## 1. Introduction

### a. Background

Depending on the body of scientific papers one reads and the emphasis of the techniques they employ, one can either conclude that tropospheric planetary wave behavior is largely understandable in terms of linear dynamics or that these waves are fundamentally nonlinear. Given this situation one might guess that both views have some merit and that it would be helpful to have a means of quantifying the relative contributions of the two types of dynamics to planetary wave conduct. With this in mind, we have carried out a data analysis study that is intended to identify in a unified fashion both linear and nonlinear signatures in the dynamics of tropospheric planetary waves and to measure the strength of each.

We have undertaken this study not simply to help settle an academic question but because the outcome has a bearing on the nature and predictability of tropospheric states. For if the planetary waves are primarily linear, then one would expect that there would be a few preferred patterns that dominate planetary wave statistics and any combination of these patterns would also be a preferred structure. On the other hand, if nonlinearities are important, planetary wave statistics will be controlled by special long-lived or recurring states with specific amplitudes, and the likelihood of any combination of such states will be in general less than the likelihood of either of its components. This contrast will in turn affect problems ranging from seasonal prediction to how the climate reacts to secular changes in forcing.

Of course, simply from an inspection of the governing equations, there is no doubt that nonlinearities play a role in tropospheric dynamics but, given the successful application of linear dynamics to many aspects of large-scale planetary wave behavior, it is an open question as to how crucial nonlinearities are to an understanding of those features that describe much of tropospheric variability. After all, characteristics of planetary wave behavior ranging from their propagation speeds (Rossby 1939; Madden 1979; Salby 1984) to their role in transmitting the effect of El Niño to midlatitudes (Hoskins and Karoly 1981) to their organization into localized teleconnection patterns (Simmons et al. 1983) have largely been explained in linear terms.

Linear planetary wave behavior is typically associated with the dynamics that results from a linearization of the governing equations about a time mean state and a spatial or temporal truncation of perturbation quantities. In such a framework the truncated scales and nonlinearities are either explicitly or implicitly assumed to be modeled by simple white noise and augmented damping (Leith 1971; DelSole and Farrell 1995). A more encompassing definition of linearity results if one also includes nonlinear processes that can be approximated in a linear fashion. For example, momentum fluxes by synoptic eddies have been shown to be linearly slaved to large-scale circulation anomalies (Branstator and Haupt 1998) and so are linearly representable, though not as a simple linearization about a mean state. In this alternative means of categorizing the linear behavior of a system linearity is defined as the part of the dynamics that can be reproduced by additive noise and a linear operator that can capture as much of the observed behavior as possible. In our study, we use this definition and consider that part of observed behavior contained in the stochastically driven, regression models of Hasselmann (1988), Penland (1989), and von Storch et al. (1995) as being the linear component of a system.

Coincident with the apparently successful development of linear theory there have been numerous studies that have suggested that nonlinearities must be taken into account to understand the planetary waves. Many of these have been motivated by the highly truncated planetary wave theory of Charney and DeVore (1979) and Wiin-Nielsen (1979) or by the behavior of low-order dynamical systems that are seen as metaphors for planetary wave dynamics (Palmer 1999). The most common evidence of planetary wave features that are indicative of nonlinear influences is multiple modes (or at least multiple non-Gaussian features) in the distribution of prominent pattern amplitudes (Kimoto and Ghil 1993a; Cheng and Wallace 1993; Corti et al. 1999). A linear model driven by additive Gaussian white noise produces Gaussian distributions of states, so this approach is consistent with our definition of linear behavior, and in the sequel to this paper (Berner and Branstator 2005, manuscript submitted to *J. Atmos. Sci.*, hereafter BB2) we report on a quantification of non-Gaussianity in planetary wave states.

Recently, building on heteroclinic orbit and Markov chain ideas of Legras and Ghil (1985), Mo and Ghil (1987), and Kimoto and Ghil (1993b), studies by Hannachi (1997a, b), Itoh and Kimoto (1999), Berner (1999, 2003), and Crommelin (2003) have presented an alternative approach for detecting nonlinear influences. They show that a richness of nonlinear behavior can be detected in realistic, high-dimensional systems if one considers the structure of their trajectories in a low-dimensional subspace. In our study, we too use trajectory information to identify and quantify signatures of nonlinearity. In particular we look for organization in short trajectory segments and designate as nonlinear that component of the organized movement that cannot be explained by linear stochastic dynamics.

### b. Example

To be more specific about the approach that we intend to use, we describe an example of applying the technique to data from nature. Details of the method and its derivation are given in section 3.

Consider the sequences of 24-hourly 500-hPa geopotential height charts *ϕ*(*t*) that have been observed in nature during northern winter for the 100 days following 1 December for each of the 40 winters beginning in 1958/59. Suppose the large-scale components of *ϕ*(*t*), denoted *ϕ̂*(*t*), are represented in terms of a basis consisting of the leading *N* empirical orthogonal functions (EOFs) of *ϕ*(*t*) so that *ϕ̂*(*t*) = [*ϕ*_{1}(*t*), *ϕ*_{2}(*t*), · · · , *ϕ*_{N}(*t*)]. For this example, fields from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis project (Kalnay et al. 1996) are used and their seasonal cycle has been removed.^{1} The leading two EOFs for these fields are structurally similar to the EOFs plotted and used in Kimoto and Ghil (1993a), Cheng and Wallace (1993), and Corti et al. (1999). Examining the plot of a segment of the time series of [*ϕ*_{1}(*t*), *ϕ*_{2}(*t*)] in Fig. 1a, we are hard-pressed to notice much organization to the sequence of states, yet our goals are 1) to determine whether this behavior is effectively that of a stochastically forced linear system and, 2) if not, to characterize the departures from such behavior.

