Ideal and Dissapative Stability of Shock Fronts in Magnetohydrodynamics
EPSRC Research Grant GR/S96609/02
Principal Investigator: Dr K.I. Ilin
Visiting Researcher: Dr Y.L. Trakhinin

Summary

In the framework of the project we have studied the stability and existence of discontinuity fronts in compressible magnetohydrodymanics (MHD). Such discontinuity fronts are often observed in astrophysical and geophysical phenomena. For example, the Earth magnetopause, which is a complex structure between the Earth magnetosphere and the solar wind, can be treated theoretically as a current-vortex sheet (a closed or nightside magnetopause) or an Alfven discontinuity (an open or dayside magnetopause). Discontinuity fronts are formally introduced within the framework of a mathematical model and sometimes do not exist in the real physical world. A necessary condition for existence of these discontinuities is their stability. We have employed the mormal mode analysis to investigate the stability of a planar Alfven discontinuity and found that, for certain values of the parameters, the Alfven discontinuity is strongly unstable and that the growth rate of the instability can be arbitrarily large. This implies that in the unstable region of the parameter space Alfven discontinuities as smooth surfaces cannot exist for any finite, however short, time interval. We have obtained a mathematically rigorous result on the existence of compressible current-vortex sheets, namely: we have proved the existence of a current-vortex sheet provided that certain conditions on initial data are satisfied. Also we have started an investigation of the effect of dissipation (due to viscosity and electrical resistivity) on the stability of MHD shock waves and identified the conditions under which viscous analogues of shock waves are stable with respect to one-dimensional perturbations.

See papers below for more details.

Publications

1. K. I. Ilin, Y. L. Trakhinin, The stability of Alfven discontinuity, Physics of Plasmas, vol. 13, 102101--102108 (2006). (pdf)

2. K. I. Ilin, Y. L. Trakhinin, On stability of Alfven discontinuities (submitted to Mathematical Methods in the Applied Sciences), 2007. (pdf)

3. A. Blokhin, Y. Trakhinin, Well-posedness of linear hyperbolic problems: theory and applications, New York: Nova Science Publishers, 2006.

4. Freist\"{u}hler, Y. Trakhinin, On viscous and inviscid stability of magnetohydrodynamic shock waves. In preparation. (pdf)

5. A. Morando, Y. Trakhinin, P. Trebeschi, Stability of incompressible current-vortex sheets. In preparation. (pdf)

6. Y. Trakhinin, Dissipative symmetrizers of hyperbolic problems and their applications to shock waves and characteristic discontinuities, SIAM J. Math. Anal., vol. 37 (2006), 1988--2024. (pdf)

7. Y. Trakhinin, Existence and stability of compressible and incompressible current-vortex sheets. In: Calgaro C., Coulombel J.-F., Goudon T. (eds.), Analysis and Simulation of Fluid Dynamics, pp. 229-246. Advances in Mathematical Fluid Mechanics, Basel: Birkhauser Verlag, 2006. (pdf)

8. Y. Trakhinin, The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics, Arch. Ration. Mech. Anal., 68 pages, to appear. (pdf)

9. Y. Trakhinin, On compressible current-vortex sheets. To appear in: Benzoni S., Serre D. (eds.) {\it Hyperbolic problems. Theory, Numerics, Applications}, 15 pages. Basel, Boston, Berlin: Birkhauser, 2007. (pdf)