Limits of predictability in the North Pacific sector of a comprehensive climate model


  • Dimitrios Giannakis,

    Corresponding author
    1. Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York, USA
    • Corresponding author: D. Giannakis, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA. (

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  • Andrew J. Majda

    1. Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York, USA
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[1] We study limits of interannual to decadal predictability of sea surface temperature (SST) in the North Pacific sector of the Community Climate System Model version 3 (CCSM3). Using a set of low-frequency and intermittent spatiotemporal SST modes acquired through nonlinear Laplacian spectral analysis (a nonlinear data manifold generalization of singular spectrum analysis), we build a hierarchy of regression models with external factors to determine which modes govern the dynamic evolution and predictability of prominent large-scale patterns, namely the Pacific Decadal Oscillation (PDO) and North Pacific Gyre Oscillation (NPGO). Retaining key triple correlations between prognostic variables and external factors, as well as the seasonality of the data, we find that the PDO and NPGO modes of CCSM3 can be described with remarkably high fidelity as an outcome of forcing by the intermittent modes (with phase demodulation by the seasonal cycle) and cubic interactions between the low-frequency modes. Our results differ from the classical picture of ENSO-driven autoregressive models for North Pacific SST variability, providing evidence that intermittent processes, such as variability of the Kuroshio current, limit long-range predictability in this climate model.

1. Introduction

[2] A key question in the dynamics of the North Pacific ocean-atmosphere system is how much of the low-frequency (interannual to decadal) variability and long-range predictability of the sea surface temperature (SST) is due to processes internal to the extratropics, as opposed to external mechanisms coupled to the tropics. Some observational and numerical studies suggest that the dominant low-frequency mode in the North Pacific SST, namely the Pacific Decadal Oscillation (PDO), is strongly correlated to the El Niño–Southern Oscillation (ENSO) due to the atmospheric bridge [Alexander et al., 2002]. As result, in conventional scalar autoregressive (AR) models, the PDO is forced additively by both atmospheric noise and an ENSO external factor, giving rise to a reddened signal with interannual variability, but little predictability beyond those timescales [Newman et al., 2003]. In multivariate models, a mode known as North Pacific Gyre Oscillation (NPGO) [Di Lorenzo et al., 2008] or Victoria pattern [Bond et al., 2003], which is obtained through the second empirical orthogonal function (EOF) of SST and interpreted as the oceanic expression of the atmospheric North Pacific Oscillation, is introduced to describe aspects of low-frequency variability which cannot be explained by the PDO alone. Tropical forcings are thought to play a role in the dynamics of that mode as well.

[3] In this work, we study limits of predictability in the North Pacific sector of a comprehensive climate model (the Community Climate System Model version 3 (CCSM3)) through external-factor regression modeling. In particular, we demonstrate that the prominent patterns for low-frequency SST variability in CCSM3 can be described in terms of a set of spatiotemporal modes which carry two orders of magnitude less variance than the ENSO signal in the North Pacific, and yet can explain the conventional PDO and NPGO modes with pattern correlation higher than 90% in decadal-scale hindcasts. These patterns were recovered using a data analysis algorithm called nonlinear Laplacian spectral analysis (NLSA) [Giannakis and Majda, 2012a],which generalizes classical singular spectrum analysis (SSA) [Aubry et al., 1991; Ghil et al., 2002] to take into account the nonlinear manifold structure of datasets generated by complex dynamical systems. Characterized by temporal intermittency, these modes depend crucially on phase relationships with the annual cycle and its harmonics, and thus cannot be detected if the data is seasonally detrended.

[4] The intermittent modes are associated with well-known features of the North Pacific, such as the boundary currents (Kuroshio, Oyashio, Alaska, and California currents), and exhibit strong triple correlations with the PDO and NPGO modes, despite being orthogonal to those patterns in both space and time. Owing to such correlations, they are used here as external factors in joint regression models of the PDO and NPGO, where cubic interactions also contribute to improved skill. Our study thus provides evidence that (at least in CCSM3) the predictability of large-scale modes like the PDO and NPGO is limited by features of the North Pacific associated with the intermittent modes, without direct reference to ENSO signals. Conversely, the low-frequency modes, suitably modulated by multiplicative couplings to the annual cycle, can explain most of the variability of the intermittent modes, suggesting that these two families of modes can potentially be used in self-contained stochastic models for North Pacific SST variability.

