How Do Stratospheric Perturbations Influence North American Weather Regime Predictions?

Simon H. Lee; Andrew J. Charlton-Perez; Steven J. Woolnough; Jason C. Furtado

doi:10.1175/jcli-d-21-0413.1

How Do Stratospheric Perturbations Influence North American Weather Regime Predictions?

Lee, Simon H.; Charlton-Perez, Andrew J.; Woolnough, Steven J.; Furtado, Jason C. 2022-09-15 00:00:00 15 SEPTEMBER 2022 LE E E T A L . 5915 How Do Stratospheric Perturbations Inﬂuence North American Weather Regime Predictions? a a b c SIMON H. LEE, ANDREW J. CHARLTON-PEREZ, STEVEN J. WOOLNOUGH, AND JASON C. FURTADO Department of Meteorology, University of Reading, Reading, United Kingdom National Centre for Atmospheric Science, Department of Meteorology, University of Reading, Reading, United Kingdom School of Meteorology, University of Oklahoma, Norman, Oklahoma (Manuscript received 27 May 2021, in ﬁnal form 8 May 2022) ABSTRACT: Observational evidence shows changes to North American weather regime occurrence depending on the strength of the lower-stratospheric polar vortex. However, it is not yet clear how this occurs or to what extent an improved stratospheric forecast would change regime predictions. Here we analyze four North American regimes at 500 hPa, con- structed in principal component (PC) space. We consider both the location of the regimes in PC space and the linear re- gression between each PC and the lower-stratospheric zonal-mean winds, yielding a theory of which regime transitions are likely to occur due to changes in the lower stratosphere. Using a set of OpenIFS simulations, we then test the effect of re- laxing the polar stratosphere to ERA-Interim on subseasonal regime predictions. The model start dates are selected based on particularly poor subseasonal regime predictions in the European Centre for Medium-Range Weather Forecasts CY43R3 hindcasts. While the results show only a modest improvement to the number of accurate regime predictions, there is a substantial reduction in Euclidean distance error in PC space. The average movement of the forecasts within PC space is found to be consistent with expectation for moderate-to-large lower-stratospheric zonal wind perturbations. Overall, our results provide a framework for interpreting the stratospheric inﬂuence on North American regime behavior. The results can be applied to subseasonal forecasts to understand how stratospheric uncertainty may affect regime predictions, and to diagnose which regime forecast errors are likely to be related to stratospheric errors. SIGNIFICANCE STATEMENT: Predicting the weather several weeks ahead is a major challenge with large poten- tial beneﬁts to society. The strength of the circulation more than 10 km above the Arctic during winter (i.e., the polar vortex) is one source of predictability. This study investigates how forecast error and uncertainty in the polar vortex can impact predictions of large-scale weather patterns called “regimes” over North America. Through statistical analy- sis of observations and experiments with a weather forecast model, we develop an understanding of which regime changes are more likely to be due to changes in the polar vortex. The results will help forecasters and researchers un- derstand the contribution of the stratosphere to changes in weather patterns, and in assessing and improving weather forecast models. KEYWORDS: Climate classiﬁcation/regimes; North America; Stratosphere; Stratosphere-troposphere coupling; Subseasonal variability; Winter/cool season 1. Introduction and quasi-stationary (e.g., Michelangeli et al. 1995) with typical time scales of weeks, well suited to the subseasonal scale where The framework of large-scale weather regimes is now in- they can manifest “windows of opportunity” for skillful ex- creasingly used in wintertime subseasonal-to-seasonal (S2S) tended-range forecasts (Mariotti et al. 2020; Robertson et al. prediction (from ∼2 weeks to 2 months ahead; White et al. 2020). 2017), although the concept of a weather “regime” is not new Unlike empirical orthogonal functions (EOFs) (e.g., Hannachi (Rex 1951). Regimes are characteristically recurrent, persistent, et al. 2007), regimes deﬁned through clustering methods are not bound by orthogonality or variance partitioning con- straints. These regimes can therefore more closely represent Denotes content that is immediately available upon publica- tion as open access. the full anomalous ﬂow conﬁguration on a given day by beneﬁting from “mode mixing” and are accordingly easier to interpret, providing a useful way to understand extended- Supplemental information related to this paper is available range ensemble forecasts. By characterizing recurrent ﬂow at the Journals Online website: https://doi.org/10.1175/JCLI-D-21- conﬁgurations, weather regimes can also be used to diagnose 0413.1.s1. ﬂow-dependent predictability (Ferranti et al. 2015; Matsueda Simon H. Lee’s current affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/ licenses/by/4.0/). Corresponding author: Simon H. Lee, simon.h.lee@columbia.edu DOI: 10.1175/JCLI-D-21-0413.1 Ó 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). 5916 J O U R N A L O F C LI MATE VOLUME 35 and Palmer 2018). From an impacts perspective, regimes have while others focus on the continent as a whole and incorpo- been used to better understand meteorological impacts on en- rate both Atlantic and Paciﬁc variability. Despite some meth- ergy demand (e.g., Grams et al. 2017; van der Wiel et al. 2019; odological differences, a growing number of studies have Garrido-Perez et al. 2020), precipitation and wildﬁre risk deﬁned a consistent and reproducible set of four wintertime (Robertson and Ghil 1999; Robertson et al. 2020), and public regimes in the 500-hPa geopotential height anomaly ﬁeld cen- health (Charlton-Perez et al. 2019; Huang et al. 2020). tered over North America (e.g., Straus et al. 2007; Vigaud A signiﬁcant source of tropospheric subseasonal predict- et al. 2018; Lee et al. 2019b; Robertson et al. 2020). The re- ability during boreal winter is variability in the Arctic strato- gimes capture both PNA-like and NAO-like behavior. spheric polar vortex, including sudden stratospheric warmings More speciﬁcally, Lee et al. (2019b) analyzed these four (SSWs; e.g., Charlton and Polvani 2007) and strong vortex North American regimes (the Arctic high, Arctic low, Alas- events (e.g., Limpasuvan et al. 2005; Tripathi et al. 2015). The kan ridge, and Paciﬁc trough) in the context of the strength of downward inﬂuence of the stratosphere can be viewed as the the lower-stratospheric polar vortex in reanalysis. They found modulation of weather regime transition and persistence. Per- signiﬁcant differences between the occurrence of three of the haps the simplest regime framework employs the two phases regimes during strong and weak stratospheric vortex states of of the North Atlantic Oscillation (NAO), which are similar to a similar magnitude to those in Charlton-Perez et al. (2018) the Northern Annular Mode (NAM) and Arctic Oscillation for the North Atlantic. The Alaskan ridge regime did not (AO) patterns and strongly inﬂuenced by the stratosphere show a relationship with the stratospheric vortex strength, but (Ambaum et al. 2001; Baldwin and Thompson 2009; Hitchcock was found to be strongly linked to North American cold and Simpson 2014). More complex regime analyses for the waves. Lee et al. (2019b) hypothesized that tropical forcing North Atlantic–European sector invoke four (e.g., Vautard (e.g., Wang et al. 2014) or stratospheric wave reﬂection 1990; Cassou 2008), six (Falkena et al. 2020), or seven (e.g., (Kodera et al. 2016; Kretschmer et al. 2018; Matthias and Grams et al. 2017) regimes depending on the method, focus, or Kretschmer 2020) may dominate driving the Alaskan ridge, purpose of the analysis. owing to the similarity of the regime to patterns associated Using four North Atlantic regimes, Charlton-Perez et al. with both. As a purely observation-based study, the results of (2018) found signiﬁcant differences in the occurrence likelihood Lee et al. (2019b) were noncausal and did not assess when or of three regimes between strong and weak lower-stratospheric how changes in the stratospheric state would change regime vortex states, while the probability of Scandinavian blocking occurrence, or whether improved stratospheric forecasts was invariant. Beerli and Grams (2019) related the strato- would yield better regime predictions. Addressing these points spheric modulation of Atlantic weather regimes to whether or is therefore a goal of the present study. not the regime projected strongly onto the NAO pattern. To diagnose the downward inﬂuence of the stratosphere on They emphasized that regimes that do not project strongly the troposphere, and changes in tropospheric forecast skill onto the NAO provide a route for a wider variety of weather arising from a correctly predicted stratosphere, model experi- patterns following anomalous stratospheric vortex states. ments in which the stratospheric state is artiﬁcially nudged or Subsequently, Maycock et al. (2020) analyzed the North relaxed to a different state (such as that from reanalysis) have Atlantic response to SSWs from the perspective of modu- been used. Most studies have focused on the seasonal-scale lation of the three eddy-driven jet regimes, ﬁnding an increase effects (Douville 2009; Hitchcock and Simpson 2014; Jung in the occurrence and persistence of the southernmost regime et al. 2010a,b). However, Kautz et al. (2020) used relaxation (corresponding to the negative NAO). Domeisen et al. (2020a) experiments on S2S time scales to quantify the role of the assessed the varying degrees of stratosphere–troposphere cou- February 2018 SSW in the predictability and onset of the sub- pling following major SSWs (e.g., Karpechko et al. 2017; White sequent Eurasian cold wave. They found an increased proba- et al. 2019) by considering the regimes present during SSW bility of surface cold extremes in forecasts with a nudged onset and in the weeks afterward, suggesting that the anteced- stratosphere, but that the evolution of the lower-stratospheric ent state of the troposphere may play an important role in NAM following the SSW}rather than simply the occurrence determining subsequent downward coupling. of the SSW}was important for more accurate tropospheric In recent years, the inﬂuence of the stratosphere on North forecasts. The importance of persistent lower stratospheric American climate variability has received increased attention, anomalies in eliciting a tropospheric response is consistent likely owing to the extreme cold-air outbreaks during winter with climate model studies (Maycock and Hitchcock 2015; 2013/14 that accompanied disruption to the polar vortex (Yu Runde et al. 2016) and the polar-night jet oscillation events of and Zhang 2015; Waugh et al. 2017). However, relatively less Hitchcock et al. (2013). attention has been given to explicitly viewing the impact of Although SSWs and their strong vortex counterpart are typi- the stratosphere on North American weather from a tropo- cally harbingers of persistent anomalous lower-stratospheric spheric regimes perspective. As North America is inﬂuenced NAM states (Baldwin and Dunkerton 2001), they do not by weather from both the Atlantic and Paciﬁc to different de- necessarily propagate into the lowermost stratosphere, and grees across the continent, a challenge with deﬁning North anomalous lower-stratospheric NAM states can occur without American regimes is the choice of domain. Some studies (e.g., a typical midstratospheric precursor. Hence, analysis of the Amini and Straus 2019; Fabiano et al. 2021) focus on up- effect of the stratosphere on the troposphere need not only stream variability in the Paciﬁc–North American (PNA) sec- focus on such extreme midstratospheric circulation events. tor (akin to the Atlantic regimes with respect to Europe), Further, the NAM in the lower stratosphere during midwinter 15 SEPTEMBER 2022 LE E E T A L . 5917 possesses a very long time scale (over 4 weeks; Baldwin et al. the daily climatology over this period. (Any trends in Z500 are 2003), key for the S2S prediction scale. In this study, we focus on found to have little impact on the regimes, so detrending is not subseasonal variability in the strength of the lower-stratospheric performed.) Then, data are weighted by the square root of co- polar vortex, diagnosed through the zonal-mean zonal wind sine latitude, and EOF analysis is performed, retaining the lead- at 100 hPa and 60 N (U100). We do not explicitly consider ing 12 EOFs that explain close to 80% of the variance; k-means SSWs or strong vortex events. clustering is then performed (Pedregosa et al. 2011) in the non- The overall goal of this study is to understand how changes standardized 12-dimensional principal component (PC) space, or uncertainty in the subseasonal lower stratospheric vortex with k set to 4. In addition to reducing the dimensionality of state can inﬂuence changes or uncertainty in predictions of the clustering problem and ﬁltering smaller-scale variability, North American weather regimes. We do this ﬁrst by a statis- performing the clustering in PC space produces a coordinate tical analysis of the regimes and their underlying EOFs in system that enables interpretation of the regimes in terms of reanalysis, and then through analyzing a set of model experi- their comprising EOFs, linking two widely used prediction ments in which the stratosphere is nudged toward reanalysis. A frameworks. After generating the clusters, each day is then as- greater understanding of the relationship between stratospheric signed to one of the four regimes by the minimum Euclidean variability and regimes will help in both the real-world under- distance to the cluster centroids in PC space. standing and interpretation of regime forecast uncertainty, and For regime assignment in the hindcasts, the model Z500 in subsequent studies of regime dynamics and predictability. It climate is ﬁrst subtracted, to account for systematic biases. would also be a useful tool to examine how model biases affect The model climate is computed for each initialization date the representation of stratosphere–troposphere coupling. and lead time over the 20-year hindcast period. Then, the The paper is organized as follows. Section 2 introduces the daily data are projected onto the 12 EOFs, and each day is data, methods, and model experiments. Section 3 deﬁnes the assigned to a regime based on these pseudo-PC loadings. As regimes and their underlying EOFs, and the relationship be- an additional forecast diagnostic in the model experiments, tween these EOFs and the lower-stratospheric polar vortex weekly mean regimes are produced by ﬁrst averaging the PCs strength. Section 4 develops a theory of how the stratosphere over a 7-day period and then assigning to a regime; these are may inﬂuence regime behavior. Section 5 presents the results found to be largely consistent with the regime occupying the of a modeling study used to test the theory. A summary and majority of days within each week (not shown). conclusion of our work follows in section 6, including implica- c. Regime bust criteria tions for S2S prediction. We select subseasonal regime “busts” from the ECMWF 2. Data and methods hindcasts where there is strong ensemble support ($7members, or approximately two-thirds) for one speciﬁc incorrect regime to a. Hindcasts and reanalysis be dominant (i.e., present on at least 8 days) during days 14–27 For historical analysis and