379
3 The 111ultilayer model
A sharp density gradient can be generated in the sea water by solar heating of the upper
layer or in a strait connecting two basins where watprs have different densitips, as is the
case in the Strait of Gibraltar. In these cases, fresher water flows over more saline, and
consequently, heavier water. It even may happen that water flows in opposite directions
in each layer. In this situation a shallow water model is not suitable: on the onp hanc\,
densities variations are important; on the otllf'r, large velocity fluctuations along the
water column can be expected.
In this section, a model considering sea water as composed of several inmiscible
layers of different constant densities is proposed. In such a model, waves appear not
only in the surface but also in the interface between the layers. It will be assumed
that phenomena to be modelled have wavelenghts large enough to made appropiatp the
shallow water approximation in each layer.
The model described here allows any finite number of layers. Nevertheless, for
the sake of clarity, to present the derivation of the model only the two layer case is
considered. In section 3.3 the equations for the generalmultilaypr mOllel are presented.
The numerical experiences carriecl out and shown in section 3.4 also correspond to the
two-layer approximation. This initial restriction to two layers is due to the difficulties
found to correctly choose suitable boundary conditions for every layer and to the fact
that the two-layer model well represents the real structure of the ZOll(' uncleI' study.
3.1 Primary equations
Let us use the notation introduced in figure 4. Two constant reference levels, ...1.1 and A.2•
have been chosen ill each layer. More precisely, A2 is the reference level for the deeper
layer and A = A I + A2 is the reference level for the upper on~. Lf't lIS aSSlllllf' that the
sea surface, the iuterface and the bottom arc given, respectively, by the eqllatiolls:
X3
SI(:L·I,.1'2,t),
:1':l
S2(·t1,X2,t),
1'3
b(xj,.Tz),
and let T71 and 1)2 be, respectively, the elevations of the surface and the interface above
their reference levels A and AI, i.p.
-S1(.rj,:r2,t)
S2(TI,.f2,t)
T/,(J'I, :r2, t) + A,
1)2( :1'1, XZ, t) + Az.
(37)
Let us denote by iTi the ith-Iayer velocity, i the pressure and pi the density, i = 1,2,
(we suppose that PI < P2)' After simplifications, a set of equations analogous to (12)(14) is obtained for each layer. The boundary conditions on the surface arc (4), (5) and
(6) and the boundary conditions on the bottom are (7). (8) and (9). Suitable bounclary
conditions for the two layers at the interface
3 The 111ultilayer model
A sharp density gradient can be generated in the sea water by solar heating of the upper
layer or in a strait connecting two basins where watprs have different densitips, as is the
case in the Strait of Gibraltar. In these cases, fresher water flows over more saline, and
consequently, heavier water. It even may happen that water flows in opposite directions
in each layer. In this situation a shallow water model is not suitable: on the onp hanc\,
densities variations are important; on the otllf'r, large velocity fluctuations along the
water column can be expected.
In this section, a model considering sea water as composed of several inmiscible
layers of different constant densities is proposed. In such a model, waves appear not
only in the surface but also in the interface between the layers. It will be assumed
that phenomena to be modelled have wavelenghts large enough to made appropiatp the
shallow water approximation in each layer.
The model described here allows any finite number of layers. Nevertheless, for
the sake of clarity, to present the derivation of the model only the two layer case is
considered. In section 3.3 the equations for the generalmultilaypr mOllel are presented.
The numerical experiences carriecl out and shown in section 3.4 also correspond to the
two-layer approximation. This initial restriction to two layers is due to the difficulties
found to correctly choose suitable boundary conditions for every layer and to the fact
that the two-layer model well represents the real structure of the ZOll(' uncleI' study.
3.1 Primary equations
Let us use the notation introduced in figure 4. Two constant reference levels, ...1.1 and A.2•
have been chosen ill each layer. More precisely, A2 is the reference level for the deeper
layer and A = A I + A2 is the reference level for the upper on~. Lf't lIS aSSlllllf' that the
sea surface, the iuterface and the bottom arc given, respectively, by the eqllatiolls:
X3
SI(:L·I,.1'2,t),
:1':l
S2(·t1,X2,t),
1'3
b(xj,.Tz),
and let T71 and 1)2 be, respectively, the elevations of the surface and the interface above
their reference levels A and AI, i.p.
-S1(.rj,:r2,t)
S2(TI,.f2,t)
T/,(J'I, :r2, t) + A,
1)2( :1'1, XZ, t) + Az.
(37)
Let us denote by iTi the ith-Iayer velocity, i the pressure and pi the density, i = 1,2,
(we suppose that PI < P2)' After simplifications, a set of equations analogous to (12)(14) is obtained for each layer. The boundary conditions on the surface arc (4), (5) and
(6) and the boundary conditions on the bottom are (7). (8) and (9). Suitable bounclary
conditions for the two layers at the interface