This paper investigates the predictive relation between the Atlantic multidecadal oscillation (AMO) and Atlantic meridional overturning circulation across different climate models. Three overturning patterns that are significantly coupled to the AMO on interannual time scales across all climate models are identified using a statistical optimization technique. Including these structures in an autoregressive model extends AMO predictability by 2–9 years, relative to an autoregressive model without these structures.
The Atlantic multidecadal oscillation (AMO) is a major component of climate variability characterized by basinwide changes in Atlantic sea surface temperatures (SSTs) that can persist for a decade or more. The AMO has been linked to a number of prominent decadal-scale climate anomalies, including African drought (Hoerling et al. 2006; Knight et al. 2006), hurricane frequency in the Atlantic (Goldenberg et al. 2001), drought in the central United States (McCabe et al. 2004), and rainfall in northeastern Brazil (Knight et al. 2006). Because of these connections to high-impact phenomena, accurate predictions of the AMO would be of tremendous socioeconomic importance.
Traditionally, the AMO index has been defined as the area average North Atlantic SST after the trend has been removed. The resulting index, shown by the black curve in Fig. 1, reveals warm periods during 1930–60 and 2000–14 and a cool period between 1960 and 2000. Although this particular index of the AMO has been criticized (Trenberth and Shea 2006; Mann et al. 2014), alternative AMO indices consistently agree that the AMO can persist in the same phase on decadal time scales.
The mechanisms responsible for AMO variability are debated vigorously. One hypothesis is that the AMO is a component of internal variability. Evidence supporting this interpretation includes instrumental data (Kushnir 1994; Mann and Park 1994; Schlesinger and Ramankutty 1994), proxy reconstructions (Delworth and Mann 2000; Gray et al. 2004), and climate models (Delworth et al. 1993; Timmermann et al. 1998; Delworth and Mann 2000; DelSole et al. 2011). Many studies attribute the internal variability to meridional advection of heat by the ocean, especially the Atlantic meridional overturning circulation (AMOC; Delworth et al. 1993; Timmermann et al. 1998; Jungclaus et al. 2005; Yeager et al. 2012). It has been argued by some that the AMO can be generated in the absence of low-frequency variations of the ocean circulation. Specifically, stochastic atmospheric heat flux forcing is believed to play a significant role in forcing low-frequency SST in the North Atlantic (Griffies and Bryan 1997b; Schneider and Fan 2012). This mechanism is further substantiated by the recent work of Clement et al. (2015), who found that the AMO could be generated in slab-ocean model experiments. Alternatively, Booth et al. (2012) proposed that AMO variability is driven primarily by aerosol emissions, with cooling periods forced by enhanced cloud reflectivity resulting from aerosol–cloud interactions. Unfortunately, none of the above mechanisms are completely satisfactory. For instance, although most climate models can capture multidecadal variability of both the North Atlantic SST and the AMOC, these models show considerable differences in the amplitude and frequency ranges of these variations (Medhaug and Furevik 2011; Zhang and Wang 2013; Ba et al. 2014; Muir and Fedorov 2015). Also, a number of climate models show an SST response to changes in the strength of the AMOC, but the phasing and strength of covariability differ between individual models. Moreover, the presence of external forcing interferes with these apparent relations (Tandon and Kushner 2015). Finally, the simulations used by Booth et al. (2012) to support the argument that aerosols play a dominant role were found to be problematic in other areas, such as inconsistencies between modeled and observed upper-ocean heat content variability (Zhang et al. 2013).
Despite unresolved questions about mechanisms, most studies agree that AMO variability is predictable on multiyear time scales (Griffies and Bryan 1997a; Boer 2004; Pohlmann et al. 2004; Collins et al. 2006; DelSole et al. 2011, 2013). Actual forecast skill for the observed North Atlantic SST is on the order of 3–5 years (Zanna 2012). Climate models show comparable skill, though forecast accuracy is model dependent (Kim et al. 2012).