veriﬁcation, we use the (weeks 3–4). These criteria are designed to pick out cases that European Centre for Medium-Range Weather Forecasts suggest a strong, but incorrect, subseasonal signal constraining (ECMWF) ERA-Interim reanalysis (Dee et al. 2011). Hind- the model analogous to a “precise but inaccurate” forecast. As casts are taken from version CY43R3 of the ECMWF such, the model conﬁdence may be erroneously interpreted extended-range prediction system (used to produce opera- as enhanced predictability and accuracy, with potentially tional forecasts from July 2017 to June 2018) as part of the large real-world impacts from subsequent decision-making. We S2S database. The hindcasts consist of an 11-member en- choose only hindcasts initialized during December–February, semble (1 unperturbed member and 10 perturbed members) as the seasonal cycle may affect week-3–4 forecasts initialized initialized from ERA-Interim twice per week. The model during March. These criteria yield 31 initialization dates. A has a resolution of Tco639 up to day 15 and Tco319 after further stipulation is applied such that the initialization dates day 15, and 91 vertical levels. All data are sampled once must be separated by at least 21 days to avoid analyzing multi- per day at 0000 UTC, and regridded to 2.58 latitude– ple instances of the same event; in these cases, the earliest ini- longitude resolution for computational efﬁciency and since tialization date is selected. This step ﬁlters the number of we are only considering large-scale ﬁelds. cases to 20 (i.e., on average 1 per winter), which are listed in Table 1. Except for forecasts of an Arctic high verifying as an b. Regime definitions Alaskan ridge, all forecast–veriﬁcation combinations are in- The deﬁnition of North American weather regimes follows cluded at least once (not by design). that of Lee et al. (2019b), extended by 1 year. We take 500-hPa No stratospheric error criteria are included in order to as- geopotential heights (Z500) in the region 1808–308W, 208–808N sess both to what extent poor subseasonal regime forecasts in all December–March days in the period 1 January 1979– are associated with stratospheric errors and the effect of 31 December 2018 in ERA-Interim (4840 days) and subtract stratospheric relaxation even in cases with a relatively well- forecast stratosphere. We ﬁnd that the majority of bust cases 1 feature ensemble-mean U100 error magnitudes $ 3m s Tco 5 cubic octahedral spectral truncation. (14 of the 20 initialization dates, including 8 week-3 and Details of the prediction system can be found on the ECMWF website https://confluence.ecmwf.int/display/S2S/ECMWF+Model. 12 week-4 forecasts), approximately the mean absolute error 5918 J O U R N A L O F C LI MATE VOLUME 35 TABLE 1. North American regime busts in ECMWF CY43R3 hindcasts (HC) from December 1997 to February 2017. The week-3–4 dominant (W3–4 dom.) regime is that which is predicted by $7 ensemble members (64%) to be present on $8 days during days 14–27 inclusive, veriﬁed against the ERA-Interim regime that is present for $8 days during the same time period. Week-3 and week-4 regimes are theregimes of theweeklymean ﬁeld with the largest ensemble support; «U is the ensemble-mean error in the 100-hPa 608N zonal-mean zonal winds averaged over each week. The data are grouped by the dominant regime prediction and then sorted by the week-4 «U. 21 21 Initialization W3–4 dom. percent (ERA) W3 HC (ERA) W3 «U (m s ) W4 HC (ERA) W4 «U (m s ) Arctic high 21 Dec 2005 64 (PT) ArH (PT) 20.5 ArH (PT) 4.2 1 Feb 2009 64 (ArL) ArH (ArL) 2.5 ArH (ArL) 3.2 8 Feb 2010 73 (PT) ArH (ArH) 0.3 ArH (PT) 24.8 29 Jan 1998 64 (PT) PT (PT) 28.5 ArH (PT) 26.7 Arctic low 29 Jan 2001 73 (AkR) ArL (ArL) 6.5 ArL (AkR) 8.5 28 Dec 2016 82 (AkR) ArL (AkR) 2.7 ArL (AkR) 3.0 8 Feb 2006 64 (ArH) ArL (ArH) 4.8 ArL (ArH) 2.3 22 Jan 1999 64 (PT) ArL (PT) 21.5 ArL (PT) 1.0 19 Feb 2011 64 (PT) ArL (PT) 20.3 ArL (PT) 20.6 4 Dec 2011 64 (PT) ArL (ArL) 0.1 ArL (PT) 21.3 Alaskan ridge 11 Dec 2001 64 (ArH) AkR (ArH) 2.3 AkR (PT) 3.1 15 Feb 2017 64 (ArL) AkR (ArL) 20.6 AkR (AkR) 2.6 4 Dec 2003 73 (PT) ArH (PT) 0.4 AkR (ArL) 23.0 Paciﬁc trough 12 Feb 1999 64 (ArH) PT (PT) 3.3 PT (ArH) 14.0 8 Jan 2010 64 (ArH) PT (ArH) 4.1 ArH (ArH) 8.7 25 Dec 2015 73 (ArH) PT (ArH) 7.7 PT (ArH) 7.7 7 Dec 2000 64 (ArH) PT (ArH) 7.3 PT (ArH) 2.8 18 Jan 2016 73 (AkR) PT (PT) 0.3 PT (AkR) 0.4 21 Dec 2014 73 (AkR) AkR (AkR) 21.7 PT (AkR) 22.1 25 Dec 2006 82 (ArL) PT (ArL) 25.8 PT (ArL) 28.7 (MAE) of the December–February week-3–4 hindcasts (see representation of initial condition uncertainty, so some differ- Fig. S1 in the online supplemental material). This suggests ences between these model runs and the equivalent hindcasts that regime busts and large lower-stratospheric vortex errors are to be expected. As we are primarily considering forecasts often co-occur. on time scales of several weeks, the initial condition un- certainty is considered less important, and the stochastic d. OpenIFS model schemes generate spread comparable to the hindcasts in the ﬁelds analyzed in this study. For model experiments, we use OpenIFS version 43r3v1} For each initialization date, two sets of ensembles are a research version of the ECMWF IFS (Integrated Forecast produced: a control (CTR) run in which the forecast freely System) model CY43R3, but without data assimilation. The evolves (comparable with the equivalent hindcast, notwith- model is initialized from ERA-Interim and run on a linear standing the model differences), and a relaxed (RLX) run in Gaussian grid with T255 resolution, 60 vertical levels (i.e., which the Arctic stratosphere is nudged toward ERA-Interim the resolution of ERA-Interim), and a time step of 45 min. using the IFS relaxation scheme (e.g., Jung et al. 2010a). The Output data are bilinearly interpolated onto a 2.58 latitude– relaxation scheme operates by applying a nonphysical ten- longitude grid. Each ensemble consists of an unperturbed dency to the model equations of the form member and 20 perturbed members, in which spread is gener- ated by the stochastically perturbed parameterization tenden- k(X 2 X), (1) obs cies (SPPT) and stochastic kinetic energy backscatter (SKEB) schemes (Leutbecher et al. 2017). The ensemble size is chosen where X is a model prognostic variable, X is the “observed” obs as a balance between the potential gain from additional value from ERA-Interim, and k [unit: (time step) ]is the relax- members compared with the 11-member hindcasts and com- ation coefﬁcient controlling the strength of the forcing [following, putational expense. The OpenIFS runs differ from the ope- e.g., Jeuken et al. (1996) and Magnusson (2017)]. The term X obs rational model in both resolution and in that there is no at each model time step is generated by linear interpolation be- tween 6-hourly reanalysis ﬁles. A relaxation time scale of 12 h is used in this study, corresponding to k 5 0.0625 per time step Specific details of the model can be found at https://confluence. ecmwf.int/display/OIFS/Release+notes+for+OpenIFS+43r3v1. given the 45-min model time step, which can be interpreted 15 SEPTEMBER 2022 LE E E T A L . 5919 FIG. 1. Vertical and latitudinal proﬁle of the relaxation coefﬁcient scaling (i.e., a value of 1 de- notes full relaxation, here with a time scale of 12 h), for both pressure (left-hand ordinate) and model level number (right-hand ordinate and horizontal grid lines; labeled to level 31 for clarity). The red dashed and dotted lines denote the bounds, in latitude and height respectively, where the coefﬁcient is 0.5. The hatched area denotes the region where the scaling is at least 0.99. as nudging the model state at each time step by 6.25% of e. Significance testing the departure from the reanalysis. Vorticity, divergence, Throughout the paper, statistical signiﬁcance is assessed at and temperature are relaxed in model gridpoint space with the 95% conﬁdence level by bootstrap resampling (e.g., Wilks an exponential taper at both the latitude and model-level 2019). Random samples (with replacement) are taken from boundaries. the population and the quantity under analysis (e.g., a regres- A proﬁle of the relaxation domain is shown in Fig. 1. The sion coefﬁcient) is calculated and stored. This process is domain boundaries are chosen to both maximize constraint of repeated 10 000 times, and then a conﬁdence interval is con- the polar lower stratosphere while allowing for a sufﬁciently structed from the appropriate percentiles of this distribution smooth taper to minimize negative numerical effects, and to (2.5th–97.5th percentiles for two-sided 95% conﬁdence). remain largely poleward and upward of the subtropical jet to reduce directly constraining the tropical upper-tropospheric 3. Regimes and EOFs waveguide. The choice of domain is also limited by the verti- cal level spacing of the model in the upper troposphere and The centroids of the four regimes (expressed as the Z500 lower stratosphere. We employ a weaker stratospheric nudg- ﬁeld reconstructed from the sum of the centroid loading in ing than some previous studies (e.g., Jung et al. 2010a; Kautz the leading 12 EOFs), along with the percent of days assigned et al. 2020), but note that the relaxation in our study extends to each (the occupation frequency), are shown in Figs. 2a–d. further into the lower stratosphere. Analysis of the output In terms of both spatial patterns and the ranking of occupa- ﬁelds show this relaxation strength is enough to constrain the tion frequency, these match the regimes of Lee et al. (2019b) model. Time series of the U100 forecasts from the CTR and and so we follow their naming convention [after Straus et al. RLX experiments and the corresponding veriﬁcation from (2007)]: Arctic high (ArH), Arctic low (ArL), Alaskan ridge ERA-Interim are shown in Fig. S2. (AkR), and Paciﬁc trough (PT). The coordinates of the re- As the random seed used in the stochastic schemes is ﬁxed gime centroids in the leading 12 PCs are shown in Fig. 2e. for each ensemble member, the equivalent ensemble mem- Only the leading three PCs have large contributions to the bers in the CTR and RLX experiments differ only by the centroids; performing the same clustering analysis but retain- stratospheric nudging. In analyzing the OpenIFS runs, we as- ing only the leading three PCs yields very similar patterns, sume the model climatology is equivalent to that of the corre- with only 4% of days assigned to a different regime. There- sponding CY43R3 hindcasts. fore, we now focus our analysis on these leading three EOFs. 5920 J O U R N A L O F C LI MATE VOLUME 35 FIG.2.(a)–(d) Centroids of the four regimes, expressed as 500-hPa geopotential height anomalies with respect to daily 1979–2018 climatol- ogy in ERA-Interim, and the percent of days assigned to each regime in all December–Marchdays in the period1 Jan1979–31 Dec 2018. (e) Coordinates of the regime centroids in raw (nonstandardized) 12-dimensional principal component space. (f)–(h) The leading three EOFs (multiplied by the square root of the eigenvalue) of daily 500-hPa geopotential height anomalies in the domain 1808–308W, 208–808N, and the percent of total variability explained by each EOF. Maps of the EOFs and the percent of the total variance ex- under consideration. The EOFs presented here}with the plained are shown in Figs. 2f–h. In total, these three EOFs ex- most NAM-like pattern in EOF2, while the leading EOF con- plain close to 40% of the daily variance within the domain, tains NAM/NAO and PNA-like characteristics}agrees well and are well separated according to the criterion of North with the upper-tropospheric EOF analysis of Baldwin and et al. (1982). The sign of the EOFs is here deﬁned such that a Thompson (2009). For all three North American EOFs, the positive loading produces an anomalous trough in the north- e-folding time scales of the PC time series are 5–7 days, which east Paciﬁc. EOF1 is similar to the PNA (Wallace and Gutzler is similar to the median number of consecutive days with the 1981) but slightly eastward shifted. It also bears some similar- same regime assignment. However, a quarter of the individual ity to the tropical–Northern Hemisphere (TNH) pattern (Mo blocks of consecutive regime days persist for more than 1 week and Livezey 1986; Liang et al. 2017). Furthermore, there is a (including one instance of 39 days of ArL up to and including meridional dipole in the North Atlantic in the eastern edge of 22 February 1990), motivating their utility for extended-range the domain, reminiscent of NAO-like variability. EOF2 has a prediction. meridional dipole in Z500 anomalies, and thus some similarity To understand the relationship between regime occurrence to the surface-based NAM/AO, but with a center of action and the lower-stratospheric vortex presented in Lee et al. over Alaska that is not characteristic of the surface NAM (2019b), we examine the relationship between U100 and the (e.g., Thompson and Wallace 1998). EOF3 is characterized by leading EOFs which deﬁne the clusters. We perform linear re- gression between each PC time series and the contemporane- a wavenumber-2 pattern across the domain. Comparison of these regional EOFs with the leading three ous U100 to see how changes in U100 may modulate the EOFs for the Northern Hemisphere poleward of 208N location of a point within the 3D PC space and thus its regime (Figs. S3–S5) shows a high degree of similarity in both the cor- attribution. The instantaneous relationship is used since we relation of the PC time series (Pearson’s correlation r $ 0.77; are considering the lower stratosphere as an upper boundary p , 0.05) and spatially (area-weighted pattern correlation condition to the troposphere, with both a much longer mem- $ 0.87 over the North American domain). We can therefore ory (e.g., Baldwin et al. 2003) and greater predictability (Son be conﬁdent that the leading three EOFs used in the cluster- et al. 2020); lagged relationships (not shown) reveal these ing are regional manifestations of hemispheric variability, and coefﬁcients are either effectively maximized at lag 0 or, con- that hemispheric variability is dominant in the smaller domain sidering uncertainty, largely invariant for 67 days (within the 15 SEPTEMBER 2022 LE E E T A L . 5921 PC e-folding time scale). Some of this relationship may relate to the vertical extension of a primarily tropospheric zonal wind signature associated with these EOFs into the lower stratosphere. However, on subseasonal scales (well beyond tropospheric decorrelation time scales) this remains the com- ponent of the structure that is potentially predictable. The regression coefﬁcients are shown in Fig. 3. Although the coefﬁcients for all three EOFs are signiﬁcantly different from zero, the linear relationship is 3–5 times stronger for EOF2. Similarly, the Pearson’s correlations between U100 and PCs 1 and 3 are small (r 520.13 and 0.10, respectively), but moderate for PC2 (r 5 0.42). Thus, the effect of the stratosphere in this 3D EOF space is mostly contained within EOF2, which is consistent with its annular-like spatial pattern and the height-dependent NAM results of Baldwin and Thompson (2009). The sign of the regression coefﬁcients is such that a decrease in U100 is associated with an increase in Z500 in the vicinity of Greenland/the northern node of the NAO, in agreement with the canonical response of the tropo- sphere to a weakened stratospheric vortex. 