*ϕ̂*(

*t*) are produced by a linear stochastic system driven by Gaussian white noise. Under this assumption, as Hasselmann (1988) and Penland (1989) have pointed out, the linear propagator 𝗟 for that system can be recovered from the expressionwhere 𝗖(

*μ, ν*) is the covariance of

*μ*and

*ν*. Next we generate an artificial sequence of states,

*ϕ̃*(

*t*), fromwhere Δ

*t*is the observation interval and

*ϵ*is centered, additive white Gaussian noise. The noise is chosen in such a way that, if the

*ϕ̂*(

*t*) really do act as if they were generated by a system of the form of (2), then the

*ϕ̃*(

*t*) will be statistically indistinguishable from

*ϕ̂*(

*t*). A segment from such an artificial sequence is shown in Fig. 1b.

*ϕ̃*

_{1}(

*t*),

*ϕ̃*

_{2}(

*t*)] that reside in each, then we produce the mean tendencies shown in Fig. 1d. These correspond to the same damped linear oscillation inherent in the projected dynamics of 𝗟, a result explained in section 3 and appendix A. When we repeat the calculation of local mean tendencies but use observations from nature we find marked departures from the simple linear signature of Fig. 1d. To highlight these departures we plot in Fig. 1c the difference between the two. Because of these large differences we conclude that planetary wave behavior cannot be completely described by linear, additive white noise, stochastic processes. Furthermore, because in a least squares sense 𝗟 from (1) is the most complete linear approximation to the dynamics that produces the observed tendencies, we take the tendencies in Fig. 1d to be our measure of the linear contribution to large-scale planetary wave behavior in nature and Fig. 1c to be our indication of the nonlinear contribution. In this way we address goal 2.

Though the results of Fig. 1 appear to indicate both substantial linear and nonlinear contributions to the 24-h trajectories of large-scale planetary waves, extensive additional work is needed to be certain of this conclusion. When derived from (1), the propagator 𝗟, or its associated tendency operator, is sometimes referred to as a linear inverse model (LIM). As Penland and Sardeshmukh (1995) point out in their LIM study of the coupled ocean–atmosphere system, model (1)–(2) dynamics might be incomplete if its state vector excludes key system variables. For example, in our case this could happen if interactions with the ocean make important contributions to planetary wave evolution. Perhaps even more troubling is that we have found that the data record from nature is too short for reliable estimates of local mean tendencies using simple methods. The problem is analogous to the problem of estimating probability density functions in more than one dimension, which Hsu and Zwiers (2001) and BB2 have reported is also a victim of sampling errors when attempted with data from nature.

To minimize these problems, we have decided to apply the approach outlined above to very long integrations of an atmospheric general circulation model (AGCM) with fixed boundary conditions. It turns out that statistically robust signatures of both linear and nonlinear planetary wave dynamics, rather like those in our example, are present in this dataset. The remainder of our paper describes these signatures, the circulation states that are instrumental in producing them, and properties of the method we use for identifying and measuring them.

## 2. Data

Data for our investigation come from integrations of the AGCM known as Community Climate Model version 0 (CCM0), which was developed at the National Center for Atmospheric Research. The formulation of this nine-level, rhomboidal-15 truncation model is described by Williamson (1983), and its climate is outlined by Williamson and Williamson (1983). Though no longer considered to be a state-of-the-art AGCM, as reported on by Branstator (1990, 1992), the variability of its planetary waves has key similarities to observations, including the scale, structure, and maintenance of the dominant patterns. Moreover, the computational efficiency of the model makes it possible to generate a dataset for analysis that is long enough to produce robust results while being more realistic than the highly simplified models often used in studies of planetary wave nonlinearities.

To provide the lengthy records of system behavior that our study needs, we use 280 integrations of CCM0. Each integration starts from a different initial condition while all integrations use the same perpetual January boundary conditions, including a fixed distribution of sea surface temperatures. Each integration produces 50 100 simulated days, and pairs of consecutive time step model states (which are separated by 1/2 h) are stored every 12 simulated hours. After expunging the influence of initial conditions by deleting the initial 100 days of each integration, we are left with 14 million simulated days. All of the results described in this paper are derived from 7 million of these days. Except when 1/2-h information is needed, all calculations use states sampled every 12 h. All of our reported calculations have been repeated using the second 7 million days of the dataset, and the features described in this paper are nearly unchanged. Thus we have decided that there is no need to perform formal statistical significance tests of our results.

