1 Introduction

In recent two decades, extreme cold events have become more intense and frequent in boreal winter (Petoukhov and Semenov 2010; Overland et al. 2015; Cohen et al. 2020), such as the snowstorm disaster over South China in 2008 (Wu et al. 2011; Kuang et al. 2019) and eastern U.S. severe cold surge in 2013/2014 (Whan et al. 2016; Cohen et al. 2018). A recent case was the extreme cold wave in East Asia in early January 2021, which has been hotly investigated (Yao et al. 2022; Zheng et al. 2022; Zhang et al. 2022a, b; Bueh et al. 2022; Yu et al. 2022). This cold surge swept across the whole eastern China and considered to be a “boss level” case, which exerted a profound impact on socioeconomic development and human life (Dai et al. 2022; Yu et al. 2022). Particularly in North China (NC), station temperature dropped by more than 10 °C (Fig. 1a), and the lowest temperatures on 7 January 2021 reached − 19.6 °C and − 19.9 °C in Beijing and Tianjin, respectively, setting new records for this day (Zhang et al. 2022a).

Fig. 1
figure 1

(a) The difference of minimum daily mean temperature (unit: ℃) over 4–9 January 2021 and that on 3 January 2021. The red box represents NC (33.5° N–43.5° N, 105° E–120° E). (b) The daily SAT (black line; unit: ℃) averaged over NC from 1 December 2020 to 31 January 2021, the climatological values (blue line; unit: ℃) for these dates averaged over 1979–2020, and the daily climatological means plus or minus its one standard deviation (red lines; unit: ℃). (c) Daily Tmin (blue solid line; unit: ℃) and Tmax (red solid line; unit: ℃) averaged over NC from 1 December 2020 to 31 January 2021. Blue and red dashed lines represent their 10th percentile values, respectively. The gray shading represents the extreme cold event episode (i.e. 4–9 January 2021)

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Previous studies suggested that atmospheric internal factors can affect boreal winter cold extremes, such as the Arctic Oscillation (AO)/North Atlantic Oscillation/Northern Hemisphere annular mode (Gong et al. 2001; Jeong and Ho 2005; Wang and Chen 2010; Ding et al. 2014; Xie et al. 2019), the stratospheric quasi-biennial oscillation (Yamazaki et al., 2020), storm tracks (Basu et al. 2013; Yamazaki et al. 2020), the Southern Hemisphere annular mode (Wu et al. 2009; Zheng et al. 2014), etc. In addition, atmospheric teleconnections like the North Pacific Oscillation (Wallace et al. 1981; Linkin et al. 2008), the Pacific/North American pattern (Zhou et al. 2012), and the Eurasian teleconnection (EU) (Liu and Chen 2012; Luo et al. 2021) may also lead to wintertime cold extremes. In particular, the Ural high (Zhang et al. 2012) and East Asian trough (Zhang et al. 1997; Wang et al. 2009; Song et al. 2016) act as crucial synoptic systems affecting Eurasian extreme cold events. Moreover, the features of cold extremes are also associated with external forcing including La Niña (Kim and Ahn 2012; Basu et al. 2013), the Pacific Decadal Oscillation (Wang et al. 2008), the Atlantic Multidecadal Oscillation (Hao et al. 2016), Eurasian snow cover (Yu et al. 2018), and Arctic sea ice (Tang et al. 2013; Gao et al. 2015; Chen and Luo 2017; McKenna et al. 2018; Zhang et al. 2020), etc. For a specific cold extreme in East Asia, its causes can be quite complex.

As for the “boss level” cold extreme in NC in early January 2021, some studies found that the stratospheric sudden warming and its corresponding downward propagation of planetary wave played a role (Zhang et al. 2022a, b), or believed that the synergistic effect of the warm Arctic and La Niña provided an indispensable background to intensify the atmospheric circulation anomalies in mid-high latitudes (Zheng et al. 2022; Yao et al. 2022). In fact, the warm Arctic and La Niña pattern occurred every a few years while boreal wintertime cold extremes did not always coincide with it (Zhang et al. 2022a). Therefore, the effect of warm Arctic and La Niña pattern on a specific cold event (e.g. 2021 case) is hard to explain due to their difference in timescale. On the other hand, some scholars also analyzed the roles of the Siberian high, the Ural blocking, large-scale tilted ridge and trough over Asian continent from the perspective of tropospheric circulation causing cold events directly (Yao et al. 2022; Bueh et al. 2022). Furthermore, researchers also studied the effect of Arctic sea ice on this cold event (Zhang et al. 2022a, b; Zheng et al. 2022; Yu et al. 2022). Some of them held that the Arctic sea ice can affect East Asia through the Polar–Eurasian teleconnection (POL) by stimulating the eastward Rossby wave (Xu et al. 2020; Han et al. 2021; Yang et al. 2021). However, in addition to the POL at high latitude, the subtropical wave train also has an important impact on East Asian weather and climate such as rainfall in southern China and haze over NC (Wallace and Gutzler 1981; Peng et al. 1995; Czaja and Frankignoul 2002; Li and Betas 2007; Li and Sun 2015; An et al. 2020; An et al. 2022a).

