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We conduct a survey of numerical simulations to probe the structure and appearance of non‐radiative black hole accretion flows like the supermassive black hole at the Galactic Centre. We find a generic set of solutions, and make specific predictions for currently feasible rotation measure (RM) observations, which are accessible to current instruments including the Expanded Very Large Array (EVLA), Giant Metrewave Radio Telescope (GMRT) and Atacama Large Millimeter Array (ALMA). The slow time variability of the RM is a key quantitative signature of this accretion flow. The time variability of RM can be used to quantitatively measure the nature of the accretion flow, and to differentiate models. Sensitive measurements of RM can be achieved using RM synthesis or using pulsars.  Our energy conserving ideal magnetohydrodynamical simulations, which achieve high dynamical range by means of a deformed‐mesh algorithm, stretch from several Bondi radii to about one‐thousandth of that radius, and continue for tens of Bondi times. Magnetized flows which lack outward convection possess density slopes around −1, almost independent of physical parameters, and are more consistent with observational constraints than are strongly convective flows. We observe no tendency for the flows to become rotationally supported in their centres, or to develop steady outflow.  We support these conclusions with formulae which encapsulate our findings in terms of physical and numerical parameters. We discuss the relation of these solutions to other approaches. The main potential uncertainties are the validity of ideal magnetohydrodynamic and the absence of a fully relativistic inner boundary condition. The RM variability predictions are testable with current and future telescopes.

Numerical parameter survey of non‐radiative black hole accretion: flow structure and variability of the rotation measure