Stratospheric Sudden Warming (SSW) events are typically, but not always, accompanied by negative Northern Annular Mode anomalies in the troposphere. However, large uncertainties remain as to which dynamical processes are responsible for those anomalies. In order to highlight sources of variability in stratosphere-troposphere coupling among SSW events, we present a case study of three selected events and show detailed Transformed Eulerian Mean diagnostics for momentum changes in the stratosphere and troposphere in the course of those events. Our results suggest that planetary-scale waves, especially the zonal wave number 2 component, may play an important role not only for the onset of tropospheric anomalies in response to SSW events but also for introducing variability in the vertical coupling, i.e., whether the tropospheric circulation anomalies lag, lead, or occur simultaneous to the weakening of the vortex. Particularly, the meridional propagation of those waves in the upper troposphere could be an important factor that determines whether SSW events lag or lead tropospheric anomalies.
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 Abrupt breakdowns of the stratospheric polar vortex, so-called Stratospheric Sudden Warming (SSW) events, often present signs of coupling between the stratosphere and the troposphere. While the stratospheric polar vortex decelerates, the typical tropospheric response is manifest in the dominant mode of Northern Hemisphere extratropical variability, the Northern Annular Mode (NAM) [Thompson and Wallace, 2000]. More specifically, negative phase of NAM is present for up to 2 months after the event [Baldwin and Dunkerton, 2001]. This downward coupling is robustly documented both in the climate models and mechanical models [e.g., Gerber and Polvani, 2009; Gerber et al., 2010]. Due to their implication to medium-range weather forecast, SSW events and the associated tropospheric responses have also been extensively examined using operational models [Kuroda, 2008].
 However, SSW events exhibit a significant variability among individual events. This is true not only in the deformation of the stratospheric vortex [Charlton and Polvani, 2007; Mitchell et al., 2013] but also in the characteristics of the tropospheric response. Although composite analyses show gradual downward propagation of NAM index anomalies from the stratosphere to troposphere, individual events exhibit large variations in their persistence, coupling timescale, and structure of coupling. Most importantly, Baldwin and Dunkerton  documented that not all SSW events present time-lagged tropospheric response: While some events are accompanied by negative NAM index anomalies in the troposphere, others are not or even exhibit positive NAM index anomalies, opposing the persistent negative NAM index anomalies in the stratosphere. Mitchell et al.  particularly indicated that downward coupling is pronounced only in vortex split events. In this regard, it is also important to note that SSW events are occasionally led by tropospheric anomalies before the events instead of the time-lagged response usually observed in composites [Baldwin and Dunkerton, 2001].
 To understand the stratosphere-troposphere coupling and its variability associated with SSW events, various theories have been proposed. They include downward control [Haynes et al., 1991; Thompson et al., 2006], potential vorticity inversion [Hartley et al., 1998; Black and McDaniel, 2004], and wave-mean flow interactions [Matsuno, 1971; Zhou et al., 2002]. However, it is still unclear what determines the nature of vertical coupling observed during the SSW events. Among others, Nakagawa and Yamazaki  suggested that tropospheric response to SSW events in a timescale of weeks is largely determined by planetary-scale waves of zonal wave number 2, mainly through their variations in vertical propagation. However, their Figure 3 also shows significant difference in meridional wave propagation in the upper troposphere between downward-coupled versus non-coupled events. Since they have not directly compared the forcing resulting from the vertical and meridional propagation of waves to observed wind tendencies, the relative importance of those remain uncertain.
 The present study revisits the nature of vertical coupling during the SSW events by examining different types of SSW-related tropospheric variability: whether tropospheric circulation anomalies are leading, lagging, or simultaneous to the onset of stratospheric circulation anomalies. To highlight their difference, case studies of three reference events are performed. Although case study can be hardly generalized, it is beneficial as it can reveal detailed temporal and spatial structures of SSW-related circulation anomalies. This contrasts composite analyses which are common but often obscure or dilute the detailed characteristics. Unlike in previous studies that have examined tropospheric response up to 3 months, only the short-term coupling within 10 days of the maximum stratospheric deceleration is emphasized in this study, therefore focusing more on the mechanisms responsible for the onset of circulation anomalies than their maintenance. Evaluating the dynamical features responsible for the onset of the tropospheric response to SSW events on such short timescale will improve our understanding of the predictability of SSW events in operational weather forecasts.
