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

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28 CHAPTER 5. DYNAMICS OF THE OCEAN
In such situation the last term in equation (5.34) has to be dropped.
Exercise 17: suppose (x,y) = (R cos(ωt),Rsin(ωt)) for a fluid particle in the fluid corresponding
to fig. 5.4, without exterior forces acting. Calculate ω. Such kind of motion, that is,
anti-cyclonic rotation with a period which is half the local rotation period, is indeed often observed
in oceanic and atmospheric motion and is called inertial oscillation and their frequency
(−f) is called inertial frequency.
tel-00545911, version 1 - 13 Dec 2010
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Figure 5.4: Cylinder in rotation with a free surface; two fluid particles with centrifugal force
(green) and pressure gradient force (red).
On earth the same thing happens, the centrifugal force changes the geopotential of the
earth, flattens it a little bit, makes it an ellipsoid.
5.7 The Shallow Water Equations in a Rotating Frame
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If we take the results from the previous section we see that we only have to add the Coriolis
force in the shallow water equations to obtain the shallow water equations in a rotating frame:
∂ t u + u∂ x u + v∂ y u − fv + g∂ x η = ν∇ 2 u (5.38)
∂ t v + u∂ x v + v∂ y v + fu + g∂ y η = ν∇ 2 v (5.39)
∂ t η + ∂ x [(H + η)u] + ∂ y [(H + η)v] = 0 (5.40)
+boundary conditions.
The nonlinear terms can be neglected if Rossby number ǫ = u/(fL) is small. The Rossby
number compares the distance a fluid particle has traveled in the time f −1 to the length scale
of the phenomenon considered. The linear (small Rossby number) version the shallow water
equations in a rotating frame is:
∂ t u − fv + g∂ x η = 0 (5.41)