A dynamical model for the low-high (L-H) confinement mode transitions consisting of three ordinary differential equations (3-ODE model) for the essential state variables is proposed. The model is derived from the energy balance equations for the resistive pressure-gradient-driven turbulence and describes temporal evolutions of three characteristic variables (u, k, f), the potential energy contained in the pressure gradient, the turbulent kinetic energy and the shear flow energy. The energy input to the peripheral plasma region is included as an external control parameter in the model. The model equations have stationary solutions corresponding to the L- and H-modes. The L to H and H to L transitions are obtained by varying the energy input parameter. The type of L-H transition, whether like a first- or second-order transition, is shown to be determined by the sheer flow damping. At a higher level of the energy input parameter the H-mode stationary solution becomes unstable and bifurcates to a limit cycle which shows periodic oscillations characteristic of the H-localized mode (ELM) confinement state.
The ELMy H mode as a limit cycle and the transient responses of H modes in tokamaks