Abstract

Predictability and variability of the tropical Atlantic Meridional Mode (AMM) is investigated using linear inverse modeling (LIM). Analysis of the LIM using an “energy” norm identifies two optimal structures that experience some transient growth, one related to El Niño–Southern Oscillation (ENSO) and the other to the Atlantic multidecadal oscillation (AMO)/AMM patterns. Analysis of the LIM using an AMM-norm identifies an “AMM optimal” with similar structure to the second energy optima (OPT2). Both the AMM-optimal and OPT2 exhibit two bands of SST anomalies in the mid- to high-latitude Atlantic. The AMM-optimal also contains some elements of the first energy optimal (ENSO), indicating that the LIM captures the well-known relationship between ENSO and the AMM.

Seasonal correlations of LIM predictions with the observed AMM show enhanced AMM predictability during boreal spring and for long-lead (around 11–15 months) forecasts initialized around September. Regional LIMs were constructed to determine the influence of tropical Pacific and mid- to high-latitude Atlantic SST on the AMM. Analysis of the regional LIMs indicates that the tropical Pacific is responsible for the AMM predictability during boreal spring. Mid- to high-latitude SST anomalies contribute to boreal summer and fall AMM predictability, and are responsible for the enhanced predictability from September initial conditions. Analysis of the empirical normal modes of the full LIM confirms these physical relationships. Results indicate a potentially important role for mid- to high-latitude Atlantic SST anomalies in generating AMM (and tropical Atlantic SST) variations, though it is not clear whether those anomalies provide any societally useful predictive skill.

1. Introduction

The dominant mode of covariability between the atmosphere and ocean in the tropical Atlantic is the Atlantic Meridional Mode (AMM; Servain et al. 1999; Chiang and Vimont 2004; Xie and Carton 2004). The AMM is characterized by a meridional gradient in sea surface temperature (SST) near the location of the mean intertropical convergence zone (ITCZ), a shift of the ITCZ toward the warmer hemisphere, and cross-equatorial boundary layer winds that blow toward the warmer hemisphere (Hastenrath and Heller 1977; Hastenrath and Greischar 1993; Nobre and Shukla 1996). Although the AMM is typically defined via statistical analysis of observed data, it also emerges as the most rapidly growing structure in simple coupled ocean–atmosphere models (Xie 1999; Vimont 2010), highlighting the dynamical origin of AMM variations. Destabilization of meridional modes occurs through a positive feedback between wind, evaporation, and SST—the so-called WES feedback (Chang et al. 1997; Xie 1997).

Despite dominating coupled ocean–atmosphere covariability in the observed record, the AMM may not be in a linearly unstable state in nature. Instead, analysis of simple models indicates that meridional mode variations emerge as structures that experience transient growth over a finite period (Vimont 2010). These disturbances require some sort of external (to the tropical ocean–atmosphere system) or stochastic forcing to generate meridional mode variations. Vimont (2010) finds that subtropical SST variations play a key role in generating tropical meridional mode variations in a simple coupled model. Observational analyses of Czaja et al. (2002) show that a similar mechanism operates in observations, as boreal winter subtropical trade wind variations that are caused by the North Atlantic Oscillation (NAO) lead to subtropical and tropical SST variations that can excite AMM variations during the ensuing boreal spring. The NAO (Xie and Tanimoto 1998; Chiang and Vimont 2004) and ENSO (Curtis and Hastenrath 1995; Nobre and Shukla 1996; Enfield and Mayer 1997; Giannini et al. 2000) have been identified as sources of AMM and tropical Atlantic variability during boreal spring. It has also been shown that aerosol variations contribute to the long-term variability of tropical Atlantic SST (Evan et al. 2009; Chang et al. 2011).

Some attempts have been made to predict tropical Atlantic SST variations, especially those associated with variations in the meridional SST gradient, which we interpret here as AMM-related variations. Penland and Matrosova (1998) use linear inverse modeling (LIM, the same technique used herein) to show that predictions of northern tropical Atlantic and Caribbean SST anomalies show skill that exceeds that of persistence for long lead times. Chang et al. (1998) use a hybrid coupled model to show that decadal variations in the tropical Atlantic are more predictable than the year-to-year variations, though this result may be biased by the enhanced instability of AMM variations in that hybrid coupled model. Stockdale et al. (2006) review Atlantic predictability by coupled models and find skill in the northern tropical Atlantic that beats persistence, but they also find that serious biases in coupled general circulation models’ simulations of the mean state limit predictability of SST variations in boreal summer. All of these aforementioned studies identify ENSO as an essential contributor to tropical Atlantic SST predictability. Huang et al. (2009) also find that ENSO is essential but highlight the importance of persistent SST anomalies in the tropical Atlantic from boreal winter into spring.

While the AMM exhibits maximum variance during boreal spring (Chiang and Vimont 2004; Czaja 2004), it also plays an important role in tropical Atlantic climate variability during boreal summer and fall. The structure of SST variations that accompany the AMM is shown in Fig. 1 for boreal spring (March–May) and late summer/early fall [August–October (ASO)]. The AMM spatial structure during boreal summer is similar to its spring counterpart, though summer SST anomalies in the Southern Hemisphere tend to be more muted than their spring counterparts, and summer SST anomalies in the northern midlatitudes tend to be like-signed (especially around 45°N). Vimont and Kossin (2007) and Kossin and Vimont (2007) show that the AMM plays an important role in seasonal hurricane activity through its effect on large-scale environmental conditions in the tropical Atlantic. Kossin and Vimont (2007) also show, using a similar model to the one used here, that boreal summer and fall AMM variations may be predictable up to a year in advance. We further explore the predictability of AMM variations in section 4.

Fig. 1.

AMM composite SST (shading), 925-mb geopotential height (contour 2 m), and 925-mb winds for (a) March–May and (b) August–October averaged conditions. Composites are taken around the 8 yr with the largest positive and negative values of the AMM index (SST based) for a given season, and plots represent one-half the mean difference between the two sets of years. Note that the years for (a),(b) are not the same. For geopotential height, solid contours denote positive values, dashed contours denote negative values, and the zero contour has been omitted. Winds are shown where the composite difference exceeds the 90% confidence level based on a bivariate t test.

Fig. 1.

AMM composite SST (shading), 925-mb geopotential height (contour 2 m), and 925-mb winds for (a) March–May and (b) August–October averaged conditions. Composites are taken around the 8 yr with the largest positive and negative values of the AMM index (SST based) for a given season, and plots represent one-half the mean difference between the two sets of years. Note that the years for (a),(b) are not the same. For geopotential height, solid contours denote positive values, dashed contours denote negative values, and the zero contour has been omitted. Winds are shown where the composite difference exceeds the 90% confidence level based on a bivariate t test.

