Non-linear and chaotic dynamo regimes

Axel Brandenburg, brandenb@nordita.org, NORDITA, Sweden

Abstract
The objective of IAU Symposium 294 is to provide an update since IAU Symposium 157 on "The Cosmic Dynamo" in Potsdam, 1992. The suggested title of the present talk reflects the common thinking at that time that nonlinearity and chaos tend to come together. This is highlighted by the realization that a simple mean-field dynamo model for poloidal and toroidal fields needs to be supplemented by a third equation to produce chaos, and the idea that this third equation is the equation of magnetic helicity conservation. To my mind, these developments have contributed to the unfortunate perception of mean-field dynamo theory being just as a toy rather than a quantitatively predictive theory. With the advent of numerous computer simulations of hydromagnetic turbulence exhibiting large-scale dynamo action, a new field of computer astrophysics has emerged where the objective is to understand simulated dynamos, where one has a chance to resolve all time and length scales. This approach has helped making mean-field theory quantitatively reliable and predictable. Various predictions have emerged and have been tested. Firstly, dynamos must transport magnetic helicity to escape catastrophic quenching. They do this through coronal mass ejections and through turbulent exchange across the equator. The resulting field is bi-helical, with opposite signs of magnetic helicity at large and small length scales. The signs depend on the sign of kinetic helicity and on the relative importance of turbulent diffusion. This has meanwhile been confirmed observationally using Ulysses data. Secondly, mean-field theory also predicts the formation of local magnetic flux concentrations as a result of strong density stratification. Simulations have now confirmed this remarkable theoretical prediction. This has opened the floor for suggestions that active regions and sunspots might be shallow phenomena operating near the surface at some 40 Mm depth. Returning to the topic of the talk about nonlinear and chaotic dynamo regimes, one can say in summary that chaotic variability is an integral part of generalized mean-field theory. Its correct implementation requires however proper inclusions of nonlocality in space and time.