Minimal model for transport barrier dynamics based on ion-temperature-gradient turbulence

Low-order mode coupling equations are derived to describe recent computer simulations of the toroidal ion-temperature-gradient turbulent convection with steady and pulsating sheared mass flows in the transport barrier zone. The three convective transport states are identified with the tokamak confinement regimes called low mode (L-mode), high mode (H-mode), and barrier localized modes (BLMs) when the transport barrier is in the core plasma. The L-mode limit cycle is analytically derived and a bifurcation diagram showing L to H and H to BLM transitions in confinement is constructed numerically. Markovian closure procedures are sought to further reduce the dimensionality of the nonlinear system. First an exact expression is given for the energy transfer rate from the fluctuations to the sheared mass flow through the triplet velocity correlation function. Then the time scale expansion required to derive the Markovian closure formula is given. Markovian closure formulas form the basis for the thermodynamic-like L–H bifurcation models.