Quasi-geostrophic omega
analyses
John Gyakum
ATOC-541
January 4, 2006
Outline
• The QG omega equation and its physical
interpretation
The quasi-geostrophic omega
equation:
(s2 + f022/∂p2) =
f0/p{vg(1/f02 + f)}
+ 2{vg  (- /p)}+
2(heating)
+friction
Before considering the physical effects of the quasigeostrophic forcing, consider the thermal vorticity:
Thermal= geostrophic Top- geostrophic Bottom
= (g/f) times (2zTop - 2zBottom)
= (g/f) times (2h)
Where h = zTop - zBottom = thickness
Therefore, the thermal vorticity is the algebraic difference
between the top and bottom geostrophic vorticity
A cold pool of air is also a thermal
vorticity maximum
Top
pressure
cold
cold
Bottom
Pressure
(high)
Top
Pressure
(low)
Bottom
Pressure
Therefore, a cold trough is a
thermal vorticity maximum
Consider the ‘vorticity advection’ forcing for
The quasi-geostrophic vorticity equation:
(s2 + f022/∂p2) =
f0/p{vg(1/f02 + f)} + ...
An upward increase in cyclonic vorticity advection
(or an upward decrease in anticyclonic vorticity advection)
produces an increase in cyclonic thermal vorticity, which
cools the column locally
How does this cooling occur?
Local change in temperature=
Horizontal temperature advection + vertical motions
+ diabatic changes
Thus, the only means of cooling the column is through ascent
in a hydrostatically stable atmosphere
1.
2.
3.
4.

The thickness is decreasing
The heights are falling at all levels, but more so aloft
Convergence is responsible for the vorticity increase below
‘PVA’ overwhelms the effects of divergence aloft
top
z/t
V
-/t
bottom
-
0
+
-
0
+
Now consider the effect of warm advection in the quasigeostrophic omega equation
(s2 + f022/∂p2) =
+ 2{vg  (- /p)}+...
Warming produces an increase in thickness
and a warm ridge locally;
Therefore, Thermal/t < 0
And…
1. The thickness is increasing
2. Divergence is responsible for all
Vorticity changes…because of mass
Continuity, the vertical integral of
Vorticity change must be zero
geostrophic upper/t < geostrophic lower/t
So,…how is this done???
The vorticity changes must occur
Only through divergence/
Convergence, since there is no
Vorticity advection.
Therefore, we must have the
Divergence structure seen here:
top V

z/t
-/t
bottom
-
0
+
Note that the diabatic effects of heating have the same
mathematical structure as for the horizontal
temperature advection effect:
(s2 + f022/∂p2) =
+ 2{vg  (- /p)}+ 2(heating)+...
Therefore:
And…
1. The thickness is increasing
2. Divergence is responsible for all
Vorticity changes…because of mass
Continuity, the vertical integral of
Vorticity change must be zero
geostrophic upper/t < geostrophic lower/t
The vorticity changes must occur
Only through divergence/
Convergence, since there is no
Vorticity advection.
Therefore, we must have the
Divergence structure seen here:
top
V

z/t
-/t
bottom
Diabatic heating effects
• Regions of surface sensible heat flux
(particularly strong in cold-air outbreaks
over relatively warm waters; e. g. Great
Lakes in late autumn, early winter; cold
outbreaks over the Gulf Stream current of
the North Atlantic)
• Regions of latent heating (stratiform and
moist convection)
Current synoptic conditions (1200 UTC, 4
January 2006; Eta model SLP; 1000-500
hPa thickness)
The 300-hPa heights and isotachs for 1200 UTC,
4 January 2006
Reference:
• Bluestein, H. B., 1992: Synoptic-dynamic
meteorology in midlatitudes. Volume I:
Principles of kinematics and dynamics.
Oxford University Press. 431 pp.