Geography Reference
In-Depth Information
Fig. 2.4
Dependence of unit vector
j
on
latitude.
We next turn to the component expansion of the force terms in (2.8). The Coriolis
force is expanded by noting that
has no component parallel to
i
and that its
components parallel to
j
and
k
are 2 cos φ and 2sinφ, respectively. Thus, using
the definition of the vector cross product,
i
j
k
−
2
×
U
=−
2
0
cos φ
sin φ
u
v
w
=−
(2w cos φ
−
2v sin φ)
i
−
2u sin φ
j
+
2u cos φ
k
(2.15)
The pressure gradient may be expressed as
i
∂p
j
∂p
k
∂p
∂z
∇
=
∂x
+
∂y
+
p
(2.16)
and gravity is conveniently represented as
g
=−
g
k
(2.17)
where g is a positive scalar (g
=
9.8ms
−
2
at the earth's surface). Finally, recall
from (1.5) that
k
F
rz
(2.18)
Substituting (2.14)-(2.18) into the equation of motion (2.8) and equating all
terms in the
i
,
j
, and
k
directions, respectively, we obtain
F
r
=
i
F
rx
+
j
F
ry
+
Du
Dt
−
uv tan φ
a
uw
a
1
ρ
∂p
∂x
+
+
=−
2v sin φ
−
2w cos φ
+
F
rx
(2.19)
u
2
tan φ
a
Dv
Dt
+
vw
a
1
ρ
∂p
∂y
−
+
=−
2u sin φ
+
F
ry
(2.20)
