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Dynamics of nonlinear Alfvén waves in the shallow water magnetohydrodynamic equations

Martin Magill*

Aaron Coutino, Benjamin A. Storer, Marek Stastna, and Francis J. Poulin

  • Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St N, Oshawa, Ontario, Canada L1G 0C5

  • Department of Applied Mathematics, University of Waterloo, 200 University Ave W, Waterloo, Ontario, Canada N2L 3G1

  • *martin.magill1@uoit.net
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Phys. Rev. Fluids 4, 053701 – Published 1 May, 2019

DOI: https://doi.org/10.1103/PhysRevFluids.4.053701

Abstract

The large-scale hydrodynamics of the solar tachocline have been modeled by a version of the well-known shallow water equations augmented to include the effects of magnetism. Due to the conservation of magnetic flux, these shallow water magnetohydrodynamic equations have an additional constraint that the classical shallow water equations (without magnetism) do not. This restricts the set of possible initial conditions, but in itself does not provide information about the dynamics. The linear theory of wave motion on both the f plane and β plane is relatively well documented. We thus seek to explore the dynamics of nonlinear waves on the magnetohydrodynamic f plane. By initializing with a circular vortex and no free surface displacement, we make contact with a substantial literature in geophysical fluid dynamics, satisfy the magnetohydrodynamic constraint, and provide a means to generate all possible wave species through the adjustment process. We find that for Cowling numbers typical of the solar tachocline, the response is dominated by zonally propagating Alfvén waves, which take the form of vortices. These vortices are expressed in the velocity, free surface, and magnetic field variables. We explore how nonlinearity and variation in the Cowling number affect these dynamical structures, finding that nonlinearity warps the axially symmetric Alfvén wave modes of linear theory. Moreover, while the waves maintain a fixed shape during pure propagation, we find that wave shape is significantly modified during collisions. The extent of warping during collisions is amplitude dependent, and hence a nonlinear effect, albeit one that is fundamentally different from soliton models of nonlinear waves in shallow water. We explore parameter ranges outside those relevant to our Sun's tachocline and find that our results are robust over significant portions of parameter space. In the limit of rapidly rotating stars (i.e., rates 100 times larger than those associated with our Sun) we find that the Alfvén waves destabilize over time.

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Phys. Rev. Fluids 4, 053701– Published 1 May, 2019
  • Received 29 October 2018
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