## 1. Introduction

The leading modes of variability of the extratropical circulation are usually referred to as the “annular modes” (e.g., Thompson and Wallace 2000; Thompson et al. 2000) and are derived from empirical orthogonal function (EOF) analysis of meteorological fields such as geopotential height or zonal-mean zonal wind (Kidson 1988; Lorenz and Hartmann 2001, 2003). The leading EOF of zonal-mean zonal wind takes the form of a dipolar structure centered on the time-mean jet and describes north–south fluctuations in time of the jet. The second EOF peaks at the maximum of the time-mean jet, and, independently, it describes intensification and narrowing (“pulsing”) of the midlatitude jet. However, the leading EOFs of zonal wind are not necessarily independent of each other and, together, the two leading EOFs may describe the poleward propagation of zonal wind anomalies. Such propagation has been noted in models (e.g., James and Dodd 1996; Lee et al. 2007) and in both Northern and Southern Hemisphere observations (Feldstein 1998) and is illustrated for Southern Hemisphere wind anomalies by the examples shown in Fig. 1. Periods of both propagation and nonpropagation are evident, in all seasons; propagating anomalies typically appear in low latitudes and migrate generally poleward over the course of 2–3 months, although there are some instances of equatorward propagation. In terms of the leading EOFs, propagation is manifested through nonzero lag correlations between the two corresponding principal components (Lorenz and Hartmann 2001; Son and Lee 2006; Watterson 2007; Sparrow et al. 2009). For the examples shown in Fig. 1, reconstructions based on the projections onto the two leading EOFs of zonal-mean zonal wind are shown in Fig. 2; most of the midlatitude variability, including propagation when it occurs, is reproduced by these two components, but much of the variability in low and high latitudes is not. Among other things, Son and Lee (2006) explored the circumstances under which the leading modes of variability of a simplified dynamical core model take the form of nonpropagating or propagating anomalies. When the climatology exhibits a single jet, they found nonpropagating behavior, for which the variance is dominated by a single EOF; with a double jet, zonal wind anomalies propagate poleward, the variance associated with the second EOF exceeding about one-half of that associated with the first.

In this paper we explore the characteristics of variability in the propagating regime. We shall argue that under such circumstances the leading, coupled, EOFs form a single underlying mode of the system, such that individual EOFs cannot be properly regarded as “modes.” One consequence of this is that the autocorrelations of the leading principal components (PCs) corresponding to the individual EOFs do not decay exponentially but show some oscillatory characteristics. This behavior, in turn, cautions against using simple paradigms based on stochastically forced, single component, systems as a basis for estimating eddy feedback strength (e.g., Lorenz and Hartmann 2001, 2003; Watterson 2007; Chen and Plumb 2009) or for estimating the sensitivity of the system to climate perturbations (Gerber et al. 2008; Ring and Plumb 2008).

Such questions are addressed here by application of theory to results from a dry, dynamical core, global model run in a simplified solsticial configuration. The model setup and climatology are described briefly in section 2. In section 3 it is shown that the leading modes of variability of the zonal jets are nonpropagating in the “summer” hemisphere and propagating in the “winter” hemisphere. These two differing types of anomaly are then used to illustrate the theoretical development of the paper. The dynamical modes of the zonal-mean flow system are identified in section 4 through calculation of the principal oscillation patterns (POPs; von Storch et al. 1988; Penland 1989). In the nonpropagating case, the leading POP is essentially identical to the dominant EOF, with a real eigenvalue corresponding to the decorrelation rate of the dominant PC, as found by Ring and Plumb (2008). In the propagating regime, on the other hand, the leading POP and its corresponding eigenvalue are complex; the structure of this single, propagating, mode involves at least two EOFs. The theoretical implications are explored in section 5, in which the characteristics of PC auto- and cross correlations in a stochastically forced system are described, with some remarks made about the implications for the response of the jets to climate perturbations. Finally, in section 6 we discuss the characteristics of the observed Southern Hemisphere zonal flow in the light of these results.

## 2. Model details

The dynamical core model is similar to that of Polvani and Kushner (2002), with some modifications as described below. The model is dry and hydrostatic, solving the global primitive equations with T42 resolution in the horizontal and 40 sigma levels in the vertical. There is no surface topography. Radiation and convection schemes are replaced by relaxation to a zonally symmetric equilibrium temperature profile identical to Held and Suarez (1994) in the troposphere. In the stratosphere (above 200 hPa), the equilibrium temperature profile is a perpetual-solstice version of the equilibrium temperature specifications used in Sheshadri et al. (2015), with winter conditions in the Southern Hemisphere. As will be seen below, these choices result in a model configuration with reasonable annular mode time scales in the Southern Hemisphere. The model was run for 13 000 days, and the last 10 000 days are used in the analysis.

The time-averaged zonal-mean zonal wind is shown in Fig. 3. There is a single deep tropospheric jet near 35°N in the Northern Hemisphere; in the Southern Hemisphere, an upper-tropospheric subtropical jet near 30°S, a deeper midlatitude jet near 45°S, and a polar night jet near 60°S extending through the stratosphere. Note the hemispheric asymmetry of the zonal-mean zonal winds even in the troposphere, where the equilibrium temperature is symmetric about the equator. For simplicity, we shall refer to the Northern and Southern Hemispheres as summer and winter, respectively, even though there is no such contrast in the tropospheric equilibrium temperature.

## 3. EOFs of zonal-mean wind variability

**Σ**being the diagonal matrix

^{1}containing the corresponding singular values

*σ*. While the EOFs compactly and efficiently describe the fluctuations (the contribution of each to the total variance being proportional to

*σ*

^{2}) and are frequently referred to as modes (such as the annular modes that are our focus here), they are not necessarily modes in the usual physical sense of being eigenmodes of the underlying dynamical system. Rather, just as the variability they describe, the EOFs and corresponding PCs depend on how the system is forced to produce the variability, and not just on the properties of the system modes themselves. Moreover, they are by design orthogonal (

^{T}

^{T}

We consider here the EOFs of the zonal-mean zonal wind fluctuations below 100 hPa and poleward of 20° latitude. To weight each data point appropriately, the variable used to perform (1) is *δp* is the pressure thickness of the model level,

This behavior should be contrasted with that of the fluctuations of zonal wind in the summer hemisphere of the model. Figure 6 shows the leading EOFs for that case. The structures of the EOFs show characteristics similar to those in the winter hemisphere, with EOF1 dominated by a dipole straddling the climatological jet at 35°; EOF2 is also dipolar but with one of its extrema nearly coincident with the jet. In this case, however, EOF1 is dominant: it accounts for 62% of the variance, compared to 12% for EOF2, and the decorrelation times of the corresponding PCs are very different (68 vs 20 days). Further, the cross correlation between the two EOFs is very weak, implying that they act independently. As Fig. 5b illustrates, one consequence of this is the absence of propagation of the wind anomalies; Fig. 5d shows that EOF1 is sufficient to capture the evolution of zonal-mean zonal wind anomalies. Also evident in Fig. 5b is the remarkable longevity of zonal wind anomalies, consistent with the long decorrelation time of EOF1. Such unrealistically long decorrelation times are often found in this kind of model (Gerber et al. 2008) and, indeed, Chan and Plumb (2009) found them to be especially long when the climatological subtropical and midlatitude jets are merged, as is the case here (Fig. 3).

Before leaving this section, it should be emphasized that the distinction between propagating and nonpropagating anomalies is not necessarily one associated with the difference between winter and summer, even though this is the case here. For example, in a troposphere-only model, Ring and Plumb (2007, 2008) found a single dominant mode in each hemisphere. Rather, the two cases illustrated here as winter and summer should be seen as illustrative of two distinct patterns of behavior that can be found under different conditions. Note also that the separation into the propagating and nonpropagating regimes is consistent with the results of Son and Lee (2006), who found that propagation occurs whenever the subtropical and midlatitude jets are separated, and when the variance described by EOF2 exceeds about one-half of that described by EOF1.

## 4. Principal oscillation patterns

### a. Theory

**f**can be determined. However, in practice

*u*are the responses of a stable system described by (2) to white-noise forcing, the eigenvectors of the matrixwhereis the lag covariance of the zonal wind anomalies, are the eigenvectors of

*λ*. Hence,Thus, taking this approach yields the POPs and their eigenvalues without the need to perform an explicit inversion of the covariance matrix and (as we shall see below) permits their evaluation without the need to truncate the data. One does need to invert

One advantage of POPs over EOFs is that the latter are dependent on how the data are weighted before applying (1), whereas it is straightforward to show that the POPs are independent of such weighting (appendix A). (This is no longer true if one uses a truncated series of EOFs to filter the data.) This fact can become especially important for a deep system (such as the troposphere and stratosphere) if, say, pressure weighting is applied.

*Z*are the same as the EOFs. In such a case, therefore, the EOFs and the PC decorrelation rates describe the modes of the system. Clearly, however, this is not generally true of a system in which the PCs have nonzero cross correlations, as we shall now illustrate, using results from the dry dynamical core model.

### b. POPs in the dynamical core model

Following the outlined procedure, the POPs were calculated for the winter and summer hemispheres. It is first necessary to choose a value of lag *τ* at which to evaluate the PC lag covariances. For reasons noted above, it is desirable to avoid short lags (*τ* < 10 days); lags much longer than one decorrelation time scale are also best avoided, as the correlations then become weak. Accordingly, the calculation was done for *τ* = 20 days for the winter hemisphere and *τ* = 60 days for summer. (Robustness of the results with respect to the chosen lag will be discussed below.)

One further consideration concerns selection of the POPs (of many) that are of interest. For the EOFs, the association of their singular values with variance explained lends itself to a natural (and conventional) ranking by decreasing variance. No such obvious ranking exists for the POPs; their eigenvalues determine the time scales for each mode but bear no direct relationship to explained variance.^{2} Since it is the two leading EOFs that, by definition, explain the most variance, we focus attention here on those POPs onto which the first two EOFs project the most strongly.

*Z*) and Im(

*Z*) strongly reflect the structure of the EOFs, with only minor differences. The corresponding eigenvalues are

The summer case, on the other hand, is more straightforward. The real part of *Z* is almost identical to EOF1, while its imaginary part is very small. Similarly, its corresponding eigenvalue *λ* = 0.0134 ± 0.0033*i* day^{−1} is dominated by its real part. To a first approximation, then, this mode describes a decaying disturbance, with very weak propagation and decay time

The robustness of the calculation to chosen lag *τ* for evaluation of the cross correlation is illustrated in Fig. 8, which shows how the eigenvalue *λ* for the propagating case depends on *τ*. While there is some sensitivity, there is a wide range of lags 20 ≤ *τ* ≤ 50 days over which the eigenvalue changes little. (The calculation becomes problematic near *τ* = 60 days, where the cross correlations pass through zero.)

## 5. Lag covariances under stochastic forcing

### a. Covariances of the POP coefficients

*m*,

*n*= 1 or 2,

*N*is the total number of modes, and

*p*th element of the diagonal matrix

*N*= 2, in which case we havewhere “c.c.” denotes complex conjugate. Then it is shown in appendix B that the autocorrelations areand the cross correlation iswhere

Several points of interest become evident from (14)–(16). First, the decay of the autocorrelations is not exponential, except in the limit of a single uncoupled mode for which *τ*. We shall now assess the impact of these characteristics on the example from the model’s winter hemisphere.

### b. Application to model data

For the propagating case of the model’s winter hemisphere discussed in sections 3 and 4 and shown in Figs. 4 and 7, reconstructed lag correlations of the PCs given by (11) are shown in Figs. 9a and 9b. The separation of the two PC autocorrelations is noticeable, as is the asymmetry of the cross correlation. Comparing these predictions with the PC lag correlations determined from the model and shown in Fig. 4, the similarity is obvious. It should be noted that although there is some circularity in this comparison—the coefficients were themselves derived from the model results—all the information was obtained from the PC covariances at a single lag: the lag dependence shown in Fig. 9 derives entirely from the theoretical analysis.

Lag correlations calculated from the truncated results given in (14)–(16) are shown in Figs. 9c and 9d. Agreement with the model results shown in Figs. 4c and 4d is now more qualitative. Nevertheless, from (14) and (15) and Fig. 9c, the more rapid decay of PC2 compared with PC1 evident in Fig. 4c and in its shorter “decorrelation time” can now be seen as a consequence of the propagation of the mode. Although the decay time of the mode

The cross correlations determined from the truncated result in (16), while qualitatively satisfactory, are too large in magnitude [as compared with the model results of Fig. 4d and with the untruncated theoretical prediction of (11) and Fig. 9b] by a factor of about 1.7. The discrepancies between the truncated and untruncated results indicate that, despite some indications to the contrary, contributions to the leading mode from EOFs other than the first two are not quantitatively negligible.

### c. Implications for the climate perturbation problem

**f**is not just the imposed forcing that appears in the momentum equation but is the “Eliassen response” of the zonal wind tendency to any imposed mechanical or thermal forcing.]

*Z*and their eigenvalues

*λ*are complex. The relationship can also be expressed in terms of the projection of the response and forcing onto the EOFs

*U*, using (8) to givewhere

As discussed earlier, in circumstances in which the PCs are all independent, the POPs are just the EOFs, the eigenvalues are real and equal to the inverse of the PC decorrelation time, and *n*th EOF as

*μ*is defined in appendix B. Hence there is cross fertilization—a nonzero response in EOF2 (EOF1) to a forcing purely in EOF1 (EOF2)—whenever

## 6. Propagation in Southern Hemisphere data

In this section, we investigate the applicability of the model behavior to observations of the Southern Hemisphere zonal flow. We use daily data from the ERA-Interim (Dee et al. 2011) for the years 1979–2013 on 21 vertical levels between 1000 and 100 hPa. Examples of the evolution of zonal-mean zonal wind anomalies were shown in Fig. 1. As with the model analysis in section 2, we consider fluctuations of zonal-mean zonal wind poleward of 20°S, over the entire record (i.e., including all seasons). The first two EOFs of zonal-mean zonal wind are shown in Fig. 11. These two leading EOFs contribute 37% and 19%, respectively, to the total variance and have decorrelation times of 11 and 8 days. Figure 11 also shows the PC auto- and cross correlations associated with these two EOFs. Note that the autocorrelations appear to asymptotically approach a nonzero positive value at large lag, which may indicate the influence of interannual variability (Keeley et al. 2009; Byrne et al. 2016). As in the winter hemisphere in the idealized model, the two PCs show some cross correlation peaking near 12 days, but with smaller magnitudes. The POPs were calculated and chosen based on the procedure outlined in section 4a, with the PC covariances calculated at *τ* = 10 days (similar to the decorrelation time of the first PC). The results (Fig. 12) bear a strong resemblance to those of the model’s winter hemisphere, with the real and imaginary parts of the POP primarily reflecting the structures of EOFs 1 and 2 (Fig. 11). The corresponding eigenvalues are

## 7. Discussion and conclusions

These theoretical and modeling results have several implications for our understanding of annular modes and their role in climate variability. Perhaps the most important conclusion is that whenever the leading mode is propagating (revealed by finite lag correlations between the two leading PCs) one cannot regard the two leading EOFs as being independent; rather, they are simply two projections into EOF space of a single mode [as is made clear by the analyses of Son and Lee (2006) and Sparrow et al. (2009) and by the fact that the corresponding POPs form a conjugate pair]. Propagation is not just a characteristic of this kind of model; Feldstein (1998) described poleward-propagating zonal wind anomalies in both hemispheres (Southern Hemisphere propagation was also shown in Fig. 1 here), and Lorenz and Hartmann (2001) noted that the characteristics of the two leading Southern Hemisphere PCs are consistent with propagation. Reasons for the propagation are not addressed here; Robinson (1994, 2000) and Lee et al. (2007) have described how the eddy flux response to zonal wind anomalies can lead to poleward propagation in the atmosphere, while Chan et al. (2007) have described similar processes leading to equatorward propagation in a simple ocean model.

One important consequence of the propagation is that the PC autocorrelations decay nonexponentially and faster than the mode itself decays. In the winter example discussed here, the apparent decorrelation time, obtained by fitting an exponential to the PC1 autocorrelation function, of 19 days is much shorter than the POP decay time of 39 days. Since the persistence of the mode, beyond time scales of dissipation by surface drag, can be taken as indicative of eddy feedback acting to maintain the anomalous winds, it follows that the eddy feedback may be underestimated if the PC decorrelation time is interpreted as the relevant decay time (e.g., Chen and Plumb 2009). A second consequence is that the autocorrelation functions of the two leading PCs separate and have different apparent decorrelation times. Lorenz and Hartmann (2001) interpreted such a separation between the PCs corresponding to the two leading Southern Hemisphere EOFs as indicative of eddy feedback acting on EOF1 but not EOF2, an interpretation that clearly would not be valid in the winter case analyzed here. In a propagating regime, decay of the autocorrelations occurs through conversion between PCs, and not just dissipation of the mode. Moreover, since the two modes share a conjugate pair of eigenvalues, they must be subject to the same eddy feedback strength.

Analysis of Southern Hemisphere winds from ERA-Interim produces results that are consistent with the propagating regime in the winter hemisphere of the dynamical core model: the two leading EOFs are manifestations of a single dynamical mode with complex conjugate eigenvectors and eigenvalues. Poleward propagation of zonal-mean zonal wind anomalies is clearly evident in reanalysis data and is not obviously confined to any given season, but unlike in the propagating regime in the idealized model, this feature is not ubiquitous, consistent with the criterion of Son and Lee (2006), which, given the ratio of variances explained by EOFs 1 and 2, places the observed Southern Hemisphere as marginal for propagation, at least over the multiyear and all-season average used here.

Finally, we note that the “climate” problem of the steady response to anomalous steady forcing is more complex in the propagating regime than in the nonpropagating case. For the nonpropagating case with a single dominant EOF, the eigenvalue *λ* is just the inverse of the decorrelation time of the primary PC, and thus the response in the leading mode depends on the projection of the forcing onto the EOF, multiplied by a time scale given by the PC decorrelation time. In such a regime, Ring and Plumb (2008) found some success, though with quantitative limitations, applying (18) to a model similar to that used here, and using the PC decorrelation time as the time scale. Nevertheless, it is clear that, in the propagating regime, the relevant time scale may differ significantly from the PC decorrelation time (as is the case in the example analyzed here). Moreover, since the leading mode comprises (at least) two EOFs, a forcing that projects purely onto EOF1 will produce a response in EOF2 as well as EOF1, and vice versa. We note that this might explain the findings of Black et al. (2006) and Black and McDaniel (2007), from Northern and Southern Hemisphere observations, and Sheshadri et al. (2014), from a dynamical core model, that the tropospheric response to the stratospheric final warming does not project very well onto the leading EOF of zonal wind variability. In particular, responses are likely to project onto at least two EOFs [this is consistent with Sheshadri and Plumb (2016), who found that the tropospheric response to lower-stratospheric cooling in a very similar idealized model as is used here contained a substantial contribution from EOF2, in addition to EOF1]. Further, the coefficient that determines the sensitivity of the system is not just the decorrelation time of the leading PC, but involves at least two time scales: that governing the rate of decay of the dynamical mode and the period associated with the propagation.

This work was partially supported by Junior Fellow Award 354584 from the Simons Foundation to AS and by the National Science Foundation through Grant OCE-1338814 to MIT. Part of the work was done during an extended visit by RAP to the University of Reading, with support provided by the Department of Meteorology Visitors Programme. We thank Erik Lindgren for his help with running the model experiment with an imposed torque and acknowledge helpful discussions and communications with Michael Blackburn, Daniela Domeisen, Ted Shepherd, Sarah Sparrow, Seok-Woo Son, and David Thompson. We also thank Ian Watterson and an anonymous reviewer for their constructive comments, which greatly improved the manuscript.

# APPENDIX A

## Independence of POPs under Weighting

So the eigenvalues are the same, and the deweighted eigenvectors

# APPENDIX B

## Covariances When Truncated to Two Modes and Two EOFs

## REFERENCES

Black, R. X., and B. A. McDaniel, 2007: Interannual variability in the Southern Hemisphere circulation organized by stratospheric final warming events.

,*J. Atmos. Sci.***64**, 2968–2974, doi:10.1175/JAS3979.1.Black, R. X., B. A. McDaniel, and W. A. Robinson, 2006: Stratosphere–troposphere coupling during spring onset.

,*J. Climate***19**, 4891–4901, doi:10.1175/JCLI3907.1.Byrne, N., T. G. Shepherd, T. Woolings, and R. A. Plumb, 2016: Annular modes and apparent eddy feedbacks in the Southern Hemisphere.

,*Geophys. Res. Lett.***43**, 3897–3902, doi:10.1002/2016GL068851.Chan, C. J., and R. A. Plumb, 2009: The response of the troposphere to stratospheric perturbations and its dependence on the state of the troposphere.

,*J. Atmos. Sci.***66**, 2107–2115, doi:10.1175/2009JAS2937.1.Chan, C. J., I. Cerovecki, and R. A. Plumb, 2007: Annular modes in a migrating zonal jet regime.

,*J. Atmos. Sci.***64**, 4053–4068, doi:10.1175/2007JAS2156.1.Chen, G., and R. A. Plumb, 2009: Quantifying the eddy feedback and the persistence of the zonal index in an idealized atmospheric model.

,*J. Atmos. Sci.***66**, 3707–3720, doi:10.1175/2009JAS3165.1.Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system.

,*Quart. J. Roy. Meteor. Soc.***137**, 553–597, doi:10.1002/qj.828.Feldstein, S. B., 1998: An observational study of the intraseasonal poleward propagation of zonal mean flow anomalies.

,*J. Atmos. Sci.***55**, 2516–2529, doi:10.1175/1520-0469(1998)055<2516:AOSOTI>2.0.CO;2.Gallagher, F., H. von Storch, R. Schnur, and G. Hannoschöck, 1991: The POP manual. Deutches KlimaRechenZentrum Tech. Rep. 1, 64 pp. [Available online at http://www.hvonstorch.de/klima/pdf/POP-manual.dkrz.1991.pdf.]

Gerber, E. P., S. Voronin, and L. M. Polvani, 2008: Testing the annular mode autocorrelation time scale in simple atmospheric general circulation models.

,*Mon. Wea. Rev.***136**, 1523–1536, doi:10.1175/2007MWR2211.1.Gritsun, A., and G. Branstator, 2007: Climate response using a three-dimensional operator based on the fluctuation–dissipation theorem.

,*J. Atmos. Sci.***64**, 2558–2575, doi:10.1175/JAS3943.1.Hassanzadeh, P., and Z. Kuang, 2016: The linear response function of an idealized atmosphere. Part II: Implications for the practical use of the fluctuation–dissipation theorem and the role of operator’s nonnormality.

,*J. Atmos. Sci.***73**, 3441–3452, doi:10.1175/JAS-D-16-0099.1.Hasselmann, K., 1976: Stochastic climate models. Part I. Theory.

,*Tellus***28**, 473–485, doi:10.3402/tellusa.v28i6.11316.Held, I. M., and M. J. Suarez, 1994: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models.

,*Bull. Amer. Meteor. Soc.***75**, 1825–1830, doi:10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2.James, I. N., and J. P. Dodd, 1996: A mechanism for the low-frequency variability of the mid-latitude troposphere.

,*Quart. J. Roy. Meteor. Soc.***122**, 1197–1210, doi:10.1002/qj.49712253309.Keeley, S. P. E., R. T. Sutton, and L. C. Shaffrey, 2009: Does the North Atlantic Oscillation show unusual persistence on intraseasonal timescales?

,*Geophys. Res. Lett.***36**, L22706, doi:10.1029/2009GL040367.Kidson, J. W., 1988: Interannual variations in the Southern Hemisphere circulation.

,*J. Climate***1**, 1177–1198, doi:10.1175/1520-0442(1988)001<1177:IVITSH>2.0.CO;2.Lee, S., S.-W. Son, K. Grise, and S. B. Feldstein, 2007: A mechanism for the poleward propagation of zonal mean flow anomalies.

,*J. Atmos. Sci.***64**, 849–868, doi:10.1175/JAS3861.1.Leith, C. E., 1975: Climate response and fluctuation dissipation.

,*J. Atmos. Sci.***32**, 2022–2026, doi:10.1175/1520-0469(1975)032<2022:CRAFD>2.0.CO;2.Lorenz, D. J., and D. L. Hartmann, 2001: Eddy–zonal flow feedback in the Southern Hemisphere.

,*J. Atmos. Sci.***58**, 3312–3327, doi:10.1175/1520-0469(2001)058<3312:EZFFIT>2.0.CO;2.Lorenz, D. J., and D. L. Hartmann, 2003: Eddy–zonal flow feedback in the Northern Hemisphere winter.

,*J. Atmos. Sci.***16**, 1212–1227, doi:10.1175/1520-0442(2003)16<1212:EFFITN>2.0.CO;2.Lutsko, N. J., I. M. Held, and P. Zurita-Gotor, 2015: Applying the fluctuation–dissipation theorem to a two-layer model of quasigeostrophic turbulence.

,*J. Atmos. Sci.***72**, 3161–3177, doi:10.1175/JAS-D-14-0356.1.Majda, A. J., R. Abramov, and M. Grote, 2005:

American Mathematical Society, 133 pp.*Information Theory and Stochastics for Multiscale Nonlinear Systems.*Martynov, R. S., and Yu. M. Nechepurenko, 2004: Finding the response matrix for a discrete linear stochastic dynamical system.

,*J. Comput. Math. Phys.***44**, 771–781, doi:10.1134/S0965542506070062.North, G. R., R. E. Bell, and J. W. Hardin, 1993: Fluctuation dissipation in a general circulation model.

,*Climate Dyn.***8**, 259–264, doi:10.1007/BF00209665.Penland, C., 1989: Random forcing and forecasting using principal oscillation pattern analysis.

,*Mon. Wea. Rev.***117**, 2165–2185, doi:10.1175/1520-0493(1989)117<2165:RFAFUP>2.0.CO;2.Polvani, L. M., and P. J. Kushner, 2002: Tropospheric response to stratospheric perturbations in a relatively simple general circulation model.

,*Geophys. Res. Lett.***29**, doi:10.1029/2001GL014284.Ring, M. J., and R. A. Plumb, 2007: Forced annular mode patterns in a simple atmospheric general circulation model.

,*J. Atmos. Sci.***64**, 3611–3626, doi:10.1175/JAS4031.1.Ring, M. J., and R. A. Plumb, 2008: The response of a simplified GCM to axisymmetric forcings: Applicability of the fluctuation–dissipation theorem.

,*J. Atmos. Sci.***65**, 3880–3898, doi:10.1175/2008JAS2773.1.Robinson, W. A., 1994: Eddy feedbacks on the zonal index and eddy–zonal flow interactions induced by zonal flow transience.

,*J. Atmos. Sci.***51**, 2553–2562, doi:10.1175/1520-0469(1994)051<2553:EFOTZI>2.0.CO;2.Robinson, W. A., 2000: A baroclinic mechanism for the eddy feedback on the zonal index.

,*J. Atmos. Sci.***57**, 415–422, doi:10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2.Sheshadri, A., and R. A. Plumb, 2016: Sensitivity of the surface responses of an idealized AGCM to the timing of imposed ozone depletion-like polar stratospheric cooling.

,*Geophys. Res. Lett.***43**, 2330–2336, doi:10.1002/2016GL067964.Sheshadri, A., R. A. Plumb, and D. I. V. Domeisen, 2014: Can the delay in Antarctic polar vortex breakup explain recent trends in surface westerlies?

,*J. Atmos. Sci.***71**, 566–573, doi:10.1175/JAS-D-12-0343.1.Sheshadri, A., R. A. Plumb, and E. P. Gerber, 2015: Seasonal variability of the polar stratospheric vortex in an idealized AGCM.

,*J. Atmos. Sci.***72**, 2248–2266, doi:10.1175/JAS-D-14-0191.1.Son, S.-W., and S. Lee, 2006: Preferred modes of variability and their relationship with climate change.

,*J. Climate***19**, 2063–2075, doi:10.1175/JCLI3705.1.Sparrow, S., M. Blackburn, and J. D. Haigh, 2009: Annular variability and eddy–zonal flow interactions in a simplified atmospheric GCM. Part I: Characterization of high- and low-frequency behavior.

,*J. Atmos. Sci.***66**, 3075–3094, doi:10.1175/2009JAS2953.1.Thompson, D. W. J., and J. M. Wallace, 2000: Annular modes in the extratropical circulation. Part I: Month-to-month variability.

,*J. Climate***13**, 1000–1016, doi:10.1175/1520-0442(2000)013<1000:AMITEC>2.0.CO;2.Thompson, D. W. J., J. M. Wallace, and G. C. Hegerl, 2000: Annular modes in the extratropical circulation: Part II: Trends.

,*J. Climate***13**, 1018–1036, doi:10.1175/1520-0442(2000)013<1018:AMITEC>2.0.CO;2.von Storch, H., T. Bruns, I. Fischer-Burns, and K. Hasselmann, 1988: Principal oscillation pattern analysis of the 30- to 60-day oscillation in general circulation model equatorial troposphere.

,*J. Geophys. Res.***93**, 11 022–11 036, doi:10.1029/JD093iD09p11022.Watterson, I. G., 2002: Wave–mean flow feedback and the persistence of simulated zonal flow vacillation.

,*J. Atmos. Sci.***59**, 1274–1288, doi:10.1175/1520-0469(2002)059<1274:WMFFAT>2.0.CO;2.Watterson, I. G., 2007: Southern “annular modes” simulated by a climate model—Patterns, mechanisms, and uses.

,*J. Atmos. Sci.***64**, 3113–3131, doi:10.1175/JAS4014.1.

^{1}

Here, and throughout this paper, we adopt the notation that diagonal matrices are represented by uppercase Greek characters.

^{2}

A methodology for ascribing variance explained to the POPs has been outlined in Gallagher et al. (1991).