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  • View in gallery

    Climatological properties of (a) TAUX, (b) SSH, and (c) SST anomalies in the North Atlantic basin (25°N–65°N, 80°W–0°); the SST is taken to equal the temperature in the upper 5 m of the SODA TEMP dataset. Colors correspond to interannual variability (standard deviation), with extreme values given in the legend (max). Contour lines correspond to the TAUX, SSH, and SST mean fields, with a contour interval (CI) of 0.015 N m−2, 0.14 m, and 2.5°C, respectively. Solid and dashed lines correspond to positive and negative anomaly values in (a) and (c), and to differences from the mean sea level in (b).

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    Spectral properties of the North Atlantic TAUX anomalies. The M-SSA analysis here uses a window length of yr; the subsequent varimax rotation uses ST-EOFs 2–40, while ST-EOF 1 is excluded. (a) The estimated variance in each mode is shown as black dots, plotted as a function of their corresponding frequency. Lower and upper ticks on the error bars correspond to the 1% and 99% quantiles from a Monte Carlo test that uses the scaled target-rotation algorithm of ST-EOFs; the test ensemble has 1000 members. In the composite null hypothesis, the leading ST-EOF (target dot) is excluded from the test and the remaining ST-EOFs are tested against AR(1) noise. (b)–(e) Reconstruction of the TAUX field with ST-EOF 1, and three oscillatory pairs at a period length of 2.7, 7.7, and 2.2 yr, respectively; each mode is projected onto its leading S-EOF. The variance captured by each mode is given in the legend of each panel (in percent).

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    As in Fig. 2, but for TEMP and SSH anomalies from a joint M-SSA analysis; the subsequent varimax rotation uses ST-EOFs 3–50. In the composite null hypothesis, the leading ST-EOFs 1–4 (target dots) are excluded from the test and the remaining ST-EOFs are tested against AR(1) noise.

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    Spatial structure of interannual variability in the three joint oscillatory modes with periods of (a)–(c) 7.7, (d)–(f) 2.7, and (g)–(i) 2.2 yr. Shown is the standard deviation of the corresponding reconstructed components (RCs) resulting from an M-SSA analysis with a window length of yr and a subsequent varimax rotation of ST-EOFs 3–50; see appendixes A and B for a review of the methodology. The standard deviations in the anomalies of (left) TAUX, (middle) SSH, and (right) SST are plotted in color, and they are transformed back into physical units. Extreme values are given in the legend (max). The contour lines in all nine panels correspond to the SSH mean field, with a CI of 0.2 m.

  • View in gallery

    Phase composites of the SST anomalies for the 7.7-yr oscillatory mode from a joint M-SSA analysis of TAUX, SSH, and TEMP anomalies (cf. Fig. 4). The eight phase categories are labeled by their midpoint (in years) in the legend of each panel and are shown in color; also listed in the legends are the minimum and maximum (min and max) amplitude of the anomalies (in K). The contours in all eight panels correspond to the SSH mean field, with CI = 0.2 m, as in Fig. 4.

  • View in gallery

    As in Fig. 5, but for TAUX anomalies (in color). The superimposed phase composites of the vector wind stress (arrows) are reconstructed using a projection of the TAUY anomalies onto the joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies; see text for details.

  • View in gallery

    Phase maps for the TAUX, SSH, and SST anomalies in the three oscillatory modes with periods of (a)–(c) 7.7, (d)–(f) 2.7, and (g)–(i) 2.2 yr. The color coding reflects the cyclical character of the phase, with the interval being mapped onto each of the three periods [0, T], with and 2.2 yr, respectively. M-SSA parameters as in Fig. 4; grid points of low variance are masked, and the SSH CI = 0.2 m, as before.

  • View in gallery

    Vertical phase coherence in the TEMP anomalies of the oscillatory modes with periods of (a) 7.7, (b) 2.7, and (c) 2.2 yr. The mean phase-coherence index (mean) is indicated in the legend, as well as a 99% level of significance (sig), against the null hypothesis of randomly shuffled phases. M-SSA parameters as in Fig. 4, and CI = 0.2 m for the SSH contours (solid black).

  • View in gallery

    Schematic diagram of interannual variability in SST and TAUX. The SST map (lower level of the figure) and the TAUX map (upper level) show phase 3 from the phase composite analyses in Figs. 5 and 6, respectively.

  • View in gallery

    Differences in the participation index (in color) between a projection of the TEMP anomalies onto the SSH anomalies and onto the TAUX anomalies, respectively. Shown are the square root of the differences in (a)–(c) the 7.7-yr mode, (d)–(f) the 2.7-yr mode, and (g)–(i) the 2.2-yr mode, at the depths of (left) 5 m, (middle) 171 m, and (right) 466 m. Extreme values are given in the panel legends (min and max) (in K). M-SSA parameters for TAUX and SSH as in Figs. 2 and 3, respectively. SSH CI = 0.2 m.

  • View in gallery

    Phase composites in TAUX (in color) from the 7–8-yr oscillatory mode of a joint M-SSA analysis of TAUX, SSH, and TEMP anomalies, during the second, 1929–2008 subinterval; see text for details. Four phases that correspond to one-half of the cycle are shown. The phase categories are labeled in the panel legends by their midpoint (in years); also shown are extreme values (min and max). Superimposed phase composites of the vector wind stress (arrows) are reconstructed from a projection of the TAUY anomalies onto the joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies. M-SSA parameters as in Fig. 4; SSH CI = 0.2 m.

  • View in gallery

    As in Fig. 11, but in the SST anomalies.

  • View in gallery

    Reconstruction of spatiotemporal oscillations in the presence of strong correlated noise. (a) Noise-free reference signal, channels of length . (b)–(g) The noise-contaminated signal is first projected onto the leading L = {3, 7, 12, N} S-EOFs from a conventional PCA analysis, which captures 45%, 73%, 87%, and 100% of the total variance, respectively. The resulting PCs are analyzed via M-SSA with a window length M = 40 and the first 20 ST-EOFs varimax rotated. (b),(d),(f) and (h) show average reconstruction from an ensemble of 200 repetitions of the experiment; (c),(e),(g) and (i) show corresponding channel-wise standard deviation of the full signal (red), the reference signal (green), and the average reconstruction (blue). The signal-to-noise ratio is 1/12. For illustration purpose, the standard deviation of the full signal is scaled by 12−1/2.

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Interannual Variability in the North Atlantic Ocean’s Temperature Field and Its Association with the Wind Stress Forcing

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  • 1 Department of Atmospheric and Oceanic Sciences, and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California
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Abstract

Spectral analyses of the North Atlantic temperature field in the Simple Ocean Data Analysis (SODA) reanalysis identify prominent and statistically significant interannual oscillations along the Gulf Stream front and in large regions of the North Atlantic. A 7–8-yr oscillatory mode is characterized by a basinwide southwest-to-northeast–oriented propagation pattern in the sea surface temperature (SST) field. This pattern is found to be linked to a seesaw in the meridional dipole structure of the zonal wind stress forcing (TAUX). In the subpolar gyre, the SST and TAUX fields of this mode are shown to be in phase opposition, which suggests a cooling effect of the wind stress on the upper ocean layer. Over all, this mode’s temperature field is characterized by a strong equivalent-barotropic component, as shown by covariations in SSTs and sea surface heights, and by phase-coherent behavior of temperature layers at depth with the SST field. Recent improvements of multivariate singular spectrum analysis (M-SSA) help separate spatiotemporal patterns. This methodology is developed further and applied to studying the ocean’s response to variability in the atmospheric forcing. Statistical evidence is shown to exist for other mechanisms generating oceanic variability of similar 7–8-yr periodicity in the Gulf Stream region; the latter variability is likewise characterized by a strongly equivalent-barotropic component. Two other modes of biennial variability in the Gulf Stream region are also identified, and it is shown that interannual variability in this region cannot be explained by the ocean’s response to similar variability in the atmospheric forcing alone.

Permanent affiliation: Department of Mathematics, Israel Institute of Biological Research, Ness Ziona, Israel.

Additional affiliation: Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure, Paris, France.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-16-0370.s1.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: A. Groth, andreasgroth@atmos.ucla.edu

Abstract

Spectral analyses of the North Atlantic temperature field in the Simple Ocean Data Analysis (SODA) reanalysis identify prominent and statistically significant interannual oscillations along the Gulf Stream front and in large regions of the North Atlantic. A 7–8-yr oscillatory mode is characterized by a basinwide southwest-to-northeast–oriented propagation pattern in the sea surface temperature (SST) field. This pattern is found to be linked to a seesaw in the meridional dipole structure of the zonal wind stress forcing (TAUX). In the subpolar gyre, the SST and TAUX fields of this mode are shown to be in phase opposition, which suggests a cooling effect of the wind stress on the upper ocean layer. Over all, this mode’s temperature field is characterized by a strong equivalent-barotropic component, as shown by covariations in SSTs and sea surface heights, and by phase-coherent behavior of temperature layers at depth with the SST field. Recent improvements of multivariate singular spectrum analysis (M-SSA) help separate spatiotemporal patterns. This methodology is developed further and applied to studying the ocean’s response to variability in the atmospheric forcing. Statistical evidence is shown to exist for other mechanisms generating oceanic variability of similar 7–8-yr periodicity in the Gulf Stream region; the latter variability is likewise characterized by a strongly equivalent-barotropic component. Two other modes of biennial variability in the Gulf Stream region are also identified, and it is shown that interannual variability in this region cannot be explained by the ocean’s response to similar variability in the atmospheric forcing alone.

Permanent affiliation: Department of Mathematics, Israel Institute of Biological Research, Ness Ziona, Israel.

Additional affiliation: Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), Ecole Normale Supérieure, Paris, France.

