Xiuping YAO ,Ruoying LI ,Xiaohong BAO3, ,and Qiaohua LIU3,

1China Meteorological Administration Training Centre, Beijing 100081, China

2State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China

3College of Atmospheric Sciences, Nanjing University of Information Science and Technology, Nanjing 210044, China

ABSTRACT An extreme torrential rain (ETR) event occurred in Henan Province,China,during 18-21 July 2021.Based on hourly rain-gauge observations and ERA5 reanalysis data,the ETR was studied from the perspective of kinetic energy (K),which can be divided into rotational wind (VR) kinetic energy (KR),divergent wind kinetic energy (KD),and the kinetic energy of the interaction between the divergent and rotational winds (KRD).According to the hourly precipitation intensity variability,the ETR process was divided into an initial stage,a rapid increase stage,and maintenance stage.Results showed that the intensification and maintenance of ETR were closely related to the upper-level K,and most closely related to the upperlevel KR,with a correlation coefficient of up to 0.9.In particular,the peak value of hourly rainfall intensity lagged behind the KR by 8 h.Furthermore,diagnosis showed that K transformation from unresolvable to resolvable scales made the ETR increase slowly.The meridional rotational wind (uR) and meridional gradient of the geopotential (φ) jointly determined the conversion of available potential energy (APE) to KR through the barotropic process,which dominated the rapid enhancement of KR and then caused the rapid increase in ETR.The transportation of K by rotational wind consumed KR,and basically offset the KR produced by the barotropic process,which basically kept KR stable at a high value,thus maintaining the ETR.

Key words: extreme torrential rain,rotational kinetic energy,kinetic energy generation and transport,barotropic process

1.Introduction

During 18–22 July 2021,a rarely seen extreme torrential rainfall (ETR) event occurred in Henan Province,China,where the human population density and levels of agricultural activity are high (hereafter referred to as the “21·7”ETR),the maximum hourly precipitation of which set a new record for the Chinese mainland,reaching 201.9 mm (Su et al.,2021;Zhang et al.,2021;Chyi et al.,2022).The 21·7 ETR caused heavy human casualties and losses of property,resulting in 398 deaths or missing persons and a direct economic loss in Henan Province exceeding RMB 120 billion(Disaster Investigation Group of the State Council,2022).Therefore,the causes of the 21·7 ETR have become a key focus of meteorologists.

Previous studies have indicated that the 21·7 ETR was generated by the combined action of multiscale weather systems at different heights of the troposphere superimposed with topographic effects (Ran et al.,2021;Zhang et al.,2021;Cai et al.,2022;Chyi et al.,2022;Deng et al.,2022;Duan et al.,2022;Fu et al.,2022;Liang et al.,2022;Xu et al.,2022a;Zhu et al.,2022) in the context of extreme daily circulation (Xu et al.,2022b;Zhang et al.,2022),while the south–north shifting of the rainfall was related to the varying direction of low-level jets in the boundary layer(Luo and Du,2023).Warming may lead to significant intensification of both regional-scale and station-scale precipitation extremes,and squall-line convection results in much higher precipitation extremes at both regional and station scales than unorganized convection (Qin et al.,2022).The abnormal northward shift of the western Pacific subtropical high along with the binary typhoon system of Typhoon In-Fa(2021) and Typhoon Cempaka (2021) jointly drove water vapor transportation and convergence in Henan (Sun et al.,2021;Bueh et al.,2022;Wang et al.,2022),where water vapor mainly originates from southern China and the western North Pacific (Nie and Sun,2022),with the contribution of the former (52.51%) double that of the latter (25.51%) (Cui and Yang,2022).At the same time,there were also large quantities of hydrometeors transported to the updraft area,which may have greatly accelerated the microphysical process of water vapor transformation into cloud water droplets and,ultimately,precipitation (Chen et al.,2022).Furthermore,the updrafts of the arc-shaped convergence zone may also have attracted all associated precipitation that was overlaid and concentrated into the same trailing region of the convective system to generate the extreme hourly rainfall over Zhengzhou (Yin et al.,2022).The abnormally strong vertical motion (Zhang et al.,2021,2023) was caused by dynamic forcing,diabatic forcing,and topographic forcing,of which diabatic forcing contributed the most,while topographic forcing contributed to the vertical motion of the lower layer(Zhao et al.,2022).

