Large-scale Model of the Axisymmetric Dynamo with Feedback Effects

Abstract

A dynamo model is presented, based on a previously introduced kinematic model, in which the effect of the magnetic field on the mass flow through the Lorentz force is included. Given the base mass flow corresponding to the case with no magnetic field, and assuming that the modification of this flow through the Lorentz force can be treated as a perturbation, a complete model of the large-scale magnetic field dynamics can be obtained. The input needed consists of the large-scale meridional and zonal flows, the small-scale magnetic diffusivity, and a constant parameter entering the expression of the $\alpha$-effect. When applied to a Sun-like star, the model shows realistic dynamics of the magnetic field, including cycle duration, consistent field amplitudes with the correct parity, progression of the zonal magnetic field toward the equator, and motion toward the poles of the radial field at high latitudes. In addition, the radial and zonal components show a correct phase relation, and at the surface level, the magnetic helicity is predominantly negative in the northern hemisphere and positive in the southern hemisphere.

Appendix: Expressions of \({\boldsymbol{S}}\) Terms

The explicit expression of the term $S_{\phi }$ that enters Equation 11 is

On the other hand, it is convenient to express the terms depending on ${\boldsymbol{S}}$ in Equation 12 by separating them into a part associated with the differential rotation and another part associated with the meridional flow. The part corresponding to the differential rotation is written as

where

$$\begin{aligned} F_{r} =&3\cos \theta \partial _{\theta }\Omega +\sin \theta \bigl( \partial _{\theta \theta }\Omega +r\partial _{r}\Omega -2r^{2}\partial _{rr}\Omega \bigr) , \\ F_{\theta } =&\frac{3+\cos 2\theta }{2}\csc \theta \partial _{\theta } \Omega -r\sin \theta \partial _{r\theta }\Omega -r^{2}\cos \theta \partial _{rr}\Omega , \\ F_{r\theta } =&r^{2}\sin \theta \partial _{r\theta }\Omega , \\ F_{\theta r} =&r\cos \theta \partial _{r}\Omega -\sin \theta \partial _{\theta }\Omega +r\sin \theta \partial _{r\theta }\Omega , \\ F_{\theta \theta } =&\cos \theta \partial _{\theta }\Omega +\sin \theta \partial _{\theta \theta }\Omega -r^{2}\sin \theta \partial _{rr} \Omega . \end{aligned}$$

The part associated with the meridional flow is given by

where

$$\begin{aligned} G_{r} =&U_{r}+U_{\theta }\cot \theta +r\partial _{r}U_{r}+r^{2} \partial _{rr}U_{r}+r \partial _{r\theta }U_{\theta }, \\ G_{\theta } =&U_{r}\cot \theta +U_{\theta }\csc ^{2}\theta + \partial _{\theta \theta }U_{\theta }-r\partial _{r}U_{\theta }+r \partial _{r\theta }U_{r}, \\ G_{r\theta } =&-U_{\theta }+\partial _{\theta }U_{r}+r \partial _{r}U _{\theta }, \\ G_{rr} =&\partial _{r}U_{r}, \\ G_{\theta \theta } =&U_{r}\partial _{\theta }U_{\theta }. \end{aligned}$$