High beta plasmas exhibit kinetic behaviors due to large ion gyro-radii compared to the scale length. The existence of a field-null point makes it difficult to predict the plasma behavior because the particle motion near the point is complicated. A fast ion with the larger canonical angular momentum draws the betatron orbit and contributes to generate the diamagnetic ion current. On the other hand, a slow ion near the surface shows a stochastic behavior and drifts along the paramagnetic direction. Therefore, obtaining velocity distributions is a key issue for the high-beta plasmas. The particle simulation or particle and fluid electron simulation (i.e., hybrid simulation) are useful tools to describe the high-beta plasma numerically. A physically valid weighting of superparticle is needed in the particle simulation modeling. Another kinetic method is to solve the Vlasov and Maxwell's equations. It is not easy, however, to obtain the velocity distribution functions that are described by the variables in the six-dimensional phase space. If the finite difference method is applied, the number of grid meshes becomes enormous. In the present study, the velocity distribution functions are fitted to the exponential function of a quadratic expression of the velocity space variables. The Boltzmann's H theorem that predicts the distribution functions are relaxed to the shifted-Maxwellian due to collisions is implicitly assumed in this expression. This fitting is effective to reduce the number of grid meshes in the velocity space significantly and to estimate precisely the first and second derivatives with respect to the velocity variables. The time evolutions of the distribution functions are simulated by solving the Vlasov equation. The fields are calculated from the wave equation for the scalar and vector potentials. The objectives of this study are to predict a high-beta plasma structure of an electromagnetically relaxed state and to demonstrate kinetic effects in the plasma with field-null surface by the plasma kinetic theory.
Preliminary results are obtained for infinitely long and cylindrically symmetric plasma whose averaged beta value is about 0.9. In this case the real space variable is the radial position alone, and the velocity space variables are the radial and azimuthal velocity components. According to Spivey's work, the only shifted-Maxwellian distributions that satisfy the Vlasov equation for systems with cylindrical symmetry are rigid rotors with constant temperature.[1] The comparison between Spivey's theory and numerical results are made here. This study will be applied in near future to the Field-Reversed Configurations (FRCs) with several unknown physical issues.

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