The 21·7 ETR event was accompanied by drastic variations in the horizontal wind field (Zhang et al.,2022;Luo and Du,2023),which can be decomposed into the divergent and rotational winds,thereby providing more information on the relationship between the structural changes in the horizontal wind fields and strengthening of the rainstorm (Deng et al.,2012).Diagnosing the kinetic energy budget is an effective way to analyze the sources and sinks of horizontal wind variations.Correspondingly,the kinetic energy can be decomposed into divergent and rotational kinetic energies and the kinetic energy of interaction between the divergent wind and the rotational wind (Fuelberg and Browning,1983;Buechler and Fuelberg,1986).Through this decomposition method,we can understand the role of each component of kinetic energy in the kinetic energy balance and the specific process of their respective generation,transport,and interconversion.In this way,we can obtain a deeper understanding of the kinetic energy of rainstorms.

Diagnosis of the divergent and rotational kinetic energies has been applied in the study of many systems related to heavy rainfall,such as tropical cyclones (Ding and Liu,1985;Wang et al.,2016),extratropical cyclones (Pearce,1974),the mei-yu system (Xie et al.,1980;Wang and Liu,1994;Fu et al.,2016),upper-and lower-level jets (Zhong et al.,2021),and low vortexes (Sun et al.,1993;Fu et al.,2011),with many meaningful results having been obtained.Rotational kinetic energy is generally dominant in the formation and development stages of weather systems associated with precipitation (Duan et al.,1997;Zhong et al.,2021).There is a certain relationship between the horizontal flux divergence of rotational kinetic energy in the area of a typhoon and the enhancement of its heavy rainfall (Li and Shou,1995).In addition,divergent kinetic energy acts as a catalyst in the conversion between available potential energy (APE) and rotational kinetic energy (Wang,1993,Wang and Liu,1994;Yu and Yao,1999;Cheng et al.,2014).The conversion of divergent kinetic energy to rotational kinetic energy can contribute to an increase in rotational kinetic energy (Sha et al.,2018),which is conducive to the development of torrential rain (Jin et al.,2020;Bao and Yao,2022).

However,the relationship between the divergent and rotational kinetic energies and the development and maintenance mechanism in the 21·7 ETR has not yet been studied.Accordingly,we studied the 21·7 ETR from the perspective of kinetic energy to explore the sources and sinks of the wind variations surrounding it,the aim being to improve our understanding of the development and maintenance mechanism of the 21·7 ETR.

The remainder of the paper is organized as follows.The data and methods employed in our study are introduced in section 2.Section 3 presents the relationship of the ETR event with the kinetic energies and the development and maintenance mechanism for this particular case.Finally,conclusions and some further discussion are provided in section 4.

2.Data and methods

2.1.Data

This study used hourly precipitation data from 0800 LST [Local Standard Time,which is eight hours ahead of coordinated universal time (UTC)] on 18 July to 0800 LST on 21 July 2021 provided by the National Meteorological Information Center of the China Meteorological Administration.In addition,the hourly horizontal wind,vertical velocity,and geopotential height during the same period were also used,which were derived from fifth major global reanalysis produced by ECMWF (ERA5).The hourly reanalysis dataset had a horizontal resolution of 0.25° × 0.25° and 27 layers from 1000 hPa to 100 hPa in the vertical direction (Hersbach et al.,2020).

2.2.Methods

The horizontal wind (V) was decomposed into the rotational wind (VR) and divergent wind(VD) by adopting the Endlich method,and the derivatives were calculated using centered finite differences (Endlich,1967).The method works easily and has high precision,as it does not use the relationships between the stream function and vorticity function to solve Poisson equations and is independent of boundary conditions.The main idea of this method is to iteratively adjust the rotational wind to make its horizontal divergence gradually tend towards zero,and its vertical component of the relative vorticity (ζ) should meet the requirements of theζof the original wind field,to successfully obtain the rotational wind,and then the rotational wind is subtracted from the original wind field to obtain the divergent wind.In this iterative process,the divergence error at each grid point is less than or equal to 1 × 10-8s-1;that is,less than or equal to 0.001% of the maximum divergence of the original horizontal wind field.

