The quasi-geostrophic omega equation (see Bluestein, Volume I; eq (  2 + (f 0 2 /  )  2 /∂p 2 )  = (-f 0 /  )  /  p{-v g  p (  g + f)} + (-R/  p)  2 p {v g   p T} + (-R/  p)  2 p {1/C p (dQ/dt)} + (-f 0 /  )  /  p{k   x F}

Where  = - (RT/p)(  ln  /  p) (4.3.6)

Consider the following form of omega forced by friction:  F = (f 0 /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) V 1000

Therefore,  F = - (f 0 /K  )  /  p{  (p) k   x V 1000 } = -{(f 0  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(z z 1000 ) < 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.

Thus, the only means of cooling the column is through ascent in a hydrostatically stable atmosphere top bottom  V FF  z/  t -  /  t 1. The thickness is decreasing 2. The heights are rising at all levels, but less so aloft 3.Convergence at low levels is overwhelmed by the effect of frictional dissipation. 4.Aloft, divergence is responsible for the vorticity decreases responsible for the vorticity increase below

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 and 300-mb levels (solid, 1000-mb isobars; dashed, mb thicknesses) Warm Low 1000 High 300

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.