Hopf bifurcation for tearing instability in magnetohydrodynamics, theory and numerical results

Abstract

Fusion plasmas inside a Tokamak may be represented using magnetohydrodynamic equations. An equilibrium solution consists in a stationary solution of these equations, generally depending only on some radial coordinate. In Tokamak experiments, we try to obtain such equilibrium solutions; however, some instabilities may appear and destroy such an equilibrium configuration. For instance, the so-called tearing instability, which destroys the equilibrium magnetic configuration, is really observed in experiments. Above a critical value of some physical parameter, the instability appears first as a stationary solution; then, increasing again some physical parameter, oscillatory behaviour can be found, before more complicated states and exponential growth of the solutions. These physically observed phenomena have also been numerically computed (new results using the DEMA code concern the so-called double-tearing instability). The mathematical framework of bifurcation theory is well suited for such a study; we first recall the existence of a stationary branch of bifurcated solutions; then, using mainly a Center Manifold Theorem, we prove existence of Hopf bifurcation. Most of the results are obtained with given diffusion coefficients (viscosity, resistivity) but new results are also obtained when these coefficients depend nonlinearly of the unknowns. We also give a new mathematical justification of the decomposition of the velocity and magnetic fields used in the previously cited DEMA code. Finally, we also recall some results concerning the existence of a global attractor (in dimension 2).