  Strongly magnetized accretion discs around black holes have attractive
features that may explain enigmatic aspects of X-ray binary behaviour. The
structure and evolution of these discs are governed by a dynamo-like mechanism,
which channels part of the accretion power liberated by the magnetorotational
instability (MRI) into an ordered toroidal magnetic field. To study dynamo
activity, we performed three-dimensional, stratified, isothermal, ideal
magnetohydrodynamic shearing box simulations. The strength of the
self-sustained toroidal magnetic field depends on the net vertical magnetic
flux, which we vary across almost the entire range over which the MRI is
linearly unstable. We quantify disc structure and dynamo properties as a
function of the initial ratio of mid-plane gas pressure to vertical magnetic
field pressure, $\beta_0^{\rm mid} = p_{\rm gas} / p_B$. For $10^5 \geq
\beta_0^{\rm mid} \geq 10$ the effective $\alpha$-viscosity parameter scales as
a power-law. Dynamo activity persists up to and including $\beta_0^{\rm mid} =
10^2$, at which point the entire vertical column of the disc is magnetic
pressure-dominated. Still stronger fields result in a highly inhomogeneous disc
structure, with large density fluctuations. We show that the turbulent steady
state $\beta^{\rm mid}$ in our simulations is well-matched by the analytic
model of Begelman et al. (2015) describing the creation and buoyant escape of
toroidal field, while the vertical structure of the disc can be broadly
reproduced using this model. Finally, we discuss the implications of our
results for observed properties of X-ray binaries.
