1. ECMWF Training Course 2009 – NWP-PR
Atmospheric Variability: Extratropics
Mark Rodwell
19 March 2009

2. Talk Outline Free Barotropic Rossby Waves The Rossby Wave Source
Observations
Theory
The Rossby Wave Source
Explaining the extra-tropical response to the aerosol change
Diabatic Processes
Potential Vorticity
Explosive growth of cyclones
Causes of forecast “busts”
Precipitation
Deterministic verification
Combined prediction systems

3. Free Barotropic Rossby Waves

4. Rossby waves. Upper tropospheric vΨ, vχ & RWS
Contour 8ms-1
Phase Velocity
Group Velocity
24 May
25 May
Often we see wave-trains within the extratropics that tend to propagate eastward. Here, the contours highlight meridional flow anomalies in the upper troposphere (strictly, what is contoured is the rotational flow component of the total meridional flow). By drawing a line through some of these features it is clear that they are propagating eastward. Notice also that the head of the wave-train appears to move more quickly than the individual anomalies move themselves. This faster effect happens at the “group velocity” of the wave packet.
26 May
27 May
2008

5. The Vorticity Equation
Motivation (2D flow) : x z y
is the unit “vertical” vector and
is the horizontal curl operator
Curl of the 3D momentum equation in absolute frame of reference:
Vorticity is a key component of this talk. This slide introduces the concept of vorticity and the figures hopefully give more physical insight into the mathematical terms.
Note that there is some cancellation between the divergence and tilting terms in the vorticity equation. This cancellation is associated with “stretching”. Indeed, stretching only affects vorticity if it is accompanied by convergence into the axis of rotation: ballerinas spin faster because they bring their hands in, not because they stand up. The figures under the vorticity equation illustrate the four terms after this cancellation has been removed.
The baroclinic term is non-zero if the gradients of pressure and density are not parallel. This term explains the “sea-breeze” effect when the daytime air density over the warmer land is smaller than over the cooler ocean. The frictional term may simply act to damp the flow (as illustrated).
Interestingly, the same form of the vorticity equation also applies if we measure the 3D wind and the change in absolute vorticity relative to the rotating Earth. This is because the component removed from the Lagrangian derivative to take account of the Earth’s rotation exactly cancels the component removed from the tilting term. Other terms are unaffected.
Having discussed the mechanisms in which the vorticity of a parcel of air can be modified, we will now assume that these mechanisms are negligible! (For mid-latitude synoptic scales). This is the case if the flow is non-divergent, horizontal, barotropic and frictionless. The equation at the bottom of the slide shows the remaining terms in the vorticity equation. Basically, the Lagrangian derivative has been split into an Eulerian time derivative and an advection part. The advection is that associated with the rotational component of the flow since we have assumed that the flow is non-divergent.
Shallow atmosphere approximation & assuming non-divergent, horizontal, barotropic, frictionless flow:

6. Free Barotropic Rossby Waves
Non-divergent barotropic
vorticity equation
Where
Seeking wave-like
solutions …
We obtain the
“dispersion relation”
Where
Rossby waves get advected downstream and propagate upstream
The larger the spatial scale of the wave, the faster the upstream propagation
Here the barotropic vorticity equation (the last equation on the previous slide) has been linearised about a mean flow and re-written with the advection term split into its two main components. These are the advection of relative vorticity by the zonal mean flow and the advection of planetary vorticity by the anomalous meridional flow.
As in the tropical talk, we can look for wave-like solutions to this equation, here of the form:
where
Inserting this solution into the barotropic vorticity equation we obtain the dispersion equation shown on the slide. This dispersion relation shows how the wave’s angular frequency and wavenumber are related.
We see that the phase speed of the wave is the difference between the down-stream advection speed and the upstream propagation speed
Notice that the larger the spatial scale of the wave, the smaller the value of , and faster the upstream propagation speed. At a certain spatial scale, the upstream propagation speed could equal the down-stream advection speed. At that scale, a wave will be stationary.
For stationary waves
Mid-latitude stationary
zonal wavenumbers

7. Rossby waves. Upper tropospheric vΨ, vχ & RWS
Contour 8ms-1
Phase Speed
24 May
Group Speed
25 May
Going back to our first example, we now see that the wave packet is a Rossby wave packet. It’s phase speed (10ms-1) is in excellent agreement with the theory. The group velocity of the wave is given by For waves with similar zonal and meridional scales, this simplifies to Again the observed group velocity is in good agreement with the theory.
26 May
27 May
10ms-1 agrees well with theory
2008