 By applying the Transformed Eulerian Mean (TEM) diagnostics to high-resolution reanalysis data, it is shown that the evolution of tropospheric circulation anomalies during the three SSW events are largely determined by planetary-scale waves of zonal wave number 2 in the upper troposphere and lower stratosphere as in Nakagawa and Yamazaki  but through their meridional propagation rather than vertical propagation. To our best knowledge, detailed TEM diagnostics on daily timescale for the selected events have not been reported in the literature. In section 2, data and methodology are described. It is followed by spatiotemporal evolution of NAM index and zonal mean zonal wind anomalies for the selected cases. Individual terms of TEM momentum equation are also examined. Lastly, overall findings are summarized in section 4.
2 Data and Methodology
 This study uses ERA-Interim reanalysis data from the European Centre for Medium-Range Weather Forecasts [Dee et al., 2011] from 1979 to 2011. Four times daily wind, temperature, and geopotential height data are obtained on a 1.5°×1.5° latitude-longitude grid on 37 pressure levels. Most of the results are presented as anomalies calculated by removing a 90 day low pass filtered daily climatology of the 33 years available. We then apply a 1-2-1 filter to daily averaged anomalies to remove spurious noise. Values presented as an average over a latitude band are weighted by the cosine of latitude.
 To quantify stratosphere and troposphere variabilities, the NAM index is analyzed. Throughout this paper the NAM index is computed as the negative of daily anomalies in area-averaged geopotential height north of 65°N [Cohen and Salstein, 2002]. Values are then normalized by one standard deviation independently at each pressure level. This simple method is known to be well correlated to Empirical Orthogonal Function (EOF) based methods [Baldwin and Thompson, 2009]. Some details of the physical implication of the NAM index are described in section 3.
2.2 SSW Events
 The SSW events are detected using the time derivative of the NAM index and zonal wind reversal. We first seek maximum tendency of negative NAM index at 10 hPa ensuring at least a decrease of 0.6σ over 3 days. This leaves only rapidly decelerating vortex events. Then, a reversal of the zonal mean zonal wind at 10 hPa and 60°N (per the WMO definition of major SSW) is checked within 10 days after the maximum tendency. As such, our selection of SSW events is more strict than WMO definition. To avoid counting the same event multiple times, only the events that are spaced at least 20 days apart are considered. In addition, only midwinter events are kept (December-January-February), excluding final warming events.
 In this study, the onset of SSW events is defined by the date of maximum decrease of NAM index at 10 hPa. This approach differs from previous ones that have often used the date of zonal wind reversal [e.g., Charlton and Polvani, 2007], the date when NAM index anomaly becomes smaller than a given reference value [e.g., Baldwin and Dunkerton, 2001], or the mid-date of the time period when NAM index anomaly, smaller than a reference value, is maintained [Limpasuvan et al., 2004]. Tendency of polar-cap temperature was used to detect SSW events in Nakagawa and Yamazaki , but unlike our definition, they did not enforce wind reversal in the stratosphere.
 The tendency-based index has a couple of advantages in defining the onset of SSW event with respect to traditional ones. By detecting maximum decrease of stratospheric NAM index or strongest deceleration of zonal mean zonal wind, the onset of SSW event is objectively determined. This allows us to obtain more robust tropospheric flow evolution on daily timescales. All analyses presented in this study are in fact performed on daily timescales with a minimal smoothing. This contrasts previous studies that have examined tropospheric anomalies averaged over weeks or months after SSW events. However, if tropospheric response over extended time period is the main interest, overall result would be insensitive to the choice of index as described in a companion paper (not shown). By using tendency-based index, this study can be also directly compared with composite analyses of Nakagawa and Yamazaki .