An additional, though related, source of variability in the tropical and northern Atlantic comes from the so-called Atlantic multidecadal oscillation (AMO; Kushnir 1994; Mann and Park 1994; Kerr 2000; Enfield et al. 2001). There is still considerable debate about the source of AMO variations. Some studies suggest that the AMO arises in response to variations in the Atlantic meridional overturning circulation (AMOC; Kushnir 1994; Delworth and Mann 2000; Knight et al. 2005; Dong and Sutton 2005; Sutton and Hodson 2005). Indeed, model simulations of AMOC variations show that a reduction in the strength of the AMOC results in a large-scale cooling over the entire North Atlantic (NA) and warming over the southern tropical Atlantic (Vellinga and Wood 2002; Dong and Sutton 2005; Dahl et al. 2005; Knight et al. 2005; Zhang and Delworth 2005), consistent with the AMM structure. Still, others argue that the AMO is a reflection of changes in radiative forcing (Andronova and Schlesinger 2000; Mann and Emanuel 2006). The task of distinguishing between externally forced AMO variations and variations that arise due to processes internal to the climate system is not trivial, though attempts have been made to distinguish between the two contributions through the use of signal-to-noise maximizing EOF analysis (Ting et al. 2009) or extended EOF analysis (Guan and Nigam 2009). Regardless of the cause of the AMO, the tropical structure of AMO variations bears a strong resemblance to tropical AMM variations in both observations and in models, which leads to the question of whether the AMO and AMM are related. Vimont and Kossin (2007) show that the AMO and AMM are very highly correlated, and argue based on lagged correlations between indices of the boreal summer and fall (July–November) AMO and AMM that the AMO is a source of external forcing for AMM variations on decadal time scales. The link between mid- and high-latitude SST variations and tropical AMM variations is explored further in this study.

What is the mechanism connecting mid- to high-latitude SST variations and tropical AMM variations? The time lag between high-latitude and tropical variability (on the order of one year; Vimont and Kossin 2007), and the basin-scale spatial structure of the response suggest that decadal-frequency changes in tropical ocean circulation may not be the dominant connection. Johnson and Marshall (2002) and Johnson and Marshall (2004) show that variations in the meridional overturning circulation induce a relatively rapid response in the same hemisphere and basin due to wave propagation within the basin. Model simulations of the Last Glacial Maximum (Chiang and Bitz 2005) show that the presence of ice sheets in high latitudes can excite tropical variability in the Atlantic (after a lag of a few years) through thermodynamic coupling between the atmosphere and a motionless “slab” ocean model. In that model, cool and dry conditions in high latitudes alter latent heat fluxes and SST in high latitudes, which eventually propagate equatorward, generating an AMM-like response. Dahl et al. (2005) find a similar high-latitude influence on tropical meridional mode–like variations in an atmospheric model coupled to a slab ocean model, though they explain the tropical response as that which is required to balance changes in higher-latitude oceanic heat transports. A similar argument is used by Kang et al. (2009), who find that the ITCZ and atmospheric meridional overturning circulation responds to changes in high-latitude oceanic heat fluxes, even in the absence of ocean–atmosphere coupling. Finally, Mahajan et al. (2010) shows that meridional mode–like variations can be generated even in the absence of a WES feedback, though in that case the amplitude of meridional mode variation is considerably reduced. Finally, it is noted that these mechanisms are not mutually exclusive. Although the present study does not establish a specific mechanism for linking mid- or high-latitude SST variations with the AMM, the rapid time scale associated with the connection suggests that atmospheric or coupled thermodynamical feedbacks are fundamental in linking the two regions.

The current study uses a LIM to diagnose the source and predictability of AMM variations, and highlights a connection between the AMM and mid- to high-latitude SST variations. This paper is organized as follows. SST data and the LIM methodology are described in section 2. In section 3 the LIM is developed and optimal growth structures are analyzed. AMM predictability and the source of AMM variability is investigated in section 4, including the relationship between the AMM and SST variations in different regions of the Atlantic and Pacific. Section 5 presents conclusions and a discussion of the results.

2. Data

a. Data

SST data are taken from the Met Office Hadley Centre Sea Ice and SST version 1.1 (HadISST1.1) (Rayner et al. 2003). The SST data are available on a 1° × 1° grid. For all analysis in the present study, the SST data are spatially smoothed using one pass of a 1–4–6–4–1 smoother in the meridional and zonal direction, detrended, and temporally smoothed using a 3-month running mean. Land points are removed, as are any locations that ever experience ice cover (the latter eliminates spurious variability associated with ice). The analysis in this study uses SST over the years 1950–2008. We repeated some of our analyses on the period 1900–49 and were unable to construct a robust LIM over that period (indeed, even ENSO variability did not appear predictable using LIM over that period). The lack of consistency between the periods 1900 and 1949 and 1950 and 2008 may result from data quality, or from differences in dynamical processes occurring during those periods. We note that Huang et al. (2005) show that the relationship between ENSO and the AMM during boreal winter and spring emerges more robustly using data from the entire twentieth century, rather than the latter half alone.

A time series for the AMM is constructed using the SST structure of the AMM as defined in Chiang and Vimont (2004), except using data over the period 1950–2005. The smoothed SST data described above are interpolated to the spatial resolution and region used in Chiang and Vimont (2004; i.e., 21°S–32°N, 74°W–15°E), and projected onto the spatial SST structure of the AMM. The resulting time series will be referred to as the AMM time series in this analysis.

Composite maps in Fig. 1 are constructed using the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis 925-mb winds and 925-mb height, and the smoothed HadISST1.1 SST as described above.

b. LIM

In this analysis, we will use a linear inverse model to determine structures that experience transient growth over a finite period. We briefly describe the methodology here; a more complete description of LIM can be found in Penland and Sardeshmukh (1995). Linear inverse modeling assumes that the evolution of the state of a system x can be approximated as a multivariate Markov model,

 
formula

where is the linear system matrix and ξ is a white noise forcing. The least biased prediction is the solution to the homogeneous part of (1) at time τ,

 
formula

where is the Green’s function for a given lag τ. The matrix can be estimated by

 
formula

where and are the contemporaneous and τ-lag covariance matrices for x. The linear system matrix is obtained by taking the logarithm of (3) and dividing by τ. Estimates of , and hence of , may be sensitive to the choice of lag over which (3) is evaluated. In particular, lags near the half-period of the shortest oscillatory time scale of the empirical normal modes (the eigenvectors of ) can lead to unstable estimates of (Penland and Sardeshmukh 1995). We have conducted a “τ test” (details in the  appendix) that shows that a lag of 6 months provides a stable estimate of for the various analyses herein. Subsequent estimates of are calculated as .

Growth of an initial structure x(0) into a final structure x(τ) over time τ can be calculated via

 
formula

where or indicates the norm of a vector under the initial or final norm kernel or , respectively. For growth under the Euclidean norm (referred to here as the energy norm), the initial and final norm kernels are the identity matrix, and γ(τ) is calculated from the eigenvalue problem (Farrell 1988; Penland and Sardeshmukh 1995; Tziperman and Ioannou 2002) using

 
formula

Here, the eigenvector vτ is an initial condition that produces growth equal to the eigenvalue γ(τ) over a period τ. The eigenvalues may be ordered such that the first eigenvector produces the maximum growth under the energy norm, the second eigenvector produces the next-most growth subject to a constraint of orthogonality with the first eigenvector, and so forth. Herein, the eigenvectors are referred to as “optimal” structures as they optimally grow under the chosen norm.