8. Stationary Rossby Waves: Vorticity advection
Non sig.
Sig. 10%
Advection by Anomalous Rotational Wind
Advection of Anomalous Vorticity
Upstream Propagation
Downstream Advection
The top-left picture shows schematically how the downstream advection and upstream propagation can cancel to leave a wave stationary.
The bottom plots show the mean values of the two advection terms for the aerosol change experiment for the June—August season (discussed in the previous talk). Outside the African region, the cancellation of these terms is very clear. We can conclude that the extratropical response seen in the previous talk is indeed a stationary Rossby wave response to the aerosol change.
Notice that the wavelength of the Rossby wave response in the Southern Hemisphere is longer than in the Northern Hemisphere. This is because the Jet is stronger in the Southern (winter) hemisphere: the stationary wave must have a larger scale in order to propagate upstream faster.
How do these extratropical Rossby waves get excited? For this, we will now discuss the “Rossby wave source”.
40-year mean response to change in aerosol climatology deduced using seasonal-mean data. Results are very similar when daily data are used. Anomalies integrated hPa.

9. The Rossby Wave Source

10. Upper Troposphere Divergent Wind Anomaly
If you remember, the change (reduction) in Saharan soil-dust aerosol led to a reduction in the strength of the north African monsoon. Here we can see that the weaker monsoon convection has a very large-scale effect on the upper-tropospheric divergent winds. These divergent wind anomalies extend well into the subtropics and we will see that it is this effect that can lead to the excitation of the extratropical Rossby waves.
New minus Old aerosol. Anomaly is integrated between 100 and 300 hPa

11. The “Rossby Wave Source”
When divergent winds
are not neglected in the
vorticity equation
Application to barotropic models:
Sardeshmukh and Hoskins (1988)
Since the divergent flow anomalies extend into the subtropics, we will now not neglect the divergence term of the vorticity equation. This leads to the equation at the top on the slide. There are two components on the right-hand side of this equation, these are the divergence term itself and the advection of vorticity by the divergent flow. These can both be combined into a single term as shown.
The vorticity equation has therefore been written in a form that places the purely rotational terms on the left-hand side (involving only vΨ and ζ≡ vΨ) and the terms involving the divergent flow (vχ) on the right-hand side. The equation shows that the divergent flow can lead to vorticity generation. The previous slide suggests that this generation is predominantly associated with the (anomalous) tropical convective outflow. The left-hand-side is then traditionally thought of as the non-divergent extra-tropical response to this tropical forcing. To emphasize this “cause and effect”, the right-hand side of the equation is known as the “Rossby wave source”.
While this equation has been extensively used to understand tropical forcing of the extra-tropics in highly simplified barotropic models, there are complications to using it with highly complex GCMs. One issue is that the level of convective outflow can be very sensitive to the formulation of the convection scheme (amongst other things). If the vorticity equation is analysed on a single level then a small change in the convection scheme could lead to apparent large changes in the Rossby wave source. On the other hand, the equivalent barotropic response in the extra-tropics will not be very sensitive to this small change in outflow level. To avoid this issue, all terms in the vorticity equation are integrated from 100 to 300 hPa.
For use in complex GCMs, it is found here to be useful to vertically integrate this equation between 100 and 300 hPa

12. JJA Balance in Vorticity Equation New-Old
Rossby Wave Source
The Rossby Wave Source is indeed seen as the tropically induced source of the extratropical stationary Rossby wave response
Wave Initiators
10-11 s-2
Non sig.
Sig. 10%
Advection by Anomalous Rotational Wind
Advection of Anomalous Vorticity
Upstream Propagation
Downstream Advection
Here we repeat the two vorticity advection terms but we have now also plotted the Rossby wave source. Notice how the Rossby wave source has large magnitudes in the subtropics to the north and south of Africa. This occurs where the anomalous divergent flow meets the subtropical jet (a region of strong vorticity gradients). Interestingly however, it is the divergence term that contributes most to the Rossby wave source with the advection of vorticity by the divergent flow being somewhat smaller.
The anomalous Rossby wave source closes the vorticity balance and can be seen as the ‘initiator’ of the extratropical Rossby wave response to the aerosol change.
40-year mean response to change in aerosol climatology deduced using seasonal-mean data. Results are very similar when daily data are used. Anomalies integrated hPa.

