The Butterfly Effect

The Butterfly Effect:
Myth or Reality?
A Million Dollar
Problem with Trillion-
Dollar Implications!
Tim Palmer, ECMWF

“The Butterfly Effect is a phrase that
encapsulates the more technical
notion of sensitive dependence on
initial conditions in chaos theory”
(Wikipedia)

dX
dt
dY
dt
dZ
dt
=− σX
+ σY
=− XZ + rX −Y
= XY −bZ
Lorenz 1963
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.
I believe this is not what Lorenz had in mind by “The
Butterfly Effect” – he had in mind systems without
continuous dependence on initial conditions – these
exhibit a much more radical type of unpredictability.

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…..
…Before the Washington meeting I had sometimes
used a sea gull as a symbol for sensitive dependence.
The switch to a butterfly was actually made by the
session convener .. who was unable to check with me
when he had to submit the program titles”

“The following is the text of a 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...”
Lorenz: The Essence of Chaos

Tellus 1969

“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

Let Ek ( ) denote the kinetic energy per unit
wave number of the system at wave number
k
−3/2 −1/2
Eddy "turn over" time τ ( k) : k E ( k)
The time taken for some initial small-scale error
to grow and nonlinearly infect some large-scale
component k , N powers of 2 larger in scale,
N
n
( N) τ ( 2 k )
Ω =∑
n=
0
L
L
:
:

−5/3 2/3
If Ek ( ) k then τ
−
: ( k)
: k an
N
∑
n
Ω ( N) = τ (2 k )
n=
0
L
N
2 /3
( ) ∑
− n
L
2 : τ
L)
n=
0
= τ k ( k as
N →∞
d

The “Real Butterfly Effect”
Error
Increasing scale
The Predictability of a Flow Which Possesses Many
Scales of Motion. E.N.Lorenz (1969). Tellus.
Cf G.D.Robinson 1967

Ek ( )
:
3
k −
Ek ( )
:
5/3
k −
Atmospheric Wavenumber Spectra

“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

Clay Mathematics Institute
Millenium Million-Dollar Prize
Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory

Clay Mathematics Institute
Millenium Million-Dollar Prize
Problems
• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincaré Conjecture
• Riemann Hypothesis
• Yang-Mills Theory

Navier-Stokes Equations
For smooth initial conditions
and suitably regular
boundary conditions
do there exist smooth,
solutions at all future
times?

The Millenium Navier-Stokes
problem concerns the finite-time
downward cascade of energy to
arbitrarily small scales.
An equivalent description of the
problem is in terms of the finitetime
upscale cascade of error
from arbitrarily small scales, ie
the ”Real Butterfly Effect”!

Climate Change is the
“defining issue of our era*”
*Ban Ki Moon UN Secretary General

….. and a trillion-dollar problem
(Stern Review on the Economics
of Climate Change)

Standard Paradigm for a Weather/ Climate Model
⎛ ∂ ⎞
ρ⎜
+ ∇ ⎟ = ρ −∇ + ν ∇
⎝∂t
u ⎠
u g u
2
. p , ...
X X ...
1
X
2 3
...
X
n
Increasing scale
Eg Clouds, radiation,
small-scale topography..
Local bulk-formula
parametrisation
P X
( )
n ;
α
to represent
unresolved processes

20-km resolution

80-km resolution

320-km resolution

The “Real Butterfly Effect” raises fundamental
unanswered questions about convergence of
climate simulations with increasing resolution:
1. How much will uncertainties in climate-change
predictions reduce if models are run at 20km, 2km or
even 0.2km resolution rather than say 200km
resolution?
2. Is there an irreducible level of uncertainty in
predictions of climate change associated with the
underlying PDEs of climate?
3. What is the most efficient way of using finite
computer resources for climate prediction – eg how to
partition resources between resolution, earth-system
complexity and ensemble size?

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
high-dimensional fluid turbulence in PDEs
• Myth or Reality? The Real Butterfly Effect is
unproven and the subject of a million-dollar
mathematics prize.
• The “Real Butterfly Effect” is relevant to assessing
our ability to predict climate change – a trilliondollar
problem!