Based on the above analysis, many studies have revealed the impact of Arctic sea ice on this cold wave in early January 2021, but the potential effect of synoptic Arctic sea ice variation on this event has not been clarified. In addition, although the subtropical westerly jet and associated wave train contributed to the East Asian extreme cold event, whether they have synergistic effect with Arctic sea ice on this event has not been revealed. To illuminate these issues, we performed this study and organized the paper as follows. Section 2 describes the datasets and methods used in this paper. Section 3 mainly describes features of the extreme cold event and associated atmospheric circulation. Section 4 introduces evolution of the subtropical wave train and POL. Section 5 reveals principal mechanisms that the two Rossby waves affect the extreme cold event over NC, which is simulated using the linear baroclinic model (LBM). Conclusions and discussion are given in Sect. 6.

2 Data and method

2.1 Data

In this study, we used daily-mean reanalysis data from the National Centers for Environmental Prediction (NCEP) and National Center for Atmospheric Research (NCAR) (Kalnay et al. 1996) covering the period from 1979 to 2021, including air temperature, minimum temperature (Tmin), maximum temperature (Tmax), geopotential height, horizontal winds (u and v) and vertical velocity (omega) with vertical levels from 1000 to 10 hPa and a horizontal resolution of 2.5° longitude × 2.5° latitude. Besides, surface air temperature (SAT), net long-wave radiation (LW), net sensible heat flux (SH), and net latent heat flux (LH) were also obtained from the NCEP/NCAR reanalysis 1. The LW, SH and LH are on T62 Gaussian grids while SAT is at 2.5° × 2.5° grids. For sea ice concentration (SIC), we obtained the daily-mean data from the National Oceanic and Atmospheric Administration (NOAA) which spans 1981 to 2021 with a 0.25° × 0.25° spatial resolution. The daily-mean precipitation data was provided by the fifth generation European Center for Medium Range Forecasting Reanalysis (ERA-5) with a spatial resolution of 0.5° × 0.5° over 2000–2021.

In addition, we also used daily AO index provided by the Climate Prediction Center (CPC) from 1 December 2020 to 31 January 2021 (Thompson and Wallace 2000). To accurately present the change of SAT in NC, the station-based daily SAT observations of 645 stations in NC covering 1980–2021 provided by the National Meteorological Information Center, China Meteorological Administration were also used. All of the daily anomalies were relative to the climate mean during 1979–2020 except for precipitation and the station-based daily SAT which were relative to the climatological mean during 2000–2020 and 1981–2010, respectively.

2.2 Methodology

To obtain the dominant spatiotemporal mode of wave train influencing the NC cold extreme over Eurasia, empirical orthogonal function (EOF; Lorenz 1956) analysis was applied to daily 300-hPa meridional wind anomaly over the Northern Hemisphere during January 2021, which was tested according to North et al. (1982).

To explore the propagation of anomalous Rossby waves along mid-high latitudes in the Northern Hemisphere, the horizontal stationary wave activity flux (WAF) was calculated as follows (Takaya and Nakamura 2001):

$${\varvec{W}}=\frac{pcos\phi }{2\left|\mathbf{U}\right|}\bullet \left(\begin{array}{c}\frac{U}{{a}^{2}{cos}^{2}\phi }\left[{\left(\frac{\partial {\psi }^{^{\prime}}}{\partial \lambda }\right)}^{2}-{\psi }^{^{\prime}}\frac{{\partial }^{2}{\psi }^{^{\prime}}}{\partial {\lambda }^{2}}\right]\\ +\frac{V}{{a}^{2}cos\phi }\left[\frac{\partial {\psi }^{^{\prime}}}{\partial \lambda }\frac{\partial {\psi }^{^{\prime}}}{\partial \phi }-{\psi }^{^{\prime}}\frac{{\partial }^{2}{\psi }^{^{\prime}}}{\partial \lambda \partial \phi }\right]\\ \begin{array}{c}\frac{U}{{a}^{2}cos\phi }\left[\frac{\partial {\psi }^{^{\prime}}}{\partial \lambda }\frac{\partial {\psi }^{^{\prime}}}{\partial \phi }-{\psi }^{^{\prime}}\frac{{\partial }^{2}{\psi }^{^{\prime}}}{\partial \lambda \partial \phi }\right]\\ +\frac{V}{{a}^{2}}\left[{\left(\frac{\partial {\psi }^{^{\prime}}}{\partial \phi }\right)}^{2}-{\psi }^{^{\prime}}\frac{{\partial }^{2}{\psi }^{^{\prime}}}{{\partial \phi }^{2}}\right]\end{array}\end{array}\right)$$
(1)

