1. Introduction
Earth’s atmosphere is characterized in general by motions of continuous temporal and spatial scales. The interactions among different scales often form the foundations of various weather and climate phenomena. For example, subweekly, synoptic-scale disturbances constituting storm tracks are known to play an important role in the development of atmospheric blocks, that is, quasi-stationary, vertically coherent high pressure features in the extratropical atmosphere (e.g., Green 1977; Nakamura and Wallace 1993; Maeda et al. 2000; Park et al. 2015; Ma and Liang 2017). Initial baroclinic development at long synoptic scales followed by increasingly important barotropic growth often characterizes the life cycle of persistent (thus low-frequency) negative height anomalies over the North Pacific in boreal winter that project effectively onto the Pacific–North America (PNA) teleconnection pattern (Dole and Black 1990). Lau and Holopainen (1984) showed in the quasigeostrophic (QG) framework that vorticity and heat fluxes associated with both high-frequency (synoptic-scale) and low-frequency eddies act together to maintain the winter monthly mean flow in the Northern Hemisphere. In a broader sense, there exists a symbiotic relationship between extratropical synoptic- and planetary-scale disturbances: the former extract energy from the zonal flow to compensate for their own energy dissipation and supply energy to the latter through barotropic inverse energy cascade; the latter form regions of enhanced baroclinicity where the former preferentially grow (Cai and Mak 1990; Cai and van den Dool 1991, 1992). As the dominant mode of low-frequency variability in the northern extratropical atmosphere, the northern annular mode (NAM) in the troposphere is characterized by meridional meandering of the jet in the zonal-mean zonal wind and such movement is also largely driven by westerly momentum fluxes of subweekly synoptic-scale disturbances of baroclinic origins (e.g., Robinson 1991, 1996; Yu and Hartmann 1993; Lorenz and Hartmann 2001).
The investigation of atmospheric scale-interaction processes in the past has relied on a combination of bandpass or spatial filtering of relevant fields with dynamical diagnoses based upon the evaluation of local geopotential tendency, vorticity and/or energy budget (e.g., Hayashi 1980; Lau and Holopainen 1984; Cai and Mak 1990; Sheng and Hayashi 1990; Sheng and Derome 1991, 1993; Cai and van den Dool 1994; Cuff and Cai 1995; Deng and Jiang 2011; Jiang et al. 2013a,b). These approaches are easy to comprehend and when applied to observational or model data often provide excellent depictions of where and how strongly active scale interactions occur in the physical or frequency domain. One of the biggest drawbacks of the temporal/spatial filtering, as recently argued in a series of papers, is that multiscale energy in physical sense cannot be appropriately defined with just any filters (e.g., Liang 2016a; Xu and Liang 2017; Ma and Liang 2017). Two minor drawbacks of the filtering are that these approaches do not automatically reveal the exact structure of significantly interacting disturbances given the predefined temporal or spatial filtering, and an examination of the temporal evolution of the interaction of interest often demands case compositing requiring prior knowledge of the occurrences of events, for example, a vorticity budget analysis applied to multiple blocking events to understand the contribution of synoptic-scale disturbances to the blocking development.
Some research groups have taken an entropy-based approach. Liang and Kleeman (2007a,b) proposed a rigorous formalism of information transfer (flow) between dynamical system components for both discrete mapping and continuous flow, and the transfer is measured by comparing the entropy increases between an original system under consideration and a modified system where the source component is instantaneously frozen. Liang (2014) derived from first principles (entropy) a measure of causality based on the notion of information flow and used this measure to unravel the cause–effect relation between time series, and specifically, to understand the causal relation between two modes of tropical SST variability (i.e., ENSO and the Indian Ocean dipole). Liang (2016b) further provided a comprehensive discussion of the concept of information flow (transfer) and demonstrated information flow and causality as rigorous notions ab initio.
In this study we seek to bring a novel approach to understanding scale-interaction processes in the atmosphere, namely, the framework of causal discovery and specifically structure learning for temporal probabilistic graphical models (e.g., Pearl 1988, 2000; Spirtes et al. 1993, 2000; Neapolitan 2003; Koller and Friedman 2009). Probabilistic graphical models (PGMs) have been used in multiple areas of atmospheric sciences. In a first wave of related research PGMs were mainly used for prediction tasks, such as the predictions of severe weather (Abramson et al. 1996), daily pollution levels (Cossention et al. 2001), and precipitation (Cofino et al. 2002). In the second wave structure learning of PGMs and similar tools (e.g., graphical Granger models) have been used to identify potential cause–effect relationships from data (e.g., Chu et al. 2005; Strong et al. 2009; Chen et al. 2010; Bahadori and Liu 2011; Ebert-Uphoff and Deng 2012a,b; Runge et al. 2012; Hlinka et al. 2013; Zerenner et al. 2014; Deng and Ebert-Uphoff 2014; Kretschmer et al. 2016). Runge (2018) provides an excellent overview of the practical aspects of constructing PGMs from time series. The authors’ earlier work (e.g., Ebert-Uphoff and Deng 2012a) provides to the climate research community an introduction of structure learning using probabilistic graphical models. It shows how structure learning can be used to derive hypotheses of causal relationships among four prominent modes of atmospheric low-frequency variability in the northern extratropics. The idea was further expanded to obtain a climate network that emphasizes flow of information (defined by the directions of directed edges in a PGM) in the 500-hPa geopotential field and the strength of information flow (in terms of the number of directed edges coming out of one geographical location) in such a network was shown to be decreasing in the future climate with enhanced forcing from greenhouse gases (Ebert-Uphoff and Deng 2012b; Deng and Ebert-Uphoff 2014). Note that compared to more rigorous quantitative definitions such as those adopted in Liang (2016b), the concept of “information flow” is used here more qualitatively. Namely existence of information flow here indicates that the algorithm identifies directions/time delays of statistically significant edges along which information flow occurs, but it does not provide measures of the actual amount of information transferred, that is, the quantitative strength of the information flow.
