Forced versus Coupled Dynamics in Earth System Modelingand Prediction

Brigitte Knopf1, HermannHeld1, andHansJoachimSchellnhuber1,2

1PotsdamInstitutefor ClimateImpactResearch(PIK), P.O. Box 601203,14412Potsdam,Germany

2TyndallCentrefor ClimateChangeResearch,Norwich,UK

Manuscriptsubmittedto

Nonlinear Processes in Geophysics

Manuscript-No.12345

Offset requests to:B. Knopf

PotsdamInstitutefor ClimateImpactResearchPotsdam,Germany

NonlinearProcessesin Geophysics(2004)0000:0001–10SRef-ID:1607-7946/npg/2004-0000-0001 LOGO

Forced versus Coupled Dynamics in Earth System Modeling andPrediction

Brigitte Knopf1, Hermann Held1, and Hans Joachim Schellnhuber1,2

1PotsdamInstitutefor ClimateImpactResearch(PIK), P.O. Box 601203,14412Potsdam,Germany2TyndallCentrefor ClimateChangeResearch,Norwich,UK

Abstract. We comparecouplednonlinearclimate modelsandtheir simplified forcedcounterpartswith respectto pre-dictability andphasespacetopology. Varioustypesof uncer-tainty plagueclimate changesimulation,which is, in turn,a crucialelementof EarthSystemmodelling.Sincethecur-rentlypreferredstrategy for simulatingtheclimatesystem,ortheEarthSystematlarge,is thecouplingof sub-systemmod-ules(representing,e.g.,atmosphere,oceans,global vegeta-tion), this paperexplicitly addressesthe errorsandindeter-minaciesgeneratedby thecouplingprocedure.Thefocusison a comparisonof forceddynamicsasopposedto fully, i.e.intrinsically, coupleddynamics.Theformerrepresentsapar-ticular type of simulation,wherethe time behaviour of onecomplex systemscomponentis prescribedby dataor someotherexternalinformationsource.Sucha simplifying tech-niqueis oftenemployedin EarthSystemmodelsin ordertosavecomputingresources,in particularwhenmassivemodelinter-comparisonsneedto be carriedout. Our contributionto thedebateis basedon theinvestigationof two representa-tive modelexamples,namely(i) a low-dimensionalcoupledatmosphere-oceansimulator, and(ii) a replica-likesimulatorembracingcorrespondingcomponents.

Whereasin generaltheforcedversion(ii) is ableto mimicits fully coupledcounterpart(i), we show in this paperthatfor a considerablefraction of parameter- and state-space,the two approachesqualitatively differ. Herewe take up aphenomenonconcerningthepredictabilityof coupledversusforcedmodelsthat wasreportedearlier in this journal: theobservationthatthetime seriesof theforcedversiondisplayartificial predictiveskill. We presentanexplanationin termsof nonlineardynamicaltheory. In particularwe observe anintermittent versionof artificial predictive skill, which wecall on-off synchronization,andtraceit backto the appear-anceof unstableperiodicorbits. We alsofind it to be gov-ernedby a scalinglaw that allows us to estimatethe prob-ability of artificial predictive skill. In addition to artificial

Correspondence to: Brigitte Knopf (knopf@pik-potsdam.de)

predictabilitywe find artificial bistability for theforcedver-sion,which hasnot beenreportedsofar. Theresultssuggestthatbistability andintermittentpredictability, whenfoundinaforcedmodelset-up,shouldalwaysbecross-validatedwithalternativecouplingdesignsbeforebeingtakenfor granted.

1 Introduction

In theclimatemodellingcommunityit is commonpracticetoestablisha modularstructure,consistingof ecosphere,bio-sphere,vegetation,ocean,atmosphere,etc.,thatbuildsupanEarthSystemModel (cf. theClimateSystemModel projectCSM (Boville andGent,1998)).Someof thesecomponentsare also modelledby external forcing, describedfrom ob-served data. This is donee.g. in the AtmosphericModelIntercomparisonProjectAMIP (Gateset al., 1992), wherean atmosphericgeneralcirculation model (AGCM) is con-strainedby realisticseasurfacetemperatureandseaice andtheoutputis usedfor diagnosticresearch.Althoughthis ex-perimentis not meantto beusedfor climatechangepredic-tions, diagnosticsubprojectshave beenestablished,thoughit is not quiteclearto whatextent the forcedAGCM outputis comparableto thesystemwith complex ocean-atmospherefeedbacks.Thesecoupledsystemsare investigatede.g. inthe CoupledModel IntercomparisonProjectCMIP (Meehlet al., 2000;Covey et al., 2003a). The comparisonof cou-pled ocean-atmospheremodelswith simulationsusingpre-scribedseasurfacetemperaturesshows that thereareindeedsomeimportant differencesconcerninge.g. temperaturesnearthepoleandtropicalprecipitation(Covey etal.,2003b).Otherpublicationsmentionastrongeffectof thecouplingonthe midlatitudevariability of the ocean-atmospheresystem(BarsugliandBattisti, 1998)or on thedecadalvariability ofoceanicvariablesin theNorthPacific (Pierceetal., 2001).

Hence,thesubjectof investigationis theeffectof prescrib-ing a modulethroughdatainsteadof implementingthe dy-namicalmodule.This hasalreadybeeninvestigatedby Wit-tenberg andAnderson(1998),but herewe will focuson po-

2 Knopf et al.: ForcedversusCoupledDynamics

tential constraintspreventingtheconsistency of forcing andcoupling. With this paperwe want to emphasizethatwhenforcing onemoduleby anotherinsteadof couplingthe twocomponents,onehasto keepin mind that inherentlynonlin-earphenomenacanoccurthat leadto qualitatively differentfeaturesthanexpected.This typeof analysisthatwe areun-dertakingcanbeassignedto many othercasesof investigat-ing forcedversuscoupledmodelruns.