From where does the predictability of the North Atlantic SST arise? As mentioned above, many modeling studies suggest that the predictability arises from predictable fluctuations of the AMOC. Initialized model experiments find improved predictive skill of North Atlantic SST (e.g., Keenlyside et al. 2008; Pohlmann et al. 2009; van Oldenborgh et al. 2012; Ho et al. 2013; Matei et al. 2012; Doblas-Reyes et al. 2013; Hazeleger et al. 2013; Yang et al. 2013; Ham et al. 2014), but it is unclear exactly how the ocean is providing this skill. However, Branstator and Teng (2014) found that there are some highly predictable components of upper-ocean heat content that are unrelated to AMOC. Moreover, Tandon and Kushner (2015) found that external forcing can inflate the predictive AMOC–AMO relation in some models. Potential predictability in the SST–AMOC relation is often inferred from cross correlations between representative indices of the large-scale North Atlantic temperature and AMOC fluctuations. Inferring a predictive relation between the AMO and AMOC in terms of cross-correlated indices is problematic for a number of reasons. First, the indices are often heavily smoothed prior to analysis; as a result, the lead–lag relation between the AMO and AMOC is distorted. Consequently, the inferred predictability time scales may appear to be greater than they are. In the appendix, we demonstrate this effect through Monte Carlo simulations and cross-correlation analysis of smoothed bivariate autoregressive processes. Second, the index representing the AMOC (i.e., the maximum streamfunction value) may not accurately represent the component of the oceanic overturning most related to the AMO.
In this study we present a rigorous methodology for isolating the AMOC contribution to AMO variability that takes into account all of the aforementioned weaknesses. The principal goal of this work is to understand the source of AMO predictability in long control simulations from phase 5 of the Coupled Model Intercomparison Project (CMIP5) archive. Rather than use a single index to represent the AMOC variability, a novel optimization technique is used to identify structures of oceanic meridional overturning streamfunction most related to AMO (DelSole and Tippett 2009; Jia and DelSole 2011). This method maximizes the squared correlation between the AMO and the leading principal components of oceanic meridional streamfunction, when integrated over positive and negative lags. This maximized quantity has units of time and is called the average predictability time (APT). The APT analysis yields a set of AMOC patterns most related to the AMO across all time scales. The corresponding APT time series are used as representative indices of the AMOC. In section 4c, we will demonstrate that these indices can explain more of the AMO variance than the standard maximum streamfunction. To aid in our investigation of AMO predictability, an empirical model of the AMO is constructed for each climate model.
In the following sections, the APT optimization technique is explained (section 2) and the datasets utilized in this paper are described (section 3). In section 4, the results of applying APT analysis to climate model simulations are described. In particular, we find three AMOC structures that are related to the AMO on interannual-to-multidecadal time scales across the eight climate models considered in this study. We find that across all climate models, the evolution of the AMO can be modeled as a first-order autoregressive [AR(1)] process forced by the AMOC, or equivalently as an AR(1) model with exogenous inputs, called an ARX(1) model. Further, we find that for most climate models, the AMOC both leads (by one year) and simultaneously forces the AMO. Relative to an AR(1) model, the AMOC is found to substantially improve AMO predictability for all of the climate models investigated here.
a. Generalized APT and decomposition
The APT optimization technique developed by DelSole and Tippett (2009) finds the linear combination of variables that maximize a predictability measure when integrated over all time lags. This technique contrasts with other techniques such as predictable component analysis (Schneider and Griffies 1999), signal-to-noise EOFs (Venzke et al. 1999), and canonical correlation analysis (Barnett and Preisendorfer 1987) by taking into account more than one lag. DelSole and Tippett (2009) applied this technique to 1000-hPa zonal velocities and showed that it seamlessly diagnosed predictability on time scales ranging from weather to climate. This technique has been applied to climate model output to diagnose the most predictable components of annual mean sea surface temperatures on multiyear time scales (DelSole et al. 2011). APT has also been used to diagnose the components of land-based surface temperature and precipitation that are predictable across multiple climate models (Jia and DelSole 2011).