To achieve robust results and to satisfy requirements for the stochastic models that we construct it is necessary to represent the states of the AGCM in a reduced phase space. To arrive at a reduced system we represent the AGCM states in terms of 500-hPa geopotential height, and to further reduce the dimensionality we project the departures of these fields from a long-term mean onto their leading EOFs. Thus we concentrate on the largest scale, most frequently occurring, and highest amplitude structures. As has been pointed out by those whose aim is to approximate the dynamical behavior of a system with as few degrees of freedom as possible (e.g., Kwasniok 1996; Achatz and Schmitz 1997; Crommelin and Majda 2004), the patterns that explain the most variance are not necessarily the most crucial for capturing important dynamical behavior. So by studying the leading EOFs we are not necessarily including all of the important dynamics. But we choose to study them because they are the patterns in which most observational studies have been interested. Furthermore, since they have the largest amplitudes, everything else being equal, they are the most likely candidates for displaying any nonlinear behavior that may be present.

Of the leading EOFs it is projections onto the first four that we concentrate on in our investigation. As shown in Fig. 2, these are all large-scale patterns with the first resembling the “Northern Annular mode” (Wallace and Thompson 2002) and the second having similarities to the Pacific North American pattern (Wallace and Gutzler 1981). These EOFs are based on 12-hourly data and together represent 21% of AGCM variance (and 38% of monthly variance). For display purposes we often find it advantageous to plot projections onto an EOF in terms of standardized values (i.e., the projections are divided by their standard deviation), but any calculations involving projections are done in terms of raw values, commonly referred to as principal components (PCs).

## 3. Methodology

The description in the introduction of the approach that we use for analyzing planetary wave dynamics omitted various details concerning the stochastic model. Here we fill in this omission. (Readers who are not interested in these details may wish to skip this section.)

The key to the approach is to compare the actual behavior of large-scale planetary waves in a system **Σ** (nature or CCM0) to the behavior of planetary waves in a model whose operator is based on the assumption that planetary wave variability in **Σ** is the result of a stochastically driven linear process. As explained below, various properties of this operator make it possible to find those characteristics of this stochastic system required for our study without explicitly knowing its dimensionality or the structure of the forcing that drives it and without actually integrating it in time.

As explained in the introduction, the large-scale planetary waves in **Σ** are assumed to behave like the resolved components of a system of the form (2), which is a first-order, linear Markov model. From studies of Markov models (e.g., Chandrasekhar 1943), this can only be true if there is a sufficient time-scale separation between the components of **Σ** that are resolved in (2) and the components that are treated as noise. The EOF basis that we employ is a way to automatically impose this separation. There tends to be an inverse relationship between 500-hPa height EOF index and the time scale of the associated PC (Branstator et al. 1993). So, if we assign all EOFs with index greater than or equal to some value *N* to be resolved components and all others to be represented by noise, we will approximately achieve the required time-scale separation. For instance, in the example of the sample trajectory given in Fig. 1b, the first ten EOFs formed the basis of the stochastic model. These explain 54% of half-daily 500-hPa height variance, and, according to their lag-correlations, they typically have time scales at least twice as long as most of the remaining EOFs.

*N*, the stochastic model’s propagator is easily calculated from (1), so the only ingredient missing for examining planetary wave behavior under the assumption of stochastic linearity is the structure of the noise. We have already stated that we assume the noise is white, additive, and Gaussian. These are all reasonable attributes for the noise in our problem. As just explained, our choice of basis means the unresolved components have relatively short time scale and thus effectively no memory; multiplicative noise would be an indication of nonlinear interactions between the resolved and unresolved components; as the average effect of short time-scale phenomena, the forcing of resolved long time-scale components by unresolved nonlinearities should tend to be Gaussian. As far as the horizontal structure of the noise is concerned, we use noise with covariance structure 𝗤 consistent with the assumption that the large-scale planetary waves in

**Σ**act as if produced by (2) (Penland 1989):withThis choice has the advantage of guaranteeing that (2) will produce states whose lag-0 covariances match those of

**Σ**.

^{2}

The mean local tendencies of this stochastic linear system, which are needed for comparison to mean local tendencies of **Σ**, can be found by subdomain conditional averages, as in our introductory example. Basic texts (e.g., Brockwell and Davis 1987) show that solutions of (2) with additive Gaussian white noise forcing are stationary, provided 𝗟 is stable. By construction, our 𝗟 is stable (Penland 1989), so considering such averages makes sense. But inspection of (2) and (3) makes it apparent that, because the noise has zero mean, the outcome of this averaging process is equivalent to evaluating (𝗟 − 𝗜)/Δ*t*, where 𝗜 is the identity, at the mean position of states in each subdomain. In fact, to find the average tendency in a neighborhood of any phase space position, it is sufficient to evaluate (𝗟 − 𝗜)/Δ*t* at that position. So we do not actually integrate (2). Note that because it is unnecessary to integrate (2) to arrive at the linear component of local mean tendencies, it is also unnecessary to explicitly calculate 𝗤.