For our studywe usea conceptualmodelthat, comparedto GCMs,hastheadvantagethat themodelitself aswell astheoutputcanbeanalysedalongthelinesof dynamicalsys-tems’ theory. This makesit easierto realizepathfollowingof solutionsin parameterspace.

The structureof the paper is as follows: The coupledocean-atmospheremodel we are analysingis describedinsection2 and the phenomenonof locking for coupledandforcedtrajectoriesispresented.In section3 themathematicalbackgroundfor replicasystemsis introduced.In section4 weinvestigatethe statisticsof the locking anddeducea powerlaw scalingfor the lengthof the locking phases.In section5 we analysethemodelin dependenceon its parametersandhighlightsomefundamentaldifferencesbetweenforcingandcoupling.Theroleof unstableperiodicorbitsconcerningthelocking phenomenonis alsoinvestigated.Thepaperwill fin-ish with theconclusionsin section6.

2 Coupled and forced model

To investigatethe differencebetweena forced and a fullycoupledset-up,a coupledatmosphere-oceansystemis cho-sen becausethe predictability of the Earth’s climate de-pendsstronglyonthevariability inducedby theinteractionofthesetwo components.As a very instructive exampleof thecoupledatmosphere-oceansystem,the following low-ordermodelis examined:������������ ������������������������ �"!�#�$&%&% (1)��'�(�)�*��+,�*�.-/�0�1��2(��354 (2)��*�(�0�6��+7�8�9-/�5�6��3;: (3)�4<���=>:8�@?�� (4)�:6��=A4'�B?�� (5)

with ���DC5E�FG�IH , �J�DKLEMH , +N�DC5E H , -'�PO , 3Q�P?R�DCLESF ,2��(CLEM�IH , #T�UFVCXW"K Y HEM�IH , =��(�Z!�#AW"O , where# is ascalingfactorwith oneunit of system’s time referringto 10 days.

This model is taken from Wittenberg and Anderson(1998). The atmospheresystemmodel (Eq.(1)-(3)) is apotentially chaoticLorenz system(Lorenz,1984), that de-scribesthemidlatitudequasi-geostrophicflow. While $ rep-resentsthetime,thevariable� representstheintensityof thewesterlywind currentor the meridionaltemperaturegradi-ent.Thevariables� and � aretheamplitudesof thesineandcosinecomponentsof a large travelling wave, which trans-ports heatpoleward. � and 2 are forcing termsbasedonthe averagenorth-southtemperaturecontrastandthe earth-seatemperaturecontrast,wherethe seasonalvariationof �

68 69 70 71 72 73 74 75 76 77 78−0.5

0

0.5

1

1.5Fully coupled run

x

68 69 70 71 72 73 74 75 76 77 78−0.5

0

0.5

1

1.5

time / yr

Forced run

x

Fig. 1. Comparisonof thefully coupledsystem(top)andtheforcedsystem(bottom). A fully coupledrun is taken asreferencetrajec-tory. Additional to thereferencetrajectory, in eachsubfiguretherearerunsfrom slightly varying initial conditionsin atmosphericco-ordinates.In the upperfigure the curvesarefrom a fully coupledrun. In the lower figure, the trajectoriesare forced by the oceanfrom the referencetrajectory. In this figure,we have reproducedamajorfindingby Wittenberg andAnderson(1998).

is expressedthroughthe sine. The oceansystemis a sim-ple harmonicoscillator, with an oscillation frequency = offour years,wherep andq representzonalasymmetriesin seasurfacetemperature.Thecouplingbetweenoceanandatmo-sphereproceedsthroughtheinteractionof theseasymmetrieswith themodelatmosphere’seddyfield (y andz).

Wittenberg andAnderson(1998)carriedout two differentsetsof simulations. One set of simulationsrepresentstheoutcomeof the fully coupledsystemwith little variationinthe initial statevectors. In the other set the output of theoceanfrom onespecialrun is usedto force theatmosphere.Again this is undertakenfor slightly perturbedinitial condi-tions. Sotherearetwo ensembles:onefrom a fully coupledsystemandonefrom a forcedsystemthat includesno feed-backfrom theatmosphereto theocean.

As canbe seenfrom Fig. 1, which wasreproducedfromWittenberg and Anderson(1998), the forced ensembleismorecompact,but doesnotmirror thetruesolution.Further-more,Wittenberg andAnderson(1998)show that thestatis-tics of the forcedvariability, like spatialand temporaldis-tributions,aresignificantlydifferent from thoseof coupledvariability. For modellingissuesthismeansthataprescribedforcing (e.g.prescribingtheseasurfacetemperature)cannotemulatethe fully coupledsystem.The interestingeffect oftheforcing is thatall trajectoriessometimeslock on thetruesolutionfor a short time andthenseparateagain,so partialsynchronizationcan be observed. One can concludefromthis that on the onehandfully coupledandforcedsystemsdonotshow thesamebehaviour but ontheotherhandthatintruly forcedsystemstheremayexist a region in phasespace,

Knopf et al.: ForcedversusCoupledDynamics 3

wherepredictability is very high. The questionto be fol-lowedis whatmechanismis responsiblefor thelocking phe-nomenonandwhattypesof couplingshow suchbehaviour.

3 Mathematical framework

In orderto explain thelockingphenomenon,astabilityanal-ysisof thesystemappearsthemostnaturalapproach.In thetraditionof Wittenberg andAnderson(1998)andSmithetal.(1999),onewould expectthatthelocal linearstability prop-ertiesgoverntheobservedphenomenon.In orderto establishsucha link we derive the JacobianJ [ of the driven atmo-sphere.For this,weconsidertheODEsystem

�x [ � F [ � x [\ 4;�]$&% \ :�]$&% \ $&% ^I�5_ x [ �`�]� \ � \ �X%ba (6)

with F [ beingmadeup from Eqs.(1)-(3). The Jacobianreads(Wittenberg andAnderson,1998)

J [�c �Dd F [d x [ � ef �g� �8�Z�h�8�I��*�.-,�6�N��+��8-,��1��-,�i-/�j�N�B+kl E (7)

NotethattheJacobiandoesnot dependexplicitly on time,henceonecouldhopeto link lockingto certainregionsin thephasespaceof x [ .

