= Coupled-hemisphere dynamo parity threshold
{title2=$F_d=\Omega^2(\alpha_0-\epsilon_1)^2/4-\epsilon_2(\alpha_0-\epsilon_1)$}
In a two-hemisphere <mean-field dynamo>, dipole and quadrupole subspaces can reduce to $\dot A=aB-A$, $\dot B=bA-B$, with $(a,b)=(\alpha_0-\epsilon_1,i\Omega-\epsilon_2)$ and $(\alpha_0+\epsilon_1,i\Omega+\epsilon_2)$ respectively. Their <eigenvalues> satisfy $(\lambda+1)^2=ab$. Strict decay is equivalent to $\operatorname{Re}(ab)+[\operatorname{Im}(ab)]^2/4<1$. Thus dipole decay requires $F_d<1$, while quadrupole decay requires $F_q=\Omega^2(\alpha_0+\epsilon_1)^2/4+\epsilon_2(\alpha_0+\epsilon_1)<1$. Their difference is $F_q-F_d=\alpha_0(\Omega^2\epsilon_1+2\epsilon_2)$, which selects the first unstable parity when starting from a state stable in both subspaces.
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