SNAPSI experiments consist of retrospective forecasts of three case study stratospheric polar vortex disruption events: the boreal major SSW of 12 February 2018 (which we refer to as SSWFeb18), the boreal major SSW of 2 January 2019 (SSWJan19), and the austral minor warming of 18 September 2019 (SSWSep19). SSWFeb18 and SSWJan19 are referred to as major warmings as the zonal-mean zonal wind at 10 hPa, 60° N reversed to become easterly, and their central dates are those on which the winds reversed. In contrast, this threshold (at 60° S) was not crossed for SSWSep19, although the zonal winds did decelerate rapidly; in SSWSep19 the central date is taken to be that of minimum zonal wind at 10 hPa, 60° S. For each case study we consider three experiments:  

free A standard forecast ensemble, which is free-running following initialization.

The observed evolution and climatology used for nudging are derived from the ERA5 reanalysis . Each experiment is initialized at two different dates for each case study (given in Table ), and each consists of a 50-member ensemble, generated according to a methodology that varies by forecast system. For example, the IFS generates oceanic and atmospheric perturbations separately, the atmospheric component combining singular vectors and ensemble data assimilation. The same set of perturbations is applied to the free, control, and nudged experiments for fair comparisons. We analyze the output of 7 forecasting systems contributing to SNAPSI, which at the time of analysis had the data necessary archived on the JASMIN facility; for our study available. These systems are listed in Table . A detailed description of the observed dynamical evolution of the three events and of the SNAPSI experimental design is given by , the impact of nudging on upward wave activity is investigated by , and an analysis of surface predictability within the SNAPSI experiments is given by . However, to allow for interpretation of our later results, we illustrate the ensemble-mean impact of nudging on the polar mid-stratospheric zonal-mean zonal wind (at 10 hPa, 60° N/S) in Fig. .

SSWFeb18 (Fig. a) is seen to be poorly predicted by the free experiment at the early initialization, but well predicted at the late initialization. Indeed, for most models, the early initialization free experiment has a stronger polar vortex than the control. For SSWJan19 (Fig. b) , there is a smaller difference between early initialization free and nudged experiments, with some free models successfully predicting an SSW despite being initialized 20 d before it occurred. The late initialization for SSWJan19 comes after the SSW central date (this was chosen due to the longer lag between the SSW and surface anomalies, described further in ), and the subsequent recovery of the polar vortex from this time is well predicted by the free experiments, with a large difference between these and control until early February. Similarly, for SSWSep19 (Fig. c), the differences between free and nudged experiments are smaller than with either of those and control, at least until November, indicating relatively good skill in the prediction of this event.

One motivation for the choice of these three case study events for SNAPSI was the range of surface weather extremes that were observed in the months following them. Here we will describe some key features of these events, with a focus on surface temperature extremes. In the period following SSWFeb18, from late-February to early-March 2018, negative Northern Annular Mode (NAM) and North Atlantic Oscillation (NAO) patterns were associated with anomalously cold temperatures extending over much of Eurasia . Concurrent weather events included snow over the UK (which we analyze in Sect. ) , as well as enhanced precipitation over Iberia , ending an extended period of drought. An analysis of precipitation within SNAPSI experiments of SSWFeb18 is presented in .

In contrast, during the period following the SSW of SSWJan19 the NAM remained near neutral, and temperature anomalies over Eurasia were comparably muted . However, more significant temperature anomalies occurred in North America, with an extensive cold spell in late January 2019 covering much of the Midwestern US and eastern Canada, with several daily cold temperature records being set in this region . The dynamical evolution of this event has been described on synoptic scales as relating to an equatorward deviation of the extratropical jet stream as well as a tropopause polar vortex , and on larger scales as linked to an Alaskan Ridge weather regime. However, in an analysis of SNAPSI experiments, find little stratospheric impact on the development of this weather regime. Indeed, by contrasting stratospherically- and tropically-nudged seasonal hindcast experiments, proposed a stronger tropical influence. Nonetheless, a direct attribution of the stratospheric role in the extreme cold spell has not been carried out.