where W denotes the horizontal WAF (unit: m2 s−2). U (\(=\) (U, V); m s−1) represents the basic flow. \(\psi (=\) Φ \(/f)\) is the geostrophic stream function. Φ (m) is the geopotential height; \(f (=2\mathrm{\Omega sin}\phi )\) is the Coriolis parameter. \(a\) is Earth’s radius; \(\Omega\) is earth’s rotation rate. \(\phi\) and \(\lambda\) represent latitude and longitude, respectively. \(p\) is pressure/(1000 hPa). W was calculated using daily reanalysis data.

To analyze the generation of Rossby wave, the linearized Rossby wave source following Sardeshmukh and Hoskins (1988) was calculated as follows:

$$\mathrm{S}=-{\nabla }_{H}\bullet \left\{{{\varvec{u}}}_{\chi }^{^{\prime}}\left(f+\overline{\zeta }\right)\right\}-{\nabla }_{H}\bullet \left\{{{\varvec{u}}}_{\chi }^{^{\prime}}{\zeta }^{^{\prime}}\right\}.$$
(2)

In this equation, \({\varvec{u}}=(\mathrm{u},\mathrm{ v})\) denotes the horizontal wind velocity, \({\nabla }_{H}\) is the horizontal gradient, and \(\upchi \left(\psi \right)\) represents the divergent (rotational) component.\(\upzeta\) and \(f\) are the relative vorticity and the geostrophic vorticity, respectively. Overbars indicate climatological mean and primes signify anomalies.

In order to determine the distribution of anomalous heating released by rainfall, we calculated the atmospheric apparent heat source (Q1) according to the equation obtained by Yanai et al. (1973). The equation can be described as follows:

$$\mathrm{q}1= \frac{\partial T}{\partial t}-\left(\omega \sigma -{\varvec{V}}\bullet \nabla T\right)$$
(3)
$$\mathrm{Q}1={c}_{p}\frac{\partial T}{\partial t}-{c}_{p}\left(\omega \sigma -{\varvec{V}}\bullet \nabla T\right)$$
(4)

where \({c}_{p}\) denotes the specific heat at constant pressure, T, t and \(\upomega\) represent the air temperature, time and vertical pressure velocity, respectively.\(\upsigma =\left(\mathrm{RT}/{c}_{p}p\right)-\left(\partial T/\partial p\right)\) is the static stability. R and p are the gas constant and pressure, respectively. V is the horizontal wind vector. ∇ represents the horizontal gradient operator. q1 is the net heat flux and Q1 represents total diabatic heating (including radiation, latent heating and surface heat flux) and subgrid-scale heat flux convergence (Yanai et al. 1973).

In addition, the LBM (Watanabe and Kimoto 2000) was used to verify the observed forced circulation anomalies. This model has a horizontal resolution of T42 (roughly equivalent to 2.8°) and 20 vertical sigma levels and has been widely used to examine the steady linear response to idealized diabatic heating (Lu and Lin 2009; An et al. 2022a). It consists of basic equations linearized with respect to the mean state of December (0)–January (1) climatology from the NCEP/NCAR reanalysis for 1981–2010. The LBM was integrated for 30 days. The variable (e.g. zonal wind, meridional wind, and geopotential height) on the last day (e.g. day 30) was taken as the steady response to the prescribed heating over North Atlantic or Barents Sea.

3 Features of the 2021 extreme cold event and associated atmospheric circulation

The extreme cold event hit China during 4–9 January 2021, bringing a wide range of extremely low temperatures. The difference of the minimum daily mean temperature during the event and daily mean temperature on 3 January (i.e. the day before the outbreak of this event), representing the temperature drop, is shown in Fig. 1a. Most regions of China to the east of 100° E experienced a significant cooling, and the temperature drop in NC (33.5° N–43.5° N, 105° E–120° E) was more than 10 °C (Fig. 1a).