Supplemental information related to this paper is available at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-16-0370.s1.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: A. Groth, andreasgroth@atmos.ucla.edu

1. Introduction and motivation

The North Atlantic Oscillation (NAO) is one of the most prominent large-scale patterns of atmospheric variability over the middle and high latitudes of the Northern Hemisphere. It exerts a dominant influence upon climate variability on various time scales, from weeks to decades, over the Atlantic sector and across wide regions of North America, the Arctic, Eurasia, and the Mediterranean.

Interannual oscillatory modes in the North Atlantic region in both the ocean and the atmosphere have been identified in many observational studies (e.g., Bjerknes 1964; Deser and Blackmon 1993; Dettinger et al. 1995; Hurrell and Van Loon 1997; Moron et al. 1998; Da Costa and Colin de Verdière 2002; Feliks et al. 2010; Jajcay et al. 2016). The extent to which these oscillations are due to purely oceanic (Dijkstra and Ghil 2005; Berloff et al. 2007, and references therein), purely atmospheric (Frankignoul and Hasselmann 1977; Frankignoul et al. 1997), or coupled (Vannitsem et al. 2015, and references therein) processes is still a matter of controversy. Whatever the origin of these oscillations, understanding the complex interactions between the ocean and the overlying atmosphere is crucial for the prediction of low-frequency climate variability (Kushnir et al. 2002; Czaja et al. 2003; Hurrell et al. 2003; Visbeck et al. 2003; Liu 2012; Vannitsem et al. 2015; Pierini et al. 2016).

In this paper, we use the Simple Ocean Data Assimilation (SODA) reanalysis (Carton and Giese 2008; Giese and Ray 2011) to address the question of possible shared mechanisms of interannual variability between the North Atlantic Ocean and the atmosphere above and near it. In particular, we will address the following two questions:

  1. Can we extract statistically significant modes—either individual or joint—in both the atmosphere and the ocean by using advanced multivariate methods of spectral analysis?
  2. To what extent may such modes be associated with shared mechanisms of interannual variability?
Question 1 will be investigated in sections 3 and 4 of the paper, while question 2 will be the object of sections 5 and 6.

To identify interannual variability in the extensive reanalysis dataset, we rely here on multivariate singular spectrum analysis (M-SSA), an advanced method for the analysis of spatiotemporal datasets; please see Ghil et al. (2002) for a comprehensive review of M-SSA and Groth and Ghil (2011, 2015) for recent improvements thereof that help separate distinct spatiotemporal patterns. These improvements developed further herein study the ocean’s response to variability in the atmospheric forcing.

A series of papers have studied the wind-driven double-gyre problem across a hierarchy of models, from simple rectangular-shaped ocean basins and 1.5-layer reduced gravity models (Jiang et al. 1995; Speich et al. 1995; Chang et al. 2001) to more realistic topographies and several layers Simonnet and Dijkstra (2002); Simonnet et al. (2003a,b); Dijkstra (2005); Dijkstra and Ghil (2005); Simonnet et al. (2005). These papers consistently found a relaxation-oscillation–like response of the ocean on interannual time scales to time-independent wind forcing, either idealized or climatological.

In this paper, an objective test against red noise is therefore discussed to distinguish deterministic behavior from random fluctuations. The present analysis of oceanic and atmospheric datasets extends the information gained from this hierarchy of models insofar as the atmospheric forcing comes from an atmospheric reanalysis (20CR; Compo et al. 2011), with its own inherent spatiotemporal complexity, and the ocean models are replaced by the oceanic general circulation model that is used in the SODA reanalysis.

In the SODA reanalysis, the atmosphere is just a prescribed time-and-space–dependent boundary condition for the ocean model. In this setting, variability in the ocean model cannot instantaneously feed back into the atmospheric model used for 20CR, and that in turn prevents the atmosphere and the ocean from mutually adjusting their spatiotemporal behavior, as they would in a fully coupled ocean–atmosphere model. In this two-step approach, however, mutual adjustment is possible through assimilation of instrumental data.

Carton and Giese (2008) have, in fact, shown that the assimilation of ocean observations further improves the quality of the reanalysis, and that it reduces mismatches in eddy kinetic energy and sea level variability between independent observations and the ocean model’s output in the presence of atmospheric forcing alone.

In the absence of ocean data assimilation, though, these authors have shown that such mismatches are more substantial. Despite a demonstrable improvement of the reanalysis that the assimilation of oceanic data produces, discrepancies between the temporal patterns in the atmospheric forcing and in the ocean dynamics may still persist. It is precisely these residual discrepancies that we aim to identify and understand in the present work by applying our improved M-SSA methodology.

To this end, we have devised a protocol related to the classical M-SSA null-hypothesis test (Allen and Smith 1996; Groth and Ghil 2015) to test the hypothesis of interannual variability in the ocean being a response to similar variability in the atmospheric forcing. The present method therefore complements the common approach of comparing multiple realizations of an ocean general circulation model, driven with different atmospheric forcings (e.g., Grégorio et al. 2015; Sérazin et al. 2015).

The remainder of the paper is organized as follows: In section 2, we present the SODA dataset and give a brief introduction to M-SSA. In section 3, we analyze the atmospheric and oceanic datasets in two separate M-SSA analyses and identify patterns of similar interannual variability. Then, in section 4, we undertake a joint analysis of both datasets to obtain further insight into the spatiotemporal behavior of the coupled ocean–atmosphere dynamics. Discrepancies between the temporal patterns that we find in the atmospheric forcing and in the ocean’s response are ascertained and analyzed in section 5. A summary of the results concludes the paper in section 6, and four appendixes provide technical details.

2. Dataset and methods

a. The Simple Ocean Data Assimilation dataset

The Simple Ocean Data Assimilation reanalysis of ocean climate variability used herein is version 2.2.4 of Giese and Ray (2011), and it extends over the 138-yr interval 1871–2008. Note that the original SODA dataset provided by Carton and Giese (2008) only extended over 50 yr and it used forcing from the ERA-40 atmospheric reanalysis (Uppala et al. 2005).

In our study we focus on SODA’s oceanic temperature field (TEMP), with a vertical resolution of 40 levels, and the sea surface height (SSH). The global ocean model used in SODA has an eddy-permitting horizontal resolution of 0.25° × 0.4°, and the output variables are then mapped onto a uniform grid with a resolution of 0.5° × 0.5°. The SODA reanalysis assimilates hydrographic and SST data into this model with surface boundary conditions from the atmospheric 20CRv2 reanalysis (Compo et al. 2011), from which we include the zonal surface wind stress (TAUX) in our analysis.

The monthly TEMP, SSH, and TAUX data fields are first converted into anomalies (i.e., we remove at each grid point the average value over the entire 138-yr interval). Since we are interested in variability on interannual time scales, we further eliminate variability at frequencies cycles per year with a Chebyshev type-I low-pass filter from which we then take all July values.

Feliks et al. (2013) showed in their appendix that such a Chebyshev filter comes very close to an ideal low-pass filter, with practically no distortions of either phase or amplitude in the passband (i.e., below the Nyquist frequency of cycles per year) and a highly effective damping in the stopband of cycles per year. The more customary 12-month or 3-month averages, on the other hand, are strongly frequency dependent and therefore influence any subsequent spectral analysis; see the discussion of Figs. A1 and A2 in Feliks et al. (2013) for additional details.

Figure 1 displays the spatial patterns of the mean field (contour lines) and of the standard deviation of interannual variability (color) in the North Atlantic region of interest, taken here to be (25°–65°N, 80°W–0°). The Gulf Stream front is very prominent in the two ocean variables of SSH and SST variability (Figs. 1b,c); it appears as a strong gradient in the mean field and also as a narrow, meandering ridge of interannual variability.

Fig. 1.
Fig. 1.

Climatological properties of (a) TAUX, (b) SSH, and (c) SST anomalies in the North Atlantic basin (25°N–65°N, 80°W–0°); the SST is taken to equal the temperature in the upper 5 m of the SODA TEMP dataset. Colors correspond to interannual variability (standard deviation), with extreme values given in the legend (max). Contour lines correspond to the TAUX, SSH, and SST mean fields, with a contour interval (CI) of 0.015 N m−2, 0.14 m, and 2.5°C, respectively. Solid and dashed lines correspond to positive and negative anomaly values in (a) and (c), and to differences from the mean sea level in (b).

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

The front is aligned with the Florida Current, which flows along the east coast of North America until the separation point at Cape Hatteras; it then extends into the open ocean to 40°W, beyond which longitude it weakens and the mean-field contour lines diverge. This divergence is associated with the birth of the Azores Current that flows toward the southeast and the North Atlantic Drift flowing northeastward.

The contour lines of the TAUX mean field (Fig. 1a), on the other hand, show a typical pattern of midlatitude westerlies and lower-latitude easterlies. The pattern of interannual variability on top of it (color) reaches its maximum at about 60°N and contributes to a strengthening or weakening of the westerlies.

We have also analyzed the meridional component of the 20CRv2 surface wind stress (TAUY), but found its interannual variability (not shown) to possess much smaller variance over the North Atlantic basin. To simplify the M-SSA analysis, we therefore focus in this paper on TAUX, and only discuss TAUY whenever necessary.

b. Multivariate singular spectrum analysis

To identify complex patterns of spatiotemporal behavior in the extensive SODA reanalysis datasets, we rely here on M-SSA, which provides an efficient and robust tool to extract dynamics from short, noisy time series. M-SSA relies on the classical Karhunen–Loève decomposition of stochastic processes into data-adaptive orthogonal functions. Broomhead and King (1986b) introduced M-SSA into dynamical system analysis as a robust version of the Mañé–Takens idea to reconstruct the underlying dynamics from a time-delayed embedding of time series; see Ghil et al. (2002) and Alessio (2016, chapter 12) for a comprehensive overview of the methodology and of related spectral methods. Appendix A here presents some of the requisite mathematical details.

