Forced versus coupled dynamics in Earth system modelling and prediction

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(1)Forced versus coupled dynamics in Earth system modelling and prediction B. Knopf, H. Held, H. J. Schellnhuber. To cite this version: B. Knopf, H. Held, H. J. Schellnhuber. Forced versus coupled dynamics in Earth system modelling and prediction. Nonlinear Processes in Geophysics, European Geosciences Union (EGU), 2005, 12 (2), pp.311-320. �hal-00302558�. HAL Id: hal-00302558 https://hal.archives-ouvertes.fr/hal-00302558 Submitted on 17 Feb 2005. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) Nonlinear Processes in Geophysics (2005) 12: 311–320 SRef-ID: 1607-7946/npg/2005-12-311 European Geosciences Union © 2005 Author(s). This work is licensed under a Creative Commons License.. Nonlinear Processes in Geophysics. Forced versus coupled dynamics in Earth system modelling and prediction B. Knopf1 , H. Held1 , and H. J. Schellnhuber1,2 1 Potsdam 2 Tyndall. Institute for Climate Impact Research (PIK), P.O. Box 601203,14412 Potsdam, Germany Centre for Climate Change Research, Norwich, UK. Received: 20 October 2004 – Revised: 10 February 2005 – Accepted: 11 February 2005 – Published: 17 February 2005. Abstract. We compare coupled nonlinear climate models and their simplified forced counterparts with respect to predictability and phase space topology. Various types of uncertainty plague climate change simulation, which is, in turn, a crucial element of Earth System modelling. Since the currently preferred strategy for simulating the climate system, or the Earth System at large, is the coupling of sub-system modules (representing, e.g. atmosphere, oceans, global vegetation), this paper explicitly addresses the errors and indeterminacies generated by the coupling procedure. The focus is on a comparison of forced dynamics as opposed to fully, i.e. intrinsically, coupled dynamics. The former represents a particular type of simulation, where the time behaviour of one complex systems component is prescribed by data or some other external information source. Such a simplifying technique is often employed in Earth System models in order to save computing resources, in particular when massive model inter-comparisons need to be carried out. Our contribution to the debate is based on the investigation of two representative model examples, namely (i) a low-dimensional coupled atmosphere-ocean simulator, and (ii) a replica-like simulator embracing corresponding components. Whereas in general the forced version (ii) is able to mimic its fully coupled counterpart (i), we show in this paper that for a considerable fraction of parameter- and state-space, the two approaches qualitatively differ. Here we take up a phenomenon concerning the predictability of coupled versus forced models that was reported earlier in this journal: the observation that the time series of the forced version display artificial predictive skill. We present an explanation in terms of nonlinear dynamical theory. In particular we observe an intermittent version of artificial predictive skill, which we call on-off synchronization, and trace it back to the appearance of unstable periodic orbits. We also find it to be governed by a scaling law that allows us to estimate the probability of artificial predictive skill. In addition to artificial preCorrespondence to: B. Knopf ([email protected]). dictability we observe artificial bistability for the forced version, which has not been reported so far. The results suggest that bistability and intermittent predictability, when found in a forced model set-up, should always be cross-validated with alternative coupling designs before being taken for granted.. 1. Introduction. In the climate modelling community it is common practice to establish a modular structure, consisting of ecosphere, biosphere, vegetation, ocean, atmosphere, etc., that builds up an Earth System Model (cf. the Climate System Model project CSM Boville and Gent, 1998). Some of these components are also modelled by external forcing, described from observed data. This is done e.g. in the Atmospheric Model Intercomparison Project AMIP (Gates et al., 1992), where an atmospheric general circulation model (AGCM) is constrained by realistic sea surface temperature and sea ice and the output is used for diagnostic research. Although this experiment is not meant to be used for climate change predictions, diagnostic subprojects have been established, though it is not quite clear to what extent the forced AGCM output is comparable to the system with complex ocean-atmosphere feedbacks. These coupled systems are investigated e.g. in the Coupled Model Intercomparison Project CMIP (Meehl et al., 2000; Covey et al., 2003a). The comparison of coupled ocean-atmosphere models with simulations using prescribed sea surface temperatures shows that there are indeed some important differences concerning e.g. temperatures near the pole and tropical precipitation (Covey et al., 2003b). Other publications mention a strong effect of the coupling on the midlatitude variability of the ocean-atmosphere system (Barsugli and Battisti, 1998) or on the decadal variability of oceanic variables in the North Pacific (Pierce et al., 2001). Hence, the subject of investigation is the effect of prescribing a module through data instead of implementing the dynamical module. This has already been investigated by Wittenberg and Anderson (1998), but here we will focus on po-.