4. Theory of regime transitions and the stratosphere FIG. 3. Linear regression coefﬁcients between the 100-hPa 60 N zonal-mean zonal wind and the raw PC time series of the leading In this section, we develop a theory of which regime transi- three EOFs, in all December–March days in ERA-Interim tions may be possible solely due to a stratospheric perturbation 1979–2018. Error bars indicate 95% conﬁdence intervals obtained by jointly considering the linear relationship between U100 by bootstrapping with replacement (see section 2e for details). and the three PCs, and the location of the regimes within the space spanned by the three PCs. The theory can be interpreted The angle u between b and g follows as as an idealized framework where all else is instantaneously equal and only the stratosphere is changed, retaining potential b · g u(b, g) 5 arccos , (4) predictability arising from other tropospheric processes. bg Using the regression coefﬁcients between U100 and the PC time series, we deﬁne the stratospheric perturbation vector b. 2 2 2 where x 5 x + x + x denotes the Euclidean norm of a 1 2 3 This vector represents the movement within the 3D PC space 3D vector x. arising from a perturbation to U100, DU, that is explained by We use this framework to model which regime transitions the linear regression coefﬁcients: are possible solely with stratospheric forcing by considering ⎛ ⎞ whether the vectors b (either positive or negative) and g ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ point in a similar direction, known as “cosine similarity” (e.g., ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b 5DU⎜ 91 ⎟ : (2) ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ Han et al. 2012). If u $ 908 (cosu # 0), then no component of ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 20 the regime transition or movement within the 3D PC space can be explained by the linear relationship between the PCs Note that b is not a function of the position within PC space and U100, since the contribution of b would be0(in thecaseof and is thus constant for a given DU. While the truncation to a maximally dissimilar vectors, u 5 908)oroppose g (cosu , 0). 3D PC space was earlier motivated by the coordinates of the A smaller angle indicates the transition is more likely since the regime centroids, the linear relationship between the leading projection of b in the direction of g is larger (as cos u is larger), three EOFs and U100 also accounts for nearly all of the linear thus requiring a smaller DU. We focus on angles, rather than relationship with Z500 (Fig. S6). explicit distances, since the distances between regimes for any The transition vector g between two points (e.g., two clus- point are dependent on the initial location. ter centroids) within this space is then deﬁned as the respec- Figure 4 presents a 3D depiction (in the space spanned by the tive distances between the coordinates in the three PCs: leading three EOFs) of b (both positive and negative; i.e., for a ⎛ ⎞ strengthening or weakening stratospheric vortex) applied to DPC ⎜ ⎟ ⎜ ⎟ each regime centroid and the transition vector g between the ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ g 5 ⎜⎜ DPC ⎟⎟ , (3) ⎜ ⎟ ⎜ ⎟ centroids. The regime centroids form a tetrahedron in this space. ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Some of the transition vectors lie closer to b than others owing DPC to their relative locations within this space. For example, the where DPC 5 PC (B) 2 PC (A) for the transition from positive b vector and the transition vector from the ArH to PT k k k point A to point B. Hence, inverse transitions have an equal centroids are close, while the transition vectors from the AkR but opposite transition vector: g(A, B) 5 g(B, A). centroid are almost perpendicular to either sign of b. 5922 J O U R N A L O F C LI MATE VOLUME 35 and b are all relatively large for AkR (Fig. 5c), as previously suggested by the 3D depiction in Fig. 4.For b , 0, only a transition to ArH has an angle , 908. Transitions to ArL and PT are possible with b . 0, but the angles are relatively large and thus more unlikely. We next extend our analysis beyond points initiating at the centroids and incorporate the effect of spread around the PC space spanned by each regime. First, we consider all the assigned regime days in ERA-Interim. The leading three PCs are then per- turbed by b in the range 230 #DU # 30 m s , and subse- quently reassigned to a regime by minimum Euclidean distance. The maximum magnitude of DU is chosen here to be close to the maximum observed variability in U100; the largest U100 errors in individual CY43R3 ensemble members are close to 620 m s . Note that in reality, the tropospheric response may be larger for a smaller DU as a consequence of the linear framework. Figure 6 depicts the conditional probability, for each initial regime, of either remaining in the same regime or transitioning to each of the other regimes for each DU. Only those transition pathways with u , 908 occur, and the relative likelihood mani- FIG. 4. Visualization of the regimes in the space occupied by the fests the degree of similarity (i.e., the angle) between b and g. leading three EOFs. Colored markers indicate the regime cent- There are no transitions away from ArH for DU , 0(Fig. 6a) roids. Colored arrows represent the transition vectors from each or away from ArL for DU . 0(Fig. 6c). For DU , 0, the domi- centroid to the other centroids, scaled to 0.253. The black arrows nant transition forall regimesistoArH. For DU . 0, transitions show the stratospheric perturbation vector, scaled to a 610 m s from ArH to PT dominate (Fig. 6a) while transitions to ArL perturbation (solid positive; dashed negative), which is the same at all points. dominate for AkR and PT (Figs. 6b,d). Transitioning into AkR from any other regime is unlikely even for large |DU|, while tran- sitioning out of AkR is the least likely for any of the regimes The angles between the centroid g vectors and b are quanti- where a transition pathway exists (despite its unique approxi- ﬁed in the protractor-like polar plots in Fig. 5. The angles are ex- mately equal sensitivity for either sign of DU). Although not ex- pressed such that both positive and negative b are aligned with plicitly shown, there is also evidence of multiple transitions 08 (thus, the angle between each g and b , 0is a reﬂection of occurring as |DU| increases. For example, the probability of tran- that to b . 0 about 908). For a point starting at the ArH centroid sitioning into AkR from each of the other regimes reaches a (Fig. 5a), there is substantial cosine similarity between b . 0and peak for |DU| between 10 and 20 m s before declining. transition vectors to all otherregimes(forall three, u , 608). The As a general diagnostic of the sensitivity of each initial similarity is strongest for the transition vectors to PT and ArL, regime state to a lower-stratospheric perturbation, we can which have approximately equal cosine similarity. The angles be- consider the probability of transitioning out of the regime for tween b , 0 and all three transition vectors are .908;thus, the DU 5610 m s (approximately equal to the maximum theory does not allow a transition away from ArH given DU , 0. week-3–4 ensemble-mean U100 error magnitude in CY43R3 Overall, ArH has the largest number of transition vectors with hindcasts). For DU 5 10 m s , 58% of ArH days transition small angles/high cosine similarity. Equally, the minimum angle into a new regime, while only 17% of AkR days and 6% of between either sign of b and any g vector is between b , 0and PT days do so. For DU5210 m s , the sensitivity of PT and transitions to ArH (Figs. 5b–d). This is consistent with the ArL is approximately equal, with 39% of PT and 38% of ArL observed probability of transitions into, and the persistence of, days transitioning into a new regime. Only 15% of AkR days ArH/NAO-, which is the most sensitive of both the North transition into a new regime. American and North Atlantic regimes to the strength of U100 Overall, the results presented in Figs. 4–6 are in agreement (Charlton-Perez et al. 2018; Lee et al. 2019b). with the observed differences in regime occurrence in strong and For the PT regime (Fig. 5b), there is a small angle between weak stratospheric vortex states in Lee et al. (2019b).The theory the negative b vector and the transition vector to ArH (i.e., also gives results consistent with the relationship between the re- equal and opposite to the positive b and the transition from gimes (particularly ArH and ArL) and the concurrent NAO in- ArH to PT). While transitions are possible to both AkR with dex (Fig. S7), given the strong modulation of the NAO by the b , 0, and to ArL with b . 0, the angles are close to 908, sug- stratosphere. Further, the proposed framework yields insight gesting that these are unlikely. Considering the ArL regime into speciﬁc regime transitions under different vortex states that (Fig. 5c), transitions to all three other regimes are possible are not limited by the observational sample size. In summary: with b , 0. The smallest angle is to the ArH transition vector, while the angles to the PT and AkR transitions are large. No DU , 0 moves the majority of points within PC space to- regime transitions from ArL are possible in this framework ward only ArH, consistent with this regime being the only with DU . 0. Last, the angles between the transition vectors one more likely under weak vortex conditions. 15 SEPTEMBER 2022 LE E E T A L . 5923 FIG. 5. Polar plots showing angles between the stratospheric perturbation vector (solid positive; dashed negative) and the centroid transition vector for each of the four regimes in 3D EOF space, as visualized in Fig. 4. • • DU . 0 does little to changing the regime assignment for What is the effect of stratospheric relaxation on regime days initially assigned to ArL or PT, while these are fa- forecast accuracy in these cases? vored transitions for initial ArH and AkR states. This is Regardless of the forecast accuracy, is the change in the forecast consistent with the theory in section 4? consistent with ArL and PT being more likely under strong vortex conditions. Very large DU is required to shift toward and away from a. Regime predictions AkR, with a similar proportion of transitions resulting from A comparison between the weekly mean regimes in the CTR both positive and negative perturbations. This behavior is and RLX ensembles, for weeks 3 and 4, is shown in Fig. 7.The consistent with the observed statistically equal occurrence improvement in the total number of ensemble members with a of this regime in strong and weak vortex states. correctly assigned weekly mean regime is modest: 13% in week 3 and 15% in week 4. Therefore (recalling that these cases were These conclusions are highly idealized, requiring both a per- selected as particularly poor forecasts), the overall fraction of fectly linear response and the sole (or dominant) change being to correctly assigned regimes remains low in the RLX experiment: U100. It is also possible that b may be sensitive to the initial posi- 40% in week 3 and 25% in week 4. Any improvement is also tion within PC space. However, the corroboration with observa- case dependent. The greatest improvement in week 3 is in the tions suggests the potential use of this framework in interpreting 11 December 2001 case (7 more members correctly assigned to the regime response to changes and uncertainty in the strato- ArH), and in the 29 January 1998 case (5 more members cor- sphere on subseasonal time scales. The analysis in the next sec- rectly assigned to PT) in week 4. The latter was a case with a tion considers whether imposing stratospheric relaxation yields a very large U100 error (cf. Table 1). In several cases, there is a tropospheric response consistent with this simple but novel theory. decrease in the number of correctly assigned ensemble members. Thus, constraining the stratospheric state is not enough to ﬁx these regime bust cases}which may be unsurprising given that 5. Model experiments only a selection of these cases have very large stratospheric er- In analyzing the results of the relaxation experiments, we rors, while all have largely inaccurate regime predictions. This seek to answer the following two questions: result indicates that the stratospheric state should not be viewed 5924 J O U R N A L O F C LI MATE VOLUME 35 FIG.6. (a)–(d) Given each initial regime, the conditional probability of either remaining in the same regime or tran- sitioning to each of the other regimes, when all days assigned to each regime in ERA-Interim are perturbed by the stratospheric perturbation vector in the range 230#DU # 30 m s . as exerting simple control on the subseasonal tropospheric ﬂow study; we instead focus on the general results across this set of over North America. forecasts. Figure 7 also shows that there are changes to the number of b. Error reduction in PC space ensemble members assigned to the incorrect regimes, regard- less of whether there is a change to the number assigned to Despite the small and case-dependent regime improve- the correct regime. On a member-by-member basis, 34% and ment, for almost all cases the mean Euclidean distance error 57% of the total ensemble members in weeks 3 and 4 respec- of the ensemble in 3D PC space is reduced (Fig. 8a). This di- tively are assigned to a different regime in the RLX experi- agnostic is useful because it incorporates changes to forecasts ments. Thus, by week 4, the stratospheric nudging has shifted that maintain the same regime attribution and is proportional the majority of ensemble members into a new regime} to the root-mean square error (RMSE) of the Z500 ﬁeld re- suggesting signiﬁcant movement within the PC space in which constructed from the leading three EOFs (see the online the regimes are assigned. For example, in week 4 of the 11 supplemental material; note that because non-normalized December 2001 case, there is no increase in the number of PCs are used, the total error on subseasonal time scales is members correctly assigned to PT, but there is a gain of eight dominated by the EOFs with the largest eigenvalues). Hence, ensemble members assigned to AkR (with ArH and ArL los- in the space in which regimes are assigned, the RLX forecasts ing four members each). While a full case-by-case analysis are almost entirely closer to the veriﬁcation. The improve- may yield further speciﬁc insight, it is beyond the scope of this ment is maximized in week 3 (median 14%), with only two 15 SEPTEMBER 2022 LE E E T A L . 