Evolution of the plasma rotation and the radial electric field for a toroidal plasma in the Pfirsch--Schlueter and plateau regimes subject to a biased electrode

In this paper, the fluid equation approach is used to analyze the time evolution of the plasma rotation and the ambipolar electric field in a nonsymmetric toroidal plasma subject to an external biasing voltage induced by a probe. Under consideration is a plasma with low rotation speed in the Pfirsch--Schlueter or the plateau regime that includes the effects of a background neutral gas. A time-dependent charge conservation equation is used to determine the ambipolar electric field as a function of time. It is found that, after the application of the biasing voltage, the electric field and the plasma rotation change quickly and reach steady-state after a time inversely proportional to the sum of the momentum damping rates due to parallel viscosity and ion--neutral collisions. The steady state is characterized by a radial electric field and a plasma rotation that are proportional to the electric current flowing through the biasing probe. The direction of the plasma flow is determined by the relative magnitude of the momentum damping rates on the flux surface. From the steady-state solution, an expression for the radial electric conductivity is obtained, which includes the effect of collisions with neutrals as well as viscosity. Axisymmetric systems without neutrals are also discussed, which is a special case since there is no momentum damping in the toroidal direction. Here, the toroidal velocity increases continuously in time with the bias and never reaches steady state. Finally, a model for nonsymmetric magnetic fields is presented and the viscous damping rate, the radial conductivity and the spin-up rate for a plasma in the Pfirsch--Schlueter regime are calculated. As examples, the cases of the rippled tokamak and the classical and helically symmetric stellarators are evaluated