(3) Knopf et al.: Forced versus Coupled Dynamics. 312. 1 0.5 0 −0.5 68. oupled bed in ed and ematiection rs and ng and ng the we deer law er will. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 2 Coupled and forced model. Forced run 1.5 1. x. comMs, has ut can al sysntinuasented certain om the those we do GCMs d also owing ass of n quite. highlight some fundamental differences between forcing and coupling. The role of unstable periodic orbits concerning the locking phenomenon is also investigated. In Sect. 5 we determine the statistics of the locking and deduce a power law scaling for the length of the locking phases. The paper will finish with the conclusions in Sect. 6.. Fully coupled run 1.5. x. ng and when he two onlinfferent re unstigat-. 0.5 0 −0.5 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. time / yr. Fig. of the coupled systemsystem (top) and theand forced Fig.1.1.Comparison Comparison offully the fully coupled (top) the system (bottom). A fully Acoupled run is taken reference trajecforced system (bottom). fully coupled run isastaken as reference tory. Additional to the to reference trajectory, in each there trajectory. Additional the reference trajectory, in subfigure each subfigure are runs slightly varying initialinitial conditions in atmospheric cothere arefrom runs from slightly varying conditions in atmospheric coordinates. upperfigure figurethe thecurves curvesare arefrom from aa fully coupled ordinates. In In thetheupper run. In In the the lower lower figure, figure, the the trajectories are forced by the ocean run. from the the reference reference trajectory. trajectory. In In this figure, we have reproduced a from major finding finding by by Wittenberg Wittenberg and and Anderson (1998). major. tential theWittenberg consistency and of forcing and This constraints model is preventing taken from Anderson coupling. With this paper we want to emphasize that when (1998). The atmosphere system model (Eq.(1)-(3)) is a forcing onechaotic moduleLorenz by another instead of coupling the two potentially system (Lorenz, 1984), that decomponents, one has to keep in mind that inherently scribes the midlatitude quasi-geostrophic flow. Whilenonlinrepear phenomena occur that represents lead to qualitatively different resents the time, can the variable the intensity of the features than expected. This type of analysis that we unwesterly wind current or the meridional temperatureare gradidertaking can be assigned to many other cases of investigatent. The variables and are the amplitudes of the sine and ing forced versus coupled model runs. wave, which transcosine components of a large travelling For our study we use a conceptual ports heat poleward. and are model forcingwhich, terms compared based on to more sophisticated climate models like GCMs, adthe average north-south temperature contrast andhas thethe earthvantage that the model itself as well as the output can transsea temperature contrast, where the seasonal variation of be analysed along the The linesocean of dynamical systems’ isparently expressed through the sine. system is a simtheory. This makes it easier to realise path continuation ple harmonic oscillator, with an oscillation frequency of of solutions parameter The zonal resultsasymmetries presented ininthis four years,inwhere p and space. q represent sea paper reveal the underlying mechanisms forocean certainand artificial surface temperature. The coupling between atmophenomena produced in forced systems. sphere proceeds through the interaction of these asymmetries From the knowledge of theeddy mechanism we then with the model atmosphere’s field (y and z). conclude that those shortfalls of forced models are generic. ThereWittenberg and Anderson (1998) carried out two different fore we do not so much suggest to perform path continuation sets of simulations. One set of simulations represents the for GCMs (although has successfully impleoutcome ofasthewell fully coupleditsystem with little been variation in mented also for GCMs, Dijkstra, 2000) but we understand the initial state vectors. In the other set the output of the the following sections a motivation cross-validate a cerocean from one specialasrun is used to to force the atmosphere. tain class of forced GCM results by alternative coupling deAgain this is undertaken for slightly perturbed initial condisigns in quite a classical manner. tions. So there are two ensembles: one from a fully coupled The and structure of the paper system is as follows: The no coupled system one from a forced that includes feedocean-atmosphere model we are analysing is described in back from the atmosphere to the ocean. Sect. 2 and the phenomenon of locking for coupled and As can be seen from Fig. 1, which was reproduced from forced trajectories is presented. In Sect. 3 the mathematiWittenberg and Anderson (1998), the forced ensemble is cal background for replica systems is introduced. In Sect. 4 more compact, but does not mirror the true solution. Furtherwe analyse the model in dependence on its parameters and more, Wittenberg and Anderson (1998) show that the statis-. $. . . . 2. . =. a fully s chote detion of of the -order (1) (2) (3) (4) (5). SCFEHG ,. caling ays.. B. Knopf et al.: Forced versus coupled dynamics. To investigate the difference between a forced and a fully coupled set-up, a coupled atmosphere-ocean system is chosen because the predictability of the Earth’s climate depends strongly on the variability induced by the interaction of these two components. As a very instructive example of the coupled atmosphere-ocean system, the following low-order model is examined: x˙ = −y 2 − z2 − ax + a(F + sin(2πγ t)) y˙ = xy − cy − bxz + G + αp. (1) (2). z˙ = xz − cz + bxy + αq p˙ = −ωq − βy q˙ = ωp − βz. (3) (4) (5). with a=0.125, F =3.5, c=0.5, b=4, α=β=0.1, G=0.25, γ =10/365.25, ω=2π γ /4, where γ is a scaling factor with one unit of system’s time referring to 10 days. This model is taken from Wittenberg and Anderson (1998). The atmosphere system model (Eqs. 1–3) is a potentially chaotic Lorenz system (Lorenz, 1984), that describes the midlatitude quasi-geostrophic flow. While t represents the time, the variable x represents the intensity of the westerly wind current or the meridional temperature gradient. The variables y and z are the amplitudes of the sine and cosine components of a large travelling wave, which transports heat poleward. F and G are forcing terms based on the average north-south temperature contrast and the earthsea temperature contrast, where the seasonal variation of F is expressed through the sine. The ocean system is a simple harmonic oscillator, with an oscillation frequency ω of four years, where p and q represent zonal asymmetries in sea surface temperature. The coupling between ocean and atmosphere proceeds through the interaction of these asymmetries with the model atmosphere’s eddy field (y and z). Wittenberg and Anderson (1998) carried out two different sets of simulations. One set of simulations represents the outcome of the fully coupled system with little variation in the initial state vectors. In the other set the output of the ocean from one special run is used to force the atmosphere. Again this is undertaken for slightly perturbed initial conditions. So there are two ensembles: one from a fully coupled system and one from a forced system that includes no feedback from the atmosphere to the ocean. As can be seen from Fig. 1, which was reproduced from Wittenberg and Anderson (1998), the forced ensemble is more compact, but does not mirror the true solution. Furthermore, Wittenberg and Anderson (1998) show that the statis-.

(4) upled Dynamics B. Knopf et al.: Forced versus coupled dynamics. 313. norm of error vector |x−x’|. tics of and the forced variability, like 2.5 ike spatial temporal dis-spatial and temporal distributions, are significantly different from those of coupled ferent from those of coupled variability. For modelling issues this means that a prescribed 2 forcing (e.g. prescribing the sea surface temperature) cannot es this means that a prescribed emulate the fully coupled system. The interesting effect of the temperature) forcing is that all trajectories sometimes lock on the true a surface cannot 1.5 solution for a short time and then separate again, so partial em. The interesting effect of can be observed. One synchronization, so called “locking” can conclude from this that on the one hand fully coupled 1 es sometimes lock on the true and forced systems do not show the same behaviour but on the other hand that inso trulypartial forced systems there may exist a hen separate again, 0.5 region in phase space, where predictability is very high. The ing can question be observed. One can to be followed is what mechanism is responsible for the locking phenomenon and what types of coupling show 0 e one hand fully coupled and 50 60 70 80 90 100 such behaviour. time / yr he same behaviour but on the 0 | between the fully couFig. 2. Norm of the error vector δx=|x−x. Mathematical systems3 there mayframework exist a rethe forced (a=0.09). x

(5) x Fig. 2. Normpled ofandthe errorrun;vector x between the edictability is very high. The In order to explain the locking phenomenon, a stability analcoupled and the forced run; (

(6) ). ysis of the system appears the most natural approach. In the mechanism is responsible for ematical form. With this formalization “reality” – prescribed tradition of Wittenberg and Anderson (1998) and Smith et al. through data – is being represented through a perfect model (1999),of one would expect that the local linear stability propwhat types coupling show scenario in the model-world. erties govern the observed phenomenon. Empirically, howIn Fig. 2 the norm of the error vector δx=|x−x0 | is plotever, we find that the explanation for lockingform. given in WitWith this formalization “reality” prescribed thro ted. Sometimes the two systems synchronize-but then sudtenberg and Anderson (1998) does not hold. We find that denlyrepresented the system shows through long-lasting bursts, where themodel two data - isinbeing a perfect scen locking shows no correlation to the trajectory’s residence trajectories seem to evolve independently. the “locking region” identified in Wittenberg and Anderson in the model-world. Generally, this type of coupling between two identical sys(1998) and, in particular, that locking persists an order of k tems be written as magnitude longer than the trajectory resides in the Inlocking Fig. 2 thecannorm of the error vector x x x region. Contrary to a local linear stability analysis, we will x˙ = F(x) x˙0 = F(x0 ) + K(x − x0 ), (11) plotted. relate the locking period to extended invariant manifolds, Sometimes the two systems synchronize but phenomenon, a stability analwhere K is the coupling function. emerging from the nonlinear dynamics of the system. This suddenly the system shows long-lasting bursts where the By transforming Eq. (11) to the transversal coordinates will allow us to introduce meaningful time-averaged characmost natural approach. In the 0 x⊥ =x−xto andevolve considering only small perturbations, so that teristics. To frame a discussion of potential nonlinear causes trajectories seem independently. derson (1998) Smith et al. of Pecora x≈x0 and F(x0 )≈F(x)+J(x)(x0 −x) the equation can be apof locking,and we follow the concepts and Carroll Generally, this type proximated by of coupling between two identical and Pecora et al. (1997). Different from Fig. 1, where the local(1990) linear stability propwe looked at a set of several forced trajectories, we investitems can be x˙written F(x0 ) − K(x⊥ ) ≈ J(x)x⊥ − K(x⊥ ), (12) ⊥ = F(x) −as gate here just the fully coupled run and one forced run. The nomenon. Empirically, how-. . %. %. forced system can be written as a so-called replica system. %,E. . where J(x) is the Jacobian matrix of F evaluated on the ion for (Pikovsky locking given in Witet al., 2001), where a replica of onexor more synchronization AK linearisation Fequax x Fmanifold. x x x of the function tions is made. Together with Eqs. (1)–(5) we have a replica K(x) around zero, where we assume that K(0)=0 and neglect does not hold. We find that system of the following form: higher order terms of x⊥ , leads to where K is the coupling function. o the trajectory’s residence in ˜ ⊥, x˙0 = −y 02 − z02 − ax 0 + a(F + sin(2π γ t)) (6) x˙ ⊥ ≈ (J(x) − K)x (13) 0 0 0 By transforming Eq. (11) to the transversal coordin in Wittenberg and y˙0 = x 0 y 0 − cy − bxAnderson z + G + αp (7) with 0

