An Eddy-Zonal Flow Feedback Model for Propagating Annular Modes
GGenerated using the official AMS L A TEX template—two-column layout. FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
An Eddy-Zonal Flow Feedback Model for Propagating Annular Modes S ANDRO
W. L
UBIS ∗ AND P EDRAM H ASSANZADEH
Rice University, Houston, Texas, USA
ABSTRACTThe variability of the zonal-mean large-scale extratropical circulation is often studied using individualmodes obtained from empirical orthogonal function (EOF) analyses. The prevailing reduced-order modelof the leading EOF (EOF1) of zonal-mean zonal wind, called the annular mode, consists of an eddy-meanflow interaction mechanism that results in a positive feedback of EOF1 onto itself. However, a few studieshave pointed out that under some circumstances in observations and GCMs, strong couplings exist betweenEOF1 and EOF2 at some lag times, resulting in decaying-oscillatory, or propagating, annular modes. Here,we introduce a reduced-order model for coupled EOF1 and EOF2 that accounts for potential cross-EOF eddy-zonal flow feedbacks. Using the analytical solution of this model, we derive conditions for the existence of thepropagating regime based on the feedback strengths. Using this model, and idealized GCMs and stochasticprototypes, we show that cross-EOF feedbacks play an important role in controlling the persistence of theannular modes by setting the frequency of the oscillation. We find that stronger cross-EOF feedbacks leadto less persistent annular modes. Applying the coupled-EOF model to the Southern Hemisphere reanalysisdata shows the existence of strong cross-EOF feedbacks. The results highlight the importance of consideringthe coupling of EOFs and cross-EOF feedbacks to fully understand the natural and forced variability of thezonal-mean large-scale circulation.
1. Introduction
At the intraseasonal to interannual time scales, the vari-ability of the large-scale atmospheric circulation in themid-latitudes of both hemispheres is dominated by the“annular modes”, which are usually defined based onempirical orthogonal function (EOF) analysis of zonal-mean meteorological fields (e.g., Kidson 1988; Thomp-son and Wallace 1998, 2000; Lorenz and Hartmann 2001,2003; Thompson and Woodworth 2014; Thompson and Li2015). The barotropic annular modes are often derived asthe first (i.e., leading) EOF (EOF1) of zonal-mean zonalwind, which exhibits a dipolar meridional structure anddescribes a north-south meandering of the eddy-driven jet.Note that in this paper, the focus is on the barotropic an-nular modes, hereafter simply called annular modes (seeThompson and Woodworth 2014; Thompson and Barnes2014, and Thompson and Li 2015 for discussions aboutthe “baroclinic annular modes”). The second EOF ofzonal-mean zonal wind (EOF2) has a tripolar meridionalstructure centered on the jet, describing a strengtheningand weakening of the eddy-driven jet (i.e., jet pulsation).By construction, EOF1 and EOF2 (and any two EOFs) areorthogonal and their associated time series (i.e., principal ∗ Corresponding author address: components, PCs), sometimes called zonal index, are in-dependent at zero time lag.The persistence of the annular mode (EOF1) and its un-derlying dynamics have been the subject of extensive re-search and debate in the past three decades (e.g., Robinson1991; Branstator 1995; Feldstein and Lee 1998; Robinson2000; Lorenz and Hartmann 2001, 2003; Gerber and Vallis2007; Gerber et al. 2008b; Chen and Plumb 2009; Simp-son et al. 2013; Zurita-Gotor 2014; Nie et al. 2014; Byrneet al. 2016; Ma et al. 2017; Hassanzadeh and Kuang 2019).Many of the aforementioned studies have pointed to a pos-itive eddy-zonal flow feedback mechanism as the source ofthe persistence: The zonal wind and temperature anoma-lies associated with the annular mode (EOF1) modify thegeneration and/or propagation of the synoptic eddies at thequasi-steady limit (greater than 7 days) in such a way thatthe resulting eddy fluxes reinforce the annular mode (seeHassanzadeh and Kuang (2019) and the discussion andreferences therein). Most notably, Lorenz and Hartmann(2001) developed a linear eddy-zonal flow feedback model(LH01 model hereafter) for the annular modes by regress-ing the anomalous eddy momentum flux divergence ontothe zonal index of EOF1 ( z ) and interpreting correlationsbetween z ( t ) and regressed momentum flux divergence( m ( t ) ) at long lags (greater than 7 days) as evidence foreddy-zonal flow feedbacks, i.e., feedbacks of EOF1 ontoitself. Lorenz and Hartmann (2001) developed a similar Generated using v4.3.2 of the AMS L A TEX template a r X i v : . [ phy s i c s . a o - ph ] J u l J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S model, separately, for EOF2 and found, respectively, pos-itive and weak eddy-zonal flow feedbacks for EOF1 andEOF2, respectively, consistent with the longer persistenceof EOF1 compared to EOF2. Such single-EOF eddy-zonalflow feedback models have been used in most of the subse-quent studies of the annular modes (e.g., Lorenz and Hart-mann 2003; Simpson et al. 2013; Lorenz 2014; Robertet al. 2017; Ma et al. 2017; Boljka et al. 2018; Hassan-zadeh 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 twoEOFs can be correlated at long lags (e.g., greater than10 days), and that in fact the combination of these twoleading EOFs represents coherent meridional propaga-tions of the zonal-mean flow anomalies. Such propagat-ing regimes have been observed in both hemispheres inreanalysis data (e.g., Feldstein 1998; Feldstein and Lee1998; Sheshadri and Plumb 2017). Anomalous polewardpropagation of zonal wind typically emerges in low lati-tudes and mainly migrate poleward over a few months, al-though non-propagating regimes can also appear in someinstances (see Fig. 1 of Sheshadri and Plumb 2017 andFig. 6 in this paper). Similar behaviors have also beenreported by in general circulation models (GCMs) (e.g.,James and Dodd 1996; Son and Lee 2006; Son et al. 2008;Sheshadri and Plumb 2017). Son and Lee (2006) foundthat the leading mode of variability in an idealized dryGCM can be either the propagating or non-propagatingregime depending on the details of thermal forcing im-posed in the model. They also found that unlike thenon-propagating regimes, the z and z of the propagat-ing regimes are strongly correlated at long lags, peakingaround 50 days (see their Fig. 3; also Figs. 4b of thepresent paper). Furthermore, Son and Lee (2006) reportedthat non-propagating regimes are often characterized bya single time-mean jet with a dominant EOF1 (in termsof the explained variance) while the propagating regimesare characterized by a double time-mean jet in the mid-latitudes with the variance associated with EOF2 being atleast half of the variance of EOF1. Furthermore, Son et al.(2008) found that the e -folding decorrelation time scaleof z in the propagating regime to be much shorter thanthat of the non-propagating regime. The long e -foldingdecorrelation time scales for the annular modes in the non-propagating regime were attributed to an unrealisticallystrong positive EOF1-onto-EOF1 feedback, while the rea-son behind the reduction in the persistence of the annularmodes in the propagating regime remained unclear.More recently, Sheshadri and Plumb (2017) presentedfurther evidence for the existence of propagating and non-propagating regimes and strong lagged correlations be-tween z and z in reanalysis data of the Southern Hemi-sphere (SH) and in idealized GCMs. Moreover, theyelegantly showed, using a principal oscillation patterns(POP) analysis (Hasselmann 1988; Penland 1989), that EOF1 and EOF2 are in fact manifestations of a single,decaying-oscillatory coupled mode of the dynamical sys-tem. Specifically, they found that EOF1 and EOF2 are,respectively, the real and imaginary parts of a single POPmode, which describes the dominant aspects of the spatio-temporal evolution of zonal wind anomalies. Sheshadriand Plumb (2017) also showed that in the propagatingregime, the auto-correlation functions of z and z decaynon-exponentially.Given the above discussion, a single-EOF model isnot enough to describe a propagating regime because theEOF1 and EOF2 in this regime are strongly correlated atlong lags and that the auto-correlation functions of the as-sociated PCs do not decay exponentially (but rather showsome oscillatory behaviors too). From the perspectiveof eddy-zonal flow feedbacks, one may wonder whetherthere are cross-EOF feedbacks in addition to the previ-ously studied EOF1 (EOF2) eddy-zonal flow feedbackonto EOF1 (EOF2) in the propagating regime. In cross-EOF feedbacks, EOF1 (EOF2) changes the eddy forcingof EOF2 (EOF1) in the quasi-steady limit. Therefore,there is a need to extend the single-EOF model of LH01and build a model that includes, at a minimum, both lead-ing EOFs and accounts for their cross feedbacks. The ob-jective of the current study is to develop such a model andto use it to estimate effects of the cross-EOF feedbacks onthe variability of propagating annular modes.The paper is structured as follows: Section 2 com-pares the characteristics of z , z , m , and m for the non-propagating and propagating annular modes in reanalysisand idealized GCMs. In Section 3 , we develop a lineareddy-zonal flow feedback model that accounts for cross-EOF feedbacks, validate the model using synthetic datafrom a stochastic prototype, discuss the key properties ofthe analytical solution of this model, and apply this modelto data from reanalysis and an idealized GCM. The paperends concluding remarks in Section 4.
