## 1. Introduction and motivation

We are interested here in the nonlinear dynamics of planetary waves in fairly realistic, high-dimensional atmospheric models. A promising way of better understanding the dynamics is to investigate the properties of state-averaged trajectories in a low-dimensional subspace of the full model’s phase space. The mean (i.e., state-averaged) phase-space tendencies describe long-term preferred motion in the observed or simulated atmosphere and can possibly be related to transitions between distinct weather regimes (Selten and Branstator 2004). These tendencies depend on the climate state; they can be used to identify signatures of linear and nonlinear dynamics in intermediate models (Selten and Branstator 2004; Franzke et al. 2007), as well as in more highly resolved atmospheric general circulation models (AGCMs; Branstator and Berner 2005).

Branstator and Berner (2005) studied mean phase-space tendencies in version 0 of the Community Climate Model (CCM0). They projected the results of 280 integrations of CCM0, each of which was 50–100 simulated days long, onto a basis of empirical orthogonal functions (EOFs) and considered the dynamics in the six planes, each spanned by two of the four leading EOFs. These authors found that linear inverse models (LIMs; Penland 1996) can describe relatively well the mean tendencies in some of these planes. For other planes, though, there were strong departures from linearity, such as a distinctive signature of “double swirls” in the diagnosed, two-dimensional (2D) tendencies, and LIMs were clearly not adequate.

Franzke et al. (2007) analyzed the mean phase-space tendencies of Marshall and Molteni’s (1993) global baroclinic, quasigeostrophic, three-level (QG3) model with topography. They projected the model equations onto a finite vector space spanned by the energy-norm EOFs and defined the “resolved” modes as the EOF pair spanning any 2D plane under consideration, while the rest of the EOFs were designated as the “unresolved” modes.

By using time series from a long integration of the full QG3 model, Franzke et al. (2007) decomposed the phase-space tendencies into several components; they did so by computing state averages of the instantaneous tendencies due to linear and nonlinear terms in the full QG3 model’s equations. The nonlinear tendencies were further divided into those contributed either by the “bare truncation” (i.e., resolved modes) or by the unresolved modes; the latter contributions were interpreted as being due to additive or multiplicative noise, along the lines of the Majda–Timofeyev–Vanden-Eijnden (MTV) classification (Majda et al. 2003, 2008). Franzke et al. (2007) concluded that the contributions of the unresolved modes are crucial in forming the double-swirl patterns of mean tendencies in the QG3 model.

In this study, we aim to revisit the significance of various contributions—from resolved and unresolved modes—that lead to signatures of nonlinear dynamics by using an alternative, empirical way of defining the nonlinear interactions between the resolved and unresolved modes. Our approach differs in both its spirit and its methodology from state averaging the full QG3 model’s instantaneous tendencies, as in Franzke et al. (2007). We will also quantify the sensitivity of the mean phase-space tendency attribution to the number of resolved modes considered.

Our procedure involves as its first preliminary step the construction of a low-order model in the phase space of the full QG3 model’s EOFs by using the empirical model reduction (EMR) methodology of Kravtsov et al. (2005, 2009) and Kondrashov et al. (2005, 2006). The EMR methodology models statistically the daily increments (hereafter called daily tendencies) of the reduced state vector as a quadratic function of the state vector itself; it includes stochastic forcing to parameterize the effect of the modes not explicitly present in the deterministic EMR operator.

We will show first that long EMR simulation produces an excellent match to the QG3 model’s mean daily tendencies [see also Kondrashov et al. (2006)]. We are thus in a position to explicitly identify the terms in the EMR model’s dynamical operator that dominate contributions to nonlinear, double-swirl patterns in the QG3 model’s mean tendencies. In particular, we will investigate various ways to define the resolved modes among the QG3 model’s leading EOFs. Subsequently, we will give a dynamic interpretation of the mean phase-space tendencies of the EMR model as resulting from interactions either between the resolved modes or between them and the unresolved ones. The analysis of state-averaged, EMR-parameterized daily tendencies will lead to very different conclusions than those obtained from Franzke et al.’s (2007) state-averaging of the QG3 model’s instantaneous tendencies.

In section 2, we briefly review the EMR formulation. In section 3, we present the detailed budget analysis of the estimated tendencies. The results are summarized in section 4.

## 2. Empirical model reduction

EMR models formally belong to a class of multivariate, parametric stochastic models. They extend LIMs by including quadratic—and higher-order polynomial, if necessary—combinations of predicted variables in the empirical dynamical operator and by adopting a multilevel approach in modeling the noise; the latter approach allows us to deal efficiently with correlated residual noise (Kravtsov et al. 2005, 2009).

