Fast dynamo action in a steady flow

Abstract

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

On the mean-field theory of the Karlsruhe Dynamo Experiment

“…This decay results from magnetic flux expulsion out of the rotating inner parts of each cell. By the way, in agreement with results of asymptotic studies (Soward, 1987) it was found that φ behaves like Rm −3/2 ⊥ and, therefore, α ⊥ like Rm…”

Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

“…Thus the solution 542 follows closely Childress (1979) and Soward (1987) and is detailed here for completeness…”

“…and is determined by matching a non-trivial solution in the interior of the flow cells with problem for the whole of Regime I turns out to be identical to that arising in the ho-mogenisation approach and solved by Soward (1987). However, it is only in the limit Introducing the expansions…”

Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations

“…Because the boundary layers occur along the streamlines, ψ(x, y) provides a good basis. We define ψ(x, y) as one coordinate and then define a family of lines orthogonal to the streamlines [Childress (1979); Soward (1987)] by φ(x, y) = C b · dl, where dl is the tangent to the streamlines. This is an "angle" variable that measures the distance along a given streamline.…”