  (Abridged) We analyse the stability and 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 power-law functions of
cylindrical radius. We employ two different hydrodynamic codes to perform
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 viscosity. We
present a stability analysis of the problem that we use to interpret the
simulation results. We find that 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| >> 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 operates when the dimensionless disc kinematic viscosity nu <
10^{-6} (Reynolds numbers Re>H c_s/nu > 2500). In 3D the instability generates
a quasi-turbulent flow, and the Reynolds stress produces a fluctuating
effective viscosity coefficient whose mean value reaches alpha ~ 6 x 10^{-4} by
the end of the simulation. The vertical shear instability in disc models which
include realistic thermal physics has yet to be examined. Should it occur,
however, our results suggest that it will have significant consequences for
their internal dynamics, transport properties, and observational appearance.
