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\begin{center}
{\bf \LARGE 
Documentation for LMDZ, Planets version

\vspace{1cm}
\Large
The upper boundary sponge layer
}

\vspace{1cm}
S\'ebastien Lebonnois , Ehouarn Millour

\vspace{1cm}
Latest version: \today
\end{center}

\section{Theoretical aspects}
Because of the inevitable numerical boundary at the top of the model,
upward travelling waves are found to non-physically reflect down into the
atmosphere.
A common remedy to this unwanted behaviour is to apply a sponge layer near
the top of the model in order to quench these waves and avoid significant
reflection thereof.
In practice such quenching is done by adding a dissipative term which forces
a relaxation of potential temperature and/or winds of the form:
\[
 A(t)=A_m+A_0 \exp(-\lambda t )
\]
Where $A_m$ is the value towards which $A$ is to asymptotically relax, and
$\lambda$ is the inverse of the characteristic relaxation time scale.
As there is no obvious value of $A_m$ towards which to relax, in practice
it is often chosen to be either the zonal average of $A$ (evaluated at time $t$,
i.e. conveniently ignoring that $A_m$ then is in fact not time-independent),
or zero (at least for winds, since this would have little physical meaning for
potential temperature).

\section{Pratical aspects in the code}

%The sponge layer is applied at the upper boundary when the \textsf{ok\_strato}
%flag is set to {\em True} in \textsf{gcm.def} 
%(this parameter also controls the application of a second step in the 
%determination of vertical variation of coefficients for
%the horizontal dissipation, see \textsf{inidissip.F} and
%\textsf{disspi\_horiz.pdf} document).

The tendencies for the upper boundary sponge layer are computed separately in
the \textsf{top\_bound.F} routine (called from \textsf{leapfrog.F}) and
added in place. 
The resulting sponge tendency \textsf{dutop}, in m/s, is also given as an output for
diagnostics.

Three parameters may be adjusted in the \textsf{gcm.def} file: 
\begin{itemize}
\item \textsf{iflag\_top\_bound}: selects the affected layers. 
  \begin{itemize}
  \item 1: only the top 4 layers are affected. In this case, the damping rate 
  is divided by 2 in the second layer, 4 in the third and 8 in the fourth. 
  \item 2: layers with pressure lower than 100 times the top pressure. 
  In this case, the damping rate depends linearly on the pressure.
  \end{itemize}
\item \textsf{mode\_top\_bound}: selects how the fields are affected.
  \begin{itemize}
  \item 0: No sponge layer is applied.
  \item 1: Zonal and meridional winds are damped to zero.
  \item 2: Zonal and meridional winds are damped to their zonally averaged value.
  \item 3: Temperature, zonal and meridional winds are damped to their zonally 
  averaged value.
  \end{itemize}
\item \textsf{tau\_top\_bound}: damping rate ($\lambda$ in equation above,
expressed in Hz) in the topmost layer.
\end{itemize}

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