Cosmic Ray propagation in sub-Alfvenic magnetohydrodynamic turbulence

This work has the main objective to provide a detailed investigation of cosmic ray propagation in magnetohydrodynamic turbulent fields generated by forcing the fluid velocity field at large scales. It provides a derivation of the particle mean free path dependences in terms of the turbulence level described by the Alfv\'enic Mach number and in terms of the particle rigidity. We use an upgrade version of the magnetohydrodynamic code {\tt RAMSES} which includes a forcing module and a kinetic module and solve the Lorentz equation for each particle. The simulations are performed using a 3 dimension periodical box in the test-particle and magnetostatic limits. The forcing module is implemented using an Ornstein-Uhlenbeck process. An ensemble average over a large number of particle trajectories is applied to reconstruct the particle mean free paths. We derive the cosmic ray mean free paths in terms of the Alfv\'enic Mach numbers and particle reduced rigidities in different turbulence forcing geometries. The reduced particle rigidity is $\rho=r_L/L$ where $r_L$ is the particle Larmor radius and $L$ is the simulation box length related to the turbulence coherence or injection scale $L_{inj}$ by $L \sim 5 L_{inj}$. We have investigated with a special attention compressible and solenoidal forcing geometries. We find that compressible forcing solutions are compatible with the quasi-linear theory or more advanced non-linear theories which predict a rigidity dependence as $\rho^{1/2}$ or $\rho^{1/3}$. Solenoidal forcing solutions at least at low or moderate Alfv\'enic numbers are not compatible with the above theoretical expectations and require more refined arguments to be interpreted. It appears especially for Alfv\'enic Mach numbers close to one that the wandering of field lines controls perpendicular mean free path solutions whatever the forcing geometry.