ELSEVIER

1 September 1997

Physics Letters A 233 ( 1997) 347-354

Predictability in the Lorenz low-order circulation model

J.M. GonzBlez-Miranda

PHYSICS LETTERS A

general atmospheric

Departamento de Fikca Fundamental, Universidad de Barcelona, Avenida Diagonal 647. 08028 Barcelona, Spain

Received 19 August 1996; revised manuscript received 5 June 1997; accepted for publication 16 June 1997

Communicated by A.R. Bishop

Abstract

A technique devised to characterize and measure predictability recently introduced [L.W. Salvino et al., Phys. Lett. A 209 ( 1995) 3271 is applied to study predictability of the general circulation of the atmosphere by means of a numerical study of a new model introduced by Lorenz for the time evolution of the global wind current and chains of eddies that are the main features of the global behavior of the atmosphere [ E.N. Lorenz, Tellus 42 A ( 1984) 981. In particular, attention

is focused on the effect on predictability of the choice of the signal to study, of the seasonal nature of this signal, and of the presence of statistical errors affecting the data points. @ 1997 Elsevier Science B.V.

PACS: 05.45.fb; 07.05.Kf; 92.60.Bh

Keywords: Deterministic chaos; Time series; Predictability; General circulation of the atmosphere

Since deterministic chaos has become an object of

major attention for the scientific community, the prob-

lem of characterizing chaotic evolutions has been at the core of the research activity in this field [ 1,2].

An idea that is central to deterministic chaotic evolu-

tion of a physical system is that of the predictability of the future state of the system from its past evo- lution. In particular, deterministic chaos is associated with a loss of predictability, in contrast with deter- ministic non-chaotic evolutions which are predictable.

Recently Salvino et al. [3] have developed a tech-

nique aimed to measure the degree of predictability of a given time series. This technique allows the possi- bility of discussions on the predictability of the future behavior of a system in quantitative terms. A field in the physical sciences where predictability is a factor of outstanding importance is that of the evolution of the atmosphere. Moreover, research in this field has

played an important role in the rise of interest towards

what it is today known as chaos theory [4]. An inter- esting model for the general circulation of the atmo- sphere, which describes the combined time evolution

of the large scale symmetric globe-encircling west- erly wind current and the strength of the large scale

eddies which transport heat poleward, has been pro- posed by Lorenz [5]. This has been useful to show

how the existence of two possible coexisting climates combined with variations of the solar heating may give rise to seasons with inter-annual variability [ 61, how

the asymmetry between oceans and continents is ba- sic for the system to exhibit complex behaviors [ 71, and how the climate is affected by the interactions between the atmosphere and the oceans [ 81. Further- more, seen in general terms as a dynamical systems, it has proven to be a system of interest as it exhibits a rich variety of chaotic and non-chaotic behaviors,

03759601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved

PII SO375-9601 (97)00541-O

348 JM. Gonzdlez-Mirunda/Physics Letrers A 233 (1997) 347-354

as well as transition mechanisms among them, when control parameters are varied [ 5-81. In this context, the purpose of this Letter is to report the first results

of an application of the concepts and techniques on

predictability introduced by Salvino et al. [ 31 to time

series generated by means of the model of Lorenz for the general circulation of the atmosphere [ 51. In par-

ticular, I will report results for various coexisting and non-coexisting regimes, for the model with seasons,

and for the effect on predictability of the presence of

statistical errors in the data. The general atmospheric circulation model of

Lorenz is a low-order non-linear dynamical system

described by the following set of ordinary differential

equations [ 51,

dX -=--Y*-2%aX+aF, dt

dY -=XY-bZX-Y+G, dt

dZ -=bXY+XZ-Z, dt

(1)

