1. Introduction
At the intraseasonal to interannual time scales, the variability of the large-scale atmospheric circulation in the midlatitudes of both hemispheres is dominated by the “annular modes,” which are usually defined based on empirical orthogonal function (EOF) analysis of zonal-mean meteorological fields (e.g., Kidson 1988; Thompson and Wallace 1998, 2000; Lorenz and Hartmann 2001, hereafter LH01, 2003; Thompson and Woodworth 2014; Thompson and Li 2015). The barotropic annular modes are often derived as the first (i.e., leading) EOF (EOF1) of zonal-mean zonal wind, which exhibits a dipolar meridional structure and describes a north–south meandering of the eddy-driven jet. Note that in this paper, the focus is on the barotropic annular modes, hereafter simply called annular modes [see Thompson and Woodworth (2014), Thompson and Barnes (2014), and Thompson and Li (2015) for discussions about the “baroclinic annular modes”]. The second EOF of zonal-mean zonal wind (EOF2) has a tripolar meridional structure centered on the jet, describing a strengthening and narrowing of the eddy-driven jet (i.e., jet pulsation). By construction, EOF1 and EOF2 (and any two EOFs) are orthogonal and their associated time series [i.e., principal components (PCs)], sometimes called zonal index, are independent at zero time lag.
The persistence of the annular mode (EOF1) and its underlying dynamics have been the subject of extensive research and debate in the past three decades (e.g., Robinson 1991; Branstator 1995; Feldstein and Lee 1998; Robinson 2000; Limpasuvan and Hartmann 1999; LH01; Lorenz and Hartmann 2003; Gerber and Vallis 2007; Gerber et al. 2008b; Chen and Plumb 2009; Simpson et al. 2013; Zurita-Gotor 2014; Nie et al. 2014; Byrne et al. 2016; Ma et al. 2017; Hassanzadeh and Kuang 2019). Many of the aforementioned studies have pointed to a positive eddy–zonal flow feedback mechanism as the source of the persistence: The zonal wind, meridional wind, and temperature anomalies associated with the annular mode (EOF1) modify the generation and/or propagation of the synoptic eddies at the quasi-steady limit (on long time scales) in such a way that the resulting eddy fluxes reinforce the annular mode (see Hassanzadeh and Kuang 2019, and the discussion and references therein). Most notably, LH01 developed a linear eddy–zonal flow feedback model for the annular modes by regressing the anomalous eddy momentum flux divergence onto the zonal index of EOF1 [z 1(t)] and interpreting positive correlations between z 1(t) and regressed momentum flux divergence [m 1(t)] at long lags (greater than 7 days) as evidence for positive eddy–zonal flow feedbacks, i.e., positive feedbacks of EOF1 onto itself. 1 LH01 developed a similar model separately for EOF2 and found weak eddy–zonal flow feedbacks for EOF2, consistent with the longer persistence of EOF1 compared to EOF2. Such single-EOF eddy–zonal flow feedback models have been used in most of the subsequent studies of the annular modes (e.g., Lorenz and Hartmann 2003; Simpson et al. 2013; Lorenz 2014; Robert et al. 2017; Ma et al. 2017; Boljka et al. 2018; Hassanzadeh and Kuang 2019; Lindgren et al. 2020).
While EOF1 and EOF2 are independent at zero lag, a few previous studies have pointed out that these two EOFs can be correlated at long lags (i.e., greater than 7 days), and that in fact the combination of these two leading EOFs represents coherent meridional propagations of the zonal-mean flow anomalies. Such propagating regimes have been observed in both hemispheres in reanalysis data (e.g., Feldstein 1998; Feldstein and Lee 1998; Sheshadri and Plumb 2017). Anomalous poleward propagation of zonal wind typically emerges in low latitudes and mainly migrate poleward over a few months, although nonpropagating regimes can also appear in some instances [see Fig. 1 of Sheshadri and Plumb (2017) and Fig. 6 in this paper]. Similar behaviors have also been reported by in general circulation models (GCMs) (e.g., James and Dodd 1996; Son and Lee 2006; Son et al. 2008; Sparrow et al. 2009; Sheshadri and Plumb 2017). Son and Lee (2006) found that the leading mode of variability in an idealized dry GCM can be either the propagating or nonpropagating regime depending on the details of thermal forcing imposed in the model. They also found that unlike the nonpropagating regimes, the z 1 and z 2 of the propagating regimes are strongly correlated at long lags, peaking at around 48 days (see their Fig. 3 and Fig. 4b of the present paper). The statistically significant cross correlation between z 1 and z 2 at long lags indicates that positive PC1 leads to positive PC2, and positive PC2 leads to negative PC1. Given the relative structure of the EOFs, this cross correlation describes the poleward propagation of zonal wind anomalies. Furthermore, Son and Lee (2006) reported that nonpropagating regimes are often characterized by a single time-mean jet with a dominant EOF1 (in terms of the explained variance) while the propagating regimes are characterized by a double time-mean jet in the midlatitudes with the variance associated with EOF2 being at least half of the variance of EOF1. Such differences in the structure and position of the jet can affect the propagation of Rossby waves by changing the critical latitude dynamics (e.g., Lee et al. 2007; Ronalds et al. 2018), resulting in different eddy–mean flow interaction characteristics and hence, different characteristics of the annular modes. Furthermore, Son et al. (2008) found the e-folding decorrelation time scale of z 1 in the propagating regime to be much shorter than that of the nonpropagating regime. The long e-folding decorrelation time scales for the annular modes in the nonpropagating regime were attributed to an unrealistically strong positive EOF1-onto-EOF1 feedback, while the reason behind the reduction in the persistence of the annular modes in the propagating regime remained unclear.
