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\AtBeginDocument{{\noindent\small 
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 109--116.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}
\setcounter{page}{109}


\begin{document} 

\title[\hfilneg EJDE/Conf/11 \hfil Shallow water problem]
{On a problem of shallow water type} 

\author[M. ElAlaoui Talibi \& M. H. Tber\hfil EJDE/Conf/11 \hfilneg]
{Mohamed ElAlaoui Talibi, Moulay Hicham Tber}  % in alphabetical order

\address{Departement de Math\'ematiques, 
Facult\'e des Sciences Semlalia, Maroc}
\email[M. ElAlaoui]{elalaoui@ucam.ac.ma}
\email[M. H. Tber]{tber.hicham@eudoramail.com}

\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{35Q30, 76D03, 76B15, 76M10, 65M25, 65M60} 

\keywords{Shallow water equations; friction in the bottom; existence result;
\hfill\break\indent
Galerkin method; finite element method; characteristics method} 

\begin{abstract}
 In this paper we present an existence theorem for a problem of
 shallow water kind. We take into account a general friction
 term depending on water depth and the norm of velocity, which is
 the main difficulty. We present also a numerical study in the case
 which we consider the above problem as a perturbation of shallow
 water equations in the non conservative dept-mean velocity form.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and setting of the problem}

The two-dimensional shallow water equations (briefly SWE)
are deduced by integrating, with respect to depth, the continuity
and  the momentum equations of the three-dimensional
incompressible Navier-Stokes system, neglecting the influence of
the vertical component of acceleration, the pressure is then
supposed hydrostatic \cite{Bernardi-Pironneau}. They provide a
model allowing to describe the flows of water in domains
characterized by small ratio between vertical and horizontal
length scales, therefore typical physical situations modelled are:
tidal waves, currents in portual basins, lagoon, ..etc. But their
use is surprisingly extended to very different phenomena even with
discontinuous behavior, like the "dam break"
problem \cite{Hervouet}.

The shallow-water system we are studying in this work reads
\begin{gather}
\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot
\mathbf{\nabla })\mathbf{u-}\nu_1 \mathbf{ \mathbf{\Delta u}+}C(h
)\left|\mathbf{u}\right| \mathbf{u}+\mathbf{l}\times
\mathbf{u}+g\mathbf{\nabla }h = \mathbf{f} \quad\text{on }\Omega ,
\label{e1} \\
\frac{\partial h} {\partial t}-\nu_2\triangle \mathbf{h} + \nabla
\cdot(h \mathbf{u})=f \quad\text{on }\Omega , \label{e2}
\end{gather}
where $\mathbf{u=(}u_1,u_2)^{\perp }$ is the velocity vector and
$h$ is the depth of studied layer, it can be considered as sum of
the bottom topography which is given and the topography of the
free surface. $\Omega \in \mathbb{R}^2$ is the projection of the
domain of the study on the horizontal plane. $\Gamma$ denotes its
boundary. $\mathbf{l}$ is the Coriolis force defined by
$(0,0,2\omega sin(\phi))$, where $\omega$ is the rotation rate of
the earth and $\phi$ the latitude. g denotes the acceleration of
the gravity. The bottom friction effect is presented by the term
$C(h)\left|\mathbf{u}\right| \mathbf{u}$ where $C(.)$ is a
continuous function satisfying the condition 
$0\leq C(.)<\overline{\varepsilon}$ which physically justified by the
Manning-Strickler's formula and by the Chezy's one if the free
surface elevation remain larger than minimal level. $\nu_1$,
$\nu_2$ are respectively the eddy viscosity and diffusivity
coefficients which we consider as an artificial viscosity taken,
numerically, equal to zero to have the shallow-water equations in
the nonconservative depth-mean velocity form. The right-hand side
terms $\mathbf{f}$ and $f$ represent, respectively, the outside
stress and the fluid
exchanges (rain, evaporation, etc.).