2. Propagating annular modes in an idealized GCMand reanalysis
In this section, we will examine the basic characteris-tics and statistics of propagating annular modes in an ide-alized GCM (the dry dynamical core) and reanalysis. Wefocus on the southern annular mode, which makes it eas-ier to compare the results of the reanalysis and the ideal-ized aquaplanet GCM simulations. We will start with theidealized GCM to demonstrate the characteristics of thepropagating and non-propagating 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 witha flat, uniform lower boundary (i.e., aquaplanet) with T63spectral resolution and 40 evenly spaced sigma levels in
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 1. One-point lag-correlation maps of the vertically averagedzonal-mean zonal wind anomalies (cid:104) u (cid:105) , reconstructed from projectionsonto the two leading EOFs of (cid:104) u (cid:105) for the (a) non-propagating regimeand (b) propagating regime in two setups of an idealized GCM. Thebase latitude is at 30 ◦ S and the contour interval is 0.1. Regions enclosedby contour lines denote values significant at the 95% level. the vertical for 50000-day integrations after spinup. Thephysics of the model is based on Held and Suarez (1994),an idealized configuration for generating a realistic globalcirculation with minimal parameterization (Held 2005;Jeevanjee et al. 2017). All diabatic processes are repre-sented by Newtonian relaxation of the temperature fieldtoward a prescribed equilibrium profile, and Rayleigh fric-tion is included in the lower atmosphere to mimic the in-teractions with the boundary layer.The non-propagating and propagating regimes are pro-duced in two slightly different setups of this model. Forthe setup with non-propagating regime, we use the stan-dard configuration of Held and Suarez (1994), which em-ploys an analytical profile approximating a troposphere inunstable radiative-convective equilibrium and an isother-mal stratosphere for Newtonian relaxation. For the setupwith propagating regime, we follow an approach similarto the one used by Sheshadri and Plumb (2017). In thissetup, for the equilibrium temperature profile in the tro-posphere and stratosphere, we use the perpetual-solstice version of the equilibrium temperature specifications usedin Lubis et al. (2018a), calculated from a rapid radiativetransfer model (RRTM), with winter conditions in the SH.As will be seen later, these choices result in a large-scalecirculation with reasonable annular mode time scales inthe SH.In Fig. 1, we show, following Son and Lee (2006), theone-point lag-correlation maps for the vertically averagedzonal-mean zonal wind anomalies (cid:104) u (cid:105) reconstructed fromprojections onto the two leading EOFs of (cid:104) u (cid:105) for the twosetups (hereafter, angle brackets and overbars denote thevertical and zonal averages, respectively). The anomaliesare defined as the deviations from the time mean. Thenon-propagating and propagating regimes are clearly seenin Figs. 1a and 1b, respectively. In the latter, the prop-agating anomalies emerge in low latitudes and propagategenerally poleward over the course of 3-4 months. In con-trast, the non-propagating regime is characterized by per-sistence zonal flow anomalies in the mid-latitude (Fig. 1a).To understand the relationship between zonal-meanzonal wind and eddy forcing in the non-propagating andpropagating annular modes, the vertically averaged zonal-mean zonal wind anomalies ( (cid:104) u (cid:105) ) and vertically averagedzonal-mean eddy momentum flux convergence anomalies( (cid:104) F (cid:105) ) are projected onto the leading EOFs of (cid:104) u (cid:105) follow-ing Lorenz and Hartmann (2001). The time series of zonalindex ( z ) and eddy forcing ( m ) associated with EOF1 andEOF2 are formulated as: z , ( t ) = (cid:104) u (cid:105) ( t ) We , (cid:113) e T , We , , (1) m , ( t ) = (cid:104) F (cid:105) ( t ) We , (cid:113) e T , We , , (2)where z , ( m , ) denotes the component of the field (cid:104) u (cid:105) ( (cid:104) F (cid:105) ) that projects onto the latitudinal structure of thetwo leading EOFs. (cid:104) u (cid:105) ( t ) and (cid:104) F (cid:105) ( t ) are (cid:104) u (cid:105) ( φ , t ) and (cid:104) F (cid:105) ( φ , t ) with their latitude dimension vectorized, W is adiagonal matrix whose elements are the cos ( φ ) weightingused when defining the EOF structure e , and φ is latitude(Simpson et al. 2013; Ma et al. 2017). Here, the verticallyaveraged zonal-mean eddy momentum flux convergence (cid:104) F (cid:105) is calculated in the spherical coordinate as: (cid:104) F (cid:105) ( φ , t ) = − φ ∂ ( (cid:104) u (cid:48) v (cid:48) cos φ (cid:105) ) a ∂ φ (3)where u (cid:48) and v (cid:48) are deviations of zonal wind and merid-ional wind from their respective zonal means, and a isEarth’s radius.Figure 2 shows lagged-correlation analysis between z and m in the GCM setup with non-propagating regime.The auto-correlation of z , as discussed in past studies J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 2. Lagged-correlation analysis of the GCM setup with non-propagating regime. (a) Auto-correlation of z (blue) and z (red), (b)cross-correlation z z , (c) cross-correlation m z , (d) cross-correlation m z , (e) cross-correlation m z , and (f) cross-correlation m z . Thetwo leading EOFs contribute 60.2% and 19.2%, respectively, to the totalvariance. The e -folding decorrelation time scales of z and z are 64 . . (e.g., Chen and Plumb 2009; Ma et al. 2017), has a no-ticeable shoulder at around 5-day lags and shows an unre-alistically persistence annular mode, well separated fromthe faster decaying z , which is consistent with the consid-erable difference in the contribution of the two EOFs to thetotal variance (60.2% versus 19.2%). The e -folding decor-relation time scales of z and z are 64 . . m z and insignificant cross-correlations of m z at large pos-itive lags suggest the existence of a positive eddy-zonalflow feedback for EOF1 (from EOF1) but not for EOF2(see Son et al. (2008) and Ma et al. (2017)). Figure 2bshows that the z z cross-correlations are weak at positiveand negative lags, which consistently with the one-pointlag-correlation map of Fig. 1a and Fig. 3 (shown later), areindicative of a non-propagating regime, as reported pre-viously for a similar setup (Son and Lee 2006; Son et al.2008). The m z and m z cross-correlations are small andoften insignificant, suggesting the absence of the cross-EOF feedbacks in the non-propagating regime (Figs. 2e-f). All together, the above analysis shows that for thenon-propagating regime, single-EOF reduced-order mod-els such as LH01 are sufficient. F IG . 3. Anomalous zonal-mean zonal wind ( ¯ u ) regressed onto z and z in the GCM setup with non-propagating regime: (a, b) simultaneous,(c) z leads by 20 days, and (d) z leads by 20 days. The contours arethe climatological zonal-mean zonal wind with interval of 5 ms − . The weak cross-correlations between z and z in theGCM with non-propagating regime (Fig. 2b) can be alsoseen by regressing the zonal-mean zonal wind anomalieson the zonal index at 0- and 20-day time lag. Figures 3aand 3b show the wind anomalies regressed on z and z at lag 0, yielding approximately the EOF1 and EOF2 pat-terns, respectively. Twenty days after z leads zonal windanomalies, the anomalies do not drift poleward or decay,but rather persist (Fig. 3d). In contrast, 20 days after z leads zonal wind anomalies, the anomalies decay and dis-appear (Fig. 3c). These observations are consistent withthe long and short persistence of z and z , respectively,consistent with the weak cross-correlations of z and z atpositive or negative lags, and as become clear below, con-sistent with the non-propagating nature of this setup.Figure 4 shows lagged-correlation analysis between z and m in the GCM setup with propagating regime. Theauto-correlation of z , its persistence compared to that of z , and the explained variance by the two EOFs (40.4%versus 32.5%) are much more similar to what is observedin the SH (shown later in Fig. 7). The e -folding decorrela-tion time scales of z and z are 14 . . z and z are strongly corre-lated at long lags peaking at around ±
