1 Introduction

Atmospheric teleconnection patterns play crucial roles in large-scale and regional climate variability and climate change, which are of great importance to the socioeconomic development, ecosystem balance, crop growth, and people's daily life (e.g. Ogi et al. 2003; Stott et al. 2004; Beniston 2004; Matsueda 2011; Barriopedro et al. 2011; Otomi et al. 2013; IPCC 2013; Ye and Wu 2017; Li et al. 2018; Chen and Song 2019; Chen et al. 2019; Xu et al. 2019a). For example, studies indicated that the anomalous positive phase of the North Atlantic Oscillation (NAO), an important atmospheric internal mode over the North Atlantic (Hurrell and van Loon 1997), contributes largely to extremely high temperatures in summer of 2003 over most parts of Europe (Beniston 2004; Ogi et al. 2005) that led to substantial economic loss, extensive forest fire and many casualties (Stott et al. 2004; Beniston 2004). Studies demonstrated that the large negative (positive) phase of the Arctic Oscillation (AO; Thompson and Wallace 1998, 2000; Cheung et al. 2012) in the winter of 2009/10 (summer of 2010) resulted in record-breaking cold winter (hot summer) over many parts of Eurasia (Matsueda 2011; Barriopedro et al. 2011). Deng et al. (2018) reported that the concurrent summer heat waves over Northern China and East Europe are resulted from an atmospheric teleconnection over Eurasia. Miao et al. (2019) found that atmospheric wave train over the mid-high latitudes of Eurasia has a considerable contribution to the occurrence of a persistent heavy rainfall event over South China during the first rainy season (April–May–June).

Many studies have investigated atmospheric wave trains over Eurasia during boreal winter and summer. These studies indicated that there exists an atmospheric wave train propagating eastward from Europe to East Asia along the subtropical Asian jet in boreal winter (Branstator 2002; Watanabe 2004; Li and Sun 2015; Li and Zhou 2016; Hu et al. 2018). The atmospheric circulation anomalies related to this wintertime wave train have significant impacts on surface air temperature (SAT) and rainfall variations over Eurasia. In boreal summer, there exist two prominent atmospheric wave trains over Eurasia in the upper troposphere along the subtropical jet, i.e., the Silk Road pattern and Circumglobal teleconnection (Lu et al. 2002; Enomoto et al. 2003; Ding and Wang 2005; Zhou et al. 2019). Zhou et al. (2019) indicated that the Silk Road pattern and the Circumglobal teleconnection are highly similar in term of the spatial structure over the Eurasian region. In addition, they found that centers of the atmospheric circulation anomalies over Eurasia related to the Silk Road pattern are much stronger than those related to the Circumglobal teleconnection. A number of studies reported that the Silk Road pattern and Circumglobal teleconnection have notable impacts on the climate variability over Eurasia (Ding and Wang 2005; Huang et al. 2011; Chen and Huang 2012; Sato and Takahashi 2006; Hong et al. 2018; Zhou et al. 2019). Atmospheric wave trains have also been observed over the mid-high latitudes of Eurasia, propagating along the polar front jet during boreal winter and summer (Wang and Yasunari 1994; Sato and Takahashi 2007; Iwao and Takahashi 2008; Xu et al. 2019b).

By contrast, few studies have been conducted to investigate the atmospheric wave trains over Eurasia during boreal spring. Spring is the transitional season from winter to following summer with plant recovery, snow melting and crop growing. Abnormal spring SAT and rainfall have pronounced impacts on the restoration of ecosystem and agriculture over Eurasia. A recent study of Chen et al. (2016) investigated the dominant modes of spring SAT anomalies over the mid-high latitudes of Eurasia. They found that the first Empirical Orthogonal Function (EOF) mode of spring Eurasian SAT anomalies features a same-sign anomaly pattern, which is closely linked to the spring AO. The second EOF mode of the spring Eurasian SAT anomalies is featured by a tripole pattern, with significant same-sign SAT anomalies over western and eastern parts of Eurasia and opposite sign SAT anomalies over central Eurasia (Fig. 1a). The second EOF mode is found to be closely related to an atmospheric wave train extending from the North Atlantic through Europe to East Asia (Fig. 1b, c). However, several relevant issues remain to be explored regarding this spring atmospheric wave train. Is this atmospheric wave train one of the favored modes of spring atmospheric anomalies over the North Atlantic and Eurasian region? What are the structure and maintenance mechanisms of this atmospheric wave train? Do the formation and maintenance of this wave train have a relation with external forcings (such as sea surface temperature anomalies)? In addition, can this wave train exert significant influences on the Eurasian spring precipitation variation in addition to the SAT? This study aims to address these important issues.

Fig. 1
figure 1

Regression maps of spring (March–April–May-mean) (a) SAT (unit: °C), (b) 500-hPa geopotential height (shadings; unit: m) and wave activity flux (vectors; unit: m2 s− 2), and (c) 250-hPa meridional wind (unit: m s− 1) anomalies onto the normalized principal component (PC) time series of spring SAT anomalies over mid-high latitudes of Eurasia (i.e. 0°–140°E, 40°–70°N) during 1948–2017. Stippling regions in (a–c) denote SAT, geopotential height, and meridional wind anomalies significant at the 5% level. Wave activity flux anomalies less than 0.1 m2 s− 2 in both directions are omitted

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The structure of this paper is as follow. Section 2 describes the data and methods used in this analysis. Section 3 presents the definition of the springtime wave train over the North Atlantic-Eurasian region and examines its spatial structure. Section 4 discusses internal dynamics of this spring atmospheric wave train via an analysis of the vorticity and energy budgets. Section 5 shows relation of this wave train with the North Atlantic sea surface temperature (SST). Section 6 examines impacts of the spring atmospheric wave train on the Eurasian spring SAT and rainfall variations. Section 7 provides a summary and discussion.

