### 3.1 Poloidal to toroidal

Let us begin by expressing the (steady) large-scale flow field u as the sum of an axisymmetric
azimuthal component (differential rotation), and an axisymmetric “poloidal” component
(), i.e., a flow confined to meridional planes:
where and is the angular velocity (). Substituting this expression into
Equation (5) and into the -components of Equation (1) yields
Advection means bodily transport of by the flow; globally, this neither creates nor destroys magnetic
flux. Resistive decay, on the other hand, destroys magnetic flux and therefore acts as a sink of magnetic
field. Diamagnetic transport can increase B locally, but again this is neither a source nor sink of magnetic
flux. The compression/dilation term is a direct consequence of toroidal flux conservation in a meridional
flow moving across a density gradient. The shearing term in Equation (12), however, is a true source term,
as it amounts to converting rotational kinetic energy into magnetic energy. This is the needed
production mechanism.
However, there is no comparable source term in Equation (11). No matter what the toroidal component
does, will inexorably decay. Going back to Equation (12), notice now that once is gone, the
shearing term vanishes, which means that will in turn inexorably decay. This is the essence of
Cowling’s theorem: An axisymmetric flow cannot sustain an axisymmetric magnetic field against resistive
decay.