Quasi-geostrophic theory
(Continued)
John R. Gyakum
The quasi-geostrophic omega
equation:
(s2 + f022/∂p2) =
f0/p{vg(1/f02 + f)}
+ 2{vg  (- /p)}+
2(heating)
+friction
The Q-vector form of the quasigeostrophic omega equation
(p2 + (f02/s)2/∂p2) =
(f0/s)/p{vgp(1/f02 + f)}
+ (1/s)p2{vg  p(- /p)}
= -2p  Q - (R/sp)b(T/x)
Excepting the b effect for
adiabatic and frictionless
processes:
• Where Q vectors converge, there is forcing
for ascent
• Where Q vectors diverge, there is forcing
for descent
5340 m
The beta effect:
Warm 5400 m Cold
X
-(R/sp)b(T/x)>0
Warm
-(R/sp)b(T/x)<0
Advantages of the Q-vector
approach:
• Forcing functions can be evaluated on a constant
pressure surface
• Forcing functions are “Galilean Invariant” (the
functions do not depend on the reference frame in
which they are being measured)…although the
temperature advection and vorticity advection terms
are each not Galilean Invariant, the sum of these two
terms is Galilean Invariant
• There is not partial cancellation between terms as
there typically is with the traditional formulation
Advantages of the Q-vector
approach (continued):
• The Q-vector forcing function is exact, under the
adiabatic, frictionless, and quasi-geostrophic
approximation; no terms have been neglected
• Q-vectors may be plotted on analyses of height and
temperature to obtain a representation of vertical
motions and ageostrophic wind
However:
• One key disadvantage of the Q-vector
approach is that Q-vector divergence is not
as physically meaningful as is seen in either
horizontal temperature advection or
vorticity advection
• To remedy this conceptual difficulty,
Hoskins and Sanders (1990) have proposed
the following analysis:
Q = -(R/sp)|T/y|k x (vg/x)
where the x, y axes follow
respectively, the isotherms, and the
opposite of the temperature gradient:
isotherms
cold
y
X
warm
Q = -(R/sp)|T/y|k x (vg/x)
Therefore, the Q-vector is oriented 90
degrees clockwise to the geostrophic
change vector
To see how this concept works, consider the case of only horizontal
thermal advection forcing the quasi-geostrophic vertical motions:
Q = -(R/sp)|T/y|k x (vg/x)
(from Sanders and Hoskins 1990)
Now, consider the case of an equivalent-barotropic atmosphere (heights
and isotherms are parallel to one another, in which the only forcing for
quasi-geostrophic vertical motions comes from horizontal vorticity
advections:
Q = -(R/sp)|T/y|k x (vg/x)
(from Sanders and Hoskins 1990)
(from Sanders and Hoskins 1990):
Q = -(R/sp)|T/y|k x (vg/x)
Q-vectors in a zone of geostrophic
frontogenesis:
Q-vectors in the entrance region
of an upper-level jet
Static stability influence on QG
omega
• Consider the QG omega equation:
(s2 + f022/∂p2) =
f0/p{vg(1/f02 + f)} +2{vg
 (- /p)}
+
2(heating)+friction
• The static stability parameter
s=-T ln/p
Static stability (continued)
1. Weaker static stability produces more
vertical motion for a given forcing
2. Especially important examples of this
effect occur when cold air flows over
relatively warm waters (e.g.; Great Lakes
and Gulf Stream) during late fall and winter
months
3. The effect is strongest for relatively
short wavelength disturbances
Static stability (Continued)
1. The ‘effective’ static stability is reduced for
saturated conditions, when the lapse rate is
referenced to the moist adiabatic, rather than the dry
adiabat
2. Especially important examples of this effect occur
in saturated when cold air flows over relatively
warm waters (e.g.; Great Lakes and Gulf Stream)
during late fall and winter months
3. The effect is strongest for relatively short
wavelength disturbances and in warmer temperatures
Static stability
• Conditional instability occurs when the
environmental lapse rate lies between the
moist and dry adiabatic lapse rates:
Gd > g > Gm
• Potential (or convective) instability occurs
when the equivalent potential temperature
decreases with elevation (quite possible for
such an instability to occur in an inversion
or absolutely stable conditions)
Cross-sectional analyses:
temperature (degrees C)
The shaded zone illustrates the
transition zone between the upper
troposphere’s weak stratification
and the relatively strong
stratification of the lower
stratosphere
(Morgan and Nielsen-Gammon
1998).
theta (dashed) and wind speed
(solid; m per second)
What is the shaded zone? Stay
tuned!
References:
• Bluestein, H. B., 1992: Synoptic-dynamic
meteorology in midlatitudes. Volume I:
Principles of kinematics and dynamics. Oxford
University Press. 431 pp.
• Morgan, M. C., and J. W. Nielsen-Gammon, 1998:
Using tropopause maps to diagnose midlatitude
weather systems. Mon. Wea. Rev., 126, 25552579.
• Sanders, F., and B. J. Hoskins, 1990: An easy
method for estimation of Q-vectors from weather
maps. Wea. and Forecasting, 5, 346-353.