Geography Reference
In-Depth Information
Fig. 3.6
Relationship between the change in angular direction of the wind δβ and the radius of
curvature R.
the streamlines instead of the curvature of the trajectories in the gradient wind
equation, it is necessary to investigate the relationship between the curvature of
the trajectories and the curvature of the streamlines for a moving pressure system.
We let β(x, y, t) designate the angular direction of the wind at each point on an
isobaric surface, and R
t
and R
s
designate the radii of curvature of the trajectories
and streamlines, respectively. Then, from Fig. 3.6, δs
=
Rδβso that in the limit
δs
→
0
Dβ
Ds
=
1
R
t
∂β
∂s
=
1
R
s
and
(3.20)
where Dβ/Ds means the rate of change of wind direction along a trajectory (posi-
tive for counterclockwise turning) and ∂β/∂s is the rate of change of wind direction
along a streamline at any instant. Thus, the rate of change of wind direction fol-
lowing the motion is
Dβ
Dt
=
Dβ
Ds
Ds
Dt
=
V
R
t
(3.21)
or, after expanding the total derivative,
Dβ
Dt
=
∂β
∂t
+
V
∂β
∂β
∂t
+
V
R
s
∂s
=
(3.22)
Combining (3.21) and (3.22), we obtain a formula for the local turning of the
wind:
V
1
∂β
∂t
=
1
R
s
R
t
−
(3.23)
Equation (3.23) indicates that the trajectories and streamlines will coincide only
when the local rate of change of the wind direction vanishes.
