Detailed mathematical and numerical analysis of a dynamo model

School of Mathematics and Statistics, University of Sheffield, S3 7RH, UK
e-mail: smp11as@sheffield.ac.uk; e.kim@sheffield.ac.uk

Received: 25 May 2013
Accepted: 31 December 2013

Abstract

We investigate the role of nonlinear feedback by α-quenching, flux losses, and feedback by differential rotations in dynamo. Specifically, by studying the nonlinear dynamo model analytically and numerically, we unfold how frequency p of magnetic field, magnetic field strength |B|, and phase φ are influenced by different types of nonlinear feedback in the limit of a very weak mean and/or fluctuating differential rotation. We find that p and φ are controlled by both flux losses with no influence by α-quenching when there is no back reaction because of fluctuating differential rotation. We find a similar effect of poloidal flux loss and toroidal flux loss on p and |B| in the absence of a back reaction of shear. Their effect becomes totally different with the inclusion of this back reaction. Detailed investigations suggest that toroidal flux loss tends to have more influence than poloidal flux loss (with or without α-quenching) in the presence of fluctuating shear. Furthermore, the effect of α-quenching is boosted when combined with toroidal flux loss, indicating that the dynamic balance of dynamo is optimized in the presence of both α-quenching and flux loss. These results highlight the importance of nonlinear transport coefficients and differential rotation in the regulation of a dynamo.