the butterfly effect

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The real butterfly effect

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2014 Nonlinearity 27 R123

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| London Mathematical Society Nonlinearity

Nonlinearity 27 (2014) R123R141 doi:10.1088/0951-7715/27/9/R123

Invited Article

The real butterfly effect

T N Palmer1, A Doring1,2 and G Seregin3

1 Department of Physics, University of Oxford, Parks Rd, Oxford, OX1 3PU, UK2 Department of Physics, University of Erlangen-Nurnberg, Staudstrasse 7, 91058, Germany3 Mathematical Institute, University of Oxford, Woodstock Rd, Oxford, OX2 6GG, UK

E-mail: [email protected]

Received 4 February 2014, revised 13 July 2014Accepted for publication 14 July 2014Published 19 August 2014

Recommended by K Julien

AbstractHistorical evidence is reviewed to show that what Ed Lorenz meant by theiconic phrase the butterfly effect is not at all captured by the notion of sensitivedependence on initial conditions in low-order chaos. Rather, as presented inhis 1969 Tellus paper, Lorenz intended the phrase to describe the existence ofan absolute finite-time predicability barrier in certain multi-scale fluid systems,implying a breakdown of continuous dependence on initial conditions for largeenough forecast lead times. To distinguish from mere sensitive dependence,the effect discussed in Lorenzs Tellus paper is referred to as the real butterflyeffect. Theoretical evidence for such a predictability barrier in a fluid describedby the three-dimensional NavierStokes equations is discussed. Whilst it isstill an open question whether the NavierStokes equation has this property,evidence from both idealized atmospheric simulators and analysis of operationalweather forecasts suggests that the real butterfly effect exists in an asymptoticsense, i.e. for initial-time atmospheric perturbations that are small in scale andamplitude compared with (weather) scales of interest, but still large in scale andamplitude compared with variability in the viscous subrange. Despite this, thereal butterfly effect is an intermittent phenomenon in the atmosphere, and itspresence can be signalled a priori, and hence mitigated, by ensemble forecastmethods.

Keywords: butterfly effect, finite-time predictability, chaos, surface quasi-geostrophic equationsMathematics Subject Classification: 37L99

1. Introduction

The butterfly effect is one of the most iconic phrases in 20th century science. As described innumerous textbooks and as understood by almost all those who work in nonlinear dynamics,

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Nonlinearity 27 (2014) R123 Invited Article

the expression encapsulates the more technical notion of sensitive dependence on initialconditions in chaos theory. The expression is generally attributed to Ed Lorenz as a metaphorfor the unpredictability of low-order chaotic systems, as described in his seminal 1963paper [18].

As is often the case, the truth of the matter has become distorted by the passage oftime. The phrase the butterfly effect was coined by Gleick in his popular book on Chaos[6]. Gleicks inspiration was the title of a presentation which Lorenz gave at a scientificmeeting in 1972, based on relatively recent research [19] on loss of predictability in multi-scale fluid systems. Crucially, the type of unpredictability described in this research wasmuch more radical than mere sensitive dependence on initial conditions. In particular,this research led Lorenz to speculate on the existence of an absolute horizon for loss ofpredictabilitya horizon which could not be extended in time by reducing uncertainty in theinitial conditions. The Lorenz 1963 system cannot exhibit this type of unpredictability because,although sensitive to initial conditions, solutions nevertheless depend continuously on initialconditions.

In section 2 these historical developments are described in more detail. In section 3results are presented from an idealized nonlinear simulator of three-dimensional turbulencewhich provides some evidence for the finite-time loss of predictability described by Lorenz.In section 4 evidence is discussed for whether the effect about which Lorenz speculated is aprovable property of the three-dimensional NavierStokes equations. We conclude it is stillan open question.

Since the butterfly effect is almost universally considered synonymous with sensitivedependence on initial conditions, in this paper we introduce the expression the real butterflyeffect to describe what Lorenz really had in mind. In section 5 we reformulate the realbutterfly effect in terms of what is described as asymptotic ill conditioning and conclude thatit is a real effect and extremely relevant to modern-day weather prediction. On the other hand,it appears only intermittently and its effect and can be largely mitigated through ensembleforecast techniques. Some concluding remarks are made in section 6.

2. The real butterfly effect: an historical analysis

The opening chapter of James Gleicks influential book popularizing the science of chaosis entitled The Butterfly Effect. Gleick describes the now well-known story of howmeteorologist and founding father of chaos theory, Ed Lorenz, discovered the sensitivedependence on initial conditions of solutions of his seminal 1963 model of low-orderchaos

x = (y x)y = x(r z) yz = xy bz. (2.1)

Here X = (x, y, z) is the state vector of the Lorenz system, whilst , r and b are positiveparameters. What was Gleicks inspiration for this iconic phrase? It came from the title of apresentation which Lorenz gave at a meeting of the American Association for the Advancementof Science (AAAS) in December 1972. The title of the presentation was Does the Flap of aButterflys Wings in Brazil set off a Tornado in Texas?.

That Ed Lorenz himself saw the phrase the butterfly effect as describing the phenomenadescribed in this AAAS presentation is consistent with the fact that the first appendix to hispopular 1993 book The Essence of Chaos [20], which reproduces in original form the text of

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this AAAS presentation, is also entitled The Butterfly Effect 4. Below, a key part of thispresentation is reproduced verbatim. From this it will be seen that what Lorenz understood bythe butterfly effect is in fact something much more radical than mere sensitive dependenceon initial conditions.

