An efficient method for computing the eigenfunctions of the dynamo equation

TL;DR

This paper introduces an efficient eigenfunction computation method for the dynamo equation that avoids differentiation of coefficients, enabling accurate solutions even with tabulated data, and validates it through various comparisons.

Contribution

The paper presents a novel biorthogonal method for eigenfunction calculation in dynamo theory that simplifies computations and improves accuracy with tabulated coefficients.

Findings

Method produces reliable eigenfunctions without differentiating coefficients.

Test calculations agree with classical and numerical results.

Applicable to mean-field dynamo coefficients from simulations.

Abstract

We present an elegant method of determining the eigensolutions of the induction and the dynamo equation in a fluid embedded in a vacuum. The magnetic field is expanded in a complete set of functions. The new method is based on the biorthogonality of the adjoint electric current and the vector potential with an inner product defined by a volume integral over the fluid domain. The advantage of this method is that the velocity and the dynamo coefficients of the induction and the dynamo equation do not have to be differentiated and thus even numerically determined tabulated values of the coefficients produce reasonable results. We provide test calculations and compare with published results obtained by the classical treatment based on the biorthogonality of the magnetic field and its adjoint. We especially consider dynamos with mean-field coefficients determined from direct numerical simulations of the geodynamo and compare with initial value calculations and the full MHD simulations.