1. Introduction
Arctic sea ice is a sensitive component of the climate system, with dynamics and variability that are strongly coupled to the atmosphere and ocean. This sensitivity is evident in the recent precipitous decline in September sea ice extent, of roughly 9% per decade since 1979 (Stroeve et al. 2007; Serreze et al. 2007). Trends in sea ice extent are negative for all months of the year and all Arctic regions except for the Bering Sea (Cavalieri and Parkinson 2012). In addition to these strong trends, Arctic sea ice also exhibits large internal variability. Studies using comprehensive climate models have estimated that 50%–60% of recent Arctic sea ice changes can be attributed to externally forced trends, with the remainder resulting from internal variability in the climate system (Kay et al. 2011; Stroeve et al. 2012). Therefore, the challenge of making accurate projections of future Arctic sea ice conditions crucially hinges on 1) quantifying the sea ice response to changes in external forcing (i.e., greenhouse gas forcing) and 2) understanding the nature and magnitude of internal variability in the coupled ice–ocean–atmosphere system. This study will focus on the latter.
The Arctic regions of interest in this study are shown in Fig. 1. The leading empirical orthogonal function (EOF) of observational Arctic sea ice concentration (SIC) exhibits strong out-of-phase anomalies between the Labrador and Greenland–Barents Seas and weaker out-of-phase anomalies between the Bering Sea and Sea of Okhotsk (Deser et al. 2000). Regression of sea level pressure (SLP) onto the corresponding principal component (PC) yields a spatial pattern that closely resembles the Arctic Oscillation (AO; Thompson and Wallace 1998), the leading pattern of SLP variability north of 20°N. Deser et al. (2000) observe a connection between the low-frequency (interannual to decadal) variability of the atmosphere and the low-frequency variability of sea ice. In particular, they find that the AO and its associated geostrophic winds are physically consistent with the ice anomalies of the leading SIC mode, suggesting that atmospheric circulation anomalies force sea ice anomalies. These winds have thermodynamic and dynamic effects on sea ice via advection of surface air temperature and ice advection. Many other studies have analyzed sea ice variability in the context of the AO, finding that the AO affects sea ice on a wide range of time scales ranging from seasonal (Serreze et al. 2003) to decadal (Rigor et al. 2002; Rigor and Wallace 2004; Zhang et al. 2004). These studies suggest that a “high index” AO produces an Ekman divergence, leading to reductions in sea ice thickness and concentration. This process has been proposed as a mechanism for the recent decline in Arctic sea ice.
The regions of interest in this study: the Barents–Kara Seas (BK), the Labrador Sea (LS), the Greenland Sea (GS), the Bering Sea (BER), and the Sea of Okhotsk (OK). The Arctic domain is defined as all grid points north of 45°N.
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Others have questioned the efficacy of the AO as a predictor for sea ice changes (Maslanik et al. 2007), suggesting that other patterns of large-scale atmospheric variability may play a more important role. In particular, an SLP pattern known as the Arctic dipole anomaly (DA) has drawn considerable recent attention (Wu et al. 2006; Wang et al. 2009; Tsukernik et al. 2010; Overland and Wang 2005, 2010; Watanabe et al. 2006). The DA exhibits opposite-signed SLP anomalies between the eastern and western Arctic, which drive strong meridional winds. These winds act to enhance (reduce) sea ice export from the Arctic basin through Fram Strait when the DA is in positive (negative) phase. Recent record lows in summer sea ice extent generally correspond to years in which the DA index was positive (Wang et al. 2009). DA-like SLP patterns have also been associated with the large internal variability observed in the sea ice component of the Community Climate System Model, version 3 (CCSM3; Collins et al. 2006; Wettstein and Deser 2014). Other studies have suggested that the location and frequency of storms (Screen et al. 2011), and the phase of the Pacific–North America (PNA) pattern (L’Heureux et al. 2008) also play an important role in setting the summer sea ice minimum.
The PCs corresponding to large-scale atmospheric patterns, such as the AO and DA, are quite noisy and contain significant spectral power at time scales ranging from monthly to decadal. A typical approach has been to initially low-pass filter the atmospheric component (by forming annual or winter means), as a way of smoothing out these PCs and emphasizing interannual to decadal variability. Besides the studies already cited, a large number of works have analyzed the impact of this low-frequency atmospheric variability on Arctic sea ice (Walsh et al. 1996; Proshutinsky and Johnson 1997; Mysak and Venegas 1998; Yi et al. 1999; Johnson et al. 1999; Deser et al. 2000; Polyakov and Johnson 2000; Moritz et al. 2002). These studies emphasize that sea ice regimes are modulated by low-frequency atmospheric circulation regimes.
The variability of Arctic sea ice is also strongly coupled to sea surface temperature (SST) variability (e.g., Francis and Hunter 2007). Blanchard-Wrigglesworth et al. (2011) proposed a mechanism for sea ice–SST covariability, in which sea ice and SST anomalies trade off, allowing for unexpected “memory” effects in sea ice. These memory effects were termed “sea ice reemergence,” inspired by the similar North Pacific and North Atlantic SST phenomena (Alexander et al. 1999; Timlin et al. 2002; de Coëtlogon and Frankignoul 2003). Sea ice reemergence is a lagged correlation phenomenon, in which spring sea ice anomalies are positively correlated with fall sea ice anomalies, despite a loss of correlation over the intervening summer months. There is also a similar, but weaker, reemergence between fall sea ice anomalies and anomalies the following spring. The spring–fall mechanism of Blanchard-Wrigglesworth et al. (2011) suggests that spring sea ice anomalies imprint SST anomalies of opposite sign, which persist over the summer months. During the fall, ice grows southward and interacts with these SST anomalies, reproducing ice anomalies of the same sign as the spring. This reemergence mechanism has been observed in the North Pacific sector in the CCSM3 model output and observations (Bushuk et al. 2014). Deser et al. (2002) note a similar winter-to-winter persistence of sea ice anomalies in the Labrador Sea and propose an atmospheric mechanism in which sea ice anomalies persist because of persistent large-scale atmospheric circulation regimes.
