1. Introduction
As the planet continues to warm under rising greenhouse gas (GHG) concentrations, we strive for an improved assessment and understanding of how the large-scale atmospheric circulation and associated regional climate is expected to change in the future. While general circulation models (GCMs), in general, simulate a poleward shifting of the zonal-mean midlatitude westerlies and associated storm tracks as the planet warms (Yin 2005; Kidston and Gerber 2010; Swart and Fyfe 2012; Wilcox et al. 2012; Barnes and Polvani 2013; Chang et al. 2012), there is considerable spread among models in the magnitude of this response (Harvey et al. 2012; Woollings and Blackburn 2012; Delcambre et al. 2013). Regionally, changes to the large-scale stationary waves lead to deviations from this zonal-mean poleward-shifting view (Stephenson and Held 1993; Joseph et al. 2004; Simpson et al. 2014) and enhanced uncertainty resulting from the varied model representation of stationary waves and their climate change response.
Some of this uncertainty in the climate change response will be irreducible (Hawkins and Sutton 2009), arising from internal variability of the climate system. The remaining uncertainty, under a consistent forcing scenario, arises from structural differences among models in how they represent the processes of relevance for the large-scale circulation. It is this uncertainty that we can hope to reduce, through improved understanding and representation of the relevant processes.
The shifting of the midlatitude westerlies is thought to arise primarily as a response to altered temperature gradients produced via the thermodynamic effects of increasing GHG concentrations [see the review paper by Shaw et al. (2016) and references therein]. The warming of the tropical upper troposphere and the cooling of the stratosphere are each thought to shift the westerlies poleward while this is partially offset, in the Northern Hemisphere (NH), by the influence of amplified Arctic warming during boreal winter (Lorenz and DeWeaver 2007; Butler et al. 2010; Harvey et al. 2014). Model differences in the representation of the various feedback processes that modify the large-scale temperature gradients, such as cloud and water vapor radiative effects (Voigt and Shaw 2015; Ceppi and Hartmann 2016) or sea ice loss and albedo changes (Barnes and Screen 2015), likely contribute substantial uncertainty (Harvey et al. 2014; Wenzel et al. 2016; Ceppi and Shepherd 2017). In addition, the varied model representation of the mean-state circulation may give rise to differences in the climate change response through modification of the dynamics of eddy-mean flow feedbacks or other processes (Kidston and Gerber 2010; Barnes and Hartmann 2010; Sigmond and Scinocca 2010; Simpson and Polvani 2016).
In the NH wintertime, a potential source of uncertainty in tropospheric circulation change is the representation of the stratosphere and the downward influence of stratospheric circulation changes, as models do not agree on how the vortex will change in the future (Manzini et al. 2014; Butchart et al. 2010). Earlier studies on this topic suggested a dependence of the stratospheric response on vertical resolution with resulting tropospheric impacts but did not agree on the sign of this influence (Shindell et al. 1999; Scaife et al. 2012; Karpechko and Manzini 2012), while Gillett et al. (2002) found no significant influence of vertical resolution on the stratospheric response to climate change. As the number of high-top models available within multimodel intercomparisons has increased (Gerber et al. 2012; Charlton-Perez et al. 2013), it has become clear that there is actually no consistent link between a model’s top or stratospheric resolution and how it responds to increasing GHGs (Butchart et al. 2010; Manzini et al. 2014). The processes that give rise to strengthening, weakening, or no change in the NH wintertime polar vortex remain an open question. Past studies have argued that the way in which a model’s stratospheric circulation is tuned in the presence of parameterized gravity waves can impact future changes in stratospheric wave propagation (Sigmond et al. 2008), while others have shown a link between a model’s vortex response and the altered source of stratospheric planetary waves from the troposphere below (Karpechko and Manzini 2017).
