1 Introduction

An extremely rare heavy rainstorm occurred in Henan Province from 17 to 22 July 2021; it exhibited significant extremes in terms of duration, heavy rainfall coverage, cumulative rainfall amount, and daily and hourly rainfall intensity (Shi et al. 2021; Sun et al. 2021; Zhang et al. 2021; Yin et al. 2021). According to statistics by the National Meteorological Center of China Meteorological Administration, the cumulative rainfall in most north-central regions of Henan Province exceeds 500 mm, and the region with accumulated rainfall greater than 250 mm accounts for 32.5% of the total land area of the province. Particularly, the cumulative rainfall reached 993.1 mm at the meteorological station of Baizhai town in Zhengzhou city and 1122.6 mm at the meteorological station of the Science and Technology Center in Hebi city. The daily rainfall exceeded 100 mm for approximately half of the meteorological stations and exceeded the historical maximum for twenty national meteorological stations in Henan Province. Particularly, the maximum daily rainfall recorded by the national meteorological station of Zhengzhou reached 624.1 mm, which is 3.3 times the historical maximum daily rainfall. Due to heavy rainfall disasters, extraordinary floods occurred in various cities and regions of Henan Province, resulting in critical economic and social losses (Shi et al. 2021; Sun et al. 2021; Zhang et al. 2021; Yin et al. 2021). Particularly, 302 people died, 50 people were reported missing, and the direct economic loss has exceeded one hundred billion yuan (Shi et al. 2021). Therefore, the underlying causes of this extremely rare heavy rainfall in Henan Province should be investigated.

Regarding the critical socio-economic losses of this extreme precipitation, many efforts have been made to investigate the underlying causes. For example, Shi et al. (2021) suggested that the extreme precipitation was caused by the enhanced easterly jet, which was induced by the generation of typhoon “In-fa” and westwards extending of subtropical high. Zhang et al. (2021) claimed that the anomalous precipitation was closely related to topography. Sun et al. (2021) proposed that the rapid enhancement of the anomalous precipitation was caused by transportation of hydrometeors into the uplifting zone of large-scale precipitation system. However, the analysis from the perspective of moisture budget has not been conducted. According to previous studies (Seager et al. 2010; Seager and Vecchi 2010; Han et al. 2019a, b, 2020; Qiao et al. 2021), the moisture budget equation can be written as \(\delta P\approx \delta \mathrm{MCD}+\delta \mathrm{TH}+\delta R\), where \(\delta P\) is the anomalous precipitation, \(\delta \mathrm{MCD}\) is the dynamic term, \(\delta \mathrm{TH}\) is the thermodynamic term, and \(\delta R\) includes the contributions of anomalous evaporation, high-frequency variability of transient eddies, nonlinear effects, and surface boundary terms. The moisture budget equation shows that anomalous precipitation can be decomposed into the contributions of the dynamic term, thermodynamic term, and residual terms. Thus, the quantitative contributions of these terms on the right hand of the moisture budget equation to the extremely rare heavy rainfall in Henan Province in July 2021 can be investigated.

Although the quantitative contributions can be investigated from the perspective of the moisture budget, the underlying physical mechanism cannot be clearly clarified by the moisture budget equation. Fortunately, a novel three-pattern decomposition of the global atmospheric circulation (3P-DGAC) method has been developed to investigate the complex interactions of atmospheric circulations between low latitudes and middle–high latitudes (Liu et al. 2008; Hu et al. 2017, 2018a, b, 2020; Cheng et al. 2018). Hu et al. (2017) proposed that the global atmospheric circulation can be decomposed into the sum of the horizontal, meridional, and zonal circulations, which can be viewed as the global generation of the Rossby wave in the middle–high latitudes and the Hadley and Walker circulations in the low latitudes. Han et al. (2021) combined the 3P-DGAC method and the moisture budget equation to develop a novel moisture budget equation of three-pattern circulations. By using this novel moisture budget equation, the relative contributions of horizontal, meridional, and zonal circulations to extremely rare heavy rainfall and the underlying physical mechanism can be explored.