The station-based daily SAT was used to calculate the daily regional average temperature anomalies over NC from 1 December 2020 to 31 January 2021 (Fig. 1b). SAT over NC was consecutively below –1 standard deviation from 4 to 9 January 2021 (gray shading in Fig. 1b). The daily Tmin and Tmax during this period were also lower than their 10th percentile, respectively (Fig. 1c). Based on the analysis above, we selected the period 4–9 January 2021 as the study period in the following analysis.

Figure 2 showed the atmospheric circulation anomalies during 4–9 January 2021. At 300 hPa, the meridional wind was characterized by an alternating northerly and southerly anomalies within the mid–high latitudes westerly jet (Fig. 2a). The northerly and southerly anomalies at middle latitudes corresponded to the geopotential height anomalies which were similar to the subtropical wave train along westerly jet (Hoskins et al. 1993; Li and Sun 2015; An et al. 2020, 2022a, b). At high latitudes, the northerly and southerly anomalies resembled the POL pattern (Chen et al. 2022).

Fig. 2
figure 2

(a) The average meridional wind anomalies (shaded; unit: m s–1) and westerly jet (green contours; faster than 20 m s–1) at 300 hPa during the extreme cold event. (b) The average geopotential height anomalies (shaded; unit: m) at 500 hPa and wind anomalies (arrows; unit: m s–1) at 850 hPa during the extreme cold event. The red box represents NC

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At 500 hPa (Fig. 2b), the POL teleconnection was located between polar region and East Asian. The positive geopotential height anomaly over Siberia might strengthen the Ural ridge, and the negative anomaly over East Asian might deepen the East Asian trough. According to Luo et al. (2021), such a circulation pattern usually made the Arctic cold air flow to East Asia easily and strengthened the East Asian winter monsoon. Thus, the geopotential height anomalies related to the POL pattern in the high latitudes might have a significant impact on this cold extreme over NC. In the lower to mid latitudes, there was a clear Rossby wave train along the subtropical westerly jet propagating eastward from the North Atlantic (Fig. 2a, b). Both the subtropical wave train and the POL had a cyclonic anomaly over East Asia, which might strengthen the East Asian trough. Furthermore, the wind vector at 850 hPa also corresponded to the geopotential height anomaly at 500 hPa, and NC located in the southwest of the anomalous cyclone over East Asia was occupied by northerly anomaly in the lower troposphere (Fig. 2b), which supported the Arctic cold air outbreak to NC.

To obtain the dominant pattern of atmospheric circulation, the leading EOF mode of the 300-hPa meridional wind in January 2021 is shown in Fig. 3. The first mode of the meridional wind was a Rossby wave train manifested by the northerly and southerly anomalies of the 300-hPa meridional wind (Fig. 3a), consistent with Li et al. (2017), which explained 16.6% of total variance. The subtropical westerly jet was located along the strong subtropical wave train at mid latitudes (Fig. 3a), suggesting that the subtropical westerly jet might serve as a waveguide for this wave train (Hoskins and Ambrizzi 1993). In addition, the meridional wind anomalies over the Barents Sea and Kara Sea indicated an anticyclonic anomaly of POL (Fig. 3a). The principal component (PC1) was positive (about 0.6 to 1.6) during the extreme cold event 4–9 January 2021 (Fig. 3b). Therefore, NC was characterized by a northerly wind anomaly during this period. The results in Figs. 1 and 2 further elucidated that the subtropical wave train along westerly jet and POL were important in causing the extreme cold event over NC in early January 2021.

Fig. 3
figure 3

(a) The spatial pattern and (b) time series (i.e. PC1) of the leading EOF mode for 300-hPa meridional wind anomalies in January 2021 (shaded) and westerly jet (green contours for winds faster than 20 m s−1) at 300 hPa during the extreme cold event. The value in the upper-right corner of (a) is the percentage of variance explained by the first mode. The gray shading represents the extreme cold event during 4–9 January 2021

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4 Spatial characteristics and evolutions of the subtropical wave train and POL

To further understand the spatial characteristics of subtropical wave train and POL, three-dimensional geopotential height anomalies were displayed based on composite analysis. Figure 4a showed the average geopotential height anomalies at 300 hPa during the period of this cold extreme. As can be see, the northern POL and southern wave train both acted on a cyclonic anomaly over East Asia deepening the low pressure anomaly there (Fig. 4a).

