Two-dimensional transport of tokamak plasmas

Abstract

A reduced set of two‐fluid transport equations is obtained from the conservation equations describing the time evolution of the differential particle number, entropy, and magnetic fluxes in an axisymmetric toroidal plasma with nested magnetic surfaces. Expanding in the small ratio of perpendicular to parallel mobilities and thermal conductivities yields as solubility constraints one‐dimensional equations for the surface‐averaged thermodynamic variables and magnetic fluxes. Since Ohm’s law E +u×B =R′, where R′ accounts for any nonideal effects, only determines the particle flow relative to the diffusing magnetic surfaces, it is necessary to solve a single two‐dimensional generalized differential equation, (∂/∂t) ∇ψ. (∇p − J×B) =0, to find the absolute velocity of a magnetic surface enclosing a fixed toroidal flux. This equation is linear but nonstandard in that it involves flux surface averages of the unknown velocity. Specification of R′ and the cross‐field ion and electron heat fluxes provides a closed system of equations. A time‐dependent coordinate transformation is used to describe the diffusion of plasma quantities through magnetic surfaces of changing shape.