Rigorous Bounds on the Fast Dynamo Growth Rate Involving Topological Entropy

The fast dynamo growth rate for a C k+l map or flow is bounded above by topological entropy plus a 1/k correction. The proof uses techniques of random maps combined with a result of Yomdin relating curve growth to topological entropy. This upper bound implies the following anti-dynamo theorem: in C ~ systems fast dynamo action is not possible without the presence of chaos. In addition topological entropy is used to construct a lower bound for the fast dynamo growth rate in the case Rm = oo. 1. The Kinematic Fast Dynamo Problem Magnetic dynamo theory involves the study of the generation of magnetic field in astrophysical objects such as planets and stars. One star of particular interest, of course, is the sun, which exhibits vigorous magnetic field activity on time scales much shorter than the magnetic diffusive time. Kinematic fast dynamo theory attempts to gain some understanding of the non-diffusive processes that might be involved by addressing the question of what sort of fluid motions can induce exponential growth of magnetic field at high magnetic Reynolds number. This is one of a large class of singular problems with important physical implications for which there is a need for a better understanding of the limiting behavior of complicated processes. Natural questions that arise are: what if any relation holds between the limiting and singular limit solutions, and what information about the limiting process can be gained from the singular limit problem? In this paper we consider a conjectured growth rate bound of Finn and Ott (1988) based on stretching properties of the fluid flow and prove a slightly generalized version of it. The equations of dynamo theory are those of incompressible MHD (see Roberts (1967)). Denoting the magnetic field by B and the fluid velocity by u the magnetic * This author is supported by an NSF postdoctoral fellowship ** This author is partially supported by an NSF grant 624 I. Klapper, L.S. Young