The Jacobiangovernsthe time evolution of an infinitesi-mal perturbationof theatmospherictrajectorym x [ :

m �x [ � J [ �n$&% m x [ \ (8)

wherebythe time-dependenceof J [ is inducedby the dy-namicsof the unperturbedtrajectory. Below it will proveusefulto introducethesquarednormof theperturbation

o c �`� m x [p\qm x [ % \ (9)

where“( , )” denotesthestandardscalarproduct.Traditionalstability analysisaddressestwo extremetypes

of stability: the(instantaneous)growth rater �n$&% c � d o W d $7�n$&%and the asymptoticbehaviour of the perturbation, s c �t �Su'vnwyx o �n$&% . If we requirethatat the beginningof a lock-ing period,in �]z&% c �`{ $}| \ $�~q� , theerror

oneedsto shrink,and

lateron, in �]z�z&% c �`{ $�~ \ $ � � , oshouldat leastnot grow, thenit

is sufficient to askfor

�5�Z��� ~�� aN� v������� o �n$&%�� o �n$ | % e� ��v � v]���

and (10)� v�����]��� o �n$&%�� o �n$�~7% \ (11)

where� denotesthedurationof region(I).For condition(11), it is sufficient to requirethat

� v�����]��� r �]$&%��CLE (12)

For the convergenceregion (I), the caseis slightly morecomplicated,andwewill comebackto thisbelow.

Weareawareof thefactthattheabovesufficientconditionfor region (II) might be too restrictive aslocking is a time-averagedphenomenon,i.e., it would be enoughto requirethat the errordoesnot grow on average.If we followedthelatter logic, however, we would refer to trajectory-specificproperties,while in the traditionalansatzthefirst attemptisto link phenomenato local, trajectory-independentproper-ties. This thennaturallyleadsto conservative conditionsofthetypeEq.(12).

As is well known (Farrell andIoannou,1996), r links totheJacobianJ [ in thefollowing way:

r ��d od $ ��� m �

x [ \�m x [ %���� m x [ \qm �x [ %�`� m x [ \ J a[ m x [ %��(� m x [ \ J [ m x [ %�R�y� m x [�\ J ���7����m x [ % (13)

with J �]�7���(c � F� � J [ � J a[ %,EAs J ���7��� is symmetric,r ��C wouldbeestablishedby the

requirementthat the largesteigenvalue � of J ���7��� is non-positive. A simplecalculationtransferstheseideasalongthelines of Lyapunov theoryinto a practicalcounterpartof thecondition(10):

� �L�"����� ~���� �qa � � v������� � �n$&%� �0�g� (14)  � � v�����¡� o �]$&%�� o �]$}|¢% e� � � ��v � v � � � (15)

with � �n$&% againbeing the largesteigenvalue of J ���7��� �]$&% .Note that both equationscan be taken as counterparts,butEq.(15) follows from Eq.(14).

In summary, the sufficient conditionsfor the regions (I)and(II) bothlink to thelargesteigenvalueof thesymmetrisedJacobianwhich shallbenon-positive. Empirically, however,we find theopposite:a look at Fig. 2(c) revealsthat � is al-wayspositive. Obviously, theabove reasoninginvolvessuf-ficientconditionswhicharetoo restrictive, i.e. tooconserva-tive.

Often,shrinkingof errorshasbeenexplainedin linear, lo-cal terms,but with respectto theeigenvaluesof theoriginalJacobian(seee.g. Holton (1992)), rather than to its sym-metrisedcounterpart(Yanchuket al., 2001). For thesystemconsideredhere,in fact,a so-called“binding region” canbeestablishedin which all eigenvaluesof theJacobianareneg-ative. We find, however, that the existenceof sucha bind-ing regiondoesnot correlatewith theparticulartypeof errorshrinking – locking – we are consideringhere, for severalreasons.

First, empirically, we find (seeFig. 2(b)) that the lockingperiodby far exceedsthe typical residencetime of the sys-tem’s trajectoryin the binding region. On the other hand,the systemmay visit the binding region without displayinglocking.

4 Knopf et al.: ForcedversusCoupledDynamics

32 34 36 38 40 42 44 46 480

1

2

norm

of e

rror

vect

or |x

−x’

|

32 34 36 38 40 42 44 46 48

−1

0

1

Eig

enva

lues

of J

A

32 34 36 38 40 42 44 46 48−5

0

5

Eig

enva

lues

of

(JA+

J AT)

time / yr

Eigenvalues smaller than zero, but no locking

a.)

b.)

c.)

Fig. 2. Eigenvaluesof J £ and J £T¤ J ¥£ . (a) Norm of the errorvectorbetweencoupledandforcedrun. (b) Eigenvaluesof theJaco-bianmatrixJ £ . (c) Eigenvaluesof thesymmetrisedJacobianmatrixJ £¦¤ J ¥£ . This figureshows that thereareregionswhereall eigen-valuesarenegative but whereno locking canbedetected.(For thisfiguretheparameter§ is setto 0.09insteadof 0.125).

Second,to our knowledge,it is in generalnot possibletodirectlyextractlocalandfinite-timestabilitypropertiesfromthe eigenvaluesof the Jacobian. Rather the symmetrisedcounterpartis needed(seeabove). Theeigenvaluesof theJa-cobianaretypically employedto establishasymptoticprop-ertieslike s �¨C . Even if onecould interpretlocking asatime-asymptoticproperty, however, the Jacobianwould im-plicatea stablesolutiononly if it wastime-independent.Fortime-dependentJacobianslike in our caseone would stillcomebackto themoredemandingcondition(14) involvingthesymmetrisedJacobian.

In summary, anapproachdifferentfrom local linearstabil-ity analysisis clearlyneeded.We will relatethelocking pe-riod to invariantmanifolds,emergingfrom thenon-lineardy-namicsof thesystem.This will allow usto introducemean-ingful time-averagedcharacteristicsstill without having toreferto eachindividualmastertrajectoryseparately.

