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# Tim Palmer - Fall 2012 - Lecture I

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The phrase “The Butterfly Effect” is almost universally used to describe sensitive dependence on initial conditions in chaotic systems (be they high or low order). However, this is not what Lorenz originally had in mind by this phrase. Rather he postulated the existence of something much more radical: that a high dimensional system like the atmosphere may have a finite predictability horizon which cannot be extended in time, no matter how small the initial uncertainties are. Is there evidence for “The Real Butterfly Effect” in the real world, and is “The Real Butterfly Effect” a property of the Navier-Stokes equations? In this seminar, I will review some of these issues and then conclude that an understanding of the “The Real Butterfly Effect” is of crucial practical importance as we aim to provide reliable weather and climate predictions to a range of real-world applications from health to agronomy to hydrology. Thanks to Tim Palmer for sharing his presentation.

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### Tim Palmer - Fall 2012 - Lecture I

1. 1. The "real" butterfly effect: A study of predictability in multi-scalesystems, with implications for weather and climate by T.N.Palmer University of Oxford ECMWF
2. 2. “The Butterfly Effect is a phrase thatencapsulates the more technicalnotion of sensitive dependence oninitial conditions in chaos theory”(Wikipedia)
3. 3. Journal of the Atmospheric Sciences 1963 X = -s X + s Y Y = -XZ + rX -YZ = XY - bZ
4. 4. dX   X   Y dt dY   XZ  rX  Y dt dZ  XY  bZ dtExhibits sensitive but nevertheless continuousdependence on initial conditions – you tell me howaccurately you want to know the forecast state, I’ll tell youhow accurately you need to know the initial conditions.This is not what Lorenz had in mind by “The ButterflyEffect” – he had in mind systems which might not exhibitcontinuous dependence on initial conditions – theseexhibit a much more radical type of unpredictability.
5. 5. Hurricane Katrina: Semi-Predictable Hurricane Nadine: Unpredictable Cyclone Sidr : Predictable 20050825 0 UTC 20120920 0 UTC 20071112 0 UTC Probability that NADINE will pass within 120km radius during the next 120 hours Probability that KATRINA will pass within 120km radius during the next 120 hours Probability that 06B will pass within 120km radius during the next 120 hours tracks: black=OPER, green=CTRL, blue=EPS numbers: observed positions 60°W at t+..h 60°W black=OPER, green=CTRL, blue=EPS numbers: observed positions at t+..h tracks: 40°W 20°W 0° tracks: black=OPER, green=CTRL, blue=EPS numbers: observed positions at t+..h 100°W 80°W 80°E 100°E 100 1 1 50°N 50°N 9 9030°N 30°N 9 80 8 8 40°N 40°N 40°N 40°N70 7 7 0 -12 6 -24 6020°N 20°N 6 -36 -48 -60 5 -108 -96 -72 -72 50 5 30°N -120 -84 -84 -84 -84 30°N 4 30°N 30°N -132 40 4 -144 3 3010°N -6 10°N 0 3 -156 2 -12 20°N -168 -168 -168 -168 20°N -180 20 2 1 20°N 20°N 10 1 5 80°E 100°E 5 5 60°W 40°W 20°W 0° 100°W 80°W 60°W
6. 6. 20121025 0 UTC 20121028 0 UTC Probability that SANDY will pass within 120km radius during the next 120 hours Probability that SANDY will pass within 120km radius during the next 120 hours tracks: black=OPER, green=CTRL, blue=EPS numbers: observed positions 40°Wt+..h 100°W 80°W 60°W at50°N 50°N100 tracks: black=OPER, green=CTRL, blue=EPS numbers: observed positions 40°Wt+..h 100°W 80°W 60°W at 100 90 50°N 50°N90 8040°N 40°N 80 70 40°N 40°N70 60 6030°N 30°N 50 30°N 0 0 0 0 0 0 0 0 0 30°N50 -12 -24 -36 40 -48 4020°N 20°N -60 0 30 20°N 20°N30 -12 -72 -24 -84 -84 -84 -84 -36 -36 -36 -36 20 -96 -96 -96 -96 20 -48 -48 -48 -48 -108 -12010°N 10°N10 10°N 10°N10 5 5 100°W 80°W 60°W 40°W 100°W 80°W 60°W 40°W sea. The GFS model also has an out to sea track, but has shifted an absolutely devastating storm for the northern mid-Atlantic and On the other hand, the Canadian model - which had conjured up Northeast in earlier runs - has shifted the storm’s track out to a bit closer to the coast compared to yesterday. www.washingtonpost.com
7. 7. Lorenz. The Essence of Chaos(1993)“The expression (The Butterfly Effect) has asomewhat cloudy history: It appears to havearisen following a paper that I presented at ameeting in Washington in 1972, entitled: Doesthe Flap of a Butterfly’s Wings in Brazil Set Offa Tornado in Texas..”
