The Role of Rossby Wave Breaking in Shaping the Equilibrium Atmospheric Circulation Response to North Atlantic Boundary Forcing

The role of Rossby wave breaking (RWB) is explored in the transient response of an atmospheric general circulation model to boundary forcing by sea ice anomalies related to the North Atlantic Oscillation (NAO). When theNCAR Community Climate Model,version3, was forced by an exaggerated sea ice extentanomaly correspondingtoonearisingfromapositiveNAO,alocalizedbaroclinicresponsedevelopedandevolvedinto a larger-scale equivalent barotropic pattern resembling the negative polarity of the NAO. The initial baro- clinic response shifted the phase speeds of the dominant eddies away from a critical value equal to the background zonal ﬂow speed, resulting in signiﬁcant changes in the spatial distribution of RWB. The forcing of the background zonal ﬂow by the changes in RWB accounts for 88% of the temporal pattern of the response and 80% of the spatial pattern of the zonally averaged response. Although results here focus on one experiment, this ‘‘RWB critical line mechanism’’ appears to be relevant to understanding the equilibrium response in a broad class of boundary forcing experiments given increasingly clear connections among the northern annular mode, jet latitude shifts, and RWB.


Introduction
The response of atmospheric general circulation models to boundary forcing is quite varied across models and experiments (Kushnir et al. 2002) and has been challenging to interpret because of nonlinearities with respect to the sign of the forcing anomaly (Kushnir and Lau 1992;Peng et al. 2002Peng et al. , 2003Ferreira and Frankignoul 2005) and the location of the forcing anomaly relative to the storm track (Peng et al. 1997). Despite these complexities, studies of transient eddy feedback have provided some clues as to the physical processes underlying these varied response patterns, particularly when the response projects onto the northern annular mode or North Atlantic Oscillation (NAO). In such cases, it has been suggested that the response is in balance with patterns of eddy vorticity flux divergence (e.g., Robinson 2000;Kushnir et al. 2002;Peng et al. 2003).
The purpose of the present paper is to illustrate the role of Rossby wave breaking (RWB) in the time evolution of the atmospheric response to boundary forcing, where RWB refers to the rapid, irreversible overturning of potential vorticity (PV) on isentropic surfaces (McIntyre and Palmer 1983). By explicitly examining how RWB contributes to the development of NAO-like response patterns, we aim to add dynamic specificity to the ''transient eddy feedback'' to which such response patterns are often attributed. Although we anticipate that RWB is relevant to a broad class of boundary forcing experiments, we base the presentation here on a particular set of experiments with one type of forcing that was conducted with the National Center for Atmospheric Research Community Climate Model, version 3 (CCM3).
Specifically, Magnusdottir et al. (2004) and Deser et al. (2004) found a strong winter mean response when forcing CCM3 with sea ice extent anomalies corresponding to a strong positive NAO. In a recent paper, Deser et al. (2007, hereinafter referred to as DTP) extended this study by considering daily output from a large ensemble of experiments forced in the same way, each extending from November through April. They documented the daily evolution of the response from an initial baroclinic response, restricted in horizontal extent to the area of the forcing and nearby regions, to an equivalent barotropic response that is hemispheric in scale. DTP further showed, using a linear baroclinic model, that the equilibrium response is maintained by nonlinear transient eddy fluxes of vorticity (and, to a lesser extent, heat). DTP did not suggest a physical mechanism bridging the initial baroclinic, horizontally restricted response and the subsequent, large-scale equivalent barotropic response.
Here, we present evidence that RWB is the bridging mechanism that shapes the equilibrium response in the ''ICE'' experiment. 1 Since the equilibrium response is NAO-like, the idea that RWB is important follows logically from previous studies establishing the role of RWB, or RWB proxies such as eddy momentum flux, in maintaining NAO-like variability (Franzke et al. 2004;Abatzolglou and Magnusdottir 2006;Riviere and Orlanski 2007;Strong and Magnusdottir 2008). Since polarity changes of the NAO are associated with shifts in the latitude of the eddy-driven jet, our results are also related to recent research connecting jet shifts to changes in the phase speed of dominant eddies or the background zonal flow (Chen et al. 2007;Chen and Zurita-Gotor 2008).
Analyzing the output of the ICE experiment in the context of RWB, we find that 1) the initial, baroclinic response shifts dominant eddy phase speeds so as to alter the spatial distribution of Rossby wave breaking, and 2) the changes in RWB relative frequency force the background zonal flow in a manner accounting for more than 80% of the temporal and spatial patterns of the response. We present methods in section 2, followed by two results sections describing RWB variability driven by the ICE forcing (section 3a) and its effect on the atmospheric response (section 3b).

