1. Background
Extratropical atmospheric variability is characterized by highly nonlinear and complex interactions between very many degrees of freedom. It exhibits a wide range of spatiotemporal scales from rapidly developing synoptic weather systems or storms (with synoptic time scales) to low-frequency modes of variability. Whereas the former systems are relatively well understood, as they can be reasonably well explained by linear growth theory, the latter part, mostly dominated by complex interactions between tropospheric planetary waves and other coherent large-scale structures (e.g., blocking) remains not fully understood (Hannachi et al. 2017). This part is embedded in what is referred to as climate, which can be defined as the collection of long-term statistical properties of the system (Lorenz 1970). Therefore, climate can be described as including not only the geographical distributions of physical variables (e.g., sea level pressure) but also variances and covariances and other high-order moments between those variables (Hannachi et al. 2017).
A substantial part of low-frequency variability (LFV) is dominated by the presence of extensive and robust intraseasonal variability on time scales roughly from several days to about 40 days. It encompasses, in particular, changes in the background circulation affecting location and intensity of storms and includes in particular ridges and blocking patterns in addition to changes in jet streams in the extratropical troposphere (e.g., Hannachi et al. 2017).
The topic of large-scale flow dynamics has attracted the attention of atmospheric and climate scientists since the early 1940s (e.g., Rossby 1940) in large part because of the merits it bears, including the long-range predictability of tropospheric flow in addition to the academic and didactic virtue it has. The existence of intraseasonal large-scale flow structures, particularly in the midlatitudes, has been recognized since the 1950s (e.g., Namias 1950). These patterns can summarize important dynamical processes, which tend to persist longer than typical synoptic time scales. These modes of variability, also referred to as teleconnections, have been pioneered since the 1950s by Namias (1950, 1978), Bjerknes (1969), Wallace and Gutzler (1981), and others. For example, Rogers and van Loon (1979) describe the teleconnections in surface temperature between Greenland and northern Europe and the effect of oceanic forcing as well as its link to lower latitudes (Meehl and van Loon 1979). A recent description of teleconnection patterns can be found in Feldstein and Franzke (2017). This has led to the emergence of two main lines of thought, namely, that extratropical tropospheric planetary wave behavior is essentially linear or that it is fundamentally nonlinear.
In the linear paradigm, the origins and mechanisms of large-scale and LFV have been discussed extensively in the literature in terms of Rossby waves (e.g., Hoskins and Karoly 1981; Newman et al. 2003). In this framework, the effects of small scales and nonlinearity are assumed to be represented by white noise forcing. This yields theoretically a system with multivariate Gaussian probability density functions (PDFs; Penland and Sardeshmukh 1995; Toth 1991; Nitsche et al. 1994). When nonlinearity is not prominent and is primarily decorrelating, it may be approximated by a multiplicative noise. In this case, even though the system is linear (i.e., its deterministic component), the state-dependent (or multiplicative) noise yields non-Gaussian statistics (e.g., Sura and Hannachi 2015, and references therein).
The nonlinear paradigm is based on the theory of multiple equilibria first mentioned by Rossby (1940) and given prominence in a seminal paper by Charney and Devore (1979) and Wiin-Nielsen (1979) [see, e.g., Sura and Hannachi (2015) for details]. This paradigm may suggest ideally a multimodal PDF associated with preferred flow states; however, non-Gaussian PDFs can also stem from nonlinear interaction between the processes of different scales. A detailed review of the different theories of atmospheric LFV can be found in Hannachi et al. (2017).
PDF of a long simulation of the Lorenz model shown by shaded and solid contours within (a) the (x, z) plane and (b) the flow tendency plane plotted in terms of magnitude (shaded) and direction (normalized vectors). A chunk of the model trajectory is also shown in both panels along with the fixed points. Note that the variables are scaled by 10 and the value z0 = 25.06 of the fixed point is subtracted from z.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDF of a long simulation of the Lorenz model shown by shaded and solid contours within (a) the (x, z) plane and (b) the flow tendency plane plotted in terms of magnitude (shaded) and direction (normalized vectors). A chunk of the model trajectory is also shown in both panels along with the fixed points. Note that the variables are scaled by 10 and the value z0 = 25.06 of the fixed point is subtracted from z.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDF of a long simulation of the Lorenz model shown by shaded and solid contours within (a) the (x, z) plane and (b) the flow tendency plane plotted in terms of magnitude (shaded) and direction (normalized vectors). A chunk of the model trajectory is also shown in both panels along with the fixed points. Note that the variables are scaled by 10 and the value z0 = 25.06 of the fixed point is subtracted from z.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
While linear models have been used satisfactorily to explain teleconnection patterns and atmospheric response to tropical heating (Hoskins and Karoly 1981) or large-scale circulation control of synoptic eddy momentum fluxes (Branstator and Haupt 1998), the multiple equilibria idea has a number of attractive features. It may explain the recurrence of weather patterns, also referred to as weather regimes, in addition to mirroring low-order chaotic models that have been used as a simple analog of the midlatitude atmosphere (Ambaum 2008). Also, nonlinear phenomena may lead to extended limits of atmospheric predictability (Koo et al. 2002).
Persistent flow patterns have been interpreted as analogs of multiple quasi-stationary states or regimes in the highly truncated barotropic model of, for example, Charney and Devore (1979). These states are meant to represent metastable fixed points of the full nonlinear equations hence mimicking the behavior of tropospheric large-scale variability. If this is the case, it would mean that these metastable points will necessarily attract intermittently the system trajectory hence triggering the recurrent behavior and accounting for the longer residence time in their vicinity compared to neighboring flows within the large-scale state space [e.g., Legras and ;Ghil 1985; Haines and Hannachi 1995; Hannachi 1997a,b; see also Hannachi et al. (2017) for more references]. Furthermore, this should yield a higher probability of occurrence of the system trajectory in the vicinity of those metastable or quasi-stationary states in a similar manner to the simple low-order Lorenz model (Fig. 1).
The existence of atmospheric multiple equilibria, however, is not clear. To the best of our knowledge, no study has shown, without a shadow of a doubt, large-scale extratropical tropospheric multiple equilibria. In addition, the number of these structures is also debatable [see, e.g., Christiansen (2007), Corti et al. (1999), and Hannachi et al. (2017) for more discussion]. The above interpretation of multiple equilibria is not completely supported by studies carried out with more realistic models (see, e.g., Hannachi et al. 2017). While a number of authors suggest the existence of multiple equilibria, others point out a lack of significance of those studies [see, e.g., Stephenson et al. (2004), Ambaum (2008), and references therein]. Weather regimes on regional scales, however, have been found in a number of papers, particularly over the North Atlantic and North Pacific Ocean sectors. These sectorial regimes seem quite robust; they resemble teleconnection patterns such as the Pacific–North America (PNA) pattern and the North Atlantic Oscillation (NAO). They largely reflect variation in the jet stream [e.g., Woollings et al. 2010a,b; Hannachi et al. 2012; see also Hannachi et al. (2017) for a complete review and more references]. Interestingly, Hannachi (2010) suggests synchronization between sectorial flow regimes as a mechanism contributing to the occurrence of hemispheric circulation regimes.