Equation (5) is valid for producing optimal growth for an energy norm that gives equal weight to all elements of x. The use of a different norm is motivated by a desire to identify the growth of defined structures, such as the AMM. In that case, γ(τ) is calculated via solving the generalized eigenvalue problem (Farrell 1988; Tziperman and Ioannou 2002; Zanna et al. 2011) as shown:

 
formula

where and are the initial and final norm kernels. We define an AMM norm kernel following the method of Zanna and Tziperman (2005) and Tziperman et al. (2008). First, we use linear regression to define a vector of regression coefficients RAMM that define the AMM time series in our EOF/principal component (PC) space. The vector RAMM is then normalized to have unit length; this normalizes the growth to be unity for lag τ = 0. Finally, the AMM norm kernel is defined using the normalized RAMM as

 
formula

As we have no a priori expectation for an initial structure, we use the energy norm for the initial norm, that is, , the identity matrix (Hawkins and Sutton 2009). For convenience, we will refer to optimal initial structures under the energy norm as energy-optimals, and optimal initial structures under the AMM norm as AMM-optimals.

3. Analysis of the LIM

In this section we develop a linear inverse model over the full domain (tropical Pacific, tropical Atlantic, and midlatitude Atlantic), and examine the optimal structures and growth under the energy and AMM norms.

For the full region, we use SST over the tropical Pacific (30°S–30°N, 120°E–75°W), the tropical Atlantic (30°S–30°N, 75°W–15°E), and the midlatitude Atlantic (30°–75°N, 75°W–15°E). Results are relatively insensitive to the use of SST anomalies in the midlatitude Pacific and tropical Indian Oceans, so data from those regions are not used [this is in contrast to Penland and Sardeshmukh (1995), who highlight the importance of the Indian Ocean; differences may be due to the dataset used, or the domain used herein]. An EOF/PC prefilter is applied to the SST field prior to analysis, and the state vector x is a subset of the dominant principal components. We retain enough PCs to retain at least 80% of the SST variance over the region of analysis—in this case, 15 PCs over the full region. Note that the LIM is seasonally independent, in that it is calculated using data from all seasons; nonetheless, predictive skill may have seasonal dependence, as shown in section 5. A more detailed analysis of the LIM is presented in the  appendix.

The growth curves for the leading two energy-optimals are plotted in Fig. 2a, their spatial structures (initial conditions) for τ = 9 months are shown in Figs. 3a and 3c, and their associated final structures [final structures are calculated as (9 months) times the energy-optimal] are shown in Figs. 3b and 3d. The amplification curve shows that the leading energy-optimal for lag τ = 9 months (OPT1) grows by e1.5 ≈ 4.5 through about a 7–9-month lag, and that it stays greater than 1 (indicating growth) through the full 18-month analysis. The OPT1 spatial structure and associated final structure show that this growth is related to ENSO. This structure has been discussed extensively elsewhere and is presented here only for comparison.

Fig. 2.

Growth curves under the (a) energy and (b) AMM norms. Shown in (a) is growth under the energy-optimal of OPT1 (solid line with squares), OPT2 (solid line with circles), and AMM-optimal (dashed line with ’s). Shown in (b) is growth under the AMM-optimal of OPT2 (solid line with circles), the AMM-optimal (dashed line with ’s), and what would be expected via damped persistence of the AMM (gray dashed line; see text for explanation).

Fig. 2.

Growth curves under the (a) energy and (b) AMM norms. Shown in (a) is growth under the energy-optimal of OPT1 (solid line with squares), OPT2 (solid line with circles), and AMM-optimal (dashed line with ’s). Shown in (b) is growth under the AMM-optimal of OPT2 (solid line with circles), the AMM-optimal (dashed line with ’s), and what would be expected via damped persistence of the AMM (gray dashed line; see text for explanation).

Fig. 3.

Nine-month (a),(c),(e) optimal initial and (b),(d),(f) associated final conditions of (a),(b) OPT1, (c),(d.) OPT2, (e),(f.) AMM-optimal. By construction the optimal structures have unit length (final conditions do not); as such, units are arbitrary but may be compared between initial and final conditions.

Fig. 3.

Nine-month (a),(c),(e) optimal initial and (b),(d),(f) associated final conditions of (a),(b) OPT1, (c),(d.) OPT2, (e),(f.) AMM-optimal. By construction the optimal structures have unit length (final conditions do not); as such, units are arbitrary but may be compared between initial and final conditions.

The second energy-optimal amplification curve shows weaker energy growth through about τ = 6 months, followed by decay through τ = 18 months (Fig. 2a). If only 14 PCs are retained for the LIM, then the initial growth does not appear; however, the spatial structures of the optimals do not significantly change. The spatial structure of the second energy-optimal for lag τ = 9 months (OPT2; Fig. 3c) exhibits positive anomalies across the entire Atlantic north of the equator, with the largest amplitude anomalies in two relatively zonal bands extending along about 40° and 60°N. Though OPT2 has amplitude in the tropical Atlantic, that amplitude is relatively weak compared to the mid- to high-latitude Atlantic. The final structure that OPT2 develops into after 9 months is shown in Fig. 3d, and in the tropical Atlantic it resembles boreal summer AMM variations (cf. Fig. 1b). The final structure in the tropical North Atlantic (along about 15°N) is characterized by positive SST anomalies with larger amplitude than those seen in the initial OPT2 structure. In the final structure, tropical SST anomalies are associated with an anomalous meridional SST gradient near the equator with positive anomalies to the north and negative anomalies to the south of the equator. The overall structure of OPT2 and its final condition are relatively consistent for lags 5–18 months (not shown).

The temporal evolution of the AMM, OPT2, NA SST (averaged over 45°–75°N, 35°–15°W), and global mean SST without the North Atlantic (G-NA; global coverage, except data over the North Atlantic are masked) are plotted in Fig. 4. The time series OPT2 is generated by projecting the 9-month optimal structure onto the state vector x. The OPT2 time series closely follows the evolution of North Atlantic SST variability on longer time scales, consistent with the spatial structure in Fig. 3c. The time series of the AMM bears some resemblance to OPT2, though there appears to be much more high-frequency (interannual) variability in the AMM time series that is not correlated with the second optimal. To quantify the shared variance on high and low frequencies, we construct low-pass-filtered and high-pass-filtered versions of the OPT2 and AMM time series. We use consecutive 25- and 37-month running means for a low-pass filter, which produces half power at about 8.3 yr (Vimont 2005); high-pass-filtered data are constructed by subtracting the low-pass-filtered data from the raw data. The cross correlation between the low-pass-filtered OPT2 time series and the low-pass-filtered AMM maximizes at lag −7 months (OPT2 leads AMM by 7 months) at r = 0.77, while the corresponding correlation for the high-pass-filtered data maximizes at only r = 0.31 at a lag of −2 months (OPT2 leads AMM by 2 months). The raw NA time series, in turn, is correlated with the OPT2 time series at r = 0.79. This points to the North Atlantic as a potentially important source of variability for the tropical Atlantic.