SSWSep19 was the most significant disturbance to the austral stratospheric polar vortex since the major warming of 2002 . Following the central date of 18 September on which zonal mean winds at 10 hPa, 60° S reached a minimum (in the ERA5 data set), a negative Southern Annular Mode (SAM) pattern slowly descended from the mid-stratosphere, reaching the surface from approximately mid-October. The negative SAM pattern has been associated with hot and dry conditions over eastern Australia during late October to December, which in turn contributed to unprecedented destructive wildfires in the region . Stratosphere-troposphere coupling within the SSWSep19 SNAPSI experiments is discussed in detail by , including an analysis of wildfire risk.

Motivated by these observed events we restrict our analysis to broad regions that capture the dominant observed temperature extremes for each case study (Fig. ). For SSWFeb18 (Fig. a) this region matches exactly that of , for SSWJan19 (Fig. b) we select a region containing the most extreme cold anomalies identified by , and for SSWSep19 (Fig. c) we select a region containing the entire continent of Australia, in all cases considering only land areas for extreme temperature analysis. We also define a time window for each case study in which to determine temperature extremes. In doing so, we balance a desire to capture events similar in timing to that observed, while also not overly penalizing models that capture the magnitude of events well but have errors in timing. As such, we define periods that follow the central date of each SSW and bracket the most extreme temperatures observed in reanalysis by a margin of at least 5 d, or as wide as possible while remaining fully inside the common forecast period of both early and late initializations – with an exception of SSWSep19, where we extend the horizon beyond the range of the early forecast in order to bracket the highest extremes seen in the late ensemble. The SSWFeb18 ensembles bracket the extremes comfortably, but the SSWJan19 ensembles end before the greatest extremes and the late SSWSep19 ensembles start just barely before the greatest extremes, making them more sensitive to the choice of time interval. Expanded forecast horizons may be a goal for future experiments, and more systematic sensitivity analysis may be a goal for future attribution methods, but the current setup suffices to demonstrate our core methodology. The spatial and temporal boundaries of our below analysis are shown in Table , and Fig.  shows large and relatively spatially coherent anomalies of temperature extremes within the chosen regions, motivating their suitability for area-averaged analysis. The following spatial maps will focus only on the selected sub-regions.

For the purposes of our analysis we define severity as the spatially averaged and then temporally minimized (for SSWFeb18 and SSWJan19) or maximized (for SSWSep19) temperature. We notably consider absolute temperature, not anomaly, which is better connected to some major societal impacts , and on which several downstream impacts (e.g. snow) depend. Crucially, we quantify statistical responses to different nudging interventions using two quantities:  

relative risk (RR), a ratio of probabilities of exceeding some fixed severity, and

These two quantities have complementary strengths and weaknesses that allow us to discern robust causal signals that partially overcome model bias.

We need some notation to summarize our statistical approach. Let g be an index for the data-generating system, either a forecast system or reanalysis, ∈{CESM2-CAM6,…,IFS,ERA5}; e be an index for the experimental nudging protocol ∈{control,free,nudged} (not applicable to ERA5); i be an index for the initialization date ∈{early,late} (not applicable to ERA5); t be a timestamp within the temporal window [A,B] in which the extremes can occur; m denote the ensemble member, ∈{1,…,M=50} (for forecast systems) or {1980,…,2019} (for ERA5); and Sg,e,i,m(t) be the intensity simulated by system g, experiment e, initialized from date i, as realized by the mth ensemble member, on day t. When g=ERA5, the subscripts (e,i) are not applicable, and are replaced by (⋅,⋅). For all three SSWs, S represents a land-only area-averaged 2 m temperature 〈T(t)〉 that is maximised daily over four six-hourly samples. For SSWFeb18 and SSWJan19, 〈T(t)〉 is multiplied by -1 before maximising, such that a more positive S represents colder temperature minima for these events, while a more positive S represents warmer temperature maxima for SSWSep19. The areas for averaging and the time intervals of interest are described in the previous section and shown in Table .

