Dynamo action in turbulent accretion discs around black holes – II. The mean magnetic field

The presence of strong differential rotation and vertical density gradients in thin accretion discs allows the generation of a large-scale, mean magnetic field by $\alpha\enspace \omega$ dynamo action. Analytic expressions for the preferred steady-state solutions are found, which when matched on to an exterior vacuum field show that the class of quadrupole modes ($B_\phi$ and Br even, and Bz odd functions of z) are most easily excited. The critical dynamo number of this mode is $|Q_\text {e,c}^{\star\enspace 1/3}| = 2.4$, corresponding to a turbulent Mach number of $M_\text t^\text e = 0.19$.