Fig. 4.

Temporal evolution of the AMM, OPT2, NA SST (see text), and G-NA. All time series are standardized.

Fig. 4.

Temporal evolution of the AMM, OPT2, NA SST (see text), and G-NA. All time series are standardized.

Finally, G-NA is plotted in Fig. 4 to allow comparison between OPT2 and the long-term global SST variations. The G-NA time series is correlated with the OPT2 time series at r = 0.21, and the associated low-pass-filtered time series are only correlated at r = 0.17. This suggests that the low-frequency variation in OPT2 is not merely a reflection of a global SST variation [see also the results of Ting et al. (2009)].

Despite its AMM-like final condition, OPT2 does not necessarily optimally grow into the AMM. To investigate optimal AMM growth, we use the AMM norm for calculating the AMM-optimals. Growth of the AMM-optimal under the AMM norm can be calculated via (6), and growth of the energy-optimal (OPT2) under the AMM norm is calculated via

 
formula

where v(L)τ is OPT2, the second energy-optimal (Fig. 3c). Growth curves of the AMM-optimal and OPT2 under the AMM norm are plotted in Fig. 2b. The growth curve for the AMM-optimal under the AMM norm is less than one and nearly linear under the log transformation, indicating a nearly steady exponential decay of initial conditions. Still, the decay is considerably less than what would be expected via damped persistence of the AMM (calculated as , where r1 = 0.92 is the one-lag autocorrelation of the AMM) indicating that AMM variance is being maintained via other processes. The growth curve for OPT2 under the AMM norm is also less than one for all lags, with maximum growth (minimum decay) for a lag of 5–6 months. Growth of the AMM-optimal under the AMM norm is always greater than growth of OPT2 under the AMM norm.

The spatial structure of the τ = 9 months AMM-optimal is shown in Fig. 3e, and its associated final condition after 9 months is shown in Fig. 3f (the spatial structure is relatively consistent for lags 5–18 months; not shown). The AMM-optimal bears a strong resemblance to OPT2, especially in the Atlantic. Most notable are the two zonal bands of positive SST anomalies that extend along about 40° and 60°N. In the Atlantic, the final condition associated with the AMM-optimal also bears a strong resemblance to the OPT2’s final structure, with positive SST anomalies extending along about 15°N, and an anomalous northward meridional SST gradient near the equator. In the equatorial Pacific, the final conditions associated with the AMM-optimal shows positive SST anomalies that bear some resemblance to El Niño–like conditions. These anomalies are absent from the OPT2 final structure as the growth of ENSO is captured by OPT1 and its final condition, which by construction are orthogonal to OPT2 and its associated final condition. The AMM-optimal is not affected by this orthogonality constraint (the projection of the AMM-optimal onto OPT1 and OPT2 is 0.34 and 0.75, respectively). The growth of the AMM-optimal under the energy norm (Fig. 2a, dashed line with triangles) shows energy growth through about 15 months; this growth is due to the amplification of ENSO anomalies.

The empirical normal modes (ENMs) of the system are the eigenvectors (uj) of and have associated eigenvalues βj. The ENMs evolve as (Penland and Sardeshmukh 1995)

 
formula

where βj = σj + j is the eigenvalue, and aj and bj are the real and complex components of the eigenvector rotated so that aj has unit length, aj and bj are orthogonal, and bj has length >1. From (9) it is clear that the real component of the eigenvalue indicates the decay rate (all eigenvalues have negative real parts) and that the imaginary component of the eigenvalue indicates the frequency of oscillation. The decay time scale and period of the various ENMs are presented in Table 1.

Table 1.

ENMs and their projections (proj) onto various optimals. Mo = month. Inf = Infinity.

ENMs and their projections (proj) onto various optimals. Mo = month. Inf = Infinity.
ENMs and their projections (proj) onto various optimals. Mo = month. Inf = Infinity.

Growth over a finite period can occur if the mode is underdamped or through the nonnormality of the ENMs. For this model, only one pair of modes is underdamped—modes 10/11 (complex normal modes come in pairs with the same damping time scale and with oscillatory time scales that differ only in sign). These two modes have oscillatory time scales near 58 months and an e-folding decay time of 14.5 months. The real and imaginary components of modes 10/11 (not shown) bear strong resemblances to the transition and peak phases of ENSO (Penland and Sardeshmukh 1995) and as such will be referred to as the “ENSO mode” (of course, nonnormality of the system implies that other modes play a role in ENSO development; however, modes 10/11 appear to dominate ENSO evolution).

The contribution of individual normal modes to OPT2 and the AMM-optimal is shown in Table 1 as the amplitude of the complex projection of the associated optimal structure onto the eigenvectors of . The projections explain some of the differences between the growth rates of the different optimal structures. OPT1 projects strongly onto the ENSO mode, which does experience energy growth as it evolves from its real to complex phase. Still, the growth of the OPT1 structure is greater than the energy growth of the ENSO mode in isolation, indicating that nonnormal processes are responsible for the growth of the OPT1 structure. The second optimal structure (OPT2) has a large projection onto mode 1, the least damped (46-month damping time scale) stationary normal mode of the system, as well as modes 4/5 (damping time scale of 15 months and oscillatory time scale of 30 yr; the oscillatory time scale is clearly approximate given the length of the record). As mode 1 is damped and modes 4/5 are overdamped, growth of OPT2 must also occur through nonnormal processes. Finally, the AMM-optimal projects onto a number of real and complex normal modes, including modes 1–7 and the ENSO mode. Also of note is the projection onto modes , which will be discussed further in section 4a.

4. Seasonal AMM predictions and regional influences

The predictability of the AMM using the LIM is investigated in this section. The purpose of this analysis is twofold—to identify potential predictability and to diagnose the source of AMM variations during various seasons. The seasonal forecast skill is shown in section 4a, regional influences on the AMM are investigated in section 4b, and the contribution of individual normal modes to the forecast skill is shown in section 4c.

a. Forecast skill

A cross-validated predicted AMM index is constructed for each lag τ in the following manner. First, 10 yr of data (e.g., 1950–59) are set aside as a “verification” dataset and a EOF/PC prefilter is applied to the remaining 49 yr (1960–2008) of “training” data. Enough PCs are retained so that the estimate of M using a 6-month lag does not violate a simple τ test (see the  appendix). This results in less variance retained (about 70%–75%) when constructing the state vector , where indicates an estimate from the training dataset. Next, the 6-month propagator is estimated from (3) using the subset of PCs from the 49 yr of training data. The linear system matrix is estimated from as described in section 2, and the propagator for other lags is then calculated as . AMM predictions for the remaining decade are made by first projecting the verification data onto the spatial EOF structures of the training data, which results in an estimate of the state vector for the 10 independent years, where indicates an estimate from the verification data. Then, (2) is used to predict for the verification time. Finally, a cross-validated AMM prediction is constructed by first calculating the regression coefficients (as was done for the AMM norm) from the training data (we do not scale to have unit length), and then projecting onto . The whole process is then repeated for each of the next 4 decades of verification data, with the last 9 yr of data (2000–08) being predicted from the first 50 yr (1950–99).

