Nonlin. Processes Geophys., 13, 275–285, 2006 www.nonlin-processes-geophys.net/13/275/2006/ © Author(s) 2006. This work is licensed under a Creative Commons License.
Nonlinear Processes in Geophysics
A conceptual ENSO model under realistic noise forcing J. Saynisch, J. Kurths, and D. Maraun Nonlinear Dynamics Group, University Potsdam, Germany Received: 16 January 2006 – Revised: 20 April 2006 – Accepted: 26 April 2006 – Published: 13 July 2006
Abstract. We investigated the influence of atmospheric noise on the generation of interannual El Ni˜no variability. Therefore, we perturbed a conceptual ENSO delay model with surrogate windstress data generated from tropical windspeed measurements. The effect of the additional stochastic forcing was studied for various parameter sets including periodic and chaotic regimes. The evaluation was based on a spectrum and amplitude-period relation comparison between model and measured sea surface temperature data. The additional forcing turned out to increase the variability of the model output in general. The noise-free model was unable to reproduce the observed spectral bandwidth for any choice of parameters. On the contrary, the stochastically forced model is capable of producing a realistic spectrum. The weakly nonlinear regimes of the model exhibit a proportional relation between amplitude and period matching the relation derived from measurement data. The chaotic regime, however, shows an inversely proportional relation. A stability analysis of the different regimes revealed that the spectra of the weakly nonlinear regimes are robust against slight parameter changes representing disregarded physical mechanisms, whereas the chaotic regime exhibits a very unstable realistic spectrum. We conclude that the model including stochastic forcing in a parameter range of moderate nonlinearity best matches the real conditions. This suggests that atmospheric noise plays an important role in the coupled tropical pacific ocean-atmosphere system.
1 Introduction Over thirty years ago Bjerknes (1969) described the basic feedback mechanisms underlying El Ni˜no/Southern OscillaCorrespondence to: J. Saynisch (
[email protected])
tion (ENSO). These mechanisms are based on coupling between the ocean and the atmosphere. The observed interannual variability of ENSO can in principle be explained by deterministic dynamics: Strong nonlinear interactions between ocean and atmosphere lead to low order deterministic chaos (Timmermann and Jin, 2003). Nonlinear interactions of ENSO modes with the seasonal cycle can also lead to deterministic chaos (Tziperman, 1994, 1995). The latter concept is able to explain the locking of El Ni˜no to the seasonal cycle. Recently, a discussion has arisen about the influence of atmospheric noise on the ocean and the coupled oceanatmosphere system (Thompson and Battisti, 2001; Kessler, 2002; Fedorov, 2003). Models based solely on chaotic behavior cannot explain the full amount of ENSO variability, e.g. all models failed to predict the devastating 1997/98 El Ni˜no (Fedorov, 2003). For this event, Lengaigne (2004) studied the effect of measured high- and low-frequency winds, based on GCM simulations. These investigations revealed that tropical wind anomalies (especially westerly wind bursts) are well able to influence the onset and growth of El Ni˜nos and even trigger a warm event. More conceptual examinations of this question consist of driving simple models with Gaussian noise (Stone and Price, 1998). Here the models are easily influenced by noise and are able to show stochastic resonance. However, it is still under debate as to whether wind bursts are capable of causing ENSO variability or rather are the effect of the latter. Also, there is disagreement about whether typical atmospheric noise may at all be strong enough to influence the onset, progression, strength and period of El Ni˜nos. Our approach to this controversial discussion is on a level of complexity between time consuming GCM studies and overly simple conceptual models disregarding essential physical mechanisms. We investigate the possible effects of – undoubtedly present – atmospheric noise, by utilizing
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Table 1. Parameter list of the GT model. Parameter a, b AWinde AWindn A∗ b0 Co τwind ε m T h H1 µ ρ re/w τ1,2 T Tsub ω¯
Description scaling parameters of additional noise drive equatorial additional noise drive northern additional noise drive relates non-equatorial windstress to equatorial SST mean ocean-atmosphere coupling strength Kelvin wave propagation speed wind affected fraction of basin crossing time strength of seasonal variation damping coefficient of the ocean thermal damping coefficient thermocline depth anomaly upwelling layer depth seasonal varying coupling density of ocean water coastal boundary wave reflection coefficients basin crossing time (1 Rossby, 2 Kelvin) temperature anomaly at eastern boundary temperature anomaly at depth H1 mean upwelling
a conceptual model of considerable complexity and adding realistic stochastic forcing to the internal (i.e. deterministic) windstress. For this purpose we construct surrogates from measured windspeed data, in such a way that the low frequency variability is conserved while the high frequency part is randomized. This allows us to investigate the influence of realistic windstress on the spectrum, strength and period of ENSO on a conceptual basis. These investigations include chaotic and quasiperiodic model regimes. The criteria to evaluate the quality of this stochastic extension are also based on measured data. We calculate spectra and amplitude-period relations (APR) of the model runs with the additional forcing and compare them with the same diagnostic measures from the NINO3 region of the Kaplan (1998) reconstruction and noise free model runs, respectively. Taking into account the natural origin of most model parameters, we performed a stability analysis. The model has to reproduce its spectrum and APR in a reasonable parameter space. A measure of robustness was introduced for this purpose. The paper is organized as follows: In the second section the model and our stochastic forcing are introduced in more detail. The influence of the additional forcing is presented in Sect. 3. The last section presents our conclusions.
