| 78 |
\subsubsection{Boundary conditions} |
\subsubsection{Boundary conditions} |
| 79 |
|
|
| 80 |
The upper and lower boundary conditions are : |
The upper and lower boundary conditions are : |
| 81 |
\begin{eqnarray*} |
\begin{eqnarray} |
| 82 |
\mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\ |
\mbox{at the top:}\;\;p=0 &&\text{, }\omega =\frac{Dp}{Dt}=0 \\ |
| 83 |
\mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo} |
\mbox{at the surface:}\;\;p=p_{s} &&\text{, }\phi =\phi _{topo}=g~Z_{topo} |
| 84 |
\label{eq:boundary-condition-atmosphere} |
\label{eq:boundary-condition-atmosphere} |
| 85 |
\end{eqnarray*} |
\end{eqnarray} |
| 86 |
In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega |
In $p$-coordinates, the upper boundary acts like a solid boundary ($\omega |
| 87 |
=0 $); in $z$-coordinates and the lower boundary is analogous to a free |
=0 $); in $z$-coordinates and the lower boundary is analogous to a free |
| 88 |
surface ($\phi $ is imposed and $\omega \neq 0$). |
surface ($\phi $ is imposed and $\omega \neq 0$). |
| 95 |
is not dynamically relevant and can therefore be subtracted from the |
is not dynamically relevant and can therefore be subtracted from the |
| 96 |
equations. The equations written in terms of perturbations are obtained by |
equations. The equations written in terms of perturbations are obtained by |
| 97 |
substituting the following definitions into the previous model equations: |
substituting the following definitions into the previous model equations: |
| 98 |
\begin{eqnarray*} |
\begin{eqnarray} |
| 99 |
\theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\ |
\theta &=&\theta _{o}+\theta ^{\prime } \label{eq:atmos-ref-prof-theta} \\ |
| 100 |
\alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha}\\ |
\alpha &=&\alpha _{o}+\alpha ^{\prime } \label{eq:atmos-ref-prof-alpha}\\ |
| 101 |
\phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi} |
\phi &=&\phi _{o}+\phi ^{\prime } \label{eq:atmos-ref-prof-phi} |
| 102 |
\end{eqnarray*} |
\end{eqnarray} |
| 103 |
The reference state (indicated by subscript ``0'') corresponds to |
The reference state (indicated by subscript ``0'') corresponds to |
| 104 |
horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi |
horizontally homogeneous atmosphere at rest ($\theta _{o},\alpha _{o},\phi |
| 105 |
_{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi |
_{o}$) with surface pressure $p_{o}(x,y)$ that satisfies $\phi |
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