2 Data and methods

2.1 Data

This study uses monthly mean sea surface temperature data from the National Oceanic and Atmospheric Administration (NOAA) Extended Reconstructed SST version 3b (ERSSTv3b) dataset (Smith et al. 2008). This SST dataset has a horizontal resolution of 2° × 2° and is available since 1854. The monthly mean atmospheric variables, precipitation rate, surface latent and sensible heat fluxes, and surface shortwave and longwave radiations are extracted from the National Centers for Environmental Prediction-National Center for Atmospheric Research (NCEP-NCAR) reanalysis dataset (Kalnay et al. 1996). The NCEP-NCAR atmospheric data span from 1948 to the present and has a horizontal resolution of 2.5° × 2.5°. Surface heat fluxes and the precipitation rate data from the NCEP-NCAR reanalysis dataset are on T62 Gaussian grid. The present study also uses monthly mean atmospheric data from the Japanese 55-year (JRA55) reanalysis (Ebita et al. 2011) to verify the results obtained from the NCEP-NCAR reanalysis. Atmospheric data from the JRA55 have a horizontal resolution of 1.25° × 1.25° and are available from January 1958 to the present. The monthly mean SAT and precipitation data provided by the University of Delaware Air Temperature & Precipitation version v5.01 are available from January 1900 to December 2017 with a horizontal resolution of 0.5° × 0.5° (Matsuura and Willmott 2009).

This study focuses on interannual variations. Thus, all variables are subjected to a 2–9 year band pass filter to remove their long-term trends and interdecadal variations (Duchon 1979). Statistically significance levels of correlation and regression coefficients are assessed using a two-tailed Student's t test.

2.2 Wave activity flux

The wave activity flux (\(W\)) is calculated following Takaya and Nakamura et al. (2001) and is expressed as follows:

$$W=\frac{1}{2\left|\mathbf{U}\right|}\left(\begin{array}{c}\stackrel{-}{u}\left({\psi }_{x}^{{\prime }2}-{\psi }^{{\prime }}{\psi }_{xx}^{{\prime }}\right)+\stackrel{-}{v}\left({\psi }_{x}^{{\prime }}{\psi }_{y}^{{\prime }}-{\psi }^{{\prime }}{\psi }_{xy}^{{\prime }}\right)\\ \stackrel{-}{u}\left({\psi }_{x}^{{\prime }}{\psi }_{y}^{{\prime }}-{\psi }^{{\prime }}{\psi }_{xy}^{{\prime }}\right)+\stackrel{-}{v}\left({\psi }_{y}^{{\prime }2}-{\psi }^{{\prime }}{\psi }_{yy}^{{\prime }}\right)\\ \frac{{f}^{2}}{R\sigma /p}\left\{\stackrel{-}{u}\left({\psi }_{x}^{{\prime }}{\psi }_{p}^{{\prime }}-{\psi }^{{\prime }}{\psi }_{xp}^{{\prime }}\right)+\stackrel{-}{v}\left({\psi }_{y}^{{\prime }}{\psi }_{p}^{{\prime }}-{\psi }^{{\prime }}{\psi }_{yp}^{{\prime }}\right)\right\}\end{array}\right),$$
(1)
$$\sigma =\frac{R\stackrel{-}{T}}{{C}_{p}p}-\frac{d\stackrel{-}{T}}{dp},$$
(2)

where \(R\), \({C}_{p}\), and \(f\) denote gas constant associated with dry air, specific heat at constant pressure, and the Coriolis parameter, respectively. \(\mathbf{U}=\left(u,v\right)\) and \(\psi\) denote the geostrophic horizontal winds and stream function, respectively. Overbar and prime indicate climatological mean for the period of 1948–2018 and anomalies. In addition, subscripts x and y are the derivative in the zonal and meridional directions, respectively. Takaya and Nakamura (2001) has demonstrated that the wave activity flux (\(W\)) is parallel to the local group velocity of stationary Rossby wave and can be used to describe propagation of the atmospheric wave train.

2.3 Linearized Rossby wave source

Following Sardeshmukh and Hoskins (1988), the linearized Rossby wave source can be expressed as follows:

$$S=-{\nabla }_{H}\cdot \left\{{\varvec{u}}_{\chi }^{\prime}\left(f+\stackrel{-}{\zeta }\right)\right\}-{\nabla }_{H}\cdot \left\{{\stackrel{-}{\varvec{u}}}_{\chi }{\zeta }^{\prime}\right\},$$
(3)

where \(\varvec{u}=\left(u,v\right)\) represents the horizontal wind velocity. \({\nabla }_{H}\)is horizontal gradient. Subscript \(\chi\)(\(\psi\)) represents divergent (rotational) component. Overbar indicates climatological mean and prime represents anomalies.

2.4 Linearized vorticity equation

The linearized vorticity equation under the geostrophic approximation proposed by Kosaka and Nakamura (2006) could be expressed as:

$$S - \underbrace {{ - \bar{u}_{\psi } \frac{{\partial \zeta ^{\prime}}}{{\partial x}}}}_{{ZA}} - \underbrace {{ - \bar{v}_{\psi } \frac{{\partial \zeta ^{\prime}}}{{\partial y}}}}_{{{\text{MA}}}}\underbrace {{ - \bar{u}_{\psi } \frac{{\partial \bar{\zeta }}}{{\partial x}} - v^{\prime}_{\psi } \frac{{\partial \left( {f + \bar{\zeta }} \right)}}{{\partial y}}}}_{{{\text{Beta}}}} - Residuals = 0.$$
(4)

Here, \(\zeta\) is the relative vorticity and \(S\) is the linearized Rossby wave source. In Eq. (4), ZA term indicates the advection of vorticity anomalies induced by climatological zonal winds. MA term denotes the advection of vorticity anomalies due to climatological meridional winds. Term Beta (i.e. \(-{\stackrel{-}{u}}_{\psi }\frac{\partial \stackrel{-}{\zeta }}{\partial x}-{{v}^{\prime}}_{\psi }\frac{\partial \left(f+\stackrel{-}{\zeta }\right)}{\partial y}\)) indicates the advection of mean vorticity induced by horizontal wind anomalies. Furthermore, term \(Residuals\) is the combination of contributions of nonlinear effect, vertical advection, dissipation and data uncertainty. A positive (negative) value of a given term in Eq. (4) denotes a positive (negative) contribution of cyclonic vorticity tendency.