2 2.1
Model and methodology The model
The investigations within this paper are based on the ENSO model of Galanti and Tziperman (2000), hereafter GT. This Nonlin. Processes Geophys., 13, 275–285, 2006
delay oscillator consists of two zonal ocean stripes and describes the eastern thermocline depth anomaly h as a function of wind-excited Kelvin and Rossby waves (with parameters listed in Table 1): h(t) = e−m (τ1 +τ2 ) rw re h(t − τ1 − τ2 ) τ1 A∗ τ1 −e−m (τ2 + 2 ) rw τWind τ1 µ(t − τ2 − ) βρ 2 τ1 ∗b0 T (t − τ2 − ) 2 τ τ2 τ2 −m 22 τWind τ2 +e µ(t − )b0 T (t − ). ρCo 2 2
(1)
A change of zonal, equatorial windstress over the central Pacific simultaneously generates upwelling (respectively downwelling) Kelvin waves and downwelling (respectively upwelling) Rossby waves (see Eq. 1). These waves travel to and from the Pacific ocean and are reflected at the coastal boundaries. In this way, they carry the upwelling and downwelling signal to the Ecuadorean coast and determine the eastern tropical pacific thermocline depth. Since the wave reflection and propagation weakens and delays the traveling signal, a nonintuitive and complex interplay of forces arises. For a detailed description of the model, see Galanti and Tziperman (2000). The strength of the zonal windstress (τx ) is assumed to be in balance with the eastern sea surface temperature (SST) and the ocean-atmosphere coupling (see Eq. 2). The latter varies according to the seasonal cycle (µ) around a mean value (b0 ): τx = b0 µT µ = 1 + ε cos(2πt/12 − 5π/6).
(2)
The SST at the eastern boundary is given as a function of thermal damping (T ) and upwelling (ω): ¯ ∂t T = −T T − γ
ω¯ (T − Tsub (h)) H1
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The temperature of the upwelling water is modeled as the hyperbolic tangent of the thermocline depth anomaly and represents the main nonlinearity in the GT (Galanti and Tziperman, 2000). 2.2
Stochastic forcing
To implement stochastic forcing, we extended the model with two additive terms: AWindn and AWinde . These terms were included in the part of the model that relates windstress to SST and thus represent additional windstress: h(t) = e−m (τ1 +τ2 ) rw re h(t − τ1 − τ2 ) τ1 1 −e−m (τ2 + 2 ) rw τWind τ1 βρ τ1 τ1 ∗ ∗A {µ(t − τ2 − )b0 T (t − τ2 − ) 2 2 +AWindn (t − τ2 −
τ1 )} 2
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Fig. 1. measurement (left) The phase phaseand andamplitude amplitude conservation of lower frequencies including the Fig.Windstress: 1. Windstress: measurement (left)and andsurrogates surrogates (right). (right). The conservation of lower frequencies including the annualannual cyclecycle can be TheThe phase of of higher uniformrandom random distribution. canseen. be seen. phase higherfrequencies frequencies follows follows aa uniform distribution. τ2
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at a time. The calculation of V was repeated for the four regimes (periodic, modelock, SOR, chaotic). As expected, www.nonlin-processes-geophys.net/13/275/2006/ the periodic regime results to be the most stable while the spectrum of the chaotic regime turns out to react very sensitively even to minor parameter changes. In the latter regime,
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spectral band representing the annual (+/− a half year) variI ability has been kept to conserve the phase and shape of the 5 II annual cycle of the real data. Lower frequencies have been (4) 4 erased to ensure that ENSO variability present in the wind 3 III data does not affect our analysis and thus avoiding circular To study the influence of the additional forcing terms on the IV reasoning. To account for stochastic variability, we randommodel 2behavior in general, we performed model runs with ized the phases of high frequency contributions. These sursimple 1concepts of additional forcing (constant-, gaussian-, rogates are added to the model’s internal windstress, with an sinusoidal-gaussian forcing). From these model realiza0 additional amplitude of less than 10 percent (for further detions, we calculated the corresponding spectra, attractors and tails, see Appendix A). -1 amplitude-period relations (For details see Appendix B). By The correct construction of surrogates proved to be crucomparison of spectra and attractors of the stochastic runs -2 cial for the success of this method. In this surrogate driven with those of the standard deterministic model, we obtained -3 model (SGT), generation waves no longer the following results: 1940 1950 1960 1970 1980 1990 0 0.1 0.2the0.3 0.4 0.5of ocean 0.6 0.7 0.8 is 0.9 1 bound strictly to the internal state of the oscillator, especially Forcing with different constants results in different model t [years] frequency [1/year] the slowly changing SST, but is also subject to external inattractors. One can interpret additional noise as continued fluences. The latter have a stochastic component but are not change to these constants, thus causing jumps between the Fig. 2. SST (left)Therefore, and powerthe spectrum (right) of the Kaplan In (see the time thecorresponds multiscale variability entirely random Fig. series, 1). This to findings of of ENSO attractors. reconstructed attractor underNINO3 the reconstruction. can beinfluence seen. This results in a broad spectrum with at least four characteristic peaks of the following periods: I: 3.6 years; II: 5.8 years; III: Eisenman and Tziperman (2005). of noise is similar to the undisturbed chaotic at10-20tractor years; (see VI: 2.8 years. Fig. 4) where the system jumps between different resonance frequencies (Tziperman, 1995). 