Sea ice reemergence may also have implications for sea ice predictability. Day et al. (2014) found that sea ice forecast skill was strongly dependent on initialization month, with certain months exhibiting a slower decay of forecast skill than others. The authors suggested that this initialization month dependence was attributable to sea ice reemergence mechanisms. Day et al. (2014) also examined sea ice reemergence in five global climate models (GCMs) and observations, finding robust reemergence signals, of varying strength, across all models and a weaker reemergence signal in the observational record.
In this study, we examine the coupled variability of Arctic SIC, SST, and SLP using nonlinear Laplacian spectral analysis (NLSA), a recently developed data analysis technique for high-dimensional nonlinear time series (Giannakis and Majda 2012a,b, 2013, 2014). The NLSA algorithm is a nonlinear manifold generalization of singular spectrum analysis (SSA; Broomhead and King 1986; Vautard and Ghil 1989; Ghil et al. 2002). SSA is also commonly referred to as extended empirical orthogonal function (EEOF) analysis. Here, we apply the multivariate version of the NLSA algorithm, coupled NLSA (Bushuk et al. 2014), which provides a scale-invariant analysis of multiple variables with different physical units. Coupled NLSA yields spatiotemporal modes, analogous to EEOFs, and temporal modes, analogous to PCs. These modes are constructed using a set of empirically derived Laplacian eigenfunctions on the nonlinear data manifold and, unlike linear approaches, do not maximize explained variance. Compared to linear techniques, NLSA (and other related nonlinear methods; Berry et al. 2013) provides superior time-scale separation and is able to effectively capture low-variance modes that may have important dynamical significance. These low-variance modes are known to be crucial in producing accurate representations of nonlinear dynamical systems (Aubry et al. 1993; Giannakis and Majda 2012b) and, in the present context, are efficient in explaining reemergence phenomena (Bushuk et al. 2014).
We use coupled NLSA modes to study the basinwide and regional characteristics of Arctic sea ice reemergence in a comprehensive climate model and observations. We compute modes using CCSM3 model output from a 900-yr equilibrated control integration. Modes are also obtained for the 34-yr observational record, using SIC and SST data from the Met Office Hadley Centre Sea Ice and Sea Surface Temperature dataset (HadISST) and the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Re-Analysis (ERA-Interim) SLP data. No preprocessing of the data is required, enabling simultaneous extraction of interannual, annual, and semiannual patterns of variability. Using these modes, we identify low-dimensional families that efficiently describe sea ice reemergence. These families capture a significant portion of the reemergence signal, and have the surprising property of being relatively low variance. The families also reveal time-dependent aspects of reemergence, which were not accessible in previous studies. The SST and SIC modes of each family exhibit an SST–sea ice reemergence mechanism consistent with that of Blanchard-Wrigglesworth et al. (2011). Interannual components of large-scale SLP variability, which emerge objectively from this analysis, are found to be related to coherent sea ice reemergence events in geographically distinct regions and suggest an SLP–sea ice reemergence mechanism.
This paper is organized as follows: In section 2, we summarize the coupled NLSA algorithm. In section 3, we describe the CCSM3, HadISST, and ERA-Interim datasets used in this study. In section 4, we study the SIC, SST, and SLP spatiotemporal modes obtained via coupled NLSA. In section 5, we examine the regional and temporal characteristics of sea ice reemergence, and in section 6 we investigate oceanic and atmospheric reemergence mechanisms. We conclude in section 7. (Movies, illustrating the spatiotemporal evolution of NLSA modes, are available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-14-00354.s1.)
2. Coupled NLSA methodology
In this study, we apply the coupled NLSA approach, as developed in Bushuk et al. (2014), to Arctic SIC, SST, and SLP. This technique is an extension of the recently developed NLSA algorithm (Giannakis and Majda 2012b, 2013) and provides a scale-invariant approach for multivariate time series analysis. Unlike other multivariate data analysis approaches, coupled NLSA does not require initial normalization of the input fields to unit variance. Rather, the coupled NLSA algorithm implicitly selects the variance ratio between different physical fields, without requiring a choice of normalization by the user. Here, we briefly summarize the method and refer the reader to the more thorough description of Bushuk et al. (2014). Figure 2 is a schematic that summarizes the flow of data in the coupled NLSA algorithm.
Schematic summarizing the flow of data in the coupled NLSA algorithm.
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Singular value decomposition (SVD) of the operator for the kth variable 
3. Dataset description
a. CCSM3 model output
This study analyzes model output from a 900-yr equilibrated control integration (model run b30.004) of CCSM3 (Collins et al. 2006). These data were downloaded from the Earth System Grid website. We use monthly averaged data for SIC, SST, and SLP, which come from the Community Sea Ice Model (CSIM; Holland et al. 2006); the Parallel Ocean Program (POP; Smith and Gent 2004); and the Community Atmosphere Model, version 3 (CAM3; Collins et al. 2004), respectively. The model uses a T42 spectral truncation for the atmospheric grid (roughly 2.8° × 2.8°), and the ocean and sea ice variables are defined on the same grid, of 1° nominal resolution. This study focuses on a pan-Arctic domain, which we define as all grid points north of 45°N. Note that the seasonal cycle has not been removed from this dataset. This is crucial for capturing intermittent patterns associated with reemergence. In particular, intermittent modes, described in section 4, are not recoverable in datasets that have been deseasonalized (Giannakis and Majda 2013). As will be shown in section 5, these modes are essential in low-dimensional descriptions of sea ice reemergence.