2.3 TEM Diagnostics
 We present diagnostics of zonal momentum using the primitive form of the TEM equation set on pressure coordinate [e.g., Andrews et al., 1987]
where overbar denotes zonal mean and is a residual term that includes internal diffusion, surface friction as well as parameterized forcing such as gravity wave drag. All the other symbols are standard. Equation (1) indicates that zonal mean zonal wind tendency is determined by the residual circulation and wave driving. The residual circulation is defined as
where prime denotes deviation from zonal mean. Physically, it is an approximation to the isentropic mass-weighted circulation [e.g., Vallis, 2006]. Note here that is defined on pressure coordinate and negative value denotes upward circulation. In (1), the terms related to the residual circulation are the Coriolis force acting on the residual circulation (first term on right hand side), the meridional advection of zonal momentum (second term), and vertical advection of zonal momentum (third term) by the residual circulation. Generally, the first term is much stronger than the latter two terms. All local effects of wave activity are expressed as the divergence of the EP flux vector (fourth term), except the effect of unresolved waves such as orographic gravity waves which are parameterized in ERA-Interim. The EP flux vector is written as follows:
and is parallel to fluxes of wave activity under quasi-geostrophic linear assumptions [e.g., Vallis, 2006]. The meridional and vertical components of the EP flux are largely dominated by momentum flux (3B) and form stress (3D) respectively, and only these terms appear in the quasi-geostrophic equation.
 As a graphical convention to represent EP flux as arrows in function of pressure and latitude, we follow the recommendations of Edmon et al. . The meridional and vertical length of the arrows () on the axes are related to the EP flux by
where K is a constant chosen for visualization purpose. Such scaling insures that arrows that look nondivergent correspond to . The graphical divergence is related to zonal wind tendency by a factor of (a cosφ)−1. When the divergence of EP flux () is reported, it is scaled by (a cosφ)−1in order to express it as a zonal wind tendency. Detailed pros and cons of using such scaling for EP flux representation are discussed more extensively in Edmon et al.  and Baldwin et al. .
 The NAM is the leading mode of variability in the Northern Hemisphere and represents a zonally symmetric seesaw of atmospheric mass between midlatitudes and the polar region. The characteristic wind anomalies associated with the NAM are in the form of a dipole with centers of maximum variability located around 30°N and 60°N [Thompson and Wallace, 2000]. Although circulation anomalies are known to usually project on the same phase of NAM within the stratosphere or troposphere, NAM signature is known to vary with height, especially when contrasting tropospheric and stratospheric circulations. In this regard, evaluating NAM phase at each level independently can provide a measure of coupling between circulation at different atmospheric levels [Baldwin and Thompson, 2009]. The NAM is a good indicator of the strength of the polar vortex in the stratosphere while it represents a meridional shift of the jet position in the troposphere (equatorward shift for negative phase of NAM, or negative NAM index). Figure 1 illustrates the winter variability of the NAM index and its evolution in the stratosphere and the troposphere for the last decade. Although the stratospheric and tropospheric NAM index anomalies fluctuate at different timescales, usually shorter in the troposphere [Gerber et al., 2010], there is an apparent connection between the two layers: the NAM index anomalies are often of the same sign between the stratosphere and troposphere, especially during more dramatic events such as the SSW events [Baldwin and Dunkerton, 1999]. Specifically, SSW events, indicated by red triangles in Figure 1, are often, but not always, characterized by a strong decrease of the NAM index in both layers.
