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 st rongly damped by viscosity, v, and magnetic fields below a scale l(R) are s trongly 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 magne tic 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 initi al growth of the field happens at scales below l(D). In this paper we exami ne numerically and analytically the growth and saturation of magnetic field on scales less than l(D) in two dimensions. If the initial seed field stre ngth 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 fiel d 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)/e ta). The small-scale velocities are insufficient to unwind the small-scale field and produce any kind of inverse cascade. When the scale of the satura tion field reaches l(R), the field field strength begins to decay. In the i nitial phase of decay, coherent loops of magnetic field that are formed dur ing saturation persist and act to limit the rate of decay of magnetic energ y. Only after these loops have resistively decayed can the final, rapid, ki nematic decay take place.