The key of structure learning is to detect direct connections and eliminate indirect ones through the use of conditional independence tests (Pearl 1988, 2000; Spirtes et al. 1993, 2000). The identified direct connections, either between different teleconnection indices (e.g., Ebert-Uphoff and Deng 2012a), or between geographical locations for a selected atmospheric variable (e.g., geopotential height; Ebert-Uphoff and Deng 2012b) provide a straightforward view of potential causal pathways in the field of interest. Figure 1a indicates the process of causal discovery in grid space, as used in the latter type of study. Ebert-Uphoff and Deng (2017) tested the performance and accuracy of this approach for dynamic systems using advection–diffusion simulations as a test bed. Others have studied the general algorithms thoroughly for other settings; see, for example, Ramsey and Andrews (2017).
Basic block diagrams of (a) the process of causal discovery in grid space and (b) the process of causal discovery in spectral space. Here “SH” stands for spherical harmonics.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Basic block diagrams of (a) the process of causal discovery in grid space and (b) the process of causal discovery in spectral space. Here “SH” stands for spherical harmonics.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Basic block diagrams of (a) the process of causal discovery in grid space and (b) the process of causal discovery in spectral space. Here “SH” stands for spherical harmonics.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Here we apply causal discovery to the time series of the daily coefficients of spherical harmonics obtained from a spectral decomposition of the Northern Hemisphere 500-hPa geopotential field to identify the most prominent direct connections between different spherical harmonics components, and use these potential causal pathways to build a causality-based view of the interaction between atmospheric disturbances of different scales (i.e., spherical harmonics components). Figure 1b shows the proposed approach of causal discovery in spectral space. There have been a few past and ongoing efforts that study information flow and causality between different scales by the geophysical fluid dynamics community (most notably, Liang 2013; Materassi et al. 2014; Liang and Lozano-Duran 2016; Liang 2019). However, to the best of our knowledge, only one other research group has combined structure learning with spherical harmonics, namely, Zerenner et al. (2014). Zerenner et al. (2014) discussed the physical basis why modeling atmosphere as a set of oscillators is more justified in spectral domain compared to in spatial domain: spherical harmonics are associated with large-scale propagating Rossby–Haurwitz waves in the atmosphere. The nature of the connections identified was also elaborated as arising through nonlinear advection of the extratropical quasigeostrophic flow. However, they only performed a static analysis on monthly mean atmospheric fields, and the insights they obtained using this approach were limited, as their graphs are very sparse with connections primarily between neighboring nodes. Zerenner et al. (2014) thus cautioned about deriving a network via thresholding and the potential issue of multivariate Gaussian assumption of the spherical harmonics coefficients. Our results, on the other hand, show many additional connections compared to those reported in Zerenner et al. (2014). This different outcome is likely due to several differences between their approach and ours; for example, we derive a temporal model, use a different data preprocessing scheme, and a different structure learning method. Thus, expanding on the original idea proposed by Zerenner et al., we develop a method that succeeds in identifying many interesting connections in spectral space. The initial focus of this work is on the interaction between synoptic and planetary scales, which have received most attention in the past. Following this introduction, section 2 describes the data used and detailed analysis steps. The main findings are provided in section 3. Section 4 gives some concluding remarks.
2. Data and methods
In this study, we use the daily 500-hPa geopotential height data at 2.5° × 2.5° horizontal resolution from the NCEP–NCAR reanalysis (Kalnay et al. 1996; Kistler et al. 2001) to derive daily fields of atmospheric disturbances of various spatial scales. The study focuses on the Northern Hemisphere and covers all boreal winter months [December–February (DJF)] in the period 1948–2015. The winter months are chosen due to the prominence of eddy–mean flow interaction and scale interactions in the winter season. To focus on the Northern Hemisphere, the time series at locations north of the equator are mirrored along the equator onto the Southern Hemisphere (Blackmon 1976). This yields the time series at grid points, which is shown in block 1 in Fig. 1b.
