= Unequal-hemisphere dynamo rotating wave
{title2=$|B_{1,2}|^2=(a\pm\sqrt{a^2-4\epsilon_1^2})/(2p)$}
For nonzero coupling $\epsilon_1$, an <alpha-quenching rotating wave> can be made real in its toroidal amplitudes by a common phase rotation. Set $a=\alpha_0-2\omega/\Omega$. The amplitude equations are $(a-pB_1^2)B_1+\epsilon_1B_2=0$ and $(a-pB_2^2)B_2+\epsilon_1B_1=0$. Subtracting after cross multiplication gives $(B_1^2-B_2^2)(pB_1B_2+\epsilon_1)=0$. Unequal magnitudes therefore require $B_1B_2=-\epsilon_1/p$ and $B_1^2+B_2^2=a/p$. They exist exactly when $a>2|\epsilon_1|$ and have the displayed squared amplitudes. Exchanging hemispheres yields the companion solution. This breaks equatorial reflection symmetry and combines both <dipole and quadrupole parity in a mean-field dynamo>. Existence does not alone prove nonlinear stability.
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