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  • View in gallery

    (a) The linear trend of the sea surface height anomaly in the study domain and (b) the coefficient of determination, which is a measure of the goodness of linear fit (1-residual sum of squares/total sum of squares). The 1000-, 2000-, and 3000-m isobaths are shown in solid gray lines.

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    The standard deviation of the sea surface height anomaly.

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    The temporal averages over the period of 1993–2010 of (a) original SSHA, (b)–(g) the IMFs at increasing time scales, and (h) the residual (long-term trend).

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    The variance explained by the signals over different time scales corresponding to components in Figs. 3b–h.

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    The IMFs and the residual decomposed from the spatially averaged SSHA over the subpolar gyre. Units are cm.

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    The Hilbert transforms of the IMFs (C1–C8). (a) The instantaneous energy density level (IE) in cm2, (b) the Hilbert spectrum, and (c) marginal spectral density (MSD) of each frequency (in cm).

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    Similar to Fig. 6, but only applying the Hilbert transforms to the selected IMFs (C5–C8). (a) The instantaneous energy density level (IE; in cm2), (b) the Hilbert spectrum, and (c) marginal spectral density (MSD) of each frequency (in cm).

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    The altimeter SSHA and steric height anomaly calculated from temperature and salinity observations in the SPG.

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    The IMFs and the residuals decomposed from the heat content anomalies estimated from surface heat flux (gray dotted line; right y axis) and the SSHA (solid black line; left y axis). The depth-integrated heat content anomaly (dashed black line; left y axis) is also shown for comparison. Units are GJ m−2.

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    Close relationships among the heat content of the central Labrador Sea (dashed black line), the sea surface height anomaly of the central Labrador Sea (solid black line), the meridional overturning circulation transport at (gray dotted line), and the cumulative NAO index (black line with circles). The central Labrador Sea is defined by a box bounded by 56°–61°N and 48°–58°W. The unit and range of each index is labeled with y axis of same color. Note that the y axis of the CumNAO index is reversed.

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    Domains of areas 1, 2, and 3 used to spatially average the SSHA and examine the lagged correlation between each other. Arrows indicate possible propagation pathway of AMOC variations. The 1000-, 2000-, and 3000-m isobaths are shown in solid gray lines.

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    The IMFs (C1–C6) and the residual (R) decomposed from a SSHA time series (input) at 60°N, 42°W. Units are cm.

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Climate Signals in the Mid- to High-Latitude North Atlantic from Altimeter Observations

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  • 1 College of Earth, Ocean and Environment, University of Delaware, Newark, Delaware
  • | 2 College of Earth, Ocean and Environment, University of Delaware, and University of Delaware–Xiamen University Joint Institute of Coastal Research and Management, Newark, Delaware
  • | 3 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California
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Abstract

The variability of the sea surface height anomaly (SSHA) in the mid- to high-latitude North Atlantic for the period of 1993–2010 was investigated using the ensemble empirical mode decomposition to identify the dominant time scales. Sea level variations in the North Atlantic subpolar gyre (SPG) are dominated by the annual cycle and the long-term increasing trend. In comparison, the SSHA along the Gulf Stream (GS) is dominated by variability at intraseasonal and annual time scales. Moreover, the sea level rise in the SPG developed at a reduced rate in the 2000s compared to rates in the 1990s, which was accompanied by a rebound in SSHA variability following a period of lower variability in the system. These changes in both apparent trend and low-frequency SSHA oscillations reveal the importance of low-frequency variability in the SPG. To identify the possible contributing factors for these changes, the heat content balance (equivalent variations in the sea level) in the subpolar region was examined. The results indicate that horizontal circulations may primarily contribute to the interannual to decadal variations, while the air–sea heat flux is not negligible at annual time scale. Furthermore, the low-frequency variability in the SPG relates to the propagation of Atlantic meridional overturning circulation (AMOC) variations from the deep-water formation region to midlatitudes in the North Atlantic, which might have the implications for recent global surface warming hiatus.

Corresponding author address: Xiao-Hai Yan, College of Earth, Ocean and Environment, University of Delaware, 261 S. College Ave., Newark, DE 19716. E-mail: xiaohai@udel.edu

Current affiliation: Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University, Durham, North Carolina.

Current affiliation: Department of Oceanography, Pusan National University, Busan, South Korea.

Abstract

The variability of the sea surface height anomaly (SSHA) in the mid- to high-latitude North Atlantic for the period of 1993–2010 was investigated using the ensemble empirical mode decomposition to identify the dominant time scales. Sea level variations in the North Atlantic subpolar gyre (SPG) are dominated by the annual cycle and the long-term increasing trend. In comparison, the SSHA along the Gulf Stream (GS) is dominated by variability at intraseasonal and annual time scales. Moreover, the sea level rise in the SPG developed at a reduced rate in the 2000s compared to rates in the 1990s, which was accompanied by a rebound in SSHA variability following a period of lower variability in the system. These changes in both apparent trend and low-frequency SSHA oscillations reveal the importance of low-frequency variability in the SPG. To identify the possible contributing factors for these changes, the heat content balance (equivalent variations in the sea level) in the subpolar region was examined. The results indicate that horizontal circulations may primarily contribute to the interannual to decadal variations, while the air–sea heat flux is not negligible at annual time scale. Furthermore, the low-frequency variability in the SPG relates to the propagation of Atlantic meridional overturning circulation (AMOC) variations from the deep-water formation region to midlatitudes in the North Atlantic, which might have the implications for recent global surface warming hiatus.

Corresponding author address: Xiao-Hai Yan, College of Earth, Ocean and Environment, University of Delaware, 261 S. College Ave., Newark, DE 19716. E-mail: xiaohai@udel.edu

Current affiliation: Division of Earth and Ocean Sciences, Nicholas School of the Environment, Duke University, Durham, North Carolina.

Current affiliation: Department of Oceanography, Pusan National University, Busan, South Korea.

1. Introduction

Vigorous large-scale oceanic circulations and intense air–sea interactions in the mid- to high-latitude North Atlantic make this region an important component in global climatic studies. Here, tropical warm and saline seawaters, transported northward by the Gulf Stream (GS) and the North Atlantic Current (NAC), meet the cold and fresh Arctic waters that are transported southward by the Greenland and the Labrador Currents. The strong boundary currents constitute the cyclonic North Atlantic subpolar gyre (SPG), whose variability, in both intensity and shape, is critical in the redistribution of heat and salt in the North Atlantic (Hátún et al. 2005; Sarafanov 2009; Häkkinen et al. 2011). The SPG is also known as the formation region of intermediate water due to intense air–sea interaction, especially during severe winters (Dickson et al. 1996; Marshall et al. 1998). Warm and saline waters become dense after releasing heat to the atmosphere and thus sink at certain locations in the SPG, generating an equatorward return flow at depth. This return flow coupled with surface northward western boundary currents together form the lower and the upper limbs of the Atlantic meridional overturning circulation (AMOC), which may have significant impacts on abrupt climate changes (Clark et al. 2002; McManus et al. 2004; Lynch-Stieglitz et al. 2007, 2011).