(7) x 0 z0 − cz0 +an bx 0 yorder + αq of x x(8) x and lockingz˙0 =persists dK

(8)

(9) considering only small perturbation 0 0 ˜ 0 ˙ K = (14) = −ωq − βy (9)

(10) x . dxF that x x and F x J x x x the equa x =0 jectory pqresides in the locking 0 0 ˙0 = ωp − βz , (10) In our case, where can be approximated by we have linear coupling, the matrix K˜ ear stability analysis, we will where the primed system x0 =(x 0 , y 0 , z0 , p 0 , q 0 )T is identical is original fullymanifolds, coupled system x=(x, y, z, p, q)T exxtendedto the invariant   0 0 0 0 0 cept for slightly different initial conditions and the substi J x x x Fx Fx0 0 0 α 0K K x  x dynamics ofvariables the system. This tuted p and q instead of p0 and q 0 , that emulate   ˜ = 0 0 0 0 α . K (15) forcing through prescribed data. In this system p0 and   ningful the time-averaged charac  0 0 0 0 0 0 J x is the Jacobian matrix of F evaluated on the q have no influence on the dynamics of thewhere other primed 0000 0 variables nonlinear and are only introduced to allow for a closed mathof potential causes chronization manifold. A linearisation of the function K ncepts of Pecora and Carroll around zero, where we assume that K 0 0 and neg . Different from Fig. 1, where. . . %. % % %. % %7 %. %. %. %. %<.

(11) Knopf et al.: Forced versus Coupled Dynamics. 314. the two trajectories are totally independent, reads. <τ> / arb. units. a.) 1. t∗ ≈. 1E−2 1E−4 1E−6 0.1 1.02. 0.15. 0.2. 0.25. 0.3. 0.35. 0.4. 0.45. 0.5. b.) stable periodic orbit unstable periodic orbit. 1. x. . This 1983), (1996) del inlargest Several et al., el none rol pa-. 0.98 0.96 0.1 c.). 0.15. 0.2. 0.25. 0.3. 0.35. 0.4. 0.45. 0.5. 1.02. stable periodic orbit unstable periodic orbit. 1. x′. nent is al synis lost. B. Knopf et al.: Forced versus coupled dynamics. 0.98. 1 L ln , λ δx(0). (16). where λ is the Lyapunov exponent, L denotes the characteristic length of the attractor and δx(0) the error that cannot be dissolved by a given accuracy (Argyris et al., 1995). Here t ∗ is found to be about 1.3 years. Nevertheless locking can be observed over much longer timescales, as can be deduced from Fig. 2. This demonstrates that in the period of locking, a non-average, non-standard situation is present. Below we will link it to phase-space structures of low measure, yet of a noticeable domain of attraction.. 0.96 0.94 0.1. 0.15. (16). racternnot be . Here ng can educed ocking, ow we yet of a. d to the as well for the domain nd only. 0.2. 0.25. 0.3. 0.35. 0.4. 0.45. 0.5. coupling strength α. Fig. 3. 3. Relative Relativemean meanlocking lockingtime time relation to the domiin in relation to the mostmost dominant nant invariant sets (periodic the system without seasonal invariant sets (periodic orbits)orbits) for thefor system without seasonal cycle

(12) (a) Relative ). (a) Relative (see cycle mean time locking time (with (with a=0.12). mean locking <τ > (seeEq. 17) ! . Forpoint everyinpoint Eq. 17) in dependence of the coupling in dependence of the coupling strength strength α. For every this in this the figure theconditions initial conditions for chosen x were randomly chosen randomly and figure initial for x were and the trathe trajectories were integrated overyears, 500 years, according 730500 jectories were integrated over 500 according to 730to 500 time time settled an attractor. forced trajecsteps,steps, after after they they settled downdown on anonattractor. The The forced trajectory was was started on the attractor with slightly tory started on the attractor with slightlyperturbed perturbedinitial initialcondiconditions, chosen from a gaussian gaussian distribution distribution with with aa standard standarddeviation deviation of 0.01. For every value of the coupling coupling strength strength α! the the integration integration was performed several times. times. As As aa mean mean length length of of zero zero cannot cannot be be depicted in a logarithmic plot, plot, we we added added an an offset offsetof of10 (b)BiBi −6 "$# ; ;(b) furcation diagram for the variable variable x% of of the the five five dimensional dimensional (5-D) (5-D) driving system; for the periodic periodic orbits, orbits, just just one one point point referring referring toto the maximum of the orbit is is plotted; plotted; (c) (c) Bifurcation Bifurcationdiagram diagramfor forthe the combined drive drive and and response responsesystem. system. AAfilled filled variable x% 0 of the 8-D combined circle symbol represents a saddle saddle node node bifurcation, bifurcation,an anunfilled unfilledcircle circle stands for a torus bifurcation, an an upward-pointing upward-pointing triangle triangledenotes denotesaa period doubling bifurcation and and aa downward-pointing downward-pointing triangle trianglesymsymbolizes a branch point.. 1.5. (17). is eraged e norm ories x ropper. ng time served.. To achieve complete synchronization, it is required that 1 for t→∞, x⊥ goes to zero. From the linearised Eq. (13) 0.5 expect that the two systems will synchronize if one would the transverse Lyapunov exponents, that are the Lyapunov exponents0 associated with Eq. (13), are all negative. This criterion was first proposed by Fujisaka and Yamada (1983), −0.5 but in contrast to this, e.g. Gauthier and Bienfang (1996) observe only incomplete synchronization in their model instead −1 of the proposed full synchronization, when the largest transverse Lyapunov exponent is smaller than zero. Several crite−1.5 0.1 0.2were developed 0.3 0.4 ria for synchronization (Blakely et al.,0.5 2000), coupling strength α but it was also shown there that for their model none of these criteria exactly predicts the range of the control parameter, Fig. 4. Bifurcation diagram for % in dependence of the coupling where full canseasonal be observed. ! insynchronization the system without cycle and

(13) . strength x’. ed and n order ely the easonal easonal n lockg. 3(a),. In our case the largest transverse Lyapunov exponent is positive with λ⊥ ≈0.084 but we also observe partial synchroThe fully The coupled of two totally independent nization. timesystem t ∗ afterconsists which all information is lost and systems x and x , where the coupling matrix K of Eq. (15). 4. Comparison of forced and coupled system. In this section we systematically compare the coupled to the forced system with respect to time-series properties as well as phase space topology. As a necessary condition for the forced system to emulate the coupled one in the time-domain we require that the forced system shows locking if and only if its coupled counterpart does. 4.1. System without seasonal cycle. An important structural difference between coupled and forced systems will be discussed in this section. In order to separate the two forcing effects in this model, namely the ocean forcing through the variables p and q and the seasonal forcing, the model is firstly investigated without the seasonal cycle. We analyse the dependence of the relative mean locking time <τ > on the coupling strength α, see Fig. 3a, where τ=. tlocking , T. (17). where T is the length of the whole time series and tlocking is the time, where locking can be observed. This is averaged over many locking periods. Locking is defined by the norm of the error vector δx=|x−x0 | of the two trajectories x and x0 being smaller than a critical threshold . For a propper choice of see below, here we took =0.01. A significant difference in the relative mean locking time for a fully coupled run and a forced run can be observed. The fully coupled system consists of two totally independent ˜ of Eq. (15) systems x and x0 , where the coupling matrix K is zero. The forced system is the 8-D combined drive and response system, consisting of Eqs. (1)–(5) and (6)–(8), the Eqs. (9) and (10) are neglected in this case as they have no influence on the system’s dynamics. Whereas in the fully coupled system the relative mean locking time <τ >, as a function of α, is always zero (not plotted in the diagram), in the forced system there are small parameter ranges, where the trajectories always show locking (for α∈[0.1, 0.177]), or where locking never appears, e.g. for α∈[0.277, 0.4]. Additionally, in the forced system there are also regions, where we observe intermittent synchronization as shown in Fig. 2, e.g. for α>0.4 – like in the original system which includes a.

