To identify cold spells and their magnitudes, the cold wave magnitude index daily CWMId; is applied at each horizontal grid point. The daily cold spells index is based on at least 3 consecutive days below a daily threshold, defined as the 10th percentile of anomalies relative to the climatology of minimum daily air temperature smoothed with a 31 d running mean filter. The climatology is based on the reference period 1981–2020. The background trend is small relative to the daily anomalies, and we have verified that it does not affect the overall conclusions drawn in the paper (not shown).

To select cold spell regions, we first identified historical winter extreme 2 m temperature days on the hemispheric scale. To do so, a non-standardized version of the mid-latitude extreme MEX; index is applied for land in four latitude bands (35–45, 45–55, 55–65° N, and the entire mid-latitudes: 35–65° N). Dividing the mid-latitudes into three sub-bands enables an identification of extreme cold days in the southern, central, and northern sections of the mid-latitudes. The MEX provides an overall measure of mid-latitude temperature daily variance by averaging over each grid point in the mid-latitudes the squared anomaly of the temperature. Here, “non-standardized” means the MEX index is used without subtracting its time-averaged mean and dividing by its standard deviation, as done in . The top 1 % of daily MEX values – a criteria to classify cold extreme days – within all the winters during the period 1978–2024 yields 125, 117, 119, and 141 d for the bands 35–45, 45–55, 55–65, and 35–65° N, respectively. To identify the areas providing the largest cold anomaly contribution to the top 1 % MEX days of each latitude band, the CWMId (Fig. ) and temperature anomaly (Fig. S1 in the Supplement) of the extreme temperature days are calculated at each grid point in the mid-latitudes. Since these two diagnostics yield similar results, we focus on CWMId in the analysis.

Figure  shows the composite of the gph anomaly at 500 hPa and the cold-wave magnitude index for cold extreme days for each latitude band. For the top 1 % MEX days over the whole midlatitude (35–65° N; Fig. a), cold spells are more intense in the northwestern part of North America and northwestern Eurasia. The intensity decreases toward the south and east in both continents. The gph anomaly is positive over Greenland and the North Pacific Ocean, the two primary blocking regions . Over the North Atlantic Ocean, a negative gph anomaly is encountered that may indicate the negative phase of the North Atlantic Oscillation during these cold events, a phenomenon that promotes European winter cold spells e.g.,.

For the top 1 % MEX days over 35–45° N, little or no positive gph anomaly is encountered over Greenland, which is consistently associated with a weak intensity of CWMId over Europe (Fig. b). The main gph anomaly consists of a positive height anomaly over Alaska and a negative height anomaly downstream over West North America. Consistently, in this latitude band, CWMId is more pronounced over northwest North America, the central US, and Central Asia. Intense CWMId over Central Asia appears to differ from that of the central US, as it is likely not linked with higher latitudes, leading us to speculate that the cold spells in this region do not originate from Arctic cold air. Therefore, this region is not included in our study.

For the top 1 % MEX days over 45–55° N, in contrast to 35–45° N, there is a strong positive gph anomaly upstream of Europe, centered over Iceland, while the positive gph anomaly is weak upstream of North America (Fig. c). For this latitude band, the CWMId is strong over the middle and east of Europe and over the west of Canada. For the top 1 % MEX days over 55–65° N, CWMId is stronger at higher latitudes, encompassing Scandinavia, the west of Canada, and Northern Siberia (Fig. d). For this latitude band, there are two strong positive gph anomaly centers over Baffin Bay and the Bering Sea. The top 1 % MEX over the 45–55 and 55–65° N, similar to MEX across the entire midlatitude, exhibits a negative gph anomaly over the North Atlantic Ocean.

Based on the results of the composite of days with the top 1 % MEX over these four latitude bands, 9 regions are selected for the study of the impact of large-scale upper atmospheric circulation on cold spells, shown as hatched areas in Fig. . The regions include Northwest Canada (R01), Northwest US and Southwest Canada (R02), Central US (R03), British Isles (R04), Scandinavia (R05), Europe (R06), Northwest Russia (R07), Southwest Russia (R08), and Northern Siberia (R09). If 50 % or more of the grid points within each region experience cold spells on a given day, it is considered a cold spell day in that region. Applying grid-point thresholds between 20 % and 70 % are also tested to ensure that the drawn overall conclusions remain unchanged for an altered threshold. The number of days with 50 % or more of the grid points experiencing cold spells in each region, as based on the CWMId, is 239, 184, 164, 157, 170, 171, 195, 150, and 152, respectively, for R01 to R09.

