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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:
u(r,h) = u (r,h) + p_O_(r, h)^e (10) p f
where p = rsinh and _O_ is the angular velocity (rad s-1). Substituting this expression into Equation (5View Equation) and into the f-components of Equation (1View Equation) 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 (12View Equation), 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 (11View Equation). No matter what the toroidal component does, A will inexorably decay. Going back to Equation (12View Equation), 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.


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