The next step is to apply spectral decomposition (block 2 in Fig. 1b). The daily geopotential height data is decomposed into disturbances of various spatial scales in terms of spherical harmonics, which form a complete set of orthonormal basis functions capable of modeling functions on a sphere (e.g., Blackmon 1976). The properties of the spherical harmonic basis functions characterizing spatial scales are described by the total wavenumber L and the zonal wavenumber M. The daily geopotential height data are projected onto these basis functions to obtain daily time series of complex spherical harmonics coefficients (block 3 in Fig. 1b) as a function of L and M (Samarasinghe et al. 2017). The coefficients for which L + M is an odd number vanish automatically due to the mirroring of the Northern Hemisphere. This leaves only half of the original number of spherical harmonics coefficients for the study.
To apply structure learning of graphical models we first have to define the nodes of the graph (block 4 in Fig. 1b). To represent the complex coefficient of each spherical harmonics component there are two primary alternatives; namely, we can choose the nodes to represent magnitude and phase, or real and imaginary parts, of each coefficient. Either alternative results in each spherical harmonics component being represented by two separate nodes in the graphical model. We first tried the magnitude and phase representation, because it appears to be more physically meaningful, as those quantities can be related individually to the power and direction of propagation of atmospheric waves. However, the distribution of the phase variables are cyclic and non-Gaussian, which violates a key assumption of many structure learning algorithms. (Algorithms that do not assume Gaussian distributions exist, but they are of higher computational complexity, and thus generally not feasible for such a large number of variables as required here). Thus, we switched to using the real and imaginary part of the spherical harmonics coefficients instead, which matches the choice by Zerenner et al. (2014), whose algorithms also assume Gaussianity. The statistically significant edges in the graphical model identified in the next step among all the nodes (corresponding to different spherical harmonics) thus represent potential causal interactions among these spherical harmonics, that is, atmospheric disturbances of different spatial scales.
A customized truncation scheme is adopted to determine how many projection coefficients should be used for structure learning (also block 4 in Fig. 1b). A triangular truncation with 0 < L ≤ Lmax = 50 and 0 ≤ M ≤ L is applied first. We then impose an additional constraint that the temporal-mean magnitude of the projection coefficient needs to be greater than a specific threshold (0.4 m in our case) to ensure a good signal to noise ratio such that the structure learning techniques perform well. This truncation results in a total of N = 570 nodes for a static model (no time lags considered).
Finally, we apply structure learning (block 5 in Fig. 1b) to the time series of the selected nodes to find potential causal interactions between atmospheric disturbances of different spatial scales. We use a constraint-based structure learning technique that is based on probabilistic graphical models, in contrast to the study by Zerenner et al. (2014) that uses a Graphical LASSO (GLASSO) approach. Namely, we use the PC stable algorithm developed by Colombo and Maathuis (2012, 2014), which is a modification of the classic PC algorithm (Spirtes and Glymour 1991) for improved robustness and speed. [For its use in climate science, see also Ebert-Uphoff and Deng (2012a,b).] The algorithm starts with a fully connected graph, where each node is assumed to have a direct causal interaction with every other node. Then a procedure based on conditional independence tests identifies and eliminates indirect interactions to obtain a final set of direct causal interactions. We want to emphasize here that using this causal discovery approach, we can only detect potential causal interactions, since there can always be latent confounding variables, that is, hidden common causes that were not included in the model. Because of that possibility every relationship found can either be a true causal connection, due to a latent variable or both. All identified relationships must thus be treated as hypotheses of causal relationships, rather than taken as a fact. The only way to find out whether these causal hypotheses represent real causal relations is to check whether the identified cause–effect connections can be interpreted in terms of distinct dynamical and physical processes.
In this study, we use Fisher’s Z test for partial correlation with a significance level of α = 0.05 to determine statistically significant edges. Furthermore, to understand the directionality and time delay of the causal interactions among different nodes, a temporal model is developed using an approach first suggested by Chu et al. (2005) and applied in grid space in Ebert-Uphoff and Deng (2012a,b). Specifically, we consider S = 11 time slices with neighboring time slices being 2 days apart. This allows us to consider causal interactions occurring at a maximum time lag of 20 days. The temporal model thus consists of N × S = 570 × 11 = 6270 nodes. To find the direction of the edges (i.e., direction of the flow of information), we adopt the temporal constraint that nodes can only influence other nodes at the same or a later time. To handle the initialization problem of temporal Bayesian networks obtained from structure learning, we dropped the first time slice, leaving 10 of the original time slices for analysis. This process finally yields the graph of dependencies (block 6 in Fig. 1b). In this graph, we identify interactions that are consistently repeating over the temporal model. If an edge does not show a repetitive nature and pops up arbitrarily, it may indicate a false discovery, so is not shown.