To set the scene, Lorenz is discussing the predictability of weather systems in a sessiondevoted to an international weather research programme called The Global AtmosphericResearch Programme. Consider a cyclonic weather system with an overall scale of a thousandkilometres. A key question is to ask how far ahead could such a weather system be predicted:one day, one week, one month? In considering the answer to this question, Lorenz noted thatembedded within this weather system may be mesoscale structures whose scales are hundredsof kilometres or less and embedded within these mesoscale structures may be individual cloudsystems with scales of kilometres. Within a cloud, the saturated air can be highly turbulentwith sub-cloud eddies having scales of metres or less. Lorenz asks the question: what is crucialfor determining the overall predictability of the cyclonic weather system? Is it uncertainties inthe initial state on the scale of the weather system itself, or uncertainties in the initial state onthe scale of the sub-cloud turbulence. At this point it is worth quoting directly from appendix 1of The Essence of Chaos:

(1) Small errors in the coarser structure of the weather patternthose features whichare readily resolved by conventional observing networkstend to double inabout three days. As the errors become larger, the growth rate subsides. Thisinformation alone would allow us to extend the range of acceptable predictionby three days every time we cut the observation error in half, and would offer thehope of eventually making good forecasts several weeks in advance.

(2) Small errors in the finer structuree.g. the positions of individual cloudstendto grow much more rapidly, doubling in hours or less. This limitation alone wouldnot seriously reduce our hopes for extended-range forecasting, since ordinarilywe do not forecast the finer structure at all.

(3) Errors in the finer structure, having attained appreciable size, tend to induceerrors in the coarser structure. This result, which is less firmly established thanthe previous ones, implies that after a day or so there will be appreciable errorsin the coarser structure, which will thereafter grow just as if they had beenpresent initially. Cutting the observation error in the finer structure in halfa formidable taskwould extend the range of acceptable prediction of even thecoarser structure only by hours or less. The hopes for predicting two weeks ormore in advance are thus greatly diminished.

The cognoscenti in the audience would have known that in this presentation Lorenz wasactually giving a highly simplified version of his technical paper The predictability of a flowwhich possesses many scales of motion, published in 1969 (and hence only a few years beforethe AAAS meeting) in the Scandinavian journal Tellus. The following words are taken directlyfrom the abstract of the 1969 paper.

It is proposed that certain formally deterministic fluid systems which possess manyscales of motion are observationally indistinguishable from indeterministic systems;

4 As discussed by Lorenz in [20], this title was actually created by the Session Chair, meteorologist Phil Merilees(who was unable to contact Lorenz at the time the programme titles had to be submitted). However, Merliees titlewas quite appropriate because Lorenz himself had used the symbolism of flaps of seagulls wings as characterizing thepotential unpredictability of weather. If Merilees title had actually used the symbolism of seagulls and the seagulleffect had instead become common currency in describing sensitive dependence in low-order chaos, the discussionbelow would still be just as relevant.

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specifically that two states of the system differing initially by a small observationalerror will evolve into two states differing as greatly as randomly chosen states ofthe system within a finite time interval, which cannot be lengthened by reducing theamplitude of the initial error.

The crucial words which distinguish the type of systems considered in the 1969 paper fromthose for whom the 1963 model is a prototype, occur at the endfor these systems a reductionin amplitude of initial error will not reduce the amplitude of the forecast error, after a certainfinite time.

Low-order chaos cannot have this property of finite-time loss of predictability: for example,even though the Lorenz 1963 system exhibits sensitive dependence on initial conditions, itnevertheless exhibits continuous dependence on initial conditions. Put simply, one can predictas far ahead as one wants in the Lorenz 1963 system, providing uncertainty in the initialconditions is small enough. To show that the Lorenz 1963 system has the property of continuousdependence on initial conditions, consider first the energy identity:

1

2

t|X|2 + x2 + y2 + bz2 = ( + r)xy,

where |X| is the length of the state vector X. Since the rate of growth of |X| is determined byterms which are no more than quadratic in the state variables, very rough arguments show thatthere exists a positive constant c1, depending on , r , and b such that

|X(t)| |X0| exp (c1t)for all t 0, where X0 = (x0, y0, z0) denote initial conditions. Of course, Lorenz showedsomething much strongeri.e. that trajectories cannot escape to infinity as t . Moreprecisely, there exists a ball of R3 centred at the origin such that the rest of the trajectory X(t),starting from some t0 0, remains in this ball. The radius of this ball is determined by , rand b only. In the language of dynamical systems, (2.1) has a bounded absorbing set. Now letX(t) + X(t) be a solution to the Lorenz system with perturbed initial data X0 + X0 so thatX(t) satisfies the perturbed system:

x = (y x),y = x(r z) xz zx y,z = xy + yx + xy bz.

The energy identity for the perturbed system can be written as

1

2

t|X|2 + (x)2 + (y)2 + b(z)2 = ( + r)xy + (yz zy)x.

Consider a time t in the interval [0, T ], then, taking into account the first energy estimate, wecan state that there exists a positive constant c2(, r, b,X0, T ) such that

|X(t)| |X0| exp (c2t) (2.2)for all t [0, T ]. The latter means that solutions to the Lorenz system depend continuouslyon the initial data on the closed interval [0, T ].

A simple scaling argument can be used to understand why the type of nonlinear multi-scale systems described by Lorenz in [19] may not necessarily be of this type. Consider athree-dimensional multi-scale turbulent fluid. Let k denote horizontal eddy wavenumber andE(k) the corresponding eddy kinetic energy per unit wavenumber. From simple dimensionalarguments a generic timescale (k) k3/2E1/2 can be defined; it is supposed that (k)

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characterizes the time it takes for errors at wavenumber k to grow and infect nonlinearly theaccuracy of simulations at the larger scale k/2.

