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 2π l , where l −1 = R ½ β 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 .