 Figure 1 shows that eight events meet our definition of a SSW from 2000 to 2010. Events display a great amount of variability in strength, duration, and timing. More importantly, the nature of vertical coupling varies among the events. This is readily seen in NAM index tendency (Figure 1, right). For instance, the SSW event in 2001 exhibits upward propagation of the NAM index signal before the onset of the event then downward signal propagation afterward. This contrasts SSW events in 2003 and 2007 which, respectively, show only downward and upward propagating NAM index signal about the onset day. Tropospheric lead and tropospheric lag will hereafter be used to refer to the upward and downward propagating signals of the NAM index or zonal wind anomalies characterizing the SSW events. Here it is important to note that tropospheric lead or lag do not necessarily represent causal relationship as tropospheric anomalies could occur independently from the stratospheric anomalies.
 To demonstrate variability of vertical coupling and related dynamics, three representative SSW events are selected. In terms of onset date, they are the ones that occurred on 16 January 2003 (third row in Figure 1, hereafter 2003 event), 22 February 2007 (seventh row, 2007 event), and 21 January 2009 (ninth row, 2009 event). Each of the three events will be inspected individually. Here the 2003 event is an example of tropospheric lag, whereas the 2007 event exhibits tropospheric lead characteristics in contrast to composites of SSW events shown in Baldwin and Dunkerton . The last case, 2009 event, is a record-breaking event characterized by a transition from a very strong vortex to a very weak vortex, which also lead to long-lasting NAM index in the lower stratosphere [Manney et al., 2009; Harada et al., 2010; Martineau and Son, 2010]. Unlike the 2003 and 2007 events, this event has an extended persistence that is similar to the composites of SSW events shown in Baldwin and Dunkerton . Subjective inspection reveals that the 2003 event was characterized by a weak split of the vortex while the 2009 event experienced strong splitting of the vortex. The latter was shown clearly in Figure 3 of Harada et al. . On the other hand, the 2007 event was characterized by a displacement of the vortex, in terms of SSW classification presented by Charlton and Polvani .
 Figure 2 illustrates the evolution of NAM index and zonal mean zonal wind anomalies during the three events along with vertical component of planetary-scale EP flux anomalies. Zonal wind and EP flux anomalies are averaged over 60°N–80°N, to be related with NAM index anomalies. Vertical component of EP flux is presented because SSW events are often preceded by strong planetary-scale wave propagation into the stratosphere, where they break and deposit easterly momentum [Polvani and Waugh, 2004; Limpasuvan et al., 2004]. By definition, lag 0 denotes the date of maximum decrease of NAM index at 10 hPa.
 As for the onset of 2003 event, negative NAM index anomalies appear first in the stratosphere with a peak at lag 2 (first row). They are connected with negative anomalies in the troposphere that are observed from lag 2 and reach minimum at lag 6. This time-lagged downward coupling is clearer in the NAM index tendency (second row), and similar structure is also observed in zonal wind anomalies (third row) and their tendencies (fourth row). As hinted in Figure 2(bottom row), this event is likely caused by zonal wave number 1 forcing in the stratosphere. The vertical component of the EP Flux for zonal wave number 1 is strong around lag −10 in the troposphere and maximizes in the stratosphere around lag −1. The wave number 2 component has rather small peaks around lags 1 and 7 in the troposphere. Although the first peak at lag 1 is accompanied by a small hint of upward propagation into the stratosphere, this is much weaker than wave number 1 component.
 In contrast to the 2003 event, the 2007 event shows the characteristics of tropospheric lead (Figure 2, middle column). The NAM index anomalies appear first in the troposphere (lag 0) with strong negative anomalies observed around lag 3 in the stratosphere. This decreasing trend in stratospheric NAM index is accompanied by increasing trend in tropospheric NAM index. Zonal wind anomalies also show tropospheric lead with anomalous easterlies at lags −2 to −1 in the troposphere and at lag 3 in the stratosphere. In terms of vertical wave propagation, this event is characterized by a strong vertical component of EP flux by zonal wave number 1 appearing at lag −5 in the troposphere and reaching a maximum around lag 2 in the stratosphere. There are also two episodes of upward wave number 2 propagation at lags −6 and −1 in the troposphere but they are confined to that layer.