Now suppose we are only interested in predicting large-scale aspects of the flow (inLorenzs terminology, the cyclonic weather pattern, but not the detailed cloud structuresembedded within the cyclone). Let kL denote a characteristic wavenumber of these large-scale weather patterns. It can then be asked how long, P , it will take before initial errors atlarge wavenumbers 2NkL, N 1, will affect large-scale simulations of the flow. A plausibleestimate of this is given by [17]

P (N) = (2NkL) + (2N1kL) + (2N2kL) + . . . (20kL) =Nn=0

(2nkL). (2.3)

Now the kinetic energy E(k) of nonlinear multi-scale systems often exhibits power-lawstructure [21]. For rapidly rotating almost two-dimensional fluid systems forced at large-scale, e.g. as described by the quasi-geostrophic equations, E(k) k3 implying that (k) isactually independent of k. In such systems, P (N) diverges as N . This is consistentwith continuous dependence on initial conditions: providing the initial error is at a sufficientlysmall scale and hence is sufficiently small in amplitude, it can take an arbitrarily long timebefore the initial error infects scales greater than the minimum scale of interest kL.

On the other hand, for a fully three-dimensional fluid system, thenE(k) k5/3 consistentwith the famous Kolmogorov scaling. For such a system (k) k2/3 and P (N) isconvergent (to about 2.7(kL)) as N . This is precisely the situation described morequalitatively in Lorenzs AAAS paperthe paper which spawned the phrase the butterflyeffect. In particular, if the weather scale of interest is equal to 1000 km and an inherentpredictability time (kL) 3 days, then a cloud scale with a length scale of 1 km will have ainherent predicability time3/4 h. Going further, the same scaling estimate implies a sub-cloudeddy with scale 1 m would have an inherent predictability time of less than a minute. Hence,according to this scaling argument, if we invested in an observing system which measuredthe initial state perfectly down to scales of 1 m, we would add less than half a minute to thepredictability time of the weather scale of interest, compared with an observing system withresolution 1 km.

Figure 1, from [19], illustrates this graphically. To produce this result, Lorenz considersthe two-dimensional vorticity equation

t + J (, ) = 0 2 = , (2.4)which is linearized about a reference solution. An equation for the evolution of error variance isthen derived with expectations over ensembles of errors and over reference states. Assumptionsof isotropy and homogeneity are made, implying that solutions depend only on k. Also,fourth order moments, which arise as products of quadratic functions of the reference solutionmultiplied by quadratic functions of the error terms, are assumed to be factorizable as productsof expectations of quadratics. The reader is referred to [19] for details of the derivation of thisquasi-empirical equation. However, the key equation for error growth which results from thisanalysis is given by

d2Zkdt2

=n

l=1CklZl, (2.5)

where Zk is the ensemble-mean of the kinetic energy of the error fields as a function ofwavenumber k, and Ckl is a (highly asymmetric) array of spectral interaction coefficients.Nonlinearity is represented in the model by simply cutting off the growth of error when itreaches saturation, i.e. the value of the background kinetic energy. This sudden imposition of

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4000

0

2000

0

1000

0

5000

Kilometres

2500 62

5

156 39 1.2

0.4 0

5 days

1 day

5 hrs

1 hr

15 m

Figure 1. The key figure from [19] illustrating what Lorenz himself meant by thebutterfly effect. The bounding curve at the top shows the saturation energy as a functionof horizontal scale. The other curves show the evolution of error, where initial error iscontained in very small scales only. The error growth curves coincide with the boundingcurve at their intersection with the bounding curve. Areas are proportional to energy.According to the figure, a further halving of initial error would add minutes or less to theoverall predictability time of the large scales. Reproduced with permission from [19].

nonlinearity has recently been improved by Durran and Gingrich [4], without any dramaticchange in results.

Figure 1 shows a bounding curve which describes an assumed background kinetic energyspectrum as a function of horizontal scale. The other curves show the growth in amplitudeand spatial scale of the energy of forecast error, with initial error restricted to very largewavenumbers. The error is shown at forecast times of 15 min, 1 h, 5 h, 1 day and 5 days.The error curves coincide with the thick bounding curve to the right of their intersection withthe bounding curve. On the basis of this model, a typical weather scale would have anabsolute predictability of about 1 day, whilst the largest planetary scales in the atmospherewould have an absolute predictability of about 10 days. According to the real butterfly effect,these predicability estimates would not be extended in any significant way, by making theinitial error smaller.

Over the years, as numerical weather prediction systems have become more and moresophisticated, Lorenzs predictability estimates have been shown to be unduly pessimisticthese days individual weather systems can often be predicted with skill a week or moreahead [30]. This implies either that one or more of the assumptions used by Lorenz in his 1969paper must be incorrect, or that the phenomenon is not as ubiquitous as Lorenz may have thoughtit was. Some of these issues are discussed in more detail below. However, for now, as a resultof the fact that Lorenzs analysis was not rigorous, and because estimates of predictabilityappeared unduly pessimistic, the 1969 paper is not so well known to recent generations ofmeteorologists as it was in the early 1970s. On top of this, with the mathematicians discoveryof the 1963 paper in the 1970s, the 1969 paper has been utterly eclipsed by the 1963 paper.However, as will be discussed in the next section, with minor modifications, the basic conceptillustrated in the 1969 paper is not at all discredited. The real butterfly effect is still alive andflapping!

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Figure 2. Variance power spectra of wind and potential temperature based on aircraftobservations. The spectra of meridional wind and temperature are shifted by one andtwo decades to the right, respectively. Lines with slopes 3 and 5/3 are entered at thesame relative coordinates for each variable, for comparison. Reproduced from [24]. American Meteorological Society. Used with permission.

