Gravitational instabilities (GI) are important in the early stages of disk evolution. A gravitationally unstable disk settles into a system of stable angular momentum transport by generating spiral arms under slow cooling. The underlying dynamics of these spirals determine the observable disk morphology and small-scale turbulence, and can affect the efficiency of planet formation from fragmentation scenarios. We study gravitational turbulence in 2D using a parameterized β-cooling law. We characterize and explain the disk structure, compare it to previous 3D analyses consisting of localized spirals, and examine the propagation of the flow and its correlation to viscous disk models. We perform 2D global numerical simulations of self-gravitating accretion disks using the finite-volume code PLUTO in cylindrical geometry. We vary disk mass and the cooling parameter β, and evolve our models sufficiently long to trigger the GI and the formation of spirals. We find that the spirals show self-similarity and orient themselves into pitch angles that are constant in radius and time. The radial and azimuthal wavenumbers of the spirals match those predicted by linear perturbation theory. The values of gravitational stress match those of 3D simulations, and in higher resolution we see an increase in hydrodynamic stress. Our models are significantly faster and computationally efficient compared to 3D global models of self-gravitating disks using grid and SPH methods, and highlight the caveats of treating self-gravity in 2D.
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