More recently, Sheshadri and Plumb (2017) presented further evidence for the existence of propagating and nonpropagating regimes and strong lagged correlations between z 1 and z 2 in reanalysis data of the Southern Hemisphere (SH) and in idealized GCMs. Moreover, they elegantly showed, using a principal oscillation pattern (POP) analysis (Hasselmann 1988; Penland 1989), that EOF1 and EOF2 are in fact manifestations of a single, decaying-oscillatory coupled mode of the dynamical system. Specifically, they found that EOF1 and EOF2 are, respectively, the real and imaginary parts of a single POP mode, which describes the dominant aspects of the spatiotemporal evolution of zonal wind anomalies. Sheshadri and Plumb (2017) also showed that in the propagating regime, the autocorrelation functions of z 1 and z 2 decay nonexponentially.
Given the above discussion, a single-EOF model is not enough to describe a propagating regime because the EOF1 and EOF2 in this regime are strongly correlated at long lags and the autocorrelation functions of the associated PCs do not decay exponentially (but rather show some oscillatory behaviors as well). From the perspective of eddy–zonal flow feedbacks, one may wonder whether there are cross-EOF feedbacks in addition to the previously studied EOF1 (EOF2) eddy–zonal flow feedback onto EOF1 (EOF2) in the propagating regime. In cross-EOF feedbacks, EOF1 (EOF2) changes the eddy forcing of EOF2 (EOF1) in the quasi-steady limit. Therefore, there is a need to extend the single-EOF model of LH01 and build a model that includes, at a minimum, both leading EOFs and accounts for their cross feedbacks. The objective of the current study is to develop such a model and to use it to estimate effects of the cross-EOF feedbacks on the variability of propagating annular modes.
The paper is structured as follows: Section 2 compares the characteristics of z 1, z 2, m 1, and m 2 for the nonpropagating and propagating annular modes in reanalysis and idealized GCMs. In section 3, we develop a linear eddy–zonal flow feedback model that accounts for cross-EOF feedbacks, validate the model using synthetic data from a stochastic prototype, discuss the key properties of the analytical solution of this model, and apply this model to data from reanalysis and an idealized GCM. The paper ends with concluding remarks in section 4.
2. Propagating annular modes in an idealized GCM and reanalysis
In this section, we will examine the basic characteristics and statistics of propagating annular modes in an idealized GCM (the dry dynamical core) and reanalysis. We focus on the southern annular mode, which makes it easier to compare the results of the reanalysis and the idealized GCM simulations. We will start with the idealized GCM to demonstrate the characteristics of the propagating and nonpropagating annular modes.
a. An idealized GCM: The dry dynamical core
We use the Geophysical Fluid Dynamics Laboratory (GFDL) dry dynamical core GCM. The GCM is run with a flat, uniform lower boundary with T63 spectral resolution and 40 evenly spaced sigma levels in the vertical for 50 000-day integrations after spinup. The physics of the model is based on Held and Suarez (1994), an idealized configuration for generating a realistic global circulation with minimal parameterization (Held 2005; Jeevanjee et al. 2017). All diabatic processes are represented by Newtonian relaxation of the temperature field toward a prescribed equilibrium profile, and Rayleigh friction is included in the lower atmosphere to mimic the interactions with the boundary layer.
The nonpropagating and propagating regimes are produced in two slightly different setups of this model. For the setup with the nonpropagating regime, we use the standard configuration of Held and Suarez (1994), which employs an analytical profile approximating a troposphere in unstable radiative–convective equilibrium and an isothermal stratosphere for Newtonian relaxation. For the setup with the propagating regime, we follow an approach similar to the one used by Sheshadri and Plumb (2017). In this setup, for the equilibrium temperature profile in the troposphere and stratosphere, we use the perpetual-solstice version of the equilibrium-temperature specifications used in Lubis et al. (2018a), calculated from a Rapid Radiative Transfer Model (RRTM), with winter conditions in the SH. As will be seen later, these choices result in a large-scale circulation with overall reasonable annular mode time scales in the SH.
In Fig. 1, we show, following Son and Lee (2006), the one-point lag-correlation maps for the zonal-mean zonal wind anomalies integrated across the depth of the troposphere (1000–100 hPa)
One-point lag-correlation maps of the zonal-mean zonal wind anomalies integrated across the depth of the troposphere (1000–100 hPa)
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Figure 2 shows lagged-correlation analysis between z and m in the GCM setup with nonpropagating regime. The autocorrelation of z 1, as discussed in past studies (e.g., Chen and Plumb 2009; Ma et al. 2017), has a noticeable shoulder at around 5-day lags and shows an unrealistically persistent annular mode, well separated from the faster decaying z 2, which is consistent with the considerable difference in the contribution of the two EOFs to the total zonal wind variance (60.2% vs 19.2%). The e-folding decorrelation time scales of z 1 and z 2 are 64.5 and 4.8 days, respectively. The strong, positive cross correlations of m 1 z 1 and insignificant cross correlations of m 2 z 2 at large positive lags suggest the existence of a positive eddy–zonal flow feedback for EOF1 (from EOF1) but not for EOF2 (from EOF2) (see Son et al. 2008; Ma et al. 2017). Figure 2b shows that the z 1 z 2 cross correlations are weak at positive and negative lags, which are consistent with the one-point lag-correlation map of Figs. 1a and 3 (shown later), and are indicative of a nonpropagating regime, as reported previously for a similar setup (Son and Lee 2006; Son et al. 2008). The m 1 z 2 and m 2 z 1 cross correlations are small and often insignificant, suggesting the absence of the cross-EOF feedbacks in the nonpropagating regime (Figs. 2e,f). All together, the above analysis shows that for the nonpropagating regime, single-EOF reduced-order models such as LH01 are sufficient.