20 days. This behav-ior along with the one-point lag-correlation map of Fig. 1band regression map of wind anomalies (Fig. 5, shownlater) suggests the existence of a propagating regime, asnoted by few previous studies (e.g., Son and Lee 2006;Sheshadri and Plumb 2017). It should be noted that Sonand Lee (2006) have proposed a rule of thumb based onthe ratio of the explained variance of EOF2 to EOF1: Anon-propagating (propagating) regimes exists if the ratio
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 4. Lagged-correlation analysis of the GCM setup with propa-gating regime. (a) Auto-correlation of z (blue) and z (red), (b) cross-correlation z z , (c) cross-correlation m z , (d) cross-correlation m z ,(e) cross-correlation m z , and (f) cross-correlation m z . The two lead-ing EOFs contribute 40.4% and 32.5%, respectively, to the total vari-ance. The e -folding decorrelation time scales of z and z are 14 . . is smaller (larger) than 0.5. The regime of our two se-tups are consistent with this rule of thumb as the ratios are ∼ . ∼ . m z cross-correlations are positive at long positive lags (5-20 days)and then negative but small. Fig. 4d indicates smalland negative cross-correlations between z and m at thetimes scale of longer than 20 days (Fig. 4c). Overall, theshape of the m z and m z cross-correlation functionsare similar between the non-propagating and propagatingregimes, although the m z cross-correlations are largerand more persistent in the non-propagating regime. Incontrast, the m z and m z cross-correlations are substan-tially different between the two regimes (Figs. 4e-f). Thereare statistically significant and large positive m z cross-correlations at large positive lags ( > m z cross-correlationsat positive lags up to 30 days. Note that as emphasized inthe figures, positive lags here mean that z ( z ) is leading m ( m ). Therefore, these cross-correlations, as discussedlater, indicate the existence of cross-EOF feedbacks in thepropagating regime. F IG . 5. Anomalous zonal-mean zonal wind ( ¯ u ) regressed onto z and z in the GCM setup with propagating regime: (a, b) simultaneous, (c) z leads by 20 days, and (d) z leads by 20 days. The contours are theclimatological zonal-mean zonal wind with interval of 5 ms − . Figure 5 shows anomalous zonal-mean zonal wind re-gressed on z and z at 0- and 20-day time lag in theGCM setup with propagating regime. Figures 5a and 5bshow the wind anomalies regressed on z and z at lag0, again yielding approximately the EOF1 and EOF2 pat-terns, respectively. As shown in Fig. 5c, 20 days after z leads zonal wind anomalies, the anomalies have driftedpoleward and project strongly onto the structure of windanomalies associated with EOF1 (Figs. 5a,c, pattern corre-lation = 0.93). This is consistent with positive correlationof z z at lag +20 days when z leads z (Fig. 4b). Like-wise, twenty days after z leads zonal wind anomalies, theanomalies (of Fig. 5a) have drifted poleward and projectstrongly onto the structure of anomalies associated withEOF2, but with an opposite sign (Figs. 5b,d, pattern corre-lation = -0.85). This is consistent with negative correlationof z z when z leads z by 20 days (Fig. 4b).Overall, these results suggest the existence of cross-EOF feedbacks in the propagating annular mode. In Sec-tion 3, we will developed a model to quantify these fourfeedbacks and understand the effects of their magnitudeand signs on the variability (e.g., persistence) of z and z .But first, we will examine the variability and characteris-tics of z and m in reanalysis. In particular, we will see thatthe z and m cross-correlations in the GCM’s propagatingregime well resemble those in the SH reanalysis data. b. Reanalysis We use the 1979-2013 data from the European Centrefor Medium-Range Weather Forecasts (ECMWF) interimreanalysis (ERA-Interim; Dee et al. 2011). Zonal andmeridional wind components ( u , v ) are 6 hourly, on 1.5 ◦ J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 6. One-point lag-correlation maps of the vertically averagedzonal-mean zonal wind anomalies from year-round ERA-Interim dataintegrated across the depth of the troposphere (1000-100 hPa) ( (cid:104) u (cid:105) ) inthe Southern Hemisphere. (a) Total anomaly fields and (b) reconstructedfrom projections onto the two leading EOFs of (cid:104) u (cid:105) . The base latitudeis at 30 ◦ S and the contour interval is 0.1. Regions enclosed by contourlines denote values significant at the 95% level according to the t -test. latitude × ◦ longitude grid, and on 21 vertical levelsbetween 1000 and 100 hPa. Anomalies used for comput-ing correlations and EOF analyses are defined as the devi-ations from the climatological seasonal cycle. The meanseasonal cycle is defined as the annual average and the firstfour Fourier harmonics of the 35-yr daily climatology.Figure 6 shows a one-point lag-correlation map of ver-tically averaged zonal-mean zonal wind (cid:104) u (cid:105) in the SH,where the base latitude is 30 ◦ S. Comparing this figurewith Fig. 1, it can be seen that there is an indication ofpoleward-propagating anomalies in SH, which appear inlow latitudes and migrate poleward over the course of 2-3 months (Fig. 6a). However, the poleward-propagatingsignals are not as clearly as those observed in the GCMsetup with the propagating regime (Fig. 1b, or Fig. 2 ofSon and Lee 2006). This is consistent with previous stud-ies (e.g. Feldstein 1998; Feldstein and Lee 1998; She-shadri and Plumb 2017), showing that both propagatingand non-propagating anomalies exist in all seasons in theSH, which somehow obscure the propagating signals. Re-constructions based on the projections onto the two lead- F IG . 7. Lagged-correlation analysis for the Southern Hemisphere,calculated from year-round ERA-Interim data. (a) Auto-correlations of z (blue) and z (red), (b) cross-correlation z z , (c) cross-correlation m z , (d) cross-correlation m z , (e) cross-correlation m z , and (f)cross-correlation m z at different lags. The two leading EOFs con-tribute to 45.1% and 23.2% of the total variance, respectively. The e -folding decorrelation time scales of z and z are 10 . . ing EOFs of zonal-mean zonal wind further show thatmost of the mid-latitude SH wind variability can be ex-plained by the two leading EOF modes (Fig. 6b). Theratio of the fractional variance of EOF2 (23.2%) to thatof EOF1 (45.1%) is 0.51, which is right at the boundaryfrom the rule of thumb. Overall, as already pointed out bySheshadri and Plumb (2017), a propagating annular modeexists in the SH and is largely explained by the two leadingEOF modes.Figure 7a shows the auto-correlations of z and z . Con-sistent with Lorenz and Hartmann (2001), the estimateddecorrelation time scales of these two PCs are 10.3 and 8.1days, respectively. Figure 7b depicts the cross-correlation z z , showing statistically significant and relatively strongcorrelations that peak around ±
10 days. As discussed inearlier studies, such lagged correlations are a signature ofthe propagating annular modes (Feldstein and Lee 1998;Son and Lee 2006; Son et al. 2008; Sheshadri and Plumb2017), implying that the period of the poleward propaga-tion is about 20-30 days in the SH (Fig. 7b), consistentwith Sheshadri and Plumb (2017) and with Fig. 6.
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S z and z on m and m ,we also examine the cross-correlations between z and m atdifferent lags (Figs. 7c-f). The shape and the magnitude ofthe m z and m z cross-correlations (Figs. 7c-d) are sim-ilar to those originally shown by Lorenz and Hartmann(2001) (see their Figs. 5 and 13a) and later by many oth-ers using different reanalysis products and time periods.As discussed in Lorenz and Hartmann (2001), the statis-tically significant positive m z cross-correlations at longpositive lags ( ∼ −
20 days) and the insignificant m z cross-correlations for time scales longer than ∼ m z and m z cross-correlations at differ-ent lags. The m z cross-correlations show statistical sig-nificant positive correlations at large positive lags, signi-fying that a cross-EOF feedback, i.e, z modifying m ,is present. Note that the magnitude of the m z cross-correlations at positive lags is overall larger than those of m z (Fig. 7c). There are also statistically significant butnegative m z correlations at large positive lags, again sug-gesting the existence of a cross-EOF feedback, i.e, z mod-ifying m . These results indicate that in the presence ofpropagating regime in the SH, there are indeed cross-EOFfeedbacks; however, these feedbacks were always ignoredin the previous studies and reduced-order models of theSH extratropical large-scale circulation.