A similar cross-validation procedure is used to construct a first-order autoregression (AR1) prediction for the AMM. In that case, (2) is used in a univariate sense: the state vector x is simply the AMM time series, the LIM is reduced to univariate linear regression, and is simply the AMM’s autocorrelation function. Note that the AR1 forecast used here differs from a standard persistence forecast, in that the amplitude of the AR1 forecast time series is reduced by the autocorrelation function.

The seasonal forecast skill of the AMM as predicted from the LIM is depicted as the correlation between the AMM prediction and the actual AMM time series for a given lag and predicted month in Fig. 5a. The cross-validated seasonal forecast skill for a given month can be inferred for various lags along a horizontal line in Fig. 5. For example, the forecast skill of the September AMM decreases relatively rapidly to r = 0.4 for a 5-month lead and then improves to nearly r = 0.5 for a 12-month lead. Recall that a 3-month running mean has been applied to both the predicted SST and verified AMM index; so, for example, a lag 12-month forecast for September represents a 1 November real-time prediction using conditions during the previous 3 months (August–October) The skill of a AR1 forecast (also cross validated) is depicted in Fig. 5b. Because of the EOF/PC prefilter used in the LIM, even a zero-lag prediction using the LIM would not be perfectly correlated with the actual AMM time series (the seasonal correlation between an AMM time series constructed from the EOF/PC prefiltered SST data and the AMM time series used here varies between r = 0.96 and r = 0.99).

Fig. 5.

Cross-validated seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) the LIM, and (b) an AR1 forecast. For each panel, the y axis (month) indicates the month for which the AMM is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time [e.g. in (a), moving left along a horizontal line for September indicates that a 5-month LIM AMM forecast (March–May average initial condition) correlates with the actual September AMM at r ≈ 0.4]. The diagonal dashed line indicates a September (ASO average) initial condition.

Fig. 5.

Cross-validated seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) the LIM, and (b) an AR1 forecast. For each panel, the y axis (month) indicates the month for which the AMM is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time [e.g. in (a), moving left along a horizontal line for September indicates that a 5-month LIM AMM forecast (March–May average initial condition) correlates with the actual September AMM at r ≈ 0.4]. The diagonal dashed line indicates a September (ASO average) initial condition.

The LIM forecast is not necessarily more skillful than the AR1 forecast. For short lag (<5 months) predictions of the boreal winter (November–April) AMM, the AR1 forecast correlation is larger than the LIM forecast correlation. Beyond a few months, however, there are two periods of enhanced correlation for the LIM forecasts: 6–12-month forecasts of the boreal spring and summer AMM (April–August) and forecasts that are initialized in early boreal fall (around September; see the dashed line in Fig. 5a). The timing of the boreal spring predictability is consistent with the timing of ENSO teleconnections, which have maximum amplitude during boreal winter and hence generate the maximum tropical Atlantic response in boreal spring (Czaja 2004). The timing of the September initial conditions will be explored below.

The skill of the LIM at forecasting the AMM can also be expressed using the mean-square error (MSE). A standard skill score (SS) can be generated as the difference between the mean-square error of the LIM forecast and a reference (REF) forecast (e.g., climatology or persistence), normalized by the difference between a perfect forecast (no error) and the reference forecast (see, e.g., Wilks 2006), as shown:

 
formula

If climatology is used as a reference forecast, then the right-hand side of (5) is simply one minus the ratio of the variance of the forecast error to the background variance, and as such it represents the reduction of error variance due to the LIM.

The LIM forecast is compared to two different reference forecasts in Fig. 6: climatology (Fig. 6a) and the AR1 forecast (Fig. 6b). When compared to climatology, the LIM forecast skill bears a strong resemblance to the correlation map in Fig. 5a; structural differences exist because of the seasonality in AMM variance. It is noteworthy that the two regions of enhanced correlation—forecasts of late boreal spring AMM variations and forecasts that are initialized in September—also emerge in the skill metric. Still, values of the skill score using September initial conditions are only around SS = 0.2, indicating only a modest (20%) reduction in error variance. When compared to the AR1 forecast (Fig. 5b), the skill scores are negative for short lead times, and for forecasts of late boreal fall and early boreal winter AMM variations (a negative skill score means that the LIM does not perform as well as the AR1 forecast). The lack of LIM forecast skill for very short leads (when compared to the AR1 forecast) may occur because the LIM uses an EOF/PC prefilter, while the AR1 forecast uses the same full AMM time series that is used for verification. As with the correlation maps, the two regions of enhanced skill discussed earlier emerge, with the LIM exhibiting about a 20% reduction in error variance over the AR1 forecast for long-lead (6–12 months) predictions of late boreal spring AMM variations, and a modest (about 10%–20%) reduction in error variance for forecasts initialized in September. Figure 6b indicates that the LIM provides useful additional forecast skill over persistence for seasonal forecasts of boreal spring AMM variations, while the utility of long-lead (>10 months) forecasts is questionable.

Fig. 6.

Cross-validated seasonal forecast skill score (see text) for the LIM forecast AMM, compared to (a) climatology and (b) the persistence forecast. For each panel, the y axis (month) indicates the month that is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time (see Fig. 5 caption). The diagonal dashed line indicates a September (ASO average) initial condition.

Fig. 6.

Cross-validated seasonal forecast skill score (see text) for the LIM forecast AMM, compared to (a) climatology and (b) the persistence forecast. For each panel, the y axis (month) indicates the month that is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time (see Fig. 5 caption). The diagonal dashed line indicates a September (ASO average) initial condition.

b. Regional sources of AMM variability

To investigate the source of AMM variability, we develop a set of “regional” LIMs; note that the results presented for the regional LIMs are not cross validated, and as such the phrase “prediction” is used herein to indicate the reconstructed final state at time t + τ, and not a true prediction. For the regional LIMs, the state vector x contains the leading principal components of SST over 1) the tropical Atlantic only (TA-only; 30°S–30°N, 75°W–15°E), 2) the tropical Pacific and Atlantic only (TP-TA; 30°S–30°N, 120°–15°E), and 3) the midlatitude and tropical Atlantic only (MA-TA; 30°S–75°N, 75°W–15°E). For the TA-only and TP-TA regions, enough PCs are kept to retain 80% of the SST variability; for the MA-TA region, we eliminate the last PC (thus, only explaining 79% of the SST variability) based on results from the τ test. For each regional LIM, a predicted AMM time series is obtained for each lag τ by projecting the reconstructed state vector x(t + τ) [from (3)] onto the respective AMM regression coefficients (the unscaled AMM norm) for the region. We note that processes that originate outside each regional LIM’s domain (e.g., ENSO, in the case of the MA-TA LIM) are not necessarily eliminated because of the covariance with modes in each basin. However, the results below suggest that the regional LIMs are not strongly affected by processes outside of their respective region. Other studies have attempted to isolate regional influences (Shin et al. 2010) or specific dynamical processes (Newman et al. 2011) through examination or alteration of the elements of itself. We attempted a similar analysis (not shown) using PCs from each region to form a state vector; although the model used in that analysis did not pass the “tau test,” the results were similar to those presented below.

