Three-dimensional numerical simulation of the multi-helicity magnetohydrodynamic relaxation process in low-q spheromaks

Three-dimensional magnetohydrodynamic computer simulations have been made on the dynamic behaviour of the high temperature spheromak plasma whose conductivity profile is peaked at the magnetic axis. On the assumption of stationary spatial profiles of the plasma conductivity, these simulations examine the transient process from the current peaking to the subsequent relaxation. They reveal low-q relaxations caused by multi-helicity (current driven) kink modes. The low-q relaxations are classified into two types: the fast-type relaxation without n = 2 mode saturation and the slow-type relaxation with n = 2 mode saturation. In these simulations, resistive current loss in the outer region of the plasma causes a peaking of the current density profile, resulting in a departure from the initial Taylor state to a low-q state. As the q value at the magnetic axis, q0, decreases to <0.5, the internal kink mode with a toroidal mode number n = 2 is first destabilized. The feature of the subsequent relaxation process depends on the degree of peaking of the conductivity profile at the magnetic axis, including its change during the relaxation phase. When the peaking of the conductivity profile is strong enough to decrease q0 to much less than 0.5, the higher mode (the n = 3 mode) is destabilized, which is found to trigger the subsequent relaxation. During the relaxation phase, the non-linear coupling of these n = 2 and 3 (and sometimes 4) modes leads to flux conversion from poloidal to toroidal and the configuration with the excessive poloidal flux can relax back to a state close to the Taylor state with a balanced ratio of the poloidal flux to the toroidal flux. This is the scenario of the fast-type relaxation. On the other hand, when the conductivity profile is weakly peaked, the slow decrease in q0 causes a slow growth of the n = 3 mode, resulting in the saturation of the n = 2 mode. Even if the coupling of the n = 2 and 3 modes triggers a relaxation, the relaxation event is weak, unclear and incomplete. This is the scenario of the slow-type relaxation. It is also found that if the peaked conductivity proflle is maintained during the relaxation phase, relaxation back to the Taylor state is less complete.