Lagged-correlation analysis of the GCM setup with nonpropagating regime. (a) Autocorrelation of z 1 (blue) and z 2 (red), (b) cross-correlation z 1 z 2, (c) cross-correlation m 1 z 1, (d) cross-correlation m 2 z 2, (e) cross-correlation m 1 z 2, and (f) cross-correlation m 2 z 1. The two leading EOFs contribute 60.2% and 19.2%, respectively, to the total variance. The e-folding decorrelation time scales of z 1 and z 2 are 64.5 and 4.8 days, respectively. Gray shading represents 5% significance level according to the test of Bartlett (appendix).
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Anomalous zonal-mean zonal wind (
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
The weak cross correlations between z 1 and z 2 in the GCM with the nonpropagating regime (Fig. 2b) can be also seen by regressing the zonal-mean zonal wind anomalies on the zonal index at 0- and 20-day time lags. Figures 3a and 3b show the wind anomalies regressed on z 1 and z 2 at lag 0, yielding approximately the EOF1 and EOF2 patterns, respectively. Twenty days after z 1 leads zonal wind anomalies, the anomalies do not drift poleward or decay, but rather persist (Fig. 3d). In contrast, 20 days after z 2 leads zonal wind anomalies, the anomalies decay and disappear (Fig. 3c). These observations are consistent with the long and short persistence of z 1 and z 2, respectively, consistent with the weak cross correlations of z 1 and z 2 at positive or negative lags, and as it becomes clear below, consistent with the nonpropagating nature of this setup.
Figure 4 shows lagged-correlation analysis between z and m in the GCM setup with propagating regime. The autocorrelation of z 1, its persistence compared to that of z 2, and the explained variance by the two EOFs (40.4% versus 32.5%) are much more similar to what is observed in the SH (shown later in Fig. 7). The e-folding decorrelation time scales of z 1 and z 2 are 14.1 and 9.2 days, respectively. Figure 4b shows that z 1 and z 2 are strongly correlated at long lags peaking at around ±20 days. This behavior along with the one-point lag-correlation map of Fig. 1b and regression map of wind anomalies (Fig. 5, shown later) suggests the existence of a propagating regime, as noted by few previous studies (e.g., Son and Lee 2006; Sheshadri and Plumb 2017). It should be noted that Son and Lee (2006) have proposed a rule of thumb based on the ratio of the explained variance of EOF2 to EOF1: a nonpropagating (propagating) regimes exists if the ratio is smaller (larger) than 0.5. The regimes of our two setups are consistent with this rule of thumb as the ratios are ~0.3 and ~0.8 in our nonpropagating and propagating regimes.
Lagged-correlation analysis of the GCM setup with propagating regime. (a) Autocorrelation of z 1 (blue) and z 2 (red), (b) cross-correlation z 1 z 2, (c) cross-correlation m 1 z 1, (d) cross-correlation m 2 z 2, (e) cross-correlation m 1 z 2, and (f) cross-correlation m 2 z 1. The two leading EOFs contribute 40.4% and 32.5%, respectively, to the total variance. The e-folding decorrelation time scales of z 1 and z 2 are 14.1 and 9.2 days, respectively. Gray shading represents 5% significance level according to the test of Bartlett (appendix).
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Anomalous zonal-mean zonal wind (
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Furthermore, Fig. 4c shows that the m 1 z 1 cross correlations are positive at long positive lags (5–20 days) and then negative but small. Figure 4d indicates weak positive cross correlations at lags 5–10 days and weak negative cross correlations at the times scale of longer than 20 days between z 2 and m 2 (Fig. 4c). Overall, the shape of the m 1 z 1 and m 2 z 2 cross-correlation functions are similar between the nonpropagating and propagating regimes, although the m 1 z 1 cross correlations are larger and more persistent in the nonpropagating regime. In contrast, the m 1 z 2 and m 2 z 1 cross correlations are substantially different between the two regimes (Figs. 4e,f). There are statistically significant and large positive m 1 z 2 cross correlations at large positive lags (>5 days) and statistically significant and large negative m 2 z 1 cross correlations at positive lags up to 30 days. Note that as emphasized in the figures, positive lags here mean that z 1 (z 2) is leading m 2 (m 1). Therefore, these cross correlations, as discussed later, indicate the existence of cross-EOF feedbacks in the propagating regime.
Figure 5 shows anomalous zonal-mean zonal wind regressed on z 1 and z 2 at 0- and 20-day time lags in the GCM setup with propagating regime. Figures 5a and 5b show the wind anomalies regressed on z 1 and z 2 at lag 0, again yielding approximately the EOF1 and EOF2 patterns, respectively. As shown in Fig. 5c, 20 days after z 2 leads zonal wind anomalies, the anomalies have drifted poleward and project strongly onto the structure of wind anomalies associated with EOF1 (Figs. 5a,c; pattern correlation = 0.93). This is consistent with positive correlation of z 1 z 2 at lag +20 days when z 1 leads z 2 (Fig. 4b). Likewise, 20 days after z 1 leads zonal wind anomalies, the anomalies (of Fig. 5a) have drifted poleward and project strongly onto the structure of anomalies associated with EOF2, but with an opposite sign (Figs. 5b,d; pattern correlation = −0.85). This is consistent with negative correlation of z 1 z 2 when z 2 leads z 1 by 20 days (Fig. 4b).
Overall, these results suggest the existence of cross-EOF feedbacks in the propagating annular mode. In section 3, we develop a model to quantify these four feedbacks and understand the effects of their magnitude and signs on the variability (e.g., persistence) of z 1 and z 2. But first, we will examine the variability and characteristics of z and m in reanalysis. In particular, we will see that the z and m cross correlations in the GCM’s propagating regime well resemble those in the SH reanalysis data.
b. Reanalysis
We use the 1979–2013 data from the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim; Dee et al. 2011). Zonal and meridional wind components (u, υ) are 6 hourly, on 1.5° latitude × 1.5° longitude grid, and on 21 vertical levels between 1000 and 100 hPa. Anomalies used for computing correlations and EOF analyses are defined as the deviations from the climatological seasonal cycle. The mean seasonal cycle is defined as the annual average and the first four Fourier harmonics of the 35-yr daily climatology.