APT analysis begins with seeking the linear combination of the elements of a vector y that maximizes the integrated square correlation with a variable x (which is a scalar). Both y and x are realizations of a stochastic process. We will denote the linear combination of vector y by a new quantity z = qTy, where q contains the weights. The standard square correlation coefficient for z and x at lag τ is given by the following formula:
where Cxx and Czz are the variances of x and z, respectively, and Cxz is the time-lagged covariance between x and z. The APT of z is defined as follows:
We seek the weights q that maximize APT. Substituting the relations
into (1) gives
Equation (4) is a Rayleigh quotient, and it is well known that optimization of this quotient leads to the generalized eigenvalue problem:
As discussed in DelSole and Tippett (2009) and Jia and DelSole (2011), the eigenvalues of (6) equal the APT, the eigenvectors give the corresponding weights, and the time series qTy corresponding to different eigenvectors are uncorrelated. It is convention to order the eigenvalues in descending order, in which case the leading eigenvalue is the maximum APT, the second eigenvalue gives the maximum APT out of all weights whose time series are uncorrelated with the first, and so on. The loading vector for the APT is obtained by projecting the component times series qTy onto the original data (i.e., the loading vector is a regression map between the component time series and y). As in Jia and DelSole (2011), APT analysis is generalized to allow the predictor to differ from the predictand. Our analysis expands upon Jia and DelSole (2011) by allowing integration to span both negative and positive lags, including zero. We purposely include both positive and negative lags, to allow for the possibility that the AMOC forces the AMO, and vice versa.
b. APT significance
The statistical significance of the APT is tested relative to the null hypothesis of white noise, or no predictability. To do this, APT analysis is repeated using random variables drawn independently from a Gaussian distribution and sampled to have the same dimensions as x and y (section 4a). One thousand Monte Carlo experiments are performed, and the 95th percentile for each eigenvalue is selected. The null hypothesis of no predictability is rejected if the APT values exceed the 95th percentile.
We examine climate model simulations from CMIP5. Because we are interested in diagnosing the relation between the AMO and AMOC in the absence of global warming, the preindustrial control runs from the CMIP5 archive are analyzed. Eight models were selected for analysis and are described in Table 1. For the sake of simplicity, we refer to model-specific results using the label of the associated modeling center name acronym. It should be noted that different modeling centers use various model versions, and our naming convention applies only to the experiments listed in Table 1. To be selected for study, model output must contain at least 500 years of simultaneous time series of SST and oceanic mass overturning streamfunction. Studies have shown that the global SST from some CMIP5 and CMIP3 preindustrial control experiments are contaminated by incomplete spinup, the effects of which are most pronounced in the first 50 years of simulation (Boer and Lambert 2008; Sen Gupta et al. 2013). To minimize this effect, the first 50 years were discarded from each model. To allow for intermodel comparison, the streamfunction data from each model are interpolated to a common latitude–depth grid of dimension 154 × 41. Latitude has a 1° resolution and spans the Atlantic from 75°S to 75°N. Depth has a nonuniform grid, with higher resolution in the upper ocean, down to a maximum depth of 5.42 km. Prior to analysis the data are split into a training period covering the first 225 years and a test period of the last 225 years. The data are subdivided to allow for verification of results. A majority of the analysis is based on annual mean data. Annual anomalies are found by detrending and centering the yearly data separately with respect to the training or test period. An index for the AMO is defined as the area-weighted annual sea surface temperature anomalies (SSTAs) over the North Atlantic from 0° to 60°N.
An estimate of the observed AMO index is computed from the NOAA ERSST.v3b dataset, which is a global monthly sea surface temperature product based on the International Comprehensive Ocean–Atmosphere Data Set (ICOADS) covering the period from January 1854 to the present (Smith et al. 2008).
APT analysis is performed using the AMO index and a vector of multimodel principal components (PCs) of the Atlantic meridional mass overturning streamfunction. In the notation of section 2, the AMO index corresponds to scalar x, and the vector of multimodel PCs of the Atlantic meridional mass overturning streamfunction corresponds to vector y. Because of the presence of a nonuniform vertical grid, streamfunction data are weighted by the square root of the layer thickness prior to EOF analysis (Baldwin et al. 2009). APT, like all optimization techniques, is subject to overfitting when the number of parameters is not a small fraction of the sample size. To mitigate this effect, the number of retained PCs is dependent upon the temporal resolution of the data; if annual data are used, 10 PCs are retained, while for decadal data the number is reduced to 2. These truncations were chosen based on robustness of results between two independent segments of the control simulation.