a. Model output
We analyze the same experiments as in DTP. For the ICE case, the sea ice areal extent forcing varied slightly through the season, but the mid-January anomalies in Fig. 1 are representative (complete details on the model configuration and forcing are given in DTP). Briefly, a 240-member ensemble of integrations was developed by forcing CCM3 (T42 with 18 vertical levels) with the ICE anomaly pattern, with each member running from 1 December to 30 April with unique initial conditions taken from a long, unforced run. To provide a control, a second 240-member ensemble of integrations was developed using the same series of initial conditions as the ICE experiment but with climatological sea ice extent.
For each of the 240 members, the forced and unforced runs provide a pair of integrations with the same initial conditions. The response to the forcing on a particular day of the experiment is obtained by subtracting each control result from its paired anomaly experiment for that day and then averaging across all the ensemble members for that day. This response is a function of space and time, is denoted by a prefix d, and may have a subscript to indicate that it has been averaged over specific days of the integration. For example, du 1-14 is the zonal wind response averaged over the first 14 days of the experiment and has values at each grid point over the hemisphere.

b. RWB analysis
For the control and forced ensembles, we calculated PV on the 350-K isentropic surface using daily average 2 fields of pressure, temperature, and horizontal velocity v [ (u, y). We then identified all instances of anticyclonic RWB for the Northern Hemisphere following the method detailed in Strong and Magnusdottir (2008, hereinafter SM8). Using Fig. 2 to briefly describe the method, we identify regions where circumpolar PV . We then quantify how often anticyclonic or cyclonic RWB centroids are observed within N 5 400 equal-area bins over the Northern Hemisphere. For each bin centered on (l, f) n , n 5 1, 2, . . . , N, we calculate the relative frequency of anticyclonic RWB centroids, which is a dimensionless quantity given by where t is the number of daily observations in the period for which g is calculated and the event parameter b takes the value 1 when an anticyclonic centroid is located in the bin at time t and 0 otherwise. A corre-sponding variable for cyclonic RWB denoted by g c is also calculated. We refer to regions of frequent RWB as ''surf zones,'' defined by the threshold g a , g c $ 0.04. The surf zones in the CCM3 simulations (not shown) correspond well to the observational surf zones shown in Fig. 2 of SM8.

c. Forcing of background flow by RWB
We calculated the composite forcing exerted on the background flow by anticyclonic and cyclonic RWB following the method used to generate Fig. 3 in SM8. Briefly, the forcing of the background zonal wind speed is given by the divergence of the vector E [ (y9 2 À u9 2 , Àu9y9) (see appendix A in Hoskins et al. 1983), where a prime indicates deviation from an average taken for a particular day of year and location over all ensemble members. We randomly selected 200 cases of RWB from different regions in the model, aligned them at their centroids, calculated $ Á E for each regional set, and then zonally averaged the results from 308 west of the centroid to 308 east of the centroid. The results from the control and forced ensembles (not shown) closely resemble those in Figs. 3a and 3b of SM8, indicating that anticyclonic RWB tends to accelerate the zonal flow on its poleward flank and decelerate the zonal flow on its equatorward flank, whereas cyclonic RWB exerts the opposite forcing. We will refer in section 3 to these fundamental composite $ Á E patterns as the the RWB u-forcing patterns. The magnitude of the forcing of the zonal flow by RWB shown in SM8 is approximately 64 3 10 25 m s 22 , meaning that the presence of RWB near a grid point for one day could change the background zonal flow by approximately 3.5 m s 21 .

d. Statistics
To test the significance of the model's response to forcing, we compared the mean of the forced ensemble with the mean of the control ensemble using a t test that accounts for serial and cross correlation in the samples (e.g., Wilks 1995, chapter 5). To develop multiple linear regression models, we used a backward stepwise bootstrap regression based on the Akaike information criterion. Regressors were included in the model if they were retained in at least 60% of the 1000 resampled iterations-a procedure shown to produce parsimonious models with excellent predictive capability (Austin and Tu 2004). An empirical orthogonal function (EOF) is the unit-length spatial eigenvector of the area-weighted covariance matrix of a scalar field. The associated principal component time series is the standardized (zero mean, unit standard deviation) projection of the corresponding data onto the EOF.