To solve these equations we take homogeneous boundary
conditions and we set the initial data as
\begin{gather*}
(\mathbf{u},h)=(0,0) \quad\text{on } \Gamma,\\
(\mathbf{u},h)(t=0)=(\mathbf{u}_0,h_0) \quad\text{in } \Omega
\end{gather*}

\begin{remark} \label{rmk1} \rm
To be compatible with the physical situation for which the
friction formulation is justified, we assume that
$h=h_B\geq h_{min}>0$ on $\Gamma$ and $h(0)\geq h_{min}>0$ in
$\Omega$. However by setting $h:=h+h_L$ where $h_L$ is the
solution of the problem
\begin{gather*}
\frac{\partial h_L}{\partial t}-\nu_2\triangle h_L =0 \quad\text{in } \Omega \\
h_L(0)= 0 \quad\text{in }\Omega \\
h_L=h_B \quad\text{on } \Gamma\,.
\end{gather*}
(As shown in \cite{Zuazua}, this problem has a solution in
$L^2(0,T,H^1(\Omega))\cap L^{\infty}(0,T,L^{\infty}(\Omega))$ for
$h_B\in L^2(0,T,H^{\frac{1}{2}}(\Gamma))\cap
L^{\infty}(0,T,L^{\infty}(\Gamma))$) we find again the homogeneous
boundary conditions modulo a constant in the momentum equation and
a linear term in the continuity one changing quit the reasoning
done below. Therefore we will consider, for convenience, only the
homogeneous case.
\end{remark}


\section{Notation and variational formulation}

We introduce the following functional spaces:
$V_1=(H^1_0(\Omega))^2$ ,$H_1=(L^2(\Omega))^2$,
$V_2=H_0^1(\Omega)$, $H_2=L^2(\Omega)$, $V=V_1\times{V_2}$,
$H=H_1\times{H_2}$. 
The norm and semi-norm defined on $H^1(\Omega)$ are equivalent in
$V_1$, $V_2$, and $V$.\\
Then we set $\|\mathbf{u}\|=\|\mathbf{u}\|_{V_1}$,
$\|h\|=\|h\|_{V_2}$ and $\|X\|=\|X\|_V$ for $\mathbf{u}\in{V_1}$,
$h \in{V_2}$, and $X\in{V}$. $|\cdot|$ denotes the norm in
$L^2(\Omega)$, $|\cdot |_2$ denotes the Euclidean norm in
$\mathbb{R}^2$, $(\cdot,\cdot)$ is the scalar product in $H_1$,
$H_2$ or $H$ and $(\cdot,\cdot)_2$ the scalar product in
$\mathbb{R}^2$.
We define
\begin{gather*}
a_1(\mathbf{u},\mathbf{v})=\nu_1(\nabla\mathbf{u},\nabla\mathbf{v}),\\
a_2(h,\beta)=\nu_2(\nabla h,\nabla \beta),\\
a(X,Y)=a_1(\mathbf{u},\mathbf{v})+a_2(h,\beta)
\end{gather*}
 with
$X=(\mathbf{u},h)$ and $Y=(\mathbf{v},\beta)$. Note that
 $a_1$, $a_2$ and $a$ are bilinear continuous coercive forms,
respectively, on $V_1$, $V_2$ and $V$.

