Geography Reference
In-Depth Information
Fig. 1.11
Slope of pressure surfaces in the x, z plane.
evaluated holding
p
constant. Transformation of the horizontal pressure gradi-
ent force from height to pressure coordinates may be carried out with the aid of
Fig. 1.11. Considering only the x, z plane, we see from Fig. 1.11 that
(p
0
+
(p
0
+
δz
δx
δp)
−
p
0
δp)
−
p
0
z
=
δx
δz
p
where subscripts indicate variables that remain constant in evaluating the differ-
entials. Thus, for example, in the limit δz
x
→
0
(p
0
+
−
δp)
−
p
0
∂p
∂z
x
→
δz
x
where the minus sign is included because δz < 0 for δp > 0.
Taking the limits δx, δz
0 we obtain
5
→
∂p
∂x
∂p
∂z
∂z
∂x
z
=−
p
which after substitution from the hydrostatic equation (1.18) yields
x
∂p
∂x
g
∂z
∂x
∂
∂x
1
ρ
−
z
=−
p
=−
(1.25)
p
Similarly, it is easy to show that
∂p
∂y
z
=−
∂
∂y
1
ρ
−
(1.26)
p
5
It is important to note the minus sign on the right in this expression!