X being the intensity of the westerly wind current, Y and 2 the cosine and sine phases of the strength of

the large scale eddies, aF and G the symmetric and asymmetric external thermal forcings, and a and b pa-

rameters of the model that modulate the dumping of the eddies and its displacement by the westerly cur- rent. The variable t represents time, and the equations are scaled so that the time unit is five days. Among the several regimes studied in the literature I have

chosen for the purposes of this Letter those selected by Lorenz [6] to discuss the basic properties of the model, i.e. seasons and inter-annual variability. These

are determined by the parameter values a = 0.25, b = 0.4, G = 1.0 and various selections of F that corre- spond to the following cases: (i) coexistence of two limit cycles, F = 6.0; (ii) deterministic chaos without seasons, F = 8.0; (iii) chaos with seasons and inter- annual variability, F(t) = 7.0+2.Ocos( cot), with w = 0.0860121. So we have four attractors that, following Lorenz [ 61, will be designated through the paper as the weak summer limit cycle, the strong summer limit cycle, the perpetual winter attractor, and the system with seasons, respectively. To obtain the time evolu- tion of X, Y and Z, I have integrated the above sys- tem of equations by means of a fourth-order Runge-

Kutta algorithm with a time step of 0.025, which is equivalent to 3 hours. Except when explicitly stated, the time evolutions studied extent up to 187008 time

steps, what means that the samples used in the calcu- lations correspond to an observational time window of 64 years.

On the other hand, the main output in the pre-

dictability algorithm of Salvino et al. [3] is the un-

certainty on the future evolution of the system, U, as

a function of the prediction (or lead) time, 7. In the present paper, it has been obtained from the follow- ing algorithm. Given a time-dependent signal, x(t), recorded as a set of evenly spaced points between t = 0 and t = iMax, one starts using standard embedding methods to reconstruct the phase space of the corre- sponding system [ 1,9- 111. Once the dimension of the

embedding space, nd, and the delay time A have been obtained, one proceeds defining the past function vec-

tarp(t) = [x(t), x(t-A), x(t-2A>,..., x(t- ndd) 1. This function is defined for elements of the

time series corresponding to times such that ndA < t<t ~~~ - T, T being the maximum prediction time to be studied (0 < 7 6 T) . The embedding space is then divided in nd-dimensional cells of size 8, each past function point is assigned to the cell where it lies, and

the cells containing more than one point are selected to be used in the calculation. To obtain reliable statis-

tics, the box length E has to be chosen such that the number of cells containing just one point, N,, and the number of cells containing two or more points, NC, verify N1/2 < NC < 4N1 [ 31. If indices i and j are used to design past function points in a cell and cells

containing more than one point, respectively, one will have 11, points in cell j E { 1,2,. . , NC}, each point corresponding to a time tji. So, one can compute a probability measure of the jth box pj = nj/N, N be- ing the total number of points used in the calculation

[ 31. For any prediction time 7, and any cell occupied by more than one point, one computes the statistical average value of x( tji + T), (x(T))~, and its fluctu- ations around this value given by the corresponding statistical variance, g;(7). Moreover, for all the data points used in the calculation one computes the global

average of x(t), (x)~, and the corresponding global variance us. Then, the lack of predictability for the time series at the prediction time 7 is measured by the uncertainty function, U(T), computed as a weighted average over box uncertainties as

J. M. Gonzdlez-Miranda/Physics Letters A 233 (1997) 347-354 349

U(7) = ( f$JdjU~(T)/O+)"2. ,;=I

From the function U(r), one can compute estima- tors, useful to characterize predictability by means of a single number. Salvino et al. [ 31, have proposed the

mean prediction time defined as

T

5=

J [I - U2(r>] dr

0

provided that the upper limit in the integral is large

enough to have U(T) M 1. This is the simplest imple-

mentation of the method of Salvino et al. [ 31. More

sophisticated definitions for the past functions and of

the above averages and variances, that define the fu- ture function, may be used for special purposes [ I] ; however, as the aim of this Letter is to present a first

test of the ability of the method to make asserts on a physical system rather than a comprehensive study, they will not be considered here.

Most of the numerical results presented here have been obtained with values of nd = 3 and of A = 16.