One of the controversial examples that attracted attention is the wave amplitude index suggested by Hansen and Sutera (1986) in an attempt to show bimodality in the midlatitude 500-hPa geopotential height amplitude (see also Christiansen 2005a). The example was revisited later by Ambaum (2008), who constructed a local wave amplitude index at each longitude using Hilbert transform. He found no sign of bimodality at each longitude, but different amplitudes are realized at different longitudes. He then concluded that the bimodality suggested by Hansen and Sutera (1986) is not a reflection of genuine bimodality at any location. Rather, it was an artifact of aliasing two different mean amplitudes at different longitudes into a single index. Nonlinear principal component (PC) analysis based on neural networks have also been used to study the nonlinearity of Northern Hemisphere atmospheric variability (Monahan et al. 2000). Christiansen (2005b), however, recommends that nonlinear PCs should not be used for multimodality detection and regime behavior. Christiansen (2009) also investigated circulation regimes in the Northern Hemisphere winter using projection pursuit. He found strong evidence for non-Gaussianity but no evidence for bi- or multimodality in the stratosphere on monthly and daily time scales. The only bimodality was on interannual time scales of stratospheric extended winter. Moreover, in the troposphere, only evidence for non-Gaussianity is found.
A proper way to identify and quantify signatures of nonlinearity is by studying the system trajectory projected onto a low-dimensional state space. This can be achieved by considering the local (or conditional) phase space tendencies, which can be compared to those resulting from linear stochastic dynamics as in Hannachi (1997a,b) and Branstator and Berner (2005), who used the leading empirical orthogonal functions (EOFs), or, similarly, PCs, as a low-dimensional state space. The local tendencies of a long simulation of the National Center for Atmospheric Research (NCAR) general circulation model (GCM) performed by Branstator and Berner (2005) and also Berner and Branstator (2007) revealed clear signatures of nonlinearity with two fixed points. The structure of the PDF, however, did not show any signature of bimodality, a result that looks inconsistent with the theory outlined above. Integrated statistics, such as the case of (linear) principal components, which looks mainly for directions maximizing variance (e.g., Hannachi et al. 2007), suffer from the fast convergence to near-Gaussian statistics of the central limit theorem [e.g., Sura and Hannachi 2015; Ambaum 2008; see also Christiansen (2009) for a discussion on this].
In this manuscript, we follow the above lines by analyzing the trajectory and flow tendencies of a long simulation of a three-level quasigeostrophic (3LQG) model on the sphere. To overcome the weakness of PCs, and in consistency with the nonlinear character of the dynamics, we use kernel EOFs as a low-order state space to investigate the multimodality of the system and provide hence a consistent picture. We then apply this procedure to a number of reanalyses fields for both hemispheric and sectorial scales. The paper builds on Hannachi and Iqbal (2018, manuscript submitted to Tellus), which applies kernel PCs to the hemispheric sea level pressure from the Japanese reanalysis, by providing more details on the analysis method and extending the application to two reanalysis datasets in the troposphere and stratosphere. The manuscript is organized as follows. Section 2 briefly describes the model. The methodology is outlined in section 3. Section 4 discusses the results obtained from the 3LQG model. The application to the reanalyses data is presented in section 5. A summary and discussion are given in the last section.
2. Three-level quasigeostrophic model
a. Model description
Although the model can be considered as an intermediate-order model (Vannitsem 2017), it is comparable in terms of complexity to a simplified GCM regarding extratropical dynamics. It has been shown that the model reproduces faithfully midlatitude dynamical processes and is competitive to full GCMs in this regard but with a substantially smaller number of degrees of freedom (e.g., Marshall and Molteni 1993; Gritsun 2013; D’Andrea and Vautard 2001; D’Andrea 2002; Molteni 1996a,b; Molteni and Corti 1998; Kondrashov et al. 2004; Peters and Kravtsov 2012; Sempf et al. 2007a,b).
The model is an extension of the barotropic model of Gritsun (2013) obtained by including two more levels. The model variables are nondimensionalized by scaling the spatial and temporal dimensions by Earth’s radius and the inverse of Earth’s angular velocity as in Gritsun (2013) and Vannitsem (2017). The model is integrated using the Galerkin method for spatial differentiation and leapfrog in the time domain with a time step of 36 min. The model is run forward in time for more than one million days. Only one million days are kept for the analysis after discarding the initial spinup.
b. Literature of model regime and quasi-stationarity studies
A number of studies have investigated the regime and quasi-stationarity question in the 3LQG model. The results are disparate. D’Andrea and Vautard (2001) identified five regimes using a clustering procedure from simulations of the full T21 model as well as a low-dimensional truncated version of it. D’Andrea (2002) computed the quasi-stationary states of the low-order truncation of the full model and reported three states. These states compare to some of the clusters of the full model. The three states were identified, respectively, as the Arctic high and the NAO/PNA and its opposite phase.
Molteni (1996a) associated cluster centroids with generalized neutral vectors, which correspond closely to quasi-stationary states. Five cluster regimes were identified from the 3LQG model (Molteni 1996a). In a subsequent paper, Molteni (1996b) considered three truncated versions of the 3LQG. The model was truncated by considering only antisymmetric spherical harmonics and with only zonal wavenumber 3 (labeled QG231 model). In addition, two further truncated models are obtained from the QG231 (with 9 and 18 degrees of freedom, respectively) obtained by keeping the lowest one or two meridional wavenumbers. Three quasi-stationary solutions are found in the latter two truncated models. For the former truncated QG231 model, and as a further reduction, the 500-hPa streamfunction at 50°N was meridionally filtered along the sine or cosine of (zonal) wavenumber 3. The PDF of the 10-day low-pass-filtered time series showed bimodality. The procedure followed to obtain bimodality is quite reminiscent of Hansen and Sutera’s (1986) method, and it is not inconceivable that the obtained bimodality could be an artifact of aliasing as discussed in section 1.
The same T21 3LQG model of Marshall and Molteni (1993) was considered by Kondrashov et al. (2004). They run the model for 54 000 winter days and identified four clusters within the three-dimensional EOF space of the 500-hPa streamfunction using mixture modeling and k-means clustering. The obtained regimes represent both phases of the NAO and the Arctic Oscillation (AO).