The format of the ensemble lends itself naturally to extreme value analysis with the method of block maxima . For each (g,e,i,m), we define the severity as the peak intensity attained, and call it Sg,e,i,m∗: 1Sg,e,i,m∗=max⁡A≤t≤BSg,e,i,m(t), and fit a generalized extreme value (GEV) distribution to each ensemble {Se,i,m∗:m=1,…,M}, using the method of probability-weighted moments, which tends to be more stable than maximum-likelihood for small sample sizes . In other words, we consider Sg,e,i∗ to be a random variable whose cumulative distribution function (CDF) is defined for any value s as 2P{Sg,e,i∗≤s}=Fs;μg,e,i,σg,e,i,ξg,e,iwhereF(s;μ,σ,ξ)=exp⁡[-(1+ξs-μσ)+-1/ξ]ifξ≠0exp⁡[-exp⁡(-s-μσ)+]ifξ=0, where the subscript + denotes the positive part of the expression, i.e. x+:=max⁡(x,0). We will also refer below to the complementary CDF, or CCDF, which is the exceedance probability: CCDF=1-CDF. The three parameters of the GEV (μ,σ,ξ) are called the location, scale, and shape, respectively, which we abbreviate as a vector θ=[μ,σ,ξ].

We emphatically acknowledge that the GEV is not a priori guaranteed to be appropriate, as this would require S∗ to be the maximum of a set of random variables that is large in number (much more than 2 weeks), independent (unlike an ensemble of skillful forecasts), and identically distributed (devoid of any seasonal cycle). For this reason, we note that confidence intervals calculated with the GEV are likely to be underestimates. However, the GEV was found to capture the distribution of temperature extrema better than a Gaussian, and we use it only as a regularizing tool for the relatively small ensemble size of M=50. It is worth exploring other distribution choices more thoroughly – RR and QS are equally well-defined for any parametric family – but GEV was a natural, illustrative, and sufficiently fitting distribution family for the present demonstration.

To provide some context and aid the interpretation of the GEV results to follow, we provide in Fig. a a visual summary of distributional parameters derived from ERA5. Specifically, we consider the 16 d time interval (as in Table ) from 21 February (ending on 7 or 8 March depending on the year, inclusive) each year from 1980 to 2019 as a different ensemble member (thus, M=40 in the ERA5 case). We compute the severity for each year, but for geographic illustration, the area averages are performed separately on each of the boxes tiling the domain (restricting the average to land pixels in all boxes). The mean severity and its standard deviation are shown in Figs. a(i,ii). Similar maps for SSWJan19 and SSWSep19, over their respective areas of interest, can be found in the Supplement.

We then fit a GEV distribution to each grid box, and plot the corresponding parameter maps in Fig. a(iii, iv, and v). The mean and GEV location have a clearly similar spatial pattern as do the standard deviation and GEV scale, confirming that these two GEV parameters represent what their names suggest. The shape parameter (Fig. a(v)) stands apart as a subtler-to-interpret, but highly influential parameter: a negative shape parameter imposes a bound on severity equal to μ-σ/ξ, whereas a positive shape parameter allows for unbounded severity (see Eq. ). The map indicates negative shape nearly everywhere, which is usually the case in GEV analysis of temperature extremes , despite the lack of any a priori bounds on surface temperature . Recent work has, however developed theoretical frameworks for upper bounds temperature in terms of geopotential height and soil moisture , and fruitfully applied this framework to data-driven statistical models .

We also illustrate equivalent distribution parameters for the 25 January 2018 initialization of the free ensemble of the IFS model in Fig. b. In general, the IFS model parameters are similar to those from reanalysis indicating that the ensemble is a close match to the climatology of extremes, but has little precision from this initialization date.

