1. PV Diagnostics Meteorological Training Course 25 April 2006
Mark Rodwell
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

2. Structure of Talk What is PV? Insight into Dynamics Verification
Momentum
Vorticity
Potential Vorticity
Conservation
Lagrangian Sources
Inversion
Insight into Dynamics
Tropopause
“Induced flow”
Cut-off lows and blocking highs
Cyclogenesis
Verification
Deterministic forecasts
Probabilistic forecasts
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

3. What is PV?
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

4. Momentum
In an absolute frame of reference, the momentum equation can be written as:
(u is 3D absolute wind, p is pressure and  is density)
Often spherical polar coordinates are used (k being the local vertical unit vector), winds are relative to the rotating planet and the shallow atmosphere approximations‡ are made
‡ Replacing “r” with a constant Earth radius “a” in the spherical metrics and neglecting some of the curvature terms and the local horizontal component of the Earth’s rotation.
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

5. Vorticity I
Making the shallow atmosphere approximations, using pressure surfaces and taking of the horizontal momentum equations we get an equation for the local vertical component of absolute vorticity,
(v = horizontal wind relative to the planet and   dp/dt)
For barotropic, frictionless, horizontal, non-divergent flow, this reduces to the “non-divergent barotropic vorticity equation”:
i.e. absolute vorticity is conserved following the flow
EG: A PARCEL OF AIR STARTS SPINNING (RELATIVE VORTICITY) IF MOVED TO A DIFFERENT LATITUDE
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

6. Vorticity II
The non-divergent barotropic vorticity equation (3) helps:
Understand (and predict‡) mid-tropospheric flow
Understand tropical-to-extratropical teleconnections
Introduce the concept of “potential vorticity”:
Conservation of absolute vorticity, 
Inversion of  to determine all other fields:
(Can define a streamfunction  such that
Knowing P means we can determine  and thus the wind)
Can we define a vorticity-like quantity that is conserved and (nearly) invertible without making approximations?
‡ The non-divergent barotropic vorticity equation was used in the first numerical prediction models
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

7. Ertel Potential Vorticity (EPV) I
For barotropic frictionless flow, (curl of (1))
Using Stoke’s theorem‡, this implies that nS is constant
Assuming adiabatic flow (potential temperature, , is constant),
we can write
Hence
Since , must be perpendicular to
and so we do not need to assume barotropic flow.
‡ The behaviour of the material circuit contributes nothing.
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

8. Ertel Potential Vorticity (EPV) II
For adiabatic, frictionless flow, is conserved even for 3D, non-hydrostatic motions :
EPV is conserved because the creation of vorticity by 2D convergence into the axis of rotation is balanced, within EPV, by the accompanying‡ increase in distance between isentropes (“stretching”).
Generalisation to include diabatic and frictional “sources” gives:
‡ Some 2D convergence may be accompanied by an increase in density rather than stretching and the  (in EPV) takes account of this.
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

9. Isentropic Potential Vorticity (IPV) I
Using the shallow atmosphere approximation (as models often do) and assuming adiabatic, frictionless flow we can use (2) and calculate derivatives on isentropic surfaces (replace p with  and  with ):
Combining with the continuity equation for adiabatic flow:
We get:
Where is conserved following the horizontal
flow on an isentropic surface.
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

10. Insight in Dynamics
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

11. Importance of Static Stability at the Tropopause
LARGE INCREASE IN /p (AND THUS PV) AT THE TROPOPAUSE
CAN DEFINE THE TROPOPAUSE BY A VALUE OF PV
Copyright Federation Francaise de Parachutisme (1999)
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

12. Dynamic Tropopause at 2PV units
STRATOSPHERE
(HIGH PV)
TROPOPAUSE
(PV=2 ISO-SURFACE)
TROPOPAUSE FOLD
TROPOSPHERE
(LOW PV)
Massacand (1996)
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

13. Flow “Induced” by an IPV Anomaly (Cyclonic)
MORE STABLE
COLD
WARM
LESS STABLE
DYNAMIC TROPOPAUSE ⊙ ⊗
IPV’
VORTICITY AND STABILITY ANOMALIES TYPICALLY HAVE THE SAME SENSE AS IPV
TONGUE OF STRATOSPHERIC AIR
COLD ADVECTION, LESS STATIC STABLITY AND ANTICYCLONIC IPV ON SURFACE
WEAKER TROPOSPHERIC STABILITY ALLOWS CONVECTION WHICH CAN LOWER OR DAMP THE IPV ANOMALY
TO MAINTAIN THERMAL WIND BALANCE
Hoskins, McIntyre and Robertson (1985) Fig. 9a.  in K, PV in PV units
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