One of the most challenging aspects in studying the variability of the North Atlantic is the involvement of physical processes acting over a wide range of temporal and spatial scales, from the short time scale local tip jets events (e.g., Våge et al. 2008) to eddy activities with strong seasonality and interannuality (e.g., Lilly et al. 2003; Volkov 2005) to variations in horizontal gyre circulations at intraseasonal to interannual time scales whose spatial scales could have an impact on climate change. The subpolar region is particularly climate-relevant as variability in the SPG intensity in past decades has been linked to variations in the deep-water formation rate in the Labrador Sea and the AMOC (Böning et al. 2006; Bersch et al. 2007; Rhein et al. 2011). The observed spinup (Belkin 2004) and the subsequent spindown of the SPG (Häkkinen and Rhines 2004) and the recent weakening of the AMOC (Latif et al. 2006) have revealed the dramatic variability of the North Atlantic at interannual to decadal time scales. Moreover, the apparent warming of subsurface ocean waters in the subpolar region has been suggested to be a major contributor for accelerated glacier melting (Bindschadler 2006; Holland et al. 2008), implying a potential relationship between the SPG intensity that determines the amount of heat transported along Greenland coast and the melting ice mass along the margin of the SPG. It is, however, still not well understood whether the ongoing warming is part of low-frequency variations or a long-term trend. Therefore, distinguishing the relative importance of different modes of variability and possible related causes has become of great necessity and has increasingly captured the interest and attention of both scientists and policymakers.

Sea surface heights, which have been measured by the repeat coverage of altimeters, provide one of the most important oceanic signals related to the climate variability at both global and regional scales. The sea level responds to high-frequency wind forcing, for example, during storms and hurricanes, and has been used for eddy studies (e.g., Volkov 2005; Chelton et al. 2011). In addition, the sea surface height anomaly (SSHA) can be interpreted as either the variations of upper ocean water properties due to changes of temperature and salinity in the water columns or the changes in the oceanic circulation given that the dynamic sea level variations balance geostrophic velocity anomalies (e.g., Häkkinen and Rhines 2004; Böning et al. 2006). Therefore, the sea level is a good indicator for large-scale oceanic variability particularly at annual and longer time scales. Since the launch of TOPEX/Poseidon and ERS-1 and -2 in the early 1990s, followed by Jason-1 and Jason-2, the precision of an individual sea surface height measurement based on these missions has now reached the 1–2-cm level (e.g., Beckley et al. 2010). This continuous sampling of sea level with increasing accuracy using altimetry measurements provides a powerful tool to study the variability of the sea level in the context of climate variability and climate change.

Sea level variations in the North Atlantic have been investigated by previous studies for certain time scales. Ferry et al. (2000) attributed the seasonality of the SSHA in the North Atlantic to the steric changes of heating by examining in situ data and a numerical simulation. At interannual time scales, the SSHA changes were found to coincide with the shifts in the North Atlantic Oscillation (NAO) index (Häkkinen 2001; Esselborn and Eden 2001; Volkov and van Aken 2003). Based on numerical simulations, Häkkinen (1999, 2001) concluded that the low-frequency SSHA variability along the GS and in the SPG is related to the meridional overturning process. Zhang (2008) further suggested that the SSHA pattern could be used as a “fingerprint” of AMOC variations at middle to high latitudes using a 1000-yr model simulation. Li et al. (2012), based on altimeter observations alone, indicated that the low-frequency SSHA variability in the North Atlantic responds to the cumulative NAO forcing and its distinct reduction of variance between the 1990s and 2000s might relate to the propagation of AMOC variations. However, one should be cautious about using SSHA to represent the AMOC change given the existence of a high-frequency wind-driven response as suggested by Lorbacher et al. (2010), based on a sequence of global ocean-ice model experiments. The authors also pointed out that a careful interpretation of the SSHA patterns is needed since its causes depend on the time scales of interest. Given that the changes associated with eddies and winds are large and the length of altimeter record is relatively short, it is possible that small shifts in the short-term variability will get aliased into the trend. In the present paper, we primarily analyzed 18-yr records of the SSHA (1993–2010) using the merged products of multiple altimetry missions that were or currently are on operation. We explored the characteristic features of the SSHA patterns in the North Atlantic at various time scales and then investigated the possible causes. This analysis helps to advance our understanding of the North Atlantic variability, especially in the SPG, in terms of the SSHA variations. The Hilbert–Huang transform (HHT) was applied to examine different modes of variability as well as the long-term trends in the SSHA. Based on the separation of sea level modes, we intended to find the dominant time scales of variability and factors that may be responsible for each mode of variability.

Contents are organized as follows. Section 2 offers a brief description on the data and methods used. In section 3, the altimeter data and the decomposed components at various time scales are analyzed. Spatial and temporal differences of the SSHA variability are discussed with more emphasis on the SPG. The possible causes of the SSHA variability at corresponding time scales are explored in section 4 by primarily investigating the upper layer heat content anomaly. The conclusions appear in section 5.

2. Data and method

a. Data used

Weekly and monthly altimeter products with ⅓° spatial resolution were produced by SSALTO–Data Unification and Altimeter Combination System (DUACS) and distributed by Archiving, Validation and Interpretation of Satellite Oceanographic data (AVISO), with support from the Centre National d’Etudes Spatiales (CNES). The delayed time (DT) products combine data from different missions, significantly improving the estimation of mesoscale features (Le Traon and Dibarboure 1999).

Quality-controlled subsurface ocean temperature and salinity data were obtained from the Ensemble3 (EN3) dataset (Ingleby and Huddleston 2007). This dataset combines data from the World Ocean Database 2005 (WOD05), the Global Temperature and Salinity Profile Project (GTSPP), Argo, and the Arctic Synoptic Basinwide Oceanography (ASBO) project. The quality control procedure includes the use of altimeter data. Objectively analyzed gridded data at monthly intervals at 1° × 1° resolution were used.

