130 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 20 Deterministic Nonperiodic Flow1 EDWARD N.LORENZ Massachusetts Islilute of Technology (Manuscript received 18 November 1962， in revised form 7 January 1963) ABSTRACT Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider- ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic. The feasibility of very-long-range weather prediction is examined in the light of these results. 1.Introduction Thus there are occasions when more than the statistics Certain hydrodynamical systems exhibit steady-state of irregular flow are of very real concern. In this study we shall work with systems of deter- flow patterns, while others oscillate in a regular periodic ministic equations which are idealizations of hydro- fashion. Still others vary in an irregular, seemingly haphazard manner, and, even when observed for long dynamical systems. We shall be interested principally in periods of time, do not appear to repeat their previous nonperiodic solutions, i.e., solutions which never repeat history. their past history exactly, and where all approximate repetitions are of finite duration. Thus we shall be in- These modes of behavior may all be observed in the volved with the ultimate behavior of the solutions, as familiar rotating-basin experiments, described by Fultz, opposed to the transient behavior associated with et al. (1959)and Hide (1958). In these experiments, a arbitrary initial conditions. cylindrical vessel containing water is rotated about its A closed hydrodynamical system of finite mass may axis, and is heated near its rim and cooled near its center ostensibly be treated mathematically as a finite collec- in a steady symmetrical fashion. Under certain condi- tion of molecules-usually a very large finite collection tions the resulting flow is as symmetric and steady as the -in which case the governing laws are expressible as a heating which gives rise to it. Under different conditions finite set of ordinary differential equations. These equa- a system of regularly spaced waves develops, and pro- tions are generally highly intractable, and the set of gresses at a uniform speed without changing its shape. molecules is usually approximated by a continuous dis- Under stil different conditions an irregular flow pattern tribution of mass. The governing laws are then expressed forms, and moves and changes its shape in an irregular as a set of partial differential equations, containing such nonperiodic manner. quantities as velocity, density, and pressure as de- Lack of periodicity is very common in natural sys- pendent variables. tems, and is one of the distinguishing features of turbu- It is sometimes possible to obtain particular solutions lent flow. Because instantaneous turbulent flow patterns of these equations analytically, especially when the are so irregular, attention is often confined to the sta- solutions are periodic or invariant with time, and, in- tistics of turbulence, which, in contrast to the details of deed, much work has been devoted to obtaining such turbulence, often behave in a regular well-organized solutions by one scheme or another. Ordinarily, how- manner. The short-range weather forecaster, however, ever, nonperiodic solutions cannot readily be deter- is forced willy-nilly to predict the details of the large- mined except by numerical procedures. Such procedures scale turbulent eddies-the cyclones and anticyclones- involve replacing the continuous variables by a new which continually arrange themselves into new patterns. finite set of functions of time, which may perhaps be the values of the continuous variables at a chosen grid of IThe research reported in this work has been sponsored by the points, or the coefficients in the expansions of these Geophysics Research Directorate of the Air Force Cambridge variables in series of orthogonal functions.The govern- Research Center, under Contract No. AF 19(604)-4969. ing laws then become a finite set of ordinary differential
131EDWARD N.LORENZMARCH1963an M-dimensional Euclidean spaceTwhose coordinatesequations again, although a far simpler set than the oneareXi,.,Xm.Eachpointinphasespacerepresentsawhich governs individual molecular motions.In any real hydrodynamical system, viscous dissipa-possible instantaneous state of the system. A statewhich is varying in accordancewith (1)is representedtion is always occurring, unless the system is moving asby a moving particle in phase space, traveling along aa solid, and thermal dissipation is always occurring,trajectory in phase space. For completeness, the positionunless the system is at constant temperature. For cer-of a stationary particle, representing a steady state, istain purposes many systems may be treated as conserva-tive systems, in which the total energy, or some otherincluded