5925 FIG. 7. For (a) week 3 and (b) week 4, values denote the number of ensemble members assigned to each regime in the RLX experiment, with the number in parentheses indicating the difference from CTR. Bold font indicates the ERA-Interim regime. Color shading indicates the difference in ensemble-mean U100 between the experiments (RLX-CTR). Grouping is as in Table 1. cases showing an increase in error (21 December 2005 and a U100 change of only 1 m s while the largest error increase 8 February 2010, both of which had negligible week-3 U100 occurs with a U100 change of 2.6 m s (8 February 2010). errors in the CTR run). The median improvement in week 4 The large relative error reduction for small DU suggests a is 12%, but with much greater spread than week 3. There was potential role of zonally asymmetric corrections or other a 30% improvement in a single case (21 December 2014), while changes to the vortex that do not project strongly onto U100 four cases show no change or increased error (7 December 2000, (and thus fall outside the framework proposed here). How- 11 December 2001, 8 February 2010, and 15 February 2017). ever, across this set of 20 cases, for DU exceeding 3 m s , Also shown in Fig. 8a is the mean change in Euclidean dis- there is a systematic error reduction. We revisit this apparent tance error obtained by perturbing the PCs of the CTR en- threshold in the analysis below. semble by b multiplied by DU between the CTR and RLX experiments. This shows that a simple statistical nudge of the c. Movement within PC space PCs using the known linear relationships also yields an error reduction of on average ∼50% of that obtained by running We now investigate whether the movement of the forecasts the full dynamical relaxation experiment. Thus, a substantial within 3D PC space is consistent with what might be expected component of the dynamical effect of imposing a different from the theory established in section 4. For this analysis, we stratospheric state on these EOFs can be explained by the ob- analyze three vectors and three different angles within PC served linear relationship between the PCs and U100. space. Figure 9 shows a schematic of this approach. The To understand whether larger stratospheric forcing yields vectors are deﬁned as follows: larger error reduction, Fig. 8b shows the case-by-case change • CTR-ERA: the vector between the CTR forecast and the in ensemble-mean Euclidean distance error against the mag- nitude of the U100 change between the CTR and RLX ex- veriﬁcation from ERA-Interim (i.e., the error in the CTR periments for weeks 3 and 4. There is no immediately clear forecast). relationship, with the greatest error reduction occurring with CTR-RLX: the vector between the CTR and RLX forecasts. 5926 J O U R N A L O F C LI MATE VOLUME 35 FIG. 8. (a) Boxplots of the ratio between the ensemble-mean Euclidean distance error in 3D PC space between the weekly averaged RLX and CTR ensembles for the 20 cases. Red lines denote the median, and notches show 95% conﬁdence intervals obtained by 10 000 bootstrap resamples (with replacement). Black triangles denote the mean. Blue circles represent the average ratio obtained by statistically perturbing the CTR PCs by the stratospheric pertur- bation vector multiplied by the change in U100 between the CTR and RLX ensembles. Whiskers extend to 1.5 times the interquartile range or extremes (whichever is smaller); outliers shown as open circles. (b) Scatterplot of the week- 3 (green squares) and week-4 (maroon circles) error ratio against the magnitude of the ensemble-mean change in U100 between CTR and RLX. CTR-STAT: the vector between the CTR forecast and the Figure 10b assesses whether the stratospheric perturbation CTR forecast statistically perturbed by b multiplied by DU vector outlined in section 4 is a good representation of the ef- between CTR and RLX ensembles (STAT). fect of a dynamically applied stratospheric perturbation. For |DU| , ∼3m s , the points are scattered across almost the Then, the size of the three angles can be used to answer the full range of angles, indicating no clear relationship between following questions: the theory and the movement of these forecasts in PC space. However, although the sample is smaller, for |DU| . ∼3m s , • u 5 u(CTR-ERA, CTR-RLX): Does stratospheric relaxa- the angles are systematically much smaller than 908}especially tion move the CTR forecast toward the veriﬁcation? for week-4 forecasts, which feature larger DU.Hence, we con- • u 5 u(CTR-RLX, CTR-STAT): Does stratospheric relaxa- clude that on average, these forecasts moved in PC space in the tion move the CTR forecast in the direction expected from b? general direction expected from the theory. • u 5 u(CTR-ERA, CTR-STAT): Does statistical nudging Finally, Fig. 10c assesses whether the simple statistical per- by b move the CTR forecast toward the veriﬁcation? turbation moves the CTR forecast toward the veriﬁcation A scatter of the week-3 and week-4 angles versus the mag- without running a full dynamical experiment (cf. Fig. 10a). As nitude of DU between the CTR and RLX experiments is in Fig. 10b, but unlike in Fig. 10a, there is no clear evidence of shown in Fig. 10. To focus on the overall shift of the ensemble vector similarity for small DU, but there is evidence of a sys- in the relaxed experiments, and since b is deﬁned from linear tematic shift for DU exceeding ∼3m s in magnitude. As a best-ﬁt regression coefﬁcients, we perform this analysis on the result, for larger U100 errors the tropospheric forecast can be perturbations to the PCs and U100 averaged across the en- partially corrected statistically (as indicated by Fig. 8a), but semble. Nevertheless, similar results are obtained when con- there is evidently additional gain from a dynamically cor- sidering the results across all individual ensemble members rected stratosphere even for small DU. (not shown). Figure 10a shows that in the majority of cases The 3 m s threshold is most apparent for angles involving and in both weeks 3 and 4, the stratospheric relaxation gener- b, although there is some suggestion for the behavior of the ally moved the predictions toward the veriﬁcation. Only two RLX experiment (in terms of both angles and Euclidean dis- cases in week 3 and six cases in week 4 do not exhibit any sim- tance error). It is not clear why 3 m s should be a threshold; ilarity (i.e., u . 90 ). These results are consistent with the re- it may be related to the signal magnitude required to emerge duction in Euclidean distance error and its relationship with above the typical ensemble-mean variability, and thus may the magnitude of DU (Fig. 8). be sensitive to ensemble size. Across the CY43R3 hindcasts, 15 SEPTEMBER 2022 LE E E T A L . 5927 FIG. 9. Schematic of the angle-based approach (here in a 2D PC space). There are three vectors: the vector from the control fore- cast to the ERA-I veriﬁcation (CTR-ERA; red), the vector from the control forecast to the relaxed forecast (CTR-RLX; purple), and the stratospheric perturbation vector to the statistically nudged forecast (CTR-STAT; gray). Here u denotes the angle between CTR-ERA and CTR-RLX, u the angle between CTR-RLX and CTR-STAT, and u the angle between CTR-ERA and CTR-STAT. 3m s is approximately two-thirds of the standard deviation of the ensemble-mean U100 in weeks 3–4(∼4.5 m s ), al- though these are not directly comparable owing to the smaller hindcast ensemble size. As mentioned in section 2,3 m s is also approximately the MAE of the ensemble-mean week-3–4 U100 in the CY43R3 hindcasts, and so errors of this magni- tude are a reasonably frequent occurrence. In week 4 (when DU is generally largest), the magnitude of the correlations between the ensemble-mean change in the PCs and the ensemble-mean DU from CTR to RLX (and thus the individual components of b) are maximized. These corre- lations are largest for EOF2 (r 5 0.60, p , 0.05) and EOF3 (r 5 0.48, p , 0.05) but the correlation is small and insigniﬁ- cant for EOF1 (r 520.19, p 5 0.40; although it is similar to that in ERA-Interim). Furthermore, we can ﬁnd the “effective” FIG. 10. Scatterplots of the magnitude of the ensemble-mean vector in the model by computing the regression coefﬁcients weekly mean U100 change between the CTR and RLX experi- between DU and each DPC across all ensemble members. For ments, vs the angle between (a) CTR-ERA and CTR-RLX (u ), weeks 3–4, these are not signiﬁcantly different from the com- (b) CTR-RLX and CTR-STAT (u ), and (c) CTR-ERA and CTR- ponents of b in ERA-Interim, except slightly for EOF1 in STAT (u ), in 3D-PC space. week 3. As a result, the angles between this effective vector and b are small (268 in the week-3 forecasts and 128 in the week-4 forecasts), conﬁrming that b is a good approximation of the response to an imposed stratospheric change. 5928 J O U R N A L O F C LI MATE VOLUME 35 Nevertheless, across the range of cases studied here, the relaxation generally moved the forecasts toward the ERA- response of EOF1 to stratospheric perturbations is not well Interim veriﬁcation and in the direction of that expected approximated by linear regression. This may be due to non- from the theory, while statistically nudging the CTR ensem- linearity, or that the relationship between the EOF and U100 bles by the corresponding stratospheric perturbation vector is not causal (recalling the similarity between the EOF and also generally moved the forecasts toward the veriﬁcation. patterns related to tropical forcing). Sample size may be an is- For |DU| . ∼3m s , this effect was particularly pronounced. sue, given that the small expected response in EOF1. There Consequently, the model experiments support the proposed may also be limitations in the representation of stratosphere– theory of which regime transitions may be possible solely be- troposphere coupling in the model, such as the overestimation cause of changes to the stratospheric state (Fig. 5). of the NAO response reported by Kolstad et al. (2020) using Overall, our results provide evidence that, all else being equal: a similar but more recent ECMWF forecast model (CY45R1). The average shift of an ensemble of subseasonal North The relatively low vertical resolution employed here, parti- American weather regime forecasts in response to changes cularly in the upper troposphere and lower stratosphere, may in the strength of the lower-stratospheric vortex is broadly also have limited the downward coupling and forecast im- generic and predictable. provement arising from the stratosphere (Kawatani et al. Correcting the stratospheric state leads to an improvement 2019; Domeisen et al. 2020c). in the large-scale subseasonal tropospheric forecast over North America, but it does not necessarily correct the re- 6. Summary and conclusions gime assignment (likely due to other sources of error). • Some tropospheric regime states are more likely to change Understanding and exploiting stratospheric variability is a regime assignment for a given stratospheric perturbation key way in which the accuracy and usefulness of S2S forecasts than others. This arises due to the location of the regimes and the ﬁdelity of stratosphere–troposphere coupling within in PC space relative to the linear tropospheric response to models can be increased. In this study, we investigated how the stratosphere. perturbations to the strength of the lower-stratospheric polar vortex can inﬂuence North American weather regime predic- We therefore propose that this vector-based approach can be tions. Our novel technique involved jointly considering the used to identify, a priori, the regime forecast-veriﬁcation scenar- linear relationship between the vortex strength and the lead- ios in which lower-stratospheric errors are more likely to have ing EOFs that contribute to the regimes (Fig. 3), and the rela- played a substantial role}and thus toward understanding the tive location of the regimes within the EOF space (Fig. 4). We overall contribution to subseasonal North American weather re- used an angle-based approach to quantify which transitions gime forecast accuracy. Further, it is possible that in certain cir- are likely to occur (using cosine similarity) for a given regime cumstances when stratospheric uncertainty is dominant that the and stratospheric perturbation (Fig. 5). These results agree method could be used in real time to qualitatively interpret re- with the observed changes in regime occurrence under differ- gime forecast uncertainty owing to stratospheric uncertainty. ent stratospheric vortex states reported in Lee et al. (2019b) This approach is likely to be most useful 2–3 weeks before and provide an explanation for the regime behavior. How- SSWs or strong vortex events, when abrupt forecast shifts (e.g., ever, both the regime framework and EOFs are deﬁned pri- Lee et al. 2019a) are more likely due to the current predictability marily from a mathematical, rather than physical, standpoint, limit of these phenomena (Domeisen et al. 2020b). It may also and therefore the results of this work largely focus on the be plausible to use the technique on-the-ﬂy to linearly impose al- mathematics of regime attribution. ternate regime “storylines” arising from a different stratospheric We then performed a set of stratospheric relaxation model evolution without running additional dynamical forecasts. experiments, selecting 20 cases from the ECMWF hindcasts Moreover, the dominantly linear and apparently generic re- in which there was strong, coherent ensemble support for an sponse to the lower-stratospheric forcing on these time scales incorrect regime to dominate during weeks 3–4. The majority is somewhat similar to the long-lag response following SSWs in (14) of these cases featured U100 errors approximately equal the model experiments of White et al. (2020). The idea that the to or greater than the MAE in either week 3 or 4 or both, sug- tropospheric ﬂow conﬁguration following an imposed strato- gesting a link to the erroneous tropospheric forecasts. We spheric change depends on the state of the troposphere is not a found that the stratospheric relaxation is not enough to elimi- new idea (e.g., Gerber et al. 2009), but as a result, potential gains nate the regime errors, but the relaxation does lead to shifts in subseasonal regime prediction skill from the stratosphere in the ensemble distribution of the regimes within each fore- may be minimal if the tropospheric forecast otherwise drifts too cast indicating substantial movement within PC space (Fig. 7). far from the truth [also recently suggested by Charlton-Perez The results also showed an overall 10%–20% improvement in et al. (2021)]. This potential limitation is consistent with the re- the accuracy of the forecasts in terms of Euclidean distance gime forecasts remaining largely inaccurate even in cases where error/RMSE, which was most consistent in cases where the large lower-stratospheric errors were corrected, notwithstanding stratospheric error was larger (Fig. 8). the imperfections of the model experiment. Analysis of the transition vectors between the CTR and Employing a stronger stratospheric nudging in the model RLX forecasts in PC space provided insight into the effect of experiments presented in this paper may produce greater im- stratospheric relaxation in the space in which regimes are provement in the regime forecasts. On the other hand, con- assigned. The results (Fig. 10) illustrated that stratospheric straining the prediction too strongly would exceed a 15 SEPTEMBER 2022 LE E E T A L . 5929 realistically achievable level of stratospheric forecast accuracy data used in this study are available at https://doi.org/10.5281/ on these scales. It is also plausible that the nudging may have zenodo.4818044. EOF and k-means clustering analysis were limited potential tropospheric forecast accuracy (when com- performed using the freely available Python packages “eofs” (Dawson 2016) and “scikit-learn” (Pedregosa et al. 2011), pared with a true perfect stratosphere forecast) by inducing respectively. unrealistic wave behavior or generation on the boundary of the nudging domain (Hitchcock et al. 2022). Also, model ex- periments with a greater horizontal and vertical resolution REFERENCES may also yield better results, with evidence supporting a link between increased resolution and better representation of Ambaum, M. H., B. J. Hoskins, and D. 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Publisher: American Meteorological Society
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DOI: 10.1175/jcli-d-21-0413.1
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Abstract

