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

\begin{frame}

\frametitle{Scale decomposition of the conservation equation}

\begin{tabular}{l|l}
{\bf Conservation equation } & $\V$ : wind field\\
                             & $\q$ : conserved quantity
\end{tabular}


\begin{eqnarray}\nonumber
\mbox{\bf Lagrangian form :} && \frac{d\q}{dt}=0\\\nonumber
\mbox{\bf Advective form :} && \dt{\q}+\V\grad{\q}=0\\\nonumber
\mbox{\bf Flux form :} && \dt{\rho\q}+\div{\rho\V\q}=0
\end{eqnarray}


\begin{tabular}{l|l}
{\bf Scale decomposition} & \\
$\ave{X}$ : "average" or "large scale" variable & $\implies$ $\ave{\V\q}=\ave{\V}\ \ave{\q} + \ave{\V'\q'}$ \\
$X'=X-\ave{X}$ : turbulent fluctuation &\\
\end{tabular}

\begin{equation}\nonumber
\dt{\ave{q}}+{\ave{V}}.\grad\ {\ave{q}}
+\frac{1}{\rho}\div{\ave{\rho\V'\q'}} =0
\end{equation}

\end{frame}