2. Spatiotemporal Modes Recovered by NLSA

[5] The dataset used in this study is the monthly-averaged SST field extracted from a 500 y control integration of CCSM3 with a T85 atmosphere (experiment b30.009 [seeCollins et al., 2006]) sampled on the model's native ocean grid (1° nominal horizontal resolution) in the region 20°N–65°N and 120°E–110°W. This dataset was part of the cross-model comparative study byGiannakis and Majda [2012b] (dataset designation C85), where more details of the data and NLSA algorithms can be found. Here, we note that of key importance was to analyze the data without annual cycle subtraction or partitioning into seasonal averages. In separate calculations, we found that removing the monthly climatology distorts the nonlinear data manifold in a manner that eliminates the dynamically important intermittent modes, which are central to the regression framework developed here.

[6] NLSA produces a set of temporal patterns vk(t) (analogous to principal components) and a corresponding set of orthogonal spatial patterns ukin lagged-embedding space (analogous to extended EOFs), which can be grouped in families of periodic, low-frequency, and intermittent modes. Thevk(t) patterns are orthogonal with respect to a weighted inner product associated with square-integrable (L2) spaces of scalar functions on the nonlinear data manifold [Giannakis and Majda, 2012a]. The regression models introduced below are based on various combinations of vk(t), which are distinguished using the notations Pi(t), Li(t), and Ii(t) for the i-th periodic, low-frequency and intermittent modes, respectively, in order of explained variance. Specifically, we employ the two-fold degenerate set of annual modes {P1, P2} ↔ {v1, v2}; the PDO, NPGO, and ENSO modes {L1, L2, L3} ↔ {v5, v6, v7}; the leading four intermittent modes {I1, I2, I3, I4} ↔ {v10, v11, v12, v13}. The time dependence and frequency spectra of these modes are shown in Figure 1; Animation S1 in the auxiliary material displays the dynamic evolution of the projected SST anomaly field onto the corresponding spatial patterns uk [see also Giannakis and Majda, 2012b]. Certain noteworthy features of the modes are as follows.

Figure 1.

Example timeseries, shown for years 420–470 of CCSM3 experiment b30.009, and power spectral densities (PSDs) fk(ν) of the temporal modes vk(t) used for regression modeling. The PSDs were estimated via the multitaper method [Thomson, 1982; Ghil et al., 2002] with time-bandwidth productp = 6 and K = 2p− 1 = 11 Slepian tapers. The effective half-bandwidth resolution for theN = 5376 available monthly samples is Δν = p/(N δt) = 1/75 y−1, where δt = 1 m is the sampling interval.

[7] The annual periodic modes, {P1, P2}, are sine and cosine waves evolving in temporal quadrature, and are not equivalent to monthly climatology [Giannakis and Majda, 2012b]. They provide crucial phase information for the interaction between the low-frequency and intermittent modes. ModesL1 and L2correlate strongly with the leading two EOFs of seasonally-detrended North Pacific SST. Specifically, the maximum spatial correlation coefficients of SST EOFs 1 and 2 with the correspondingu5 and u6modes within a 2 y lagged-embedding window are 0.97 and 0.83, respectively. In light of these correlations, we refer toL1 and L2 as PDO and NPGO modes, respectively, following the conventional terminology in the literature [Di Lorenzo et al., 2008]. Characterized by significant power over interannual to decadal timescales and absence of significant spectral peaks, modes of this type have been employed in bivariate indices for low-frequency SST variability [Bond et al., 2003; Di Lorenzo et al., 2008]. Mode L3 is the manifestation of ENSO in the North Pacific sector of CCSM3, involving a coastal Kelvin wave pattern off the California coast; see Animation S1. The spatial correlation coefficient between this mode and the leading mode of seasonally-detrended global SST is 0.93 (maximum value in a 2 y lagged-embedding window). It is a known issue with CCSM3 that the extratropical ocean is less strongly correlated to the tropics than in nature. Moreover, as manifested by the concentration of spectral power for ModeL3 in 2–4 y timescales (Figure 1), the model's ENSO exhibits a degree of biennial quasi-periodicity which is not observed in the real ocean [Collins et al., 2006].

[8] The intermittent modes come in near-degenerate pairs, {I1, I2} and {I3, I4}, but do not evolve in exact quadrature. Prominent spatial features of pair {I1, I2} are SST anomalies in the North Pacific current (flowing eastward along ∼40°N) and subtropical gyre accompanied with anomalies of opposite sign in the Alaska and California currents. Pair {I3, I4} is characterized by SST anomalies shed eastward from the coast of Japan in the Kuroshio–Oyashio extension region, as well as activity in the subpolar gyre. Modes I1I4 all have the temporal structure of modulated signals with a carrier frequency of 1 y−1. This means that they can be demodulated to extract their low-frequency amplitude and phase components via linear combinations of products with the annual periodic modes of the formPi(t)Ij(t).