Our focus here is not on what gives rise to model diversity in the stratospheric circulation response to rising GHGs, but rather, what is the downward influence of this diversity on the troposphere below? Our modeling study is heavily influenced by the analysis based on phase 5 of the Coupled Model Intercomparison Project (CMIP5) of Manzini et al. (2014, hereinafter M2014). They presented regressions, across models, of tropospheric circulation change onto a measure of polar vortex change under the representative concentration pathway 8.5 (RCP8.5) forcing scenario and showed that models that exhibit a greater weakening of the stratospheric polar vortex in the future exhibit less of a reduction in Arctic sea level pressure (SLP) and a reduced poleward shifting of the tropospheric westerlies. A problem with such analyses of multimodel ensembles is that the diagnosed influence may in fact be caused by a multitude of other factors, such as differences in model resolution, tuning, representation of physical processes, and so on, in addition to the phenomenon of interest. In addition, while stratospheric circulation variability has an influence on the troposphere below (e.g., Baldwin and Dunkerton 2001), the primary driver of stratospheric variability in the first place is variations in the wave activity propagating upward from the troposphere. So, identified links between aspects of the stratospheric and tropospheric circulation could represent a causal connection in either direction. While some attempt at establishing cause and effect has been made through lagged regressions (M2014), idealized experiments, designed to unambiguously test and quantify the influence of stratospheric circulation changes on the troposphere below, are needed. This is what we provide here.
The methodology we use is to nudge the stratospheric zonal-mean climatological circulation, within one model, to states that span the range of CMIP5 projections of the zonal-mean stratospheric polar vortex under the RCP8.5 scenario. In this way, we can assess the influence of changes in the zonal mean, climatological, stratospheric boundary conditions on the troposphere below. We stress that these experiments cannot tell us everything about the potential role the stratosphere may play in tropospheric circulation change. What they can tell us is the climatological circulation changes that are produced in response to climatological zonal-mean changes in the stratosphere. For example, the mean meridional circulation produced as a “downward control” (Haynes et al. 1991) response to the altered climatological stratospheric forcings that drive the vortex changes should be represented in this framework (Hitchcock and Haynes 2014) along with mechanisms whereby changes in the lower-stratospheric state affect tropospheric transient eddies or larger-scale planetary waves, with ensuing impacts on the tropospheric zonal-mean circulation (Song and Robinson 2004; Kushner and Polvani 2004; Wittman et al. 2007; Lorenz and DeWeaver 2007; Simpson et al. 2009). What these experiments cannot tell us is the influence of nonlinear effects associated with large-amplitude events. For example, if nonlinearities associated with a change in the number of sudden stratospheric warmings (SSWs) or with planetary wave reflection from anomalous polar vortex configurations (Perlwitz and Harnik 2003; Shaw et al. 2010) were important to the time-averaged response, the present approach of relaxing the zonal mean toward a climatological mean state would not capture such effects. Nevertheless, it will be shown that the experiments demonstrate a similar stratospheric influence on climatological tropospheric circulation change to that inferred from the regression analysis of M2014, suggesting that the mechanisms that can be captured with this methodology dominate in the stratosphere’s role in intermodel spread of future wintertime climatological change.
We begin with an initial analysis of the CMIP5 intermodel spread in future predicted changes in the NH wintertime stratospheric polar vortex in section 2. This motivates the design of the model experiments, which will be described, along with our CMIP5 analysis methodology, in section 3. Results are presented in section 4, followed by discussion and conclusions in section 5.
2. The intermodel spread in NH winter stratospheric circulation change
The motivation for these experiments is the complete lack of agreement among CMIP5 models on the future of the NH winter zonal-mean stratospheric polar vortex under a rising GHG scenario. While M2014 already discussed this wide spread among models, we illustrate this here again in Fig. 1 for the set of CMIP5 models that we use to design the perturbation experiments. This makes use of the 35 models and ensemble members listed in Table 1 under the historical and RCP8.5 forcing scenarios (Taylor et al. 2012) with the ensemble mean for each model calculated prior to the multimodel mean. The “past” is the 27-yr period between 1979 and 2005 of the historical simulations, and “future” is the 30-yr period between 2070 and 2099 of the RCP8.5 simulations.
