Dynamo Action in a Family of Flows with Chaotic Streamlines

The kinematic dynamo problem is investigated for the class of flows u = (A sin z + C cos y, B sin x + A cos z, C sin y + B cos x) which in general have chaotic streamlines. Numerical results are reported for magnetic Reynolds numbers R(m) up to 450 and various choices of A, B and C. For A = B = C = 1 dynamo action is present in at least two windows in R(m) the first extending from approximate to 9 to approximate to 17.5 and the second beyond approximate to 27. Certain symmetries implied by the flow are preserved in the lower window but are broken in the upper. The fastest growing mode shows concentrated cigar-like structures centered on stagnation points in the flow. When A, B and C are varied, windows of dynamo action may or may not be present. When one of the coefficients vanishes, the flow becomes two-dimensional and non-chaotic, but with three-dimensional magnetic fields, dynamo action is still possible and has been investigated for R(m) up to 1500. In the two-dimensional example studied the growth rate achieved a maximum near R(m) = 300 and then behaved in a way appropriate for a slow dynamo (one whose growth rate tends to zero as R(m) -> infinity). It is not clear yet whether or not in the three-dimensional case the opposite can happen (fast dynamo). The alpha-effect that is produced by these helical flows acting on very large-scale magnetic fields is calculated. Surprisingly, it can remain finite even when dynamo action is present at the scale of the flow, as long as certain symmetries are not broken.