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 up (≡ ur(r,πœƒ )ˆer + u πœƒ(r,πœƒ)ˆeπœƒ), i.e., a flow confined to meridional planes:
u (r,πœƒ) = up (r,πœƒ) + ϖ Ω (r,πœƒ)ˆeΟ• (10 )
where ϖ = rsinπœƒ and Ω is the angular velocity (−1 rad s). Substituting this expression into Equation (5View Equation) and into the Ο•-components of Equation (1View Equation) yields
( ) ∂A- = η ∇2 − 1-- A − up-⋅ ∇ (ϖA ), (11 ) ∂t ϖ2 ϖβ—Ÿ----β—β—œ----β—ž β—Ÿ------β—β—œ------β—ž advection resistive decay ( ) ( ) ∂B-- 2 1-- -1 ∂(ϖB--)∂η- B- ∂t = η ∇ − ϖ2 B + ϖ ∂r ∂r − ϖup ⋅ ∇ ϖ − Bβ—Ÿ∇◝ ⋅β—œ upβ—ž + ϖβ—Ÿ-(∇-×-(A◝ ˆeβ—œΟ•)) ⋅-∇Ωβ—ž. (12 ) β—Ÿ------β—β—œ------β—ž β—Ÿ-----β—β—œ----β—ž β—Ÿ-----β—β—œ-----β—ž 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 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 decay2.

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