MHD theory of Earth's magnetosheath for an axisymmetric model

An analysis is made of the Earth's magnetosheath along the Sun-Earth line under conditions that the IMF (Interplanetary Magnetic Field) is nearly parallel or anti-parallel to the solar wind flow. MHD conservation equations in temporally-averaged steady-state form for the mass, momentum and energy density are combined with the magnetic divergence and induction equations, a hard conducting-sphere model for the magnetopause, and an adiabatic equation of state in the magnetosheath. The equations are integrated from the nose of the bow shock to the magnetopause and reduced to a set of nonlinear-coupled equations for the magnetosheath thickness and average magnetosheath parameters, which are then used to obtain a new equation for the thickness of the magnetosheath. This is the first analytical equation for the magnetosheath thickness that has been derived, and it exhibits an interesting functional dependence on Alfven and sonic Mach number M(A) and M(s), the angle theta(o) between the bow shock normal and the IMF, the Chapman-Ferraro constant k(o) at the magnetopause, the polytropic index gamma, and the thermal conductivity Q(r) at the bow shock. The thickness is found to decrease for decreasing M(s), but increase with decreasing M(A). It exhibits the qualitative feature found in both gasdynamic and theta(o) greater than or equal to 45 degrees MHD simulations of an approximate linear variation of the magnetosheath thickness with the density jump ratio X across the bow shock, but it also exhibits a unique negative slope and an offset that is a function both of M(A) and of k(o).