ELMy-H mode as limit cycle and transient responses of H-modes in tokamaks

A model of Edge Localized Modes (ELMs) in tokamaks is presented. The model of the L/H transition is extended to include the temporal evolution and the spatial structure. The existence of the electric field bifurcation infers a hysteresis curve between the plasma gradient and the loss flux (flow). The time-dependent Ginzburg-Landau equation is formulated for the electric field development. The self-generated oscillation of edge density appears, associated with periodic bursts of the loss, under the condition of constant particle flux from the core. This is attributed to the small and frequent ELMy activity in H-mode. Periodic decay and re-establishment of the transport barrier occur. This oscillation appears near the L/H transition boundary in the parameter space. It is found that in H- and ELMy-H- states, the edge region forms an intermediate radial structure of diffusion coefficient (mesophase) between H-phase and L-phase. The diffusion Prandtl number, the ratio of the viscosity to the diffusivity, is found to determine the thickness of the barrier. Phase diagram of L-, ELMy-H, H- and L/H bistable-states is obtained on plasma parameter space. To consider other types of ELMs, transient responses to the perturbations of the flux from the core are studied in each phase. In low collisionality plasma a sudden decrease of the flux can cause the transient bursting of a few ELMs before the L to H transition is completed. Large scale pulse from the core causes the transient return to L-phase and a giant burst occurs in high collisionality plasma. In ELMy-H state, external oscillations cause the mode-locking and other complicated dynamics. The perturbation in the flux widens the parameter regime of the occurrence of burstings. (J.P.N.)