[9] A key property of vk(t) is that they exhibit non-trivial higher-order lagged correlations; in particular, triple and quadruple correlations. For instance, the PDO and NPGO modes have negligible double correlations with the intermittent modes, but rich lead–lag relationships emerge when triple correlations with the periodic modes are considered. Such relationships are shown inFigure 2, where, among other features, a time asymmetry between 〈I3(τ)P2(τ + t1)L1(τ + t2)〉 (mode I3 leading) and 〈L1(τ)P2(τ + t1)I3(τ + t2)〉 (mode I3 lagging) is clear. More generally, as shown in Figures S1–S3 in the auxiliary material, the structure of the triple correlation coefficients ρijk at δt = 1 m lag (the regression timestep used in this work) exhibits groups of modes vi(t)vj(t)vk(t + δt) which “stand out” in absolute correlation value (in other words, lie in high percentiles of the correlation coefficient distribution) compared to other triple pairs associated with small |ρijk|. It is natural to incorporate the associated products, vivj, as predictor variables in regression models for vk. The quadruple correlations, ρijkl, also exhibit a similar behavior, suggesting that certain cubic interactions, vivjvk, may also be dynamically important.

Figure 2.

Selected triple lagged correlations ρijk(t1, t2) = 〈vi(τ)vj(τ + t1)vk(τ + t2)〉 of the temporal modes from NLSA. Here, angle brackets denote temporal averaging with respect to τ, so that the first argument in 〈vi(τ)vj(τ + t1)vk(τ + t2)〉 is leading. The correlation functions were estimated from the empirical covariance of the N′ = N/2 = 2688 monthly samples in the training dataset for each vi(t). All modes were standardized to zero sample mean and unit variance, inline imagevi(tl) = 0, inline imagevi2(tl)/N′ = 1.

3. Regression Methodology

[10] Following the red noise paradigm of Frankignoul and Hasselmann [1977], the canonical autoregressive models for low-frequency SST variability in the North Pacific have the form

display math

In the above, the i-th low-frequency mode at timet + δt depends linearly on a vector Li(t) of low-frequency modes at timet (also governed by prognostic equations similar to (1)), a vector of prescribed external factors Fi(t), and a scalar Gaussian process ϵ(t). For instance, a model of this type for mode L1 with L1 = [L1], F1 = [L3], and 1 y timestep corresponds to the ENSO-forced scalar regression model for the PDO ofNewman et al. [2003]. Dropping F1 and assuming that ϵi(t) is uncorrelated in time leads to a standard red noise model for regime behavior in the North Pacific [Rudnick and Davis, 2003]. Here, in light of the more recent paradigm involving both the PDO and NPGO [Bond et al., 2003; Di Lorenzo et al., 2008], our baseline regression setup is a bivariate model for modes L1 and L2 with autoregressive components L1 = L2 = [L1, L2], and, optionally, an ENSO external factor F1 = F2 = [L3]. Note that the timestep δt in these models is 1 m rather than 1 y.

[11] Motivated by the correlation results of Figure 2 and Figures S1–S3, we put forward an alternative model for the low-frequency SST modes:

display math

In equation (2), the terms [P * I]i(t) and [L * L * L]i(t) are vectors whose elements are given by double and triple products of the form Pj(t)Ik(t) and Lj(t)Lk(t)Ll(t), respectively. Thus, compared to the canonical case in equation (1), this class of models features (i) forcing of the low-frequency PDO and NPGO modes by the intermittent modes, suitably demodulated via products with the annual periodic modes; (ii) cubic interactions between the PDO and NPGO modes. The Gaussian noisesϵi(t) are not necessarily uncorrelated but are assumed to be Markovian; i.e., 〈ϵi(t)ϵj(τ)〉 may be nonzero but the increments Δϵi(t) = ϵi(τ + t) − ϵi(t) are uncorrelated for all τ > 0. Continuing this line of argument, we write down the corresponding regression model for the intermittent modes,

display math

where the periodic and low-frequency modes are treated as external factors.