The local mean tendencies of **Σ** with which the linear behavior is compared must be arrived at by actually conditionally averaging trajectories of **Σ**. In addition to accumulating tendencies in a regular network of subdomains, we also find the mean value of states in each subdomain. Then the linear tendencies are evaluated for those same mean state values, so comparison of linear dynamics and the dynamics of **Σ** can be made at the identical phase space locations. We could carry out this comparison as a function of position in the *N*-dimensional space in which the stochastic model is formulated. We find, however, we have insufficient data to calculate mean tendencies for **Σ** for values of *N* greater than about four provided the mesh of the subdomains is fine enough to resolve features of interest. Therefore it is necessary to make comparisons in a subspace with dimension lower than *N*. For example, the results for the section 1 example were given in a two-dimensional subspace defined by EOF1 and EOF2.

Evaluating mean total tendencies in a subspace presents no problems, though, because it involves averaging of states with very different planetary structure, it may mask interesting behavior [e.g., Selten and Branstator (2004) give an example of this]. Furthermore, as explained in appendix A, the full dimensional propagator 𝗟 need not be calculated in order to evaluate linear mean tendencies in the reduced space. Rather, it is sufficient to just consider tendencies produced by 𝗟 projected in the subspace. This in turn means that the dimension *N* of the underlying stochastic system need never be chosen.

## 4. Mean tendencies

In applying our methodology to our AGCM dataset we first choose Δ*t* = 24 h in (3) to calculate tendencies. We reason that this will be sufficient to resolve the evolution of large-scale patterns like those in the four EOFs (Fig. 2) that we plan to study. We also decide to consider tendencies in planes since these are easiest to depict. Through trial and error we find that the features of the mean tendency fields can be captured if we divide each plane into boxes whose sides are 1/3 standard deviation long. And we find that, if we restrict ourselves to only considering boxes that have at least 1000 contributions from 12-hourly observations, we need only accumulate and display mean tendencies in the 18 × 18 boxes whose centers have PCs in the range −3 to +3 standard deviations.

When we find the linear propagators from (1) for each of the six planes defined by pairs of our four EOFs [or equivalently (appendix A) one propagator for the four-dimensional system] and evaluate them at box centers to produce 24-h tendencies, we find rather different behavior depending on which plane we examine. Figure 3 shows examples for four planes chosen for display because they turn out to include good examples of each of the types of behavior we find during our investigation. Because 𝗟 must be stable, the results correspond to motion that is either pure exponential damping (planes 1–2, 1–3, 1–4, and 3–4) or a damped oscillation about the origin (planes 2–3, 2–4). Treating the planes individually, we use eigenanalysis to characterize the time scales of the motion and find *e*-damping rates ranging from 1/(11 d) to 1/(36 d), depending on the plane and direction, and periods of 119 and 123 days. These are denoted in Fig. 3.

Next, we find mean tendencies for AGCM states in the same 18 × 18 boxes in the same planes. Comparing these with the linear results we find that particularly in planes 2–3, 2–4, and 3–4 the linear component is a reasonable approximation to most of the features in the total tendencies. But, in the other three planes some features depart substantially from what can be produced by our stochastic linear model. This contrasting behavior is easily discerned in the Fig. 4 plots of total mean tendencies. One way to recognize the nonlinear influence on these fields is to recall that linear dynamics must be antisymmetric for reflections through the origin. But in both planes 1–3 and 1–4 tendencies along many lines through the origin show marked departures from antisymmetry. Another easy way to notice departures from linearity is to compare the mean tendency speeds in Figs. 3 and 4, which are denoted by shading. Under the linear assumption, locations with the same speed necessarily lie along ellipses, but in planes 1–2, 1–3, and 1–4 the boundaries between shaded regions are far from elliptical. Note how the linear regression model is forced into compromises that do not fit the observed mean tendencies.

To highlight the departures from linearity we subtract the linear tendencies from the total tendencies producing plots (Fig. 5) of what we designate as the nonlinear component of the dynamics. In the 1–3 and 1–4 planes the prominence of the nonlinearities in terms of tendency directions is striking with rotation about two separate locations being evident in both planes. When fine contours of speed are examined (not shown), it is seen that mean tendency speeds for each of these features is essentially zero at its center and increases radially outward. The patterns of nonlinear tendencies in the other planes are more complicated and in terms of speed they are weaker, but the patterns are reproducible when corresponding plots for the second half of our dataset are calculated.

v

_{ij}is the mean tendency vector in box (

*i*,

*j*), subscripts “tot,” “lin,” and “nl” stand for the total tendency (as in Fig. 4), the linear component of the tendency (as in Fig. 3), and the nonlinear component of the tendency (as in Fig. 5), respectively, and where the sums are taken over boxes meeting our sampling requirement. Here

## 5. Time-interval dependence

Further calculations indicate that the relative strength of the linear and nonlinear components derived in section 4 are sensitive to what one would have hoped to be an unimportant parameter in the calculation, namely Δ*t*. This sensitivity is evident in Fig. 6, which shows total mean tendencies for the 1–4 plane for three different values of Δ*t*, namely 1/2 h, 12 h, and 96 h. (The 24-h tendency plot of Fig. 4c constitutes a fourth member of this sequence.) These examples, as well as similar plots for every plane that we have considered, indicate that the larger Δ*t* is, the more directed toward the origin are local mean tendencies. Additional calculations show that the inward-directed components are well approximated by linear functions of phase space position. This means that in each plane the linear component of our decomposition is systematically more prominent, the higher the value of Δ*t* used.