Monthly surface flux data are provided by the National Centers for Environmental Prediction (NCEP) Reanalysis datasets from Physical Sciences Division (PSD) (Kalnay et al. 1996). Net heat flux is obtained by summing up the sensible and latent heat fluxes and the shortwave and longwave radiations with positive values denoting heat gain by the ocean.

b. Hilbert–Huang transform

The combination of the Hilbert spectral analysis and the empirical mode decomposition (EMD), namely Hilbert–Huang transform, is designed specifically for analyzing nonlinear and nonstationary data (Huang et al. 1998; Huang and Wu 2008). This method, to a large extent, overcomes the difficulties of the more commonly used Morlet wavelet analysis that include the leakage problem due to limited length and the nonadaptive nature of the basic wavelet function. The Hilbert transform is a useful tool to estimate the amplitude and frequency of a signal over time, allowing localization of specific events in time–frequency space. However, the Hilbert transform requires data to be band-limited in order to produce meaningful results, and it is not obvious, a priori, what bands should be selected. To resolve this issue, the EMD method was proposed by Huang et al. (1998), which decomposes any complicated dataset into a finite and, often, small number of intrinsic mode functions (IMFs) that admit well-behaved Hilbert transforms (see the appendix for more details).

The essence of the EMD method is to identify the IMFs by their characteristic time scales in the data empirically, and then decompose the data accordingly. Each IMF must satisfy two conditions: 1) in the whole dataset, the number of extrema and the number of zero crossings in each must either be equal or differ at most by one, and 2) at any data point, the mean value of the envelope defined using the local maxima and the envelope defined using the local minima is zero. In the present study, we applied the improved white noise–assisted decomposition method of ensemble EMD (EEMD). The EEMD method can eliminate, to a large degree, the frequent appearance of the mode mixing associated with traditional EMD (Wu and Huang 2009). This approach defines the true IMF components as the mean of an ensemble of trials, each consisting of the data plus a white noise of finite amplitude. By eliminating the problem of mode mixing, it also produces a set of IMFs that can be physically interpreted and a time–frequency distribution without transitional gaps (Huang and Wu 2008). The IMFs isolate physical processes of various time scales and also give the temporal variation with the processes in their entirety without resorting to the linear assumption. The removal of certain frequency components will therefore not affect other information, either linear or nonlinear, included in the signal. Statistical significance testing of the components is performed based on the distribution of energy as a function of mean period of the IMFs relative to that of pure white noise (Wu and Huang 2005). It is designed to objectively determine the information content of the IMF components from a noisy dataset derived from EEMD relative to the white-noise statistics (see the appendix for description on the statistical significance test of IMFs).

Having obtained the IMFs, the Hilbert transform is applied to each component. The resulting frequency–time distribution of the amplitude is designated as the Hilbert amplitude spectrum, , or simply Hilbert spectrum. Two useful parameters can be defined based on the Hilbert spectrum: the first is the marginal spectrum density (MSD),
e1
and the second is the instantaneous energy density level (IE),
e2

The MSD is a function of frequency, offering a measure of total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the entire data span in a probabilistic sense, which means a higher likelihood for an oscillation at a frequency have appeared locally in the whole time span of the data (Huang et al. 1998). The IE depends only on time and can be thus used to check the energy fluctuation.

3. SSHA in the North Atlantic

a. Dipole pattern

We investigated the characteristic patterns of the SSHA over the middle to high latitudes of North Atlantic using multisatellite altimeter measurements from 1993 to 2010. The SSHA evolution in the North Atlantic reveals a positive sea level trend centered over the SPG (Fig. 1a), which corresponds to a scenario of weakening SPG surface circulation after mid-1990s that has been seen in previous studies (Verbrugge and Reverdin 2003; Häkkinen and Rhines 2004; Böning et al. 2006; Li et al. 2012). Such sea level changes could also indicate shifts in the SPG shape as suggested by Hátún et al. (2005). The SSHA trends in the other regions, especially in the GS region (weakly negative trend), may not represent a true long-term trend given their small coefficients of determination (Fig. 1b; the coefficient of determination is given by 1-residual sum of squares/total sum of squares). Hence, the SSHA changes in the SPG could be characterized by a linear trend while they may fluctuate without a simple trend in the GS region. The former signal could highlight the persistence of the sea level changes in the SPG since the 1990s.

Fig. 1.
Fig. 1.

(a) The linear trend of the sea surface height anomaly in the study domain and (b) the coefficient of determination, which is a measure of the goodness of linear fit (1-residual sum of squares/total sum of squares). The 1000-, 2000-, and 3000-m isobaths are shown in solid gray lines.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Figure 2 indicates high standard deviation (STD) of the SSHA over the GS region and low STD over the SPG, which is consistent with the low and high determination coefficients, respectively, as shown in Fig. 1. The GS, the NAC, and the Azores Current embrace most of the SSHA variance, in which the highest STD values (>10 cm) are constrained by 1000-m isobath to the west and the east and by the sub-Arctic front (SAF) to the north. The SPG interior has more variability than its periphery, which reflects larger variations of the horizontal circulations. The contrast between the SPG and the GS region in the linear trend as well as in the STD suggests different dominant time scales of variability due to different physical processes. Given that the linear trend might be a part of long time scale variability, a more general conclusion could be made here: the SSHA variability in the SPG has very important low-frequency components compared to that in the GS. This is tested in following sections.

Fig. 2.
Fig. 2.

The standard deviation of the sea surface height anomaly.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

b. SSHA variability

Volkov and van Aken (2003) examined the SSHA variability at annual time scales and interannual time scales in the North Atlantic. The authors used a harmonic shape with a frequency of 1 cycle per year (cpy) to determine the annual sea level cycle and applied a running mean with a window width equal to about 1 year to estimate the interannual signal. However, given that even mean sea level fluctuates from year to year, estimation of a signal with a fixed frequency window may remove physically meaningful nonstationary information carried by the signal. A sine function, therefore, may not be suitable to analyze the SSHA variability, especially when the study area has undergone a clear nonstationary warming process. Volkov and van Aken (2003) also noted the difficulties in estimating the characteristic period of the interannual change or in obtaining meaningful long-term trend with only 8 years of measurements. With such a short record, small shifts in the timing of the annual cycle (for example) may be aliased into the trend.

