Dynamo convergence tests, and application to a planar velocity dynamo

Abstract

Bachtiar, Ivers and James (2006, BIJ), showed that the proof of the long standing planar velocity antidynamo theorem fails when the volume of the conducting fluid is a finite sphere. BIJ also found a planar velocity that appeared to support growth of the magnetic field B, but an unequivocal conclusion was prevented by inadequate convergence of the growth rate λ near the critical magnetic Reynolds number. This follow-up article revisits the BIJ model, with a revised numerical code, attaining much higher truncation levels [J, N]. Given the convergence difficulties, we are led to compare various tests of convergence based on normalized differences of λ, its poloidal-toroidal eigenvector (S, T), the vector B and surface and volume root mean square (SRMS, VRMS) averages of B. We have ranked these tests with respect to sensitivity to changes in [J, N], by applying them to various established kinematic dynamos. Contrary to expectations, we find that λ is more sensitive than S, T, and often even more sensitive than B. The SRMS test is more convenient and usually more sensitive than the S, T test, but is not as sensitive as λ or B. The VRMS test is least sensitive. All these tests imply conclusively that the BIJ planar flow does support growing magnetic fields. However, because of its sensitivity, high accuracy for λ has still not been achieved, and probably requires an alternative approach to the BIJ spectral representations.