 The 2009 event (Figure 2, right column) presents a marked transition from stronger-than-usual NAM index to weaker-than-usual NAM index around lag 0 from the surface to the stratosphere [see also Harada et al., 2010; Martineau and Son, 2010]. Tropospheric lead or lag characteristics are not obvious in this case although downward propagation of NAM index anomalies is evident within the stratosphere. In contrast to 2003 and 2007 events, this event is characterized by a relatively weak wave number 1 forcing. The upward propagation of planetary-scale waves is dominated by zonal wave number 2: anomalous upward propagation becomes stronger at lag −5 in the troposphere and also peaks at around lag −3 in the stratosphere. Strong upward propagation in the troposphere lasts until about lag 3 and is then also connected to another stratospheric maximum around lag 5. The two peaks of upward stratospheric EP flux were well documented by Harada et al. , who also found the 2009 event to present a record amount of upward wave number 2 fluxes.
 The above results highlight considerable differences in zonal mean flow and wave activities among the SSW events. However, in all three events, regardless of the wave activity in the stratosphere, zonal wind deceleration in the troposphere occur concurrently with enhanced upward propagation of zonal wave number 2 waves in the troposphere (compare fourth and sixth rows in Figure 2: i.e., ∼lag 1 in 2003, ∼lag −5 in 2007, and ∼lag 0 in 2009). This may suggest that the direction of stratosphere-troposphere coupling (upward versus downward) is largely determined by the timing of zonal wave number 2 component in the troposphere. To confirm this, the relationship between wave activity and the onset of tropospheric anomalies is further investigated in the TEM framework (Figure 3).
 Figure 3 shows that the observed wind tendency in the free atmosphere is remarkably well explained by the sum of forcing terms of the TEM momentum diagnostic in all events (top two rows in Figure 3). Significant mismatch is observed at the surface because surface friction is not included in our diagnostic (i.e., in equation (1)).
 The 2003 event presents a good match between EP flux divergence (Figure 3, third row) and wind tendencies (second row) in the stratosphere, the main feature being a strong forcing for deceleration around lag 0 at 10 hPa. It also matches roughly in the upper tropospheric region with forcing for acceleration around lags −4 to 0 and deceleration from lags 0 to 6. The meridional EP flux divergence (Figure 3, fourth row) generally contributes to the acceleration in the stratosphere except during a short period from lag −1 to lag 1, while in the troposphere, it contributes to deceleration, especially from lag 0 to 5. The vertical EP flux convergence (Figure 3, fifth row) shows a great contribution to deceleration in the stratosphere from lags −10 to 5, indicating that the breaking of vertically propagating waves is responsible for vortex deceleration around the onset of SSW events. It also contributes to the acceleration around the tropopause region (≈300 hPa). This term is however largely canceled by the effects of the residual circulation both in the stratosphere and troposphere around the onset of SSW event (compare fifth and sixth rows of Figure 3).
 The 2007 event also shows a good match between stratospheric EP flux divergence and observed wind tendency. The meridional EP flux convergence contributes greatly to the total stratospheric wave forcing with a contribution to deceleration from lags −5 to 2, while vertical EP flux convergence is important in lags −2 to 5. The former also contributes strongly to the tropospheric deceleration from lags −7 to −1 and an episode of acceleration around lags 1 to 4, which matches reasonably well with the observed wind tendencies. The vertical EP flux convergence presents forcing for deceleration and acceleration around the tropopause region for lags −5 to 0 and lags 0 to 5, respectively. The effect of the residual circulation is again to roughly oppose the forcing by the EP flux divergence in the stratosphere. Cancelation of the vertical component of EP flux divergence is especially strong in the upper troposphere, resulting in zonal wind change mostly due to meridional EP flux convergence.