Kinetic energy per unit mass can be decomposed into

where

The kinetic energy of an atmospheric volume in isobaric coordinates (Abeing the horizontal limited computational area) is calculated as follows:

where

Here,Kis the kinetic energy of a limited region (hereinafter simply referred to as “kinetic energy”),KDis the divergent kinetic energy,KRis the rotational kinetic energy,andKRDis the kinetic energy of the interaction between the divergent and rotational winds.KD,KR,andKRDare all defined in a limited region.

The budget equation forKR(Buechler and Fuelberg,1986) is expressed as follows:

Here,uRandvRare the zonal and meridional components of the rotational wind,respectively.Similarly,uDandvDare the zonal and meridional components of the divergent wind,respectively.Further,ζis the vertical component of the relative vorticity,ωis the vertical velocity (units: Pa s–1),fis the Coriolis parameter,φis the geopotential,andFis the frictional force.

The sum of terms Af,Az,B,and C is denoted as C(KD,KR).Therefore,Eq.(5) can be simplified as DKR=IR+C(KD,KR)+GR+HFR+FR.The term on the left-hand side denotes the local change inKR.The term IRrepresents the change inKRcaused by the nonlinear interaction between the rotational and divergent winds.Term Af is to satisfy the conservation of angular momentum on the tangential motion and is called the geostrophic effect term.Term Az is to satisfy the conservation of angular momentum on the rotational motion.Both Af and Az are affected by the relative magnitudes and orientations ofVRandVD.Term B describes the vertical exchange ofKR,while term C is related to the configuration ofVDwithVRand the vertical distribution ofVD.The term C(KD,KR) denotes the conversion betweenKDandKR,where a positive value indicates a conversion fromKDtoKRand a negative value indicates the opposite.Term GRis the generation term forKR,indicating the conversion between the APE andKRdue to the cross-contour flow ofVR.Term HFRis the transportation term forKR,indicating the horizontal transportation ofKbyVR.Term FRis the friction term related toVR,including frictional processes and the energy transfer between the sub-grid scale and the grid scale of motion.As FRis calculated as the residual,it includes possible errors from other terms in Eq.(5).For analyzing the balance of the budget equation forKR,we calculated the average ratio of the right-hand side terms (except term FR) to the left-hand side term DKRas 3.46,which is reasonable compared with an earlier study using the same equation (Buechler and Fuelberg,1986).

3.Results

3.1.Stages of the ETR event

During 18–21 July 2021,torrential rainfall occurred in most areas of Henan Province,with the accumulated precipitation exceeding 800 mm (3 d)-1(Fig.1a).From the day-byday distribution of precipitation in Fig.1,the rainstorm range was concentrated and relatively static,but the intensity was increasing.On 18 July,the heavy rainfall was mainly distributed in the northern part of Henan Province (Fig.1b).On 19 July,the range of heavy rainfall expanded,with the maximum daily precipitation exceeding 400 mm d-1(Fig.1c);and on 20 July,the range of heavy rainfall continued to expand,with the range of daily precipitation exceeding 250 mm d-1,reaching a maximum,and the maximum daily precipitation exceeding 600 mm d-1(Fig.1d).From Fig.1,the precipitation during the ETR event was concentrated in the region of (32.5°–37°N,111.5°–115.5°E).Therefore,this region was identified as the key region in this research,which we refer to as the torrential-rain area (red dashed frame in Fig.1).

Fig.1.Distribution of accumulated precipitation during (a) 0800 LST 18 July to 0800 LST 21 July 2021[shaded;units: mm (3 d)-1],(b) 0800 LST 18 July to 0800 LST 19 July 2021 (shaded;units: mm d-1),(c)0800 LST 19 July to 0800 LST 20 July 2021 (shaded;units: mm d-1),and (d) 0800 LST 20 July to 0800 LST 21 July 2021 (shaded;units: mm d-1).The red dashed frame represents the torrential-rain area (32.5°–37°N,111.5°–115.5°E;the same in subsequent figures).

Fig.2.Temporal evolution of the regional-mean hourly precipitation within the torrentialrain area from 0800 LST 18 July to 0800 LST 21 July 2021 (units: mm h-1).