3. Methodology
a. Flow tendencies and low-dimensional PDFs
b. Kernel EOFs
Conventional (linear) EOF analysis (Hannachi et al. 2007) is one of the methods extensively used in weather and climate for dimensionality reduction and exploration. EOF analysis finds directions maximizing variance regardless of the underlying dynamics generating the data or the manifold containing the trajectory. However, the nonlinearity of the extremely complex climate system may not be revealed by such linear methods.
A proper way to disentangle these complex structures, such as clusters, is to embed the data into another space referred to as feature space where structures become emphasized in some way (e.g., linear) and where it becomes easier to detect complex relationships within data (e.g., to delineate clusters). Figure 2 shows a schematic of such a transformation from the data space into the feature space involving a nonlinear transformation 
Schematic of a nonlinear transformation from the initial space into a feature space where the structure becomes discernable.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Schematic of a nonlinear transformation from the initial space into a feature space where the structure becomes discernable.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Schematic of a nonlinear transformation from the initial space into a feature space where the structure becomes discernable.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
There are two main classes of reproducing kernels: those with finite-dimensional feature spaces such as polynomial kernels and those with infinite-dimensional feature spaces involving nonpolynomial kernels as outlined above. One main weakness of polynomial kernels is the lack of a prior knowledge of the dimension of the feature space. And trying different dimensions is not practical and useful as the dimension increases as a power law of the degree of the polynomial. Another weakness is that polynomial kernels are not localized, which can be problematic given the importance of locality and neighborhood in nonlinear dynamics.
These weaknesses are overcome via the use of kernels with localized structures such as kernels with compact support. A typical localized kernel is the Gaussian kernel. Despite being not “truly” compact support, the Gaussian kernel is well localized, because of the exponential decay, a nice and very useful feature. Furthermore, compared to other kernels (e.g., tangent hyperbolic used in neural networks), the Gaussian kernel has only one single parameter σ. Last, we should add that polynomial kernels are normally obtained by constructing polynomials of scalar products 
The kernel EOF method is applied to test examples of various complexity before its application to model outputs or reanalysis data using the Gaussian kernel [Eq. (10)]. The application to multivariate Gaussian data and data uniformly distributed on the sphere always yields one spherical cluster. For a mixture of a few multivariate Gaussian distributions, the kernel PCA yields separation along the leading two kernel PCs. This is the case even for quite small separation between clusters and for a quite wide range of the smoothing parameter 
As a last tough test example, we generate three concentric clusters. Two clusters are distributed on concentric spheres of radii 50 and 30, respectively, and the third one is a spherical cluster of radius 10. Figure 3a shows the data projected onto the first two coordinates. Note that the outer two clusters are distributed near the surface of the associated spheres. The conventional PC analysis (Fig. 3b) does not help as no projective technique can delineate these clusters. The kernel PC analysis with a Gaussian kernel, however, is able to discriminate these clusters (Fig. 3c). Figure 3c is obtained with 
Scatterplot of (a) the first two coordinates of three concentric clusters, two of them distributed near the surface of spheres of radii 50 and 30 and one spherical cluster of radius 10, (b) the first two PCs, and (c) the scatter within the first two Gaussian kernel PCs; (d) the associated PDF of the kernel PC1–PC2; and scatter within the first two polynomial kernel PCs with degrees (e) 4 and (f) 9.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Scatterplot of (a) the first two coordinates of three concentric clusters, two of them distributed near the surface of spheres of radii 50 and 30 and one spherical cluster of radius 10, (b) the first two PCs, and (c) the scatter within the first two Gaussian kernel PCs; (d) the associated PDF of the kernel PC1–PC2; and scatter within the first two polynomial kernel PCs with degrees (e) 4 and (f) 9.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Scatterplot of (a) the first two coordinates of three concentric clusters, two of them distributed near the surface of spheres of radii 50 and 30 and one spherical cluster of radius 10, (b) the first two PCs, and (c) the scatter within the first two Gaussian kernel PCs; (d) the associated PDF of the kernel PC1–PC2; and scatter within the first two polynomial kernel PCs with degrees (e) 4 and (f) 9.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
4. Flow tendencies and nonlinearity of the QG model
a. Model performance
Climatology and variability are computed from the long simulation of the T21 three-level QG model. Figure 4 shows the climatology (Fig. 4a) and the variance (Fig. 4b) of the 500-hPa quasigeostrophic daily streamfunction. The climatology reproduces quite well the main observed features, namely, the confluent flow, with a high meridional gradient of the streamfunction associated with strong zonal wind speed around the east coasts of the continents of North America and Asia. Associated with this confluence, there is also a diffluent flow around the eastern part of the North Atlantic and North Pacific Oceans and western boundaries of the continents. The variance map (Fig. 4b) shows two regions of high variability located in the midlatitude North Atlantic and North Pacific.
(a) Climatology and (b) variance of daily midlevel streamfunction from a long simulation of the three-level quasigeostrophic model. Note that the fields are scaled as described in the text. Units are (a) 29.8 × 108 and (b) 88.8 × 1016 m4 s−2.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a) Climatology and (b) variance of daily midlevel streamfunction from a long simulation of the three-level quasigeostrophic model. Note that the fields are scaled as described in the text. Units are (a) 29.8 × 108 and (b) 88.8 × 1016 m4 s−2.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a) Climatology and (b) variance of daily midlevel streamfunction from a long simulation of the three-level quasigeostrophic model. Note that the fields are scaled as described in the text. Units are (a) 29.8 × 108 and (b) 88.8 × 1016 m4 s−2.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
The traditional EOF analysis (not shown) indicates that the two leading EOFs are well separated. The modes of variability (not shown) indicate that the leading two EOFs are well separated with 9% and 6.5% explained variance, respectively. The leading EOF is reminiscent of the AO (Thompson and Wallace 1998). It has two main centers of action, with opposite polarity, respectively, over the north polar region centered over east Greenland and over the midlatitudes centered mostly over the North Pacific, North America, and North Atlantic. The second EOF (not shown) shows a wavy pattern emanating from the north central Pacific around 25°N and propagating northeastward. Its signature over North America projects somewhat onto the PNA pattern.
b. Flow tendencies and multimodality
1) Case of EOF state space
The flow tendencies as well as the PDF of the trajectory of the model simulation are explored first within the prominent PCs. Figure 5a shows an example of the flow tendencies of the midlevel (500 hPa) streamfunction within the PC1–PC5 state space. Note that, in this manuscript, we do not discuss the dependence of the flow tendencies with the sampling interval, a point that was discussed in Branstator and Berner (2005). The flow tendencies (Fig. 5a) reveal clear nonlinearities similar to those of Branstator and Berner (2005) and Berner and Branstator (2007) and also Selten and Branstator (2004). The nonlinear signature can be identified by examining both the tendencies and their amplitudes. In linear dynamics, the tendencies are antisymmetric with respect to the origin, and the tendency amplitudes are normally elliptical, as shown in Fig. 5b. Figure 5a clearly shows that both of these features depart markedly from linearity. The result of subtracting the linear tendencies from the total tendencies is shown in Fig. 5c. This figure clearly reveals two singular (or fixed) points representing (quasi-)stationary states. The flow structures associated with these stationary states are discussed in the next section.