Although each grid box sub-region has its own severity distribution, the following quantitative risk analysis only considers full-region severities. Note the full-region GEV parameters are not simply the average of the sub-region GEV parameters, and a legitimate concern with the full-region average is lumping together many different regional climatologies, each of which merits a separate localized analysis. We deem the fits shown in the next section sufficiently good to provide robust results and establish the methodology, but a follow-on paper will quantify the extremal behavior across sub-regions and averaging scales.

We now aim to apply our statistical analysis to compare temperature extremes in the three SNAPSI experiments and in ERA5. Figure  summarizes full-region risk analysis on the IFS model for SSWFeb18. The left and right Fig.  panel groups (i and ii) display results for the early and late initializations, respectively, and the three rows display in order results for free, nudged, and control experiments. For each experiment (in red) and for ERA5 (in black) we show, on the left Fig. (i), the empirical risk function consisting of ordered pairs of points at (1/m, T(m)) for m=1,…,M, where T(m) is the temperature (area-averaged and then time-minimized) of the mth ensemble member (or year, for ERA5) in order of increasing severity (i.e., the severity with a 1m chance, empirically, of being equalled or exceeded). We also plot as solid lines the GEV-fitted risk functions and their 50th percentile-bootstrapped confidence intervals (resampling with replacement 1000 times, re-calculating GEV parameters for each, and shading between the 25th and 75th percentiles). The ±1 standard deviation error bars on the GEV parameters themselves are indicated in the legends.

We next explain how each forecast ensemble is boiled down to two numbers for later inter-model comparison. Starting at the ERA5 SSWFeb18 severity (∼ 250 K), we read off the horizontal position of the black curve to get a baseline risk level, regularized by the GEV fit, which we denote as AR0. Then we draw a vertical line through this risk level and find the intersection with the early, free ensemble to get a temperature value, EQ0, against which all other forcing experiments will be compared. The rationale is to substitute the observed ERA5 climatology with the model's early, free ensemble, which is very similar for IFS but is markedly different for some other models, in a form of bias-correction that treats the model as a “fixed-effect” and enables a clearer comparison between forcings, rather than a comparison between raw model fidelity. This baseline pairing is marked as a cyan dot. Then, for all six (forcing, initialization date) pairs, the corresponding ensemble's absolute risk (AR), shown by the vertical red dashed line, is the probability of exceeding EQ₀ (noting that, although we use the term “absolute”, AR remains conditional on the forecast initialization and any nudging). Similarly, each pair's equivalent-risk quantile (EQ) is shown by the horizontal red dashed line, as the severity at the same quantile as the baseline ERA5 value. In the (free, early) ensemble, these coincide with the baseline values EQ0 and AR0. Mathematically, following the notation in Sect. , let g denote the model, e denote the forcing experiment, and i∈{1,2} denote the initialization. We define for the (free, early) ensemble 3ARERA5,2018=P{SERA5∗≥SERA5,2018∗}=1-F(SERA5,2018∗;θERA5)4ARg,free,1=ARERA5,20185EQg,free,1=F-1(1-ARERA5,2018;θg,free,1) where F-1 is the inverse function of F, also known as the quantile function.

Then, for all (e,i) pairs, 6ARg,e,i=P{Sg,e,i∗≥EQg,free,1}=1-F(Sg,free,1∗;θg,e,i)7EQg,e,i=F-1(1-ARg,free,1;θg,e,i)

In the following, RR is a ratio of two absolute risks while QS is a difference in quantile. For example, 8RR=ARIFS,nudged,1ARIFS,free,1and QS=EQIFS,nudged,1-EQIFS,free,1 measure two aspects of the response to nudging in the IFS model, initialized on 25 January 2018 (the early date, i=1). RR and QS have complementary advantages and drawbacks: RR is interpretable on a universal scale, no matter the event type or the physical units, but it may be zero or infinite in the presence of model bias. QS is always finite, but has physical units and characteristic magnitudes that depend, for example, on the spatial scale used for averaging. As we will find below, RR and QS should be interpreted with consideration for the underlying forecast skill of the event in question. For instance, if an event is well-predicted in free, nudging will be unlikely to influence its risk or severity (even if the event is stratospherically-influenced), hence the RR of nudged relative to free might be close to unity. In such a case, a better indicator of stratospheric influence might be the RR of control relative to free.

