 Alpha-quenching rotating wave

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 ${ϵ}_2=0$, eliminating the poloidal amplitudes givesThe matrix is Hermitian, so its eigenvalue is real and a nonzero rotating wave with $\Omega \ne 0$ has $\omega =\pm 1$. Writing $a={\alpha }_0-2\omega /\Omega$, pure dipole and quadrupole waves have $|B_j{|}^2=(a-{ϵ}_1)/p$ and $(a+{ϵ}_1)/p$, respectively, when those quantities are positive.