3 The noise driven oscillator The complexity of the results, i.e. the number of different ble toreached the Kaplan data and simultaneously behaves robustly From this point of view, the chaotic regime, despite showattractors, depends on the amplitude and mean value The of the variability, model outputseems is evaluated the SSTto repunderofnatural i.e.asmall parameter changes. ingquality reasonable not toagainst be suitable the noiseconditions, forcing. Here, moderate mean value added toTo a reconstruction done by Kaplan (1998). This global data set this end, we slightly altered parameters and resent the real ENSO system. On the contrary, under palow amplitude noise has thethe same effect as a of zeroTable mean1noise covers the years 1856–1991. From the time series, integrated utilized measure (V amplitude. ) to evaluate resulting spectral rameter changes of 5-10% the SOR shows a broad spectrum witha asimple considerably larger Zero mean noise with over the NINO3 region of this data set, we derived the power small amplitude has only a negligible effect. changes: with characteristic and an Elrelation Ni˜no-like time spectrum and a simple peaks amplitude-period (APR). Theseries. Since the model is driven by seasonal modulation of the In Fig. 6 a sample comparison between the SOR and the derivation of these diagnostics are explained in Appendix B. N ocean-atmosphere impact 1 X (scoupling, (p ± δp))2of the additional n (p) − snthe chaotic regime of the SGT is shown. Each graph consists of V± (p,forcing δp) =also varies throughout the (5) N n=1 sn (p)2 year. Consequently, the three spectra that correspond to slightly different parameterexistence and the right phase relation of annual variability in 3.1 sets Spectrum (−δp ;± 0; +δp). The varied parameters in this figthesnadditional have proven benth of great importance where (p) is theforcing spectral power oftothe frequency proure are: ocean-damping (m ), western wave reflection coincrease modelset variability. The GT undergoes the quasiperiodic route to chaos (Eccles ducedtoby parameter p. The spectral power sn (p ± δp) is efficient (r mean upwelling strength (¯ ω ).ofItthis becomes w ) and In order to provide a realistic forcing, we constructed and Tziperman, 2004). For a detailed explanation produced by a slightly different parameter set. evident the SOR(1995). spectrum is robustonunder the applied nonzero mean windstress surrogates from windspeed data route, seethat Tziperman Depending the strength of paWe applied variations of a few percent to one parameter rameter variations. In can some parameters, spectrum measured by central pacific buoys in the following way: The the seasonal forcing, one distinguish threethe regimes of the of the
chaotic regime is equally or even more robust (see end of Table 2) than the SOR, but in general the chaotic regime tends Nonlin. Processes Geophys., 13, 275–285, 2006 to change dramatically under slight parameter variations.
Fig. 1. Windstress: measurement (left) and surrogates (right). The phase and amplitude conservation of lower frequencies including the annual cycle can be seen. The phase of higher frequencies follows a uniform random distribution. 278 J. Saynisch et al.: A ENSO model under realistic noise forcing 6
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Fig. 2.Fig. SST (left)(left) and and power spectrum (right) NINO3reconstruction. reconstruction.In In time series, the multiscale variability of ENSO 2. SST power spectrum (right)ofofthe theKaplan Kaplan NINO3 thethe time series, the multiscale variability of ENSO can becan seen. This This results in a in broad spectrum with at at least four the following followingperiods: periods:I: I:3.63.6years; years; years; III: be seen. results a broad spectrum with least fourcharacteristic characteristic peaks peaks of of the II: II: 5.85.8 years; 10-20III: years; VI:years; 2.8 years. 10–20 VI: 2.8 years.
modelock, chaotic). It isrobustly evident ble toGT the(periodic/quasiperiodic, Kaplan data and simultaneously behaves that all regimes lack the observed ENSO variability when under natural conditions, i.e. small parameter changes. one To compares GT spectra (see left panel of Fig. 3) with observathis end, we slightly altered the parameters of Table 1 and tions (see Fig. 2). utilized a simple measure (V ) to evaluate resulting spectral All GT regimes are able to reproduce the main El Ni˜no changes:
peak but fail to produce the spectral bandwidth of the Kaplan N is not surprising that the 2broadest spectrum reconstruction. 1 XIt (s n (p) − sn (p ± δp)) V± (p,and δp)therefore = (5) the most variability2is produced by the chaotic N n=1 sn (p) regime. addition of stochastic forcing with in the profrewhere sThe is the spectral power of the nthpower frequency n (p) quency band of one per year or higher as defined in Sect. duced by parameter set p. The spectral power sn (p ± δp)2.2is increases general (see produced by a model slightlyvariability differentinparameter set. right panel of Fig. 3). This especially affects the El Ni˜ spectral main We applied variations of a few percent tonoone parameter band, i.e. frequencies of interannual variability. Here, addiat a time. The calculation of V was repeated for the four tional oscillations are excited and the lower frequencies gain regimes (periodic, modelock, SOR, chaotic). As expected, more power in general. This is very astonishing since the the periodic to isbeonly the amost stable while the additionalregime forcingresults strength few percent of model spectrum of the chaotic regime turns outintothe react veryENSO sensigenerated windstress and has no power affected tivelyspectral even toband minor changes. In the latter regime, at parameter all. Estimation of correlation dimensions a Kaplan-like is only reproduced for anisunreal(not shown)spectrum revealed that the dynamics of a regime not altered by our additional forcing. It is of interest that the istically strongly confined parameter region (see Tableperi2). odic regime is not able to show much variability even under noise forcing. Therefore, we regard the periodic regime as less able to reflect natural conditions. By making small changes to the parameters we were able to find a regime (hereafter called the spectral optimized regime, SOR) which is well able to reproduce the Kaplan spectrum and the variability of the SST. The similarity is especially remarkable when keeping the conceptual nature of the model in mind (Fig. 5). The parameterization of this regime has a stronger seasonal cycle and therefore has a stronger nonlinearity than the modelock model runs. The SOR may therefore show weakly chaotic behavior. Nonlin. Processes Geophys., 13, 275–285, 2006
