The fast dynamo growth rate for aCk+1 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: inC∞ 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 caseRm=∞.

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Rigorous bounds on the fast dynamo growth rate involving topological entropy

Exploring the origins and evolution of magnetic fields in planets, stars and galaxies, this book gives a basic introduction to magnetohydrodynamics and surveys the observational data, with particular focus on geomagnetism and solar magnetism. Pioneering laboratory experiments that seek to replicate particular aspects of fluid dynamo action are also described. The authors provide a complete treatment of laminar dynamo theory, and of the mean-field electrodynamics that incorporates the effects of random waves and turbulence. Both dynamo theory and its counterpart, the theory of magnetic relaxation, are covered. Topological constraints associated with conservation of magnetic helicity are thoroughly explored and major challenges are addressed in areas such as fast-dynamo theory, accretion-disc dynamo theory and the theory of magnetostrophic turbulence. The book is aimed at graduate-level students in mathematics, physics, Earth sciences and astrophysics, and will be a valuable resource for researchers at all levels.

Self-Exciting Fluid Dynamos

Abstract Numerical simulations of dynamo action are presented for a fluid stirred by a body force such that in the absence of the magnetic field, an ABC flow (to be viewed as the superposition of three helical modes of same wavenumber k 0), establishes for kinetic Reynolds numbers below a critical value. For the kinematic dynamo, we show that keeping a fixed magnetic Reynolds number and decreasing the period of the ABC flow with respect to the size of the system, generally amplifies the growth rate of the magnetic field. This effect is especially significant when changing k 0 = 1 to k 0 = 2, while for larger k 0, it scales like the rate of strain of the flow. A correlation between the rate of growth of the dynamo and the largest Lyapunov exponent is also observed when the mode amplitudes of the ABC flow are varied. In the dynamical regime where the flow is subject to magnetic feed-back through the Lorentz force, we always observe that decreasing the spatial period of the stirring force strongly enhances t...

Linear and Non-linear Dynamos Associated with ABC Flows (統計数理研究所 研究活動 (研究会報告 乱流の統計理論とその応用))

We review cellular space-periodic dynamos without scale separation, starting with early work in the 1980s on ABC flows with prescribed steady velocity fields u  = (A sin z + C cos y, B sin x + A cos z, C sin y + B cos x). These naturally led to work done in the 1990s together with Mike Proctor on 2-D time-dependent versions which gave strong numerical evidence for the existence of fast dynamos growing on the flow turnover timescale. Similar calculations were subsequently performed for a spherical shell geometry jointly with Rainer Hollerbach. Also in the 1990s other studies began to take into account the back reaction of the Lorentz force when the flow rather than being prescribed was instead allowed to evolve in response to a forcing of the above ABC form. The dynamos that resulted were mostly filamentary and showed a disconcerting tendency to equilibrate with total magnetic energy much less than total kinetic energy in the low diffusivity limit relevant for astrophysics. The remarkable discovery by Arch...

ABC flows then and now

Dynamo action in a fully helical Arn'old-Beltrami-Childress flow is studied using both direct numerical simulations and subgrid modeling. Sufficient scale separation is given in order to allow for large-scale magnetic energy buildup. Growth of magnetic energy obtains down to a magnetic Prandtl number ${P}_{M}={R}_{M}∕{R}_{V}$ close to 0.005, where ${R}_{V}$ and ${R}_{M}$ are the kinetic and magnetic Reynolds numbers. The critical magnetic Reynolds number for dynamo action ${R}_{M}^{c}$ seems to saturate at values close to 20. Detailed studies of the dependence of the amplitude of the saturated magnetic energy with ${P}_{M}$ are presented. When ${P}_{M}$ is decreased, numerical experiments are conducted with either ${R}_{V}$ or ${R}_{M}$ kept constant. In the former case, the ratio of magnetic to kinetic energy saturates to a value slightly below unity as ${P}_{M}$ decreases. Examination of energy spectra and structures in real space reveals that quenching of the velocity by a large-scale magnetic field takes place, with an inverse cascade of magnetic helicity and a force-free field at large scale in the saturated regime.

Inverse cascades and α effect at a low magnetic Prandtl number

We consider the evolution of a magnetic field in a spherical shell of highly conducting fluid surrounded by an insulator. We impose an axisymmetric, time-dependent flow, having large regions of chaotic particle paths. This flow appears to yield fast dynamo action, in which the field grows on the fast advective, rather than on the slow diffusive time scale. We demonstrate that the field adjusts to the Bondi-Gold theorem, according to which the field in the insulators inside and outside the shell cannot grow on the fast time scale, by becoming increasingly self-contained within the shell.

Numerical evidence of fast dynamo action in a spherical shell.

We present a numerical solution of the magnetic induction equation in a spherical fluid shell, with an insulator inside and outside. Prescribing an axisymmetric, time-dependent, chaotic flow, we find that the magnetic field appears to grow on the fast advective, rather than on the slow diffusive time scale. We demonstrate how this may be reconciled with the theorem of Bondi and Gold (1950), that the potential field in these insulators inside and outside the shell cannot be amplified on the fast time scale, by having the field become increasingly contained within the shell with increasing magnetic Reynolds number. Thus, as the Bondi-Gold theorem becomes more and more applicable, there is indeed less and less external field being amplified. This is in precise agreement with the conjecture of Radler (1982) that the resolution would be to have an “invisible dynamo,” one having no external field. Finally, we consider some of the implications of this adjustment for the different symmetries of the field...

On the adjustment to the Bondi-Gold theorem in a spherical-shell fast dynamo

All known fast dynamos where diffusion has been explicitly included exhibit eigenfunctions with fine structures whose characteristic sizes apparently scale as R m −1/2 , where R m is the magnetic Reynolds number Results for a number of such calculations are summarised, including cases with and without stagnation points, in Cartesian and spherical geometries In all cases there is agreement with the conclusion of Moffatt and Proctor (1985) that fast dynamos can only occur where such fine structures are present However, the fraction of space occupied by these structures tends to zero as R m →∞, and the associated Lorentz force becomes infinite There are thus paradoxes in considering the kinematic problem in this limit, and an astrophysically meaningful calculation must embody the back reaction of the magnetic field on the flow dynamics

D. J. Galloway

Papers

Numerical evidence of fast dynamo action in a spherical shell.

On the adjustment to the Bondi-Gold theorem in a spherical-shell fast dynamo

Fine Structure in Fast Dynamo Computations