new nonlinear interaction appears due to the Hall term alone at the lowest non-trivial order, even when k ⊥ d i ∼ 1. However, at the second non-trivial order, we discover that the Hall physics leads to a slow but resonant nonlinear interaction between co-propagating Alfvén waves, an inherently 3D effect. Including the effects of finite temperature, finite frequency, and electron inertia, the two-fluid Alfvén wave also becomes dispersive once one or more of k ⊥ ρ s , k ⊥ d e , or k d i becomes significant: for oblique waves at low β as studied here, this can be at a much smaller scale than d i . We show that the timescale for one-dimensional steepening of two-fluid Alfven waves is only significant at these smaller dispersive scales, and also derive an expression for the amplitude of driven harmonics of a primary wave. Importantly, both new effects are absent in gyrokinetics and other commonly used reduced two-fluid models. Our calculations have relevance for the interpretation of laboratory Alfvén wave experiments, as well as shedding light on the physics of turbulence in the solar corona and inner solar wind, where the dominant nonlinear interaction between counter-propagating waves is suppressed, allowing these new effects to become important.

Nonlinear dynamics of small-scale Alfvén waves

“…It should be noted that all of these works did not provide a fully systematic derivation of their equations, and implicitly used a small-amplitude approximation. In a previous paper, we have systematically (re)derived 45 the equations used by Seyler & Lysak, providing support for their conclusion that small-amplitude kinetic and/or inertial Alfvén solitons with continuous density profiles do not exist.…”

“…It is only at first order that non-trivial evolution of the waves appears, dependent on the dispersive parameters k 2 ρ 2 s and k 2 ∥ d 2 i . Second, in an extension of the theory resulting in the modified Korteweg-de Vries (mKdV) equation 46,47 , we have shown that non-singular AW solitons still exist even when d i ∂ x ∼ 1, taking the form of full rotations of the transverse magnetic field: these solutions cannot be found if one takes a small-amplitude approximation, even with strong nonlinearity 44,45 . If the structure has a large width compared to d i , our dynamical equation reduces to the mKdV equation, with both dip and spike solitons.…”