*N*regularly sampled observations of the state vector

**x**(

*t*) = {

*x*(

_{i}*t*):

*i*= 1, … ,

*M*;

*t*= 1, … ,

*N*};

**x**(

*t*) is represented in this study by

*M*leading principal components of the QG3 model’s streamfunction. The main level of the EMR equations for modeling the increments Δ

*x*is

_{i}*N*,

_{ijk}*F*that fit the predictand variables Δ

_{i}*x*. The regression residuals

_{i}*r*

_{0,i}define the time-dependent stochastic forcing; these residuals typically have long-tailed autocorrelations. Additional model levels are thus included to simulate correlated noisefor 0 ≤

*l*≤

*L*. The linear maps

**b**

*are estimated recursively, and the number of levels*

_{l}*L*is chosen so that the last level

**r**

_{L+1}is well approximated by a spatially correlated white-noise process.

## 3. Mean phase-space tendencies of the QG3 model

### a. The quasigeostrophic, three-level model

Marshall and Molteni’s (1993) QG3 model describes the evolution of winds at each of its three pressure-coordinate levels, which represent the lower, middle, and upper troposphere. The model is formulated in spherical harmonics and, at a T21 truncation, it has more than a thousand degrees of freedom. Despite its simple form, the QG3 model has a fairly realistic climatology and complex variability; the latter compares favorably with observed atmospheric behavior (D’Andrea and Vautard 2001; Kondrashov et al. 2004). Franzke and Majda (2006) refer to it as a “prototypical GCM,” although it is more commonly referred to as an intermediate model between highly idealized “toy” models and full GCMs (Ghil and Robertson 2000).

The EMR methodology has been quite successful in reducing this QG3 model to a nonlinear stochastic model with quadratic nonlinearity and a much smaller number of variables on its main level; see Eq. (1). This EMR model accurately reproduces non-Gaussian features of the full QG3 model’s probability density function (PDF), as well as the intraseasonal oscillations that characterize the full model’s low-frequency variability (Kondrashov et al. 2006; Kravtsov et al. 2009).

### b. Mean phase-space tendencies

We combine here several elements from the previous studies of Branstator and Berner (2005) and Franzke et al. (2007) with our EMR methodology and arrive thereby at a distinctly different approach for the detailed budget analysis of mean phase-space tendencies. First, we define our resolved modes as the leading four EOFs of the daily 500-hPa QG3 field, {*x _{i}*(

*t*): 1 ≤

*i*≤ 4}, using the streamfunction norm. These EOFs account for more than 30% of variance in the unfiltered daily data (Fig. 1), have the most pronounced deviations from Gaussianity in terms of skewness and kurtosis, and have longer tails in their autocorrelation function (Strounine et al. 2010). Moreover, Kondrashov et al. (2006) established that these EOFs describe the most interesting aspects of the QG3 model’s dynamical behavior, oscillatory as well as episodic (cf. Ghil and Robertson 2002), namely both its intraseasonal oscillations and multiple regimes.

Second, we derive a nonlinear EMR model based on Eqs. (1) and (2), using the 15 leading streamfunction-norm EOFs, computed from daily 500-hPa streamfunction fields of a 300 000-day-long integration of the QG3 model. This EMR model is essentially the same as that of Kondrashov et al. (2006) and it has *L* = 2 levels. We study two-component vectors of the mean (daily) tendencies (〈Δ*x _{j}*〉, 〈Δ

*x*〉), for the six pairs {

_{k}*j*,

*k*: 1 ≤

*j*,

*k*≤ 4}; the averaging operator 〈·〉 was applied to all the states that reside in a particular cell of our rectangular partition of each 2D (

*x*,

_{j}*x*) plane. These two vectors of tendencies were estimated from the simulations of the full QG3 model, as well as the reduced EMR. To ensure robust results, we estimated tendencies on a 20 × 20 grid, and only if there were more then 100 states in the particular cell.

_{k}Figures 2 and 3 show the mean tendencies of the full QG3 and the EMR model, respectively, in all six planes spanned by the possible pairs of our resolved EOFs. The strikingly good agreement between each pair of corresponding panels in the two figures confirms the EMR model’s success in capturing the full QG3 model’s mean temporal behavior in phase space. Therefore, to understand the linear and nonlinear contributions to the mean tendencies in the full QG3 model, we can safely use the EMR simulation and the EMR model’s empirical dynamics.

