Ballooning instabilities in tokamaks with sheared toroidal flows

The ballooning-mode eikonal representation is applied to the linearized incompressible magnetohydrodynamic (MHD) equations in axisymmetric systems with toroidal mass flows to obtain a set of initial value partial differential equations in which the time t and the poloidal angle theta are the independent variables. To derive these equations, the eikonal function S is assumed to satisfy the usual condition B.gradS=0 to guarantee that the modes vary slowly along the magnetic field. In addition, to resolve the V.grad operator acting on perturbed quantities, the eikonal must also satisfy the condition dS/dt=0. this induces a Doppler shift in S. This description of the instability, however, is incompatible with normal mode solutions of the MHD equations because the wave vector gradS becomes time dependent when the velocity shear is finite. Nevertheless, the author is able to investigate the effects of the sheared toroidal flows on localized ballooning instabilities because the initial value formulation of the problem developed does not constrain the solutions to evolve as exp (i omega t). Fixed boundary MHD equilibria with isothermal toroidal flows that model the JET device are generated numerically with a variational inverse moments code. As the initial value evaluations are evolved in time, periodic burst of ballooning activity are observed which are correlated with the formation of a ballooning structure at the outside edge of the torus that becomes displaced by 2 pi in the extended poloidal angle domain from one burst to the next. The velocity shear has a stabilizing influence on plasma ballooning.