[12] Potentially serious drawbacks of models with nonlinear terms such as equations (2) and (3)is a superlinear scaling of the number of regression coefficients with the number of modes (here, quartic scaling), as well as finite-time blowups [Majda and Yuan, 2012]. Blowups may occur independently because of structural model error in the regression model (if the parametric family of the proposed regression models does not include the true model), or statistical error in the estimated model coefficients (e.g., if the design matrix of the regression model is ill-conditioned due to collinearity of the predictor variables). In such situations, imposing sparsity via variable selection can significantly improve skill, at the expense of some bias being introduced if variables which are formally part of the true model are rejected.

[13] Here, we sparsify our regression models via a simple univariate thresholding scheme, which is also effective in ensuring dynamical stability when the models are integrated to produce decadal or longer hindcasts. Specifically, for some a priori chosen parameters θA, θB, and θC, we retain only those terms of the form vi(t), vi(t)vj(t), and vi(t)vj(t)vk(t) in the right-hand side ofequations (2) and (3) meeting the conditions |〈vi(τ)vl(τ + δt)〉| ≥ θA, |〈vi(τ)vj(τ)vl(τ + δt)〉| ≥ θB, and |〈vi(τ)vj(τ)vk(τ)vl(τ + δt)〉| ≥ θC, respectively, where vl is the response variable in the model under consideration. Hereafter, we set θA = 0.02, θB = 0.2, and θC = 0.75. As illustrated in Figures S1–S3, the admissible predictor variables for these threshold values typically lie in high percentiles of the cumulative distributions of the corresponding univariate correlation coefficients with the response variable. In separate calculations, we have verified that comparable hindcast skill results can be achieved by more conservative (smaller) θ values, increasing the complexity of each model by ∼5 coefficients.

[14] Our variable selection scheme based on marginal contributions to the model response is neither intended to be applicable in every regression scenario, nor always capable of identifying the optimal model. However, the approach is similar in spirit to screening methods analyzed formally in the statistics literature [Fan and Lv, 2008]. In the applications studied here, univariate thresholding was particularly useful in homing in on candidate models which are highly skillful in explaining the variability of independent test data. Without discounting the importance of model selection, for the purpose of identifying limits of predictability (the main goal of this work) the availability of any sufficiently skillful model is sufficient; how the structure of that model was arrived at is a secondary issue.

4. Results and Discussion

[15] We trained bivariate regression models for the PDO (L1) and NPGO (L2) modes specified in equations (1) and (2) by applying ordinary least squares to the first half of the available timeseries for vk(t). Here, we present results obtained from a suite of five regression models. Model 1 is a standard autoregressive model from equation (1) with no external factors, model 2 incorporates the ENSO mode (L3) as an additive external factor, and models 3–5 are all forced by products of annual periodic and intermittent modes, in accordance with equation (2). The coefficients of model 3 were evaluated without including any of the linear (Ai) and cubic (Ci) terms. Model 4 augments model 3 by including Ai terms, and both Ai and Ciare incorporated in model 5. After variable selection, there a total of six forcings from the intermittent modes and two cubic interactions per low-frequency mode. Using the same portion of the available data for training, we also built four models for the intermittent modes based onequation (3). In this case, models 1 and 2 contain only Ai or Bi terms, respectively, model 3 contains both Ai and Bi, and model 4 features all of Ai, Bi, and Ci coefficients.

[16] The properties of the models, including the interaction coefficient values, condition number κof the design matrices, and the Durbin-Watson (DW) statisticd of the residuals are listed in Tables S1–S5 in the auxiliary material. Further information about the residuals is provided in the histograms, quantile-quantile plots, and time autocorrelation functions inFigures S4–S9. In all cases, the design matrices are well conditioned (κ ≲ 45), indicating that the models are not adversely affected by predictor collinearity. The residuals exhibit a high degree of Gaussianity, but, according to the DW and autocorrelation results for ϵi(t), are not time independent. On the other hand, the residual increments Δϵi(t) are independent at 1 m lags to a good approximation (apart from models 2 and 4 for the intermittent modes), as indicated by the corresponding d≈ 2 DW statistics. The experiments presented here were performed with a standard Gaussian white noise model, which led to excellent hindcast skill. However, in an operational forecast setup, a time-correlated noise process may be appropriate. We have performed a posteriori tests for blowups by integrating the models with external factors prescribed for the full second half of the available data (224 years, or 2688 monthly timesteps). No blowups occurred in those tests. We used the same test interval to evaluate the pattern correlation score and root mean square error (RMSE) of 10 y hindcasts. Example realizations and skill scores are shown inFigure 3.