To differentiate the SSHA signals with limited frequency bands and take into account the nonlinear and nonstationary nature of the process, we decomposed the monthly SSHA time series into components with variable periods (the IMFs and residual) by using the EEMD method (see the appendix for description and an example of EEMD decomposition of SSHA time series). The decomposition was applied to every spatial point over the study domain. Figure 3 shows the temporal averages of the IMFs with high to low frequency (C1 to C6) and of the residual (R) resulting from the EEMD method, respectively. Given that the SSHA was obtained referencing to the 7-yr mean sea level from 1993 to 1999 (AVISO procedure), the time-averaged SSHA over the entire period of this study (1993–2010) represents the sea level changes relative to the reference period. Therefore, Fig. 3 reflects the 1993–2010 average relative to the 1993–99 average using monthly mean data. It is worth mentioning that the residual (or the long-term trend) differs from a linear trend because it can still have time-dependent fluctuations yet at a very low frequency. Significance testing was performed and only those values significant at the 95% significance level were kept. C1 represents the intraseasonal sea level variation, which may be associated with mesoscale eddy activity (e.g., Volkov and van Aken 2003). C2 and C3 have periods around 1 year, representing an annual or quasi-annual cycle. C4–C6 correspond to interannual variations. It is notable that the residual and the original SSHA data have the same order of amplitude, which is greater than that of the other components. This again suggests the importance of the long-term SSHA variability in the North Atlantic, whose actual period may not be well resolved because of the limited time span of data used. Moreover, SSHA variations are found to mainly localize in the SPG and the GS regions while the variations in the regions beyond are generally negligible but with increasing trends. The increases and decreases of sea level in the SPG and the GS respectively as shown in Figs. 3a and 3h occurred mainly in the 2000s. For the other components, however, the sea level in the SPG changed slightly (Figs. 3b–d) or even decreased somehow at longer time scales (Figs. 3e,f) in the 2000s. Caution should be taken in the GS region, because in this region only the intraseasonal and annual signals (C1 and C2) are statistically significant. Both of these two modes showed increased sea level in this region, in contrast to the decreasing trend (Fig. 3h).

Fig. 3.
Fig. 3.

The temporal averages over the period of 1993–2010 of (a) original SSHA, (b)–(g) the IMFs at increasing time scales, and (h) the residual (long-term trend).

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

After the decomposition of the sea level variability, we examined the contribution from each component to the total variance, showing the percentage of the variance explained by sea level variability over certain time scales (Fig. 4). Sea level variations in the SPG are mostly explained by the long-term trend (R, about 50%; Fig. 4g), confirming a persistent change in the SPG sea level from the 1990s to the 2000s. Besides the long-term trend, a significant part of the apparent sea level variance in the SPG is due to shifts in the seasonal cycle (C2, about 30%; Fig. 4b). In contrast, the total variance of the sea level in the GS region is dominated by the intraseasonal (C1, about 40%) and the annual signals (C2, about 30%), while less than 20% is explained by the residual. The intraseasonal SSHA signals may relate to direct wind forcing and current meandering and resultant persisting eddy generation. The annual signals could be related to shifts in the seasonal cycle. It was also noted that in the regions beyond the SPG and the GS, small SSHA variations with amplitude of ±1 mm are dominated by the annual signal.

Fig. 4.
Fig. 4.

The variance explained by the signals over different time scales corresponding to components in Figs. 3b–h.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

c. Characteristic SSHA variations in the subpolar gyre

We focused on the SPG given its crucial role in linking the upper and the lower limbs of the AMOC (Bersch et al. 2007; Rhein et al. 2011) and its undergoing changes in terms of both its intensity and its shape (Hátún et al. 2005; Sarafanov et al. 2008; Sarafanov 2009). Weekly SSHA was spatially averaged over the SPG domain (52°–66°N, 65°–20°W) for analysis. The EEMD results are presented in Fig. 5, which includes the full decomposition to illustrate the high- to low-frequency progression. In total eight IMFs and a residual were obtained due to the improved temporal resolution of the dataset. All IMF components are statistically significant at 99% confidence level. C1, C2, and C3 are the intraseasonal signals with periods ranging from weeks to months. They correspond to fast sea level variations that might relate to regional events such as storm surge occurrences. Given that the contributions from the intraseasonal signals are generally small in comparison to the total SSHA variance (Fig. 4), we did not perform further interpretation on this time scale. C4 has the largest amplitude of variations with clear seasonality, which is associated with the solar radiation in this region. At interannual to decadal time scales (C5, C6, C7, and C8), signals have irregular cycles in which a depressed sea level was observed in the 2000s. The long-term trend (R) shows an overall sea level rise at the rate of approximately 4 mm yr−1, suggesting a regional exaggeration of the sea level rise in the SPG (a global mean rate is 3.2 ± 0.6 mm yr−1 reported by AVISO; http://www.aviso.oceanobs.com/en/news/ocean-indicators/mean-sea-level/). However, there is variable rate of the sea level rise in the long-term trend and a reduction of the rates was observed after around 2002.

Fig. 5.
Fig. 5.

The IMFs and the residual decomposed from the spatially averaged SSHA over the subpolar gyre. Units are cm.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Hilbert transforms were then performed on all IMFs in order to examine not only the contribution from each mode to the total SSHA variations but also how this contribution changes with time. The resultant Hilbert amplitude spectrum provides a unique amplitude–frequency–time plot in which the amplitude is a function of both frequency and time. The MSD and IE were computed and displayed along for illustration. Figure 6 shows the results after applying Hilbert transform to all IMFs (C1–C8). The amplitudes with frequency higher than 2 cpy are not displayed. A maximum in the Hilbert spectrum was found at the frequency of 1 cpy, corresponding to the annual signal dominating over the time span of this study. This annual signal is also reflected in a rather broad peak in the MSD given that small fluctuations still exist in the annual signal. Another persistent maximum appeared at very low frequencies around approximately 0.1 cpy. This maximum yields a relatively sharp peak in the MSD, indicating its periodic nature. Besides the dominant annual signal, the appearance of signals within a small band of low frequencies may be of more interest given their potential connection to long time scale processes in the North Atlantic. There are also some local maxima at the interannual time scales, revealing more complex and nonperiodic phenomena associated with the sea level variations in the SPG. Furthermore, the IE plot shows a decreasing spectrum density for the period from the 1990s to the early 2000s but this is followed by a period of increasing energy. While the decrease may be related to reduction in kinetic energy due to weakening of the SPG surface circulation since the 1990s (Häkkinen and Rhines 2004), the following regaining of energy might be a result of a major “shift in the system” that occurred in the beginning of the 2000s (Sarafanov et al. 2012). The latter may relate to the changes in the production of the intermediate layer water mass in the Labrador Sea (van Aken et al. 2011) as well as to the changes in the SPG intensity indicated by the first principal component of the sea surface heights (Häkkinen and Rhines 2009).

Fig. 6.
Fig. 6.

The Hilbert transforms of the IMFs (C1–C8). (a) The instantaneous energy density level (IE) in cm2, (b) the Hilbert spectrum, and (c) marginal spectral density (MSD) of each frequency (in cm).