 We denote by
$\overline{\varepsilon}$, $\nu$, $A$, $B$, $C$, $\lambda$ and
$\theta$ constants such that:
\begin{gather*} %3
0 \leq C(.) \leq \overline{\varepsilon} \\
a(X,Y)\geq \nu \|X\|\|Y\| \text{ for } (X,Y)\in V \times V\\
\lambda > 0, \quad C_g=\mbox{constant}\cdot g, \quad B=2\nu-C_g-\lambda,
\end{gather*}
and $C=\mbox{constant}\cdot C_G$ where $C_G$ is the best constant of the
Gagliardo-Nirenberg inequality \cite{Brezis}:
\begin{equation}
\|\mathbf{u}\|^2_{L^4(\Omega)^2}\leq C_G \|\mathbf{u}\|
|\mathbf{u}|.
\end{equation}
In what follows we take homogeneous boundary conditions and we
write
 $$
 (\mathbf{u}.\nabla)\mathbf{u}=\frac{1}{2}\mathop{\rm grad}(|\mathbf{u}|_2^2)
 +\mathop{\rm curl}(\mathbf{u})\alpha({u})
 $$
where
$\mathop{\rm curl}(\mathbf{u})=\frac{\partial{u_2}}{\partial{x_1}}
-\frac{\partial{u_1}}{\partial{x_2}}$
and $\alpha(\mathbf{u})=(-u_2,u_1)$.
Now we can set the weak formulation  of the problem:
\begin{itemize}
\item[($\mathcal{V}$)]
 Find $(\mathbf{u},h)\in L^2(0,T,V)\cap L^{\infty}(0,T,H)$ such
 that
\begin{gather}
\begin{aligned}
\big(\frac{\partial{\mathbf{u}}}{\partial{t}},\mathbf{v}\big)
+a_1(\mathbf{u},\mathbf{v})
+(curl(\mathbf{u})\alpha(\mathbf{u}),\mathbf{v})+\frac{1}{2}(\mathop{\rm grad}|u|^2_2,\mathbf{v})&
\\
+(C(h)|\mathbf{u}|_2\mathbf{u},\mathbf{v})+
(\mathbf{l}\land\mathbf{u},\mathbf{v})-g(\mathop{\rm div}(\mathbf{v}),h)&=(\mathbf{f},\mathbf{v})
\end{aligned} \label{VF}\\
\big(\frac{\partial{h}}{\partial{t}},\beta\big)
+a_2(h,\beta)+(\mathop{\rm div}(h\mathbf{u}),\beta)=(f,\beta) \quad
\forall(\mathbf{v},\beta)\in V\,, \\
(\mathbf{u},h)(t=0)=(\mathbf{u_0},h_0).
\end{gather}
\end{itemize}

\section{Existence theorem}

\begin{theorem} \label{thm1}
Assume that $\mathbf{F}=(\mathbf{f},f)\in L^2(0,T,H),
 X_0=(\mathbf{u_0},h_0)\in V\cap L^{\infty}(\Omega)^3$.
Also assume the following conditions are satisfied,
\begin{enumerate}
  \item $B=2\nu-C_g-\lambda>0$
  \item $|X_0|< B/C$
  \item $(B/C)^2> |X_0|^2+\frac{1}{\lambda}|\mathbf{F}|$
\end{enumerate}
where constants are defined above.
Then the variational problem ($\mathcal{V}$) admits at last one
solution $(\mathbf{u},h)$ in $L^2(0,T,V)\cap L^{\infty}(0,T,H)$.
\end{theorem}

The proof of the theorem is based on the three next lemmas.

\begin{lemma} \label{lem1}
Let $X=(\mathbf{u},h)$ be a classic solution of the problem ($V$),
on $[0,T]$. Under the same hypothesis in the theorem, we have
\begin{gather*}
\|X\|_{L^{\infty}(0,T,H)}+(B-C\|X\|_{L^{\infty}(0,T,H)})\|X\|_{L^2(0,T,V)}\leq
\frac{1}{\lambda}\|\mathbf{F}\|_{L^2(0,T,H)}+ \left|X_0\right| \\
(B-C\|X\|_{L^{\infty}(0,T,H)})>0. \\
\|X\|_{L^{\infty}(0,T,H)}+\|X\|_{L^2(0,T,V)}\leq \mbox{constant}
\end{gather*}
\end{lemma}

\begin{proof}
By writing the energy inequality and using the hypothesis above, we
find the result via Green's formula and Gagliardo-Neirenberg
inequality.
\end{proof}

