MHD turbulence and the dynamo problem

Finding the origin of global magnetic fields around stars and planets is called ‘the dynamo problem.’ These global magnetic fields arise from layers of turbulent plasma that can be modeled as a spherical shell of magnetofluid in which magnetohydrodynamic (MHD) turbulence occurs due to convection. Ignoring compressibility, the problem reduces to examining incompressible MHD turbulence in a spherical shell. Velocity and magnetic fields and represented by orthogonal function expansions whose coefficients form a dynamical system. Assuming that the magnetofluid is ideal (no dissipation), we can apply equilibrium statistical mechanics based on the ideal invariants of incompressible MHD. For a rotating system, the ideal invariants are energy and magnetic helicity; if there is no rotation, cross helicity is also an invariant. Expectation values such as means and variances of the dynamical variables are then calculated and tested numerically using Fourier codes, since the statistical mechanics for a spherical shell and a periodic box are equivalent. Expectation values mostly agree with numerical time averages, but there is a surprise: Although all mean values are predicted to be zero, the largest-scale modes have dynamical mean values that are large and quasi-stationary. This ‘broken ergodicity’ is due to a dynamically broken symmetry. Here, we review the theory and computation that lead to this solution of the dynamo problem, as well as discuss its applicability to real MHD turbulence.