The seasonal forecast correlations for the full LIM and for the regional LIMs are shown in Fig. 7. The correlation values for the full LIM (Fig. 7a) are larger than those in Fig. 5 because no cross validation is used for Fig. 7. The full LIM correlation values show a very similar structure to the cross-validated forecast correlation in Fig. 5a, with enhanced correlations for the boreal spring AMM predictions and when forecasts are initialized around September. When the LIM is developed around data in the tropical Atlantic only (Fig. 7b), both the 3–12-month predictions of boreal spring AMM variability and the correlations for September initial conditions are reduced from the full LIM correlations. The effect of tropical Pacific SST variability is clearly seen in the TP-TA LIM (Fig. 7c), where the enhanced correlations for predictions of boreal spring AMM variability are clearly present. In contrast, the enhanced correlations for September initial conditions—especially forecasts with lead times of 9 or more months—are not evident in the TP-TA LIM. The enhanced forecast correlations for September initial conditions and for long-lead correlations of boreal fall AMM variability are higher in the MA-TA LIM (Fig. 7d) than either the TP-TA LIM or the TA-only LIM. Together with the structure of OPT2 and the AMM-optimal in Fig. 3, the seasonal correlations for the MA-TA LIM suggest a role for mid- to high-latitude SST variability in generating tropical Atlantic AMM variations.

Fig. 7.

Seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) the full LIM (not cross validated), (b) the TA-only LIM, (c) the TP-TA LIM, and (d) the MA-TA LIM. For each panel, the y axis (month) indicates the month for which the AMM is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time [e.g. in (a), moving left along a horizontal line for September indicates that a 5-month LIM AMM forecast (March–May average initial condition) correlates with the actual September AMM at r ≈ 0.5]. The diagonal dashed line indicates a September (ASO) initial condition.

Fig. 7.

Seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) the full LIM (not cross validated), (b) the TA-only LIM, (c) the TP-TA LIM, and (d) the MA-TA LIM. For each panel, the y axis (month) indicates the month for which the AMM is being predicted and the x axis indicates the model lag (τ), which is equivalent to the forecast lead time [e.g. in (a), moving left along a horizontal line for September indicates that a 5-month LIM AMM forecast (March–May average initial condition) correlates with the actual September AMM at r ≈ 0.5]. The diagonal dashed line indicates a September (ASO) initial condition.

To further investigate the regional source of AMM variations, the AMM-optimals for the TP-TA LIM and the MA-TA LIM are shown in Fig. 8 (the TA-only optimal bears a strong resemblance to the AMM structure, indicating that damped persistence is dominating AMM growth in that LIM). The TP-TA AMM-optimal (Fig. 8a) bears the AMM structure in the tropical Atlantic, suggesting that some of the AMM predictability is simply persistence. In the Pacific, the AMM-optimal looks similar to OPT1 from the full LIM (Fig. 3a) and the AMM-optimal from the full LIM (Fig. 3e), both of which lead to positive SST anomalies in the tropical Pacific. This projection explains some of the growth of positive SST anomalies in the equatorial Pacific for the TP-TA LIM (Fig. 8b). It has been shown that ENSO contributes to warming in the northern tropical Atlantic during boreal winter and spring (Curtis and Hastenrath 1995; Giannini et al. 2001); hence, the structure of the TP-TA AMM-optimal in the tropical Pacific explains some of the boreal spring AMM predictability in the TP-TA LIM. The AMM-optimal for the MA-TA LIM (Fig. 8c) has the same features as the Atlantic portion of OPT2 and the AMM-optimal for the full LIM (Figs. 3c and 3e). In particular, the MA-TA AMM-optimal contains the same two bands of positive SST anomalies extending along about 40° and 60°N. These anomalies develop into positive SST anomalies in the northern tropical Atlantic for the 9-month final condition (Fig. 8d), similar to the full LIM.

Fig. 8.

Nine-month (a),(c) AMM-optimal initial and (b),(d) associated final conditions. (a),(b) TP-TA LIM, and (c),(d) MA-TA LIM. By construction the optimal structures have unit length (final conditions do not); as such, units are arbitrary but may be compared between initial and final conditions.

Fig. 8.

Nine-month (a),(c) AMM-optimal initial and (b),(d) associated final conditions. (a),(b) TP-TA LIM, and (c),(d) MA-TA LIM. By construction the optimal structures have unit length (final conditions do not); as such, units are arbitrary but may be compared between initial and final conditions.

The seasonal variances of OPT2, the AMM-optimal from the full model, and the TP-TA and MA-TA AMM-optimals are shown in Fig. 9. Each optimal structure is projected onto the state vector for the associated model, resulting in a time series for that optimal structure. The time series is standardized, and the monthly variance is calculated and plotted in Fig. 9a. The same procedure is used to calculate the seasonality of the associated final conditions in Fig. 9b. Penland (1996) shows that the seasonality of the optimal or of the stochastic forcing can be calculated from the flux dissipation relationship as well; this was not tried in this analysis.

Fig. 9.

Seasonality of the variance of (a) optimal and (b) associated final structures. Curves represent OPT2 (solid line with ○’s), the AMM-optimal for the full LIM (dashed line with ’s), the AMM-optimal for the TP-TA LIM (dotted line with □’s), and the AMM-optimal for the MA-TA LIM (dotted line with Δ’s).

Fig. 9.

Seasonality of the variance of (a) optimal and (b) associated final structures. Curves represent OPT2 (solid line with ○’s), the AMM-optimal for the full LIM (dashed line with ’s), the AMM-optimal for the TP-TA LIM (dotted line with □’s), and the AMM-optimal for the MA-TA LIM (dotted line with Δ’s).

