Current sheets and nonlinear growth of the <i>m</i>=1 kink-tearing mode

A calculation is presented that accounts for rapid nonlinear growth of the m=1 kink-tearing instability. The equilibrium analysis contained in the Rutherford theory [Phys. Fluids 16, 1903 (1973)] of nonlinear tearing-mode growth is generalized to islands for which the constant-ψ approximation is not valid. Applying the helicity-conservation assumption introduced by Kadomtsev [Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1977), Vol. I, p. 555], the presence of a current-sheet singularity is shown that gives rise to a narrow tearing layer and rapid reconnection. This rapid reconnection, in turn, justifies the use of the helicity conservation assumption. The existence of a family of self-similar m=1 equilibrium islands is demonstrated. The formalism introduced here is shown to apply both to the case of the m=1 kink-tearing mode and to the case of forced reconnection. These two cases are compared and contrasted.