  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 EVLA, GMRT and 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 magneto-hydrodynamical 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 MHD and the absence of a fully relativistic inner boundary
condition. The RM variability predictions are testable with current and future
telescopes.
