
\def\V{{\bf v}}
\def\ave#1{\overline{#1}}
\def\mave#1{\tilde{#1}}
\def\q{c}


\def\P{-\overline{w'u'}\der{u}{z}-\overline{w'v'}\der{v}{z}}
\def\D{\frac{g}{\theta}\overline{w'\theta'}}
%\end{document}
\begin{frame}
 \frametitle{Turbulent diffusivity $K_z$}


\begin{itemize}
\item Prandlt (1925) mixing length : $K_z=l \ave{|w'|}$ or
 $K_z=l^2\frac{\partial ||\V||}{\partial z}$

\item Accounting for static stability (Ex. Louis 1979)
\begin{equation}\label{eq:KzRi}
K_z=f(Ri) l^2\left|\frac{\partial \V}{\partial z}\right|,
\hspace{0.1\textwidth} \mbox{with } Ri=\frac{g}{\theta}
\frac{\frac{\partial \theta}{\partial z}}{\dep{\frac{\partial\V}{\partial z}}^2}
\end{equation}


\item Turbulent kinetic energy 
$\ave{{w'}}^2\simeq e=\frac{1}{2}\depb{\overline{u'^2}+\overline{v'^2}+\overline{w'^2}}$
\begin{equation}\nonumber
\der{e}{t}=\P+\D-\frac{1}{\rho}\der{\ave{w'p'}}{z}-\der{\ave{w'e}}{z}-\epsilon
\end{equation}
Ex : Mellor and Yamada 
$\overline{w'\phi'}=-K_\phi \der{\phi}{z}$ with
 $K_\phi=l \sqrt{2e} S_\phi(Ri)$ 

Note : $\der{e}{t}=0$ (stationarity) $\implies$ $K_z$ of form Eq. \ref{eq:KzRi}
\end{itemize}




\end{frame}