To frame a discussionof potential nonlinearcausesoflocking,wefollow theconceptsof PecoraandCarroll (1990)andPecoraet al. (1997). Different from Fig. 1, wherewelooked at a setof several forcedtrajectories,we investigateherejustthefully coupledrunandoneforcedrun. Theforcedsystemcanbewrittenasaso-calledreplicasystem(Pikovskyet al., 2001), wherea replica of one or more equationsismade.Togetherwith Eqs. (1)-(5) we have a replicasystemof thefollowing form:��0©������ © � �B� © � �B��� © ����������������� �"!�#�$&%&% (16)��©0�(� © � © ��+,� © �.-/� © � © ��2(��354 (17)�� ©)�(� © � © ��+7� © ��-/� © � © ��3;: (18)�4)©0���=>: © �B?A� ©

(19)

50 60 70 80 90 1000

0.5

1

1.5

2

2.5

time / yr

norm

of e

rror

vec

tor

|x−

x’|

Fig. 3. Norm of the error vector ª x «P¬ x ­ x ® ¬ betweenthe fullycoupledandtheforcedrun; ( §¯«.°Z± °V² ).

�:"©0��=A4 © �B?�� © E (20)

wherethe primedsystemx © �³�]� © \ � © \ � © \ 4 © \ : © % a is identi-cal to the original fully coupledsystemx �´�]� \ � \ � \ 4 \ :Z% aexceptfor slightly differentinitial conditionsandthesubsti-tutedvariables4 and : insteadof 4 © and : © , thatemulatetheforcingthroughprescribeddata.In thissystem4 © and : © haveno influenceon the dynamicsof the otherprimedvariablesandareonly introducedto allow for a closedmathematicalform. With this formalization“reality” - prescribedthroughdata- is beingrepresentedthrougha perfectmodelscenarioin themodel-world.

In Fig. 3 the norm of the error vector m x �¶µ x � x © µ isplotted. Sometimesthe two systemssynchronizebut thensuddenlythesystemshowslong-lastingburstswherethetwotrajectoriesseemto evolve independently.

Generally, this typeof couplingbetweentwo identicalsys-temscanbewrittenas�x � F � x % �

x ©�� F � x © %�� K � x � x © %,E (21)

whereK is thecouplingfunction.By transformingEq. (21) to the transversalcoordinates

x · � x � x © andconsideringonly small perturbations,sothat x ¸ x © andF � x © % ¸ F � x %¹� J � x %7� x © � x % the equationcanbeapproximatedby�x · � F � x %º� F � x © %º� K � x · % ¸ J � x % x · � K � x · % (22)

whereJ � x % is theJacobianmatrix of F evaluatedon thesyn-chronizationmanifold. A linearisationof the functionK � x %aroundzero,wherewe assumethat K � 0 %<� 0 andneglecthigherordertermsof x · , leadsto

�x ·.¸ � J � x %º�P»K % x ·\ (23)

with

»K � d Kd x ¼¼¼¼ x ½ | E (24)

Knopf et al.: ForcedversusCoupledDynamics 5

In our case,wherewe have linearcoupling,thematrix »Kis

»K �e¾¾¾¾f C6C6C¿C@CC6C6C*3�CC6C6C¿C¿3C6C6C¿C@CC6C6C¿C@C

k�ÀÀÀÀl E (25)

To achieve completesynchronization,it is requiredthatfor $�Áà, x · goesto zero. From the linearisedequation(23)onewouldexpectthatthetwo systemswill synchronizeif thetransverseLyapunov exponents,thataretheLyapunovexponentsassociatedwith Eq. (23), are all negative. Thiscriterionwasfirst proposedby FujisakaandYamada(1983),but in contrastto this, e.g. Gauthierand Bienfang (1996)observe only incompletesynchronizationin their model in-steadof theproposedfull synchronization,whenthe largesttransverseLyapunov exponentis smallerthanzero. Severalcriteria for synchronizationweredeveloped(Blakely et al.,2000),but it wasalsoshown therethat for their modelnoneof thesecriteriaexactly predictstherangeof thecontrolpa-rameterwherefull synchronizationcanbeobserved.

In our casethe largesttransverseLyapunov exponentispositive with � · ¸ C5E C ÄZO but we alsoobserve partial syn-chronization.Thetime $ � afterwhich all informationis lostandthetwo trajectoriesaretotally independent,reads$ �1¸ F� t � Åm �;�nC % \ (26)

where � is theLyapunov exponent,Å denotesthecharacter-istic lengthof theattractorand m �;�nC�% theerrorthatcannotbedissolved by a given accuracy (Argyris et al., 1995). Here$ � is found to be about1.3 years.Neverthelesslocking canbeobservedovermuchlongertimescales,ascanbededucedfrom Fig. 3. This demonstratesthatin theperiodof locking,a non-average,non-standardsituationis present.Below wewill link it to phase-spacestructuresof low measure,yetof anoticeabledomainof attraction.

4 On-off Synchronization

Thealternationbetweenregions,wheretheerrorbetweentheforcedandcoupledtrajectoryis nearlyzero,andbetweenre-gionswith largeburstsasshown in Fig. 3, resemblethoseofon-off intermittency. On-off intermittency, asintroducedbyPlattet al. (1993),refersto asituationwherethevariablesofa chaoticdynamicalsystemexhibit two distinctstateswhereat the“off ” statethesystemis nearlyconstantonaninvariantmanifold,andat the“on” statelargeburstsfrom theselami-narphasesoccur. Thefrequency of burstsis controlledby acharacteristicparameter� of thesystemandapproacheszero,whenthe so-calledblowout bifurcation(Ott andSommerer,1994) is reachedas � attainsa critical value �Æ . To assignthe phenomenonof on-off intermittency to our objectof in-terest,intermittency is notseenaslaminarphasesinterruptedby turbulent burstsbut as locking of the fully coupledand

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

length of the locking region / yr

prob

abili

ty p

(τ)

Fig. 4. Probabilitydistribution for thelengthof thelocking region.1,500locking phasesweretakeninto account.

