From exact solutions in MHD to chaos

Ideal MHD is the most basic single-fluid model for determining the macroscopic equilibrium and stability properties of a plasma, such as the small scale laboratory plasma in experiments for magnetic fusion, or, the grandiose scale plasma of cosmic experiments taking place in our astrophysical universe. This review article focus on the state of the art of two related problems in MHD. The first deals with a systematic method for constructing symmetric ideal MHD equilibria in various geometries; representative examples of such equilibria for incompressible fluids, as in Tokamaks or convective rolls, and compressible fluids, as in solar/stellar winds, cosmic jets and solar prominences will be presented. The second deals with the simple but fundamental question on the existence of nonsymmetric equilibria. We shall discuss the Taylor-Proudman and Parker theorems in hydrostatics and magnetostatics, respectively, and their equivalent in MHD. In this spirit we shall explore the formal analogy between the field lines of symmetric equilibria that spatially symmetric ideal MHD equilibria are topologically unstable to finite amplitude perturbations that do not share their symmetry properties. Such resulting ergodicity of perturbed symmetric equilibria may have important consequences in the transport properties of magnetically dominated plasmas. (author). 18 refs.; 7 figs