In addition to the above analysis, we performed a separate analysis using a modified index of North Atlantic SST variability. The new index is found by regressing out the lag-1 relation from the annual mean AMO index [i.e., we fit the AMO index to an AR(1) model and then investigate the residuals]. Optimization is repeated using the residual AMO index and the multimodel PCs of the Atlantic meridional mass overturning streamfunction. The intent of this separate analysis is to isolate the component of the AMO variability that cannot be captured by an autoregressive model. The two separate analyses yield consistent results; hence, we show only one of these—namely, those obtained from optimization of the standard AMO index.
b. Interannual-to-multidecadal relation between AMO and AMOC
To identify the common features in the AMO–AMOC relation across all climate models, we adopt a multimodel optimization, where APT is maximized across multiple models by pooling the covariance matrices. We also conducted separate analysis in which optimization was performed separately for each model. This latter analysis found that the leading four APT components of each model were similar in spatial structure to the multimodel components (except with different orderings) and were separately statistically significant in each model. Since multimodel and individual-model APT analysis give similar significant structures, there is no loss of generality in simply using structures derived from multimodel analysis. This choice also simplifies interpretation since the structures being analyzed are not model dependent. Given that both the observed and simulated AMO and AMOC display high-frequency variations that might mask low-frequency relations, APT analysis is repeated for decadal means. Of the eight models analyzed, only MRI has statistically significant APT values on multidecadal time scales. We conclude that there is no robust multidecadal relation between AMO and AMOC in these models, so we restrict our analysis to annual means.
The APT values derived by maximizing across all models are shown in Fig. 2. This figure shows that the eigenvalues associated with the first five components are significant. However, separate analysis reveals that the fourth and fifth variates are not significant in the verification data in all of the models and have relatively weak correlation with the AMO index; as such, they are excluded from further analysis.
The structures associated with the leading three APT components are shown in Fig. 3. The patterns of the first two leading APT components show some similarities in structure to leading EOFs of the low-pass-filtered North Atlantic streamfunction (MacMartin et al. 2013). The spatial pattern of the leading APT component (i.e., the structure most related to the AMO) is characterized by a single overturning cell (Fig. 3, top), with a peak variation centered about 40°N and a depth of 1000 m. This pattern is indicative of localized changes in AMOC strength and is consistent with the hypothesis that changes in northward heat transport are linked to AMO variability (Delworth et al. 1993; Delworth and Mann 2000; Knight et al. 2005; Zhang et al. 2007). The structure of the second APT component (Fig. 3, middle) is characterized by a tripole of oppositely signed overturning anomalies, with the largest changes occurring in the subtropical and subpolar regions. Modeling studies have identified these opposing overturning circulation variations in the North Atlantic as the fast AMOC response to a negative phase of the North Atlantic Oscillation (NAO; Eden and Willebrand 2001; Deshayes and Frankignoul 2008; Gastineau and Frankignoul 2012). The wind stress associated with the negative NAO leads to warming over much of the North Atlantic through changes in the surface heat flux (Cayan 1992), while Ekman transport associated with this pattern tends to reinforce the warm SST at higher latitudes and cool the subtropical SST (Eden and Willebrand 2001). The delayed response of the AMOC to a negative-phase NAO is a spindown of the subpolar gyre and a decrease in the AMOC transport (Eden and Willebrand 2001). The third APT component captures peak streamfunction variations between 40° and 60°N at depths 1000–3000 m. This structure shows some similarities to the low-frequency AMOC pattern previously identified by Msadek and Frankignoul (2009). Collectively, the leading APT components are consistent with AMOC structures linked to AMO variability and likely capture aspects of the AMOC–AMO dynamics.
