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 $u p$ ( $=_ ur(r,h)^er + uh(r,h)^eh$ ), i.e., a flow confined to meridional planes:

where $p = rsinh$ and $_O_$ is the angular velocity ( $rad s-1$ ). Substituting this expression into Equation (5

) and into the $f$ -components of Equation (1

) yields

$( ) @A- = j \ ~/ 2 - -1- A - up-. \~/ (pA) , (11) @t p2 p---- ---- ------ ------ advection resistive decay ( ) ( ) @B-- 2 -1- 1-@(pB)--@j- B- @t = j \ ~/ - p2 B + p @r @r - pup .\ ~/ p - B\ ~/ .up + p(\ ~/ -×-(A^ef)) . \~/ _O_ . (12) ------ ------ ----- ----- ----- ----- compression shearing resistive decay diamagnetic transport advection$

Advection means bodily transport of $B$ 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 $P --> T$ production mechanism.

However, there is no comparable source term in Equation (11 ). No matter what the toroidal component does, $A$ will inexorably decay. Going back to Equation (12 ), notice now that once $A$ is gone, the shearing term vanishes, which means that $B$ will in turn inexorably decay. This is the essence of Cowling’s theorem: An axisymmetric flow cannot sustain an axisymmetric magnetic field against resistive decay³ .