3. The surface quasi-geostrophic equations

As discussed by Rotunno and Snyder [29] at the time Lorenz was writing his 1969 paper,it was only just becoming apparent that the observed energy spectrum of the atmosphericcirculation on weather (and larger timescales) is much closer to a k3 power-law behaviourthan a k5/3 behaviour [31], consistent with the energy spectrum of an almost two-dimensionalsystem forced at the large-scale. Hence, Lorenzs use of a two-dimensional vorticity equationto represent k5/3 behaviour is inconsistent with these observations. Moreover, as discussedabove, in a k3 system, the predictability time can be made arbitrarily long by making initialerror sufficiently small. In particular, the predictability time of a k3 system can be muchlonger than that of a k5/3 system. This would seem to explain the pessimistic predicabilityestimates in Lorenzs 1969 paper.

However, as estimates of atmospheric energy spectra were obtained over a range ofhorizontal scales, from thousands of kilometres to kilometres, it was clear (see figure 2) thatthe observed power-law structure makes a transition from k3 to k5/3 at scales of around100 km. At this scale the effects of Earths rotation become less dominant on circulations,and their structure becomes increasingly three dimensional. These days, the truncation scalesof global numerical weather prediction models (typically around 1020 km) lie well withinthe k5/3 range. One can therefore speculate that the real butterfly effect remains relevant tomodern-day prediction, in the sense that a decrease in initial error made by decreasing modelresolution may not increase the predictability of weather scales of interest.

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But how can this be tested? The so-called surface quasi-geostrophic (SQG) equations [1]have been studied intensely since the mid-90s, see e.g. [7]. The equations provide an interestingtoy model that exhibit multi-scale physics and turbulence-like behaviour with a k5/3 energyspectrum. These equations describe the evolution of some boundary temperature field underadvection by an internal 3D fluid, which itself satisfies a strong (zero potential vorticity)constraint. Hence, although the SQG equations are formally 2D and therefore computationallytractable, they describe the energy-cascade dynamics of a 3D turbulent fluid (in particular,therefore, they should not be confused with the more familiar quasi-geostrophic equations)which mimics the 3D Euler equations [22]. The SQG equations are

D

Dt=

t+ v = 0, (3.6)

where the 2D velocity, v = (v1, v2), is determined by by a stream function ,

(v1, v2) =( x2

,

x1

)= , (3.7)

and satisfies

() 12 = . (3.8)Thus can be inferred from the scalar by

= R2

1

|y|(x + y, t) dy. (3.9)

In the presence of topography h, we replace by + ch in (3.6), where c is a constant, so that

D

Dt=

t+ v ( + ch) = 0. (3.10)

There are a number of physical, geometric and analytic analogies between the SQG equationsand the 3D Euler equations which make numerical investigations with the SQG equationsappropriate. In particular, the vector field is analogous to the vorticity in 3Dincompressible flow, and the level sets of are analogous to vortex lines for 3D Euler equations.For certain initial conditions, the size of | | grows fast, and it is an open question whethera singularity can form in finite time. For more details, the reader is referred to [22].

Rotunno and Snyder repeated the analysis in Lorenzs 1969 paper, but now using the SQGequations. Key results from that study are shown in figure 3. In figure 3(a) parametrizederror growth for the SQG system is shown. Figure 3(b) shows the same for two-dimensionalturbulence with a k3 spectrum. Unlike figure 1 for the Lorenz 1969 system, the error growthis shown with linear forecast time increments. For SQG, consistent with the real butterflyeffect, the convergence to a finite-time horizon at t = 2 is clear. By contrast in the 2D systemthere is no convergence at t = 2or indeed any other timeand hence no real butterfly effect,even though the 2D system still exhibits sensitive dependence on initial conditions, and hencechaos.

Rotunno and Snyders study continued to use Lorenzs parametrized error growth model.It is of interest to assess evidence for the real butterfly effect in the SQG system withoutresorting to such a parametrized error growth model. Here the growth of small initial-conditionperturbations in the SQG model is shown 5. To do this, a long reference integration was madewith a truncation wavenumber k = 127 with fractal surface topography h defined by a simplemidpoint displacement algorithm. Perturbations were added to states from the reference run

5 Here we use the spectral implementation www.cims.nyu.edu/shafer/tools/index.html of the SQG equations byShafer Smith at the Courant Institute.

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http://www.cims.nyu.edu/~shafer/tools/index.html


(a) SQG

K K

5/3

t=0

t=2

1 104 1 104

(b) 2DV

3

t=0

t=2

Figure 3. A reworking of figure 1, using the surface quasi-geostrophic equationsas well as the two-dimensional vorticity equations. Evolution of error energy perunit wavenumber as a function of time, for linear increments in time (a) surfacequasi-geostrophic turbulence and (b) two-dimensional turbulence. The heavy solidline indicates the base-state kinetic energy spectra per unit wavenumber. Reproducedfrom [29]. American Meteorological Society. Used with permission.

states at four different times ti(0) along the trajectory (defined by ti(0) = 1, 1.5, 2 and 2.5million timesteps from the start of the reference integration). For each ti(0), three differentlysized perturbations pj were formed by taking the difference between the reference integrationat ti(0) and the reference integration at ti(0) + 2 million timesteps, and filtering this differenceso that only components with wavenumber greater than kj remain. Here k1 = 15, k2 = 31and k3 = 63. As an example, figure 4 shows and at t0 = 1 million timesteps. In orderto define perturbation amplitude, all fields were first filtered to retain only wavenumbers 15 orless (these, it is assumed, correspond to the larger-scale weather patterns of interest). Then,for each grid point, the square of the difference between the perturbed and the unperturbedstreamfunction field was taken, the values averaged over all grid points, and the square roottaken. Results are shown in figure 5.