2.5 Conversions of kinetic energy and available potential energy

Following previous studies (Kosaka and Nakamura 2006; Kosaka et al. 2009), conversion of kinetic energy (available potential energy) from the mean flow, denoted by CK (CP), can be written as:

$$CK=\frac{{v}^{{\prime }2}-{u}^{{\prime }2}}{2}\left(\frac{\partial \stackrel{-}{u}}{\partial x}-\frac{\partial \stackrel{-}{v}}{\partial y}\right)-{u}^{{\prime }}{v}^{{\prime }}\left(\frac{\partial \stackrel{-}{u}}{\partial y}+\frac{\partial \stackrel{-}{v}}{\partial x}\right),$$
(5)
$$CP=-\frac{f}{\sigma }\left({v}^{{\prime }}{T}^{{\prime }}\frac{\partial \stackrel{-}{u}}{\partial p}-{u}^{{\prime }}{T}^{{\prime }}\frac{\partial \stackrel{-}{v}}{\partial p}\right).$$
(6)

It is noted that positive values of CK (CP) correspond to conversions of kinetic energy (available potential energy) from basic mean flow to the atmospheric wave train.

2.6 Time scales related to the CK and CP

Following previous studies (Kosaka and Nakamura 2006; Kosaka et al. 2009), the time scales related to the CK (\({\tau }_{CK}\)) and CP (\({\tau }_{CP}\)) can be estimated as follows:

$${\tau }_{CK}=\frac{\left[KE\right]}{CK},$$
(7)
$${\tau }_{CP}=\frac{\left[APE\right]}{CP}.$$
(8)

Here, \(KE=\frac{{u}^{{\prime }2}+{v}^{{\prime }2}}{2}\) denotes kinetic energy integrated from surface to 100-hPa, \(APE=\frac{R{T}^{\prime}}{2\sigma p}\) represents available potential energy integrated from surface to 100-hPa. The bracket "[]" denote regional mean.

2.7 Geopotential height tendency

The synoptic-scale eddy generated geopotential height tendency can be written as follows (Lau 1988; Cai et al. 2007; Lau and Nath 2014; Chen et al. 2015):

$${\left(\frac{\partial Z}{\partial t}\right)}_{eddy}=\frac{f}{g}{\nabla }^{-2}\left[-\nabla \cdot \stackrel{-}{\left({V}^{\prime}{\zeta }^{\prime}\right)}\right].$$
(9)

Here, Z, g, and f represent geopotential height, acceleration of gravity, and Coriolis parameter, respectively. \({\zeta }^{\prime}\) and \({V}^{\prime}\)denote the synoptic-scale vorticity and horizontal winds. Synoptic-scale eddies are obtained by subjecting the original daily fields to a 2–8 day band pass Lanczos filter (Duchon 1979). Studies indicated that the wave-mean flow interaction and feedback of synoptic scale eddies to the mean flow can be quantitatively measured by the geopotential height tendency.

3 The leading atmospheric wave train over the North Atlantic and Eurasia

In this section, we extract the leading atmospheric wave train in boreal spring and analyze its structure. For this purpose, we apply an EOF analysis to spring 250-hPa meridional wind anomalies over the region of 80°W–120°W, 35°–75°N (denoted by the black box in Fig. 2). The above region is selected for the EOF analysis because it is the domain corresponding to the largest standard deviation of 250-hPa meridional winds (Fig. 2). Large values of standard deviations are seen to extend eastward from the mid-high latitudes of the North Atlantic to east Eurasia, implying strong wave activities over these regions. It is noted that the meridional wind field has been weighted by the cosine of latitude to account for the change of area with the latitude prior to the EOF analysis (North et al. 1982a).

Fig. 2
figure 2

Standard deviation (unit: m s− 1) of spring 250-hPa meridional winds over 1948–2018. The black box in the figure denotes the region (80°W–120°E and 35°–75°N) employed to EOF analysis in Fig. 3

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Figure 3a, b show regressions of spring meridional wind anomalies at 250-hPa onto the normalized principal component (PC) time series of the first and second EOF modes of spring 250-hPa meridional wind anomalies for the period of 1948–2018. Similar results are obtained when the EOF analysis is performed for the regions of 30°–80° N, 50° W–120° E and 35°–70° N, 60° W–120° E. Spatial patterns of the first two EOF modes of spring 250-hPa meridional winds obtained based on the JRA55 reanalysis for the period of 1958–2018 (not shown) are very similar to those shown in Fig. 3a, b. EOF1 and EOF2 explain about 23% and 18% of the total interannual variance of spring meridional wind anomalies, respectively. They are well separated from each other and from the other EOF modes according to the criterion of North et al. (1982b). EOF1 is featured by a wave-like structure over the mid-high latitudes of the North Atlantic and Eurasia, with centers of action over the North Atlantic around 45° N, west Europe, north coast of East Europe, central Eurasia, and East Asia, respectively (Fig. 3a). A subtropical wave train is also apparent in the EOF1 pattern, with northerly anomalies over northwest Africa, the Arabian Peninsula, and southerly anomalies around Egypt, and the Indian subcontinent, respectively (Fig. 3a). It is interesting to note that the spatial structure of EOF1 of the spring meridional wind anomalies (Fig. 3a) bear a close resemblance to that related to the EOF2 of the spring SAT anomalies over the mid-high latitudes of Eurasia (Fig. 1c). The correlation coefficient between PC1 of spring 250-hPa meridional winds and PC2 of spring Eurasian SAT is as high as 0.8 for the period of 1948–2017, significant at the 1% level. The EOF2 of spring 250-hPa meridional wind anomalies is also characterized by a wave train structure with the action centers shifting westward compared to those of EOF1 (Fig. 3a, b). In the following, we concentrate on investigating the wave train related to EOF1 because it explains the largest faction of the spring 250-hPa meridional wind anomalies and it has a close relation with the spring tripole SAT anomaly pattern over Eurasia.

