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On Linear Instability and Stability of the Rayleigh–Taylor Problem in Magnetohydrodynamics

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Abstract

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh–Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving a priori estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Fei Jiang

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Song Jiang

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  1. Fei Jiang
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  2. Song Jiang
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Correspondence to Fei Jiang.

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Communicated by S. Friedlander

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Jiang, F., Jiang, S. On Linear Instability and Stability of the Rayleigh–Taylor Problem in Magnetohydrodynamics. J. Math. Fluid Mech. 17, 639–668 (2015). https://doi.org/10.1007/s00021-015-0221-x

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  • DOI: https://doi.org/10.1007/s00021-015-0221-x

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