Geoscience Reference
In-Depth Information
A
A
p
1
p
2
Figure 6.5. The pressure
gradient force.
n
n
1
n
2
+
n
direction
(
force acting at
n
)
−
(
force acting at
n
)
v
1
2
t
F
=
n
.
(6.29)
PGF
mass of parcel
But since pressure is force per unit area,
force acting at
n p
,
1
1
and likewise for the right face at
n
2
. Therefore,
A
ppn
1
(
pp
n
−
)
v
t
2
1
t
F
=−=−
(
)
,
(6.30)
PGF
m
1
2
ρ
(
nn
−
)
2
1
since
m
m
m
A
1
.
ρ
== =
=
(6.31)
&
V
An
∆
An n
(
−
)
m
ρ
(
nn
−
)
1
2
1
2
Taking the differential limit of Eq. (6.30), we obtain the final expression for the
pressure gradient force:
1
2
p
n
v
t
F
=−
.
(6.32)
PGF
ρ
2
n
Note the negative sign in Eq. 6.32, which indicates that the direction of the
pressure gradient force is down the gradient, that is, from high to low values.
In local Cartesian coordinates, the
horizontal pressure gradient force
is
1
2
p
i
1
2
p
j
v
t
t
xy
,
PGF
F
=−
−
(6.33)
ρ
2
x
ρ
2
y
and the vertical pressure gradient force is
1
2
p
k
v
t
z
F
=−
.
(6.34)
PGF
ρ
2
z
6.3 HYDROSTATIC BALANCE
Because pressure decreases with height in the earth's atmosphere and oceans,
/
pz
22
< 0 and the vertical pressure gradient force (Eq. 6.34) is directed upward