The OPT2, AMM-optimal from the full LIM, and the MA-TA AMM-optimals all exhibit the same seasonal variance with maximum variance during boreal summer (July–September) and minimum variance during boreal spring (March–May). This corresponds well with the enhanced correlations for September initial conditions seen in Figs. 7a and 7d. In contrast, the AMM-optimal for the TP-TA LIM does not exhibit a strong seasonality. The final conditions for OPT2 and the MA-TA LIM (Fig. 9b) exhibit a similar seasonality as their optimal initial conditions, with maxima during boreal summer (July–September). The final conditions associated with the AMM-optimal from both the full LIM and the TP-TA LIM exhibit a seasonal cycle with maximum variance during the end of the calendar year (November–January), consistent with ENSO seasonality. The presence of positive tropical Pacific SST anomalies in these final conditions, together with a separate analysis of SST variance in the Atlantic only, confirms that the seasonality in Fig. 9b is dominated by ENSO instead of the AMM.

c. Normal mode contribution to AMM variability

The physical processes that contribute to seasonal AMM variability can be further illuminated by examining the contribution of individual ENMs to the seasonal correlations. We do this by reducing the effect of a particular ENM (or pair of ENMs) in the linear system matrix . As eigenvectors of the dynamical system matrix, the ENMs also diagonalize , so that

 
formula

where contains the eigenvectors (ENMs) of and is a diagonal matrix of (complex) eigenvalues. As discussed earlier, the real and imaginary components of the eigenvalue indicate the decay rate and period, respectively, of the particular ENM. The effect of a particular ENM can be effectively removed by increasing the damping rate of the particular ENM and recalculating using (11). For the full LIM, damping time scales range from about 2.5 to 46 months (Table 1). We arbitrarily set the damping time scale for a particular ENM (or pair of ENMs) to 0.1 month, recalculate , and determine the seasonal correlations for the new AMM predictions. We note that while this process does not completely remove the ENM, the mode is so rapidly damped that it does not contribute substantially to growth in the model (other modified damping rates were tested with little effect on the results). This process was repeated for each of the nine sets of ENMs (six pairs of complex ENMs and three real ENMs). Two sets of ENMs were found to affect the seasonal correlations strongly: modes (8-month damping time scale and 130-month period) and mode 1 (46-month damping time scale). The resulting modified LIMs will be referred to as LIM-6/7 and LIM-1.

The seasonal correlations between the actual AMM and the AMM that is predicted by LIM-6/7 or LIM-1 are plotted in Fig. 10. For reference, the spatial structures of ENMs 1 and 7 are plotted in Fig. 11. Removal of modes has a large impact on the seasonal correlations for boreal winter and spring AMM variability (Fig. 10a). In particular, the period of high correlations for predictions of January–May AMM variations, with lead times of 3–8 months, is largely gone for predictions made with LIM-6/7. The imaginary component of ENM7 (Fig. 11b) exhibits positive SST anomalies in the central equatorial Pacific and a positive AMM-like structure in the tropical Atlantic. This mode represents the relationship between tropical Pacific ENSO variability and the AMM (Covey and Hastenrath 1978; Giannini et al. 2001). In contrast, the seasonal correlations from LIM-1 (Fig. 10b) are still large for predictions of the boreal spring AMM, but they are reduced for predictions of the late boreal summer through early winter (August–January). The spatial structure of ENM1 (Fig. 11c) is very similar to the final condition associated with OPT2 (see Table 1) and the final condition associated with the AMM-optimal from the MA-TA LIM (Fig. 8d), highlighting the role of ENM1 in long-term AMM predictability.

Fig. 10.

Seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) LIM-6/7 (ENMs are artificially damped) or (b) LIM-1 (ENM1 is artificially damped). Plotting conventions are as in Fig. 7.

Fig. 10.

Seasonal correlation between the actual AMM time series and the forecast AMM time series using (a) LIM-6/7 (ENMs are artificially damped) or (b) LIM-1 (ENM1 is artificially damped). Plotting conventions are as in Fig. 7.

Fig. 11.

Spatial ENM structures: (a) real component of ENM7, (b) imaginary component of ENM7, and (c) ENM1 (ENM1 is purely damped, so there is no imaginary component). Both real and imaginary components of the ENMs are scaled to have unit length prior to plotting, so units are arbitrary.

Fig. 11.

Spatial ENM structures: (a) real component of ENM7, (b) imaginary component of ENM7, and (c) ENM1 (ENM1 is purely damped, so there is no imaginary component). Both real and imaginary components of the ENMs are scaled to have unit length prior to plotting, so units are arbitrary.

5. Conclusions and discussion

Atlantic Meridional Mode (AMM) predictability and variability is investigated using linear inverse modeling (LIM). A LIM is constructed over a region that includes the tropical Pacific, tropical Atlantic, and mid- to high-latitude Atlantic. The two structures that experience optimal growth (OPT1 and OPT2) under an energy-norm are related to tropical Pacific ENSO variations (Penland and Sardeshmukh 1995) and variations that resemble the atlantic multidecadal oscillation (AMO)/AMM. Of note for the second structure are two bands of initial SST anomalies in the mid- to high-latitude North Atlantic, along about 40° and 60°N that lead to AMM-like anomalies in the tropics some 3–12 months later. The same structure of initial conditions in the mid- to high-latitude North Atlantic emerges when optimal growth structures are calculated relative to an AMM-norm. The LIM shows reasonable skill over a modified persistence forecast for late boreal spring AMM predictions (about a 25% reduction in error variance) and some skill for long-lead (12 months) forecasts of the ASO AMM (about 10%–15% reduction in error variance). It is not clear whether the reduced error variance for long-lead forecasts of ASO AMM variations will provide any societally useful skill.

The analysis of AMM predictability highlighted two periods with enhanced AMM predictability: forecasts of late boreal spring AMM variability and forecasts initialized around September. The roles of the tropical Pacific and mid- to high-latitude Atlantic in generating AMM variability were investigated using a set of regional LIMs [a tropical Pacific/tropical Atlantic (TP-TA) LIM and a midlatitude Atlantic/tropical Atlantic (MA-TA) LIM], and by changing damping rates for specific empirical normal modes in the full LIM. The TP-TA LIM showed that tropical Pacific ENSO variability is responsible for predictability of the boreal spring AMM, consistent with numerous other studies (Enfield and Mayer 1997; Penland and Matrosova 1998; Chang et al. 1998; Stockdale et al. 2006; Huang et al. 2009). The MA-TA LIM also identified SST variations in the mid- to high-latitude Atlantic as important contributors to late boreal summer/boreal fall (August–November) AMM predictability. Notably, the mid- to high-latitude Atlantic appears to be an especially important contributor to late boreal summer/boreal fall forecasts that are initialized in the previous September.

This study has focused on evaluating the seasonality of tropical Atlantic AMM predictability and on diagnosing the source of that predictability. As such, numerous questions remain about the cause of AMM predictability or AMM variations. In particular, it is not clear how mid- to high-latitude SST variations can impact the tropical Atlantic, though similar equatorward influences are found in modeling studies of the Atlantic thermohaline circulation (Johnson and Marshall 2002; Vellinga and Wood 2002; Dong and Sutton 2005; Dahl et al. 2005; Knight et al. 2005; Sutton and Hodson 2005; Zhang and Delworth 2005), analyses of Pacific variability (Vimont et al. 2001, 2003a,b, 2009; Alexander et al. 2010), and in theoretical models of tropical meridional modes (Liu and Xie 1994; Vimont 2010).

Additional evidence for the mid- to high-latitude influence on AMM variability comes from a related general circulation model study (Smirnov and Vimont 2012). Smirnov and Vimont (2012) use the Atlantic component of OPT2 as initial heat content anomalies in an ensemble of model simulations using the National Center for Atmospheric Research Community Atmospheric Model, version 3.1, coupled with a slab ocean model (CAM+SOM). When CAM+SOM is initialized on 1 November with only the mid- to high-latitude component (north of 23°N) of OPT2, the model develops SST anomalies that span the Atlantic along about 15°N by the following boreal summer and fall, in the same manner as the LIM herein.

