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  • View in gallery
    Fig. 1.

    Latitude–time composites of (a) the Eliassen–Palm flux divergence (EPFD; units: m s−1 day−1) anomalies filtered for planetary waves 1–3 and vertically integrated over 100–0.85 hPa for 70 control SSWs (see section 2a), (b) the residual streamfunction (Ψυ¯*; units: kg m s−1) vertically integrated over 100–0.85 hPa, and (c) u¯ anomalies at 970 hPa. Green contours in (a) and (b) indicate the vertically integrated u¯ anomalies with a contour interval of 5 m s−1 and the zero contour omitted. Note that anomalies that are not statistically significant different from zero at the 95% level using a standard Student’s t test are not shaded. Black vertical line indicates lag 0, and the horizontal black line in (c) indicates the climatological December–February jet maximum.

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    Fig. 2.

    (a) EPFD anomalies (shading; units: m s−1 day−1) for the 70 control SSWs averaged over lags −30 to −1 (i.e., over the forcing stage). Black contours show u¯ anomalies with contours at ±0.5, 1, 2, 3, 5, 10, 15, 20, … m s−1. (b) Time series during the forcing stage for the 100–0.85-hPa pressure-weighted and 40°–90°N area-averaged EPFD anomalies from (a). Black line shows the daily composite mean whereas gray shading shows the minimum and maximum daily anomalies of the 70 SSW events in control. The cumulative sum of the daily EPFD anomalies over this lag interval is inset into (b).

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    Fig. 3.

    (a) Spatial and (b) temporal profiles of an example imposed momentum forcing with MF = −5 m s−1 day−1 and Nd = 12 days [i.e., the longer-impulse (PTRBl) run]. Units in (a) are m s−1 day−1, whereas (b) is unitless. Horizontal white lines in (a) show the levels between which the torque is linearly decreased to zero (i.e., pt = 60 hPa and pb = 100 hPa) and vertical line shows the lowest latitude at which a torque is applied (i.e., φL = 40°N). Blue line in (b) shows a temporal profile with sudden jumps in the torque on days 0 and Nd = 12, whereas the red line shows a smooth profile with a gradual increase and decrease of the torque centered on midnight of day Nd/2, respectively. Inset into (a) is the 100–0.85-hPa pressure-weighted and 40°–90°N area-averaged values of the daily imposed torque, and inset into (b) is the time-integrated forcing (i.e., the “impulse”).

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    Fig. 4.

    Time series of (a) u¯ (m s−1) at 60°N and 10 hPa and (b) T¯ anomalies (K) area averaged over 50°–90°N and pressure weighted over 150–1 hPa, for the 70 control SSWs (black line) and for various PTRB runs with pt = 60 hPa (colored lines). Note that there are 49 ensemble members for each PTRB run. Gray shading in (a) shows the maximum and minimum daily u¯ values for the control SSWs. Shading in (b) is similar but for integrated T¯ anomalies.

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    Fig. 5.

    Latitude–time composites of u¯ anomalies for (a) the 70 control SSWs (repeated from Fig. 1c), (b) the MF = −5 m s−1 day−1, Nd = 12 days PTRBl run, and (c) the MF = −20 m s−1 day−1, Nd = 3 days PTRBs run. Note that there are 49 ensemble members for each PTRB run. Units are m s−1. Also, note that only u¯ anomalies that are statistically significant at the 95% level are shaded. Dashed black horizontal line indicates the climatological jet maximum (∼43°N), and the dashed black vertical line indicates lag zero. The thin dashed red lines in (b) and (c) show the corresponding forcing duration for each of the PTRB runs. Inset into each panel are the minimum values.

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    Fig. 6.

    Scatterplot of u¯ anomalies at 100 hPa against u¯ at 970 hPa, both averaged over 60°–87°N and the recovery stage, for different 49-ensemble-member PTRB experiments (see legend) and the 70 control SSWs. Note that the recovery stage is defined as the 90-day period after the torque has been switched off (see section 3b for details). Filled colored squares indicate the corresponding ensemble means for each experiment. Note that a vortex-strengthening PTRB run is included (MF = 5 m s−1 day−1, Nd = 12 days; cyan) to better test the linearity of the tropospheric response to an imposed torque. The black line shows the best-fit line calculated using a least-squares fit. The slope of the linear regression line (along with the confidence interval) and the correlation coefficient (r) are included in the bottom right. Note that the regression slopes for the control SSWs and the PTRB SSWs are the same, hence one line is plotted.

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    Fig. 7.

    Latitude–height composites of (left) the anomalous residual-mean meridional circulation Ψυ¯* along with contributions from (center) the Eulerian-mean circulation Ψυ¯ and (right) the eddy heat flux convergence Ψυθ¯. The anomalies are averaged over the “forcing” lags which for (top) the 70 control SSWs are defined as lags −20 to +3. For the two 49-ensemble-member PTRB runs, the forcing lags are defined to be the duration of the imposed forcing, which for (middle) the PTRBl run is Nd = 12 days and (bottom) the PTRBs run is Nd = 3 days. Units are kg m s−1. Black contours show u¯ anomalies averaged over the corresponding lags with contours at ±0.5, 1, 2.5, 5, 10, … m s−1. Thick black contours indicate statistically significant differences from zero at the 95% level.

  • View in gallery
    Fig. 8.

    As in Fig. 7, but for the “recovery” lags defined as the 90-day period following the forcing lags. In particular, (top) for the 70 control SSWs this is defined to be the lag-4–93-day period after the SSW onset, (middle) for the PTRBl run it is defined as lags 13–102, and (bottom) for the PTRBs run it is defined as lags 4–93 (see section 3b for details).

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    Fig. 9.

    Contributions from each term in the Eulerian zonal-mean momentum budget [Eq. (7)] for the longer-impulse (PTRBl) run without topography. Units are m s−1 day−1. The budget is vertically integrated over 925–125 hPa. Green contours indicate the vertically integrated u¯ anomalies. Thick black contours indicate statistically significant differences from zero at the 95% level using a standard Student’s t test. Black and red vertical lines indicate lag 0 and lag 12, respectively, the latter being the lag at which the forcing is switched off in the PTRBl run.

  • View in gallery
    Fig. 10.

    Time series showing each of the terms in the Eulerian zonal-mean momentum budget [Eq. (7)] area averaged over 60°–90°N and vertically integrated over 925–125 hPa, for (a) the short-impulse PTRBs run (MF = −20 m s−1 day−1, Nd = 3 days) and (b) the long-impulse PTRBl run (MF = −5 m s−1 day−1, Nd = 12 days). Units are m s−1 day−1. Thick line segments indicate statistically significant differences from zero at the 95% level using a standard Student’s t test. Black and red vertical lines indicate lag 0 and the lag on which the torque is switched off, respectively.

  • View in gallery
    Fig. 11.

    Experiments using a zonally symmetric version of the Reading IGCM (Hoskins and Simmons 1975) forced with an impulse of MF = −5 ms−1 day−1 for Nd = 12 days (i.e., equivalent to the PTRBl run in MiMA, see Fig. 3), with (left) an experiment with no radiative damping or near-surface friction and (right) an experiment with both processes switched on. The damping time scale is 10 days whereas the time scale for the friction on the lowest model level is 1 day. (top),(middle) The Eulerian-mean streamfunction (Ψυ¯; shading; units: kg m s−1) during the forcing and recovery stages, respectively, with the zonal-mean zonal wind (u¯) shown with contours at ± 0.5, 1, 2.5, 5, 10, 15, … m s−1. (bottom) Latitude–time projections of the 970–125-hPa vertically integrated Coriolis torque term (fυ¯; shading; units: m s−1 day−1) with u¯ shown with contours at ±0.5, 1, 2, 3, … m s−1. Dashed red line in bottom row indicates the lag at which the torque is switched off. Note that the color bars in the top and middle rows are different than those in Figs. 7 and 8 and the color bar in the bottom row is different from that in Fig. 9.

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    Fig. 12.

    As in Fig. 9, but for a thermally forced PTRB run with a 15-K transient high-latitude heating perturbation being switched on for 3 days in the stratosphere, following White et al. (2020). Note the topography has been removed to avoid issues with calculating the Eulerian-mean momentum budget [Eq. (7)] at lower levels. Black and red vertical lines indicate lag 0 and lag 3, respectively, the latter being the lag at which the heating is switched off.