3.2 Stability From this point of view, the chaotic regime, despite show-
ing reasonable variability, seems not to be suitable to rep-
Many parameters of the GT are idealized parameterizations resent the real ENSO system. On the contrary, under paof complex processes that are subject to slow and rapid rametere.g. changes of 5-10% shows a broad spectrum changes, it is unlikely thatthe theSOR latitude of Rossby wave with characteristic peaks and an El Ni˜ n o-like time propagation is constant as represented by the second model series. In Fig. 6 a sample comparison between the of SOR and the stripe. Fluctuations of this latitude result in a change propchaoticspeed regime the SGT is shown.A Each graphmodel consists of agation and of reflection coefficient. conceptual naturally disregards physical mechanisms. However, three spectra that certain correspond to slightly different parameteritsets should react to reasonable parameter variations. (−δp ;±robustly 0; +δp). The varied parameters in this figTherefore, did a pragmatic stability analysis of the SGT coure are: we ocean-damping (m ), western wave reflection and the deterministic GT to study the robustness of these efficient (rw ) and mean upwelling strength (¯ ω ). It becomes models tothat suchthe parameter variations. aimunder was tothe find the paevident SOR spectrum is The robust applied parameter regime that produces realistic variability compararameter variations. In some parameters, the spectrum of the ble to the Kaplan data and simultaneously behaves robustly chaotic regime is equally evenparameter more robust (see end under natural conditions, i.e.orsmall changes. To of Table 2) than the SOR, but in general the chaotic regime this end, we slightly altered the parameters of Table 1 and tends to change dramatically parameter utilized a simple measure under (V ) to slight evaluate resulting variations. spectral changes: V± (p, δp) =
N 1 X (sn (p) − sn (p ± δp))2 N n=1 sn (p)2
(5)
where sn (p) is the spectral power of the nth frequency produced by parameter set p. The spectral power sn (p±δp) is produced by a slightly different parameter set. We applied variations of a few percent to one parameter at a time. The calculation of V was repeated for the four regimes (periodic, modelock, SOR, chaotic). As expected, the periodic regime results to be the most stable while the spectrum of the chaotic regime turns out to react very sensitively even to minor parameter changes. In the latter regime, a Kaplan-like spectrum is only reproduced for an unrealistically strongly confined parameter region (see Table 2). www.nonlin-processes-geophys.net/13/275/2006/
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Fig. 3.Fig. Quasiperiodic route to chaos: Power spectrum model(GT, (GT, left) stochastically forced 3. Quasiperiodic route to chaos: Power spectrumofofthe the deterministic deterministic model left) andand thethe stochastically forced modelmodel (SGT,(SGT, right). right). The weakly nonlinear deterministic regimes theENSO ENSOmain main frequency. deterministic chaotic The weakly nonlinear deterministic regimesexhibit exhibitaasingle single oscillation oscillation atatthe frequency. The The deterministic chaotic regimeregime generates a small spectral band around that frequency. theapplied appliedstochastic stochastic forcing broadens the spectrum and excites generates a small spectral band around that frequency.InInall all regimes, regimes, the forcing broadens the spectrum and excites strong strong additional oscillations. additional oscillations.
this point of view, the chaotic regime, despite show3.3 From Amplitude-period relation
ing reasonable variability, seems not to be suitable to represent the real ENSO system. On the contrary, under parameter changes of 5–10% the SOR shows specIt is still unclear whether a strong El Ni˜ no hasa abroad longer or
shorter duration compared to a weaker one, or even if El Ni˜no period and amplitude are rather independent of each other. www.nonlin-processes-geophys.net/13/275/2006/ Eccles and Tziperman (2004) obtained the following results in their study of the GT model: For the chaotic regime, they inferred an inversely proportional relation, i.e. the stronger
trum withi.e. characteristic peaks andlast an El Ni˜no-like se- ones. instead, stronger El Ni˜nos longer than time weaker ries. In Fig. 6 a sample comparison between the SOR and Since the latter regimes have an constant intrinsic period, the chaotic regime of the SGT is shown. Each graph conthe investigations had to be rather indirect. By parameter sists of three spectra that correspond to slightly different
variations, period changes were induced. An additional approach to this question is possible for models with noise (see Appx. B): Nonlin. As the stochastic forcing of13, the275–285, SGT results Processes Geophys., 2006in amplitude and period variations in all regimes, we could investigate the APR directly even in the weakly nonlinear regimes. The left panel in Fig. 7 shows the results of the deterministic
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Fig. 4. Quasiperiodic route to chaos: Attractor model (GT, (GT,left) left)and andthethe stochastically forced model (SGT, Fig. 4. Quasiperiodic route to chaos: Attractorofofthe thedeterministic deterministic model stochastically forced model (SGT, right).right). The The route route to chaos results in aingrowing attractor ofofincreasing Theapplied appliedstochastic stochastic forcing results an increased complexity. to chaos results a growing attractor increasingcomplexity. complexity. The forcing alsoalso results in aninincreased complexity.