DJF zonal-mean zonal wind: multimodel-mean (a) past climatology and (b) future − past difference. (c) The across-model σ of the future − past difference. (d) As in (c), but after regressing out the component of the zonal wind difference that is linearly related to the globally averaged surface temperature increase. (e) The future − past difference at 10 hPa averaged from 60° to 75°N. Models are shown in order of increasing zonal wind difference. Black bars show the 2.5th–97.5th percentile range of bootstrap sample differences from a model’s preindustrial control simulation (see section 2), and solid/hatched bars depict anomalies that are/are not significantly different from zero at the 5% level. The letters H and L depict whether a model is considered high or low top, determined by whether or not the model lid is above 1 hPa (Charlton-Perez et al. 2013). (f) As in (e), but after removing the component that is linearly related to the globally averaged surface temperature increase. The solid green lines in (e) and (f) show the zonal wind anomaly for CESML46 FREE4x − FREE1x, and the dashed lines in (f) show CESML46 FREE4x − FREE1x ±5 m s−1 (i.e., the magnitude of 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
List of models and historical and RCP8.5 members used in the CMIP5 analysis. The subset of eight models used to define the stratospheric perturbation above 10 hPa are highlighted in boldface. (Expansions of acronyms are available online at http://www.ametsoc.org/PubsAcronymList.)
This wide spread among models is further illustrated in Figs. 1e and 1f, which show the future − past difference in 
Next, we assess whether the change in zonal-mean zonal wind for an individual model is significant when compared with an equivalent sampling of the preindustrial control simulation for that model. Consider CESM1(WACCM), which has seven past samples and three future samples, as an example. Seven chunks of length 27 yr and three chunks of length 30 yr are sampled at random from the preindustrial control simulation (these chunks may overlap) to represent the seven past samples and three future samples, respectively. The mean climatology for each chunk is calculated followed by the mean over the seven chunks that mimic the past and the three chunks that mimic the future. The difference between these means is then calculated, and this represents one sample of the difference that could be obtained between three 30-yr climatologies and seven 27-yr climatologies when sampled from that model’s control simulation. This is then repeated 5000 times to build up a distribution of zonal wind anomalies that could be obtained with a sample of this size. This procedure is then followed for each model with the sample size that is equivalent to the number of past and future members used (Table 1). The black bars in Figs. 1e and 1f show the 2.5th–97.5th percentile range of these preindustrial control samples, and note that intermodel variations in this range can arise from both differences in the sample size and differences in stratospheric variability between the models. For a model where the zonal wind response lies outside of this range we can conclude that there is less than a 2.5% chance of obtaining an anomaly that big (or that small) from sampling alone (equivalent to the response being significantly different from zero at the 5% level by a two-sided test). This assumes that the stratospheric variability does not change substantially from the preindustrial to the historical or RCP8.5 climates, which is reasonable given the lack of consensus on this point (Rind et al. 1998; McLandress and Shepherd 2009; Bell et al. 2010; Karpechko and Manzini 2012; Mitchell et al. 2012; Ayarzagüena et al. 2018).
This analysis then reveals that, at 10 hPa and 60°–75°N, nine models exhibit a significant weakening of the zonal-mean zonal wind and six [seven if the globally averaged Ts contribution is first regressed out (Fig. 1f)] exhibit a significant strengthening, while 20 models exhibit a response that is not larger than expected from the sampling of internal variability (Fig. 1e). As in M2014, there is no clear link between a model’s lid height and the response (the letters H and L in Figs. 1e,f).
While our results are similar to M2014, the multimodel-mean weakening of the high-latitude winds is slightly reduced in our group of models (note the lack of a −1 m s−1 contour in Fig. 1b compared to Fig. 2a in M2014, where the ensemble-mean response surpasses −1 m s−1). In addition, M2014 concluded that around 70% of models in CMIP5 exhibit a weakening of the polar vortex, while here we find it is roughly 50%. This is likely partly due to the increased number of models included here (35 compared to 24) and is also partly due to the different latitude region considered (60°–75°N here, compared to 70°–80°N in M2014). Three of the significantly strengthening models in Fig. 1e (and four in Fig. 1f) were not included in the analysis of M2014, and if we use the 70°–80°N measure for the current set of models, we find 60% of the models exhibit a weakening.