Figure 6 shows a one-point lag-correlation map of vertically averaged zonal-mean zonal wind
One-point lag-correlation maps of the vertically averaged zonal-mean zonal wind anomalies from year-round ERA-Interim data integrated across the depth of the troposphere (1000–100 hPa) (
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Figure 7a shows the autocorrelations of z 1 and z 2. Consistent with LH01, the estimated decorrelation time scales of these two PCs are 10.3 and 8.1 days, respectively. Figure 7b depicts the cross-correlation z 1 z 2, showing statistically significant and relatively strong correlations that peak around ±10 days. As discussed in earlier studies, such lagged correlations are a signature of the propagating annular modes (Feldstein and Lee 1998; Son and Lee 2006; Son et al. 2008; Sheshadri and Plumb 2017), implying that the period of the poleward propagation is about 20–30 days in the SH (Fig. 7b), consistent with Sheshadri and Plumb (2017) and with Fig. 6.
Lagged-correlation analysis for the Southern Hemisphere, calculated from year-round ERA-Interim data. (a) Autocorrelations of z 1 (blue) and z 2 (red), (b) cross-correlation z 1 z 2, (c) cross-correlation m 1 z 1, (d) cross-correlation m 2 z 2, (e) cross-correlation m 1 z 2, and (f) cross-correlation m 2 z 1 at different lags. The two leading EOFs contribute to 45.1% and 23.2% of the total variance, respectively. The e-folding decorrelation time scales of z 1 and z 2 are 10.3 and 8.1 days, respectively. Gray shading represents 5% significance level according to the test of Bartlett (appendix).
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
To understand the effects of z 1 and z 2 on m 1 and m 2, we also examine the cross correlations between z and m at different lags (Figs. 7c–f). The shape and the magnitude of the m 1 z 1 and m 2 z 2 cross correlations (Figs. 7c,d) are similar to those originally shown by LH01 (see their Figs. 5 and 13a) and later by many others using different reanalysis products and time periods. As discussed in LH01, the statistically significant positive m 1 z 1 cross correlations at long positive lags (~8–20 days) and the insignificant m 2 z 2 cross correlations for time scales longer than ~5 days are indicative that a positive eddy–zonal flow feedback exists only for EOF1, but not for EOF2 (also see Byrne et al. 2016; Ma et al. 2017). We emphasize that this positive feedback is from EOF1 onto itself.
To see if there are cross-EOF feedbacks, in Figs. 7e and 7f we plot the m 1 z 2 and m 2 z 1 cross correlations at different lags. The m 1 z 2 cross correlations show statistically significant positive correlations at large positive lags, signifying that a cross-EOF feedback, i.e., z 2 modifying m 1, is present. Note that the magnitude of the m 1 z 2 cross correlations at positive lags is overall larger than those of m 1 z 1 (Fig. 7c). There are also statistically significant but negative m 2 z 1 correlations at large positive lags, again suggesting the existence of a cross-EOF feedback, i.e., z 1 modifying m 2. These results indicate that in the presence of the propagating regime in the SH, there are indeed cross-EOF feedbacks; however, these feedbacks were always ignored in the previous studies and reduced-order models of the SH extratropical large-scale circulation.
3. Eddy–zonal flow feedbacks in the propagating annular modes: Model and quantification
In this section, an eddy–zonal flow feedback model that accounts for the coupling of the leading two EOFs and their feedbacks, including the cross-EOF feedbacks, is introduced. Then this model is validated using synthetic data from a simple stochastic prototype, and from its analytical solution, we derive conditions for the existence of the propagating regime. Finally, we use this model to estimate the feedback strengths of the propagating annular modes in data from the reanalysis (SH) and the idealized GCMs.
a. Developing an eddy–zonal flow feedback model for propagating annular modes
Assuming a linear representation for the feedback of an EOF onto itself, LH01 and later studies wrote
Note that if one attempts to find b
11 using a single-EOF approach such as LH01, then, from Eq. (7), one would be implicitly assuming that
The basic assumptions of our model, Eqs. (4)–(10), are similar to those of the LH01 model: (i) a linear representation of the feedbacks is sufficient, and (ii) the eddy forcing m does not have long-term memory independent of the variability in the jet (represented by z
1 and z
2). The second assumption means that at sufficiently large positive lags (beyond the time scales over which there is significant autocorrelation in
In the following section, we will present a proof of concept for this eddy–zonal flow feedback model using synthetic data obtained from a simple stochastic prototype and show that using Eqs. (9) and (10), the prescribed feedbacks can be accurately backed out.
b. Validation using synthetic data from a simple stochastic prototype
Synthetic time series of z 1, z 2, m 1, and m 2 are produced by numerically integrating Eqs. (4) and (5), (7) and (8), and (11) and (12) forward in time with two different sets of prescribed b 11, b 22, b 12, and b 21. In the first set, b 12 > 0, b 22 = 0, and there is no cross-EOF feedback, i.e., b 12 = b 21 = 0 (Table 3). In the second set, b 11 and b 22 are the same as those in the first set, but here, there is cross-EOF feedback, i.e., b 12 and b 21 ≠ 0 (Table 3). For both sets, we use τ 1 = τ 2 = 8 days. The values of b and τ are reasonably chosen based on the observed values in the SH (see Table 4).