Forparameter-sets the periodic (−δp; and modelock of theparameters GT, our ap±0; +δp).regime The varied in thisrevealed figure are:a ocean-damping wave reflection proach one and a two(point relation (c.f. correm ), western coefficient (rw ) and mean in upwelling ω). ¯ It becomes sponding attractors shown Fig. 4).strength Our (result for the evident that the SOR spectrum is robust under the applied pachaotic regime confirms the inversely proportional relation rameter variations. In some parameters, the spectrum of the found by Eccles and Tziperman (2004). chaotic regime is equally or even more robust (see end of TaThe periodic stochastic regime shows a obvious proportional relation of period and amplitude. While in the stochastic modelock case a proportional still recognizNonlin. Processes Geophys., 13,tendency 275–285,is2006 able, no obvoius APR seems to exist in the stochastically disturbed chaotic case. Further increase of the coupling to the seasonal cycle results in an inversely proportional relation
(2004) hold the but stochastically forced model too.tends ble 2) than thefor SOR, in general the chaotic regime to change dramatically under slight parameter variations. The fundamental difference between the model APR of strong (chaotic) and weakly nonlinear (quasiperiodic) regimes can be utilized for a comparison with measured data: For the NINO3 region SST data, we obtained a rather proportional relation (See Fig. 8). This result is neither matched by the chaotic regime nor by the SOR in deterministic or stochasticwww.nonlin-processes-geophys.net/13/275/2006/ model runs. On the other hand, the stochastically forced model runs show more conformity with the observations. Within the SGT runs, the modelock and the modelock/weakly chaotic SOR resemble the observed relation
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Fig. 5.Fig. SST and power spectrum (right) ofof thethespectral (SOR)ofofthe thestochastic stochastic model (ε = 0.7). The model exhibits 5. (left) SST (left) and power spectrum (right) spectraloptimized optimized regime regime (SOR) model (ε=0.7). The model exhibits El Ni˜ nos amplitude and Kaplandata data(see (seeFig. Fig. This results a model spectrum with recognizable El Ni˜nofosvarious of various amplitude andperiod periodsimilar similar to to the the Kaplan 2).2). This results in a in model spectrum with recognizable ENSO ENSO main main peakspeaks (I-IV). Obviously the the high frequency bythe themodel. model. (I–IV). Obviously high frequencyvariability variabilityisis not not captured captured by
Amplitude-period Table3.3 2. Regime sensitivity torelation variations in parameter-set It is still unclear a strong El Ni˜ no has a longer Parameter SOR (Vwhether Chaotic regime (V− + V+ ) or − + V+ ) shorter duration compared to a weaker one, or even if El Ni˜no period and amplitude are rather independent of each other. rwEccles ± 1%and Tziperman 0.48 (2004) obtained the 394.41 following results rein±their 1% study of the 0.23GT model: For the chaotic 11.79 regime, they minferred ± 5% an inversely 0.22 proportional relation, 96.71 i.e. the stronger tthe ± 5% 0.22 20.68 El Ni˜no, the shorter its duration. For the modelock and ω ¯periodic ± 1% regime, however, 0.20 36.34 they found a proportional relation H1 ± 1% i.e. stronger 0.28El Ni˜nos last longer31.96 instead, than weaker ones. ρSince ± 1%the latter regimes 1.13 have an constant1.17 intrinsic period, Lthe ± 1% 0.55 By parameter investigations0.38 had to be rather indirect. b0variations, ± 1% period0.56 changes were induced. 0.29 An additional approach to this question is possible for models with noise (see Appendix B): As the stochastic forcing of the SGT results in amplitude and period variations in all regimes, we could investigate the APR directly even in the weakly nonlinear 4 Summary and conclusions regimes. The left panel in Fig. 7 shows the results of the deterministic GT, the right panel those of the stochastic SGT. We investigated the influence of additional surrogate forcing For the periodic and modelock regime of the GT, our apon theproach variability of aa conceptual ENSO revealed one and a two pointmodel. relationThe (cf.forcing correnoisesponding shares the power spectrum as well as the shape attractors shown in Fig. 4). Our result for and the phasechaotic of the regime annualconfirms cycle with windspeed data. theTOGA/TAO inversely proportional relation The quality ofEccles the model runs was evaluated found by and Tziperman (2004). in the following way: We the modelregime spectrashows and amplitude-period Thecompared periodic stochastic a obvious proporrelation (APR) to the corresponding counterparts tional relation of period and amplitude. While inof theobserved stochastic modelock case a proportional tendency is still recognizNINO3 region SST data. As complex physical processes able, no obvoius APR seems to exist in stochastically disare represented by simple parameters in the conceptual models, turbed chaotic case. Further increase of the coupling to the these parameters are assumed to be subject to considerable seasonal cycle results in an inversely proportional relation fluctuations. Thus, as a further criterion, we investigate the (not plotted). Hence, both findings of Eccles and Tziperman model’s ability to produce a robust and realistic spectrum (2004) hold for the stochastically forced model too.