In summary, there is no clear consensus on the response of the stratospheric polar vortex to increasing GHGs over the coming century. Models predict that the vortex may significantly strengthen, significantly weaken, or exhibit no significant change. This motivates the following model experiments, which aim to assess to what extent this model spread, in future changes in the strength of the stratospheric zonal-mean polar vortex, may impact the troposphere below.
3. Model experiments and CMIP5 analysis
a. The model
The experiments are performed using a modified version of the Community Earth System Model (CESM), version 1.2, which consists of the Community Atmosphere Model, version 5 (CAM5), coupled to the Parallel Ocean Program model, version 2 (POP2), and the Community Land Model, version 4 (CLM4). The atmosphere model uses the finite-volume dynamical core at approximately 0.9° × 1.25° latitude–longitude resolution but, in contrast to the default configuration of CAM5 with 30 levels and a model top at 2 hPa, we use a 46-level configuration that extends to 0.3 hPa, described in Richter et al. (2015). This version also contains the nonorographic gravity wave drag parameterization described in Richter et al. (2010), which results in the free-running model exhibiting an internally generated quasi-biennial oscillation along with reasonable SSW statistics. A similar version, but making use of the spectral-element dynamical core with prescribed sea surface temperatures, has been previously used in the studies of Richter et al. (2015) and Polvani et al. (2017). We will refer to the model configuration used here as CESML46.
b. Simulations
The model simulations are summarized in Table 2. These consist of a 260-yr-long free-running control simulation (FREE1x) with GHG concentrations specified at preindustrial levels (284.7 ppm), along with a 272-yr-long 4 × CO2 simulation (FREE4x) in which carbon dioxide (CO2) concentrations are elevated to 4 times preindustrial levels (1138.8 ppm) from the beginning of the simulation.
A description of the CESML46 model experiments.
The remaining four experiments employ a relaxation/nudging of the zonal-mean state of the stratosphere to various target climatologies following a similar methodology to Simpson et al. (2011) and Hitchcock and Simpson (2014). For a given field X, with zonal-mean component 
The first of these experiments (NUDG1x) is a preindustrial simulation in which the zonal-mean state of the stratosphere is relaxed toward the seasonally varying climatology of FREE1x (specifically, the first four harmonics of the seasonal cycle averaged over years 10 to 260 of FREE1x). The second (NUDG4x) is a 4 × CO2 simulation that is branched off from year 50 of FREE4x. Year 50 was chosen as it is after the initial rapid warming in response to elevated CO2 has slowed, with only the slower ocean adjustment occurring throughout the remainder of the simulation (Fig. 2).1 In NUDG4x, the zonal-mean state of the stratosphere is nudged toward the first four harmonics of the seasonally varying climatology from years 50 to 272 of FREE4x. This pair of simulations is, therefore, analogous to FREE1x and FREE4x, but rather than having a freely evolving stratosphere, the zonal-mean stratospheric state is nudged toward the climatologies from FREE1x and FREE4x, respectively. This allows us to confirm that the relaxation does not substantially alter the tropospheric response to increased CO2.
(a) DJF globally averaged temperature anomaly in FREE4x relative to the FREE1x climatology. The gray-shaded region and dashed black line show the CMIP5 range and multimodel mean of the globally averaged DJF future − past difference.
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
The final two experiments, WEAK4x and STRONG4x, are also 4 × CO2 simulations branched from year 50 of FREE4x with the zonal-mean stratospheric state nudged toward the seasonally varying climatology of years 50 to 272 of FREE4x but with added perturbations that are designed to span the CMIP5 model spread in the zonal-mean stratospheric polar vortex response to climate change (Fig. 1).
Each of the nudged experiments are run for 222 years so that they are sampling the same period of response to 4 × CO2 as years 50–272 of the FREE4x simulation, which is the period used to define the nudging target state and the period used for comparison.
