1. Potential Vorticity (PV) Analysis
Advanced Synoptic
M. D. Eastin

2. PV Analysis Outline: History and Definition
Synoptic-Scale Distribution
Equations and Practical Concepts
“PV Non-conservation”
“PV Impermeability”
“PV Inversion”
Advantages / Disadvantages
Advanced Synoptic
M. D. Eastin

3. PV Analysis: History Introduction into Obscurity:
Concept first introduced by Rossby (1936) for a barotropic ocean
Adapted by Rossby (1940) for a barotropic atmosphere
Expanded by Ertel (1942) for a baroclinic atmosphere
Eliassen and Kleinschmidt (1957) first developed /applied the concepts of
“PV conservation” and “PV inversion” for a baroclinic atmosphere
Rise to Popularity:
Hoskins et al. (1985) wrote a long paper demonstrating the applicability
of PV-based interpretation and techniques to a broad spectrum of geophysical
problems by emphasizing the succinct and powerful manner in which PV relates
to both theoretical and observational aspects of atmospheric science
Rossby-wave propagation
Barotropic / baroclinic instability and cyclogenesis
Structure and evolution of mid-latitude cyclones
Use of PV analysis has been “en vogue” ever since…
Advanced Synoptic
M. D. Eastin

4. PV Analysis: Definitions
Basic Idea:
Potential vorticity represents the absolute vorticity an air column would have
if it were brought isentropically to a standard latitude and stretched/shrunk
to a standard depth
Analogous to “potential temperature” for an air parcel
Multiple Formulations:
Valid for Full Governing Equations
Equation [4.1] in Lackmann
Valid for Isentropic Analysis
Equation [4.2] in Lackmann
Valid for QG Analysis
Equation [2.38] in Lackmann
Valid for Shallow-water (Barotropic) Analysis
Equation [4.26] in Holton
Advanced Synoptic
M. D. Eastin

5. PV Analysis: Synoptic-Scale Distribution
Basic Concepts:
Related to the product of absolute vorticity and static stability
Largest in polar regions (large f) and the stratosphere (large ∂θ/∂p)
Potential Vorticity Unit (PVU) → K kg-1 m2 s-1
→ Troposphere PVU < 1.0
→ Stratosphere PVU > 2.0
Dynamic Tropopause → Definition of the tropopause using a PV isosurface
→ Often 1.5 or 2.0 PVU
Dynamic
Tropopause
Advanced Synoptic
M. D. Eastin

6. PV Analysis: Synoptic-Scale Distribution
Basic Concepts:
PV Anomalies → Defined relative to a climatological average
Positive anomalies → Low pressure / Troughs
Negative anomalies → High pressure / Ridges
→ Stratosphere is a “PV reservoir”
→ Tropospheric synoptic-scale troughs are produced by “injections”
of stratospheric PV anomalies down into the troposphere
Dynamic
Tropopause
Advanced Synoptic
M. D. Eastin

7. PV // Potential Temperature
PV Analysis: Synoptic-Scale Distribution
Comparison to Isobaric Analyses:
Regions of low geopotential heights correspond to regions with large PV values
All troughs (even weak ones) show some evidence of PV > 1.5 (stratospheric air)
Cross-sections demonstrate the downward extrusion of large PV air associated
with each tropospheric trough
500mb Heights // mb PV
PV // Potential Temperature S N
Advanced Synoptic
M. D. Eastin

8. PV Analysis: Synoptic-Scale Distribution
Comparison to Isobaric Analyses:
Notice how locally strong geopotential height gradients (i.e., geostrophic jet maxima)
correspond to strong lateral gradients in stratospheric PV
This “double stair step” PV pattern is indicative of two distinct westerly jet maxima
[northern ↔ polar front jet southern ↔ subtropical jet]
500mb Heights // mb PV
PV // Zonal winds S N W E W
Advanced Synoptic
M. D. Eastin

9. Potential Temperature and Winds on 2-PVU Surface
PV Analysis: Synoptic-Scale Distribution
Dynamic Tropopause Maps:
A convenient way to plot the relevant features of all upper-air jet streams
Select a PV surface → usually 1.5 PVU or 2.0 PVU
Plot potential temperature, pressure, and winds on the PV surface
Provides a “topographic map” of the tropopause
500mb Heights // mb PV
Potential Temperature and Winds on 2-PVU Surface
Advanced Synoptic
M. D. Eastin