The results presented in section 3 focus on the directed edges among the nodes with nonzero time lags. In other words, we are seeking causal interactions that occur over a finite time period among disturbances of different spatial scales. The initial effort is devoted to the connection between the Northern Hemisphere planetary-scale and synoptic-scale disturbances as many features of the interactions between the two have been well documented (see discussions in section 1). We define two regimes in the zonal wavenumber space representing, respectively, the planetary-scale (0 ≤ M ≤ 3) and synoptic-scale disturbances (6 ≤ M ≤ 10). The primary reason for excluding zonal wavenumbers 4 and 5 is that these two wavenumbers sit in between classic planetary scales and classic synoptic scales, with the former distinctly tied to large-scale orographic and thermal (e.g., land–sea contrast) forcing and the latter associated mainly with baroclinic instability. These two wavenumbers are often affected by disturbances related to low-frequency variability in the extratropics such as blocking and other “persistent anomalies,” which are in turn partly connected to synoptic-scale disturbances. Furthermore, the PC stable algorithm used in this study can only robustly detect linear causal effects due to the use of partial correlations in the conditional independence test. This limitation has been demonstrated by applying the algorithm to identify the known linear and nonlinear causal effects in the Lorenz model and the relevant results are summarized in the appendix. Given such limitations, we choose to focus the discussion on two groups of disturbances with distinct physics and sufficiently separated in scales (to create a “quasi-linear” condition) by excluding zonal wavenumbers 4 and 5 in all the figures even though these two components are included in the actual process of structure learning.
As our graphical model has separate nodes for the real and imaginary components of each spherical harmonic, we need a suitable mapping to the (M, L) space to be able to interpret these results. Based on our mapping, for the two defined regimes, when one or more of the four conditions listed below are met, we claim that a node in regime I influences (causes changes in) a node of regime II:
Regime I real-part node → Regime II real-part node
Regime I real-part node → Regime II imaginary-part node
Regime I imaginary-part node → Regime II real-part node
Regime I imaginary-part node → Regime II imaginary-part node
Directed edges between regime I and regime II with respect to the total wavenumber L and the zonal wavenumber M. (a) Regime I to regime II. (b) Regime II to regime I. Black dots denote the nodes used for the analysis. A blue circle around a node indicates that the node has more than two incoming or outgoing edges. The size of each blue circle is proportional to the number of edges associated with the node. The gray-shaded circles are proportional to the winter-mean magnitude of the coefficient of the corresponding spherical harmonics component. The number alongside each interacting node denotes the winter-mean magnitude of the spherical harmonics coefficient scaled by 10.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Directed edges between regime I and regime II with respect to the total wavenumber L and the zonal wavenumber M. (a) Regime I to regime II. (b) Regime II to regime I. Black dots denote the nodes used for the analysis. A blue circle around a node indicates that the node has more than two incoming or outgoing edges. The size of each blue circle is proportional to the number of edges associated with the node. The gray-shaded circles are proportional to the winter-mean magnitude of the coefficient of the corresponding spherical harmonics component. The number alongside each interacting node denotes the winter-mean magnitude of the spherical harmonics coefficient scaled by 10.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Directed edges between regime I and regime II with respect to the total wavenumber L and the zonal wavenumber M. (a) Regime I to regime II. (b) Regime II to regime I. Black dots denote the nodes used for the analysis. A blue circle around a node indicates that the node has more than two incoming or outgoing edges. The size of each blue circle is proportional to the number of edges associated with the node. The gray-shaded circles are proportional to the winter-mean magnitude of the coefficient of the corresponding spherical harmonics component. The number alongside each interacting node denotes the winter-mean magnitude of the spherical harmonics coefficient scaled by 10.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
3. Results
Figure 2 provides an overview of the interaction between planetary- and synoptic-scale disturbances in regime I and regime II, respectively. A directed edge (a red arrow in Fig. 2) indicates that changes in the disturbance with wavenumbers (M, L) at the beginning of the edge (arrow) tends to cause changes in the disturbance with wavenumbers at the end of the edge (arrow). In other words, the directed edge depicts the direction of the flow of information in the wavenumber space. The presence of a blue circle around a node means that there are more than 2 incoming (or outgoing) edges toward (or out) of this node, suggesting a relatively active role played by this node in the interaction between regime I and regime II disturbances. The size of the blue circle is proportional to the number of edges associated with the node. The size of the gray shaded circle around each node indicates the winter-mean magnitude of the corresponding spherical harmonics component. Based on these magnitudes, it is evident that the identified causal structure is not only driven by the magnitudes of the coefficients. Further, to reduce the impact of false discoveries, we do not show edges that only occur once in the temporal model. This information, viewed together with the directed edges of the node, provides a more concrete idea how physically important an identified connection is, as a connection of statistical significance does not always mean that the connection is “substantial” and the number of edges is not equivalent to the magnitude of the flow of information. Figure 2a shows all edges pointing from regime I to regime II nodes, representing the influence of planetary-scale disturbances on synoptic-scale disturbances. The total number of such edges is large (~80). However, the energy of disturbances with large meridional wavenumbers (L − M > 6) is rather small (Cai and Mak 1990) and this naturally places more weights on the edges identified at the lower-right corner of the (M, L) parameter space in Fig. 2b. Judged by the size of both the blue circle and gray shaded circle, synoptic-scale disturbances (6, 10), (6, 12), (6, 14), (7, 17), and (8, 16) are under the most obvious influence of planetary-scale disturbances with zonal wavenumbers ranging from 1 to 3. Figure 2b displays the edges characterizing the impact of synoptic-scale on planetary-scale disturbances (regime II to regime I). Disturbances (6, 10), (6, 14), (6, 16), (7, 11), (7, 13), and (8, 16) stand out as those providing most active feedback to planetary-scale disturbances. Disturbance (2, 4) has the largest mean magnitude among all planetary-scale disturbances influenced by feedbacks from synoptic-scale disturbances and this verifies the traditional picture that zonal wavenumber-2 planetary wave actively interacts with the two Northern Hemisphere storm tracks in boreal winter (e.g., Deng and Mak 2006; Mak and Deng 2006).