The mean flow patterns in the planes spanned by the EOF pairs 1–3 and 2–3 are characterized by the tendencies being antisymmetric for reflections through the origin and having approximately constant speed along ellipsoids. As noted by Branstator and Berner (2005), these features correspond to linear behavior. On the other hand, strongly nonlinear behavior is observed in the four other planes considered, and especially in the EOF 1–4 plane (top right panels of Figs. 2 and 3), which we will study below in detail. Branstator and Berner (2005) also noted that the mean tendencies 〈**x**(*t* + *τ*)〉 − 〈**x**(*t*)〉 depend strongly on the lag time *τ* used for their computation; in particular, the linear component increases with *τ*. In the present study, however, we are particularly concerned with EMR-modeled daily tendencies (*τ* = Δ*t* = 24 h), which match nearly perfectly the daily tendencies of the full QG3 model; see again Figs. 2 and 3. As the EMR dynamical operator is explicitly available, the decomposition of the daily tendencies into their various components is explicit and unambiguous.

### c. Decomposition of the EMR tendencies

When projected onto the subspace spanned by the four resolved modes, the mean tendencies in Figs. 2 and 3 contain contributions from both the 4 resolved and the 11 unresolved modes. Given the explicit form of our EMR model’s deterministic dynamics, Eq. (1), it is quite straightforward to estimate the relative contributions of the various dynamic terms, both resolved and unresolved, from the time series of the EMR simulations.

*x*

_{i}*x*

_{i}*V*

_{Li}The individual tendencies Δ*r*_{0,j}(*t*) of the residual stochastic forcing in Eq. (2) do depend on **x**. But the contribution to the mean tendencies from the residual stochastic forcing (〈*r*_{0,j}(*t*)〉, 〈*r*_{0,k}(*t*)〉) is very small in amplitude and largely random. This is true both cellwise and when averaged over all states, for all six planes spanned by the possible pairs of our resolved EOFs, as shown in Fig. 4. We can thus neglect these unresolved contributions safely in the following analysis.

*F*necessarily equal the average of the nonlinear dynamical contributionsfor all

_{i}*j*and

*k*. Note that averaging in Eq. (4) reduces to a single sum over the diagonal elements, since the simulated PCs are close to being orthogonal in time, provided the EMR model integration is sufficiently long.

*i*≤ 4, we can identify the contribution

*V*(

_{Ri}*t*) to its tendencies 〈Δ

*x*〉 that are due solely to the nonlinear interactions within the resolved-mode set Ω of

_{i}*j*and

*k*; here Ω = {1 ≤

*j*,

*k*≤ 4} and we sum over it below:The remaining nonlinear terms involve interactions either between a resolved and an unresolved mode or between two unresolved modes, that is, within the set

*j*,

*k*such that either

*j*or

*k*or both are not in Ω. The corresponding contribution

*V*(

_{Ui}*t*) can be defined accordingly, in a way that is similar to

*V*(

_{R}*t*) in Eq. (5)by summing over the

*j*,

*k*in

*x*〉 themselves, but it ensures a clean interpretation of the results. In particular—by using Eqs. (5), (6), and (7)—one can obtain that the time mean of the full, as well as of the partitioned nonlinear tendencies, is zero, similar to Eq. (3):

_{i}### d. Further analysis of the EMR tendencies

*x*,

_{j}*x*) planes of interest. Four of the six mean flow fields exhibit the characteristic double-swirl pattern indicative of strongly nonlinear behavior; the two planes that do not are those spanned by EOFs 1–3 and 2–3, in agreement with the total tendencies shown in Figs. 2 and 3.

_{k}Figure 6 presents the fully resolved nonlinear tendencies (〈*V _{Rj}*(

*t*)〉, 〈

*V*(

_{Rk}*t*)〉). The difference between it and Fig. 5 is very small for all planes where nonlinear swirls are strong, namely the planes spanned by EOFs 1–2, 1–4, 2–4, and 3–4. Clearly, the effect of the nonlinear tendencies

*V*that involve unresolved modes is also very small for these pairs.