c. APT components versus ψmax
The meridional overturning of the ocean plays an important role in maintaining the mean climate of the North Atlantic region through poleward heat advection. It has been suggested that fluctuations in the strength of this circulation strongly impact SST variability of the region (e.g., Bjerknes 1964). Intensity changes in meridional overturning strength are often represented by the maximum value of the meridional circulation ψmax (e.g., Delworth et al. 1993). Presumably, the ψmax index captures the portion of the meridional circulation most associated with northward heat transport and SSTs. To evaluate the predictive skill of the ψmax index versus the time series of the leading three APT components, we compare the amount of AMO variance that can be explained by both sets of predictors as a function of lag (i.e., we compare the R2 values; Fig. 4).The square multiple correlation coefficient between the three APT components and the AMO index (Fig. 4, red line) is greater than the square correlation between ψmax and the AMO index (Fig. 4, blue line) for most of the climate models. Depending upon the climate model, the three APT components collectively explain 10%–25% more of the AMO variance than the ψmax index, indicating that AMOC structures not captured by the ψmax index are important for the modeled North Atlantic SST variability. Across the models, the relation between the APT components (Fig. 4, red line) and the AMO (or ψmax and AMO, for that matter) are quite diverse.
The relation between APT components and AMO captured during the training period (Fig. 4, red line) may exaggerate the true relation between these variables, as a result of overfitting. To obtain an independent estimate of the strength of this relation, we compute R2 for the independent verification period using the APT components computed from the training data. As noted above, the lead–lag squared correlations between the three APT components and the AMO for the training period are shown as the red curves in Fig. 4; the black curves are for the same analysis but for the verification period. Comparing these two curves, it is evident that the relation between the AMO and APT components is not robust. Not only does the structure of R2 change between the two periods, so too does the magnitude the correlation. Differences between the R2 for the verification and training periods indicate that the relation between the AMO and AMOC is noisy and sensitive to sampling.
It should be noted that we perform a multimodel-based APT analysis because it allows us to investigate the AMOC–AMO relation across models using a consistent set of AMOC structures. While common AMOC structures are considered, the individual model APT component time series are found by projecting the individual model AMOC PCs onto the common APT loading vectors. This projection ensures that both the recovered APT component time series and the relations between these time series and a given model’s AMO are unique to that particular model. The unique model-specific relations are clearly demonstrated by the R2 shown in Fig. 4. Moreover, the relation between the APT components and the AMO is greater than the relation between ψmax and the AMO index, as indicated by the larger R2 values. The ψmax index is strongly related to the first APT component, with R2 values ranging from 0.3 to 0.7, depending on the model. The relation between ψmax and the second and third APT components is significantly reduced across all models. Though for most climate models, correlations between the third APT component and ψmax tend to be greater than those found for the second. Therefore, application of the multimodel optimization allows us to capture more meaningful AMOC–AMO relations across the set of models than would have been possible from standard ψmax–AMO relation.
d. Empirical model of the AMO
The cross-correlation analysis above reveals that the AMOC is significantly related to the AMO across the climate models, but in diverse ways. These relations are not robust (i.e., they are dependent on sampling), with correlations largest at the shortest lags. Given these results, we question whether the AMOC contributes significantly to the AMO variability on longer time scales. To examine AMO predictability in relation to the AMOC, we construct an empirical model for the AMO for each climate model and evaluate its skill as a function of lead time.
Traditionally, the null hypothesis for SST variability is defined as the upper-ocean response to stochastic atmospheric forcing (Hasselmann 1976). According to this mechanism, the weather noise forces the ocean mixed layer through changing surface fluxes, and these random changes are integrated by the mixed layer, resulting in a reddened SST spectrum. In addition to stochastic forcing, SST variability has been linked to heat transport via the ocean. To account for both processes, we model the evolution of the AMO as a forced AR(1), or equivalently as an ARX(1) model:
where AMO(t − 1) is the lag-1 AMO index; ψi* is a time series of the ith APT component; ϕ, βi, and γi are regression coefficients to be estimated from the coupled climate models separately; and ε is noise. Recall that APT components four or greater were inconsistent across the models and account for less than 10% of the explained variance for each model, so these are a priori excluded from the ARX(1) model.
