Riccati method for numerical tearing layers

We present a method for numerical solution of the tearing mode dispersion relation based on a technique for integrating stiff equations due to Forman Acton. The method is robustly stable for the tearing layers studied, and should be broadly applicable. The tearing layer equations are first Fourier transformed (FT), followed by a change of dependent variable using the Riccati-like transform w=kφ'(k)/φ(k) where φ(k) is the FT stream function. The solution is obtained via backwards integration from large k to zero. The Riccati transformation separates the `good' and `bad' solutions (as k-->∞) in the range of the Riccati variable w. The tearing layer dispersion function is related to the slope of the Riccati variable at k=0, and found by least squares fitting to a Pade approximant there.

Acton's method is generalized to higher order equations by a matrix Riccati transformation found using a Lie symmetry. We use the generalized matrix Riccati solver to study a 3rd order nonlinear system which reproduces the Glasser Effect (and real frequency tearing layers) despite neglecting classical transport and the divergence of the EXB drift in the tearing layer.