Figure 7 gives some insight into the behavior that this time-interval dependence represents. The first panel of that figure shows the equilibrium distribution of states in the 1–4 plane as given by a count of states in each of the 18 × 18 boxes used earlier and expressed as a density. If we sample the states in the heavily outlined box and plot their density 24 h after they were observed there, the heavy contours of Fig. 7b result. Even though we are dealing with states from a deterministic system, they spread in all directions through phase space because each state has rather different values in directions orthogonal to the plane. Similar spreading is seen after 24 h for states in the lightly outlined box of Fig. 7a. Their density is plotted in Fig. 7b with light contours. Following these states to their destinations 10 days after being observed in the boxes of Fig. 7a, we find that they have the distributions shown in Fig. 7c. They have begun to spread across much of the domain. This figure also displays the positions of the centroids of the distributions every two days as the clouds of states evolve. For both clouds the centroid comes closer and closer to the origin as time passes. Assuming the AGCM is ergodic this is exactly what one would expect because any subpopulation of states should eventually be indistinguishable from the general population; in particular the mean of a subpopulation will asymptote to the general population mean. Figure 7d, which shows the two distributions after 30 days, indicates this is what is happening in that by this time both clouds have nearly the same distributions as the entire population.

Though the influence of ergodicity provides an explanation of why local mean tendencies have components directed toward the climatological state increase with increasing Δ*t*, the state dependence of this component is not given by this consideration. One piece of information about its structure, however, can be found by comparing the nonlinear components of mean tendencies in the 1–4 plane given by different values of Δ*t*. It turns out that in this plane the linear component of the 1/2 h tendencies is essentially zero so that Fig. 6a describes not only the total mean tendency field but also the nonlinear tendency field. But, comparing this figure with Fig. 5c, we see that the two are very similar. Examination of other planes and other values of Δ*t* shows that this is a general property; provided Δ*t* is small enough, the Δ*t* dependence only affects the linear component. Furthermore, we find it is only the radial part of the linear component that depends strongly on Δ*t*; oscillations like those in planes 2–3 and 2–4 are independent of this parameter if it is small.

*t*-dependent component of our mean tendency fields match the lag-dependence of linear Markov models that DelSole (2000) has pointed out will result when they are fit to finite-dimensional dynamical systems. He argues that on short time scales covariances in any linear Markov model cannot have the same lag-dependent structure as the covariances of a finite-dimensional dynamical system, basically because on short enough time scales processes represented by white noise in a Markov model have memory in a dynamical system. So, though (2) can be trained to match the lag-0 and lag-Δ

*t*covariances of CCM0, its covariance structure at other lags will not match. So, if (2) is trained to match CCM0 covariances at lag-0 and lag-Δ

*t*′, Δ

*t*′ different from Δ

*t*, then its propagator will change. Thus when the 𝗟 in (2) is derived from (1), it should more correctly be denoted 𝗟(Δ

*t*). Moreover, DelSole points out that for sufficiently small Δ

*t*, the Δ

*t*dependence of lag covariances for finite-dimensional dynamical systems has a specific functional form, namelywhere 𝗞 is a skew symmetric matrix and 𝗦 is a symmetric matrix. Hence the linear component of our mean tendency distributions isConsidering that the decomposition of a real matrix into symmetric and skew symmetric matrices is unique, (7) says that the skew symmetric component of [𝗟(Δ

*t*) − 𝗜]/Δ

*t*is not affected by the choice of Δ

*t*while the symmetric component is proportional to Δ

*t*. When we investigate the linear components of our mean tendencies from the perspective of (7), we find that it does well in approximating the Δ

*t*dependence.

^{3}A skew matrix produces a pure oscillation, which is indeed the characteristic of linear tendencies that we find is not affected by Δ

*t*. A symmetric matrix produces a pure damped motion. This is the component that we find to depend on Δ

*t*and further calculations show the dependence to be linear in Δ

*t*.

Equation (7) suggests an alternative linear/nonlinear decomposition that does not suffer from sensitivity to Δ*t*. One can reason that, since the symmetric component of linear mean tendencies is not uniquely defined from data, one should only use the skew symmetric component when comparing the contributions of linearity with nonlinearity. As expected, when we carry out calculations in which the symmetric component of [𝗟(Δ*t*) − 𝗜]/Δ*t* is removed from the linear and total mean tendencies (or equivalently when very small Δ*t* is used), then these fields are no longer dependent on Δ*t*, as long as Δ*t* is less than about 2 days. Furthermore, because the skew component of linear mean tendencies (Fig. 8) is much weaker than the total mean linear 24-h tendencies (Fig. 3) in most planes,