The M-SSA method essentially diagonalizes the lag-covariance matrix to yield a set of orthogonal eigenvectors and the corresponding eigenvalues. The eigenvectors are also referred to as space–time empirical orthogonal functions (ST-EOFs), while the eigenvalues equal the variance that is captured in the direction of a given ST-EOF. Oscillatory behavior is reflected in so-called oscillatory pairs, that is, ST-EOFs with approximately equal eigenvalues and fundamental frequencies (Vautard and Ghil 1989; Plaut and Vautard 1994). To improve the separability of distinct frequencies, we rely here on a subsequent varimax rotation of the ST-EOFs (Groth and Ghil 2011).

The presence of such oscillatory pairs, though, is not sufficient to reliably identify deterministic behavior in the underlying dynamics. Allen and Smith (1996) provided, therefore, an objective Monte Carlo–type hypothesis test against red noise for the statistical significance of eigenvalues in the single-channel case, and Allen and Robertson (1996) suggested a multichannel extension thereof to M-SSA.

The latter authors, however, already stated that this test may become too lenient and less specific as a consequence of the variance-compression problem. In this context, Groth and Ghil (2015) found the Procrustes algorithm helpful in comparing the data eigendecomposition with that of the surrogate data generated by the red-noise null hypothesis, and showed that application of this algorithm substantially improves the explanatory power of the test. These authors further provided a rigorous mathematical extension of a composite null hypothesis test for M-SSA: for instance, a strong trend component can be excluded from the raw data when estimating the parameter of the null hypothesis.

In the analysis of extensive datasets, such as the SODA dataset at hand, it is common practice to first project the dataset onto a subset of spatial EOFs (S-EOFs) by means of a conventional principal component (PC) analysis. This compression of the dataset into a few leading PCs is meant to reduce the computational costs of a subsequent M-SSA analysis (e.g., Dettinger et al. 1995; Allen and Robertson 1996; Robertson 1996); see also Moron et al. (1998) and Table 1 therein.

Groth and Ghil (2015), though, have demonstrated possible negative effects of this variance compression on the detection rate of weak signals when the number of retained PCs becomes too small. In appendix B, we demonstrate that this practice can further affect negatively the judicious reconstruction of the spatiotemporal patterns of such weak signals, and recommend retaining a maximum number of PCs. Moreover, we show that the M-SSA analysis on PCs is mathematically equivalent to that on the full gridded dataset, when the full number of PCs is retained.

To combine different physical quantities in a single M-SSA analysis, we normalize the annually sampled TAUX, SSH, and TEMP anomalies to unit variance (i.e., we divide each dataset by its corresponding standard deviation). Furthermore, to account for geographical variations in the grid size resolution, we multiply the anomalies at each grid point by the square root of the cosine of its latitude. Throughout the whole paper, physical units are restored by an inverse transformation when presenting patterns of interannual variability.

3. Individual oscillatory modes

First, we study the atmospheric and oceanic datasets in two separate M-SSA analyses. Doing so will give us a first look at whether there are significant similarities among the interannual variabilities of the TAUX, SSH, and TEMP anomalies, but also highlight possible differences.

a. Wind stress modes

We start with an M-SSA analysis of the TAUX anomalies. The annually sampled dataset of length N = 138 is first projected onto the full set of S-EOFs, which gives a total of L = 138 PCs as input channels for the analysis.

The M-SSA results are shown in Fig. 2, with the spectrum of eigenvalues and the corresponding significance levels in Fig. 2a. The latter levels have been derived from a composite null hypothesis of the leading ST-EOF 1 plus AR(1) noise; see Groth and Ghil (2015) for more methodological details.

Fig. 2.
Fig. 2.

Spectral properties of the North Atlantic TAUX anomalies. The M-SSA analysis here uses a window length of yr; the subsequent varimax rotation uses ST-EOFs 2–40, while ST-EOF 1 is excluded. (a) The estimated variance in each mode is shown as black dots, plotted as a function of their corresponding frequency. Lower and upper ticks on the error bars correspond to the 1% and 99% quantiles from a Monte Carlo test that uses the scaled target-rotation algorithm of ST-EOFs; the test ensemble has 1000 members. In the composite null hypothesis, the leading ST-EOF (target dot) is excluded from the test and the remaining ST-EOFs are tested against AR(1) noise. (b)–(e) Reconstruction of the TAUX field with ST-EOF 1, and three oscillatory pairs at a period length of 2.7, 7.7, and 2.2 yr, respectively; each mode is projected onto its leading S-EOF. The variance captured by each mode is given in the legend of each panel (in percent).

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

In the significance test, we have chosen to exclude ST-EOF 1 as it clearly exceeds the other ST-EOFs in its variance. The corresponding reconstruction of ST-EOF 1, which is projected onto its leading S-EOF for illustration, shows a strong trend that is clearly visible early in the dataset (cf. Fig. 2b). To which extent this trend could be attributed to changes in the availability of data in the atmospheric reanalysis or to natural variability, though, remains an open question.

The remainder of the ST-EOFs is then tested against AR(1) noise, in which two oscillatory pairs clearly emerge as significant in our Monte Carlo test at the 99% level; their frequencies are indicated by the dashed lines in Fig. 2a. These two pairs represent quasi-biennial variability, with a period of 2.7 and 2.2 yr, respectively. The reconstruction of the 2.7-yr mode (Fig. 2c) shows that this mode is most active in the second half of the dataset, , and that it almost vanishes in the first half, . The reconstruction of the 2.2-yr mode (Fig. 2e), on the other hand, shows activity throughout the whole time interval, although it is slightly modulated and reaches its maximum near 1950.

It appears that no further ST-EOFs exceed the 99% significance threshold. There is, however, one oscillatory pair with a period of 7.7 yr that appears close to the upper significance level, and that is significant at the 95% level. This alone is not sufficient to pay it greater attention, but we do have prior reason to focus on atmospheric and oceanic variability in the 7–8-yr frequency band for the North Atlantic (e.g., Moron et al. 1998; Da Costa and Colin de Verdière 2002; Gámiz-Fortis et al. 2002; Paluš and Novotná 2004; Feliks et al. 2013); hence we reran the analysis so that it enhances the signal-to-noise ratio in this particular case.

The reconstruction of the 7.7-yr mode in Fig. 2d shows a decline in variability as time advances past the mid-twentieth century, and we therefore restrict the analysis in a second run to the time interval 1871–1950. This modification increases the relative variance that this mode captures and the null hypothesis can be indeed rejected at a higher level of 98% (not shown).

In summary, we are thus able to identify three oscillatory modes in the TAUX anomalies, with periods of 7.7, 2.7, and 2.2 yr, respectively. These findings are consistent with numerous other publications that have presented interannual variability in the atmospheric fields over the North Atlantic basin (see e.g., Rogers 1984; Hurrell 1995; Hurrell and Van Loon 1997; Robertson 2001; Feliks et al. 2010). In particular, these findings are consistent with the global presence of the 7–8-yr mode in the interannual climate variability for the North Atlantic, North America, Europe, and eastern Africa (see e.g., Kondrashov et al. 2005; Hanrahan et al. 2009; Feliks et al. 2013; Jajcay et al. 2016).

b. Oceanic modes

To which extent is the interannual variability in the TAUX anomalies also reflected in ocean dynamics? To find out, we analyze the TEMP and SSH anomalies in a joint M-SSA analysis and first combine both annually sampled datasets into a multivariate time series with a much larger number of channels. Note that each dataset has been normalized and latitude-weighted as explained in section 2, prior to a projection onto the full set of S-EOFs, yielding finally L = 138 PCs as input channels for the M-SSA analysis.

The combined M-SSA results for the TEMP and SSH anomalies are plotted in Fig. 3. The eigenvalue spectrum and the corresponding significance levels are given in Fig. 3a; these levels are based on a composite null hypothesis of low-frequency variability in ST-EOFs 1–4 plus AR(1) noise.

Fig. 3.
Fig. 3.

As in Fig. 2, but for TEMP and SSH anomalies from a joint M-SSA analysis; the subsequent varimax rotation uses ST-EOFs 3–50. In the composite null hypothesis, the leading ST-EOFs 1–4 (target dots) are excluded from the test and the remaining ST-EOFs are tested against AR(1) noise.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

The reconstruction with ST-EOFs 1–4 (Fig. 3b) shows low-frequency behavior with a minimum at around 1910 and an irregular increase thereafter. In the group of ST-EOFs 1–4, we observe a pairing in ST-EOFs 2–3, which could be linked to multidecadal behavior with a 60–70-yr period. The chosen window length of M = 40 yr, however, together with the presence of two cycles only, limits the reliability of detecting such long-term behavior.

Still, it is interesting to remark that this mode contributes to a decline in the trend at the end of the time interval, roughly after 2005. Such low-frequency variability in turn can limit our ability to predict the persistence of the secular warming trend, as stated already by Ghil and Vautard (1991).

In Fig. 3a, we identify several modes that emerge above their upper significance level. Among them are three modes that agree rather well in period with the three modes found to be significant in TAUX, namely 7.7, 2.6, and 2.3 yr, although the ordering by variance of the first two differs. The reconstructions of these three oscillatory modes in Figs. 3c–e resemble to some extent those of the TAUX anomalies, although interesting differences are visible. We will return to these differences in greater detail in section 5.

The present analysis yields, furthermore, ST-EOFs that are significant in the combined TEMP-SSH anomalies but that lack an equivalent significant mode in TAUX. A 13-yr oscillation, for example, agrees well in periodicity and geographical extent (not shown), with decadal variability found in the North Atlantic SST field; see Moron et al. (1998) for an overview. Deser and Blackmon (1993) found a similar pattern of decadal variability in the SST and TAUX fields; this pattern, however, was not statistically significant in their study.

The number of significant interannual variability modes in the SODA dataset provides additional food for thought in dealing with the question—already mentioned at this paper’s beginning—of whether these modes arise from intrinsic ocean dynamics alone (Jiang et al. 1995; Dijkstra and Ghil 2005; Berloff et al. 2007; Simonnet et al. 2005; Pierini et al. 2016), through the ocean’s linear (Frankignoul and Hasselmann 1977; Frankignoul et al. 1997) or nonlinear (Dewar 2003) response to atmospheric fluctuations, or from genuinely coupled modes (Vannitsem et al. 2015, and references therein).