The MEX metric may be somewhat biased because it accounts for both warm and cold anomalies. The simultaneous occurrence of cold anomalies in Eurasia and Northwest America, alongside warm anomalies in Greenland and Beringia, might have contributed to a higher MEX value for the selected days (see Fig. S1). Nevertheless, our selected regions cover all of Europe and the western part of North America, with the only significant area not included in our analysis being the eastern part of North America, which could be addressed in future research.

To test the null hypothesis that anomalies associated with the cold spells are zero, a Monte Carlo simulation with random samples is employed. Specifically, the average number of cold spell days across each region is compared with the average of an equivalent number of randomly selected days. Instead of selecting random days from the entire winter season, the selection is weighted based on the frequency distribution through the season of cold spell days, since cold spells appear to occur more frequently in some parts of winter. This Monte Carlo test is repeated at least 2000 times to ensure a sufficient sample size, and significant differences from zero are identified at the 95 % confidence level.

To enhance the trackability of atmospheric waves, a zonal Fourier decomposition of the gph field is employed to separate it into distinct harmonic waves based on the zonal wave number. The combination of the first five waves captures the large-scale ridges and troughs of atmospheric waves (Z1–5; see the solid line in Fig. a for an example). These ridges and troughs represent planetary waves in low latitudes and both planetary and synoptic-scale waves in high latitudes. The longitude position of these ridges and troughs at high latitudes aligns with the location of unfiltered gph data (Fig. a). However, in lower latitudes, the locations may not accurately represent the true positions of ridges and troughs in the unfiltered gph data. Nevertheless, since the main focus of this study is on high latitudes, where the wave structures are more accurately captured, the overall conclusions remain robust. In addition, the intention here is not to detect all ridges and troughs, but just those associated with the longest waves (Z1–5).

We estimate the speed of atmospheric wave zonal propagation by utilizing a ridge and trough tracking algorithm (Fig. a). Small amplitude ridges and troughs are ignored since they are often short lived (red stars in Fig. a). This is accomplished by considering a threshold for the gph at each local maximum (ridge) and local minimum (trough): at each time step (3-hourly) and latitude, the gph at the ridge (trough) position must be greater (less) than the gph at 30° to the west and east of the ridge (trough) position by at least 8 % of the Z1–5 amplitude (see blue texts and arrows in Fig. a). The Z1–5 amplitude is defined as the difference between the highest ridge and the lowest trough at all longitudes around a latitude circle at each time step and latitude. These criteria were carefully chosen after multiple experiments to ensure that all large waves are tracked while minor waves are neglected. Nonetheless, in all the experiments, as long as the thresholds do not suppress large waves, the drawn overall conclusions regarding wave speed changes remain unchanged for other parameter settings. We track each local ridge and trough in time over longitudes to find the ridge and trough speed, respectively. Also, we save all local ridge latitude and longitude (RLL) and trough (TLL) positions. At each latitude, ridges and troughs are labeled according to their positions (longitude) and are tracked to the next time step by searching for the nearest ridge or trough within a 12° limit both to the west and east of their location (Fig. b). If no corresponding feature is detected within this range, the ridge or trough is considered to have decayed. Conversely, if a new ridge or trough appears without a corresponding feature in the previous time step, it is considered to have formed.

Figure  displays the probability of the winter climatological positions of ridges and troughs, as well as the decay and formation of these features. Throughout winter, ridges and troughs tend to appear with greater frequency in certain areas compared to others (Figs.  and S4). This study focuses on three primary locations for ridges and three for troughs, assigning specific names to the ridges and troughs that occur in these regions. The three main ridges are located over the west coast of North America (hereafter, WNA ridge), the western flank of the Tibetan Plateau (hereafter, TP ridge), and the North Atlantic Ocean (hereafter, NaO ridge, at low latitude over the middle part and at high latitude over the eastern part of the ocean). The three main troughs are located over Eastern North America (hereafter, ENA trough), the eastern Mediterranean (hereafter, EMed trough), and East Asia (hereafter, EAsia trough). The daily speed of these six major ridges and troughs is determined by summing their 3-hourly values across their respective longitudinal ranges (Fig. c). The speeds of the ridges over 180–90° W, 60° W–30° E, and 30–120° E are added to calculate the daily speeds of the WNA ridge, the NaO ridge, and the TP ridge, respectively. The daily speeds of the ENA trough, EMed trough, and EAsia trough are also found by taking the sum of the speeds of the troughs over 120–30° W, 0–90° E, and 90–180° E, respectively (Fig. c).

