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d D 1.2 92/09/02 15:18:57 hourdin 2 1
c sauvegarde
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d D 1.1 91/11/29 19:51:24 hourdin 1 0
c date and time created 91/11/29 19:51:24 by hourdin
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\subsection{Conservation properties}
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\chapter[Conservation du moment cin\'etique]
{Conservation du moment cin\'etique dans le Mod\`ele de
Circulation G\'en\'erale du LMD
}
\def\localdir{\these/conserv}
\begin{center}
Note interne LMD $n^o175$
\end{center}
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\input{\localdir/intro}
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\def\psk{P_s^\kappa}
\def\ucov{\tilde{u}}
\def\vcov{\tilde{v}}
\def\ucont{\tilde{\ucov}}
\def\vcont{\tilde{\vcov}}
\def\cu{c_u}
\def\cv{c_v}
\def\h{\theta}
\def\pext{\tilde{p}_s}
\def\fext{{\em f}}
\def\dx{\delta_X}
\def\dy{\delta_Y}
\def\dz{\delta_Z}
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\input{\localdir/eqorig}
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Horizontally, equations are projected on a cylindrical $(x,y)$
coordinates system in which the  variables are distributed on
a rectangular staggered, Arakawa's C-grid \cite{Arak:77}.
For the vertical we used the sigma coordinates
which is the pressure normalized by its surface value ($\sigma=p/p_s$).
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\input{\localdir/erreur}
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The independant varibles of the model are the $x$ and $y$ component
of the horizontal velocity $u$ and $v$, the potential hentalpie
($\h = c_p T/\psk$)
and the surface pressure $p_s$.
Equations are derived using the covariant forms of
the horizontal velocities
($\ucov$ and $\vcov$)
and math flux components

\begin{equation}
U=\dz \av{p}{X} \ucont ,\  V=\dz \av{p}{Y} \vcont \  and \  W
\end{equation}

were $\ucont $ and $\vcont$ are the contravariant forms of the
velocity components.
The covariant and contravariants component are given by

\begin{equation}
\begin{array}{llllllllll}
\ucov = \cu u & and & \ucont = u / \cu & with & 
\cu = a \cos{\phi} \frac{d\lambda}{dX} \\
\vcov = \cv v & and & \vcont = v / \cv & with &
\cv = a \frac{d\phi}{dY}
\end{array}
\end{equation}

The two horizontal components of the equation of motion are derived
in the following way:

\begin{equation} \label{eq:u1}
\dt{\ucov} =
\av{Z}{Y} \av{V}{X,Y} 
- \dx \left( \Phi + E_{cin} \right)
-s \av{\h}{X} \dx \psk
- \frac{\av{\ucov}{Y,Y} \dz \av{W}{X} }
{\av{\pext}{X} \dz{\sigma}}
-\frac{\dz \left( \av{W}{X} \av{\ucov}{Z} \right) }
{\av{\pext}{X} \dz{\sigma}}
\end{equation}

\begin{equation} \label{eq:v1}
\dt{\vcov} =
\av{Z}{X} \av{V}{X,Y} 
- \dy \left( \Phi + K \right)
-s \av{\h}{Y} \dy \psk
- \frac{\av{\vcov}{X,X} \dz \av{W}{Y} }
{\av{\pext}{X} \dz{\sigma}}
-\frac{\dz \left( \av{W}{Y} \av{\ucov}{Z} \right) }
{\av{\pext}{X} \dz{\sigma}}
\end{equation}

where $Z$ is the absolute vorticity

\begin{equation}
Z=\frac{\dx \vcov - \dy \ucov + \fext}{\av{\pext}{X,Y}}
\end{equation}

and K the kinetic energy derived as

\begin{equation}
K= \frac{1}{2}
\left( \av{\ucov \ucont}{X} + \av{\vcov \vcont}{Y} \right)
\end{equation}

$\fext$ is the extensive coriolis factor (i.e. Coriolis factor
multiplied by the local area).
$\Phi$ is the geopotential which is derived from the temperature field
using the hydrostatic equation.

The continuity equation reduces to

\begin{equation}
\left\{
\begin{array}{lll}
\dz \Omega & = & \dz \sigma \left( \dx U + \dy V \right) \\
W & = & \Omega - \sigma \Omega_s \\
\dt{\pext} & = & \Omega_s
\end{array}
\right.
\end{equation}

The particular form of the horizontal advective terms
in equations~\ref{eq:u1} and~\ref{eq:v1}
in which the rotationnal and divergent parts are separated
was used to insure exact conservation of potential enstrophy
for divergent barotropic flow (see papers by Sadourny
\cite{Sado:75,Sado:75a} for details of the derivation).
It was found at this time that the conservation of
enstrophy reinforced geatly the model stability.
It is also more and more clear than enstrophy conservation 
is of first importance for the representation of the
midlatitude motion at least and that the enstrophy budget
is a good diagnostic....


The rather complicated form of the two last terms
of the same equations which acounts for the
vertical advection where derived by the autors
conserve exactly angular momentum in particular conditions which
will be presented after.
It had been found, in fact, that the old formulation 
in which the advection term of angular momentum was taken in a
classical flux divergence form:

\begin{equation}
\left( \dt{\ucov} \right)_{vert. adv.} =
- \frac{\av{ \av{W}{X} \dz \ucov}{Z}}{\av{\pext}{X}\dz \sigma}
\end{equation}

could lead to systematic errors on the angluar momentum budget
tending to reinforce angular momentum in low latitude
regions especially in divergent meridionnal circulation
such as occure in the upper part of the ascendin motions of the
Haddley circulation. Even if not too strong, this
nn-conservation produced some
systematic errors on the mean zonal wind field
even in terrestrial condition which have disapeared with
the new formulation. 
In the case of the study of superrotation, the exact conservation
of angular momentum would be of course of first importance
to get some interresting results.
As we will show now, this new formulation insures an
exact conservation of angular momentum  for zonallly
axi-symetric stationnary flow.

For this we have to derive the evolution of angular momentum
due to the horizontal advection as represented in equation~\ref{eq:u1}.
The angular momentum

\begin{equation}
M=a \cos{\phi} \left( u + a \Omega_P \cos{\phi} \right)
\end{equation}

becomes with our notations

\begin{equation}
M=\frac{dX}{d\lambda} \left( \ucov +  \cu a \Omega_P \cos{\phi} \right)
\end{equation}

The horizontal advection of angular momentum as produced
by the numerical scheme is given by:

\begin{equation}
\frac{d\lambda}{dX} \dt{M}=
\end{equation}
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\input{\localdir/newdyn}
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