Figure 2 shows the temporal evolution of hourly precipitation (also called the hourly rain intensity) within the torrential-rain area during 18–21 July,from which we can see that the hourly precipitation in that period enhanced with time.On 18 July,the hourly average precipitation increased by 0.05 mm h-1and the average hourly precipitation was 0.93 mm h-1.On 19 July,the hourly average precipitation increased by 0.09 mm h-1and the average hourly precipitation was 2.02 mm h-1.On 20 July,the hourly average precipitation increased by 0.01 mm h-1and the average hourly precipitation was 3.43 mm h-1.The average hourly precipitation was largest on 20 July,second largest on 19 July,and smallest on 18 July.Also,the hourly average precipitation increased the most on 19 July,second most on 18 July,and least on 20 July.

To comprehensively assess the average hourly precipitation and hourly rainfall intensity variability,the ETR event was divided into three stages: the initial stage,from 0800 LST 18 to 0800 LST 19 July 2021;the rapid increase stage,from 0800 LST 19 to 0800 LST 20 July 2021;and the maintenance stage,from 0800 LST 20 to 0800 LST 21 July 2021.

3.2.Spatial and temporal distribution of kinetic energy

Figure 3 shows the vertical profile of the regional averageK,KR,KD,andKRDin the torrential-rain area.In the torrential-rain area,the vertical totals ofKwere closely related to the hourly precipitation,with their correlation coefficients being greater than 0.6 (figure omitted).The vertical distributions ofK,KR,KD,andKRDshow a bimodal pattern in the initial,rapid increase,and maintenance stages of the ETR event,with the main peak and secondary peak at around 200 hPa and 800 hPa,respectively (Fig.3).Also,as the main and secondary peak values increase,the ETR strengthens accordingly (Fig.3).In Fig.3,KandKRare basically the same,and the values ofKDandKRDare small and mostly around zero.The peaks ofKandKRat 200 hPa are 43 J m-2and 36 J m-2,respectively,during the initial stage (Fig.3a),increasing to 96 J m-2and 88 J m-2during the rapid increase stage (Fig.3b),and both exceeding 100 J m-2during the maintenance stage (Fig.3c),withKRincreasing to 120 J m-2.The reason whyKRis greater thanKduring the maintenance stage is thatKRDdecreases to -30 J m-2and thusKis weakened.During the ETR process,although theKandKRvalues at 800 hPa also increase,their changes are less than half of those at 200 hPa.

Fig.3.Vertical profiles of area-averaged K,KR,KD,and KRD (units: J m-2) in the (a) initial stage,(b) rapid increase stage,and (c) maintenance stage.

It can be seen thatKis enhanced during the ETR,and the ETR thus subsequently enhanced,withKplaying a major role at 200 hPa butKRalways determining the vertical distribution ofK.The contribution ofKDandKRDtoKis small,andKRDmakes a negative contribution toK.The distribution and variation ofKandKRare greatest at 200 hPa;therefore,we focused onKandKRat 200 hPa.

Figure 4 shows the horizontal distribution ofKandKRat 200 hPa.As can be seen from Figs.4a–c,the horizontal distribution ofKis very similar to that ofKR.Both are continuously enhanced,indicating the upper-level shortwave trough in the west of the torrential-rain area and the upper-level jet in the east were strengthened (figure omitted).This promoted the vertical motion of the torrential-rain area (figure omitted),and the ETR was subsequently enhanced.During the initial stage (Fig.4a),bothKandKRare 40 J m-2,which both then increase to 120 J m-2during the rapid increase stage (Fig.4b),and to 200 J m-2and 240 J m-2,respectively,during the maintenance stage (Fig.4c).

Fig.4.Horizontal distributions of K (shaded) and KR (contours) at 200 hPa (units: J m-2) in the (a) initial stage,(b)rapid increase stage,and (c) maintenance stage.

In summary,during this ETR process,Kwas enhanced,and subsequently so too was the ETR.Also,the distribution and variation ofKwere greatest at 200 hPa,whereKwas mainly derived fromKR.The enhancement ofKandKRat 200 hPa in the torrential-rain area enhanced the vertical motion and then favored the enhancement of ETR.

