Abstract:
The behaviour of magnetic eld in the stretch{fold{shear (SFS) dynamo map is considered for zero magnetic diusion. It is shown by a mixture of analytical and numerical approaches that the SFS map is a perfect dynamo: for suciently large shear, the adjoint operator has smooth, growing eigenfunctions and so smooth
ux averages grow exponentially with time for zero diusion. In the paper rst a number of numerical discretisations are presented that give differing results for growth rates, and indicate the need to develop systematic theory. Then magnetic elds that are only required to be square integrable are considered, and the spectral properties of the SFS dynamo operator and its adjoint are discussed, as operators in L 2. Adjoint eigenfunctions are typically not smooth, however. To obtain smooth, growing adjoint eigenfunctions attention is restricted to a subset of magnetic elds that are analytic in a disc in the complex plane. Restricted to this subset and using a supremum norm, the SFS adjoint operator is compact and this allows a numerical treatment of eigenvalues and eigenfunctions with systematic error estimates. These estimates show that for suciently large shear there are smooth growing adjoint eigenfunctions and so perfect dynamo action is established.
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