Fig. 3
figure 3

Spring 250-hPa meridional wind anomalies (unit: m s− 1) regressed upon the normalized PC time series of the (a) first and (b) second EOF modes of interannual variation of spring 250-hPa meridional winds over 80°W–120°E and 35°–75°N for the period of 1948–2018. Stippling regions denote meridional wind anomalies significant at the 5% level

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Figure 4 displays spring geopotential height and wave activity flux anomalies at 850-hPa, 500-hPa, and 250-hPa in association with the PC1 of the spring 250-hPa meridional wind anomalies described above. Spatial structures of the geopotential height and wave activity flux anomalies are similar at 850-hPa, 500-hPa, and 250-hPa (Fig. 4). This suggests that the spring atmospheric circulation anomalies associated with the leading mode of spring 250-hPa meridional winds over the North Atlantic and Eurasia have a barotropic structure. The geopotential height anomalies display a wave-like structure, with three centers of negative anomalies over subtropical North Atlantic, north Europe, and East Asia, and two centers of positive anomalies over the North Atlantic around 60°N and central Eurasia, respectively (Fig. 4). The wave activity fluxes are mainly started from the subtropical North Atlantic, propagate northeastward to north Europe, turn eastward to central Eurasia, and eventually turn southeastward to East Asia. Note that wave activity fluxes can also be observed to propagate from the North Pacific but with much smaller amplitudes compared to those over the North Atlantic and Eurasia. This implies that the atmosphere-ocean conditions over the North Pacific may also partly contribute to the formation of the spring atmospheric wave train.

Fig. 4
figure 4

Regression maps of spring geopotential height (shadings; unit: m) and wave activity flux (vectors; unit: m2 s− 2) anomalies at (a) 850 hPa, (b) 500-hPa, and (c) 250-hPa onto the normalized PC1 time series of spring 250-hPa meridional winds anomalies over the North Atlantic and Eurasia. Stippling regions denote geopotential height anomalies significant at the 5% level

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Figure 5 presents longitude-height cross section of spring relative vorticity and air temperature anomalies averaged between 50°N and 60°N regressed upon the normalized PC1 of the spring 250-hPa meridional wind anomalies during 1948–2018. 50°N–60°N is selected as it corresponds to the central latitudes of action centers of the atmospheric wave train over the mid-high latitudes of Eurasia (Fig. 4). To the east of 30°W, there are four pronounced centers of vorticity anomalies, which correspond to the positive geopotential height anomalies over the mid-latitudes of the North Atlantic, central Eurasia, and negative geopotential height anomalies over North Europe and East Asia (Fig. 5a). The vertical distribution of vorticity anomalies confirms the quasi-barotropic structure with maximum values between 500-hPa and 250-hPa (Fig. 5a). From surface to 400-hPa, significant positive air temperature anomalies are seen around 30°W and 60°E, and negative anomalies are found around 15°E and 120°E (Fig. 5b). Air temperature anomalies above 300 hPa are nearly opposite to those below 400 hPa. The vertical change in vorticity anomalies is well connected with the air temperature anomalies via the thermal wind relation.

Fig. 5
figure 5

Longitude-Height cross-section (50°–60°N-mean) of spring (a) relative vorticity (unit: 10− 6 s− 1), and (b) air temperature (unit: °C) anomalies regressed upon the normalized PC1 time series of spring 250-hPa meridional winds anomalies over the North Atlantic and Eurasia. Stippling regions in (a, b) denote anomalies of vorticity, and air temperature that are significantly different from zero at the 5% level, respectively

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4 Internal dynamics of the atmospheric wave train

4.1 Vorticity budget analysis

In this subsection, we perform a vorticity budget analysis to understand the formation and persistence of the atmospheric wave train over the North Atlantic and Eurasia. Figure 6a–e display the spatial distribution of each term at 250-hPa in the linearized vorticity equation (i.e., Eq. 4) in association with the leading mode of the spring meridional wind anomalies over the North Atlantic and Eurasian region. Alternately positive and negative Rossby wave sources can be observed along the atmospheric wave train (Figs. 4 and 6a). The largest negative Rossby wave source is found over the North Atlantic around 30°–60°N (Fig. 6a) where large positive precipitation and strong upward motion anomalies are located (Fig. 7). Previous studies have demonstrated that the convergence/divergence anomalies in the upper troposphere can be regarded as an effective Rossby wave source for the stationary Rossby wave (Branstator 2002; Watanabe 2004). Hence, the divergence anomalies in the upper troposphere (not shown) associated with positive precipitation and upward motion anomalies (Fig. 7) play a crucial role in the formation of the strong negative Rossby wave source over the North Atlantic. The positive precipitation and upward motion anomalies may be partly related to the SST anomalies in the North Atlantic, which will be examined later.

Fig. 6
figure 6

Vorticity budget of the EOF1 of spring 250-hPa meridional wind anomalies over North Atlantic and Eurasia. a Rossby wave source term, b zonal advection term (ZA), c meridional advection term (MA), d beta effect term, and e the residual term. Units are 10− 11 s− 2. [may added a figure of precipitation and omega over the North Atlantic]

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Fig. 7
figure 7

Regression maps of spring 500-hPa omega (unit: 10− 3 Pa s− 1) and precipitation (mm day− 1) onto the normalized PC1 time series of spring 250-hPa meridional winds anomalies over the North Atlantic and Eurasia. Stippling regions denote anomalies significant at the 5% level

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ZA and Beta terms (Fig. 6b, d) are much larger than the MA term (Fig. 6c). However, the sign of the ZA term tends to be opposite to that of the Beta term. This implies that the advection of vorticity anomalies due to climatological horizontal winds tend to be offset by the Beta term (i.e., advection of mean vorticity by the perturbed winds). Note that centers of the negative (positive) vorticity anomalies in Fig. 5a are located around 22.5°W and 65°E (10°E and 110°E). In comparison, centers of the negative (positive) values of the Beta term in Fig. 6d appear around 35°W and 45°E (10°W and 100°E). This indicates that the Beta term tends to shift westward compared to the atmospheric wave train related to the EOF1 pattern (Figs. 5a and 6d). This result is in good agreement with the traditional notion that the advection of mean vorticity due to perturbed winds shifts the wave train westward. Moreover, centers of the negative (positive) values of the ZA term in Fig. 6b are observed around 5°W and 90°E (35°E and 130°E), which are located eastward compared to maximum centers of the spring wave train (Figs. 5a and 6b). This suggests that the advection of the perturbed vorticity by climatological mean winds shift the spring atmospheric wave train eastward. From the above analysis, a possible picture of the spring atmospheric wave train over the North Atlantic and Eurasia can be summarized as follows. The Rossby wave sources over the mid-latitude North Atlantic are induced by the upper troposphere divergence related to positive precipitation and upward motion anomalies. The zonal advection of the perturbed vorticity by mean rotational wind drives the spring atmospheric wave train propagating eastward, while the advection of climatological mean vorticity by perturbed winds shifts the spring wave pattern westward. The MA term is much smaller in amplitude compared to the ZA and Beta terms (Fig. 6b–d). In addition, the\(Residuals\) term is less organized in spatial structure and smaller in amplitude (Fig. 6e). These suggest that these two terms play smaller roles in the formation and maintenance of the leading atmospheric wave train over the North Atlantic and Eurasia.