\begin{lemma} \label{lem2}
Let $(X_n)$ be a sequence of classic solution of ($V$) on $[0,T]$
satisfying
\begin{equation}\label{e1l2}
\|X_n\|_{L^{\infty}(0,T,H)}+\|X_n\|_{L^2(0,T,V)}\leq C'.
\end{equation}
where $C'$ is a constant independent of $n$. Then there
exist a subsequence also denoted by $X_n$ and $X=(\mathbf{u},h)\in
L^{\infty}(0,T,H)\cap L^2(0,T,V)$ such that
\begin{gather}\label{e2l2}
  X_n \longrightarrow X \quad\text{weakly  in } L^\infty (0,T,H), \\
\label{e3l2}
  X_n \longrightarrow X \quad\text{weakly in } L^2(0,T,V), \\
\label{e4l2}
  X_n \longrightarrow X \quad\text{strongly in } L^2(0,T,H).
\end{gather}
\end{lemma}

\begin{proof}
Statements (\ref{e2l2}) and (\ref{e3l2}) are immediate consequences of
(\ref{e1l2}).
On the other hand we can show that the sequence $X_n$ is uniformly
bounded in the set 
$$
Y=\big\{{\mathbf{v}\in L^2(0,t,V),\frac{\partial
{\mathbf{v}}}{\partial{t}}\in L^1(0,T,V') }\big\}.
$$
According to \cite{Temam}, the injection of $Y$ into $L^2(0,T,H)$
is compact. Then we can extract from $X_n$ a subsequence also
denoted by $X_n$ such that we have $(\ref{e4l2})$
\end{proof}

\begin{lemma} \label{lem3}
let $(\mathbf{u_n},h_n)$ be a sequence converging toward $(u,h)$
in $L^2(0,T,H)$ strongly and $L^2(0,T,V)$ weakly. Then for any
$\varphi(t)\in \mathcal{C}^1(0,T)$ and $(\mathbf{v},\beta)\in
V\cap L^{\infty}(\Omega)^3$ we have
\begin{gather*}
\int^T_0
(C(h_n)|\mathbf{u_n}|\mathbf{u_n},\varphi(t)\mathbf{v})dt\longrightarrow
\int^T_0 (C(h)|u|u,\varphi(t)\mathbf{v})dt\,,
\\
\int_0^T
(\mathop{\rm div}(h_m\mathbf{u_n}),\varphi(t)\beta)dt\longrightarrow\int_0^T
(\mathop{\rm div}(h\mathbf{u}),\varphi(t)\beta)dt\,,
\\
\int^T_0
(curl(\mathbf{u_n})\alpha(\mathbf{u_n}),\varphi(t)\mathbf{v})dt\longrightarrow
\int^T_0 (curl(\mathbf{u})\alpha({u}),\varphi(t)\mathbf{v})dt\,,
\\
\frac{1}{2}\int^T_0
(\mathop{\rm grad}|u|^2_2,\varphi(t)\mathbf{v})dt\longrightarrow
\frac{1}{2}\int^T_0 (\mathop{\rm grad}|u|^2_2,\varphi(t)\mathbf{v})dt\,.
\end{gather*}
\end{lemma}

We can proof this lemma  using Shwartz inequality and
appropriated Sobolev injections.


\begin{proof}[Proof of the Theorem \ref{thm1}]
 The proof is based on the
construction of sequence of finite dimensional Problems 
$(\mathcal{V}_n)$ of which the solutions $(X_n)$ 
(by using lemmas \ref{lem1} and \ref{lem2}) converge strongly in $H$ 
and weakly in $V$ to $X\in(\mathbf{u},h)\in L^2(0,T,V)\cap L^{\infty}(0,T,H)$. 
Then by
third Lemma we can show that $X$ is a solution of the problem.
\end{proof}

\section{Numerical studies}

The goal of this numerical studies  is to know how
the solution of the problem varies when  the
included artificial diffusivity coefficients $\nu_2$ tend to zero. 
The approach we are using here is based on the finite
elements for the space discretization and on the discretization of
the Lagrangian derivative along the characteristics. This method
provides a centred scheme which have the advantage of stabilizing
the convection and allow large time steps to be taken when
compared to standard time-stepping methods
\cite{Dawson}.

