= Alpha-quenching rotating wave
{title2=$\omega=\pm1$}
A rotating wave in a complex <dynamo quenching> model has constant amplitude and a common phase factor $e^{i\omega t}$. In the two-hemisphere model with toroidal coupling parameter $\epsilon_2=0$, eliminating the poloidal amplitudes gives
$$
\frac{2\omega-i(1-\omega^2)}{\Omega}\begin{pmatrix}B_1\\B_2\end{pmatrix}=\begin{pmatrix}\alpha_0-p|B_1|^2&\epsilon_1\\\epsilon_1&\alpha_0-p|B_2|^2\end{pmatrix}\begin{pmatrix}B_1\\B_2\end{pmatrix}.
$$
The matrix is <Hermitian>, so its <eigenvalue> is real and a nonzero rotating wave with $\Omega\ne0$ has $\omega=\pm1$. Writing $a=\alpha_0-2\omega/\Omega$, pure dipole and quadrupole waves have $|B_j|^2=(a-\epsilon_1)/p$ and $(a+\epsilon_1)/p$, respectively, when those quantities are positive.
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