Geography Reference
In-Depth Information
where we have used (6.15) to write the geostrophic vorticity and its tendency
in terms of the Laplacian of geopotential. Thus, since by (6.7) the geostrophic
wind can be expressed in terms of , the right-hand side of (6.21) depends on the
dependent variables , and ω alone. An analogous equation dependent on these
two variables can be obtained from the thermodynamic energy equation (6.13b)
by multiplying through by f
0
σ and differentiating with respect to p. Using the
definition of χ given above, the result can be expressed as
f
0
σ
f
0
σ
∂
∂p
κJ
σp
∂
∂p
∂χ
∂p
∂
∂p
∂ω
∂p
−
∂
∂p
=−
V
g
·∇
−
f
0
f
0
(6.22)
where σ was defined below (6.13a).
The ageostrophic vertical motion, ω, has equal and opposite effects on the left-
hand sides in (6.21) and (6.22). Vertical stretching
∂ω
∂p > 0
forces a positive
tendency in the geostrophic vorticity (6.21) and a negative tendency of equal mag-
nitude in the term on the left side in (6.22). The left side of (6.22) can be interpreted
as the local rate of change of a normalized
static stability anomaly
(i.e., a measure
of the departure of static stability from S
p
, its standard atmosphere value). This
can be demonstrated by substituting from the hydrostatic equation (6.2) to give for
the left side of (6.22)
f
0
σ
1
σp
1
S
p
f
0
S
p
∂
∂p
∂χ
∂p
∂
∂p
∂T
∂t
∂
∂p
∂T
∂t
∂
∂t
∂T
∂p
=−
Rf
0
=−
f
0
≈−
pσ
R varies only slowly with height
in the troposphere. Because T is the departure of temperature from its standard
atmosphere value, the expression
where we have used the fact that S
p
=
∂T
∂p
f
0
S
−
1
p
is proportional to the local static stability anomaly divided by the standard atmo-
sphere static stability. Multiplication by f
0
gives this expression the same units as
vorticity.
The left side of (6.22) is negative when the lapse rate tendency is positive
(i.e., when the static stability tendency is negative). As was shown in Fig. 4.7,
an air column that moves adiabatically from a region of high static stability to a
region of low static stability is stretched vertically
∂ω
∂p > 0
so that the upper
portion of the column cools adiabatically relative to the lower portion. Thus, the
relative vorticity in (6.21) and the normalized static stability anomaly in (6.22)
are changed by equal and opposite amounts. For this reason the normalized static
stability anomaly is referred to as the
stretching vorticity.
Purely geostrophic motion (ω
0) is a solution to (6.21) and (6.22) only in
very special situations such as barotropic flow (no pressure dependence) or zonally
=