Ridges and troughs typically form upstream of their climatological positions, sometimes wobble around these locations for several time steps (or days), and subsequently decay downstream (Fig. ). The WNA ridge, NaO ridge, and ENA trough exhibit stronger formation and decay at high latitudes compared to mid-latitudes, indicating that these ridges and troughs tend to persist longer in the midlatitudes. The TP ridge and EMed trough show large formation and decay at mid-latitudes, likely due to their interaction with the complex orography. Nonetheless, the likelihood of their decay and formation remains significantly lower than their climatological existence, indicating persistency of the ridges and troughs (Fig. ). The EAsia trough forms more strongly at midlatitudes, while its decay is more pronounced at higher latitudes. Over the Pacific Ocean, there is one ridge and one trough formation center, both of which decay downstream from their formation center. As can be seen from the climatological position of ridges and troughs, there is no strong ridge or trough in these locations, suggesting that these features are short-lived.

The probabilities of climatological ridge and trough positions here differ significantly from those in , mainly due to differing detection methods. identify ridges and troughs by comparing nearby grid points, while our method compares values across all longitudes at each latitude. Consequently, their probabilities are higher at high latitudes, where ridges and troughs are more distinct, but lower at low latitudes due to local gph flattening. Our method, however, still indicates ridges and troughs at subtropical and tropical latitudes, but these could rather be assigned to patterns such as subtropical highs than Rossby waves. Additionally, the probabilities estimated by our method represent the likelihood of ridges or troughs occurring at a specific location, rather than over an area defined by exceeding a curvature threshold of the gph field as in ; hence, the probabilities in our study are naturally much lower than those reported by .

Several metrics have been developed to study the waviness of the atmosphere e.g.,, many of which rely on the meandering of the 500 hPa isohypses (lines of constant gph) as a proxy. Considering the variations of wave amplitude with different isohypses – e.g., associated with the seasonal variability and poleward shift due to global warming – selecting an appropriate isohypses is critical. To address these challenges, derived isohypses from the daily average of the gph at 500 hPa between 30 and 70° N, hereby varying in time. They regarded the meandering of these isohypses as an indicator of the waviness of the atmospheric flow at 50° N.

Here we focus on identifying changes in waviness at each latitude and at 300, 500, and 850 hPa, which requires some modification of the approach: For a given study region, the most common isohypse at each time step and latitude is taken as the zonal average of the gph over half of the hemisphere centered on the center longitude of each study region. Then, the meridional wave extent for that latitude is defined as the meridional extent of the associated isohypse determined by calculating the difference between the maximum latitudinal location and the minimum location of that isohypse (total gph isohypse extent; TIE). By focusing on half of the hemisphere, it is ensured that at least one full ridge and trough are included within the region. In a similar way, the meridional extent of the first five Fourier decomposition wave numbers of gph (Z1–5) is calculated (planetary isohypse extent; PIE). During cold spells in the defined regions, both TIE and PIE demonstrate the same significant pattern of changes (not shown). This shows the modest impact of small waves on the wave amplitude changes during cold spells. Therefore, we only consider the results of the meridional extent of Z1–5.

Some isohypses have unconnected subparts (see Fig. a), referred to as cut-off low and cut-off high, which might lead to interpretation errors in the wave amplitude. For instance, in Fig. a, at high latitudes (around 85° N), the most common isohypse is 8300 m, and at these latitudes, the waviness of this isohypse is small. But, due to a cut-off low of this isohypse at midlatitudes, the estimated wave amplitude at 85° N is large. Moreover, the PIE does not specify the separate contribution to the wave amplitude from ridges and troughs. Understanding these contributions is critical, especially during cold spells, as it helps determine which parts of the wave – ridge or trough – have the greater impact.

Hence, we here propose an alternative metric where the meridional extent of the Z1–5 ridges and troughs are measured separately (Fig. b). The meridional extent of the ridge is given as the latitudinal difference between the RLL and the isohypse determined by the zonal average of the Z1–5 over a 90° longitude range centered at the RLL (Fig. b, red line); hence, the meridional extent of a ridge at a given latitude is obtained as the difference between that latitude and the latitude of the first cross of the isohypse when moving northward along the longitude of the RLL. The employed method does not take into account the possible meridional tilting of waves. To address this, at 300 hPa, the wave amplitude is calculated for all longitudes within the 90° longitude range. The highest value obtained is then considered the meridional amplitude of the ridge. While the highest value often yields slightly larger wave amplitudes than that obtained at the RLL, the overall conclusions regarding changes in wave amplitude during cold spells remain the same for both approaches (not shown). Therefore, to reduce computational usage, we used the amplitude at the ridge position. The meridional extent of troughs is obtained in a similar way but moving southward from TLL.