10. Pressure and Winds on 2-PVU Surface
PV Analysis: Synoptic-Scale Distribution
Dynamic Tropopause Maps:
A convenient way to plot the relevant features of all upper-air jet streams
Select a PV surface → usually 1.5 PVU or 2.0 PVU
Plot potential temperature, pressure, and winds on the PV surface
Provides a “topographic map” of the tropopause
500mb Heights // mb PV
Pressure and Winds on 2-PVU Surface
Advanced Synoptic
M. D. Eastin

11. PV Analysis: Equations
Derivations and Interpretations:
The full derivations of the PV-conservation and PV-tendency equations for isentropic
coordinates are provided in the Lackmann text (Section 4.3.1)
PV-Conservation:
where
Valid for adiabatic, frictionless flow along isentropic surfaces
In such situations → PV remains constant, however, the relative vorticity, corioils
parameter, and/or static stability may change
→ PV can be used as a “tracer” to track air parcel motions and
determine a parcels origin(s) at a previous time
→ Evaluate non-conservative processes by documenting any
PV changes (which must have resulted from either
diabatic or frictional processes) **
Equation (4.16) in Lackmann text
Advanced Synoptic
M. D. Eastin

12. PV Analysis: Equations
Derivations and Interpretations:
The full derivations of the PV-conservation and PV-tendency equations for isentropic
coordinates are provided in the Lackmann text (Section 4.3.1)
PV-Tendency:
where: and
Term A → Vertical Diabatic Forcing
→ Relevant for vertically-stacked systems
Term B → Sheared Diabatic Forcing
→ Relevant for vertically-tilted systems (developing cyclones / fronts)
Term C → Frictional Forcing (often neglected…we will too!!)
Equation (4.17)
in Lackmann text
Term B
Term A
Term C
Advanced Synoptic
M. D. Eastin

13. PV Analysis: Non-Conservation
How do Diabatic Processes change PV? Term A: Vertical Diabatic Forcing
Assume isentropic surfaces are horizontally-uniform (equivalent to geopotential height)
Heating maximum (due to condensation) is centered in the lower troposphere
Above heating max → local heights ascend → divergence
Below heating max → local isentropes / heights descend → convergence
As the height anomalies amplify, local height gradient accelerations will produce
convergence (divergence) below (above) the heating maximum (just like in QG theory…)
The heating maximum also alters the local static stability:
Above heating max → reduced static stability → PV decreases (-)
Below heating max → increased static stability → PV increases (+)
Advanced Synoptic
M. D. Eastin

14. PV Analysis: Non-Conservation
How do Diabatic Processes change PV? Term B: Sheared Diabatic Forcing
Assume isentropic surfaces are tilted with height (as is often the case near fronts)
Heating maximum (due to condensation) is centered in the lower troposphere
In this case, the heating maximum alters the (1) local horizontal (isentropic) gradients,
(2) local static stability, and (3) local vertical shear (due to thermal wind balance),
producing a complex response, but…
Above heating max → PV always decreases (-) → horizontally displaced
Below heating max → PV always increases (+) → horizontally displaced
The magnitude and direction of horizontal displacement are functions of both
the vertical shear and the local heating rate
Advanced Synoptic
M. D. Eastin

15. PV Analysis: Impermeability
Implications for Cases of Significant Mass Removal:
If we integrate the PV-conservation equation over an isentropic volume bounded laterally
by a streamline on which flow is adiabatic and frictionless, one can easy show
This is the “PV impermeability theorem” from Haynes and McIntyre (1987)
Powerful constraint as to how PV can change
PV is not “created” nor “destroyed, but rather “redistributed”
Any process that results in the significant movement of mass across an isentropic surface
will alter the local potential vorticity structure:
Heavy precipitation (thunderstorms and tropical cyclones)
Dry deposition of large particles (sand storms)
Equation (4.24) in Lackmann text
Advanced Synoptic
M. D. Eastin

16. PV Analysis: Inversion
Invertibility Principle:
Allows the user to “recover” the balanced wind and thermodynamic fields associated with
any given PV anomaly
The balanced flow (red) and related temperature and pressure structures (not shown)
extend to spatial locations far removed from that of the anomaly itself (green)
Analogous to the far electric fields associated with point charges
(…this is one reason why you are required to take Physics-2)
Advanced Synoptic
M. D. Eastin