## 6. Leading circulation structures

Having found both prominent linear and nonlinear signatures, it is of interest to identify the circulation patterns associated with these. To isolate the linear signature, we employ principal oscillation pattern (POP) analysis (Hasselmann 1988; von Storch et al. 1995) to find the structures associated with the dominant linear behavior that we have seen in planes 2–3 and 2–4. [Recall that POPs are simply the eigenmodes of the tendency operator 𝗔 associated with the lag-regression propagator 𝗟 defined in Eq. (1).] To allow for the possibility that the prominent oscillatory signatures that we have seen in these planes are two manifestations of a single multidimensional oscillation, we calculate the POPs of 24-h 500-hPa heights using a four-dimensional operator. When we do this, we find that there is only a single oscillating mode. This mode is displayed in Fig. 9 in terms of its structure at two phases separated by 90 degrees. These structures have substantial projections onto EOF2, EOF3, and EOF4, but not EOF1, and represent a progression of states that are very reminiscent of the prominent westward propagating feature that Branstator (1987) and Kushnir (1987) found in nature. When we look at the time evolution of these projections as a function of POP phase, we find that in planes 2–3 and 2–4 the sense of rotation matches the mean tendency oscillations in planes 2–3 and 2–4 (e.g., Figure 8d). This similarity supports the notion that this eigenmode is the four-dimensional counterpart to these mean planar oscillations.

Assigning a time scale to this POP based on its eigenfrequency is problematic given that we find this frequency to be sensitive to the truncation used in the analysis. (This sensitivity can be anticipated by considering the opposing trajectories that can occur when projecting a high-dimensional oscillation onto a subspace that is not orthogonal to the direction of the rotation axis.) On the other hand, via cross-spectral analysis we are able to arrive at a period that is not sensitive to truncation. Figure 10 shows the results of a cross-spectral analysis of projections onto the two phases of the four-dimensional POP of Fig. 9. The coherency in that diagram has a maximum at roughly 25 days, a period for which variations in the two patterns are 90 degrees out of phase. Similar results are found for POPs when the calculation is repeated with truncations between 2 and 20. Interestingly, Branstator (1987) found a similar period for the apparent counterpart to this mode in nature.

Given the success of associating the planar linear oscillations with a single higher dimensional oscillation, when finding circulation states associated with the distinctive nonlinearities in planes 1–3 and 1–4, we would like to allow for the possibility that the signatures in both planes have a common, higher dimensional origin. Identifying four-dimensional nonlinear mean tendency features by eye is not practical, so we need an alternative means of dealing with these fields. For guidance in this endeavor we first do a three-dimensional analysis.

Calculating the nonlinear component in three dimensions is a straightforward generalization of the procedure used in planes, with the domain being divided into cubes, conditional mean 24-h tendencies being calculated for states that reside in each cube, and departures from the tendencies implied by a three-dimensional LIM being derived. As a way to visualize the resulting three-dimensional field of vectors, we treat them as defining the dynamics of a system, insert particles into the system, and use a simple integration technique to follow how the particles evolve under the influence of the mean nonlinear tendencies. An example of the trajectories produced by following several states inserted into the mean nonlinear tendencies for the 1–2–3 subspace is shown in Fig. 11a.^{4} We see that the tendencies are organized into two features corresponding to rotations about two well-separated axes. We find pairs of oppositely rotating features in other cubes as well. For example, Fig. 11b shows several trajectories in the 1–3–4 subspace.

Next, to extend our analysis to four dimensions, we first generalize our mean tendency decomposition to the 1–2–3–4 subspace. This is done in the obvious way using a four-dimensional LIM and mean four-dimensional 24-h tendencies associated with a network of hypercubes. When we apply (6) to this subspace, with the sum in that expression being taken over the hypercubes, we find

To associate these four-dimensional nonlinearities with specific states, we note that the pairs of rotating features that we saw in most three-dimensional subspaces (Fig. 11) are consistent with there being two regions of contrasting, but locally quasi-linear, dynamics in some higher dimensional representation of the system. This being the case, the natural choices for circulation structures of interest are the dynamical centers of these two regions, that is, the points where the locally linear dynamics is at rest. As described in appendix B, we have estimated these two centers of rotation in the four-dimensional field of mean tendencies by locally fitting linear models to the nonlinear mean tendencies in various subregions of the domain and clustering the centers of rotation of these models into two groups. When we do this, the centroids of the two clusters, plotted as large dots in Fig. 5, match up well with the positions of the double swirls in planes 1–3 and 1–4. This is consistent with these planar features corresponding to a single pair of higher-dimensional nonlinear features. Interestingly, in the 1–2 plane the centers of the four-dimensional nonlinear rotations also correspond to key nonlinear features, namely the foci of local divergence and convergence.

Constructing the circulation anomalies for the two centers of nonlinear rotation and displaying them in Figs. 12a,b, we find that they each correspond to a wave train whose central lobe is over Alaska. At 150 m, the maximum value of each of these anomalies is substantial enough that, when each is added to the climatological mean state, the resulting total fields are markedly different from each other (Figs. 12c,d). These differences are especially pronounced over Alaska where the state in Fig. 12c (which corresponds to the negative PC1 lobe in Figs. 5b, 5c and 11) is essentially zonal, while the state in Fig. 12d (which corresponds to the positive PC1 lobe) is blocked. Other features of the circulation are also different in the two states in such a way that, throughout the mid- and high-latitude Pacific, North America, and the Atlantic, climatological ridges and troughs are weakened in one state and enhanced in the other.