8. 8. “The following is the text of the talk I presented …in Washington..on 1972…in its original form Predictability:Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?…The most significant results are the following:1. Small errors in the coarser structure of the weather patterns…tend to double in about three days..2. Small errors in the finer structure, eg the positions of individual clouds- tend to grow much more rapidly, doubling in hours or less…3. Errors in the finer structure, having attained appreciable size, tend to induce errors in the coarser structure. This result...implies that after a day or so there will be appreciable errors in the coarser structure. Cutting the observational error in the finer structure in half – a formidable task - would extend the range of acceptable prediction of even the coarser structure only by hours or less...”
9. 9. Tellus 1969
10. 10. “It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationallyindistinguishable from indeterministic systems; specifically that two states of the system differing initially by a small “observational error” will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error…..” Lorenz 1969 Tellus
11. 11. Atmospheric Wavenumber Spectra
12. 12. The “Real” Butterfly Effect: A problem in PDEs, not ODEs ?Let E (k ) denote the kinetic energy per unitwave number of the system at wave number k
13. 13. Suppose we are only interested in predicting somelow wavenumber (ie large-scale) k L .How long before small-scale errors, confined to Nwavenumbers greater than 2 k L , affect k L ?Let the time taken for a small-scale initial error,to grow and nonlinearly infect k L be given by( N )   (2 k L )   (2 N N 1 k L )  ... (2 k L ) 0 N =  (2n k L ) n 0
14. 14. The “Real Butterfly Effect”Error Increasing scaleThe Predictability of a Flow Which Possesses Many Scales of Motion. E.N.Lorenz (1969). Tellus.
15. 15. Most of the time, small (egconvective) scales are controlled by large (eg synoptic scales) and hence L69 is an overly pessimistic estimate of predictability. Butintermittently the opposite occurs…
16. 16. EgThis is when the real butterfly effect is most active.
17. 17. For such cases, could it literally betrue that errors propagate up to the large scale from arbitrarily small scales in finite time?
18. 18. “We have not been able to prove or disprove ourconjecture, since in order to render theappropriate equations tractable we have beenforced to introduce certain statisticalassumptions which cannot be rigorouslydefended.”Lorenz 1969
19. 19. Lifted from Wikipedia:• The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that• A solution exists• The solution is unique• The solution depends continuously on the data, in some reasonable topology. If the “real” butterfly effect is true as N then the initial value problem for , the Navier-Stokes equations is not well posed. Is it literally true?
20. 20. Clay Mathematics Millenium Problems• Birch and Swinnerton-Dyer Conjecture• Hodge Conjecture• Navier-Stokes Equations• P vs NP• Poincaré Conjecture• Riemann Hypothesis• Yang-Mills Theory
21. 21. Clay Mathematics Millenium Problems• Birch and Swinnerton-Dyer Conjecture• Hodge Conjecture• Navier-Stokes Equations• P vs NP• Poincaré Conjecture• Riemann Hypothesis• Yang-Mills Theory
22. 22. MNSNavier-Stokes EquationsFor smooth initial conditionsand suitably regularboundary conditionsdo there exist smooth,bounded solutions at allfuture times?
23. 23. Is the initial value problem for the 3D Navier-Stokes problem well posed? 1. Because MNS is an open problem, we formally don’t know. Certainly one can choose to work with function spaces where the initial value-problem is not well posed. However, such function spaces would probably not be considered “physical” and the corresponding topologies not “reasonable”. 2. However, it is known that if we assume a “sufficiently smooth” global solution and perturb the initial data of the basic solution in some “reasonable” way, then the perturbed solution converges to the basic solution on any finite time interval, as long as the perturbed initial data converges to the basic initial data. The question of what “sufficiently smooth” means is problematic. Itis unknown whether finite-energy solutions are “sufficiently smooth” (Gregory Seregin - personal communication).
24. 24. Asymptotic Ill Posedness The question of strict ill-posedness is not physically relevant to weather and climateprediction: trunction scales in weather prediction models are many orders of magnitude larger than the viscous scale. Consider, the weaker but more physicallyrelevant conjecture where the predictability time Ω(N) diverges as N→∞, but nevertheless asymptotes to some finite value as initial errors are confined to smaller and smaller scales (larger and larger N), each still larger than the viscous scales.