13. JJA New-Old RWS, vχ,Ψ and mean ζ
This plot summarises how the upper-tropospheric anomalous divergent flow (vectors) can interact with the mean absolute vorticity (thin grey contours) to produce a Rossby wave source (coloured). The anomalous streamfunction associated with the induced extratropical Rossby waves is also shown (thick contoured). There are, in fact, two Rossby wave trains in the Southern Hemisphere that occur on subtropical and extratropical jets, respectively.
Rossby wave paths agree beautifully with those predicted by Hoskins and Ambrizzi (1995)

14. JJA Precipitation, v925 and Z500. New-Old
mm day-1. 10% Sig.
The upper tropospheric flow anomalies associated with the stationary Rossby wave feature on the southern extratropical jet are in excellent agreement with the final response featured at the beginning of the previous talk, that of the 500 hPa geopotential height anomaly (highlighted in this slide).
In the extratropics, the flow is much more barotropic and this is why the same structure features throughout the depth of the troposphere. We have now explained all the features seen in at the beginning of the first talk that were associated with the aerosol change.
Hopefully, this makes us feel that we have gained considerable insight into how the atmosphere can respond to changes (or errors) in our forecast model.

15. DJF New-Old RWS, vχ,Ψ and mean ζ
Rossby wave path agrees with that shown by Hoskins and Ambrizzi (1993)
Extratropical RWS anomaly coincides with precipitation changes. Is upper tropospheric divergent wind directly related to local physics?
For the December—February season, the model also responds to the change in aerosol in much the same way. There is a reduction in precipitation over equatorial Africa and the Gulf of Guinea. There is also a Kelvin-wave-like response over the tropical Indian Ocean which is accompanied with increased precipitation.
Here we see the anomalous upper-tropospheric divergent flow vectors for this season. Again, there is a forcing of the Rossby wave source term (coloured) and this excites a stationary extratropical Rossby wave (thick contours show anomalous upper-tropospheric stream function) in the Northern (winter) Hemisphere. The path and wave-length of this wave are in good agreement with theoretical studies. Importantly, the wave acted to cancel some long-standing biases in the model such as a high bias over the central northern Pacific.
Notice that the “Rossby wave source” has quite strong values in the extratropics too, particularly over the northern Pacific. Such values mean that the term “Rossby wave source” can be a little miss-leading. This term can also arise from adiabatic flow anomalies in the extratropics if potential temperature surfaces are getting closer or further apart. One way to get around this issue would be to think about “Potential Vorticity” rather than vorticity (see next slide).
Interestingly, there are also precipitation anomalies that coincide with these extratropical “Rossby wave source” anomalies. Is it possible that diabatic processes are also playing a role in the “Rossby wave source” (see later).

16. Diabatic Processes

17. Adiabatic & Diabatic Contributions to RWS
θ=340K
300
ADIABATIC
+ DIABATIC
ADIABATIC
hPa
PV ‘absorbs’ the adiabatic stretching
600
θ=320K
30oN
60oN
90oN
Account for adiabatic stretching by considering Potential Vorticity (P):
“Stretching”, Tilting & Advection
by Diabatic Processes (+ Friction)
The RWS may be reflecting adiabatic stretching rather than (or in addition to) diabatic forcing. To isolate the diabatic forcing, Potential Vorticity (PV) can be used instead. PV is unaltered by adiabatic stretching.
The PV equation shows that any long-term mean advection of PV on an isentrope by the horizontal wind must be balanced by a diabatic (or frictional) PV ‘source’.
When averaging over a long period

18. SYNOP Precipitation Anomaly Summer 2007-3 -2 -1 1 2 3
mm/day
Total rainfall was double the climatological mean
To investigate the role of diabatic processes within the extratropical weather and climate, we first look at the case of the wet summer of Here, rainfall data are based on SYNOP observations. Much of Europe received twice as much rainfall as normal during this “summer” season!
Based on 24hr accumulations and our new global SYNOP climatology for the years

19. Components of a Predictable Signal-3 -2 -1 1 2 3
mm/day
SCHEMATIC
DYNAMICAL WAVES?
LOCAL
PHYSICS?
PHYSICS
ALONG WAVE?
TROPICAL
PHYSICS?
Of course, a large component of any one season’s anomaly will be down to luck (or bad luck). If there is a predictable component of a seasonal-mean anomaly then the previous slides suggest a framework for trying to understand this anomaly.
Europe’s wet summer of 2007 could have been the unlucky mean of unpredictable variability. But if not, then this schematic shows some likely building blocks to predictability