c. Perturbation design
We will refer to the zonal wind anomaly at 10 hPa and 60°–75°N, after regressing out the component related to globally averaged Ts [u′ in (1); Fig. 1f] as the “polar vortex index” 
d. CMIP5 analysis
e. Assessing the stratospheric contribution to intermodel spread
We also wish to assess the magnitude of the potential influence of the stratosphere relative to the total CMIP5 model spread. For this purpose, we define the CMIP5 model spread as 4σ, where σ is the across-model standard deviation; that is, for a normal distribution, this is the range within which 95% of samples lie. We show two measures of stratospheric influence. The first, measure 1, consists of assessing, within CMIP5, the reduction in 4σ that arises from regressing out the contribution that is linearly related to 
A comparison of measures of the influence of stratospheric polar vortex changes on different measures of tropospheric change: 
f. Significance testing
Bootstrapping tests are used to assess the significance of, and uncertainties on, regression coefficients or differences. For the CMIP5 regression coefficient, this involves randomly resampling, with replacement, 35 models from the 35 available, and recomputing the regression coefficient βX 1000 times. For the difference between two experiments, 1 and 2, with length Ny1 and Ny2, Ny1 years are resampled with replacement from experiment 1, and Ny2 years are resampled with replacement from experiment 2 and the difference in their means is calculated. This is repeated 1000 times. For both the regression coefficients and the differences, the uncertainty is taken as the 2.5th–97.5th percentile range and the quantity is considered significant if this range does not encompass zero (equivalent to significance at the 5% level for a two-sided test).
In Table 3 (see also Fig. 4e), confidence intervals are provided on the two measures of stratospheric influence on intermodel spread. In all cases, the reduction in spread is expressed as a percentage of the CMIP5 spread, but the uncertainties in the CMIP5 spread itself (i.e., the denominator) are not considered when providing this uncertainty estimate. For the second measure of stratospheric influence, that is, the difference between the WEAK4x and STRONG4x experiments (section 3e), the confidence interval is simply derived from the confidence interval on the difference between these two experiments as described above. For the first measure, that is, the reduction in spread obtained by regressing out the component related to 
4. Results
We first show the overall anomalies in zonal-mean zonal wind for the NUDG4x − NUDG1x, STRONG4x − NUDG1x, and WEAK4x − NUDG1x differences in Figs. 3a–c. These can be compared with the anomalies for the CMIP5 multimodel mean and the mean of the three most strengthening and three most weakening models in Figs. 3d–f. Aside from the fact that the 4 × CO2 experiments in CESML46 warm more than the CMIP5 RCP8.5 multimodel mean (because the CO2 perturbation is bigger), resulting in a greater strengthening of the subtropical jet, these experiments are successful in mimicking the range of high-latitude stratospheric anomalies that are seen in CMIP5. We now proceed to examine the influence of these stratospheric perturbations on the troposphere below and compare with the CMIP5 across-model regressions for various fields.
DJF-averaged zonal-mean zonal wind anomalies for CESML46 (a) NUDG4x − NUDG1x, (b) STRONG4x − NUDG1x, (c) WEAK4x − NUDG1x, (d) the CMIP5 multimodel-mean future − past difference, (e) the future − past difference for the mean of the three models with the greatest increase in 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
a. Zonal-mean zonal wind
The influence that this range in polar vortex responses may have on the zonal-mean zonal wind in the troposphere can be assessed from Fig. 4. Note that here, and in all subsequent analyses, we present results from the perspective of a weakening of the polar vortex. The regression, across models, of the future − past difference in 
DJF-averaged zonal-mean zonal wind. (a) The regression of the future − past difference onto 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
This can be compared with the difference between WEAK4x and STRONG4x where extreme weakening and strengthening anomalies of the polar vortex have been artificially imposed (Fig. 4b). By construction, Figs. 4a and 4b look very similar above the nudging level (green horizontal lines in Fig. 4b). In addition to this, in agreement with the hypothesis that the tropospheric anomalies are produced as a response to the stratospheric anomalies, the imposition of these polar vortex anomalies within CESML46 produces quantitatively similar anomalies in the troposphere as well. In WEAK4x, relative to STRONG4x, there is an easterly anomaly of about −0.5 m s−1 extending to the surface on the poleward side of the jet and a westerly anomaly farther equatorward. Although the anomalies in WEAK4x − STRONG4x are slightly weaker than those from the CMIP5 regression and the easterly anomalies do not extend to as high latitudes, the WEAK4x − STRONG4x difference lies well within the uncertainty range of the CMIP5 regression (Fig. 4c). Given that, within one model, quantitatively similar results to that inferred from the regression across CMIP5 models can be obtained by imposing anomalies in the stratospheric polar vortex, it can be concluded that the downward influence inferred from such CMIP5 regressions likely does represent a true downward influence of the stratosphere on the troposphere below.