A number of points can be made. Firstly it can be seen that the growth of perturbations isdependent on the initial condition ti(0). This is to be expected in any nonlinear system. (LetX = F [X] denote a nonlinear dynamical system. Small perturbations evolve according to thelinearized equation X = dF/dX X. Since F is at least quadratic in X, then J is at leastlinear, and hence dependent on X.) Secondly, it can be seen that for all four initial conditions,a reduction in the amplitude of the initial perturbation from k1 = 15 to k2 = 31 does leadto a reduction in the amplitude of the evolved perturbation, at least to about 4000 timesteps.However, in three of the four cases (figure 5 top left, top right and bottom left), a furtherreduction in the amplitude of the initial perturbation from k2 = 31 to k2 = 63 leads to almostno further drop in evolved perturbation amplitude. For the fourth initial condition (figure 5bottom right), this further reduction in amplitude does lead to a long-term reduction in theamplitude of the perturbation. Overall, these results are consistent with behaviour expectedas a result of the real butterfly effect. However, in order to provide results which can be

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Figure 4. Model state after 1M timesteps. Top: field. Bottom: field.

compared more directly with those from [29], a much larger number of initial conditions andwith a much higher resolution version of this model will be needed. It is intended to performsuch experiments and report on the results in due course. Nevertheless, a consequence offigure 5, which could not have been obtained from the parametrized error model, is that thereal butterfly effect is likely to be an intermittent phenomenon in practice. This in turn raisesan important practical question as to whether it is possible to mitigate the impact of suchsituations by flagging a priori the possibility that a given initial state is likely to be particularlysensitive to rapidly growing small-scale perturbations. This issue is addressed further insection 5.

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0 5000 100000

0.002

0.004

0.006

0.008

0.01

0 5000 100000

0.002

0.004

0.006

0.008

0.01

0 5000 100000

0.002

0.004

0.006

0.008

0.01

0 5000 100000

0.002

0.004

0.006

0.008

0.01

Figure 5. Growth of perturbations pj , j = 1 . . . 3 from initial conditions ti (0),i = 1 . . . 4. Solid: j = 1 corresponding to initial perturbations truncated to wavenumber15. Dotted: j = 2 corresponding to initial perturbations truncated to wavenumber 31.Dashed: j = 3 corresponding to initial perturbations truncated to wavenumber 63.

4. The NavierStokes equations

Referring back to the quote in section 2 from the abstract of [19], the real butterfly effect positsthat two states of the atmosphere, differing initially by a small perturbation field, will evolveinto two states differing as much as randomly chosen states of the system within a finite timeinterval, and that this time interval cannot be lengthened by reducing the amplitude of the initialerror. One could imagine that this small perturbation projects entirely onto scales within theviscous subrange. Surely, when the perturbation is this small, the real butterfly effect cannotbe literally true. Can it?

Lorenz was aware that his parametried analysis of the real butterfly effect was not rigorous,commenting the following.

We have not been able to prove or disproof our conjecture, since in order to render theappropriate equations tractable we have been forced to introduce certain statisticalassumptions which cannot be rigorously defended.

The question therefore arises: is the real butterfly effect a property of the full NavierStokesequations? To answer this, recall that a mathematical model of some physical phenomenon iswell posed if it has the properties that

solutions exist, solutions are unique, solutions depend continuously on the initial data in some reasonable topology.

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As discussed above, the real butterfly effect in its literal sense can be associated with abreakdown of continuous dependence, and hence a breakdown of well-posedness. Is theinitial-boundary value problem for the 3D NavierStokes equations ill posed?

The classical initial-boundary value problem for the NavierStokes system describing theflow of a viscous incompressible fluid in a domain of the three-dimensional Euclidian spaceR

3 can be formulated as follows: find the velocity field u(x, t) = (u1(x, t), u2(x, t), u3(x, t))and the pressure field p(x, t) that satisfy the NavierStokes equations

tu(x, t) + u(x, t) u(x, t) u(x, t) = p, divu(x, t) = 0 (4.11)for all x and for all instances of time t > 0, subject to the homogeneous Dirichletboundary condition

u(x, t) = 0 (4.12)for all x belonging to the boundary of the domain and for all t > 0, and the initialcondition

u(x, 0) = u0(x) (4.13)for all x . It is supposed that u0 is a given smooth divergence-free field vanishing on theboundary and the viscosity is a positive parameter.

Before discussing continuous dependence on initial conditions, there is a more basic issueto consider: whether or not there exists a unique flow u starting with the initial velocity u0 andsmoothly evolving in time from t = 0 to . This is one of seven millennium problems statedby the Clay Mathematical Institute in 2000, see [5]. It is still an open problem.

On the other hand, providing

|u0(x)|2 dx < , (4.14)then it can be shown that there exists a positive time T , depending on u0 and , and a pair ofsmooth functions u and p satisfying the NavierStokes equations in (0, T ). For t < Tthe smooth function u is known as a strong solution to the NavierStokes equations.

Denote by T the longest time that such a strong solution u to (4.11)(4.13) exists. Thenthe millennium problem may be reformulated by asking whether T = or T < . It canbe shown that T = , implying a unique global strong solution, providing the quantity on theleft-hand side of inequality (4.14), is sufficiently small. Conversely, the necessary conditionfor T < is that

limtT0

supx

|u(x, t)| = . (4.15)

The time T is also referred to as a blow-up time since it is known that (4.15) is equivalent to

limtT0

| u(x, t)|2 dx = ,where u is the vorticity. The short-term existence of strong solutions was proven for theCauchy problem, which is the problem defined by (4.11)(4.13) for the case = R3, in thecelebrated paper [16]. For a bounded domain , the same type of result was established in [11]some time later.