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On the Tropospheric Response to Transient Stratospheric Momentum Torques

Ian P. WhiteaInstitute of Earth Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

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Chaim I. GarfinkelaInstitute of Earth Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

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Peter HitchcockbDepartment of Earth and Atmospheric Sciences, Cornell University, Ithaca, New York

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Abstract

An idealized model is used to examine the tropospheric response to sudden stratospheric warmings (SSWs), by imposing transient stratospheric momentum torques tailored to mimic the wave-forcing impulse associated with spontaneously occurring SSWs. Such an approach enables us to examine both the ∼2–3-week forcing stage of an SSW during which there is anomalous stratospheric wave-activity convergence, as well as the recovery stage during which the wave forcing abates and the stratosphere radiatively recovers over 2–3 months. It is argued that applying a torque is better suited than a heating perturbation for examining the response to SSWs, due to the meridional circulation that is induced to maintain thermal-wind balance (i.e., the “Eliassen adjustment”); an easterly torque yields downwelling at high latitudes and equatorward flow below, similar to the wave-induced circulation that occurs during spontaneously occurring SSWs, whereas a heating perturbation yields qualitatively opposite behavior and thus cannot capture the initial SSW evolution. During the forcing stage, the meridional circulation in response to an impulse comparable to the model’s internal variability is able to penetrate down to the surface and drive easterly-wind anomalies via Coriolis torques acting on the anomalous equatorward flow. During the recovery stage, after which the tropospheric flow has already responded, the meridional circulation associated with the stratosphere’s radiative recovery appears to provide the persistent stratospheric forcing that drives the high-latitude easterly anomalies, whereas planetary waves are found to play a smaller role. This is then augmented by synoptic-wave feedbacks that drive and amplify the annular-mode response.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian White, ian.white2009@hotmail.co.uk

Abstract

An idealized model is used to examine the tropospheric response to sudden stratospheric warmings (SSWs), by imposing transient stratospheric momentum torques tailored to mimic the wave-forcing impulse associated with spontaneously occurring SSWs. Such an approach enables us to examine both the ∼2–3-week forcing stage of an SSW during which there is anomalous stratospheric wave-activity convergence, as well as the recovery stage during which the wave forcing abates and the stratosphere radiatively recovers over 2–3 months. It is argued that applying a torque is better suited than a heating perturbation for examining the response to SSWs, due to the meridional circulation that is induced to maintain thermal-wind balance (i.e., the “Eliassen adjustment”); an easterly torque yields downwelling at high latitudes and equatorward flow below, similar to the wave-induced circulation that occurs during spontaneously occurring SSWs, whereas a heating perturbation yields qualitatively opposite behavior and thus cannot capture the initial SSW evolution. During the forcing stage, the meridional circulation in response to an impulse comparable to the model’s internal variability is able to penetrate down to the surface and drive easterly-wind anomalies via Coriolis torques acting on the anomalous equatorward flow. During the recovery stage, after which the tropospheric flow has already responded, the meridional circulation associated with the stratosphere’s radiative recovery appears to provide the persistent stratospheric forcing that drives the high-latitude easterly anomalies, whereas planetary waves are found to play a smaller role. This is then augmented by synoptic-wave feedbacks that drive and amplify the annular-mode response.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Ian White, ian.white2009@hotmail.co.uk

1. Introduction

Sudden stratospheric warmings (SSWs), whereby the westerly polar vortex substantially weakens over the course of a week or so, are understood to have an appreciable influence on the tropospheric flow below. In particular, SSWs are often followed by an equatorward shift of the tropospheric midlatitude jet that can persist for more than 2 months [e.g., see Fig. 1c created from a composite of 70 SSWs taken from the control run used in this study described in section 2, and Baldwin and Dunkerton (2001), Simpson et al. (2011), and Kidston et al. (2015)]. Given that this jet determines weather over North America and Europe, understanding the downward coupling has the potential to improve subseasonal to seasonal weather forecasts (e.g., Sigmond et al. 2013). However, the mechanism(s) which govern the downward influence are not well understood.

Fig. 1.
Fig. 1.

Latitude–time composites of (a) the Eliassen–Palm flux divergence (EPFD; units: m s−1 day−1) anomalies filtered for planetary waves 1–3 and vertically integrated over 100–0.85 hPa for 70 control SSWs (see section 2a), (b) the residual streamfunction (Ψυ¯*; units: kg m s−1) vertically integrated over 100–0.85 hPa, and (c) u¯ anomalies at 970 hPa. Green contours in (a) and (b) indicate the vertically integrated u¯ anomalies with a contour interval of 5 m s−1 and the zero contour omitted. Note that anomalies that are not statistically significant different from zero at the 95% level using a standard Student’s t test are not shaded. Black vertical line indicates lag 0, and the horizontal black line in (c) indicates the climatological December–February jet maximum.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

The evolution of SSWs can be approximately split into a forcing and a recovery stage. During the forcing stage, the polar vortex is weakened, and eventually breaks down due to easterly torques applied by an anomalously large convergence of upward-propagating wave activity at mid- to high latitudes [indicated by the Eliassen–Palm flux divergence (EPFD) in Fig. 1a]. This forcing stage of the SSW generally lasts around 2–3 weeks up until lag ∼3–5, and to maintain thermal wind balance, the easterly torque applied by the converging waves is balanced by a poleward flow of higher-angular-momentum air from lower latitudes (Fig. 1b), which then adiabatically descends at high latitudes, warming the polar cap. Conversely, during the recovery stage, both the wave driving and the associated mechanically driven meridional circulation have abated (Figs. 1a,b), and the vortex radiatively recovers over the next 2–3 months as wave activity is suppressed throughout the atmospheric column (e.g., Hitchcock and Simpson 2016).

Both the forcing and recovery stages can exhibit a tropospheric jet shift (e.g., Fig. 1c). The response during the latter is somewhat better understood with a consensus on the role of synoptic-wave feedbacks necessary for amplifying the tropospheric jet shift (e.g., Kushner and Polvani 2004; Limpasuvan et al. 2004; Song and Robinson 2004; Gerber et al. 2009; Domeisen et al. 2013; Garfinkel et al. 2013; Hitchcock and Simpson 2014; White et al. 2020). However, the nature of the continuous stratospheric forcing required to maintain the jet shift and thus trigger the continuous synoptic-wave feedbacks is still under debate, with suggestions for the role of the radiatively driven meridional circulation once the wave driving has shut off (Thompson et al. 2006) and planetary-wave suppression throughout the entire atmospheric column (Hitchcock and Simpson 2016).

During the forcing stage, however, things are more complicated because it is not clear, unlike in the recovery stage, if the tropospheric anomalies are unequivocally of stratospheric origin. In particular, the cause of the SSW and the subsequent response are inextricably linked in spontaneously occurring SSWs; anomalous upward-propagating planetary wave activity is often generated in the troposphere (being associated with tropospheric precursors; Garfinkel et al. 2010; Cohen and Jones 2011; White et al. 2019), while at the same time, the wave-driven circulation required to maintain thermal-wind and hydrostatic balance extends downward and can influence the tropospheric flow (e.g., Haynes et al. 1991; Ambaum and Hoskins 2002). This “downward control” was shown by Song and Robinson (2004) to play an important role in the tropospheric response, although the effect was shown to be weaker if planetary waves were damped. Figure 1c highlights the conundrum; it is not clear if the jet shift found at lags −20 to ∼5 days is related to the presence of precursory amplifying wave activity or to some downward influence from aloft. Because of this, our understanding of the tropospheric evolution during the forcing stage, and potentially therefore the initial tropospheric response, is quite poor. We here focus on trying to better understand the tropospheric response during the entire SSW evolution.