Spectral analysis of z 1,2 and m 1,2 shows that the synthetic data indeed have characteristics similar to those of the observed SH. For example, for the case with cross feedbacks (Fig. 8), we find that consistent with observations [see Fig. 4 of LH01 or Fig. 3 of Ma et al. (2017)], the time scales of z 1 and z 2 are much longer (i.e., slower variability) than m 1 and m 2, and the power spectra of z can be interpreted, to the first order, as reddening of the power spectra of eddy forcing m (LH01; Ma et al. 2017). The power spectra of eddy forcings m 1 and m 2 have in general broad maxima centered at the synoptic frequency (with m 1 also peaking at low frequency), consistent with observations. Given that the characteristics of the synthetic data mimic the key characteristics of the observed annular modes, we use this idealized framework to validate the lagged-correlation approach of Eqs. (9) and (10) for quantifying eddy–zonal flow feedbacks.
Spectra of z 1,2 and m 1,2 from the synthetic data with cross-EOF feedbacks. Black lines show the power spectra of (a) z 1, (b) z 2, (c) m 1, and (d) m 2. The red-noise spectra are indicated by the smooth solid red curves, and the smooth dashed blue lines are the 5% and 95% a priori confidence limits.
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Figure 9 shows the lagged-correlation analysis of the synthetic data without cross-EOF feedbacks. It is clearly seen that the only noticeable cross correlations are that of m 1 z 1, and there are no (statistically significant) cross correlations between z 1 z 2, m 1 z 2 and m 2 z 1 at any lag, consistent with a nonpropagating regime and the absence of cross-EOF feedbacks (Fig. 2). Using Eqs. (9) and (10) and lag l = 8–20 days, we can closely estimate the prescribed feedback parameters, i.e., b 11 = 0.04 day−1 and b 22 = b 12 = b 21 = 0 (see Table 3).
Lagged-correlation analysis of synthetic data without cross-EOF feedbacks. (a) Autocorrelation of z 1 (blue) and z 2 (red), (b) cross-correlation z 1 z 2, (c) cross-correlation m 1 z 1, (d) cross-correlation m 2 z 2, (e) cross-correlation m 1 z 2, and (f) cross-correlation m 2 z 1. The e-folding decorrelation time scales of z 1 and z 2 are 18.6 and 9.2 days, respectively. Gray shading represents 5% significance level according to the test of Bartlett (appendix).
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Figure 10 shows the lagged-correlation analysis of the synthetic data with cross-EOF feedbacks. First, we see that there are statistically significant and often large cross correlations in z 1 z 2, m 1 z 1, m 1 z 2, and m 2 z 1, with the shape of the cross-correlation distributions not that different from that of the SH reanalysis and the idealized GCM setup with propagating regime (Figs. 4 and 7). The positive m 1 z 1 and near zero m 2 z 2 cross correlations at large positive lags signify a positive z 1-onto-z 1 feedback through m 1, but no z 2-onto-z 2 feedback through m 2, consistent with the prescribed b 11 > 0 and b 22 = 0. In addition, Figs. 10e and 10f also show that there are statistically significant and large correlations in m 1 z 2 and m 2 z 1 at positive lags, consistent with the introduction of cross-EOF feedbacks by setting b 12 = 0.06 day−1 and b 21 = −0.025 day−1. The positive m 1 z 2 cross correlations at positive lags are higher than those of m 1 z 1 (note that b 12/b 11 ≈ 1.5), and the sign of m 2 z 1 cross correlations is opposite to the sign of m 1 z 2 cross correlations (note that b 12 b 21 < 0). Using Eqs. (9) and (10) and lag l = 8–20 days, we can again closely estimate the prescribed feedback parameters, including the strength of the cross-EOF feedbacks (see Table 3).
Lagged-correlation analysis of synthetic data with cross-EOF feedbacks. (a) Autocorrelation of z 1 (blue) and z 2 (red), (b) cross-correlation z 1 z 2, (c) cross-correlation m 1 z 1, (d) cross-correlation m 2 z 2, (e) cross-correlation m 1 z 2, and (f) cross-correlation m 2 z 1. The e-folding decorrelation time scales of z 1 and z 2 are 13.9 and 6.5 days, respectively. The regions outside the gray shading indicate 95% significance level according to the test of Bartlett (appendix).
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
The above analyses validate the approach using Eqs. (9) and (10) for quantifying the feedback strengths b jk in data from both propagating and nonpropagating regimes. Furthermore, a closer examination of z 1 and z 2 autocorrelations in Figs. 9a and 10a show that both z 1 and z 2 in the case without cross-EOF feedbacks are more persistent than those in the case with cross-EOF feedbacks; e.g., the e-folding decorrelation time scale of z 1 is 18.6 days in Fig. 9a while it is 13.9 days in Fig. 10a. This observation might be counterintuitive because both cases have the same b 11 > 0 while the case with cross-EOFs feedback has b 12 > 0, which might seem like another positive feedback that should further prolong the persistence of z 1. Finally, we notice that b 12 b 21 < 0 in Table 3 and in the SH reanalysis and idealized GCM setup with the propagating regime (Tables 4 and 5). Synthetic data generated with the same parameters as in Table 2 but with the sign of b 21 flipped results in cross-correlation distributions that are vastly different from those of Fig. 10 and what is seen in the SH reanalysis and idealized GCM. Inspired by these observations, next we examine the analytical solution of the deterministic version of Eqs. (4) and (5) and Eqs. (7) and (8) to better understand the impacts of the strength and sign of b jk on the variability and in particular the persistence of z 1 and z 2.
Prescribed and estimated feedback strengths (day−1) in synthetic data for the case without cross-EOF feedbacks. The imposed damping rates of friction are τ 1 = τ 2 = 8 days. The values of b and τ are motivated by the observed ones; see Table 4.