In this study, we aim to analyze the extreme precipitation in Henan Province from 17 to 22 July 2021 by using the moisture budget equation and the 3P-DGAC method. The remainder of this paper is organized as follows. Section 2 provides the data and methods used in this study. In Sect. 3, we investigate the quantitative contributions of the terms on the right side of the moisture budget equation to extreme rainfall. The relative contributions of the horizontal, meridional, and zonal circulations to extreme rainfall and the underlying physical mechanism are also explored. Finally, a summary is provided in Sect. 4.

2 Data and methods

2.1 Data

In this study, we use hourly rainfall data and associated atmospheric variables, including horizontal wind, vertical velocity, specific humidity, and surface pressure, derived from the European Center for Medium Range Weather Forecasts Reanalysis 5 (ERA5, Hersbach et al. 2020) with a horizontal resolution of \(0.5^\circ \times 0.5^\circ\). The time period analyzed in this study is from 17 to 22 July in 2021, and the climatological mean of variables is defined as the climatological mean from 1981 to 2010.

2.2 Novel moisture budget equation of three-pattern circulations

In this study, the following moisture budget equation is adopted (Seager et al. 2010; Seager and Vecchi 2010):

$$\delta P\approx -\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({\overrightarrow{V}}_{0}\delta q\right)dp-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({q}_{0}\delta \overrightarrow{V}\right)dp+\delta R$$
(1)

where \({\rho }_{w}\) is the density of water, \(g\) is the gravitational acceleration, \(\overrightarrow{V}\) is the horizontal wind, \(q\) is the specific humidity, \(p\) is the pressure, and \({p}_{s}\) is the surface pressure. \(\delta\) is the difference in variables between the period of 17 July to 22 July 2021 and the climatological mean of 1981–2010. Variables with subscript 0 represent the climatological mean of 1981–2010. \(\delta \mathrm{TH}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({\overrightarrow{V}}_{0}\delta q\right)dp\) is the thermodynamic term that is induced by anomalous specific humidity. The thermodynamic term can further be decomposed into two components, i.e., \(\delta \mathrm{TH}=\delta \mathrm{THA}+\delta \mathrm{THD}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\left({\overrightarrow{V}}_{0}\cdot \nabla \delta q\right)dp-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\left(\delta q\nabla \cdot {\overrightarrow{V}}_{0}\right)dp\), where \(\delta \mathrm{THA}\) and \(\delta \mathrm{THD}\) represent the thermodynamic term induced by advection and divergence, respectively. \(\delta \mathrm{MCD}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({q}_{0}\delta \overrightarrow{V}\right)dp\) is the dynamic term that is induced by anomalous circulation. The dynamic term can be decomposed into two components, i.e., \(\delta \mathrm{MCD}=\delta {\text{MCDA}}+\delta {\text{MCDD}}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\left(\delta \overrightarrow{V}\cdot \nabla {q}_{0}\right)dp-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\left({q}_{0}\nabla \cdot \delta \overrightarrow{V}\right)dp\), where \(\delta {\text{MCDA}}\) and \(\delta {\text{MCDD}}\) represent the dynamic term induced by advection and divergence, respectively. The residual term \(\delta R\) includes the contributions of anomalous evaporation, high-frequency variability of transient eddies, nonlinear effects, and surface boundary terms.

The 3P-DGAC method (Liu et al. 2008; Hu et al. 2017, 2018a, b, 2020; Cheng et al. 2018) can be used to decompose the global atmospheric circulation \(\overrightarrow{V}\) into the sum of the horizontal circulation \({\overrightarrow{V}}_{H}={u}_{H}\overrightarrow{i}+{v}_{H}\overrightarrow{j}\), meridional circulation \({\overrightarrow{V}}_{M}={v}_{M}\overrightarrow{j}+{\omega }_{M}\overrightarrow{k}\), and zonal circulation \({\overrightarrow{V}}_{Z}={u}_{Z}\overrightarrow{i}+{\omega }_{Z}\overrightarrow{k}\), i.e.,

$$\overrightarrow{V}={\overrightarrow{V}}_{H}+{\overrightarrow{V}}_{M}+{\overrightarrow{V}}_{Z}$$
(2)

with the following components:

$$\left\{\begin{array}{c}u={u}_{Z}+{u}_{H}=a\left({p}_{s}\frac{\partial Z}{\partial p}-\frac{\partial H}{\partial \theta }\right)\\ v={v}_{H}+{v}_{M}=a\left(\frac{1}{sin\theta }\frac{\partial H}{\partial \lambda }-{p}_{s}\frac{\partial M}{\partial p}\right)\\ \omega ={\omega }_{M}+{\omega }_{Z}=\frac{{p}_{s}}{sin\theta }\frac{\partial \left(sin\theta M\right)}{\partial \theta }-\frac{{p}_{s}}{sin\theta }\frac{\partial Z}{\partial \lambda }\end{array}\right.$$
(3)

where \(a\) is the Earth radius and \(H\), \(M\), and \(Z\) are the stream functions of horizontal, meridional, and zonal circulations, respectively.

Han et al. (2021) combined the 3P-DGAC method and the moisture budget equation to develop a novel moisture budget equation of three-pattern circulations. Specifically, by combining Eqs. (1) and (3), the novel moisture budget equation of three-pattern circulations can be obtained as follows:

$$\begin{gathered} \delta P \approx - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (\vec{V}_{H0} \delta q)} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (q_{0} \delta \vec{V}_{H} )} dp \hfill \\ - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (\vec{V}_{M0} \delta q)} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (q_{0} \delta \vec{V}_{M} )} dp \hfill \\ - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (\vec{V}_{Z0} \delta q)} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {\nabla \cdot (q_{0} \delta \vec{V}_{Z} )} dp + \delta R \hfill \\ \end{gathered}$$
(4)

where \(\delta \mathrm{TH}\_\mathrm{H}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({\overrightarrow{V}}_{H0}\delta q\right)dp\) and \(\delta \mathrm{MCD}\_\mathrm{H}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({q}_{0}\delta {\overrightarrow{V}}_{H}\right)dp\) are the thermodynamic term and dynamic term induced by horizontal circulation, respectively;\(\delta \mathrm{TH}\_\mathrm{M}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({\overrightarrow{V}}_{M0}\delta q\right)dp\) and \(\delta \mathrm{MCD}\_\mathrm{M}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({q}_{0}\delta {\overrightarrow{V}}_{M}\right)dp\) are the thermodynamic term and dynamic term induced by meridional circulation, respectively; and \(\delta \mathrm{TH}\_\mathrm{Z}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({\overrightarrow{V}}_{Z0}\delta q\right)dp\) and \(\delta \mathrm{MCD}\_\mathrm{Z}=-\frac{1}{{\rho }_{w}g}\underset{0}{\overset{{p}_{s}}{\int }}\nabla \cdot \left({q}_{0}\delta {\overrightarrow{V}}_{Z}\right)dp\) are the thermodynamic term and dynamic term induced by zonal circulation, respectively.

Furthermore, the thermodynamic term induced by the three-pattern circulations can be decomposed into the thermodynamic term induced by advection and divergence. Namely,

$$\left\{ \begin{gathered} \delta {\text{TH\_H = }}\delta {\text{THA\_H + }}\delta {\text{THD\_H = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\vec{V}_{H0} \cdot \nabla \delta q)dp} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta q\nabla \cdot \vec{V}_{H0} )dp} \hfill \\ \delta {\text{TH\_M = }}\delta {\text{THA\_M + }}\delta {\text{THD\_M = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\vec{V}_{M0} \cdot \nabla \delta q)dp} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta q\nabla \cdot \vec{V}_{M0} )dp} \hfill \\ \delta {\text{TH\_Z = }}\delta {\text{THA\_Z + }}\delta {\text{THD\_Z = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\vec{V}_{Z0} \cdot \nabla \delta q)dp} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta q\nabla \cdot \vec{V}_{Z0} )dp} \hfill \\ \end{gathered} \right.$$
(5)

where \(\delta \mathrm{THA}\_\mathrm{H}\), \(\delta \mathrm{THA}\_\mathrm{M}\), and \(\delta \mathrm{THA}\_\mathrm{Z}\) (\(\delta \mathrm{THD}\_\mathrm{H}\), \(\delta \mathrm{THD}\_\mathrm{M}\), and \(\delta \mathrm{THD}\_\mathrm{Z}\)) represent contributions of the change in advection (divergence) to the thermodynamic term induced by the horizontal, meridional, and zonal circulations, respectively. Similarly, the dynamic term induced by the three-pattern circulations can be decomposed into the dynamic term induced by advection and divergence. Namely,

