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

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5.13. A FEW WORDS ABOUT WAVES 33
leading to:
∂ t ζ − f∂ z w + βv = 0. (5.59)
which states, that the vorticity ζ is changed by the vertical gradient of the vertical velocity
(vortex stretching) and the planetary change, due to β and the latitudinal velocity).
Exercise 33: what is the sign of f and β on the northern and southern hemisphere, respectively?
Exercise 34: what is the value of f and β on the equator, north and south pole?
Exercise 35: discuss the importance of f and β for equatorial dynamics.
5.13 A few Words About Waves
tel-00545911, version 1 - 13 Dec 2010
As mentioned in the preface we do not explicitly consider wave dynamics in this introductory
text. I like to make, nevertheless, some “hand waving” arguments about the role of waves in
the ocean.
The ocean and atmosphere dynamics at large scales are always close to a geostrophic balance.
There are, however, different sources of perturbations of the geostrophically balanced state:
• variation of the Coriolis parameter f
• non-linearity
• topography
• instability
• forcing (boundary conditions)
• friction
• other physical processes (convection, ..)
As the geostrophic adjustment process happens on a much faster time scale than the
geostrophic dynamics, these perturbations lead not so much to a departure from the geostrophic
state but more to its slow evolution. In this adjustment process, discussed in section 5.10,
(gravity) waves play an important role. It is an important part of research in geophysical
fluid dynamics (GFD) (DFG, en français) to find equations that reflect the slow evolution of
the geostrophic state, without explicitly resolving the geostrophic adjustment process. Such
equations are called balanced equations, and are based on the evolution of PV. The best known
system of balanced equations are the quasi-geostrophic equations. The problem in constructing
such equations is how to calculate the velocity field from PV, a process usually referred to as
inversion. The fast surface gravity waves influenced by rotation, Poincaré waves have no PV
signature and thus do not appear in the balanced equations, which leads to a large simplification
for analytical and numerical calculations. Balanced equations such rely on the assumption that
the ocean dynamics can be separated into fast wave motion and slow vortical motion with no
or negligible interactions between the two. They describe the dynamics on time-scales longer
than the period of gravity waves, typically several inertial periods f −1 . The balanced equations
are not valid when approaching the equator, as f −1 → ∞. The dynamics described by the
balanced equations is said to represent the slow dynamics or to evolve on the slow manifold .