368
Some of the simplifications made to obtain the one-layer model are not admissible
in the case of stratified water seas, as is the case in the Alboran Sea and the Strait of
Gibraltar. In this zone, at least two different water layers can be distinguished: the
upper layer is the fresher Atlantic water that enters through the Strait of Gibraltar
and exits into the western Mediterranean basin, while the lower layer is the outflowing
saltier Mediterranean water pouring out from the eastern Mediterranean into the Atlantic. Some authors divide the Mediterranean layer into two more different layers (cf.
[14]). In any case, this vertical structure avoids the constant density hypothesis in all
the domain. Instead of considering a full 3D model with variable density, a multi-layer
model can be considered. The model presented in chapter 3 derives from a stratified
tridimensional multilayer flow. In each layer, the motion is governed by the 3D incompressible Navier-Stokes equations with constant density (density differs from one layer
to another). Considering this 3D vertical structure, the same process of vertical integration in each layer is applied, taking into account the usual shallow-water simplifying
hypotheses.
Other shallow-water models applied to the study of the Alboran Sea's dynamics
can be found in [l1J and [15J. These authors propose a two-layer model: an active
upper layer and a deeper inertial layer with infinite depth. They also work with a
non-realistic domain approximation (a rectangular approximation of the Alboran Sea
is considered). Herburn et al. [7J propose a more realistic three-layer model and the
domain approximation is obtained from a real coast geometry. The discretization in
all of these models has been made using finite difference methods. As far as we know,
this is the first finite element multi-layer shallow-water model proposed to simulate the
water exchanges through the Strait of Gibraltar and the Alboran Sea's dynamics. In
chapter 3, some numerical results in both regions are presented.
2 Shallow water equations
In this section, we present a shallow water model. Sections 2.1 and 2.2 show the 3D
primary equations and some simplifications usually made in Oceanography. The simplified equations obtained are vertically integrated in section 2.3, resulting in a 2D system
of Partial Differential Equations: these are known as shallow water or St.Venant equations. The set of equations obtained is a non-linear first-order hyperbolic system of
P.D.E. Therefore, their properties are similar to those of the compressible Euler equations. In 2.4, we briefly discuss these properties in order to introduce some vocabulary
to readers not familiar with Oceanography but who are familiar with the main features
of hyperbolic systems. In this section, we follow Stoker's classical book [13J. This book
may be consulted for further details in the study of 1D shallow water equations. For a
more oceanographic approach, see [8J, [10J, [9J.
The numerical resolution of the equations is presented in 2.5. The algorithm used,
developed at the University of Santiago de Compostela (Spain), is briefly described (see
the chapter of Prof. Bermudez in this volume and the references therein for detail).
Finally, in section 2.6 the boundary conditions allowed by the model are presented.
Some of the simplifications made to obtain the one-layer model are not admissible
in the case of stratified water seas, as is the case in the Alboran Sea and the Strait of
Gibraltar. In this zone, at least two different water layers can be distinguished: the
upper layer is the fresher Atlantic water that enters through the Strait of Gibraltar
and exits into the western Mediterranean basin, while the lower layer is the outflowing
saltier Mediterranean water pouring out from the eastern Mediterranean into the Atlantic. Some authors divide the Mediterranean layer into two more different layers (cf.
[14]). In any case, this vertical structure avoids the constant density hypothesis in all
the domain. Instead of considering a full 3D model with variable density, a multi-layer
model can be considered. The model presented in chapter 3 derives from a stratified
tridimensional multilayer flow. In each layer, the motion is governed by the 3D incompressible Navier-Stokes equations with constant density (density differs from one layer
to another). Considering this 3D vertical structure, the same process of vertical integration in each layer is applied, taking into account the usual shallow-water simplifying
hypotheses.
Other shallow-water models applied to the study of the Alboran Sea's dynamics
can be found in [l1J and [15J. These authors propose a two-layer model: an active
upper layer and a deeper inertial layer with infinite depth. They also work with a
non-realistic domain approximation (a rectangular approximation of the Alboran Sea
is considered). Herburn et al. [7J propose a more realistic three-layer model and the
domain approximation is obtained from a real coast geometry. The discretization in
all of these models has been made using finite difference methods. As far as we know,
this is the first finite element multi-layer shallow-water model proposed to simulate the
water exchanges through the Strait of Gibraltar and the Alboran Sea's dynamics. In
chapter 3, some numerical results in both regions are presented.
2 Shallow water equations
In this section, we present a shallow water model. Sections 2.1 and 2.2 show the 3D
primary equations and some simplifications usually made in Oceanography. The simplified equations obtained are vertically integrated in section 2.3, resulting in a 2D system
of Partial Differential Equations: these are known as shallow water or St.Venant equations. The set of equations obtained is a non-linear first-order hyperbolic system of
P.D.E. Therefore, their properties are similar to those of the compressible Euler equations. In 2.4, we briefly discuss these properties in order to introduce some vocabulary
to readers not familiar with Oceanography but who are familiar with the main features
of hyperbolic systems. In this section, we follow Stoker's classical book [13J. This book
may be consulted for further details in the study of 1D shallow water equations. For a
more oceanographic approach, see [8J, [10J, [9J.
The numerical resolution of the equations is presented in 2.5. The algorithm used,
developed at the University of Santiago de Compostela (Spain), is briefly described (see
the chapter of Prof. Bermudez in this volume and the references therein for detail).
Finally, in section 2.6 the boundary conditions allowed by the model are presented.