$$\left\{ \begin{gathered} \delta {\text{MCD\_H = }}\delta {\text{MCDA\_H + }}\delta {\text{MCDD\_H = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta \vec{V}_{H} \cdot \nabla q_{0} )} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(q_{0} \nabla \cdot \delta \vec{V}_{H} )} dp \hfill \\ \delta {\text{MCD\_M = }}\delta {\text{MCDA\_M + }}\delta {\text{MCDD\_M = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta \vec{V}_{M} \cdot \nabla q_{0} )} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(q_{0} \nabla \cdot \delta \vec{V}_{M} )} dp \hfill \\ \delta {\text{MCD\_Z = }}\delta {\text{MCDA\_Z + }}\delta {\text{MCDD\_Z = }} - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(\delta \vec{V}_{Z} \cdot \nabla q_{0} )} dp - \frac{1}{{\rho_{w} g}}\int\limits_{0}^{{p_{s} }} {(q_{0} \nabla \cdot \delta \vec{V}_{Z} )} dp \hfill \\ \end{gathered} \right.$$
(6)

where \(\delta {\text{MCDA}}\_\mathrm{H}\), \(\delta {\text{MCDA}}\_\mathrm{M}\), and \(\delta {\text{MCDA}}\_\mathrm{Z}\) (\(\delta {\text{MCDD}}\_\mathrm{H}\), \(\delta {\text{MCDD}}\_\mathrm{M}\), and \(\delta {\text{MCDD}}\_\mathrm{Z}\)) represent contributions of the change in advection (divergence) to the dynamic term induced by the horizontal, meridional, and zonal circulations, respectively.

By using the novel moisture budget equation of the three-pattern circulation and Eqs. (5) and (6), the relative contributions of the horizontal, meridional, and zonal circulations to the extremely rare heavy rainfall and the underlying physical mechanism can be investigated.

3 Results

Figure 1a shows that the accumulated precipitation center between 17 and 22 July in 2021 from ERA5 reanalysis datasets is located around southern Shanxi, northern Henan, and western Hebei. The location and intensity of the precipitation anomaly center are similar to those of the precipitation center (comparing Fig. 1a,c), implying that the precipitation in 2021 is clearly stronger than the climatological precipitation from 1981 to 2010 (comparing Fig. 1a,b). Figure 2 displays the changes in the moisture budget components of 2021 minus the climatological mean (1981–2010), which reflect the sum of dynamic and thermodynamic terms (\(\delta \mathrm{MCD}\) and \(\delta \mathrm{TH}\)), the dynamic term (\(\delta \mathrm{MCD}\)), and the thermodynamic term (\(\delta \mathrm{TH}\)), the contributions of changes in divergence (\(\delta {\text{MCDD}}\)) and advection (\(\delta {\text{MCDA}}\)) to the dynamic term, and the contributions of changes in divergence (\(\delta \mathrm{THD}\)) and advection (\(\delta \mathrm{THA}\)) to the thermodynamic term. Figures 1c and 2a show that the spatial distribution of the sum of \(\delta \mathrm{MCD}\) and \(\delta \mathrm{TH}\) is similar to that of the precipitation anomaly in the region where extreme precipitation occurs (purple box in figure; 111°E–115°E, 33°N–39°N). The quantitative results in Fig. 4a show that the sum of dynamic and thermodynamic terms (121.7 mm) can explain 97% of the anomalous precipitation (125.5 mm), while the residual term that includes the anomalous evaporation, high-frequency variability of transient eddies, nonlinear effects, and surface boundary terms (3.8 mm) only explains 3% of the anomalous precipitation, indicating that the dynamic and thermodynamic terms are the main contributors to the anomalous precipitation. Specifically, the regionally averaged \(\delta \mathrm{MCD}\) and \(\delta \mathrm{TH}\) are 94 mm and 27.7 mm, respectively, which can explain 75% and 22% of the anomalous precipitation. Figures 2b, d, and 4a (Figs. 2c, g, and 4a) show that the spatial distribution of \(\delta \mathrm{MCD}\) (\(\delta \mathrm{TH}\)) and that of \(\delta {\text{MCDD}}\) (\(\delta \mathrm{THA}\)) are similar in the region where the extreme precipitation occurs, and the regionally averaged \(\delta {\text{MCDD}}\) (\(\delta \mathrm{THA}\)) is 100.8 mm (27.8 mm), which can explain 80% (22%) of the anomalous precipitation, suggesting that changes in divergence (advection) are the main contributor to the dynamic (thermodynamic) term that further contributes to the anomalous precipitation.

