Transport coefficients for the shear dynamo problem at small Reynolds numbers

Abstract

We build on the formulation developed in Sridhar & Singh (JFM, 664, 265, 2010), and present a theory of the \emph{shear dynamo problem} for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter. Specializing to the case of a mean magnetic field that is slowly varying in time, explicit expressions for the transport coefficients, ${\alpha }_{il}$ and ${\eta }_{iml}$, are derived. We prove that, when the velocity field is non helical, the transport coefficient ${\alpha }_{il}$ vanishes. We then consider forced, stochastic dynamics for the incompressible velocity field at low Reynolds number. An exact, explicit solution for the velocity field is derived, and the velocity spectrum tensor is calculated in terms of the Galilean--invariant forcing statistics. We consider forcing statistics that is non helical, isotropic and delta-correlated-in-time, and specialize to the case when the mean-field is a function only of the spatial coordinate $X_3$ and time $\tau$; this reduction is necessary for comparison with the numerical experiments of Brandenburg, R{ä}dler, Rheinhardt & Käpylä (ApJ, 676, 740, 2008). Explicit expressions are derived for all four components of the magnetic diffusivity tensor, ${\eta }_{ij}(\tau )$. These are used to prove that the shear-current effect cannot be responsible for dynamo action at small $\re$ and $\rem$, but for all values of the shear parameter.

27 pages, 5 figures, Published in Physical Review E

References in corpus (10)

• Large-scale dynamos in turbulent convection with shear
• Generation of Magnetic Field by Combined Action of Turbulence and Shear
• Alpha effect and turbulent diffusion from convection
• Large-scale Dynamo Action Driven by Velocity Shear and Rotating Convection
• Numerical experiments on dynamo action in sheared and rotating turbulence
• Do mean-field dynamos in nonrotating turbulent shear-flows exist?
• The shear dynamo problem for small magnetic Reynolds numbers
• Nonperturbative quasilinear approach to the shear dynamo problem
• Nonhelical mean-field dynamos in a sheared turbulence
• Dynamo efficiency with shear in helical turbulence

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• Coherent nonhelical shear dynamos driven by magnetic fluctuations at low Reynolds numbers
• Enhancement of small-scale turbulent dynamo by large-scale shear
• Numerical studies of dynamo action in a turbulent shear flow - I
• Non-linear Galactic Dynamos and the Magnetic Rädler Effect
• On large-scale dynamos with stable stratification and the application to stellar radiative zones
• On the shear-current effect: toward understanding why theories and simulations have mutually and separately conflicted
• On the existence of shear-current effects in magnetized burgulence
• Mean field dynamo action in renovating shearing flows
• Moffatt drift driven large scale dynamo due to $\alpha$ fluctuations with nonzero correlation times
• Unified treatment of mean-field dynamo and angular-momentum transport in magnetorotational instability-driven turbulence
• Mean field dynamo action in shearing flows. II: fluctuating kinetic helicity with zero mean
• Mean field dynamo action in shear flows. I: fixed kinetic helicity
• Effect of flow shear on the onset of dynamos
• Calculating turbulent transport tensors by averaging single plume dynamics and application to dynamos