(14) d r e l l ,. stands for a torus bifurcation, an upward-pointing triangle denotes a period doubling bifurcation and a downward-pointing triangle symbolizes a branch point. B. Knopf et al.: Forced versus coupled dynamics 1.5. 0.5. 0. −0.5. e .. coupling strength α∈. x. x0. locking?. [0.1, 0.177]. SPO. SPO. locking. [0.177, 0.277]. SPO. two SPOs. locking, when x and x 0 are on the same PO; else no locking. [0.277, 0.3]. SPO. same SPO and torus. locking, when x 0 is on the PO; no locking, when x 0 is on the torus. [0.3, 0.4]. SPO. (differing) SPO and torus. no locking. [0.4, 0.5]. UPO. UPOs. intermittent locking (on-off synchronization). −1. ). s d m x r. 315. Table 1. Overview over the different regions in parameter space. SPO stands for stable periodic orbit, UPO for unstable periodic orbit.. 1. x’. n y. −1.5 0.1. 0.2. 0.3 coupling strength α. 0.4. 0.5. Fig.Bifurcation 4. Bifurcation diagram for for x% 0 in of the Fig. 4. diagram independence dependence of coupling the coupling strength α in the system without seasonal cycle and a=0.12.. strength ! in the system without seasonal cycle and

(15) . seasonal cycle. This suggests that the seasonal forcing is not. The fully coupled system consists of two totally independent the main cause of the observed intermittent behaviour. the coupling matrix K ofofEq. systemsThe x presented and x , where result does not depend on the choice the (15) threshold , provided that is not too small. The relative mean locking time <τ > grows only slowly with the , so there would be only a slight shift for the value of <τ > in Fig. 3a. On the other hand it is important to choose a threshold that is not too small (Lai, 1996), because it then takes a long time until the trajectory falls below the threshold and therefore one would need very long runs to calculate a reliable value for the mean locking time <τ >. In the remaining part of this section, we empirically correlate time-series properties and phase-space topology in order to explain the locking phenomenon of the forced system. To get an impression of the phase space topology of the system in dependence of the parameter α, a bifurcation analysis is performed with the bifurcation analysis program AUTO (Doedel, 1981). In Fig. 3b the bifurcation diagram for the variable x of the fully coupled system – which simultaneuosly plays the role of the forced system’s master trajectory – is plotted, in Fig. 3c the same is done for the variable x 0 of the 8-D combined drive and response system. (Remember that for locking to occur, x and x 0 must coincide in the time series.) For α<0.177, both diagrams display the identical bifurcation diagram that simply consists of one stable periodic orbit. We propose that locking occurs in the forced system if both the master and the perturbed trajectory end up on the same periodic orbit. For α>0.177, the difference of the bifurcation diagrams is amazing: while for any α<0.4 a stable periodic orbit exists in both the coupled (Fig. 3b) and the forced system (Fig. 3c), this orbit either lives on a different branch (for α∈[0.3, 0.4]) or another stable periodic coexists (for α∈[0.177, 0.277]), born in a saddle node bifurcation at α=0.177. In the latter situation (α∈[0.177, 0.277]), full synchronization can be observed if the perturbed trajectory starts in the domain of. attraction of that periodic orbit which mimics the orbit of the master system (upper branch); otherwise the perturbed trajectory gets trapped by the concurring (lower branch) periodic orbit and locking never occurs. In the former case (for α∈[0.3, 0.4]), the attracting orbits differ, hence synchronization should be impossible. This is in fact consistent with the relative mean locking time observed in Fig. 3a. In Fig. 4 another bifurcation diagram for x 0 is obtained by numerical integration to capture the movement on a torus, that cannot be depicted in the other diagram. In fact, the torus bifurcation is found in the upper Fig. 3c for α=0.277, but the torus cannot be followed with this method. For each parameter value α we let the system settle down to an attractor and then plotted the forced variable x 0 , when the trajectory crosses the y 0 axis at 0.5. From left to right we see again the stable periodic orbit and at α=0.177 the generation of a second orbit. From these stable periodic orbits a quasiperiodic motion on a torus emerges at α=0.277. The dynamics on the torus is sometimes adjourned by the stable periodic orbit that can be found in Fig. 3c for α<0.3. The torus disappears at α=0.4 through a period doubling bifurcation and passes into chaotic motion. For the driving system x no quasiperiodic dynamics are found, so that for α∈[0.277, 0.3], before the branch point at α=0.3 emerges, the system shows locking when x0 is on the stable limit cycle or shows no locking when x0 is on the torus. Which state will be adopted depends on the initial conditions. For α larger than the bifurcation value α=0.3, there is scarcely any locking because the forced system is on the torus and the fully coupled system on the stable periodic orbit. An overview of the different classifications in dependence on the parameter space is given in Table 1..

(16) Knopf et al.: Forced versus Coupled Dynamics. 316. B. Knopf et al.: Forced versus coupled dynamics. tion, hence no synchronizing drive is present and no locking will occur. This emphasizes that full synchronization can in this case be explained by a stable periodic orbit (or a stable equilib1 rium point) that drives the forced system to the same dynamic behaviour. After the stability has been lost in a bifurcation point, the 0.5 unstable periodic orbit embedded into the attractor influences the system so that intermittently locking occurs even in a re0 gion, where the transversal Lyapunov exponent is positive and – naively, “on average” – no synchronization is expected. Therefore the concept of (unstable) periodic orbits seems to −0.5 be crucial for the locking phenomenon and the loss of synchronization can be traced back to the transition from stable −1 to unstable periodic orbits. Ott and Sommerer (1994) call −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 x this a “nonhysteretic” blowout bifurcation, where for a<ac the system is on an attractor and for a>ac on-off intermit5. Phase space thefull full attracting attracting set region) Fig. 5.Fig.Phase space of ofthe set(non-locking (non-locking region)tency can be detected. The role of unstable periodic orbits and of two exemplary locking regions (1 and 2), whereas here reand ofgion tworefers exemplary locking regions (1 and 2), whereas here re-(UPOs) for synchronization is also pointed out by Pazó et al. to a period in time. For comparison the unstable periodic (2003) and by Pikovsky et al. (1997). gion refers to a period in For comparison orbit for this parametertime. constellation is plotted. the unstable periodic The interpretation with regard to UPOs can be stressed orbit for this parameter constellation is plotted. through Fig. 5, where we analysed the phase space of the locking regions in comparison to the full attracting set. It can So far we can conclude that if the system is on the same clearly be seen that the locking region is in good coincidence stable periodic orbit for x and x0 , we get full synchronizaand the trajectory becomes repelled, reminiscent of intermit-with the unstable periodic orbit, whereas the non-locking retion. On the other hand, when x runs on a periodic orbit tency.notHence, we suggest that the locking can begion covers a much larger part of the whole phase space. concurring to the replicate of intermittent x’s periodic orbit, locking tracedcannot backbetoobserved. a co-existence of two identical unstable That means there can be intrinsic ob-peri- Just beyond a bifurcation point, where a periodic orbit has stacles that system performs as one the fully odic orbits, one ainforced the fully coupled, and in thecoupled combinedbecome unstable, the Monodromy matrix of the related mapping will display a long timescale on the unstable manifold, system. For modelling issues this is a crucial outcome, as for 8D fully coupled and replica system. and generically shorter time-scales for the remaining stable more sophisticated models the calculation of the state space In summary, there are two major differences in the coupledmanifold. Therefore, the unstable periodic orbit still has a of the forced and the coupled model is very costly so there fair chance to attract on the stable manifold and synchronize and the forced dynamics: the forced is hardly anysystem’s way to decide if forcingfirst, is suitable, above system all the trajectory. This will reveal itself as locking. After a while, displays a richer bifurcation diagram coexisting because normally the fully coupled modelincluding is not known. For users of forced models is therefore of central importance to stable invariant manifolds where the coupled system wouldthe long timescale on the unstable manifold manifests itself, note that synchronization might if beboth artefacts the andand the trajectory becomes repelled, reminiscent of intermitnot (“artificial bistability”),effects and even thefrom master tency. Hence, we suggest that the intermittent locking can be forced set-up. the perturbed trajectory would end up on the same periodictraced back to a co-existence of two identical unstable periIn order to obtain a full understanding of the fundamenorbit,talthediscrepancies coupled system lacks a synchronizer, hencesyscannotodic orbits, one in the fully coupled, and one in the combined between the forced and fully coupled display At leastwethe second effect is independent tem,locking. in the following will focus on the phenomenon of of8D fully coupled and replica system. intermittent synchronization that arises for α>0.4, where no the complexity of the model and should occur in GCMs as In summary, there are two major differences in the coupled and the forced system’s dynamics: first, the forced system well. stable periodic orbit is detected. displays a richer bifurcation diagram including coexisting One potential practical consequence could be within stable invariant manifolds, where the coupled system would 4.2 The role of (un)stable periodic orbits weather forecast: if locking occurred already at the begin-not (“artificial bistability”), and even if both the master and observationsininthe the “forecast previous section we can ning From of thethedynamics, period”, theconensem-the perturbed trajectory would end up on the same periodic clude that if the system is in a region, where it is on a stable ble error along the trajectory could be vastly underestimatedorbit, the coupled system lacks a synchronizer, hence cannot periodic orbit, the forced system shows locking all the time, display locking. At least the second effect is independent of in forced ensembles. This could imply that e.g., a hurricanethe complexity of the model and should occur in GCMs as provided that we are in a region in parameter space, where x couldand affect a region on its way much earlierThe than anticipated.well. x0 show the same bifurcation diagram. argumentation reads as follows: as the orbit of the 8-D system is stable, that One potential practical consequence could be within Furthermore, artificial bistability could bring about the trajectories of the driving system and the driven system climate policy becomes overly conservative as society triesweather forecast: if locking occurred already at the beginend upcrossing on the same periodic orbit, but they could have afromning of the dynamics, in the “forecast period”, the ensemto prevent a threshold which is just anstill artefact phase shift. If there were a phase shift, then this shift would ble error along the trajectory could be vastly underestimated the forced also bemodel seen inset-up. the ocean coordinates. But this is excluded in forced ensembles. This could imply that e.g. a hurricane through the replica approach, where only the atmosphere cocould affect a region on its way much earlier than anticipated. 4.3 ordinates System are with seasonal cycleto this, for the coupled sysvaried. Contrary Furthermore, artificial bistability could bring about that tem the ocean coordinates are also subject to the perturbaclimate policy becomes overly conservative as society tries full attracting set (non−locking) unstable periodic orbit locking region 1 locking region 2. y. 1.5. The assertion of the role of stable and unstable periodic orbits can also be endorsed by Fig. 6, where the system with seasonal cycle in dependence on the coupling strength is. 3.

