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??)

PV = g(-/p)a

g is gravity, a is the component of absolute vorticity

normal to an isentropic surface, and-/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:

QuickTime™ and aGIF decompressor

are needed to see this picture.

What are the units of PV?

PV = g(-/p)a

typical tropospheric values:-/p = 10K/100 hPa

a≈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

The shaded zone illustrates that the1-3PVU band lies within thetransition zone between the uppertroposphere’s weak stratificationand the relatively strong stratifica-tion of the lower stratosphere(Morgan and Nielsen-Gammon1998).

Now, we are prepared to appreciate the cross sectionsthat we viewed at the end of this morning’s lecture!

1 PVU= 10-6 m2 s-1K kg-1

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

Consider the comparison of the cross sections we have been

viewing:

temperature cross section

potential temperature crosssection:isentropes slope up to cold airand downward to warm air

high/low pressure on a thetasurface corresponds to warm/cold temperature on a pressuresurface

700 hPa heights (m; solid) andTemperature (K; dashed)

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

Potential vorticity structures

• surface cyclone

• surface anticyclone

• upper-tropospheric trough

• upper-tropospheric ridge

Surface cyclone (warm ‘anomaly’) PV = g(-/p)a

• 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 (gu- gl=0-gl<0

cold coldwarmmore stable

0 distance (km) 4000

Pressure(hPa)

1000

200

Surface anticyclone (cold ‘anomaly’) PV = g(-/p)a

• 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 (gu- gl=0-gl>0

warm warmcoldless stable

0 distance (km) 4000

Pressure(hPa)

1000

200

Upper-tropospheric trough (positive PV ‘anomaly’) PV = g(-/p)a

• 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 (gu- gl= gu->0

warm warmcoldless stable

0 distance (km) 4000

Pressure(hPa)

1000

200

cold coldwarmmorestable

Upper-tropospheric ridge (negative PV ‘anomaly’) PV = g(-/p)a

• 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 (gu- gl= gu-<0

cold cold

0 distance (km) 4000

Pressure(hPa)

1000

200

less stablewarm

morestable

coldwarm warm

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

N

Consider a PV reference state:

• Consider the PV contours at right with increasing PV northward (owing primarily to increase of the Coriolis parameter)

N

larger PV

PV-PV

PV

PV+PV

PV+2PV

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

N

larger PV

PV

PV+PV

PV+2PV

+ - +

L

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; aSteeply-sloping tropopause in theMiddle latitudes

Tropopause potentialtemperatures (contour intervalof 5K from 305 K to 350 K) at12-h intervals (from Morgan andNielsen-Gammon 1998)

The appearance of the 330 K closed contour in panel c is produced by the large values ofequivalent potential temperatureascending in moist convectionand ventilated at the tropopauselevel;as discussed earlier, this is anexcellent means of showing theeffects of diabatic heating, andverifying models

the sounding shows a tropopausefold extending from 500 to 375hPa at 1200 UTC, 5 Nov. 1988for Centerville, AL,with tropospheric air above and extending to 150 hPa.

The fold has descended intoCharleston, SC by 0000 UTC,6 November 1988 to the 600-500hPa layer. The same isentropiclevels are associated with each fold

Coupling index:Theta at the tropopauseMinus the equivalentPotential temperature atLow levels(a poor man’s lifted index)

December 30-31, 1993 SLPAnd 925 hPa theta

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

QuickTime™ and aGIF decompressor

are needed to see this picture.

QuickTime™ and aGIF decompressor

are needed to see this picture.

QuickTime™ and aGIF decompressor

are needed to see this picture.

The dynamic tropopause animation during the 11 May

1999 hailstorm:

QuickTime™ and aGIF decompressor

are needed to see this picture.

An animation of the dynamic tropopause for the period from

December 1, 1998 through February 28, 1999:

QuickTime™ and aGIF decompressor

are needed to see this picture.

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:

+ PV anomaly

0 x

z

Now, consider a reference frame of the PV anomaly in which the anomaly is fixed:

0 x

z

+ PV anomaly

CVA; >0 AVA; <0

<0>0

Consider the quasi-geostrophicVorticity equation in the referenceFrame of the positive PV anomaly

0= -vg(g + f)-f0

Now, consider the same PV anomaly in which the anomaly is fixed from the perspective of the

thermodynamic equation:

+ PV anomaly

x0

z

cool

x

z

0

+ PV anomaly

coolCA WA>0 <0

0 = -vg T + (p/R)

Consider vertical motions in the vicinity of a warm surface potential temperature anomaly (surrogate PV anomaly) from

the vorticity equation:

x

z

0

AVA<0

>0

CVA>0

<0

+ PV+

0= -vg(g + f)-f0

Consider vertical motions in the vicinity of a warm surface potential temperature anomaly (surrogate PV anomaly) from

the thermodynamic equation:

0 = -vg T + (p/R)

>0

cold

warm

<0

CAWA

+ PV+

z

y

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 processesSince =0 at the surface, potential temperature changesOccur 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 regionsOf 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 potentialTemperature surface

z

x

+

-

Note that cyclones and anticyclones move with higher terrain to their right, in the absence of any

other effects.

N

MountainRange

+ -

-

+

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.