In the present M-SSA analysis, we focus on the conceptually simplest hypothesis of a linear relationship and continue to examine the three modes that we found to have similar periods in both media: the 7–8-yr mode and the two quasi-biennial modes (cf. Figs. 2 and 3). The restriction to linearity is for the sake of simplicity, and not because of an essential limitation of the methodology to the analysis of linear systems alone. Thus, for instance, Groth and Ghil (2011) found M-SSA helpful in the synchronization analysis of coupled chaotic oscillators.

4. Joint ocean–atmospheric modes

Next, we proceed with a joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies to better understand the spatiotemporal structure associated with interannual variability in the wind stress and the North Atlantic temperature field. As before, we combine the three annually sampled and normalized datasets into a single multivariate time series; projection onto its full set of S-EOFs yields the L = 138 PCs as input channels for our M-SSA analysis (cf. appendix B).

In this sequential approach—starting with separate M-SSA analyses of TAUX and TEMP-SSH anomalies, and then combining all anomalies into a single M-SSA analysis—the limitations of a decomposition into spatially orthogonal functions become clearly visible. Just as the number of EOFs is limited by the rank of , so is the capacity of M-SSA to separate between different spectral components. This limitation emerges in particular when the number D of channels is much larger than the length N of the time series.

It is therefore not surprising that—in the M-SSA analysis on the combined fields, atmospheric and oceanic—we found again three significant oscillatory modes, which exhibit the periods of 7.7, 2.7, and 2.2 yr that were identified in the two separate M-SSA analyses above. This result shows that differences between the ST-EOFs of the TAUX anomalies and those of the TEMP and SSH anomalies are sufficiently small for their joint M-SSA analysis to indicate joint interannual variability in all three fields.

We will return to this limitation of M-SSA, and to a more detailed analysis of the differences in the oscillatory behavior of the atmospheric and oceanic fields, in section 5. The joint M-SSA analysis, though, is mathematically equivalent to a total-least squares regression of the three fields onto common oscillatory modes, in which the joint RCs (not shown) resemble an average of the separate RCs in Figs. 2 and 3.

a. Spatial structure

The spatial structure of interannual variability in the three joint modes with periods of 7.7, 2.7, and 2.2 yr is shown in Fig. 4. The variability is plotted in physical units that correspond to the standard deviation of the reconstructed components (RCs) resulting from the M-SSA.

Fig. 4.
Fig. 4.

Spatial structure of interannual variability in the three joint oscillatory modes with periods of (a)–(c) 7.7, (d)–(f) 2.7, and (g)–(i) 2.2 yr. Shown is the standard deviation of the corresponding reconstructed components (RCs) resulting from an M-SSA analysis with a window length of yr and a subsequent varimax rotation of ST-EOFs 3–50; see appendixes A and B for a review of the methodology. The standard deviations in the anomalies of (left) TAUX, (middle) SSH, and (right) SST are plotted in color, and they are transformed back into physical units. Extreme values are given in the legend (max). The contour lines in all nine panels correspond to the SSH mean field, with a CI of 0.2 m.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

In the standard deviations of the TAUX anomalies (left column), we observe in all three modes a fairly homogeneous pattern, with a single large maximum extending over much of the subpolar gyre. This pattern is most intense and lies farthest north in the 2.7-yr mode, and it is weakest and lies farthest south, astride the North Atlantic Drift, in the 2.2-yr mode. In all three modes, this pattern is located in fairly high latitudes (i.e., in a region in which interannual variability in TAUX is generally high; cf. Fig. 1a).

In the standard deviations of the SSH and SST anomalies, plotted in the middle and right column of Fig. 4, respectively, we observe much smaller-scale features lying mainly along the Gulf Stream front. The scale of these features seems to correspond to the meander-and-eddy pattern associated with the Gulf Stream’s instabilities, as shown also by Figs. 1b and 1c.

For the 2.7-yr and 2.2-yr modes, the intense variability is limited by-and-large to the Gulf Stream region and extends only little farther, along the North Atlantic Drift (cf. Figs. 4e,f and 4h,i). But in the 7.7-yr mode—in the SSHs and especially in the SSTs, in Figs. 4b and 4c, respectively—we observe, beside the intense variability along the Gulf Stream front, a larger-scale pattern of high variability that covers large areas south and east of the front.

This spatiotemporal pattern starts in the Florida Current and continues farther east and north of Cape Hatteras, especially in the SST anomalies (Fig. 4c), along the North Atlantic Drift, to finally cover most of the subpolar gyre. Although the amplitude of the SST standard deviation in this mode only reaches about 0.15°C in absolute value, it captures up to 13% of the total interannual variability in this active region; when excluding the low-frequency behavior in ST-EOFs 1–4, the variance captured is as high as 25%. This finding shows that, especially off the Gulf Stream, the fraction of variance associated with the 7.7-yr mode can be much larger than the roughly 2.7% that this mode captures on average in the entire North Atlantic TEMP field (cf. Fig. 3d).

b. Dynamical behavior

To understand the dynamical properties that underlie the spatial structure seen in Fig. 4, we construct next the so-called phase composites of the oscillatory modes (Moron et al. 1998). That is, the successive maps associated with a given mode’s RCs are divided into eight phase categories, and one averages in time over all the maps in each category to obtain the phase composite of that category; see appendix C for the specific procedure used herein.

Figure 5 shows the corresponding eight phases for the 7.7-yr mode in the SST anomalies. Along the Gulf Stream front, we observe localized coherent structures of large amplitude, up to roughly ±0.6°C. The small characteristic scale and the large amplitude of this pattern are in agreement with the generally high variability in this region. In addition, we observe the southwest–northeast propagation of a wave off the front. The propagating wave starts near the Florida Current, attains 0.2°C in absolute value, covers large areas south and east of the front, and reaches the subpolar gyre after a full cycle.

Fig. 5.
Fig. 5.

Phase composites of the SST anomalies for the 7.7-yr oscillatory mode from a joint M-SSA analysis of TAUX, SSH, and TEMP anomalies (cf. Fig. 4). The eight phase categories are labeled by their midpoint (in years) in the legend of each panel and are shown in color; also listed in the legends are the minimum and maximum (min and max) amplitude of the anomalies (in K). The contours in all eight panels correspond to the SSH mean field, with CI = 0.2 m, as in Fig. 4.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

Similar oscillatory patterns have already been reported in the analysis of other instrumental SST records for the North Atlantic basin. Sutton and Allen (1997), for example, have identified in their analysis of shipboard observations a northward propagation of SST anomalies with a propagation speed similar to that of our analysis. Their dominant spectral peak, though, was centered at 12–14 yr in the SST power spectrum [i.e., similar to that of Deser and Blackmon (1993)].

Moron et al. (1998) carried out an M-SSA analysis of North Atlantic SST anomalies from the UK Met Office and were able to identify two separate standing-wave modes, with periods of 13 and 7.5 yr, respectively. In our analysis, though, only the 13-yr mode shows a standing wave with a similar geographical extent as in Moron et al. (1998) (not shown). The apparent discrepancy between the 7.7-yr traveling wave found here and the standing-wave character of the 7.5-yr mode of Moron et al. (1998) may be due to their strong compression of the UK dataset onto a few leading S-EOFs prior to M-SSA analysis. (This possibility is discussed in appendix B here, in connection with Fig. B1).

Still, the region occupied by the 7.5-yr mode of Moron et al. (1998) agrees well with the region in which our 7.7-yr mode is active. Moreover, in an analysis of the same UK dataset, Simonnet et al. (2005) identified a northward-propagating mode with a period of 7–8 yr, while Da Costa and Colin de Verdière (2002) and Jamison and Kravtsov (2010) found a similar mode in the dataset of Kaplan et al. (1998).

Figure 6 shows the corresponding eight phases of the 7.7-yr mode in the TAUX anomalies (in color). The oscillatory pattern for the zonal wind stress is essentially characterized by a meridional-dipole structure, with a much stronger northern pole. The eight phases reflect a standing wave, with in-phase behavior over large regions, and the dipole’s up-and-down seesaw leads to an alternative weakening and strengthening of the westerlies north of 40°N.

Fig. 6.
Fig. 6.

As in Fig. 5, but for TAUX anomalies (in color). The superimposed phase composites of the vector wind stress (arrows) are reconstructed using a projection of the TAUY anomalies onto the joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies; see text for details.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

Phase composites of the vector wind stress (arrows) have been reconstructed by a projection of the TAUY anomalies onto the joint M-SSA analysis of TAUX, SSH, and TEMP anomalies; see appendix D for the details of the algorithm. This analysis clearly illustrates the great intensity of coherent TAUX variations over the subpolar gyre, but it also shows that the wind pattern in our 7.7-yr mode resembles to some extent the atmospheric dipole pattern identified by Deser and Blackmon (1993). In agreement with their analysis, a comparison of the phase composites in the SST and TAUX anomalies of Figs. 5 and 6, respectively, shows that, at higher latitudes, positive TAUX anomalies correspond to negative SST anomalies; see also the animation in the supplementary online material. Sutton and Allen (1997) have also identified similar patterns of sea surface cooling by the wind stress in higher latitudes.

This cooling effect is physically quite plausible but ascertaining its exact mechanism will require a study of the ocean–atmosphere heat fluxes—sensible and latent—that goes beyond the scope of the current paper. In the meantime, it is interesting to note that the corresponding phase composites in the SSH anomalies (not shown) resemble in many of their details the SST anomalies; that is, we observe covariations between SST and SSH in this 7–8-yr traveling wave.

To illustrate the dynamical structure of an oscillatory mode in a single picture, we continue next with the calculation of phase maps, as described in greater detail in appendix C. For each of the three oscillatory modes that we found, it is possible to identify in a unique way a spatial phase that puts—for each cycle of the oscillation, each variable (TAUX, SSH, and SST) and each channel —the standard interval in one-to-one correspondence with the period of the oscillation [0, T]. As done for the phase composites, and for each oscillation, the origin of the phase is chosen arbitrarily, and phase maps are drawn with respect to this arbitrary origin and for one of the variables (TAUX, SSH, or SST), which is kept fixed.

