Geography Reference
In-Depth Information
Fig. 2.3
Dependence of unit vector
j
on
longitude.
From Fig. 2.4 it is clear that for northward motion
|
δ
j
|=
δφ,butδy
=
aδφ, and
δ
j
is directed downward so that
∂
j
∂y
=−
k
a
Hence,
D
j
Dt
=−
u tan φ
a
v
a
i
−
k
(2.12)
Finally, by similar arguments it can be shown that
D
k
Dt
=
i
u
j
v
a
a
+
(2.13)
Substituting (2.11)-(2.13) into (2.10) and rearranging the terms, we obtain the
spherical polar coordinate expansion of the acceleration following the relative
motion
Du
Dt
−
i
Dv
Dt
+
j
u
2
tan φ
a
D
U
Dt
=
uv tan φ
a
uw
a
vw
a
+
+
+
(2.14)
Dw
Dt
−
k
u
2
v
2
+
+
a
