Geography Reference
In-Depth Information
overestimate of the balanced wind in a region of cyclonic curvature and an under-
estimate in a region of anticyclonic curvature. For midlatitude synoptic systems,
the difference between gradient and geostrophic wind speeds generally does not
exceed 10-20%. [Note that the magnitude of V /(f R) is just the Rossby number.]
For tropical disturbances, the Rossby number is in the range of 1-10, and the gra-
dient wind formula must be applied rather than the geostrophic wind. Equation
(3.17) also shows that the antibaric anomalous low, which has V
g
< 0, can exist
only when V /(f R) <
1. Thus, antibaric flow is associated with small-scale
intense vortices such as tornadoes.
−
3.3
TRAJECTORIES AND STREAMLINES
In the natural coordinate system used in the previous section to discuss balanced
flow, s(x, y, t) was defined as the distance along the curve in the horizontal plane
traced out by the path of an air parcel. The path followed by a particular air parcel
over a finite period of time is called the
trajectory
of the parcel. Thus, the radius
of curvature
R
of the path
s
referred to in the gradient wind equation is the radius
of curvature for a parcel trajectory. In practice,
R
is often estimated by using the
radius of curvature of a geopotential height contour, as this can be estimated easily
from a synoptic chart. However, the height contours are actually
streamlines
of
the gradient wind (i.e., lines that are everywhere parallel to the instantaneous wind
velocity).
It is important to distinguish clearly between streamlines, which give a “snap-
shot” of the velocity field at any instant, and trajectories, which trace the motion
of individual fluid parcels over a finite time interval. In Cartesian coordinates,
horizontal trajectories are determined by the integration of
Ds
Dt
=
V (x, y, t)
(3.18)
over a finite time span for each parcel to be followed, whereas streamlines are
determined by the integration of
dy
dx
=
v(x, y, t
0
)
u(x,y,t
0
)
(3.19)
with respect to
x
at time t
0
. (Note that since a streamline is parallel to the velocity
field, its slope in the horizontal plane is just the ratio of the horizontal velocity
components.) Only for
steady-state
motion fields (i.e., fields in which the local
rate of change of velocity vanishes) do the streamlines and trajectories coincide.
However, synoptic disturbances are not steady-state motions. They generally move
at speeds of the same order as the winds that circulate about them. In order to
gain an appreciation for the possible errors involved in using the curvature of
