The influence of wave trains in mid-high latitudes on persistent heavy rain during the first rainy season over South China

Abstract

Based on daily precipitation data from the Chinese Meteorological Administration and reanalysis data from the National Centers for Environmental Prediction-Department of Energy, the character of low-frequency precipitation variability during the first rainy season (April–June) over South China and its corresponding atmospheric circulations in the mid-high latitudes are investigated. The results show that the precipitation anomalies during this period exhibit obvious quasi-biweekly oscillation (QBWO) features, with a period of 8–24 days. The influence of wave trains in the mid-high latitudes to low-frequency persistent heavy rain event (PHR-LF event, the 8–24-day filtered precipitation larger than one standard deviation of filtered time series and persisting at least three days over South China) is further discussed. During the first rainy season over South China, there are two low-frequency wave trains in the mid-high latitudes associated with the PHR-LF event—the wave train crossing the Eurasian continent and the wave train along the subtropical westerly jet. Analysis of wave activity flux indicates that the wave energy disperses toward eastern China along these two low-frequency wave trains from north to south and from west to east, and then propagates downward over South China. Accordingly, the disturbance of the relative vorticity of the cyclonic anomalies over eastern China is strengthened, which enhances the meridional gradient of relative vorticity. Owing to the transport of low-frequency relative vorticity and geostrophic vorticity by meridional wind, the ascending motion over South China intensifies and lasts for a long time, triggering a PHR-LF event. In addition, the tropical system is also a key factor to PHR-LF event. The QBWO of the convection over the South China Sea provide moisture for PHR-LF events, maintaining persistent rainfall and vertical ascending motion over South China.

1 Introduction

Heavy rain in China can be sudden, frequent and persistent. Persistent heavy rain (PHR) is the most likely to trigger large and severe floods, and thus it is more prone to cause disasters. Since the 1990s, PHR events in South China have increased significantly (Bao 2007; Qian 2012), manifesting as longer duration and larger spatial coverage as well as greater intensity (Chen and Zhai 2013). In China, the occurrence of PHR events is greatest in South China, and most of these events occur during the first rainy season (April–June) (Qian 2012; Chen and Zhai 2013). In South China, owing to the complex terrain, the thermal difference between land and sea, and the influence of the summer monsoon, rainfall is influenced by the synergistic effects of multiple factors at different latitudes and scales, making the precipitation forecast very challenging in this region.

The most immediate cause of persistent extreme events is the stable and persistent anomalous atmospheric circulation (Higgins and Mo 1997), and persistent extreme precipitation is caused by the interaction of several key circulations (Ding and Reiter 1982; Chen and Zhai 2014a, b). As the linkage and bridge between various anomalous circulations, the atmospheric teleconnection wave train has multi-factor synergy, exerting a powerful influence on the anomalous changes of weather and climate in East Asia. Frequent teleconnection patterns give rise to long-term circulation anomalies, leading to persistent extreme weather (Archambault et al. 2008). There are three teleconnection wave trains associated with summer rainfall in East Asia (Huang et al. 2013), namely the East Asia–Pacific teleconnection (EAP) (Huang and Li 1987; Nitta 1987), the Silk Road teleconnection (Lu et al. 2002; Enomoto et al. 2003; Wang et al. 2017), and the Eurasia–Asia teleconnection (EU) (Wallace and Gutzler 1981). During the strong and negative phase of the Silk Road pattern, both the westerly jet in the mid-latitude and the convergence zone of the upper troposphere stretch to the south, causing the subtropical high over the western Pacific to move southward, and finally leading to the negative precipitation anomalies in South China (Yang and Zhang 2007). The EU pattern propagating from the Nordic region along the great circle route to the downstream also effects summer rainfall in South China (Jin et al. 2010).

It has been documented extensively in the literature that the precipitation during the first rainy season over South China has obvious quasi-biweekly oscillation (QBWO) characteristics (Tang et al. 2007; Cao et al. 2012; Hong and Ren 2013; Li et al. 2014; Miao et al. 2017), which are closely related to the low-frequency oscillations of circulation. Low-frequency signals propagate in different directions from different origins, resulting in the occurrence of PHR. The intraseasonal oscillation of the upper tropospheric circulation in the mid-high latitudes has an impact on the low-frequency variation of PHR (Zhang et al. 2003; Li et al. 2010; Kong et al. 2017). Most studies focus only on the interannual and interdecadal variability of the atmospheric teleconnection wave train. In fact, studies have shown that the typical timescale of the atmospheric teleconnection is 1–2 weeks (Feldstein 2000), and some studies have suggested that the large-scale teleconnection mode in the Northern Hemisphere varies periodically with a period of 10–30 days (Sun and Li 2011). For example, as one of the low-frequency variability patterns, the total energy pattern of the Silk Road pattern can be replenished within a week or so (Kosaka et al. 2009). The impact of the EAP pattern on the PHR in the Yangtze–Huai River Valley is more remarkable on a 10–30 day timescale (Li et al. 2016). The interaction between the EU pattern and the EAP pattern on the same time scale affects the low-frequency activity of the Northeast Cold Vortex, while the conversion of the low-frequency oscillation phase modulates the precipitation over eastern China (Liu et al. 2012). A case study has shown that the low-frequency EU-type wave train in the mid-high latitudes can influence the low-frequency circulation over South China by the dispersion of Rossby waves during the first rainy season (Miao et al. 2017).

Despite the influence of the teleconnection patterns on the rain over South China during the first rainy season and the linkage between the QBWO of the wave trains and the rain over eastern China descriptively presented in some previous studies, there is rarely a systematic and unified view on the effect of the teleconnection patterns on PHR over South China. Which wave train is the most important to the PHR during the first rainy season over South China? How does it affect the duration and intensity of the precipitation? What is its dynamic mechanism? Moreover, a few studies has focused on the subseasonal scale. Therefore, this study examines the influence of the wave trains on the PHR during the first rainy season over South China on the subseasonal scale and associated dynamic mechanism, to provide a theoretical basis for the formation mechanism of PHR and a reference for the extended-range weather forecast of severe precipitation during the first rainy season over South China.

The remainder of this paper is organized as follows. The data and methods used in this study are introduced in Sect. 2. Characteristics of persistent precipitation during the first rainy season over South China are examined in Sect. 3, including the low-frequency characteristics of precipitation and the definition of a PHR-LF event. The influence of low-frequency circulations on the PHR-LF event is discussed in Sect. 4, including the influence of the low-frequency wave trains in Sect. 4.1 and diagnostic analysis of the vertical motion in South China in Sect. 4.2. A discussion is presented in Sects. 5 and conclusions follow in Sect. 6.