Prescribed and estimated feedback strengths (day−1) in synthetic data for the case with cross-EOF feedbacks. The imposed damping rates of friction are τ 1 = τ 2 = 8 days. The values of b and τ are motivated by the observed ones; see Table 4.
c. Analytical solution of the two-EOF eddy–zonal flow feedback model
To understand the effects of the feedback strength b jk on the persistence of z j , we compute the analytical solutions for 5 systems that have the same b 11 > 0 and b 22 = 0 (Table 3): in EXP1, there is no cross-EOF feedback (b 12 = b 21 = 0), while in EXP2–EXP5, b 12 > 0 and b 21 < 0 and they have been doubled from experiment to experiment. Figure 11 shows the autocorrelation coefficients of z 1 and their e-folding decorrelation time scales for EXP1–EXP5. EXP1, corresponding to nonpropagating regimes, has the slowest-decaying autocorrelation function, i.e., longest e-folding decorrelation time scale (Figs. 11a,b). EXP2–EXP5, which all satisfy the condition of Eq. (20), have faster-decaying autocorrelation functions, i.e., shorter e-folding decorrelation time scale, consistent with our earlier results in idealized GCM and stochastic prototype (Figs. 4 and 10). As discussed above, in the propagating regime, the eigenvectors and the corresponding eigenvalues are complex and thus, z 1,2 do not decay just exponentially, but rather show some oscillatory characteristics too [Fig. 11a, Eqs. (26) and (27)]. Since the frequency of these oscillations ω [Eq. (24)] increases as the cross-EOF feedback strengths increase, shorter time scales in z 1 are expected in the experiment with stronger b 12 b 21 (Fig. 11b).
Prescribed feedback strengths (day−1) used to analyze the impact of cross-EOF feedbacks on the decorrelation time scales of z 1 and z 2. The imposed damping rates of friction are τ 1 = τ 2 = 8 days.
(a) Autocorrelation functions of z 1 and (b) their corresponding e-folding decorrelation time scales from the analytical solutions [Eqs. (22)–(25)] for the experiment with no cross-EOF feedback (EXP1) and the experiments with increasing cross-EOF feedback strength (EXP2–EXP5). The prescribed feedback strength b jk are shown in Table 3.
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
The dependence of the e-folding decorrelation time scales of z 1 and z 2 on the feedback strengths, and in particular the cross-EOF feedback strengths, is further evaluated in Fig. 12. In Fig. 12a, it is clearly seen that the impact of increasing b 11 > 0 in the propagating regime (filled symbols) is to increase the persistence, i.e., decorrelation time scale, of z 1, consistent with increasing the positive eddy–zonal flow feedback (z 1-onto-z 1 through m 1). However, when the feedback is further increased to twice the control value, the condition of Eq. (20) for the existence of a decaying-oscillatory solution is not satisfied anymore, and consistent with this, we see that the system undergoes a transition to the nonpropagating regime. Further increasing b 11 leads to substantially more persistent z 1 and less persistence z 2. Note that in nonpropagating regimes when b 12 b 21 ≠ 0, the decay of z 2 depends on b 11 too [see Eq. (18)].
The computed e-folding decorrelation time scale (days) of z 1 (blue circles) and z 2 (red squares) as a function of feedback strengths (day−1). The impact of varying (a) b 11, (b) b 12, and (c) b 12 and b 21 on the decorrelation time scale (the y axis) while all other b jk are kept the same. The x axis shows the value of varied b jk as fraction of the value in EXP2 (Table 3); the vertical dashed line indicates the control values. (d) The impact of varying b 11 in EXP1 (Table 3). The filled circles and squares indicate that the parameters satisfy the condition for propagating regimes, i.e., existence of decaying-oscillatory solutions [Eq. (20)].
Citation: Journal of the Atmospheric Sciences 78, 1; 10.1175/JAS-D-20-0214.1
Figure 12b shows that in the propagating regime, unlike increasing b 11 > 0, increasing b 12 > 0 leads to reduction in the persistence of z 1. This is the counterintuitive behavior we had observed earlier in the stochastic prototype (section 3b). Now we understand that this is because increasing b 12 increases the frequency of the oscillation ω [Eq. (24)] in the system, resulting in reduction in the decorrelation time scale of z 1 (and z 2); also see Fig. 11. Such impact can even be more pronounced when both cross-EOF feedbacks b 12 and b 21 are increased (Fig. 12c), leading to shorter decorrelation time scales. Because a positive b 12 decreases the persistence of z 1, we do not refer to is as a “positive feedback.” To understand this behavior, we have to keep in mind that in the eddy forcing of z 1 (z 2), i.e., m 1 in Eq. (7) [m 2 in Eq. (8)], b 12 > 0 (b 21 < 0) is the coefficient of z 2 (z 1). When z 2 leads z 1, they are negatively correlated (Figs. 4b, 7b, and 10b), thus z 2 multiplied by b 12 > 0 reduces m 1 that is forcing z 1, decreasing the persistence of z 1. Similarly, when z 1 leads z 2, they are positively correlated, thus z 1 multiplied by b 21 < 0 reduces m 2 and thus the persistence of z 2.
Finally, for the sake of completeness, we also examine the effect of increasing b 11 in the absence of cross-EOF feedback (Fig. 12d). As expected, increasing b 11 leads to increasing the persistence of z 1 and has no impact on the persistence of z 2 as now z 1 and z 2 are completely decoupled.
d. Quantifying eddy–zonal flow feedbacks in reanalysis and idealized GCM
The results of sections 3b and 3c show the importance of carefully quantifying and interpreting the eddy–zonal flow feedbacks, including the cross-EOF feedbacks, to understand the variability of the zonal-mean flow.