The spatial dimensions (number of spatial grid points) of these datasets are dSIC = dSST = 13 202 and dSLP = 2048. Using a 2-yr embedding window with q = 24 (Giannakis and Majda 2012b; Bushuk et al. 2014), this yields lagged-embedding dimensions (the product of the number of spatial grid points and the embedding window) of qdSIC = qdSST = 316 848 and qdSLP = 49 152. These data are monthly averaged and consist of s = 10 800 time samples for the 900-yr simulation period. The value Δt = 24 months was used as the time lag because this embedding window is longer than the seasonal cycle, which is a primary source of non-Markovianity in this dataset. A number of different embedding windows were tested, yielding qualitatively similar results for Δt ≥ 12 months and qualitatively different results for Δt < 12 months.
b. HadISST observations
We also analyze HadISST (Rayner et al. 2003), which consists of monthly averaged SIC and SST data on a 1° latitude–longitude grid. The spatial dimension of the Arctic domain is dSIC = dSST = 9453. As with the CCSM3 data, we use an embedding window of Δt = 24 months, which yields lagged-embedding dimensions of qdSIC = qdSST = 226 872. In this study we use the satellite era data from January 1979 to August 2013. Note that all ice-covered grid points in HadISST were assigned an SST value of −1.8°C, the freezing point of saltwater at a salinity of 35 parts per thousand. Also, the trend in the dataset was removed by computing a long-term linear trend for each month of the year and removing the respective linear trend from each month. The seasonal cycle has not been removed from this dataset.
c. ERA-Interim data
Finally, we also study monthly averaged SLP data from ERA-Interim (Dee et al. 2011). These data are defined on a 0.75° latitude–longitude grid, of considerably higher resolution than the CCSM3 SLP data. The spatial dimension of the Arctic domain is dSLP = 29 280, corresponding to a lagged-embedding dimension of qdSLP = 702 720. These data have been detrended by subtracting the monthly trend from each month, but the seasonal cycle has not been subtracted.
4. Coupled SIC–SST–SLP spatiotemporal modes of Arctic variability
We utilize the coupled NLSA algorithm outlined in section 2 to study the spatiotemporal evolution of (i) SIC, SST, and SLP in CCSM3 and (ii) SIC and SST from HadISST and SLP from ERA-Interim. Hereafter, we refer to the joint HadISST and ERA-Interim datasets as observations. For both the model and observational data, we use a lagged-embedding window of Δt = 24 months.
a. CCSM3 modes
We choose ϵ, the Gaussian locality parameter, as ϵ = 0.90. Using the spectral entropy criterion of Giannakis and Majda (2012a, 2013), we select a truncation level of l = 27 eigenfunctions and express the data matrices 
1) Temporal modes
Figures 3–5 show selected temporal patterns for SIC, SST, and SLP, respectively. For each variable, we observe three distinct types of temporal modes: periodic, low-frequency, and intermittent modes, indicated by P, L, and I in the figures.
Snapshots of the time series, power spectral density, and autocorrelation functions for the CCSM3 SIC PCs (υk) from coupled NLSA. Shown here for 50-yr portions of the 900-yr time series are the annual periodic (
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Snapshots of the time series, power spectral density, and autocorrelation functions for the CCSM3 SST PCs from coupled NLSA. Shown here are the annual periodic (
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Snapshots of the time series, power spectral density, and autocorrelation functions for the CCSM3 SLP PCs from coupled NLSA. Shown here are the annual periodic (
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The periodic temporal modes closely resemble sinusoids, with frequencies given by integer multiples of 1 yr−1. These modes appear as doubly degenerate pairs, with a phase offset of π/2. The leading periodic modes, representing the annual and semiannual cycles, capture more variance than the low-frequency and intermittent modes of the system. Higher harmonic periodic modes are found later in the mode spectrum. The low-frequency modes are characterized by significant interannual variability and have a typical decorrelation time of approximately 3 years. These modes carry significant spectral power at frequencies below 1 yr−1 and exhibit a sharp decline in spectral power at frequencies above this.
The intermittent modes are characterized by periods of intense activity followed by periods of quiescence. Each intermittent mode has a base frequency of oscillation and a broadband peak in spectral power centered upon this frequency. These modes carry lower variance than their periodic and low-frequency counterparts, yet they have potentially high dynamical significance. For example, annual and semiannual intermittent modes are crucial components in low-dimensional descriptions of sea ice reemergence phenomena (Bushuk et al. 2014). Note that the leading low-frequency and intermittent modes are insensitive to truncation level, whereas increasing l will eventually disrupt the temporal character of some intermittent modes.
Intermittent modes closely resemble a periodic signal modulated by a low-frequency envelope. We find that nearly all intermittent modes can be directly associated with a particular low-frequency mode, which provides this modulating envelope (Bushuk et al. 2014). To determine this association, we compare the envelope function of the intermittent modes to the low-frequency modes. We find the envelope function via the Hilbert transform (von Storch and Zwiers 1999). Let I(t) be a given intermittent mode and let H(I)(t) be the Hilbert transform of I. Then the envelope function e(t) is given by e(t) = [I(t)2 + H(I)(t)2]
Correlations between low-frequency modes and envelope functions for intermittent modes. Mode pairs with large positive correlations indicate that the low-frequency mode provides the modulating envelope for the intermittent mode.