2 Data and methods

According to the World Meteorological Organization (WMO), 1981–2010 is the current standard climatic background. To study the climatological characteristics of low-frequency oscillations, we take 1981–2010 as our target period. The daily precipitation data used in this study were obtained from the Chinese Meteorological Administration (CMA), which has 2474 observation stations (available online http://data.cma.cn/). Although the starting and ending times of first rainy season vary year to year, in average, the rainy season spans from 6 April to 4 July (Gu et al. 2018). So the first rainy season in the study is defined to be the period from 1 April to 30 June (AMJ). The standard deviation distribution of 30 year (1981–2010) daily precipitation anomalies over eastern China during the first rainy season is shown in the left column of Fig. 1. The domain of South China is defined as 21.5°–26°N, 106.5°–120°E, which includes 230 CMA stations, and is shown in the right column of Fig. 1. The area-averaged precipitation over this domain is defined as the South China precipitation series.

Fig. 1

Standard deviation of the 30 year (1981–2010) daily precipitation anomalies (shadings) over eastern China (left column, the red box highlights South China), and distribution of CMA stations (dots) in South China (right column)

To explore the corresponding atmospheric activities, we used the daily atmospheric reanalysis data from the National Centers for Environmental Prediction-Department of Energy (NCEP-DOE) (R-2; Kanamitsu et al. 2002). In addition, the daily outgoing longwave radiation (OLR) data (Liebmann and Smith 1996) from the National Oceanic and Atmospheric Administration (NOAA) was used to detect the features of tropical convection. Both the NCEP-DOE R-2 and OLR datasets span from 1981 to 2010 on a 2.5° × 2.5° horizontal resolution.

To detect the contribution of the subseasonal oscillations to PHR, we used the Lancozs filtering method (Duchon 1979) to obtain the band-pass filtered time series of precipitation and atmospheric anomalies (after removing the corresponding climatological daily mean values). A two-tail student t-test was used to determine the significance of the composite variables using the formula:

$${\text{t}}=\frac{{\left| {\mathop \sum \nolimits_{{i=1}}^{N} F_{i}^{\prime }/N} \right|}}{{\sigma \sqrt {1/N^{\prime}} }},$$

(1)

where ${F}_{i}^{{\prime}}$ is the anomaly of a variable at time i, N is the sample size and $\sigma$is the standard deviation of ${F}_{i}^{{\prime}}$. For the filtered series, the effective sample size is approximate by${N}^{{\prime}}\cong$$\text{N}\frac{1-\rho }{1+\rho }$ (Wilks 2006; Jia et al. 2011) and $\rho$ is the lag-1 autocorrelation coefficient of ${F}_{i}^{{\prime}}$.

3 Persistent characteristics of precipitation during the first rainy season over South China

It is known that precipitation during the first rainy season over South China has a significant relationship with low-frequency oscillations (Tang et al. 2007; Cao et al. 2012; Hong and Ren 2013; Li et al. 2014; Miao et al. 2017). But how does the low-frequency oscillation affect the duration of precipitation? What is the relationship between the oscillation and PHR? To answer these questions, we need to investigate the low-frequency characteristics of precipitation and obtain a definition of the typical PHR-LF events.

3.1 Low-frequency characteristics of precipitation

To examine the low-frequency characteristics of precipitation during the first rainy season over South China, the power spectrum of the 5-day running-mean daily precipitation anomalies (Fig. 2a) is analyzed. The result shows that the precipitation anomalies during this period have remarkable QBWO characteristics, with a period of 8–24 days, while 30–60 days is the second significant period. By calculating the variance contribution of the low-frequency component to the precipitation anomalies (the ratio of the variance of the low-frequency component to that of the precipitation anomalies), the annual mean variance contribution of the 8–24 day component is 30% (Fig. 2b), which is significantly larger than that of the 30–60 day component (7%, figure omitted). Furthermore, we compare the daily precipitation anomaly time series and its 8–24 day filtered subseasonal component (Fig. 3). Here we can see that there is good correspondence between PHR and the QBWO, 83% of the real PHR events (the precipitation anomalies larger than one standard deviation of the climate state and last 3 days or more) appear at the strong positive phase of QBWO (the low-frequency precipitation larger than one standard deviation of the climate state), among which 76% appear when the strong positive phase of QBWO is persistent at least 3 days. That is, when the amplitude of low-frequency precipitation is strong and the duration is long, the real PHR events often occur. Therefore, the low-frequency oscillations can explain PHR well and are a good indicator for the prediction of PHR. Correspondingly, the 8–24 day QBWOs of the atmospheric circulations are focused on in the present study. It should be noted that the low-frequency precipitation and variables we mention below are all 8–24 day filtered.

Fig. 2

a Power spectrum of the 5-day running means of daily precipitation anomalies during AMJ (unit: W² m^− 4). The thick black, blue, dotted green, and red lines denote the power spectrum, 95% confidence level, 5% confidence level, and red noise, respectively. b Interannual variance contribution of 8–24 day low-frequency component to precipitation anomalies. The dotted black line denotes the 30 year (1981–2010) mean value

Fig. 3

The 30 year (1981–2010) daily precipitation anomalies (grey bars) and the 8–24-day filtered precipitation anomalies (red lines, unit: mm). The blue and black lines indicate one positive or negative standard deviation of precipitation anomalies and low-frequency precipitation, respectively. The black arrows indicate the real PHR events

3.2 Definition of low-frequency PHR (PHR-LF) event

First, we define a low-frequency heavy rain event as 8–24-day filtered precipitation greater than one standard deviation of the 30 year (1981–2010) filtered time series. Then, by counting the number of low-frequency heavy rain events and their total days at different durations (Fig. 4a), it can be seen that low-frequency heavy rain events lasting at least 3 days account for a large proportion of all events in terms of both number of cases (69.0%) and total precipitation days (83.6%).

Fig. 4

a Numbers and total precipitation days of low-frequency heavy rain events at different durations, b composites of geopotential height anomalies at 200 hPa for low-frequency heavy rain events (unit: gpm). Shadings are the areas with statistical significance exceeding the 95% level. And c spatial correlation coefficients between the composite representative field (geopotential height anomalies at 200 hPa over 0–80°N, 0–120°E) of low-frequency heavy rain events at different durations and that of all low-frequency heavy rain events

Our previous studies have shown that the maintenance of anomalous circulation at 200 hPa over 0–80°N, 0–120°E is a necessary condition for the occurrence of a persistent rain anomaly over South China (Miao et al. 2017). The composite geopotential height anomalies at 200 hPa for all low-frequency heavy rain events are showed in Fig. 4b. To determine the reasonability of the definition of the duration (3 days) of a persistent event, we calculate the spatial correlation coefficient between the composite geopotential height anomaly at 200 hPa over this domain of low-frequency heavy rain events at different durations and that of all low-frequency heavy rain events, respectively (Fig. 4c). It is found that there is a noticeable difference between the composite circulations of low-frequency heavy rain events lasting 1–2 days and those of all low-frequency heavy rain events. However, the composite circulations of low-frequency heavy rain events lasting 3 days or more are highly correlated with those of all low-frequency heavy rain events, among which the spatial correlation of events lasting only 3 days is the highest. In summary, when low-frequency heavy rain events persist at least 3 days, the corresponding circulations are more stable. Therefore, we define a PHR-LF event as the 8–24-day filtered precipitation larger than one standard deviation of filtered time series and persisting at least 3 days during the first rainy season over South China.

