Electron temperature anisotropy modeling and its effect on anisotropy-magnetic field coupling in an underdense laser heated plasma

The laser interaction with an underdense plasma leads to an anisotropic laser heating of electrons. This temperature anisotropy gradient in turn is the source of an early magnetic field, which has an important effect on the plasma evolution, due to the thermal flux reduction. We describe the temperature anisotropy by an evolution equation including the anisotropy-magnetic field coupling and observe a rather efficient magnetic field generation. However at high anisotropy levels, a small-scale instability emerges, leading to a serious problem in numerical calculations. We introduce the kinetics effects, which fix the problem by the anisotropy diffusion through the heat flux tensor. A constant-coefficient Fokker-Planck model in the 2-D geometry allows us to derive an anisotropy diffusion term. The diffusion coefficient is fitted from the kinetic theory of the collisional anisotropic (Weibel) instability growth rate. Such an anisotropy diffusion term wipes out the unphysical instability without any undesirable smoothing. This diffusion along with the viscosity term leads also to a quite good restitution of the Weibel instability growth rate and to the short wavelength cutoff, even in a weakly collisional situation. This allows us to use such a model to predict emergence the Weibel instability as well as its saturation