Scalar dynamo models

The equation [formula omitted] is a scalar analogue of the magnetic induction equation. If the velocity field u(x, t) and the ‘stretching’ function R(x, t) are explicitly given, then we have the analogue of the dynamo problem. The scalar problem displays many of the same features as the vector kinematic dynamo problem. The fastest growing modes have growth rates that approach a finite limit as k→0 while the eigenfunctions develop more and more complex structure at smaller and smaller length scales. Some insight is provided by an analysis which finds a lower bound on the growth rate that is asymptotically independent of the diffusivity.