This relation expresses the Richardson criterion in a medium with a / gradient and heat losses. The various solutions of this equation are functions of r (see details in [364]). For the case relevant to stellar interiors with r C 1, one has from (12.58) r « -0/(1 + 0) « -0. This leads to the following local diffusion or viscosity coefficient

Again, the average over (dV/dz) should be taken differently for vshear and Dshear as in (12.41) and (12.40). In absence of a ^ gradient, one is brought back to (12.38).

The above criterion shows that diffusion due to shear turbulence occurs if i.e., it is sufficient that the excess energy in the shear overcomes the stabilizing effect of the ^ gradient. In absence of a ^ gradient, any shear is unstable. However, in order the instabilities are not damped by viscous effects, the system must also satisfy the Reynolds criterion (12.69, see Appendix B.5.1). The stabilizing effect of the thermal structure does not intervene in the stability criterion, but only in the value of the diffusion coefficient (12.59).

If the above condition is applied to stellar interiors of differentially rotating models [408], it essentially kills the mixing in regions where there is a ^ gradient and easily allows mixing in homogeneous regions where it has no effect! The observations of massive stars show evidences of a significant mixing (Sect. 27.4.2). Thus, the above expression of the Richardson criterion, although formally correct, gives too stringent conditions. Several solutions to this problem were proposed [349, 562]. Below, we examine the effect of the horizontal turbulence, which considerably reduces the inhibiting effect of the ^ gradient on the shear instability.

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