The Onset of Dissipation in the Kinematic Dynamo

The kinematic regime of the magnetic dynamo neglects the backreaction of the magnetic field on the flow. For small magnetic diffusivity, in the early stage of evolution, there is an ideal phase where dissipative effects can also be neglected. We estimate the magnitude of the energy dissipation term (Ohmic heating), taking into account differential constraints on chaotic flows. We find that the period of ideal evolution is roughly doubled over an estimate without constraints. The helicity generation terms are exponentially smaller than the energy dissipation, so that large quantities of energy are dissipated before any helicity can be created. Helicity flow is exponentially larger than net helicity generation. The constraints also lead to the existence of a singular initial condition for the magnetic field for which sizable amounts of helicity can potentially be created.