Global Stability and Local Bifurcations in a Two-Fluid Model for Tokamak

We study a two-fluid description for high and low temperature components of the electron velocity distribution in an idealized tokamak plasma evolving on a cylindrical domain, taking into account nonlinear drift effects only. We refine previous results on the laminar steady state stability and include viscosity. Taking the temperature difference as the primary parameter, we show that linear instabilities and bifurcations occur within a finite interval and for small enough viscosity only, while the steady state is globally stable for parameters sufficiently far outside the interval. We find that primary instabilities always stem from the lowest spatial harmonics for aspect ratios of poloidal versus radial extent below some value larger than 2. Moreover, we show that any codimension-one bifurcation of the laminar state is supercritical, yielding spatio-temporal oscillations in the form of traveling waves, and hence locally stable for such bifurcations destabilizing the laminar state. In the degenerate case, where the instability region in the temperature difference is a point, these solutions form an arc connecting the bifurcation points. We also provide numerical simulations to illustrate and corroborate the analysis and find additional bifurcations of the traveling waves