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
As a comparison, we also performed SSA on the concatenated and unit-variance normalized SIC–SST–SLP dataset. Similar to the findings of Bushuk et al. (2014), SSA produces periodic modes, many low-frequency modes, and some modes that loosely resemble the intermittent modes of NLSA, with a spectral maximum at a certain base frequency. We find that the SSA modes do not share the same intermittent–low-frequency mode relationships as the NLSA modes. These relationships will be important for explaining reemergence, as they reflect the interaction of large-scale low-frequency modes of variability with the familiar annual and semiannual cycles in the climate system.
2) SIC spatiotemporal patterns
Figure 7 shows spatial patterns of selected modes at a snapshot in time. Movies 1 and 2 in the online supplementary material show the spatiotemporal evolution of these modes and others. Below, we describe the prominent features of the spatiotemporal modes recovered for SIC, SST, and SLP.
Spatial patterns of selected sea ice, SST, and SLP NLSA modes. For each mode, we plot the spatial pattern with largest variance (of the q spatial patterns that make up the spatiotemporal pattern): (a)–(i) CCSM3 modes and (j)–(l) observational modes, indicated by an O subscript. The fields have been normalized to have a maximum absolute value of 1.
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The annual periodic SIC modes 
The low-frequency modes closely resemble the leading EOFs of Arctic SIC in the CCSM3 model. Mode 
Each intermittent mode has a natural association with a certain low-frequency mode, which acts as a modulating envelope for the intermittent mode. There is also a clear spatial connection, as the intermittent modes are active in the same parts of the domain as their low-frequency counterpart. The annual and semiannual intermittent mode pairs 
3) SST spatiotemporal patterns
Mode 
The intermittent modes associated with 
4) SLP spatiotemporal patterns
Mode 
Mode 
Mode 
b. Observational modes
We compute the coupled NLSA observational modes using a locality parameter of ε = 1.20 and a truncation level of l = 21 eigenfunctions. A primary difference between the observational modes and CCSM3 modes is the variables used for the eigenfunction computation. We find that computing SIC–SST–SLP eigenfunctions from the observational datasets yields temporal modes which are significantly noisier (more high-frequency power) than the corresponding modes from CCSM3. This corruption occurs because of the inclusion of the SLP data in the eigenfunction computation. We find that the eigenfunctions are substantially cleaner when computed using SIC and SST, and we use this as the base case for this study. On the other hand, the CCSM3 results are insensitive to the inclusion of SLP, with SIC–SST–SLP and SIC–SST eigenfunctions yielding very similar modes and conclusions regarding sea ice reemergence. We obtain SLP observational modes by projecting the SLP data onto the SIC–SST eigenfunctions and performing an SVD of the resulting operator. Note that the observational SLP data is roughly 4 times finer spatial resolution than the CCSM3 SLP data. This discrepancy in resolution may explain the corruption in observational modes compared with CCSM3 modes. An NLSA kernel that incorporates an initial spatial smoothing of the input data (making them comparable to the T42 resolution of the CCSM3 data) could alleviate these issues, but we elected not to carry out these calculations since we are able to identify reemergence families ahead using SIC and SST only as inputs to the kernel.
The observational temporal modes have a similar character to those obtained from CCSM3. For each variable, we find periodic, low-frequency, and intermittent modes, and in many cases the low-frequency modes act as modulating envelopes for the intermittent modes. (The temporal modes for SIC, SST, and SLP are shown in the figures available as supplemental material at the Journals Online website: http://dx.doi.org/10.1175/JCLI-D-14-00354.s2.)
Next, we provide a brief description of the spatiotemporal modes that will be discussed later in the paper. Movies 3 and 4 in the online supplementary material provide a more revealing spatiotemporal evolution of these modes and others. Mode 
One feature which is conspicuously absent from the observational SLP modes is a DA-like mode. Other fields, such as 850-hPa geopotential height and surface winds, and smaller domains were tested, but a low-frequency DA mode analogous to the CCSM3 results was not found. Certain modes obtained were quite transient and resembled the DA pattern at certain snapshots in time but not persistently.
c. Interpretation of low-frequency SLP modes
The low-frequency SLP modes have spatial patterns that closely resemble the familiar spatial patterns obtained via EOF analysis. However, their temporal behavior differs substantially. The low-frequency NLSA temporal modes have significant 1-yr autocorrelation and carry most of their power at frequencies below 1 yr−1. This lies in sharp contrast to the PCs obtained via EOF analysis, which have a nearly white power spectrum and decorrelate very rapidly, losing all memory after 1–2 months. Despite these extremely different temporal characteristics, there is natural connection between the two: the low-frequency NLSA modes closely resemble a low-pass-filtered version of the noisy PCs from EOF analysis.
This is illustrated in Fig. 8, which shows temporal behavior for 
Snapshots of the time series, power spectral density, and autocorrelation functions for (top) 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The observational SLP modes also display a similar correspondence, with a correlation of 0.83 between 
5. Arctic sea ice reemergence in models and observations
Sea ice reemergence is a time-lagged correlation phenomenon. SIC anomalies decorrelate over a 3–6-month time scale; however, at some time lag in the future, an increase in correlation occurs. Sea ice reemergence is observed in two forms: a spring–fall reemergence, in which spring anomalies are reproduced the following fall, and a fall–spring reemergence, in which fall anomalies are reproduced the following spring. Both forms are observed in CCSM3 model output and HadISST observations, with the spring–fall reemergence being the significantly stronger signal in both cases.
