Stability of Discontinuous States in Magnetohydrodynamics
EPSRC Research Grant GR/R79753/01
Principal Investigator: Dr K.I. Ilin
Postdoctoral Research Assistant: Dr Y.L. Trakhinin

Summary

To study the linear stability of steady MHD flows with current-vortex sheets, we have employed the energy principle first proposed by Bernstein et al for magnetostatic equilibria and later generalized by Frieman and Rotenberg to the case of steady MHD flows. We have extended the method to steady MHD flows with surfaces of tangential discontinuities across which the tangent velocity or the tangent magnetic field or both of them have jump discontinuities. We have shown that there are non-trivial steady flows for which the energy of the linearized problem is positive semi-definite. >From this fact we have concluded that these flows are linearly stable and formulated the corresponding sufficient conditions for stability to small three-dimensional perturbations. For a general magnetostatic equilibrium with a current sheet, we proved that non-negative definiteness of the energy is not only sufficient, but also necessary for stability. Explicit stability criteria have been formulated for a number of simple examples. We have found that Bogoyavlenskij's transformation (which is a symmetry transformation of the steady equations of incompressible magnetohydrodynamics) has the following remarkable property: if a steady MHD flow is stable by the energy method, a `half' of all flows which can be obtained from this flow by the transformation are also stable. Two new families of stable MHD flows which depend on all three spatial coordinates have been constructed. For each family, we have identified the conditions under which all members of the family are stable. Then, we have demonstrated that the similarity in the stability properties of steady MHD flows connected by Bogoyavlenskij's transformation holds for flows with current-vortex sheets, so that the known examples of stable discontinuous flows can also be generalized in the same manner as for continuous MHD flows. Finally, we have partially justified our stability results by showing that the corresponding linearized problems are mathematically well-posed (this has been proved for both compressible and incompressible fluids). This result may lead to a proof of the local existence theorem for the exact nonlinear problem as well as to examples of nonlinear stability.

See papers below for more details.

Publications

1. K. I. Ilin, Y. L. Trakhinin and V. A. Vladimirov, The stability of steady magnetohydrodynamic flows with current-vortex sheets, Physics of Plasmas, 2003, Vol. 10, 2649-2658. (pdf)

2. K. I. Ilin and V. A. Vladimirov, Energy principle for magnetohydrodynamic flows and Bogoyavlenskij's transformation, accepted for publication in Phys. Plasmas, 2004 (Vol. 11, No 7). (pdf)

3. Y. L. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, 2004 (submitted to Mathematical Methods in the Aplied Sciences).

4. Y. L. Trakhinin, On the existence of compressible current-vortex sheets: variable coefficient linear analysis, 2004 (submitted to Archive for rational mechanics and analysis).