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Abstract: The non-linear Hall term present in the induction equation in theelectron-magneto-hydrodynamics limit is responsible for the Hall drift of themagnetic field and, in some cases, for the so-called Hall instability. Weinvestigate whether or not the growth rates and eigenfunctions found in thelinear analysis are consistent with the results of non-linear numericalsimulations. Following the linear analysis of Rheinhardt and Geppert, we studythe same cases for which the Hall instability was predicted by solving thenon-linear Hall induction equation using a two-dimensional conservative anddivergence-free finite difference scheme that overcomes intrinsic difficultiesof pseudo-spectral methods and can describe situations with arbitrarily highmagnetic Reynolds numbers. We show that unstable modes can grow to the level ofthe background field without being overwhelmed by the Hall cascade, and cause acomplete rearrangement of the field geometry. We confirm both the growth ratesand eigenfunctions found in the linearized analysis and hence the instability.In the non-linear regime, after the unstable modes grow to the backgroundlevel, the naturally selected modes become stable and oscillatory. Later on,the evolution tends to select the modes with the longest possible wavelengths,but this process occurs on the magnetic diffusion timescale. We confirm theexistence of the Hall instability. We argue against using the misleadingterminology that associates the non-linear Hall term with a turbulent Hallcascade, since small-scale structures are not created everywhere. The fieldevolves instead in a Burgers-like manner, forms local structures with stronggradients which become shocks in the zero resistivity limit, and Hall waves arelaunched and propagated through the entire domain.

Autor: Jose A. Pons, Ulrich Geppert


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