The green lines in Fig. 4d show the FREE4x − FREE1x and NUDG4x − NUDG1x differences. The only region where nudging has a significant influence on the 4 × CO2 minus 1 × CO2 difference is poleward of 80°N. Elsewhere, and in all subsequent analyses, an influence of nudging on the response is not apparent. In any case, we will always be comparing one nudged run with another so any discrepancies related to the presence of nudging should be largely cancelled out.2
Figures 4d and 4e also allow us to assess the magnitude of the stratospheric influence relative to the overall CMIP5 spread (see section 3e). The 4σ range of the CMIP5 models is around 2 m s−1 (thin solid lines in Fig. 4d). After removing the component of the 
b. Latitude–longitude 700-hPa zonal wind
For this and subsequent latitude–longitude fields we use a uniform nine-panel format in the figures. Considering 700-hPa zonal wind u700 (Fig. 5), CESML46 under 4 × CO2 (Fig. 5b) exhibits a generally similar response to the CMIP5 multimodel mean (Fig. 5a), albeit with a greater magnitude as a result of greater warming. For example, the enhanced westerlies at the extension of the Atlantic jet over Europe and the easterlies over North Africa (Woollings and Blackburn 2012; Simpson et al. 2014; Zappa et al. 2015) are present in both, along with enhanced westerlies west of California (Neelin et al. 2013; Seager et al. 2014b) and east of Japan. The CESML46 response to 4 × CO2 is extremely similar in the nudged and free configurations (cf. Figs. 5b and 5c). Next, the CMIP5 regression, 
DJF 700-hPa zonal wind. (a) CMIP5 multimodel-mean future − past difference, (b) FREE4x − FREE1x, and (c) NUDG4x − NUDG1x differences. (d) The CMIP5 regression onto 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
Figures 5g–i provide indications of the magnitude of the stratospheric influence relative to the overall CMIP5 spread. Over Scotland and the North Sea, the CMIP5 4σ range (Fig. 5g) is around 5 m s−1, which can be compared with the anomalies in Fig. 5d and 5e of around 1–1.5 m s−1. After removing the component of the u700 anomalies in CMIP5 that are related to 
Overall, the agreement between the WEAK4x − STRONG4x difference and the CMIP5 regression indicates that the u700 anomalies found in the CMIP5 regression, at least in the North Atlantic and over northern Europe, are likely produced as a response to the different zonal-mean climatological changes in the stratosphere. Removal of the stratospheric influence by linear regression removes around 5%–10% of the CMIP5 model spread in the North Atlantic but, all else being equal, we should expect the difference between the polar vortex states on the extreme weakening and strengthening ends of the CMIP5 range to result in zonal wind anomalies over the North Atlantic and Europe that are on the order of 20% of the CMIP5 spread.
c. Sea level pressure
The SLP response can be examined through Fig. 6. The main features of the CMIP5 predicted response in SLP (Fig. 6a) are reproduced in the CESML46 response to 4 × CO2, both in the free-running (Fig. 6b) and nudged (Fig. 6c) configurations, albeit with a greater magnitude. The CMIP5 regression, 
As in Fig. 5, but for sea level pressure.
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
d. Precipitation
Scaife et al. (2012) argued for an important role for stratospheric circulation on future European precipitation (pr) change in their high-top–low-top comparisons. This can be examined through Fig. 7. The CMIP5 multimodel mean shows a drying over the Mediterranean Sea and southern Mediterranean land regions along with a wetting over northern Europe (Fig. 7a). Similar features but with enhanced amplitude are seen in the free (Fig. 7b) and nudged (Fig. 7c) 4 × CO2 experiments. The CMIP5 regression, 
As in Fig. 5, but for precipitation. Note the different method of signifying significance that allows the patterns of precipitation change to be seen more clearly given the small patchy regions of significance. Stippled regions in (a)–(f) are not significant at the 5% level.