Given the relatively short time scale of the apparent influence of mid- to high-latitude Atlantic initial conditions on the tropical Atlantic (on the order of a year), results from this analysis suggest that processes such as oceanic adjustment or advection are not essential in this connection. Of course, this latter statement does NOT apply to the source of mid- to high-latitude SST variations in the Atlantic in the first place; this is another unanswered question in this study. The apparently long time scale of the leading optimal structure (Fig. 4) does point to low-frequency oceanic processes in the generation of mid- to high-latitude SST variations. Research is underway to diagnose the source of high-latitude SST variations associated with the second optimal structure and to better understand the physical mechanisms that link the mid- to high-latitude SST anomalies with tropical Atlantic AMM variability.

Acknowledgments

This work was supported by NSF Grants ATM-0735030 and ATM-0849689. The HadISST SST data were provided by the Met Office through the British Atmospheric Data Centre. NCEP reanalysis data and CPC precipitation data were provided by NOAA/OAR/ESRL PSD, Boulder, Colorado, from its website (at http://www.cdc.noaa.gov/). Thanks to D. Smirnov for his comments on an earlier version of the manuscript. Thanks also to M. Alexander, C. Penland, and two anonymous reviewers for their comments; the review process led to a substantially improved manuscript, and the author is very grateful to the editor and reviewers for their thorough reviews.

APPENDIX

Tests for the Assumption of Linear Dynamics

Penland and Sardeshmukh (1995) outline a series of tests for linearity of the dynamical system. In particular, they argue that if the dynamics are linear, then 1) its statistics will be Gaussian, 2) the estimated parameters of the linear system (i.e., ) will be independent of the lag used to define them (Penland et al. 2000), 3) the covariance matrix of the forcing will be positive definite, 4) the forecasts will be “good,” and 5) the actual forecast errors grow as the trace of the expected error covariance matrix, as shown:

 
formula

We test these assumptions herein.

The assumption of Gaussian statistics is investigated using quantile plots of relevant time series used in this analysis. Figure A1 shows a quantile plot of the time series of ENMs 1 and 10 constructed by projecting the ENM1 or ENM10 structure onto the state vector x for the full LIM. Also shown is the quantile plot of the AMM as represented by the state vector . In general, the quantile plots fall on a straight line (indicating Gaussian statistics), except at the extremes. Still, there are some notable differences from Gaussian statistics. Figure A1a indicates that ENM1 is slightly positively skewed (skewness 0.23) and platykurtic (kurtosis 2.6). ENM10 (the ENSO mode) has real and imaginary components that represent the transition and peak of an ENSO event. The imaginary component deviates more strongly from a Gaussian distribution than the real component, with positive skewness (0.50) and a leptokurtic (kurtosis 3.4) structure. Finally, the AMM fits a normal distribution very well, with only a slight negative skewness (−0.15). The deviations from Gaussian statistics are a caveat on the use of the LIM in this study.

Fig. A1.

Quantile plots of standardized time series for various structures in this analysis. (a) ENM1 (Table 1); (b) ENM10, the ENSO mode (Table 1), with solid dots representing the real (transition phase) structure and open circles representing the imaginary (peak phase) structure; and (c) AMM time series as reconstructed by .

Fig. A1.

Quantile plots of standardized time series for various structures in this analysis. (a) ENM1 (Table 1); (b) ENM10, the ENSO mode (Table 1), with solid dots representing the real (transition phase) structure and open circles representing the imaginary (peak phase) structure; and (c) AMM time series as reconstructed by .

To test whether the estimated parameters of the linear system are independent of the lag used to define them, we perform two tests. First, we repeat the test of Penland and Sardeshmukh (1995) and calculate different estimates of using different lags τ0 as shown:

 
formula

Next, the Euclidean norms of the various submatrices of are plotted as a function of τ0 in Fig. A2. The bottom curve in Fig. A2 is the upper-left element of , the second-to-the-bottom curve is the norm of the top 2 × 2 matrix of , and so forth until the top curve, which is the norm of the full matrix . While the norms are not flat, they do not show strong peaks until lag τ0 = 7 months, which may exist because of the near half-period of ENM14/15 [16-month period; lags of one-half the period of the ENMs can lead to large uncertainties in the calculation of parameters, as discussed in Penland and Sardeshmukh (1995)]. We chose to use a lag of τ0 = 6 months when calculating the full LIM to avoid possible uncertainties in calculations of . A second “tau test” consists of checking whether the expected error variance is independent of lag τ0 (Penland et al. 2000). We found that the trace of the error covariance matrices were nearly indistinguishable when was calculated using lags τ0 that ranged from 3 to 6 months (all curves were nearly identical to the “perfect prediction” curve, shown in Fig. A3). Furthermore, we restricted our LIM to only 14 EOFs and found indistinguishable error growth for lags that ranged between 3 and 9 months; beyond 9 months, error growth curves started to differ and criteria 5 above was violated (not shown). This confirms that the use of a constant is justified.

Fig. A2.

Thick solid curve: Euclidean norm of ; thin curves: Euclidean norm of the 14 submatrices of ( has rank 15). The norms are plotted as a function of τ0, the lag used in (3) to define .

Fig. A2.

Thick solid curve: Euclidean norm of ; thin curves: Euclidean norm of the 14 submatrices of ( has rank 15). The norms are plotted as a function of τ0, the lag used in (3) to define .

Fig. A3.

Normalized root-mean-square error growth. Shown are the expected error growth as the trace of (A1) (solid line with •’s), actual error growth (solid line with □’s), error growth of AR1 forecast (dashed line), and persistence forecast error growth (dashed line with ○’s). All error growth is calculated in, and relative to, the PC subspace used to define the LIM.

Fig. A3.

Normalized root-mean-square error growth. Shown are the expected error growth as the trace of (A1) (solid line with •’s), actual error growth (solid line with □’s), error growth of AR1 forecast (dashed line), and persistence forecast error growth (dashed line with ○’s). All error growth is calculated in, and relative to, the PC subspace used to define the LIM.

The criteria that the forcing be positive definite is tested by examining the eigenvalues of the noise covariance matrix using

 
formula

We found one negative eigenvalue of when was calculated using 15 PCs and a lag of τ0 = 6 months. When we restricted the number of PCs to 14, all the eigenvalues were positive definite. For reference, the optimal structures did not change when only 14 PCs were used.

Findings that the forecasts are good and that the actual error growth grows as (A2) are shown in Fig. A3, which depicts the normalized root-mean-square error growth of the state vector x for the LIM, for a persistence forecast, and for an AR1 forecast (not cross validated). The actual error growth (squares in Fig. A3) is much lower than persistence, the AR1, and a climatology forecast, confirming that the LIM forecasts are good. Furthermore, the actual error growth is nearly identical to the growth that is predicted by (A2), as shown by the solid line with solid dots.

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