Potential vorticity and the dynamic tropopause John R. Gyakum Department of Atmospheric and Oceanic Sciences McGill University E-mail: gyakum@zephyr.meteo.mcgill.ca Phone: 514-398-6076

Outline Motivation (why use potential vorticity??) Isentropic coordinates Potential vorticity structures Potential vorticity invertability Dynamic tropopause analyses Comparison of potential vorticity analyses with traditional quasi-geostrophic analyses

Motivation (why use potential vorticity Motivation (why use potential vorticity??) PV = g(-q/p)zaq g is gravity, zaq is the component of absolute vorticity normal to an isentropic surface, and -q/p is the static stability It is conserved in adiabatic, frictionless three-dimensional flow

Consider the following animation of PV on the 325 potential temperature surface:

What are the units of PV? PV = g(-q/p)zaq typical tropospheric values: -q/p = 10K/100 hPa zaq≈f=10-4 s-1 and PV=10 m s-2(10K/100 hPa)(1 hPa(100 kg m s-2m-2)-1)10-4s -1 =10-6 m2 s -1 K kg-1= 1.0 Potential Vorticity Unit (PVU) Values of PV less than 1.5 PVUs are typically associated with tropospheric air and values greater than 1.5 PVUs are typically associated with stratospheric air

Now, we are prepared to appreciate the cross sections that we viewed at the end of this morning’s lecture! 1 PVU= 10-6 m2 s-1K kg-1 The shaded zone illustrates that the 1-3PVU band lies within the transition zone between the upper troposphere’s weak stratification and the relatively strong stratifica- tion of the lower stratosphere (Morgan and Nielsen-Gammon 1998).

Non-conservation of PV is often associated with interesting diabatic effects in explosive cyclones (Dickinson et al. 1997)

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

potential temperature cross section: isentropes slope up to cold air 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

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

Potential vorticity structures surface cyclone surface anticyclone upper-tropospheric trough upper-tropospheric ridge

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

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

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

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

Potential vorticity invertability If we know the distribution of isentropic potential vorticity, then we also know the wind field The wind field is ‘induced’ by the PV anomaly field The amplitude of the induced wind increases with size of the anomaly and with a reduction in static stability

Potential vorticity inversion may be used to understand the motions of troughs and ridges: Potential vorticity maxima and minima instantaneous winds max min max min N

Consider a PV reference state: Consider the PV contours at right with increasing PV northward (owing primarily to increase of the Coriolis parameter) larger PV PV+2dPV N PV+dPV PV PV-dPV

Consider the introduction of alternating PV anomalies: The sense of the wind field that is induced by the PV anomalies There will be a propagation to the left or to the west (largest effecct for large anomalies This effect is opposed by the eastward advective effect larger PV N PV+2dPV L + - PV+dPV + PV East

The application of PV inversion to the problem of cyclogenesis (Hoskins et al. 1985)

Dynamic tropopause analysis; What is the dynamic tropopause? A level (not at a constant height or pressure) at which the gradients of potential vorticity on an isentropic surface are maximized Large local changes in PV are determined by the advective wind This level ranges from 1.5 to 3.0 Potential vorticity units (PVUs)

Consider the cross sections that we have been viewing: Our focus is on the isentropic cross section seen below the opposing slopes of the PV surfaces and the isentropes result in the gradients of PV being sharper along isentropic surfaces than along isobaric surfaces

Dynamic tropopause pressure: A Relatively high (low pressure) Tropopause in the subtropics, and a Relatively low (high pressure) Tropopause in the polar regions; a Steeply-sloping tropopause in the Middle latitudes

Tropopause potential temperatures (contour interval of 5K from 305 K to 350 K) at 12-h intervals (from Morgan and Nielsen-Gammon 1998) The appearance of the 330 K closed contour in panel c is produced by the large values of equivalent potential temperature ascending in moist convection and ventilated at the tropopause level; as discussed earlier, this is an excellent means of showing the effects of diabatic heating, and verifying models

the sounding shows a tropopause fold extending from 500 to 375 hPa at 1200 UTC, 5 Nov. 1988 for Centerville, AL, with tropospheric air above and extending to 150 hPa. The fold has descended into Charleston, SC by 0000 UTC, 6 November 1988 to the 600-500 hPa layer. The same isentropic levels are associated with each fold

Coupling index: Theta at the tropopause Minus the equivalent Potential temperature at Low levels (a poor man’s lifted index)

December 30-31, 1993 SLP And 925 hPa theta

An example illustrates the detail of the dynamic tropopause (1 An example illustrates the detail of the dynamic tropopause (1.5 potential vorticity units) that is lacking in a constant pressure analysis

250 and 500-hPa analyses showing the respective subtropical and polar jets: 250-hPa z and winds 500-hPa z and winds

Dynamic tropopause map shows the properly-sharp troughs and ridges and full amplitudes of both the polar and subtropical jets

The dynamic tropopause animation during the 11 May 1999 hailstorm:

An animation of the dynamic tropopause for the period from December 1, 1998 through February 28, 1999:

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

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

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

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 x z + PV anomaly cool >0 <0 CA WA x

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 x + PV +

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 +

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

Consider that air parcels are displaced 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  +  

The previous slide shows the maximum 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.

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

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

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.