3. Eddy-zonal flow feedbacks in the propagating annu-lar modes: Model and quantification
In this section, an eddy-zonal flow feedback model thataccounts for the coupling of the leading two EOFs andtheir feedbacks, including the cross-EOF feedbacks willbe introduced. Then this model will be validated usingsynthetic data from a simple stochastic prototype, andfrom its analytical solution, we will derive conditions forthe existence of the propagating regime. Finally, we willuse this model to estimate the feedback strengths of thepropagating annular modes in data from the reanalysis(SH) and the idealized GCM. a. Developing an eddy-zonal flow feedback model forpropagating annular modes
With the same notations as in Lorenz and Hartmann(2001), the time series of zonal indices ( z and z ) andeddy forcing ( m and m ) associated with the first twoleading EOFs are calculated by projecting the verticallyaveraged zonal-mean zonal wind (cid:104) u (cid:105) and eddy momen-tum flux convergence (cid:104) F (cid:105) anomalies onto the patterns of the first and second EOFs of (cid:104) u (cid:105) (see Eqs. (1)-(2)). Equa-tions for the tendency of z and z can be then formulatedas: dz dt = m − z τ , (4) dz dt = m − z τ , (5)where t is time and the last term on the right-hand side ineach equation represents damping (mainly due to surfacefriction) with time scale τ . As discussed in Lorenz andHartmann (2001), Eqs. (4)-(5) can be interpreted as thezonally and vertically averaged zonal momentum equa-tion: ∂ (cid:104) u (cid:105) ∂ t = − φ ∂ ( (cid:104) u (cid:48) v (cid:48) cos φ (cid:105) ) a ∂ φ − D , (6)projected into EOF1 and EOF2, respectively. In the aboveequation, D includes the effects of surface drag and ismodeled as Rayleigh drag in Eqs. (4)-(5).Assuming a linear representation for the feedback of anEOF onto itself, Lorenz and Hartmann (2001) and laterstudies wrote m ( t ) = ˜ m ( t )+ b z ( t ) and m ( t ) = ˜ m ( t )+ b z ( t ) , where b and b are the feedback strengths (with b j > z j ). ˜ m is the random, zonal flow-independentcomponent of the eddy forcing that drives the high-frequency variability of z (Lorenz and Hartmann 2001; Maet al. 2017).Here, to account for the cross-EOF feedbacks, i.e., theeffect of z on m and z on m , we extend the LH01 modeland write m = ˜ m + b z + b z , (7) m = ˜ m + b z + b z . (8)With j , k = , b jk is the strength of the linearizedfeedback of z k onto z j through modifying m j in the quasi-steady limit; thus the cross-EOF feedbacks are representedby the terms involving b and b . To find the val-ues of b jk , we can use the lagged-regression method ofSimpson et al. (2013), which assumes that reg l ( ˜ m j , z j ) = sum ( ˜ m j ( t + l ) z j ( t )) ≈ l . By lag-regressing each term in Eqs. (7) onto z and then onto z ,we find (cid:20) reg l ( z , z ) reg l ( z , z ) reg l ( z , z ) reg l ( z , z ) (cid:21) (cid:20) b b (cid:21) = (cid:20) reg l ( m , z ) reg l ( m , z ) (cid:21) (9)and similarly, from Eq. (8) we find (cid:20) reg l ( z , z ) reg l ( z , z ) reg l ( z , z ) reg l ( z , z ) (cid:21) (cid:20) b b (cid:21) = (cid:20) reg l ( m , z ) reg l ( m , z ) (cid:21) , (10)where we assumed reg l ( ˜ m j , z k ) ≈ j , k = , J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S T ABLE
1. Prescribed and estimated feedback strengths (in day − ) insynthetic data for the case without cross EOF-feedbacks. The imposeddamping rates of friction are τ = τ = 8 days. The values of b and τ aremotivated by the observed ones, see Table 4.Feedback b b b b Prescribed 0.040 0.000 0.000 0.000Estimated (Eqs. (9)-(10)) 0.042 0.001 -0.0006 0.0005
Note that if one attempts to find b using a single-EOF approach such as LH01, then, from Eq. (7), onewould be implicitly assuming that reg l ( ˜ m + b z , z ) = reg l ( ˜ m , z ) + b reg l ( z , z ) ≈ b reg l ( z , z ) is zero.However, as shown earlier, in the propagating regime, the z z cross-correlations can be large at long lags, and as dis-cussed below, the range of time lags needed to be used inEqs. (9)-(10) and the lags at which z z cross-correlationspeaks are often comparable. Consequently, if b (cid:54) = b jk should be determined together bysolving the systems of equations (9)-(10).The basic assumptions of our model, Eqs. (4)-(10), aresimilar to those of the LH01 model: i) A linear represen-tation of the feedbacks is sufficient, and ii) The eddy forc-ing m does not have long-term memory independent ofthe variability in the jet (represented by z ). The second as-sumption means that at sufficiently large positive lags (be-yond the time scales over which there is significant auto-correlation in ˜ m ) the feedback component of the eddy forc-ing will dominate the m j z k cross-correlations Lorenz andHartmann (2001); Chen and Plumb (2009); Simpson et al.(2013); Ma et al. (2017)), i.e., reg l ( ˜ m j , z k ) ≈ reg l ( z j , z j ) would be smalland inaccurate. To find the appropriate lag to use, onemust look for non-zero m j z k cross-correlations at positivelags beyond an eddy lifetime. In this study, the strengths ofthe individual feedbacks are averaged over positive lags of8 to 20 days for both GCM and reanalysis (e.g., Simpsonet al. 2013; Burrows et al. 2016). We choose this range inorder to avoid the high-frequency variability at short lags(indicated by impulsive and oscillatory characters of the˜ m auto-correlation) and strong damping at the very longlags.In the following section, we will present a proof of con-cept for this eddy-zonal flow feedback model using syn-thetic data obtained from a simple stochastic prototype andshow that using Eqs. (9)-(10), the prescribed feedbackscan be accurately backed out. T ABLE
2. Prescribed and estimated feedback strengths (in day − )in synthetic data for the case with cross EOF-feedbacks. The imposeddamping rates of friction are τ = τ = 8 days. The values of b and τ aremotivated by the observed ones, see Table 4.Feedback b b b b Prescribed 0.040 0.060 -0.025 0.000Estimated (Eqs. (9)-(10)) 0.043 0.067 -0.026 -0.002 b. Validation using synthetic data from a simple stochasticprototype
We begin by constructing a simple stochastic system toproduce synthetic time series z and m in the presence orabsence of cross-EOF feedbacks. The equations of thissystem are the same as Eqs. (4)-(5) and (7)-(8). FollowingSimpson et al. (2013), we generate a synthetic time seriesof the random component of the eddy forcing (cid:101) m , using asecond-order autoregressive (AR2) noise process:˜ m ( t ) = . m ( t − ) − . m ( t − ) + ε ( t ) , (11)˜ m ( t ) = . m ( t − ) − . m ( t − ) + ε ( t ) , (12)where t denotes time (in days) and ε is white noise dis-tributed uniformly between -1 and +1.Synthetic time series of z j and m j are produced by nu-merically integrating Eqs. (4)-(5), (7)-(8), and (11)-(12)forward in time with two different sets of prescribed b jk .In the first set, there is no cross-EOF feedback, i.e., b = b = b and b = b and b (cid:54) = τ = τ = b and τ are reasonablychosen based on the observed values in the SH (see Table4).Spectral analysis of z , and m , shows that the syn-thetic data indeed have characteristics similar to those ofthe observed SH. For example, for the case with cross-feedbacks (Fig. 8), we find that consistent with observa-tions (see Fig. 4 of Lorenz and Hartmann (2001) or Fig. 3of Ma et al. (2017)), the time scales of z and z are muchlonger (i.e., slower variability) than m and m , and thepower spectra of z can be interpreted, to the first order, asreddening of the power spectra of eddy forcing m Lorenzand Hartmann (2001); Ma et al. (2017). The power spectraof eddy forcings m and m have in general a broad max-imum centered at the low and synoptic frequency, consis-tent with observations. Given that the characteristics ofthe synthetic data mimic the key characteristics of the ob-served annular modes, we use this idealized framework to O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 8. Spectra of z , and m , from the synthetic data with cross-EOF feedbacks. Black lines show the power spectra of (a) z , (b) z ,(c) m , and (d) m . The red-noise spectra are indicated by the smoothsolid red curves, and the smooth dashed blue lines are the 5% and 95%a priori confidence limits. validate the lagged-correlation approach of Eqs. (9)-(10)for quantifying eddy-zonal flow feedbacks.Figure 9 shows the lagged-correlation analysis of thesynthetic data without cross-EOF feedbacks. It is clearlyseen that the only noticeable cross-correlations are thatof m z , and there are no (statistically significant) cross-correlations between z z , m z and m z at any lag, con-sistent with a non-propagating regime and the absenceof cross-EOF feedbacks (Fig. 2). Using Eqs. (9)-(10)and lag l =8-20 days, we can closely estimate the pre-scribed feedback parameters, i.e., b = .