 The 2009 event shows multiple episodes of anomalous EP flux convergence in the stratosphere at lags −9, −4, and 5, the latter being the strongest of all. Somewhat surprisingly, the effect of the residual circulation is primarily responsible for the peak in deceleration observed at lag 0. This is in stark contrast to 2003 and 2007 events. A strong equatorward residual circulation at 10 hPa from lags −3 to 1 is likely formed in response to EP flux convergence above that level (not shown). Similar residual circulation anomalies are observed in the transient response to EP flux divergence aloft in numerical model experiments [e.g., Haynes et al., 1991]. EP flux divergence shows a rather weak match with wind tendencies in the troposphere around lag −5 to 0. Tropospheric wind change is largely explained by meridional EP flux convergence alone.
 The above result suggests that stratospheric wind tendency during the SSW events is driven not only by local wave forcings as seen in the EP flux divergence (2003 and 2007 events) but also by the residual circulation which results from the cumulative effect of diabatic and wave forcing in the whole atmosphere (2009 event). Their anomalies often balance or cancel each other in rather complex ways, resulting in a small total forcing compared to their individual values. In contrast, tropospheric wind tendency is mostly explained by meridional component of EP flux convergence. In general, total EP flux convergence tends to show an ill fit to the observed wind tendency in the troposphere. This may be explained in part by the fact that the effect of the residual circulation is typically heavily opposed to the vertical component of EP flux divergence, allowing the meridional component of EP flux divergence, mostly eddy momentum flux convergence, to act on its own. Note that, in equation (1), the Coriolis force acting on the residual circulation and the vertical component of EP flux divergence share the term
with an opposite sign.
 One major interrogation is the nature of the waves responsible for the tropospheric wind tendencies: tropospheric response is sometimes assumed to result from feedback between synoptic-scale waves and background flow in the lower stratospheric and upper troposphere [e.g., Limpasuvan et al., 2004]. To identify the scale of waves that are responsible for vertical coupling, meridional component of EP flux divergence is decomposed into different wave numbers and presented in Figure 4. In all cases, stratospheric EP fluxes are dominated by wave numbers 1 and 2 components. Synoptic-scale waves play only a minor role. Although not shown, this is true even in the vertical component of EP fluxes (see Figure 2, bottom two rows).
 In the troposphere, meridional component of EP flux divergence is predominantly explained by zonal wave number 2 component at high latitudes. Although synoptic-scale waves also affect tropospheric wind tendency (e.g., at lag −2 in 2007 event and at lags −5 and 2 in 2009 event), they are relatively minor around the onset of the SSW event. This result is at first sight consistent with Nakagawa and Yamazaki  in the sense that wave number 2 plays an important role in vertical coupling through enhanced upward and poleward fluxes of wave activity. However, we have shown that the meridional propagation of planetary-scale waves (mostly momentum flux) could be more important than vertical propagation (mostly heat flux) in some cases for the tropospheric wind change because of a heavy cancelation of the vertical component of wave forcing by the effects of the residual circulation.
 Figures 5 and 6 show, respectively, the vertical and latitudinal structure of the TEM forcing terms and the corresponding EP flux anomalies for the three selected events during episodes of tropospheric zonal wind deceleration. Those periods correspond to lags 0 to 3, −7 to −1, and −4 to 1 for the 2003, 2007, and 2009 events, respectively (Figure 3, first row). In all cases, tropospheric zonal wind decelerates on the poleward side of 60°N with small latitudinal variations amongst events. These tropospheric deceleration episodes are all coincident with forcing by the meridional convergence of EP flux (Figure 5, fourth row). Meridional EP flux convergence in the polar region is usually the result of anomalous poleward EP fluxes around 60°N, with a great contribution by the wave number 2 component (Figure 6, third row). Other wave numbers bring a minor contribution to the poleward fluxes, such as wave number 3 in the 2003 event and wave numbers 1 and 4, and greater in the 2009 event.
