A two-layer $\alpha\omega$ dynamo model, and its implications for 1-D dynamos

I will discuss an attempt at representing an interface dynamo in a simplified, essentially 1D framework. The operation of the dynamo is broken up into two 1D layers, one containing the $\alpha$ effect and the other containing the $\omega$ effect, and these two layers are allowed to communicate with each other by the simplest possible representation of diffusion, an analogue of Newton's law of cooling. Dynamical back-reaction of the magnetic field on $\omega$ is included. I will show extensive bifurcation diagrams, and contrast them with diagrams I computed for a comparable purely 1D model. The bifurcation structure shows remarkable similarity, but a couple of subtle changes imply dramatically different physical behaviour for the model. In particular, the solar-like dynamo mode found in the 1-layer model is not stable in the 2-layer version; instead there is an (apparent) homoclinic bifurcation and a sequence of periodic, quasiperiodic, and chaotic modes. I argue that the fragility of these models makes them effectively useless as predictors or interpreters of more complex dynamos.