Etudes et évaluation de processus océaniques par des hiérarchies ...

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Chapter 7
Multi-Layer Ocean dynamics
tel-00545911, version 1 - 13 Dec 2010
7.1 The Multilayer Shallow Water Model
The models employed so far to study the ocean dynamics consisted of a single layer, which
represented the dynamics of a single vertically homogeneous (in speed and density) layer above
a solid bottom or above a infinitely deep inert layer of higher density (reduced gravity model).
We also saw that these type of models are very successful in explaining the main features of
the large scale ocean circulation. There are, however, important phenomena of the circulation
which can not be explained by such one-layer models. We thus move on to the dynamics
of several layers of homogeneous (in speed and density) fluid layers of different density and
velocity, stacked one above the other. We will here restrict the analysis to a model with two
active layers, the generalisation to more layers is strait forward. The equations governing the
dynamics of such a hydrostatic two-layer shallow water model are:
∂ t u 1 + u 1 ∂ x u 1 + v 1 ∂ y u 1 − fv 1 + g∂ x (η 1 + η 2 ) = ν∇ 2 u 1 (7.1)
∂ t v 1 + u 1 ∂ x v 1 + v 1 ∂ y v 1 + fu 1 + g∂ y (η 1 + η 2 ) = ν∇ 2 v 1 (7.2)
∂ t η 1 + ∂ x [(H 1 + η 1 )u 1 ] + ∂ y [(H 1 + η 1 )v 1 ] = 0 (7.3)
∂ t u 2 + u 2 ∂ x u 2 + v 2 ∂ y u 2 − fv 2 + g ′′ ∂ x (η 1 + η 2 ) + g ′ ∂ x η 2 = ν∇ 2 u 2 (7.4)
∂ t v 2 + u 2 ∂ x v 2 + v 2 ∂ y v 2 + fu 2 + g ′′ ∂ y (η 1 + η 2 ) + g ′ ∂ y η 2 = ν∇ 2 v 2 (7.5)
∂ t η 2 + ∂ x [(H 2 + η 2 )u 2 ] + ∂ y [(H 2 + η 2 )v 2 ] = 0 (7.6)
+boundary conditions .
Where the index 1 and 2 denote the upper and the lower layer, respectively. It is interesting
to note that the two layers interact only through the hydrostatic pressure force caused by
the thicknesses of the layers. Indeed, the upper layer (layer 1) is subject to the hydrostatic
pressure of the surface which has a total anomaly of η 1 + η 2 . Whereas the the lower layer
(layer 2) is subject to the same pressure plus the pressure at the interface g ′ η 2 due to the
increased density in the lower layer, where g ′ = g(ρ 2 − ρ 1 )/ρ 2 is the reduced gravity, that is,
the weight of the lower-layer fluid in the upper layer environment and g ′′ = gρ 1 /ρ 2 . In the
Boussinesq approximation g ′′ is set equal to g, thus neglecting the density differences in the
inertial mass but keeping it in the weight. Equations (7.1) – (7.6) are the mathematical model
for the investigations of the present chapter.
Exercise 43: What happens to equations (7.1) – (7.6) if ρ 1 = ρ 2 ?
Exercise 44: Write down the linearised version of eqs. (7.1) – (7.6).
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