4 Influence of low-frequency circulations on the PHR-LF events

To discuss the low-frequency circulations (8–24-day filtered circulations) related to PHR-LF events, we show the composite circulation anomalies and their low-frequency components for PHR-LF events. Comparing the composites of the anomalous (Fig. 5b) and low-frequency (Fig. 5d) fields of geopotential height at 200 hPa, it can be clearly seen that during a PHR-LF event, there are two obvious low-frequency wave trains in the mid-high latitudes: one is the wave train crossing the Eurasian continent and the other is the wave train along the subtropical westerly jet. These results imply that both wave trains play an important role in PHR-LF events during the first rainy season over South China. It is known from the climatology of the zonal wind at 200 hPa (figure omitted) during the first rainy season that South China is located beneath the southern section of the entrance of the jet over Japan, providing favorable conditions for upper-level divergence. Eastern China is under control of the cyclonic anomalies, while South China is located at the south flank of the cyclonic anomalies. The negative anomaly of the cyclonic anomalies over eastern China strengthens the height gradient, strengthening the jet and further enhancing the upper-level divergence over South China, which is conducive to the occurrence of PHR-LF events. Furthermore, the anomalous and low-frequency fields in Fig. 5b, d show similar spatial distributions. Figure 6a, b show the longitude-height sections of composites geopotential height anomalies and low-frequency geopotential height anomalies along 20°–40°N associated with PHR-LF events. We can find almost equivalent structures in both raw anomalies and low-frequency anomalies, and they are similar to each other, the spatial correlation coefficient between the anomalous fields and low-frequency fields are all significant and increases with height (Fig. 6c). The above findings suggest that the low-frequency circulations contribute greatly to the anomalous circulation related to PHR-LF events. The spatial correlation coefficient increases with height and the notable centers of composite anomalies are located above 400 hPa, indicating that the PHR-LF events may be associated with such upper-level circulation pattern. Correspondingly, we also compare the composite geopotential height at 200 hPa for low-frequency heavy rain events lasting only 1–2 days (Fig. 5a, c). There is no obvious wave-like pattern, and the spatial correlation between the low-frequency and anomalous fields is insignificant. This indicated that the low-frequency oscillations contribute largely to the anomalous circulations related to the PHR-LF events, and the superposition of these two wave trains in the mid-high latitudes is crucial to the persistence of heavy rain.

Fig. 5

a, b Composites of geopotential height anomalies at 200 hPa for low-frequency heavy rain events lasting a only 1–2 days and b 3 days or more, and c, d composites of 8–24-day filtered geopotential height anomalies at 200 hPa for low-frequency heavy rain events lasting c only 1–2 days and d 3 days or more. The black contour interval is 2 gpm. Shadings are the areas with statistical significance exceeding the 95% level. The black dash lines and red contours in b and d show the wave trains and zonal wind greater than 20 m s^− 1, respectively

Fig. 6

Longitude-height sections of composites a geopotential height anomalies and b low-frequency geopotential height anomalies along 20°–40°N for PHR-LF events. Contour intervals are 2 gpm. c Vertical profile of the spatial correlation coefficient between the anomalous field in Fig. 5b and the low-frequency field in Fig. 5d

4.1 Influence of low-frequency wave trains on PHR-LF events

To further illustrate the role of low-frequency wave trains in the anomalous circulations related to PHR-LF events, we calculate the three-dimensional wave activity flux (WAF) to depict the pattern of wave activity (Takaya and Nakamura 2001; Tam and Li 2006) using the following formula:

$${\mathbf{W}}=\frac{\gamma }{{2\left| {\mathbf{U}} \right|}}\left( {\begin{array}{*{20}{c}} {U\left( {\psi _{x}^{2} - \psi {\psi _{xx}}} \right)+V\left( {{\psi _x}{\psi _y} - \psi {\psi _{xy}}} \right)} \\ {U\left( {{\psi _x}{\psi _y} - \psi {\psi _{xy}}} \right)+V\left( {\psi _{y}^{2} - \psi {\psi _{yy}}} \right)} \\ {\frac{{f_{0}^{2}}}{{{S_p}}}\left[ {U\left( {{\psi _x}{\psi _p} - \psi {\psi _{xp}}} \right)+V\left( {{\psi _y}{\psi _p} - \psi {\psi _{yp}}} \right)} \right]} \end{array}} \right),\;\;\left( {~\begin{array}{*{20}{c}} {\gamma =1,U>0} \\ {\gamma = - 1,U<0} \end{array}~} \right),$$

(2)

where $\psi$ is the perturbed stream-function, the subscripts of which represent partial derivatives, and $\mathbf{U}$ is the horizontal mean flow with U and V as its zonal and meridional components, respectively. We define the mean flow as the 30 year (1981–2010) climatological mean from April to June in this study. The WAF vector is parallel to the group velocity of Rossby wave in the WKB (Wentzel-Kramers-Brillouin) approximation. Its convergence leads to amplification of the wave pseudo-moment, that is, when the WAF vectors converge, wave energy accumulates and it is conducive to the disturbance enhancement, while wave energy disperses when the WAF vectors diverge (Sobel and Bretherton 1999). In this expression, f₀ is the Coriolis parameter and S_p is the static stability parameter and defined by ${S}_{p}=-\hspace{0.17em}\alpha (\partial ln\theta /\partial p)$, where $\theta$ denotes potential temperature and $\alpha$ represents specific volume. Figure 7 shows the WAF and its divergence for the composite low-frequency fields of PHR-LF events in the troposphere. In addition, the perturbation stream-function of composite fields is shown to delineate the circulation of wave trains. The low-frequency wave train, which is similar to the EU pattern in the high latitudes, has an equivalent barotropic structure, while the wave train pattern along the subtropical jet is less obvious at the lower troposphere, which may be due to the influence of the Tibetan Plateau and baroclinic effect. The wave energy disperses toward South China along these two wave trains. In the upper troposphere, the cyclonic anomalies over eastern China is where the wave energy converges. However, the propagation of wave energy is mainly concentrated in the upper and middle levels, with weaker propagation in the lower level. In the lower troposphere, the South China Sea becomes the origin of the wave energy, which spreads north to South China. Figure 8a, b show the vertical cross-sections of the WAF along the latitude and longitude range of the domain where the aforementioned cyclonic anomalies over eastern China is located. The results reveal that the wave energy disperses toward eastern China along these two wave trains respectively from north to south and from west to east, and sinks in South China, with maximum value mainly in the lower troposphere. The downward WAF are located where the center of positive vorticity anomalies is found and the cold air and warm air converge, because the center of cyclonic anomalies is located over South China (Fig. 8c, d). In addition to the way of WAF to describe the wave trains, the energy conversion from the mean flow to perturbations is calculated (Kosaka and Nakamura 2006). It seems that the wave train in the high latitudes gains available potential energy from the mean flow, while the wave train in the mid-latitude gains kinetic energy from the mean flow and loses available potential energy (figure ommitted).

