Advected fields in maps: I. Magnetic flux growth in the stretch–fold–shear map

The behaviour of magnetic field in the stretch–fold–shear (SFS) dynamo map is considered for zero magnetic diffusion. It is shown by a mixture of analytical and numerical approaches that the SFS map is a perfect dynamo; for sufficiently large shear, the adjoint operator has smooth, growing eigenfunctions and so smooth flux averages grow exponentially with time for zero diffusion. In the paper first a number of numerical discretisations are presented that give differing results for growth rates, and indicate the need to develop systematic theory. Then magnetic fields 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 L2. Adjoint eigenfunctions are typically not smooth however. To obtain smooth, growing adjoint eigenfunctions attention is restricted to a subset of magnetic fields 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 sufficiently large shear there are smooth growing adjoint eigenfunctions and so perfect dynamo action is established.