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

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24 CHAPTER 5. DYNAMICS OF THE OCEAN
which we combine to:
∂ tt η = gH∂ xx η (5.25)
+boundary conditions.
This is a one dimensional linear non-dispersive wave equation. The general solution is given
by:
η(x,t) = η − 0 (ct − x) + η + 0 (ct + x) (5.26)
u(x,t) = c H (η− 0 (ct − x) − η + 0 (ct + x)), (5.27)
tel-00545911, version 1 - 13 Dec 2010
where η − 0 and η + 0 are arbitrary functions of space only. The speed of the waves is given by
c = √ gH and perturbations travel with speed in the positive or negative x direction. Note
that c is the speed of the wave not of the fluid!
Rem.: If we choose η − 0 (˜x) = η + 0 (−˜x) then initially the perturbation has zero fluid speed,
and is such only a perturbation of the sea surface! What happens next?
An application of such equation are Tsunamis if we take: g = 10m/s 2 , H = 4km and
η 0 = 1m, we have a wave speed of c = 200m/s= 720km/h and a fluid speed u 0 = 0.05m/s.
What happens when H decreases? Why do wave crests arrive parallel to the beach? Why do
waves break?
You see this simplest form of a fluid dynamic equation can be understood completely. It
helps us to understand a variety of natural phenomena.
Exercise 11: does the linearized one dimensional shallow water equation conserve energy?
Exercise 12: is it justified to neglect the nonlinear term in eq. (5.21) for the case of a
Tsunami?
5.3 Reduced Gravity
Suppose that the layer of fluid (fluid 1) is lying on a denser layer of fluid (fluid 2) that is
infinitely deep. H 2 → ∞ ⇒ c 2 → ∞, that is perturbations travel with infinite speed. This
implies that the lower fluid is always in equilibrium ∂ x P = ∂ y P = 0. The lower fluid layer is
passive, does not act on the upper fluid but adapts to its dynamics, so that η 1 = ρ1−ρ2
ρ 1
η 2 . If we
set η = η 1 − η 2 then η =
ρ2
ρ 2−ρ 1
η 1 and the dynamics is described by the same sweqs. 5.18, 5.19
and 5.20 with gravity g replaced by the reduced gravity g ′ = ρ2−ρ1
ρ 2
g (“sw on the moon”).
Example: g ′ = 3 · 10 −3 g, H = 300m, η 0 = .3m we get a wave speed c = √ g ′ H = 3m/s and
a fluid speed of u = 1m/s.
Comment 1: when replacing η by g ′ η it seems, that we are changing the momentum
equations, but in fact the thickness equation is changed, as we are in the same time replacing
the deviation of the free surface η (which is also the deviation of the layer thickness in notreduced-gravity
case) by the deviation of the layer thickness η , which is (ρ 2 − ρ 1 )/ρ 2 times
the surface elevation in the reduced gravity case. This means also that every property which
is derived only from the momentum equations not using the thickness equation is independent
of the reduced gravity.
Comment 2: fig. 5.3 demonstrates, that the layer thickness can be measured in two
ways, by the deviation at the surface (η 1 ) or by density structure in the deep ocean (η 2 ). For
ocean dynamics the surface deviation for important dynamical features, measuring hundreds
of kilometers in the horizontal, is usually less than 1m whereas variations of (η 2 ) are usually