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

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32 CHAPTER 5. DYNAMICS OF THE OCEAN
5.11 Potential Vorticity (non-linear)
tel-00545911, version 1 - 13 Dec 2010
Similar calculations for the non-linear equations (5.38) – (5.40) lead to
( )
d ζ + f
= 0. (5.57)
dt H + η
This means that every fluid parcel, or in this case every fluid column, conserves its potential
vorticity Q sw = (ζ+f)/(H+η), that is, potential vorticity is transported by the two dimensional
flow. The part f/(H + η) which does not depend explicitly on the velocity is called planetary
potential vorticity, while ζ/(H + η) is called the dynamical part.
Example: Eddy over sea mount.
Exercise 30: derive eq. (5.57).
Exercise 31: If you make the assumption of linearity, can you obtain Q lin
sw from Q sw ?
Exercise 32: The moment of inertia of a cylinder of mass m, radius r and height H is given
by I = mr 2 /2 the angular momentum is given by L = Iω. If a cylinder stretches or flatness
without any forces acting from the outside its angular momentum is conserved:
H
Show, that during this process ω/H is conserved.
The previous exercise demonstrates, that the conservation of potential vorticity is nothing
else than the conservation of angular momentum applied to a continuum in a rotating frame.
5.12 The Beta-plane
H
So far we supposed the earth to be flat! The dominant difference, induced by the spherical
shape of the earth, for the large scale ocean dynamics, at mid- and low latitudes, is the change
of the (locally) vertical component of the rotation vector.
For the large scale circulation a major source of departure from geostrophy is the variation
of f = f 0 with latitude. So far we have considered f to be constant we will now approximate it
by f = f 0 + βy, where f 0 = 2|Ω| sin(θ 0 ) and β = 2(|Ω|/R) cos(θ 0 ) are constant where R is the
radius of the earth, it takes its maximum value β max = 2.3 × 10 −11 m −1 s −1 at the equator. The
geometry with a linearly changing Coriolis parameter is called the β-plane. The change of f
with latitude, the so called β-effect, compares to the effect of constant rotation for phenomena
with horizontal extension L ≈ f/β = R tan(θ) or larger.
Replacing f by f 0 + βy in equations (5.41), (5.42) and (5.43):
∂ t (∂ x v − ∂ y u) + f(∂ x u + ∂ y v) + βv = 0 (5.58)