under reasonable parameter changes. A measure of robustness was introduced for this purpose. www.nonlin-processes-geophys.net/13/275/2006/
The results of our study were the following: From the stochastic and the deterministic periodic regime are
ily. While the deterministic modelock regime also lacks complexity, the stochastic forcing leads to a drastically increased Parameter SOR (V− + V+ ) Chaotic regime (V− + V+ ) variability. The spectrum and the APR match the observarw ±1% 0.48 output was found 394.41 tions well. The model to be robust under re ±1% 0.23 11.79 realistic parameter variations. m ±5% 0.22 96.71 The deterministic t ±5% 0.22chaotic regime shows 20.68 a rather complex ω±1% ¯ 36.34bandwidth of the behavior but still0.20 does not match the H 1±1% 31.96 Kaplan spectrum. 0.28 Adding the stochastic forcing results in a ρ±1% 1.13 1.17 further increment 0.38 of variability and therefore spectral bandL±1% 0.55 width. Nonetheless, the resemblance with the observations is b0 ±1% 0.56 0.29 easily destroyed by slight parameter changes. Furthermore, the deterministic chaotic APR and its stochastic counterpart show a strong inversely proportional tendency that does not fitThe the observations at all. fundamental difference between the model APR
Table 2. Regime sensitivity to variations in parameter-set
of strong (chaotic) and weakly nonlinear (quasiperiodic) regimes can be utilizedthe for spectrum, a comparison withand measured data:analyTo summarize, APR stability For the NINO3 region SST data, we obtained a rather prosis favor regimes with medium nonlinearity and additional portional relation (see Fig. 8). This result is neither matched stochastic forcing. In particular, a modelock/weakly chaotic by the chaotic regime nor by the SOR in deterministic or regime disturbed by surrogate data (SOR) representing tropstochastic model runs. On the other hand, the stochastically ical winds wellconformity matchingwith our criteria could be forcedpacific modelsurface runs show more the obserfound: This regime resembles the spectrum of NINO3 vations. Within the SGT runs, the modelock and the mod-region SST measurement the variability within the ENSO elock/weakly chaotic data, SOR i.e. resemble the observed relation spectral band and the four ENSO main spectral peaks were best.
reproduced. Additionally, the APR is consistent with the observed one. Finally, this regime turned out to be stable under 4reasonable Summaryparameter and conclusions changes. The inability of all deterministic regimes to express obWe investigated the influence of additional surrogate forcing served ENSO variability, and the result that additional forcon the variability of a conceptual ENSO model. The forcing raises the spectral power within the ENSO main band ing noise shares the power spectrum as well as the shape and (2-7 years), suggest that atmospheric noise plays a significant role for the ENSO dynamics. This external influence Nonlin. Processes Geophys., 13, 275–285, 2006 includes high frequency weather noise as well as low frequency windstress variations that interact with the seasonal cycle presented by ocean-atmosphere coupling. However, all
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phase of the annual cycle with TOGA/TAO windspeed data.
repeatThe thisquality studyofwith more complex models. in the following the model runs was evaluated way: We compared the model spectra and amplitude-period relation (APR) to the corresponding counterparts of observed NINO3 region SST data. As complex physical processes Appendix A Construction of the surrogates are represented by simple parameters in conceptual models,
The construction of the surrogates turned out to be a cruNonlin. Processes 13, 275–285, 2006 upon zonal cial part of this work. Geophys., The construction is based windspeed data taken from the TOGA/TAO buoy array. After removal of the mean value, we applied a Fourier transformation. In the frequency domain, we performed two essential
these parameters are assumed to be subject to considerable riod longer than a half years from the spectrum. fluctuations. Thus,one as aand further criterion, we investigate the This operation eliminates thea robust main El no time scales unfrom the model’s ability to produce andNi˜ realistic spectrum der reasonable parameter measuredistributed of robustness time series. Second, wechanges. added a Auniform phase to was introduced for thisperiods purpose.shorter than 100 days. Note that all oscillations with
this second procedure does not alter the power spectrum of the data set. After inverse Fourier transformation and addition of thewww.nonlin-processes-geophys.net/13/275/2006/ mean value, we now have a new windspeed time series. This time series has exactly the same power spectrum as the original one except for the deleted low frequencies. Furthermore, the oscillations between 1.5 years and
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the seasonal cycle of the model in a natural way. Finally, www.nonlin-processes-geophys.net/13/275/2006/ the phases of quick oscillations faster than 100 days are dis-
turbed randomly to represent the stochastic component of the Nonlin. Processes Geophys., 13, 275–285, 2006 atmosphere. A simple drag relation turns this time series into
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2
quency windstress variations that interact with the seasonal cycle presented by ocean-atmosphere coupling. However, all References these results have to be interpreted in the light of the conceptual origin of this model. Thus, it would interesting Bjerknes, J.: Atmospheric teleconnections frombethe equatorial to Parepeat this study with more complex models. cific. Mon. Weather Rev., 97, 163–172, 1969.
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Fig. relation of of the the Kaplan KaplanNINO3 NINO3reconstrucreconstrucFig. 8. 8. Amplitude-period Amplitude-period relation tion. A rather proportional relation is visible. tion. A rather proportional relation is visible.