We study sea ice reemergence via the time-lagged pattern correlation methodology of Bushuk et al. (2014). For each month of the year, pattern correlations are computed between the SIC anomaly field of the given month and the SIC field at lags of 0–23 months into the future. This is done for all pairs (month, month + lag) in the time series, and we report the average of these correlation values. Note that the pattern correlations are performed on anomalies from the seasonal cycle, are area weighted, and are uncentered (global mean has not been subtracted from the anomaly field). This differs from the approach of Blanchard-Wrigglesworth et al. (2011), where the lagged correlations were performed using a time series of total sea ice area. Performing correlations using the full SIC field, as opposed to its total area, allows for inclusion of the spatial distribution of sea ice. The pattern correlation approach is able to detect opposite-signed anomaly features, such as sea ice dipoles, which would be integrated away in the total area approach. It also enforces a notion of locality, since anomalies must be spatially coincident in order to yield a significant pattern correlation. This ensures that a reported sea ice reemergence signal represents recurrent anomalies at the same spatial location.
In this paper, we focus on the regions defined in Fig. 1: a pan-Arctic domain (45°–90°N over all longitudes), the Barents and Kara Seas (65°–80°N, 30°–90°E), the Labrador Sea and Baffin Bay (45°–80°N, 70°–40°W), the Greenland Sea (55°–80°N, 40°W–0°), the Bering Sea (55°–65°N, 165°E–160°W), and the Sea of Okhotsk (45°–65°N, 135°–165°E).
a. Regional sea ice reemergence in models and observations
We begin with a regional study of sea ice reemergence using raw SIC data from HadISST observations and CCSM3 output: the results of which are shown in Fig. 9. This figure shows time-lagged pattern correlations, computed for all initial months and lags of 0–23 months. All correlations plotted in color are greater than 0.1 and are significant at the 95% level, based on a t-distribution statistic, which tests for the statistical significance of the time-mean pattern correlation values against a null hypothesis that there is no correlation.
Time-lagged pattern correlations of Arctic sea ice in different regions: (left) results from CCSM3 model output and (right) results from HadISST observations. All colored boxes are significant at the 95% level, based on a t test.
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Over a pan-Arctic domain, in both the model and observations, we observe a clear “summer limb” of positive correlations corresponding to sea ice anomalies that originate in the melt season (March–August) and reemerge in the growth season (Figs. 9a,b). The “winter limb” of fall–spring reemergence, corresponding to anomalies originating in September–February, is weak over the Arctic domain, except for a small hint of the limb in the CCSM3 data. An interesting consequence of the time-lagged pattern correlation approach is the striking similarity of pan-Arctic lagged correlations in CCSM3 and observations. This lies in contrast to the total area lagged correlation methodology of previous studies, which reveal a clearly enhanced reemergence signal in the model relative to observations (Blanchard-Wrigglesworth et al. 2011; Day et al. 2014). This indicates that, despite differences in memory of total sea ice area anomalies, the model and observations are quite similar in their memory of sea ice spatial patterns.
The pan-Arctic reemergence signal is similar in the model and observations; however, a regional analysis reveals significant differences between the two. Both CCSM3 and HadISST have strong summer limb signals in the Barents–Kara Seas (Figs. 9g,h) and the Greenland Sea (Figs. 9k,l). The CCSM3 data also exhibit a winter limb in the Barents–Kara Seas, which is not significant in observations. A striking difference is found in the Labrador Sea, with a strong summer limb and a significant winter limb in observations: neither of which is found in the model (Figs. 9i,j). Conversely, the strong summer limbs in the Bering Sea and Sea of Okhotsk found in the model data are absent in the observations (Figs. 9c–f). Note that the winter limb signal in the Bering Sea and Sea of Okhotsk should not be overinterpreted, as these domains are essentially sea ice free during the summer and early fall. Therefore, the North Pacific winter limb lagged correlations are performed using an extremely low-variance signal and are not robust.
b. Sea ice reemergence revealed via coupled NLSA
Given the nontrivial lagged correlation structures in the CCSM3 and HadISST sea ice datasets, we seek a low-dimensional representation of sea ice reemergence via the coupled NLSA modes obtained in section 4. We aim to answer two main questions: 1) Can the reemergence signal of the raw data be efficiently reproduced by low-dimensional families of modes? 2) Can these mode families reveal possible mechanisms for Arctic sea ice reemergence? To answer the former, we perform time-lagged pattern correlations using small subsets of reconstructed spatiotemporal fields from coupled NLSA. Our approach here is to first construct families of SIC modes and then to augment these families with SST and SLP modes, based on correlations.
1) CCSM3 reemergence families
Based on the associations between low-frequency and intermittent modes identified in section 4a(1), we construct two families of SIC modes, each consisting of a low-frequency mode and annual and semiannual intermittent modes. These families, which we refer to as 
In Fig. 10d, we show time-lagged pattern correlations computed over the Arctic domain using NLSA family 
Time-lagged pattern correlations of sea ice computed over the Arctic domain, using NLSA families 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
This NLSA family has a qualitatively similar correlation structure to the raw data, yet it is natural to ask whether this family is capturing the portion of the signal responsible for the summer limb in the raw data. As a method for addressing this question, we compute time-lagged cross correlations between the raw data and the NLSA subspaces, shown in Figs. 10b,c. To explain Figs. 10b,c, we introduce LC(A, B), a function that computes time-lagged pattern correlations, with the dataset B lagging A. Using this notation, Fig. 10a shows LC(Raw, Raw) and Fig. 10d shows LC(
If the reemergence signal of 
In Figs. 10e–h, we plot the same quantities as Figs. 10a–d but for family 
2) HadISST reemergence families
The observational modes also admit a mode family that is able to reproduce the reemergence signal of the raw HadISST. This family is given by 
Time-lagged pattern correlations of sea ice computed over the Arctic domain, using HadISST family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
c. Variance explained by reemergence families
Another way to test the effectiveness of the families in capturing the reemergence signal is to directly subtract the families from the raw sea ice data and compute time-lagged pattern correlations on the resulting dataset. Figure 12c shows 
Time-lagged pattern correlations of sea ice computed over the Arctic domain. Lagged correlations for CCSM3 data are shown for (a) the raw data, (c) 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The variance explained by these families is lower if one considers the full (nonintegrated) sea ice anomaly field. Over the full Arctic domain family 
The comparison with SSA, a variance greedy algorithm, is also illuminating. The leading two SSA modes capture 2.5% and 2% of the variance, respectively, and the leading 10 nonperiodic SSA modes capture 14% of the variance. By comparison, the leading 10 nonperiodic NLSA modes capture 10% of the variance, which is modestly less than SSA. The main reason for this discrepancy is that the intermittent modes of NLSA carry less variance than low-frequency modes. Despite being low variance, these intermittent modes are crucial components of the reemergence families and illustrate an important point: low-variance modes can play an important role in explaining dynamical phenomena.