−5 −4 −3 −2 −1 0 1−8

−7

−6

−5

−4

−3

−2

log (τ)

log

p(τ)

slope = −1.5

Fig. 5. Probabilitydistribution of the lengthof the locking region.Doublelogarithmicversionof Fig. 4 to calculatethescalingcoeffi-cient.

the forcedtrajectories(off-state)interruptedby non-locking(on-state),which we will call on-off synchronization. Thetransferfrom on-off intermittency to on-off synchronizationbecomesclearerwhenthedifferencebetweenx andx © is un-derstoodasa new variable.

Systemsthat generateon-off intermittency show charac-teristicscalinglaws for theintermittentphases(Heagyetal.,1994; Lai, 1996), the distribution of the amplitudesof theburstsandfor thepowerspectrumof thetrajectories(seeref-erencescitedby Johnet al. (2002)). Heagyet al. (1994)in-vestigatea certainclassof drivensystems,thatconsistsof adiscretemapanda randomdriving variablewith a smoothdensity. They show thatfor theprobabilitydistributionof thelengthof the laminarphases4�� � % a power law holdswith4;� � %¦Ç � �0È

, where � is the lengthof the laminarphasesand # the scalingexponentthat attainsa universalvalueof3/2 in thevicinity of thethresholdfor on-off intermittency.

This scalinglaw with the sameexponentcanalsobe ap-provedfor ourcaseof on-off synchronization,ascanbecon-cludedfrom thelogarithmicscalingin Fig.5,wheretheprob-ability distribution 4;�]É5% of thelengthof thelocking region Éis plotted.Thehistogramfor 1,500locking phasesis plottedin Fig. 4, wherelocking is definedby the norm of the errorvector m x �ʵ x � x © µ of the two trajectoriesx andx © being

6 Knopf et al.: ForcedversusCoupledDynamics

smallerthana critical thresholdË . To determinethestatisticalsofor the long laminarphasesit is importantto chooseathresholdË thatis not toosmall(Lai, 1996).For ouranalysiswe took Ë ��CLE CLF .

In this sectionwe have shown that thescalinglaw for thedurationof the laminarphasesin systemswith on-off inter-mittency holdsalsofor asystemwith on-off synchronizationand could be extendedto continuoussystemswith a driv-ing systemthat is not randombut chaotic. This meansthatthepower law scalingis moreuniversalthanproposedwhenit was introduced. It canalsobe interpretedin this regard,that the underlyingmechanismsof on-off intermittency andon-off synchronizationareanalogous.As intermittency is of-tentracedbackto the”almostexistence”of a stableperiodicorbit, lateronouranalysiswill concentrateon stableandun-stableperiodicorbitsaspotentialcausesfor locking.

5 Comparison of forced and coupled system

5.1 Systemwithout seasonalcycle

An important structural difference betweencoupled andforced systemswill be discussedin this chapter. In orderto separatethetwo forcing effectsin this model,namelytheoceanforcing throughthevariables4 and : andtheseasonalforcing, themodelis firstly investigatedwithout theseasonalcycle. We analysethedependenceof therelativemeanlock-ing time Ì ÉÎÍ on the coupling strength 3 , seeFig. 6(a),whereÉ¿� $}Ï Ð,Ñ�Ò,Ó Ô�Õ� \ (27)

where � is thelengthof thewholetime seriesand $ Ï Ð,Ñ�Ò,Ó Ô�Õ isthe time, wherelocking canbe observed. This is averagedoverseveralruns.

A significantdifferencein the relative meanlocking timefor a fully coupledrun and a forced run can be observed.Thefully coupledsystemconsistsof two totally independentsystemsx andx © , wherethecouplingmatrix K of Eq.(25)iszero. The forcedsystemis the 8-D combineddrive andre-sponsesystem,consistingof Eqs.(1)-(5) and(16)-(18),theEqs.(19) and(20) areneglectedin this caseasthey have noinfluenceon the system’s dynamics. Whereasin the fullycoupledsystemthe relative meanlocking time Ì ÉÎÍ , as afunctionof 3 , is alwayszero(not plottedin thediagram),intheforcedsystemtherearesmallparameterrangeswherethetrajectoriesalwaysshow locking (for 3×ÖD{ C5E�F \ CLESFGØ Ø¢� ), orwherelocking never appears,e.g. for 3�Ö�{ CLEM� Ø Ø \ CLE OZ� . Ad-ditionally, in theforcedsystemtherearealsoregionswherewe observeon-off synchronizationasdefinedabove,e.g.for3UÍÙCLE O – like in the original systemwhich includesa sea-sonalcycle. Thissuggeststhattheseasonalforcing is not themaincauseof theobservedintermittentbehaviour.

In the remainingpart of this section,we empirically cor-relatetime-seriespropertiesandphase-spacetopology. Toget an impressionof the phasespacetopology of the sys-temin dependenceof theparameter3 , abifurcationanalysis

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51E−6

1E−4

1E−2

1

<τ>

/ ar

b. u

nits

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.96

0.98

1

1.02

x

stable periodic orbitunstable periodic orbit

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.94

0.96

0.98

1

1.02

coupling strength α

x′

b.)

stable periodic orbitunstable periodic orbit

c.)

a.)