In order to further illustrate the calculation of the empirical risk function, the right-hand panels (a)–(f)(ii) of Fig.  show time series of intensity (daily region-averaged temperature) from the model ensemble and from all years of ERA5. The red and black crosses show the location of the minimum temperature during the period of interest, lining up with the locations of the points shown in the empirical risk function. The black dashed line shows the ERA5 time series for 2018. We see that the 2018 minimum temperature during this period, reached on 27 February, is the coldest on record for that day of year, and is the third coldest temperature reached over the whole period of interest. It is also apparent that in all 6 model experiment cases this observed severity lies within the ensemble spread, indicating the IFS model's ability to capture events of this magnitude. The same is true for SSWJan19 and SSWSep19, for which plots equivalent to Fig.  are shown in the Supplement.

At the earlier initialization, the risk function for all three experiments matches relatively closely that for ERA5 (Fig. , left column). This both illustrates that the forecasts from this initialization date have little precision beyond climatology (as previously noted in Fig. ), and also that the “IFS climatology” – interpreted in the sense of an ensemble that has run just long enough to disperse, not a long multi-year single run – is relatively unbiased. The control climatology matches ERA5 climatology even better, which makes some heuristic sense because the control forcing makes the stratosphere match the ERA5 climatological mean by construction; however, nudging toward a single mean value does not imply matching the full distribution better, and might equally be expected to reduce the variance across members as a result of the lack of dispersion in the stratosphere and its influence on the troposphere. In other words, it is not obvious whether free or control experiments should better match the climatological distribution, especially when the climatological mean stratosphere is a sum over both SSW and non-SSW years and might not even be physically realizable. This is an important caveat for interpretation, and a potential improvement area for experimental design. For the later initialization (Fig. , right column), the IFS risk function is notably flatter than that of ERA5, indicative of greater forecast precision at the later initialization.

The exceedance probabilities (CCDFs) for the three experiments, and both initialization dates of SSWFeb18 (retaining IFS as our example model), are overlaid in Fig. . RR is shown in panels Fig. (ii) as a function of severity, and QS is shown in panels Fig. (iii) as a function of exceedance probability, which are simply the horizontal ratio and vertical differences between curves in panels Fig. (i). The special thresholds given by ERA5 are marked by black dashed lines, and the equivalent-risk quantile in the early, free experiment is marked by green dashed lines. The intersection of these lines with the QS and RR curves, respectively give the nominal QS and RR values plotted as ordered pairs in panels Fig. (iv), and later used for inter-model comparison.

In the case of IFS, there is a clear and consistent pattern of influence: nudging toward observations both increases the RR and intensifies the QS relative to the control simulation, and the effect is larger for the earlier initialization. Figure iv captures this as a downward and rightward displacement of red relative to blue, which shrinks in magnitude at the later date. In the special case that ξ=0 for both experiments and the nominal severity S∗ is far into the tail, RR and (log) QS are proportional, since the Gumbel distribution (GEV with ξ=0) is asymptotically exponential and thus the CCDF on a log scale is asymptotically linear: 9F(s;μ,σ,ξ=0)=exp⁡[-exp⁡(-s-μσ)]∼1-exp⁡(-s-μσ)⟹log⁡(1-F)∼-s-μσass→∞.