04 day − and b = b = b = z z , m z , m z , and m z , with theshape of the cross-correlation distributions not that dif-ferent from that of the SH reanalysis and the idealizedGCM setup with propagating regime (Figs. 4 and 7).The positive m z and near zero m z cross-correlationsat large positive lags signify a positive z -onto- z feed-back through m , but no z -onto- z feedback through m ,consistent with the prescribed positive value of b and b =
0. In addition, Figs. 10e-f also show that thereare statistically significant and large correlations in m z and m z at positive lags, consistent with the introductionof cross-EOF feedbacks by setting b = .
06 day − and b = − .
025 day − . The positive m z cross-correlationsare positive lags are higher than those of m z (note that F IG . 9. Lagged-correlation analysis of synthetic data without cross-EOF feedback. (a) Auto-correlation of z (blue) and z (red), (b) cross-correlation z z , (c) cross-correlation m z , (d) cross-correlation m z ,(e) cross-correlation m z , and (f) cross-correlation m z . The e -foldingdecorrelation time scales of z and z are 18 . . b / b ≈ . m z cross-correlations isopposite to the sign of m z cross-correlations (note that b b < l =8-20 days, wecan again closely estimate the prescribed feedback param-eters, including the strength of the cross-EOF feedbacks(see Table 2).The above analyses validate the approach usingEqs. (9)-(10) for quantifying the feedback strengths b jk indata from both propagating and non-propagating regimes.Furthermore, a closer examination of z and z auto-correlations in Figs. 9a and 10a show that both z and z in the case without cross-EOF feedbacks are morepersistence than those in the case with cross-EOF feed-backs; e.g., the e -forcing deccorelation time scale of z is18 . . b > b >
0, which might seem like another pos-itive feedback that should further prolong the persistenceof z . Finally, we notice that b b < b flipped results in cross-correlation distributions0 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 10. Lagged-correlation analysis of synthetic data with cross-EOF feedback. (a) Auto-correlation of z (blue) and z (red), (b) cross-correlation z z , (c) cross-correlation m z , (d) cross-correlation m z ,(e) cross-correlation m z , and (f) cross-correlation m z . The e -foldingdecorrelation time scales of z and z are 13 . . that are vastly different from those of Fig. 10 and what isseen in the SH reanalysis and idealized GCM. Inspired bythese observations, next we examine the analytical solu-tion of the deterministic version of Eqs. (4)-(5 and (7)-(8)to better understand the impacts of the strength and signof b jk on the variability and in particular the persistence of z and z . c. Analytical solution of the two-EOF eddy-zonal flowfeedback model We focus on the deterministic (i.e., ˜ m j =
0) version ofEqs. (4)-(5) and (7)-(8), which can be re-written as the fol-lowing system of ordinary differential equations (ODEs):˙ z = Az , (13)where z = (cid:20) z z (cid:21) , and A = (cid:34) b − τ b b b − τ (cid:35) . (14)The solution to this system is z ( t ) = e A t z ( ) = (cid:2) V e Λ t V − (cid:3) z ( ) , (15) where V and Λ are the eigenvector and eigenvalue matri-ces of A : V = [ v v ] = (cid:20) v v v v (cid:21) , and Λ = (cid:20) λ λ (cid:21) . (16)To find the eigenvalues λ , we set the determinant of A equal to zero and solve the resulting quadratic equation toobtain: λ , = − (cid:18) τ + τ − b − b (cid:19) ± (cid:115)(cid:26)(cid:18) τ − τ (cid:19) − ( b − b ) (cid:27) + b b , (17)which, in the limit of τ ≈ τ (reasonable given their esti-mated values in Tables 4 and 5), simplifies to: λ , = − (cid:18) τ − b − b (cid:19) ± (cid:113) ( b − b ) + b b . (18)The solution (Eq. (15)) can be re-written as z = c e λ t v + c e λ t v , (19)where c and c depend on the initial condition.This system has a decaying-oscillatory solution, i.e., isin the propagating regime, if and only if the eigenvalues(18) have non-zero imaginary parts, which requires, as anecessary and sufficient condition: ( b − b ) < − b b . (20)Equation (20) also implies that a necessary condition forthe existence of propagating regimes is b b < . (21)Thus, non-zero cross-EOF feedbacks of opposite signs areessential components of the propagating regime dynamics.The propagating regimes in the stochastic prototype (Ta-ble 2), SH reanalysis (Table 4), and idealized GCM (Ta-ble 5) satisfy the conditions of Eqs. (20)-(21), while thenon-propagating regimes (Tables 1 and 5) do not.In the non-propagating regime, λ , = − σ , < v , are real and in this regime, z , just decay exponen-tially according to z = c e ( − σ t ) v + c e ( − σ t ) v . (22)In the propagating regime, λ , = − σ ± i ω and v , arecomplex where σ = (cid:18) τ + τ − b − b (cid:19) , (23) ω = (cid:115)(cid:26)(cid:18) τ − τ (cid:19) − ( b − b ) (cid:27) + b b . (24) O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S z , decay and oscillate according to z = c e ( − σ t ) e ( i ω t ) v + c e ( − σ t ) e ( − i ω t ) v . (25)Realizing that in this case v = v are real, and v = v ∗ and c = c ∗ = c , where ∗ means complex conjugate, wecan re-write the above equations as z = (cid:104) c e ( i ω t ) v + c ∗ e ( − i ω t ) v (cid:105) e ( − σ t ) , (26) z = (cid:104) c e ( i ω t ) v ∗ + c ∗ e ( − i ω t ) v (cid:105) e ( − σ t ) . (27)These equations show that z and z have the same de-cay rate ( σ ) but different oscillatory components with fre-quency ω . These results are consistent with the POPanalysis of Sheshadri and Plumb (2017) who showed thatEOF1 and EOF2 are, respectively, the real and imaginaryparts of a single decaying-oscillatory POP mode (see theirSection 4b). As a results, the two modes have the same de-cay rate and frequency, but have different auto-correlationfunction decay rates and have strong lag cross-correlationsbecause the oscillations are out of phase. A key contribu-tion of our work is to find the decay rate σ and frequency ω as a function of b jk and τ j (Eqs. (23)-(24)).To understand the effects of the feedback strength b jk on the persistence of z j , we compute the analytical solu-tions for 5 systems that have the same b > b = b = b = b > b < z andtheir e -folding decorrelation time scales for EXP1-EXP5.EXP1, corresponding to non-propagating regimes, has theslowest-decaying auto-correlation function, i.e., longest e -folding decorrelation time scale (Figs. 11a,b). EXP2-EXP5, which all satisfy condition (Eq. 20), have faster-decaying auto-correlation functions, i.e., shorter e -foldingdecorrelation time scale, consistent with our earlier resultsin idealized GCM and stochastic prototype (Figs. 4 and10). As discussed above, in the propagating regime, theeigenvectors and the corresponding eigenvalue are com-plex and thus, z , do not decay just exponentially, butrather show some oscillatory characteristics too (Fig. 11a,Eqs. (26)-(27)). Since the frequency of these oscilla-tions ω (Eq. (24)) increases as the cross-EOF feedbackstrengths increase, shorter time scales in z are expectedin the experiment with stronger b b (Fig. 11b).The dependence of the e -folding decorrelation timescales of z and z on the feedback strengths, and in par-ticular the cross-EOF feedback strengths, is further eval-uated in Fig. 12. In Fig. 12a, it is clearly seen that theimpact of increasing b > z (Fig. 12a), consistent with in-creasing the positive eddy-zonal flow feedback ( z -onto- z through m ). However, when the feedback is further T ABLE