The estimated meridional amplitudes of a given ridge (trough) are highly influenced by the longitudinal range over which the zonal average of the Z1–5 is taken for determining the isohypse. Using a smaller longitude range yields an isohypse value closer to the value of Z1–5 at RLL (TLL), which leads to small amplitudes. Using a large longitude range might yield an isohypse that cannot be found near the RLL (TLL), resulting in a missing value for the amplitude. Therefore, after conducting multiple experiments, the 90° threshold is chosen. Nonetheless, despite the arbitrariness of the meridional wave amplitude when defined this way, its magnitude exhibits a consistent latitude-dependent pattern and comparable bands of significance in both the climatology and cold-spell cases, which is important and sufficient for this study.

To compare the results of the refined method with PIE, considering the total wave amplitude, the average of the amplitude of the ridges plus those of the troughs over half of the hemisphere centered at each study region is calculated (ridge-trough isohypse extent; RTIE). For climatology, the RTIE amplitude (Fig. S2, gold lines) is nearly half of the PIE amplitude. During cold spells, there are likewise many similarities between the RTIE and the PIE, yet they have some important differences. During cold spells, the PIE amplifies over a wider latitude range compared to the RTIE. This could be attributed to the influence of cut-off lows or cut-off highs on the amplitude of distant locations.

To investigate the impact of atmospheric waves on the formation of winter cold spells, nine regions with a high likelihood of cold spells are selected (Fig. ). These regions are chosen based on their significant contribution to the historical extreme 2 m temperature days on a hemispheric scale (see Sect. 2.2). During cold spells over North America (R01, R02, and R03), large-scale gph anomaly patterns show an anticyclonic gph anomaly over the North Pacific and a cyclonic gph anomaly over North America (Fig. S3a, b, and c), similar to the findings of other studies e.g.,. During cold spells, the WNA ridge and the ENA trough are located to the west of their corresponding climatological positions (Fig. a, b, and c). This repositioning is more pronounced for R02 and R03 than for R01. The NaO ridge extends from northwest Africa to northwest Europe, specifically exhibiting an eastward relocation at lower latitudes. For cold spells over western regions of Europe (R04, R05, and R06), the gph anomaly at 500 hPa shows a positive anomaly located north of a negative anomaly (Fig. S3d, e, and f). This anomaly resembles a high-over-low North Atlantic blocking situation, which is similar to the findings of other studies e.g.,. For R04 and R05, at high latitudes, the locations of the NaO ridge and EMed trough tend to be to the west of their respective climatological centers (Fig. d and e). During cold spells over Eastern Europe (R08), the EMed trough and the TP ridge at high latitudes are located to the east of their respective climatological locations (Fig. h). The cold spell over Northern Siberia (R09) primarily aligns with a deep trough downstream over northwest Asia, as depicted in Figs. S3i and i. Despite the high latitude of this region, it is sufficiently far from the pole to be affected by cold advection from higher latitudes during cold spells . For R09 (Fig. i), ridges form over the western flank of the Ural Mountains, and the EAsia trough at high latitudes is located in the west of its climatological location. These findings indicate that a distinct positioning pattern of ridges and troughs on a hemispheric scale occurs during cold spells over each region. These patterns can also be observed at 500 hPa (not shown) and 850 hPa (not shown), indicating that during cold spells the entire troposphere is located in a rather barotropic manner. These findings suggest an interaction between atmospheric dynamics and surface temperatures.

Figure  shows the time-development of ridges and troughs before and during cold spells for the composites of regions in North America and Europe. This choice is based on the similarity of the upstream ridge and downstream trough in the selected regions for each composite. In North America, cold spell regions are predominantly located in the vicinity of the WNA ridge. In North America, before cold spells occur, the ridge encompasses a broad area, including the cold-spell regions (Fig. a). At the start of cold spells, the upstream ridge becomes more localized in the western side of the cold-spell regions (Fig. e). This pattern provides northerly wind flow from the Arctic toward the cold-spell area (Fig. S4). In Europe, most of the cold spell regions are primarily located in the vicinity of the EMed trough. Similar to North America, before cold spells occur, the ridge encompasses a wide area (Fig. b), indicating that cold or warm anomalies are not concentrated to specific locations. However, with the onset of cold spells (Fig. f), ridges and troughs become more concentrated, with a ridge positioned upstream and a trough located within the cold-spell regions (Fig. f). Hence, associated with cold spells, roughly the same type of regional-scale circulation pattern is found in both North America and Europe. These findings highlight the critical role of the nearest upstream ridge and downstream trough in the formation of cold spells.