Fig. 4
figure 4

(a) The average geopotential height anomalies (shaded and contours; unit: m) at 300 hPa during the extreme cold event. The purple dashed line is the propagation path of the subtropical wave train, and the blue dashed line represents the propagation path of the POL. The red box represents NC. (b) The average vertical structures of the anomalous geopotential height (contours; unit: m) and air temperature (shaded; unit: ℃) along the blue line during 4–9 January 2021. (c) The same as (b), but along the purple line

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Furthermore, we calculated the average vertical structures of the anomalous geopotential height and air temperature along the blue and purple lines (Fig. 4a) during 4–9 January 2021, respectively (Fig. 4b). Both the POL (Fig. 4b) and subtropical wave train (Fig. 4c) had a quasi-barotropic vertical structure. Their anomaly centers were mainly located in the upper layer (i.e. 300 hPa) of the troposphere, and the wave-like geopotential height anomalies were strongest at 200–300 hPa (Fig. 4b, c). And a strong negative temperature anomaly throughout the whole troposphere over NC (Fig. 4b, c). For the northern branch, temperature in the Arctic region (40° E–65° E) showed a positive anomaly below 300 hPa while a negative anomaly above 300 hPa (Fig. 4b). For the southern branch, temperature in the Atlantic region (55° W–35° W) also showed a positive anomaly below 300 hPa and a negative anomaly above 300 hPa (Fig. 4c). Similar results can also be obtained with ERA-5 reanalysis data (not shown).

To investigate the evolutions of the subtropical wave train and POL, horizontal WAF was calculated (Fig. 5). All data used here has been conducted a 5-day running mean. At lower latitudes, there was a positive geopotential anomaly over North Atlantic on 25 December 2020 (Fig. 5a) and then propagated eastward along subtropical westerly jet which served as a waveguide (Fig. 5b−f). Around 4 January 2021, the subtropical wave train spread to East Asia on 4 January 2021 (Fig. 5f). At higher latitudes, a Rossby wave propagated southward from the Barents Sea since 2 January 2021 (Fig. 5e, f), and reached East Asia at around 4 January 2021 (Fig. 5f). Meanwhile, the positive geopotential height anomaly over Barents–Kara Seas strengthened (Fig. 5a−f). Noting that the prominent negative geopotential height anomaly (i.e. the cyclonic anomaly) appeared in East Asia on 4–6 January 2021 (Fig. 5f, g) contributed by both the subtropical wave train and POL. Then, as the two Rossby waves continued to spread eastward, the anomalous cyclone over East Asia weakened and this extreme cold event decayed (Fig. 5h−j).

Fig. 5
figure 5

The 5-day running mean geopotential height anomalies (shaded; unit: m) and WAF (arrows; unit: m2 s−2) at 500 hPa on (a) 25, (b) 27, (c) 29, (d) 31 December 2020 and (e) 2, (f) 4, (g) 6, (h) 8, (i) 10, (j) 12 January 2021, respectively. The red box represents NC

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5 Physical mechanisms triggering the Rossby wave trains

5.1 The role of Atlantic precipitation anomalies on the subtropical wave train

To explain the origin of the subtropical wave train, Fig. 6a showed the average Rossby wave source at 300 hPa during 21–23 December 2020. A significant negative Rossby wave source was identified over North Atlantic around 30° N–55° N, 35° W–15° W (Fig. 6a). Furthermore, we analyzed the distribution of precipitation anomalies during this period (Fig. 6b). Large values of precipitation anomalies mainly distributed around 40° N–60° N, 70° W–40° W, which were located in the west of the Rossby wave source (Fig. 6b). The ascending motion related to the upper divergence and lower convergence was strong over the positive anomalous precipitation region of North Atlantic (Fig. 6c). Similarly, the descending motion related to the upper convergence and lower divergence was also strong over Rossby wave source region (Fig. 6c). These two vertical branches contributed to form a vertical circulation over North Atlantic (Fig. 6c), which was a possible reason that the Rossby wave source was located in the east of the precipitation area.

Fig. 6
figure 6

(a) The average Rossby wave source (shaded; CI (contour interval) = 10−10 s−2) at 300 hPa during 21–23 December 2020. (b) The positive precipitation anomalies (shaded; unit: mm day−1) during 21–23 December 2020. (c) Longitude–height section of geopotential height anomalies (shaded; unit: m), divergence anomalies (contours; CI = 10−7 m s−1; the solid contours represent divergence and dashed contours represent convergence) and wind vector (arrows; zonal and vertical velocity; unit: m s−1) averaged over 40° N–60° N

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To estimate the diabatic heating produced by the rainfall anomalies over North Atlantic, we calculated q1 quantity using Eq. (3). Figure 7 showed the average q1 anomalies during 21−23 December 2020. An obvious positive q1 anomaly appeared in the North Atlantic (30° N−50° N, 50° W−30° W) (Fig. 7a). Additionally, shortwave radiation from the sun reaching the lower troposphere decreased during rainfall because it was blocked by clouds (not shown), therefore, the heating characterized by the positive anomalous q1 was mainly contributed by latent heat component released from precipitation. To investigate the vertical profile of q1, we further calculated the values of q1 between 1000 and 100 hPa in 40° N, 40° W (Fig. 7b). And the maximum value of q1 (i.e. 6 K day−1) occurred at 700 hPa (Fig. 7b).