In an effort to understand the mechanisms governing the downward impact, stratospheric vortex events have often been artificially forced in general circulation models by applying either zonally symmetric (e.g., Polvani and Kushner 2002; Kushner and Polvani 2004; Williams 2006; Lorenz and DeWeaver 2007; Gerber and Polvani 2009; Yang et al. 2015; White et al. 2020) or zonally asymmetric (e.g., White et al. 2021) heating perturbations to the high-latitude stratosphere. A number of these studies were motivated by observations of the Southern Hemisphere ozone hole and not SSWs, and although useful for examining the longer-lag response in the recovery stage of SSWs, pure heating perturbations are not suitable for studying the forcing stage of an SSW and potentially the initial tropospheric response. This is because they do not capture the wave-driven meridional circulation response that occurs during spontaneously occurring SSWs. To see this, consider separately the quasigeostrophic (QG) transformed Eulerian mean (TEM) zonal-mean momentum and thermodynamic energy budgets. The latter takes the form
T¯t+Γw¯*=Q¯
(e.g., Andrews et al. 1987), where T¯, w¯*, Γ, and Q¯ are the zonal-mean temperature, residual vertical velocity, static stability, and diabatic heating rate. From this equation, an imposed diabatic heating perturbation (i.e., Q¯>0) will be partitioned between a positive temperature tendency (T¯/t>0) and upward adiabatic motion (w¯*>0). The upward motion at high latitudes drives an equatorward branch aloft and a poleward branch below by mass conservation (e.g., see Fig. 10 of White et al. 2020). Note that the diabatic heating cannot just be balanced by an increase in temperature as otherwise thermal wind balance would eventually be violated—the anomalous meridional circulation acts to mitigate this and is often referred to as the “Eliassen adjustment” to a thermal forcing after Eliassen (1951). Hence, the response to an imposed high-latitude stratospheric heating is a circulation cell with upward motion at high latitudes, equatorward motion aloft, sinking motion at lower latitudes, and poleward motion in the troposphere.
However, the meridional circulation cell that occurs during the wave-forcing stage of spontaneously occurring SSWs is in the opposite sense. To see this, now consider the zonal-mean momentum budget:
u¯tfυ¯*=G¯,
where u¯, υ¯*, and f are the zonal-mean zonal wind, residual meridional circulation, and Coriolis parameter, and G¯ is some zonal-mean torque. A negative/easterly torque (G¯<0) such as that which is found during the wave-forcing stage of an SSW, would be partitioned between a deceleration of the zonal-mean flow (u¯/t<0) and a poleward circulation (fυ¯*>0). The poleward circulation draws high angular momentum air from lower latitudes to partially balance the loss of westerly momentum by the imposed torque. Note that G¯<0 contributes to both u¯/t and fυ¯* as a pure deceleration of u¯ would drive the atmosphere away from thermal wind balance; fυ¯* ensures this does not happen and this is the Eliassen adjustment to a momentum torque. During an SSW, this torque-induced meridional circulation subsequently leads to descent over the polar cap, adiabatic warming, and equatorward motion below. The circulation that results from a negative momentum torque is therefore opposite to that resulting from a heating perturbation, and as such, an imposed heating perturbation would not be able to capture the onset and immediate aftermath of an SSW. Given that the wave-driven circulation is associated with equatorward motion in the troposphere, which via Coriolis torques and/or angular momentum advection can yield easterly anomalies, a tropospheric response could develop during the forcing stage.

Despite most studies applying heating perturbations, a handful of studies have imposed momentum torques with the aim of understanding the mechanisms behind the downward coupling from the stratosphere (Song and Robinson 2004; Chen and Zurita-Gotor 2008). Aside from differences in the modeling setup (model complexity, lack of planetary waves, torque location, etc.), the major difference from the current study is that both Song and Robinson (2004) and Chen and Zurita-Gotor (2008) applied steady torques to the polar stratosphere in an idealized dry dynamical core and only examined the time-mean tropospheric response rather than the transient evolution. Given that an SSW event is highly spontaneous with the anomalous wave torque only persisting for at most 2–3 weeks (Fig. 1), a continuously applied, weak torque is somewhat unrealistic. Further, by applying a steady forcing, the radiative relaxation to climatology and associated meridional circulation that is observed during the recovery stage of an SSW cannot be captured. In an effort to isolate the meridional circulation that develops during both the forcing stage (wave-driven circulation) and recovery stage (radiatively driven circulation), Thompson et al. (2006) relaxed a zonally symmetric QG model to the transient EPFD and radiative-heating profiles associated with a weak polar vortex in observations. They found that the meridional circulation during both stages is capable of driving the high-latitude easterly anomalies, but cannot account for the full jet shift or near-surface amplification. Rather, other processes involving eddies must be at work, but the use of a purely zonally symmetric model precluded further investigation.

In our study, we combine the approaches by Song and Robinson (2004), Chen and Zurita-Gotor (2008), and Thompson et al. (2006) in order to understand the tropospheric development during the entire SSW evolution (both the forcing and recovery stages). In particular, using a more realistic (albeit still idealized) model than that used by the three aforementioned studies, we apply a transient easterly momentum torque to the polar vortex, tailored to mimic the wave forcing (in terms of magnitude, duration, and spatial extent) that occurs in free-running SSWs. This approach allows us to capture the “sudden” nature of an SSW, and by applying a tailored torque to the vortex without the preceding upward-propagating planetary waves, we are isolating the tropospheric response to a “pure” stratospheric forcing that mimics the wave forcing in observed SSW events, but without the preceding large-scale planetary wave anomalies that forced the SSW in the first place. The paper is organized as follows: section 2 describes the model and setup used in this study, section 3 describes our rationale for choosing the tunable parameters of the forcing described in section 2, section 4 presents our results, and the final section provides a summary and discussion.

2. Model setup

Our model setup closely follows that of White et al. (2021) with use of the same model of an idealized moist atmosphere (MiMA) developed by Jucker and Gerber (2017) and the same ensemble of control runs. The key difference is in the imposed stratospheric forcing used here, as we discuss below. Briefly, however, MiMA includes a more realistic radiation scheme than that which has been used in previous studies on this topic using dry dynamical cores (e.g., Polvani and Kushner 2002; Song and Robinson 2004; Chen and Zurita-Gotor 2008; Domeisen et al. 2013). The inclusion of a mixed-layer ocean and topography allows for stationary waves to be simulated that are at least as realistic as those in CMIP5 models (Garfinkel et al. 2020a,b). For more details on MiMA, we refer the reader to Jucker and Gerber (2017).

a. Control runs and SSW identification

As aforementioned, the control runs used here are the same as those used in White et al. (2021, see their section 2a). In short, we utilize an ensemble of five control runs, each run for 49 years after an initial 10-yr spinup period that gives the mixed layer ocean ample time to equilibrate. The model is run with a T42 horizontal resolution (i.e., 2.8° × 2.8°) along with 40 vertical levels up to ∼0.01 hPa. The “main” control run, from which the perturbation experiments described in the next section are spun off, is referred to herein as CTRL.

In contrast to White et al. (2021), we identify SSWs in our five control runs using the definition of Charlton and Polvani (2007). This definition states that the zonal-mean zonal wind (u¯) at 60°N and 10 hPa must reverse to easterly sometime between 1 November and 30 March (30-day months are simulated in this study) and it is this date of the reversal that is referred to as the onset or central date. In addition to this, to ensure that final warmings are not wrongly identified as SSWs, u¯ must become westerly for 10 consecutive days prior to 30 April. Finally, to avoid counting the same event twice, a separation of 20 days is mandated to occur between consecutive events. In total, this yields 70 events over all five control runs of which 22 are the same as those identified in White et al. (2020). A ratio of 0.29 SSWs per year (70 events in 245 years) is smaller than that seen in observations (∼0.67 yr−1, e.g., Butler et al. 2015) but this is in part due to the strong vortex bias present in MiMA (see online supplementary Fig. S1 of White et al. 2021).

b. Momentum-forced perturbation experiments

Taking as initial conditions every midnight on 1 January in the 49-yr CTRL, we spin off a perturbation experiment wherein a zonally symmetric momentum torque is applied to the stratosphere and switched on for a prescribed duration, similar to the thermal perturbation setup used by White et al. (2020). Note that we do not utilize the extra four control runs in these perturbation experiments. This momentum torque acts as an additional forcing to the model’s zonal momentum budget and takes the form
F(φ,p,t)=τ(t)Φ(φ)Λ(p),
where
τ(t)=(1,if0<tt0Nddays,0,otherwise,
Φ(φ)=MFsin(πφφLφHφL),
and
Λ(p)=(ppbptpb,ifpt<p<pb,1,ifppt,0,p>pb
In these equations, the model variables are t, φ, and p indicating the model time, latitude, and pressure. All other parameters represent the tunable parameters of the forcing. In particular, MF is the zonally symmetric momentum forcing rate (units of m s−1 day−1 although hereafter we drop the units for brevity), Nd is the prescribed duration (in days) for which the torque is switched on past the reference time (t0) of midnight on 1 January, φL and φH are the lower and upper latitudinal limits of the imposed torque, and pt and pb are the upper and lower levels between which the torque decreases linearly. The overall time-integrated and spatially averaged torque corresponding to a chosen configuration of the parameters above is herein referred to as the “impulse.” An example impulse is shown in Fig. 3, which is tailored to match the wave forcing in the control SSWs, although we defer the reader to section 3 for details.