The reconstructed daily 500-hPa geopotential height field using only spectral components that are interacting according to Fig. 2 provides a direct view of the spatial structure of the actively interacting planetary-scale and synoptic-scale disturbances. Once the interacting spectral components are identified, the specific disturbance can be reconstructed as a linear combination of the spectral components (spherical harmonics bases). For a specific reconstruction, we use each identified spectral component only once, regardless of the number of interactions it has with other spectral components. Figure 3 displays the total (Fig. 3a) as well as the reconstructed geopotential height field for 16 February 2001. Figures 3b and 3c are, respectively, the regime I disturbance and the regime II disturbance that is being actively influenced by the regime I disturbance on 16 February 2001. In the extratropics, the planetary-scale flow is characterized by a roughly zonal wavenumber-1 structure with a major trough (negative geopotential height anomalies) extending from the east coast of Asia toward North America and a major ridge (positive geopotential height anomalies) over western Europe (Fig. 3b). This wavenumber-1 planetary-scale disturbance modulates the activity of synoptic-scale disturbances shown in Fig. 3c, which manifest themselves as a classic circumglobal wave train consisting of meridionally elongated disturbances. The amplitude of these shortwave disturbances peaks over the North Pacific and North Atlantic, indicating significant baroclinic growth and subsequent downstream development over these two ocean basins and thus the presence of the two major storm tracks (e.g., Chang et al. 2002). The regime I to regime II disturbance structure revealed here is consistent with the classic theory of storm-track dynamics where storm-track disturbances (cyclones/anticyclones) preferentially develop downstream of the planetary-scale troughs (and time-mean jets) as a result of locally enhanced baroclinicity at the location of the jets and farther downstream development associated with ageostrophic flux of geopotential (e.g., Cai and Mak 1990; Chang et al. 2002).
Reconstruction of the daily geopotential height (m) on 16 Feb 2001 using only the significantly interacting spectral components and excluding the interactions with M = 0. (a) The 500-hPa geopotential height field. (b),(c) Interacting disturbances from (b) regime I to (c) regime II. (d),(e) Interacting disturbances from (d) regime II to (e) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Reconstruction of the daily geopotential height (m) on 16 Feb 2001 using only the significantly interacting spectral components and excluding the interactions with M = 0. (a) The 500-hPa geopotential height field. (b),(c) Interacting disturbances from (b) regime I to (c) regime II. (d),(e) Interacting disturbances from (d) regime II to (e) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Reconstruction of the daily geopotential height (m) on 16 Feb 2001 using only the significantly interacting spectral components and excluding the interactions with M = 0. (a) The 500-hPa geopotential height field. (b),(c) Interacting disturbances from (b) regime I to (c) regime II. (d),(e) Interacting disturbances from (d) regime II to (e) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Figures 3d and 3e depict the influence of synoptic-scale disturbance on the planetary-scale disturbance. Specifically, the synoptic-scale disturbances in Fig. 3d are producing a zonal wavenumber-2 structure with negative height anomalies off the east coasts of Asia and North America and positive height anomalies over the eastern North Pacific and western Europe. This wavenumber-2 structure locally enhances the trough off the east coast of Asia and the ridge over western Europe as shown in Fig. 3b. Therefore, the Regime I disturbance is responsible for the excitation/propagation of the regime II disturbance whose dynamical feedback to the regime I disturbance locally reenforces the trough and ridge pattern in the original wavenumber-1 structure of the regime I disturbance. This result again verifies the notion that winter planetary-scale flow and the associated zonal wind jets tend to be “self-sustained.” The feedback of synoptic-scale to planetary-scale disturbances is achieved through the vorticity (momentum) and heat flux by synoptic-scale disturbances (e.g., Lau and Holopainen 1984). In Fig. 3d, synoptic-scale disturbances change the alignment of their major axis from being southwest–northeast (SW–NE) oriented over the central North Pacific to being northwest–southeast (NW–SE) oriented over western North America. The associated meridional flux of zonal momentum by these disturbances subsequently changes from a poleward to an equatorward direction. The poleward (equatorward) flux of zonal momentum is consistent with a poleward (equatorward) shift of an eddy-driven westerly jet and thus the formation of a negative (positive) height anomaly on the poleward side of the disturbance over the central North Pacific (western North America) following the argument of geostrophic balance (Fig. 3e). The same mechanism also applies to the North Atlantic and western Europe, responsible for the formation of the negative and positive height anomalies there, respectively. Comparing the reconstructed height fields with the total height field (Fig. 3a), we note that the planetary-scale disturbances depicted in Figs. 3b and 3e miss another major trough over central Eurasia. This is likely the result of excluding zonal wavenumbers 4 and 5 from the figures, for reasons described in section 2.