(17) Knopf Forced versus Coupled Dynamics Knopf et et al.:al.: Forced versus Coupled Dynamics B. Knopf et al.: Forced versus coupled dynamics System seasonal cycle System withwith seasonal cycle. <τ> / arb. units <τ> / arb. units. forced system forced system coupled system fullyfully coupled system. a.) system with nonlinear coupling (without seasonal cycle) a.) 0 system with nonlinear coupling (without seasonal cycle). 0 10 10. <τ> / arb. units. 0a.) a.). <τ> / arb. units. 0. 10 10. 3177 7. −2. −2. −4. −4. 10 10. −5 −5. 10 10. 0.08 0.10.1 0.12 0.120.14 0.140.16 0.160.180.18 0.2 0.2 0.220.220.240.24 0.08. 10 10 0.120.12. 0.160.16. 0.180.18. 0.2 0.2. 0.220.22. stable periodic stable periodic orbitorbit unstable periodic unstable periodic orbitorbit. 1.2 1.2. 1 1 0.95 0.95 0.08 0.08 0.10.1 0.12 0.120.14 0.140.16 0.160.180.18 0.2 0.2 0.220.220.240.24 coupling strength α α coupling strength. 0.160.16 0.180.18 parameter a a parameter. 0.2 0.2. 0.220.22. Fig. 6.Variation Variation ofthe the coupling α! in system with seaFig. 6. 6. Variation of of the coupling strength ! in the system with seaFig. coupling strength inthe the system with seasonal cycle with Relative mean locking time <τ

(18) a=0.27. > (see sonal cycle with (a) Relative mean locking time

(19) . (a) sonal cycle with . (a) Relative mean locking time Eq. 17) onon theon coupling strength α. Again added (see Eq.Eq. 17)in independence dependence the coupling strength we we ! . ! Again (see 17) in dependence the coupling strength . we Again −6 to depict a mean length of zero; (b) Bifurcation an offset of 10 added an an offset of of a mean length of zero; (b)(b) Bifur " # " to# depict added offset to depict a mean length of zero; Bifurdiagram of the system. As forAsthis constellation x and x 0 cation diagram of of thethe system. forparameter thisthis parameter constellation cation diagram system. As for parameter constellation same bifurcation diagram,diagram, just one just bifurcation diagram is % and % % the show thethe same bifurcation oneone bifurcation show % show and same bifurcation diagram, just bifurcation plotted. diagram is plotted. diagram is plotted.. 1 1 x, x’. x x. 1.05 1.05. 0.8 0.8 0.6 0.6 0.120.12. O( 5*5.*. +,+,

(20) -,

(21) )-, 8) 8 22 3)4 3)W4 W O( ( ) 6.

