Magnetic field growth and saturation in plasmas with large magnetic Prandtl number. I. The two-dimensional case

In turbulent plasmas, velocities at scales smaller than a scale l(D) are strongly damped by viscosity, v, and magnetic fields below a scale l(R) are strongly dissipated by resistivity, eta. In galaxies and protogalaxies, l(D) much greater than l(R); i.e., the magnetic Prandtl number, P-M = v/eta = l(D)(2)/l(R)(2), is very large. The limit of high magnetic Prandtl number in two dimensions is the focus of this paper. In the kinematic phase of magnetic field growth, when the field is too weak to affect the flow, the field strength grows and the field scale length l(B) decreases. Much of the initial growth of the field happens at scales below l(D). In this paper we examine numerically and analytically the growth and saturation of magnetic field on scales less than l(D) in two dimensions. If the initial seed field strength is very weak, the field grows and l(B) decreases down to the resistive scale l(R) before the Lorentz force can affect the magnetic field : once l(B) similar to l(R) it damps out rapidly. However, if the initial seed field is large enough the field grows until it saturates at a scale l(B) with l(D) > l(B) > l(R). In saturation the field strength remains constant and l(B) decreases on the resistive time of the saturation scale (i.e., l(B)(2)/eta). The small-scale velocities are insufficient to unwind the small-scale field and produce any kind of inverse cascade. When the scale of the saturation field reaches l(R), the field field strength begins to decay. In the initial phase of decay, coherent loops of magnetic field that are formed during saturation persist and act to limit the rate of decay of magnetic energy. Only after these loops have resistively decayed can the final, rapid, kinematic decay take place.