20. Terms in PV equation @330K Quadratic Diabatic (residual)
UNIT = 10-13Km2kg-1s-2 26 22 18 14 10
Clearer view of Rossby wave? 6 2 -2 -6
-10
-14
-18
-22
-26
Quadratic
Diabatic (residual)
Agreement with rainfall anomalies.
Important for sustaining wave?
Here the PV advection (on the 330K surface) for the wet summer of 2007 is decomposed in a somewhat similar fashion to that done for the vorticity advection. We can see that there does appear to be a Rossby wave-train associated with the wet summer. PV anaomlies are advected downstream and propagate upstream. The total PV advection should equal the diabatic forcing. A small negative diabatic forcing is seen in the region of strong rainfall over northern Europe but it remains to be seen if this plays an important role in the maintaining the flow anomaly associated with the season.
Results are based on 0 and 12Z analyses. An over-bar indicates the climatological mean and a prime indicates the instantaneous 2007 departure from the climatological mean. (June 1 to August 13)

21. Analysis of winter storm “Lothar”
18Z, 25 DEC1999
TROPOPAUSE FOLDING ASSOCIATED WITH KURT AND ISENTROPIC DOWN-GLIDING(?)
CYCLONE “KURT”
PV=2 SURFACE
LOW-LEVEL CYCLONE “LOTHAR”
V850
Diabatic processes clearly played an important role in the explosive growth of cyclone “Lothar”. Over the next two slides, we see a PV tower developing in the vicinity of Lothar. This is a response to convective heating. This PV tower appears to interact with an upper-tropospheric PV “tongue” (tropopause fold) associated with the pre-existing cyclone “Kurt”.
Wernli et al. (2002) Fig. 7a

22. Analysis of winter storm “Lothar”
0Z, 26 DEC1999
TROPOPAUSE FOLD NOW ALSO ASSOCIATED WITH LOTHAR
CYCLONE “KURT”
PV=2 SURFACE
LOW-LEVEL CYCLONE “LOTHAR”
V850
Wernli et al. (2002) Fig. 7b

23. 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

24. RWS, vχ and Meridional wind anomalies
30 May
Classic example that led to a forecast “bust” over Europe a few days later
What is the diabatic forcing?
How well does the (first guess) forecast represent this forcing?
What are the implications of observation rejection?
Look at developments like this from PV perspective.
5 June
This slide shows a Rossby wave approaching North America in early summer When it reaches the US, it leads to strong poleward flow of moist air from the Gulf of Mexico into the Great Plains. It seems that the ECMWF model is not able to represent well the convection associated with this moisture flow. Errors in parametrizing the diabatic processes involved can leave the first guess forecast a long way from the observations and quality control then effectively throws-away the observations. This can lead to analysis errors. In the case shown here, ECMWF suffered a bad forecast “bust” for Europe a few days later. Understanding how diabatic processes interact with such Rossby wave features is important for addressing this forecasting issue.
6 June
RWS shade interval s-2. Meridional wind contour interval 8 ms hPa integrals

25. Precipitation

26. Deterministic Scores: Z500, θPV=2 & Precip
This slide shows deterministic spatial anomaly correlation scores of 500 hPa geopotential heights, a PV quantity (actually potential temperature on the PV=2 surface, the dynamic tropopause) and precipitation. The details of the precipitation scoring are given in the next slide.
It is clear that smaller-scale fields are more difficult to predict than smooth fields such as 500 hPa geopotential height.
Notice, however, that the forecasts appear to be getting better for all these quantities year-on-year.

27. Extratropical Deterministic Precip Scores
24h Accumulated Precipitation Forecast Scored against SYNOP Observations
2008
1995
ECMWF may soon incorporate a weather-related score into it long-term planning. Such a score needs to be insensitive to year-to-year flow variations. One possibility is to score extratropical precipitation against SYNOP observations. There are about 2000 such SYNOP stations available although half of these are in Europe. The score shown here is based on all extratropical SYNOP observations available on any given day. To avoid emphasising rainy (mountainous) regions, observations and forecast data are first divided by the climatological mean precipitation rate for the month in question. (If this climatological rate is less than a given threshold, the station is igored).
The plot shows quite a good trend to improved precipitation scores over the 14 years shown. Further tests of this score are planned shortly.
D+5 forecast in 2008 as good as D+1 forecast in 1995
Area = [SP--30oS & 30oN--NP]