Fig. 7

Stream-functions (contours) and wave activity flux (arrows; unit: m² s^− 2) and its divergence (shadings; unit: 10^− 6m s^− 2) for composite anomalous fields at a 200 hPa, b 500 hPa, c 850 hPa for PHR-LF events. Contour intervals are 3 × 10⁶ m² s^− 1 in a, 2 × 10⁶ m² s^− 1 in b, and 1 × 10⁶ m² s^− 1 in c

Fig. 8

Vertical cross-sections of a meridional WAF (arrows; unit: m² s^− 2) and vertical WAF (contours; unit: Pa² s^− 2) along 100°–115°E, and b zonal WAF (arrows; unit: m² s^− 2) and vertical WAF (contours; unit: Pa² s^− 2) along 20°–26°N for composite anomalous fields. The red shading indicates the downward flux, and the black shading indicates topography. Vertical cross-sections of low-frequency air temperature (contours; unit: K) and vorticity anomalies (shadings; unit: 10^− 6 s^− 1) along c 100°–115°E and d 20°–26°N

To analyze the specific impact of the aforementioned low-frequency wave trains on the PHR-LF events, their temporal evolutions are examined. Figure 9 presents the evolution of the composite of the precipitation anomaly and its different components at different timescales for PHR-LF events, including the synoptic component (3–6 days), quasi-biweekly component (8–24 days), and intraseasonal component (30–60 days). Day 0 represents the occurrence of PHR-LF events, and d refers to the dth day prior to (negative) or after (positive) occurrence of PHR-LF events. The result reveals that the contribution of the QBWO dominants the precipitation anomalies for PHR, while the synoptic and intraseasonal components have little or even negative contribution. Figure 10 shows the temporal evolution of 8–24-day filtered WAF and its divergence and the perturbation stream-function of the composite low-frequency fields. From the distribution of the WAF divergence, it can be seen that Western Europe and the northern Mediterranean is the origin of the Rossby waves, which propagates along two paths in the mid-high latitudes respectively. One branch of the wave energy propagates first northeastward to West Siberia and then southeastward to the southeast coast of China; the other branch propagates first southeastward to the Iranian Plateau and then eastward to South China through the Tibetan Plateau. The path of wave energy dispersion is exactly the aforementioned two wave trains.

Fig. 9

Temporal evolution of composite anomalous precipitation (grey bars), 8–24-day filtered anomalous precipitation (red bars), 3–6-day filtered anomalous precipitation (blue bars), and 30–60-day filtered anomalous precipitation (yellow bars) averaged over South China (unit: mm). Day 0 represents the occurrence of PHR, and d refers to the dth day prior to (negative) or after (positive) occurrence of PHR

Fig. 10

Same as Fig. 7, but for temporal evolution. The number on the upper left of each panel refers to the number of days prior to (negative) or after (positive) the occurrence of the PHR-LF events

From day − 8 to − 2, the wave train in the mid-latitude appears before the occurrence of PHR-LF events, and its anomalous circulation centers gradually strengthens. On day 0, divergence of the waves originating over Western Europe and the northern Mediterranean reaches its strongest, and the pattern of the wave train becomes very clear. Then, the propagation of wave energy in the eastern part of the Tibetan Plateau becomes enhanced with the enhancement of low-frequency precipitation, which is conducive to a strengthening of the disturbance of the cyclonic anomalies over eastern China. South China is the sink of the wave energy. On the second day after the beginning of precipitation, that is, when the intensity of precipitation is the strongest (Fig. 9), the anomaly of the cyclonic anomalies over eastern China reaches its maximum, and the WAF convergence over South China also reaches a peak, which may be caused by the upper-level divergence associated with the rainfall. During the PHR-LF events, from day 0 to day 2, the source and sink of wave trains are consistent in the distribution of divergence of WAF and wave source calculated by the equation of Sardeshmukh and Hoskins (1988), which are Western Europe and the northern Mediterranean and South China, respectively. The wave train pattern in the mid-latitude weakens rapidly after the significant decrease of precipitation, so the wave train may be a trigger for heavy rain. As for the wave train in the high latitudes, each circulation center is weak and the pattern is not clear before the heavy rain occurs. However, with the occurrence and maintenance of the heavy rain, the pattern and its energy propagation are significantly enhanced. The southward energy dispersion of the wave train is also conducive to the enhancement of the cyclonic anomalies disturbance over eastern China, so this may be the reason for the maintenance of precipitation. It can be seen that the convergence of wave energy enhances the disturbance of the cyclonic anomalies over eastern China, and is conducive to development and maintenance of the disturbance, which in turn leads to the occurrence and maintenance of the PHR-LF events. It is noted that the evolution of the two wave trains and their effects on the precipitation are clearly different.

4.2 Diagnostic analysis of the vertical motion in South China

The effect of wave trains on the disturbance of the cyclonic anomalies over eastern China is mentioned earlier, and taking into account the importance of the vertical motion to the rain anomalies, we use the quasi-geostrophic $\omega$ equation to diagnose the formation mechanism of the vertical motions related to PHR-LF events during the first rainy season over South China. According to Holton (1992), ω is contributed by three parts: the vertical difference of vorticity horizontal advection, horizontal temperature advection and the diabatic heating. In the study, we didn’t discuss the term of diabatic heating, which always has similar change to rainfall and hence to omega. Here we diagnosed anomalous vertical motion that could be induced by adiabatic processes (Tam and Li 2006; Kosaka et al. 2011; Hu et al. 2017). So the quasi-geostrophic $\omega$ equation can be taken as:

$${{\left( {\sigma {\nabla ^2}+{f_0}^{2}\frac{{{\partial ^2}}}{{\partial p}}} \right)\omega=} \atop {\phantom {\left( {\sigma {\nabla ^2}+{f_0}^{2}\frac{{{\partial ^2}}}{{\partial p}}} \right)\omega=}}} \frac{{f_0}\frac{\partial }{{\partial p}}{{\left[ {\overrightarrow {{V_g}} ~ \cdot ~\nabla \left( {{\zeta _g}+f} \right)} \right]}}}{{\text{A}}} {{+} \atop {\phantom{+}}}\frac{{{\nabla ^2}\left[ {\frac{R}{p}\overrightarrow {{V_g}~} ~ \cdot ~\nabla T} \right]}}{{\text{B}}},$$

