Transcript The quasi-geostrophic omega equation (see Bluestein

The quasi-geostrophic omega equation
(see Bluestein, Volume I; eq. 5.6.14
(2 + (f02 /)2/∂p2) =
(-f0 /) /p{-vgp(g + f)}
+ (-R/p) 2p{vg  pT}
+ (-R/p) 2p{1/Cp (dQ/dt)}
+ (-f0 /) /p{k   x F}
Where
 = - (RT/p)(ln/p)
(4.3.6)
Consider the following form of
omega forced by friction:
F = (f0 /K) /p{k   x F}
Now, consider that the friction
force, F, is proportional to, and
opposite in direction from, the
1000-mb geostrophic wind:
F = -(p) V1000
Therefore,
F = -(f0 /K) /p{(p) k   x V1000}
= -{(f0 1000)/ (K)} (/p)
Physically, we assume that (p) is a
maximum at the ground and vanishes
near the tropopause
Therefore, /p)
is greater than zero, and
F represents ascent or descent, when 1000
is cyclonic or anticyclonic, respectively.
Consider that when 1000 is cyclonic
(positive in the N. Hemisphere), the
surface friction is trying to reduce
this vorticity by producing a
negative vorticity change, while at
the top of the troposphere (e. g., 300
mb), there would be no friction force,
and therefore no vorticity change.
Therefore,
300 /t - 1000 /t > 0;
And the thermal vorticity increases
TH /t > 0;
And
the
tropospheric
thickness
decreases:
/t(z300 - z1000) < 0
Since there is no temperature advection or
diabatic heating, the necessary thickness
decrease (tropospheric column cooling) must
be accomplished by adiabatic cooling in
association with ascent. The justification for
the ascent is the same as for the forcing of
upward increase in cyclonic vorticity advection
however, the vertical profiles are different,
because of the vertical structure of friction.
1.
2.
3.
4.
Thus, the only means of cooling the column is through
ascent in a hydrostatically stable atmosphere
The thickness is decreasing
The heights are rising at all levels, but less so aloft
Convergence at low levels is overwhelmed by the effect of
frictional dissipation.
Aloft, divergence is responsible for the vorticity decreases
responsible for the vorticity increase below
top
z/t
V
F
-/t
bottom
-
0
+
-
0
+
Thermal lows
There are often surface-based circulation systems, particularly
equatorward of the middle-latitude belt of strong meridional
temperature gradient and eastward-migrating cyclones and
anticyclones, in which lows and highs change very little from day
to day, and in which little or no geostrophic advection of
temperature or vorticity is apparent.
Most prominent of such lows occur over western India, much of
the Middle East, and the Sahara during the warm season. Other
slightly weaker lows are found over northern Mexico, South Africa,
and Australia during the respective warm seasons.
Such systems are often called,
‘Thermal Lows’, because of
their cyclonic character of
circulation in the lower
troposphere, yet their
anticyclonic character in the
upper troposphere is also just
as prominent.
An example of a ‘thermal low’, showing the
circulations at both the 1000- and 300-mb
levels (solid, 1000-mb isobars; dashed, 1000300 mb thicknesses)
High300
Warm
Low1000
Such thermal lows are driven primarily by the
intense diabatic heating of the lower
troposphere by the arid ground below. The
large-scale ascent associated with this
heating, and due to surface friction, produces
adiabatic cooling, that might act together with
radiative cooling to maintain a stead-state
warm core. (The radiative cooling causes a
small contribution to descent, which reduces
the ascent associated with other caused, but
does not eliminate it)
However, the steady-state nature of the vorticity and
height change, cannot be explained on the basis of
quasi-geostrophic theory.