Fig. 7
figure 7

(a) The average q1 anomalies (shaded; unit: K day−1) during 21–23 December 2020 at 700 hPa. (b) The vertical profile of actual anomalous q1 (blue solid line; unit: K day−1) and imposed heat forcing (red dashed line; unit: K day−1) in the experiment at the location marked by the black star in (a). (c) The heat forcing at 700 hPa (shaded, unit: K day−1) and the steady response of geopotential height anomalies (contours; unit: m) and wind anomalies (arrows; unit: m s−1) at 300 hPa. The red box represents NC

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To further prove that the heat source can excite the eastward propagation Rossby wave, a numerical experiment based on LBM was conducted to verify the observational results and test the potential role of North Atlantic precipitation in inducing the subtropical wave train. This numerical experiment simulated the atmospheric response to heat forcing induced by increased rainfall in the North Atlantic (Fig. 7c). The experiment was initialized with a diabatic heating centered at 40° N, 40° W which matched the maximum value of q1 (black star in Fig. 7a). The maximum heating with an amplitude of 8 K day−1 at 700 hPa was shown as the red dashed line in Fig. 7b, which was used to force the experiment. Figure 7c illustrated the response of 300-hPa geopotential height and wind anomalies to the diabatic heating over North Atlantic. The LBM results showed that the subtropical wave train appeared in the mid-high latitudes, and there was an anticyclonic anomaly over North Atlantic and a cyclonic anomaly over NC (Fig. 7c). These results confirmed that the anomalous precipitation in the North Atlantic and its related heat source can stimulate the Rossby wave propagating eastward along the mid-high latitudes which might produce a cyclonic anomaly over NC.

5.2 The role of the Arctic sea ice on the POL

To illustrate the physical mechanisms by which the POL influenced the cold extreme over NC, we analyzed the change of the Arctic sea ice and its relationship with this extreme cold event. Figure 8a showed the distribution of the SIC change, SIC standard deviation and 1000-hPa wind anomalies from 25 December 2020 to 10 January 2021. The area where sea ice changed significantly in this period was mainly located in the Barents Sea and Kara Sea in terms of SIC standard deviation (Fig. 8a). The sea ice was decreasing in the northern Barents Sea, while it was increasing over a small area of the southwestern Kara Sea (Fig. 8a). The corresponding wind at 1000 hPa was anomalous southerly wind (Fig. 8a).

Fig. 8
figure 8

(a) The average SIC trend (shaded; unit: % day−1), SIC standard deviation (contours; %) and average 1000-hPa wind anomalies (arrows; unit: m s−1) from 25 December 2020 to 10 January 2021. (b) Standardized daily SIC in Barents–Kara Seas (red line) and the AO index (blue line) from 25 December 2020 to 10 January 2021. (c) The lead-lag correlations (black line, the positive values indicate that AO index leads sea ice change) between the Barents–Kara Seas SIC and the AO index from 25 December 2020 to 10 January 2021. The red dashed lines represent the 95% confidence level. (d, e) the same as (b, c), respectively, but for SIC of Barents Sea and the AO index. The purple box in (a) represents the Barents Sea (77° N–83° N, 20° E–80° E), and the gray box represents the Kara Sea (70.5° N–75° N, 57° E–69° E). The gray shading in (b, d) represents the extreme cold event episode over 4–9 January 2021

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The above analysis of atmospheric circulations during the extreme cold event in Sects. 3 and 4 showed that the distributions of circulation anomalies in the Northern Hemisphere were similar to the negative AO pattern. Therefore, we presented the evolutions of sea ice indices of Barents–Kara Seas (Fig. 8b) or Barents Sea (Fig. 8d) and their relationships with the daily AO index, respectively. The two sea ice indices all showed decreasing trends in general, and the reduction rates in the early stage were relatively slow but significantly accelerated in the cooling period 4–7 January 2021 (Fig. 8b, d). The AO index also changed from positive to negative phase on January 4, and the cooling period corresponded to strong AO negative phase (Fig. 8b, d). Moreover, we calculated the lead-lag correlations between sea ice indices of Barents–Kara Seas (Fig. 8c) or Barents Sea (Fig. 8e) and the AO index, respectively. The AO index led sea ice change by 1–2 days, which mainly reflected the impact of the AO on sea ice (Fig. 8c, e). The atmospheric circulation anomalies corresponding to the AO negative phase were conducive to the exchange of cold and warm air masses between high and middle latitudes, then the warmer and humid air from lower latitudes was transported to the Arctic which might play a certain role in the melting of sea ice.