Fig. 1
figure 1

a Spatial distribution of the accumulated precipitation from 17 to 22 July in 2021. (b) and (c) Same as (a) except for the climatological precipitation (1981–2010) and the precipitation anomaly in 2021. The purple box (111°E–115°E, 33°N–39°N) indicates the region where extreme precipitation occurs. The precipitation anomaly is computed based on the climatological mean of 1981–2010. The units for the precipitation and precipitation anomalies are mm

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Fig. 2
figure 2

The mean (from 17 to 22 July) changes in the moisture budget components of 2021 minus the climatological mean (1981–2010), which reflects (a) the dynamic and thermodynamic terms (\(\delta \mathrm{MCD}\) and \(\delta \mathrm{TH}\)), (b) the dynamic term (\(\delta \mathrm{MCD}\)), and (c) the thermodynamic term (\(\delta \mathrm{TH}\)). (d) and (e) Same as (b) except for the contributions of changes in divergence (\(\delta {\text{MCDD}}\)) and advection (\(\delta {\text{MCDA}}\)) to the dynamic term. (f) and (g) Same as (c) except for the contributions of changes in divergence (\(\delta \mathrm{THD}\)) and advection (\(\delta \mathrm{THA}\)) to the thermodynamic term. The units for the changes in the moisture budget components are mm

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To investigate the underlying physical mechanism, the contribution of the horizontal, meridional, and zonal circulations to \(\delta {\text{MCDD}}\) and \(\delta \mathrm{THA}\) was decomposed using the novel moisture budget equation of three-pattern circulations. Figure 3b,c shows that the contributions of meridional circulation and zonal circulation to \(\delta {\text{MCDD}}\) (\(\delta {\text{MCDD}}\_\mathrm{M}\) and \(\delta {\text{MCDD}}\_\mathrm{Z}\)) in the region where extreme precipitation occurs are positive, and \(\delta {\text{MCDD}}\_\mathrm{Z}\) is larger than \(\delta {\text{MCDD}}\_\mathrm{M}\) (Fig. 4). However, the contribution of horizontal circulation to \(\delta {\text{MCDD}}\) (\(\delta {\text{MCDD}}\_\mathrm{H}\)) is negligible (Fig. 3a). Specifically, the regionally averaged \(\delta {\text{MCDD}}\_\mathrm{H}\), \(\delta {\text{MCDD}}\_\mathrm{M}\), and \(\delta {\text{MCDD}}\_\mathrm{Z}\) are − 3.1 mm, 16.3 mm, and 87.6 mm, which can explain − 3%, 13%, and 70% of the anomalous precipitation (Fig. 4b). Figures 2g and 3d–f show that the spatial distribution of \(\delta \mathrm{THA}\) and that of the contribution of horizontal circulation to \(\delta \mathrm{THA}\) (\(\delta \mathrm{THA}\_\mathrm{H}\)) are similar, while the contributions of meridional circulation and zonal circulation to \(\delta \mathrm{THA}\) (\(\delta \mathrm{THA}\_\mathrm{M}\) and \(\delta \mathrm{THA}\_\mathrm{Z}\)) are small. Specifically, the regionally averaged \(\delta \mathrm{THA}\_\mathrm{H}\), \(\delta \mathrm{THA}\_\mathrm{M}\), and \(\delta \mathrm{THA}\_\mathrm{Z}\) are 19.6 mm, 0.7 mm, and 7.5 mm, which can explain 16%, 0%, and 6% of the anomalous precipitation (Fig. 4b). Generally, combining the results of \(\delta {\text{MCDD}}\) and \(\delta \mathrm{THA}\) reveals that the contributions of the horizontal, meridional, and zonal circulations to the anomalous precipitation caused by \(\delta {\text{MCDD}}\) and \(\delta \mathrm{THA}\) are 13%, 13%, and 76%, respectively (Fig. 4b), suggesting the dominant role of zonal circulation in causing this anomalous precipitation.