Note that herein, we refer to the perturbation experiments with the imposed torque as PTRB runs. Anomalies in these PTRB runs are calculated relative to CTRL (i.e., the “main” 49-yr control run) and so by construction, composite plots of anomalies are zero before 1 January. It is important to stress that the initial atmospheric state before switching on the stratospheric torque is different between each PTRB ensemble member. Thus, the signal after 1 January in our PTRB experiments represents the deterministic response to the stratospheric forcing.

In addition to running these experiments with realistic topography, they are repeated in an aquaplanet setting without topography. This is to remove issues associated with calculating terms in the Eulerian budget at levels close to the surface. See section 4c for details.

3. Choice of model parameters

In this section, motivation for estimates of the spatial extent and time duration of the imposed torque is provided using Eliassen–Palm flux divergence (F/ρ0a cosφ, hereafter EPFD, where F is the Eliassen–Palm flux in log-pressure spherical coordinates) anomalies during the forcing stage for the 70 control SSWs.

a. Spatial profile

Figure 2a shows the latitude–pressure profile of the EPFD averaged over lags −30 to −1. As expected, there is anomalous convergence of wave activity throughout the extratropical stratosphere, dominated by planetary waves 1–3 (not shown) and peaking at higher altitudes and around 60°–70°N. This convergence drives easterly u¯ anomalies in the stratosphere associated with the SSW itself. There are also considerable negative NAM tropospheric precursors with a dipole of u¯ anomalies straddling the maximum climatological tropospheric jet (∼45°N: see later Fig. 5). Based on this spatial structure, we choose to impose a torque extending latitudinally from φL = 40°N to φH = 90°N (i.e., centered at 65°N) and with a linear decrease between pt = 60 hPa and pb = 100 hPa (see Fig. 3a). Despite this choice, the results are insensitive to changes in latitude. In terms of vertical extent, moving the lowest level of maximum torque higher in the stratosphere (i.e., setting pt = 10 hPa but still with pb = 100 hPa) but with an equivalent magnitude total impulse, yields quantitatively similar results to those presented herein.

Fig. 2.
Fig. 2.

(a) EPFD anomalies (shading; units: m s−1 day−1) for the 70 control SSWs averaged over lags −30 to −1 (i.e., over the forcing stage). Black contours show u¯ anomalies with contours at ±0.5, 1, 2, 3, 5, 10, 15, 20, … m s−1. (b) Time series during the forcing stage for the 100–0.85-hPa pressure-weighted and 40°–90°N area-averaged EPFD anomalies from (a). Black line shows the daily composite mean whereas gray shading shows the minimum and maximum daily anomalies of the 70 SSW events in control. The cumulative sum of the daily EPFD anomalies over this lag interval is inset into (b).

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

Fig. 3.
Fig. 3.

(a) Spatial and (b) temporal profiles of an example imposed momentum forcing with MF = −5 m s−1 day−1 and Nd = 12 days [i.e., the longer-impulse (PTRBl) run]. Units in (a) are m s−1 day−1, whereas (b) is unitless. Horizontal white lines in (a) show the levels between which the torque is linearly decreased to zero (i.e., pt = 60 hPa and pb = 100 hPa) and vertical line shows the lowest latitude at which a torque is applied (i.e., φL = 40°N). Blue line in (b) shows a temporal profile with sudden jumps in the torque on days 0 and Nd = 12, whereas the red line shows a smooth profile with a gradual increase and decrease of the torque centered on midnight of day Nd/2, respectively. Inset into (a) is the 100–0.85-hPa pressure-weighted and 40°–90°N area-averaged values of the daily imposed torque, and inset into (b) is the time-integrated forcing (i.e., the “impulse”).

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

b. Time profile

To determine an estimate of the total impulse to be applied, Fig. 2b shows time series of the vertically and latitudinally integrated EPFD anomalies for the control SSWs, pressure weighted over 100–1 hPa and area averaged over 40°–90°N. It is clear that the integrated EPFD anomalies become negative at around lag −20 and peak close to the onset date. The maximum daily, vertically and latitudinally integrated EPFD is −2.3 m s−1 day−1 with a cumulative sum (i.e., the “impulse”) of −22.6 m s−1 over 30 days. It is this impulse that we try to match in our forcing experiments. The overall impulse is kept the same in all experiments with MF and Nd varying to accomplish this. Note that we have limited our time integration to lags −30 to −1 even though the EPFD anomalies remain negative for 3–4 days after lag 0. Our applied torque therefore slightly underestimates the integrated wave forcing found in free-running SSWs.

To approximate the aforementioned impulse during the control SSWs, we apply the torque shown in Fig. 3. In particular, we set MF = −5 m s−1 day−1, which when integrated over 100–0.85 hPa and 40°–90°N applies a torque each day of −1.8 m s−1 day−1 to the stratosphere. This imposed daily torque is comparable to the maximum composite EPFD anomalies in Fig. 2b (see black line). This is applied for Nd = 12 days yielding a total impulse of −21.5 m s−1 over 12 days, comparable to that in the control SSWs. We refer to this experiment as the “longer-impulse” PTRB run (PTRBl). Note that our results are insensitive to whether the forcing has a jump switch-on (blue line in Fig. 3b) or is gradually switched on and then off (red line in Fig. 3b).

We also apply a stronger, more sudden forcing with MF = −20 m s−1 day−1 which yields an imposed daily torque of −7.2 m s−1 day−1 to the stratosphere and which is still smaller than the maximum daily torque in the control SSWs (see gray shading in Fig. 2b). To maintain the same impulse as in the control SSWs and in the long-impulse (PTRBl) experiment aforementioned, the torque is switched on for Nd = 3 days. This experiment is herein referred to as the “shorter-impulse” PTRB experiment (PTRBs). A final experiment is performed in which a MF = −3.5 m s−1 day−1 torque is switched on for Nd = 17 days which again yields approximately the same impulse as other runs and obtains qualitatively similar results (not shown).

To summarize, the magnitude of the daily imposed torque is constrained by the spread of the latitudinally and vertically integrated EPFD anomalies in the control SSWs. The duration of the forcing is then determined by attempting to approximate the total impulse in the control events. To test the sensitivity of the tropospheric response, the magnitude of the daily imposed torque is varied, and in order to keep the same total impulse, the duration is changed accordingly.

In the control and PTRB runs, we refer to the “forcing” and “recovery” stages. The former is chosen to represent the duration of the applied torque for the PTRB runs and the time it takes for the EPFD anomalies in Fig. 2b to no longer be negative for the control SSWs. The recovery stage is then defined as the 90-day period thereafter. For control, the forcing and recovery stages are lags −20 to +3, and lags 4–93, respectively, for the long-impulse run (hereafter PTRBl), lags 1–12 and 13–102, respectively, and for the short-impulse run (PTRBs hereafter), lags 1–3 and 4–93, respectively. Nevertheless, the results are not sensitive to changes in these defined lag stages and one could, for instance, define lags −20 to −1 as the forcing stage for the control SSWs.

