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Astron. Astrophys. 364, 876-878 (2000)

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3. Conditions for vertical shear instability

3.1. Richardson criterion

In a stellar radiation zone, the stable stratification of entropy tends to inhibit any instability arising from a vertical shear [FORMULA], where V is the amplitude of the horizontal velocity and z the vertical coordinate. In the adiabatic limit, i.e. when radiative losses are negligible, the shear instability is suppressed whenever:

[EQUATION]

The strength of the stratification is measured by the Brunt-Väisälä frequency N:

[EQUATION]

with the usual notations for the gravity g, the pressure scale height [FORMULA], the logarithmic temperature gradients [FORMULA] and [FORMULA], P, [FORMULA] and T being pressure, density and temperature. The critical Richardson number [FORMULA] is of order unity for typical flow profiles; in the following we shall take [FORMULA].

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:

[EQUATION]

in this expression v and l are the velocity and the size of turbulent elements and K the thermal diffusivity:

[EQUATION]

[FORMULA] is the Stefan constant, [FORMULA] the Rosseland mean opacity and [FORMULA] the first adiabatic exponent.

When [FORMULA] the Richardson criterion takes the modified form (Dudis 1974; Zahn 1974):

[EQUATION]

from which one infers what are the largest turbulent scales that can survive in a stratified shear flow.

3.2. Critical Reynolds number

However turbulence will be maintained only if the turnover rate of the eddies is faster than their viscous decay rate:

[EQUATION]

[FORMULA] is the viscosity. This critical Reynolds number [FORMULA] is smaller than the classical one which governs the onset of instability in a shear flow whose velocity varies by U between two boundaries separated by the distance L, namely [FORMULA]. The reason is of course that [FORMULA] and [FORMULA].

For the value of this [FORMULA] 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 [FORMULA], where [FORMULA] is the classical Kolmogorov length, with [FORMULA] being the energy injection rate per unit mass. In the inertial cascade the velocities scale as [FORMULA], and therefore:

[EQUATION]

from which we draw the critical Reynolds number in that experiment: [FORMULA], a value we shall adopt for the present purpose.

The conditions (4) and (5) can be simultaneously fulfilled when:

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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