(3)

where $\sigma$ is the static stability parameter, p is pressure, ${\zeta }_{g}$ is geostrophic vorticity, $\overrightarrow{{V}_{g}}$ is geostrophic wind, R is dry air specific gas constant, and T is temperature. The term on the left-hand side is the Laplacian of $\omega$, which is approximately equivalent to $\omega$ multiplied by a negative coefficient. The terms on the right-hand side are the vertical variation of geostrophic absolute vorticity advection (called term A) and the Laplacian of geostrophic temperature advection (called term B), respectively. To diagnose the effect of low-frequency component and discuss the interaction between the basic state and the perturbation, all variables are decomposed into a basic state and its perturbation (Wen et al. 2010, Hsu and Li 2011; Wei et al. 2014). Then terms A and B in Eq. 3 can be written as:

$$\begin{aligned} {f_0}\frac{\partial }{{\partial p}} & \left[ {\overrightarrow {{V_g}} ~ \cdot ~\nabla \left( {{\zeta _g}+f} \right)} \right] \\ & ={f_0}\frac{\partial }{{\partial p}}\left[ {\left( {\overline {{\overrightarrow {{V_g}} }} +{{\overrightarrow {{V_g}} }^{~^{\prime}}}} \right) \cdot ~\left[ {\nabla \left( {\overline {{{\zeta _g}}} +{\zeta _g}^{{^{\prime}}}} \right)+\frac{{\partial f}}{{\partial y}}} \right]} \right] \\ & ={f_0}\frac{\partial }{{\partial p}}\left[ {\overline {{{u_g}}} \frac{{\partial \overline {{{\zeta _g}}} }}{{\partial x}}+{u_g}^{{^{\prime}}}\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial x}}+\overline {{{u_g}}} \frac{{\partial {\zeta _g}^{{~^{\prime}}}}}{{\partial x}}+{u_g}^{{~^{\prime}}}\frac{{\partial {\zeta _g}^{{^{\prime}}}}}{{\partial x}}+\overline {{{v_g}}} \frac{{\partial \overline {{{\zeta _g}}} }}{{\partial y}}+{v_g}^{'}\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial y}}+\overline {{{v_g}}} \frac{{\partial {\zeta _g}^{{^{\prime}}}}}{{\partial y}}} \right. \\ & \;\;\left. {+{v_g}^{'}\frac{{\partial {\zeta _g}^{{^{\prime}}}}}{{\partial y}}+\left( {\overline {{{v_g}}} \frac{{\partial f}}{{\partial y}}+{v_g}^{{^{\prime}}}\frac{{\partial f}}{{\partial y}}} \right)} \right], \\ \end{aligned}$$

(4)

$$\begin{aligned} {\nabla ^2} & \left[ {\frac{R}{p}\overrightarrow {{V_g}~} ~ \cdot ~\nabla T} \right] \\ & \;\;=\frac{R}{p}{\nabla ^2}\left[ {\left( {\overline {{\overrightarrow {{V_g}} }} +{{\overrightarrow {{V_g}} }^\prime }} \right)~ \cdot \nabla \left( {\overline {T} +{T^\prime }} \right)} \right] \\ & \;\;=\frac{R}{p}{\nabla ^2}\left( {\overline {{{u_g}}} \frac{{\partial \overline {T} }}{{\partial x}}+{u_g}^{\prime }\frac{{\partial \overline {T} }}{{\partial x}}+\overline {{{u_g}}} \frac{{\partial {T^\prime }}}{{\partial x}}+{u_g}^{\prime }\frac{{\partial {T^\prime }}}{{\partial x}}+\overline {{{v_g}}} \frac{{\partial \overline {T} }}{{\partial y}}+{v_g}^{\prime }\frac{{\partial \overline {T} }}{{\partial y}}+\overline {{{v_g}}} \frac{{\partial {T^\prime }}}{{\partial y}}+{v_g}^{\prime }\frac{{\partial {T^\prime }}}{{\partial y}}} \right). \\ \end{aligned}$$

(5)

We define the variables with an over bar as the basic state, which are the climatological means from 1 April to 30 June during 1981–2010, and the variables with a prime as the perturbation, which are the 8–24-day filtered composite fields for PHR-LF events. Considering that the basic state variables themselves satisfy Eq. 3, the $\omega$ equation of low-frequency component can be written as, which is similar to previous studies (e.g., Tam and Li 2006; Kosaka et al. 2011; Hu et al. 2017):

$$\begin{aligned} \left( {\sigma {\nabla ^2}+{f_0}^{2}\frac{{{\partial ^2}}}{{\partial p}}} \right)\omega ^{\prime} \\ & \frac{={f_0}\frac{\partial }{{\partial p}}\left[ {{{\overrightarrow {{V_g}} }^\prime } \cdot \nabla \left( {{\zeta _g}^{\prime }+\overline {{{\zeta _g}}} +f} \right)+\overline {{\overrightarrow {{V_g}} }} \cdot \nabla {\zeta _g}^{\prime }} \right]}{\text{A}} { \vphantom{\frac{={f_0}\frac{\partial }{{\partial p}}\left[ {{{\overrightarrow {{V_g}} }^\prime } \cdot \nabla \left( {{\zeta _g}^{\prime }+\overline {{{\zeta _g}}} +f} \right)+\overline {{\overrightarrow {{V_g}} }} \cdot \nabla {\zeta _g}^{\prime }} \right]}{\text{A}}} + \atop \vphantom{\frac{\frac{R}{p}{\nabla ^2}\left[ {{{\overrightarrow {{V_g}} }^\prime } \cdot \nabla \left( {\bar {T}+{T^\prime }} \right)+\overline {{\overrightarrow {{V_g}} }} \cdot \nabla {T^\prime }} \right]}{\text{B}}}} \frac{\frac{R}{p}{\nabla ^2}\left[ {{{\overrightarrow {{V_g}} }^\prime } \cdot \nabla \left( {\bar {T}+{T^\prime }} \right)+\overline {{\overrightarrow {{V_g}} }} \cdot \nabla {T^\prime }} \right]}{\text{B}} \\ & \frac{{{=f_0}\frac{\partial }{{\partial p}}\left( {{u_g}^{\prime }\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial x}}} \right.}}{{{\text{A1}}}}{ {\vphantom{=\frac{{{f_0}\frac{\partial }{{\partial p}}\left( {{u_g}^{\prime }\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial x}}} \right.}}{{{\text{A1}}}}}+} \atop {\vphantom{=\frac{{{f_0}\frac{\partial }{{\partial p}}\left( {{u_g}^{\prime }\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial x}}} \right.}}{{{\text{A1}}}}}} }\frac{{\;\;\overline {{{u_g}}} \frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A2}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;{v_g}^{\prime }\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial y}}}}{{{\text{A4}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;\overline {{{v_g}}} \frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}}}{{{\text{A5}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;{v_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}}}{{{\text{A6}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\left. {\;\;{v_g}^{\prime }\frac{{\partial f}}{{\partial y}}} \right)}}{{{\text{A7}}}} \\ &\;\;{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;\frac{R}{p}{\nabla ^2}\left( {{u_g}^{\prime }\frac{{\partial \bar {T}}}{{\partial x}}} \right.}}{{{\text{B1}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;\overline {{{u_g}}} \frac{{\partial {T^\prime }}}{{\partial x}}}}{{{\text{B2}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;{u_g}^{\prime }\frac{{\partial {T^\prime }}}{{\partial x}}}}{{{\text{B3}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;{v_g}^{\prime }\frac{{\partial \bar {T}}}{{\partial y}}}}{{{\text{B4}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\;\;\overline {{{v_g}}} \frac{{\partial {T^\prime }}}{{\partial y}}}}{{{\text{B5}}}}{ \vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}+ \atop {\vphantom{\frac{{\;\;{u_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial x}}}}{{{\text{A3}}}}}} }\frac{{\left. {\;\;{v_g}^{\prime }\frac{{\partial {T^\prime }}}{{\partial y}}} \right)}}{{{\text{B6}}}}. \\ \end{aligned}$$