Fig. 3
figure 3

The contributions of the change in divergence to the dynamic term induced by the (a) horizontal, (b) meridional, and (c) zonal circulations. (d)–(f) Same as (a)–(c) except for the contributions of changes in advection to the thermodynamic term induced by the (d) horizontal, (e) meridional, and (f) zonal circulations. The units for the changes in the moisture budget components are mm

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Fig. 4
figure 4

a The regionally averaged precipitation anomaly over the region of the purple box (111°E–115°E, 33°N–39°N) and the calculated contributions of changes in the moisture budget components to the anomalous precipitation. (b) The calculated contributions of horizontal, meridional, and zonal circulations to the anomalous precipitation. The bars represent the regionally averaged precipitation anomaly derived from reanalysis datasets and the changes in the moisture budget components. The percentages are the ratios between changes in the moisture budget components and the precipitation anomaly derived from reanalysis datasets. The units for the precipitation anomaly and changes in the moisture budget components are mm

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Figure 5a–f displays the spatial distribution of the horizontal circulation and the anomaly of horizontal circulation from 17 to 22 July in 2021. Figure 5a–c shows that typhoon “In-Fa” was formed in the northwest Pacific, which caused the low-pressure center in the northwest Pacific. The anomalous east wind caused by the convergence of typhoon “In-Fa” and the high-pressure center north of typhoon “In-Fa” can bring a large amount of water vapor from the northwest Pacific to the Chinese inland area, which can provide an abundant water vapor source for anomalous precipitation (Fig. 5d–f). The high-pressure center north of typhoon “In-Fa” and continental high caused two high-pressure centers in the middle latitudes at 500 hPa, which favored the formation of the inverted trough in the region where extreme precipitation occurs (Fig. 5b). In the upper troposphere, there also exists a low-pressure trough, and the region where extreme precipitation occurs is in front of the low-pressure trough at 200 hPa (Fig. 5c). Additionally, the anomalous vertical vorticity of the horizontal circulation from the lower troposphere to the upper troposphere is positive, which can provide dynamic lifting conditions for the anomalous precipitation (Fig. 5g–i).

Fig. 5
figure 5

Spatial distribution of the horizontal circulation at (a) 850 hPa, (b) 500 hPa, and (c) 200 hPa represented by the stream function (shading) and velocity fields (vectors) from 17 to 22 July in 2021. (d)–(f) Same as (a)–(c) except for the anomaly of horizontal circulation in 2021. (g)–(i) Spatial distribution of the vertical vorticity anomaly of horizontal circulation at (g) 850 hPa, (h) 500 hPa, and (i) 200 hPa. Anomalies are computed based on the climatological mean of 1981–2010. The units for the stream function, velocity fields, and vertical vorticity are 10−6 s−1, m s−1, and 10−5 s−1

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The vertical motion was further induced under the background of the spatial distribution of the horizontal circulation (Fig. 6d). According to Eq. (3), the contribution of the meridional and zonal circulations to the vertical velocity can be decomposed using the 3P-DGAC method (Fig. 6). Namely, the vertical velocity \(\omega\) can be decomposed into the sum of the vertical velocity of the meridional circulation (\({\omega }_{M}\)) and zonal circulation (\({\omega }_{Z}\)). In terms of climatic mean, the spatial distributions of \(\omega\) and \({\omega }_{M}\) are similar, and the regionally averaged \(\omega\), \({\omega }_{M}\), and \({\omega }_{Z}\) are − 0.015 Pa s−1, − 0.012 Pa s−1, and − 0.003 Pa s−1, implying that meridional circulation plays the dominant role in the climatic mean (Fig. 6a–c). In 2021, the regionally averaged \(\omega\) was − 0.126 Pa s−1 (Fig. 6d), which is an order of magnitude greater than the climatic mean. Additionally, the regionally averaged \({\omega }_{Z}\) (− 0.082 Pa s−1) was about twice as large as the \({\omega }_{M}\) (− 0.044 Pa s−1) in 2021 (Fig. 6e,f), and the anomalies of the regionally averaged \(\omega\), \({\omega }_{M}\), and \({\omega }_{Z}\) are − 0.111 Pa s−1, − 0.032 Pa s−1, and − 0.079 Pa s−1, respectively, suggesting that zonal circulation plays the dominant role, which is different from the climatic mean.