The model (7) was selected from a number of potential models by evaluating the increase in skill gained by the incremental inclusion of higher-order terms for AMO and ψi* separately. Specifically, we incrementally added higher-lagged predictors to the empirical model and then tested the hypothesis that the corresponding regression coefficients are zero. Introduction of these higher-order terms is considered to significantly increase the model skill if the F statistic exceeds the critical F value at a 95% level. Following this methodology, it was found that the lag 1 was sufficient for the AMO term, across all climate models. The models showed some differences in terms of the number of lagged APT components to retain. The predictive skill for most climate models showed the greatest increase in explained variance when the ψi*(t − 1) terms were included. Predictive models for GFDL and MRI showed a slight increase in skill when higher-order lag terms of ψi* were added. The introduction of these higher-order lag terms, however, did not substantially improve the predictive skill of the empirical models. Moreover, the inclusion of each new higher-order term makes the model susceptible to overfitting. To mitigate this possibility, we selected the lowest-order model [i.e., (7)] since it yields the greatest gain in explained variance. Moreover, inclusion of higher-order terms only marginally improves the skill at the long lead times for all climate models.
e. Evaluating AMO predictability
To quantify the predictive relation between the AMO, oceanic forcing, and year-to-year persistence in temperature anomalies, we evaluate the empirical model at multiple lead times τ. The skill as a function of lead is quantified by the multivariate R2 and is shown by the black curves in Fig. 5. Squared correlations are considered significant at a 5% level when the estimate exceeds the critical R2 value, which is designated by the dashed black curve. Ignoring the red and blue curves for the moment, it is evident that AMO predictability varies significantly between climate models. At the lower end of predictive skill, the CCC and NCAR models captured a maximum AMO predictability at a 2-yr lead (at higher lead times the correlations are insignificant). The MPI model possesses the greatest predictive skill, with significant skill out to a 9-yr lead. For the remaining models, predictive skill is significant out to leads of 3–7 years.
As noted above, the ψmax index is often used to represent intensity changes in meridional overturning strength. With the correlation analysis presented in Fig. 4, we demonstrate that there are some elements of the AMOC as it relates to the AMO that are not captured by ψmax. These additional features of the streamfunction are captured by the APT components. It is likely, then, that predictability of the AMO as it relates to ψmax will be less skillful than prediction based on APT components. To test this hypothesis, we construct an ARX(1) model using lag terms of ψmax as the exogenous inputs. As before, the ARX model is evaluated at multiple leads and skill quantified in terms of the R2 values, which are shown as the red curves in Fig. 5. Comparing the R2 values shown by the black and red curves, collectively we can see that a model utilizing the APT components always outperforms a model utilizing ψmax.
By construction, the AMO predictability derived from empirical model (7) results from both oceanic forcing and year-to-year persistence in SST. To measure the relative importance of oceanic forcing on AMO predictability, we fit a first-order autoregressive model to the AMO of each climate model and then evaluate the resulting AR(1) model at multiple lead times. As before, skill is measured by the squared correlation at different leads and is shown by the blue curve in Fig. 5. Correlations are significant at a 5% level if the estimates are greater than values given by the blue dashed curve in Fig. 5. Comparing the blue and black curves, it is clear that for all climate models AMO predictability is improved by including subsurface circulation indices. The exclusion of the AMOC terms from (7) greatly reduces the AMO predictability to less than a year for CCC, GFDL, MRI, and NCAR. For all other climate models, predictability vanishes in the year.
Replacing the APT components in (7) with the leading PCs yields comparable predictability across the climate models, though the R2 values tend to be slightly lower at short leads for a number of the models. Although the same level of predictability can be obtained using the leading PCs, one knows this only after the fact; that is, had we used only EOFs as predictors, the question would have arisen as to whether other structures could have given even more predictability. Since the APT methodology finds the linear combination of PCs that maximizes the integrated square correlation R2, it takes the guesswork out of the predictor selection problem. In essence, the APT components compress the predictability as much as possible into the first few components. Another conclusion that can be drawn is that the combination of leading EOFs captures not only variance (by definition) but also the AMOC structures most related to the AMO.
