Potential Vorticity (PV) Analysis

Advanced Synoptic M. D. Eastin

Potential Vorticity (PV) Analysis


PV Analysis

Outline:

• History and Definition

• Synoptic-Scale Distribution

• Equations and Practical Concepts

• “PV Non-conservation”• “PV Impermeability”• “PV Inversion”

• Advantages / Disadvantages


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


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


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


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


DynamicTropopause


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

SS

N

N


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

SS

N

N

E W

W


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


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


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


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


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


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


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


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


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**)


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

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



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)


PV Analysis: Websites


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.

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Potential Vorticity (PV) Analysis

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