Table 4 presents the feedback strengths obtained from applying (9) and (10) with l = 8–20 days to the year-round SH reanalysis data. We find b 11 = 0.038 day−, a positive feedback from z 1 onto z 1, consistent with the findings of LH01 in their pioneering work. This estimate of b 11 is slightly higher than what we find using the single-EOF approach (b 11 = 0.035 day−1), which is the same as what LH01 found using their spectral cross-correlation method. We also find nonzero cross-EOF feedbacks: b 12 = 0.059 day−1 and b 21 = −0.020 day−1. We estimate b 22 = 0.017 day−1, which is slightly higher from what the single-EOF approach yields (Table 4). The estimated feedback strengths and friction rates (τ) in Table 4 satisfy the condition for the propagating regime [Eq. (20)]. It should be noted that we also extended our approach to include the leading 3 EOFs and quantified the 9 feedback strengths; however, we found the effects of EOF3 on EOF1 and EOF2 negligible, which suggests that a two-EOF model [Eqs. (9) and (10)] is enough to describe the current SH extratropical large-scale circulation (not shown).
Feedback strengths (day−1) estimated for year-round ERA-Interim. The damping rates of friction are estimated as τ 1 = 8.3 days and τ 2 = 8.4 days following the methodology in the appendix of LH01.
Table 5 presents the feedback strengths obtained from applying (9) and (10) with l = 8–20 days to the two setups of the idealized GCM. In the nonpropagating regime, we find b 11 = 0.133 day−1, and small b 22 and negligible b 12 and b 21, indicating the absence of cross-EOF feedbacks, consistent with insignificant m 1 z 2 and m 2 z 1 cross correlations (Figs. 2e,f). The values of b jk do not satisfy the condition for propagating regime, which is consistent with weak cross correlation between z 1 and z 2 at long lags (Fig. 2b). These results suggest that a strong z 1-onto-z 1 feedback dominates the dynamics of the annular mode in this setup (the standard Held–Suarez configuration), which leads to an unrealistically persistent annular mode, similar to what is seen in Fig. 12d, and consistent with the findings of previous studies (Son and Lee 2006; Son et al. 2008; Ma et al. 2017). Using the linear response function (LRF) of this setup computed by Hassanzadeh and Kuang (2016a, 2019) showed that this eddy–zonal flow feedback is due to enhanced low-level baroclinicity [as proposed by Robinson (2000) and LH01] and estimated, from a potential vorticity budget analysis, that the positive feedback is increasing the persistence of the annular mode by a factor of 2.
Feedback strengths (day−1) estimated for the idealized GCM setups with nonpropagating and propagating regimes. The estimated damping rates of friction are τ 1 = 7.4 days and τ 2 = 7.6 days for the GCM setup with nonpropagating regime, and τ 1 = 7.1 days and τ 2 = 7.4 days for the GCM setup with propagating regime (estimated using the methodology in the appendix of LH01).
In the propagating regime, we find b 11 = 0.101 day−1, which is slightly lower than b 11 of the nonpropagating regime. However, in the propagating regime, we also find strong cross-EOF feedbacks b 12 = 0.075 day−1, b 21 = −0.043 day−1 as well as b 22 = 0.023 day−1. These feedback strengths satisfy the condition for the propagating regime, consistent with strong cross correlation between z 1 and z 2 at long lags (Fig. 4b). Comparing the two rows of Table 5 and Figs. 2a and 4a with Table 4 and Fig. 7a suggests that while it is true that the b 11 of the idealized GCM’s nonpropagating regime is larger than that of the SH reanalysis (by a factor of 3.5), the unrealistic persistence of z 1 in this setup (time scale ≈65 days) compared to that of the reanalysis (time scale ≈ 10 days; compare Figs. 2a and 7a) could be, at least partially, due to the absence of cross-EOF feedbacks (thus oscillations). As we showed earlier in section 3c, these oscillations reduce the persistence of the annular modes. The GCM setup with propagating regime has b 11 that is around 2.7 times larger than that of the SH reanalysis, yet their z 1 e-folding decorrelation time scales are comparable (14 days vs 10 days).
Note that like past studies, we have used the e-folding decorrelation time scale of z 1 as the main measure of the time scale of the annular modes (e.g., Gerber et al. 2008b; Sheshadri and Plumb 2017; Ma et al. 2017). The e-folding decorrelation time scale of z 1 measures how long an anomalous structure stays the same (with respect to the stationary pattern of EOF1), which is an appropriate measure of lifetime for the nonpropagating annular modes. However, for the propagating annular modes, in which anomalies propagate but may not change in structure and amplitude, a better measure of lifetime might be needed. We suggest that the one-point lag-correlation maps (Figs. 1 and 6) can provide one example of such measure. For instance, we can track the contour line of −0.4 correlation in Fig. 1b, which spans lag day −35 to lag day +25. This suggests a lifetime of around 60 days for anomalies that propagate poleward and remain anticorrelated with an anomalous structure at the base latitude of 30°S with a magnitude of 0.4 or higher [we choose 0.4, motivated by exp(−1) ≈ 0.37, but other cutoff values could be used too]. Applying the same definition of lifetime to the nonpropagating regime (Fig. 1a) yields ~100 days. Similar analysis of the year-round ERA-Interim data (Fig. 6b) suggests a lifetime of around 20 days for the propagating anomalies. The similarity between the decorrelation time scales of the reanalysis and GCM’s propagating regime and the substantial difference between the lifetime of propagating anomalies in these two systems (at least based on one-point lag-correlation maps) suggest that future studies should carefully examine different measures of time scale for the propagating regimes.
In summary, these findings show the importance of quantifying and examining cross-EOF feedbacks to fully understand the dynamics and variability of the annular modes and to better evaluate how well the GCMs simulate the extratropical large-scale circulation.
