Nonlinear Periodic Solutions for Isothermal Magnetostatic Atmospheres.

Magnetohydrodynamic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential
⁠, known as the Grad–Shafranov equation. Specifying the arbitrary functions in the latter equation, one obtains three types of nonlinear elliptic equations (a Liouville equation, a sinh Poisson equation, and a generalization of those with a sum of exponentials). Analytical solutions are obtained using the tanh method; this is elaborated in the Appendix. The solutions are adequate to describe an isothermal atmosphere in a uniform gravitational field showing parallel filaments of diffuse, magnetized plasma suspended horizontally in equilibrium.