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Given the importance of low-frequency SSHA variations, we then isolated the signals at longer than annual time scales (C5, C6, C7, and C8) and applied the Hilbert transform to them. Figure 7 shows the Hilbert spectrum of the selected low-frequency components. After removing the intraseasonal and annual signals, the progression of IE becomes smoother with a reversal around the early 2000s (Fig. 7a). This suggests that such a reversal is mainly influenced by the low-frequency signals, and the high-frequency signals only produce fluctuations along this trend. Therefore, the importance of the low-frequency signals and their contributions to the total SSHA variability should be especially considered in an analysis of ongoing processes in the North Atlantic from the 1990s to the 2000s. Maximum amplitudes appear in the frequency range of 0.05–0.1 cpy, corresponding to signals with periods between 10 and 20 years (Figs. 7b,c). Furthermore, there are several local maxima of amplitude existing in 0.1–1 cpy frequency range, suggesting more complex interannual components of the sea level variations in the SPG. However, these maxima were generally shifted toward smaller frequency bands between 0.1 and 0.2 cpy (periods of 5–10 yr) in the late 2000s. The very low-frequency component decomposed from the SSHA may be spurious given the limited length of data record and therefore should be interpreted with caution. Moreover, sudden drops of the IE in 1995/96 and 2009/10 (Fig. 7a) were found to be concomitant with drops in the wintertime NAO index, which indicates the close relationship between the SSHA and the sign changes of the NAO index that has been previously observed (Esselborn and Eden 2001; Volkov and van Aken 2003).

Fig. 7.
Fig. 7.

Similar to Fig. 6, but only applying the Hilbert transforms to the selected IMFs (C5–C8). (a) The instantaneous energy density level (IE; in cm2), (b) the Hilbert spectrum, and (c) marginal spectral density (MSD) of each frequency (in cm).

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

4. Possible causes of the variability in the SPG

The total SSHA can be split into two components including contributions from steric effects and variations in bottom pressure, respectively (Gill and Niiler 1973). We first tested the steric effects in the subpolar region. To make the steric height anomalies comparable to the SSHA, we derived relatively to the mean compatible with the mean used to derive the SSHA (both reference periods are 1993–99). The results are shown in Fig. 8. We regarded both the SSHA and as a composition of two parts: a mixture of modes of variability and a linear trend. The former can be obtained by detrending the datasets. Then, we were able to compare the variance of the steric height anomalies and the SSHA, thus showing the percentage of the total SSHA variance explained by each of the steric components. Regardless of long-term trends, , as a sum of the thermosteric height anomaly and the halosteric height anomaly , explains 60% of the total variance in the SSHA. In comparison, explains 63% of the SSHA variance, while explains 5% of the total variance. Removing high-frequency variations in the time series, after applying a 3-month smoothing, barely changes the results. Moreover, if the trends were added back, the variance explained by increases to 73%, but the determination coefficient for becomes negative. This is due to the opposite long-term trends between and as shown in Fig. 8 in which the trend of largely represents the SSHA trend. Therefore, most of the SSHA variability in the SPG at annual and longer time scales is dominated by the steric effects, and especially by the thermosteric effect due to either heating or advection of heat. In addition, at low frequency (<1 cpy), the thermosteric and halosteric height variations effectively cancel each other. The contribution from salinity is therefore not trivial at these time scales. The results showed a consistent scenario as has been suggested by others (Antonov et al. 2002; Volkov and van Aken 2003; Ivchenko et al. 2008; Cazenave and Llovel 2010; Stammer et al. 2013). The limited period of the study may not allow for resolution of variability at time scales longer than decadal. However, the leading mode of altimeter SSHA data is known to covary with the subsurface temperature anomalies in the North Atlantic, which is likely a part of multidecadal time scales linked to the AMOC strength (Zhang 2008; Mahajan et al. 2011).

Fig. 8.
Fig. 8.

The altimeter SSHA and steric height anomaly calculated from temperature and salinity observations in the SPG.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Direct measurements of the bottom pressure anomalies are very scarce and the data are usually from numerical modeling. The bottom pressure anomalies have amplitudes of less than 1 cm over most parts of the deep ocean and of several centimeters over shallow boundary regions at high latitudes and short time scales (Ponte 1999; Ferry et al. 2000; Vinogradova et al. 2007; Bingham and Hughes 2008). Quinn and Ponte (2012) demonstrated coherence between ocean bottom pressure and sea level anomalies at middle and high latitudes at periods less than 100 days based on observations from the Gravity Recovery and Climate Experiment (GRACE) and altimetry. On longer time scales, the sea level and bottom pressure variability are essentially different with the former prevailing (Vinogradova et al. 2007; Ivchenko et al. 2008; Stammer et al. 2013). Jayne et al. (2003) found that combining altimetry observations with satellite measurements of the time-varying bottom pressure improves the ocean heat storage estimates well on time scales of the annual cycle and shorter. However, in the SPG, inclusion of bottom pressure anomalies contributed to a reduction in RMS error of about 0.05 GJ m−2 at semiannual and annual cycles and in the linear trends [see Figs. 4–6 in Jayne et al. (2003)]. Piecuch et al. (2013) demonstrated that considerable sea level variance could be explained by bottom pressure variance at interannual scales in several regions by using the newly updated GRACE data along with altimetry. Nevertheless, their results identified the northeastern corner of the North Atlantic as the region that bottom pressure is important at interannual scales [see Fig. 5 in Piecuch et al. (2013)].

We are then concerned with the sea level variations induced by changes in heat exchange between ocean and atmosphere and by advective processes (i.e., changes at annual and longer time scales). Recall that the SSHA variability in the SPG is dominated by annual cycles and the long-term increasing trend, which can explain 50% and 30% of the total SSHA variance, respectively. At these dominant time scales, the SSHA and heat content anomaly estimated from the temperature are strongly correlated with correlation coefficients of 0.89 for the annual cycles and 0.97 for the long-term trends.

We obtained the total heat content anomaly, , from the SSHA by using a linear relationship, , after Chambers et al. (1997), where is the seawater density, is the heat capacity of the seawater, is the thermal expansion coefficient, and is the SSHA. The use of a linear relationship to derive changes in the heat content from the SSHA has its advantages for using continuous and consistent satellite measurements with high accuracy and high resolution to infer changes in ocean heat content and taking into account changes take place through the whole water columns without the need of a known depth to perform the temperature integration. The specific heat and density vary by less than 1% over most of the ocean, and so assuming that they are constant introduces little relative error (Chambers et al. 1997). However, the thermal expansion coefficient varies with time and locations especially at high latitudes, which may cause large error in the heat content anomaly estimation. For example, the in the SPG varies between months with a maximum difference of , which yields an error in the of (assume a value of , , and ). This is even greater than the heat content caused solely by surface heating. Therefore, monthly values of were determined for each grid using World Ocean Atlas 2009 temperature and salinity data (http://www.nodc.noaa.gov/OC5/WOA09/pr_woa09.html; interpolated to the same resolution as the altimeter data). The total heat content anomaly in the upper oceans is balanced by air–sea heat flux, , horizontal heat transport, , and diffusion (Kelly 2004; Kelly and Dong 2004). Diffusion is found to be relatively small and thus heat transport variations, , can be inferred as a residual between heat content changes and air–sea fluxes (Kelly 2004). The surface heating term, , can be calculated based on the NCEP heat flux reanalysis data and was tested in this paper. We converted the surface heat flux into changes in the upper ocean heat content by multiplying monthly surface net heat flux by the seconds of a month.