The analysis above also indicated that the results of the sea ice indices of Barents–Kara Seas and Barents Sea exhibited a good consistency. Therefore, we selected the Barents Sea as the key sea ice area for the following analysis.

To further quantify the ice–air interaction in the key sea ice area, we calculated the average surface net heat flux anomalies over the Barents Sea. Figure 9a showed the change of net surface heat flux and its three components LW, SH, and LH from 25 December 2020 to 10 January 2021. Before the extreme cold event, the net heat flux was negative from the end of December 2020 to 5 January 2021, that is, the atmosphere heats the ocean in this period (Fig. 9a). While during the cooling period 4–7 January, the net surface heat flux gradually changed from negative to positive value and reached the peak, suggesting that the influence of ocean on atmosphere was dominant during this period (Fig. 9a). Figure 9b showed the distribution of net surface heat flux anomalies averaged over the cooling period 4–7 January. The total anomalies including LW, SH and LH in the northern Barents Sea (purple box in Fig. 8a) were positive meaning that the ocean heated the atmosphere, which was conducive to stimulating the anticyclone anomaly over Arctic and strengthening the negative phase of AO. Thus, it was beneficial to the exchange of air masses between middle and high latitudes. Similar results can also be obtained with ERA-5 reanalysis data (not shown).

Fig. 9
figure 9

(a) The evolutions of surface net heat flux anomalies (black line) and its components, including LW (green line), LH (red line), and SH (blue line) from 25 December 2020 to 10 January 2021. The gray shading represents period of 4–7 January 2021. (b) The distribution of average net heat flux anomalies during 4–7 January 2021 (shaded; unit: W m−2) in the Arctic region

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To further prove the effect of sea ice loss in the Barents Sea and the upward net heat flux on the anticyclonic anomaly over Arctic, a numerical experiment was conducted based on the LBM. This numerical experiment simulated the atmospheric response to heat forcing associated with sea ice loss in the Barents Sea (Fig. 10a). The experiment was initialized and forced with diabatic heating centered at 81° N, 23° E which essentially matched the region with maximum net surface heat flux in the Barents Sea (Fig. 9b), which was prescribed at 925 hPa in the experiment (Fig. 10a). Figure 10a illustrated the responses of 300-hPa geopotential height and wind anomalies to the imposed heating over Barents Sea. The result of the LBM showed that the Rossby wave propagating downstream did appear in the mid-high latitudes, and it was an anticyclonic anomaly near the Barents–Kara Seas and a cyclonic anomaly over East Asia (Fig. 10a), which was similar to the results of Yang et al. (2021). Moreover, Xu et al. (2020) also found similar results from a linearized barotropic model. Therefore, the sea ice loss in the Barents Sea and its accompanying upward net heat flux can stimulate the Rossby wave propagating downstream along the mid-high latitudes, and the wave train (i.e. POL) had a cyclonic anomaly over East Asia.

Fig. 10
figure 10

(a) The heat forcing (shaded; unit: K day−1) centered at (81° N, 23° E, 925 hPa), the steady responses of geopotential height (contours; unit: m) and wind anomalies at 300 hPa (arrows; unit: m s−1). The inner panel shows the vertical profiles of q1 anomalies (blue solid line; unit: K day−1) and the initial heat forcing (red dashed line; unit: K day−1) at 81° N, 23° E. (b) Sum of steady responses of geopotential height anomalies (contours; unit: m) forced by North Atlantic and Arctic heat anomalies. The red box represents NC

