Critical Richardson numbers and gravity waves

Free access article

Issue		A&A Volume 384, Number 3, March IV 2002


Page(s)		1119 - 1123
Section		Physical and chemical processes
DOI		http://dx.doi.org/10.1051/0004-6361:20011773

A&A 384, 1119-1123 (2002)
DOI: 10.1051/0004-6361:20011773

Critical Richardson numbers and gravity waves

V. M. Canuto

NASA, Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA Dept. of Applied Physics and Mathematics, Columbia University, New York, NY 10027, USA

(Received 23 July 2001 / Accepted 7 December 2001 )

Abstract
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:

$\begin{displaymath}Ri {\rm (cr)} = Ri^\ell {\rm (cr)} = 1/4. \end{displaymath}$

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):

$\begin{displaymath}Ri {\rm (cr)} = Ri^{\rm t} {\rm (cr)} \approx 4 Ri^\ell {\rm (cr)}\approx 1. \end{displaymath}$

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):

$\begin{displaymath}Ri {\rm (cr)} = Ri^{\rm r} {\rm (cr)} \sim (1+Pe)Pe^{-1}Ri^{\rm t} {\rm (cr)} \end{displaymath}$

which, for Pe<1, allows turbulence to survive far longer than (2). Finally, turbulent convection generates gravity waves that propagate into the radiative region and act as an additional source of energy. This further changes Eq. (3) to (gw stands for gravity waves):

$\begin{displaymath}Ri {\rm (cr)} = Ri^{\rm gw} {\rm (cr)} =Ri^{\rm r} (1+\eta_{\rm gw}) \end{displaymath}$

where $\eta_{\rm gw}>1$ . In conclusion, the successive inclusion of relevant physical processes leads to a chain of increasing values of $Ri {\rm (cr)}$ :

$\begin{displaymath}Ri {\rm (cr)} = Ri^{\ell} {\rm (cr)} \rightarrow Ri^{\rm t} {... ...row Ri^{\rm r} {\rm (cr)}\rightarrow Ri^{\rm gw} {\rm (cr)}. \end{displaymath}$

The second result concerns the dependence of the diffusivity D on $\Omega$ . We show that the commonly used expression

$\begin{displaymath}D_{\chi^{-1}}\sim (\Omega/N)^2 \end{displaymath}$

is not correct for the regime Pe<1 that characterizes a stably stratified regime. The proper $\Omega$ -dependence is:

$\begin{displaymath}D_{\chi^{-1}}\sim (\Omega/N)^4. \end{displaymath}$

Key words: Sun: general -- Sun: interior -- convection -- turbulence

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