Chaotic flows and fast magnetic dynamos

Abstract

Given a prescribed flow of an initially unmagnetized conducting fluid, one can ask if a small seed magnetic field will amplify exponentially with time. This is called the kinematic dynamo problem. In many cases of interest (particularly in astrophysics), very high electrical conductivity of the fluid (high magnetic Reynolds number R_m ) is relevant. In this paper the kinematic dynamo problem is considered in the R_m →∞ limit (the ‘‘fast’’ kinematic dynamo). It appears that an important ingredient for a kinematic dynamo in this limit is that the orbits of fluid elements in the flow be chaotic. In this paper it is shown that the magnetic field tends to concentrate on a zero volume fractal set, and, in addition, tends to exhibit arbitrarily fine‐scaled oscillations between parallel and antiparallel directions. Idealized analyzable examples exhibiting these properties are presented, along with numerical computations on more typical examples. For the latter a numerical technique for treating fast dynamos is developed and its properties are discussed. The relation of the dynamo growth rate to quantitative measures of chaos, namely, the Lyapunov exponent and topological entropy, is also discussed.