(a) Total flow tendency of the midlevel streamfunction within the conventional PC1–PC5 state space, (b) linear tendency based on a AR(1) fitted to the same data, (c) difference between the two tendencies of (a) and (b) showing the departure of the total tendencies from the linear part, and (d) kernel PDF of the same data within the same 2D state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a) Total flow tendency of the midlevel streamfunction within the conventional PC1–PC5 state space, (b) linear tendency based on a AR(1) fitted to the same data, (c) difference between the two tendencies of (a) and (b) showing the departure of the total tendencies from the linear part, and (d) kernel PDF of the same data within the same 2D state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a) Total flow tendency of the midlevel streamfunction within the conventional PC1–PC5 state space, (b) linear tendency based on a AR(1) fitted to the same data, (c) difference between the two tendencies of (a) and (b) showing the departure of the total tendencies from the linear part, and (d) kernel PDF of the same data within the same 2D state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Following the hypothesis discussed above, the region near the fixed points are expected to be associated with a slowing down of the trajectory, resulting in higher frequency of occurrence compared to neighboring locations. The PDF of the system trajectory is shown in Fig. 5d and is clearly unimodal. As discussed above, by construction, the (linear) PCs provide a measure of the variability in the data and therefore may not be suitable to provide the desired information about the system PDF.
2) Kernel PC space
Here, we use the kernel EOFs as an alternative to conventional EOFs. The Gaussian kernel is used here [see Eq. (10)] because of the nice properties of the Gaussian function. Various tests have been carried out to find the appropriate smoothing parameter. Notice, for example, that for large σ, the kernel matrix becomes close to singular, and for small σ, the matrix becomes close to the identity matrix. Note that the smoothing parameter is not universal but is data dependent, making the kernel EOF data adaptive.
In the application to the model simulation and reanalyses below, we found that a wide range of values of 
As in Fig. 5, but for the kernel PC1–PC4 state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
As in Fig. 5, but for the kernel PC1–PC4 state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
As in Fig. 5, but for the kernel PC1–PC4 state space.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
In addition to its local character plus the fact that kernel PC analysis enables delineation of possible complex structures, as pointed out earlier, one expects a different picture of the system PDF. Figure 6d shows the PDF of the midlevel streamfunction within kernels PC1 and PC4. The figure now reveals strong bimodality where the modes are also shown. By comparing the location of the PDF modes with the regions of low tendencies, a clear feature stands out, namely, a clear correspondence between the PDF modes and the fixed points. There is a slight departure between the right mode and the corresponding singular point, which can be explained by the instability of the quasi-stationary state. The obtained result corroborates the theory/hypothesis mentioned above (see also Fig. 1) regarding the association between low tendencies or slow trajectory and high occurrence frequency of states.
Figure 7 displays the two circulation flows corresponding to the PDF modes of Fig. 6d showing the anomalies (top) and the total (bottom) flows. These maps are obtained by compositing over states within the neighborhood of each PDF mode of Fig. 6d. It is clear that the anomalous stationary states are mostly localized over the North Pacific, the North Atlantic, and North America and stretch to the Mediterranean region. The first stationary state shows a low over the North Pacific associated with a dipole over the North Atlantic reflecting the negative NAO phase (Woollings et al. 2010a). The second anomalous stationary solution represents approximately the opposite phase, with a high pressure over the North Pacific associated with an approximate positive NAO phase. In both cases, the anomalies over the North Atlantic are shifted slightly poleward compared to the NAO counterparts.
(a),(b) Anomalies and (c),(d) total flows of midlevel streamfunction field obtained by compositing over states within the neighborhood of the modes of the bimodal PDF. Contour intervals are (a),(b) 29.8 × 108 and (c),(d) 29.8 × 106 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a),(b) Anomalies and (c),(d) total flows of midlevel streamfunction field obtained by compositing over states within the neighborhood of the modes of the bimodal PDF. Contour intervals are (a),(b) 29.8 × 108 and (c),(d) 29.8 × 106 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(a),(b) Anomalies and (c),(d) total flows of midlevel streamfunction field obtained by compositing over states within the neighborhood of the modes of the bimodal PDF. Contour intervals are (a),(b) 29.8 × 108 and (c),(d) 29.8 × 106 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
The total flow of the stationary solutions, obtained by adding the climatology to the anomalous stationary states, are shown in the bottom panels of Fig. 7. The first solution shows a ridge over the western coast of North America associated with a diffluent flow over the North Atlantic with a strong ridge over the eastern North Atlantic. This latter flow is reminiscent of a blocked flow over the North Atlantic. Note the stronger North Atlantic ridge compared to that of the western North American continent. The second stationary state (Fig. 7) shows a clear zonal flow over both basins. The flow obtained by compositing over states within the neighborhood of the singular (fixed) points obtained from the flow tendencies of Fig. 6c (not shown) are quite similar to the circulation regimes obtained from the PDF modes (Fig. 7). These solutions are reminiscent of the zonal and blocked flows of Charney and Devore (1979). Similar fixed stationary solutions were also obtained by Haines and Hannachi (1995) from a 10-yr perpetual-January simulation of a GCM. Note also that the flows obtained from the tendencies within the (conventional) PC space of Fig. 5c (not shown) are similar to those of Fig. 7.
It is of interest to mention here the relation of our results to those of Vannitsem (2001), where clustering was based on the dominant instability as measured by the local Lyapunov exponent. His obtained clusters from a similar model yielded PNA patterns. This may suggest a difference between states based on stability/instability and those based on singularity (or zero instability). States based on maximum instability or stability will cluster around the unstable or stable manifolds, respectively. Those states are expected to be different from the singular (or fixed) points, located at the crossings between both manifolds. This is reminiscent of the difference between clusters based on persistence and those based on quasi stationarity (Legras and Ghil 1985). Here, we attempted to bring together stationarity and probability of occurrence to define our teleconnections. It should be noted here though that the model of Vannitsem (2001) is not identical to the one used here, with, for example, different dissipation and numerical schemes. For example, our model has around 70 positive Lyapunov exponents versus 100 for Vannitsem’s (2001).