4. Results

a. Zonal wind and temperature response

We start by highlighting the effect of the applied impulse on the vortex strength. Figure 4 shows the zonal-mean zonal wind u¯ at 60°N and 10 hPa (Fig. 4a) and the polar-cap integrated zonal-mean temperature T¯ (Fig. 4b, area averaged over 50°–90°N and pressure weighted over 150–1 hPa, although the results are insensitive to the choice of upper pressure level in the integral) for the control SSWs (black lines) and for the longer-impulse PTRBl and shorter-impulse PTRBs runs (red and blue lines, respectively). Neither of the two aforementioned PTRB runs weaken the vortex or warm the polar cap as strongly as during the control SSWs. In particular, the PTRB runs do not lead to a wind reversal which may occur for a number of reasons. For instance, the spatial and temporal profile of the PTRB torques do not exactly match the EPFD anomalies in the control events and thus it is possible that the discrepancies are important (e.g., the latitudinal extent of the torque is not as wide as the EPFD in the control events). Further, the imposed torque is partially compensated for by resolved and/or unresolved waves acting to restrengthen the vortex (e.g., Cohen et al. 2013; Watson and Gray 2015; Garfinkel and Oman 2018). Nevertheless, increasing the overall applied impulse, (pink and green lines) further weakens the vortex and warms the polar cap with a doubled impulse (MF = −10 m s−1 day−1 for 12 days) leading to a better agreement with the control SSWs in terms of magnitude. Indeed, it could be argued that matching the response in u¯ is more appropriate than matching the EPFD (as described in section 3), but given that it is not entirely clear, we have opted to err on the side of caution and apply a weaker torque based on the EPFD associated with an SSW that directly corresponds to a torque in the model’s momentum budget. Note that the peak in PTRBl (Nd = 12 days) occurs later than the peak in PTRBs (Nd = 3 days), a result of the forcing duration itself.

Fig. 4.
Fig. 4.

Time series of (a) u¯ (m s−1) at 60°N and 10 hPa and (b) T¯ anomalies (K) area averaged over 50°–90°N and pressure weighted over 150–1 hPa, for the 70 control SSWs (black line) and for various PTRB runs with pt = 60 hPa (colored lines). Note that there are 49 ensemble members for each PTRB run. Gray shading in (a) shows the maximum and minimum daily u¯ values for the control SSWs. Shading in (b) is similar but for integrated T¯ anomalies.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

To examine the near-surface response to the imposed torques, Fig. 5 shows latitude–time profiles of 970-hPa u¯ for the control SSWs (Fig. 5a), and the PTRBl and PTRBs runs (Figs. 5b,c). Both the control SSWs and the PTRB runs exhibit a clear dipole straddling the December–February climatological jet maximum near 43°N with u¯<0 on the poleward flank and u¯>0 on the equatorward flank, i.e., shifting the climatological jet equatorward. In the control events, tropospheric precursors are also present before the onset date. For the forced events, the PTRBs run exhibits an immediate near-surface response which becomes significantly different from zero at lags of 3 + days. In the PTRBl run, however, it takes ∼10 days before the u¯<0 anomalies at high latitudes become significantly different from zero. It should also be noted that the u¯>0 anomalies on the equatorward flank of the jet are delayed relative to the u¯<0 anomalies on the poleward flank. This delayed projection onto the model’s tropospheric annular mode appears to be related to the onset of the synoptic-wave feedback that takes 1–2 weeks to initialize (see Fig. 9).

Fig. 5.
Fig. 5.

Latitude–time composites of u¯ anomalies for (a) the 70 control SSWs (repeated from Fig. 1c), (b) the MF = −5 m s−1 day−1, Nd = 12 days PTRBl run, and (c) the MF = −20 m s−1 day−1, Nd = 3 days PTRBs run. Note that there are 49 ensemble members for each PTRB run. Units are m s−1. Also, note that only u¯ anomalies that are statistically significant at the 95% level are shaded. Dashed black horizontal line indicates the climatological jet maximum (∼43°N), and the dashed black vertical line indicates lag zero. The thin dashed red lines in (b) and (c) show the corresponding forcing duration for each of the PTRB runs. Inset into each panel are the minimum values.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

The linearity of the tropospheric response to the lower-stratospheric anomalies is tested in Fig. 6 where polar-cap (60°–87°N) averaged u¯ anomalies, time averaged over the “recovery” period are plotted at 100 hPa, against 970 hPa. We use u¯ rather than T¯ or geopotential height Z¯ as it is u¯ that is directly modified by the applied torque, and although correlations are slightly weaker if T¯ or Z¯ is used, they are dynamically consistent via thermal wind balance. Extra runs are included in this figure in order to test the linearity. In particular, the PTRBl run (MF = −5 m s−1 day−1 for Nd = 12 days) is used as the base run and further runs are conducted with MF = +5, −10, −15 m s−1 day−1 for Nd = 12 days. The PTRBs run (MF = −20 m s−1 day−1 for Nd = 3 days) is also included to compare the strength of the surface response to the PTRBl run.

Fig. 6.
Fig. 6.

Scatterplot of u¯ anomalies at 100 hPa against u¯ at 970 hPa, both averaged over 60°–87°N and the recovery stage, for different 49-ensemble-member PTRB experiments (see legend) and the 70 control SSWs. Note that the recovery stage is defined as the 90-day period after the torque has been switched off (see section 3b for details). Filled colored squares indicate the corresponding ensemble means for each experiment. Note that a vortex-strengthening PTRB run is included (MF = 5 m s−1 day−1, Nd = 12 days; cyan) to better test the linearity of the tropospheric response to an imposed torque. The black line shows the best-fit line calculated using a least-squares fit. The slope of the linear regression line (along with the confidence interval) and the correlation coefficient (r) are included in the bottom right. Note that the regression slopes for the control SSWs and the PTRB SSWs are the same, hence one line is plotted.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

Overall, a strong linear relationship is found (correlation coefficient of 0.91), similar to that found by White et al. (2020) for thermally forced SSW events. Hence, a stronger lower-stratospheric vortex anomaly generally leads to a stronger near-surface response no matter how the vortex weakening is achieved. The linear relationship also extends to vortex strengthening events (see cyan markers that represent a PTRB run with an impulse of MF = −5 m s−1 day−1 for 12 days). A final point is that the PTRBl (red) and the PTRBs runs (blue) yield similar composite means, i.e., the magnitude of the surface response depends on the overall impulse applied and not on the suddenness with which the torque is applied. The results in Fig. 6 are insensitive to changes in the latitudes and lags averaged over.

b. Meridional streamfunction response

Here we compare the structure of the meridional circulation for free-running SSWs with those that occur in response to a transient mechanical forcing. The left column of Fig. 7 shows the transformed Eulerian mean (TEM) streamfunction Ψυ¯* for the “forcing” stage of the SSWs with contributions from the Eulerian mean circulation Ψυ¯ and eddy heat flux Ψυ ′θ¯ shown in the middle and right columns, respectively [e.g., see Eq. (10) in White et al. 2020]. The control SSWs are shown in the top row, PTRBl in the middle row, and PTRBs in the bottom row.

Fig. 7.
Fig. 7.

Latitude–height composites of (left) the anomalous residual-mean meridional circulation Ψυ¯* along with contributions from (center) the Eulerian-mean circulation Ψυ¯ and (right) the eddy heat flux convergence Ψυθ¯. The anomalies are averaged over the “forcing” lags which for (top) the 70 control SSWs are defined as lags −20 to +3. For the two 49-ensemble-member PTRB runs, the forcing lags are defined to be the duration of the imposed forcing, which for (middle) the PTRBl run is Nd = 12 days and (bottom) the PTRBs run is Nd = 3 days. Units are kg m s−1. Black contours show u¯ anomalies averaged over the corresponding lags with contours at ±0.5, 1, 2.5, 5, 10, … m s−1. Thick black contours indicate statistically significant differences from zero at the 95% level.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

During the forcing stage, the circulation response to the naturally occurring wave forcing in the control SSWs and the applied torque is the same with a Ψυ¯*>0 cell. Three things should be noted here: the first is that the cell in the PTRBs run is much stronger than in the PTRBl run, despite the overall applied impulse being the same. This does not yield an appreciable difference in the magnitude of the tropospheric wind response (cf. contours in middle and bottom rows), and instead only yields a delay in the onset of the near-surface response in the PTRBl run (Fig. 5). Second, it should be noted that the peak in Ψυ¯*>0 for the control SSWs is located in the middle to lower troposphere and decreases aloft, whereas in the PTRB runs, the peak is located in the upper troposphere–lower stratosphere. The lower peak in the control SSWs is partly due to an anomalous convergence of wave activity in the troposphere before the onset date (not shown, although see Fig. 6 of White et al. 2020), and is hence at least partly due to tropospheric precursors that are independent of the stratosphere. Nevertheless, the PTRB runs indicate that there is a lower-tropospheric extension associated with the stratospheric circulation cell that is independent of the precursors. Third, the balance of terms giving rise to the Ψυ¯* anomalies is very different between the control and PTRB events: the control SSWs are associated with upward-propagating wave activity (Ψυθ¯>0) being in large part balanced by an equatorward overturning Eulerian circulation (Ψυ¯<0), whereas the PTRB events are associated with an Eulerian circulation cell in the opposite direction (Ψυ¯>0) occurring as the Eliassen response to the imposed torque (Eliassen 1951).