(6)

Terms A1–A7 are dynamic terms related to vorticity advection, and terms B1–B6 are thermal terms related to temperature advection. These 13 terms represent 13 factors that affect the low-frequency variation of vertical motion over South China during the first rainy season. By calculating each term on the right-hand side of Eq. 6, we can determine the main contributors to the low-frequency vertical motion over South China. Subsequently, we choose 500 hPa, which is deemed to be a non-divergence level, to diagnose the vertical motion by calculating A1–A7 and B1–B6 in South China.

As shown in Fig. 11, the results suggest that terms A5 $\left( {{f_0}\frac{\partial }{{\partial p}}\left( {\overline {{{v_g}}} \frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}} \right)} \right)$, A6 $\left( {{f_0}\frac{\partial }{{\partial p}}\left( {{v_g}^{\prime }\frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}} \right)} \right)$ and A7 $\left( {{f_0}\frac{\partial }{{\partial p}}\left( {{v_g}^{\prime }\frac{{\partial f}}{{\partial y}}} \right)} \right)$ are primary contributors to the updraft anomalies. Furthermore, terms B2 $\left( {\frac{R}{p}{\nabla ^2}\left( {\overline {{{u_g}}} \frac{{\partial T^{\prime}}}{{\partial x}}} \right)} \right)$ and A4 $\left( {{f_0}\frac{\partial }{{\partial p}}\left( {{v_g}^{\prime }\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial y}}} \right)} \right)$ are less important to the updraft anomalies. Terms A5 and A6 denote the vertical difference of advections of the low-frequency relative vorticity by the basic meridional flow and low-frequency meridional wind, respectively. Term A7 denotes the advections of vorticity of Earth by low-frequency meridional wind. Term B2 denotes the advection of low-frequency temperature by the basic zonal flow and term A4 denotes the advection of the basic relative vorticity by the low-frequency meridional wind.

Fig. 11

Composites of dynamic terms denoted by A1–A7 (red bars) and thermal terms denoted by B1–B6 (blue bars) at 500 hPa averaged over South China (unit: 10^− 19 m s^− 1 kg^− 1)

Figure 12 shows the composites of low-frequency fields for PHR-LF events and the basic-state fields to illustrate the impact of the aforementioned terms on the vertical motion. From the distribution of the low-frequency relative vorticity, South China is located below the center of the positive low-frequency relative vorticity, and the SCS is located in the region of negative low-frequency relative vorticity (Fig. 12a, c, e), thus the meridional gradient of the low-frequency relative vorticity $\left( {\frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}} \right)$ is positive. Terms A5 and A6 are closely related to the advection by the basic and low-frequency southerly winds, respectively, and to the negative low-frequency relative vorticity over SCS. Because the westerly basic winds prevail in the eastern China at upper level, the basic southerly winds ($\stackrel{-}{{v}_{g}}$) decrease with height. From the distribution of the low-frequency winds (Fig. 12a, c, e), it can be seen that the low-frequency southerly winds (${{v}_{g}}^{{\prime}}$) also decrease with height. During PHR-LF events, South China is located in the center of a low-frequency cyclonic anomalies related to the anomalous cyclone in the lower troposphere (Fig. 12c, e), while in the upper level, South China lies to the southeast of the low-frequency cyclonic anomalies associated with the anomalous cyclone in the eastern China (Fig. 12a), so the meridional gradient of the low-frequency relative vorticity $\left( {\frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}} \right)$ decreases with height. In addition, the northward propagation of wave energy over SCS is mainly focused on the lower levels (Fig. 7c), strengthening the low-frequency relative vorticity of the cyclonic anomalies over South China in the lower levels, which is another reason why the meridional gradient of the low-frequency relative vorticity $\left( {\frac{{\partial {\zeta _g}^{\prime }}}{{\partial y}}} \right)$ decreases with height. For term A7, the higher the latitude, the larger the Coriolis parameter f. From the basic state of relative vorticity (Fig. 12b, d, f), it can be seen that South China is located on the northwest flank of the western Pacific subtropical anticyclone in the lower troposphere, where the southwest winds are prevalent, while in the upper troposphere there is a strong anticyclone over the Indo-China Peninsula and on the southern flank of the Tibetan Plateau, and South China is located on the northern flank of the anticyclone, where the westerly winds are prevalent. Therefore the meridional gradient of the basic state of relative vorticity $\left( {\frac{{\partial \overline {{{\zeta _g}}} }}{{\partial y}}} \right)$ is positive, and decreases with height. In summary, the advection of relative vorticity associated with basic and low-frequency southerly winds as well as the advection of geostrophic vorticity associated with the low-frequency southerly winds all decrease with height. According to the relationship between the vertical motion and vorticity advection term in Eq. 3, that is $-\omega \propto \frac{\partial }{\partial p}\left(-\overrightarrow{{v}_{g}}\bullet \nabla \left({\zeta }_{g}+f\right)\right)$, terms A4, A5, A6, and A7 are all contributors to the ascending motion over South China. However, the meridional gradient of the basic relative vorticity is small and shows little change with height because of the large spatial scale of the basic-state circulations; as a result, term A4 contributes less to the updraft anomalies than the other three dynamic terms. The above discussion about low-frequency relative vorticity is consistent with what show in Fig. 13b. In addition, the strong upper-level divergence and lower-level convergence (Fig. 13c) also contribute to the strong vertical motion over South China (Fig. 13a).

Fig. 12

Composite of low-frequency relative vorticity (shadings; unit: 10^− 6 s^− 1) and winds (arrows; unit: m s^− 1) at a 200 hPa, c 500 hPa, e 850 hPa. Basic-state fields of relative vorticity and winds at b 200 hPa, d 500 hPa, f 850 hPa averaged in AMJ. The low-frequency relative vorticity fields with statistical significance exceeding the 95% level are highlighted by thick lines, and only low-frequency winds passing the 95% confident level have been plotted. The red box highlights South China

Fig. 13

Vertical cross-sections of composites of low-frequency a vertical velocity (unit: Pa s^− 1), b relative vorticity (unit: s^− 1) and c divergence (unit: s^− 1) along 106.5°–120°E for persistent heavy rain events. Shadings are the areas with statistical significance exceeding the 95% level

Figure 14 shows the composite of low-frequency temperature and basic winds at 500 hPa for PHR-LF events. The westerly basic winds prevail in the mid-latitudes, and there is a strong warm sector over the Tibetan Plateau and its southern flank, thus there are warm advections to South China. According to the relationship between the vertical motion and temperature advection term in Eq. 3, that is $-\omega \propto \frac{R}{p}\left(-\overrightarrow{{v}_{g}}\bullet \nabla T\right)$, the low-frequency warm advection by the basic westerly winds leads to the ascending motion over South China as denoted by term B2.

