28 Pierre-Yves Le Traon
There will be the same velocity over the total water column. These are the socalled barotropic motions which do not depend on the density field.
The density actually depends on x, y, z and t [p = p(x,y,z,t»). Let us define p' as
the density anomaly given by p = is the water density at a
given temperature and salinity (e.g. T = O°C , S = 350/00). From equation 3, we
derive:
A,v = _.K.~
az
poax
(4)
This is the thermal wind equation. Horizontal density variations are associated
with vertical shear (so-called baroc1inic motions). Integrating (4) from zo, the reference level, to z 10 the surf ace, yields:
v(z,) = v(zo)_8 -~dz
f
Zll a I
f zoPo ax
(5)
If density is known, we only need to know the velocity at the reference level, Zo,
to calculate the velocity over the total water column. From equation (5), we can
easily obtain equation (6):
8 0Tls
~ v(z,) = v(zo) + af
ZI I
f x
with TI s (zo, z,) = - E:.. dz and TI s is the steric height.
a
Zo Po
v(z,) = 8~a where TI is the sea level and from (6):
f x
"aTl s "a11
v(zo) + 2 . - = 2 . -
fax
fax
(6)
Thus, at the surface,
1 apz
Let's now write v(zo) = -~where Pz is the pressure at the reference level,
fpo aX
o
usually defined as the bottom.
The same expression holds for u (with y derivative) which means that:
PZo
~11=11+s
Pog
The dynamic topography 11 (measured by altimetry) is the sum of the steric
height (baroc1inic component) and of a bottom pressure term, sometimes incorrectly referred to as the barotropic component. The exact definition of a barotropic
motion is a motion which does not depend on z; a barotropic motion will induce