Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems pp 515-539 | Cite as

# Large Magnetic Reynolds Number Dynamo Action in Steady Spatially Periodic Flows

## Abstract

Dynamo action at large magnetic Reynolds number R is very sensitive to the streamline topology. A starting point for a systematic study is the 2-dimensional steady spatially periodic flow u’ = (∂*ψ*’/∂y, -∂*ψ*’/∂x, K*ψ*’), *ψ*’ = sinx siny + *δ* cosx cosy, where *δ*,K = constants. Slow dynamo action is possible within the spiralling vortices, and individual modes are localized in the neighbourhood of particular streamsurfaces. When *δ* = 0, almost fast dynamo action occurs in boundary layers of width R^{-1/2} containing the streamline separatrices. These connect the stagnation points and bound adjacent vortices. This streamline pattern is structurally unstable, and so, when 0 < *δ* ≪ 1, channels emerge between the vortices. With the addition of mean motion ū = (M,N,O)/(M^{2} +N^{2})^{1/2}, where M,N = relatively prime integers, ∈ ≪ 1, but δ = 0, the streamlines connected to the X-type stagnation points bound channels with multiplicity of order L = M + N, and are dense in the *irrational* limit L→ ∞. Dynamo action in all these systems is discussed. Results for the latter case shed light on dynamo processes, which may occur in the fully 3-dimensional ABC-flows. There the flow downstream of stagnation points on certain 2-dimensional manifolds provides the sites of possible fast dynamo action. The manifolds form dense subregions, where the fluid particles paths are chaotic.

## Keywords

Boundary Layer Stagnation Point Magnetic Induction Equation Dynamo Action irrationaL Limit## Preview

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