Geography Reference
In-Depth Information
2.1.1
Total Differentiation of a Vector in a Rotating System
The conservation law for momentum (Newton's second law of motion) relates
the rate of change of the absolute momentum following the motion in an inertial
reference frame to the sum of the forces acting on the fluid. For most applications
in meteorology it is desirable to refer the motion to a reference frame rotating
with the earth. Transformation of the momentum equation to a rotating coordinate
system requires a relationship between the total derivative of a vector in an inertial
reference frame and the corresponding total derivative in a rotating system.
To derive this relationship, we let
A
be an arbitrary vector whose Cartesian
components in an inertial frame are given by
i
A
x
+
j
A
y
+
k
A
z
A
=
and whose components in a frame rotating with an angular velocity
are
A
=
i
A
x
+
j
A
y
+
k
A
z
Letting D
a
A
/
Dt
be the total derivative of
A
in the inertial frame, we can write
DA
y
Dt
DA
x
Dt
DA
z
Dt
D
a
A
Dt
i
j
k
=
+
+
i
DA
x
Dt
j
DA
y
Dt
k
DA
z
D
a
i
Dt
D
a
j
Dt
D
a
k
Dt
=
+
+
Dt
+
A
x
+
A
y
+
A
z
The first three terms on the line above can be combined to give
DA
Dt
≡
i
DA
x
Dt
j
DA
y
Dt
k
DA
z
Dt
+
+
which is just the total derivative of
A
as viewed in the rotating coordinates (i.e.,
the rate of change of
A
following the relative motion).
The last three terms arise because the directions of the unit vectors (
i
,
j
,
k
) change
their orientation in space as the earth rotates. These terms have a simple form for
a rotating coordinate system. For example, considering the eastward directed unit
vector:
∂
i
∂λ
δλ
∂
i
∂φ
δφ
∂
i
∂z
δz
δ
i
=
+
+
For solid body rotation δλ
=
δt , δφ
=
0, δz
=
0, so that δ
i
/δt
=
(∂
i
/∂λ)(δλ/δt)
and taking the limit δt
→
0,
D
a
i
Dt
=
∂
i
∂λ