3. Prescribed feedback strengths (in day − ) used to analyzethe impact of cross-EOF feedbacks on the decorrelation time scales of z and z . The imposed damping rates of friction are τ = τ = 8 days.Feedback b b b b Exp1 0.040 0.000 0.000 0.000Exp2 0.040 0.060 -0.025 0.000Exp3 0.040 0.120 -0.050 0.000Exp4 0.040 0.240 -0.100 0.000Exp5 0.040 0.480 -0.200 0.000 increased to twice the control value, condition (20) for theexistence of a decaying-oscillatory solution is not satisfiedanymore, and consistent with this, we see that the systemundergoes a transition to the non-propagating regime. Fur-ther increasing b leads to substantially more persistent z and less persistence z . Note that in non-propagatingregimes when b b (cid:54) =
0, the decay of z depends on b too (see Eq. (18)).Figure 12b shows that in the propagating regime, unlikeincreasing b >
0, increasing b > z . This is the counter-intuitive be-havior we had observed earlier in the stochastic prototype(Section 3b). Now we understand that this is because in-creasing b increases the frequency of the oscillation ω in the system, resulting in reduction in the the decorrela-tion time scale of z (and z ); also see Fig. 11. Such im-pact can even be more pronounced when both cross-EOFfeedbacks b and b are increased (Fig. 12c), leading toshorter decorrelation time scales. Because a positive b decreases the persistence of z , we do not refer to is as a”positive feedback”. To understand this behavior, we haveto keep in mind that in the eddy forcing of z ( z ), i.e., m in Eq. (7) ( m in Eq. (8)), b > b <
0) is the coef-ficient of z ( z ). When z leads z , they are negativelycorrelated (Figs. 4b, 7b, and 10b), thus z multiplied by b > m that is forcing z , decreasing the per-sistence of z . Similarly, when z leads z , they are posi-tively correlated, thus z multiplied by b < m and thus the persistence of z .Finally, for the sake of completeness, we also examinethe effect of increasing b in the absence of cross-EOFfeedback (Fig. 12d). As expected increasing b leads toincreasing the persistence of z and has no impact on thepersistence of z as now z and z are completely decou-pled. d. Quantifying eddy-zonal flow feedbacks in reanalysisand idealized GCM The results of Sections 3b and 3c show the importanceof carefully quantifying and interpreting the eddy-zonalflow feedbacks, including the cross-EOF feedbacks, to un-derstand the variability of the zonal-mean flow.2 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S F IG . 11. Auto-correlation functions of z (a) and their corresponding e -folding decorrelation time scales (b) from the analytical solutions for theexperiment with no cross-EOF feedback (EXP1) and the experiments with increasing cross-EOF feedback strength (EXP2-EXP5). The prescribedfeedback strength b jk are shown in Table 3.F IG . 12. The computed e -folding decorrelation time scale (day) of z (blue circles) and z (red squares) as a function of feedback strengths(day − ). The impact of varying (a) b , (b) b , and (c) b and b 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) Theimpact of varying b in EXP1 (Table 3). The filled indicates that the parameters satisfy the condition for propagating regimes, i.e., existence ofdecaying-oscillatory solutions (Eq. (20)). Table 4 presents the feedback strengths obtained fromapplying (9)-(10) with l = −
20 days to the year-roundSH reanalysis data. We find b = .
038 day − , a pos-itive feedback from z onto z , consistent with the find-ings of Lorenz and Hartmann (2001) in their pioneer-ing work. This estimate of b is slightly higher thanwhat we find using the single-EOF approach ( b = .
035 day − ), which is the same as what Lorenz and Hart-mann (2001) found using their spectral cross-correlation method. We also find non-zero cross-EOF feedbacks: b = .
059 day − and b = − .
020 day − . We also es-timate b = .
017 day − that is slightly higher from whatthe single-EOF approach yields (Table 4). The estimatedfeedback strengths and friction rates ( τ ) in Table 4 satisfythe condition for propagating regime (Eq. 20). It should benoted that we also extended our approach to include theleading 3 EOFs and quantified the 9 feedback strengths;however, we found the effects of EOF3 on EOF1 and O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S T ABLE
4. Feedback strengths (in day − ) estimated for year-roundERA-Interim reanalysis. The damping rates of friction are estimatedas τ = . τ = . b b b b Eqs. (9)-(10) 0.038 0.059 -0.020 0.017LH01 0.035 - - 0.002
EOF2 negligible, which suggests that a two-EOF model(9)-(10) is enough to describe the current SH large-scalecirculation (not shown).Table 5 presents the feedback strengths obtained fromapplying (9)-(10) with l = −
20 days to the two setupsof the idealized GCM. In the non-propagating regime,we find b = .
133 day − , and small b and negligi-ble b and b , indicating the absence of cross-EOF feed-backs, consistent with insignificant m z and m z cross-correlations (Figs. 2e-f). The values of b jk do not satisfythe condition for propagating regime, which is consistentwith weak cross-correlation between z and z at long lags(Fig. 2b). These results suggest that a strong z -onto- z feedback dominates the dynamics of the annular mode inthis setup (the standard Held-Suarez configuration), whichleads to an unrealistically persistent annular mode, similarto what is seen in Fig. 12d, and consistent with the findingsof previous studies (Son and Lee 2006; Son et al. 2008; Maet al. 2017). Using the linear response function (LRF) ofthis setup (Hassanzadeh and Kuang 2016b, 2019) showedthat this eddy-zonal flow feedback is due to enhanced low-level baroclinicity (as proposed by Robinson (2000) andLorenz and Hartmann (2001)) and estimated, from a bud-get analysis, that the positive feedback is increasing thepersistence of the annular mode by a factor of two.In the propagating regime, we find b = .
101 day − ,which is slightly lower than b of the non-propagatingregime. However, in the propagating regime, we also findstrong cross-EOF feedbacks b = .
075 day − , b = − .
043 day − as well as b = .
023 day − . Thesefeedback strengths satisfy the condition for propagatingregime, consistent with strong cross-correlation between z and z at long lags (Fig. 4b). Comparing the two rowsof Table 5 and Figs. 2a and 4a with Table 4 and Fig. 7a sug-gests that while it is true that the b of the the idealizedGCM’s non-propagating regime is larger than that of theSH reanalysis (by a factor of 3.5), the unrealistic persis-tence of z in this setup (time scale ≈
65 days) comparedto that of the reanalysis (time scale ≈
10 days; compareFigs. 2a and 7a) could be, at least partially, due to the ab-sence of cross-EOF feedbacks (thus oscillations), whichas we showed earlier in Section 3c, reduce the persistenceof the annular modes. The GCM setup with propagatingregime has b that is around 2.7 times larger than that of T ABLE
5. Feedback strengths (in day − ) estimated for the idealizedGCM setups with non-propagating and propagating regimes. The esti-mated damping rates of friction are τ =7.4 days and τ =7.6 days for theGCM setup with non-propagating regime, and τ =7.1 days and τ =7.4days for the GCM setup with propagating regime (estimated using themethodology in Appendix A of Lorenz and Hartmann (2001)).Feedback b b b b Non-propagating 0.133 0.003 0.002 0.021Propagating 0.101 0.075 -0.043 0.023 the SH reanalysis, yet their z e -folding decorrelation timescales are comparable (14 days vs. 10 days).These findings show the importance of quantifying andexamining cross-EOF feedbacks to fully understand thedynamics and variability of the annular modes and to bet-ter evaluate how well the GCMs simulate the extratopicallarge-scale circulation.