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
The CMIP5 4σ range (Fig. 7g) shows relatively greater intermodel spread in southern Europe around the zero line of the multimodel-mean future − past difference (Fig. 7a) as well as over the western portion of Scandinavia. In southern Europe, after regressing out the component related to 
e. Sensitivity to methodology
Throughout this analysis we have made subjective choices. In particular, our choice of polar vortex index (60°–75°N and 10-hPa 
Our measure of model spread was 4σ. M2014 instead used variance σ2 and also first regressed out the component of intermodel spread associated with tropical upper-tropospheric warming and Arctic amplification before assessing the subsequent reduction in σ2. These are, again, subjective choices and the impact on the conclusions can be assessed from Table 3. The prior regressions onto tropical upper-tropospheric warming and Arctic amplification have very little impact on the reduction in spread achieved by regressing out the stratospheric contribution (cf. columns A and B in Table 3). The choice of measure of model spread has a bigger impact, with the reduction in σ2 being somewhere between 1.5 and 2 times the reduction in 4σ (cf. the first four columns with the second four columns in Table 3). This difference is to be expected from these different measures and does not reflect any particular properties of the response.
Finally, while we have used our CESML46 experiments to provide a measure of the difference between models on opposite ends of the CMIP5 range, an equivalent measure could be obtained from the CMIP5 regression directly by using the CMIP5 regression coefficient multiplied by 
5. Discussion and conclusions
Prior evidence, both from observational and model assessments of stratospheric influence on the troposphere and from comparison of the climate change response between different models, has indicated that the simulation of stratospheric change represents a potential source of uncertainty in projections of tropospheric climate change.
A number of previous studies have inferred, either from comparison of different model versions (Shindell et al. 1999; Sigmond et al. 2008; Scaife et al. 2012; Karpechko and Manzini 2012) or from multimodel intercomparisons (M2014), that the way in which the stratospheric polar vortex responds in the future is connected to aspects of NH winter tropospheric circulation change. If this connection were to represent a downward influence of the stratosphere on the troposphere below then, given the wide spread among models in their predictions of stratospheric vortex change shown here and elsewhere (M2014; Butchart et al. 2010), the simulation of stratospheric change may represent a potential source of uncertainty on tropospheric climate change.
Our aim has been to complement these existing studies by performing idealized experiments, within one model, where stratospheric zonal-mean vortex states that mimic the CMIP5 range have been artificially imposed alongside an increase in GHGs, using a nudging methodology. The advantage of this is that it is only the stratospheric zonal-mean vortex anomalies that differ between model experiments, by construction. This allows for a clean demonstration and quantification of the influence of differences in the stratospheric zonal-mean climatology on the troposphere below and an unambiguous demonstration of the presence of a causal link between stratospheric vortex change and the tropospheric circulation. These experiments should capture the influence of the climatological zonal-mean stratospheric boundary conditions on the troposphere below, but, given their design, there will necessarily be aspects of the stratospheric influence that they cannot capture. In particular, the nudging of the zonal-mean stratospheric winds and temperatures toward a seasonally varying climatological state means that transient large-amplitude events such as SSWs or reflection of planetary waves from anomalous vortex configurations (Perlwitz and Harnik 2003; Shaw et al. 2010) cannot be captured. This means that the influence of these aspects on the wintertime climatology will not be fully represented and it also precludes an investigation of subseasonal variability in these simulations. It should also be noted that we performed our investigation under a 4 × CO2 climate which is warmer than the end of the century under RCP8.5 in the CMIP5 models that we compare against. This would not be a fair comparison if the influence of the stratospheric perturbations changed as warming progresses, but the similarity between the stratospheric influence in our experiments and that inferred from CMIP5 suggests this is not so.
In terms of the stratospheric influence on wintertime climatological circulation change, good agreement is found between these experiments and the stratospheric influence inferred from linear regression across different models (M2014). This confirms that (i) the stratospheric influence inferred from such linear regressions is indeed a true downward influence of the stratospheric vortex change on the troposphere below and (ii) a substantial component of the stratospheric contribution to intermodel spread in the wintertime climatological change in the troposphere arises from the change in the zonal-mean climatological stratospheric boundary conditions.
