A Vlasov-Hybrid Code with Hermite Expansion of the Distribution Function for the Study of Low Growth Rate Instabilities

Abstract

Within turbulence there are many phenomena which are currently unsolved. In the solar wind temperature anisotropies and low growth rates instability have a dominant role in de ning the turbulent behaviour of plasma. Due to the non linearity of the equations involved in the description of the physics of plasmas numerical simulations are a fundamental tool to study the dynamics of turbulent phenomena. In particular, hybrid codes are widely used in space plasma applications due to their ability to simulate large regions of volume maintaining some kinetic e ects. However, due to the sensitivity to the initial level of noise in the simulation, low growth rate instabilities are particularly di cult to simulate. Particle in Cell-hybrid simulations require too many particles to reduce the initial noise, while Vlasovhybrid simulations require too many grid points to fully discretize spatial and velocity phase spaces. We present here a Vlasov-hybrid algorithm and code implementation where the distribution function is expanded in series of Hermite functions. Thanks to the properties of these it is possible to project the Vlasov equation to nd an equation for each coe cient of the expansion. These coe cients are advanced in time using a Current Advance Method algorithm with splitting method for the Vlasov operator. The former is treated explicitly, while the latter is treated implicitly with a GMRES solver. The current is advanced with a temporal ODE derived taking moments of the Vlasov equation. A 1D3V code is implemented, tested and used to study low growth rate instabilities such as a proton cyclotron instability and a ion/ion right hand resonant instability with small relative velocity drift between beam and core populations. The results are compared with existing hybrid algorithms that we implemented. A 2D3V parallelized version of the code is implemented and described here. Initial results are presented and future improvements are discussed.