According to Cowling's theorem, an axi - symmetric, localised dynamo cannot sustain itself. In order to prove this theorem, let us consider a system with axial symmetry around the $ z $ axis. Next, we decompose the velocity and magnetic field into poloidal and toroidal components

$ \mathbf{v} = \Omega r \hat{\varphi} + \nabla \times \left( \frac{\psi}{r} \hat{\varphi} \right) $

$ \mathbf{B} = B \hat{\varphi} + \nabla \times \left(A \hat{\varphi} \right ) $

Substituting into the equation of motion

$ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left( \mathbf{v} \times \mathbf{B} \right ) - \eta \nabla^2 \mathbf{B} $

yields two equation: one for the poloidal component, and one for the toroidal component

$ \frac{\partial A}{\partial t} + \frac{1}{r} \left( \mathbf{u}_p \cdot \nabla \right ) A = \eta \left( \nabla^2 - \frac{1}{r^2} \right ) A $

$ \frac{\partial B}{\partial t} + r \left( \mathbf{v}_p \cdot \nabla \right ) \left( \frac{B}{r} \right ) = \eta \left(\nabla^2 - \frac{1}{r^2} \right )B + r \mathbf{B}_p \cdot \nabla \Omega $

The first equation only describes advection and diffusion. Since it does not have any source term, it is bound to decay with time. The second equation also describes diffusion and advection, but also has a source term. However, that source term depends on the poloidal component, which was shown to decay, so the ultimate fate of the second equation is no different from the first.