Let us explore in more detail the issue of how solutions depend on the initial data. To thisend, consider two strong solutions u1 and u2 with the corresponding initial data u10 and u

20

and with blow-up times T1 and T2. We let v = u1 u2 and T = min (T1, T2). Then directlyfrom (4.11) one can deduce that

tv v + (p1 p2) = div(u1 u1 u2 u2

). (4.16)

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Similar to the analysis in section 2, let us now derive an energy equation. Specifically, if wemultiply (4.16) by v and integrate the product over , the following is found as a result ofintegration by parts:1

2t

|v|2 dx +

|v|2 dx =

(u1 u1 u2 u2

): v dx

=

(u2 v

): v dx =

(v v

): u2 dx

(

|v|4 dx) 1

2(

|u2|2 dx) 1

2.

Using the following multiplicative inequality(

|v|4 dx) 1

2 c(

|v|2 dx) 1

4(

|v|2 dx) 3

4.

together with the Young inequality, it can be shown

t

|v|2 dx +

|v|2 dx c3

|v|2 dx(

|u2|2 dx)2.

Applying Grownwalls lemma, a key inequality

u1(, t) u2(, t)2, u10 u202, exp( c3

t0

(

|u2(x, )|2 dx)2

d)

(4.17)

valid for all t (0, T ) and for some universal positive constant c is obtained. Here,f 2, :=

(

|f (x)|2 dx) 1

2.

Inequality (4.17) is the equivalent for the NavierStokes equation of what (2.2) is for theLorenz 1963 equations. Now, from (4.17), asu10u202, 0 so doesu1(, t)u2(, t)2,,implying unique solutions before the blow-up time. The question we may now ask is whetherinequality (4.17) also implies well posedness on long timescales, i.e. in the sense of the realbutterfly effect. As will be seen, this is a delicate issue, dependent on the choice of distancefunction, as discussed below.

As mentioned above, the existence of strong solutions is known only on a short timeinterval whose length tends to zero as the Reynolds number goes to infinity. However, asdescribed more explicitly below, the notion of what is meant by a solution can be weakened,and existence of so-called weak solutions can be proven over arbitrarily long time intervals.The existence of such weak LerayHopf solutions to the problem (4.11)(4.13) has been provenby Leray in [16] for = R3 and by E. Hopf in [9] for a bounded domain . Weak LerayHopf solutions are also called energy solutions, since the fluid kinetic energy and fluid energydissipation are bounded. Moreover, the velocity field u obeys the energy inequality

|u(x, t)|2 dx + 2 t

0

|u(x, )|2 dx d

|u0(x)|2 dx (4.18)for all t 0. Ideally, one would like expression (4.18) to be an equality rather than aninequality, but u is not smooth enough to allow an equality to be derived from the NavierStokes equations directly; this is why such solutions are called weak solutions. In particular,we do not know whether or not a weak solutionu has second derivatives in spatial variables andfirst derivatives in time; such derivatives are needed simply to write down the classical system(4.11). To be precise, weak solutions satisfy the NavierStokes equations in the followingsense:

0

( u tw + u : w u u : w

)dx dt = 0 (4.19)

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for any smooth test function w which is divergence free and vanishing in a neighbourhood ofthe boundary of the spacetime domain (0,). The reason why the function u is stillreferred to as a solution to (4.11) is that any smooth solution to (4.11) satisfies the identity(4.19) and conversely any smooth function satisfying (4.19) also satisfies the classical system(4.11). We should emphasize that it is not known if a weak solution u is smooth. In otherwords, the Millennium NavierStokes problem can be described as that of either showing thatu is smooth, or presenting a counter-example.

For the NavierStokes equations, the ideal choice of phase space (from the physical point ofview) is the space of solenoidal (i.e. divergence free) vector-valued functions which are squareintegrable over the domain . We denote this space by H(). Each function u0 from thisspace can be approximated by smooth divergence-free functions vanishing in a neighbourhoodof with respect to the distance of the Lebesgue space L2(). A metric in this space isdefined with the help of the norm 2, so that the distance between two functions f and g inL2() is given by f g2,. The physical relevance of such a space is that it consists of allsolenoidal vector-valued functions having finite kinetic energy. However, it does not appearpossible to prove uniqueness results with this choice. In particular, although there exists atleast one weak LerayHopf solution for each initial data from the space H(), uniquenessof solutions in the energy class cannot in general be proved even on a short time interval. Infact, there is a strong belief motivated by very recent result in [10] that there is no continuousdependence on initial data in H(). That is to say, there might be initial data belonging toH() providing non-uniqueness on any interval (0, T ) implying, essentially, instantaneousnon-uniqueness.

In summary, we cannot, even formally, define a dynamical system with unique solutions inthe space H() for the three-dimensional NavierStokes equations. However, due essentiallyto Leray, the following can be proved. Assume that we have two weak LerayHopf solutionsu1 and u2 starting from initial data u10 and u

20, respectively. Suppose in addition that u

2 isa strong solution to the initial-boundary value problem (4.11)(4.13) on the interval (0, T )with good initial data u20 satisfying (4.14). Then the estimate (4.17) still holds. This impliescontinuous dependence of solutions on the initial data in H(). However, again, this resultis practically useless for large t and small viscosity . Hence, we cannot disprove the realbutterfly effect.