During the recovery stage (Fig. 8), the circulation anomalies are in remarkable quantitative agreement between the control and PTRB runs. In particular, the signature extratropical three-cell Ψυ¯* structure associated with the tropospheric NAM (e.g., Martineau et al. 2018; White et al. 2020) is found in all three runs, indicating the generic nature of the tropospheric response at longer lags. This structure, with Ψυ¯*>0 at ∼45°–65°N flanked by Ψυ¯*<0 either side (although the high-latitude cell is very weak) results from a part cancellation between larger Ψυ¯>0 and Ψυθ¯<0 cells. The latter is associated with wave suppression throughout the atmospheric column after an SSW. However, the reason for the Ψυ¯>0 cell at high latitudes is not as clear and is likely due to the radiative cooling associated with the vortex recovery (as in Thompson et al. 2006), or related to the wave suppression indicated by Ψυθ¯<0. Note that the results are insensitive to varying the time-average window, with similar results being found even if the window is only taken to be the week after the forcing shuts off (not shown).

Fig. 8.
Fig. 8.

As in Fig. 7, but for the “recovery” lags defined as the 90-day period following the forcing lags. In particular, (top) for the 70 control SSWs this is defined to be the lag-4–93-day period after the SSW onset, (middle) for the PTRBl run it is defined as lags 13–102, and (bottom) for the PTRBs run it is defined as lags 4–93 (see section 3b for details).

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

c. Analyzing the Eulerian-mean momentum budget

The previous section showed that the meridional circulation reaches down to the lower troposphere, and here we quantify its role in the surface response. In particular, we quantify the contributions of various terms in the Eulerian zonal-mean momentum budget:
u¯t=fυ¯fv1acos2φφ(cos2φuυ¯k3)UVy131acos2φφ(cos2φuυ¯k>3)UVy4+[w¯u¯z+1ρ0z(ρ0uw¯)]υ¯acosφφ(u¯cosφ)ageo+X¯+res
(e.g., Andrews et al. 1987; Hitchcock and Simpson 2016), where the acceleration of the zonal-mean zonal wind on the left-hand side is contributed to by processes associated with (from left to right on the right-hand side): Coriolis torques acting on meridional motion (fv), eddy momentum flux convergence due to planetary waves (UVy1–3), eddy momentum flux convergence due to synoptic waves (UVy4+), ageostrophic eddy and mean flow terms (ageo), and parameterized processes including the zonal wind tendency due to vertical and horizontal diffusion and gravity wave drag in the model (X¯). All variables are standard (e.g., see Andrews et al. 1987). The final term (res) is the budget residual and is contributed to by issues associated with resolution and sampling and is here calculated as the difference between the left- and right-hand sides of the budget. Note that the residual is calculated using the eddy flux terms outputted directly from the model and not using the synoptic and planetary wave decomposition presented above.

An issue in calculating this budget for the long- and short-impulse PTRB runs presented up until this point, is that the topography strongly influences the lowest few atmospheric levels in MiMA. In particular, interpolating from the model’s hybrid sigma–pressure levels to pressure levels results in spurious values at any grid points that are influenced by topography and for which the pressure level intersects the surface. Levels below 700 hPa are particularly problematic and hence we cannot reliably calculate the eddy-flux terms in the budget at such levels. To address this, a second pair of PTRB experiments (as well as a “new” CTRL run) are conducted in which the applied torque is identical, but without topography and with zonally symmetric albedo and q-flux profiles. Note that retaining the asymmetric albedo and q-flux profiles from CTRL does not quantitatively change the results. The budget [Eq. (7)] is then calculated for these two experiments. In these two extra runs, despite stationary waves associated with topography and land–sea contrast being absent, transient waves are still present.

Figure 9 shows latitude–time panels of each of the terms in Eq. (7) vertically integrated over 925–125 hPa for the PTRBl run without topography. Such a vertical integral over the depth of the troposphere removes meridional circulations associated with purely tropospheric waves and thus, ostensibly leaves only the meridional circulation that is driven by stratospheric processes (in this case the applied torque and subsequent weakened vortex). It is immediately clear that at lags of less than ∼2 weeks, the major balance is between the deceleration of the tropospheric westerlies (u¯/t<0) and the Coriolis torque acting on the equatorward flow (fυ¯<0). Thus, the meridional circulation associated with the stratospheric torque penetrates down to the surface and the equatorward return flow drives the tropospheric easterly anomalies at early lags.

Fig. 9.
Fig. 9.

Contributions from each term in the Eulerian zonal-mean momentum budget [Eq. (7)] for the longer-impulse (PTRBl) run without topography. Units are m s−1 day−1. The budget is vertically integrated over 925–125 hPa. Green contours indicate the vertically integrated u¯ anomalies. Thick black contours indicate statistically significant differences from zero at the 95% level using a standard Student’s t test. Black and red vertical lines indicate lag 0 and lag 12, respectively, the latter being the lag at which the forcing is switched off in the PTRBl run.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

Nevertheless, after lag ∼14, other terms become important with the high-latitude fυ¯<0 being balanced by surface stress (X¯>0), and divergence of both synoptic- and planetary-scale momentum fluxes (i.e., −UVy4+ < 0 and −UVy1–3 < 0) with opposite-signed anomalies farther equatorward. There is also evidence for some unexplained residual effect that opposes the surface stress at high latitudes, although this effect becomes significant only after lag ∼20. The dipole in the synoptic-scale momentum fluxes has a narrower meridional scale than the Coriolis term and thus appear to give rise to the jet shift associated with the tropospheric annular mode. This difference in the scales of the momentum fluxes and the zonal winds is agreeable with Black and McDaniel (2009). Further, the fυ¯<0 at these lags agrees with the arguments of Thompson et al. (2006) in that the meridional circulation associated with the lower-stratospheric anomalous radiative cooling appears to provide the exogenous forcing to encourage the persistent tropospheric jet shift. The zonally symmetric model in the next section will be used to confirm this.

Note that qualitatively similar results are found in the PTRBs run (see supplementary Fig. S1), except that the surface response is stronger and occurs earlier, as well as in the weakest MF = −3.5 m s−1 day−1 for Nd = 17 days experiment (see supplementary Fig. S2) for which the surface response is delayed.

To better quantify the roles of the various terms in the budget, Figs. 10a and 10b show time series of the area-averaged (60°–90°N) and vertically integrated (925–125 hPa) anomalies for the PTRBs and PTRBl runs with no topography, respectively. It is clear that at lags of less than 12–14 days, the decelerating winds (u¯/t<0) are dominantly contributed to by the Coriolis torque acting on the equatorward flow (i.e., fυ¯<0). Although insignificantly different from zero, the balance is provided by the surface stress (X¯>0) at these (as well as later) lags. Note that changing the forcing to linearly decrease between pt = 10 hPa and pb = 100 hPa (as opposed to between pt = 60 hPa and pb = 100 hPa) and keeping the same overall impulse does not quantitatively change the results.

Fig. 10.
Fig. 10.