Some calculations performed with other values for nd

and A close to those indicate that the conclusions pre- sented in this paper will not be altered by reasonable

changes in the values used for those parameters. The uncertainty functions have been computed up to a pre-

diction time of two years (except in a especial case, which is indicated) ; however, only the results up to one year are displayed in most of the figures for the

sake of clarity. As there are three variables X, Y, and Z, we can

consider three different choices for the observed sig-

nal, x(t) . Using the above algorithm, I have obtained the global uncertainty functions and mean prediction

times shown in Fig. la for the case of the winter at- tractors, which is chaotic. From this figure it is clear that, for the same system, predictability is affected by the signal that one chooses to be studied. In this particular case, one observes a faster growth of un- certainty for the signals describing the eddies (Y, Z) than for the signal describing the current (X). The mean prediction times obtained are: F(X) = 129 days, 7(Y) = 96 days, and T(Z) = 93 days. In Fig. lb ap- pear the results of computing the uncertainty function, for each of the signals, using the past function p(t) =

[X(t) , Y(t) , Z(t) 1, constructed from the real state points, rather that the reconstructed ones. The over- all behavior is the same than using the reconstructed dynamics, predictability is slightly improved and one obtains somewhat larger prediction times for the vari-

ables describing the eddies (T(Y) = 112 days, and 5( 2) = 110 days) and a close value for the variable

describing the current (7(X) = 122 days). Moreover,

the X signal is still the one with an slower increase of uncertainty of the three. In what follows, I will concentrate on the Z signal, which having relatively

rapidly convergent functions is a case of more easy study. Analogous conclusions would be drawn from

the system with seasons case or even for other math-

ematical models. I will note that I have observed sig-

nal dependences of this sort in some calculations per-

formed for the classical Lorenz model [ 41 and for the Rossler model [ 121 at the parameter values studied

by Salvino et al. [3]. Uncertainty functions for the four attractors con-

sidered in this Letter are displayed in Fig. 2. For the perpetual summer attractors, which are limit cycles, u(r) takes very small values that oscillate around a small constant, this constant and the fluctuations be-

ing larger in the strong summer case (Fig. 2a), which is a more complicated attractor [6,7]. For the per-

petual winter (Figs. 2b and 2c) and the system with

seasons (Figs. 2d-2f) attractors, which are chaotic, the uncertainty functions increase rapidly from zero

during the first six months to approach asymptotically to a constant value, which is practically achieved at a prediction time of one year. This constant value is

u = 1.00 for the perpetual winter case and a yuan- tity slightly smaller than one for the system with sea- sons (U “N 0.98) (Fig. 2e). More detailed analyses of

U( 7) in both chaotic cases shows that one can distin- guish three time regimes in the uncertainty function. There is a short time behavior, which roughly corre- sponds to the first two weeks (insets of Figs. 2b and 2d), in which after a very fast growth during the first two days, uncertainty practically stops growing and

fluctuates around a value close to three or six, depend- ing on the attractor considered. Then, up to roughly the sixth month the uncertainty function experiences a fluctuating but systematic increase whose overall be- havior can be quite well characterized by a function of the type U( 7) E [ 1 - A exp( -_(yr)] iI2 in the win- ter case (Fig. 2c), and V( 7) E 1 - BrP in the sys-

350 J.M. Gonzdlez-Miranda/Physics Letters A 233 (1997) 347-354

1.00

0.75

3 0.50

0.25

0.00 U

(a) _-___ _,’

-0

.7

I

I,’ /r’;-:- ,f’ 0.4

11’ ,( .. ‘., ..;

/” I

‘_ ’ _ ’ I’

“,2 ,f,,--

rZl

/ i I

0.0 0.0 0.3 0.6

(b)

7 I I ’

;_ -

,’

,I

1 0.4

;I;-

.I’ I

,’ o,2 ‘f,__/

!’ ,,---‘-

0.0 0.0 0.3 0.6

3 6 Y I.! 0 3 6 Y 12 7 (months) z (months)

