Quasi-geostrophic theory (Continued)

John R. Gyakum

The quasi-geostrophic omega equation:

(2 + f022/∂p2) =

f0/p{vg(1/f02 + f)}

+ 2{vg (- /p)}+ 2(heating)

+friction

The Q-vector form of the quasi-geostrophic omega equation

(p2 + (f0

2/)2/∂p2) =

(f0/)/p{vgp(1/f02 + f)}

+ (1/)p2{vg p(- /p)}

= -2p Q - (R/p)T/x)

Excepting the 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

5400 m

X-(R/p)T/x)>0

-(R/p)T/x)<0

WarmColdWarm

The beta effect:

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/p)|T/y|k x (vg/x)where the x, y axes follow

respectively, the isotherms, and the opposite of the temperature gradient:

Xy

isothermscold

warm

Q = -(R/p)|T/y|k x (vg/x)

Therefore, the Q-vector is oriented 90 degrees clockwise to the geostrophic

change vector

(from Sanders and Hoskins 1990)

To see how this concept works, consider the case of only horizontalthermal advection forcing the quasi-geostrophic vertical motions:Q = -(R/p)|T/y|k x (vg/x)

(from Sanders and Hoskins 1990)

Now, consider the case of an equivalent-barotropic atmosphere (heightsand isotherms are parallel to one another, in which the only forcing forquasi-geostrophic vertical motions comes from horizontal vorticityadvections:

Q = -(R/p)|T/y|k x (vg/x)

(from Sanders and Hoskins 1990):

Q-vectors in a zone of geostrophicfrontogenesis:

Q-vectors in the entrance regionof an upper-level jet

Q = -(R/p)|T/y|k x (vg/x)

Static stability influence on QG omega

• Consider the QG omega equation: (2 + f0

22/∂p2) = f0/p{vg(1/f02 + f)} +2{vg (- /p)} + 2(heating)+friction

• The static stability parameter 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 adiabat2. 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 months3. 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: d > > m

• 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)

The shaded zone illustrates thetransition zone between the uppertroposphere’s weak stratificationand the relatively strong stratification of the lower stratosphere(Morgan and Nielsen-Gammon1998).

temperature (degrees C)

Cross-sectional analyses:

theta (dashed) and wind speed(solid; m per second)What is the shaded zone? Staytuned!

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, 2555-2579.

• Sanders, F., and B. J. Hoskins, 1990: An easy method for estimation of Q-vectors from weather maps. Wea. and Forecasting, 5, 346-353.