4. Concluding remarks
The low-frequency variability of the extra-tropicallarge-scale circulation is often studied using a reduced-order model of the leading EOF of zonal-mean zonalwind. The key component of this model (LH01) is aninternal-to-troposphere eddy-zonal flow interaction mech-anism which leads to a positive feedback of EOF1 ontoitself, thus increasing the persistence of the annular mode(Lorenz and Hartmann 2001). However, several studieshave showed that under some circumstances, strong cou-plings exist between EOF1 and EOF2 at some lag times,resulting in decaying-oscillatory, or propagating, annularmodes (e.g. Son and Lee 2006; Son et al. 2008; She-shadri and Plumb 2017). In the current study, follow-ing the methodology of Lorenz and Hartmann (2001) andusing data from the SH reanalysis and two setups of anidealized GCM that produce circulations with a domi-nant non-propagating or propagating regime, we first showstrong cross-correlations between EOF1 (EOF2) and theeddy forcing of EOF2 (EOF1) at long lags, suggestingthat cross-EOF feedbacks might exist in the propagatingregimes. These findings together demonstrate that there isa need to extend the single-EOF model of LH01 and builda model that includes, at a minimum, both leading EOFsand accounts for their cross feedbacks.With similar assumptions and simplifications used inLorenz and Hartmann (2001), we have developed a two-EOF model for propagating annular modes (consisting ofa system of two coupled ODEs, Eqs. (4)-(5) with (7)-(8))that can account for the cross-EOF feedbacks. In thismodel, the strength of the feedback of k th EOF onto j thEOF is b jk ( j , k = , J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S is shown that the propagating regime, which requires adecaying-oscillatory solution of the coupled ODEs, canexist only if the cross-EOF feedbacks have opposite signs( b b < ( b − b ) < − b b . These criteria showthat non-zero cross-EOF feedbacks are essential compo-nents of the propagating regime dynamics.Using this model and the idealized GCM and a stochas-tic prototype, we further show that cross-EOF feedbacksplay an important role in controlling the persistence of thepropagating annular modes (i.e., the e -folding decorrela-tion time scale of the zonal index, z j ) by setting the fre-quency of the oscillation ω (Eq. (24)). Therefore, in thisregime, the persistence of the annular mode (EOF1) doesnot only depend on the feedback of EOF1 onto itself, butalso depends on the cross-EOF feedbacks. We find thatas a result of the oscillation, the stronger the cross-EOFfeedbacks, the less persistent the annular mode.Applying the coupled-EOF model to the reanalysis datashows the existence of strong cross-EOF feedbacks in thecurrent SH extratropical large-scale circulation. Annularmodes have been found to be too persistent compared toobservations in GCMs including IPCC AR4 and CMIP5models (Gerber and Vallis 2007; Gerber et al. 2008a;Bracegirdle et al. 2020). This long persistence has beenoften attributed to a too strong positive EOF1-onto-EOF1feedback in the GCMs. The dynamics and strength of thisfeedback depends on factors such as the mean flow andsurface friction (Robinson 2000; Lorenz and Hartmann2001; Chen and Plumb 2009; Hassanzadeh and Kuang2019). External (to troposphere) influence, e.g., from thestratospheric polar vortex, has been also suggested to af-fect the persistence of the annular modes (Byrne et al.2016; Saggioro and Shepherd 2019). Our results show thatthe cross-EOF feedbacks play an important role in the dy-namics of the annular modes, and in particular, that theirabsence or weak amplitudes can increase the persistence,offering another explanation for the too-persistent annularmodes in GCMs.Overall, our findings demonstrate that to fully under-stand the dynamics of the large-scale extratropical circu-lation and the reason(s) behind the too-persistent annularmodes in GCMs, the coupling of the leading EOFs andthe cross-EOF feedbacks should be examine using modelssuch as the one introduced in this study.An important next step is to investigate the underly-ing dynamics of the cross-EOF feedbacks. So far wehave pointed out that cross-EOF feedbacks are essentialcomponents of the propagating annular modes; however,the propagation itself is likely essential for the existenceof cross-EOF feedbacks. In fact, our preliminary resultshows that the cross-EOF feedbacks result from the out-of-phase oscillations of EOF1 (north-south jet displace-ment) and EOF2 (jet pulsation) leading to an orches-trated combination of equatorward propagation of wave activity (a baroclinic process) and nonlinear wave break-ing (a barotropic process), which altogether act to reducethe total eddy forcings (not shown). In ongoing work,we aim to explain and quantify the propagating annularmodes dynamics using the LRF framework of Hassan-zadeh and Kuang (2016a,b) and finite-amplitude wave-activity framework (Nakamura and Zhu 2010; Lubis et al.2018a,b) that have been proven useful in understandingthe dynamics of the non-propagating annular modes (Nieet al. 2014; Ma et al. 2017; Hassanzadeh and Kuang 2019). Acknowledgments.
We thank Aditi Sheshadri, DingMa, and Orli Lachmy for insightful discussions. Thiswork is supported by National Science Foundation (NSF)Grant AGS-1921413. Computational resources wereprovided by XSEDE (allocation ATM170020), NCAR’sCISL (allocation URIC0004), and Rice University Centerfor Research Computing.APPENDIX A
Standard Errors of Cross-Correlations usingBartlett’s Formula
Assuming two stationary normal time series { X t } and { Y t } ( t ∈ [ T ] ) with the corresponding auto-correlationfunctions ρ X ( l ) and ρ Y ( l ) and zero true cross-correlations,the standard error of the estimated cross-correlation at lag l ( r XY ( l ) ) can be computed as (see Bartlett 1978, page 352):var { r XY ( l ) } = T − | l | ∞ ∑ g = − ∞ [ ρ X ( g ) ρ Y ( g )] . (A1)The null hypothesis is r XY ( l ) =
0, and it is rejected atthe 5% significance level if the estimated cross-correlationvalue at lag l is larger than two times square root of the es-timated standard error, i.e., | r XY ( l ) | > × (cid:112) var { r XY ( l ) } . References
Bartlett, M. S., 1978: An introduction to stochastic processes with spe-cial reference to methods and applications.
Journal of the Institute ofActuaries ,
81 (2) , 198199, doi:10.1017/S0020268100035964.Boljka, L., T. G. Shepherd, and M. Blackburn, 2018: On the Cou-pling between Barotropic and Baroclinic Modes of Extratropical At-mospheric Variability.
Journal of the Atmospheric Sciences ,
75 (6) ,1853–1871, doi:10.1175/JAS-D-17-0370.1.Bracegirdle, T. J., C. R. Holmes, J. S. Hosking, G. J. Marshall, M. Os-man, M. Patterson, and T. Rackow, 2020: Improvements in cir-cumpolar southern hemisphere extratropical atmospheric circula-tion in cmip6 compared to cmip5.
Earth and Space Science , ,e2019EA001 065, doi:10.1029/2019EA001065.Branstator, G., 1995: Organization of Storm Track Anomalies by Re-curring Low-Frequency Circulation Anomalies. Journal of the At-mospheric Sciences ,
52 (2) , 207–226, doi:10.1175/1520-0469(1995)052 (cid:104) (cid:105)
O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S Burrows, D. A., G. Chen, and L. Sun, 2016: Barotropic and BaroclinicEddy Feedbacks in the Midlatitude Jet Variability and Responses toClimate ChangeLike Thermal Forcings.
Journal of the AtmosphericSciences ,
74 (1) , 111–132, doi:10.1175/JAS-D-16-0047.1.Byrne, N. J., T. G. Shepherd, T. Woollings, and R. A. Plumb, 2016:Annular modes and apparent eddy feedbacks in the southern hemi-sphere.
Geophysical Research Letters ,
43 (8) , 3897–3902, doi:10.1002/2016GL068851.Chen, G., and R. A. Plumb, 2009: Quantifying the Eddy Feedbackand the Persistence of the Zonal Index in an Idealized AtmosphericModel.
Journal of the Atmospheric Sciences ,
66 (12) , 3707–3720,doi:10.1175/2009JAS3165.1.Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Config-uration and performance of the data assimilation system.
QuarterlyJournal of the Royal Meteorological Society ,
137 (656) , 553–597,doi:10.1002/qj.828.Feldstein, S., and S. Lee, 1998: Is the Atmospheric Zonal IndexDriven by an Eddy Feedback?
Journal of the Atmospheric Sci-ences ,
55 (19) , 3077–3086, doi:10.1175/1520-0469(1998)055 (cid:104) (cid:105)
Journalof the Atmospheric Sciences ,
55 (15) , 2516–2529, doi:10.1175/1520-0469(1998)055 (cid:104) (cid:105)
Geophysical Research Letters ,
35 (22) ,doi:10.1029/2008GL035712.Gerber, E. P., and G. K. Vallis, 2007: Eddy-Zonal Flow Interactionsand the Persistence of the Zonal Index.
Journal of the AtmosphericSciences ,
64 (9) , 3296–3311, doi:10.1175/JAS4006.1.Gerber, E. P., S. Voronin, and L. M. Polvani, 2008b: Testing the AnnularMode Autocorrelation Time Scale in Simple Atmospheric GeneralCirculation Models.
