Linear and non-linear evolution of the vertical shear instability in accretion discs

Abstract

(Abridged) We analyse the stability and evolution of power-law accretion disc\nmodels. These have midplane densities that follow radial power-laws, and have\neither temperature or entropy distributions that are power-law functions of\ncylindrical radius. We employ two different hydrodynamic codes to perform\n2D-axisymmetric and 3D simulations that examine the long-term evolution of the\ndisc models as a function of the power-law indices of the temperature or\nentropy, the thermal relaxation time of the fluid, and the viscosity. We\npresent a stability analysis of the problem that we use to interpret the\nsimulation results. We find that disc models whose temperature or entropy\nprofiles cause the equilibrium angular velocity to vary with height are\nunstable to the growth of modes with wavenumber ratios |k_R/k_Z| >> 1 when the\nthermodynamic response of the fluid is isothermal, or the thermal evolution\ntime is comparable to or shorter than the local dynamical time scale. These\ndiscs are subject to the Goldreich-Schubert-Fricke (GSF) or `vertical shear'\nlinear instability. Development of the instability involves excitation of\nvertical breathing and corrugation modes in the disc, with the corrugation\nmodes in particular being a feature of the nonlinear saturated state.\nInstability operates when the dimensionless disc kinematic viscosity nu <\n10^{-6} (Reynolds numbers Re>H c_s/nu > 2500). In 3D the instability generates\na quasi-turbulent flow, and the Reynolds stress produces a fluctuating\neffective viscosity coefficient whose mean value reaches alpha ~ 6 x 10^{-4} by\nthe end of the simulation. The vertical shear instability in disc models which\ninclude realistic thermal physics has yet to be examined. Should it occur,\nhowever, our results suggest that it will have significant consequences for\ntheir internal dynamics, transport properties, and observational appearance.\n