Figure 4 shows the winter climatological variance [root-mean-square (RMS)] of the daily geopotential height of the actively interacting disturbances identified in Fig. 2 and presented in Fig. 3. When the flow of information from regime I to regime II (i.e., planetary-scale disturbances influencing synoptic-scale disturbances) is being considered, the dominant day-to-day variability in the planetary-scale flow peaks at the trough and ridge locations of the zonal wavenumber-1 disturbance (as shown in Fig. 3b). The corresponding variability in synoptic-scale flow maximizes along a well-defined circumglobal band in the extratropics with elevated amplitudes over the North Pacific and North Atlantic, where the two climatological Northern Hemisphere storm tracks reside (Fig. 4b). In the opposite direction of information flow, when synoptic-scale disturbance affects planetary-scale disturbances, storm tracks again characterize the day-to-day variability in the synoptic-scale flow (Fig. 4c) while the variability in the planetary-scale disturbance reflects the zonal wavenumber-2 structure that is effectively forced by synoptic-scale disturbances and locally enhances the trough and ridge of the zonal wavenumber-1 structure of the winter planetary-scale flow shown in Fig. 3b (Fig. 4d).
Variance (RMS) of the geopotential height (m2) reconstructions of the significantly interacting disturbances, excluding the interactions with M = 0. (a) Regime I to (b) regime II. (c) Regime II to (d) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Variance (RMS) of the geopotential height (m2) reconstructions of the significantly interacting disturbances, excluding the interactions with M = 0. (a) Regime I to (b) regime II. (c) Regime II to (d) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Variance (RMS) of the geopotential height (m2) reconstructions of the significantly interacting disturbances, excluding the interactions with M = 0. (a) Regime I to (b) regime II. (c) Regime II to (d) regime I.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
To see whether there is any systematic change in the amplitude of the actively interacting planetary- and synoptic-scale disturbances, we plot in Fig. 5 the seasonal-mean amplitude of the disturbances as a function of time. The amplitudes of the daily reconstructed disturbances are spatially averaged over the sphere and temporally averaged over each winter season to create these time series. Blue and orange curves in Fig. 5a are, respectively, the mean amplitude of regime I and regime II disturbances when regime I is influencing regime II. No significant trends can be identified in these time series, but substantial variations occur across interannual to interdecadal time scales. Specifically, the mid-1970s to mid-1980s are characterized by above-normal amplitudes of disturbances when regime I influence on regime II is being considered. Similar results are found in the case of the feedback of a regime II disturbance to a regime I disturbance (Fig. 5b). There appear no obvious connections between the activity levels of these wave–wave interactions and increasing greenhouse gas (GHG) forcing, nor between the activity levels and primary modes of variability in the climate system, including the ENSO and the Pacific decadal oscillation (PDO). These results suggest that the interactions identified by our approach are likely features internal to the atmosphere and the low-frequency components of the time series shown in Fig. 5 are mainly the result of nonlinearity in the system.
Annual-mean magnitude (m) time series of interacting disturbances excluding M = 0 interactions. (a) Regime I to regime II interactions. (b) Regime II to regime I interactions. A spatially averaged magnitude over the sphere is temporally averaged for each winter season from 1948 to 2015. Regime I and regime II time series are in blue and orange, respectively. The blue and orange dashed lines indicate the corresponding linear trends.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Annual-mean magnitude (m) time series of interacting disturbances excluding M = 0 interactions. (a) Regime I to regime II interactions. (b) Regime II to regime I interactions. A spatially averaged magnitude over the sphere is temporally averaged for each winter season from 1948 to 2015. Regime I and regime II time series are in blue and orange, respectively. The blue and orange dashed lines indicate the corresponding linear trends.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Annual-mean magnitude (m) time series of interacting disturbances excluding M = 0 interactions. (a) Regime I to regime II interactions. (b) Regime II to regime I interactions. A spatially averaged magnitude over the sphere is temporally averaged for each winter season from 1948 to 2015. Regime I and regime II time series are in blue and orange, respectively. The blue and orange dashed lines indicate the corresponding linear trends.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
4. Concluding remarks
This paper reports some preliminary yet encouraging results concerning the use of constraint-based structure learning to understand scale-interaction processes in the atmosphere. The analysis focuses on identifying causal pathways among atmospheric disturbances of different spatial scales. Temporal probabilistic graphical models illustrating such causal pathways are built by applying the PC stable algorithm (Colombo and Maathuis 2012, 2014) to the spherical harmonics decomposition of the boreal winter daily 500-hPa geopotential height data during the period 1948–2015. With an initial focus on interplays between planetary-scale (regime I) and synoptic-scale (regime II) disturbances, the identified directed edges (information pathways) suggest active coupling between regime I disturbances with zonal wavenumber-1–3 structures and regime II disturbances with zonal wavenumber-6–8 structures. These scales fall nicely into the classic picture depicting interactions between Northern Hemisphere planetary waves and storm tracks in boreal winter.