14. Circularly Symmetric Flows “Induced by” Simple Isolated IPV Anomalies
100
200
300
hPa
1000
600
800
400
ISENTROPES (CI = 5K) AND
AZIMUTHAL WIND (CI=INTERVAL 3ms-1)
WARM ⊙ ⊗
UPPER-LEVEL CYCLONIC IPV ANOMALY (STIPPLED)
SURFACE TEMPERATURE ANOMALY (+1OK)
100
200
300
hPa
1000
600
800
400 5 4 3 2 1 }
WARM
CAN BE INTERPRETTED AS A CYCLONIC IPV ANOMALY ⊗ ⊙
WARM
Hoskins, McIntyre and Robertson (1985)
r=1667km
r=0
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

15. Isentropic Potential Vorticity (IPV) II
The vertical penetration, magnitude and phase-speed of the flow induced by an IPV anomaly increases with its horizontal scale.
Surface warm anomalies can lead to large effective
and thus large at the surface.
This IPV also “induces” a flow field (“surface charge electric field”).
The flows “induced” by two (or more) IPV anomalies can be approximately superposed (“attribution”).
The induced velocity fields (e.g. by surface  and atmospheric IPV anomalies) can keep Rossby waves in-step.
UPPER-LEVEL +PV ADVECTED OVER A LOW-LEVEL BAROCLINIC REGION CAN LEAD TO MUTUAL AMPLIFICATION. MOIST PROCESSES COULD ENHANCE THE SURFACE DEVELOPMENT
Hoskins, McIntyre and Robertson (1985) Fig. 21
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

16. Cut-off Lows and Blocking Highs
IPV on 330K for consecutive days. 1-2PVU filled in black
CUT-OFF LOW
ADVECTION OF LOW IPV
IPV A BETTER WAY OF DEFINING “CUT-OFF” THAN 500hPa HEIGHT
BLOCKING HIGH
CUT-OFF LOWS AND BLOCKING HIGHS HAVE SIMILAR (OPPOSITE) STRUCTURES
CONVECTIVE HEATING DAMPS CYCLONE FASTER THAN RADIATIVE COOLING DAMPS ANTICYCLONE
Hoskins, McIntyre and Robertson (1985) Fig. 11
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

17. PV Modification PV and accumulated tendencies of PV
ASIAN MONSOON
PLUME OF PARTICLES INITIALLY AT 890hPa
FRICTIONAL TENDENCY
MOUNTAINS
PV and accumulated tendencies of PV
CIRCLES DENOTE CONSECUTIVE DAYS
DIABATIC TENDENCY
SHALLOW HEATING
REASONABLE BALANCE ACHIEVED PV
CAN INVESTIGATE REASONS FOR PV MODIFICATION
TOTAL TENDENCY
CAN ASK:
IS THERE AN ERROR IN MODEL’S DIABATIC OR FRICTIONAL PARAMETRIZATION?
Rodwell and Hoskins (1995) Fig. 7
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

18. Anomalies a few days after injection of Southern Hemisphere midlatitude air into Cross-Equatorial Jet
mmday-1
V850
(a)
(b)
Precipitation
TURNING IS A CONSEQUENCE OF PV ADVECTION
Based on 16 injection events (v>8.5ms-1) in JJA
V850 from ERA-40 daily. Averaged over 4-8 days after injection
Precip: Xie-Arkin pentadal. Pentads starting 2-9 days after injection
10% significance indicated
WIND ANOMALY IS VERY ROBUST FOR OTHER THRESHOLDS
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

19. Analysis of winter storm “Lothar”
TROPOPAUSE FOLDING ASSOCIATED WITH KURT AND ISENTROPIC DOWN-GLIDING(?)
18Z, 25 DEC1999
CYCLONE “KURT”
PV=2 SURFACE
LOW-LEVEL CYCLONE “LOTHAR”
V850
Wernli et al. (2002) Fig. 7a
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