Fig. 14

Basic-state fields of winds (arrows; unit: m s^− 1) averaged in AMJ and composite of low-frequency temperature (shadings; unit: K) at 500 hPa. The temperature fields with statistical significance exceeding the 95% level are highlighted by thick lines. The red box highlights South China

From what has been discussed above, it seems that the meridional transfer of low-frequency relative vorticity and geostrophic vorticity by meridional winds are the main factors affecting the low-frequency variation of vertical motion over South China, in which the meridional gradient of low-frequency relative vorticity plays an important role. Figure 15 shows the evolution of low-frequency relative vorticity and temperature at 500 hPa. Compared with the evolution of the wave trains (Fig. 10), it is found that the perturbation anomaly caused by the wave energy propagation of the two wave trains is conducive to the enhancement of the low-frequency relative vorticity anomalies in eastern China, further increasing the meridional gradient of the low-frequency relative vorticity. The vertical variation of geostrophic absolute vorticity advection in Eq. 3 itself can induce ascending motion over South China, while the enhancement of the meridional gradient of the low-frequency relative vorticity further enhances and maintains the ascending motion, which is beneficial to the development of PHR-LF events over South China. From the evolution of the low-frequency temperature in Fig. 15, we can see that the warm advection denoted by term B2 is closely related to the anticyclone near the Tibetan Plateau, which is a part of the subtropical wave train. With the development of the wave train pattern, the anticyclone is strengthened, and the warm sector over the Tibetan Plateau and its southern flank is also strengthened, which enhances the warm advection to South China, and finally leads to the enhancement of ascending motion over South China. The analyses above indicate that the relationship between the low-frequency wave trains and PHR-LF events during the first rainy season over South China is reflected in both the dynamic and thermal terms in Eq. 3.

Fig. 15

Temporal evolution of composite low-frequency relative vorticity (shadings; unit: 10^− 6 s^− 1) and temperature (contours; unit: K) at 500 hPa. The temperature fields with statistical significance exceeding the 95% level are highlighted by dots

To directly describe the relationship between the wave trains and precipitation, we define several key areas related to the precipitation over South China and explore the evolution of their responding circulations with PHR-LF events. The low-frequency relative vorticity averaged over the area 20–40°N, 100–120°E at 200 hPa is chosen to represent the anomalous variation of the cyclonic anomalies over eastern China, and its divergence of the WAF represents the convergence (divergence) of wave energy, representing the strengthening (weakening) of the disturbance in this area. The low-frequency temperature averaged over the area 15–40°N, 75–95°E at 500 hPa is chosen to represent the anomalous variation of the warm advection of the aforementioned warm sector over the Tibetan Plateau and its southern flank. The low-frequency vertical velocity averaged over the domain of South China is chosen to represent the anomalous variation of the vertical motion over South China. Figure 16 shows the evolution of composite variables over these key areas for PHR-LF events, from which we can see the close relationship between the variables and the vertical motions as well as the PHR-LF events over South China. On day − 1, there is a positive precipitation anomaly, but it does not meet the criteria of heavy rain. Meanwhile, the wave energy in the upper level begins to converge, warm advection occurs in the middle level, and weak ascending motion appears over South China. With the enhancement of precipitation, the convergence of wave energy, the warm advection, the relative vorticity, and the vertical motion all strengthen gradually. When the precipitation reaches the maximum (day 1), the convergence of the wave energy in the upper level and the warm sector over the Tibetan Plateau and its southern flank in the middle level reach a maximum simultaneously. On day 2, the precipitation remains strong, the relative vorticity in the upper level and the ascending motion reaches its strongest, and the warm sector over the Tibetan Plateau and its southern flank still remains strong, while the convergence of the wave energy in the upper level decreases. From day 3 to day 4, with the decrease of precipitation intensity, the variables of the key areas weaken, among which the WAF divergence of the cyclonic anomalies over eastern China decreases the fastest. On day 4, the heavy rain disappears but the precipitation anomaly remains positive, and the domain of the cyclonic anomalies over eastern China becomes the divergence area of the wave energy. It can be seen that the convergence of wave energy occurs before the enhancement of the relative vorticity in the upper level, indicating that the dispersion of the wave energy triggers the cyclonic anomalies circulations over eastern China related to the PHR-LF events. The warm sector over the Tibetan Plateau and its southern flank in the middle level increases (decreases) with the increase (decrease) of precipitation and the vertical motion over South China, indicating that the warm advection plays an important role in strengthening and maintenance of precipitation. In conclusion, the low-frequency circulations in the mid-high latitudes could affect the low-frequency circulations and vertical motion over South China, in which the low-frequency wave trains play an important role, and further cause the occurrence of PHR-LF events during the first rainy season over South China.

Fig. 16

Temporal evolution of composite low-frequency variables over key areas, including precipitation anomalies averaged over South China (grey bars; unit: mm), relative vorticity (blue line; unit: 10^− 6 s^− 1), and divergence of WAF (brown line; unit: 10^− 7 m s^− 2) averaged over the area 20–40°N, 100–120°E at 200 hPa, temperature averaged over the area 15–40°N, 75–95°E at 500 hPa (green line; unit: 10^− 1 K), vertical velocity averaged over South China (red line; unit: 10^− 2 Pa s^− 1)

5 Discussions

As South China is located at the southern tip of Eurasia, the occurrence of PHR during the first rainy season over South China is closely related to the large-scale circulation anomalies in the low latitudes. From the composites of anomalies and low-frequency OLR and winds for low-frequency heavy rain events (Fig. 17), it is found that there is a significant negative correlation between the low-frequency oscillations of the convection over the SCS and PHR-LF events (Fig. 17b, d); however, there is no obvious convection signal in low-frequency heavy rain events lasting only 1–2 days (Fig. 17a, c). Therefore, the convection over the SCS is another factor that affects PHR-LF events, and it is important for the persistence of heavy rain. The spatial distributions of the anomalous and low-frequency field are very similar. For example, the spatial correlation coefficient of the OLR for PHR-LF events is 0.94, higher than that of low-frequency heavy rain events lasting only 1–2 days. It is can be seen that the low-frequency oscillations in the tropics also contribute greatly to the anomalous circulations associated with PHR-LF events. From the wind fields, we can see that the PHR-LF events have stronger water vapor convergence and more water vapor channels. The water vapor can be transported from the Bay of Bengal through the SCS and then northeastward to South China, and from the northwest Pacific westward to South China, respectively. It can be seen that abundant moisture transport is a necessary condition for the formation of PHR. Studies show that continuous water vapor is an important condition of persistent heavy rain (Yuan et al. 2012; Leung et al. 2018), and intraseasonal components play a deterministic role in the changes of moisture convergence (Li and Zhou 2015).