 The 2007 event presents an episode of strong tropospheric acceleration after the peak in stratospheric deceleration (Figure 3). Such episode plays an important role in the tropospheric zonal wind evolution of this particular event and therefore merits further examination. Figure 7 presents zonal mean momentum budget and anomalous EP flux averaged over lags 0 to 5. During this period, both the meridional and vertical components of the EP flux divergence present forcing for tropospheric acceleration. Although the vertical component is largely offset by the effects of the residual circulation, it is responsible for acceleration from 60°N to 70°N, while anomalous equatorward EP flux around 65°N results in meridional EP flux divergence north of 70°N (see also Figure 4). The transition in the meridional direction of EP flux anomalies from poleward to equatorward around lag 0 (compare Figure 6 and 7) is mainly attributed to an enhanced equatorward EP flux by wave number 1 and a reduced poleward EP flux by wave number 2 while the changes of the meridional component of smaller scale waves, although less dramatic, also contributes to EP flux divergence around 65°N to 70°N after lag 0.
 The above results indicate that the synchronization of the tropospheric and stratospheric circulation (tropospheric lead, lag, or simultaneous response) during these SSW events is controlled to a great extent by meridional propagation of planetary-scale waves, mainly by those with zonal wave number 2, in the upper troposphere and lower stratosphere. When the lagged tropospheric response occurs, waves tend to propagate anomalously poleward in the extratropics, causing upper tropospheric EP flux convergence around 60°N–80°N. This contrasts to upward coupling event where anomalous poleward wave propagation occurs a couple of days before the onset of the SSW and then disappears or reverses propagation direction afterward. Although a large proportion of events show that forcing by zonal wave number 2 waves favors a joint evolution of the stratospheric and tropospheric circulations on short timescales, such mechanism cannot be generalized within the context of a case study. The occurrence of such phenomenon in the events presented could be the result of random tropospheric variability, unrelated to the weakening of the stratospheric vortex as discussed earlier. A composite analysis is required to evaluate whether the wave number 2 signal is systematic across events.
 Modification of the tropospheric zonal circulation by anomalous wave number 2 wave fluxes is not limited to the three events that were examined in detail in this study. Figure 8 shows three other cases whose evolutions differ widely. The 20 February 1979 event shows signs of tropospheric lead since it is characterized by upper tropospheric zonal wind deceleration before lag 0, when enhanced upward and poleward EP fluxes by wave number 2 occur in the troposphere. On the other hand, the 12 February 1981 event shows tropospheric lag characteristics although the deceleration signals of the stratosphere and troposphere are not strongly connected. In this case, the deceleration can be explained by meridional component of EP flux divergence with contributions from all wave scales. Finally, the 14 December 1998 event shows that no tropospheric deceleration is well connected to the stratospheric deceleration. In this event, a strong deceleration episode is found after lag 5 and coincides with forcing for deceleration by the meridional component of wave number 2 disturbances. In light of those additional events, it is appropriate to say that anomalous wave number 2 fluxes can strongly alter tropospheric zonal wind during SSW events, although it is not always the case, like in February 1981, indicating large variability among SSW events.
 It is unclear what determines the timing and direction of meridional wave propagation in the events in which zonal wind evolution is primarily forced by anomalous meridional EP flux. To identify the possible mechanism(s), we examined quasi-geostrophic refractive index in the upper troposphere and lower stratosphere for the 2003 and 2007 cases. It is known that waves tend to propagate toward the latitudes where refractive index goes to infinite, the so-called critical latitudes, but become evanescent where refractive index is negative. Since refractive index is a function of zonal mean state, wave propagation can be modified by the evolution of zonal mean flow during SSW events. However, no systematic difference is observed between tropospheric lead and lag events (not shown). This is likely because the rapid evolution of zonal mean flow during SSW events violates quasi-geostrophic assumptions.
 While EP flux vector is parallel to wave propagation direction, it is also a function of wave activity flux. Under quasi-geostrophic scaling and WKB assumptions,
where A is wave activity density and is group velocity. This relationship indicates that anomalous EP flux in the midlatitude upper troposphere and lower stratosphere could result from the changes in zonal mean PV gradient by SSW events. However, no systematic difference is found about lag 0 in the 2003 and 2007 events. This might result from the failure of quasi-geostrophic assumption. It might be also caused by the failure of zonal mean assumption. Ambaum and Hoskins  pointed out that tropospheric response to the stratospheric PV anomaly is often zonally asymmetric, indicating that zonal mean dynamics could not explain the details of the stratosphere-troposphere coupled system. Further experiments with composite analyses that can smooth noise are needed.
