  In order to explore the magnetostrophic regime expected for planetary cores,
experiments have been conducted in a rotating sphere filled with liquid sodium,
with an imposed dipolar magnetic field (the DTS setup). The field is produced
by a permanent magnet enclosed in an inner sphere, which can rotate at a
separate rate, producing a spherical Couette flow. The flow properties are
investigated by measuring electric potentials on the outer sphere, the induced
magnetic field in the laboratory frame, and velocity profiles inside the liquid
sodium using ultrasonic Doppler velocimetry. The present article focuses on the
time-averaged axisymmetric part of the flow. The Doppler profiles show that the
angular velocity of the fluid is relatively uniform in most of the fluid shell,
but rises near the inner sphere, revealing the presence of a magnetic wind, and
gently drops towards the outer sphere. The transition from a magnetostrophic
flow near the inner sphere to a geostrophic flow near the outer sphere is
controlled by the local Elsasser number. For Rossby numbers up to order 1, the
observed velocity profiles all show a similar shape. Numerical simulations in
the linear regime are computed, and synthetic velocity profiles are compared
with the measured ones. In the geostrophic region, a torque-balance model
provides very good predictions. We find that the induced magnetic field varies
in a consistent fashion, and displays a peculiar peak in the counter-rotating
regime. This happens when the fluid rotation rate is almost equal and opposite
to the outer sphere rotation rate. The fluid is then almost at rest in the
laboratory frame, and the Proudman-Taylor constraint vanishes, enabling a
strong meridional flow. We suggest that dynamo action might be favored in such
a situation.
