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.
1. The "real" butterfly effect:
A study of predictability in multi-scale
systems, with implications for weather and
climate
by
T.N.Palmer
University of Oxford
ECMWF
2. “The Butterfly Effect is a phrase that
encapsulates the more technical
notion of sensitive dependence on
initial conditions in chaos theory”
(Wikipedia)
3. Journal of the Atmospheric Sciences 1963
X = -s X + s Y
Y = -XZ + rX -Y
Z = XY - bZ
4.
5. dX
X Y
dt
dY
XZ rX Y
dt
dZ
XY bZ
dt
Exhibits sensitive but nevertheless continuous
dependence on initial conditions – you tell me how
accurately you want to know the forecast state, I’ll tell you
how accurately you need to know the initial conditions.
This is not what Lorenz had in mind by “The Butterfly
Effect” – he had in mind systems which might not exhibit
continuous dependence on initial conditions – these
exhibit a much more radical type of unpredictability.
6.
7. 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 90
30°N 30°N
9
80
8 8
40°N 40°N 40°N 40°N70
7 7
0
-12
6 -24 60
20°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 30
10°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
8. 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 at
50°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
80
40°N 40°N 80
70 40°N 40°N70
60 60
30°N 30°N
50 30°N 0
0
0
0
0
0
0
0
0 30°N50
-12
-24
-36
40 -48 40
20°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
-120
10°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
9. Lorenz. The Essence of Chaos
(1993)
“The expression (The Butterfly Effect) has a
somewhat cloudy history: It appears to have
arisen following a paper that I presented at a
meeting in Washington in 1972, entitled: Does
the Flap of a Butterfly’s Wings in Brazil Set Off
a Tornado in Texas..”
10. “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...”
12. “It is proposed that certain formally
deterministic fluid systems which possess
many scales of motion are observationally
indistinguishable 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
14. The “Real” Butterfly Effect:
A problem in PDEs, not ODEs
?
Let E (k ) denote the kinetic energy per unit
wave number of the system at wave number k
15. Suppose we are only interested in predicting some
low wavenumber (ie large-scale) k L .
How long before small-scale errors, confined to
N
wavenumbers 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
16.
17.
18. The “Real Butterfly Effect”
Error
Increasing scale
The Predictability of a Flow Which Possesses Many
Scales of Motion. E.N.Lorenz (1969). Tellus.
19. Most of the time, small (eg
convective) scales are controlled by
large (eg synoptic scales) and
hence L69 is an overly pessimistic
estimate of predictability. But
intermittently the opposite occurs…
21. For such cases, could it literally be
true that errors propagate up to the
large scale from arbitrarily small
scales in finite time?
22. “We have not been able to prove or disprove our
conjecture, since in order to render the
appropriate equations tractable we have been
forced to introduce certain statistical
assumptions which cannot be rigorously
defended.”
Lorenz 1969
23. 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?
24. Clay Mathematics Millenium
Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory
25. Clay Mathematics Millenium
Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory
26. MNS
Navier-Stokes Equations
For smooth initial conditions
and suitably regular
boundary conditions
do there exist smooth,
bounded solutions at all
future times?
27. 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. It
is unknown whether finite-energy solutions are “sufficiently smooth”
(Gregory Seregin - personal communication).
28. Asymptotic Ill Posedness
The question of strict ill-posedness is not
physically relevant to weather and climate
prediction: trunction scales in weather prediction
models are many orders of magnitude larger
than the viscous scale.
Consider, the weaker but more physically
relevant 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.
29. The real butterfly effect
Can we find “empirical evidence” from
operational NWP models?
32. 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.
33. 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
34. grid box grid box
Deterministic bulk-formula parametrisation is
based on the notion of averaging over some
putative ensemble of sub-grid processes in
quasi-equilibrium with the resolved flow (eg
Arakawa and Schubert, 1974)
35.
36. Hence reality is more consistent with
grid box grid box
which can’t be parametrised deterministically
37. 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.
38. 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.
39. • 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
40. Example of a very unreliable prediction
system: the ECMWF medium-range high
resolution deterministic forecast over
Europe!
Thomas
Haiden, personal
communication
On 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.
41. By contrast, probabilistic forecasts from the
Ensemble Prediction System are reliable
The single
most
important
verification
statistic
from a
decision
maker’s
point of view
44. PREC(1h) Summer 2011 00UTC Unreliability also a problem for
short range forecasts of intense
Reliability diagram rainfall
log (# fcst) PREC(1h) PREC(6h)
Christoph Gebhardt, personal communication
COSMO-DE-EPS verification
results
March
45. A Nonlinear Perspective on Climate
Change
Seamless Prediction
techniques allow us to
test the strength of at
least the first three links
BAMS April 2008 (Palmer, Doblas-