The results of our study were the following: windstress. Theorder stochastic the deterministic periodic regime In to payand tribute to the two ocean stripes ofare thevery GT, stable but show insufficient variability. Thus,locations the observed we derived our surrogates from two different of the spectrum and APR are reproduced onlyInrudimentarily. central pacific (0N170W, 8N170W). this way, we could While modelockof regime also lacks force the the twodeterministic stripes independently each other withcomcharplexity, thetime stochastic to a drastically increased acteristic series. forcing Model leads runs brought best results with variability. The and the APR match the observaan amplitude of spectrum the additional forcing that corresponds to tions well. The model output was found to be robust under 2.8% (respectively 9.8%) of the negative (respectively posirealistic parameterinternal variations. tive) undisturbed, model windstress in the equatorial The In deterministic chaoticstripe, regimethe shows a rather complex stripe. the off-equatorial forcing matches 0.8 % behavior but 2.8 still%)does match the bandwidth of This the (respectively of thenot model-generated windstress. Kaplan spectrum. the stochastic forcing results in a corresponds only toAdding a few thousandth of observed windstress further increment variability thereforeweak spectral bandand is because theofGT generatesand relatively windstress width. Nonetheless, the resemblance with the observations is compared to observations. easily destroyed by slight parameter changes. Furthermore, the deterministic chaotic APR and its stochastic counterpart show a strong inversely proportional tendency that does not Appendix B Diagnostics fit the observations at all. To summarize, spectrum, APRisand stability analyWhile the Kaplanthe power spectrum constructed in the sis favor regimes with medium nonlinearity and additional usual manner. To derive the APR , we determined all local stochasticthat forcing. particular, a modelock/weakly chaotic extrema were atInleast 19 months apart, and locally fitted regime disturbed by surrogate data (SOR) representing parabola to the extrema. The amplitude of an El Ni˜notropwas ical pacific winds matching could be taken to be surface the mean of a well maximum and our the criteria two neighboring found: This resembles the spectrumperiod of NINO3 region minima. Weregime defined the corresponding as the time SST measurement data, i.e. the variability within the ENSO span between those enclosing minima. spectral band and the four ENSO main spectral peaks were reproduced. Additionally, the APR is consistent with the obThe power spectra of the model are the means over five served one. Finally, this regime turned out to be stable under thousand time windows (each 140 years long and far away reasonable parameter changes. from the models spin up). Derivation of the model’s APR The inability of all deterministic regimes to express obwas done by taking duration and maximum SST of every atserved ENSO variability, and the result that additional forctractor cycle. The attractor used for this purpose was built ing raises the spectral power within the ENSO main band by taking the three terms of Eq. (4) as components of a three (2–7 years), suggest that atmospheric noise plays a signifidimensional vector. This reconstruction is isomorphic to atcant role for the ENSO dynamics. This external influence tractors gained by the embedding of model SST but period dependent folding problems are avoided. Nonlin. Processes Geophys., 13, 275–285, 2006
Boulanger, J.-P., Menkes, C., and Lengaigne, M.: Role of high- and low-frequency winds and wave reflection in the onset, growth Appendix A and termination of the 1997-1998 El Ni˜no. Climate Dyn., 22, 267–280, 2004. Construction the surrogates Eccles, F., and of Tziperman, E.: Nonlinar effects on ENSO’s period. J. Atmos. Sci., 61, 474–482, 2004. The construction the surrogates out toWind be a Bursts: cruEisenman, I., Yu, of L. and Tziperman, turned E.: Westerly cialENSO’s part of tail thisrather work.than The is based upon zonal theconstruction dog? J. Climate, in press, 2005. Fedorov, A.data V., Harper, S. L.,the Philander, S. G., buoy Winter, B., and Witwindspeed taken from TOGA/TAO array. After tenberg, El Ni˜no? Bull. Amer. Met. Soc., removal of A.: the How meanPredictable value, weisapplied a Fourier transforma2003. domain, we performed two essential tion.84,In911–919, the frequency Galanti, E.,First, and Tziperman, E.: pass ENSO’s locking operations. we used a low filterphase to erase everytope-the Seasonal Cycle in the Fast-SST, Fast-Wave, and Mixed-Mode riod longer than one and a half years from the spectrum. This Regimes. J. Atmos. Sci., 57, 2936–2950, 2000. operation eliminates the main El Ni˜no time scales from the Kaplan, A., Cane, M. A., Kushnir, Y., Clement, A. C., Blumenthal, time series. Second, we added a uniform distributed phase to M. B. and Rajagopalan, B.: Analyses of global sea surface temall oscillations with periods shorter than 10018567–18589,1998. days. Note that perature 1856-1991. J. Geophys. Res., 103, this second procedure does not alter the power spectrum of Kessler, W. S.: Is ENSO a cycle or a series of events? Geophys. the Res. dataLett., set. 29(23), After inverse Fourier transformation and addi2125–2128, 2002. tion of the mean value, we now have a newJ.windspeed timeC., Lengaigne, M., Guilyardi, E., Boulanger, P., Menkes, series. This time series exactly sameJ.:power spec-of Delecluse, P., Innes, P., has Cole, J., andthe Slingo, Triggering trum the original onewind except forinthe deletedgeneral low frequenElas Ni˜ no by westerly events a coupled circulation Climate Dyn., 2004. 