The resulting phase maps are given in Figs. 7a–i in the same 3 × 3 format as in Fig. 4, for the TAUX, SSH, and SST anomalies and for each of the three oscillatory modes with periods of 7.7, 2.7, and 2.2. In the 7.7-yr mode (top panels), the spatial structure of the phase maps matches that of the already discussed phase composites in Figs. 5 and 6. That is, in the TAUX anomalies (Fig. 7a), the strong in-phase behavior over large homogeneous regions appears as large patches of similar colors, with phase opposition across the 40°N parallel, thus highlighting the meridional-dipole structure of the wind stress pattern in this mode. The southwest–northeast-oriented propagation in the SST anomalies (Fig. 7c) is reflected in a more gradual shift that runs through all the colors, with a zero isochron that runs west-northwest to east-southeast, roughly from Newfoundland to the Canaries. The corresponding phase map in the SSH anomalies (Fig. 7b) appears remarkably close to that of the SST anomalies (Fig. 7c) and thus indicates substantial covariation between the SSH and SST fields.

Fig. 7.
Fig. 7.

Phase maps for the TAUX, SSH, and SST anomalies in the three oscillatory modes with periods of (a)–(c) 7.7, (d)–(f) 2.7, and (g)–(i) 2.2 yr. The color coding reflects the cyclical character of the phase, with the interval being mapped onto each of the three periods [0, T], with and 2.2 yr, respectively. M-SSA parameters as in Fig. 4; grid points of low variance are masked, and the SSH CI = 0.2 m, as before.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

In the 2.7-yr mode, we find a standing wave in TAUX (Fig. 7d), with a meridional dipole structure similar to the one seen in the 7.7-yr mode (Fig. 7a). The 2.7-yr mode, however, is displaced noticeably farther north, with phase opposition across a parallel that is closer to 50°N. This mode, moreover, displays no phase opposition between TAUX (Fig. 7d) and SST (Fig. 7f), nor does there appear to be noticeable covariation between the latter and SSH (Fig. 7e). Indeed, the phase dynamics in this mode’s SST and SSH fields appears to be characterized by a more complex spatial structure.

In the 2.2-yr mode, we observe a north-to-south propagation in the TAUX anomalies (Fig. 7g), with no dipole structure like the one found in the two other modes. To some extent, the corresponding phase map in the SST anomalies (Fig. 7i) can be seen to be in phase opposition over much of the northern part of the domain, although not as obviously so as in the 7.7-yr mode. A comparison of the SSH and SST phase maps in Figs. 7h and 7i, on the other hand, gives no indication of covariation.

Note that in both modes of quasi-biennial variability, the zonal component dominates the wind stress, as it does in the 7.7-yr mode in Fig. 6, while the meridional component plays only a minor role.

In the remainder of the paper, we try to gain a better understanding of the differences among the dynamical properties of the three oscillatory modes that could have caused the differences we noted in Fig. 7. In the next subsection, we examine the full TEMP field and discuss the effect of covariation across ocean layers. The whole of section 5 is then devoted to the questions of whether the oscillatory modes in the SST can be attributed entirely to variations in the wind stress forcing.

c. Vertical phase coherence

There are several mechanisms that could contribute to some degree of covariation between the SSH and SST fields [see, e.g., Jones et al. (1998) and references therein]. There is, for example, the possibility of covariation in specific geographic regions associated with strong mesoscale variability.

Lee and Cornillon (1995) had found, for instance, a dominant period of 9 months in the SST field measured by satellite infrared imagery along the Gulf Stream front. Such intraseasonal periodicities were then studied theoretically by Feliks and Ghil (1996, 1997), among others. Using an M-SSA analysis of SST observations and model-output data, Schmeits and Dijkstra (2000) identified covarying patterns of intraseasonal variability with this 9-month period along the front, specifically near the Gulf Stream’s separation point.

In the present analysis, we concentrate instead on covariations with longer interannual time scales, which also extend over longer basinwide scales, off the Gulf Stream front. To better understand dynamical properties that could underlie the observed oscillatory modes, we include all available TEMP layers at greater depths as well. The vertical coherence of the observed SST pattern is then quantified in each of the modes in Fig. 7 by calculating a phase-coherence index; see Eq. (C5) in appendix C. In regions of high phase coherence, the observed SST pattern extends into deeper layers, and increasingly equivalent-barotropic behavior is thus highlighted.

Figure 8 shows the resulting maps of vertical phase coherence for the three oscillatory modes. The statistical significance of this coherence is tested against the null hypothesis of randomly shuffled phases, as derived from a random shuffling of all grid points.

Fig. 8.
Fig. 8.

Vertical phase coherence in the TEMP anomalies of the oscillatory modes with periods of (a) 7.7, (b) 2.7, and (c) 2.2 yr. The mean phase-coherence index (mean) is indicated in the legend, as well as a 99% level of significance (sig), against the null hypothesis of randomly shuffled phases. M-SSA parameters as in Fig. 4, and CI = 0.2 m for the SSH contours (solid black).

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

Common to all modes is a concentration of high values along the Gulf Stream front, that is, in a region in which interannual variability is strong (again, cf. Fig. 4, left column). This finding shows that the interannual dynamics in the Gulf Stream region is characterized by a strong barotropic component, as found by Schmeits and Dijkstra (2000) for their 9-month mode near the Gulf Stream separation point.

Off the Gulf Stream front, we observe likewise high phase-coherence values, especially prominent in the 7.7-yr mode (Fig. 8a). It is remarkable that the region of high phase-coherence values coincides fairly well with the area in which we see strong covariation between the SSH and SST fields (Figs. 7b,c). We are thus led to the conclusion that the southwest–northeast SST propagation in this 7.7-yr mode is characterized by a strong equivalent barotropic component. Interestingly, we see that phase coherence is especially high in the subpolar gyre, in which we observe a cooling-like effect of the surface wind stress on SST; see Figs. 5 and 6.

In the 2.7- and 2.2-yr modes, on the other hand, phase-coherent behavior is much less pronounced in regions off the Gulf Stream front (cf. Figs. 8b and 8c, respectively). This finding is consistent with the more complex phase dynamics in these modes and the apparent absence of covariation in SSH and SST (cf. Fig. 7, middle and lower panels).

5. Testing the hypothesis of an ocean response to atmospheric forcing

a. Methodological considerations

We address next the question to which extent the interannual oscillatory modes found in the North Atlantic TEMP field can be attributed entirely to its response to interannual variability in the atmospheric forcing. The alternative hypotheses have already been listed in this paper before; they include the ocean’s intrinsic dynamics and coupled atmosphere–ocean modes. Various mechanisms for intrinsic variability have been proposed, such as relaxation oscillations of the Gulf Stream system in the presence of time-independent wind forcing (e.g., Simonnet et al. 2005; Dijkstra and Ghil 2005, and references therein) or the turbulent oscillator of Berloff et al. (2007, and references therein); see also Liu (2012) for a historical overview of decadal to interdecadal variability.

In section 3, several oscillatory modes of similar period have been identified in separate M-SSA analyses of the TAUX, SSH, and TEMP anomalies: a 7–8-yr mode and two quasi-biennial modes, with periodicities of 2.7 yr and 2.2 yr. In section 4, these modes have been shown, through a joint M-SSA analysis, to be consistent, in certain regions at least, with an oceanic response to atmospheric forcing.

However, despite the good match in period of the oscillatory modes, the details of their temporal behavior do exhibit certain differences. In the 7.7-yr mode, for example, we observe a constant decline in amplitude in the TAUX anomalies (cf. Fig. 2d), whereas in the SSH and TEMP anomalies, the amplitude remains fairly constant over the entire 138-yr interval (cf. Fig. 3d).

The damping of this TAUX mode in time is consistent with the similar decline of the 7.7-yr mode in the TAUY anomalies and in the sea level pressure anomalies, as ascertained from the 20CRv2 reanalysis dataset (not shown). This falling off in 7.7-yr activity in atmospheric forcing is also reflected in instrumental data, such as the NAO index (Feliks et al. 2013).

The schematic picture in Fig. 9 is meant to illustrate the mutual effects of each fluid medium on the other, as inferred here from the SODA reanalysis. In the SODA setup, the atmosphere from the 20CRv2 reanalysis is merely a prescribed, time-dependent boundary condition acting on the ocean model (i.e., path 1 in Fig. 9 is missing and with it the possibility of a positive feedback loop that could provide a truly coupled mechanism). This absence of path 1 might be one of the reasons for the differences we noted in the temporal evolution of interannual variability in SST versus that in TAUX.

Fig. 9.
Fig. 9.

Schematic diagram of interannual variability in SST and TAUX. The SST map (lower level of the figure) and the TAUX map (upper level) show phase 3 from the phase composite analyses in Figs. 5 and 6, respectively.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

To overcome the limitations of a joint M-SSA analysis that may overemphasize coupled ocean–atmosphere modes, we proceed next to apply a different technique in comparing oscillatory modes in TEMP with those in TAUX. The starting point is that of separate M-SSA analyses, similar to the one in section 3. This time, however, we also analyze the TEMP and SSH anomalies separately and not jointly, as in section 3c; that is, we carry out three separate M-SSA analyses of 1) the TAUX anomalies, 2) the SSH anomalies, and 3) the TEMP anomalies.

Next, the M-SSA results of the TEMP anomalies are compared separately with those of the SSH and TAUX anomalies, respectively. This protocol is related to the Monte Carlo–type null hypothesis test applied in the present paper, a test for which Groth and Ghil (2015) have reviewed several mathematical techniques. We rely here on a comparison of the time EOFs (T-EOFs), which describe the temporal pattern of the oscillatory modes; for more mathematical details see appendix D.