However, in general RR and (log) QS are not necessarily proportional, especially if the severity of interest – nominally, the 250.5 K level achieved by ERA5 – is near the GEV-implied severity bound where the CCDF must curve sharply. We see this already in the late-initialized control experiment in Fig. b(i, ii), where the blue curve levels off substantially before exceeding (dipping below) the ERA5 severity, and its relative risk rapidly approaches zero. The implied lower bound on control severities is not far off, at -(μ-σξ)=-(-254-2.3-0.29) = 246 K (the overall negative sign accounts for considering cold-side extremes). A small positive bias could easily have made the ERA5 severity impossible according to the control GEV fit, a conclusion which is physically reasonable but not well-described by relative risk. QS thus provides an important complementary viewpoint.

Let us point out some patterns in Fig.  which convey the effects of modulation of the stratospheric state on European cold extremes following SSWFeb18. In order from broad and robust to intricate and more uncertain: 1.

Nudging the stratosphere toward the observed SSW intensifies surface cold extremes compared to relaxing the stratosphere toward its climatological mean. The RR (nudged/free) exceeds the RR (control/free) for the full range of severities in panels Fig. a and b(ii), and QS (nudged - free) is more negative than QS (control - free) in panel Fig. a and b(iii). Both of those differences exceed the error bar width at, and around, the 2018 ERA5 values, and for both early and late initializations. At the ERA5 severity of 250.5 K, the relative risk is a factor ∼ 2 larger for nudged than control at both initializations; at the ERA5 probability of ∼ 0.1, the nudged temperature is ∼ 2 and ∼ 0.5 K colder than the control temperature based on early (Fig. a) and late (Fig. b) initializations, respectively. IFS thus delivers a clear message that SSWFeb18 disposed the troposphere towards a stronger cold-air outbreak than the climatological average stratospheric evolution would have done.

Figures equivalent to Figs. 3–5, but for SSWJan19 and SSWSep19 can be found in the Supplement. For a succinct inter-model comparison of the three events, Fig.  summarizes these results for all three case studies (Fig. a: SSWFeb18, Fig. b: SSWJan19, and Fig. c: SSWSep19) and both initialization dates (i: early and ii: late), reporting for each model and date only the slice of the RR curve evaluated at the ERA5 severity, and the slice of the QS curve evaluated at the ERA5 GEV-fitted risk, as in Fig. a and b(iv). Error bars are 50th percentile-bootstrapped confidence intervals, as in Fig. . A multi-model mean, taken by averaging these two metrics after calculating them for each model, is also shown (large hexagon). The results for SSWFeb18 are relatively consistent across models, and so with those already discussed for IFS. The early initialized nudged experiments of all models have both increased risk (on average by a factor of about 2) and increased severity (on average by about 1–2 K) compared to free forecasts. The control experiments also show (with the exception of CNRM-CM6-1) an increase in risk and severity, though this is more modest, both on average and in most individual models. This might be explained by the fact that the stratospheric polar vortex is on average stronger in the free than the control experiments for the early initialization (Fig. a). At the later initialization, nudged experiments show little change (relative to free) in RR or QS, but control experiments show a consistent reduction of both by factors of about 1/2 and 1.5 K, respectively.

The other two case studies are not as consistent across models and initializations, which highlights the diversity of SSWs: in surface impacts, in lead-time dependence of predictability, and in how robustly models represent them. In SSWJan19 late nudging towards climatology reduces the risk of cold air outbreaks over North America, and reduces their severity – the same as in SSWFeb18, with close similarity between Fig. a and b(ii) – but, unlike SSWFeb18, early nudging or relaxation to climatology in SSWJan19 has a small average effect and little consistency of sign among models. This sensitivity of the stratospheric effect to lead-time suggests state-dependence, a point that will be expanded upon in Sect. .

In SSWSep19, nudging towards observations appears to consistently strengthen the severity of hot conditions over Australia in terms of QS, but a smaller signature is seen in terms of RR. Especially for the late initialization, most RRs are not significantly different from unity, because most ensemble members of most models exceed the threshold identified from the (free, early) experiment (see Fig. S6 in the Supplement for the example of IFS), which is the very reason why RR is an erratic indicator and QS is needed to extract some meaningful signal. Also, in this case it is not as fair to compare early with late initializations because we chose a target time horizon (through 15 November) that extends beyond the range of the early forecast (which ends 14 October; see Fig. S5). Nevertheless, the larger QS for nudged than for control in the early initialization forecast, which is consistent between models, reveals the role of the SSW in increasing the temperatures over Australia in the first half of October 2019. The same result in the late initialization forecast further confirms that the SSW did play a role in the observed extremes during spring 2019.

