Laws for electron pressure variations across a collisionless shock

In order to characterize the electron pressure variations across an oblique collisionless shock, statistical methods are applied to the results of two‐dimensional (2‐D) full particle electromagnetic simulations. Local correlations are looked for between the spatial variations of the pressures p ‖ and p ⊥ (parallel and perpendicular to the local magnetic field) throughout the shock profile and the corresponding variations of the density n and the magnetic field modulus B at the same location. Different orders in regression laws are successively analyzed, including the most general 4‐D regressions p ⊥ p ‖ − u n − v B −2 w =constant, which test the degree of invariance of the quantities p ⊥ p ‖ − u n − v B −2 w , the reduced 3‐D laws p ⊥ n − a ⊥ B −2 b ⊥ = cst, p ‖ n − a ‖ B −2 b ‖ = cst (as in CGL theory), and the reduced 2‐D correlations laws p ⊥ n −γ⊥ = cst, p ‖ n −γ‖ = cst, nB − C p = cst (polytropic forms). Coefficients are determined quantitatively for each law. The use of these different regressions laws allows to check (1) where local correlations between these four quantities do exist at a given scale and, when verified, what is their effective forms; (2) when these general closure laws can be reduced to simpler ones, in particular to polytropic forms and with which polytropic indexes (γ=5/3?). This last result may have consequences concerning the fluid plasma modelizations for collisionless shocks or other nonlinear configurations; a comparison with the existing theories about the closure of fluid equations is briefly presented.