Geoscience Reference
In-Depth Information
friction and pressure gradient forces. The flow is generally directed perpendicu-
lar to the isobars or isoheights, down the pressure gradients into lows or away
from highs. More mass flows into regions of low pressure than at higher lati-
tudes, and lows are substantially weaker.
6.6 THE MOMENTUM EQUATIONS
The expressions developed for the Coriolis and pressure gradient forces can
be used to write the horizontal equations of motion (Eqs. 6.3 and 6.4) in
z
-
coordinates as
du
1
2
p
x
=− +
fv
F
(6.54)
ρ
2
x
F
dt
and
dv
2
=− −+
1
p
y
fu
F
(6.55)
ρ
2
y
F
dt
where the approximate form of the Coriolis force from Eq. 6.26 is used. If we
use the Eulerian perspective in local Cartesian coordinates (Appendix C), the
local wind or current velocity is accelerated according to
2
u
1
2
p
v
x
=− +− +
vufv
$
d
F
(6.56)
2
t
ρ
2
x
F
and
2
v
1
2
p
.
v
y
=− −− +
vvfu
$
d
F
(6.57)
2
t
ρ
2
y
F
When the local time derivatives of
u
and
v
, the advection terms, and friction
are small, the flow is geostrophic. This highlights the fact that that geostrophic
wind is steady, or nonaccelerating.
The vertical equation of motion is
2
w
1
2
p
.
v
z
=− −− +
vwg
$
d
F
(6.58)
2
t
ρ
2
z
F
Hydrostatic balance (Eq. 6.35) occurs when the local time rate of change of
w
,
advection of vertical velocity, and friction are negligible.
6.7 EXERCISES
6.1. Calculate the pressure gradient force for the system drawn in
Figure 6.8.
Write vector equations expressing your answers using the local Cartesian
coordinate system, and also draw a vector indicating the direction of the
pressure gradient force. Assume that density is 1.0 kg/m
3
.
6.2. Calculate the zonal velocity relative to the rotating earth,
u
, attained by
a parcel of air in moving from the equator to 30°N latitude conserving