This way, we get two different views of the TEMP anomalies: one that is linked to the spatiotemporal dynamics of the wind stress TAUX, and the other one linked to the dynamics of the SSH field. These two views allow the possibility of identifying regions in the TEMP anomalies that are more likely to be dynamically linked to either the TAUX or the SSH dynamics, respectively, but that cannot be distinguished from each other by a joint M-SSA analysis of all three fields.

On the one hand, projecting the TEMP anomalies onto the TAUX anomalies highlights regions in which there is a strong association between interannual variability in the wind forcing and in the ocean temperature field. On the other, projecting TEMP on SSH is likely to reflect other, possibly intrinsic, ocean dynamics.

The latter inference stands to reason, since the SODA assimilation scheme mainly corrects for differences in SST and thus has no explicit effect on the SSH field. The corrections act only implicitly on SSH, as it is linked to the vertically integrated temperature. Hence, examining closely the separate TAUX, SSH, and TEMP analyses, along with the partial joint analyses of the TAUX + TEMP fields and of the SSH + TEMP fields, can help answer the question of whether there is interannual variability in TEMP that cannot be explained by variations in TAUX alone.

b. Overall results

Figure 10 shows differences in the participation index at each grid point in the TEMP anomalies between their projection onto TAUX and SSH, respectively; see Eq. (D4) for a definition of the index . The differences are plotted for layers at different depths in the three oscillatory modes with periods of 7.7, 2.7, and 2.2 yr, respectively.

Fig. 10.
Fig. 10.

Differences in the participation index (in color) between a projection of the TEMP anomalies onto the SSH anomalies and onto the TAUX anomalies, respectively. Shown are the square root of the differences in (a)–(c) the 7.7-yr mode, (d)–(f) the 2.7-yr mode, and (g)–(i) the 2.2-yr mode, at the depths of (left) 5 m, (middle) 171 m, and (right) 466 m. Extreme values are given in the panel legends (min and max) (in K). M-SSA parameters for TAUX and SSH as in Figs. 2 and 3, respectively. SSH CI = 0.2 m.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

It turns out that in all nine panels we obtain a mixture of positive and negative differences (i.e., a stronger link of the TEMP anomalies to SSH or TAUX, respectively). These differences are comparable in absolute value to the amplitude of the three oscillatory modes (cf. Fig. 4), and it appears that positive and negative differences are geographically well structured and not just randomly distributed. This systematic character of the differences points to the existence of mechanisms that yield a stronger link of TEMP to either SSH or TAUX over well-defined regions.

Common to all oscillatory modes in Fig. 10, we observe at the surface (left column) a concentration of small spots of positive differences along the Gulf Stream front. This concentration indicates a stronger link between the spectral properties of the TEMP and SSH anomalies in the Gulf Stream region.

It is striking that this link becomes stronger with increasing depth (middle and right column); that is, the spots of positive differences increase in size and absolute value with depth, as shown also by the increase in maximum value (max) in the panel legends. The persistence and even enhancement of SSH covariance with ocean temperatures at depth is to some extent expected, as we have already identified strongly phase-coherent behavior in this region (cf. Fig. 8).

It is plausible to attribute such a strong link between TEMP and SSH to equivalent-barotropic behavior in this region and at long periods. Equivalent-barotropic behavior, in turn, could also point to the existence of intrinsic variability along the Gulf Stream front, as predicted, for example, in the setting of gyre-mode instability, in which barotropic gyre modes emerge on interannual time scales.

Beside the strong link between TEMP and SSH in all three oscillatory modes along the Gulf Stream front, we observe in Fig. 10 regions off the Gulf Stream front that show negative differences (i.e., in which interannual variability in TEMP is strongly linked to the TAUX variability). In particular, in the 7.7-yr mode at the surface (Fig. 10a), there is an area of strong negative differences in the subpolar gyre, which coincides with the region of maximum variability in TAUX (cf. Fig. 4a). This region is remarkably consistent with the one in which we have already identified a cooling effect.

According to Figs. 10b and 10c, the area of negative differences gradually vanishes with depth, as one might expect for a surface effect. At the surface (Fig. 10a), we also observe smaller negative values that cover a large region off the Gulf Stream front, in which we have observed a southwest-to-northeast propagating wave. This finding suggests an ocean response in SST due to variations in TAUX.

For the 2.7-yr mode, the middle panels of Fig. 10 show a qualitatively different picture. In all three layers, the mix of positive and negative differences and the association with the atmospheric forcing become less clear. Specifically, negative values are present in many regions of the North Atlantic and they are not limited to regions of strong TAUX variability in this mode (cf. Fig. 4d). Still, a tendency of generally more negative values at the surface and gradually more positive values along the Gulf Stream front at greater depth becomes apparent in Fig. 10; see also the min and max values in the panel legends.

The picture in the 2.2-yr mode (Fig. 10, lower panels) resembles qualitatively that of the 7.7-yr mode; that is, negative values prevail at the sea surface where TAUX variations are strong (cf. Fig. 4g) and where TEMP and TAUX anomalies are in phase opposition (cf. Figs. 7f,g). The influence of TAUX variations acting on the ocean’s TEMP fields (cf. Figs. 10h,i) likewise seems to vanish with increasing depth, where at the same time gradually more positive values along the Gulf Stream front emerge and indicate an increasingly stronger link with SSH.

Although the present analysis alone cannot provide definitive conclusions about the underlying dynamics, it is consistent with the view that interannual variability in the North Atlantic basin cannot be explained by the ocean’s response to similar variability in the atmospheric forcing alone (e.g., Dijkstra and Ghil 2005; Berloff et al. 2007; Penduff et al. 2011; Sérazin et al. 2015).

c. Changing patterns in the 7.7-yr mode

In this subsection, we are interested in a better understanding of the declining amplitude in the 7.7-yr mode in the TAUX anomalies (cf. Fig. 2d). The passive-ocean point of view would suggest a similar damping of the corresponding oscillatory pattern in the SST anomalies.

We thus divide the 138-yr SODA dataset into two shorter time intervals, 1871–1950 and 1929–2008, respectively. The small 20-yr overlap is necessary insofar as we keep a window length of yr and therefore need a minimal length of yr for each subinterval. In each of the two segments, we then run a joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies, with the same parameters as in section 4.

In the first, 1871–1950 segment, we obtain a 7–8-yr mode (not shown) that resembles in many of its spatiotemporal details the 7.7-yr mode in the full 138-yr interval, as seen in section 4 and in the supplementary animation.

The picture changes, however, when we analyze the second, 1929–2008 segment; see Figs. 11 and 12. There is still a 7–8-yr oscillatory mode with a similar period, but the corresponding phase-composite analysis reveals different dynamics in both TAUX and SST. In the phase composites in the TAUX anomalies in Fig. 11, we observe a shift in the geographical extension, with the maximum now displaced farther east. An analysis of TAUY (not shown separately, but see arrows in the figure) indicates that the decline in TAUX is not compensated by an increase in TAUY, and that TAUX still plays a dominate role in the wind stress.

Fig. 11.
Fig. 11.

Phase composites in TAUX (in color) from the 7–8-yr oscillatory mode of a joint M-SSA analysis of TAUX, SSH, and TEMP anomalies, during the second, 1929–2008 subinterval; see text for details. Four phases that correspond to one-half of the cycle are shown. The phase categories are labeled in the panel legends by their midpoint (in years); also shown are extreme values (min and max). Superimposed phase composites of the vector wind stress (arrows) are reconstructed from a projection of the TAUY anomalies onto the joint M-SSA analysis of the TAUX, SSH, and TEMP anomalies. M-SSA parameters as in Fig. 4; SSH CI = 0.2 m.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

Fig. 12.
Fig. 12.

As in Fig. 11, but in the SST anomalies.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

In the corresponding phase composites of the SST anomalies (Fig. 12), this eastward displacement of the stationary pattern is accompanied by a vanishing of the southwest–northeast propagating pattern previously seen in Fig. 5. Remarkably, though, the interannual variability along the Gulf Stream front remains quite high in Fig. 12, if not even higher in amplitude than in the 7.7-yr mode of the full dataset in Fig. 5.

In our separate M-SSA analysis of the TAUX anomalies alone for 1929–2008, it turns out that the 7–8-yr period of interannual variability is no longer significant (not shown). This finding is consistent with section 3, in which we found the 7.7-yr mode significant in TAUX in the first, 1871–1950 segment only.

The overall agreement between basinwide changes in the atmospheric TAUX and in the oceanic SST fields leads us to the conclusion that the observed traveling wave with a 7–8-yr periodicity off the Gulf Stream can, indeed, be attributed to the ocean’s response to interannual variability in the atmospheric forcing. In the Gulf Stream region, though, this change shows that 7–8-yr variability cannot merely be attributed to the atmospheric forcing; see also the concluding discussion in section 6b below about the possibility of intrinsic ocean dynamics.

The cause of the changes in the atmospheric forcing pattern, however, remains unclear from the present analysis. The changes themselves, though, appear to be consistent with the observed decadal fluctuations noted by Deser and Blackmon (1993). Moreover, we are able to identify a similar change (not shown) in the SST dataset of Kaplan et al. (1998).

6. Concluding remarks

a. Summary

By analyzing spectrally the evolution in time of the North Atlantic temperature field, denoted here by TEMP, in the Simple Ocean Data Analysis (SODA) reanalysis (Carton and Giese 2008; Giese and Ray 2011), we were able to identify and describe prominent and statistically significant interannual oscillations along the Gulf Stream front and in large regions of the North Atlantic. The extensive reanalysis dataset has been decomposed into different patterns of spatiotemporal variability with the advanced spectral method of multivariate singular spectrum analysis (M-SSA); see Ghil et al. (2002) and Alessio (2016) for a comprehensive review of M-SSA, and Groth and Ghil (2011, 2015) for recent improvements thereof that help separate distinct spatiotemporal patterns.