17. PV Analysis: Inversion
Invertibility Principle:
Allows the user to “recover” the balanced wind and thermodynamic fields associated with
any given PV anomaly
The PV field can be sub-divided into as many “PV pieces” as desired
Each PV piece can then be inverted separately (called “piecewise inversion”)
to determine its partial contribution to the total structure of a given system
You could also explore partial
contributions from different
atmospheric constituents:
water vapor
ozone
pollution 1 3 2
Advanced Synoptic
M. D. Eastin

18. PV Analysis: Advantages / Disadvantages
Synoptic-scale dynamic tropopause maps allows one to easily see all
relevant upper-level jet streaks and system structure on one map
Through piecewise inversion, one can diagnose which physical processes
were responsible for the “observed” PV distribution.
Post-event analysis of poorly forecast cases
Evaluate and quantify contributions from non-conservative processes
Evaluate and quantify numerical model errors in system structure
Learn limitations of numerical models in certain forecast situations
Allows forecasters to assign confidence to each numerical model
(see examples on next few slides…)
Disadvantages:
Computations must be performed to interpolate pressure, wind, and
moisture data onto isentropic surfaces
Nearly impossible to conduct piecewise inversion from only observations
(must use numerical model analysis and forecast fields**)
Advanced Synoptic
M. D. Eastin

19. Potential Temperature (5-K interval) // PV (1-PVU interval)
PV Analysis: Advantages / Disadvantages
Diagnosing contributions to System Structure:
Potential Temperature (5-K interval) // PV (1-PVU interval)
Extra-tropical cyclone
Large deep stratospheric source
Smaller low-level diabatic source
Tropical cyclone
Minimal stratospheric source
Large low-level diabatic source
Sub-tropical (hybrid) cyclone
Some stratospheric source
Equal low-level diabatic source
Advanced Synoptic
M. D. Eastin

20. Radar // 24-hr Eta precipitation forecast
PV Analysis: Advantages / Disadvantages
Diagnosing contributions to Model Error:
January 2000 snowstorm across the Southeast
One model (RUC) provided good forecasts
Other popular models (AVN/GFS and NAM/Eta)
did NOT forecast the event well…Why?
Underestimated diabatic PV production
from two regions of heavy precipitation
which eventually merged over SC
Human forecasters could see the errors
SLP // mb PV
RUC
Eta
Analysis
24-hr
Radar // 24-hr Eta precipitation forecast
Advanced Synoptic
M. D. Eastin

21. PV Analysis: Websites Real-time and Archived Analyses:
SUNY Albany:
University of Reading:
MIT:
University of Washington:
(personal webpage)
University of Oklahoma:
Advanced Synoptic
M. D. Eastin

22. References Advanced Synoptic M. D. Eastin
Bishop, C. H. and A. J. Thorpe, 1994: Potential vorticity and the electrostatics analogy: Quasi-geostrophic theory.
Quarterly Journal of the Royal Meteorological Society, 120,
Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume I: Principles of Kinematics and Dynamics.
Oxford University Press, New York, 431 pp.
Bluestein, H. B, 1993: Synoptic-Dynamic Meteorology in Midlatitudes. Volume II: Observations and Theory of Weather
Systems. Oxford University Press, New York, 594 pp.
Brennan, M. J., G. M. Lackmann, and K. A. Mahoney, 2008: Potential vorticity (PV) thinking in operations: The utility
of non-conservation. Weather and Forecasting, 23,
Davis, C. A., 1992b: Piecewise potential vorticity inversion. Journal of Atmospheric Science, 49,
Eliassen A., and E. Kleinschmidt, 1957: Dynamic Meteorology, Encyclopedia of Physics, Springer Publishing, 1-154
Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic
heating and frictional or other forces. Journal of Atmospheric Science, 44,
Hoskins, B.J., McIntyre, M.E. and Robertson, A.W., 1985: On the use and significance of isentropic potential vorticity
maps. Quarterly Journal of the Royal Meteorological Society, 111,
Lackmann, G., 2011: Mid-latitude Synoptic Meteorology – Dynamics, Analysis and Forecasting, AMS, 343 pp.
Rossby, C. G., 1940: Planetary flow patterns in the atmosphere. Quarterly Journal of the Royal Meteorological Society,
66,
Samuelson, R. M., 2003: Rossby, Ertel, and potential vorticity. University of Princeton, 9 pp.
Schubert W., and co-authors, 2004: English translations of twenty-one of Ertel’s papers on geophysical fluid dynamics,
Meteorologische Zeitschrift, 13,
Advanced Synoptic
M. D. Eastin