Fig. 17

a, b Composites of anomalous OLR (shadings; unit: W m^− 2) and horizontal winds (arrows; unit: m s^− 1) at 850 hPa for low-frequency heavy rain events lasting a only 1–2 days and b 3 days or more, and c, d composites of 8–24-day filtered anomalous OLR and horizontal winds (arrows; unit: m s^− 1) at 850 hPa for low-frequency heavy rain events lasting c only 1–2 days and d 3 days or more. The OLR fields with statistical significance exceeding the 95% level are highlighted by thick lines, and only winds passing the 95% confident level have been plotted

To directly describe the relationship between the convection over the SCS and PHR-LF events, the low-frequency OLR averaged over the area 5–15°N, 97.5–120°E is chosen to represent the intensity of the convection over the SCS. Figure 18 shows the evolution of composites of the convection over the SCS and the low-frequency specific humidity averaged over South China as well as the low-frequency vertical vorticity over South China. It can be seen that convection over the SCS is closely related to the water vapor over South China. Before the beginning of precipitation, the convection over SCS is strong, which first strengthens and then weakens, while the water vapor anomalies over South China are negative. On day − 2, the convection over the SCS is inhibited, while the water vapor anomalies over South China are positive. During the persistent precipitation, both the positive OLR anomalies over the SCS and the water vapor anomalies over South China first increase and then decrease with the variation of the precipitation, but the water vapor decays more rapidly. In summary, the circulation anomaly in the tropics is another signal that affects PHR-LF events during the first rainy season over South China. On the quasi-biweekly timescale, the convection over the SCS is closely related to the water vapor as well as vertical motion over South China; Therefore in the low latitudes, the low-frequency oscillations also contribute greatly to the anomalous circulations associated with PHR.

Fig. 18

Temporal evolution of composite low-frequency variables over key areas, including OLR averaged over the area 5–15°N, 97.5–120°E (purple line; unit: W m^− 2), precipitation anomalies (grey bars; unit: mm), and specific humidity (yellow bars; unit: 10^− 4 kg kg^− 1), as well as vertical velocity (red line; unit: 10^− 2 Pa s^− 1) averaged over South China

6 Conclusions

PHR events mainly occur in South China during the first rainy season, under certain long-lasting large-scale atmospheric circulations in the mid-high latitudes and the tropical zone. This study examines the low-frequency features of the precipitation anomalies during the first rainy season over South China, and investigates the role of the wave trains in the mid-high latitudes in the intensification and persistence of PHR-LF events during the first rainy season over South China on the quasi-biweekly timescale.

The results of power spectrum analysis show that the precipitation anomaly during the first rainy season over South China has obvious quasi-biweekly oscillation characteristics, to which the 8–24 day low-frequency components contribute the most. The statistics of different low-frequency heavy rain events at different durations indicate that when the low-frequency heavy rain events last more than 2 days, the corresponding circulations start to stabilize. Therefore, this study mainly focuses on the low-frequency heavy rain events lasting at least 3 days.

The low-frequency variations of precipitation are mainly affected by the atmospheric low-frequency oscillations. The long-lasting anomalous activities of the large-scale circulations are an important reason for persistent precipitation anomalies. Teleconnection wave trains can provide a stable background, of which the disturbance of the circulation in the upstream can induce anomalous variations of the circulation in the downstream. The results of this study show that during the first rainy season, the anomalous circulations in the mid-high latitudes associated with PHR over South China featured two wave trains: one crossing the Eurasian continent in the mid-high latitudes and one along the East Asian subtropical westerly jet. The low-frequency oscillations contribute greatly to the anomalous circulations related to PHR-LF events. WAF analysis indicates that the origin of Rossby waves is Western Europe and the northern Mediterranean, and the wave energy disperses along the two wave trains to the downstream, sinking in South China, which further causes the enhancement of the disturbance and the anomalous variation of the circulations. The evolution of the two wave trains shows that the wave train in the mid-latitudes before the anomalous variation of the precipitation may trigger the heavy rain, while the wave train in high latitudes varying with the occurrence and development of the heavy rain may maintain the heavy rain. The confluence of the two wave trains enhances the cyclonic anomalies disturbance and is conducive to the development and maintenance of the circulation anomalies.

The diagnostic analysis of the vertical motion using the $\omega$ equation shows that the wave trains can affect the vertical motion in South China through both dynamic and thermal effects. The results reveal that over South China, the low-frequency variation of the vertical motion is mainly caused by the advection of the low-frequency relative vorticity by the meridional wind, and the disturbance of the relative vorticity anomalies over eastern China caused by the wave energy dispersions can induce the variation of the relative vorticity, increasing the meridional gradient of the relative vorticity, thus triggering the ascending motion over South China. With the enhancement of the wave trains, the meridional gradient of the relative vorticity increases again, which in turn enhances and maintains the anomalous ascending motion. In addition, the enhancement of the subtropical wave train intensifies the warm sector near the Tibetan Plateau, and further strengthens the warm advection to South China, also strengthening the ascending motion over there.

The persistent anomaly of atmospheric circulation is the main cause of PHR, and atmospheric teleconnection wave trains can significantly affect the weather and climate in the East Asia. The analyses in this study indicate that the wave trains can provide good indicative significance for the formation and maintenance of the PHR-LF events during the first rainy season over South China on the quasi-biweekly timescale. However, it is necessary to further study the relationship and interaction between the two low-frequency wave trains. Of course, the formation of PHR involves more than one factor, and is related to the superposition of low-frequency signals from the mid-high latitudes and the tropical areas. In addition, because the onset of the South China Sea summer monsoon occurs in the middle of the first rainy season, the contributions to the frontal rainfall and South China Sea summer monsoon rainfall are different during this period. The mid-high latitude wave trains may be found during two stage, especially before the South China Sea summer monsoon onset, where most precipitation is frontal rainfall. After the onset of the South China Sea summer monsoon, the tropical system will become stronger and exert the key effect. Besides, there are other different factors responsible for persistent heavy rain over South China, such as South Asia high which can provide favorable conditions for upper-level divergence (Wang et al. 2011), western Pacific subtropical high and East Asia summer monsoon which can affect water vapor conditions (Bao 2007; Ding 1992), North Atlantic Oscillation (Gu et al. 2009). In additional, the Indian Ocean dipole and the basin-wide Indian Ocean sea surface temperature anomaly can affect the Asian summer monsoon, which will further have an impact on the precipitation pattern over east China (Yuan et al. 2008a, b). The influence of the atmospheric circulation in the tropics on PHR during the first rainy season over South China and the physical mechanism have not been analyzed in detail in this study, but will be discussed in future research.