Fig. 6. Relative meanlocking time in relation to the mostdomi-nantinvariantsets(periodicorbits) for thesystemwithout seasonalcycle (with §<«U°I±ÛÚ7Ü ). (a) Relative meanlocking time ݹÞ5ß (seeEq. 27) in dependenceof thecouplingstrengthà . For every pointin this figuretheinitial conditionsfor x werechosenrandomlyandthetrajectorieswereintegratedover500years,accordingto 730500timesteps,afterthey settleddown onanattractor. Theforcedtrajec-tory wasstartedontheattractorwith slightly perturbedinitial condi-tions,chosenfrom agaussiandistributionwith astandarddeviationof 0.01. For every valueof thecouplingstrengthà the integrationwasperformedseveral times. As a meanlengthof zerocannotbedepictedin a logarithmicplot, we addedanoffsetof Ú,°IáXâ ; (b) Bi-furcationdiagramfor the variable ã of the five dimensional(5-D)driving system;for the periodicorbits, just onepoint referring tothemaximumof theorbit is plotted;(c) Bifurcationdiagramfor thevariableã ® of the8-D combineddriveandresponsesystem.A filledcirclesymbolrepresentsasaddlenodebifurcation,anunfilledcirclestandsfor atorusbifurcation,anupward-pointingtriangledenotesaperioddoublingbifurcationandadownward-pointingtrianglesym-bolizesa branchpoint.

0.1 0.2 0.3 0.4 0.5−1.5

−1

−0.5

0

0.5

1

1.5

coupling strength α

x’

Fig. 7. Bifurcation diagramfor ã ® in dependenceof the couplingstrengthà in thesystemwithout seasonalcycleand §¯«.°I±ÛÚ,Ü .

is performedwith the bifurcation analysisprogramAUTO(Doedel,1981). In Fig. 6(b) thebifurcationdiagramfor the

Knopf et al.: ForcedversusCoupledDynamics 7

variable� of thefully coupledsystemis plotted,in Fig. 6(c)the sameis donefor the variable � © of the 8-D combineddrive andresponsesystem.Thedifferenceis amazing:bothvariables � and � © are on a stableperiodic orbit for thesameparameters,but partlystabilityswitchesthebranch(for39Ö�{ CLE K \ CLE OZ� ) or a secondstablebranchemergesat thesad-dle nodebifurcationfor 3���C5E�F"ØIØ . In the lattersituationitdependson the initial conditionswhetherfull synchroniza-tion canbe observed; in the former casesynchronizationisimpossible,which canalso be concludedfrom the relativemeanlocking time in Fig. 6(a). This meansthat for theseparametercombinationstheforcedsystemwill never mirrorthe fully coupledsystem.Full synchronizationcanonly beobservedwhen � and � © areonthesamestableperiodicorbit.

In Fig. 7 anotherbifurcation diagramfor � © is obtainedby numericalintegrationto capturethemovementonatorus,thatcannotbedepictedin theotherdiagram.In fact,thetorusbifurcation is found in the upperfigure Fig. 6(c) for 3J�C5E ��ØIØ , but the torus cannotbe followed with this method.For eachparametervalue 3 we let thesystemsettledown toanattractorandthenplottedtheforcedvariable� © , whenthetrajectorycrossesthe � © axisat0.5.Fromleft to right weseeagainthe stableperiodicorbit andat 3U�DC5E�F"ØIØ the gener-ation of a secondorbit. From thesestableperiodicorbits aquasiperiodicmotionon a torusemergesat 3���CLEM� Ø Ø . Thedynamicson the torusis sometimesadjournedby thestableperiodic orbit that can be found in Fig. 6(c) for 3 Ì CLE K .The torusdisappearsat 3R�DC5E O througha perioddoublingbifurcationandpassesinto chaoticmotion. For the drivingsystemx no quasiperiodicdynamicsare found, so that for3�ÖB{ CLEM� Ø Ø \ CLE K"� , beforethebranchpointat 3��(C5E K emerges,thesystemshowslockingwhenx © is onthestablelimit cycleor showsnolockingwhenx © is onthetorus.Whichstatewillbe adopteddependson the initial conditions. For 3 largerthanthebifurcationvalue3B�(CLE K , thereisscarcelyany lock-ing becausethe forcedsystemis on the torusandthe fullycoupledsystemon thestableperiodicorbit. An overview ofthe differentclassificationsin dependenceon the parameterspaceis givenin Table1.

So far we canconcludethat if the systemis on the samestableperiodicorbit for x andx © , we get full synchroniza-tion. This will beanalysedin detail in thenext section.Onthe otherhand,whenx runson a periodicorbit not concur-ring to the replicateof x’s periodicorbit, locking cannotbeobserved. That meanstherecanbe intrinsic obstaclesthata forcedsystemperformsasthe fully coupledsystem. Formodelling issuesthis is a crucial outcome,as for moreso-phisticatedmodelsthe calculationof the statespaceof theforcedandthecoupledmodelis verycostlysothereis hardlyany way to decideif forcing is suitable,above all becausenormallythefully coupledmodelis not known.

Despite these fundamentaldiscrepanciesbetween theforcedandfully coupledsystem,in thefollowing wewill fo-cuson thephenomenonof on-off synchronizationthatarisesfor 3.�CLE O , whereno stableperiodicorbit is detected.

couplingstrengthàTä ã ã ® locking?

[0.1, 0.177] SPO SPO locking[0.177,0.277]

SPO two SPOs locking, when ã and ã ®are on the same PO;elseno locking

[0.277,0.3] SPO same SPOandtorus

locking, when ã ® ison the PO; no locking,when ã ® is on thetorus

[0.3, 0.4] SPO (differing)SPO andtorus

no locking

[0.4, 0.5] UPO UPOs intermittent locking(on-off synchroniza-tion)

Table 1. Overview over the different regions in parameterspace.SPOstandsfor stableperiodicorbit, UPOfor unstableperiodicor-bit.

5.2 Therole of (un)stableperiodicorbits

Fromnow on we will restrictourselvesto theregionsin pa-rameterspacewherex andx © show thesamebifurcationdia-gram.Fromtheobservationsin thepreviouschapterwe canconcludethatif thesystemis in a regionwhereit is on a sta-ble periodicorbit, the forcedsystemshows locking all thetime. Theargumentationreadsasfollows: astheorbit of the8-D systemis stable,the trajectoriesof the driving systemandthedrivensystemendup on thesameperiodicorbit, butthey couldstill haveaphaseshift. If therewereaphaseshift,thenthis shift would alsobe seenin the oceancoordinates.But thisis excludedthroughthereplicaapproach,whereonlytheatmospherecoordinatesarevaried. Contraryto this, forthecoupledsystemtheoceancoordinatesarealsosubjecttotheperturbation,hencenosynchronizingdriveis presentandno locking will occur.