5. Application to reanalyses
a. Data description
Based on the obtained consistency between the results of the flow tendencies and the PDF within the low-dimensional kernel EOF state space, we apply in this section the above PDF estimation procedure to daily hemispheric reanalyses of various mass fields in the troposphere and stratosphere during the Northern Hemisphere winter. The averaged flow tendencies are not useful with the reanalyses, as it requires a quite large sample size. The data consist of hemispheric as well as sectorial reanalyses in the troposphere and stratosphere. Daily sea level pressure (SLP) and 20- and 500-hPa (Z20 and Z500, respectively) geopotential heights from the European reanalyses—ERA-40 (Uppala et al. 2005)—as well as the Japanese reanalyses—JRA-55 (Kobayashi et al. 2015; Harada et al. 2016)—are considered. The data span the period 1958–2001, and we focus mainly on the December–February (DJF) period. We note that the original JRA-55 starts from 1958 and goes beyond 2012, but for consistency and comparison with ERA-40, we restrict the analysis to the period 1958–2001. Daily anomalies are computed by subtracting the climatological daily seasonal cycle. To avoid any (possible) side effect, no filtering has been applied to the daily anomalies (Proistosescu et al. 2016). For the sake of simplicity and flow of the paper, we discuss below mainly the results from ERA-40, and the difference with JRA-55 is discussed in the next section.
b. Northern Hemispheric
1) Stratospheric 20-hPa geopotential heights
We look for possible identification of any nonlinear signature in the stratospheric flow field. We analyze here the PDF of 20-hPa geopotential height within the low-dimensional kernel EOF space. For the 20-hPa heights, we found quite a wide range of the smoothing parameter 
Figure 8 (top left) shows the kernel PDF within the space spanned by the leading kernel EOF1–EOF2. The kernel PCs are obtained using 
Kernel PDF of daily winter (DJF) of 20-hPa geopotential height anomalies of ERA-40 data within (top left) kernel PC1–PC2, and (top right),(bottom) composite over 20-hPa geopotential height anomaly states within the neighborhood of the PDF modes. Units in the height anomalies are 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Kernel PDF of daily winter (DJF) of 20-hPa geopotential height anomalies of ERA-40 data within (top left) kernel PC1–PC2, and (top right),(bottom) composite over 20-hPa geopotential height anomaly states within the neighborhood of the PDF modes. Units in the height anomalies are 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Kernel PDF of daily winter (DJF) of 20-hPa geopotential height anomalies of ERA-40 data within (top left) kernel PC1–PC2, and (top right),(bottom) composite over 20-hPa geopotential height anomaly states within the neighborhood of the PDF modes. Units in the height anomalies are 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
The dynamics of the northern winter polar vortex involve these broad states in a quite complex manner. As it can be seen from Fig. 8, although the modes are clear and well separated, the system trajectory, while residing on average within the neighborhood of one mode, can intermittently be attracted toward another mode (the modes here may be seen as potential wells of the system) and execute occasional and sporadic excursions. For instance, the split and displaced events are not categorically separated (e.g., Kwasniok et al. 2018; Hannachi et al. 2011; Mitchell et al. 2013). An example is displayed in Fig. 9, which shows the state of the vortex from December 1961 to beginning January 1962, and corresponds overall to a displaced state. In the midst of this displaced state, however, an intermittent wave-2 (or split) structure develops within a few days, around 24 December, and lasts for up to several days before it decays and returns back to its original (split) state. We recall here that the reanalysis daily data we used were not filtered. A time filter, such as 5- or 10-day low-pass filter can help reduce these intermittent bursts.
The 20-hPa ERA-40 geopotential height anomalies during a displaced event (mode A) in December 1961–January 1962. The events correspond to (top left) 20, (top center) 22, (top right) 26, (bottom left) 28, and (bottom center) 30 Dec 1961 and (bottom right) 1 Jan 1962. Units are 10 m (i.e., contour interval: 100 m).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
The 20-hPa ERA-40 geopotential height anomalies during a displaced event (mode A) in December 1961–January 1962. The events correspond to (top left) 20, (top center) 22, (top right) 26, (bottom left) 28, and (bottom center) 30 Dec 1961 and (bottom right) 1 Jan 1962. Units are 10 m (i.e., contour interval: 100 m).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
The 20-hPa ERA-40 geopotential height anomalies during a displaced event (mode A) in December 1961–January 1962. The events correspond to (top left) 20, (top center) 22, (top right) 26, (bottom left) 28, and (bottom center) 30 Dec 1961 and (bottom right) 1 Jan 1962. Units are 10 m (i.e., contour interval: 100 m).
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
2) Sea level pressure
The application to NH winter daily SLP anomalies reveals strong bimodality that cannot be obtained when conventional PCs are used. Various experiments have been carried out by investigating the SLP anomalies over the whole NH as well as north of 20° and 27°N. The results are quite similar, although the bimodality is stronger for the latter case compared to the other two choices. This reflects again the stronger nonlinearity in the winter extratropics SLP anomalies compared to when the subtropics are included. The optimal smoothing parameter for SLP anomalies are close to 
Figure 10 (top) displays the PDF within the leading kernel PC1/PC2 and shows strong bimodality of the SLP anomalies poleward of about 27°N, obtained with smoothing parameter 
As in Fig. 8, but for daily SLP anomalies over NH winter north of 27°N. Contour interval is 1 hPa.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
As in Fig. 8, but for daily SLP anomalies over NH winter north of 27°N. Contour interval is 1 hPa.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
As in Fig. 8, but for daily SLP anomalies over NH winter north of 27°N. Contour interval is 1 hPa.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
3) Z500 geopotential heights and streamfunction
A similar analysis has been carried out with the hemispheric 500-hPa geopotential height reanalyses. The PDF of Z500 height anomalies within the leading kernel PCs reveals strong bimodality again, particularly for the extratropics poleward of about 20°, 27°, and 30°N, in consistency with the SLP anomalies. The PDF is also quite robust to changes in the smoothing parameter around the value 
Figure 11 shows the PDF of the Z500 geopotential height anomalies along with a composite of states in the neighborhood of both modes. The modes represent, respectively, a polar low and a polar high with opposite anomalies over the midlatitudes around Canada, western Russia, and western Europe/east North Atlantic. The regimes are not opposite of each other where the polar high regime is stronger than the other regime by about 30 m. This also reflects the asymmetry between the two regimes.