(22) 7 + 1. / 5 6 ; 3 : () 6

(23) +7 1 -/563;: E E 4 4 : :. 0.140.14. Fig. 7.7.Influence of parameter system without seasonal Fig. ofthe thethe parameter a in the system without seasonal Fig. 7.Influence Influence of parameter in the system without seasonal cycle and with nonlinear coupling as described through Eqs. (18) cycle and with nonlinear coupling as through Eqs.Eqs. (18)(18) cycle and with nonlinear coupling as described through and (19). (a) (a) Relative mean locking time <τ (see 17)Eq. in de> (see Eq. 17) in in and (19). Relative mean locking time and (19). Relative mean locking time Eq. (see 17) pendence ofofthe a; (b) diagram of of thethe 5-D dependence the parameter ; (b) Bifurcation diagram 5- 5 0;Bifurcation dependence ofparameter the parameter (b) Bifurcation diagram of the system. Here xHere and x% and show same bifurcation dia% and % show Dand and the8-D 8-D system. Here the the same bifurcation % the show D the and the 8-D system. same bifurcation gram. The unfilled cycle stands for for a torus diagram. The unfilled cycle stands a torus bifurcation. diagram. The unfilled cycle stands for abifurcation. torus bifurcation.. of analysing thethe influence of the coupling strength Instead of analysing influence of the coupling strength 3 , Instead 3 here we focus on the effect of varying the parameter , here we focus on the effect of varying the parameter. . The system analysed so far is a system with linear coupling. Again wewe have fullfull locking when thethe system in on thethe same Again have locking when system in on same As this is acycle veryfor special case of coupling thattoisintermittent not often stable limit and , and transitions stable limit cycle for and , and transitions to intermittent used inwhen truly an coupled models, we see analyse a7.system without locking UPO is reached, Fig.Fig. By varying thethe locking when an UPO is reached, see 7. By varying a seasonal cycle and with a nonlinear coupling to determine 3 3 ainrange from 0.00.0 to 0.65 we we discover coupling strength a range from to 0.65 discover coupling strength in the influence of the type of coupling. coupling has the (not The(not a region of artificial bistability forfor shown here) as we a region of artificial bistability shown here) as we 4.4. to in prevent crossing asystem threshold which justmanifold an artefact valid this case. The needs a stable to from bevalid in this case. The system needs aisstable manifold to bethe forced model set-up. come fully synchronized. ForFor modelling issues, thatthat means come fully synchronized. modelling issues, means that it does notnot depend onon thethe strength of of coupling butbut on on thethe that it does depend strength coupling 4.3 System with seasonal cycle state in in the phase space if forcing cancan substitute coupling. state the phase space if forcing substitute coupling. AA significant difference to to thethe situation without seasonal significant difference situation without seasonal The assertion of the role of coupled stable andsystem periodic orforcing is is that here thethe fully shows lockforcing that here fully coupled unstable system shows lockbits can also be endorsed by Fig. 6, where the system with inging when thethe system is on a stable limit cycle, seesee Fig. 6(a), when system is on a stable limit cycle, Fig. 6(a), seasonal cycle in dependence on theis coupling strength reα is where the relative mean locking time 1 in the locking where the relative mean locking time is 1 in the locking reanalysed. As before, the bifurcation diagram and thesynrelagions, which means there is always locking. Intermittent gions, which means there is always locking. Intermittent syntive mean locking time are plotted. Again we can detect inchronization cancan also sometimes be be observed butbut lessless often chronization also sometimes observed often termittent synchronization and it can be seen that there is a than in in thethe forced system (Fig. 6(a)). This is due to to thethe seathan forced system 6(a)). This is due seatransition from locking to (Fig. intermittent locking. sonal forcing, that determines thethe frequency of of thethe periodic sonal forcing, that determines frequency periodic Figure 6 makes it clear that the presumption that with orbit, so so thisthis locking bears onon an an external forcing andand notnot orbit, locking external forcing stronger coupling the bears two systems will synchronize, is not onon thethe intrinsic phenomenon of of locking through prescribed intrinsic phenomenon locking through prescribed valid in this case. The system needs a stable manifold to beforcing byby variables. ButBut as as the seasonal cycle is also a means kind forcing variables. the seasonal cycle also a kind come fully synchronized. For modelling issues,isthat of of forcing, the chance of locking through an additional “synforcing, through an additional that it doesthe notchance dependofonlocking the strength of coupling but on“synthe chronizer” increases. chronizer” increases. state in the phase space if forcing can substitute coupling. A significant difference to the situation without seasonal 4.44.4 Influence of thethe type of of coupling Influence type couplingsystem shows locking forcing is that of here the fully coupled when theanalysed system isso onfar a stable limit cycle, see Fig.coupling. 6a, where The system is aissystem with linear The system analysed so far a system with linear coupling. the relative mean locking time is 1 in the locking regions, AsAs this is is a very special case of of coupling thatthat is not often this a very special case coupling is not often which means there is always locking. Intermittent synchroused in in truly coupled models, wewe analyse a system without used truly coupled models, analyse a system without nization can also sometimes be observed but less often than a seasonal cycle andand with a nonlinear coupling to to determine a in seasonal cycle nonlinear the forced systemwith (Fig.a 6a). This iscoupling due to the determine seasonal thethe influence of the type of coupling. The coupling has thethe influence the type the of coupling. forcing, that of determines frequency The of thecoupling periodichas orbit, following form: following form:bears on an external forcing and not on the inso this locking through prescribed forcing trinsic phenomenon of locking by (18) (18) variables. But as the seasonal cycle is also a kind of forcing, (19) (19) the chance of locking through an additional “synchronizer” increases. instead of of Eqs. (2)(2) andand (3).(3). AsAs andand vary approximately instead Eqs. vary approximately between -1 and 1, the introduced term bears strong nonlinbetween -1 and 1, the introduced term bears strong nonlinearity. earity.. stable periodic stable periodic orbitorbit unstable periodic unstable periodic orbitorbit. x, x’. 1.11.1. 0.140.14. b.) b.). b.) b.). Influence of the type of coupling. following form: have seen before in the system with linear coupling (Fig. 3). 3). have seen before in the system with linear coupling (Fig. 3 linear So all features found in the linear coupled system can also So all features found in the coupled system can y˙ = xy − cy − bxz + G + αp x (18) also be be discovered in the system with nonlinear coupling. This 3 discovered in the system z˙ = xz − cz + bxy + αq x. with nonlinear coupling. (19)This demonstrates that the type of coupling (linear or nonlinear) demonstrates that the type of coupling (linear or nonlinear) has no no decisive on on thethe Quite has locking phenomenon. Quite instead ofdecisive Eqs. influence (2)influence and (3). As plocking and q phenomenon. vary approximately the contrary, as periodic orbits appear frequently in nonlinthe contrary, orbits appear frequently in nonlinbetween −1 andas1,periodic the introduced term bears strong nonlinear systems, locking may occur generically in forced systems ear systems, locking may occur generically in forced systems earity. andand is much likely in influence their fully counterparts. is much less likely in their fully counterparts. Instead of less analysing the of coupled thecoupled coupling strength This stresses thatthat for the locking phenomenon linear stabilα,This here we focus onfor thethe effect of varying theaparameter a. stresses locking phenomenon a linear stability analysis does notnot hold. Again we have full locking ity analysis does hold.when the system in on the same stable limit cycle for x and x 0 , and transitions to intermittent locking when an UPO is reached, see Fig. 7. By varying the 5coupling Synchronization 5On-off On-off Synchronization strength α in a range from 0.0 to 0.65 we discover a region of artificial bistability for x 0 (not shown here) as we The alternation between regions, where thecoupling error between thethe The alternation between regions, where the error between have seen before in the system with linear (Fig. 3). forced andfeatures coupled trajectory nearly zero, andand between re- reforced and coupled trajectory is nearly zero, between So all found in the is linear coupled system can also gions with large as shown Fig. 2, resemble those of of with large bursts as shown in Fig. 2,coupling. resemble those begions discovered inbursts the system withinnonlinear This on-off intermittency. as introduced by by demonstrates that theOn-off type ofintermittency, coupling (linear or on-off intermittency. On-off intermittency, as nonlinear) introduced Platt et al. (1993), refers toonatothe situation where thethe variables has no decisive influence locking phenomenon. Quiteof of Platt et al. (1993), refers a situation where variables athe chaotic dynamical system exhibit two distinct states where contrary, as periodic orbits appear frequently in nonlina chaotic dynamical system exhibit two distinct states where atear the “off” state the system is nearly constant on an invariant systems, locking may occur generically in forced systems at the “off” state the system is nearly constant on an invariant manifold, andand at the “on” large bursts from these lamiand is much less instate their fully coupled counterparts. manifold, atlikely the “on” state large bursts from these laminar phases occur. The frequency of bursts is controlled by by a a This stresses that for the locking phenomenon a linear stabilnar phases occur. The frequency of bursts is controlled itycharacteristic analysis does not hold. characteristic parameter of the system and approaches zero, parameter of the system and approaches zero, when thethe so-called blowout bifurcation (Ott(Ott andand Sommerer, when so-called blowout bifurcation Sommerer, a critical value assign 1994) is reached as as attains attains a critical value . To . To assign 1994) is reached. 4 4. 4 4. 4 4.