4.2 Energetic budget analysis

In this subsection, we examine whether the leading spring atmospheric wave train over the North Atlantic and Eurasia can efficiently extract energy from the mean flow though barotropic and baroclinic conversion processes (Kosaka and Nakamura 2006; Sato and Takahashi 2006; Kosaka et al. 2009). From Fig. 5a, the leading atmospheric wave train extends vertically up to 100-hPa. Thus, we calculate the vertically integrated CK (CP) from 1000-hPa to 100-hPa to indicate the energy exchange between the kinetic energy (available potential energy) and basic flow. Figure 8a, b display spatial distributions of the CK and CP integrated from 1000-hPa to 100-hPa in association with the PC1 of the spring 250-hPa meridional wind anomalies. Positive values of CK are observed over the subtropical western North Atlantic around 30°N, east coast of North America around 40°–50°N, East European plain with a northward extension, and around the Lake Baikal (Fig. 8a). Large negative values of CK are also seen over the mid-high latitudes of the North Atlantic. As such, the regional mean CK over the Northern Hemisphere (25°–90°N, 0°–360°E) is small (about 3.8 × 10− 5 W m− 2). This suggests that the barotropic energy conversion has a limited contribution to the formation and maintenance of the spring atmospheric wave train over the North Atlantic and Eurasia.

Fig. 8
figure 8

The EOF1 of spring 250-hPa meridional wind anomalies-related (a) barotropical energy conversion (CK, unit: W m− 2) and (b) baroclinic energy conversion (CP, unit: W m− 2) integrated from 1000-hPa to 100 hPa. Vectors in the figures indicate spring 250-hPa wave activity flux anomalies in association with the EOF1 of spring 250-hPameridonal wind anomalies. [may be better define a name for the wave train]

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The CP has alternatively positive and negative values along the atmospheric wave train. The amplitudes of positive CP values are much larger than those of the adjacent negative CP values. The regional mean CP over the Northern Hemisphere is approximately 1.0 × 10− 3 W m− 2, which is much larger than the counterpart of CK. This implies that the spring atmospheric wave train can efficiently extract available potential energy from the basic flow. Hence, CP plays a more important role than CK in maintaining the leading spring atmospheric wave train over the North Atlantic and Eurasia.

To estimate quantitatively contributions of the CK and CP to the maintenance of the atmospheric wave train, we further calculate the time scales with which the energy associated with the atmospheric wave train can be entirely replenished by the CK (\({\tau }_{CK}\)) and CP (\({\tau }_{CP}\)). As indicated by previous studies (Kosaka and Nakamura 2006; Kosaka et al. 2009; Hu et al. 2018), if the \({\tau }_{CK}\) (\({\tau }_{CP}\)) is shorter than a season, the CK (CP) is considered to be efficient to fuel the atmospheric wave train. The\({\tau }_{CP}\)averaged over 25°–90°N, 0°–360°E is only about 0.83 days. This indicates that the spring atmospheric wave train can efficiently gain energy from the mean flow via baroclinic process. The \({\tau }_{CK}\) is about 79.7 days, also shorter than a season. However, the \({\tau }_{CK}\)increases largely (longer than 200 days) if the spring atmospheric wave pattern is shifted by westward or eastward 10° relative to the basic flow as in Kosaka and Nakamura (2010) and Hu et al. (2008). The \({\tau }_{CP}\) increases to 3 days when the wave pattern is shifted westward or eastward by 10°. It indicates that \({\tau }_{CP }\)is not very sensitive to the longitudinal shift of the wave pattern. This evidence suggests that the baroclinic energy conversion plays an important role in maintaining the spring atmospheric wave train over the North Atlantic and Eurasia. In addition, it suggests that the energy conversions may help anchor the geographical locations of the centers of the atmospheric wave train over the North Atlantic and Eurasia related to the EOF1 of spring meridional wind anomalies, consistent with previous studies (Kosaka et al. 2009; Hu et al. 2018).

4.3 Interaction between synoptic-scale eddies and the atmospheric wave train

Previous studies indicated that the interaction between synoptic-scale eddies and low frequency mean flow is an important source of the atmospheric wave train over the mid-high latitudes (Lau and Holopainen 1984; Lau 1988; Cai et al. 2007; Chen et al. 2014). In the following, we demonstrate that there are active interactions between synoptic-scale eddies and the leading spring atmospheric wave train. Figure 9a shows regression map of the spring geopotential height tendency anomalies at 250-hPa onto the normalized PC time series corresponding to EOF1 of spring 250-hPa meridional wind anomalies. Cyclonic forcings are seen over the mid-latitudes of the North Atlantic, west Europe, and around the Lake Baikal, and anticyclonic forcings appear over the high latitudes of the North Atlantic and central Eurasia (Fig. 9a). The spatial structure of the eddy-induced geopotential height tendency anomalies bears a close resemblance to the spring atmospheric wave train pattern seen in Fig. 4. It should be mentioned that spatial patterns of the geopotential height tendencies associated with other time scales variations (for example 10–20 days and 30–60 days) (not shown) are quite different from the spatial structure of the wave train shown in Fig. 4. In addition, amplitudes of the geopotential height tendencies related to other time scales variations are also much weaker (not shown) compared to those induced by the synoptic-scale eddies. Figure 9b further shows the spring storm track anomalies at 250-hPa associated with the spring atmospheric wave train. As used in previous studies (Lee et al. 2012; Chen et al. 2014, 2018), storm track is defined as the root mean square of 2–8 day band pass filter geopotential height anomalies. Significant decreases in storm track activities are seen over the North Atlantic around 40°N (Fig. 9b), corresponding well to the easterly wind anomalies there (Fig. 4). Previous studies have demonstrated that decrease (increase) in local storm track activity is accompanied by easterly (westerly) wind anomalies as well as negative (positive) geopotential height tendency to its south and positive (negative) geopotential height tendency to its north (Lau 1988; Cai et al. 2007; Chen et al. 2014, 2015). Hence, the negative storm track anomalies are coupled with the easterly wind anomalies around 50°N over the North Atlantic, which plays an important role in maintaining the anomalous cyclone and anticyclone over the North Atlantic (Figs. 4, 9b). Note that the easterly wind anomalies related to the anomalous cyclone and anticyclone over the North Atlantic can in turn help to maintain the negative storm track anomalies as they are tightly coupled with each other (Lau and Holopainen 1984; Lau 1988; Cai et al. 2007). In conclusion, above evidences suggest that the interaction between synoptic-scale eddies and low frequency mean flow also play a crucial role in maintaining the spring atmospheric wave train over the North Atlantic and Eurasia.

