Dielectric Tensor Operator of Hot Plasmas in Toroidal Axisymmetrical Systems
Kinetic theory is used to develop equations describing dynamics of small-amplitude electromagnetic perturbations in toroidal axisymmetric plasmas. The closed Vlasov-Maxwell equations are first solved for a hot stationary plasma using the expansion in the small parameter epsilon(e) = rho/L, where rho is the Larmor radius and L a characteristic length scale of the stationary state. The ordering and additional assumptions are specified so as to obtain the well-known Grad-Shafranov equation. The dielectric tensor of such a plasma is then derived. The Vlasov equation for the perturbed distribution function is solved by the expansion in the small parameters epsilon(e) and epsilon(p) = rho/lambda, where lambda is a characteristic wavelength of the perturbing electromagnetic field. The solution is obtained up to the first order in epsilon(e) and the second order in epsilon(p). By integrating the resulting distribution function over velocity space, an explicit expression for the tensor is derived in the form of a two-dimensional partial differential operator. The operator is shown to possess the proper symmetry corresponding to the energy conservation law.