(24) 8. Knopf et al.: Forced versus Coupled Dyna. 318. B. Knopf et al.: Forced versus coupled dynamics. predicted because one be approaches the Thisthe scaling law withdependence, the same exponent can also approvedscale for our case of on-off synchronization. The probaof the simulation timestep. For the long locking p slope: −3/2 0 bility distribution p(τ ) for the length of the locking region the distribution for curve A falls off exponentially, wh τ for the parameter settings given in Fig. 1 (with a=0.125) the two regions are connected by a shoulder, which is −2 is plotted as curve A in Fig. 8. As the −3/2 power-law disB seen in other systems displaying on-off intermittency ( tribution is deduced theoretically exactly only for the critical −4 al.,additionally 1993)). We assume that these imperfections fro point aet plotted the probability distribution c , we for a=0.075 inscaling curve B.law This is choice of the parameter exact in our case due tois very the distance o A −6 close toparameter the bifurcationfrom pointthe from a stable to an unstable bifurcation point. periodic orbit at ac =0.073, that was determined via bifurcation In this section we have shown that the scaling law fo −8 analysis. duration the laminar phases in systems with on-off As both curvesofindicate, when the parameter a is close −10 holds a system with on-off to the mittency bifurctationalso point the for probability distribution obeysynchroniz the predicted scalingbe lawextended and differstofrom the exact relation and could continuous systems with a −12 −6 −4 −2 0 2 4 when the parameter is further away from the critical value. ing system that is not random but chaotic. This mean log(τ) For the very short locking phases both curves deviate from the power law scaling is more universal than proposed the predicted dependence, because one approaches the time Fig. 8. Probability distribution of the length of the locking region was introduced. It For canthe also interpreted Fig. 8. Probability distribution of the length of the locking regionscale ofitthe simulation timestep. longbe locking phases in this re for more than 14 000 locking phases. Curve A is for a parameter for more than 14,000 locking phases. Curve A is for a parameterthe distribution that the for underlying mechanisms of on-off intermittenc curve A falls off exponentially, whereas setting as given following Eq. 5 with a=0.125, for curve B the pasettingrameter as given following Eq. 5 with , for curve B the

(25) the two regions are connected by a shoulder, which is also on-off synchronization are analogous. Intermittency is a is changed to 0.075, being closer to the bifurcation. The other systems displaying on-off intermittency (Platt parameter is changed 0.075, beingforcloser to the bifurcation.seen intraced dotted curve indicates ato −3/2 power-law comparison. back to the “almost existence” of a stable period et al., 1993). We assume that these imperfections from the The dotted curve indicates a -3/2 power-law for comparison. bit, and in a similar way we discovered stable and uns exact scaling law is in our case due to the distance of the periodic as potential parameter a fromorbits the bifurcation point. causes for locking. 5 On-off synchronization The knowledge of that the the above power In this section we have shown scaling law forlaw the does not the phenomenon on-offregions, intermittency to our objecttheof in-duration The alternationof between where the error between of thethe laminar phases innature systemsofwith on-off but inter-will also he stress underlying locking terest,forced intermittency not seenisas laminar interrupted and coupledistrajectory nearly zero, phases and between remittency also holds fortoa system withthe on-off synchronization applications estimate relative importance of the gions withbursts large bursts as shown in Fig. 2, resemble those of andand could be extended to continuous systems with a drivby turbulent but as locking of the fully coupled ing phenomenon. on-off intermittency. On-off intermittency, as introduced by ing system that is not random but chaotic. This means that the forced trajectories (off-state) interrupted by non-locking Platt et al. (1993), refers to a situation, where the variables of (on-state), which we will call on-off synchronization. Thethe power law scaling is more universal than proposed when a chaotic dynamical system exhibit two distinct states, where it was introduced. It can also be interpreted in this regard, transfer from intermittency to constant on-off synchronization at the “off”on-off state the system is nearly on an invariConclusion that the6underlying mechanisms of on-off intermittency and becomes clearer when between andthese x is un-on-off synchronization are analogous. Intermittency is often ant manifold, and at the the difference “on” state large bursts xfrom laminar The frequency of bursts is controlled traced back to the “almost existence”the of aeffect stable of periodic or- coupling o derstood as aphases new occur. variable. In this paper we consider module by a characteristic parameter p of the system and approaches bit, and in a similar way we discovered stable and unstable Systems that generate on-off intermittency show characoverall dynamical uncertainty for a paradigmatic nonzero, when the so-called blowout bifurcation (Ott and Somperiodic orbits as potential causes for locking. teristic scaling laws for the intermittent phases (Heagy et al., system. merer, 1994) is reached as p attains a critical value pc . To The atmosphere-ocean knowledge of the above powerWe law identify does not phase only space as 1994;assign Lai, the 1996), the distribution of the amplitudes of thestress the phenomenon of on-off intermittency to our object as time-series features with respect to which underlying nature of locking but will also help in a forced m of interest, intermittency is not seen as laminar phases inbursts and for the power spectrum of the trajectories (see ref-applications estimate the relative of the coupled lockset-uptoqualitatively differsimportance from its fully counte terrupted but as locking of the fully cou- in-ing phenomenon. erences cited by byturbulent John etbursts al. (2002)). Heagy et al. (1994) for systematic reasons. On the one hand, in accordance pled and the forced trajectories (off-state) interrupted by nonvestigate a certain class of driven systems, that consists of a the general belief, the forced and the coupled model ve locking (on-state), which we will call “on-off synchronizadiscrete map and a random driving variable with a smooth coincide in various main features, in particular in terms tion”. The transfer from on-off intermittency to on-off syn6 Conclusions density. They show that clearer for thewhen probability distribution of the erage predictive skill and the existence of the same dom chronization becomes the difference between x we consider is understood a new variable. a power law holds withIn this paper lengthandofx0the laminar as phases periodic orbit. the effect of module coupling on the Systems that generate overall dynamical uncertainty for a paradigmatic non-linear on-off intermittency show charac, where is the length of the laminar phases On the other hand, in fact we identify a considerable atmosphere-ocean system. We identify phase space as well teristic scaling laws for the intermittent phases (Heagy et al., and the scaling exponent that attains a universal value of tion in parameter whicha the phase spaces of th as time-series features with space respectfor to which forced model 1994; Lai, 1996), the distribution of the amplitudes of the 3/2 inbursts the vicinity of the threshold for on-off intermittency. model versions differ: the phase space o set-up qualitatively differsfundamentally from its fully coupled counterpart, and for the power spectrum of the trajectories (see refThis scaling withetthe can alsoinbe ap-for systematic reasons. model On the one hand, in accordance with stable per erences citedlaw by John al., same 2002). exponent Heagy et al. (1994) fully coupled is dominated by a single the general belief, the forced and the coupled model version vestigate a certain class of driven systems, that consists of a proved for our case of on-off synchronization. The probaorbit, while the forced set-up allows for the existence in various main features, in particular in terms of avmap and a random a smooth bilitydiscrete distribution for thedriving lengthvariable of the with locking region coincide additional stable periodic orbit. Since this kind of bis erage predictive skill and the existence of the same dominant density. They show that for the probability distribution of the for the parameter settings given in Fig. 1 (with )periodicityorbit. is not found in the fully coupled model, which the f length of the laminar phases p(T ) a power law holds with is plotted as curve A in Fig. 8. As the -3/2 power-law dis−γ set-up supposed toidentify emulate, we call itfrac“artificial bis p(T )∼T , where T is the length of the laminar phases and On the otheris hand, in fact we a considerable tribution deduced theoretically only forofthe These to spaces contradict conventional tion in ity”. parameter spacefinding for whichseems the phase of the two γ theisscaling exponent that attainsexactly a universal value 3/2critical in model versions fundamentally differ: the phase space of the the vicinity of the threshold for on-off intermittency. point , we additionally plotted the probability distribution dom in the Earth System modelling community stating log p(τ). 2. 4; %. . 4 %. #. 4 %. CFE C[J for . B CFEHGI J. in curve B. This choice of the parameter is very close to the bifurcation point from a stable to an unstable periodic orbit at , that was determined via. 1 CFE C"L. a fully coupled model is more a complicated entity t forced derivate, hence the coupled version is expected t play more complicated features. However, in terms of re.