The M-SSA results on the temperature field were compared with the M-SSA results on the zonal wind stress forcing, denoted by TAUX, from the atmospheric 20CRv2 reanalysis (Compo et al. 2011). In the interannual frequency band, three oscillatory modes with periods of 7.7, 2.7, and 2.2 yr and high statistical significance that are similar in both TAUX and TEMP were identified (cf. section 3). These findings are consistent with those of numerous other publications that studied interannual variability in the atmospheric field over the North Atlantic (e.g., Rogers 1984; Hurrell 1995; Hurrell and Van Loon 1997; Robertson 2001; Feliks et al. 2010) and in the North Atlantic SSTs (e.g., Moron et al. 1998; Da Costa and Colin de Verdière 2002; Jamison and Kravtsov 2010).

To understand patterns of interannual variability in the coupled ocean–atmosphere system, these three modes were furthermore analyzed in a joint M-SSA analysis (cf. section 4). It turned out that the 7.7-yr oscillatory mode is characterized by a basinwide southwest-to-northeast–oriented propagation pattern in SSTs.

This SST pattern is similar to the one identified in several previous papers (Sutton and Allen 1997; Da Costa and Colin de Verdière 2002; Simonnet et al. 2005; Jamison and Kravtsov 2010) and it was shown here to be linked to alternations in the meridional dipole pattern in TAUX. In the subpolar gyre, the SST and TAUX fields are in phase opposition, which indicates a cooling effect of the wind stress on the upper ocean layer, in agreement with Deser and Blackmon (1993) and Sutton and Allen (1997); see also our supplementary online animation. In our analysis, this propagating SST pattern is characterized, furthermore, by a strong equivalent-barotropic component, which leads to covariations in the SST and SSH fields, and phase-coherent behavior of the observed SST pattern with the temperature layers in the deep ocean that are available in the TEMP field of the SODA reanalysis.

In the two modes of quasi-biennial variability, though, the dynamics turned out to be more complex: there is no clear indication of covariation between SST and SSH off the Gulf Stream region, and the association with the wind stress forcing is less obvious.

Finally, in section 5, we analyzed to what extent the observed patterns of interannual variability in the North Atlantic temperature field can be attributed exclusively to variations in the atmospheric forcing (Bjerknes 1964; Hasselmann 1976), as opposed to or in combination with intrinsic ocean dynamics. We have shown that in large regions off the Gulf Stream front, the wind stress forcing has a substantial influence on the interannual variability of the underlying ocean in all three modes—namely the 7.7-yr mode and the two quasi-biennial modes—but also that this influence gradually decreases with depth.

Along the Gulf Stream front, our analysis supports the existence of other, internal mechanisms in all three modes; furthermore, the influence of the latter mechanisms on interannual variability increases with depth. Differences in the oscillatory modes between the wind stress forcing and the ocean temperature field were most prominent in the 7–8-yr mode: we observe a pronounced decline in amplitude of the atmospheric forcing, while the amplitude of this mode in the ocean dynamics is quite constant (cf. Figs. 2c and 3c). This decline in amplitude is accompanied by a change in the spatial pattern of the atmospheric forcing in the latter part of the dataset (i.e., 1929–2008; cf. Fig. 12), as well as in the SST pattern.

In fact, the southwest–northeast SST propagation across the North Atlantic basin disappears in the latter time interval, while 7–8-yr variability along the Gulf Stream front persists. This persistence is quite remarkable and it supports even more strongly the idea that variability in the 7–8-yr frequency band in the Gulf Stream region is substantially influenced by other mechanisms, and is not merely a response to 7–8-yr variability in the atmospheric forcing. Our findings in this respect are consistent with the view that interannual variability in the North Atlantic basin cannot be explained by the ocean’s response to similar variability in the atmospheric forcing alone.

b. Discussion

The M-SSA analyses of sections 35 suggest that different types of mechanisms participate in the variability of the oceanic TEMP field; these mechanisms are well characterized by their distinct spatiotemporal patterns.

On the one hand, our analysis of the SODA dataset shows the substantial influence that midlatitude atmospheric perturbations exert on the underlying ocean, away from the regions of intense oceanic currents, (cf. Frankignoul and Hasselmann 1977; Frankignoul et al. 1997; Sérazin et al. 2015), and that large-scale interannual variability in the atmospheric forcing is indeed reflected in the underlying ocean’s large-scale SST response (Bjerknes 1964); see again Fig. 9 here, in which this influence is schematically indicated as path 2. Several studies of ocean–atmosphere interaction over the North Atlantic basin have highlighted the role of an SST tripole pattern (e.g., Czaja and Frankignoul 1999; Marshall et al. 2001; Czaja and Frankignoul 2002; Czaja et al. 2003; Kravtsov et al. 2008; Jamison and Kravtsov 2010). This large-scale pattern (not shown) resembles to some extent that of our 7.7-yr mode. The latter, though, involves southwest-to-northeast propagation that connects the pole in the Gulf Stream region with the one in the subpolar gyre. Determining the extent to which these two patterns describe the same physical mechanism is left for future studies.

On the other hand, there is strong variability in highly localized spots along the Gulf Stream front, which can be plausibly linked to internal ocean dynamics; see also Penduff et al. (2011) and Sérazin et al. (2015). The present analysis does not ascertain the extent to which this oceanic variability emerges through gyre-mode instabilities and homoclinic bifurcations or from the turbulent oscillator of Berloff et al. (2007, and references therein).

Still, the statistical evidence of equivalent-barotropic behavior in this region is entirely consistent with the gyre-mode hypothesis, such as confirmed by several studies with more or less idealized double-gyre ocean models (Jiang et al. 1995; Speich et al. 1995; Simonnet and Dijkstra 2002; Simonnet et al. 2003a,b, 2005; Sushama et al. 2007). Across a hierarchy of models—from simple rectangular-shaped ocean basins and 1.5-layer reduced-gravity models (Jiang et al. 1995; Speich et al. 1995; Chang et al. 2001; Simonnet et al. 2003a) to more realistic North Atlantic basin shapes, several layers, and eddy-permitting resolutions (Simonnet et al. 2003b, 2005)—it was found that the idealized double-gyre flow becomes unstable to two types of barotropic modes: basin modes and gyre modes. The time scales of the former are a couple of months (Chang et al. 2001), whereas the latter instabilities have interannual time scales of 7–8 yr in the North Atlantic (Speich et al. 1995; Simonnet et al. 2005), similar to the one identified here in the SODA reanalysis. On the other hand, the turbulent-oscillator hypothesis depends in an essential way on the vigorous activity of baroclinic eddies.

Whatever the underlying mechanisms of interannual variability, it is interesting to note that, in a series of increasingly detailed atmospheric models, Feliks et al. (2004, 2007, 2011, 2016), Brachet et al. (2012), and Deremble et al. (2012) have shown that sharp SST gradients, like the Gulf Stream or Kuroshio fronts, can have a strong influence on the overlying atmosphere and its low-frequency variability. Variations in frontal strength, for example, were shown to modulate the atmospheric jet’s intensity; see also Schneider and Qiu (2015) and Kilpatrick et al. (2016, and references therein) for details on the Ekman pumping mechanism that mediates the SST front effect on the midlatitude atmosphere.

This modeling result on the atmospheric effects of ocean fronts (Small et al. 2008) is consistent with statistical evidence (Mosedale et al. 2006) for an SST effect, especially south of Cape Hatteras, on the NAO; see again Fig. 9 here, in which this potential spinup mechanism for atmospheric interannual variability is indicated by path 1. Thus, one might be able to anticipate low-frequency fluctuations in the atmosphere above and downstream of the North Atlantic basin by monitoring of the Gulf Stream.

Such monitoring could involve its total transport across a meridional section off Cape Hatteras—as done systematically by the Japan Meteorological Agency for the Kuroshio (Imawaki et al. 2001)—as well as the SST and SSH gradients across such a section.

Acknowledgments

It is a pleasure to acknowledge stimulating discussions with J. C. McWilliams. The authors would also like to thank three anonymous reviewers for their extremely careful and constructive reviews. The research herein was supported by Grants N00014-12-1-0911 and N00014-16-1-2073 from the Multidisciplinary University Research Initiative (MURI) of the Office of Naval Research and by National Science Foundation Grant OCE-1243175.

APPENDIX A

Singular Spectrum Analysis

In this appendix, we briefly summarize the M-SSA algorithm as originally proposed by Broomhead and King (1986a,b). For a review of the full M-SSA methodology, as used herein, see Ghil et al. (2002), Groth and Ghil (2011, 2015), Alessio (2016) and references therein.

Let be a multivariate time series with channels of length . Each channel is embedded into an -dimensional phase space by using lagged copies , , from which we form the full augmented trajectory matrix by concatenating all channels, . The matrix has DM columns of length .

The M-SSA algorithm then continues with a singular value decomposition (SVD) of ,
ea1
where is a normalization factor equal to the larger of N′ and DM. The matrix has nonvanishing diagonal elements, the singular values , where is equal to the smaller one of N′ and DM. The singular vectors and have both rank . The left-singular vectors describe pure temporal behavior, where the right-singular vectors describes spatiotemporal behavior; that is, is referred to as time EOFs (T-EOFs), and as space–time EOFs (ST-EOFs). The latter are composed of consecutive segments of length , each of which is associated with a channel in given by
A projection of onto ,
ea2
gives a set of pairwise uncorrelated principal components (PCs), of reduced length N′.
Dynamical behavior that is associated with a subset of ST-EOFs can be reconstructed from
ea3
with being a diagonal matrix of size , in which elements if and otherwise.
Upon averaging along the skew diagonals of ,
ea4
we finally obtain the reconstructed components (RCs). The normalization factor and the summation bounds and for the central part of the time series, , are simply . For either end of the series, or , these values are modified according to the length of the skew diagonal corresponding to the given ; see, for instance, Ghil et al. (2002).

