Dynamo quenching ($\alpha (B)={\alpha }_0/(1+B^2/B_{\ast }^2)$)

Dynamo quenching is the reduction or modification of magnetic-field generation by magnetic feedback on the generating flow and its correlations. The Lorentz force can change shear or turbulence, while magnetic helicity affects mean-field regeneration. A phenomenological scalar example is $\alpha (B)={\alpha }_0/(1+B^2/B_{\ast }^2)$. In the illustrative amplitude equation $\dot{B}=[c\alpha (B)-d]B$ with $c,d>0$, a growing field saturates at $B^2=B_{\ast }^2(c{\alpha }_0/d-1)$ when $c{\alpha }_0>d$. This demonstrates the negative feedback mechanism without claiming that algebraic quenching is a complete solar closure.

Algebraic alpha quenching prescribes a field-dependent alpha effect as a closure for nonlinear mean-field dynamo feedback. The polynomial law $\alpha ={\alpha }_0-p|B{|}^2$ with $p>0$ decreases alpha, so it can saturate positive alpha but makes negative alpha more negative. It therefore differs importantly from the sign-preserving rational law ${\alpha }_0/(1+|B{|}^2/B_{\ast }^2)$. Its solutions must satisfy nonnegative squared-amplitude conditions; an assertion based only on large $|{\alpha }_0|$ need not hold for both signs.

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.