For cold spells, the largest ridge and trough amplitude and speed differences relative to climatology occur for the local upstream ridge and downstream trough (Figs. S5 and S6). Generally, the amplitude and speed differences of ridges and troughs decay as their distance increases from the location of the cold spell. A composite over all cold spells and regions of the upstream ridges and downstream troughs provides a general overview of the local wave behavior during cold spells (Fig. ). The following refers to changes in ridges or troughs (such as amplification or slowing down), indicating significant alterations in their behavior during cold spells compared to the climatological average behavior of these features within their respective longitude bands. The upstream ridge significantly amplifies (Figs. a and S5), facilitating the transfer of Arctic cold air to the cold spell region. During cold spells, the northward winds entering the Arctic are increasing west of the upstream ridge, while the southward winds exiting the Arctic are increasing east of the upstream ridge (Fig. S4). The upstream ridge also significantly slows down (Figs. b and S6), implying an increase in the persistence of the warm northward wind to the Arctic and the cold southward wind to the cold spell location. The persistent warm northward wind to the Arctic might be the reason behind the correlation between cold events and a warmer Arctic found in other studies e.g.,; yet more research is needed to discover the impact of the amplified ridge on Arctic temperatures. At low latitudes, the upstream ridge moves faster during cold spells, which are mainly associated with North America and northern Siberia (Fig. S6). The underlying reasons for this pattern are not clear and are left for future research.

The downstream trough amplifies (deepens) at latitudes north of the cold spell location, while it weakens at southerly latitudes (Figs. a and S5). For the cold-spell regions in the southern as compared to the northern mid-latitudes, the increase in trough amplitude develops more to the south (Fig. S5). Amplification of this trough contributes to enhancing the southward cold wind from the Arctic into the cold spell region (Fig. S4). Similar to the upstream ridge, the downstream trough significantly slows down (Figs. b and S6), hereby increasing the persistency of the cold southward winds into the cold spell region.

In the previous section, it was shown that cold spells are associated with higher-amplitude and slowing ridges and troughs in the vicinity of the cold spells. This raises the question of the timing of these changes in relation to the occurrence of cold spells. Previous research has shown that upper-atmospheric changes – such as amplified planetary waves , blocking , and the stratospheric polar vortex – can lead to surface extreme events. Here, a day-lag analysis of composites over cold spells is provided to shed light on the causal relationship between upper-level waves and surface temperature (Figs.  and S7). Since the largest wave amplitude and speed differences relative to climatology belong to the upstream ridge and the downstream trough, their time series are plotted here.

The minimum speeds of the upstream ridge and downstream trough occur at a negative time lag several days prior to the onset of the cold spell, which is taken as 30 % of grid points in the region experiencing a cold spell and indicated by the lag zero. Both amplified ridge and trough are also encountered prior to the cold spell, but the speed reduction precedes the amplitude increase (Fig. a and b). This could potentially be attributed to the breaking of waves associated with exceeding the wave for the wave activity flux . In addition, this pattern can also be explained by the poleward excursion of low-potential vorticity subtropical air ahead of a slowly moving elongated trough . The preceding of the upstream ridge and the downstream trough amplification relative to the cold spell suggests that cold spells arise as a result of slow and amplified local upper-atmospheric waves. For Northern Siberia, the downstream trough amplifies at a positive time lag several days after the cold spells (Fig. S7i), indicating that the amplified trough cannot be responsible for the cold spell in this region. This is consistent with the formation of cold spells in Northern Siberia being due to diabatic processes . Nonetheless, the upstream ridge amplifies and slows down prior to the onset of cold spells in this region (Fig. S7i). The wave speed anomaly shows almost the same pattern for both 500 and 300 hPa (Fig. a and b), which suggests an almost barotropic wave signal through the troposphere during cold spell events. However, the wave-speed reduction occurs somewhat later at 850 hPa, indicating a downward propagation of the wave-speed signal.

The lead-lag behavior of other significant and non-significant changes in the upstream ridge and downstream trough (Figs. S5 and S6) is also analyzed. Their time lag either shows no significant changes or occurs after the upstream ridge and downstream trough's significantly positive amplitude and negative speed anomalies (not shown). However, there are some exceptions: During cold spells over North America (R01, R02, and R03), the lead-lag behavior of the upstream ridge anomaly at the lower midlatitudes (35–50° N) indicates that this ridge slows down at these latitudes before it does at other latitudes (Fig. S8a). For both Eastern European regions (R07 and R08), latitudes with a reduction in the upstream ridge amplitude (35–40° N; Fig. S7g and h) show strong negative amplitude anomalies weeks before the onset of the cold spell (Fig. S8b). This indicates that the ridge at lower latitudes tends to move zonally, which may weaken the mixing of warm tropical air with colder midlatitude air long in advance of the cold spell formation.