Dynamics: The Forces That Drive Atmospheric and Ocean Circulations - Climate Dynamics - page 113

relative to the earth. If the parcel conserves absolute angular momentum ( M is
constant) and moves to 30°N, the observer at 30°N will measure its zonal ve-
locity as 134 m/s westerly (see exercise 6.2). This large zonal velocity suggests
that the Coriolis effect is important. Because we do not observe values of the
zonal wind comparable to 134 m/s, this comparison also suggests that other
forces must be operating in the system to constrain the flow.
The equation

dM

 (6.15)

dt

expresses the principle of conservation of absolute angular momentum. (Note
the Lagrangian derivative, defined in Appendix C.) Substituting Eq. 6.13 into
Eq. 6.15 and carrying through the differentiation (see exercise 6.3), we obtain
the zonal component of the Coriolis force,

v

uv uw i

|

F = 2 Ω v sin φ + tan φ − 2 Ω w cos φ − , (6.16)

c m

M

a a

where the subscript M is a reminder that this component of the Coriolis force
arises due to conservation of angular momentum per unit mass in the absolute
frame of reference.

CENTRIFUGAL ACCELERATIONS

The other part of the Coriolis force accounts for centrifugal accelerations that
act in the rotating frame of reference and arise when a parcel's path in the ab-
solute frame of reference is curved.
For a parcel of air or ocean water in solid body rotation with the earth
( uvw 0   ), the centrifugal force per unit mass, F S v , felt by the parcel in the

frame of reference rotating with the earth is

U

2

v

ROT

t t

2

F = n = Ω rn , (6.17)

SB

r

where t is the unit vector perpendicular to and pointing away from the axis of
rotation (Figure 6.3) . Gray arrows in Figure 6.3 represent forces in the rotating
frame of reference that act on a parcel in solid body rotation. These are grav-
ity, F GRA v , and the centrifugal force, F S v . These two forces do not balance, so
some other net force is required to keep the parcel over the same location on
the earth's surface. Note that solid body rotation is the result of a particular
balance of forces and not a case of zero net force acting on a parcel.
In the formulation of the equations of motion for the atmosphere and
oceans, F S v is absorbed into F GRAV

v

. Effective gravity , g EFF , is defined as

v

v v v (6.18)
and drawn in Figure 6.3. The difference between F GRA v and g EF v is greatly exagger-
ated in Figure 6.3, and we will neglect F S v compared with v , so that F GRAV v g EF v .
Now, consider a parcel with an eastward zonal velocity, u > 0, relative to
the earth's surface. In the absolute frame of reference, this parcel is traveling

g

GRA =+

F F

EFF SB