This emphasizesthat full synchronizationcanin this casebe explainedby a stableperiodicorbit (or a stableequilib-rium point)thatdrivestheforcedsystemto thesamedynamicbehaviour.

After the stability has beenlost in a bifurcation point,the unstableperiodic orbit embeddedinto the attractorin-fluencesthe systemthroughthe oceanvariablesso that thesystemshows locking even in a region wherethe transver-salLyapunov exponentis positive andno synchronizationisexpected. Thereforethe conceptof (unstable)periodicor-bits seemsto becrucial for thelocking phenomenonandthelossof synchronizationcanbe tracedbackto the transitionfrom stableto unstableperiodicorbits. Ott andSommerer(1994)call this a nonhysteretic blowout bifurcation,wherefor � Ì � Æ thesystemis onanattractorandfor �NÍ�� Æ on-offintermittency canbedetected.Therole of unstableperiodicorbits(UPOs)for synchronizationis alsopointedoutbyPazoetal. (2003)andby Pikovsky et al. (1997).

The interpretationwith regard to UPOscan be stressed

8 Knopf et al.: ForcedversusCoupledDynamics

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2−1

−0.5

0

0.5

1

1.5

x

y

full attracting set (non−locking)unstable periodic orbitlocking region 1locking region 2

Fig. 8. Phasespaceof the full attractingset (non-lockingregion)andof two exemplarylocking regions(1 and2), whereasherere-gionrefersto aperiodin time. For comparisontheunstableperiodicorbit for this parameterconstellationis plotted.

throughFig. 8, wherewe analysedthe phasespaceof thelockingregionsin comparisonto thefull attractingset.It canclearlybeseenthatthelockingregionis in goodcoincidencewith theunstableperiodicorbit, whereasthenon-lockingre-gion coversamuchlargerpartof thewholephasespace.

Justbeyonda bifurcationpoint wherea periodicorbit hasbecomeunstable,theMonodromymatrixof therelatedmap-ping will displaya long timescaleon theunstablemanifold,andgenericallyshortertime-scalesfor the remainingstablemanifold. Therefore,the unstableperiodicorbit still hasafair chanceto attracton thestablemanifoldandsynchronizethetrajectory. Thiswill revealitself aslocking.After awhile,thelong timescaleon theunstablemanifoldmanifestsitself,andthetrajectorybecomesrepelled,reminiscentof intermit-tency. Hence,we suggestthaton-off synchronizationcanbetracedbackto a co-existenceof two identicalunstableperi-odicorbits,onein thefully coupled,andonein thecombined8D fully coupledandreplicasystem.

5.3 Systemwith seasonalcycle

The assertionof the role of stableandunstableperiodicor-bits canalsobe endorsedby Fig. 9, wherethe systemwithseasonalcycle in dependenceon the couplingstrength3 isanalysed.As before,thebifurcationdiagramandtherelativemeanlocking time areplotted. Again we candetecton-offsynchronization,asdefinedin section4, andit canbe seenthatthereis a transitionfrom locking to intermittentlocking.This is astrongargumentfor on-off intermittency andon-offsynchronizationbecausethis transitionregion from laminarto turbulentphases(in thiscasefrom lockingto non-locking)nearthe bifurcationpoint is the key mechanismassessedinall typesof intermittency.

Fig. 9 makes it clear that the presumptionthat withstrongercoupling the two systemswill synchronize,is notvalid in this case.Thesystemneedsa stablemanifoldto be-comefully synchronized.For modellingissues,that means

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

10−5

100

<τ>

/ ar

b. u

nits

System with seasonal cyclea.)

forced systemfully coupled system

0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.240.95

1

1.05

1.1

coupling strength α

x

stable periodic orbitunstable periodic orbit

b.)

Fig. 9. Variationof thecouplingstrengthà in thesystemwith sea-sonalcycle with §å«`°Z± ÜVæ . (a) Relative meanlocking time ݹÞ)ß(seeEq. 27) in dependenceon thecouplingstrengthà . Again weaddedanoffsetof Ú,°IáXâ to depicta meanlengthof zero;(b) Bifur-cationdiagramof the system. As for this parameterconstellationã and ã ® show the samebifurcationdiagram,just onebifurcationdiagramis plotted.

thatit doesnotdependon thestrengthof couplingbut on thestatein thephasespaceif forcingcansubstitutecoupling.

A significantdifferenceto the situationwithout seasonalforcing is that here the fully coupledsystemshows lock-ing whenthesystemis on a stablelimit cycle, seeFig. 9(a),wherethe relative meanlocking time is 1 in the locking re-gions,whichmeansthereis alwayslocking. On-off synchro-nizationcanalsosometimesbeobservedbut lessoften thanin the forcedsystem(Fig. 9(a)). This is dueto theseasonalforcing, thatdeterminesthe frequency of the periodicorbit,sothis lockingbearsonanexternalforcingandnotonthein-trinsicphenomenonof lockingthroughprescribedforcingbyvariables.But astheseasonalcycle is alsoa kind of forcing,the chanceof locking throughan additional“synchronizer”increases.

5.4 Influenceof thetypeof coupling

Thesystemanalysedsofar is a systemwith linearcoupling.As this is a very specialcaseof coupling that is not oftenusedin truly coupledmodels,we analysea systemwithouta seasonalcycle andwith a nonlinearcouplingto determinethe influenceof the type of coupling. The couplinghasthefollowing form:��'���)����+,�*�.-/�0�1��2���3540ç�� (28)��è���0�6�B+��1��-,�5�¯��3;: ç ��E (29)

insteadof Eqs.(2)and(3). As 4 and : vary approximatelybetween-1 and1, the introducedterm bearsstrongnonlin-earity.

