Transport properties of the Azimuthal Magnetorotational Instability

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.