(top) PDF of winter daily NH 500-hPa geopotential height anomalies within the kernel PC (KPC1–KPC6) state space, along with (middle),(bottom) the flows associated with the corresponding modes of the PDF. Contour interval is 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(top) PDF of winter daily NH 500-hPa geopotential height anomalies within the kernel PC (KPC1–KPC6) state space, along with (middle),(bottom) the flows associated with the corresponding modes of the PDF. Contour interval is 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(top) PDF of winter daily NH 500-hPa geopotential height anomalies within the kernel PC (KPC1–KPC6) state space, along with (middle),(bottom) the flows associated with the corresponding modes of the PDF. Contour interval is 10 m.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
For a comparison with the QG model, the method has been applied to the 500-hPa streamfunction from the reanalyses. The ERA-40 streamfunction field is computed from the wind fields as it is not provided. The KPC analysis of the streamfunction yields also bimodality with the modes dominated by the polar centers of action (Fig. 12). The dominant teleconnection is mostly located in the Eastern and Western Hemispheres for the first and second modes, respectively. Some asymmetry is also observed in the obtained teleconnections. Compared to the teleconnections obtained from the QG model streamfunction (Fig. 7), there are similarities mostly over the polar region and Greenland but also differences, particularly over the midlatitudes. The QG model has a more zonal structure of the teleconnection compared to more meridional structure for the reanalysis. This suggests an effect of the sea surface temperature (SST) forcing of the ERA-40 teleconnection but also a contribution from internal variability. Frederiksen and Branstator (2005) examined the effect of seasonally varying basic state on teleconnection in an attempt to separate the role of external forcing versus internal variability. Although the authors noted differences in teleconnections between the cases of varying and constant SST forcing, they also pointed out that perturbations react to seasonally varying basic state faster than the state is changing. This reflects again the importance and usefulness of time-independent basic state (i.e., internal variability). Also, Kushnir et al. (2002) reviewed the state of knowledge of the atmospheric response to SST forcing. They conclude that although the ocean does indeed influence the atmosphere, this influence remains modest compared to internal variability, hence the importance of the intrinsic atmospheric modes of variability.
Modes of the 500-hPa streamfunction anomaly from the ERA-40 data. Contour interval is 1 × 105 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Modes of the 500-hPa streamfunction anomaly from the ERA-40 data. Contour interval is 1 × 105 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Modes of the 500-hPa streamfunction anomaly from the ERA-40 data. Contour interval is 1 × 105 m2 s−1.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
c. Sectorial analysis
In the last part of the analysis, we set out to investigate sectorial circulation regimes. We focus here on the North Atlantic and the European sector in the troposphere specifically because of its direct link with the NAO and the position of the eddy-driven jet stream, which was shown to fluctuate between three latitudinal positions, namely, northern (N), central (C), and southern (S) positions (Woollings et al. 2010a). In addition, many researchers have suggested several circulation regimes taking place over this sector (e.g., Michelangeli et al. 1995; Dawson et al. 2012), though this was also questioned, as for the hemispheric case, by other researchers [e.g., Christiansen 2007; Fereday et al. 2008; see also Fereday (2017) for more discussion].
The North Atlantic–European sector is defined here as the region of the NH delimited by longitudes 70°W and 40°E. A similar analysis as above is carried out and applied to the SLP and Z500 geopotential height anomalies over the sector and also checked the sensitivity to varying the southern latitudinal boundary of the sector. The PDF of the SLP anomalies poleward of 27° (not shown) reveals a trimodal structure within the leading two KPCs: two strong modes and a third weak one. The modes correspond, respectively, to a −NAO, Scandinavian low, and a European blocking with a signature of moderate +NAO over the North Atlantic. In other two-dimensional KPCs, and also with the other two domains (poleward of 0° and 20°N), the PDF is mostly bimodal (not shown), mirroring essentially both phases of the NAO.
The jet latitude variability in Woollings et al. (2010a; see also Hannachi et al. 2012) was also analyzed in terms of midtropospheric geopotential heights, and for the sake of comparison, we analyze here the sectorial Z500 height anomalies. As for the SLP anomalies, the PDF of Z500 height anomalies is computed and is shown in Fig. 13 (top left) and reveals a trimodal structure within the leading two kernel PCs, with two strong modes and a third weak one. The trimodal structure is clearly more discernable with the extratropical domains compared to the other domain (poleward of 0°). The PDF in other two-dimensional KPCs (not shown) is mostly bimodal. The colored dots in Fig. 13 (top left) represent the states of the KPCs associated with the 300 points closest to each mode of the jet latitude PDF of Woollings et al. (2010a). The flow composites corresponding to these three modes have been computed and compared to the composites associated with the modes of the jet latitude (e.g., Woollings et al. 2010a; Hannachi et al. 2012). The two strongest modes represent, respectively, the Greenland blocking and a ridge over the North Atlantic. These flow regimes are unambiguously associated with the S and N positions of the eddy-driven jet stream. The third (weak) mode is dominated by a low center zonally stretched over the northern North Atlantic south of Greenland and a high over northern Scandinavia. This circulation regime can be associated with the C of the North Atlantic jet stream. It is clear from this analysis that the N and S positions of the jet are quite robust, unlike the C position. The three modes of the jet latitude PDF (Woollings et al. 2010a), however, were found to be equally strong. The present analysis suggests that the central position of the jet is not clearly a persistent state. Rather, it is likely to represent a transition path between the N and S positions and also projects onto the climatological jet stream.
