Three-dimensional numerical simulations of the relaxation process in spheromak plasmas

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Nonlinear evolution of three-dimensional magnetohydrodynamic (MHD) instabilities of a toroidal spheromak in a cylindrical flux conserver has been studied by numerical simulations for various initial equilibrium states with different q profiles. In spheromaks with qa>1, where qa is the safety factor on the magnetic axis, nonlinear evolution of the resonant internal kink mode dominates with a poloidal mode number m=1 and a toroidal mode number n=1 that causes the poloidal flux amplification. This process corresponds to that of the internal disruption model for tokamaks by Kadomtsev [Sov. J. Plasma Phys. 1, 389 (1975)]. In spheromaks with a very high qa, namely qa≳3, the gross n=1 kink mode grows extensively in the region including the major axis of the torus, which causes the flux conversion from the toroidal to poloidal directions. For spheromaks with a low qa, namely qa≲0.5, the internal kink mode with a toroidal mode number n∼1/qa is first destabilized, and the excitation of the modes with lower n numbers down to n=1 proceeds, while the n=2 mode saturates. Nonlinear coupling of various modes leads to the flux conversion from the poloidal to toroidal directions. When a center conductor is present in this case, a reversed-field pinch (RFP) configuration once formed is sustained. Relaxations through pressure-driven modes are also discussed. All final states obtained in our simulations are quite near the Taylor state with an excess magnetic energy less than 10% of that of the Taylor state.