Fig. 6
figure 6

Spatial distribution of the climatological (1981–2010) (a) omega of vertical circulation, (b) omega of meridional circulation, and (c) omega of zonal circulation at 500 hPa. (d)–(f) and (g)–(i) Same as (a)–(c) except for the omega and omega anomalies in 2021. The omega anomaly is computed based on the climatological mean of 1981–2010. The units for the omega and omega anomalies are Pa s−1

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Figure 7 displays the spatial distribution of the vapor flux vector anomaly and vapor flux divergence anomaly induced by the horizontal, meridional, and zonal circulations at 850 hPa in 2021. The spatial distributions of the vapor flux vector anomaly in Fig. 7a,b are almost identical, and the vapor flux vector anomaly induced by the vertical circulation (The vertical circulation includes the meridional and zonal circulations) is tiny (Fig. 7c), implying that horizontal circulation plays a dominant role in bringing water vapor from the northwest Pacific to the Chinese inland area. Although horizontal circulation plays the dominant role in the transport of vapor flux, the contribution of horizontal circulation to the anomalous precipitation is 13% (Fig. 4b). This is because whether precipitation can be formed depends on the vapor flux divergence anomaly rather than the vapor flux vector anomaly. The vapor flux divergence anomaly in the region where extreme precipitation occurs is negative (Fig. 7a), indicating the convergence of water vapor, which is beneficial to anomalous precipitation. The spatial distributions of the vapor flux divergence anomaly in Fig. 7a,c are almost identical, and the vapor flux divergence anomaly induced by the horizontal circulation is tiny (Fig. 7b). Additionally, the vapor flux divergence anomaly in the region where extreme precipitation occurs induced by zonal circulation (− 2.6 × 10–9 m−1 s−1) is approximately two and a half times that induced by meridional circulation (− 1.0 × 10−9 m−1 s−1), which is consistent with the results that zonal circulation plays the dominant role (Fig. 4b).

Fig. 7
figure 7

a Spatial distribution of the vapor flux vector anomaly (vectors) and vapor flux divergence anomaly (shading) induced by the horizontal, meridional, and zonal circulations at 850 hPa in 2021. (b) and (c) Same as (a) except for the vapor flux vector anomaly and vapor flux divergence anomaly induced by the horizontal circulation and the vertical circulation. (d) and (e) Same as (a) except for the vapor flux divergence anomaly induced by zonal circulation and meridional circulation. Anomalies are computed based on the climatological mean of 1981–2010. The units for the vapor flux vector anomaly and vapor flux divergence anomaly are 10−2 s−1 and 10−9 m−1 s−1, respectively

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4 Summary

In this study, we analyzed the extreme precipitation in Henan Province from 17 to 22 July 2021 by using the moisture budget equation and the 3P-DGAC method. The following conclusions are obtained.

  1. (1)

    \(\delta \mathrm{MCD}\) and \(\delta \mathrm{TH}\) can explain 75% and 22% of the anomalous precipitation, respectively, while the residual term only explains 3%. Changes in divergence (advection) are the main contributor to the \(\delta \mathrm{MCD}\) (\(\delta \mathrm{TH}\)), which can explain 80% (22%) of the anomalous precipitation.

  2. (2)

    The typhoon “In-Fa,” anomalous high-pressure center north of typhoon “In-Fa,” continental high, and low-pressure trough provide abundant water vapor sources and dynamic lifting conditions for anomalous precipitation.

  3. (3)

    Since precipitation formation relates to the vapor flux divergence anomaly rather than the vapor flux vector anomaly, although horizontal circulation can bring water vapor from the northwest Pacific to the Chinese inland area, the contribution of horizontal circulation to anomalous precipitation is only 13%. In contrast, zonal circulation, which is the main contributor to the vapor flux divergence anomaly, can explain 76% of the anomalous precipitation, while meridional circulation can explain 13%.