20. Analysis of winter storm “Lothar”
TROPOPAUSE FOLD NOW ALSO ASSOCIATED WITH LOTHAR
0Z, 26 DEC1999
CYCLONE “KURT”
PV=2 SURFACE
LOW-LEVEL CYCLONE “LOTHAR”
V850
Wernli et al. (2002) Fig. 7b
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

21. Analysis of winter storm “Lothar”
6Z, 26 DEC1999
CYCLONE “KURT”
PV=2 SURFACE
LOW-LEVEL CYCLONE “LOTHAR”
UPPER AND LOWER PV ANOMALIES NEARLY JOIN
V850
INTENSE WINDS KILL 50
Wernli et al. (2002) Fig. 7c
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

22. Analysis of winter storm “Lothar”
“INDUCED” TROPOPAUSE FOLD  UPPER-LEVEL +PV ANOMALY
DIABATIC PROCESSES WERE ESSENTIAL FOR LOTHAR
MOIST ASCENDING AIR  LATENT HEAT RELEASE
HEATING “PRODUCES” LOW-LEVEL +PV ANOMALY
MUTUAL AMPLIFICATION OF PHASE-LOCKED PV ANOMALIES
SIMILARITIES TO HOSKINS ET AL (1985) BUT MID-TROPOSPHERIC LATENT HEAT RELEASE TAKES THE PLACE OF SURFACE  ANOMALY
Wernli et al. (2002) Fig. 14
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

23. Verification
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

24. Z500 and PV330 European Anomaly Correlations
Z500 FORECASTS AT D+1 TO D+4 ARE EXTREMELY GOOD THESE DAYS
PV, WITH ITS INTERTABILITY, MAY EMBODY MORE OF THE IMPORTANT CIRCULATION AND BE A GOOD WAY TO VERIFY FORECASTS
BUT IS THIS REALLY THE WEATHER WE EXPERIENCE?
31 OCT
STRONG RAINFALL EVENT OVER ALPS
SON 2003, T511, ERA40 climatology
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

25. Forecast and Verifying analyses 31 Oct 2003
Z500
PV330
FC D+4
Z500 FIELDS VERY SIMILAR AND DO NOT HIGHLIGHT AN ESSENTIAL DIFFERENCE
MOIST ADVECTION
rEURO = 0.90
rEURO = 0.27 AN
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

26. Anomaly Correlation for Europe August 04-March 05
Z500
step = 10
step = 4
step = 1
Experimental
Mean correlation
Operations
Experimental
significant difference
PV MAY BE MORE DISCRIMINATING FOR NEW MODEL VERSION TESTING
Operations
Forecast day
RD esuite archives PT2000 spectrally rather than on reduced Gaussian grid.
This led to apparent large improvement at 29r1 (which was spurious)
step = 10
step = 4
step = 1
Experimental
Mean correlation
Operations
Experimental
significant difference
PV=2
Europe = [12.5oW-42.5oE, 35oN-75oN]. Significance if |avg(COR)| > stdev(COR). (AR1 model accounts for autocorrelation).
Operations is 28r1 to 2004/09/27, 28r3 thereafter. Experimental is 29r1. Comparison with own analysis using ERA40 climate.
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

27. Annual-Mean of Daily Anomaly Correlation Coefficients for Europe
Z500
PV=2
Pp/c
FORECAST IMPROVEMENTS ARE MORE APPARENT WITH PV AND PRECIPITATION
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

28. Brier Score Difference for event “> normal” DJF 2001-2004, Ensemble (T255) – Deterministic (T511)
IN PROBABILISTIC TERMS, A DETERMINISTIC FORECAST OF Z500 AT DAY+1 IS AS GOOD AS A FORECAST FROM THE ENSEMBLE PREDICTION SYSTEM
BRIER SCORE:
THIS IS NOT THE CASE FOR THE PV DIAGNOSTIC: THE ENSEMBLE SCHEME IS BETTER EVEN AT DAY+1
Z500
PV=2
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006

29. Summary What is PV? Insight into Dynamics Verification Momentum
Vorticity
Potential Vorticity
Conservation
Lagrangian Sources
Inversion
Insight into Dynamics
Tropopause
“Induced flow”
Cut-off lows and blocking highs
Cyclogenesis
Verification
Strong rainfall events
More discriminating for skill changes
Shows early benefit of ensemble forecasts
PV diagnostics, Met.TC, MJ Rodwell, 25 April 2006