In the application at hand, and the values of for the multivariate time series analyzed in sections 35 are on the order of for the two-dimensional arrays TAUX, SSH, and SST, and on the order of for the three-dimensional array TEMP, as well as for those involved in the joint M-SSA analyses that include TEMP.

APPENDIX B

Projection onto Spatial EOFs

The large values of used herein are typical nowadays in the climate sciences, whether one studies recent instrumental datasets from both in situ and satellite-borne observing systems or long climate simulations by high-end models (IPCC 2007, 2013). We discuss, therefore, now the implications of projecting a large dataset (i.e., ) onto classical S-EOFs.

Prior to the M-SSA analysis, the multivariate time series x is first transformed into a set of classical PCs via a SVD, , and the PCs, , as input channels for M-SSA.

As in the previous section, M-SSA starts with embedding each channel in into an -dimensional phase space by using lagged copies , , from which we obtain the matrix of size .

It can be seen that the relation simply translates into
eb1
with being the Kronecker product and the identity matrix of size .
Next, M-SSA continues in calculating the SVD of ,
eb2
with the normalization factor as in Eq. (A1).
Replacing and in (B1) by their SVDs, Eqs. (A1) and (B2), respectively, it follows that
eb3
The matrix is orthogonal and so is .
Since the SVD of a real-valued matrix is unique up to a pairwise change in the sign of the singular vectors, it follows that and have the same singular values, , and in case of nondegenerate singular values,
eb4a
eb4b
with a diagonal matrix of elements . That is, the left- and right-singular vectors from are (up to the sign) determined by the singular vectors of and .
Substituting in the RCs of [cf. Eq. (A3)] according to Eq. (B4) then gives
eb5a
eb5b
With , the RCs of , it follows that
eb6
Upon channel-wise averaging along skew diagonals on either side of Eq. (B6) [cf. Eq. (A4)], we finally obtain the relation
eb7
between the RCs of x and that of y; that is, the Kronecker product simplifies to when returning from to x.

Equation (B7) has important practical implications in the M-SSA analysis of extensive datasets, as it can be broken down into three smaller parts:

  1. The dataset x is first transformed into PCs via a classical SVD analysis, i.e., ;
  2. M-SSA ST-EOFs and RCs are determined from ; and
  3. these ST-EOFs and RCs are then back-transformed to get the ST-EOFs and RCs of the full dataset x by using Eqs. (B4) and (B7), respectively.
Equation (B7) also demonstrates that there is in particular no loss of information in the RCs when we retain the full set of PCs.

The only difference appears when we wish to rotate the ST-EOFs. Since the varimax criterion is a nonlinear function of the eigenelements, the rotation results of and can differ. In our experience with the SODA dataset, however, these differences are minor, and we observe likewise good results when rotating instead.

The extent to which a prior projection of x onto just a subset of S-EOFs can influence the reconstruction quality of M-SSA will be the subject of the remainder of this section. We follow the simulation experiment of Groth and Ghil (2015) and first generate a noise-free realization of our reference signal to be reconstructed (see Fig. B1, upper panel). Superimposed on the reference signal is spatiotemporally correlated red noise, generated from AR(1) processes, with a randomly drawn parameter from the interval . The nonvanishing lag-0 covariance matrix of the noise, as well as its variance modulation in space (cf. the red line in Fig. B1c) is meant to dominate the structure of the leading S-EOfs.

Fig. B1.
Fig. B1.

Reconstruction of spatiotemporal oscillations in the presence of strong correlated noise. (a) Noise-free reference signal, channels of length . (b)–(g) The noise-contaminated signal is first projected onto the leading L = {3, 7, 12, N} S-EOFs from a conventional PCA analysis, which captures 45%, 73%, 87%, and 100% of the total variance, respectively. The resulting PCs are analyzed via M-SSA with a window length M = 40 and the first 20 ST-EOFs varimax rotated. (b),(d),(f) and (h) show average reconstruction from an ensemble of 200 repetitions of the experiment; (c),(e),(g) and (i) show corresponding channel-wise standard deviation of the full signal (red), the reference signal (green), and the average reconstruction (blue). The signal-to-noise ratio is 1/12. For illustration purpose, the standard deviation of the full signal is scaled by 12−1/2.

Citation: Journal of Climate 30, 7; 10.1175/JCLI-D-16-0370.1

Based on an ensemble of 200 repetitions of the experiment, we evaluate statistical properties of the M-SSA reconstruction. To identify in each realization the signal part, we follow Groth and Ghil (2015) and select a pair of ST-EOFs that matches best the leading pair of ST-EOFs in the M-SSA analysis of the noise-free reference signal.

Figure B1 (left column) shows the M-SSA reconstruction averaged over the ensemble of repetitions of the experiment for varying parameter . For (Fig. B1b), the set of the leading three S-EOFs captures on average only 45% of the total variance. As a consequence of such a strong compression, the wave propagation in the reference signal is completely missed. Instead, the spatial structure of the reconstruction (Fig. B1c, blue line) follows that of the full signal (red line) and completely misses the linear increase and the small-scale line structure in the reference signal (green line).

As the number of S-EOFs increases, the reconstructed pattern gradually approaches that of the reference pattern. It is the projection onto the full set of S-EOFs that, finally, gives the best reconstruction quality (lower panels); that is, it perfectly reconstructs the wave propagation. We have also analyzed the full dataset without any prior projection onto S-EOFs (not shown) and it turns out that there is no significant improvement in this particular case.

The experiment clearly illustrates the risk that a projection onto a small number of S-EOFs entails (i.e., the risk of the reconstruction being biased toward other large-scale patterns). It also shows that wave propagation patterns and small-scale structures are likely to be missed, and we recommend retaining a maximum number of PCs as input channels for the subsequent M-SSA analysis.

APPENDIX C

Phase Composites and Phase Maps

A helpful tool for the understanding of the spatiotemporal dynamics of reconstructed oscillatory behavior is that of a phase-composite analysis [cf. Plaut and Vautard (1994) and Moron et al. (1998)]. Usually, given the reconstruction r, its SVD is calculated as shown,
ec1
and a phase is defined as the argument of the leading PC, expanded into the complex plane via either 1) its first derivative or 2) its Hilbert transform.

In the present work, though, we use a more direct approach, without the need of any further calculations, that expands the leading PC into the complex plane. Feliks et al. (2013), for example, have shown that upon varimax rotation of the ST-EOFs in M-SSA—as suggested by Groth and Ghil (2011)—the corresponding RCs become unimodal, and the leading pair (1, 2) of S-EOFs in Eq. (C1) captures most of the RCs’ variance.

It follows that S-EOFs (1, 2) span a two-dimensional subspace in which the oscillatory mode is sufficiently well represented, and we define an instantaneous phase as the argument
ec2
here are the components of the leading pair of left-singular vectors, defined as columns in ; see also Sauer et al. (1991) and Gibson et al. (1992), who discuss PCs in the state space reconstruction.
The usefulness of Eq. (C1) in the characterization of oscillatory behavior in r further emerges when describing phase relations between different channels . Projecting onto gives the spatial phase as the argument:
ec3
According to Eq. (C1), this can be simplified as
ec4
with the leading pair of singular values in and the dth element of the corresponding right-singular vectors.
The phase coherence of an oscillatory mode can be quantified by means of a phase-coherence index ,
ec5
the ratio between resultant and total length of the sum of complex-valued vectors, with the norm. The index becomes maximal , if all phases are equal, and otherwise. The index is related to the measure of dispersion , which quantifies how well vectors align (e.g., Fisher 1993). Note that is a measure of the participation of channel in the variance of , and it is this numerator in Eq. (C5) that gives greater weight in to channels with a strong participation (i.e., with a higher variance).

APPENDIX D

Procrustes Algorithm

In this section, we give a brief description of the M-SSA Procrustes algorithm from Groth and Ghil (2015) and its application to the comparison of different datasets. This comparison includes here the possibility of the datasets pertaining to different geographical regions.

Suppose we have two datasets and of the same length , but different dimension and , respectively. The M-SSA algorithm then continues in calculating the SVD of the corresponding trajectory matrices and , cf. Eq. (A1),
ed1
Given that , we obtain two T-EOF matrices that have both the same size .
The M-SSA Procrustes algorithm then seeks to find an orthogonal rotation that provides maximal similarity between and . An optimal solution to this target-rotation problem is given by the orthogonal Procrustes algorithm (Green 1952; Hurley and Cattell 1962; Schönemann 1966),
ed2a
which yields the SVD given by
ed2b
The rescaling of the T-EOFs by the corresponding singular value is an essential modification of the original test of Allen and Robertson (1996), which was shown by Groth and Ghil (2015) to improve the accuracy of the significance test.

Beside this so-called scaled target rotation of T-EOFs in Eqs. (D2), these authors have also discussed different possibilities of a rotation of ST-EOFs. However, in their case, both datasets are of equal size and so are and . Here this is not necessarily the case, as we are interested in a comparison of datasets that have different geographical extents. In contrast to a rotation of ST-EOFs, which compares spatial aspects at each grid point, a rotation of T-EOFs only compares temporal aspects of the two datasets.

Still, the solution in (D2) for the target rotation of T-EOFs in comes along with an equivalent rotation of ST-EOFs in ; that is, the SVD of becomes
ed3
with being the identity matrix. Note that this also involves a similarity transform of the singular values, , with nonvanishing elements off the main diagonal.
To quantify the contribution of channel to the variance of the kth principal component, Groth and Ghil (2011) introduced a participation index for M-SSA,
ed4
it is this index that we analyze in studying the effects of a rotation of temperature ST-EOFs in Eq. (D3). In section 5, we are in particular interested in differences between a rotation of the temperature T-EOFs onto the T-EOFs of the SSH field and a rotation onto the T-EOFs of the wind stress forcing, respectively.

In section 4, on the other hand, we rotate the T-EOFs of TAUY onto the T-EOFs of a joint M-SSA analysis of the TAUX, SSH, and TEMP dataset, to obtain rotated ST-EOFs as well as corresponding RCs for the meridional component TAUY of the wind stress.

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