15 SEPTEMBER 2022 LE E E T A L . 5915 How Do Stratospheric Perturbations Inﬂuence North American Weather Regime Predictions? a a b c SIMON H. LEE, ANDREW J. CHARLTON-PEREZ, STEVEN J. WOOLNOUGH, AND JASON C. FURTADO Department of Meteorology, University of Reading, Reading, United Kingdom National Centre for Atmospheric Science, Department of Meteorology, University of Reading, Reading, United Kingdom School of Meteorology, University of Oklahoma, Norman, Oklahoma (Manuscript received 27 May 2021, in ﬁnal form 8 May 2022) ABSTRACT: Observational evidence shows changes to North American weather regime occurrence depending on the strength of the lower-stratospheric polar vortex. However, it is not yet clear how this occurs or to what extent an improved stratospheric forecast would change regime predictions. Here we analyze four North American regimes at 500 hPa, con- structed in principal component (PC) space. We consider both the location of the regimes in PC space and the linear re- gression between each PC and the lower-stratospheric zonal-mean winds, yielding a theory of which regime transitions are likely to occur due to changes in the lower stratosphere. Using a set of OpenIFS simulations, we then test the effect of re- laxing the polar stratosphere to ERA-Interim on subseasonal regime predictions. The model start dates are selected based on particularly poor subseasonal regime predictions in the European Centre for Medium-Range Weather Forecasts CY43R3 hindcasts. While the results show only a modest improvement to the number of accurate regime predictions, there is a substantial reduction in Euclidean distance error in PC space. The average movement of the forecasts within PC space is found to be consistent with expectation for moderate-to-large lower-stratospheric zonal wind perturbations. Overall, our results provide a framework for interpreting the stratospheric inﬂuence on North American regime behavior. The results can be applied to subseasonal forecasts to understand how stratospheric uncertainty may affect regime predictions, and to diagnose which regime forecast errors are likely to be related to stratospheric errors. SIGNIFICANCE STATEMENT: Predicting the weather several weeks ahead is a major challenge with large poten- tial beneﬁts to society. The strength of the circulation more than 10 km above the Arctic during winter (i.e., the polar vortex) is one source of predictability. This study investigates how forecast error and uncertainty in the polar vortex can impact predictions of large-scale weather patterns called “regimes” over North America. Through statistical analy- sis of observations and experiments with a weather forecast model, we develop an understanding of which regime changes are more likely to be due to changes in the polar vortex. The results will help forecasters and researchers un- derstand the contribution of the stratosphere to changes in weather patterns, and in assessing and improving weather forecast models. KEYWORDS: Climate classiﬁcation/regimes; North America; Stratosphere; Stratosphere-troposphere coupling; Subseasonal variability; Winter/cool season 1. Introduction and quasi-stationary (e.g., Michelangeli et al. 1995) with typical time scales of weeks, well suited to the subseasonal scale where The framework of large-scale weather regimes is now in- they can manifest “windows of opportunity” for skillful ex- creasingly used in wintertime subseasonal-to-seasonal (S2S) tended-range forecasts (Mariotti et al. 2020; Robertson et al. prediction (from ∼2 weeks to 2 months ahead; White et al. 2020). 2017), although the concept of a weather “regime” is not new Unlike empirical orthogonal functions (EOFs) (e.g., Hannachi (Rex 1951). Regimes are characteristically recurrent, persistent, et al. 2007), regimes deﬁned through clustering methods are not bound by orthogonality or variance partitioning con- straints. These regimes can therefore more closely represent Denotes content that is immediately available upon publica- tion as open access. the full anomalous ﬂow conﬁguration on a given day by beneﬁting from “mode mixing” and are accordingly easier to interpret, providing a useful way to understand extended- Supplemental information related to this paper is available range ensemble forecasts. By characterizing recurrent ﬂow at the Journals Online website: https://doi.org/10.1175/JCLI-D-21- conﬁgurations, weather regimes can also be used to diagnose 0413.1.s1. ﬂow-dependent predictability (Ferranti et al. 2015; Matsueda Simon H. Lee’s current affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/ licenses/by/4.0/). Corresponding author: Simon H. Lee, simon.h.lee@columbia.edu DOI: 10.1175/JCLI-D-21-0413.1 Ó 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). 5916 J O U R N A L O F C LI MATE VOLUME 35 and Palmer 2018). From an impacts perspective, regimes have while others focus on the continent as a whole and incorpo- been used to better understand meteorological impacts on en- rate both Atlantic and Paciﬁc variability. Despite some meth- ergy demand (e.g., Grams et al. 2017; van der Wiel et al. 2019; odological differences, a growing number of studies have Garrido-Perez et al. 2020), precipitation and wildﬁre risk deﬁned a consistent and reproducible set of four wintertime (Robertson and Ghil 1999; Robertson et al. 2020), and public regimes in the 500-hPa geopotential height anomaly ﬁeld cen- health (Charlton-Perez et al. 2019; Huang et al. 2020). tered over North America (e.g., Straus et al. 2007; Vigaud A signiﬁcant source of tropospheric subseasonal predict- et al. 2018; Lee et al. 2019b; Robertson et al. 2020). The re- ability during boreal winter is variability in the Arctic strato- gimes capture both PNA-like and NAO-like behavior. spheric polar vortex, including sudden stratospheric warmings More speciﬁcally, Lee et al. (2019b) analyzed these four (SSWs; e.g., Charlton and Polvani 2007) and strong vortex North American regimes (the Arctic high, Arctic low, Alas- events (e.g., Limpasuvan et al. 2005; Tripathi et al. 2015). The kan ridge, and Paciﬁc trough) in the context of the strength of downward inﬂuence of the stratosphere can be viewed as the the lower-stratospheric polar vortex in reanalysis. They found modulation of weather regime transition and persistence. Per- signiﬁcant differences between the occurrence of three of the haps the simplest regime framework employs the two phases regimes during strong and weak stratospheric vortex states of of the North Atlantic Oscillation (NAO), which are similar to a similar magnitude to those in Charlton-Perez et al. (2018) the Northern Annular Mode (NAM) and Arctic Oscillation for the North Atlantic. The Alaskan ridge regime did not (AO) patterns and strongly inﬂuenced by the stratosphere show a relationship with the stratospheric vortex strength, but (Ambaum et al. 2001; Baldwin and Thompson 2009; Hitchcock was found to be strongly linked to North American cold and Simpson 2014). More complex regime analyses for the waves. Lee et al. (2019b) hypothesized that tropical forcing North Atlantic–European sector invoke four (e.g., Vautard (e.g., Wang et al. 2014) or stratospheric wave reﬂection 1990; Cassou 2008), six (Falkena et al. 2020), or seven (e.g., (Kodera et al. 2016; Kretschmer et al. 2018; Matthias and Grams et al. 2017) regimes depending on the method, focus, or Kretschmer 2020) may dominate driving the Alaskan ridge, purpose of the analysis. owing to the similarity of the regime to patterns associated Using four North Atlantic regimes, Charlton-Perez et al. with both. As a purely observation-based study, the results of (2018) found signiﬁcant differences in the occurrence likelihood Lee et al. (2019b) were noncausal and did not assess when or of three regimes between strong and weak lower-stratospheric how changes in the stratospheric state would change regime vortex states, while the probability of Scandinavian blocking occurrence, or whether improved stratospheric forecasts was invariant. Beerli and Grams (2019) related the strato- would yield better regime predictions. Addressing these points spheric modulation of Atlantic weather regimes to whether or is therefore a goal of the present study. not the regime projected strongly onto the NAO pattern. To diagnose the downward inﬂuence of the stratosphere on They emphasized that regimes that do not project strongly the troposphere, and changes in tropospheric forecast skill onto the NAO provide a route for a wider variety of weather arising from a correctly predicted stratosphere, model experi- patterns following anomalous stratospheric vortex states. ments in which the stratospheric state is artiﬁcially nudged or Subsequently, Maycock et al. (2020) analyzed the North relaxed to a different state (such as that from reanalysis) have Atlantic response to SSWs from the perspective of modu- been used. Most studies have focused on the seasonal-scale lation of the three eddy-driven jet regimes, ﬁnding an increase effects (Douville 2009; Hitchcock and Simpson 2014; Jung in the occurrence and persistence of the southernmost regime et al. 2010a,b). However, Kautz et al. (2020) used relaxation (corresponding to the negative NAO). Domeisen et al. (2020a) experiments on S2S time scales to quantify the role of the assessed the varying degrees of stratosphere–troposphere cou- February 2018 SSW in the predictability and onset of the sub- pling following major SSWs (e.g., Karpechko et al. 2017; White sequent Eurasian cold wave. They found an increased proba- et al. 2019) by considering the regimes present during SSW bility of surface cold extremes in forecasts with a nudged onset and in the weeks afterward, suggesting that the anteced- stratosphere, but that the evolution of the lower-stratospheric ent state of the troposphere may play an important role in NAM following the SSW}rather than simply the occurrence determining subsequent downward coupling. of the SSW}was important for more accurate tropospheric In recent years, the inﬂuence of the stratosphere on North forecasts. The importance of persistent lower stratospheric American climate variability has received increased attention, anomalies in eliciting a tropospheric response is consistent likely owing to the extreme cold-air outbreaks during winter with climate model studies (Maycock and Hitchcock 2015; 2013/14 that accompanied disruption to the polar vortex (Yu Runde et al. 2016) and the polar-night jet oscillation events of and Zhang 2015; Waugh et al. 2017). However, relatively less Hitchcock et al. (2013). attention has been given to explicitly viewing the impact of Although SSWs and their strong vortex counterpart are typi- the stratosphere on North American weather from a tropo- cally harbingers of persistent anomalous lower-stratospheric spheric regimes perspective. As North America is inﬂuenced NAM states (Baldwin and Dunkerton 2001), they do not by weather from both the Atlantic and Paciﬁc to different de- necessarily propagate into the lowermost stratosphere, and grees across the continent, a challenge with deﬁning North anomalous lower-stratospheric NAM states can occur without American regimes is the choice of domain. Some studies (e.g., a typical midstratospheric precursor. Hence, analysis of the Amini and Straus 2019; Fabiano et al. 2021) focus on up- effect of the stratosphere on the troposphere need not only stream variability in the Paciﬁc–North American (PNA) sec- focus on such extreme midstratospheric circulation events. tor (akin to the Atlantic regimes with respect to Europe), Further, the NAM in the lower stratosphere during midwinter 15 SEPTEMBER 2022 LE E E T A L . 5917 possesses a very long time scale (over 4 weeks; Baldwin et al. the daily climatology over this period. (Any trends in Z500 are 2003), key for the S2S prediction scale. In this study, we focus on found to have little impact on the regimes, so detrending is not subseasonal variability in the strength of the lower-stratospheric performed.) Then, data are weighted by the square root of co- polar vortex, diagnosed through the zonal-mean zonal wind sine latitude, and EOF analysis is performed, retaining the lead- at 100 hPa and 60 N (U100). We do not explicitly consider ing 12 EOFs that explain close to 80% of the variance; k-means SSWs or strong vortex events. clustering is then performed (Pedregosa et al. 2011) in the non- The overall goal of this study is to understand how changes standardized 12-dimensional principal component (PC) space, or uncertainty in the subseasonal lower stratospheric vortex with k set to 4. In addition to reducing the dimensionality of state can inﬂuence changes or uncertainty in predictions of the clustering problem and ﬁltering smaller-scale variability, North American weather regimes. We do this ﬁrst by a statis- performing the clustering in PC space produces a coordinate tical analysis of the regimes and their underlying EOFs in system that enables interpretation of the regimes in terms of reanalysis, and then through analyzing a set of model experi- their comprising EOFs, linking two widely used prediction ments in which the stratosphere is nudged toward reanalysis. A frameworks. After generating the clusters, each day is then as- greater understanding of the relationship between stratospheric signed to one of the four regimes by the minimum Euclidean variability and regimes will help in both the real-world under- distance to the cluster centroids in PC space. standing and interpretation of regime forecast uncertainty, and For regime assignment in the hindcasts, the model Z500 in subsequent studies of regime dynamics and predictability. It climate is ﬁrst subtracted, to account for systematic biases. would also be a useful tool to examine how model biases affect The model climate is computed for each initialization date the representation of stratosphere–troposphere coupling. and lead time over the 20-year hindcast period. Then, the The paper is organized as follows. Section 2 introduces the daily data are projected onto the 12 EOFs, and each day is data, methods, and model experiments. Section 3 deﬁnes the assigned to a regime based on these pseudo-PC loadings. As regimes and their underlying EOFs, and the relationship be- an additional forecast diagnostic in the model experiments, tween these EOFs and the lower-stratospheric polar vortex weekly mean regimes are produced by ﬁrst averaging the PCs strength. Section 4 develops a theory of how the stratosphere over a 7-day period and then assigning to a regime; these are may inﬂuence regime behavior. Section 5 presents the results found to be largely consistent with the regime occupying the of a modeling study used to test the theory. A summary and majority of days within each week (not shown). conclusion of our work follows in section 6, including implica- c. Regime bust criteria tions for S2S prediction. We select subseasonal regime “busts” from the ECMWF 2. Data and methods hindcasts where there is strong ensemble support ($7members, or approximately two-thirds) for one speciﬁc incorrect regime to a. Hindcasts and reanalysis be dominant (i.e., present on at least 8 days) during days 14–27 For historical analysis and veriﬁcation, we use the (weeks 3–4). These criteria are designed to pick out cases that European Centre for Medium-Range Weather Forecasts suggest a strong, but incorrect, subseasonal signal constraining (ECMWF) ERA-Interim reanalysis (Dee et al. 2011). Hind- the model analogous to a “precise but inaccurate” forecast. As casts are taken from version CY43R3 of the ECMWF such, the model conﬁdence may be erroneously interpreted extended-range prediction system (used to produce opera- as enhanced predictability and accuracy, with potentially tional forecasts from July 2017 to June 2018) as part of the large real-world impacts from subsequent decision-making. We S2S database. The hindcasts consist of an 11-member en- choose only hindcasts initialized during December–February, semble (1 unperturbed member and 10 perturbed members) as the seasonal cycle may affect week-3–4 forecasts initialized initialized from ERA-Interim twice per week. The model during March. These criteria yield 31 initialization dates. A has a resolution of Tco639 up to day 15 and Tco319 after further stipulation is applied such that the initialization dates day 15, and 91 vertical levels. All data are sampled once must be separated by at least 21 days to avoid analyzing multi- per day at 0000 UTC, and regridded to 2.58 latitude– ple instances of the same event; in these cases, the earliest ini- longitude resolution for computational efﬁciency and since tialization date is selected. This step ﬁlters the number of we are only considering large-scale ﬁelds. cases to 20 (i.e., on average 1 per winter), which are listed in Table 1. Except for forecasts of an Arctic high verifying as an b. Regime definitions Alaskan ridge, all forecast–veriﬁcation combinations are in- The deﬁnition of North American weather regimes follows cluded at least once (not by design). that of Lee et al. (2019b), extended by 1 year. We take 500-hPa No stratospheric error criteria are included in order to as- geopotential heights (Z500) in the region 1808–308W, 208–808N sess both to what extent poor subseasonal regime forecasts in all December–March days in the period 1 January 1979– are associated with stratospheric errors and the effect of 31 December 2018 in ERA-Interim (4840 days) and subtract stratospheric relaxation even in cases with a relatively well- forecast stratosphere. We ﬁnd that the majority of bust cases 1 feature ensemble-mean U100 error magnitudes $ 3m s Tco 5 cubic octahedral spectral truncation. (14 of the 20 initialization dates, including 8 week-3 and Details of the prediction system can be found on the ECMWF website https://confluence.ecmwf.int/display/S2S/ECMWF+Model. 12 week-4 forecasts), approximately the mean absolute error 5918 J O U R N A L O F C LI MATE VOLUME 35 TABLE 1. North American regime busts in ECMWF CY43R3 hindcasts (HC) from December 1997 to February 2017. The week-3–4 dominant (W3–4 dom.) regime is that which is predicted by $7 ensemble members (64%) to be present on $8 days during days 14–27 inclusive, veriﬁed against the ERA-Interim regime that is present for $8 days during the same time period. Week-3 and week-4 regimes are theregimes of theweeklymean ﬁeld with the largest ensemble support; «U is the ensemble-mean error in the 100-hPa 608N zonal-mean zonal winds averaged over each week. The data are grouped by the dominant regime prediction and then sorted by the week-4 «U. 21 21 Initialization W3–4 dom. percent (ERA) W3 HC (ERA) W3 «U (m s ) W4 HC (ERA) W4 «U (m s ) Arctic high 21 Dec 2005 64 (PT) ArH (PT) 20.5 ArH (PT) 4.2 1 Feb 2009 64 (ArL) ArH (ArL) 2.5 ArH (ArL) 3.2 8 Feb 2010 73 (PT) ArH (ArH) 0.3 ArH (PT) 24.8 29 Jan 1998 64 (PT) PT (PT) 28.5 ArH (PT) 26.7 Arctic low 29 Jan 2001 73 (AkR) ArL (ArL) 6.5 ArL (AkR) 8.5 28 Dec 2016 82 (AkR) ArL (AkR) 2.7 ArL (AkR) 3.0 8 Feb 2006 64 (ArH) ArL (ArH) 4.8 ArL (ArH) 2.3 22 Jan 1999 64 (PT) ArL (PT) 21.5 ArL (PT) 1.0 19 Feb 2011 64 (PT) ArL (PT) 20.3 ArL (PT) 20.6 4 Dec 2011 64 (PT) ArL (ArL) 0.1 ArL (PT) 21.3 Alaskan ridge 11 Dec 2001 64 (ArH) AkR (ArH) 2.3 AkR (PT) 3.1 15 Feb 2017 64 (ArL) AkR (ArL) 20.6 AkR (AkR) 2.6 4 Dec 2003 73 (PT) ArH (PT) 0.4 AkR (ArL) 23.0 Paciﬁc trough 12 Feb 1999 64 (ArH) PT (PT) 3.3 PT (ArH) 14.0 8 Jan 2010 64 (ArH) PT (ArH) 4.1 ArH (ArH) 8.7 25 Dec 2015 73 (ArH) PT (ArH) 7.7 PT (ArH) 7.7 7 Dec 2000 64 (ArH) PT (ArH) 7.3 PT (ArH) 2.8 18 Jan 2016 73 (AkR) PT (PT) 0.3 PT (AkR) 0.4 21 Dec 2014 73 (AkR) AkR (AkR) 21.7 PT (AkR) 22.1 25 Dec 2006 82 (ArL) PT (ArL) 25.8 PT (ArL) 28.7 (MAE) of the December–February week-3–4 hindcasts (see representation of initial condition uncertainty, so some differ- Fig. S1 in the online supplemental material). This suggests ences between these model runs and the equivalent hindcasts that regime busts and large lower-stratospheric vortex errors are to be expected. As we are primarily considering forecasts often co-occur. on time scales of several weeks, the initial condition un- certainty is considered less important, and the stochastic d. OpenIFS model schemes generate spread comparable to the hindcasts in the ﬁelds analyzed in this study. For model experiments, we use OpenIFS version 43r3v1} For each initialization date, two sets of ensembles are a research version of the ECMWF IFS (Integrated Forecast produced: a control (CTR) run in which the forecast freely System) model CY43R3, but without data assimilation. The evolves (comparable with the equivalent hindcast, notwith- model is initialized from ERA-Interim and run on a linear standing the model differences), and a relaxed (RLX) run in Gaussian grid with T255 resolution, 60 vertical levels (i.e., which the Arctic stratosphere is nudged toward ERA-Interim the resolution of ERA-Interim), and a time step of 45 min. using the IFS relaxation scheme (e.g., Jung et al. 2010a). The Output data are bilinearly interpolated onto a 2.58 latitude– relaxation scheme operates by applying a nonphysical ten- longitude grid. Each ensemble consists of an unperturbed dency to the model equations of the form member and 20 perturbed members, in which spread is gener- ated by the stochastically perturbed parameterization tenden- k(X 2 X), (1) obs cies (SPPT) and stochastic kinetic energy backscatter (SKEB) schemes (Leutbecher et al. 2017). The ensemble size is chosen where X is a model prognostic variable, X is the “observed” obs as a balance between the potential gain from additional value from ERA-Interim, and k [unit: (time step) ]is the relax- members compared with the 11-member hindcasts and com- ation coefﬁcient controlling the strength of the forcing [following, putational expense. The OpenIFS runs differ from the ope- e.g., Jeuken et al. (1996) and Magnusson (2017)]. The term X obs rational model in both resolution and in that there is no at each model time step is generated by linear interpolation be- tween 6-hourly reanalysis ﬁles. A relaxation time scale of 12 h is used in this study, corresponding to k 5 0.0625 per time step Specific details of the model can be found at https://confluence. ecmwf.int/display/OIFS/Release+notes+for+OpenIFS+43r3v1. given the 45-min model time step, which can be interpreted 15 SEPTEMBER 2022 LE E E T A L . 