## 1. Introduction

[2] The ability to detect physically-meaningful spatiotemporal patterns of variability in the atmosphere–ocean climate system is crucial for enhancing our understanding and prediction of a host of phenomena, including regime shifts in oceanic circulation [*Mantua and Hare*, 2002], tropical-extratropical interactions on intraseasonal [*Vitart and Jung*, 2010] to decadal [*Newman et al.*, 2003] timescales, and other applications of wide interest. Such phenomena are governed by nonlinear dynamical laws, and influenced strongly by external factors (e.g., solar forcing), meaning that the dynamically-important features of these systems are not necessarily those carrying a high energy content (explained variance). For instance,*Crommelin and Majda* [2004] demonstrate that the leading empirical orthogonal functions (EOFs) cannot reproduce chaotic regime transitions between zonal and blocked states in a reduced model of the atmosphere; *Overland et al.* [2010]allude to the importance of intermittent events of 1–2 yr duration in Northern Hemisphere ocean variability. The role of low-energy, short-time events has also been highlighted in the fluid dynamics literature [*Aubry et al.*, 1993]. In general, classical linear methods, such as singular spectrum analysis (SSA) [*Vautard and Ghil*, 1989; *Aubry et al.*, 1991; *Ghil et al.*, 2002] and its variants, are likely to have poor skill in extracting dynamically-important, low-energy modes from large-scale numerical or observational climate datasets.

[3] A variety of nonlinear principal components analysis (NLPCA) algorithms [e.g., *Hsieh*, 2009], including kernel- [*Lima et al.*, 2009] and neural-network-based algorithms [*Monahan*, 2000; *Hsieh*, 2007], have been proposed as ways of addressing the above shortcomings of SSA in the geosciences. However, even though the nonlinear mapping functions employed by these methods greatly increase the flexibility to describe nonlinear aspects of the data, the structure of those functions is generally *ad hoc* and susceptible to overfitting the data [*Christiansen*, 2005]. Moreover, nonlinear mapping functions preclude a straightforward determination of spatial EOFs corresponding to the principal components (PCs) [*Lima et al.*, 2009], limiting the utility of this class of algorithms in deriving bases for reduced dynamical modeling. Another important issue is that the computational cost of NLPCA tends to scale poorly with the ambient space dimension. Indeed, a number of applications in the geophysics literature either deal explicitly with low-dimensional systems [*Monahan*, 2000], or preprocess the data through subtraction of the seasonal cycle and/or projection onto linear PCs [*Hsieh*, 2007], thus removing several of their important nonlinear features.

[4] Recently, a method called nonlinear Laplacian spectral analysis (NLSA) was developed [*Giannakis and Majda*, 2012, also Nonlinear Laplacian spectral analysis: Capturing intermittent and low-frequency spatiotemporal patterns in high-dimensional data, arXiv:1202.6103v1, 2012], which addresses the aforementioned shortcomings of SSA and NLPCA by combining aspects of both approaches. Similarly to SSA, NLSA produces a spatiotemporal decomposition of the data through spectral analysis of linear maps (denoted here by ), which are defined objectively from the way the data is presented in ambient space. Thus, there is no need to specify feature maps, and the correspondence between EOFs and PCs is unambiguous and straightforward. However, the maps differ crucially from SSA in that they are tailored to the nonlinear geometry of the data manifold, *M.*Specifically, the key principle underlying NLSA is that for efficient analysis of high-dimensional complex data, temporal patterns (analogous to PCs) should belong in natural low-dimensional spaces of functions on*M*, spanned by the leading-few eigenfunctions of graph-theoretic Laplace-Beltrami operators. These eigenfunctions are evaluated efficiently in high ambient-space dimensions using sparse algorithms developed in machine learning [*Belkin and Niyogi*, 2003; *Coifman and Lafon*, 2006], thus providing natural orthonormal bases to extract spatiotemporal patterns through singular value decomposition (SVD) of . The parameter *l* controls the scale (“resolution”) on the data manifold resolved by the temporal modes, and can be set via spectral entropy criteria to prevent overfitting (Giannakis and Majda, arXiv:1202.6103v1, 2012).

[5] Here, we apply the NLSA methodology in a comparative study of the spatiotemporal variability of sea surface temperature (SST) in monthly-averaged, North Pacific data from three control experiments used in the Intergovernmental Panel on Climate Change assessment report 4 (IPCC AR4). Applying no spatial coarse-graining or seasonal-cycle subtraction, we identify a number of physically-distinct spatiotemporal processes, allowing comparison of the SST variability in these models at a significantly more refined level than possible through classical SSA. In particular, besides the familiar periodic and decadal modes of SST variability [*Mantua and Hare*, 2002; *Bond et al.*, 2003; *Di Lorenzo et al.*, 2008], NLSA reveals a family of modes with strongly intermittent behavior, describing variability in the Eastern and Western boundary currents, as well as mid-basin variability with year-to-year reemergence [*Alexander et al.*, 1999]. The bursting-like behavior of these modes, a hallmark of strongly-nonlinear dynamics, means that they carry little variance, and therefore are not captured by SSA.