d. Temporal variability of sea ice reemergence
To this point, all reported lagged correlations have been time-mean values, computed over the full time series. Next, we consider the time-dependent aspects of sea ice reemergence. Figure 12e shows lagged correlations of the raw CCSM3 sea ice data, conditional on the low-frequency modes of 
In CCSM3, about half of the record is characterized by a very weak reemergence signal (Fig. 12g), whereas other times exhibit strong reemergence (Fig. 12e). This may have important implications for sea ice predictability, since predictability resulting from reemergence will have a strong temporal dependence, dependent on the strength of the reemergence signal at a given time. The results here also demonstrate the efficacy of certain low-frequency NLSA modes as predictors for the strength of reemergence events. Therefore, these modes could be a valuable addition to statistical sea ice forecast models.
As another method to test the temporal variability of reemergence events, we measure the strength of the reemergence signal as a function of time. We define the reemergence strength as the sum of correlation values along the summer limb, compute this quantity for each year of the time series, and create a probability density function (PDF). The PDF (not shown) is close to Gaussian, with a slight skew toward large reemergence events. If we let μ be the mean of the PDF, we find that 23% of reemergence events are less than 0.5μ and 23% of events are greater than 1.5μ. This spread in event distribution demonstrates that reemergence strength fluctuates strongly in time.
This temporal characterization of Arctic reemergence events is a new result of this study, which was inaccessible in previous studies of reemergence based on time-lagged total area correlations. In the time-lagged pattern correlation methodology, correlations are performed space, rather than time, which allows for the temporal variations of these correlations to be studied. The mode families also allow for an investigation of the temporal variability of reemergence events and mechanisms, and this will be returned to later in section 6c.
e. SIC–SST–SLP reemergence families
We have identified families of coupled NLSA SIC modes that are able to reproduce the reemergence signal of the raw SIC data. Next, we focus on the spatiotemporal evolution of these families and their associated SST and SLP patterns. As noted earlier, there are strong correlations between the temporal modes of SIC, SST, and SLP. We use this fact to augment the families 
The low-frequency mode of 
6. Sea ice reemergence mechanisms
a. SST–sea ice reemergence mechanism
We now examine the sea ice reemergence mechanisms suggested by the SIC–SST–SLP families defined above. Bushuk et al. (2014) showed that low-dimensional families of NLSA modes produce an SST–sea ice reemergence mechanism in the North Pacific sector, which is consistent with that proposed by Blanchard-Wrigglesworth et al. (2011). Can a similar mechanism be observed in Arctic NLSA modes? In both the model and observations, the answer is yes.
Figure 13 shows spatial reconstructions of SIC, SST, and SLP using family 
Sea ice, SST, and SLP patterns of CCSM3 reemergence family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
In the winter months of January–March, we observe strong negative sea ice anomalies in the Barents Sea and strong positive anomalies in the Bering and Labrador Seas. These anomalies reach their maximum southerly extent in March. We observe SST anomalies of opposite sign, which are roughly spatially coincident with the sea ice anomalies but also extend farther south in each of the three seas. Note that in March the Kara Sea, the northern Bering Sea, and the northern Labrador Sea are all SST anomaly free. The ice anomalies move northward and weaken over the melt season, which begins in April. In June, the ice anomalies in the region of the Barents–Kara Seas are located primarily in the Kara Sea. Also, the Bering and Labrador anomalies have moved into the northern parts of these seas and weakened substantially.
As the ice anomalies move northward, they imprint an anomaly of opposite sign in the SST field. In particular, the previously anomaly-free Kara and northern Bering and Labrador Seas now have strong SST anomalies. The ice continues to retreat northward over the melt season, reaching its minimum extent in September. In September, the sea ice anomalies are extremely weak in the Barents–Kara, Bering, and Labrador Seas, yet each of these seas has retained an SST anomaly. The SST anomaly retained in the Barents–Kara and Bering Seas is particularly strong, with a weaker anomaly in the Labrador Sea. As the growth season begins, the ice moves southward, interacts with the SST anomalies that have been stored over the summer months, and reinherits anomalies of the same sign as the previous spring. In December, we observe that most of the summer imprinted SST anomalies have disappeared, and the sea ice anomalies have reemerged with the same sign as the spring anomalies. This reemergence family is typically active for a 2–8-yr period, during which we observe reemerging sea ice anomalies of a consistent sign (see movie 5 in the online supplementary material).
