Abstract
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.
Highlights
Appropriate corrections can be made by using the extended Grad–Shafranov equation, which was developed to model the equilibrium of mirror systems [10, 11]
To describe the high-beta equilibrium of a magnetospheric plasma, we need a consistent relation between the magnetic field and the phase-space distribution function
The Grad–Shafranov equation has an additional implication, that is, the internal relation between the magnetic field and the magnetization current, which is imposed by the magneto-fluid force–balance relation
Summary
The magnetosphere is a naturally made system confining a high-beta plasma [1]. A similar system may be created for fusion energy applications [2, 3]. At the first-order level, we may invoke the Grad–Shafranov equation [8] to analyze the magnetic field of a finitebeta plasma This equation falls short of considering the strong anisotropy of the distribution function, as the kinetic model makes predictions for the magnetospheric system. The extended Grad–Shafranov equation, which is still a macroscopic magneto-fluid model, considers an anisotropic pressure that is a function of the two-dimensional magnetic coordinates, namely, the flux function and magnetic field strength. As the magnetic moment is the relevant Casimir, the pair of chemical potential and Casimir parallels that of the magnetic field and magnetization in the well-known model of magnetic materials Such a “thermal equilibrium” yields the desired pressure tensor to be used in the generalized Grad–Shafranov equation.
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