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

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30 CHAPTER 5. DYNAMICS OF THE OCEAN
5.9 Energetics of flow in Geostrophic Equilibrium
For the shallow water dynamics the total energy is composed of kinetic and available potential
energy (the part of the potential energy which is available in the layered model by reducing the
surface anomaly η, if η = 0 everywhere the available potential energy vanishes):
E total = E kin + E pot = ρ ∫
H(u 2 + v 2 )dxdy + gρ ∫
η 2 dxdy (5.49)
2 A
2 A
= g2 ρ
H
2f
∫A
( (∂ 2 x η) 2 + (∂ y η) 2) dxdy + gρ ∫
η 2 dxdy (5.50)
2 A
tel-00545911, version 1 - 13 Dec 2010
where we used (eqs. 5.44 and 5.45). If the surface perturbation has the simple form η =
η 0 sin(x/L) then the energy is given by:
E total = E kin + E pot = gη2 0
4
∫
A
( Hg
f 2 L 2 + 1 )
dxdy (5.51)
Were the first term is the kinetic and the second term the available potential energy. We
see that in a geostrophic flow the kinetic energy is larger than the available potential energy
when the structure is smaller than the Rossby radius R = √ gH/f 2 . So for large geostrophic
structures most of the energy is in the potential part and for small structures in the kinetic
part. The Rossby radius is of the order of a few thousands of kilometers for the shallow water
dynamics of the ocean (the barotropic Rossby radius) but only several tenths of kilometers when
the reduced gravity dynamics of the layer above the thermocline are considered (the baroclinic
Rossby radius).
5.10 Linear Potential Vorticity and the Rossby Adjustment
Problem
If we take ∂ x (eq. (5.42)) - ∂ y (eq. (5.41)) we see that:
∂ t ζ + f(∂ x u + ∂ y v) = 0. (5.52)
relating vorticity ζ = ∂ x v − ∂ y u to divergence ∂ x u + ∂ y v. Using eq. (5.43) we get:
( ζ
∂ t
f − η )
= 0. (5.53)
H
One usually calls Q lin
sw = ζ H − fη
H 2 the linear shallow water potential vorticity. The above equations
show, that at every location linear shallow water potential vorticity (PV) is conserved, when
the dynamics is governed by the linearised shallow water equations.
The Rossby adjustment problem considers the adjustment of an initially step-like perturbation
(see fig. 5.10), and we would like to know the final, geostrophically balanced state of
this perturbation. To this end we use the conservation of potential vorticity and we further
require the final state to be in geostrophic equilibrium. The initial potential vorticity is given
by sgn(x)(fη 0 )/H 2 the PV of the adjusted state is the same, we thus have,
g/(Hf)∂ xx η a − fη a /H 2 = sgn(x)(fη 0 )/H 2 , (5.54)
R 2 ∂ xx η a − η a = η 0 sgn(x), (5.55)