 Similar  numerical schemes were considered in
\cite{Pironneau} for the incompressible Navier-Stokes problem.
Within the framework of the shallow water problems, this approach
combined with the method of the fractional steps, is adopted in
\cite{Goutal} to simulate transcritical flows, and applied later
in TELEMAC project \cite{Telemac}.

\subsection*{Temporal discretization}
The  characteristic methods consists in approaching the lagrangian
derivative of a function $S$ in time step $t^{n+1}$ by:
\begin{equation}\label{carac}
\frac{dS}{dt}(\mathbf{x},t^{n+1})\simeq
\frac{S(\mathbf{x},t^{n+1})-S(\mathbf{X}(\mathbf{x},t^{n+1};t^{n}),t^{n})}
{\Delta t}
\end{equation}
where $\mathbf{X}^{n}=\mathbf{X}(\mathbf{x},t^{n+1};t^{n})$ is the
position in the time step $t^{n}$ of the particle positioning at
the geometrical  point $\mathbf{x}$ in time step $t^{n+1}$ and
$\mathbf{X}^{n}(\mathbf{x},t^{n+1};\tau )$ is the solution of
\begin{gather*}
\frac{d\mathbf{X}^{n}}{d\tau }(\mathbf{x},t^{n+1};\tau ) 
 = 
\mathbf{u}^{n}( \mathbf{X}^{n}(\mathbf{x},t^{n+1};\tau )), \quad
\text{for }t^{n}\leq \tau \leq t^{n+1}, \\
\mathbf{X}^{n}(\mathbf{x},t^{n+1};t^{n+1})  =  \mathbf{x}. 
\end{gather*}
Using (\ref{carac}), the semi-implicit time discretization of
 (\ref{e1}), (\ref{e2}) is 
\begin{gather}
\begin{gathered}
\frac{\mathbf{u}^{n+1}-\mathbf{u}^{n}\circ \mathbf{X}^{n}}{\Delta
t}-\nu_1 \mathbf{\Delta u}^{n+1},\\
C(h^{n})\left| \mathbf{u}^{n}\right| \mathbf{u}^{n+1}
+\mathbf{l}\times \mathbf{u}^{n}+g\mathbf{\nabla }h^{n+1} \,, = 
\mathbf{f}^n
\end{gathered} \label{etd1} \\
\frac{h^{n+1}-h ^{n}\circ \mathbf{X}^{n}}{\Delta t}-\nu_2
\mathbf{\Delta u}^{n+1}+ h^{n}\mathbf{\nabla}\cdot\mathbf{u}^{n+1}  =  f^n, 
\label{etd2}
\end{gather}
where $h^{n}$ and $\mathbf{u}^{n}$ are the approximations of $h$
and $\mathbf{u}$ respectively in time step $t^{n}$.