In aggregate, we affirm the original hypothesis of the SNAPSI project with regard to surface extremes – that the SSWs significantly affect the predicted risk of extreme surface temperatures – in only some cases: SSWFeb18 (both early and late initializations), SSWJan19 (the late initialization only) and SSWSep19 (both initializations, but with incomplete information for the first, which are truncated too early for a completely fair comparison). The other effects are mixed, and warrant a great deal of further exploration, in particular the geography of impacts, which a following paper will explore.

As an illustration of how the attribution methodology described above might be adapted to other variables besides temperature, we will briefly describe an analysis of snow extremes. Specifically, we focus on SSWFeb18 and the extreme snow that accumulated over much of the UK during late February–March 2018 . Figure  shows snow depth averaged over land in the box [50° N, 55° N] × [10° W, 2° E], constituting the southern UK and Ireland. ERA5 reanalysis (in red) shows a maximum close to 3 cm being reached in early March. The 25 January 2018 initialized free forecasts from IFS are shown in gray. No ensemble members simulate an accurate timing and magnitude of this snow event, although there is some indication of an increased probability of high snow in early March (shown by a local maximum in the 90th percentile of the ensemble, in blue). The model also predicts a snow event in mid-February of much greater magnitude than in reanalysis, such that on 14 February all ensemble members have snow depth greater than ERA5.

Figure  shows exceedance probabilities (for the same time window as in Table ) of snow depth for the early initializations of two example models; IFS and CNRM-CM61. As could be seen in the snow depth time series, very few ensemble members simulate snow accumulation maxima close to or above the observed values of about 3 cm (about 5/50 ensemble members in IFS and 2/50 in CNRM-CM61). Second, the parametric fits are much poorer than those for temperature, a problem that is again greater for CNRM-CM61. We hypothesize that the poor GEV fit is due to a very large number of ensemble members having zero (or close to zero) snow during the whole evaluation period (as can be seen in the time series of Fig. ). Nonetheless, it is apparent that there are systematic differences between the three experiments, with nudged giving generally higher exceedance probabilities than free, and control generally lower probabilities.

Given the poor parametric fits for snow we illustrate a simpler approach to calculating RR based purely on counting ensemble members and not on the GEV fits. Here, RR is simply the ratio of the number of members exceeding a given threshold, and is shown for nudged/free and control/free in Fig.  for 5 of the 7 models for which appropriate snow depth data was available, as well as a multi-model mean. We choose a threshold of 0.8 cm, which is much lower than the observed maximum near 3 cm; this is to avoid large errors or infinite RR caused by dividing by a small number (or zero) when few ensemble members exceed the threshold. An inability to quantify RR for thresholds near or beyond the ensemble spread is an inherent drawback of this non-parametric approach. However, Fig. S7 shows RR (as a multi-model mean) for thresholds from 0.2 to 1.8 cm, with qualitatively similar results across this range.

Although there is significant spread between models, some consistent results emerge. First, in all models, improving the stratospheric forecast by nudging to reanalysis early (when the SSW is poorly predicted; Fig. a) increases the risk of snow (by on average about a factor of 2), while the sign of the effect of nudging late is mixed among models. Second, nudging to climatology early has a small and inconsistent sign effect, while nudging to climatology late (when the SSW is well predicted) reduces the risk of snow in all models (by on average about a factor of 0.5). These results are broadly consistent with our findings for temperature (noting the larger geographical region considered for temperature), and indicate that, in terms of inter-model consistency, there is a robust impact of the 2018 SSW on the probability of UK and Ireland snow accumulation.