Fig. 1. Uncertainty functions for the perpetual winter attractor computed for the different signals available: X (dashed line), Y (dotted

line), and Z (solid line). (a) Results for samples of 93504 data points (32 years), using delay time parameters nd = 3, and A = 16. and

the grid sizes being E = l/180 for X, and E = l/90 for Y and Z. (b) Results using 93504 state points (X, 1: Z) rather than reconstructed

states. The grid size used was E = I /90.

tern with seasons case (Fig. 2f). Finally, for predic- is increasing. For the intermediate region, a proper tion times beyond the sixth month, the uncertainty is choice is given by the parameters LY and /I that can usually greater than 0.9 and evolves asymptotically to- be obtained from fits of the functions in the above wards 1 in the perpetual winter attractor and towards paragraph to U( 7) (in the time interval between two

0.98 in the system with seasons. From a comparison weeks and six months). This characterizes the rate at

of Fig. la and Fig. 2b it appears that uncertainties which uncertainty is increasing in that regime. The

grow only slightly faster in the first case. This is more large time regime, represents the times at which pre-

clearly seen if we compare the results for the value dictability has been practically lost; therefore, instead

of the mean prediction time averaged over the three of defining a parameter for this case, I think it is better evolutions, T(Z) = 100 days, with the value in the to use the mean prediction time concept by Salvino et

previous paragraph. This mild dependence on the size al. [3] which characterizes the loss of certainty in a

of the samples is because they differ only by a factor global manner. In order that the mean prediction time two. This is in agreement with the results on this issue have some meaning in the system with seasons case reported by Salvino et al. [ 31, who, for the classical one may compute it by means of the following gener- Lorenz model [4], gave an increase in mean predic- alized definition, 7 = j” [Uk - U*(T)] d7, with U, tion time of 39 percent for an increase of the size of the asymptotic value, that can be approximated by the the sample by a factor 12.5. average value of in the second year.

In the present case, only in the perpetual winter case is it possible to use properly the concept of mean pre- diction time. The situation in the summer cases will be better characterized by the mean value, ??, of the fluctuations of U(T), which is a constant independent of time. In the chaotic cases, having distinguished var- ious regimes in the time evolution of U(T), it seems convenient to characterize them by means of specific quantities. For the short prediction time region, one may use the quantity U*, that measures the uncertainty level reached at a prediction time of one day. This is a somewhat arbitrary selection, but easy to compute and a clear measure of the rate at which uncertainty

These numerical results are understandable in the light of the nature of the various attractor involved.

The weak summer and strong summer attractors are limit cycles; therefore, the signals obtained from them are periodic signals whose future behavior must be completely predictable. This is why the uncertainty function does not increases with time but remains al- most constant and very small. The winter attractor is chaotic and then predictability must increase towards 1 as the prediction time increases. In the system with seasons case, we have a chaotic attractor which re- sults from periodic shifts between the winter attractor (fully chaotic), and either of the summer attractors

J.M. Gonzdlez-Miranda/Physics Letters A 233 (1997) 347-354 351

Weak Summer 0.0 . : * : :

0 3 6 9 12 T (months)

Cd)

0.00’ ’ j ’ ’ ’ ’ ’ 0 3 6 9 12

‘5 (months)

0.25

(b)

o.ooL* ’ ’ ’ ’ ’ J 0 3 6 9 12

z (months)

5 11111,,1 j ,‘,,l”l 0 1 10

T (months)

Fig. 2. Uncertainty functions computed from the Z(t) signal for the four attractors studied: (a) the weak and strong summer limit cycles; (b) the perpetual winter attractor in the large and small (inset) time windows; (c) the perpetual winter attractor in the intermediate region; (d) the system with seasons in the large an small (inset) time windows; (e) the system with seasons in a prediction time window of 6 years; (f) the system with seasons attractor in the intermediate region. Results for three different samples are presented in the case of the chaotic attractors ( (b), (c), (d), (f) ) to test the reproducibility of the results. The straight lines in the main plots of (c) and (f) are least-squares fits to an exponential law in (c) and a power law in (f). The insets show that a power law will not fit the decay in the perpetual winter, and a exponential decay will not work for the system with seasons. These results have been obtained from samples of 187008 data points (64 years), using delay time parameters nd = 3, and A = 16, and the grid sizes being E = 1 /I5000 for the weak summer attractor, E = l/3200 for the strong summer attractor, and E = I/ 110 for the perpetual winter and system with seasons attractors.