We observe a similar SST–sea ice reemergence mechanism in family 
Sea ice, SST, and SLP patterns of CCSM3 reemergence family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The observational family 
Sea ice, SST, and SLP patterns of HadISST reemergence family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
The reemergence families are able to capture the SST–sea ice mechanism of Blanchard-Wrigglesworth et al. (2011), previously only accessible via time correlation analysis of raw sea ice and SST fields. This mode-based representation of reemergence allows one to track the temporal variability and strength of the SST–sea ice reemergence mechanism, as will be done later in section 6c. Also, the low dimensionality of these families has implications for predictability, since a small number of predictors (specifically the low-frequency modes of the families) define the amplitude and sign of reemergence events.
b. Sea ice teleconnections and reemergence via low-frequency SLP variability
Movies 5–7 in the online supplementary material reveal consistent phase relationships between sea ice anomalies in the Barents–Kara, Bering, and Labrador Seas. The SST mechanism described above provides a local mechanism for sea ice reemergence but does not explain this phase locking between geographically disconnected seas. We find that the SLP patterns of 
We begin with family 
From January to March, the dipole anomaly is very active, with strong northerly winds over the Bering Sea and strong southerly winds over the Barents–Kara Seas. The northerly winds advect cold Arctic air over the Bering Sea and also push the ice edge southward and advect additional ice into the sea. Each of these effects encourages the formation of a positive sea ice anomaly in the Bering Sea. Similarly, the Barents–Kara Seas experience warm southerly winds, which melt additional ice and also push the ice edge northward, contributing to the observed negative sea ice anomaly. Also, the SLP anomaly centered over Greenland produces northerly geostrophic winds over the Labrador Sea, contributing to its positive sea ice anomalies for the same reasons. The SLP anomalies and corresponding winds weaken substantially over the summer months, as do the sea ice anomalies in each of these regions. In October, the SLP anomalies begin to reappear with the same sign and a similar spatial pattern to the previous winter. This coincides with the beginning of the sea ice growth season and the reemergence of ice anomalies from the previous spring. In December, we observe a strong dipole SLP anomaly, and, again, observe sea ice anomalies in the Bering, Barents–Kara, and Labrador Seas, which are physically consistent with this pattern.
Besides explaining the observed teleconnection in sea ice anomalies, these SLP patterns also suggest an SLP–sea ice reemergence mechanism via their winter-to-winter regime persistence. Mode 
As mentioned earlier, in section 4c, the SLP patterns of 
Similar relationships between sea ice and SLP anomalies are also observed in family 
The relationship between SLP and sea ice is somewhat less clear in the observations than in the model. The right panels of Fig. 15 show the 
Given the seemingly similar sea ice anomalies of 
It should be noted that the data analysis approach employed here is capable of identifying correlation but not causality. In particular, we have not quantified the relative importance of the ocean and the atmosphere in producing sea ice reemergence. Also, we have identified SLP modes with interannual to decadal variability but have not provided a mechanism for this observed variability. We speculate that, rather than intrinsic atmospheric variability, this low-frequency variability of the atmosphere results from SST or sea ice forcing. The generation of low-frequency atmospheric variability has been widely studied, with many authors suggesting that extratropical and tropical SST anomalies are capable of driving low-frequency variability in the atmosphere (Lau and Nath 1990; Latif and Barnett 1994; Trenberth and Hurrell 1994; Weng and Neelin 1998; Selten et al. 1999; Robertson et al. 2000; Kushnir et al. 2002; Czaja and Frankignoul 2002). Other authors (e.g., Mysak and Venegas 1998) have suggested that sea ice anomalies could drive low-frequency atmospheric patterns, but modeling studies have suggested that the atmospheric response is quite weak compared with the typical magnitude of atmospheric anomalies (Alexander et al. 2004; Magnusdottir et al. 2004). Therefore, we speculate that anomalous SST forcing is the most likely candidate for the observed low-frequency SLP patterns, but more study is required on this problem. These unanswered questions could be investigated in a future study involving a hierarchy of GCM experiments.
c. Metrics for sea ice reemergence
We now establish a set of reemergence metrics for sea ice, SST, and SLP, by which one can judge the activity of sea ice reemergence and associated mechanisms in different regions. These metrics, computed for the reconstructed fields of each family, quantify the intensity and sign of ice reemergence events. We focus on the values of these metrics in the Bering, Barents–Kara, and Labrador Seas. The sea ice metric is defined as the integrated (area weighted) SIC anomaly in a given region. We define the SLP metric as the maximum value of the meridional geostrophic wind over a given region. This is a proxy for the amount of warm/cold air advection and northward/southward ice advection over a given region. The SST metric is defined as the integrated (area weighted) SST anomalies in the portion of the seas that are imprinted by summer SST anomalies. Specifically, we compute the integrated SST anomalies in the Kara Sea (65°–80°N, 75°–100°E), the northern Bering Sea (60°–65°N, 165°E–160°W), and the northern Labrador Sea (60°–80°N, 70°–40°W). It is helpful to compare the metrics, plotted in Figs. 16–18, to movies 5–7 in the online supplementary material which show the dynamical evolution of the corresponding fields for the same time period.
Reemergence metrics for sea ice, SST, and wind of family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
As in Fig. 16, but for the family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
As in Fig. 16, but for the family 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Figure 16 shows 
Figure 17 shows the metrics for family 
We show metrics for 
d. Regional sea ice relationships conditional on SLP modes
The reemergence families suggest a number of sea ice teleconnections that are related to large-scale SLP patterns. Are these regional teleconnections visible in the raw SIC data? Are the teleconnections strengthened by conditioning on certain low-frequency SLP modes being active? To answer these questions, we select pairs of regions and compute lagged cross correlations in total sea ice area anomalies of the raw data between these regions. Note that the cross correlations are obtained by computing a time series of sea ice area anomalies for each region and performing lagged correlations between these two time series. Our choice of regions and SLP modes is guided by the reemergence families. We consider the regions and SLP pattern that display the strongest teleconnection for each family.
