
\def\V{{\bf v}}
\def\ave#1{\overline{#1}}
\def\mave#1{\tilde{#1}}
\def\q{c}
\def\qml{{\q}_{\mbox{\tiny ML}}}
\def\zi{z_i}

\begin{frame}
 \frametitle{Transilient matrices}



Numerical formalism (after Stull 1984)

$C$ : Air mass exchange rate matrices between model layers 
\medskip

{\tiny
{\bf For turbulent diffusions}
\begin{equation}\nonumber
\der{\q_l}{t}=\der{}{z}\dep{K_z\der{\q}{z}}\simeq 
\frac{
 K_{l+\dem}\dep{\q_{l+1}-\q_{l}   }
-K_{l-\dem}\dep{\q_{l}  -\q_{l-1} }
}{{\delta z}^2}
\end{equation}
\begin{equation}\nonumber
\implies C_{l,l+1}=K_{l+\dem}\frac{\delta t}{{\delta z}^2}, C_{l,l}=-(K_{l-\dem}+K_{l-\dem})\frac{\delta t}{{\delta z}^2},  C_{l,m}=0 \mbox{ for } |l-m|> 1
\end{equation}
}

\centerline{\includegraphics[width=\textwidth]{/home/hourdin/PRESENTATIONS/IMAGES/PBL/PDF/acm_en.pdf}}


\end{frame}
