Forces and dynamics - Physical Processes in Earth and Environmental Sciences - page 60

Angular speed of Earth surface is a function of latitude

Definition of vertical
component of vorticity due
to horizontal rotation

r ′

j

f

r

+ve -ve

j > 0 j < 0
Circular flow (velocity
changes direction in space)

2

Ω Ω r ′

s

g

Vertical axis of
vorticity meter at
center of rotation
of fluid

V

+ n

The small centripetal acceleration
acting at the surface due to rotation

j = 2 Ω

f = 2 Ω

Fig. 3.37 Vorticity of curved flow.

f = 2Ω sin f

f

f = 0

j = 2 Ω

f = -2 Ω

Since the rate of change of apparent displacement with
respect to time is a definition of acceleration, the mean
Coriolis acceleration is:

j = 2 Ω sin f

d 2

vt 2

( Ω sin f ) = 2 Ω sin fv

d t 2

Fig. 3.39 Planetary vorticity.

j = 0

Latitude, f

Fig. 3.38 Vorticity of a rotating hemisphere.

(see Fig. 3.36 for the various possibilities) contribute to
the vertical vorticity, z , represented by the local spin of the

horizontal flow about the vertical axis ( x ).

It is possible of course that the velocity gradients could
partially or wholly cancel each other out with resulting
reduced or even zero vorticity; the signs in the expression
take care of these possibilities. A similar argument holds
for the other two reference planes enabling us to specify
the total vorticity,

u / y v /

can cause spin along the z -axis specified (Fig. 3.36):
(1) a gradient of the horizontal streamwise velocity, u , in
the spanwise direction, y , gives the gradient, y ,

(2) a gradient of spanwise velocity, v , in the streamwise
direction, x , gives

u /

v / x . Either or both of these gradients .