_{Ui}To refine further our interpretation of nonlinear signatures, we have repeated the analysis of Figs. 5 and 6 by defining as resolved only the pair of EOFs that span a particular 2D plane and considering the other 13 modes as unresolved. For example, in the case of the EOF 1–3 plane, the nonlinear resolved tendencies would be only those that involve the *x*_{1}^{2}, *x*_{1}*x*_{3}, and *x*_{3}^{2} interactions; see again Eq. (5). Even though such a definition of the resolved modes is quite similar to that of Franzke et al. (2007), the results are entirely different: The nonlinear tendencies due to interaction between the resolved modes alone, as shown in Fig. 7, are very similar to the full nonlinear tendencies in Fig. 5, as well as to the tendencies due to the interactions between all four leading EOFs in Fig. 6. Thus, the interaction with the unresolved modes is quantitatively quite minor, even when choosing only two modes as resolved but doing so according to the EMR methodology.

Figure 8 presents a detailed analysis of linear and nonlinear contributions to the mean phase-space tendencies in the plane spanned by EOFs 1–4, in which the nonlinear double-swirl pattern—as seen in Figs. 8a and 8b—is especially strong. This pattern is strikingly absent, however, in Figs. 8c and 8d. The results in Fig. 8c correspond to a linear EMR model, with *N _{ijk}* ≡ 0 in Eq. (1), and are fairly close to being perfectly antisymmetric about the origin, in agreement with Branstator and Berner’s (2005) results. This antisymmetry can be explained by the following two LIM properties: (i) simulated tendencies are invariant under the mirror transformation

*x*→ −

_{i}*x*, when

_{i}*F*≈ 0 [cf. Eq. (1)] and (ii)

_{i}*x*is normally distributed.

_{i}On the other hand, the linear contribution in our quadratic EMR model, shown in Fig. 8d, is not perfectly antisymmetric. This can be explained by the non-Gaussianity of the PC-1 time series that is simulated by a nonlinear EMR model; the latter, in turn, is consistent with non-Gaussianity of the full QG3 model’s time series. The positive skewness of the PC-1 time series in Fig. 1b leads to violation of the mirror invariance under *x _{i}* → −

*x*in the estimated linear tendencies. Figure 8 thus further confirms the dominant contribution of the resolved nonlinear interactions to the QG3 model’s tendencies.

_{i}## 4. Concluding remarks

We used nonlinear stochastic models obtained by empirical model reduction (EMR) as a diagnostic tool for studying effects of linear and nonlinear dynamics in a fairly realistic intermediate climate model, the quasigeostrophic, three-level (QG3) model of Marshall and Molteni (1993). An EMR model with quadratic nonlinearity, driven by data-derived colored noise, was constructed in the phase subspace spanned by the 15 leading EOFs of the QG3 model at T21; see Eq. (1). This model, with the colored noise represented by a two-level parameterization [cf. Eq. (2)], reproduces nearly perfectly the daily-mean tendencies of the full, high-dimensional QG3 model.

The nonlinear features of these tendencies are dominated by the EMR-parameterized, deterministic interactions between the four leading EOF modes. The interactions with trailing EOFs are therefore linearly parameterizable as the multilevel noise and do not significantly contribute to the model’s nonlinear characteristics. This result is further buttressed by showing that for a particular 2D plane—in which the double-swirl patterns in the mean tendencies of the full QG3 model are strong—these signature patterns do occur mostly because of the nonlinear interactions between the pair of EOFs that span such a plane. The same holds for other planes of this type (see Fig. 7).

These findings should be contrasted with the results of Franzke et al. (2007), who used a different norm to define the EOFs and an entirely different strategy to compute the state-averaged mean tendencies. The EMR results are not qualitatively sensitive to the norm used for data compression; see Strounine (2007) and Strounine et al. (2010).

On the other hand, using an EMR model to define those interactions that are linearly parameterizable versus those that are not—in terms of their contribution to the daily-mean state-averaged tendencies (cf. Kravtsov et al. 2009)—is likely to be the primary reason for differences in interpretation between our study and that of Franzke et al. (2007). The latter defined the tendencies based on state-averaged instantaneous contributions from various linear and nonlinear terms of the full QG3 model’s operator. We conclude therefore that the manner of defining interactions between the resolved and unresolved modes plays a crucial role in the dynamical interpretation of the tendency-based statistical diagnostics.

## Acknowledgments

It is a pleasure to acknowledge exchanges with M. D. Chekroun, G. Branstator, C. Franzke, and A. J. Majda on topics related to this study. We also thank two anonymous reviewers for their constructive comments. This work was supported by the Office of Science (BER), U. S. Department of Energy (DOE), under Grants DE-FG02-07ER64439 and DE-FG02-07ER64429 at UCLA, and by DOE Grant DE-FG02-07ER64428 and NSF Grant ATM-0852459 at UWM.

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