Geography Reference
In-Depth Information
it is conventional to define x and y as eastward and northward distance, such that
Dx
=
a cos φDλand Dy
=
aDφ. Thus, the horizontal velocity components
are u
Dy/Dt in the eastward and northward directions,
respectively. The (x, y, z) coordinate system defined in this way is not, however,
a Cartesian coordinate system because the directions of the
i
,
j
,
k
unit vectors are
not constant, but are functions of position on the spherical earth. This position
dependence of the unit vectors must be taken into account when the acceleration
vector is expanded into its components on the sphere. Thus, we write
≡
Dx/Dt and v
≡
D
U
Dt
=
i
Du
j
Dv
k
Dw
u
D
i
v
D
j
w
D
k
Dt
Dt
+
Dt
+
Dt
+
Dt
+
Dt
+
(2.10)
In order to obtain the component equations, it is necessary first to evaluate the rates
of change of the unit vectors following the motion.
We first consider D
i
/
Dt
. Expanding the total derivative as in (2.1) and noting
that
i
is a function only of x (i.e., an eastward-directed vector does not change its
orientation if the motion is in the north-south or vertical directions), we get
D
i
Dt
=
u
∂
i
∂x
From Fig. 2.1 we see by similarity of triangles,
=
0
|
|
δx
=
δ
i
∂
i
∂x
1
a cos φ
lim
δx
→
and that the vector ∂
i
∂x is directed toward the axis of rotation. Thus, as is illus-
trated in Fig. 2.2,
∂
i
∂x
=
1
a cos φ
(
j
sin φ
−
k
cos φ)
Therefore
D
i
Dt
=
u
a cos φ
(
j
sin φ
−
k
cos φ)
(2.11)
Considering now D
j
/
Dt
, we note that
j
is a function only of x and y. Thus, with
the aid of Fig. 2.3 we see that for eastward motion
|
δ
j
|=
δx/(a/ tan φ). Because
the vector ∂
j
∂x is directed in the negative x direction, we have then
∂
j
∂x
=−
tan φ
a
i
