• turbulence
• plasma

Magnetised plasmas are found to be turbulent in a large variety of scenarios, from astrophysics to nuclear fusion. In solar wind, a collisionless environment, at scales much larger than the ion Larmor radius, the so called inertial range, turbulence is mainly due to the low frequency and strongly anisotropic Alfvènic fluctuations. Observations confirm the turbulent nature of solar wind by showing the existence of power law energy spectra for both Alfvènic fluctuating fields and the related compressive fluctuations. The latter result to be passively mixed by the Alfvènic cascade although affected by collisionless damping such as Landau damping. Indeed, it is not clear how the energy-cascade occurs in the phase space since the phase mixing consists in a transfer of energy towards smaller velocity space scales and this should prevent the observed fluid-like spectrum. In this work we investigate how the competition between phase mixing and nonlinear advection, depending on a stochastic $E \wedge B$-drift velocity, works in the Hermite-Fourier phase space. On a theoretical ground it turns out that the advection is responsible for coupling the phase mixing flow and an un-phase mixing flow coming back from smaller to larger velocity space scales. Because of this echo effect we expect to have a suppression on average of the Landau damping in some regions of the phase space. Therefore a fluid behaviour is restored.
We adopted a Kinetic Reduced MHD framework: the reduced MHD model well describes, taking the critical balance conjecture by Goldreich and Sridhar as an ordering assumption, strongly anisotropic Alfvènic fluctuations whereas a drift-kinetic equation governs the distribution function for the magnetised ions (electrons are considered as isothermal). In particular we have focused on the Batchelor limit for a passive scalar, the ion distribution function, and the obtained equation has been solved numerically by means of a Monte Carlo code. We observed the suppression effect, along with power law spectra, in the anisotropic regime. Moreover a partitioning of the phase space is now possible in order to find analytical solutions for the regions dominated by the suppression or the phase mixing.