as a trajectory.Phase space has been a useful concept in treatingquantity, does not vary with time. In seeking the ulti-mate behavior of a system, the use of conservativefinite systems, and has been used by such mathema-equations is unsatisfactory, since the ultimate value ofticians as Gibbs (1902) in his development of statisticalmechanics,Poincare (1881)in his treatment of the solu-any conservative quantitywould then havetoequal thetions of differential equations, and Birkhoff (1927) inarbitrarily chosen initial value. This difficulty may beobviated by including the dissipative processes, therebyhis treatise on dynamical systems.From the theory of differential equations (e,g., Fordmaking the equations nonconservative, and also includ-1933, ch. 6), it follows, since the partial derivativesing external mechanical or thermal forcing, thus pre-venting the system from ultimately reaching a state ofaF/ax, are continuous, that if tois any time, and ifrest. If the system is to be deterministic, the forcingXio, ...Xmo is any point in F, equations (1) possess afunctions, if not constant with time, must themselvesuniquesolutionvary according to some deterministic rule.(2)X, f(X1o,*-,Xaro,0), i1, **, M,In this work, then, we shall deal specifically withfinite systems of deterministic ordinary differentialvalid throughout some time interval containing to, andequations, designed to represent forced dissipativesatisfying the conditionhydrodynamical systems. We shall study the propertiesof nonperiodic solutions of these equations.(3)f(X10,**,Xmo,fo)-Xio, i=1, -, M.It is not obvious that such solutions can exist at allThe functions f. are continuous in Xio, .., X mo and t.Indeed, in dissipative systems governed by finite sets ofHence there is a unique trajectory through each point oflinear equations, a constant forcing leads ultimately tor. Two or more trajectories may,however, approach thea constant response, while a periodic forcing leads to asamepoint or the same curve asymptoticallyas t-→periodic response. Hence, nonperiodic flow has some-or as t-→--oo. Moreover, since the functions f: aretimes been regarded as the result of nonperiodic orcontinuous, the passage of time defines a continuousrandomforcing.deformation of any region of r into another region.The reasoning leading to these concludions is notIn the familiar case of a conservative svstem, whereapplicable when the governing equations are nonlinear.somepositivedefinitequantityQ,whichmayrepresentIf the equations contain terms representing advectionsome form of energy, is invariant with time, each trathe transport of some property of a fluid by the motionjectory is confined to one or another of the surfaces ofof the fluid itself-a constant forcing can lead to aconstant Q.These surfaces may take the form of closedvariable response.In the rotating-basin experimentsconcentric shells.already mentioned, both periodic and nonperiodic flowIf, on the other hand, there is dissipation and forcing,result from thermal forcing which, within the limits ofand if, whenever O equals or exceeds some fixed valueexperimental control, is constant.Exact periodic solu-Qi, the dissipation acts to diminishQmore rapidly thentions of simplified systems of equations, representingthe forcing can increase Q, then (-dQ/dt) has a positivedissipative flow with constant thermal forcing, havelower bound whereQ≥Qr,and each trajectory mustbeen obtained analytically by the writer (1962a).Theultimately become trapped in the region where Q<Qrwriter (1962b)has also found nonperiodic solutions ofTrajectories representing forced dissipative flow maysimilar systems of equations bynumericalmeans.therefore differ considerably from those representingconservativeflow.2.PhasespaceForced dissipative systems of this sort are typifiedConsider a system whose state may be described by Mby the systemvariables Xr, ., X. Let the system be governed by(4)dX/dt-EauxX,Xx-EbuX,+c,the set of equationsi.k7dX:/dt-F(X,..Xm),i=1,.,M,(1)whereaX,X,Xvanishes identically,buX,X,ispositive definite, and ci, .", c are constants.Ifwhere time t is the single independent variable, and theQ--E"(5)functions F,possess continuous first partial derivatives.Suchasystemmaybestudied bymeansofphasespace
