Rigorous approach to the nonlinear saturation of the tearing mode in cylindrical and slab geometry

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The saturation of the tearing mode instability is described within the standard framework of reduced magnetohydrodynamics (RMHD) in the case of an r-dependent or of a uniform resistivity profile. Using the technique of matched asymptotic expansions, where the perturbation parameter is the island width w, the problem can be solved in two ways: with the so-called flux coordinate method, which is based on the fact that the current profile is a flux function, and with a new perturbative method that does not use this property. The latter is applicable to more general situations where an external forcing or a sheared velocity profile are involved. The calculation provides a new relationship between the saturated island width and the Δ' stability parameter that involves a w/w₀ term, where w₀ is a nonlinear scaling length that was missing in previous work. It also yields the modification of the equilibrium magnetic flux function.