2.4 Axisymmetric formulation

The sunspot butterfly diagram, Hale’s polarity law, synoptic magnetograms, and the shape of the solar corona at and around solar activity minimum jointly suggest that, to a tolerably good first approximation, the large-scale solar magnetic field is axisymmetric about the Sun’s rotation axis, as well as antisymmetric about the equatorial plane. Under these circumstances it is convenient to express the large-scale field as the sum of a toroidal (i.e., longitudinal) component and a poloidal component (i.e., contained in meridional planes), the latter being expressed in terms of a toroidal vector potential. Working in spherical polar coordinates $(r,h,f)$ , one writes

$B(r, h,t) = \~/ × (A(r,h, t)^ef) + B(r, h,t)^ef. (4)$

Such a decomposition evidently satisfies the solenoidal constraint $\~/ .B = 0$ , and substitution into the MHD induction equation produces two (coupled) evolution equations for $A$ and $B$ , the latter simply given by the $f$ -component of Equation (1

), and the former, under the Coulomb gauge $\~/ .A = 0$ , by

$@A- 2 @t + (u . \~/ )(A^ef) = j \~/ (A^ef). (5)$