Results
In section 3a, we show that the initial baroclinic response is associated with changes in the background zonal flow and eddy phase speeds that lead to shifts in the relative frequency of RWB. In section 3b, we show how these changes in RWB force the background zonal flow via eddy momentum flux convergence anomalies, accounting for more than 80% of the spatial and temporal patterns of the large-scale equivalent barotropic response.
a. Response in zonal wind and eddies DTP describe the transient response in the ICE experiment in terms of geopotential height on the 1000-, 650-, and 300-hPa isobaric surfaces (their Fig. 6). We find it useful to supplement this view by examining the transient response in zonal wind (Fig. 3) and eddy phase speeds (Fig. 4). These two additional perspectives provide a useful supplementary view because RWB occurs near ''critical lines'' in the velocity field where the phase speed of incoming waves equals the background zonal wind speed (Randel and Held 1991), and RWB in turn feeds back onto the background zonal flow (SM8). The localized baroclinic response that develops during days 1-14 (see DTP) is in approximately geostrophic balance with a zonal wind anomaly pattern on the 350-K surface with centers of action approximately aligned along 428W (du 1-14 ; Fig. 3a). The 350-K u response for the remainder of the experiment (du 15-151 ; Fig. 3b) resembles the upper-tropospheric u anomaly pattern associated with the negative polarity of the NAO. Comparison of Figs. 3a and 3b from 428W to 08 illustrates that the equivalent barotropic response can be thought of as a translation or extension of the initial baroclinic response downstream.
To examine the role of the model's dominant eddies in this downstream translation, we decomposed the transient eddy momentum flux at 350 K into contributions from different zonal phase speeds as a function of latitude (Hayashi 1971) for the first two months of the experiment. The shading in Fig. 4a shows results averaged across the control ensemble. Similar to the observational results in Randel and Held (1991), the control eddy momentum flux was predominantly positive with a maximum near 358N, indicating generally equatorward wave propagation from a midlatitude source region. The contours in Fig. 4a show the response in eddy momentum flux. The response involved a shift toward slower phase speeds around 558N (contoured dipole; Fig. 4a). Further decomposing the spectrum in Fig. 4a by wavenumber following Randel and Held (1991), we see that the reduction in phase speed about 558 was associated with an increase in energy near wavenumber 4 and a decrease in energy near wavenumber 6 (Fig. 4b). The response also involved a shift toward higher phase speeds around 308N (Fig. 4a) with a less pronounced shift in dominant wavenumber (not shown).
The shift toward lower phase speeds around 558N in the forced experiment (contours; Fig. 4a) is toward the left on the phase axis, which is away from the critical line where the background zonal wind and phase speed match (thick curve; Fig. 4a). This shift away from critical phase speeds would be consistent with a reduction in the relative frequency of RWB over these latitudes. Indeed, the relative frequency of anticyclonic RWB is substantially reduced near 558N (g a response; Fig. 4c). The increase in phase speeds near 308N is separated from, but still toward, the critical line (Fig. 4a), which would be consistent with the modest collocated increase in g a (Fig. 4c). Figure 4c shows that smaller responses in cyclonic RWB also occurred. These g c anomalies made a nonnegligible contribution to the total response, as we will show in the next section. The decomposition of eddy momentum flux into phase speed-zonal wavenumber space appears less useful for elucidating the dynamics leading to the g c changes, in part because the method highlights dominant eddy processes. Rather than being dominant, the cyclonic response is more longitudinally confined as shown in section 3b, and the cyclonic surf zones over the Euro-Atlantic sector are approximately one-third as active as the anticyclonic surf zones (not shown here; see Fig. 2 in SM8).