The results are shown in Fig. 19, for months of the year with sea ice coverage in the marginal ice zones (December–May) and for lags from −23 to 23 months. Figures 19a,b show lagged cross correlations between the Barents–Kara and Bering Seas for the raw CCSM3 data and conditional on 
Lagged correlations in sea ice area anomalies between different seas: shown are CCSM3 correlations between the Barents–Kara and Bering Seas for (a) the raw data and (b) conditional on 
Citation: Journal of Climate 28, 14; 10.1175/JCLI-D-14-00354.1
Figures 19e,f show cross correlations between the Barents–Kara and Labrador Seas for the HadISST, conditional on 
7. Conclusions
We have studied Arctic sea ice reemergence (Blanchard-Wrigglesworth et al. 2011) in a comprehensive climate model and observations. This study has documented the regional and temporal details of sea ice reemergence and illustrated two potential reemergence mechanisms, involving SST and SLP persistence, respectively. We have used coupled NLSA (Giannakis and Majda 2012b, 2013; Bushuk et al. 2014), a nonlinear data analysis technique for multivariate time series, to analyze the covariability of Arctic SIC, SST, and SLP. Coupled NLSA was applied to a 900-yr-equilibrated control integration of CCSM3, yielding spatiotemporal modes, analogous to EEOFs, and temporal patterns, analogous to PCs. Modes were also extracted from 34 years of observational data, using SIC and SST observations from HadISST and SLP reanalysis from ERA-Interim. In both the model and observations, these NLSA modes capture three distinct types of temporal behavior: periodic, low-frequency, and intermittent variability. The low-frequency modes have spatial patterns that closely resemble the leading EOFs of each variable. In particular, the low-frequency SLP modes correlate strongly with the well-known Arctic Oscillation (AO; Thompson and Wallace 1998) and Arctic dipole anomaly (DA; Wu et al. 2006) patterns of SLP variability. The temporal patterns of the low-frequency SLP modes, obtained here without any preprocessing of the raw data, closely resemble a low-pass-filtered version of the corresponding PCs obtained via EOF analysis.
Performing time-lagged pattern correlations, we have found clear pan-Arctic sea ice reemergence signals in the model and observations. The lagged pattern correlation approach employed in this study reveals a stronger reemergence signal in observations than previous studies on reemergence (Blanchard-Wrigglesworth et al. 2011; Day et al. 2014). Using coupled NLSA modes, we have found low-dimensional families that are able to reproduce the reemergence signal of the raw SIC data. Intriguingly, these families explain a relatively small portion of the raw SIC variance, yet when removed from the raw data the resulting signal exhibits significantly weaker reemergence. Moreover, the associated SST and SLP patterns of these families demonstrate two possible reemergence mechanisms, consistent with those proposed by Blanchard-Wrigglesworth et al. (2011) and Deser et al. (2002). The SST–sea ice reemergence mechanism, in which spring sea ice anomalies are imprinted and stored as summer SST anomalies, is clearly active in the Barents–Kara, Bering, and Labrador Seas. The SLP–sea ice mechanism, in which sea ice anomalies reemerge because of the winter-to-winter persistence of low-pass-filtered SLP anomalies (atmospheric regimes), is also observed in these regions, with the exception of the Bering Sea in the observational record.
A key finding of this study is that these reemergence patterns are part of a pan-Arctic-scale organization involving SLP teleconnection patterns. In particular, we have found strong phase relationships between sea ice reemergence events in geographically distinct regions. Unable to explain this teleconnection in terms of purely local SST anomalies, we find clear relationships between regional sea ice anomalies and large-scale SLP variability. In CCSM3, an out-of-phase relationship between the Bering/Labrador and Barents–Kara Seas is found to be consistent with the phase and amplitude of the DA mode. Similarly, an out-of-phase relationship between the Bering/Barents–Kara and Labrador Seas is found to be consistent with the phase and amplitude of the AO mode. In observations, the AO mode is able to explain the strong out-of-phase anomalies of the Barents–Kara and Labrador Seas but cannot explain the weaker anomalies of the Bering Sea. These regional phase relationships are weakly visible in the raw SIC data and are significantly strengthened by conditioning on an appropriate SLP mode (the AO or DA) being active.
Another key aspect of this study is the regional and temporal characterization of sea ice reemergence. We have identified significant regional differences in reemergence between the model and observations, particularly in the Labrador Sea and the North Pacific sector, despite their pan-Arctic agreement. We have also found that reemergence events and mechanisms have significant temporal variability, and that the low-frequency modes of the reemergence families act as effective predictors of periods of active or quiescent reemergence. A set of reemergence metrics has been created, by which one can judge the strength and sign of sea ice reemergence events and the associated SST and SLP mechanisms.
In this study, we have demonstrated two plausible mechanisms for sea ice reemergence, involving the atmosphere and the ocean, but which mechanism is most crucial in producing ice reemergence? Is sea ice reemergence a fully coupled phenomenon, or does it also occur in more idealized situations? This data analysis study has identified correlation but not causation. An interesting subject for future work would be to perform a suite of coupled model experiments to study this question of causality.
Acknowledgments
We thank Ed Blanchard-Wrigglesworth and two anonymous reviewers for valuable comments that helped improve the manuscript. D. Giannakis and A. J. Majda wish to acknowledge support from ONR MURI Grant 25-74200-F7112. The research of D. Giannakis is partially supported by ONR DRI Grant N00014-14-1-0150. M. Bushuk is supported as a graduate student under this grant. M. Bushuk also received support from the Center for Prototype Climate Modeling at NYU Abu Dhabi Research Institute and NSERC PGS-D Award 404213.
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