(top left) PDF of daily winter 500-hPa height anomalies over the North Atlantic–European sector within kernel PC1–PC2, along with (top right),(bottom) corresponding maps obtained by compositing over the states closest to the respective modes. The colored dots represent the KPC states associated with the 300 events closest to each mode of the jet latitude index of Woollings et al. (2010a), with yellow, blue, and white for the southern, northern and central positions of the jet latitude, respectively. Note, in particular, the separation between the northern and southern positions and more concentration of the scatter around the PDF modes A and B compared to mode C and the white dots, which are broadly scattered. Units in the geopotential height anomalies are meters.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(top left) PDF of daily winter 500-hPa height anomalies over the North Atlantic–European sector within kernel PC1–PC2, along with (top right),(bottom) corresponding maps obtained by compositing over the states closest to the respective modes. The colored dots represent the KPC states associated with the 300 events closest to each mode of the jet latitude index of Woollings et al. (2010a), with yellow, blue, and white for the southern, northern and central positions of the jet latitude, respectively. Note, in particular, the separation between the northern and southern positions and more concentration of the scatter around the PDF modes A and B compared to mode C and the white dots, which are broadly scattered. Units in the geopotential height anomalies are meters.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
(top left) PDF of daily winter 500-hPa height anomalies over the North Atlantic–European sector within kernel PC1–PC2, along with (top right),(bottom) corresponding maps obtained by compositing over the states closest to the respective modes. The colored dots represent the KPC states associated with the 300 events closest to each mode of the jet latitude index of Woollings et al. (2010a), with yellow, blue, and white for the southern, northern and central positions of the jet latitude, respectively. Note, in particular, the separation between the northern and southern positions and more concentration of the scatter around the PDF modes A and B compared to mode C and the white dots, which are broadly scattered. Units in the geopotential height anomalies are meters.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
6. Summary and discussion
The results obtained so far from the three-level quasigeostrophic model or from the reanalyses show high consistency regarding the existence of hemispheric large-scale quasi-stationary weather regimes. For the intermediate-complexity model, averaged flow tendencies and kernel PDFs within different two-dimensional state spaces spanned by the kernel PCs yield consistent results with conceptual models of low-order dynamics. Two singular points are obtained corresponding to regions with low tendencies in agreement with high occurrence probability of large-scale flows. The obtained circulation regimes represent essentially zonal and blocked flows reminiscent of the high- and low-index flows, respectively, suggested by Charney and Devore (1979). The blocked state represents ridges over the western coast of North America and the eastern North Atlantic.
In the stratosphere, strong nonlinearity is obtained from daily 20-hPa geopotential height anomalies. Trimodal and bimodal PDFs are observed from both the ERA-40 and JRA-55. Two strong modes are observed in most two-dimensional PDFs, but trimodal PDFs are observed mainly in KPC1–KPC2 space. This strong nonlinearity is also more prominent for the extratropics (e.g., poleward of 27°N). The range of values of the smoothing parameter is also quite wide. The flow of the stratospheric regimes are identified as the undisturbed, split, and displaced vortex, similar to those observed in Hannachi et al. (2011). It is worth noting that this classification is not entirely exclusive. For example, the displaced state may also contain splitting events.
In the Northern Hemispheric troposphere, strong bimodality is also observed in the daily SLP and 500-hPa geopotential height anomalies particularly for the extratropical domain (north of 27°). The SLP circulation regimes represent mainly a polar low and a polar high with a −NAO signal. The 500-hPa geopotential height regimes are also consistent with their SLP counterparts, particularly over the polar region. It is appropriate to mention here the difference between the NAO and the AO, which has been discussed already in a number of papers (e.g., Ambaum et al. 2001). It is known that different methods generally focus on different aspects. Combining this with the high dimensionality/complexity of the system suggests the likelihood of the emergence of the NAO or the AO depending on the method and/or the data (Hannachi 2016). Notwithstanding this apparent difference, there are some consistencies between the different teleconnection patterns. The patterns obtained by the present method are comparable to those presented in the literature. The present method provides evidence of the relatively high probability of these flow structures, which in turn represent quasi-stationary states with low-flow tendencies.
Tropospheric PDF analysis over the North Atlantic–European sector reveals also a trimodal PDF with two strong modes and a weak third mode for both daily SLP and 500-hPa height anomalies. In terms of SLP anomalies, the modes correspond, respectively, to −NAO, Scandinavian low, and +NAO/European blocking. The 500-hPa height regimes are comparable to those of Woollings et al. (2010a) and Hannachi et al. (2012) and are associated, respectively, with northern (N), southern (S), and central (C) positions of the eddy-driven jet stream. In particular, the central position is the one with the weakest mode. This points to the fact that the central regime could be a transition path between N and S regimes and not a persistent regime in itself.
The results obtained using ERA-40 compare well with those obtained using JRA-55. Figure 14 shows an example of the PDF of daily 20-hPa height anomalies from JRA-55. This figure compares well with that of ERA-40 (see Fig. 8). In particular, the suggested nonlinearity of the manifold is shared between both datasets. It is also observed that, apart from the multimodality, the PDFs within the KPC space have a nonlinear (crescent-like) shape, shared between both datasets, suggesting a nonlinear (curved) manifold. This points once more to the strong nonlinear manifold of stratospheric dynamics. There are nonetheless a few differences worth mentioning. The hemispheric regimes based on ERA-40 500-hPa height are mostly dominated by polar low and high, and no PNA signal is obtained over the PNA region. A PNA regime is obtained, however, with JRA-55 500-hPa height anomalies (not shown). A circulation regime that projects onto the PNA was also obtained from 500-hPa heights from NCAR–NCEP reanalyses (Hannachi 2007). Another point is related to the sectorial regimes; the multimodality observed within the North Atlantic–European sector is in general stronger in ERA-40 compared to JRA-55.
PDF of the daily winter NH 20-hPa geopotential height anomalies from JRA-55 within the leading two KPCs.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDF of the daily winter NH 20-hPa geopotential height anomalies from JRA-55 within the leading two KPCs.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDF of the daily winter NH 20-hPa geopotential height anomalies from JRA-55 within the leading two KPCs.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
Overall, the analysis shows that the bimodality obtained with the hemispheric fields is quite robust and consistent between the different variables and also between the two reanalysis datasets used here. The method presented here provides support to the large-scale regimes identified in the literature by revealing clear bimodality in the feature space. Of particular interest is the result that kernel PCs represent large-scale flow patterns. The leading kernel PCs have indeed high correlations (not shown) with leading PCs. For example, KPC1 is correlated with several leading PCs with a particularly larger correlation with PC1.
Like many methods discussed in the literature, the present approach is based on using a low-dimensional state space. Prefiltering of the data can certainly affect the result, as pointed out, for example, by Lau et al. (1994). The point is that teleconnections, with their large-scale feature, contribute a substantial amount to the atmospheric variability. The main features of these patterns grant them some sort of robustness, and they emerge even when prefiltering is applied. Unlike our approach, which is exploratory in nature, it is possible to embed the filtering and dimensionality reduction in a modeling framework (Mukhin et al. 2015; O’Kane et al. 2017). For example, O’Kane et al. (2017, and references therein) propose a model-based approach in which a stochastic process is used to model the data via a nonlinear autoregressive possibly with time-dependent parameters. The model still uses, however, a sensible number of the leading EOFs and identifies spatiotemporal regime clusters along with the model parameters in a holistic optimization procedure. The method was applied to an atmospheric GCM by Franzke et al. (2009) and to reanalyses by Risbey et al. (2015). The obtained states are consistent with well-documented teleconnections and are comparable to the patterns found here with some differences though, because different models (and reanalyses) emphasize different aspects.