132JOURNAL OF THE ATMOSPHERIC SCIENCESVOLUME20and if er, *.", em are the roots of the equationsabsolute-value signs denote distance in phase space.Because F is continuously deformed as t varies, every(6)E (bu+bj)ej=ci,point on the trajectory through Po is also a limit point2of P(), and the set of limit points of P(t) forms a tra-jectory, or a set of trajectories, called the limiting traitfollows from (4)thatjectories of P(t).A limiting trajectory is obviously contained within R in its entirety.dQ/dt- bueej-Zbu(X;-ei)(X,-e).(7If a trajectory is contained among its own limiting6i.itrajectories, it will be called central; otherwise it will becalled noncentral.A central trajectory passes arbitrarilyThe right side of (7) vanishes only on the surface ofclosely arbitrarily often to any point through which itan ellipsoid E, and is positive only in the interior of Ehas previously passed, and, in this sense at least, sepa-The surfaces of constant Q are concentric spheres. If Srate sufficiently long segments of a central trajectorydenotes a particular one of these spheres whose interiorare statistically similar.Anoncentral trajectory remainsR contains the ellipsoid E, it is evident that each tra-a certain distance away from any point through whichjectory eventually becomes trapped within R.it has previously passed. It must approach its entire setof limit points asymptotically, although it need not3.Theinstability of nonperiodic flowapproach any particular limiting trajectory asymptoti-cally.Its instantaneous distancefrom its closestlimitIn this section we shall establish one of the mostpoint is therefore a transient quantity, which becomesimportant properties of deterministic nonperiodic flow,arbitrarily small as t-→ o.namely,its instability with respect to modifications ofA trajectory P(t) will be called stable at a point P(t)small amplitude. We shall find it convenient to do thisif any other trajectory passing sufficiently close to P(ti)by identifying the solutions of the governing equationsat time ti remains close to P(o) as t-→ co; i.e., P(t) iswith trajectories in phase space. We shall use suchstable at P(t) if for any e>0 there exists a (e,t)>0symbols as P() (variable argument) to denote trajec-such that if [Pi(t)-P(t)/< and tz>ti, [Pi(t2)tories, and such symbols as P or P(to) (no argument or-P(t2)/<e. Otherwise P(t) will be called unstable atconstant argument) to denote points, the latter symbolP(t). Because is continuously deformed as t varies,denoting the specific point through which P(t) passesa trajectory which is stable at one point is stable at everyat time lo.point, and will be called a stable trajectory.A trajectoryWe shall deal with a phase space in which a uniqueunstable at one point is unstable at every point, andtrajectory passes through each point, and where thewill be called an unstable trajectory.In the special casepassage of time defines a continuous deformation of anythat P(t) is confined to one point, this definition ofregion of into another region, so that if the pointsstability coincides with the familiar concept of stabilityP(to), Pa(to), ..-approach Po(to) as a limit, the pointsof steady flow.Pi(to++), P2(to++), ..-must approach Po(to++) as aA stable trajectory P(t)will be called uniformly stablelimit.We shall furthermore require that the traiectoriesif the distance within which a neighboring trajectorybe uniformly bounded as t- oo; that is, there must bemust approach a point P(t), in order to be certain ofa bounded region R, such that every trajectory ulti-remaining close to P(t) as t-→co,itself possesses amately remains with R. Our procedure is influenced bypositive lowerbound ast→co;i.e.,P(t)isuniformlythe work of Birkhoff (1927) on dynamical systems, butstable if for any e>0 there exists a s(e)>0 and a timediffers in that Birkhoff was concerned mainly with con-to(e) such that if ti>to and |P(t)-P(t)l< andservative systems. A rather detailed treatment of dy-t>ti.Pi(ta)-P(t)i<e.AlimitingtrajectoryPo()ofnamical systems has been given by Nemytskii anda uniformly stable trajectory P() must be uniformlyStepanov (1960), and rigorous proofs of some of thestable itself,since all trajectories passing suficientlytheorems which we shall present are to be found inclose to Po() must pass arbitrarily close to some pointthat source.of P(t) and so must remain close to P(t), and hence toWe shall first classify the trajectories in three differentPo(),ast-oo.manners,namely,according to the absence or presenceSince each point lies on a unique trajectory, anyof transient properties, according to the stability ortrajectory passing through a point through which it hasinstability of the trajectories with respect to smallpreviously passed must continue to repeat its past be-modifications,and according to the presence or absencehavior, and so must be periodic. A trajectory P(t) willof periodicbehavior.be called quasi-periodic if for some arbitrarily largeSince any trajectory P(t) is bounded, it must possesstime interval , P(t+-) ultimately remains arbitrarilyat least one limit point Po, a point which it approachesclose to P(t), i.e., P(t) is quasi-periodic if for any e>0arbitrarily closely arbitrarily often.More precisely,Poisand for any time interval To, there exists a t(e,ro)>roa limit point of P(t)if forany e>0and any timet,thereexists a time t(e,ti)>t such that |P(t2)-Po/<e. Hereandatimeti(e,ro)such thatif ta>tu,[P(t+)-P(t2)
