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

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46 CHAPTER 7. MULTI-LAYER OCEAN DYNAMICS
7.4 Barotropic versus Baroclinic
tel-00545911, version 1 - 13 Dec 2010
Barotropic flow means that iso-barique surfaces coincide with iso-density surfaces. This is the
case if and only if η 1 =const. in the two layer case or η i =const. for all i = 1...N − 1 in the
multi-layer case (index are counted from top to bottom, 1 being the surface layer and N the
bottom layer). Geostrophy applied to (7.1) – (7.6), without using the Boussinesq approximation
and the fact that g ′ + g ′′ = g. we see that the flow does not change with depth which is the
case for barotropic flow. In oceanography the barotropic component of the flow is a component
for which the horizontal velocity does not change in the vertical direction. Confusion often
arises because sometimes the depth average velocity is called the barotropic component and
sometimes it is the geostrophic flow corresponding to the surface elevation ( ∑ N
i=1 η i). Anyway
the differences between the flow and the barotropic flow is called the baroclinic flow. So there
is vertical shear in the horizontal velocity field if and only if the baroclinic flow is not vanishing.
Exercise 48: Give an example to show that the two definitions of “ barotropic component”
differ.
7.5 Eddies, Baroclinic instability
There is one important phenomena that we can not explain from what we have learned so far
and this is the abundance of oceanic eddies with the size of approximately the first baroclinic
Rossby radius, that is around 100km. The maximum speed in these eddies is 1ms −1 . Indeed
when the first satellite observations of the ocean where available the ocean looked like a “sea of
eddies”, a feature that is well reproduced by today’s numerical models of the ocean circulation.
Observations and numerical simulations show that at many locations in the ocean the velocity
fluctuations due to eddying motion are up to two orders of magnitude larger than the
average velocity.
Exercise 49: Search the Internet for maps of sea surface height (SSH) from observations and
numerical models. Where do you see the eddies?
Exercise 50: Estimate the vorticity ζ of an ocean eddy and its Rossby number Ro = ζ/f.
Are eddies well described close to a geostrophic equilibrium?
Comment: The Rossby number and the baroclinic Rossby radius are two different things
with no direct connection, they are just named after the same person. The Rossby number
compares the relative vorticity to the planetary vorticity, or the magnitude of the non-linear
term to the Coriolis term. While the Rossby radius is the distance a gravity wave of speed
√
g′ H has traveled in the time f −1 .
Exercise 51: Estimate the SSH anomaly at the eddy center of a Gulf Stream eddy.
The process of formation of these eddies, which is called Baroclinic instability, is not only an
oceanic phenomenon but the cyclones and anticyclones in the mid-latitudes which determine
our weather are their atmospheric counterparts and are dynamically the same process. In the
atmosphere the baroclinic Rossby radius is of the order of 1000km which explains the size
of the cyclones and anticyclones in the atmosphere. It is clear that these “eddies” are key
to our understanding of the atmospheric dynamics but in the ocean they are rather small,
do they affect the large scale ocean dynamics? YES! they do! We have seen in section 5.8
that geostrophic dynamics at scales larger than the baroclinic Rossby radius has most of its
energy stored as availabel potential energy which is constantly supplied by the wind-stress
through Ekman pumping at a scale which is roughly 30 times the baroclinic Rossby radius.