b. Role of RWB in the response
In this section, we show that changes in the relative frequency of RWB account for more than 80% of the spatial and temporal patterns of the response. Beginning with the spatial pattern, we use shading in Fig. 5 to show the principal regions of positive and negative zonal wind response from Fig. 3b and superimpose the response patterns of RWB relative frequency (dg a and dg c contouring; Fig. 5). Approximately three-quarters of the surface area bounded by dg contours in Fig. 5 have g response anomalies that are significant at the 95% con-fidence level based on a t test. The alignment of the zonally elongated du 15-151 and dg response bands follows logically from how RWB forces the background zonal flow (the RWB u-forcing patterns discussed in section 2c), and the A1, A3, A5, and C5 regions identified in Fig. 5 correspond to RWB regions studied in detail in connection with the NAO in SM8. Consistent with the u-forcing pattern of anticyclonic RWB, regions of positive dg a (e.g., the A5 region in Fig. 5a) are flanked to the north by positive du 15-151 and to the south by negative du 15-151 . Regions of negative dg a (e.g., the A3 and A1 regions in Fig. 5a) are flanked to the north by negative du 15-151 anomalies and to the south by positive du 15-151 anomalies, again consistent with the u-forcing pattern of anticyclonic RWB.
Considering the u-forcing pattern of cyclonic RWB, regions of positive dg c such as C5 in Fig. 5b contribute to the du 15-151 response, which is positive to the south of C5 and negative to the north. The g c increase in C5 acts cooperatively with the g a reduction in A3 to generate the zonal wind response pattern. The reduction in g c over the high latitudes of North America, by contrast, is analogous to the C1 region discussed in SM8. Specifically, it acts in opposition to the g a reduction over the Pacific and North America, contributing to the disruption of zonal symmetry across these longitudes.
To further illustrate the latitudinal alignment of the response and the RWB anomalies, we modeled the zonally averaged zonal wind response using the RWB u-forcing patterns from the A3, A5, and C5 regions (patterns defined in section 2c). In Fig. 6, the thick curve is the zonal average of the du 15-151 response in Fig. 3b. The dashed curve is a spatial statistical model built from the u-forcing patterns. To build the model, we aligned the centroid of the u-forcing patterns from A3, A5, and C5 at the central latitude of these three regions (288, 568, and 488N respectively). We then fit a multiple linear regression model (method in section 2d) in which the zonally averaged du 15-151 response was the predictand and the aligned u-forcing patterns from A3, A5, and C5 were the predictors. The coefficient in front of the A3 index is negative, consistent with the sign of the associated dg a , and the model accounts for slightly more than 80% of the spatial pattern of the zonally averaged response.
To show that RWB accounts for the temporal pattern of the response, particularly the downstream translation of the response, we project du onto its leading EOF for the domain 428W-608E (Fig. 7). The principal component associated with this leading du EOF is shown as a thick line in Fig. 8a and resembles the temporal evolution of the response in geopotential height analyzed in DTP. We developed a multiple linear regression model of the du principal component (method in section 2d) using time series of g anomalies corresponding to the A3, A5, and C5 regions in Fig. 5. These time series are shown in Fig. 8b and represent anomalies in the relative frequency of RWB (dg a for the A regions and dg c for the C5 region) for radii of approximately 108 of arc length around each region's largest response anomaly. The model is shown as a thin curve in Fig. 8a and accounts for 88% of the du principal component.

Summary and discussion
Results were presented in support of the idea that RWB plays a central role in the transient response of an atmospheric general circulation model to a pattern of winter sea ice extent forcing. When the CCM3 was forced by North Atlantic sea ice extent anomalies corresponding to a strong positive NAO, the model developed a localized baroclinic response over the first two weeks. As the baroclinic response intensified, the dominant eddies along 558N shifted toward lower phase speeds better separated from the critical value equal to the background zonal wind speed, and a dramatic decrease in anticyclonic RWB ensued. Meanwhile, anticyclonic RWB became more frequent where eddy phase speeds increased toward the background zonal flow along 358N, and cyclonic RWB became more frequent over eastern Europe. We showed that these changes in RWB relative frequency exert a forcing on the background zonal flow that is latitudinally aligned, and in sign agreement with, the spatial pattern of the zonal wind response, and that the response in RWB accounts for more than 80% of the spatial and temporal pattern of the model's zonal wind response.
Model equilibrium response patterns such as the one examined here are often attributed to, or shown to be in balance with, nonlinear transient eddy feedback. Here, we identify RWB as a specific nonlinear eddy phenomenon whose variability can be affected by an initial baroclinic response and translated into a larger-scale barotropic response. That RWB should be the key element shaping such a response is not surprising given the established linkage between RWB and the NAO. In a previous observational study of reanalysis data, SM8 found that the NAO is highly correlated with RWB over specific regions of the hemisphere. In these regions, RWB generates surface pressure anomalies that project onto the NAO centers of action while generating a pattern of upper-tropospheric momentum flux divergence that feeds back onto the background zonal flow so as to maintain the zonal wind pattern characteristic of the NAO spatial pattern. Comparing the SM8 results with the present study, we find close correspondence between the regions where RWB is related to the observed NAO and regions where RWB responses were induced by the sea ice forcing pattern.
We anticipate that the RWB critical line mechanism described here may clarify how the equilibrium response patterns develop in a broad class of boundary forcing experiments, particularly those with response patterns that are NAO-like or are most pronounced near the jet exit regions where RWB is frequent. Previous research has demonstrated the importance of the location of the forcing relative to the position of the storm track and the importance of eddy vorticity fluxes in the subsequent evolution of the response. Here, we draw attention to the importance of the position of the critical line relative to the location of the forcing and the resulting modification of the spatiotemporal distribution of RWB. Changes in the background zonal flow or eddy phase speeds resulting from diabatic forcing by sea ice or SST anomalies can relocate critical lines. The strength of the equilibrium response can then vary considerably depending on the spatial distribution of RWB relative frequency with respect to the relocated critical line. In surf zones where the spectrum of incoming Rossby wave phase speeds and amplitudes is conducive to frequent breaking, critical line modification can have dramatic effects on downstream height and velocity fields, as seen in the experiment considered here.