1.5 years and cies.model. Furthermore, the23(6),601–6020, oscillations between Stone, L.,keep Saparin P. I., Huppert, A., and Price, to C.:El Ni˜nowith chaos: 100 days their original phase relations interact role of cycle noise of andthe stochastic on the cycle. the the seasonal model resonance in a natural way.ENSO Finally, Letters, 25(2), 175–178, 1998. the Geophys. phases ofRes. quick oscillations faster than 100 days are disThompson, C. J., and Battisti, D. S.: A linear dynamical turbed randomly to represent the stochastic stochastic component of the model of ENSO. Part II: Analysis. J. Climate, 14, 445–466, 2001. atmosphere. A simple drag relation turns this time series into Timmermann, A., and Jin, F. F.: A Nonlinear Theory for El Ni˜no windstress. Bursting. J. Atmos. Sci., 60, 152–165, 2003. In order to pay tribute to the two ocean stripes of the GT, Tziperman, E., Stone, L., Cane, M. A., and Jarosh, H.: El Ni˜no we Chaos: derivedOverlapping our surrogates from two different locations ofCycle the of Resonances Between the seasonal central pacific (0 N 170 W, 8 N 170 W). In this way, we could and the Pacific Ocean-Atmosphere Oscillator. Science, 264, 72– force two stripes independently of each other with char74,the 1994. acteristic time Model runsZebiak, brought bestIrregularity results with Tziperman, E., series. Cane, M. A., and S. E.: and an amplitude of Seasonal the additional forcing thatPrediction corresponds Locking to the Cycle in an ENSO Modeltoas Explained by the 9.8%) Quasi-Periodicity Route(respectively to Chaos. J. posiAtmos. 2.8% (respectively of the negative 52(3), 293–306, 1995. tive)Sci., undisturbed, internal model windstress in the equatorial
stripe. In the off-equatorial stripe, the forcing matches 0.8% (respectively 2.8%) of the model-generated windstress. This corresponds only to a few thousandth of observed windstress and is because the GT generates relatively weak windstress compared to observations. Appendix B Diagnostics While the Kaplan power spectrum is constructed in the usual manner. To derive the APR , we determined all local extrema that were at least 19 months apart, and locally fitted parabola www.nonlin-processes-geophys.net/13/275/2006/
J. Saynisch et al.: A ENSO model under realistic noise forcing to the extrema. The amplitude of an El Ni˜no was taken to be the mean of a maximum and the two neighboring minima. We defined the corresponding period as the time span between those enclosing minima. The power spectra of the model are the means over five thousand time windows (each 140 years long and far away from the models spin up). Derivation of the model’s APR was done by taking duration and maximum SST of every attractor cycle. The attractor used for this purpose was built by taking the three terms of Eq. (4) as components of a three dimensional vector. This reconstruction is isomorphic to attractors gained by the embedding of model SST but period dependent folding problems are avoided. Acknowledgements. We would like to thank our colleague B. Blasius for enlightening discussions and J. Ong for his support. Edited by: M. Thiel Reviewed by: two referees
References Bjerknes, J.: Atmospheric teleconnections from the equatorial Pacific, Mon. Wea. Rev., 97, 163–172, 1969. Boulanger, J.-P., Menkes, C., and Lengaigne, M.: Role of high- and low-frequency winds and wave reflection in the onset, growth and termination of the 1997–1998 El Ni˜no, Clim. Dyn., 22, 267– 280, 2004. Eccles, F. and Tziperman, E.: Nonlinar effects on ENSO’s period, J. Atmos. Sci., 61, 474–482, 2004.
www.nonlin-processes-geophys.net/13/275/2006/
285 Eisenman, I., Yu, L., and Tziperman, E.: Westerly Wind Bursts: ENSO’s tail rather than the dog?, J. Climate, 18, 5224–5238, 2005. Fedorov, A. V., Harper, S. L., Philander, S. G., Winter, B., and Wittenberg, A.: How Predictable is El Ni˜no?, Bull. Amer. Meteorol. Soc., 84, 911–919, 2003. Galanti, E. and Tziperman, E.: ENSO’s phase locking to the Seasonal Cycle in the Fast-SST, Fast-Wave, and Mixed-Mode Regimes, J. Atmos. Sci., 57, 2936–2950, 2000. Kaplan, A., Cane, M. A., Kushnir, Y., Clement, A. C., Blumenthal, M. B., and Rajagopalan, B.: Analyses of global sea surface temperature 1856–1991, J. Geophys. Res., 103, 18 567–18 589,1998. Kessler, W. S.: Is ENSO a cycle or a series of events?, Geophys. Res. Lett., 29(23), 2125–2128, 2002. Lengaigne, M., Guilyardi, E., Boulanger, J. P., Menkes, C., Delecluse, P., Innes, P., Cole, J., and Slingo, J.: Triggering of El Ni˜no by westerly wind events in a coupled general circulation model, Climate Dyn., 23(6), 601–620, 2004. Stone, L., Saparin P. I., Huppert, A., and Price, C.: El Ni˜no chaos: the role of noise and stochastic resonance on the ENSO cycle, Geophys. Res. Lett., 25(2), 175–178, 1998. Thompson, C. J. and Battisti, D. S.: A linear stochastic dynamical model of ENSO. Part II: Analysis, J. Climate, 14, 445–466, 2001. Timmermann, A. and Jin, F. F.: A Nonlinear Theory for El Ni˜no Bursting, J. Atmos. Sci., 60, 152–165, 2003. Tziperman, E., Stone, L., Cane, M. A., and Jarosh, H.: El Ni˜no Chaos: Overlapping of Resonances Between the seasonal Cycle and the Pacific Ocean-Atmosphere Oscillator, Science, 264, 72– 74, 1994. Tziperman, E., Cane, M. A., and Zebiak, S. E.: Irregularity and Locking to the Seasonal Cycle in an ENSO Prediction Model as Explained by the Quasi-Periodicity Route to Chaos, J. Atmos. Sci., 52(3), 293–306, 1995.
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