  The torque in turbulent Taylor-Couette flows for shear Reynolds numbers Re_S
up to 3x10^4 at various mean rotations is studied by means of direct numerical
simulations for a radius ratio of \eta=0.71. Convergence of simulations is
tested using three criteria of which the agreement of dissipation values
estimated from the torque and from the volume dissipation rate turns out to be
most demanding. We evaluate the influence of Taylor vortex heights on the
torque for a stationary outer cylinder and select a value of the aspect ratio
of \Gamma=2, close to the torque maximum. The connection between the torque and
the transverse current J^\omega of azimuthal motion which can be computed from
the velocity field enables us to investigate the local transport resulting in
the torque. The typical spatial distribution of the individual convective and
viscous contributions to the local current is analysed for a turbulent flow
case. To characterise the turbulent statistics of the transport, PDF's of local
current fluctuations are compared to experimental wall shear stress
measurements. PDF's of instantaneous torques reveal a fluctuation enhancement
in the outer region for strong counter-rotation. Moreover, we find for
simulations realising the same shear Re_S>=2x10^4 the formation of a torque
maximum for moderate counter-rotation with angular velocities
\omega_o\approx-0.4\omega_i. In contrast, for Re_S<=4x10^3 the torque features
a maximum for a stationary outer cylinder. In addition, the effective torque
scaling exponent is shown to also depend on the mean rotation state. Finally,
we evaluate a close connection between boundary-layer thicknesses and the
torque.
