Nonlinear Alfvén waves in extended magnetohydrodynamics

Abstract

Large-amplitude Alfvén 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 Alfvén 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 Alfvén 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 nonlinearitydispersion interplay, which is somewhat different from that of solitons.PACS numbers: 52.35. Bj, 52.30.Cv, 52.35.Mw, 47.10.Df Alfvén waves are the most typical electromagnetic phenomena in magnetized plasmas. In particular, nonlinear Alfvén waves deeply influence various plasma regimes in laboratory as well as in space, which have a crucial role in plasma heating [1,2], turbulence [3][4][5], reconnection[6], etc. As an interesting property of the Alfvén waves, the amplitudes as well as wave forms are totally arbitrary when they propagate on a homogenous ambient magnetic field [7,8]. In fact, we often observe large-amplitude Alfvén waves in orderly propagation (for example [9]). To put it in theoretical language, the set A of Alfvén waves after an appropriate transformation (see [10]), is a closed linear subspace, i.e., every linear combination of the members of A gives solution to the fully nonlinear wave equation. Needless to say, the set of general solutions to a linear equation is, by definition, a linear subspace. However, it is remarkable that the nonlinear MHD equation has such a linear subspace A of solutions.Here we investigate the underlying mechanism producing such solutions in the context of a more accurate framework, generalized MHD. When we take into account dispersion effects (we consider both ion and electron inertial effects [11,12]), the wave forms are no longer arbitrary (remember that the ideal MHD model is dispersion free). Yet, we find that the generalized MHD system has a linear subspace of nonlinear solutions. The Casimir invariants of the system is the root cause of this interesting property [13].We start by reviewing how the nonlinear Alfvén waves are created in ideal MHD; we put the problem in the perspective of Hamiltonian mechanics. We then formulate the generalized MHD system in a Hamiltonian form. Via constructing equilibrium solutions (so-called Beltrami equilibriums) on Casimir leaves, we derive nonlinear wave solutions. The dispersion relation is exactly that of the linear theory, while the wave amplitude may be arbitrarily ...

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