Time series showing each of the terms in the Eulerian zonal-mean momentum budget [Eq. (7)] area averaged over 60°–90°N and vertically integrated over 925–125 hPa, for (a) the short-impulse PTRBs run (MF = −20 m s−1 day−1, Nd = 3 days) and (b) the long-impulse PTRBl run (MF = −5 m s−1 day−1, Nd = 12 days). Units are m s−1 day−1. Thick line segments indicate statistically significant differences from zero at the 95% level using a standard Student’s t test. Black and red vertical lines indicate lag 0 and the lag on which the torque is switched off, respectively.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

We note that even in the PTRB runs with topography for which we cannot properly calculate the vertically integrated budget (i.e., cannot include levels below 700 hPa in the integral), a qualitatively similar budget is found as in Fig. 9. Supplementary Figs. S3 and S4 show the budget at 700 hPa for the PTRBl and PTRBs runs with topography. It is clear that the high-latitude deceleration at early lags (u¯/t<0) is balanced by the Coriolis term acting on the equatorward flow (fυ¯<0), whereas other terms become important after approximately 2 weeks.

d. Zonally symmetric model

To isolate the role of the meridional circulation in the tropospheric response, a zonally symmetric model is now used. In particular, a zonally symmetric version of the Reading IGCM that solves the hydrostatic, dry primitive equations (Hoskins and Simmons 1975) is used, forced with the exact same impulse as in Fig. 3 (i.e., the PTRBl run with MF = −5 m s−1 day−1 torque for Nd = 12 days). This model was also used by Ming et al. (2016). Two experiments are conducted, both initialized with an isothermal (250-K) atmosphere: the first is run with no radiative damping or near-surface friction, whereas the second includes both processes, with time scales of 10 days for the damping and 1 day for the friction on the lowest model level.

Figure 11 shows latitude–height projections of the Eulerian mean streamfunction (Ψυ¯) during the forcing stage (Fig. 11a) and the recovery stage (Fig. 11c) for the experiment without radiative damping or surface friction, whereas Figs. 11b and 11d show the same but for the experiment with both processes included. The streamfunction during the forcing stage in both runs is structurally identical to that in the full MiMA setup shown in Fig. 7e and also has similar magnitude (note the different color bars between Figs. 78 and Fig. 11), indicating that the response to the forcing is purely due to the Eliassen adjustment and that the effects of eddies, for instance, are negligible at these lags. One key difference between the runs with and without damping and friction is that the circulation extends deeper into the troposphere in the run with damping (see ∼50°N, 500 hPa). This agrees with Haynes et al. (1991), who describe a diabatic enhancement to the circulation. This amplification of the cell in the run with damping and friction does not translate into much of a change in the tropospheric winds as surface friction approximately cancels out the amplification, although an additional experiment that includes only damping does confirm that such a diabatic enhancement leads to a stronger jet shift.

Fig. 11.
Fig. 11.

Experiments using a zonally symmetric version of the Reading IGCM (Hoskins and Simmons 1975) forced with an impulse of MF = −5 ms−1 day−1 for Nd = 12 days (i.e., equivalent to the PTRBl run in MiMA, see Fig. 3), with (left) an experiment with no radiative damping or near-surface friction and (right) an experiment with both processes switched on. The damping time scale is 10 days whereas the time scale for the friction on the lowest model level is 1 day. (top),(middle) The Eulerian-mean streamfunction (Ψυ¯; shading; units: kg m s−1) during the forcing and recovery stages, respectively, with the zonal-mean zonal wind (u¯) shown with contours at ± 0.5, 1, 2.5, 5, 10, 15, … m s−1. (bottom) Latitude–time projections of the 970–125-hPa vertically integrated Coriolis torque term (fυ¯; shading; units: m s−1 day−1) with u¯ shown with contours at ±0.5, 1, 2, 3, … m s−1. Dashed red line in bottom row indicates the lag at which the torque is switched off. Note that the color bars in the top and middle rows are different than those in Figs. 7 and 8 and the color bar in the bottom row is different from that in Fig. 9.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

Figures 11e and 11f show that the Coriolis torques acting on the equatorward flow in the troposphere (i.e., fυ¯<0 vertically integrated over 970–125 hPa) drives tropospheric easterlies during the forcing stage, and both u¯<0 and fυ¯<0 are of comparable magnitude to that in MiMA (Fig. 9b). Note that 970 hPa is included in the vertical integral here in the zonally symmetric model in contrast to 925 hPa for MiMA shown in Fig. 9. This does not change the circulation during the forcing stage but does at later lags where fυ¯>0 is found at levels above 970 hPa, a result of friction only being applied to the lowest model level here.

During the recovery stage, the streamfunction for the experiment with radiative damping is also qualitatively similar to the streamfunction during the recovery stage in MiMA (Figs. 8b,e,h) with Ψv¯>0 at high latitudes and Ψυ¯<0 at lower latitudes. Further, the fυ¯<0 in the recovery stage (Fig. 11f) is qualitatively similar to the high-latitude fυ¯<0 in MiMA (Fig. 9b). This similarity suggests that the high-latitude positive circulation cell is likely a response to the radiative recovery (i.e., anomalous cooling) in the lower stratosphere, and not purely due to wave suppression (shown in the right column of Fig. 8), in agreement with Thompson et al. (2006). The circulation and easterly winds in the run with damping slowly die out but are still present by lag 50 (Figs. 11d,f). In the absence of radiative damping, the meridional circulation abates once the forcing has switched off (Figs. 11c,e) due to the fact that there are no diabatic processes to drive the overturning. Nevertheless, the tropospheric easterlies remain constant thereafter.

5. Summary and discussion

This study has utilized an idealized model to examine the tropospheric response to transient high-latitude stratospheric momentum torques that have been tailored in terms of magnitude/location/timing to approximately match the wave-forcing anomalies (i.e., the impulse) found in spontaneously occurring sudden stratospheric warmings (SSWs) identified in a control run (Figs. 2, 3). To mimic the wave-forcing impulse that drives an SSW, the torque has been switched on for a limited duration spun off from a control run every 1 January. Keeping the total impulse the same between different experiments, the sensitivity of the tropospheric response to varying the daily magnitude and duration of the forcing has been tested. Applying a transient torque has allowed us to examine the tropospheric response during the entire SSW evolution; the forcing stage during which there is wave convergence in the stratosphere (defined here to be the period during which the torque is switched on), and the recovery stage that occurs once the wave driving has abated and the stratospheric temperatures relax back to climatology (defined here to be the period after the torque has been switched off). This should be contrasted with the steady torques used in Song and Robinson (2004) and Chen and Zurita-Gotor (2008) that did not permit study of the recovery stage, and who also only examined the longer-term, time-averaged response, precluding examination of how the tropospheric response initially develops.

In examining the tropospheric response to SSWs, we find that applying a momentum torque is more appropriate than a heating perturbation as a way to approximately capture the circulation evolution during the forcing stage of an SSW. This is because the forcing stage exhibits a wave-driven meridional circulation with poleward flow in the stratosphere, downwelling at high latitudes and equatorward motion below, a feature that also occurs in response to an imposed momentum torque [Fig. 7; referred to as the Eliassen adjustment after Eliassen (1951)], whereas the circulation in response to an imposed heating perturbation is qualitatively opposite (Fig. 10 of White et al. 2020). In fact, the impulse applied here is too weak to weaken the vortex at 60°N and warm the polar cap as much as in the control events (Fig. 4) despite it approximately matching the integrated Eliassen–Palm flux divergence anomalies. This may be due to the inevitable compensation by resolved and/or unresolved waves which act to restrengthen the vortex (e.g., Watson and Gray 2015), or to do with the nature of our idealized torque. However, doubling the impulse predominantly considered here yields a more realistic magnitude SSW in terms of the zonal wind and temperature response (see green lines in Fig. 4). Hence, we suggest that applying a transient momentum torque of suitable magnitude is enough to generate an artificial SSW with a relatively realistic evolution.

