Numerical Study of Dynamo Action at Low Magnetic Prandtl Numbers

Abstract

We present a three-pronged numerical approach to the dynamo problem at low magnetic Prandtl numbers PM . The difficulty of resolving a large range of scales is circumvented by combining Direct Numerical Simulations, a Lagrangian-averaged model, and Large-Eddy Simulations (LES). The flow is generated by the Taylor-Green forcing; it combines a well defined structure at large scales and turbulent fluctuations at small scales. Our main findings are: (i) dynamos are observed from PM = 1 down to PM = 10 −2 ; (ii) the critical magnetic Reynolds number increases sharply with P −1 M as turbulence sets in and then saturates; (iii) in the linear growth phase, the most unstable magnetic modes move to small scales as PM is decreased and a Kazantsev k 3/2 spectrum develops; then the dynamo grows at large scales and modifies the turbulent velocity fluctuations.PACS numbers: 47.27.eq,47.65.+a91.25wThe generation of magnetic fields in celestial bodies occurs in media for which the viscosity ν and the magnetic diffusivity η are vastly different. For example, in the interstellar medium the magnetic Prandtl number P M = ν/η has been estimated to be as large as 10 14 , whereas in stars such as the Sun and for planets such as the Earth, it can be very low (P M < 10 −5 , the value for the Earth's iron core). Similarly in liquid breeder reactors and in laboratory experiments in liquid metals, P M ≪ 1. At the same time, the Reynolds number R V = U L/ν (U is the r.m.s. velocity, L is the integral scale of the flow) is very large, and the flow is highly complex and turbulent, with prevailing non-linear effects rendering the problem difficult to address. If in the smallest scales of astrophysical objects plasma effects may prevail, the large scales are adequately described by the equations of magnetohydrodynamics (MHD),together with ∇ · v = 0, ∇ · B = 0, and assuming a constant mass density. Here, v is the velocity field normalized to the r.m.s. fluid flow speed, and B the magnetic field converted to velocity units by means of an equivalent Alfvén speed. P is the pressure and j = ∇ × B the current density. F is a forcing term, responsible for the generation of the flow (buoyancy and Coriolis in planets, mechanical drive in experiments). Several mechanisms have been studied for dynamo action, both analytically and numerically, involving in particular the role of helicity [1] (i.e. the correlation between velocity and its curl, the vorticity) for dynamo growth at scales larger than that of the velocity, and the role of chaotic fields for small-scale growth of magnetic excitation (for a recent review, see [2]). Granted that the stretching and folding of magnetic field lines by velocity gradients overcome dissipation, dynamo action takes place above a critical magnetic Reynolds number R c M ,Dynamo experiments engineering constrained helical flows of liquid sodium have been successful [3]. However, these experimental setups do not allow for a complete investigation of the dynamical regime, and many groups have searched to implement unconstrain...

Influence of Turbulence on the Dynamo Threshold

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Effective diffusivity of passive scalars in rotating turbulence

“…However, for a wide range of values of the Pèclet number, D x remains approximately constant. This is in good agreement with theoretical expectations that for small enough ν and κ, the effective turbulent Schmidt number should be of order one (see [35] for similar arguments in the case of the turbulent magnetic Prandtl number).…”

A solar surface dynamo

“…For smooth flows on larger scales, the artificial viscosity is much smaller than the magnetic diffusivity, so that our "effective" magnetic Prandtl number on these scales is smaller than unity. Given these conditions, our results do not provide a proper basis for commenting on the much debated question of the dependence of turbulent dynamo action on the value of Pr m (e.g., Boldyrev & Cattaneo 2004;Ponty et al 2005;Schekochihin et al 2005;Brandenburg & Subramanian 2005). Fig.…”