Vlasov code simulation of anomalous resistivity

One of the outstanding open questions in space physics is that of the resistivity of collisionless plasmas, where binary collisions are inefficient and collective wave-particle interactions determine transport and dissipation. In order to obtain macroscopic transport coefficients one has to solve kinetic equations together with the Maxwell equations for the interacting electromagnetic fields, an essentially nonlinear problem. For weak turbulence the quasi-linear and other weakly nonlinear theories provide predictions for the saturated field fluctuations and the so called anomalous resistivity, the normal mode of current dissipation in collisionless space plasmas. Unfortunately, no generic anomalous resistivity expression is found yet, which could be used for a macroscopic Ohm's law description like the Spitzer - Braginski one for the case of binary collisions. We shortly review the main results obtained for the anomalous resistivity caused by ion sound turbulence in the framework of weak turbulence theories and by Particle-In-Cell (PIC) simulations. Since PIC simulations are noisy, introducing artificial numerical collisions, we review the possibilities of a direct integration of Vlasov's kinetic equation. Indeed, the implementation of highly accurate numerical schemes on modern parallel-processor computers stimulates the revival of Vlasov-code simulations. We discuss the state of the art of this simulation approach to determine the anomalous resistivity and give an outlook toward future work toward the solution of the still open question of a whether there is a generic expression for anomalous resistivity eligible for a macroscopic Ohm's description of collisionless plasmas.