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,šœƒ,Ļ•), one writes
B (r,šœƒ, t) = ∇ × (A(r,šœƒ,t)ˆeĻ•) + B (r,šœƒ,t)ˆe Ļ•. (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 Ļ•-component of Equation (1View Equation), and the former, under the Coulomb gauge ∇ ⋅ A = 0, by
∂-(AˆeĻ•) 2 ∂t + (u ⋅ ∇ )(A ˆeĻ•) = η∇ (AˆeĻ•). (5 )

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