28. ECMWF “Meteogram” of Precipitation
Mumbai
ENSEMBLE PREDICTION SYSTEM
Highly useful product but …
… “What should I believe?”
At D+2?
At D+5?
This is an example of a “Meteogram” issued by ECMWF for precipitation in Mumbai, India. The Meteogram shows the high-resolution forecast, the ensemble control forecast, and a “box-an-whisker” symbol for the ensemble prediction system.
An issue for users and forecasters on the bench, in cases where these forecasts do not agree, is which one to “trust”?
At day 2 in these forecasts, the high-resolution forecast is clearly outside the range of the ensemble. At such a short lead-time, perhaps the user would prefer to trust the deterministic forecast. At longer lead-times, if the forecasts diverged, the user may prefer to trust the ensemble. But how can we formalise a method for such a decision?
One approach taken here is to produce a combined probabilistic forecast that puts an appropriate weight on each component in the combination (see the next slide).
ENSEMBLE CONTROL FORECAST
HIGH RESOLUTION DETERMINISTIC FORECAST

29. Combined Prediction System – Concept
This schematic shows how a single deterministic forecast (yellow rectangle) could be combined with a 10-member ensemble (orange squares) to produce a single forecast pdf. The question is, how big should the yellow rectangle be? Its size could depend on the forecast lead-time; being large at short lead-times and small at long lead-times.
Combining a 10-member ensemble of equally likely members (orange squares) with a single more accurate forecast (yellow rectangle)

30. Combined Prediction System - Theory
B = Brier Skill Score averaged over all stations
n = number of dates
Mj = set of stations reporting on date j
mj = number of stations in Mj
pij = CPS probability
vij = verification (0 or 1)
bclim = bclim(location,month) (from climatology)
K = number of forecast systems ( K ≤3 here)
wk is the weight applied to system k (independent of location)
Find weights that maximize Brier Skill Score
Apply in cross-validated mode
(date for year y applied in year y+1)
One approach would be to take the user’s own decision-making model and chose weights for the forecast system components that maximise the expected value to the user. Here, to demonstrate the process, we stick to the fluid dynamical problem and use the Brier Skill Score as a surrogate for the user’s value calculation.
The Brier Skill Score is differentiated by the weights, wk, applied to the individual forecast system probabilities, pk, to find the optimal solution.

31. Combined Prediction System - Weights
Here we see that the optimal weight to use for the deterministic forecast for precipitation events of 1, 5 and 10 mm day-1 as a function of forecast lead-time (for Europe). The weights are expressed in units of equivalent ensemble members. In general, the weights are insensitive to the precipitation threshold but are sensitive to forecast lead-time. At a lead-time of 1 day, the deterministic forecast is “worth” between 15 and 20 ensemble members. By day 10, it is worth about 1 or 2 ensemble members.

32. Combined Prediction System - Results
Optimal weights are calculated in cross-validated mode (so as not to artificially enhance the scores) and the combined forecast is then scored. Here we see that the mean Brier Skill Scores are statistically significantly improved for all precipitation thresholds.
Such a process could clearly combine more than two forecast systems. These could include lagged forecasts and even a climatological probability forecast. Lagged forecasts do improve the scores further. One could think of reducing the ensemble size, increasing the model resolution and using lagged members to maintain the overall number of forecast members. This approach has also been tried and, again, leads to further improvements with no added computational expense. Notice that the scores presently dip below zero after about a week. This dipping could (presently) be avoided by putting a large weight on climatology in the extended medium range.
Clearly some forecasters “on-the-bench” may prefer to be given the probabilities from the individual forecast systems and use the optimal weights themselves. Other users may simply prefer a single “best” forecast based on the combined system.

33. Combined “Meteogram” Mumbai
One possible “Combined Meteogram” is shown in the slide. The shapes indicate pdfs for precipitation. Notice that the relatively large optimal weight for the deterministic forecast at short lead-times leads to a bi-model pdf distribution on 2 June.
Further work is progressing on combined systems.

34. Summary Free Barotropic Rossby Waves The Rossby Wave Source
Propagate upstream and get advected downstream
Larger waves can become stationary (e.g. Blocking)
The Rossby Wave Source
How the (tropical) divergent flow can influence the extratropics
Diabatic Processes
May be important for maintaining anomalous flow over a season
Clearly important for explosively growing cyclones
Poor representation (over North America) may lead to forecast “busts”
Precipitation
Deterministic scores show improving trends
Combined prediction systems can improve probabilistic forecasts