2.5 Boundary conditions and parity

The axisymmetric dynamo equations are to be solved in a meridional plane, i.e., Ri ≤ r ≤ R ⊙ and 0 ≤ 𝜃 ≤ π, where the inner radial extent of the domain (Ri) need not necessarily extend all the way to r = 0. It is usually assumed that the deep radiative interior can be treated as a perfect conductor, so that R i is chosen a bit deeper than the lowest extent of the region where dynamo action is taking place; the boundary condition at this depth is then simply A = 0, ∂ (rB )∕∂r = 0.

It is usually assumed that the Sun/star is surrounded by a vacuum, in which no electrical currents can flow, i.e., ∇ × B = 0; for an axisymmetric B expressed via Equation (4View Equation), this requires

( 1 ) ∇2 − -2---2-- A = 0, B = 0, r∕R ⊙ > 1. (6 ) r sin 𝜃
It is therefore necessary to smoothly match solutions to Equations (1View Equation, 5View Equation) on solutions to Equations (6View Equation) at r∕R ⊙ = 1. Regularity of the solutions demands that A = 0 and B = 0 on the symmetry axis (𝜃 = 0 and 𝜃 = π in a meridional plane). This completes the specification of the boundary conditions.

Formulated in this manner, the dynamo solution will spontaneously “pick” its own parity, i.e., its symmetry with respect to the equatorial plane. An alternative approach, popular because it can lead to significant savings in computing time, is to solve only in a meridional quadrant (0 ≤ 𝜃 ≤ π ∕2) and impose solution parity via the boundary condition at the equatorial plane (π∕2):

∂A ---= 0, B = 0 → antisymmetric, (7 ) ∂𝜃 A = 0, ∂B--= 0 → symmetric. (8 ) ∂𝜃

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