On the other hand, it is very important to note that, in contrast with finite-dimensionaldynamical systems, the choice of phase space for systems generated by partial differentialequations can have a significant impact on the whole mathematical picture. What happens ifwe change the phase space? First of all, at least for bounded domains, there is no phase spacedifferent from H() where one could construct a global solution to initial boundary-valueproblem (4.11)(4.13). In the case = R3, there exist so-called local energy solutions ofLemarie-Riesset, possessing properties similar to weak LerayHopf solutions, see [14]. In thecase of = R3, there exists a literature of work on the so-called mild solutions. These existon a short time interval for a wider class of initial data. Moreover, as has been shown in [2],see theorem 1.1, there exists a global (in time) mild solution u for which a so-called norminflation happens. A bit more precisely, given > 0, there exists a mild solution whose initialdata are smooth and their norm in a certain Besov space is less than while the same Besovnorm of the solution u is greater than 1/ for 0 < t < .

Finally, it is worth remarking that in the two-dimensional case things are much better,see [12]. Weak solutions are unique and smooth for positive values of t . For them, (4.17) takesthe form

u1(, t) u2(, t)2, u10 u202, exp( c

t0

|u2(x, )|2 dx d)

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and according to the energy inequality (4.18), which in the two-dimensional case is, in fact,the identity, we have

u1(, t) u2(, t)2, u10 u202, exp(c2tu202,

)for any positive values of t and for some universal constant c. This estimate tells us that in thetwo-dimensional case, we do indeed have continuous dependence on the initial data in H().Hence, the initial-condition boundary-value problem for 2D NavierStokes is well-posed inH() and there is no real butterfly effect.

However, there are several other cases including situations with axial symmetry with noswirl where continuous dependence can be proved [13]. More relevant to the present discussion,it has been proven recently that strong solutions of the initial-boundary value problem for theso-called 3D primitive equations, derived using the hydrostatic approximation, are well posedwith H 2 initial data [3]. The primitive equations have been fundamental in the developmentof numerical weather prediction. Hence although the reality of the real butterfly effect cannotbe proven or disproven in general, perhaps it is not relevant in practice, in weather prediction.In the next section we argue otherwise.

5. Relevance of the real butterfly effect in the real world: asymptoticill-posedness

One could argue that a literal breakdown of continuous dependence is an irrelevance for ourunderstanding the real physical world; for perturbations on the scale of individual atmosphericmolecules, the classical NavierStokes equations are not the appropriate equations with whichto describe the evolution of the atmosphere. Moreover, the issue of literal ill posedness isnot directly relevant for realistic weather forecasting; not least, truncation scales of real globalweather and climate models, about 10 km, are at least seven or eight orders of magnitude largerthan scales in the viscous subrange.

However, as discussed later in this section, there is evidence that the predictability of large-scale weather patterns can be limited by initial uncertainty near these models truncation scale.It is worth noting that this truncation scale lies close to the range at which non-hydrostaticeffects are important and where (see section 4) the effects of ill-posedness cannot be ruledout. This suggests that a redefinition of the real butterfly effect is needed, one which takes intoaccount that forecast accuracy can be limited by poorly observed circulation patterns whichare much smaller in scale than the large-scale weather systems of interest, and yet are alsomuch larger than scales in the viscous subrange. In this sense let us redefine the real butterflyeffect through the concept of asymptotic ill posedness. Broadly speaking a prediction willbe said to be asymptotically ill posed if on the one hand P (N) as N (see(2.3)), whilst on the other hand P (N) 0 < as N N0 where Nv N0 0, butwhere 2Nv denotes a wavenumber in the viscous subrange. That is to say, even if the initialvalue problem is literally well posed (because of viscous dissipation), it may not be possibleto enhance predictability by reducing initial errors within a subrange comprising scales whichare small compared with the scale of interest, but large compared with scales in the viscoussubrange.

What is the evidence for such asymptotic predictability? Firstly, recall from (2.3) theargument that in a flow with a k5/3 energy spectrum, the error growth rates should scale ask2/3. Using high-resolution limited-area simulators, and comparing error growth in globalnumerical weather prediction simulators, Hohenegger and Schar [8] have shown that errorgrowth rates on about 10 km cloud scales is roughly 10 times larger than error growth rates onabout 1000 km weather scales. This is broadly consistent with the scaling exponent above.

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The second aspect of the real butterfly effect relates to the notion of nonlinear upscale errorpropagation. What is the evidence for this? In the opening chapter of Gleicks book on Chaos(whose title is The Butterfly Effect) Gleick mentions that the best weather forecasts in theworld came out of Reading, England, a small college town an hours drive from London. TheEuropean Centre for Medium-Range Weather Forecasts (ECMWF) still produces the worldsbest medium-range6 weather forecasts.

Despite producing the worlds best medium-range weather forecasts, deterministicforecasts from ECMWF are occasionally very poor indeed. In a recent paper, Rodwell et al [28]studied common factors that underly exceptionally poor day-6 forecasts over Europe. Typicallythe large-scale weather type associated with such poor forecasts comprises a quasi-stationaryanticyclone over Northern Europe and a corresponding quasi-stationary cyclonic low-pressuresystem over the Mediterranean (what meteorologists would call a block). A misplacementof the phase and/or the amplitude of this large-scale weather system in the day-6 forecastleads to serious errors in near-surface temperature and precipitation. Rodwell et al studiedfactors common to the initial conditions of these forecasts. They found that initial errors whichappeared to originate just east of the Rockies in a region of strong convective available potentialenergy (CAPE), propagated downstream to Europe in the form of Rossby waves, growing inamplitude and scale as they propagated. Rodwell et al studied a particular realization of thisgeneric situation and found that very high-resolution radar data indicated the existence ofintense convective (thunderstorm) activity in the region of strong CAPE. Several reports oftornado activity were associated with these convective systems.

