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Advanced Synoptic M. D. Eastin
Potential Vorticity (PV) Analysis
Advanced Synoptic M. D. Eastin
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
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…
PV Analysis: History
Advanced Synoptic M. D. Eastin
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
PV Analysis: Definitions
h
fP g
1
P
pgP a
p
f
pf
fP o
o 21
Advanced Synoptic M. D. Eastin
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) → 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
PV Analysis: Synoptic-Scale Distribution
pgP a
DynamicTropopause
Advanced Synoptic M. D. Eastin
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
PV Analysis: Synoptic-Scale Distribution
DynamicTropopause
Advanced Synoptic M. D. Eastin
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
PV Analysis: Synoptic-Scale Distribution
500mb Heights // 300-500mb PV PV // Potential Temperature
SS
N
N
Advanced Synoptic M. D. Eastin
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]
PV Analysis: Synoptic-Scale Distribution
500mb Heights // 300-500mb PV PV // Zonal winds
SS
N
N
E W
W
Advanced Synoptic M. D. Eastin
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
PV Analysis: Synoptic-Scale Distribution
500mb Heights // 300-500mb PV Potential Temperature and Winds on 2-PVU Surface
Advanced Synoptic M. D. Eastin
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
PV Analysis: Synoptic-Scale Distribution
500mb Heights // 300-500mb PV Pressure and Winds on 2-PVU Surface
Advanced Synoptic M. D. Eastin
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 anddetermine a parcels origin(s) at a previous time
→ Evaluate non-conservative processes by documenting anyPV changes (which must have resulted from eitherdiabatic or frictional processes) **
PV Analysis: Equations
0Dt
DP
pgP a
Equation (4.16) in Lackmann text
Advanced Synoptic M. D. Eastin
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!!)
PV Analysis: Equations
pgP a
t
y
F
px
F
pxp
v
yp
u
pg
Dt
DPa
Equation (4.17)
in Lackmann text
Term BTerm A Term C
Advanced Synoptic M. D. Eastin
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 → divergenceBelow 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 (+)
PV Analysis: Non-Conservation
Advanced Synoptic M. D. Eastin
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 displacedBelow heating max → PV always increases (+) → horizontally displaced
The magnitude and direction of horizontal displacement are functions of boththe vertical shear and the local heating rate
PV Analysis: Non-Conservation
Advanced Synoptic M. D. Eastin
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)
PV Analysis: Impermeability
0
dVt
PEquation (4.24) in Lackmann text
Advanced Synoptic M. D. Eastin
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)
PV Analysis: Inversion
Advanced Synoptic M. D. Eastin
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
PV Analysis: Inversion
n
iitotal PP
1
1
23
321 PPPPtotal
Advantages:
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
PV Analysis: Advantages / Disadvantages
Diagnosing contributions to System Structure:
Advanced Synoptic M. D. Eastin
PV Analysis: Advantages / Disadvantages
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
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
Advanced Synoptic M. D. Eastin
PV Analysis: Advantages / Disadvantages
Radar // 24-hr Eta precipitation forecast
SLP // 900-700mb PV
SLP // 900-700mb PV
RUC
Eta
Analysis
24-hr
Real-time and Archived Analyses:
SUNY Albany: http://www.atmos.albany.edu/index.php?d=wx_data University of Reading: http://www.met.reading.ac.uk/Data/CurrentWeather/
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)
Advanced Synoptic M. D. Eastin
PV Analysis: Websites
Advanced Synoptic M. D. Eastin
ReferencesBishop, 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 WeatherSystems. 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 diabaticheating 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 vorticitymaps. 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.