5919 FIG. 1. Vertical and latitudinal proﬁle of the relaxation coefﬁcient scaling (i.e., a value of 1 de- notes full relaxation, here with a time scale of 12 h), for both pressure (left-hand ordinate) and model level number (right-hand ordinate and horizontal grid lines; labeled to level 31 for clarity). The red dashed and dotted lines denote the bounds, in latitude and height respectively, where the coefﬁcient is 0.5. The hatched area denotes the region where the scaling is at least 0.99. as nudging the model state at each time step by 6.25% of e. Significance testing the departure from the reanalysis. Vorticity, divergence, Throughout the paper, statistical signiﬁcance is assessed at and temperature are relaxed in model gridpoint space with the 95% conﬁdence level by bootstrap resampling (e.g., Wilks an exponential taper at both the latitude and model-level 2019). Random samples (with replacement) are taken from boundaries. the population and the quantity under analysis (e.g., a regres- A proﬁle of the relaxation domain is shown in Fig. 1. The sion coefﬁcient) is calculated and stored. This process is domain boundaries are chosen to both maximize constraint of repeated 10 000 times, and then a conﬁdence interval is con- the polar lower stratosphere while allowing for a sufﬁciently structed from the appropriate percentiles of this distribution smooth taper to minimize negative numerical effects, and to (2.5th–97.5th percentiles for two-sided 95% conﬁdence). remain largely poleward and upward of the subtropical jet to reduce directly constraining the tropical upper-tropospheric 3. Regimes and EOFs waveguide. The choice of domain is also limited by the verti- cal level spacing of the model in the upper troposphere and The centroids of the four regimes (expressed as the Z500 lower stratosphere. We employ a weaker stratospheric nudg- ﬁeld reconstructed from the sum of the centroid loading in ing than some previous studies (e.g., Jung et al. 2010a; Kautz the leading 12 EOFs), along with the percent of days assigned et al. 2020), but note that the relaxation in our study extends to each (the occupation frequency), are shown in Figs. 2a–d. further into the lower stratosphere. Analysis of the output In terms of both spatial patterns and the ranking of occupa- ﬁelds show this relaxation strength is enough to constrain the tion frequency, these match the regimes of Lee et al. (2019b) model. Time series of the U100 forecasts from the CTR and and so we follow their naming convention [after Straus et al. RLX experiments and the corresponding veriﬁcation from (2007)]: Arctic high (ArH), Arctic low (ArL), Alaskan ridge ERA-Interim are shown in Fig. S2. (AkR), and Paciﬁc trough (PT). The coordinates of the re- As the random seed used in the stochastic schemes is ﬁxed gime centroids in the leading 12 PCs are shown in Fig. 2e. for each ensemble member, the equivalent ensemble mem- Only the leading three PCs have large contributions to the bers in the CTR and RLX experiments differ only by the centroids; performing the same clustering analysis but retain- stratospheric nudging. In analyzing the OpenIFS runs, we as- ing only the leading three PCs yields very similar patterns, sume the model climatology is equivalent to that of the corre- with only 4% of days assigned to a different regime. There- sponding CY43R3 hindcasts. fore, we now focus our analysis on these leading three EOFs. 5920 J O U R N A L O F C LI MATE VOLUME 35 FIG.2.(a)–(d) Centroids of the four regimes, expressed as 500-hPa geopotential height anomalies with respect to daily 1979–2018 climatol- ogy in ERA-Interim, and the percent of days assigned to each regime in all December–Marchdays in the period1 Jan1979–31 Dec 2018. (e) Coordinates of the regime centroids in raw (nonstandardized) 12-dimensional principal component space. (f)–(h) The leading three EOFs (multiplied by the square root of the eigenvalue) of daily 500-hPa geopotential height anomalies in the domain 1808–308W, 208–808N, and the percent of total variability explained by each EOF. Maps of the EOFs and the percent of the total variance ex- under consideration. The EOFs presented here}with the plained are shown in Figs. 2f–h. In total, these three EOFs ex- most NAM-like pattern in EOF2, while the leading EOF con- plain close to 40% of the daily variance within the domain, tains NAM/NAO and PNA-like characteristics}agrees well and are well separated according to the criterion of North with the upper-tropospheric EOF analysis of Baldwin and et al. (1982). The sign of the EOFs is here deﬁned such that a Thompson (2009). For all three North American EOFs, the positive loading produces an anomalous trough in the north- e-folding time scales of the PC time series are 5–7 days, which east Paciﬁc. EOF1 is similar to the PNA (Wallace and Gutzler is similar to the median number of consecutive days with the 1981) but slightly eastward shifted. It also bears some similar- same regime assignment. However, a quarter of the individual ity to the tropical–Northern Hemisphere (TNH) pattern (Mo blocks of consecutive regime days persist for more than 1 week and Livezey 1986; Liang et al. 2017). Furthermore, there is a (including one instance of 39 days of ArL up to and including meridional dipole in the North Atlantic in the eastern edge of 22 February 1990), motivating their utility for extended-range the domain, reminiscent of NAO-like variability. EOF2 has a prediction. meridional dipole in Z500 anomalies, and thus some similarity To understand the relationship between regime occurrence to the surface-based NAM/AO, but with a center of action and the lower-stratospheric vortex presented in Lee et al. over Alaska that is not characteristic of the surface NAM (2019b), we examine the relationship between U100 and the (e.g., Thompson and Wallace 1998). EOF3 is characterized by leading EOFs which deﬁne the clusters. We perform linear re- gression between each PC time series and the contemporane- a wavenumber-2 pattern across the domain. Comparison of these regional EOFs with the leading three ous U100 to see how changes in U100 may modulate the EOFs for the Northern Hemisphere poleward of 208N location of a point within the 3D PC space and thus its regime (Figs. S3–S5) shows a high degree of similarity in both the cor- attribution. The instantaneous relationship is used since we relation of the PC time series (Pearson’s correlation r $ 0.77; are considering the lower stratosphere as an upper boundary p , 0.05) and spatially (area-weighted pattern correlation condition to the troposphere, with both a much longer mem- $ 0.87 over the North American domain). We can therefore ory (e.g., Baldwin et al. 2003) and greater predictability (Son be conﬁdent that the leading three EOFs used in the cluster- et al. 2020); lagged relationships (not shown) reveal these ing are regional manifestations of hemispheric variability, and coefﬁcients are either effectively maximized at lag 0 or, con- that hemispheric variability is dominant in the smaller domain sidering uncertainty, largely invariant for 67 days (within the 15 SEPTEMBER 2022 LE E E T A L . 5921 PC e-folding time scale). Some of this relationship may relate to the vertical extension of a primarily tropospheric zonal wind signature associated with these EOFs into the lower stratosphere. However, on subseasonal scales (well beyond tropospheric decorrelation time scales) this remains the com- ponent of the structure that is potentially predictable. The regression coefﬁcients are shown in Fig. 3. Although the coefﬁcients for all three EOFs are signiﬁcantly different from zero, the linear relationship is 3–5 times stronger for EOF2. Similarly, the Pearson’s correlations between U100 and PCs 1 and 3 are small (r 520.13 and 0.10, respectively), but moderate for PC2 (r 5 0.42). Thus, the effect of the stratosphere in this 3D EOF space is mostly contained within EOF2, which is consistent with its annular-like spatial pattern and the height-dependent NAM results of Baldwin and Thompson (2009). The sign of the regression coefﬁcients is such that a decrease in U100 is associated with an increase in Z500 in the vicinity of Greenland/the northern node of the NAO, in agreement with the canonical response of the tropo- sphere to a weakened stratospheric vortex. 4. Theory of regime transitions and the stratosphere FIG. 3. Linear regression coefﬁcients between the 100-hPa 60 N zonal-mean zonal wind and the raw PC time series of the leading In this section, we develop a theory of which regime transi- three EOFs, in all December–March days in ERA-Interim tions may be possible solely due to a stratospheric perturbation 1979–2018. Error bars indicate 95% conﬁdence intervals obtained by jointly considering the linear relationship between U100 by bootstrapping with replacement (see section 2e for details). and the three PCs, and the location of the regimes within the space spanned by the three PCs. The theory can be interpreted The angle u between b and g follows as as an idealized framework where all else is instantaneously equal and only the stratosphere is changed, retaining potential b · g u(b, g) 5 arccos , (4) predictability arising from other tropospheric processes. bg Using the regression coefﬁcients between U100 and the PC time series, we deﬁne the stratospheric perturbation vector b. 2 2 2 where x 5 x + x + x denotes the Euclidean norm of a 1 2 3 This vector represents the movement within the 3D PC space 3D vector x. arising from a perturbation to U100, DU, that is explained by We use this framework to model which regime transitions the linear regression coefﬁcients: are possible solely with stratospheric forcing by considering ⎛ ⎞ whether the vectors b (either positive or negative) and g ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ point in a similar direction, known as “cosine similarity” (e.g., ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ b 5DU⎜ 91 ⎟ : (2) ⎜ ⎟ ⎜⎜ ⎟⎟ ⎜ ⎟ Han et al. 2012). If u $ 908 (cosu # 0), then no component of ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 20 the regime transition or movement within the 3D PC space can be explained by the linear relationship between the PCs Note that b is not a function of the position within PC space and U100, since the contribution of b would be0(in thecaseof and is thus constant for a given DU. While the truncation to a maximally dissimilar vectors, u 5 908)oroppose g (cosu , 0). 3D PC space was earlier motivated by the coordinates of the A smaller angle indicates the transition is more likely since the regime centroids, the linear relationship between the leading projection of b in the direction of g is larger (as cos u is larger), three EOFs and U100 also accounts for nearly all of the linear thus requiring a smaller DU. We focus on angles, rather than relationship with Z500 (Fig. S6). explicit distances, since the distances between regimes for any The transition vector g between two points (e.g., two clus- point are dependent on the initial location. ter centroids) within this space is then deﬁned as the respec- Figure 4 presents a 3D depiction (in the space spanned by the tive distances between the coordinates in the three PCs: leading three EOFs) of b (both positive and negative; i.e., for a ⎛ ⎞ strengthening or weakening stratospheric vortex) applied to DPC ⎜ ⎟ ⎜ ⎟ each regime centroid and the transition vector g between the ⎜⎜ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ g 5 ⎜⎜ DPC ⎟⎟ , (3) ⎜ ⎟ ⎜ ⎟ centroids. The regime centroids form a tetrahedron in this space. ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Some of the transition vectors lie closer to b than others owing DPC to their relative locations within this space. For example, the where DPC 5 PC (B) 2 PC (A) for the transition from positive b vector and the transition vector from the ArH to PT k k k point A to point B. Hence, inverse transitions have an equal centroids are close, while the transition vectors from the AkR but opposite transition vector: g(A, B) 5 g(B, A). centroid are almost perpendicular to either sign of b. 5922 J O U R N A L O F C LI MATE VOLUME 35 and b are all relatively large for AkR (Fig. 5c), as previously suggested by the 3D depiction in Fig. 4.For b , 0, only a transition to ArH has an angle , 908. Transitions to ArL and PT are possible with b . 0, but the angles are relatively large and thus more unlikely. We next extend our analysis beyond points initiating at the centroids and incorporate the effect of spread around the PC space spanned by each regime. First, we consider all the assigned regime days in ERA-Interim. The leading three PCs are then per- turbed by b in the range 230 #DU # 30 m s , and subse- quently reassigned to a regime by minimum Euclidean distance. The maximum magnitude of DU is chosen here to be close to the maximum observed variability in U100; the largest U100 errors in individual CY43R3 ensemble members are close to 620 m s . Note that in reality, the tropospheric response may be larger for a smaller DU as a consequence of the linear framework. Figure 6 depicts the conditional probability, for each initial regime, of either remaining in the same regime or transitioning to each of the other regimes for each DU. Only those transition pathways with u , 908 occur, and the relative likelihood mani- FIG. 4. Visualization of the regimes in the space occupied by the fests the degree of similarity (i.e., the angle) between b and g. leading three EOFs. Colored markers indicate the regime cent- There are no transitions away from ArH for DU , 0(Fig. 6a) roids. Colored arrows represent the transition vectors from each or away from ArL for DU . 0(Fig. 6c). For DU , 0, the domi- centroid to the other centroids, scaled to 0.253. The black arrows nant transition forall regimesistoArH. For DU . 0, transitions show the stratospheric perturbation vector, scaled to a 610 m s from ArH to PT dominate (Fig. 6a) while transitions to ArL perturbation (solid positive; dashed negative), which is the same at all points. dominate for AkR and PT (Figs. 6b,d). Transitioning into AkR from any other regime is unlikely even for large |DU|, while tran- sitioning out of AkR is the least likely for any of the regimes The angles between the centroid g vectors and b are quanti- where a transition pathway exists (despite its unique approxi- ﬁed in the protractor-like polar plots in Fig. 5. The angles are ex- mately equal sensitivity for either sign of DU). Although not ex- pressed such that both positive and negative b are aligned with plicitly shown, there is also evidence of multiple transitions 08 (thus, the angle between each g and b , 0is a reﬂection of occurring as |DU| increases. For example, the probability of tran- that to b . 0 about 908). For a point starting at the ArH centroid sitioning into AkR from each of the other regimes reaches a (Fig. 5a), there is substantial cosine similarity between b . 0and peak for |DU| between 10 and 20 m s before declining. transition vectors to all otherregimes(forall three, u , 608). The As a general diagnostic of the sensitivity of each initial similarity is strongest for the transition vectors to PT and ArL, regime state to a lower-stratospheric perturbation, we can which have approximately equal cosine similarity. The angles be- consider the probability of transitioning out of the regime for tween b , 0 and all three transition vectors are .908;thus, the DU 5610 m s (approximately equal to the maximum theory does not allow a transition away from ArH given DU , 0. week-3–4 ensemble-mean U100 error magnitude in CY43R3 Overall, ArH has the largest number of transition vectors with hindcasts). For DU 5 10 m s , 58% of ArH days transition small angles/high cosine similarity. Equally, the minimum angle into a new regime, while only 17% of AkR days and 6% of between either sign of b and any g vector is between b , 0and PT days do so. For DU5210 m s , the sensitivity of PT and transitions to ArH (Figs. 5b–d). This is consistent with the ArL is approximately equal, with 39% of PT and 38% of ArL observed probability of transitions into, and the persistence of, days transitioning into a new regime. Only 15% of AkR days ArH/NAO-, which is the most sensitive of both the North transition into a new regime. American and North Atlantic regimes to the strength of U100 Overall, the results presented in Figs. 4–6 are in agreement (Charlton-Perez et al. 2018; Lee et al. 2019b). with the observed differences in regime occurrence in strong and For the PT regime (Fig. 5b), there is a small angle between weak stratospheric vortex states in Lee et al. (2019b).The theory the negative b vector and the transition vector to ArH (i.e., also gives results consistent with the relationship between the re- equal and opposite to the positive b and the transition from gimes (particularly ArH and ArL) and the concurrent NAO in- ArH to PT). While transitions are possible to both AkR with dex (Fig. S7), given the strong modulation of the NAO by the b , 0, and to ArL with b . 0, the angles are close to 908, sug- stratosphere. Further, the proposed framework yields insight gesting that these are unlikely. Considering the ArL regime into speciﬁc regime transitions under different vortex states that (Fig. 5c), transitions to all three other regimes are possible are not limited by the observational sample size. In summary: with b , 0. The smallest angle is to the ArH transition vector, while the angles to the PT and AkR transitions are large. No DU , 0 moves the majority of points within PC space to- regime transitions from ArL are possible in this framework ward only ArH, consistent with this regime being the only with DU . 0. Last, the angles between the transition vectors one more likely under weak vortex conditions. 15 SEPTEMBER 2022 LE E E T A L . 5923 FIG. 5. Polar plots showing angles between the stratospheric perturbation vector (solid positive; dashed negative) and the centroid transition vector for each of the four regimes in 3D EOF space, as visualized in Fig. 4. • • DU . 