It is true, however, that the method presented here is simpler compared, for example, to that of O’Kane et al. (2017) and enables us to delineate complex structures and identify clusters in high-dimensional climate data. Nonetheless, the present method, like any other method, has some weaknesses. One of them is the choice of the kernel. Although we recommended the Gaussian kernel, the smoothing parameter has to be chosen, and no simple method exists to identify it, but has to be obtained from experience. The spatial patterns of the kernel EOFs are not straightforward but can be identified numerically via a descent algorithm. In general, the leading two KPCs provide the answer for the clusters, and in some cases, a few two-dimensional KPC spaces have to be explored. Finally, the significance level of the modes of the obtained PDFs can only be obtained using Monte Carlo and can be computationally heavy.
Last, we would like to discuss the issue related to climate change signals. A number of researchers have suggested that circulation regimes have gone through a change in frequency of occurrence and/or change in structure (e.g., Corti et al. 1999; Monahan et al. 2000; Christiansen 2003, 2005a; Hannachi 2007). A comparison of the PDFs between the first and second halves of the data shows robust changes in the frequency of occurrence of hemispheric regimes, especially in the troposphere. Figure 15 shows the PDFs of hemispheric daily SLP anomalies (top panels), obtained based on JRA-55, for the first half, that is, during winters of 1958–1979 (top-left panel), and the same PDF but for the second half, that is, winter 1980–2001 (top-right panel). This is associated with increasing frequency of low pressure system over the polar region and high pressure over North Atlantic midlatitude. A similar figure (not shown) is obtained with ERA-40. Therefore, over the reanalysis period, the zonal flow frequency has increased. This observation is also obtained with the 500-hPa height anomalies and for both reanalysis datasets. This suggests that this climate change signal is robust. Although this observation suggests inconsistency with the polar amplification hypothesis, which implies a reduction of meridional temperature gradient and hence a reduction of zonal wind, it is possible that the observed change represents the nonlinear component of the signal change. It is also possible that the observed change is due to a low-frequency component in the system.
PDFs of the (top) daily winter NH SLP anomalies and (bottom) 20-hPa geopotential height anomalies for the first and second halves of the record.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDFs of the (top) daily winter NH SLP anomalies and (bottom) 20-hPa geopotential height anomalies for the first and second halves of the record.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDFs of the (top) daily winter NH SLP anomalies and (bottom) 20-hPa geopotential height anomalies for the first and second halves of the record.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
In agreement with the above hemispheric shift in the frequency of occurrence of circulation regimes, a similar shift is observed with the PDF of Z500 and SLP anomalies over the North Atlantic–European sector. This is consistent with the fact that the hemispheric regimes project onto the NAO. The PDF of ERA-40 Z500 anomalies over the sector (not shown) reveals a clear shift of occurrence probability from mode A (or southern jet position; see, e.g., Fig. 13) in the first half to mode B (or northern jet position) in the second half. For the SLP anomalies, a similar shift is obtained (not shown) from −NAO to +NAO/European blocking. Similar results (not shown) are also obtained using JRA-55.
In the stratosphere, the climate change signal is not an obvious change in frequency as for the troposphere. The bottom panels of Fig. 15 show the PDFs of JRA-55 Z20 during both periods. A similar picture is also obtained with ERA-40 (not shown). There is indication that the displaced vortex state, corresponding to the frequency of the bottom-left PDF maximum (see also regime A in Fig. 8), did not change significantly, although the position has changed particularly along KPC2. The perturbed regime represented by the leftmost PDF maximum (mode B of Fig. 8) has slightly increased in frequency with a slight change of position (or structure). The central mode of the PDF (see also mode C of Fig. 8), corresponding to disturbed vortex, has witnessed a clear decrease in frequency in the last few decades. This particular change has also been obtained with ERA-40. This strongly suggests that, because of the prominent nonlinearity in the winter stratosphere future, climate change may induce changes in frequency of occurrence and structure of the winter polar vortex. Like for the tropospheric signal change, we do not rule out the possibility of a low-frequency signal in the data.
Acknowledgments
The authors thank Andrey Gritsun for sharing his three-level quasigeostrophic model and three anonymous reviewers for their constructive comments. W. I. is supported by a scholarship from the Department of Meteorology, MISU, Stockholm University. The computations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC).
APPENDIX
Significance of the Multimodality in Kernel PC Space
Statistical significance of the PDFs within the kernel PC space, and in nonlinear methods in general, is a challenging exercise. We have used a Monte Carlo test to assess the significance of the PDF multimodality observed in the reanalyses. For each field considered here, 100 surrogates are generated based on a multivariate first-order autoregressive [AR(1)] mode and the kernel PCs computed and the PDFs compared to that obtained from the reanalysis. The surrogates have the same observed leading modes of variability (EOFs) and associated variance. The procedure is computationally heavy, and we have used the leading 20 EOFs.
One of the main difficulties is the choice of the smoothing parameter for the surrogate. Another one is related to the choice of the KPCs of the surrogate data that will be compared to the KPCs of the reanalysis. We conducted various experiments to address these issues. It is found, in particular, that if we apply the smoothing parameter of the reanalysis to the surrogates, we find that the multimodality is always highly significant. To be more sensible, we chose typical smoothing parameters that produce PDFs of the surrogates close to those of the reanalysis (e.g., bimodal). Those PDFs can be unimodal, skewed, and, in some cases, bimodal. We could have chosen the smoothing parameter of the filtered (using, for example, the leading 20 EOFs) of the reanalysis, but this is not what we choose to do as we decided from the beginning to use unfiltered data. Next, we also compare the two-dimensional (2D) PDF of reanalysis KPCs with various 2D KPC PDFs of surrogate data. We found that most peaks of the observed multimodality are significant, as shown in Fig. A1. The peaks are significant at the 10% level (Fig. A1) and also at the 5% level (not shown). The significant region is not broad but is localized around the peaks of the different modes because most surrogate PDFs are unimodal and broad.
PDFs of the (a) ERA-40 20-hPa geopotential height, (b) SLP, and 500-hPa geopotential heights over the NH along with the 10% significance level (all color shading). The different colors simply mean they are on different PDF contour levels.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDFs of the (a) ERA-40 20-hPa geopotential height, (b) SLP, and 500-hPa geopotential heights over the NH along with the 10% significance level (all color shading). The different colors simply mean they are on different PDF contour levels.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
PDFs of the (a) ERA-40 20-hPa geopotential height, (b) SLP, and 500-hPa geopotential heights over the NH along with the 10% significance level (all color shading). The different colors simply mean they are on different PDF contour levels.
Citation: Journal of the Atmospheric Sciences 76, 1; 10.1175/JAS-D-18-0182.1
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