Calculation of the electron velocity distribution function in a plasma slab with large temperature and density gradients

A detailed description is given of a new method for calculating the high- velocity tail of the electron velocity distribution function in a plasma slab with large temperature and density gradients. Thermal electrons in a plasma are strongly coupled with each other and in a steady state their velocity distribution is always near to maxwellian. On the other hand, the collision frequency of an electron decreases rapidly with increasing speed (v ~ v-3), so that the coupling between the high-velocity electrons and the plasma is very weak. These electrons move almost freely through the plasma and an inhomogeneity can strongly affect the high-velocity part of the distribution function. In our method electrons are classified into two groups, depending on their velocity. The distribution function for the first group (thermal electrons) is accurately given by the Spitzer-Harm solution of the Landau-Fokker-Planck equation. For the second group (high-velocity electrons) the Spitzer-Harm solution is inaccurate and we calculate the distribution function as a solution to the high-velocity approximation of the Landau-Fokker-Planck equation (HVL). The two solutions are matched at a suitably chosen value of the normalized speed ξ. We solve the HVL equation numerically using an efficient method that we have developed. Application is made to the transition region of the quiet Sun using several data-sets for temperature and density gradients by different authors. The results exhibit large deviations from maxwellian throughout the transition region as well as a strongly anisotropic character of the high-velocity tail of the distribution function. The results are very sensitive to the gradients. Also, the non-local character of the formation of the velocity distribution is clearly seen.