Kinetic construction of the high-beta anisotropic-pressure equilibrium in the magnetosphere

A theoretical model of the high-beta equilibrium of magnetospheric plasmas was constructed by consistently connecting the (anisotropic pressure) Grad–Shafranov equation and the Vlasov equation. The Grad–Shafranov equation was used to determine the axisymmetric magnetic field for a given magnetization current corresponding to a pressure tensor. Given a magnetic field, we determine the distribution function as a specific equilibrium solution of the Vlasov equation, using which we obtain the pressure tensor. We need to find an appropriate class of the distribution function for these two equations to be satisfied simultaneously. Here, we consider the distribution function that maximizes the entropy on the submanifold specified by the magnetic moment. This is equivalent to the reduction of the canonical Poisson bracket to the noncanonical one having the Casimir corresponding to the magnetic moment. The pressure tensor then becomes a function of the magnetic field (through the cyclotron frequency) and flux function, satisfying the requirement of the Grad–Shafranov equation