Let us attempt to prove this proposition.An axisymmetric magnetic field cannot be maintained via dynamo action.

We adopt standard cylindrical polar coordinates: . The system is assumed to possess axial symmetry, so that . For the sake of simplicity, the plasma flow is assumed to be incompressible, which implies that .

It is convenient to split the magnetic and velocity fields into *poloidal*
and *toroidal* components:

(795) | |||

(796) |

Note that a poloidal vector only possesses non-zero - and -components, whereas a toroidal vector only possesses a non-zero -component.

The poloidal components of the magnetic and velocity fields are
written:

(797) | |||

(798) |

where and . The toroidal components are given by

(799) | |||

(800) |

Note that by writing the and fields in the above form we ensure that the constraints and are

The condition for the magnetic field to be maintained by dynamo currents,
rather than by currents at infinity, is

Consider the MHD Ohm's law for a resistive plasma:

It is easily demonstrated that

(804) |

(805) |

(806) |

(807) |

Multiplying the above equation by and integrating over all space,
we obtain

The second term on the left-hand side of the above expression can be integrated by parts to give

(809) |

(810) |

Thus, Eq. (808) reduces to

(811) |

Of course, we have not ruled out the possibility that the toroidal magnetic
field can be maintained via dynamo action. In the absence of a poloidal
field, the curl of the poloidal component of Eq. (802) yields

(812) |

Now

(814) |

(815) |

where

(817) |

Multiply Eq. (816) by , integrating over all space, and
then integrating by parts, we obtain

(818) |

Cowling's theorem is the earliest and most significant of a number of
*anti-dynamo theorems* which severely restrict the types of magnetic
fields which can be maintained via dynamo action. For instance, it is
possible to prove that a two-dimensional magnetic field cannot be maintained
by dynamo action. Here, ``two-dimensional'' implies that in some
Cartesian coordinate system, , the magnetic field is independent of
. The suite of anti-dynamo theorems can be summed up
by saying that successful dynamos possess a rather low degree of symmetry.