Analytic theory of the nonlinear m = 1 tearing mode

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 16, 1903 ( 1973) ] to the case when the constant-t/! approximation is invalid. As in that theory, the parallel current becomes an approximate flux function when the island size Wexceeds the linear tearing layer width. However, for the m = 1 mode, Wbecomes proportional to {)B, rather than (bB») /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.