  The magnetorotational instability (MRI) is thought to be a powerful source of
turbulence in Keplerian accretion disks. Motivated by recent laboratory
experiments, we study the MRI driven by an azimuthal magnetic field in an
electrically conducting fluid sheared between two concentric rotating
cylinders. By adjusting the rotation rates of the cylinders, we approximate
angular velocity profiles $\omega \propto r^{q}$. We perform direct numerical
simulations of a steep profile close to the Rayleigh line $q \gtrsim -2 $ and a
quasi-Keplerian profile $q \approx -3/2$ and cover wide ranges of Reynolds
($Re\le 4\cdot10^4$) and magnetic Prandtl numbers ($0\le Pm \le 1$). In the
quasi-Keplerian case, the onset of instability depends on the magnetic Reynolds
number, with $Rm_c \approx 50$, and angular momentum transport scales as
$\sqrt{Pm} Re^2$ in the turbulent regime. The ratio of Maxwell to Reynolds
stresses is set by $Rm$. At the onset of instability both stresses have similar
magnitude, whereas the Reynolds stress vanishes or becomes even negative as
$Rm$ increases. For the profile close to the Rayleigh line, the instability
shares these properties as long as $Pm\gtrsim0.1$, but exhibits a markedly
different character if $Pm\rightarrow 0$, where the onset of instability is
governed by the Reynolds number, with $Re_c \approx 1250$, transport is via
Reynolds stresses and scales as $Re^2$. At intermediate $Pm=0.01$ we observe a
continuous transition from one regime to the other, with a crossover at
$Rm=\mathcal{O}(100)$. Our results give a comprehensive picture of angular
momentum transport of the MRI with an imposed azimuthal field.
