Potential vorticity and the invertibility principle (pp. 187-195)

1. Potential vorticity and the invertibility principle (pp. 187-195) To a first approximation, the atmospheric structure may be regarded as a superposition of positive and negative anomalies. Further, the observed wind field is aproximately equal to the sum of the winds associated with anomaly separately.

2. The actual PV field is the sum of the reference state and each of the anomalies • The reference state is that of constant f, and zero relative vorticity • PV in the reference atmosphere is a function only of potential temperature

3. We impose a balance condition on the wind field Geostrophic, gradient, and hydrostatic assumptions are examples of balance

4. Problem:To demonstrate that it is possible to determine, uniquely, the distribution of both vorticity and static stability associated with a PV field given the appropriate boundary conditions

5. Given the balance condition, assumption of a reference state, and boundary conditions: • Only one set of vorticity and static stability values fit the global distribution of PV • The balance condition used must be compatible with the space and time scales of the observed air motion

6. Given gradient wind and hydrostatic balance: • The second-order PDE for the wind is shown in eq. 1.9.22. • It is highly non-linear when relative vorticity, static stability, and f are functions of either potential temperature (vertically) or r (horizontally).

7. The invertibility principle: • Winds are induced by the PV-anomaly field • Eq. 1.9.22 is similar in structure to that of the quasi-geostrophic omega equation.

8. The effect of scale: • The vertical scale of a PV anomaly is proportional to the horizontal scale, and inversely proportional to the square root of the static stability parameter (eq. 1.9.33) • Typical vertical scale is about 60K (see sample soundings), which is as deep as the typical troposphere • Therefore, a synoptic-scale PV anomaly at upper-levels can induce a wind field all the way down to the ground

9. Relationship of scale to wind strength (eq. 1.9.38) • Small-scale PV anomalies of a given strength induce weak wind fields, whose vertical influence is only in a shallow layer • Large-scale PV anomalies of the same strength induce strong wind fields, whose vertical influence is deep. • The response of the atmosphere to PV anomalies is dependent on scale

10. Isentropic coordinates(potential temperature is the vertical coordinate) • Air parcels will conserve potential temperature for isentropic processes • Vertical motions can be visualized • moisture transports can be better visualized than on pressure surfaces • Isentropic surfaces can be used to diagnose potential vorticity

11. Consider the comparison of the cross sections we have been viewing:temperature cross section potential temperature cross section: isentropes slope up to cold air and downward to warm air high/low pressure on a theta surface corresponds to warm/ cold temperature on a pressure surface

12. 292 K Montgomery stream function ((m2 s-2 /100) solid) and pressure (hPa; dashed) 700 hPa heights (m; solid) and Temperature (K; dashed)

13. Potential vorticity structures • surface cyclone • surface anticyclone • upper-tropospheric trough • upper-tropospheric ridge

14. Surface cyclone (warm ‘anomaly’)PV = g(-q/p)zaq • warm air is associated with isentropes becoming packed near the ground (more PV) • surface cyclone is associated with a warm core with no disturbance aloft (zT= zgu- zgl=0-zgl<0 200 Pressure (hPa) cold warm more stable cold 1000 0 distance (km) 4000

15. Surface anticyclone (cold ‘anomaly’)PV = g(-q/p)zaq • cold air is associated with isentropes becoming less packed near the ground (less PV and smaller static stability) • surface anticyclone is associated with a cold core with no disturbance aloft (zT= zgu- zgl=0-zgl>0 200 Pressure (hPa) warm cold less stable warm 1000 0 distance (km) 4000

16. Upper-tropospheric trough (positive PV ‘anomaly’) PV = g(-q/p)zaq • cold tropospheric air is associated with isentropes becoming more packed near the tropopause (more PV and greater static stability) • upper tropospheric trough is associated with a cold core cyclone with no disturbance below (zT= zgu- zgl= zgu-0>0 200 warm more stable cold cold Pressure (hPa) warm warm cold less stable 1000 0 distance (km) 4000

17. Upper-tropospheric ridge (negative PV ‘anomaly’) PV = g(-q/p)zaq • warm tropospheric air is associated with isentropes becoming less packed near the tropopause (less PV and smaller static stability) • upper tropospheric ridge is associated with a warm core anticyclone with no disturbance below (zT= zgu- zgl= zgu-0<0 200 warm warm cold Pressure (hPa) less stable warm more stable cold cold 1000 0 distance (km) 4000

18. Comparison of potential vorticity analyses with traditional quasi-geostrophic analyses • Focus is on the PV perspective of QG vertical motions and the movement of high and low pressure systems

19. OK, but what about PV???? Consider a positive PV anomaly (PV maximum) aloft in a westerly shear flow: z + PV anomaly x 0

20. Now, consider a reference frame of the PV anomaly in which the anomaly is fixed: Consider the quasi-geostrophic Vorticity equation in the reference Frame of the positive PV anomaly 0= -vg(g + f)-f0 z + PV anomaly >0 <0 AVA; <0 CVA; >0 x 0

21. Now, consider the same PV anomaly in which the anomaly is fixed from the perspective of the thermodynamic equation: z + PV anomaly 0 = -vgT + ws(p/R) cool 0 x z + PV anomaly cool >0 <0 CA WA x 0

22. Consider vertical motions in the vicinity of a warm surface potential temperature anomaly (surrogate PV anomaly) from the vorticity equation: 0= -vg(g + f)-f0 z CVA >0 AVA <0 >0 <0 0 x + PV +

23. Consider vertical motions in the vicinity of a warm surface potential temperature anomaly (surrogate PV anomaly) from the thermodynamic equation: 0 = -vgT + ws(p/R) z cold <0 y >0 WA CA warm + PV +

24. Movement of surface cyclones and anticyclones on level terrain: Consider a reference state of potential temperature: North  -     +  

25. Consider that air parcels aredisplaced alternately poleward and equatorward within the east-west channel. Potential temperature is conserved for isentropic processes Since =0 at the surface, potential temperature changes Occur due to advection only  -   North - +  L/4 L/4  + 

26. The previous slide shows themaximum cold advection occurs one quarter of a wavelength east of cold potential temperature anomalies, with maximum warm advection occurring one-quarter of a wavelength east of the warm potential temperature anomalies. The entire wave travels (propagates), with the cyclones and anticyclones propagates eastward. Just as with traditional quasi-geostrophic theory, surface cyclones Travel from regions of cold advection to regions of warm advection. Surface anticyclones travel from regions of warm advection to regions Of cold advection.

27. Orographic effects on the motions of surface cyclones and anticyclones Consider a statically stable reference state in the vicinity of mountains as shown below, with no relative vorticity on a potential Temperature surface z  +     -   x

28. Note that cyclones and anticyclones move with higher terrain to their right, in the absence of any other effects.  +    -    N - + Mountain Range

29. References • Bluestein, H. B., 1993: Synoptic-dynamic meteorology in midlatitudes. Volume II: Observations and theory of weather systems. Oxford University Press. 594 pp. • Dickinson, M. J., and coauthors, 1997: The Marcch 1993 superstorm cyclogenesis: Incipient phase synoptic- and convective-scale flow interaction and model performance. Mon. Wea. Rev., 125, 3041-3072. • Hoskins, B. J., M. McIntyre, and A. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877-946. • Morgan, M. C., and J. W. Nielsen-Gammon, 1998: Using tropopause maps to diagnose midlatitude weather systems. Mon. Wea. Rev., 126, 2555-2579.