133EDWARDN.LORENZMARCH1963if it is stable at all, its very stability is one of its tran-<e, Periodic trajectories are special cases of quasi-sient properties, which tends to die out as time properiodic trajectories.A trajectory which is not quasi-periodic will be calledgresses.Inviewoftheimpossibilityofmeasuringinitialconditions precisely,and thereby distinguishing betweennonperiodic.If P(0) is nonperiodic,P(t+-) may bea central trajectory and a nearby noncentral trajectory,arbitrarily closetoP(t)forsometimetandsomeall nonperiodic trajectories are effectively unstable fromarbitrarilylargetimeintervalr,but,ifthisisso,P(t+)the point of view of practical prediction.cannot remain arbitrarily close to P(t) as t→ oo, Non-periodic trajectories are of course representations of4.Numerical integration of nonconservative sys-deterministic nonperiodic flow,and form the principaltemssubject of this paper.Periodic trajectories are obviously central. Quasi-The theorems of the last section can be of importanceperiodic central trajectories include multiple periodiconly if nonperiodic solutions of equations of the typetraiectorieswithincommensurableperiods,whilequasi-considered actually exist. Since statistically stationaryperiodic noncentral trajectories include those whichnonperiodic functions of time are not easily describedapproach periodic trajectories asymptotically.Non-analytically, particular nonperiodic solutions can prob-periodic trajectories may be central or noncentral.ably befound most readily by numerical procedures.InWe can now establish the theorem that a trajectorythis section we shall examine a numerical-integrationwith a stable limiting trajectory is quasi-periodic.Forprocedure which is especially applicable to systems ofif Po()isa limiting trajectory of P(t),two distinctequations of the form (4).In a later section we shall usepoints P(t) and P(tr++), with r arbitrarily large, maythis procedure to determine a nonperiodic solution of abe found arbitrary close to any point Po(to). Since Po(0)simple set of equations.is stable, P() and P(t+-) must remain arbitrarilyTo solve (1) numerically we may choose an initialclose to Po(t+t-ti), and hence to each other, as t-→ co,time to and a time increment Zi, and letand P(t) is quasi-periodic.(8)Xi,n=X,(to+nAt).Itfollows immediatelythat a stable central trajectoryis quasi-periodic, or, equivalently, that a nonperiodicWe then introduce the auxiliary approximationscentral trajectory is unstable.(9)Xi(n+1)=Xi,n+F(Pn)A,The result has far-reaching consequences when thesystem being considered is an observable nonperiodic(10)Xi(n+2)= X (n+1)+F(P(n+1)At,system whose future state we may desire to predict. Itwhere Pn and P(n+1) are the points whose coordinates areimplies that two states differing by imperceptibleamountsmayeventuallyevolveinto two considerably(Xi,n,,XMn)and(Xi(n+1),**,XM(n+1)different states. If, then, there is any error whatever inThe simplest numerical procedure for obtainingobserving the present state-and in any real systemapproximate solutions of (1) is the forward-differencesuch errors seem inevitablean acceptable predictionprocedure,of an instantaneous state in the distant future mayXi,n+1=Xi(n+1).(11)well be impossible.As for noncentral trajectories, it follows that a uni-In many instances better approximations to the solu-formly stable noncentral trajectory is quasi-periodic, or,tions of (1) may be obtained by a centered-differenceeguivalently,a nonperiodic noncentral trajectory is notprocedureuniformly stable.The possibility of a nonperiodic non-Xi,n+1=Xi,n-1+2F(P.)At.(12)central trajectory which is stable but not uniformlystable still exists. To the writer, at least, such trajec-This procedure is unsuitable, however, when the detertories, although possible on paper, do not seem charac-ministic nature of (1) is a matter of concern, since theteristic of real hydrodynamical phenomena. Any claimvalues of Xy.n, ..., X M,n do not uniquely determine thethat atmospheric flow,for example, is represented by avalues ofXi,n+1,..",X.n+1.trajectory of this sort would lead to the improbableA procedure which largely overcomes the disadvant-conclusion that we ought to master long-range fore-ages of both the forward-difference and centered-differ-casting as soon as possible, because, the longer we wait,ence procedures is thedouble-approximation procedure,the more difficult our task will become.defined by the relationIn summary,wehave shown that, subject to theX,n+1=Xi,n+[F(P)+F(P(a+)JAl.(13)conditions of uniqueness, continuity, and boundednessprescribed at the beginning of this section, a centralHerethe coefficientof Atis anapproximation tothetimetrajectory, which in a certain sense is free of transientderivative of X,at time le+(n+)t.From (9) andproperties, is unstable if it is nonperiodic. A noncentral(10), it follows that (13) may be rewrittentrajectory, which is characterized by transient prop-erties, is not uniformly stable if it is nonperiodic, and,Xi.n+1=(Xtn+Xi(n+2).(14)