Based on model analysis, we conclude that the AMOC increases predictability of the AMO. However, the extent to which the ocean influences the AMO is highly model dependent. Unfortunately, no comprehensive three-dimensional measurements of the AMOC exist, so it is impossible to directly compare our model-based analysis with real-world estimates.
This paper identifies overturning structures in climate models that are most coupled to the AMO and shows that including these structures in an autoregressive model enhances AMO predictability relative to an autoregressive model without these structures. The relation between the AMO and AMOC was diagnosed using an optimization technique called APT analysis (see section 2), which identifies the component of meridional streamfunction that maximizes the sum-squared correlation with the AMO over positive and negative lags. When APT analysis is applied to decadal means, only one out of eight climate models possesses a robust AMO–AMOC relation. When APT analysis is applied to annual means, all eight models share a common set of AMOC structures that are significantly coupled to the AMO. The familiar large-scale overturning circulation often reported in the literature is the most coupled pattern. The second streamfunction pattern captures a tripole pattern centered around 20°N and is similar in structure to the previously identified NAO-forced streamfunction pattern. The third pattern is a deep overturning cell, with peak variations in the deep tropical ocean and the extratropics. These three components explain more variance at most lags than the maximum meridional streamfunction, whose value is often used as an index of the meridional circulation.
Predictability of the AMO as a function of lead time varies greatly across climate models. The AMOC is found to enhance AMO predictive skill in each climate model, with a range of significant predictive skill out to leads of 2–9 years, depending on the model. None of the climate models analyzed here exhibit multidecadal AMO predictability.
Last, the potential for temporal smoothing to produce misleading conclusions about predictability is illustrated in the appendix. In particular, examples of apparent predictability for a decade or longer are shown to be spuriously created by smoothing.
This research was supported primarily by the National Science Foundation (ATM1338427), the National Aeronautics and Space Administration (NNX14AM19G), and the National Oceanic and Atmospheric Administration (NA14OAR4310160). The views expressed herein are those of the authors and do not necessarily reflect the views of these agencies. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.
Smoothing and Cross Correlations
The lag correlations between AMO and AMOC examined in this paper suggest that AMO is predictable for 2–9 years. In contrast, some studies claim that the predictability could be on the order of a decade or more. However, many of these claims are based on the cross correlation of smoothed time series. The purpose of this appendix is to raise caution about inferring predictability time scales from the cross correlation of smoothed time series.
To illustrate the problem, we consider the following bivariate autoregressive processes:
where ZXt and ZYt are independent variates from a normal distribution with zero mean and unit variance. This process, including an analytic solution for the cross correlation between Xt and Yt, is discussed in detail in Jenkins and Watts (1968), which can be consulted for further details. Time series of length N = 100 were generated for Xt and Yt and used to compute sample cross correlations. This procedure was repeated 10 000 times, and the 5th and 95th percentiles of the cross correlation as a function of lag were computed. The resulting intervals are shown as the shaded regions in the left panels of Fig. A1. Also shown are the exact cross correlation (solid curve) and representative sample estimates based on N = 100 (curve with dots). The peak correlation occurs at lag/lead = 2. The 90% range of correlations clearly delineates the positive and negative correlations around zero lag, and sample estimates give reasonable information about the cross correlations.
Next, time series generated by the bivariate autoregressive processes given by (A1) and (A2) were smoothed using an 11-point running mean. To compute the exact cross correlation for the smoothed time series, let the smoothed time series for Xt be
Define Yt′ similarly. Then the cross covariance of the smoothed time series is
where the covariance cov[Xt+τ, Yt] is known analytically. Applying this relation to the appropriate variances and covariances yields the cross correlation of the smoothed time series. The result, as well as results of the Monte Carlo simulations, are shown in the right panels of Fig. A1. Figure A1 reveals that the exact peaks are diminished, but the 90% confidence interval increases substantially. Moreover, the 90% confidence interval includes zero for all lags, implying relatively little constraint on the cross-correlation structure. In addition, individual realizations are strongly distorted versions of the true cross correlation. These results suggest that it would be difficult or even misleading to infer the structure of the true cross correlation based on correlations estimated from smoothed time series.