Planar anti-dynamo theorem

In the standard two-dimensional setting, with planar velocity and magnetic field independent of the third coordinate and homogeneous energy-closed boundary conditions, the in-plane magnetic potential is an advected diffusing scalar. Its quadratic integral decreases, preventing regeneration of a dynamo field. A third field component independent of that coordinate also obeys a passive advection-diffusion equation. A flow depending on two coordinates but possessing a third velocity component is not covered by this strictly planar formulation.

For a stationary planar advected magnetic flux potential, average its advection-diffusion equation horizontally and split it into mean and fluctuation. The mean transport is $\overline{w{a}^{\prime }}=\eta {\bar{a}}_z+C$, with $C=⟨w{a}^{\prime }⟩$. Multiplying the fluctuation equation by $a^{\prime }$ and applying integration by parts gives $⟨\overline{w{a}^{\prime }}{\bar{a}}_z⟩+B_{0}C=-\eta ⟨|\nabla a^{\prime }{|}^2⟩$. A vanishing mean derivative removes the constant term in the first correlation, yielding the displayed dissipative identity. The boundary assumptions must eliminate surface terms; a periodic layer replaces, rather than accompanies, decay at vertical infinity.