\subsection*{Variational formulation}
let us introduce the spaces
\begin{gather*}
V^1_{\mathbf{\phi}}=\left\{\mathbf{v}\in H^1(\Omega)\times
H^1(\Omega); \mathbf{v}=\mathbf{\phi} \quad\text{on } \Gamma \right\} \\
V^2_{\eta}=\left\{h \in H^1(\Omega);h=\eta \quad\text{on } \Gamma\right\}\,.
\end{gather*}
Multiplying (\ref{etd1}) and (\ref{etd2}) by $\mathbf{v}\in
V_1$ and $q\in V_2$ respectively, and integrating by part on
$\Omega$ we obtain
\begin{gather}
\begin{aligned}
&\big(\frac{\mathbf{u}^{n+1}}{\Delta t},\mathbf{v}\big)
+\nu_1\left (\mathbf{\nabla u}^{n+1},\mathbf{\nabla v}\right)
+\left(C(h^n)\left| \mathbf{u}^{n}\right| \mathbf{u}^{n+1},\mathbf{v}\right)
-g\left(h ^{n+1},\nabla\cdot\mathbf{v}\right) \\
&= \big(\frac{\mathbf{u} ^{n}\circ \mathbf{X}^{n}}{\Delta
t}+\mathbf{f}^{n}-\mathbf{l}\times
\mathbf{u}^{n},\mathbf{v}\big),
\end{aligned} \label{e1fvd} 
\\
\big( \frac{h ^{n+1}}{\Delta t},q\big)
+\nu_1\left(\mathbf{\nabla u}^{n+1},\mathbf{\nabla v}\right)
+\left(h^{n}\nabla\cdot\mathbf{u}^{n+1},q\right)
=\big(f^n+\frac{h^{n}\circ \mathbf{X}^{n}}{\Delta t},q\big)\,.
\label{e2fvd}
\end{gather}
Then we write the time-discretized variational formulation as follows:
\begin{itemize}
\item[$\mathcal{(V)}^n$] 
Find  $(\mathbf{u}^{n+1},h^{n+1})$ in 
$V^1_{\mathbf{\phi}}\times V^2_{\eta}$  such that
\begin{gather*}
e(\mathbf{u}^{n+1},\mathbf{v})+b(\mathbf{v},h^{n+1})  = 
\left( \mathbf{f^n,v}\right), \quad \forall \mathbf{v}\in\mathbf{V_1}\,, \\
-b(\mathbf{u^{n+1}},h^n q) + e'(h^{n+1},q)+  =  (f^n,q)\,,  \forall q\in V_2\,.
\end{gather*}
where
\begin{gather*}
e(\mathbf{u},\mathbf{v})=\frac{1}{g\triangle t
}(\mathbf{u},\mathbf{v}) +\frac{\nu_1}{g} (\nabla
\mathbf{u},\nabla\mathbf{v})+\frac{1}{g}\left(C(h^n)\left|
\mathbf{u}^{n}\right| \mathbf{u},\mathbf{v}\right)\,,
\\
e'(h,q)=\frac{1}{\triangle t}(h,q) +\nu_2(\nabla h,\nabla q)\,,
\\
b(\mathbf{v},q)=-\left(q,\nabla\cdot\mathbf{v}\right)\,,
\mathbf{f}^{n}:=\frac{1}{g}\left(\mathbf{f}^{n}+\frac{\mathbf{u}
^{n}\circ \mathbf{X}^{n}}{\Delta t}-\mathbf{l}\times
\mathbf{u}^{n}\right)\,,
\\
f^n:=f^n+\frac{h^{n}\circ \mathbf{X}^{n}}{\Delta t}\,.
\end{gather*}
\end{itemize}

\subsection*{Finite element discretization}
Let $V^1_{\mathbf{\phi}h}$ and $V^2_{\eta h}$ (resp $V_{1h}$ and
$V_{2h}$) two finite elements spaces approaching
$V^1_{\mathbf{\phi}}$ and $V^2_{\eta}$ (resp $V_1$ and $V_2$) such
that the LBB condition is satisfied \cite{Brezzi}. Then the
discrete problem is written as
\begin{itemize}
\item[$\mathcal{(V)}_h^n$]
Find $(\mathbf{u}_h^{n+1},h_h^{n+1})$ in 
$V^1_{\mathbf{\phi}h}\times V^2_{\eta h}$ such that
\begin{gather*}
 e(\mathbf{u}_h^{n+1},\mathbf{v}_h)+b(\mathbf{v},h_h^{n+1}) 
=  \left(\mathbf{f}^n_h,\mathbf{v}_h\right), \quad \forall \mathbf{v_h}\in{V_{1h}}, \\
-b(\mathbf{u}^{n+1}_h,h^n_h q_h) + e'(h^{n+1}_h,q_h)+  =
 (f^n_h,q_h), \quad \forall q_h\in V_{2h}.
\end{gather*}
\end{itemize}
The value $X^m_h(x)$ is approximated by $X((n+1)\triangle t,x)$, the solution 
of the problem
$$
\frac{dX}{d\tau}=\mathbf{u}^n_h(X(\tau),\tau), \quad
X((n+1)\triangle \tau)=x,
$$
therefore, at each time step we have to solve the  linear
system
\[
\begin{pmatrix}
A & B \\
-\overline{B}^{\top } & -D
\end{pmatrix}
\begin{pmatrix}
\mathbf{U} \\
H
\end{pmatrix}
= \begin{pmatrix}
\mathbf{F}_{U} \\
F_{H}
\end{pmatrix}
\]
where $A$ and $D$ are two definite positive matrices, and $B$,
$-\overline{B}^{\top}$ are two matrices approaching operator of
divergence type.