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Figure 10b showed the linear accumulation of LBM results forced by heat anomalies over both North Atlantic and Arctic. Under the combined effect from the North Atlantic and Barents Sea sea-ice, there were two Rossby waves in the mid-high latitudes (Fig. 10b) which was basically consistent with the circulation situations of the extreme cold event (Fig. 2). On the one hand, the North Atlantic anomalous precipitation stimulated the Rossby wave propagating eastward along the subtropical westerly jet, resulting in a cyclonic anomaly over East Asia. On the other hand, the negative phase of AO contributed to the sea ice reduction in Barents Sea, and the resultant upward net heat flux heated the atmosphere and thus generated POL with an anticyclonic anomaly over Arctic region and a cyclonic anomaly over East Asia. The subtropical wave train and POL worked together to strengthen the cyclonic anomaly over East Asia. NC was located in the southwest of the cyclone with a stronger northerly wind, which was conducive to the occurrence of the extreme cold event. In order to quantitatively compare the effect of subtropical wave train and POL on the anomalous cyclone over East Asia, we calculated their contribution ratio of regional (30° N–60° N, 100° E–150° E, the location of East Asian anomalous cyclone in Fig. 3a) average geopotential height anomalies in the two experiments, respectively, divided by the total value of two experiments. The result showed that the contribution of the subtropical wave train reached 56%, while that of the POL was 44%. Although the results of the LBM can capture the wave pattern as a whole, its position was a little southward and the anomalous anticyclone near the Mediterranean was stronger (Fig. 10b), which might be because the LBM did not involve nonlinear coupling processes of sea-air and land-air.

6 Conclusions and discussion

This study investigated the mechanisms associated with the combined effect of the Rossby waves at mid-high latitudes on the extreme cold event over NC in early January 2021 based on diagnostic analysis and model simulations. The generation and propagation of the subtropical wave train and POL and their impacts on the extreme cold event over NC were summarized by a schematic diagram in Fig. 11.

Fig. 11
figure 11

Schematic diagram of synergistic effect of the subtropical wave train along westerly jet and POL on the NC extreme cold event in early January 2021. Upper panel: The red (blue) solid (dashed) circles with arrows represent anticyclonic (cyclonic) anomalies related to the subtropical wave train and POL at 300 hPa; the gray thick solid line with an arrow represents the subtropical westerly jet at 300 hPa; the pink and purple dashed lines with arrows indicate the propagation of WAF for the subtropical wave train and POL, respectively. Lower panel: The shading in the North Atlantic represents precipitation anomalies in Fig. 6b; the gray dashed circle in the west coast of North Atlantic represents a low pressure anomaly at 1000 hPa; the shading in the Arctic represents sea ice change in Fig. 8a; the shading in Eastern and Northern China represents the anomalous cooling in Fig. 1a; the vectors represent the wind anomalies at 850 hPa. The solid (dashed) green arrows indicate vertical ascending (descending) movement; the orange dashed arrows indicate the upward net heat flux

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As shown in Fig. 11, atmospheric circulation anomalies during the extreme cold event showed that the pivotal factor causing the NC extreme cold event was the anomalous cyclone over East Asia. The northerly wind anomalies at southwest of the cyclone can strengthen the East Asian winter monsoon, which was conducive to the invasion of polar cold air mass into NC. Furthermore, we found that the strengthening of cyclonic anomaly was mainly the combined contribution from the subtropical wave train and POL. In terms of the southern branch (i.e. the subtropical wave train), anomalous warming of sea surface temperature existed in the North Atlantic (not shown), corresponding to a convergence at lower troposphere and divergence at higher troposphere over North Atlantic (Fig. 6c). Such a vertical configuration in the troposphere was accompanied by ascending motion (green solid line arrow in Fig. 11), which was beneficial to precipitation there. As a result, the latent heat released by precipitation favored a Rossby wave source (Figs. 6a and 7a), which was generally located along the subtropical westerly jet and propagated eastward (Fig. 11). As for the northern branch (i.e. the POL), the AO negative phase in the lower troposphere was strengthened (Fig. 8b), and the southerly wind transported warm and humid air mass from the lower latitudes to the pole, leading to sea ice loss in the Barents Sea. With sea ice reduction, the warmer sea water can release upward net heat flux to heat the atmosphere, forming an anticyclonic anomaly over Arctic at 300 hPa (Figs. 9 and 10a), which was conducive to the strengthening and northward extension of the Ural ridge (Wu et al. 2017). Finally, both the subtropical wave train and POL had a cyclonic anomaly over East Asia, and their superposition resulted in an enhanced cyclonic anomaly over East Asia and the deepening of the East Asian trough, contributing to the occurrence of the extreme cold event (Song and Wu 2017; Luo et al. 2021). In addition, the results of the LBM also supported these conclusions.

In the simulation of the subtropical wave train by LBM, the results can capture the Rossby wave as a whole, but the location of the anticyclone anomaly in the North Atlantic was a little southward and the anticyclone near the Mediterranean was stronger compared to that in the observations. This might probably be because the LBM did not take the nonlinear processes and the land-sea-air coupling processes into account (Watanabe and Kimoto 2000). Additionally, the Arctic sea-air-ice coupling processes are quite complex. In the future, the couple model needs to be used to investigate and refine the physical processes in the Arctic.