Stability problem in dynamo

TL;DR

This paper investigates the stability of magnetic fields in dynamo models, showing that saturated alpha-effects can still lead to exponential growth under certain conditions, with implications for understanding magnetic field behavior in astrophysical objects.

Contribution

It demonstrates that saturated alpha-effects in nonlinear dynamo equations can produce exponential magnetic field growth even without feedback, and derives stability conditions for different regimes.

Findings

Exponential magnetic field growth possible without alpha feedback.

Stability conditions linked to spectral properties of the linear problem.

Results applicable to shell models and 3D dynamo simulations.

Abstract

It is shown, that the saturated $\alpha$-effect taken from the nonlinear dynamo equations for the thin disk can still produce exponentially growing magnetic field in the case, when this field does not feed back on the $\alpha$. For negative dynamo number (stationary regime) stability is defined by the structure of the spectra of the linear problem for the positive dynamo numbers. Stability condition for the oscillatory solution (positive dynamo number) is also obtained and related to the phase shift of the original magnetic field, which produced saturated $\alpha$ and magnetic field in the kinematic regime. Results can be used for explanation of the similar effect observed in the shell models simulations as well in the 3D dynamo models in the plane layer and sphere.