To summarize the main features of the stratospheric influence found in these experiments, a relative weakening of the stratospheric polar vortex is accompanied by an easterly anomaly in zonal-mean zonal wind 
We have quantified the magnitude of this stratospheric influence relative to the CMIP5 model spread, where “spread” here is defined as the CMIP5 4σ range, in two ways: 1) by assessing how much the CMIP5 model spread is reduced once this stratospheric influence has been regressed out and 2) by comparing the influence, in our experiments, of vortex states on opposite sides of the CMIP5 range, with the CMIP5 spread. Using measure 1, it is found that the CMIP5 spread in 
Overall, uncertainties in the future changes of the stratospheric polar vortex represent a small, but nonnegligible source of uncertainty in tropospheric climate change, particularly for the Arctic and Atlantic and European sectors. Of considerable impact is the influence on precipitation over Europe. The precipitation anomaly induced by polar vortex changes on opposite ends of the CMIP5 distribution can reach up to 0.25 mm day−1 over southern Europe and the Mediterranean countries, which is equivalent to roughly 10%–20% of their present-day wintertime precipitation climatology (Seager et al. 2014a, their Fig. 1a). For countries in this region that are likely to become increasingly water-stressed in the future, this is a big difference. If the predictions of greater strengthening of the polar vortex in the future are the correct ones, then we might expect the impacts of climate change in these regions to be considerably more severe than predicted by the multimodel mean. At present there is no reason to believe this outcome is more likely than the alternatives; that is, the vortex weakening or remaining more or less unchanged, but it is still a plausible way in which the stratosphere may change in the future. While, in the presence of other sources of uncertainty and internal variability, this is a relatively small contribution to the spread in model predictions, it may be an important one, and a tractable one to reduce through improved understanding, in contrast to the irreducible uncertainty associated with internal variability. This further motivates an improved understanding of the reasons behind the wide spread in stratospheric polar vortex responses among models and an improved constraint on which projections are most likely to occur in the real world.
Acknowledgments
The National Center for Atmospheric Research is sponsored by the National Science Foundation. This work was also funded by National Science Foundation Awards AGS-1317469 and AGS-1734760. We are grateful to Yaga Richter for providing the 46-level model and the free-running 1 × CO2 simulation used in this study, and Naftali Cohen for initial analysis that lead to an earlier version of Fig. 1. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 1 of this paper) for producing and making available their model output. For CMIP, the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. We would also like to acknowledge high-performance computing support from Yellowstone provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. This is LDEO contribution number 8226.
APPENDIX
Stratospheric Perturbation Design
The regression coefficients βu, βT, and βυ (Figs. A1a–c) are the latitude–pressure structures for the perturbations below 10 hPa. These coefficients show that models with more positive (negative) 
The linear regression of (a) 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
The latitude–pressure structure of the perturbations is based on the DJF-averaged anomalies, but we also give the perturbations an idealized seasonality S(t), intended to mimic the seasonality seen in the CMIP5 model regressions onto 
(a) Monthly variation of future − past difference in 10-hPa zonal-mean zonal wind regressed (across 35 models) onto the DJF 
Citation: Journal of Climate 31, 16; 10.1175/JCLI-D-18-0041.1
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The difference in CO2 between 2070 and 2099 of RCP8.5 and 1979–2005 of historical experiments is around 442 ppm while the CO2 anomalies imposed in CESML46 are close to double that at 854.1 ppm. The CESML46 simulations are lacking the increase in other GHGs, but the greater increase in CO2 and the more equilibrated ocean leads to an overall greater warming than found in CMIP5; however, we do not consider the different magnitude of warming to impact the conclusions regarding the stratospheric influence, as discussed further in section 5.
Comparison of the difference between the responses in WEAK4x and STRONG4x (blue and red in Fig. 4d) and that in NUDG4x may suggest some nonlinearity; that is, poleward of around 60°N, the response in STRONG4x is more different from the response in NUDG4x than the response in WEAK4x is. However, this is not robust enough to consider the two halves of the STRONG4x and WEAK4x experiments separately.