25. 25. The real butterfly effectCan we find “empirical evidence” from operational NWP models?
26. 26. NigelRoberts.Met Office
27. 27. What’s Going On?• For deterministic short-range prediction, increased model resolution will give better representations of topography, land-sea contrast etc , but this will be offset by an increase in forecast error because smaller-scale circulations with faster error- doubling times will be simulated explicitly. Overall, deterministic skill scores (RMS error, ACC etc) may not increase with increased model resolution.• The conclusion is not that high-resolution modelling is a waste of time and resources, but rather that all predictions, even for the short range, must be considered probabilistic, ie ensemble based. There is no range at which the forecast problem can be treated deterministically. The “classical” era of deterministic numerical weather prediction should be drawing to a close, even for short-range prediction.• Probabilistic skill scores will increase with model resolution, provided the underpinning ensemble prediction systems (EPSs) are statistically reliable. The Real Buttefly Effect suggests that model error can be a significant source of forecast uncertainty even in the short range and must be represented in an EPS. Stochastic parametrisation is an emerging technique for representing model error on all timescales.
28. 28. Traditional computational ansatz for weather/climate simulators   Eg    u.  u   g  p   2u  t  X 1 X 2 X 3 ... ... X n Increasing scale Eg momentum“transport” by: Deterministic local •Turbulent eddies in boundary layer bulk-formula parametrisation P  X n ;  •Orographic gravity wave drag. •Convective clouds
29. 29. grid box grid boxDeterministic bulk-formula parametrisation is based on the notion of averaging over some putative ensemble of sub-grid processes inquasi-equilibrium with the resolved flow (eg Arakawa and Schubert, 1974)
30. 30. Hence reality is more consistent with grid box grid box which can’t be parametrised deterministically
31. 31. What’s Going On?• For deterministic short-range prediction, increased model resolution will give better representations of topography, land-sea contrast etc , but this will be offset by an increase in forecast error because smaller-scale circulations with faster error- doubling times will be simulated explicitly. Overall, deterministic skill scores (RMS error, ACC etc) may not increase with increased model resolution.• The conclusion is not that high-resolution modelling is a waste of time and resources, but rather that all predictions, even for the short range, must be considered probabilistic, ie ensemble based. There is no range at which the forecast problem can be treated deterministically. The “classical” era of deterministic numerical weather prediction should be drawing to a close, even for short-range prediction.• Probabilistic skill scores will increase with model resolution, provided the underpinning ensemble prediction systems (EPSs) are statistically reliable. Model error is a significant source of forecast uncertainty even in the short range and must be represented in an EPS. Stochastic parametrisation is an emerging technique for representing model error on all timescales.• Climate models may only converge to reality slowly. We may need convectively resolved models not only for reliable short-range prediction, but also for reliable climate prediction.
32. 32. Conclusions• By the “Butterfly Effect”, Lorenz had something more radical and more unpredictable than just sensitive dependence on initial conditions.• The “Real Butterfly Effect” refers to the problem of predictability associated with high-dimensional fluid turbulence in PDEs. Formally, it seems to be an open problem.• The Real Butterfly Effect is associated with “asymptotic ill posedness”. This can be studied numerically.• Understanding the “Real Butterfly Effect” is relevant to both short-range weather prediction and climate prediction, and in particular to the representation of model error in ensemble prediction systems.
33. 33. • In order to produce reliable forecast probability distributions, it is necessary to represent the errors introduced by deterministic closure schemes in our ensemble prediction systems.• These errors may be random, but can still impact on the mean state of the model
34. 34. Example of a very unreliable predictionsystem: the ECMWF medium-range high resolution deterministic forecast over Europe! Thomas Haiden, personal communicationOn about 70% of the occasions when the day 4-5 ECMWF high- res forecast said it would rain at least 10mm/day, it didn’t! Not good for decision makers.
35. 35. By contrast, probabilistic forecasts from the Ensemble Prediction System are reliable The single most important verification statistic from a decision maker’spoint of view
36. 36. Beyondthemediumrange,precipforecastsstart toloosereliability
37. 37. Southern Asia (India) UROSIP(E0002)
38. 38. PREC(1h) Summer 2011 00UTC Unreliability also a problem for short range forecasts of intenseReliability diagram rainfall log (# fcst) PREC(1h) PREC(6h) Christoph Gebhardt, personal communication COSMO-DE-EPS verification results March
39. 39. A Nonlinear Perspective on ClimateChange Seamless Prediction techniques allow us to test the strength of at least the first three linksBAMS April 2008 (Palmer, Doblas-