0 does little to changing the regime assignment for What is the effect of stratospheric relaxation on regime days initially assigned to ArL or PT, while these are fa- forecast accuracy in these cases? vored transitions for initial ArH and AkR states. This is Regardless of the forecast accuracy, is the change in the forecast consistent with the theory in section 4? consistent with ArL and PT being more likely under strong vortex conditions. Very large DU is required to shift toward and away from a. Regime predictions AkR, with a similar proportion of transitions resulting from A comparison between the weekly mean regimes in the CTR both positive and negative perturbations. This behavior is and RLX ensembles, for weeks 3 and 4, is shown in Fig. 7.The consistent with the observed statistically equal occurrence improvement in the total number of ensemble members with a of this regime in strong and weak vortex states. correctly assigned weekly mean regime is modest: 13% in week 3 and 15% in week 4. Therefore (recalling that these cases were These conclusions are highly idealized, requiring both a per- selected as particularly poor forecasts), the overall fraction of fectly linear response and the sole (or dominant) change being to correctly assigned regimes remains low in the RLX experiment: U100. It is also possible that b may be sensitive to the initial posi- 40% in week 3 and 25% in week 4. Any improvement is also tion within PC space. However, the corroboration with observa- case dependent. The greatest improvement in week 3 is in the tions suggests the potential use of this framework in interpreting 11 December 2001 case (7 more members correctly assigned to the regime response to changes and uncertainty in the strato- ArH), and in the 29 January 1998 case (5 more members cor- sphere on subseasonal time scales. The analysis in the next sec- rectly assigned to PT) in week 4. The latter was a case with a tion considers whether imposing stratospheric relaxation yields a very large U100 error (cf. Table 1). In several cases, there is a tropospheric response consistent with this simple but novel theory. decrease in the number of correctly assigned ensemble members. Thus, constraining the stratospheric state is not enough to ﬁx these regime bust cases}which may be unsurprising given that 5. Model experiments only a selection of these cases have very large stratospheric er- In analyzing the results of the relaxation experiments, we rors, while all have largely inaccurate regime predictions. This seek to answer the following two questions: result indicates that the stratospheric state should not be viewed 5924 J O U R N A L O F C LI MATE VOLUME 35 FIG.6. (a)–(d) Given each initial regime, the conditional probability of either remaining in the same regime or tran- sitioning to each of the other regimes, when all days assigned to each regime in ERA-Interim are perturbed by the stratospheric perturbation vector in the range 230#DU # 30 m s . as exerting simple control on the subseasonal tropospheric ﬂow study; we instead focus on the general results across this set of over North America. forecasts. Figure 7 also shows that there are changes to the number of b. Error reduction in PC space ensemble members assigned to the incorrect regimes, regard- less of whether there is a change to the number assigned to Despite the small and case-dependent regime improve- the correct regime. On a member-by-member basis, 34% and ment, for almost all cases the mean Euclidean distance error 57% of the total ensemble members in weeks 3 and 4 respec- of the ensemble in 3D PC space is reduced (Fig. 8a). This di- tively are assigned to a different regime in the RLX experi- agnostic is useful because it incorporates changes to forecasts ments. Thus, by week 4, the stratospheric nudging has shifted that maintain the same regime attribution and is proportional the majority of ensemble members into a new regime} to the root-mean square error (RMSE) of the Z500 ﬁeld re- suggesting signiﬁcant movement within the PC space in which constructed from the leading three EOFs (see the online the regimes are assigned. For example, in week 4 of the 11 supplemental material; note that because non-normalized December 2001 case, there is no increase in the number of PCs are used, the total error on subseasonal time scales is members correctly assigned to PT, but there is a gain of eight dominated by the EOFs with the largest eigenvalues). Hence, ensemble members assigned to AkR (with ArH and ArL los- in the space in which regimes are assigned, the RLX forecasts ing four members each). While a full case-by-case analysis are almost entirely closer to the veriﬁcation. The improve- may yield further speciﬁc insight, it is beyond the scope of this ment is maximized in week 3 (median 14%), with only two 15 SEPTEMBER 2022 LE E E T A L . 5925 FIG. 7. For (a) week 3 and (b) week 4, values denote the number of ensemble members assigned to each regime in the RLX experiment, with the number in parentheses indicating the difference from CTR. Bold font indicates the ERA-Interim regime. Color shading indicates the difference in ensemble-mean U100 between the experiments (RLX-CTR). Grouping is as in Table 1. cases showing an increase in error (21 December 2005 and a U100 change of only 1 m s while the largest error increase 8 February 2010, both of which had negligible week-3 U100 occurs with a U100 change of 2.6 m s (8 February 2010). errors in the CTR run). The median improvement in week 4 The large relative error reduction for small DU suggests a is 12%, but with much greater spread than week 3. There was potential role of zonally asymmetric corrections or other a 30% improvement in a single case (21 December 2014), while changes to the vortex that do not project strongly onto U100 four cases show no change or increased error (7 December 2000, (and thus fall outside the framework proposed here). How- 11 December 2001, 8 February 2010, and 15 February 2017). ever, across this set of 20 cases, for DU exceeding 3 m s , Also shown in Fig. 8a is the mean change in Euclidean dis- there is a systematic error reduction. We revisit this apparent tance error obtained by perturbing the PCs of the CTR en- threshold in the analysis below. semble by b multiplied by DU between the CTR and RLX experiments. This shows that a simple statistical nudge of the c. Movement within PC space PCs using the known linear relationships also yields an error reduction of on average ∼50% of that obtained by running We now investigate whether the movement of the forecasts the full dynamical relaxation experiment. Thus, a substantial within 3D PC space is consistent with what might be expected component of the dynamical effect of imposing a different from the theory established in section 4. For this analysis, we stratospheric state on these EOFs can be explained by the ob- analyze three vectors and three different angles within PC served linear relationship between the PCs and U100. space. Figure 9 shows a schematic of this approach. The To understand whether larger stratospheric forcing yields vectors are deﬁned as follows: larger error reduction, Fig. 8b shows the case-by-case change • CTR-ERA: the vector between the CTR forecast and the in ensemble-mean Euclidean distance error against the mag- nitude of the U100 change between the CTR and RLX ex- veriﬁcation from ERA-Interim (i.e., the error in the CTR periments for weeks 3 and 4. There is no immediately clear forecast). relationship, with the greatest error reduction occurring with CTR-RLX: the vector between the CTR and RLX forecasts. 5926 J O U R N A L O F C LI MATE VOLUME 35 FIG. 8. (a) Boxplots of the ratio between the ensemble-mean Euclidean distance error in 3D PC space between the weekly averaged RLX and CTR ensembles for the 20 cases. Red lines denote the median, and notches show 95% conﬁdence intervals obtained by 10 000 bootstrap resamples (with replacement). Black triangles denote the mean. Blue circles represent the average ratio obtained by statistically perturbing the CTR PCs by the stratospheric pertur- bation vector multiplied by the change in U100 between the CTR and RLX ensembles. Whiskers extend to 1.5 times the interquartile range or extremes (whichever is smaller); outliers shown as open circles. (b) Scatterplot of the week- 3 (green squares) and week-4 (maroon circles) error ratio against the magnitude of the ensemble-mean change in U100 between CTR and RLX. CTR-STAT: the vector between the CTR forecast and the Figure 10b assesses whether the stratospheric perturbation CTR forecast statistically perturbed by b multiplied by DU vector outlined in section 4 is a good representation of the ef- between CTR and RLX ensembles (STAT). fect of a dynamically applied stratospheric perturbation. For |DU| , ∼3m s , the points are scattered across almost the Then, the size of the three angles can be used to answer the full range of angles, indicating no clear relationship between following questions: the theory and the movement of these forecasts in PC space. However, although the sample is smaller, for |DU| . ∼3m s , • u 5 u(CTR-ERA, CTR-RLX): Does stratospheric relaxa- the angles are systematically much smaller than 908}especially tion move the CTR forecast toward the veriﬁcation? for week-4 forecasts, which feature larger DU.Hence, we con- • u 5 u(CTR-RLX, CTR-STAT): Does stratospheric relaxa- clude that on average, these forecasts moved in PC space in the tion move the CTR forecast in the direction expected from b? general direction expected from the theory. • u 5 u(CTR-ERA, CTR-STAT): Does statistical nudging Finally, Fig. 10c assesses whether the simple statistical per- by b move the CTR forecast toward the veriﬁcation? turbation moves the CTR forecast toward the veriﬁcation A scatter of the week-3 and week-4 angles versus the mag- without running a full dynamical experiment (cf. Fig. 10a). As nitude of DU between the CTR and RLX experiments is in Fig. 10b, but unlike in Fig. 10a, there is no clear evidence of shown in Fig. 10. To focus on the overall shift of the ensemble vector similarity for small DU, but there is evidence of a sys- in the relaxed experiments, and since b is deﬁned from linear tematic shift for DU exceeding ∼3m s in magnitude. As a best-ﬁt regression coefﬁcients, we perform this analysis on the result, for larger U100 errors the tropospheric forecast can be perturbations to the PCs and U100 averaged across the en- partially corrected statistically (as indicated by Fig. 8a), but semble. Nevertheless, similar results are obtained when con- there is evidently additional gain from a dynamically cor- sidering the results across all individual ensemble members rected stratosphere even for small DU. (not shown). Figure 10a shows that in the majority of cases The 3 m s threshold is most apparent for angles involving and in both weeks 3 and 4, the stratospheric relaxation gener- b, although there is some suggestion for the behavior of the ally moved the predictions toward the veriﬁcation. Only two RLX experiment (in terms of both angles and Euclidean dis- cases in week 3 and six cases in week 4 do not exhibit any sim- tance error). It is not clear why 3 m s should be a threshold; ilarity (i.e., u . 90 ). These results are consistent with the re- it may be related to the signal magnitude required to emerge duction in Euclidean distance error and its relationship with above the typical ensemble-mean variability, and thus may the magnitude of DU (Fig. 8). be sensitive to ensemble size. Across the CY43R3 hindcasts, 15 SEPTEMBER 2022 LE E E T A L . 5927 FIG. 9. Schematic of the angle-based approach (here in a 2D PC space). There are three vectors: the vector from the control fore- cast to the ERA-I veriﬁcation (CTR-ERA; red), the vector from the control forecast to the relaxed forecast (CTR-RLX; purple), and the stratospheric perturbation vector to the statistically nudged forecast (CTR-STAT; gray). Here u denotes the angle between CTR-ERA and CTR-RLX, u the angle between CTR-RLX and CTR-STAT, and u the angle between CTR-ERA and CTR-STAT. 3m s is approximately two-thirds of the standard deviation of the ensemble-mean U100 in weeks 3–4(∼4.5 m s ), al- though these are not directly comparable owing to the smaller hindcast ensemble size. As mentioned in section 2,3 m s is also approximately the MAE of the ensemble-mean week-3–4 U100 in the CY43R3 hindcasts, and so errors of this magni- tude are a reasonably frequent occurrence. In week 4 (when DU is generally largest), the magnitude of the correlations between the ensemble-mean change in the PCs and the ensemble-mean DU from CTR to RLX (and thus the individual components of b) are maximized. These corre- lations are largest for EOF2 (r 5 0.60, p , 0.05) and EOF3 (r 5 0.48, p , 0.05) but the correlation is small and insigniﬁ- cant for EOF1 (r 520.19, p 5 0.40; although it is similar to that in ERA-Interim). Furthermore, we can ﬁnd the “effective” FIG. 10. Scatterplots of the magnitude of the ensemble-mean vector in the model by computing the regression coefﬁcients weekly mean U100 change between the CTR and RLX experi- between DU and each DPC across all ensemble members. For ments, vs the angle between (a) CTR-ERA and CTR-RLX (u ), weeks 3–4, these are not signiﬁcantly different from the com- (b) CTR-RLX and CTR-STAT (u ), and (c) CTR-ERA and CTR- ponents of b in ERA-Interim, except slightly for EOF1 in STAT (u ), in 3D-PC space. week 3. As a result, the angles between this effective vector and b are small (268 in the week-3 forecasts and 128 in the week-4 forecasts), conﬁrming that b is a good approximation of the response to an imposed stratospheric change. 5928 J O U R N A L O F C LI MATE VOLUME 35 Nevertheless, across the range of cases studied here, the relaxation generally moved the forecasts toward the ERA- response of EOF1 to stratospheric perturbations is not well Interim veriﬁcation and in the direction of that expected approximated by linear regression. This may be due to non- from the theory, while statistically nudging the CTR ensem- linearity, or that the relationship between the EOF and U100 bles by the corresponding stratospheric perturbation vector is not causal (recalling the similarity between the EOF and also generally moved the forecasts toward the veriﬁcation. patterns related to tropical forcing). Sample size may be an is- For |DU| . ∼3m s , this effect was particularly pronounced. sue, given that the small expected response in EOF1. There Consequently, the model experiments support the proposed may also be limitations in the representation of stratosphere– theory of which regime transitions may be possible solely be- troposphere coupling in the model, such as the overestimation cause of changes to the stratospheric state (Fig. 5). of the NAO response reported by Kolstad et al. (2020) using Overall, our results provide evidence that, all else being equal: a similar but more recent ECMWF forecast model (CY45R1). The average shift of an ensemble of subseasonal North The relatively low vertical resolution employed here, parti- American weather regime forecasts in response to changes cularly in the upper troposphere and lower stratosphere, may in the strength of the lower-stratospheric vortex is broadly also have limited the downward coupling and forecast im- generic and predictable. provement arising from the stratosphere (Kawatani et al. Correcting the stratospheric state leads to an improvement 2019; Domeisen et al. 2020c). in the large-scale subseasonal tropospheric forecast over North America, but it does not necessarily correct the re- 6. Summary and conclusions gime assignment (likely due to other sources of error). • Some tropospheric regime states are more likely to change Understanding and exploiting stratospheric variability is a regime assignment for a given stratospheric perturbation key way in which the accuracy and usefulness of S2S forecasts than others. This arises due to the location of the regimes and the ﬁdelity of stratosphere–troposphere coupling within in PC space relative to the linear tropospheric response to models can be increased. In this study, we investigated how the stratosphere. perturbations to the strength of the lower-stratospheric polar vortex can inﬂuence North American weather regime predic- We therefore propose that this vector-based approach can be tions. Our novel technique involved jointly considering the used to identify, a priori, the regime forecast-veriﬁcation scenar- linear relationship between the vortex strength and the lead- ios in which lower-stratospheric errors are more likely to have ing EOFs that contribute to the regimes (Fig. 3), and the rela- played a substantial role}and thus toward understanding the tive location of the regimes within the EOF space (Fig. 4). We overall contribution to subseasonal North American weather re- used an angle-based approach to quantify which transitions gime forecast accuracy. Further, it is possible that in certain cir- are likely to occur (using cosine similarity) for a given regime cumstances when stratospheric uncertainty is dominant that the and stratospheric perturbation (Fig. 5). These results agree method could be used in real time to qualitatively interpret re- with the observed changes in regime occurrence under differ- gime forecast uncertainty owing to stratospheric uncertainty. ent stratospheric vortex states reported in Lee et al. (2019b) This approach is likely to be most useful 2–3 weeks before and provide an explanation for the regime behavior. How- SSWs or strong vortex events, when abrupt forecast shifts (e.g., ever, both the regime framework and EOFs are deﬁned pri- Lee et al. 2019a) are more likely due to the current predictability marily from a mathematical, rather than physical, standpoint, limit of these phenomena (Domeisen et al. 2020b). It may also and therefore the results of this work largely focus on the be plausible to use the technique on-the-ﬂy to linearly impose al- mathematics of regime attribution. ternate regime “storylines” arising from a different stratospheric We then performed a set of stratospheric relaxation model evolution without running additional dynamical forecasts. experiments, selecting 20 cases from the ECMWF hindcasts Moreover, the dominantly linear and apparently generic re- in which there was strong, coherent ensemble support for an sponse to the lower-stratospheric forcing on these time scales incorrect regime to dominate during weeks 3–4. The majority is somewhat similar to the long-lag response following SSWs in (14) of these cases featured U100 errors approximately equal the model experiments of White et al. (2020). The idea that the to or greater than the MAE in either week 3 or 4 or both, sug- tropospheric ﬂow conﬁguration following an imposed strato- gesting a link to the erroneous tropospheric forecasts. We spheric change depends on the state of the troposphere is not a found that the stratospheric relaxation is not enough to elimi- new idea (e.g., Gerber et al. 2009), but as a result, potential gains nate the regime errors, but the relaxation does lead to shifts in subseasonal regime prediction skill from the stratosphere in the ensemble distribution of the regimes within each fore- may be minimal if the tropospheric forecast otherwise drifts too cast indicating substantial movement within PC space (Fig. 7). far from the truth [also recently suggested by Charlton-Perez The results also showed an overall 10%–20% improvement in et al. (2021)]. This potential limitation is consistent with the re- the accuracy of the forecasts in terms of Euclidean distance gime forecasts remaining largely inaccurate even in cases where error/RMSE, which was most consistent in cases where the large lower-stratospheric errors were corrected, notwithstanding stratospheric error was larger (Fig. 8). the imperfections of the model experiment. Analysis of the transition vectors between the CTR and Employing a stronger stratospheric nudging in the model RLX forecasts in PC space provided insight into the effect of experiments presented in this paper may produce greater im- stratospheric relaxation in the space in which regimes are provement in the regime forecasts. On the other hand, con- assigned. The results (Fig. 10) illustrated that stratospheric straining the prediction too strongly would exceed a 15 SEPTEMBER 2022 LE E E T A L . 5929 realistically achievable level of stratospheric forecast accuracy data used in this study are available at https://doi.org/10.5281/ on these scales. It is also plausible that the nudging may have zenodo.4818044. EOF and k-means clustering analysis were limited potential tropospheric forecast accuracy (when com- performed using the freely available Python packages “eofs” (Dawson 2016) and “scikit-learn” (Pedregosa et al. 2011), pared with a true perfect stratosphere forecast) by inducing respectively. unrealistic wave behavior or generation on the boundary of the nudging domain (Hitchcock et al. 2022). Also, model ex- periments with a greater horizontal and vertical resolution REFERENCES may also yield better results, with evidence supporting a link between increased resolution and better representation of Ambaum, M. H., B. J. Hoskins, and D. 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Journal