## 7. Summary and discussion

In an effort to determine whether linear, stochastic dynamics is sufficient to describe the behavior of large-scale planetary waves or whether additional processes, which we have termed “nonlinearities,” must also be invoked, we have examined the state dependence of the mean local tendencies of these waves in extended integrations of an AGCM. The large sample generated by the AGCM has freed us from the necessity of only looking at the pattern dependence of planetary wave dynamics that the short record from nature imposes on those who have examined it. We have found that, though there are strong linear signatures, there are also distinct signatures that cannot be explained in terms of linear dynamics driven by additive white Gaussian noise. This is true even though we have been conservative in what we have termed nonlinear in that processes of nonlinear origin, but which can be approximated by a linear operator, have been termed linear.

Our analysis has primarily been done in terms of projections onto planes defined by the leading EOFs of 500-hPa heights, and one of the most distinctive results is that the signatures vary a great deal from plane to plane. We see this as a consequence of the implicit filtering that occurs when viewing the hyperspace of planetary wave motions from different directions. For example, viewed from a direction orthogonal to the axis of a prominent oscillation, that oscillation will not be visible in plots of mean tendencies and other, possibly nonlinear, behavior will be evident. On the other hand, if one views the space from a direction that is parallel to an oscillation’s axis, the oscillation will become apparent.

Interestingly, it turned out that the linear and nonlinear signatures that we found were associated with circulation patterns that are well known from previous investigations. The dominant linear signature resulted from a westward propagating pattern in northern high latitudes with a structure and period very similar to the westward propagating mode that Branstator (1987) and Kushnir (1987) found to be prominent in nature. The nonlinear signature was associated with trajectories in the vicinity of a blocked Pacific state and a zonal Pacific state, much like the states that Charney and De Vore (1979) first proposed as equilibria for the atmosphere. A number of investigations (e.g., Kimoto and Ghil 1993a; Cheng and Wallace 1993; Corti et al. 1999; Hannachi and O’Neill 2001; Monahan et al. 2001) have reported an unusually high density of states near these flow configurations in nature. From the histogram of Fig. 7a, it can be seen that there are regions of our phase space where states tend to congregate. In BB2, it is shown that these, too, correspond to Pacific blocked and zonal circulations. This may be somewhat surprising considering that in our analysis these states correspond to small nonlinear tendencies, not small total tendencies, but from Fig. 4 it can be seen that these states do serve to define the boundaries of an elongated region of small total tendencies.

While summarizing these findings we must point out two consequences of studying projected states. First, as reported in section 5, mean tendency results depend on the time interval used to calculate tendencies. As we have explained, this dependence results from the spreading of neighboring states in an ergodic system that occurs because the states can differ from each other in the unresolved directions. This spreading is a function of lag and preferentially affects the linear component of the mean motion, so our measures of the relative importance of linearity and nonlinearity are not uniquely determined. For example, for 24-h tendencies we found that in planes where nonlinear signatures were strongest, the nonlinearities accounted for at most 40% of the state dependence of the mean tendencies. By contrast, when 1/2-h tendencies were used, in several planes the nonlinear component comprised over 95% of the mean tendencies. We did find that the results of this decomposition became much less dependent on time interval if we removed the symmetric component of linear tendencies, which, as follows from (7), can be automatically accomplished by estimating tendencies from centered rather than forward differences. This is not to suggest that the time-interval-dependent contributions are unphysical, only that they complicate our approach for quantifying the ratio of linearity to nonlinearity. As DelSole (2000) has remarked, linear models that incorporate red noise potentially have covariance properties more akin to a dynamical system than the white-noise-based model that we have used. Employing such a model in our approach might diminish the lag-dependence that we have had to contend with.

The second consequence of investigating projections is that their mean tendencies may not reflect the predominant evolution of individual states. Selten and Branstator (2004) have shown an example of a periodic orbit that produces very apparent signatures in individual trajectories in a plane parallel to the axis of the orbit. But, because such an orbit has similarly positioned but oppositely signed segments in such a plane, it makes little contribution to the mean tendencies in the plane. Comparing the spreading of clouds to the mean tendencies in Fig. 7, it is apparent that there are large canceling contributions to our mean tendencies. But in calculations that go beyond the approach used in the results sections of this paper, we have found that, despite these cancellation effects, the influence of the means can be seen in the behavior of individual states. This influence is especially pronounced in the direction that states move through the phase space. From the Fig. 7b examples we see that, though neighboring states spread in all directions, the spread is anisotropic. If we calculate histograms based on the two-dimensional directions that states in different subdomains of a plane move in a 24-h period and then plot the most likely direction in each subdomain, we produce the plots in Fig. 13. The correspondence between the predominant directions that individual states pick and the total mean tendencies of Fig. 4 is clear, an indication that in many regions the mean dynamics is detectable above the spreading that results from the truncated components.

Given the strong linear signatures that we have found, especially when considering tendencies over long time intervals, the successful application of linear models to large-scale planetary waves should not be a surprise. (It may be surprising, however, that a strong contributor to the linear component is simply ergodic spreading, though of course many factors contribute to the details of that spreading.) On the other hand, considering the significant signatures of nonlinearity that we have detected, particularly in the behavior of a few leading patterns, it is apparent that in certain circumstances the ability of simplified models to reproduce planetary wave behavior is likely to be markedly enhanced if nonlinearities are represented. Though, strictly speaking, we can only draw this conclusion for the AGCM we examined, the similar, though sampling-contaminated, signatures found for observations (Fig. 1) indicate that it may well also be true for nature.