Fig. 9
figure 9

Spring geopotential height tendency (unit: m day− 1) and storm track (unit: m) anomalies at 250-hPa regressed upon the normalized PC1 time series of the spring 250-hPa meridional wind anomalies. Stippling regions indicate anomalies significant at the 5% level

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It is noted that the baroclinic process and the interaction between the synoptic-scale eddy and mean flow may not be independent processes in maintaining the springtime atmospheric wave train over the North Atlantic and Eurasia. As has been demonstrated above, the baroclinic energy conversion process plays an important role in maintaining the spring atmospheric wave train. In addition, synoptic-scale eddy activity is tightly coupled with the mean flow changes (Lau and Holopainen 1984; Lau 1988; Cai et al. 2007). This implies that occurrences of the synoptic-scale eddy may have a relation with the baroclinic process. The detailed mechanism in linking the baroclinic process and the synoptic-scale eddy activity is out of the scope of this study but remain to be explored.

5 External forcing of the atmospheric wave train

Several studies indicated that the wintertime and summertime atmospheric wave train over Eurasia have a close relation with the SST anomalies in the North Atlantic region (Wu et al. 2009; Zuo et al. 2013; Hu et al. 2018; Zhao et al. 2019). From Fig. 4, the wave activity fluxes related to the spring atmospheric wave train are mainly originated over the North Atlantic. Chen et al. (2016) indicated that the EOF2 mode of the spring Eurasian SAT anomalies has a close relation with the North Atlantic SST. This suggests that the SST anomalies in the North Atlantic can contribute to the formation and maintenance of the spring atmospheric wave train. To confirm this, we calculate SST anomalies in the North Atlantic region obtained by regression upon the normalized PC time series of EOF1 of spring 250-hPa meridional wind anomalies (Fig. 10). SST anomalies in the tropical central-eastern Pacific are weak from preceding winter to simultaneous spring (Fig. 10a–c). The correlation coefficient between the PC1 time series of spring 250-hPa meridional wind anomalies and the Niño3.4 SST index in preceding winter (simultaneous spring) is about 0.16 (0.12) over 1948–2018. Here, the Niño3.4 SST index is defined as the region-mean SST anomalies over the domain of 5°S–5°N, 120°–170°W, which is usually employed to describe the ENSO variability. Above evidences imply that ENSO-related SST anomalies do not play a significant role in the formation and maintenance of the springtime atmospheric wave train over the North Atlantic and mid-high latitudes Eurasia related to the EOF1 of spring 250-hPa meridional winds. SST warming is seen in the tropical Indian Ocean in winter (Fig. 10a) and the region of SST warming shifts to the Bay of Bengal in spring (Fig. 10c). A horseshoe-like SST anomaly pattern is seen in simultaneous spring in the North Atlantic, with pronounced cooling in the subtropical western North Pacific and marked warming in the tropical northeastern Atlantic with a northwestward and southwestward extension (Fig. 10c). This SST anomaly pattern can also be observed in preceding JFM and FMA though the SST cooling is statically insignificant in the western Atlantic (Fig. 10a, b). Several previous studies have indicated that North Atlantic SST anomalies can induce an eastward propagating atmospheric wave train over Eurasian continent, which further influences East Asian summer climate (Wu et al. 2009, 2011; Choi and Ahn 2019). This suggests that the North Atlantic SST anomalies could be a potential precursor for the formation of the spring atmospheric wave train.

Fig. 10
figure 10

Anomalies (unit: °C) of SST at JFM (January-February-March-average), FMA (February-March-April-average), and MAM (March-April-May-average) regressed upon the normalized PC1 time series of spring 250-hPa meridional wind anomalies. Stippling regions indicate anomalies significant at the 5% level. Black boxes in (c) are used to define spring North Atlantic dipole (NAD) SST index in the following

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To further confirm the role of the North Atlantic SST anomalies, we define a North Atlantic SST index as the difference in the normalized SST anomalies between subtropical eastern (15°–35°W, 20°–45°N) and western (47.5°–60°W, 30o–45°N) North Atlantic according to Fig. 10c. Figure 11 displays anomalies of spring 250-hPa geopotential height and wave activity flux in association with the North Atlantic SST index for the period 1948–2018. Note that the preceding winter ENSO signal (represented by Nino3.4 index) has been linearly removed from the spring North Atlantic SST index to eliminate the potential impact of ENSO. The spatial structure of the 250-hPa geopotential height anomalies related to the spring North Atlantic SST index (Fig. 11) is very similar to those related to the EOF1 of spring meridional wind anomalies (Fig. 4). The anomalous cyclone and negative geopotential height anomalies over the mid-latitude North Atlantic are located to northwest of the North Atlantic SST warming (Figs. 4, 10 and 11). This spatial relationship implies a Rossby wave-like atmospheric response to the SST anomalies in the North Atlantic, consistent with previous studies (Czaja and Frankignoul 1999; Chen et al. 2016). The upward motion and positive precipitation anomalies accompanying the anomalous cyclone (Fig. 7) could lead to divergence anomalies in the upper troposphere, which is an important Rossby wave source (Fig. 6a) of the spring atmospheric wave train.

