## 3. Conditions for vertical shear instability## 3.1. Richardson criterionIn a stellar radiation zone, the stable stratification of entropy
tends to inhibit any instability arising from a vertical shear
, where The strength of the stratification is measured by the
Brunt-Väisälä frequency with the usual notations for the gravity However shear turbulence may still arise, and be sustained, provided that sufficient heat is lost by the turbulent eddies to lower their buyoancy. Whether this occurs is determined by the Péclet number characterizing these eddies: in this expression is the Stefan constant, the Rosseland mean opacity and the first adiabatic exponent. When the Richardson criterion takes the modified form (Dudis 1974; Zahn 1974): from which one infers what are the largest turbulent scales that can survive in a stratified shear flow. ## 3.2. Critical Reynolds numberHowever turbulence will be maintained only if the turnover rate of the eddies is faster than their viscous decay rate: is the viscosity. This critical
Reynolds number is smaller than the
classical one which governs the onset of instability in a shear flow
whose velocity varies by For the value of this we turn to an experiment performed by Stillinger et al. (1983), who measured the size of the turbulent motions downstream a flow traversing a grid. The size of the smallest turbulent eddies was found to be , where is the classical Kolmogorov length, with being the energy injection rate per unit mass. In the inertial cascade the velocities scale as , and therefore: from which we draw the critical Reynolds number in that experiment: , a value we shall adopt for the present purpose. The conditions (4) and (5) can be simultaneously fulfilled when: © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |