1. Introduction

Ever since Thompson and Wallace (2000) described the atmospheric annular modes, there has been an ongoing debate over whether or not such modes are truly fundamental structures of atmospheric variability or are artifacts of statistical analyses, empirical orthogonal function (EOF) analysis in particular (cf. Deser 2000; Wallace 2000). The complexity of the atmospheric circulation, especially in the Northern Hemisphere, makes it difficult to resolve this issue using observations. For this reason, Cash et al. (2002, henceforth CKV) turned their attention to an idealized system, the zonally homogeneous climate of an atmospheric general circulation model (GCM) bounded below by a global mixed layer ocean. They found that annular-mode “events,” periods when the amplitude of this mode is exceptionally large, are characterized by flows that are strongly zonally asymmetric, or in other words, nonannular. Based on this and related analyses, CKV concluded that the nonannularity of the annular mode events is fundamental and “… that the annular mode in our model is a statistical feature, rather than a particular zonally symmetric mode of variability.” This note takes issue neither with CKV's model nor their analyses. It is, however, argued that their results are consistent, or at least not inconsistent, with an alternative interpretation: that the annular mode in their model is indeed fundamental, and that the zonally asymmetric appearance of their annular-mode events results from a superposition of annular-mode variations with statistically and dynamically independent nonannular variations. That individual annular-mode events appear to be nonannular, then, indicates only the omnipresence of nonannular low-frequency variability. It implies nothing about the dynamics of the annular modes. To use an analogy: if the bell of a trumpet is tapped while it is played, the resulting sound is a superposition of musical tones and taps, but this does not make the musical tones, however pianissimo, any less the true physical modes of the horn. This alternative interpretation of CKV's results is extreme, and, as shall be seen, is unlikely to be entirely correct. It is, however, consistent with the results they present. By way of illustration, analyses similar to those of CKV are performed on the output from a simpler system, a two-level model on the sphere, which is described in the next section. Section 3 describes the results, and a concluding section discusses some possible differences between the two-level model and that of CKV.

2. Model

The model is a two-level primitive equation model on the sphere, with highly simplified physics and coarse horizontal resolution (rhomboidal 15 spectral truncation). Its low-frequency variability, both annular and nonannular, has been analyzed previously (Hendon and Hartmann 1985; Robinson 1991a,b). While the model is simple, its output provides a convenient test bed for illustrating the potential for near independence between annular and nonannular low-frequency variability. The present results are obtained using a 2.5-day time scale for the linear drag on lower-level (750 hPa) winds, and a 15-day time scale for linear thermal damping toward an equilibrium temperature profile with a 60-K equator- to-pole thermal contrast. The model is integrated for 11 100 days, and the final 10 100 days of output are analyzed. Except where otherwise noted, output is filtered with a 10-day low-pass filter.

3. Annular variability and events

This model supports a robust annular mode (Robinson 1991b). The zonally averaged, 750-hPa streamfunction is strongly anticorrelated between middle and high latitudes. The greatest anticorrelation (−0.83) is between latitudes 42° and 69°. The standardized time series of differences between the zonally averaged streamfunctions at these two latitudes provides a convenient index for the annular-mode variability.

Following CKV, annular-mode events are defined as periods of 7 days or longer when the index is greater than +1 (“high index” events) or less than −1 (“low index events”). The left-hand panels of Fig. 1 show four such events: two high index and two low index. The 750-hPa anomalous streamfunction is multiplied by the Coriolis parameter and divided by the acceleration of gravity, so that the field displayed is approximately equivalent to the 750-hPa geopotential height. Similar to what was found by CKV, high- (Figs. 1a,c) and low- (Figs. 1e,g) index events in the two-level model are not especially annular. These maps suggest that annular- mode events involve inherently nonannular dynamical processes, or, perhaps, only that there is an independent field of nonannular variability superposed on the annular-mode variability. If the latter is true, equally valid anomalous flow fields can be constructed by adding the annular component of the flow from one date to the nonannular component from a randomly selected different date. This is essentially how the maps in the right- hand panels are generated. The nonannular components of the flow for days 1001 to 6000 are switched with those from days 6001 to 11 001 (the resulting fields are denoted “scrambled” output). For example, Fig. 2a shows the flow for day 1728, while Fig. 2b shows its annular component on day 1728 added to the nonannular component from day 6728. Because the annular-mode index is determined solely by the annular component, this is still a high-index event. The annular-mode events from these scrambled flows (Figs. 1b,d,f,h) are entirely plausible. The time evolutions of these scrambled events (not shown) are equally plausible. In short, visual evidence is consistent with the idea that annular and nonannular components of the low-frequency flow undergo independent and unrelated variability.

Figure 2 (after CKV's Fig. 10) shows one-point correlation statistics for the two-layer model that summarize the structure of its low-frequency variability. These are obtained by averaging the separate one-point correlation statistics for each of the 48 base-point longitudes. The ordinate of Fig. 2c refers to the longitude relative to the base point. While Fig. 2 differs quantitatively from CKV's Fig. 10, they are qualitatively similar. Teleconnectivity is greatest between low and high and middle latitudes, and, except for base points between 50° and 60°, the teleconnection patterns are primarily dipoles, with nearly meridional axes.

These results are essentially identical when calculated using the scrambled output. In fact, this must be the case, for this model, for that of CKV, or for any zonally homogeneous model. In such a model, a pattern of nonannular variability can be rotated through any arbitrary angle about the pole, and, over a sufficiently long run, all such orientations are sampled equally. Correlations between annular and nonannular components of the flow appear only because of the limited size of the sample. Thus, the one-point correlations shown here and by CKV are linear combinations of the separate one-point correlations within the annular and nonannular components of the flow, each weighted by its relative contributions to the total root-mean-square variance. These one-point correlations cannot, by construction, reveal anything about the dynamical interdependence of annular and nonannular variability in a zonally homogeneous model. The same is true for the EOFs.

Significant correlations between annular variations and higher moments, such as the variance, of nonannular variations, are not excluded. The dashed curve in Fig. 3 shows the correlation, over time, of the annular mode index with the zonally averaged variance of the low- frequency streamfunction. These correlations are small, but not zero. As mentioned in the introduction, the extreme interpretation, that annular and nonannular variations are completely independent is, therefore, rejected. That the correlations are small, however, suggests that the interdependence between these types of variability is weak. Moreover, the correlations are much stronger between the annular-mode index and the zonally averaged variance of the high-frequency streamfunction, shown by the solid curve in Fig. 3. (The high-frequency streamfunction is the difference between the total and low-pass-filtered fields.) Figure 3 implies a strong dynamical interdependence between the annular mode and the high-frequency eddies, but a much weaker connection with the low-frequency eddies. These aspects of annular-mode variability in this model were described previously (Robinson 1991b). Whether similar conclusions hold for the dynamics of the annular mode in CKV's model cannot be determined from the results they display.

4. Discussion

CKV's results are consistent with the idea that the low-frequency variability in their model is a superposition of dynamically independent annular and nonannular variations. If this is true, the annular mode should be considered a real physical entity, and not merely a statistical artifact. In the present, much simpler, two- level model, annular and nonannular low-frequency variability are largely but not entirely, independent. Whether or not this is true for CKV's model is not known. It is hoped this note will motivate CKV to perform the additional analyses, corresponding, for example, to the present Fig. 3, that can provide a definitive answer to this question.

How is it that annular and nonannular variability do or do not interact dynamically? In the two-level model, nonannular low-frequency variability comprises trains of quasi-stationary Rossby waves. These occur all the time, with only weak fluctuations in their amplitudes and structures (Robinson 1991a). If low-frequency eddies occur in intermittent bursts, however, they will, according to eddy–mean flow interaction theory, modify the annular flow, and annular and nonannular variations will be interdependent. Annular and nonannular low- frequency structures may also interact through their influences on and responses to high-frequency eddies. This does not occur in the two-level model, because the high-frequency eddy feedback is nearly linear in the strength of the low-frequency flow (Robinson 1991a). The interactions of high-frequency eddies with the annular and nonannular components of the low-frequency flow may, therefore, be considered separate and independent. Two levels and rhomboidal 15 spectral truncation are not, however, sufficient to resolve the breaking of baroclinic eddies, so this linearity may be an artifact of the coarse resolution.

There is, therefore, reason to expect that annular and nonannular variability interact more strongly in CKV's model and in the atmosphere than in the two-level model. These interactions may be strong enough to support CKV's conclusion that the annular mode is not a dynamically useful concept, but this has not yet been demonstrated. It can be confirmed or refuted only by analyzing the dynamics of the variability. The final point of this note then, is to express the view that we have reached or surpassed the limit of what can be learned by purely statistical analyses of low-frequency variability. Further progress requires work that explicitly addresses its dynamics.