Fig. 11
figure 11

250-hPa geopotential height (shadings; unit: m) and wave activity flux (vectors; unit: m2 s− 2) anomalies at spring regressed upon the normalized spring North Atlantic SST index. Definition of the spring North Atlantic SST index is provided in the main text. Stippling regions indicate geopotential height anomalies significant at the 5% level

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Studies indicated that the North Atlantic is a region with a strong air–sea interaction (Czaja and Frankignoul 1999, 2002; Huang and Shukla 2005; Hu and Huang 2006; Peng et al. 2003; Pan 2005; and references therein). This suggests that the overlying atmospheric circulation anomalies could have a positive feedback on the North Atlantic SST anomalies. Figure 12 displays spring surface latent heat flux and net shortwave radiation anomalies obtained by regression upon the normalized PC1 time series of spring 250-hPa meridional wind anomalies. Spring surface wind speed and total cloud cover anomalies are also shown to explain the surface heat flux anomalies. Surface longwave radiation and sensible heat flux anomalies are not presented here as they are much smaller compared to surface latent heat flux and shortwave radiation anomalies. To facilitate comparison, values of surface heat fluxes are taken to be negative (positive) when their directions are upward (downward), which contribute to SST cooling (warming). In general, reduction (enhancement) in surface wind speed corresponds to decrease (increase) in surface upward latent heat flux, which has a positive contribution to the SST warming (cooling) (Figs. 10 and 12a). For example, the southwesterly wind anomalies to the east side of the cyclonic anomaly over the tropical northeastern Atlantic (Fig. 11) reduce northeasterly winds and contribute to maintenance of the positive SST anomalies there (Fig. 10) via reducing upward surface latent heat flux (Fig. 12a) (Czaja et al. 2002; Chen et al. 2016). The anomalous easterly winds around 60°N reduce westerly winds and contribute to the SST warming there via decrease in upward surface latent heat flux (Figs. 10, 11, 12a). The positive geopotential height anomalies over the tropical northeastern Atlantic are accompanied by decrease in total cloud cover and increase in downward shortwave radiation, which may also partly contribute to the maintenance of the positive SST anomalies there (Figs. 10, 11, 12b). This indicates that positive air-sea interaction may play an important role in the maintenance of the North Atlantic SST anomalies and the spring atmospheric wave train over the North Atlantic through the mid-high latitude Eurasia.

Fig. 12
figure 12

Anomalies of spring (a) surface latent heat flux (shading; unit: W m− 2) and surface wind speed (contour; unit: m s− 1), (b) surface net shortwave radiation (shading; unit: W m− 2) and total cloud cover (TCC; unit: %) regressed upon the normalized PC1 time series of spring 250-hPa meridional wind anomalies. Stippling regions in (a–b) indicate latent heat and shortwave radiation anomalies significant at the 5% level, respectively

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6 Impacts of the atmospheric wave train on SAT and precipitation over Eurasia

In this section, we further examine contribution of the spring atmospheric wave train to the SAT and precipitation variations over Eurasia. Figures 13a and 13b present anomalies of spring SAT and rainfall, respectively, over the mid-high latitudes of Eurasia regressed upon the PC1 of spring meridional wind variation. Consistent with Chen et al. (2016), a tripole SAT anomaly pattern is seen over the mid-high latitudes of Eurasia, with pronounced negative SAT anomalies over west Europe and extending southward from the Russian Far East to East Asia, together with significant positive SAT anomalies over central Eurasia around 40°–80°E (Fig. 13a). Amplitudes of the negative and positive SAT anomalies are approximately 0.7 °C. As has been demonstrated by Chen et al. (2016), anomalous wind induced meridional temperature advection plays a key role in the formation of the SAT anomalies. Northerly wind anomalies over west Europe and east Eurasia contribute to decrease in the SAT there via carrying colder air from higher latitudes (Figs. 13a, 14a). Southerly wind anomalies over central Eurasia lead to positive SAT anomalies through bringing warmer air from lower latitudes (Figs. 13a, 14a).

Fig. 13
figure 13

Spring (a) SAT (unit: °C) and (b) precipitation (unit: cm) anomalies regressed upon the normalized PC1 time series of spring 250-hPa meridional wind anomalies. Stippling region indicate anomalies significant at the 5% level

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Fig. 14
figure 14

Anomalies of spring (a) 850 hPa winds (unit: m s− 1), divergence (unit: 10− 6 s− 1) at (b) 850 hPa and (c) 250-hPa regressed upon the normalized PC1 time series of spring 250-hPa meridional wind anomalies. Shading regions in (a) indicate either direction of the wind anomalies significant at the 5% level. Stippling regions in (b, c) indicate divergence anomalies significant at the 5% level

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Above-normal rainfall is seen over large parts of Europe and the regions around the Lake Baikal, corresponding well to the convergence anomalies in the lower-level and divergence anomalies in the upper-level there (Figs. 13b, 14b, c). By contrast, below-normal rainfall is observed around 35°–55°N, 50°–85°E, which is related to the divergence anomalies in the lower level and convergence anomalies in the upper level (Figs. 13b, 14b, c). Rainfall is above-normal to west of the Lake Baikal and over northeast China. Rainfall is below normal over central China.

7 Summary and discussion

The present study investigates a boreal spring atmospheric teleconnection pattern over the mid-high latitudes of the North Atlantic and Eurasia based on reanalysis data from 1948 to 2018. This atmospheric teleconnection pattern is the leading EOF mode of boreal spring meridional wind anomalies at 250-hPa on the interannual timescale over the North Atlantic and Eurasian region. Spatial structure of this spring atmospheric wave train and the associated SST, SAT and precipitation anomalies are summarized by a schematic diagram in Fig. 15. The teleconnection is featured by a barotropic vertical structure and has five main centers of action, with positive geopotential height and anticyclonic anomalies over the mid-latitude North Atlantic and central Eurasia, accompanied by negative geopotential height and cyclonic anomalies over subtropical North Atlantic, west Europe and East Asia, respectively, during its positive phase (Fig. 15). This teleconnection pattern originates from the subtropical North Atlantic, propagates northeastward to west Europe and then eastward to central Eurasia, and eventually turns southeastward to East Asia (Fig. 15).

