Alpha-quenching rotating wave ($\omega =\pm 1$)

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.

For nonzero coupling ${ϵ}_1$, 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 $(a-p{B}_1^2)B_1+{ϵ}_1{B}_2=0$ and $(a-p{B}_2^2)B_2+{ϵ}_1{B}_1=0$. Subtracting after cross multiplication gives $(B_1^2-B_2^2)(p{B}_1{B}_2+{ϵ}_1)=0$. Unequal magnitudes therefore require $B_1{B}_2=-{ϵ}_1/p$ and $B_1^2+B_2^2=a/p$. They exist exactly when $a>2|{ϵ}_1|$ 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.