Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

Aims. We aim to assess whether magneto-rotational instability (MRI) can exist in a turbulent state generated by gravitational instability (GI). We investigated the magnetic field saturation and elucidated the ability of GI turbulence to act as a dynamo. Methods. The results were obtained by numerical simulations using the magnetohydrodynamics code Athena. A sub-routine to solve the Poisson equation for self-gravity using three-dimensional Fourier transforms was implemented for that purpose. A GI-turbulent state was then restarted, with a zero-net-flux type magnetic seed field being introduced. The seed field was chosen with β ≈ 1010 to make sure that the magnetic field of the stationary state is exclusively generated by the dynamo. Results. Shortly after introducing the magnetic seed field, a significant field amplification is observed, despite MRI not being active. This shows that GI acts as a kinematic dynamo. The growing magnetic field allows MRI to become active, which leads to the emergence of a butterfly diagram. The turbulent stress of the saturated state is found to be consistent with the superposition of GI stresses and MRI stresses. Moreover, the ratio of magnetic stress to magnetic pressure is found to lie in the 0.3−0.4 range, which is typical for MRI turbulence. Furthermore, it is found that the magnetic energy significantly decreases if self-gravity is turned off. This indicates, in accordance with the initial field amplification, that GI provides the dominant dynamo contribution and that MRI is not simply added but rather grows on the magnetic field provided by GI turbulence. Finally, it is shown that the combined GI-MRI-dynamo is consistent with an α − Ω model and that the observed oscillation frequency of the butterfly diagram roughly agrees with the model prediction.