Critical Richardson numbers and gravity waves

In this paper we present two new results. The first concerns the proper identification of the critical Richardson number Ri (cr) above which there is no longer turbulent mixing. Thus far, all studies have assumed that: $$ Ri {\rm (cr)} = Ri^\ell {\rm (cr)} = 1/4. $$ However, since $Ri^\ell$(cr) determines the upper limit of a laminar regime (superscript $\ell$), it has little relevance to stars where the problem is not to determine the end point of a laminar regime but the endpoint of turbulence. We show that the latter is characterized by $Ri^{\rm t} {\rm (cr)}$, where t stands for turbulence, and has a value four times larger than (1): $$ Ri {\rm (cr)} = Ri^{\rm t} {\rm (cr)} \approx 4 Ri^\ell {\rm (cr)}\approx 1. $$ We also show that use of (2) instead of (1) changes the conclusions of recent studies. Inclusion of radiative losses (characterized by the Peclet number Pe ) which weaken stable stratification and help turbulence, further changes (2) to (r stands for radiative): $$ Ri {\rm (cr)} = Ri^{\rm r} {\rm (cr)} \sim (1+Pe)Pe^{-1}Ri^{\rm t} {\rm (cr)} $$ which, for $Pe 1$. In conclusion, the successive inclusion of relevant physical processes leads to a chain of increasing values of $Ri {\rm (cr)}$: $$ Ri {\rm (cr)} = Ri^{\ell} {\rm (cr)} \rightarrow Ri^{\rm t} {\rm (cr)}\rightarrow Ri^{\rm r} {\rm (cr)}\rightarrow Ri^{\rm gw} {\rm (cr)}. $$ The second result concerns the dependence of the diffusivity D on Ω . We show that the commonly used expression $$ D_{\chi^{-1}}\sim (\Omega/N)^2 $$ is not correct for the regime $Pe Ω -dependence is: $$ D_{\chi^{-1}}\sim (\Omega/N)^4. $$