Monthly Weather Review ,
136 (4) , 1523–1536,doi:10.1175/2007MWR2211.1.Hassanzadeh, P., and Z. Kuang, 2016a: The linear response function ofan idealized atmosphere. Part II: Implications for the practical use ofthe fluctuation–dissipation theorem and the role of operator’s non-normality.
Journal of the Atmospheric Sciences ,
73 (9) , 3441–3452.Hassanzadeh, P., and Z. Kuang, 2016b: The Linear Response Functionof an Idealized Atmosphere. Part I: Construction Using Greens Func-tions and Applications.
Journal of the Atmospheric Sciences ,
73 (9) ,3423–3439, doi:10.1175/JAS-D-15-0338.1.Hassanzadeh, P., and Z. Kuang, 2019: Quantifying the Annular ModeDynamics in an Idealized Atmosphere.
Journal of the AtmosphericSciences ,
76 (4) , 1107–1124, doi:10.1175/JAS-D-18-0268.1.Hasselmann, K., 1988: Pips and pops: The reduction of complex dy-namical systems using principal interaction and oscillation patterns.
Journal of Geophysical Research: Atmospheres ,
93 (D9) , 11 015–11 021, doi:10.1029/JD093iD09p11015.Held, I. M., 2005: The Gap between Simulation and Understanding inClimate Modeling.
Bulletin of the American Meteorological Society ,
86 (11) , 1609–1614, doi:10.1175/BAMS-86-11-1609. Held, I. M., and M. J. Suarez, 1994: A proposal for the intercompari-son of the dynamical cores of atmospheric general circulation mod-els.
Bulletin of the American Meteorological Society ,
75 (10) , 1825–1830, doi:10.1175/1520-0477(1994)075 (cid:104) (cid:105)
Quarterly Journal of theRoyal Meteorological Society ,
122 (533) , 1197–1210, doi:10.1002/qj.49712253309.Jeevanjee, N., P. Hassanzadeh, S. Hill, and A. Sheshadri, 2017: Aperspective on climate model hierarchies.
Journal of Advancesin Modeling Earth Systems , , 1760–1771, doi:10.1002/2017MS001038.Kidson, J. W., 1988: Interannual Variations in the Southern Hemi-sphere Circulation. Journal of Climate , , 1177–1198, doi:10.1175/1520-0442(1988)001 (cid:104) (cid:105) Geophysical ResearchLetters ,
47 (6) , e2019GL086 585, doi:10.1029/2019GL086585.Lorenz, D. J., 2014: Understanding Midlatitude Jet Variability andChange Using Rossby Wave Chromatography: WaveMean Flow In-teraction.
Journal of the Atmospheric Sciences ,
71 (10) , 3684–3705,doi:10.1175/JAS-D-13-0201.1.Lorenz, D. J., and D. L. Hartmann, 2001: Eddy-Zonal Flow Feed-back in the Southern Hemisphere.
Journal of the Atmospheric Sci-ences ,
58 (21) , 3312–3327, doi:10.1175/1520-0469(2001)058 (cid:104) (cid:105)
J. Climate ,
16 (8) , 1212–1227,doi:10.1175/1520.Lubis, S. W., C. S. Huang, N. Nakamura, N.-E. Omrani, and M. Jucker,2018a: Role of finite-amplitude rossby waves and nonconserva-tive processes in downward migration of extratropical flow anoma-lies.
Journal of the Atmospheric Sciences , , null, doi:10.1175/JAS-D-17-0376.1.Lubis, S. W., C. S. Y. Huang, and N. Nakamura, 2018b: Role of Finite-Amplitude Eddies and Mixing in the Life Cycle of StratosphericSudden Warmings. Journal of the Atmospheric Sciences ,
75 (11) ,3987–4003, doi:10.1175/JAS-D-18-0138.1.Ma, D., P. Hassanzadeh, and Z. Kuang, 2017: Quantifying the EddyJetFeedback Strength of the Annular Mode in an Idealized GCM andReanalysis Data.
Journal of the Atmospheric Sciences ,
74 (2) , 393–407, doi:10.1175/JAS-D-16-0157.1.Nakamura, N., and D. Zhu, 2010: Finite-amplitude wave activityand diffusive flux of potential vorticity in eddy-mean flow interac-tion.
Journal of the Atmospheric Sciences ,
67 (9) , 2701–2716, doi:10.1175/2010JAS3432.1.Nie, Y., Y. Zhang, G. Chen, X.-Q. Yang, and D. A. Burrows, 2014:Quantifying barotropic and baroclinic eddy feedbacks in the persis-tence of the southern annular mode.
Geophysical Research Letters ,
41 (23) , 8636–8644, doi:10.1002/2014GL062210.Penland, C., 1989: Random Forcing and Forecasting Using PrincipalOscillation Pattern Analysis.
Monthly Weather Review ,
117 (10) ,2165–2185, doi:10.1175/1520-0493(1989)117 (cid:104) (cid:105) J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S
Robert, L., G. Rivire, and F. Codron, 2017: Positive and NegativeEddy Feedbacks Acting on Midlatitude Jet Variability in a Three-Level Quasigeostrophic Model.
Journal of the Atmospheric Sci-ences ,
74 (5) , 1635–1649, doi:10.1175/JAS-D-16-0217.1.Robinson, W. A., 1991: The dynamics of the zonal index in a simplemodel of the atmosphere.
Tellus A ,
43 (5) , 295–305, doi:10.1034/j.1600-0870.1991.t01-4-00005.x.Robinson, W. A., 2000: A Baroclinic Mechanism for the EddyFeedback on the Zonal Index.
Journal of the Atmospheric Sci-ences ,
57 (3) , 415–422, doi:10.1175/1520-0469(2000)057 (cid:104) (cid:105)
Geophysical Research Letters ,
46 (22) ,13 479–13 487, doi:10.1029/2019GL084763.Sheshadri, A., and R. A. Plumb, 2017: Propagating Annular Modes:Empirical Orthogonal Functions, Principal Oscillation Patterns, andTime Scales.
Journal of the Atmospheric Sciences ,
74 (5) , 1345–1361, doi:10.1175/JAS-D-16-0291.1.Simpson, I. R., T. G. Shepherd, P. Hitchcock, and J. F. Scinocca, 2013:Southern Annular Mode Dynamics in Observations and Models. PartII: Eddy Feedbacks.
Journal of Climate ,
26 (14) , 5220–5241, doi:10.1175/JCLI-D-12-00495.1.Son, S.-W., and S. Lee, 2006: Preferred Modes of Variability and TheirRelationship with Climate Change.
Journal of Climate ,
19 (10) ,2063–2075, doi:10.1175/JCLI3705.1.Son, S.-W., S. Lee, S. B. Feldstein, and J. E. Ten Hoeve, 2008: TimeScale and Feedback of Zonal-Mean-Flow Variability.
Journal of theAtmospheric Sciences ,
65 (3) , 935–952, doi:10.1175/2007JAS2380.1.Thompson, D. W. J., and E. A. Barnes, 2014: Periodic variability in thelarge-scale southern hemisphere atmospheric circulation.
Science ,
343 (6171) , 641–645, doi:10.1126/science.1247660.Thompson, D. W. J., and Y. Li, 2015: Baroclinic and Barotropic An-nular Variability in the Northern Hemisphere.
Journal of the Atmo-spheric Sciences ,
72 (3) , 1117–1136, doi:10.1175/JAS-D-14-0104.1.Thompson, D. W. J., and J. M. Wallace, 1998: The arctic oscilla-tion signature in the wintertime geopotential height and tempera-ture fields.
Geophysical Research Letters ,
25 (9) , 1297–1300, doi:10.1029/98GL00950.Thompson, D. W. J., and J. M. Wallace, 2000: Annular Modes in theExtratropical Circulation. Part I: Month-to-Month Variability*.
Jour-nal of Climate ,
13 (5) , 1000–1016, doi:10.1175/1520-0442(2000)013 (cid:104) (cid:105)
Jour-nal of the Atmospheric Sciences ,
71 (4) , 1480–1493, doi:10.1175/JAS-D-13-0185.1.Zurita-Gotor, P., 2014: On the Sensitivity of Zonal-Index Persistence toFriction.
Journal of the Atmospheric Sciences ,
71 (10)71 (10)