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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 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 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 Advanced Synoptic 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 Advanced Synoptic 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 Advanced Synoptic 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 Advanced Synoptic 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 Advanced Synoptic 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 Advanced Synoptic 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 Advanced Synoptic 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) ** Advanced Synoptic 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!!) Advanced Synoptic 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 (+) Advanced Synoptic 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 Advanced Synoptic 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) Advanced Synoptic 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) Advanced Synoptic 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 Advanced Synoptic M. D. Eastin PV Analysis: Advantages / Disadvantages 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: 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 Advanced Synoptic M. D. Eastin PV Analysis: Advantages / Disadvantages 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 Advanced Synoptic M. D. Eastin PV Analysis: Websites 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 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. Advanced Synoptic M. D. Eastin