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 B_r even, and B_z 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 $$M_\text t^\text e = 0.19$$.