4. Concluding remarks
The low-frequency variability of the extratropical large-scale circulation is often studied using a reduced-order model of the leading EOF of zonal-mean zonal wind. The key component of this model (LH01) is an internal-to-troposphere eddy–zonal flow interaction mechanism that leads to a positive feedback of EOF1 onto itself, thus increasing the persistence of the annular mode (LH01). However, several studies have shown that under some circumstances, strong couplings exist between EOF1 and EOF2 at some lag times, resulting in decaying-oscillatory, or propagating, annular modes (e.g., Son and Lee 2006; Son et al. 2008; Sheshadri and Plumb 2017). In the current study, following the methodology of LH01 and using data from the SH reanalysis and two setups of an idealized GCM that produce circulations with a dominant nonpropagating or propagating regime, we first show strong cross correlations between EOF1 (EOF2) and the eddy forcing of EOF2 (EOF1) at long lags, suggesting that cross-EOF feedbacks might exist in the propagating regimes. These findings together demonstrate that there is a need to extend the single-EOF model of LH01 and build a model that includes, at a minimum, both leading EOFs and accounts for their cross feedbacks.
With similar assumptions and simplifications used in LH01, we have developed a two-EOF model for propagating annular modes [consisting of a system of two coupled ODEs, Eqs. (4) and (5) with Eqs. (7) and (8)] that can account for the cross-EOF feedbacks. In this model, the strength of the feedback of the kth EOF onto the jth EOF is b jk (j, k = 1, 2). Using the analytical solution of this model, we derive conditions for the existence of the propagating regime based on the feedback strengths. It is shown that the propagating regime, which requires a decaying-oscillatory solution of the coupled ODEs, can exist only if the cross-EOF feedbacks have opposite signs (b 12 b 21 < 0), and if and only if the following criterion is satisfied: (b 11 − b 22)2 < −4b 12 b 21. These criteria show that nonzero cross-EOF feedbacks are essential components of the propagating regime dynamics.
Using this model and the idealized GCM and a stochastic prototype, we further show that cross-EOF feedbacks play an important role in controlling the persistence of the propagating annular modes (i.e., the e-folding decorrelation time scale of the zonal index z j ) by setting the frequency of the oscillation ω [Eq. (24)]. Therefore, in this regime, the persistence of the annular mode (EOF1) does not only depend on the feedback of EOF1 onto itself, but also depends on the cross-EOF feedbacks. We find that as a result of the oscillation, the stronger the cross-EOF feedbacks, the less persistent the annular mode.
Applying the coupled-EOF model to the reanalysis data shows the existence of strong cross-EOF feedbacks in the current SH extratropical large-scale circulation. Annular modes have been found to be too persistent compared to observations in GCMs including IPCC AR4 and CMIP5 models (Gerber and Vallis 2007; Gerber et al. 2008a; Bracegirdle et al. 2020). This long persistence has been often attributed to a too-strong positive EOF1-onto-EOF1 feedback in the GCMs. The dynamics and strength of this feedback depends on factors such as the mean flow and surface friction (Robinson 2000; LH01; Chen and Plumb 2009; Hassanzadeh and Kuang 2019). External (to the troposphere) influence, e.g., from the stratospheric polar vortex, has been also suggested to affect the persistence of the annular modes (Baldwin et al. 2003; Simpson et al. 2011; Byrne et al. 2016; Saggioro and Shepherd 2019). Our results show that the cross-EOF feedbacks play an important role in the dynamics of the annular modes, and in particular, that their absence or weak amplitudes can increase the persistence, offering another explanation for the too-persistent annular modes in GCMs.
Overall, our findings demonstrate that to fully understand the dynamics of the large-scale extratropical circulation and the reason(s) behind the too-persistent annular modes in GCMs, the coupling of the leading EOFs and the cross-EOF feedbacks should be examined using models such as the one introduced in this study. Note that processes other than internal (to the troposphere) eddy–zonal flow feedbacks [e.g., those suggested by Byrne et al. (2016) and Saggioro and Shepherd (2019)] could be also included in these models as external forcings to build more complete models for the large-scale extratropical variability.
An important next step is to investigate the underlying dynamics of the cross-EOF feedbacks. So far, we have pointed out that cross-EOF feedbacks are essential components of the propagating annular modes; however, the propagation itself is likely essential for the existence of cross-EOF feedbacks. In fact, our preliminary analysis shows that the cross-EOF feedbacks result from the out-of-phase oscillations of EOF1 (north–south jet displacement) and EOF2 (jet pulsation) leading to an orchestrated combination of equatorward propagation of wave activity (a baroclinic process) and nonlinear wave breaking (a barotropic process), which altogether act to reduce the total eddy forcings (not shown). In ongoing work, we aim to explain and quantify the propagating annular modes dynamics using the LRF framework of Hassanzadeh and Kuang (2016b,a) and finite-amplitude wave-activity framework (Nakamura and Zhu 2010; Lubis et al. 2018a,b) that have been proven useful in understanding the dynamics of the nonpropagating annular modes (Nie et al. 2014; Ma et al. 2017; Hassanzadeh and Kuang 2019).
Acknowledgments
We thank Aditi Sheshadri, Ding Ma, and Orli Lachmy for insightful discussions and Ebrahim Nabizadeh for helpful comments on the manuscript. We are grateful to three anonymous reviewers for providing valuable feedbacks and suggestions. This work is supported by National Science Foundation (NSF) Grant AGS-1921413. Computational resources were provided by XSEDE (allocation ATM170020), NCAR’s CISL (allocation URIC0004), and Rice University Center for Research Computing.
APPENDIX
Standard Errors of Cross Correlations using Bartlett’s Formula
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It is worth mentioning that recently, some studies have suggested external (to the troposphere) influences, e.g., from nonstationary interannual variability and stratospheric polar vortex, rather than internal (to the troposphere) eddy–zonal flow feedbacks, as the cause of these positive correlations between z 1 and m 1 (Byrne et al. 2016; Saggioro and Shepherd 2019).
Note that in general, for any two time series x(t) and y(t) and a given time lag l, we have defined reg l [x(t), y(t)] = sum[x(t + l)y(t)] where the summation is over time (i.e., regression). Later in the paper, when a range is given for l, we mean that following Simpson et al. (2013), reg l is computed for each l and then averaged over that range.