 We presented the evolution of three selected Stratospheric Sudden Warming (SSW) events that occurred on 16 January 2003, 22 February 2007, and 21 January 2009. They are chosen to represent leading, lagging, and simultaneous tropospheric response to stratospheric vortex deceleration in SSW events in a time scale of days. The detailed temporal evolutions of NAM index anomaly and zonal mean zonal wind anomaly, and the forcing responsible for the latter are illustrated by analyzing momentum budget in the TEM framework.
 While deceleration of the polar vortex during SSW events is caused by the residual circulation and meridional and vertical EP flux convergence of varying relative importance amongst the events, high-latitude wind deceleration in the troposphere before or after the SSW event is forced primarily by meridional EP flux convergence over 60°N–80°N in all three cases. The anomalous poleward EP flux in the extratropical upper troposphere and lower stratosphere, that is responsible for the convergence in high latitude, receives a major contribution from planetary-scale waves of wave number 2. This suggests that wave number 2 activity, often present during SSW and sometimes as a source of forcing of vortex deceleration itself (like in the 2009 event as described in detail by Harada et al. ), is effective during the selected SSW events in forcing a tropospheric flow through an enhanced poleward flux of wave activity.
 It is also found that, while some events are forced by wave number 2 both in the stratosphere and the troposphere, as in the 2009 event, other events can be forced by distinct wave scales in the two layers. The 2003 and 2007 events had their vortex decelerated mainly due to forcing by wave number 1 while the episodes of tropospheric deceleration were dominated by wave number 2 forcing. Those two events also presented different timing of tropospheric response relative to the weakening of the vortex, tropospheric wind tendency lagging the stratospheric deceleration in 2003 and leading the stratosphere in 2007. The differences in timing are largely explained by the presence of wave number 2 activity, with tropospheric deceleration episodes occurring at the same time as anomalous poleward wave activity fluxes.
 The diagnostics presented in this study bring more evidence of the important role of wave number 2 planetary-scale waves for the coupling of the tropospheric and the stratospheric flow during SSW events. In the cases investigated, the impact of these waves is further shown to be related to the poleward propagation of wave activity rather than vertical propagation proposed by Nakagawa and Yamazaki , resulting in zonal wind deceleration primarily due to momentum flux divergence in the polar region. It is however unclear what controls wave number 2 activity in the troposphere. Quasi-geostrophic zonal mean dynamics do not provide any convincing clue: Further studies are needed.
 The present work is solely based on a case study. Since each SSW event differs substantially in many aspects, our findings can not be generalized. This limitation could be partly resolved by performing composite analyses. Preliminary analyses show that although tropospheric lead events are less frequent than events that exhibit lag or simultaneous response characteristics, about 10 out of 18 events detected from 1979 to 2011 show tropospheric lead characteristics, which contrasts from the typical downward propagating signal observed in composites. This result indicates that short-term vertical coupling associated with SSW events is not simply one way, contrasting the extended coupling which typically exhibits downward coupling from a week to up to 2 months [Baldwin and Dunkerton, 2001]. Based on correlation analysis (not shown), lead, synchronous, and lag characteristics are not only associated to poleward zonal wave number 2 EP flux anomalies as emphasized in this case study but also to wave number 1 fluxes. This result indicates that downward coupling associated with SSW events is highly variable among the events and that extra care must be taken for the design of composite analyses to avoid obscuring significant variations in SSW-related stratosphere-troposphere coupling. The details of composite analyses will be presented in a companion paper.
 This work is funded by the Korea Meteorological Administration Research and Development Program under Grant CATER 2012-3065. We thank three anonymous reviewers for their constructive comments.