Large Magnetic Reynolds Number Dynamo Action in Steady Spatially Periodic Flows
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, Kψ’), 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)/(M2 +N2)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.
KeywordsBoundary Layer Stagnation Point Magnetic Induction Equation Dynamo Action irrationaL Limit
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