The evolution of a decaying spheromak

Abstract

The evolution of a low beta spheromak initially in a two-dimensional stable equilibrium having a constant J/B (the so-called minimum energy state) is calculated within the context of resistive magnetohydrodynamics. Since the initial equilibrium is stable, the spheromak at first resistively evolves through a sequence of stable quasiequilibria. This phase of the evolution is calculated with a transport code in which the resistivity is assumed to be largest at the wall and lowest at the magnetic axis. Resistive diffusion causes the safety factor q to decrease everywhere while decreasing fastest near the wall. The quasiequilibria through which the spheromak evolves are tested for stability with both an ideal linear stability code and a resistive one. The results of both stability codes are in basic agreement and show that when q drops to below (1)/(2) everywhere the spheromak becomes unstable to an n=2 mode. The agreement of the stability codes implies that the unstable mode is a resistively modified ideal mode. The unstable equilibrium is used as the initial condition in a 3-D nonlinear magnetohydrodynamic simulation. This simulation shows that after the unstable mode saturates, the spheromak resistively evolves through a sequence of three-dimensional quasiequilibria until it reaches another unstable configuration, after which it approaches the 2-D minimum energy state again. This evolutionary cycle can conceivably start again, unless the cycle time becomes comparable to the configuration decay time, which happens at high S. One consequence of the evolutionary cycle is that as the 3-D spheromak approaches the minimum energy state, the magnetic axis and hot plasma near it approach the wall and a new magnetic axis is formed. At high enough S, when the cycle time is comparable to the configuration lifetime, this convective heat loss mechanism is minimized. The 3-D code predicts that only the n=0 and n=2 modes are active. Simulations in which the n=1, 3, and 4 modes are present show that these modes decay while the n=2 grows and saturates. Closed flux surfaces are present over most of the domain so that heat loss along stochastic field lines should not be large. When the Hall term is included a toroidal rotation of the magnetic configuration is induced which most likely corresponds to an experimentally measured rotation. The results of this calculation are in qualitative agreement with various experimental data.