Furthermore, daily reconstruction of geopotential heights using major nodes [in the (M, L) wavenumber space] connected by the detected causal pathways suggest that the modulation of synoptic-scale disturbances by planetary-scale disturbances in the northern extratropics is best characterized by the flow of information from a zonal wavenumber-1 disturbance to a synoptic-scale circumglobal wave train whose amplitude peaks at the North Pacific and North Atlantic storm-track region. The feedback of synoptic-scale disturbances to the planetary-scale disturbances manifest itself as a zonal wavenumber-2 structure driven by synoptic-eddy momentum fluxes that locally enhances the East Asian trough and western Europe ridge of the original wavenumber-1 structure in the planetary-scale disturbances that are actively modulating the activity of the synoptic-scale disturbances. The seasonal-mean amplitude of the significantly interacting disturbances detected here exhibits pronounced variations across interannual-to-decadal time scales that are not correlated with major low-frequency modes of variability such as the PDO in the climate system.
The analysis presented here provides a new way to examine the scale-interaction processes in the atmosphere in the context of causal discovery and structure learning for probabilistic graphical models. The PC stable algorithm can be applied to identify potential causal pathways among atmospheric disturbances of various spatial scales. The structure of actively interacting pairs of disturbances can be reconstructed from nodes in the graphical models that are connected by statistically significant edges, making it easier to describe the spatiotemporal characteristics of interacting scales in the atmosphere. New metrics based on the detected edges such as the mean amplitude of the interacting disturbances can be computed for model simulations and observations to add a new dimension to the validation of dynamical properties of simulated atmospheric variability. Despite all the benefits of the approach, the application of the PC stable algorithm to real atmospheric and climate data must be pursued on a cautious note due to the assumed linearity of interactions in the conditional independence test. The scale interaction in the real atmosphere is largely realized via nonlinear advection processes. Therefore, in our study the cause–effect relations (i.e., directed edges) among spherical harmonics components identified by the PC algorithm are most robust and physically relevant only when they are between spherical harmonics components that are sufficiently different in scale. The scale separation ensures one of the components remains relatively “temporally steady” and/or “spatially uniform” during the interaction and creates a “quasi-linear” condition for the algorithm to be applicable. The implication of this limitation is that edges identified between neighboring spherical harmonics components in our analysis are subject to much greater uncertainties. Our aggregation of results into different regimes helps to create this “quasi-linear” condition and excluding the scales at the regime I and regime II interface (i.e., zonal wavenumbers 4 and 5) from the discussion serves to further reduce uncertainty in the results presented. However, the difficulty in reliably detecting nonlinear causal relationships places a major constraint on this approach’s capability of gaining a significant amount of new insights into the cross-scale interactions in the real atmosphere. Other approaches designed to identify nonlinear causal effects are being developed [e.g., convergent cross mapping (Sugihara et al. 2012) and causal network learning algorithms that use nonlinear conditional independence tests (e.g., Runge et al. 2018)], but the associated computational complexity makes it hard to apply to high-dimensional problems such as the one studied here. Another challenge we face in this approach of causal discovery is constructing a quantitative measure of the discovered causality. Compared to the entropy-based information flow, information flow described here does not carry a magnitude associated with known physical quantities. This prevents us from making a direct comparison with the conclusion drawn in Liang (2019) emphasizing “bottom-up causation” for midlatitude atmosphere in a quasigeostrophic setting. Paluš (2014) uses conditional mutual information to obtain information flow and the results based on an application to daily surface air temperature data over Europe suggest transfer of information from larger to smaller time scales, different with the “bottom-up causation” found in Liang (2019). All of these highlight the significant uncertainties in causal discovery for atmospheric processes, especially when real observations are being analyzed.
Our ongoing work includes an extension of the analysis to consider the interaction between disturbances in the synoptic- and mesoscale regimes and also to consider the interactions in boreal summer. Finally, note that the advective nonlinearities dictate that scale interactions in a fluid such as the atmosphere occur in the form of wave triads, whose wavenumber vectors must sum to zero. If zonal and meridional scales are quantified in terms of Fourier coefficients and these coefficients serve as inputs for the PC algorithm, any two or three directed edges involving three nodes should have the three corresponding wavenumber vectors (zonal and meridional wavenumber as the two components) sum to zero. However, in spherical harmonics expansion, zonal wavenumbers are integers and meridional scales are quantified through the number of nodes between the poles. The triad interaction therefore does not have a straightforward reflection in the detected edges that span a spectral space defined by the integer zonal and total wavenumber. Future work will include an investigation of the connection between the constructed graphs and wave triads potentially utilizing a Fourier expansion of the daily height field.
