1. Quasi-geostrophic omega analyses John Gyakum ATOC-541 January 4, 2006

2. Outline The QG omega equation and its physical interpretation

3. The quasi-geostrophic omega equation: (  2 + f 0 2  2 /∂p 2 )  = f 0  /  p{v g  (1/f 0  2  + f)} +  2 {v g   (-  /  p)}+  2 (heating) +friction

4. Before considering the physical effects of the quasi- geostrophic forcing, consider the thermal vorticity:  Thermal =  geostrophic Top -  geostrophic Bottom = (g/f) times (  2 z Top -  2 z Bottom ) = (g/f) times (  2 h) Where h = z Top - z Bottom = thickness

5. 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 Bottom Pressure (high) cold Top Pressure (low) Bottom Pressure cold

6. Therefore, a cold trough is a thermal vorticity maximum Consider the ‘vorticity advection’ forcing for The quasi-geostrophic vorticity equation: (  2 + f 0 2  2 /∂p 2 )  = f 0  /  p{v g  (1/f 0  2  + f)} +...

7. 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

8. Thus, the only means of cooling the column is through ascent in a hydrostatically stable atmosphere - 0 + top bottom  V  - 0 +  z/  t -  /  t 1. The thickness is decreasing 2. The heights are falling at all levels, but more so aloft 3. Convergence is responsible for the vorticity increase below 4. ‘PVA’ overwhelms the effects of divergence aloft

9. Now consider the effect of warm advection in the quasi- geostrophic omega equation (  2 + f 0 2  2 /∂p 2 )  = +  2 {v g   (-  /  p)}+... Warming produces an increase in thickness and a warm ridge locally; Therefore,  Thermal /  t < 0

10. And…  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: - 0 + top bottom   V  z/  t -  /  t 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

11. (  2 + f 0 2  2 /∂p 2 )  = +  2 {v g   (-  /  p)}+  2 (heating)+... Note that the diabatic effects of heating have the same mathematical structure as for the horizontal temperature advection effect:

12. Therefore: 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 And…  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:  z/  t -  /  t  V  top bottom

13. 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)

14. Current synoptic conditions (1200 UTC, 4 January 2006; Eta model SLP; 1000-500 hPa thickness)

15. The 300-hPa heights and isotachs for 1200 UTC, 4 January 2006

16. Reference: Bluestein, H. B., 1992: Synoptic-dynamic meteorology in midlatitudes. Volume I: Principles of kinematics and dynamics. Oxford University Press. 431 pp.