Figure 3.

Realizations and skill scores for the regression models of the (a) PDO and NPGO modes from equations (1) and (2), (b) the intermittent modes from equation (3). In Figure 3a, results from models 1–5 of Table S1 are plotted in cyan, red, magenta, green, and blue, respectively. In Figure 3b, results from models 1–4 of Tables S2–S5 are plotted in red, magenta, green, and blue, respectively. Black lines correspond to the truth signal in all cases. The RMSE score up to a maximum lead time tmax = 240 m was computed via RMSEk2(t) = inline image(yk(t; ti) − vk(t + ti))2/M, where yk(t; ti) is the predicted value from the regression model for mode vk initialized with initial data yk(0; ti) = vk(ti) at times ti ∈ [N/2 + 1, Ntmax] (i.e., the first M = N/2 − tmax samples in the test dataset). The pattern correlation score for t > 0 is ck(t) = inline image, where inline image, inline image, inline image, and inline image. Note the skill enhancement realized by including products of the seasonal cycle with intermittent modes (Figure 3a), or low-frequency modes (Figure 3b), as external factors (magenta, green, and blue lines).

[17] First, we turn to the canonical autoregressive models for low-frequency variability (models 1 and 2 inTable S1). These models fail to track the PDO and NPGO on average beyond two and five years, respectively, with the ENSO-based external factor model achieving a moderate gain of 15–20% in residual pattern correlation score. In contrast, all of the models forced by products of the periodic and intermittent modes are able to explain the truth signal with pattern correlation exceeding 90%. The models which do not permit any interaction between the prognostic variables are generally less skillful than those with both linear and cubic terms. In particular, the RMS errors of theL1 (PDO) and I2 components of the cubic models (blue lines in Figure 3) are smaller by approximately 50% and 30%, respectively, relative to the corresponding non-interacting cases (magenta lines). Other response variables, such asL2 (NPGO) and I1, accrue negligible improvement of skill compared to the non-interacting case. It is interesting that models which include linear but not cubic interactions (green lines) are somewhat overdamped, deviating from the truth signal over portions of the hindcast interval. This feature, which supports the dynamical significance of combined linear-cubic interactions, is seen in both low-frequency and intermittent-mode regression models.

[18] The main conclusion from this analysis is that the forcing by the intermittent modes, demodulated by the annual periodic modes, is able to reproduce the timeseries for the PDO and NPGO modes in CCSM3 with remarkable skill, despite the fact that these modes are orthogonal in both lagged embedding space (through the corresponding spatial patterns uk) and time (with respect to the weighted inner product in NLSA). Thus, our analysis suggests that the predictability of the prominent low-frequency North Pacific SST patterns in this climate model is limited by the predictability of the spatial features associated with the intermittent modes, such as the boundary currents and the subpolar and subtropical gyres, suitably phased by the seasonal cycle. Conversely, we have demonstrated that the low-frequency modes are highly significant external factors in tracking the behavior of the intermittent modes. As a result, the low-frequency and intermittent sets of patterns behave in a complementary manner, limiting each other's predictability. This picture appears to be at odds with the canonical models of low-frequency variability of North Pacific SST inequation (1), where predictability of low-frequency variables is governed by ENSO signals evolving at interannual timescales and the decorrelation time of a red noise process. Even though it is possible that the anomalously low correlation between the North Pacific and the tropics in CCSM3 [Collins et al., 2006] contributes to the poor skill of the ENSO-forced models studied here, the skill of the top-performing models inFigure 3should provide motivation to seek intermittent-type modes in observational data, or train regression models with the structure ofequations (2) and (3) in other climate models where such modes have already been identified [Giannakis and Majda, 2012b].

[19] The general approach developed here should be useful in finding the limits of predictability in other complex models with other forecasting scenarios. However, it is important to note we have not constructed such a scheme for self-contained forecasts. That would involve time-advancing the governing equations forLi and Ii in tandem, which may require additional interactions or prognostic variables if the model is to achieve skill beyond standard AR models. To that end, a potentially fruitful avenue of future research would be to apply statistical methods developed in the context of Bayesian spatiotemporal modeling [Wikle and Holan, 2011, and references therein] to inform the structure of self-contained forecast models.


[20] This work was supported by NSF grant DMS-0456713, ONR DRI grants N25-74200-F6607 and N00014-10-1-0554, and ONR MURI grant N-000-1412-10912.

[21] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.