Journal of Climate – American Meteorological Society

Published: Sep 15, 2022

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Lee, S. H.; Furtado, J. C.; Charlton-Perez, A. J.
Representation of the Scandinavia–Greenland pattern and its relationship with the polar vortex in S2S forecast models

Lee, S. H.; Charlton-Perez, A. J.; Furtado, J. C.; Woolnough, S. J.
Stochastic representations of model uncertainties at ECMWF: State of the art and future vision

Leutbecher, M.
Linking the tropical Northern Hemisphere pattern to the Pacific warm blob and Atlantic cold blob

Liang, Y.-C.; Yu, J.-Y.; Saltzman, E. S.; Wang, F.
Stratosphere–troposphere evolution during polar vortex intensification

Limpasuvan, V.; Hartmann, D. L.; Thompson, D. W.; Jeev, K.; Yung, Y. L.
Diagnostic methods for understanding the origin of forecast errors

Magnusson, L.
Windows of opportunity for skillful forecasts subseasonal to seasonal and beyond

Mariotti, A.
Estimates of flow-dependent predictability of wintertime Euro-Atlantic weather regimes in medium-range forecasts

Matsueda, M.; Palmer, T.
The influence of stratospheric wave reflection on North American cold spells

Matthias, V.; Kretschmer, M.
Do split and displacement sudden stratospheric warmings have different annular mode signatures?

Maycock, A. C.; Hitchcock, P.
A regime perspective on the North Atlantic eddy-driven jet response to sudden stratospheric warmings

Maycock, A. C.; Masukwedza, G. I.; Hitchcock, P.; Simpson, I. R.
Weather regimes: Recurrence and quasi stationarity

Michelangeli, P.-A.; Vautard, R.; Legras, B.
Tropical–extratropical geopotential height teleconnections during the Northern Hemisphere winter

Mo, K. C.; Livezey, R. E.
Sampling errors in the estimation of empirical orthogonal functions

North, G. R.; Bell, T. L.; Cahalan, R. F.; Moeng, F. J.
The effect of interactive ozone chemistry on weak and strong stratospheric polar vortex events

Oehrlein, J.; Chiodo, G.; Polvani, L. M.
Scikit-learn: Machine learning in Python

Pedregosa, F.
Observational evidence of a stratospheric influence on the troposphere by planetary wave reflection

Perlwitz, J.; Harnik, N.
Representation of synoptic-scale Rossby wave packets and blocking in the S2S prediction project database

Quinting, J.; Vitart, F.
The effect of Atlantic blocking action upon European climate

Rex, D. F.
Large-scale weather regimes and local climate over the western United States

Robertson, A. W.; Ghil, M.
Toward identifying subseasonal forecasts of opportunity using North American weather regimes

Robertson, A. W.; Vigaud, N.; Yuan, J.; Tippett, M. K.
Classification of stratospheric extreme events according to their downward propagation to the troposphere

Runde, T.; Dameris, M.; Garny, H.; Kinnison, D.
Relative roles of the MJO and stratospheric variability in North Atlantic and European winter climate

Schwartz, C.; Garfinkel, C. I.
Extratropical prediction skill of the Subseasonal-to-Seasonal (S2S) prediction models

Son, S.-W.; Kim, H.; Song, K.; Kim, S.-W.; Martineau, P.; Hyun, Y.-K.; Kim, Y.
Circulation regimes: Chaotic variability versus SST-forced predictability

Straus, D. M.; Corti, S.; Molteni, F.
The Arctic Oscillation signature in the wintertime geopotential height and temperature fields

Thompson, D. W.; Wallace, J. M.
Enhanced long-range forecast skill in boreal winter following stratospheric strong vortex conditions

Tripathi, O. P.; Charlton-Perez, A.; Sigmond, M.; Vitart, F.
The influence of weather regimes on European renewable energy production and demand

van der Wiel, K.; Bloomfield, H. C.; Lee, R. W.; Stoop, L. P.; Blackport, R.; Screen, J. A.; Selten, F. M.
Multiple weather regimes over the North Atlantic: Analysis of precursors and successors

Vautard, R.
Predictability of recurrent weather regimes over North America during winter from submonthly reforecasts

Vigaud, N.; Robertson, A. W.; Tippett, M. K.
Teleconnections in the geopotential height field during the Northern Hemisphere winter

Wallace, J. M.; Gutzler, D. S.
Probable causes of the abnormal ridge accompanying the 2013–2014 California drought: ENSO precursor and anthropogenic warming footprint

Wang, S.-Y.; Hipps, L.; Gillies, R. R.; Yoon, J.-H.
What is the polar vortex and how does it influence weather?

Waugh, D. W.; Sobel, A. H.; Polvani, L. M.
Potential applications of subseasonal-to-seasonal (S2S) predictions

White, C. J.
The down ward influence of sudden stratospheric warmings: Association with tropospheric precursors

White, I. P.; Garfinkel, C. I.; Gerber, E. P.; Jucker, M.; Aquila, V.; Oman, L. D.
The generic nature of the tropospheric response to sudden stratospheric warmings

White, I. P.; Garfinkel, C. I.; Gerber, E. P.; Jucker, M.; Hitchcock, P.; Rao, J.
A physical analysis of the severe 2013/2014 cold winter in North America

Yu, B.; Zhang, X.

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How Do Stratospheric Perturbations Influence North American Weather Regime Predictions?

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How Do Stratospheric Perturbations Influence North American Weather Regime Predictions?

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