 Coupled-hemisphere dynamo parity threshold

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.