(26) B. Knopf et al.: Forced versus coupled dynamics fully coupled model is dominated by a single stable periodic orbit, while the forced set-up allows for the existence of an additional stable periodic orbit. Since this kind of bistability is not found in the fully coupled model, which the forced set-up is supposed to emulate, we call it “artificial bistability”. These finding seems to contradict conventional wisdom in the Earth System modelling community stating that a fully coupled model is more a complicated entity than a forced derivate, hence the coupled version is expected to display more complicated features. However, in terms of replica systems – a point of view we put forward in this paper – we argue that in fact the forced set-up is the more complex one: its dynamics are generated in an eight-dimensional (oceandimension plus two times the atmosphere-dimension) state space, while that of the coupled version resides in a fivedimensional space. Furthermore, the systematic discrepancies of the two modelling versions extend into the time-domain. At least intermittently, the forced set-up displays artificial predictive skill. This is a direct consequence of the replica-nature of the forced set-up: we perturb the coordinates of the replica atmosphere in order to determine the predictive skill. As the perturbation cannot propagate to the five-dimensional subsystem driving the replica atmosphere, this five-dimensional sub-system potentially serves as a synchronizer. In case the replica atmosphere (“slave”) and the synchronizer (“master”) run in the vicinity of an identical periodic orbit which possesses a stable manifold, the ensemble will tend to collapse onto the master trajectory. Hence we identify the observed locking phenomenon as an almost-collapse to a periodic orbit. If the orbit is stable, locking will continue forever. If the orbit is unstable, the time-scale of locking is set by the competition of the stable and the unstable manifold of the periodic orbit, giving rise to intermittent locking. We observe a power law for the distribution of locking duration. Due to the phenomenological analogy to on-off intermittency, we call intermittent locking “on-off synchronization”. In any case, locking implies artificial predictive skill, which we explain by the existence of a partially attracting invariant set. For weather forecast this could imply that with a forced set-up, the ensemble error on the particular time, when a – potentially extreme – weather pattern will hit a certain region, can become extremely underestimated; hence an endangered region may prepare too lately. All analysed features can be observed in the original system with seasonal forcing and linear coupling, in the system without seasonal forcing and finally in the system with nonlinear coupling. As we were able to explain these empirical findings with a universal theoretical pattern the ingredients of which just draw on the nonlinearity of the system, we suggest that on-off synchronization and artificial bistability are a general characteristic of forced systems rather than being restricted to this particular model set-up. While we expect that bifurcation analysis will be too demanding as a standard procedure for GCMs over the next years, nevertheless we advise that at least it is carefully checked with alternative model versions whether intermittent predictability and also bistability. 319 could not be the result of a forced – instead of the full-fledged coupled – set-up. Acknowledgements. We would like to thank U. Feudel, J. Kurths, A. Pikovsky, and H. U. Voss for the fruitful discussions. This work was funded by the BMBF-projects 01LG0002 (“Model Validation and Ignorance Dynamics”) and 07GCH02 (“Scientific Office for IGBP-GAIM”). B. Knopf acknowledges support from the Rosa Luxemburg Stiftung. Edited by: G. Zöller Reviewed by: two referees. References Argyris, J. H., Faust, G., and Haase, M.: Die Erforschung des Chaos, Vieweg, 1995. Barsugli, J. J. and Battisti, D. S.: The Basic Effects of AtmosphereOcean Thermal Coupling on Midlatitude Variability, Journal of Atmospheric Sciences, 55, 477–493, 1998. Blakely, J. N., Gauthier, D., Johnson, G., Carroll, T., and Pecora, L.: Experimental investigation of high-quality synchronization of coupled oscillators, Chaos, 10, 738–744, 2000. Boville, B. A. and Gent, P. R.: The NCAR Climate System Model, Version One, Journal of Climate, 11, 1115–1130, 1998. Covey, C. K., AchutaRao, K. M., Cubasch, U., Jones, P., Lambert, S., Mann, M., Phillips, T., and Taylor, K.: An overview of results from the Coupled Model Intercomparison Project, Global and Planetary Change, 37, 103–133, 2003a. Covey, C. K., AchutaRao, K. M., Gleckler, P., Phillips, T., Taylor, K., and Wehner, M.: Coupled ocean-atmosphere climate simulations compared with simulations using prescribed sea surface temperature: Effect of a ”perfect ocean”, Global and Planetary Change, 41, 1–14, 2003b. Dijkstra, H. A.: Nonlinear Physical Oceanography, Kluwer Academic Publishers, Dordrecht, 2000. Doedel, E. J.: AUTO: A program for the automatic bifurcation analysis of autonomous systems, in Proc. 10th Manitoba Conf. on Num. Math. and Comp., Univ. of Manitoba, Winnipeg, Canada, pp. 265–284, 1981. Farrell, B. F. and Ioannou, P. J.: Generalized stability theory. Part I: autonomous operators, Journal of Atmospheric Sciences, 53, 2025–40, 1996. Fujisaka, H. and Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems, Progress in Theoretical Physics, 69, 32–47, 1983. Gates, W. L., Boyle, J., Covey, C., Dease, C., Doutriaux, C., Drach, R., Fiorino, M., Gleckler, P., Hnilo, J., Marlais, S., Phillips, T., Potter, G., Santer, B., Sperber, K., Taylor, K., and Williams, D.: An Overview of the Results of the Atmospheric Model Intercomparison Project (AMIP I), Bulletin of the American Meterological Society, 73, 1962–1970, 1992. Gauthier, D. J. and Bienfang, J. C.: Intermittent loss of synchronization in coupled chaotic oscillators: toward a new criterion for high-quality synchronization, Physical Review Letters, 77, 1751–1754, 1996. Heagy, J., Platt, N., and Hammel, S.: Characterization of on-off intermittency, Physical Review E, 49, 1140–1150, 1994. Holton, J. R.: An introduction to dynamic Meteorology, Academic Press, 1992..

(27) 320 John, T., Behn, U., and Stannarius, R.: Fundamental scaling laws of on-off intermittency in a stochastically driven dissipative patternforming system, Physical Review E, 65, 2002. Lai, Y.-C.: Distinct small-distance scaling behavior of on-off intermittency in chaotic dynamical systems, Physical Review E, 54, 321–327, 1996. Lorenz, E. N.: Irregularity: A fundamental property of the atmosphere, Tellus, 36A, 98–110, 1984. Meehl, G., Boer, G., Covey, C., Latif, M., and Stouffer, R.: The Coupled Model Intercomparison Project (CMIP), Bulletin of the American Meterological Society, 81, 313–318, 2000. Ott, E. and Sommerer, J.: Blowout bifurcations: the occurrence of riddled basins and on-off intermittency, Physical Letters A, 188, 39–47, 1994. Pazó, D., Zaks, M. A., and Kurths, J.: Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators, Chaos, 13, 309–318, 2003. Pecora, L. M. and Carroll, T. L.: Synchronization in chaotic systems, Physical Review Letters, 64, 821–824, 1990. Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar, D., and Heagy, J. F.: Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos, 7, 520–543, 1997.. B. Knopf et al.: Forced versus coupled dynamics Pierce, D., Barnett., T., Schneider, N., Saravanan, R., Dommenget, D., and Latif, M.: The role of ocean dynamics in producing decadal climate variability in the North Pacific, Climate Dynamics, 18, 51–70, 2001. Pikovsky, A., Zaks, M., Rosenblum, M., Osipov, G., and Kurths, J.: Phase Synchronization of Chaotic Oscillations in Terms of Periodic Orbits, Chaos, 7, 680–687, 1997. Pikovsky, A., Rosenblum, M., and Kurths, J.: Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, 2001. Platt, N., Spiegel, E. A., and Tresser, C.: On-off intermittency: A mechanism for bursting, Physical Review Letters, 70, 279–282, 1993. Smith, L. A., Ziehmann, C., and Fraedrich, K.: Uncertainty dynamics and predictability in chaotic systems, Quarterly Journal of the Royal Meteorological Society, 155, 2855–2886, 1999. Wittenberg, A. T. and Anderson, J. L.: Dynamical implications of prescribing part of a coupled system: Results from a low order model, Nonlin. Proc. Geophys., 5, 167–179, 1998, SRef-ID: 1607-7946/npg/1998-5-167. Yanchuk, S., Maistrenko, Y., and Mosekilde, E.: Loss of synchronization in coupled Rössler systems, Physica D, 154, 26–42, 2001..

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