We analyse the stability and nonlinear dynamical evolution of power-law accretion disc models .
These have midplane densities that follow radial power-laws , and have either temperature or entropy distributions that are strict power-law functions of cylindrical radius , R .
We employ two different hydrodynamic codes to perform high resolution 2D-axisymmetric and 3D simulations that examine the long-term evolution of the disc models as a function of the power-law indices of the temperature or entropy , the thermal relaxation time of the fluid , and the disc viscosity .
We present an accompanying stability analysis of the problem , based on asymptotic methods , that we use to interpret the results of the simulations .
We find that axisymmetric disc models whose temperature or entropy profiles cause the equilibrium angular velocity to vary with height are unstable to the growth of modes with wavenumber ratios |k _ { R } / k _ { Z } | \gg 1 when the thermodynamic response of the fluid is isothermal , or the thermal evolution time is comparable to or shorter than the local dynamical time scale .
These discs are subject to the Goldreich-Schubert-Fricke ( GSF ) or ‘ vertical shear ’ linear instability .
Development of the instability involves excitation of vertical breathing and corrugation modes in the disc , with the corrugation modes in particular being a feature of the nonlinear saturated state .
Instability is found to operate when the dimensionless disc kinematic viscosity \nu < 10 ^ { -6 } , corresponding to Reynolds numbers { Re } = Hc _ { s } / \nu > 2500 .
In three dimensions the instability generates a quasi-turbulent flow , and the associated Reynolds stress produces a fluctuating effective viscosity coefficient whose mean value reaches \alpha \sim 6 \times 10 ^ { -4 } by the end of the simulation .
The evolution and saturation of the vertical shear instability in astrophysical disc models which include realistic treatments of the thermal physics has yet to be examined .
Should it occur on either global or local scales , however , our results suggest that it will have significant consequences for their internal dynamics , transport properties , and observational appearance .