Energy principle with global invariants

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Abstract
A variational principle is proposed for constructing equilibria with minimum energy in a toroidal plasma. The total energy is minimized subject to global invariants which act as constraints during relaxation of the plasma. These global integrals of motion are preserved exactly for all ideal motions and approximately for a wide class of resistive motions. We assume, specifically, that relaxation of the plasma is dominated by a tearing mode of single helicity. Equilibria with realistic current density and pressure profiles may be constructed in this theory, which is also used here to study current penetration in tokamaks. The second variation of the free energy functional is computed. It is shown that if the second variation of any equilibrium constructed in this theory is positive, the equilibrium satisfies the necessary and sufficient conditions for ideal stability.

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