These mesoscale convective systems are themselves examples of organized convectivecloud systems. A proper simulation of such systems would require much higher resolutionthan a global numerical weather prediction model currently has. At the sorts of resolution(1 km) needed to resolve deep convection, the relevant equations are no longer representableby the hydrostatic primitive equations. Put another way, by extending down into the non-hydrostatic range, there is strong empirical evidence, consistent with that found in the surfacequasi-geostrophic equations, that the predictability of large-scale weather patterns is sensitiveto initial errors in convective cloud scales, consistent with the real butterfly effect as definedabove.

On the other hand, the Rodwell et al study makes it clear that these forecast busts are theexception rather than the rule. That is, not all forecast flows have such sensitive dependenceon small-scale uncertainties under the initial conditions. This intermittency is consistent withwhat was found in the analysis of the SQG equations in section 3. This raises an important issuewhen discussing the relevance of the real butterfly effect. Much of the time, the evolution oflarge-scale weather does not appear especially sensitive to small-scale initial error. However,when it is sensitive, the corresponding (deterministic) forecast can be completely misleading.From a societal point of view, trust in the science of meteorology can be undermined by asingle exceptional poor forecast. That is to say, a metric of societal usefulness of a set offorecasts may be better gauged by an L norm, than an L2 norm. On the other hand, if we canprovide some prior estimate of the likely sensitivity of the flow to initial-condition uncertainty,including small-scale initial-condition uncertainty, then these sensitive forecasts can be flaggedas potentially unreliable in any deterministic sense. For this reason, ensemble predictiontechniques have become universally used in operational numerical weather prediction centresaround the world [25, 27]. These (essentially Monte-Carlo) forecasts allow estimation of the

6 A medium-range forecast is a forecast produced by a global weather forecast mode, and it typically used forforecasting between about 2 days ahead and 2 weeks ahead. By contrast a short-range forecast is a forecast producedby a higher resolution regional modelwith lateral boundary conditions from a global modeland typically used forforecasting up to about 2 days ahead.

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flow-dependent dispersion of forecasts sampled from some initial probability distribution ofinitial state. Ensemble forecasts help nullify some of the most damaging ramifications of thereal butterfly effect in weather prediction.

The Rodwell et al study indicated that upscale error growth was important in explainingsome large-scale weather forecast busts. What about short-range forecasts of smaller-scaleweather features? In a recent paper, Durran and Gingrich studied the predictability of short-range forecasts using the Lorenz [19] model, based on data from ensembles of integrationsof a high resolution limited-area numerical weather prediction simulator. Based on the initialspread of the operational ensemble forecasts, Durran and Gingrich conclude that it is thelarger-scale initial errors that control forecast error (and hence that small-scale butterfliesare relatively unimportant). However, two caveats need to be made. Firstly, these resultswere based on two winter-storm cases. As discussed above, the dependence of large-scaleforecast error on small-scale initial error may actually be quite flow dependent. Secondly, theEnKF (ensemble Kalman filter), used to generate the ensemble of initial conditions, would nothave included any direct representation of model truncation error. Recent research has shownthat stochastic parametrization techniques can provide a useful way to represent the effects oftruncation error [26]. Including stochastic parametrization in the initial ensemble would tendto whiten the estimated initial error distribution.

6. Conclusions

In this paper, the historical background to the iconic phrase the butterfly effect has beenpresented. This background reveals that what Ed Lorenz himself meant by the phrase is quitedifferent (and much more radical) than the sense in which it is almost universally used today.That is to say, Lorenz did not intend the phrase to characterize mere sensitive dependenceon initial conditions in low-order chaos. Rather he intended it to characterize an absolutefinite-time barrier to predictability in certain multi-scale dynamical systems. As has beendiscussed, this would imply a breakdown in continuous dependence on initial conditions. Itis easily shown that the Lorenz 1963 system has the property of continuous dependence oninitial conditions. However, it is still an open question as to whether the three-dimensionalNavierStokes equation has this property. From a practical point of view, there is considerableevidence for the real butterfly effect in an asymptotic sense, implying that predictability cannotbe extended by reducing initial error within a subrange of scales that are small compared withthe larger weather scales of interest, but large compared with scales in the viscous subrange.Integrations of the surface quasi-geostrophic equations have been used to provide evidence forthe real butterfly effect.

However, there is an extremely important caveat to this conclusion. As described in thispaper, evidence both from idealized models and from operational weather forecasts indicatesthat the sensitivity of large-scale weather patterns to small-scale initial error is intermittent.The developmentin the time since Lorenzs 1969 paper was writtenof flow-dependentensemble prediction techniques has provided an important tool for weather forecasters to helpmitigate the butterfly effect (either in its traditional sense, or the sense emphasized in thispaper). In his 1969 paper Lorenz cast doubt on the possibility of being able to make reliableforecasts two weeks ahead (see section 2). However, with contemporary ensemble predictionsystems it is now possible to make reliable predictions two (and sometimes more) weeks ahead.In this sense the real butterfly effect is not a show-stopper for all long range prediction. Infact, using ensemble forecasting methods to flag the intermittent poor forecasts a priori, itshould be possible to improve weather forecasts further by reducing initial error. However, it

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is important to recognize that this can be achieved not only by improving the quality and densityof atmospheric observations, but also by improving the quality and resolution of the modelsinto which these observations are assimilated. This latter aspect requires further investment insupercomputing.

Acknowledgments

The authors thank Dr Peter Duben for providing assistance in constructing the fractaltopography used for the surface quasi-geostrophic model integrations. TNP and ADwere supported by the European Research Council Project Number 291406: Towards theProbabilistic Earth-System Model.

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1. Introduction2. The real butterfly effect: an historical analysis3. The surface quasi-geostrophic equations4. The Navier--Stokes equations5. Relevance of the real butterfly effect in the real world: asymptotic ill-posedness6. Conclusions Acknowledgments References

the butterfly effect

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