Dipole and quadrupole parity in a mean-field dynamo

In a reflection-symmetric mean-field dynamo model, dipole parity can be represented by an even poloidal-potential amplitude and an odd toroidal-field amplitude; quadrupole parity reverses these assignments. The names describe the reflection classes appropriate to the model, not arbitrary electric multipoles. A complex amplitude $P=A+iB$ allows a phase rotation by $i$ to interchange the two classes when the unperturbed equations have that phase symmetry.

In a two-hemisphere mean-field dynamo, dipole and quadrupole subspaces can reduce to $\dot{A}=aB-A$, $\dot{B}=bA-B$, with $(a,b)=({\alpha }_0-{ϵ}_1,i\Omega -{ϵ}_2)$ and $({\alpha }_0+{ϵ}_1,i\Omega +{ϵ}_2)$ respectively. Their eigenvalues satisfy $(\lambda +1{)}^2=ab$. Strict decay is equivalent to $\mathrm{Re}(ab)+[\mathrm{Im}(ab){]}^2/4<1$. Thus dipole decay requires $F_d<1$, while quadrupole decay requires $F_q={\Omega }^2({\alpha }_0+{ϵ}_1{)}^2/4+{ϵ}_2({\alpha }_0+{ϵ}_1)<1$. Their difference is $F_q-F_d={\alpha }_0({\Omega }^2{ϵ}_1+2{ϵ}_2)$, which selects the first unstable parity when starting from a state stable in both subspaces.

An antisymmetric perturbation can split the thresholds of two degenerate mean-field dynamo parity classes. At first order, project the perturbation equation onto the adjoint null mode using the Fredholm alternative. If multiplying the unperturbed complex mode by $i$ leaves the parameter-derivative matrix element unchanged but reverses the perturbation matrix element, the two first-order threshold shifts are equal and opposite.