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Asymptotic shallow models arising in magnetohydrodynamics. (English) Zbl 1491.76092

Summary: In this paper, we derive new shallow asymptotic models for the free boundary plasma-vacuum problem governed by the magnetohydrodynamic equations which are vital when describing large-scale processes in flows of astrophysical plasma. More precisely, we present the magnetic analogue of the 2D Green-Naghdi equations for water waves under a weak magnetic pressure assumption in the presence of weakly sheared vorticity and magnetic currents. Our method is inspired by ideas for hydrodynamic flows developed in [A. Castro and D. Lannes, J. Fluid Mech. 759, 642–675 (2014; Zbl 1446.76077)] to reduce the three-dimensional dynamics of the vorticity and current to a finite cascade of two dimensional equations which can be closed at the precision of the model.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
34E05 Asymptotic expansions of solutions to ordinary differential equations

Citations:

Zbl 1446.76077
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References:

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