Insteadof analysingtheinfluenceof thecouplingstrength3 , herewe focus on the effect of varying the parameter� .

Knopf et al.: ForcedversusCoupledDynamics 9

0.12 0.14 0.16 0.18 0.2 0.220.6

0.8

1

1.2

parameter a

x, x

’

stable periodic orbitunstable periodic orbit

0.12 0.14 0.16 0.18 0.2 0.2210

−4

10−2

100

system with nonlinear coupling (without seasonal cycle)

<τ>

/ ar

b. u

nits

a.)

b.)

Fig. 10. Influenceof theparameter§ in thesystemwithoutseasonalcycle andwith nonlinearcouplingasdescribedthroughEqs. (28)and (29). (a) Relative meanlocking time ݹÞ5ß (seeEq. 27) independenceof the parameter§ ; (b) Bifurcation diagramof the 5-D andthe 8-D system. Here ã and ãX® show the samebifurcationdiagram.Theunfilledcyclestandsfor a torusbifurcation.

Again we have locking whenthesystemin on thesamesta-blelimit cyclefor � and � © , andtransitionsto on-off synchro-nizationwhenanUPOis reached,seeFig.10. By varyingthecouplingstrength3 in a rangefrom 0.0 to 0.65we discovera regionof artificial bistability for � © (notshown here)aswehaveseenbeforein thesystemwith linearcoupling(Fig. 6).

Soall featuresfoundin thelinearcoupledsystemcanalsobe discoveredin the systemwith nonlinearcoupling. Thisdemonstratesthat the type of coupling(linear or nonlinear)hasno decisive influenceon thelocking phenomenon.Quitethe contrary, asperiodicorbits appearfrequentlyin nonlin-earsystems,lockingmayoccurgenericallyin forcedsystemsand is much lesslikely in their fully coupledcounterparts.Thisstressesthatfor thelockingphenomenona linearstabil-ity analysisdoesnot hold.

6 Conclusion

In thispaperweconsidertheeffectof modulecouplingontheoverall dynamicaluncertaintyfor a paradigmaticnon-linearatmosphere-oceansystem.We identify phasespaceaswellastime-seriesfeatureswith respectto which a forcedmodelset-upqualitativelydiffersfrom its fully coupledcounterpart,for systematicreasons.On theonehand,in accordancewiththegeneralbelief, theforcedandthecoupledmodelversioncoincidein variousmainfeatures,in particularin termsof av-eragepredictiveskill andtheexistenceof thesamedominantperiodicorbit.

On theotherhand,in factwe identify a considerablefrac-tion in parameterspacefor whichthephasespacesof thetwomodelversionsfundamentallydiffer: thephasespaceof thefully coupledmodelis dominatedby a singlestableperiodic

orbit, while the forcedset-upallows for the existenceof anadditionalstableperiodicorbit. Sincethis kind of bistabil-ity is not foundin thefully coupledmodel,which theforcedset-upis supposedto emulate,we call it “artificial bistabil-ity”. Thesefinding seemsto contradictconventionalwis-dom in the EarthSystemmodellingcommunitystatingthata fully coupledmodel is more a complicatedentity than aforcedderivate,hencethecoupledversionis expectedto dis-playmorecomplicatedfeatures.However, in termsof replicasystems– a point of view we put forwardin this paper– wearguethat in fact theforcedset-upis themorecomplex one:its dynamicsaregeneratedin an eight-dimensional(ocean-dimensionplus two times the atmosphere-dimension)statespace,while that of the coupledversionresidesin a five-dimensionalspace.

Furthermore,thesystematicdiscrepanciesof thetwo mod-elling versionsextend into the time-domain. At least in-termittently, the forced set-updisplaysartificial predictiveskill. This is a direct consequenceof the replica-natureofthe forcedset-up:we perturbthe coordinatesof the replicaatmospherein orderto determinethepredictiveskill. As theperturbationcannotpropagateto the five-dimensionalsub-systemdriving thereplicaatmosphere,this five-dimensionalsub-systempotentiallyservesasa synchronizer. In casethereplicaatmosphere(“slave”) andthesynchronizer(“master”)run in the vicinity of an identicalperiodicorbit which pos-sessesa stablemanifold, theensemblewill tendto collapseonto the mastertrajectory. Hencewe identify the observedlocking phenomenonasanalmost-collapseto a periodicor-bit. If theorbit is stable,locking will continueforever. If theorbit is unstable,thetime-scaleof locking is setby thecom-petition of the stableandthe unstablemanifold of the peri-odic orbit, giving rise to intermittentlocking. We observe apowerlaw for thedistributionof lockingduration.Dueto thephenomenologicalanalogyto on-off intermittency, we callintermittentlocking “on-off synchronization”. In any case,locking implies artificial predictive skill, which we explainby theexistenceof a partiallyattractinginvariantset.

All featurescanbe observed in the original systemwithseasonalforcing andlinear coupling, in the systemwithoutseasonalforcingandfinally in thesystemwith nonlinearcou-pling. As we wereableto explain theseempiricalfindingswith a universaltheoreticalpatternthe ingredientsof whichjust draw on thenonlinearityof thesystem,we suggestthaton-off synchronizationandartificial bistability area generalcharacteristicof forcedsystemsratherthanbeingrestrictedto thisparticularmodelset-up.In thatsenseweadvisethatitis carefullycheckedwith alternativemodelversionswhetherintermittentpredictability and also bistability could not betheresultof a forced– insteadof the full-fledgedcoupled–set-up.

Acknowledgements. We would like to thankU. Feudel,J. Kurths,A. Pikovsky, andH. U. Vossfor thefruitful discussions.This workwasfundedby the BMBF-projects01LG0002(“Model Validationand IgnoranceDynamics”) and 07GCH02(“Scientific Office forIGBP-GAIM”).

10 Knopf et al.: ForcedversusCoupledDynamics

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