Acknowledgments
We gratefully acknowledge Clark Glymour and Joseph Ramsey at Carnegie Mellon University for constructive discussions related to this project. We also thank the three anonymous reviewers for their thoughtful comments and suggestions that led to major improvement of the manuscript. The NCEP–NCAR reanalysis data used in this study were provided through the NOAA Climate Diagnostics Center. This research was supported by the NSF Climate and Large-Scale Dynamics (CLD) program under Grants AGS-1354402 and AGS-1445956 (Deng) and AGS-1445978 and HDR-1934688 (Ebert-Uphoff).
APPENDIX
The Lorenz System as a Test Case for the PC Stable Algorithm
The linear effects are as follows:
X(t − δt) → X(t)
Y(t − δt) → Y(t)
Z(t − δt) → Z(t)
X(t − δt) → Y(t)
Y(t − δt) → X(t)
The nonlinear effects are as follows:
Z(t − δt) → Y(t)
Y(t − δt) → Z(t)
X(t − δt) → Z(t)
We use a time step of δt = 0.01.
As shown in Fig. A1, these parameters give rise to the classic Lorenz attractor in the x–y–z space. Figure A2 shows the time series plots and histograms of X, Y, and Z.
A 3D plot of the generated data for ρ = 28, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
A 3D plot of the generated data for ρ = 28, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
A 3D plot of the generated data for ρ = 28, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
To create the temporal model, we use X(t), Y(t), Z(t), X(t − δt), Y(t − δt), and Z(t − δt) as the nodes of the graphical model. We then use Fisher’s Z test on partial correlation to determine conditional independencies. We repeat this process for different sample sizes (1000, 6000, 12 000), different levels of statistical significance (0.001, 0.01, 0.05) and different initial conditions {[x(0), y(0), z(0)] = [1, 1, 1], [6, −7, 3], [−15, −15, 3]}. The interactions identified remain unchanged for these different parameters. A summary of the interactions is provided in Fig. A3.
Summary of interactions identified by the PC stable algorithm. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions identified by the PC stable algorithm. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions identified by the PC stable algorithm. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
The results indicate that the algorithm identifies all the linear causal effects as the PC stable algorithm is supposed to do. None of the nonlinear causal effects are identified. This is consistent with the fact that the conditional independence test based on partial correlation assumes that the relationships between the variables are linear. It is also important to see that the PC stable algorithm does not mistakenly identify a Z → X edge, which is an indirect connection.
A 3D plot of the generated data for ρ = 10, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
A 3D plot of the generated data for ρ = 10, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
A 3D plot of the generated data for ρ = 10, σ = 10, and β = 8/3.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Following these steps, we generate 1000 samples with a time step of δt = 0.005. Figure A4 shows the trajectory of the solution while Fig. A5 shows the time series plots and histograms of X, Y, and Z.
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Time series and frequency distributions of X, Y, and Z.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Figures A6–A8 show the summary results based on several different levels of statistical significance (α value). As we increase the value of α, the model picks up more and more expected causal relationships and finally identifies correctly all edges (both linear and nonlinear) when α = 0.2 is used (80% level of statistical confidence). However, in contrast to the results of linear connections shown in Fig. A3, which are very robust, the results shown in Figs. A6–A8 are not robust. Namely, for different sample sizes and slightly different parameters, many of these edges appear or disappear.
Summary of interactions for α = 0.001 or 0.01. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.001 or 0.01. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.001 or 0.01. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.05 or 0.1. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.05 or 0.1. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.05 or 0.1. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.2. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.2. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
Summary of interactions for α = 0.2. Here we present 1δt as 1.
Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-18-0163.1
In summary, the test cases show that the PC stable algorithm using a conditional independence test based on partial correlation can robustly identify the linear causal relationships associated with the Lorenz system while excluding indirect connections such as Z → X. For a chaotic solution as in case 1, where the system spends an equal amount of time on each wing of the attractor, the changes in signs of the variables can cause some of the causal components to cancel out on average and become undetectable by an approach based on correlation/partial correlation that measure average linear dependence. The test case shows that when the samples are taken from a regime in the (x, y, z) space where x and y do not change sign (e.g., case 2), the nonlinear dependence that was canceled out in partial correlation calculations due to changing signs of x and y (e.g., case 1) can be identified (although not robustly) when a lower level of statistical confidence is used.
Even though a low-dimensional causal inference problem, the Lorenz system is considered a challenging problem because of the nonlinearity of the relationships as well as the cancellation of causal effects when the system is considered on average. There exist other causal inference methods that are more suited to identify causal effects of a nonlinear system such as the Lorenz system. Specifically, nonlinear state-space methods such as convergent cross mapping (Sugihara et al. 2012). However, these methods are less suited for time series of stochastic nature (Runge et al. 2019). Even though causal network learning algorithms that use nonparametric tests to assess conditional independencies will allow to identify both linear and nonlinear causal interactions, the associated computational complexity make these methods unsuitable for a high-dimensional problem as the one presented in the paper. Therefore, we resort to this simpler approach as a starting point.
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