134JOURNAL OF THE ATMOSPHERIC SCIENCESVOLUME20A convenient scheme for automatic computation is thetotically approaching a steady state.A similar resultsuccessive evaluation of X(n+1), Xi(n+2), and X.n+1holds when the double-approximation procedure (14) isaccording to (0), (10) and (14). We have used thisapplied to a conservative system.procedure in all the computations described in thisstudy.5.The convection equations of SaltzmanIn phase space a numerical solution of (1) must beIn this section we 'shall introduce a system of threerepresented by a jumping particle rather than a con-ordinary differential equations whose solutions affordtinuously moving particle. Moreover, if a digital com-the simplest example of deterministic nonperiodic flowputer is instructed to represent each number in itsof which the writer is aware.The system is a simplifica-memory by a preassigned fixed number of bits, onlytion of one derived by Saltzman (1962)to study finite-certain discrete points in phase space will ever be oc-amplitude convection. Although our present interest iscupied. If the numerical solution is bounded, repetitionsin the nonperiodic nature of its solutions, rather thanmust eventually occur, so that, strictly speaking, everyinits contributionstotheconvection problem,weshallnumerical solution is periodic. In practice this considera-describe its physical background briefly.tion may be disregarded, if the number of differentRayleigh (1916) studied the flow occurring in a layerpossible states is far greater than the number of itera-of fuid of uniform depth H,when the temperaturetions ever likely to be performed. The necessity fordifference between the upper and lower surfaces isrepetition could be avoided altogether by the somewhatmaintained at a constant value AT. Such a systemuneconomical procedure of letting the precision ofpossesses a steady-state solution in which there is nocomputation increaseas n increases.motion, and the temperature varies linearly with depth,Consider now numerical solutions of equations (4),If this solution is unstable, convection should develop.obtained by the forward-difference procedure (11).ForIn the case where all motions are parallel to thesuch solutions,x-z-plane, and no variations in the direction of they-axis occur, the governing equations may be writtenQn+1=Qn+(dQ/dt)nAI+ZF2(Pn)AR.(15)(see Saltzman,1962)aa0Let S' be any surface of constant Q whose interior Ra(,V)-(17)++gacontains the ellipsoid E where dQ/dt vanishes, and letata(x,2)aS be any surface of constant Q whose interior R con-tains S'.aa(+,0)ATaySince F? and dQ/dt both possess upper bounds in(18)0-+V20.atHaxa(x,2)R', we may choose At so small that Pn+i lies in R ifPn lies in R'. Likewise, since F possesses an upperHere is a stream function for the two-dimensionalbound and dO/dt possesses a negative upper bound inmotion, is the departure of temperature from thatR-R',wemaychooset so small that Qn+i<Q. if Puoccurring in the state of no convection, and the con-lies in R--R'.Hence Atmaybe chosen so small that anystants g,a,y,and x denote,respectively,theaccelerationjumping particle which has entered R remains trappedof gravity, the coefficient of thermal expansion, thewithin R, and the numerical solution does not blow up.kinematic viscosity,and the thermal conductivity.TheA blow-up may still occur, however, if initially theproblem is most tractable when both the upper andparticle is exterior to R.lower boundaries are taken to be free, in which caseConsider now the double-approximation procedure and vy vanish at both boundaries.(14). The previous arguments imply not only thatRayleigh found that fields of motion of the formP(n+1) lies within Rif P,lies within R,but also thatP(n+2) lies within R if P(n+1) lies within R. Since the(19)=o sin (raH-1a)sin (H-12),region Ris convex, it follows that Pn+1, as given by (14),(20)=,cos (πaH-1a)sin (-12),lies within Rif Pn lies within R.Hence if A is chosen sosmall that the forward-difference procedure does notwould develop if the quantityblow up, the double-approximation procedure also doesRagaHT-1k-1,(21)not blow up.We note in passing that if we apply the forward-now called the Rayleigh number, exceeded a critical valuedifference procedure to a conservative system whereR。=*a-2(1+a2)3(22)dQ/dt-0 everywhere,TheminimumvalueofRe,namely27/4,occurs(16)Qn+1=Qn+F2(Pn)Ae.when a?=2.Saltzman (1962) derived a set of ordinary differentialIn this case, for any fixed choice of At the numericalequations by expanding and in double Fourier seriessolution ultimately goes to infinity, unless it is asymp-in and s, with functions of t alone for coeffcients, and