Figure 9 illustrates the progression of signals with increasing periods that are decomposed from the spatially averaged and respectively. All the IMFs, except C3–C6 in the time series, are statistically significant at the 99% confidence level. In total, accounts for about 1/5 of amplitudes. This ratio becomes approximately 1/4–1/2 for the signals that have periods of less than 1 year (C1) and approximately 1 year (C2), suggesting that is more significant to at intraseasonal and annual time scales. In contrast, at longer than annual scales (C3–C6), the IMFs of are not statistically significant, nor are their amplitudes comparable to that of corresponding signals, making their contributions to the total heat budget in the SPG negligible. To conclude, the contributions from air–sea heat fluxes are important in analyzing the intraseasonal and annual regional SSHA variations in the SPG. In comparison, gyre-scale circulations, which contribute to the heat budget of the upper oceans in an advective fashion, play a dominant role in setting the SSHA variations in the SPG at interannual and longer time scales. Besides the different amplitudes, Fig. 9, for the first time, shows different time lags between and at annual and interannual time scales. Note that leads at both frequencies given known oceanic response to the overlying atmospheric forcing in the SPG. The time lead is 2 to 3 months at annual time scales (C2 in Fig. 9) and increases to about 2 years at interannual time scales (C4 and C5 in Fig. 9). The former is related to the time used to heat up the upper oceans from surface and the latter involves heat transport through the internal adjustments of the ocean.

Fig. 9.
Fig. 9.

The IMFs and the residuals decomposed from the heat content anomalies estimated from surface heat flux (gray dotted line; right y axis) and the SSHA (solid black line; left y axis). The depth-integrated heat content anomaly (dashed black line; left y axis) is also shown for comparison. Units are GJ m−2.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Figure 9 also presents heat content anomalies estimated from the temperature for comparison (only C6 in this time series is not significant). Heat content anomalies in the SPG estimated from the temperature and the SSHA agree well with each other in terms of both amplitude and phase. The strongest correlations for these two time series exist in the long-term trends (correlation coefficient r = 0.96), the annual cycles (r = 0.84), and the decadal cycles (r = 0.76). Therefore, great care should be taken in the estimation of heat content anomalies by altimetry alone when considering interannual scales. Moreover, Fig. 9 shows that the discrepancies between the two heat content anomalies have reduced from the 1990s to the 2000s at the decadal time scale (C5) and in the residuals (R), which may relate to the concomitant warming observed in the northern North Atlantic (e.g., Häkkinen and Rhines 2009; Häkkinen et al. 2011).

We then examined the relationship between the SSHA variability in the SPG and possible underlying processes at longer than annual time scales. These low-frequency changes are typically attributed to the variability in the deep ocean convection (DOC) processes in the Labrador Sea (Häkkinen and Rhines 2004; Bersch et al. 2007; Rhein et al. 2011) and large-scale AMOC variations (Zhang 2008; Lohmann et al. 2009a,b; Robson et al. 2012; Li et al. 2012). The DOC is a direct form of heat exchange between atmosphere and the intermediate to deep ocean. A large amount of heat is released to the atmosphere during strong DOC years from the ocean, resulting in a reduction of heat content in the upper to intermediate water column. The associated interannual trends reflect the atmospheric winter conditions persisting for more than a year (Lazier et al. 2002). Therefore, we chose the depth-integrated heat content of the central Labrador as a DOC index for the analysis of the interannual and longer time scale variability. The heat content has increased almost steadily since the mid-1990s. This is closely associated with the interannual changes of the DOC: after a period of enhanced deep convection with production of large volume of the classic Labrador Seawater (CLSW) in the early 1990s, there were some years of ventilation of the CLSW and formation of the upper Labrador Seawater (ULSW) due to reduced DOC intensity and activity in the 2000s (Kieke et al. 2007; Rhein et al. 2011). The SPG index is defined as the SSHA of the central Labrador Sea (see Häkkinen and Rhines 2004; Böning et al. 2006). The AMOC transport index used here is the meridional transport at 26.5°N (Cunningham et al. 2007) from the RAPID-WATCH MOC monitoring project (http://www.rapid.ac.uk/rapidmoc/). We considered the NAO forcing as a possible driving force in the low-frequency changes in the North Atlantic (e.g., Hurrell 1995; Marshall et al. 2001). Since the ocean signal reflects a time integration of the atmospheric forcing, for example, through mixed layer “memory” and Rossby wave propagation (Curry and McCartney 2001), the cumulative NAO index (CumNAO) was adopted instead, following Li et al. (2012), by integrating monthly NAO index over time (https://climatedataguide.ucar.edu/climate-data/hurrell-north-atlantic-oscillation-nao-index-station-based), which may more accurately represent the cumulative effects of oceanic signal variations. Since the NAO index includes positive or negative phases over time, this time integral could also provide a clear view of any accumulations as well as any shifts in time. A period of persistent positive NAO index becomes an increasing slope in the CumNAO index, and vice versa. The CumNAO can therefore highlight the shifts in the atmospheric conditions compared to the normal NAO index. For instance, a sign change of the NAO index that happened between 1995 and 1996 corresponds to a peak in the CumNAO. In addition, using different time origin of integration does not change the shape of the cumulative NAO index, but instead introduces a shift in magnitude for every point.

Figure 10 shows the close relationship between DOC, the SPG, the AMOC variability, and the cumulative NAO forcing especially at interannual to decadal time scales. The CumNAO index matches with the DOC and the SPG indices well and leads the AMOC variability with some time lags. A persistent high NAO index in the early 1990s lead to consecutive winters of strong DOC events accompanied by a reduction of heat content and an enhancement of the SPG circulation. After mid-1990s, a steady warming of the central Labrador Sea was observed with the exception of the periods of 1999 to 2003 and 2008 to 2010—two periods of observed large production of CLSW (Yashayaev 2007; Våge et al. 2009; Yashayaev and Loder 2009). The AMOC variability follows the changes in DOC and the SPG with a time lag of approximately 1 to 2 years, which agrees with previous findings based on numerical simulations (Zhang 2008, 2010). Therefore, at interannual to decadal time scales, the SSHA variations in the North Atlantic closely represent the AMOC variability driven by the NAO forcing.

