Magnetohydrodynamic Equilibrium with Flow in Toroidal Plasmas and its Relevance to Internal Transport Barriers

The appearance of transport barriers in plasmas is usually taken to be the result of bifurcations in the transport equations, in which the possible control parameters is the radial electric field, in association with shear flow, which may contribute to the reduction or suppression of turbulent fluctuations. Yet, most calculations start from the assumption of magnetohydrodynamic (MHD) equilibira calculated form the Grad-Shafranov equation,, in which no flows are taken into account. The purpose of this work is to study the effect of toroidal and poloidal flows in the quantities related to the build up of internal transport barriers. The force balance equation, gives rise to a modified Grad-Shafranov equation, which must be solved along with a Bernoulli equation. In order to understand the role of the input power, it is useful to understand the behaviour of the MHD conserved quantities as well, which are related to the topology of the fields. The electrostatic field potential remains a surface quantity, so far as resistivity is ignored. It is found that, while most of the equilibrium quantities show small variations, the most important changes are observed in the electric field.