The vorticity vector u is the curl

The vorticity vector u is the curl of the
velocity vector v:
u ¼ V
v ¼ iðvv=vz  vw=vyÞ
þ jðvw=vx  vu=vzÞ þ kðvv=vx  vu=vyÞ
(7.32)
Vorticity, therefore, has units of inverse time,
(sec)
1.
Fluids (and all objects) have vorticity
simply because of Earth’s rotation. This is
called planetary vorticity. The vector planetary
vorticity points upward, parallel to the rotation
axis of Earth, which has an angular rotation
rate of U:
uplanetary
¼ 2U (7.33)
where U ¼ 2p/day ¼ 2p/86160 sec ¼ 7.293

10
5 sec
1, so uplanetary ¼ 1.4586
10
4 sec
1.
The vorticity of the fluid motion relative to
Earth’s surface (Eq. 7.32) is called the relative
vorticity. The total or absolute vorticity of a piece
of fluid is the sum of the relative vorticity and
planetary vorticity.
For large-scale oceanography, only the local
vertical component of the total vorticity is
used because the fluid layers are thin compared
with Earth’s radius, so flows are nearly horizontal.
The local vertical component of the
planetary vorticity is exactly equal to the
Coriolis parameter f (Eq. 7.8c) and is therefore
maximum and positive at the North Pole (4 ¼
90

N), maximum and negative at the South
Pole (4 ¼ 90

S), and 0 at the equator.
The local vertical component of the relative
vorticity from Eq. (7.32) is
2 ¼ ðvv=vx  vu=vyÞ ¼ curlzv (7.34)
where v is the horizontal velocity vector. The
local vertical component of the absolute
vorticity is therefore (z þ f). The geostrophic
velocities calculated from Eq. (7.23) (Section 7.6)
are often used to calculate relative vorticity.
7.7.2. Potential Vorticity
Potential vorticity is a dynamically important
quantity related to relative and planetary
vorticity. Conservation of potential vorticity is
one of the most important concepts in geophysical
fluid dynamics, just as conservation of
angular momentum is a central concept in solid
body mechanics. Potential vorticity takes into
account the height of a water column as well
as its local spin (vorticity). If a column is shortened
and flattened (preserving mass), then it
must spin more slowly. On the other hand, if
a column is stretched and thinned (preserving)