Analytic theory of the nonlinear m=1 tearing mode

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Abstract
Numerical studies show that the m=1 tearing mode continues to grow exponentially well into the nonlinear regime, in contrast with the slow, ‘‘Rutherford,’’ growth of m>1 modes. A single helicity calculation is presented which generalizes that of Rutherford [Phys. Fluids 1 6, 1903 (1973)] to the case when the constant‐ψ approximation is invalid. As in that theory, the parallel current becomes an approximate flux function when the island size W exceeds the linear tearing layer width. However, for the m=1 mode, W becomes proportional to δB, rather than (δB)1/2 above this critical amplitude. This implies that the convective nonlinearity in Ohm’s law, which couples the m=0 component to the m=1 component, dominates the resistive diffusion term. The balance between the inductive electric field and this convective nonlinearity results in exponential growth. Assuming the form of the perturbed fields to be like that of the linear mode, we find that growth occurs at 71% of the linear rate.

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