Fig. 10.
Fig. 10.

Close relationships among the heat content of the central Labrador Sea (dashed black line), the sea surface height anomaly of the central Labrador Sea (solid black line), the meridional overturning circulation transport at (gray dotted line), and the cumulative NAO index (black line with circles). The central Labrador Sea is defined by a box bounded by 56°–61°N and 48°–58°W. The unit and range of each index is labeled with y axis of same color. Note that the y axis of the CumNAO index is reversed.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

5. Concluding remarks

This study has presented an analysis of 18-yr SSHA records in the mid- to high-latitude North Atlantic that examined the characteristic features of sea level variations. A dipole pattern, centered between the SPG and the GS path, appears in both the linear trends and the STD of the SSHA, indicating different modes of variability in the two regions. A simple increase in the SSHA could represent most changes in the SPG throughout the entire time span of the study. This increase corresponds to a continuous weakening of surface circulation. In contrast, the linear trend along the GS path is not representative of the overall sea level changes that are instead associated with greater variance. The separation of the SSHA time series into components with various time scales and the long-term trend showed more details about the contrast between the SPG and the GS regions. It is worth mentioning, however, that the long-term trend may have a contribution from aliasing due to the limited length of data record. The SSHA variability along the GS is dominated by intraseasonal to annual fluctuations, which together could account for 70% of the total sea level variance. These intense, high-frequency variations in the SSHA time series would effectively mask the low-frequency signals of greater interest (e.g., Lorbacher et al. 2010). In the SPG, the long-term trend and the annual signal accounts for 50% and 30% of the total variance, respectively. The long-term trend might also be a part of longer time scale variability that cannot be resolved in the limited altimeter record length of 18 years. Therefore, the dominance of the long-term trend in the SPG sea level changes may indicate the critical role of low-frequency processes in the North Atlantic. Furthermore, by applying the Hilbert transforms to the IMF components of the SPG sea level, signals at decadal and longer time scales were observed in addition to the well-known strong annual cycle, confirming the important role of the low-frequency components in the SPG variability. In the regions beyond the SPG and the GS, the SSHA variations are smaller and are dominated by annual variations with an overall increasing trend.

Although the long-term trends have similar pattern of the linear trends, the EEMD-derived trends contain more time-dependent information. This observation was not possible using most traditional time series analysis methods that assume stationary changes. In the SPG, a sea level rise rate reduction was found around the early 2000s, which is concomitant with the reduced amplitudes in the low-frequency signals. Spectral analysis indicated coinciding changes in the energy with the amplitude changes, which increases after the early 2000s following a period of energy loss from the system since the early 1990s. Such reversal of energy loss was influenced by the low-frequency SSHA signals (removal of the high-frequency signals did not change the result). The timing of such changes may correspond to a “shift in the system” around 2002, which was also found in the observations of the LSW formation and the first principal component of the altimetry-derived sea level in the northern North Atlantic (Sarafanov et al. 2012, and references therein). These changes in the SSHA from the 1990s to the 2000s might suggest a new cycle of SPG intensification as observed in part of its boundary currents from repeat hydrographic sections in the early 2000s (Han et al. 2010; Daniault et al. 2011). Actually, deep-water transport strengthened in opposition to its surface layer counterpart in the 1990s (Dengler et al. 2006; Sarafanov et al. 2010). Moreover, after years of weak or absent DOC activities, the Labrador Sea experienced a strong DOC event in the winter of 2007/08 even without a clear phase of preconditioning (Våge et al. 2009). Rapid development of convection under intensified atmospheric forcing greatly deepened the mixed layer to more than 1600 m, disrupting steady warming of the intermediate depth waters since 1994 (Yashayaev and Loder 2009). All of these recent observations cast questions about the future of the SPG under a warming scenario and its relationship to both external atmospheric forcing as well as internal oceanic dynamics. This requires further investigation on understanding the low-frequency variability in the subpolar North Atlantic.

This study also showed that, in the factors responsible for the SPG sea level variability, advection is important at interannual to decadal time scales while the air–sea heat flux is not negligible at annual time scale. The result is in line with previous studies on the interannual sea level anomaly in the North Atlantic (Reverdin et al. 1999; Häkkinen 2001; Esselborn and Eden 2001; Verbrugge and Reverdin 2003; Volkov and van Aken 2003). Moreover, it has been suggested that air–sea heat flux explains most of the sea level variance at annual scale in the northeastern North Atlantic (Ferry et al. 2000; Volkov and van Aken 2003). This, therefore, suggests different underlying mechanisms accounted for the SSHA in the western and eastern subpolar North Atlantic (Herbaut and Houssais 2009; Li et al. 2012). Such importance has been studied in the GS given that advection plays a more determinant role in regulating the heat content than air–sea fluxes on interannual-to-decadal time scales (Dong and Kelly 2004; Kelly and Dong 2004). Based on results from numerical simulations, Häkkinen (1999, 2001) attributed the low-frequency sea level changes predominately to overturning changes. The close relationship among DOC, the SPG, and the AMOC especially at interannual time scales was also presented here by comparing adequate indices of each process. Matches between these crucial physical processes, as well as their relationship to the cumulative NAO forcing, may explain changes in the advection and thus account for low-frequency variability in the SSHA in the North Atlantic. Since the AMOC index used here is available for a very limited time period, we tested the role of the AMOC in another way to see how low-frequency SSHA variability relates to AMOC variations. Li et al. (2012) suggested the potential of using low-frequency SSHA to represent changes in AMOC variations between different latitudes based on analysis of two regional SSHA covering the SPG and the GS respectively. Here, we extended their works by dividing the study domain into three adjacent regions in order to examine the meridional propagation of SSHA variations (Fig. 11). For consistency, we considered only significant low-frequency components (period >1 yr) in these three regions (regions 1, 2, and 3). Unless otherwise noted, the SSHA variations discussed hereinafter represent the low-frequency changes. The propagation of SSHA variations from high- to midlatitude regions was observed with certain time lags between two adjacent regions. The SSHA variations of region 2 lag region 1 by about 2 years with a significant correlation of 0.52. Toward the south, there is a 5-yr lead between SSHA variations in regions 2 and 3 with a significant correlation of −0.36. The propagation velocity is estimated by V = L/T, where L is the meridional distance between 50° and 30°N (about 2297 km) and T is 5 yr, which gives a propagation velocity of approximately 1.46 cm s−1. This finding is consistent with modeled climatological mean meridional velocity of 1.5 cm s−1 speed at which AMOC variations at high latitudes propagate to the midlatitudes (Curry et al. 1998; Zhang 2010) following interior pathways observed recently (Bower et al. 2009). Therefore, it confirms that the low-frequency SSHA variations in the SPG and along the GS are mostly related to AMOC variations, which might be a part of multidecadal variability as suggested in numerical models (Zhang 2008; Mahajan et al. 2011). Moreover, Chen and Tung (2014) indicate that increasing heat content in the Atlantic Ocean effectively contributes to ongoing surface warming slowdown (“climate hiatus”). The authors further propose the underlying mechanism of corresponding climate shifts in both salinity and ocean heat content in the subpolar North Atlantic with involvement of the AMOC variations. Therefore, characteristic modes of sea level variability especially in the subpolar North Atlantic might have further implications for the hiatus and its relationship to the AMOC. Observational verification is of great importance when a longer altimeter record becomes available.