Fig. 15
figure 15

Schematic diagram showing structure of atmospheric teleconnection related to the EOF1 of spring 250-hPa meridional wind anomalies over the North Atlantic and Eurasian region as well as the associated SAT, precipitation over Eurasia and SST anomalies in the North Atlantic. Blue and red contours denote geopotential height anomalies. Blue (yellow) shading over Eurasia indicates negative (positive) SAT anomalies. Green (purple) dot over Eurasia denotes above (below) normal precipitation anomalies. Blue (brown) shading in the North Atlantic indicates negative (positive) SST anomalies

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An analysis of the vorticity and energetics budgets indicates that the persistence of this teleconnection pattern is mainly attributed to the balance among the Rossby wave sources. Advection of the perturbed vorticity due to the horizontal mean flow (dominated by its zonal component) is offset by the advection of climatological vorticity by the perturbed wind (i.e., the Beta term). The zonal advection of the anomalous vorticity induced by the mean flow shifts the wave train downstream (i.e., eastward). By contrast, the Beta term tends to move the wave train upstream (i.e., westward). The wave train can efficiently extract available potential energy from the basic mean flow (\({\tau }_{CP}\) is only about 0.83 days averaged over NH). The wave train also gains kinetic energy from the mean flow, but less efficient (\({\tau }_{CK}\) is about 79.7 days averaged over NH). The time scale is only about 0.83 days with which the energy associated with the atmospheric wave train can be entirely replenished by the CK and CP. Hence, the baroclinic process plays a more important role in the maintenance of the springtime atmospheric wave train over the North Atlantic and Eurasia. A pronounced positive feedback exists between the synoptic-scale eddy activity and the atmospheric circulation anomalies related to the spring wave train over the North Atlantic and Eurasian region. The atmospheric circulation anomalies affect the synoptic-scale eddy activity via modulating the mean flow. Change in the synoptic-scale eddy activity, in turn, has feedback on the mean flow and helps to maintain the spring atmospheric teleconnection pattern.

The North Atlantic SST anomalies play an important role in the maintenance of the spring atmospheric wave train via coupling with the overlying atmospheric circulation anomalies. The significant SST warming in the subtropical northeastern Atlantic induces an anomalous cyclone over the subtropical North Atlantic (one of the centers of action of the spring atmospheric wave train) via a Rossby wave-type atmospheric response. The associated anomalous upward motion and upper tropospheric divergence serves as an efficient Rossby wave source of the atmospheric wave train. Note that the anomalous cyclone over the subtropical North Atlantic can, in turn, maintain the SST warming in the subtropical northeastern Atlantic via modulating surface heat flux. Specifically, the southwesterly wind anomalies to the southeast side of anomalous cyclone reduce surface wind speed and upward latent heat flux and contribute to the SST warming.

The spring atmospheric wave induces Eurasian SAT anomalies via wind-induced horizontal temperature advection and results in precipitation anomalies via modulating divergence anomalies in the lower and upper troposphere. During a positive phase of this wave train, significant negative SAT anomalies are found over west Europe and eastern part of Eurasia, and pronounced positive SAT anomalies are seen over central Eurasia. Significant above-normal rainfall appears over most parts of Europe and the regions around the Lake Baikal with an eastward extension to north Japan. Significant below-normal rainfall occurs over the regions to the east of the Caspian Sea and central Asia.

It should be mentioned that geopotential height anomalies related to the spring atmospheric wave train tend to be intensified around the central Eurasia, which cannot be explained by the contributions from either the atmospheric internal process (Fig. 9) or the North Atlantic SST anomalies (Fig. 10). Zhang et al. (2017) suggested that the snow cover change and related near surface thermal anomalies could enhance the Eurasian wave train. This implies that amplification of the atmospheric circulation anomalies over central Eurasia related to the spring wave train may be partly related to the surface process, which needs further investigations.

This study suggests that both internal process (i.e., baroclinic process and forcing of synoptic-scale eddies) and external forcing (i.e., the North Atlantic SST anomalies) are important in maintaining the springtime atmospheric wave train over the North Atlantic and mid-high latitudes of Eurasia. We have examined spring atmospheric circulation anomalies related to the PC1 time series of spring 250-hPa meridional winds anomalies after removing the signal of spring North Atlantic SST anomalies by means of linear regression with respect to the North Atlantic SST index (not shown). Negative geopotential anomalies over subtropical North Atlantic related to the spring atmospheric wave as shown in Fig. 4 become much weaker in the troposphere after the removal of the signal of the North Atlantic SST anomalies. However, the geopotential potential anomalies over the mid-high latitudes of Eurasia still exist though with a slightly smaller amplitude. This suggests that the atmospheric internal process may play a more important role in maintaining the springtime atmospheric wave train over the mid-high latitude Eurasia compared to the North Atlantic SST. A quantitative comparison of the impacts of atmospheric internal process and the North Atlantic SST anomalies in the formation and maintenance of the spring atmospheric wave train may need a set of numerical experiments and remains to be explored. In addition, previous studies have reported that Eurasian snow cover and Arctic sea ice anomalies could induce atmospheric wave trains over Eurasia (Wu et al. 2009a, b; Zhang et al. 2017; Chen et al. 2020). The roles of the Arctic sea ice and Eurasian snow cover anomalies in the formation and maintenance of the spring atmospheric wave train identified in this analysis would be examined in a further study.

The EOF2 of the spring 250-hPa meridional wind anomalies over the North Atlantic and Eurasia also displays a wave train structure (Fig. 3b), but with a zonally-elongated feature compared to the EOF1 (Fig. 3). The formation and maintenance mechanism of the atmospheric wave train related to the EOF2 remains to be explored. Furthermore, the combined impacts of atmospheric circulation anomalies related to the EOF1 and EOF2 on the Eurasian climate as well as the underlying mechanisms are unclear and call for further investigations.