Anti-dynamo theorem

An anti-dynamo theorem gives a geometric or analytic restriction excluding self-sustained dynamo action under stated magnetic diffusivity, domain and boundary conditions. It excludes regeneration against resistive decay, not all transient amplification or field induced by an external source. Important restrictions include Cowling anti-dynamo theorem, the toroidal-velocity anti-dynamo theorem, and the planar anti-dynamo theorem. Backus' necessary condition for dynamo action instead bounds the stretching required for possible regeneration.

For a solenoidal three-component velocity depending on two periodic coordinates and a zero-mean magnetic field with the same dependence, the in-plane flux potential satisfies a homogeneous advection-diffusion equation. The Poincaré inequality gives $⟨A^2⟩\le ⟨A_0^2⟩e^{-2\eta k^{2}t}$. Integrating the vertical stretching term by parts bounds it by $Q\sqrt{⟨A^2⟩⟨|\nabla B_z{|}^2⟩}$, where $Q=\Vert \nabla u_z{\Vert }_{\infty }$. Maximizing against resistive dissipation and integrating gives the displayed transient bound. Splitting off half the dissipation instead gives $E^{\prime }+\eta k^{2}E\le (Q^2/\eta )⟨A_0^2⟩e^{-2\eta k^{2}t}$, proving vertical energy decays. This permits transient generation by a nonzero vertical velocity while excluding sustained fields in this zero-mean, z-independent periodic class.

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.

For a bounded conducting sphere with uniform magnetic diffusivity and an insulating exterior, an incompressible flow tangent to concentric spheres cannot sustain dynamo action. Neither the velocity nor the magnetic field needs to be axisymmetric. The radial magnetic scalar has a homogeneous advection-diffusion equation, so the poloidal field decays. Once it is absent, the toroidal magnetic potential also has a homogeneous scalar advection-diffusion equation. This is a theorem about spherical flow geometry, not about a general cylindrical flow with an axial component.

An isolated axisymmetric magnetic field cannot be sustained by ordinary resistive dynamo action with scalar positive magnetic diffusivity. Its poloidal magnetic flux function satisfies a homogeneous scalar advection-diffusion equation with no toroidal-to-poloidal source. The maximum principle excludes regeneration of that flux; after it decays the toroidal field has no lasting source either. This constrains the symmetry of the field. An axisymmetric flow may still generate a nonaxisymmetric field, and a turbulent mean-field electromotive force is not covered by the unmodified scalar-induction proof.