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

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5.2. THE LINEARIZED ONE DIMENSIONAL SHALLOW WATER EQUATION̸ S 23
What are those boundary conditions? Well on the ocean floor, which is supposed to vary only
very slowly with the horizontal directions, the vertical velocity vanishes w = 0 and it varies
linearly in the fluid interior (see eq. 5.13). The ocean has what we call a free surface with a
height variation denoted by η. The movement of a fluid partical on the surface is govened by:
d H
η = w(η) (5.14)
dt
where dH
dt
= ∂ t + u∂ x + v∂ y is the horizontal Lagrangian derivation. We obtain:
or
∂ t η+ u∂ x η + v∂ y η − (H + η)∂ z w = 0 (5.15)
tel-00545911, version 1 - 13 Dec 2010
∂ t η+ u∂ x (H + η) + v∂ y (H + η) + (H + η)(∂ x u + ∂ y v) = 0. (5.16)
Using the hydrostatic approximation, the pressure at a depth d from the unperturbed free
surface is given by: P = gρ(η + d), and the horizontal pressure gradient is related to the
horizontal gradient of the free surface by:
∂ x P = gρ∂ x η and ∂ y P = gρ∂ y η (5.17)
Some algebra now leads us to the shallow water equations (sweq):
∂ t u+ u∂ x u + v∂ y u + g∂ x η = ν∇ 2 u (5.18)
∂ t v+ u∂ x v + v∂ y v + g∂ y η = ν∇ 2 v (5.19)
∂ t η+ ∂ x [(H + η)u] + ∂ y [(H + η)v] = 0 (5.20)
+boundary conditions .
All variables appearing in equations 5.18, 5.19 and 5.20 are independent of z!
5.2 The Linearized One Dimensional Shallow Water Equation̸
s
We will now push the simplifications even further, actually to its non-trivial limit, by considering
the linearized one dimensional shallow water equations. If we suppose the dynamics to be
independent of y and if we further suppose v = 0 and that H is constant, the shallow water
equations can be written as:
∂ t u+ u∂ x u + g∂ x η = ν∇ 2 u (5.21)
∂ t η+ ∂ x [(H + η)u] = 0 (5.22)
+ boundary conditions.
if we further suppose that u 2 ≪ gη that the viscosity ν ≪ gηL/u and η ≪ H then:
∂ t u + g∂ x η = 0 (5.23)
∂ t η + H∂ x u = 0 (5.24)
+boundary conditions,