Geography Reference
In-Depth Information
3.2
BALANCED FLOW
Despite the apparent complexity of atmospheric motion systems as depicted on
synoptic weather charts, the pressure (or geopotential height) and velocity dis-
tributions in meteorological disturbances are actually related by rather simple
approximate force balances. In order to gain a qualitative understanding of the
horizontal balance of forces in atmospheric motions, we idealize by considering
flows that are steady state (i.e., time independent) and have no vertical component
of velocity. Furthermore, it is useful to describe the flow field by expanding the
isobaric form of the horizontal momentum equation (3.2) into its components in a
so-called
natural
coordinate system.
3.2.1
Natural Coordinates
The natural coordinate system is defined by the orthogonal set of unit vectors
t
,
n
,
and
k
. Unit vector
t
is oriented parallel to the horizontal velocity at each point; unit
vector
n
is normal to the horizontal velocity and is oriented so that it is positive
to the left of the flow direction; and unit vector
k
is directed vertically upward. In
this system the horizontal velocity may be written
V
=
V
t
where V , the horizontal
speed, is a nonnegative scalar defined by V
Ds/Dt , where s(x, y, t) is the
distance along the curve followed by a parcel moving in the horizontal plane. The
acceleration following the motion is thus
≡
V
D
t
Dt
The rate of change of
t
following the motion may be derived from geometrical
considerations with the aid of Fig. 3.1:
D
V
Dt
=
D(V
t
)
Dt
t
DV
=
Dt
+
|
=
|
|
δs
|
δ
t
=
=|
|
δψ
δ
t
R
|
t
|
Here R is the
radius of curvature
following the parcel motion, and we have used
the fact that
1. By convention,
R
is taken to be positive when the center of
curvature is in the positive
n
direction. Thus, for R>0, the air parcels turn toward
the left following the motion, and for R<0 the air parcels turn toward the right
following the motion.
Noting that in the limit δs
|
t
|=
→
0, δ
t
is directed parallel to
n
, the above relationship
yields D
t
/Ds
=
n
/R. Thus,
D
t
Dt
=
D
t
Ds
Ds
Dt
=
n
R
V
and
n
V
2
R
D
V
Dt
=
t
DV
Dt
+
(3.8)