To briefly show that a heating perturbation cannot fully capture the tropospheric evolution during an SSW event, Fig. 12 shows the 925–125-hPa integrated Eulerian momentum budget [Eq. (7)] for a thermally forced PTRB run. The setup is the same as in White et al. (2020) with a 15-K transient high-latitude stratospheric heating perturbation switched on for three days, but with no topography to avoid issues with calculating this budget at lower levels. It is clear that the heating perturbation leads to an approximately 2-week delay in the tropospheric u¯<0 response compared to the momentum-forced events (Fig. 9). Further, the anomalous meridional circulation is poleward (fυ¯>0) lasting until lag 15, before equatorward anomalies occur (fυ¯<0), leading to the generation of easterly anomalies. Once the u¯<0 anomalies manifest at around lag 20, both synoptic and planetary waves become more important. Hence, applying a heating perturbation cannot capture the meridional circulation, and thus the possible tropospheric response that occurs during the forcing stage of free-running SSWs, even though such a setup may be useful for capturing the dynamics occurring during the recovery stage (White et al. 2020). These results also highlight the difference between the initial response to an SSW, and a vortex strengthening event and/or the Southern Hemisphere ozone hole, despite the dynamics at work at lags of 20+ days being very similar, albeit opposite in sign (e.g., Kidston et al. 2015).

Fig. 12.
Fig. 12.

As in Fig. 9, but for a thermally forced PTRB run with a 15-K transient high-latitude heating perturbation being switched on for 3 days in the stratosphere, following White et al. (2020). Note the topography has been removed to avoid issues with calculating the Eulerian-mean momentum budget [Eq. (7)] at lower levels. Black and red vertical lines indicate lag 0 and lag 3, respectively, the latter being the lag at which the heating is switched off.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0237.1

Overall, we find that the meridional circulation occurring in response to the imposed momentum torques (i.e., the Eliassen adjustment) is responsible for driving a deceleration of the near-surface westerlies (Figs. 911) via Coriolis torques acting on the equatorward tropospheric flow. This is the case using all of the torques considered here as well as using a purely zonally symmetric model with no eddies (Fig. 11). Importantly, this has implications for the possible tropospheric response during the wave-forcing stage of SSWs; the fact that the wave-induced meridional circulation during the forcing stage can decelerate the surface flow indicates that the wind anomalies found at approximately lags −20 to +5 in the control SSWs (Fig. 5) may be associated with an actual downward influence from the stratosphere and not just due to tropospheric precursors that are perhaps associated with the anomalous upward-propagating wave activity itself (e.g., Garfinkel et al. 2010; Cohen and Jones 2011; White et al. 2019).

Further, it is shown that the timing of the initial onset of the tropospheric response that happens during the forcing stage, is sensitive to the duration of the impulse due to the strength of the induced meridional circulation. In particular, ensuring that the total applied impulse is the same, a strong daily torque applied over a short period (i.e., the 3-day PTRBs experiment), yields a stronger meridional circulation and thus an earlier surface response than a weaker daily torque applied over a longer period (i.e., the 12-day PTRBl experiment), as shown in Figs. 7 and 10. However, the amplitude of the surface response averaged over the forcing period, as well as the period once the forcing is switched off, are quantitatively similar for runs with an equal impulse, with a clear linearity between the strength of the impulse and the surface response during the recovery stage (Fig. 6), similar to the thermally forced events in White et al. (2020).

Our results also provide insight into the tropospheric response during the recovery stage of SSWs. Generally, it is believed that the recovery stage consists of a two-stage process: a continued exogenous stratospheric forcing that encourages the persistent high-latitude easterly anomalies, augmented by a process that amplifies the near-surface response and gives rise to the lower-latitude westerly anomalies. In terms of the former, our experiments both with (Fig. 8) and without (Fig. 11) eddies suggest that the meridional circulation occurring in response to the anomalous stratospheric radiative cooling provides the exogenous forcing to the troposphere (Figs. 911), in agreement with Thompson et al. (2006). In terms of the latter, our results provide yet further evidence for the role of synoptic-wave feedbacks (e.g., Kushner and Polvani 2004; Limpasuvan et al. 2004; Song and Robinson 2004; Gerber et al. 2009; Domeisen et al. 2013; Garfinkel et al. 2013; Hitchcock and Simpson 2014; White et al. 2020) that become important once the forcing has switched off (Figs. 9, 10). However, it should be noted that a quantitative comparison of MiMA (Figs. 8, 9) and the zonally symmetric model (Fig. 11) must be performed with care. In the symmetric model, simple analytical expressions are used for the radiative relaxation and surface friction and whether these are substantially different from those in MiMA, will have an effect on a full quantitative comparison. Such an in-depth study is left for future work, although we note that given the plausible amplitudes during the recovery stage in Fig. 11, it is very likely that our results support Thompson et al. (2006).

Nevertheless, planetary waves also appear to play some role during the recovery stage. This is despite the details of the tropospheric response being quite similar in runs both with and without topography (i.e., with and without topographically forced stationary waves). In particular, Fig. 9 and supplementary Figs. S1–S4 all show momentum fluxes by planetary waves contributing to the persistent tropospheric high-latitude wind anomalies, in agreement with Hitchcock and Simpson (2016). There is also evidence of planetary-wave suppression in the stratosphere and troposphere during the aftermath of both the PTRB and control SSWs (Fig. 8 and supplementary Fig. S5) that was suggested by Hitchcock and Haynes (2016) and Hitchcock and Simpson (2016) to be able to provide the exogenous forcing to the troposphere. Song and Robinson (2004) also found an important role for planetary waves, finding that even though the meridional circulation associated with the imposed momentum torque was able to account for much of the downward influence, this was weakened if planetary waves were damped. However, as aforementioned, their modeling setup precludes immediate comparison as they applied a steady torque (as did Chen and Zurita-Gotor 2008) that did not allow an examination of the recovery stage.

The relative importance of the radiatively induced meridional circulation compared to planetary waves during the recovery stage is in contrast to a couple of recent studies (Hitchcock and Simpson 2014, 2016) that also artificially forced an SSW in a general circulation model to examine the mechanisms behind the tropospheric response. They found planetary waves to play a much larger role than the meridional circulation. This was also found to be tentatively comparable to the results associated with the small number of SSWs in ERA-Interim. Understanding whether the meridional circulation, planetary waves, or perhaps both, are more important is therefore still an avenue for further research.

It is important to note a couple of caveats related to our modeling setup. First, applying a torque yields a somewhat unrealistic stratospheric eddy response during the forcing stage. In particular, control SSWs are marked by upward-propagating waves that converge and weaken the vortex, and in a positive feedback, lead to more upward-propagating waves that eventually reverse the vortex. However, during the forcing stage in the PTRB runs, there is anomalously downward planetary-wave activity diverging from the region of the imposed torque (see supplementary Fig. S5). Thus, planetary waves act to try and counteract the imposed torque. The cancellation is here much weaker than that in Cohen et al. (2013, 2014) and Watson and Gray (2015), who all found a large degree of compensation by planetary waves in response to an imposed torque, although the net effect is that the eddy forcing is opposite to in the control SSWs.

Further, we have examined the tropospheric response to a “pure” stratospheric torque without the preceding tropospheric and stratospheric planetary-wave anomalies that are the cause of the SSW in the first place. Although this is advantageous in some respects, it also means direct comparison between the control SSWs and forced SSWs during the forcing stage is not immediately possible as the cause can offset, or possibly cloud the response in control events, leading perhaps, to little signal at such lags. Nevertheless, our results show that a stratospheric impulse with magnitude similar to that of internal variability in the model, has the potential to influence the troposphere via the Eliassen adjustment. Whether this potential is realized during the forcing (and recovery) stage depends on a multitude of other factors, and this manifests in observations as not all SSWs resulting in a near-surface response (e.g., Karpechko et al. 2017).

Acknowledgments.

We thank Walter Robinson and an anonymous reviewer whose comments have helped to much improve the manuscript. We acknowledge the support of a European Research Council starting grant under the European Union Horizon 2020 research and innovation program (Grant 677756). IPW would also like to thank Chen Schwartz, Edwin P. Gerber, Martin Jucker, Ofer Shamir, Daniela I. V. Domeisen, and Lantao Sun for helpful discussions.

Data availability statement.

The updated version of MiMA used in this study including the modified source code and example namelists to reproduce the experiments can be downloaded from https://github.com/ianpwhite/MiMA/releases/tag/MiMA-MomentumTorque-v1.0beta (with DOI: https://doi.org/10.5281/zenodo.6630623). It is expected that these modifications will also eventually be merged into the main MiMA repository which can be downloaded from https://github.com/mjucker/MiMA and is documented by Jucker and Gerber (2017) and Garfinkel et al. (2020a, 2020b).

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