Fig. 11.
Fig. 11.

Domains of areas 1, 2, and 3 used to spatially average the SSHA and examine the lagged correlation between each other. Arrows indicate possible propagation pathway of AMOC variations. The 1000-, 2000-, and 3000-m isobaths are shown in solid gray lines.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

It is worth mentioning that in most areas of the SPG, surface heat flux is positively correlated with the heat content, which indicates that surface flux introduces variations to the heat content (and thus the sea level anomaly) in the upper oceans. In contrast, the situation in the GS is different and has opposite values of and (not shown here). This is consistent with negative correlation between heat content and surface heat flux found in the western boundary currents, which suggests the dominance of oceanic feedback over atmospheric forcing (Dong and Kelly 2004; Kelly and Dong 2004). The warming of the eastern North Atlantic subsurface waters has been observed due to northward invasion of waters of subtropical origin along the eastern boundary in the 2000s (Sarafanov et al. 2008; Häkkinen et al. 2011). The warming of the Irminger Sea might be related to the contraction of the SPG and the westward retreat of the SAF, which needs further investigation and is beyond the topic of this present work.

Acknowledgments

The authors thank the editor and two anonymous reviewers for their helpful comments that greatly improved this manuscript. The authors also thank Autumn Kidwell for editorial assistance in English. This research was partially supported by NASA Physical Oceanography Program, NASA EPSCoR Program, NASA Space Grant, and NOAA Sea Grant.

APPENDIX

Hilbert–Huang Transform

Empirical mode decomposition

The empirical mode decomposition (EMD) method identifies the intrinsic mode function (IMF) by a “sifting” process, which decomposes a dataset into components with varying time scales (Huang et al. 1998). The sifting starts with identifying all the extrema and then connecting all the local maxima (minima) by a cubic spline line to form the upper (lower) envelope. The mean of the upper and the lower envelopes is then subtracted from the data to obtain the first guess of the first IMF. The sifting process is performed repeatedly in order to refine the IMF by eliminating background waves on which the IMF is riding and making the wave profiles more symmetric. The first IMF, which has the finest scale or the shortest-period oscillation in the data, is then subtracted from the original data to yield a residue, which is then treated as the new data and subjected to the same sifting process to obtain an IMF of lower frequency. Each IMF must satisfy two conditions: 1) in the whole dataset, the number of extrema and the number of zero crossings in each must either equal or differ at most by one, and 2) at any data point, the mean value of the envelope defined using the local maxima and the envelope defined using the local minima is zero. With these requirements, an IMF oscillates in a narrow frequency band, a reflection of quasi-periodicity and nonlinearity. The decomposition procedure uses intrinsic information derived from the dataset itself instead of prescribing frequency bands or imposing any particular basis function such as wavelets. Therefore, it is adaptive and very efficient at extracting nonlinear and nonstationary processes. The decomposition process finally stops when the residual becomes a monotonic function or a function with only one extreme from which no more IMF can be extracted. The original dataset, X(t), can be therefore expressed as
ea1
where is the jth IMF, n is the total number of the IMFs, and R(t) is the residual.

Figure A1 gives an example of the progression of the IMFs with high to low frequency (C1–C6) and the residual (R) decomposed from a SSHA time series at a location within the subpolar gyre (60°N, 42°W).

Fig. A1.
Fig. A1.

The IMFs (C1–C6) and the residual (R) decomposed from a SSHA time series (input) at 60°N, 42°W. Units are cm.

Citation: Journal of Climate 29, 13; 10.1175/JCLI-D-12-00670.1

Statistical significance test of IMFs

Wu and Huang (2005) studied the characteristics of IMFs from uniformly distributed white noise, and gave the empirical relationship between the energy density and the mean period. This leads to the establishment of the energy distribution function for each IMF and the spread function of the energy distribution for various percentiles. Based on the characteristics of the white noise, the information content of IMF components from a noisy dataset derived from EMD can be objectively determined relative to the white noise statistics. Therefore, statistical significance of the IMFs is based on the distribution of energy as a function of mean period of the IMF relative to that of pure white noise. This method allows one to differentiate true signals from components of noise with any selected statistical significance level (95%, 99%) (Wu and Huang 2004, 2005; Huang and Wu 2008). Insignificant IMFs are interpreted as noise inherent to the original signal. Significance testing shows that all IMF components shown in Fig. A1 are significant at the 95% significance level.

An alternative way of testing the significance of an IMF of data is implemented based on the Monte Carlo method (Coughlin and Tung 2004, 2005; Flandrin et al. 2004, 2005). The special characteristics of data, such as lagged autocorrelation, are used to make a null hypothesis for the type of noise process for data. One can generate a number of samples of the noise series with the same length as that of data and decompose them into a large number of sets of IMFs. By comparing the energy or variance of each IMF from data with that from random samples, one can tell whether an IMF contains any statistically significant information. A minor drawback of this approach is the demand of large computational resources to obtain an accurate distribution of energy of an IMF when the size of data to be analyzed is very large (Wu and Huang 2005).

Hilbert transforms of IMFs

The Hilbert transform is applied to each IMF. An IMF component, C(t), and its Hilbert transform, F(t), then can well define a complex function, the analytical signal,
ea2
The instantaneous amplitude of is
ea3
and the instantaneous phase of is
ea4
Then, the instantaneous frequency is defined as
ea5
The instantaneous amplitude and frequency are calculated for all IMF components. The residual (R) has been left out of Hilbert transform by purpose since the energy involved in the residual represents typically a mean offset that could be overpowering (Huang et al. 1998). After performing the Hilbert transform on each IMF component, the original data can be expressed, then, in the following form:
ea6

This enables us to represent the amplitude and the instantaneous frequency as functions of time in a three-dimensional plot, in which each point of amplitude corresponds to a pair of [t, ].

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