(periodic). Because of this, in this case, one must ex- pect the uncertainty function to exhibit a component

of predictability due to the time spent by the system in periodic attractors and this is why the uncertainty must grow towards a value smaller than 1 . This dif- ferent nature of the two chaotic attractors well might be the cause for the different laws describing the in-

crease of U(T) in the intermediate time region. The existence of different regimes in the chaotic cases is difficult to interpret with the data at hand. Possibly, it is due to the existence of different scales for the re- laxation times of the various contributions to the dy- namic of the system involved, which in some cases differ by an order of magnitude. This is the case of

the dumping of the eddies and its displacement by the

westerly current which are given by parameters a =

l/4 and b = 4, respectively. To elucidate this question will require extensive calculations of predictability as a function of the parameters of the system.

I have investigated how these results are modified by the presence of measurement errors in the data used to compute the uncertainty. To this aim I have generated signals Z (ti) by means of the integration of the equations of motion (Eq. ( 1) ) . Then I have modified the points in each signal by adding a small quantity, 6i, to each data point, with the values of Si distributed along a Gaussian function centered at zero. Therefore, the signal studied WZIS X( ti) = Z( to+ Si,

J.M. Gonzdlez-Mirunda/Physics Leriers A 233 (1997) 347-354 352

0.25

0.00

1

I ..:.. : : 0 3 6 9 12

‘5 (months)

(d)

0.25

0.00 1 1 ’ ’ ’ 0 3 6 9 12

r (months)

V.“”

0 3 6 9 12

z (months)

0.25

0.25

0 3 6 9 12

z (months)

240

180

I!+

60

Fig. 3. Effect of random errors in the data on the uncertainty function for: (a) the weak summer; (b) the strong summer; (c) the perpetual

winter; (d) the system with seasons attractors. Arrows are used to signal the direction of increasing error level. In the summer cases, only

some of the calculations performed are displayed to maintain the plots clear. The insets in (c) and (d) show details of the intermediate

region for moderate error levels (Y 2 4) Dependence with the error level for: (e) the average uncertainty in the summer cases (full

symbols), and the uncertainty at a prediction time of one day for the chaotic attractors (hollow symbols), and (f) for the mean prediction

time, and parameters 1 /a and l//3 for the intermediate region (inset). The interpretation of the symbols is as follows: weak summer

(diamonds), strong summer (triangles), winter (circles), and system with seasons (squares). The data points plotted at ga < 10-s are,

in fact, the result for the free error case (~6 = 0). that have been included to show the asymptotic behavior. The dotted lines are just

guides to the eye.

which is the result of the solution of the equations of motion, Z ( ti), modified by the measurement error, Si. This is a case of interesting study because it simu- lates the situation in which we have a real system that evolves along Z( ti) ; but, because of the uncertain-

ties inherent to the measurement process, what we ob- serve is x( ti) . A set of 12 calculations have been per- formed for 12 different values of the dispersion given by (+6 = ( XM - xm)/20’-‘), v E { 1, 2,. . . ,12}, and x,+, and x, being the maximum and minimum values of x, respectively. The results obtained, as illustrated in Fig. 3, can be summarized as follows. For both sum-

mer attractors, the uncertainty functions still behave as a fluctuation around a constant, but this constant

increases with the degree of imperfection of the data, being practically equal to 1 when the error level is of

the same size than the signal (Figs. 3a and 3b). For both chaotic attractors, the effect of imperfections in the data is to increase the rate at which uncertainty in- creases, while one still observes a fluctuating increas- ing of the uncertainty along the same three regimes. Moreover, in the system with seasons, the asymptotic constant also increases as the degree of imperfection of the data increases. The dependence of the different parameters used to characterize the uncertainty func- tion, E E {u, ?, U’, l/a, l//I}, with the disper- sion of the distribution of random errors, a&shown in Figs. 3e and 3f, presents the correct limit behaviors:

J.M. Gonza’lez-Miranda/Physics Letters A 233 (1997) 347-354 353

E = E. = E(as = 0) in the absence of imperfec- of Lyapunov exponents, which just provides us with

tions, and E = 0 or E = 1, depending on whether E a global averaged invariant number for each system.

is a prediction time or an uncertainty, when the errors The considerations in this paragraph, as well as the re-

overcome the signal. The behavior of 1 /(.y and l//3 for sults along the paper, tell us that the analysis of both

large error levels can not be properly studied because predictability in the circulation of the atmosphere [ 51

the exponential and potential approximations for the time dependence of 1 - U2 fail when there are large

and the measure of predictability [ 31 itself is not ex- hausted in this Letter, but deserve more detailed study

errors in the data (V < 4.) in several aspects.

From those numerical results it follows that small random errors in the signal do not alter the qualita-

tive behavior of U( 7). Only when the errors are of the same size as the signal, the uncertainty functions be- have like those of noisy signals as the ones reported

by Salvino et al. [ 31 for random evolutions. The ef- fect of the imperfections in the signal is in any case

a global loss of predictability, which increases rapidly with the size of the errors.

In deserves to be noted that what has been studied in this Letter is the ability to predict the value of a state variable of the system at a particular lead time, 7, from their values at particular times f in the past. There are other possible choices of future functions [ 31 which had given different results for predictabil- ity. In particular, their use may result in improving

or worsen predictability because using different future functions means asking for different items to be pre-

dicted, some of which may be easiest to predict, while

others may be harder. For example, a case of possi-

ble interest is the study of the ability to predict the value of a state variable averaged over a significative time interval [ 31, such as a day or a week. It is rea- sonable to expect that such election of future function will result in an improvement of predictability because the requirement of information on the future is less demanding. The idea is that other choices of future

functions may be more favorable for prediction, but

this has to be at the cost of being less stringent in our prediction demands. In fact, one of the virtues of the method considered here to measure predictability is its flexibility to take account, in a distinctive form, of a

variety of experimental situations of practical interest, for the same system. In this sense, we may say that us- ing this method one has the chance to introduce some of the subjectivity of the observer in the measure of predictability. In some extent this has been illustrated in this Letter by the comparison of results for differ-

ent observed signals coming from the same system. This is in contrast with the use, more conventional,

In summary, I have reported a numerical study of the predictability content of signals generated by a

reasonable model for the general circulation of the at- mosphere. I have found that even for the same system, different significative variables will exhibit different

shapes for the uncertainty functions, and then differ-

ent numerical values of the estimators that character-

ize predictability. For the particular case studied the

algorithm used to compute predictability has distin- guished unambiguously between chaotic, non-chaotic and noisy signals, and it has detected the presence

of non-chaotic components on an otherwise chaotic evolution. Furthermore, the analyses of U(r) in the

chaotic attractors has shown evidence of the existence of different regimes in the rate at which predictability is lost. Small errors in the signals increase the numer-

ical values of the uncertainty for the same prediction

time, but do not alter the qualitative behaviors for the

dependence of uncertainty with prediction time. If this

results are general, this means that a small amount of error in the data does not alter the predictive nature of the signal. At an speculative level this appears to be important for the sake of classification of time evo- lutions of dynamical systems which is an issue in a variety of fields ranging from solar and earth sciences [4,5,13-151 to neurobiology [ 16-181.

I gratefully acknowledge financial support from DGICYT (project PB93-0780) to perform this work.

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