 We can show easily (see for example \cite{Pironneau1}
\cite{IASTED}) that the problem $\mathcal{(V)}^n$ (resp
$\mathcal{(V)}_h^n$ ) is well posed if $h^n$ (resp $h^n_h$) remain
larger than one level $\xi>0$. Moreover in \cite{TAMTAM} and
\cite{Goutal}, a preconditionner of Cahouet-Chabard kind
\cite{Cahouet} are proposed for the linear system.

\subsection*{Numerical results}
The studied domain is a square with $1km$ in length with mean
water elevation of $1m$. We suppose there is no exchange with the
external medium and the surface stress is reduced to the wind
stress tensor defined by
$$
\mathbf{f}_{wind}=\frac{1}{h}\frac{\rho_{water}}{\rho_{air}}a_{wind}
|\mathbf{u}_{\rm wind}|_2\mathbf{u}_{wind},
$$
where ${\rho_{water}}$, ${\rho_{air}}$ are the density of the
water and the air respectively, and $a_{wind}$ is an adimensional
empiric coefficient.
On the other hand, if we choose the Manning-Strickler's formula
for the bottom friction we obtain
\begin{equation}\label{Manning}
  C(h)=\frac{gn^2}{h^\frac{4}{3}}
\end{equation}
where $n$ is the Manning coefficient.

\begin{table}[htb]
\caption{Physical parameters}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}\hline 
$g(m/s^{-2})$ & $\rho _{water}(kg/m^{3})$ & $\rho_{air}(kg/m^{3})$ & $n$ &
$\nu _{2}(m^{2}/s)$ \\
\hline 
1 & 999.00 & 1.225 & 0.03 & 0.1  \\
\hline
\end{tabular}
\\ \quad \\
\begin{tabular}{|l|l|l|l|l|} \hline 
 $w$ (rad/s)& $\phi$ (°)& $\nu _{1}(m^{2}/s)$ & $\Delta t(s)$ &$a_{wind}$ \\
\hline 
 0-100 & 0.1& $0.56510^{-3}$& $7.29210^{-10}$ & 45°
\\ \hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1}
\end{center}
 \caption{The section h(x,250m) for different values of $\nu2$}
\label{VarNu2}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
   \caption{Water elevation for $\nu2=0$ in $t=10s$}
    \label{H3D}
\end{figure}    
    
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3}
\end{center}
\caption{Velocity field for $\nu2=0$ in $t=10s$}
    \label{U3D}
\end{figure}

\subsection*{Conclusions}
Although  the continuous problem (\ref{VF}) requires a condition
on it, we can take the diffusion coefficient of continuity
equation $\nu_2$ numerically  as small as we want, without any
explosion of the solution (see figure 1). Then for $\nu_2=0$ and
$f=0$ we find the shallow water equations established in
\cite{Bernardi-Pironneau}. Moreover for this case we can prove
formally by characteristics that the free surface elevation remain
larger than minimal level if the initial one it is. Therefore the
choice of Manning-Strickler's formula (\ref{Manning}) is justified
and the numerical results are satisfactory.

\subsection*{Acknowledgement}
This work was supported by  the Projet WADI (5th PCRD - Programmes IST 
et INCO-MED)  and Action integr\'{e}e CMIFM AI MA/04/94. 

\begin{thebibliography}{00}

\bibitem{Bernardi-Pironneau} C. Bernardi and
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\end{document}