Axisymmetric equilibria of a gravitating plasma with incompressible flows

It is found that the ideal magnetohydrodynamic equilibrium of an axisymmetric gravitating magnetically confined plasma with incompressible flows is governed by a second-order elliptic differential equation for the poloidal magnetic flux function containing five flux functions coupled with a Poisson equation for the gravitation potential, and an algebraic relation for the pressure. This set of equations is amenable to analytic solutions. As an application, the magnetic-dipole static axisymmetric equilibria with vanishing poloidal plasma currents derived recently by Krasheninnikov, Catto, and Hazeltine [Phys. Rev. Lett. 82 , 2689 (1999)] are extended to plasmas with finite poloidal currents, subject to gravitating forces from a massive body (a star or black hole) and inertial forces due to incompressible sheared flows. Explicit solutions are obtained in two regimes: (a) in the low-energy regime β0 ≈ γ0 ≈ δ0 ≈ ǫ0 ≪ 1, where β0, γ0, δ0, and ǫ0 are related to the thermal, poloidal-current, flow and gravitating energies normalized to the poloidal-magnetic-field energy, respectively, and (b) in the high-energy regime β0 ≈ γ0 ≈ δ0 ≈ ǫ0 ≫ 1. It turns out that in the high-energy regime all four forces, pressure-gradient, toroidal-magnetic-field, inertial, and gravitating contribute equally to the formation of magnetic surfaces very extended and localized about the symmetry plane such that the resulting equilibria resemble the accretion disks in astrophysics. Permanent address: University of Ioannina, Association Euratom Hellenic Republic, Physics Department, Theory Division, GR 451 10 Ioannina, Greece