At this point it is not clear whether incorporation of nonlinearities in planetary wave models should take the form of multiplicative noise, as suggested by studies by Sura (2002) and Sardeshmukh et al. (2001), or whether systems that also incorporate the possibility of nonlinear deterministic terms, as for example in the work of Kwasniok (1996) and Majda et al. (2001, 2003), are required. One avenue for incorporating the nonlinearities into stochastic planetary wave models is suggested by the fact that our mean tendencies are identical to the potentially nonlinear drift terms in a Fokker–Planck equation derived from data (Siegert et al. 1998). Berner (2005) has pursued this idea.

## Acknowledgments

The authors appreciate beneficial conversations with F. Selten, R. Saravanan, J. von Storch, J. Tribbia, A. Hense, P. Sardeshmukh, C. Tebaldi, and D. Nychka. Especially useful has been correspondence with T. DelSole, who pointed out the relevance of his paper [DelSole (2000)] to the question of Δt dependence of LIM operators and who clarified remarks made in that paper. A. Mai formatted the AGCM data and produced some of our figures. This study was partially funded by NOAA Grant NA17GP1376 and NASA Grant S-44809-G. The support of the second author by NCAR’s Advanced Study Program is also gratefully acknowledged.

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## APPENDIX A

### Projected Mean Linear Tendencies

Evaluating the linear components is simplified by the fact that the distribution of states produced by (2) is centered and multivariate Gaussian (Brockwell and Davis 1987). Moreover, because (2) uses as a basis the EOFs of its own population, this distribution will be equal to the product of its marginals. Together these facts imply that if we conditionally average states according to their values in a subset of the stochastic model’s *N* directions, their average value in the unrestricted directions will be zero. So when evaluating the linear tendency (𝗟 **−** 𝗜)/Δ*t* in the subspace, only elements involving interactions between the components in the subspace will be nonzero. This in turn means that only the elements of 𝗟 pertaining to these interactions need be evaluated via (1). But since (2) is formulated using the EOFs of **Σ**, the denominator of (1) is diagonal meaning that these elements will be independent of *N*. Hence it is unnecessary to choose a particular value of *N* so long as it can be assumed to be larger than any of the EOFs that define the very low dimensional subspace in which the comparison of mean tendencies is carried out.

## APPENDIX B

### Local Linear Fitting

To identify four-dimensional counterparts to the two regions of local quasi-linear behavior seen in Fig. 11, we fit linear functions to mean nonlinear tendencies in various regions of the four-dimensional space. This is done by first finding nonlinear mean tendencies in a network of four-dimensional hypercubes, each side of which is 1/3 PC standard deviation long. These are calculated in the same manner employed at lower dimensions, namely accumulating average four-dimensional mean 24-h tendencies in each hypercube from observations of the long AGCM trajectory and then subtracting the corresponding tendency of a four-dimensional LIM from each of these means. Next we examine every four-dimensional block of these hypercubes consisting of five hypercubes on a side. If each of the 625 hypercubes is well sampled, within the block we least squares fit a four-dimensional linear system that maps state into nonlinear mean tendency. Unlike in LIM construction, when performing the fit we weight each hypercube, rather than each observation, equally. Moreover, the fitting allows for an arbitrary constant offset so that the assumed local linear system is not necessarily centered on the center of the block. If subtracting the local fit linear tendencies from the actual 625 nonlinear mean tendencies reduces the mean square of these tendencies by at least 85%, then we conclude that we have located a region of the complete space that is locally linear. We find in our four-dimensional space that there are 195 locally linear regions. The offset values represent the dynamical centers of the locally linear dynamics, which is the point in four-dimensional space that all eigenmodes of the local linear system revolve around or are damped toward. Perusing the 195 centers we find that most of them are near two positions, which we capture from a two cluster *k*-means cluster analysis of offsets.

The reduction in mean square two-dimensional mean tendencies produced when their nonlinear component is removed, as calculated using (6). The left column is for total 24-h tendencies and the right column is for 24-h tendencies with the symmetric linear component removed.

^{1}

The seasonal cycle is found by calculating 40-yr averages for each calendar date and then taking 15-day running means of these averages.

^{2}

Penland and Magorian (1993) point out that such noise may not be realizable, but for our data from nature and the AGCM, we find that 𝗤 has positive eigenvalues, so this is not an issue here.

^{3}

By contrast, when we investigate whether the Δ*t* dependence of 𝗟 may result from the systematic errors produced by using finite differences to estimate drift terms in stochastic systems (Berner 2003; Sura and Barsugli 2002), we find these errors are small. Indeed, they are too small and of the wrong structure to explain the Δ*t* dependence in our results.

^{4}

Note such trajectories are not trajectories of observed states, or even the mean trajectory of clusters of states, but are only used as a means of portraying the local mean nonlinear tendencies.