  We employ a variety of numerical simulations in the local shearing box system
to investigate in greater depth the local hydrodynamic stability of Keplerian
differential rotation. In particular we explore the relationship of Keplerian
shear to the nonlinear instabilities known to exist in simple Cartesian shear.
The Coriolis force is the source of linear stabilization in differential
rotation. We exploit the formal equivalence of constant angular momentum flows
and simple Cartesian shear to examine the transition from stability to
nonlinear instability. The manifestation of nonlinear instability in shear
flows is known to be sensitive to initial perturbation and to the amount of
viscosity; marginally (linearly) stable differentially rotating flows exhibit
this same sensitivity. Keplerian systems, however, are completely stable; the
strength of the stabilizing Coriolis force easily overwhelms any destabilizing
nonlinear effects. In fact, nonlinear effects speed the decay of applied
turbulence by producing a rapid cascade of energy to high wavenumbers where
dissipation occurs. Our conclusions are tested with grid resolution experiments
and by comparison with results from a code that employs an alternative
numerical algorithm. The properties of hydrodynamic differential rotation are
contrasted with magnetohydrodynamic differential rotation. The kinetic stress
couples to the vorticity which limits turbulence, while the magnetic stress
couples to the shear which promotes turbulence. Thus magnetohydrodynamic
turbulence is uniquely capable of acting as a turbulent angular momentum
transport mechanism in disks.
