# PV-Conservation

```Potential Vorticity (PV) Analysis
M. D. Eastin
PV Analysis
Outline:
• History and Definition
• Synoptic-Scale Distribution
• Equations and Practical Concepts
• “PV Non-conservation”
• “PV Impermeability”
• “PV Inversion”
M. D. Eastin
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…
M. D. Eastin
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:
P
1

  
P   g a
P

p
1 2
  f o  


  f 
fo
p   p 
P
g  f
h
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
M. D. Eastin
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)
P   g a

p
Potential Vorticity Unit (PVU) → 10-6 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
M. D. Eastin
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
M. D. Eastin
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 // 300-500mb PV
PV // Potential Temperature
N
S
N
S
M. D. Eastin
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 // 300-500mb PV
PV // Zonal winds
N
W
E
S
N
W
S
M. D. Eastin
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 // 300-500mb PV
Potential Temperature and Winds on 2-PVU Surface
M. D. Eastin
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 // 300-500mb PV
Pressure and Winds on 2-PVU Surface
M. D. Eastin
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:
DP
0
Dt
where
P   g a

p
Equation (4.16) in Lackmann text
• 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) **
M. D. Eastin
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:
    u  v     F  F 
DP
  
  

  g   a


Dt

p

p

y

p

x

p

x

p

y

 
 

Term A
where:

P   g a
p
Term B
and
 
Equation (4.17)
in Lackmann text
Term C

t
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!!)
M. D. Eastin
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 (+)
M. D. Eastin
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
M. D. Eastin
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
P
 t dV  0
Equation (4.24) in Lackmann text
• 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)
M. D. Eastin
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)
M. D. Eastin
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
n
Ptotal   Pi
i 1
Ptotal  P1  P2  P3
• You could also explore partial
contributions from different
atmospheric constituents:
1
2
3
• water vapor
• ozone
• pollution
M. D. Eastin
 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…)
 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**)
M. D. Eastin
Diagnosing contributions to System Structure:
Potential Temperature (5-K interval) // PV (1-PVU interval)
Extra-tropical cyclone
Tropical cyclone
Sub-tropical (hybrid) cyclone
Large deep stratospheric source
Minimal stratospheric source
Some stratospheric source
Smaller low-level diabatic source
Large low-level diabatic source
Equal low-level diabatic source
M. D. Eastin
Diagnosing contributions to Model Error:
RUC
Analysis
• 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
Radar // 24-hr Eta precipitation forecast
SLP // 900-700mb PV
Eta
24-hr
SLP // 900-700mb PV
M. D. Eastin
PV Analysis: Websites
Real-time and Archived Analyses:
SUNY Albany:
http://www.atmos.albany.edu/index.php?d=wx_data
MIT:
http://wind.mit.edu/~reanal/pv.html
University of Washington: http://www.atmos.washington.edu/~hakim/tropo/info.html
(personal webpage)
University of Oklahoma:
http://weather.ou.edu/~scavallo/real_time_plots.html
(personal webpage)
M. D. Eastin
References
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, 713-731.
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, 168-182
Davis, C. A., 1992b: Piecewise potential vorticity inversion. Journal of Atmospheric Science, 49, 1397-1411
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, 828-841
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, 877-946.
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, 68-87.
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, 527-576.
M. D. Eastin
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