3.3.Relationship between K and hourly precipitation intensity

As can be seen in Fig.5,the magnitude and evolutionary trend ofKat 200 hPa are close to those ofKR,and the enhancement of both promotes the enhancement of ETR.Using Pearson correlation coefficients,we calculated the simultaneous temporal correlation coefficients ofKandKRfor the regional average of the torrential-rain area at 200 hPa with the hourly precipitation intensity,respectively,which revealed the correlation coefficient ofKRwith the hourly precipitation intensity to be much larger,at up to 0.9,which was statistically significant at the 0.01 confidence level,based on a Student’st-test.This shows thatKRhas a stronger correlation with the hourly precipitation intensity.During the initial stage,bothKRand the hourly precipitation intensity slowly;during the rapid increase stage,they both intensify sharply;and during the maintenance stage,they both show fluctuating changes and maintain high values.The evolution ofKRis ahead of the hourly precipitation intensity in the ETR process,and this advance is largest (～8 h) in the maintenance stage,which indicates thatKRcould perhaps serve as a predictor of ETR development.

Fig.5.Temporal evolution of the area-averaged hourly precipitation (black line;units:mm h-1),K (blue line,units: J m-2),and KR (red line,units: J m-2) at 200 hPa in the torrentialrain area.

In conclusion,KRwas most closely related to the hourly precipitation intensity at 200 hPa,with a correlation coefficient as high as 0.9.TheKRat 200 hPa could perhaps be used to predict the hourly precipitation intensity 8 h in advance at the earliest,and thus could serve as a predictor of the ETR.Therefore,the rotational kinetic energy equation was used to diagnose the ETR process.

3.4.Rotational kinetic energy budget

According to Fig.6a and Table 1,during the initial stage,DKRpresents a distribution of “positive in the west and negative in the east”,with a regional average value of 0.4 × 10-4W m-2Pa-1.During the rapid increase stage,the positive-value area of DKRexpands and covers the whole torrential-rain area,with a regional average value of 11.46 ×10-4W m-2Pa-1(Fig.6b and Table 1).During the maintenance stage,the absolute values of DKRare relatively small over the whole torrential-rain area,with a regional average value of -1.9 × 10-4W m-2Pa-1(Fig.6c and Table 1).This shows thatKRincreased slowly in the initial stage,rapidly increased in the rapid increase stage,and maintained a high value in the maintenance stage.

Fig.6.Horizontal distribution of the local variation in KR (DKR) (contours;units: 10-4 W m-2 Pa-1): (a–c) the conversion between APE and KR (GR);(d–f) the horizontal flux divergence of K by VR (HFR);(g–i) the conversion between KR and KD [C(KD,KR)];and (j–l) the friction term related to VR (FR) (shaded;units: 10-4 W m-2 Pa-1) at 200 hPa.Panels (a,d,g,j) present the initial stage,(b,e,h,k) present the rapid increase stage,and (c,f,i,l) present the maintenance stage.

Diagnosis shows that the regional average values of the conversion term GRbetween available energy andKR,FR,and IRare always positive,which is beneficial to the enhancement ofKR,and the regional average values of HFR,C (KD,KR),are always negative,which is beneficial to the reduction ofKR(Table 1).During the initial stage,the positive-value area of FRcovers the torrential-rain area (Fig.6j),and its regional average value of 3.47 × 10-4W m-2is conducive to the enhancement ofKR.Meanwhile,GR(Fig.6a) and IR(figure omitted) have less beneficial effects,indicating kinetic energy transfer from the sub-grid to grid scale,leading to the slow increase inKRin the initial stage.During the rapid increase stage,the positive-value range of GRexpands and concentrates in the north-central part of the torrentialrain area (Fig.6b),and its regional average value increases to 26.5 × 10-4W m-2Pa-1,which is conducive to the enhancement ofKR.Meanwhile,the beneficial effects of FR(Fig.6k)and IR(figure omitted) are far less than those of GR,indicating that the pressure gradient force does positive work,so that the APE can be converted intoKRthrough the barotropic process,leading to the enhancement ofKR.During the maintenance stage,the absolute values of GR,HFR,and C(KD,KR)increase significantly (Figs.6f,i and l).As can be seen from Table 1,the regional average value of GRincreases to 44.6 ×10-4W m-2Pa-1,which can generateKR,indicating that the conversion of the APE toKRthrough the barotropic process is enhanced.However,the regional average value of HFRdecreases to -40.44 × 10-4W m-2Pa-1,indicating that the horizontal transportation ofKbyVRis a net output,which will consumeKR(Table 1).Therefore,under the joint action of GRand HFR,the high value ofKRis maintained.

In conclusion,during the initial stage,FRdominated the slow enhancement ofKR.During the rapid increase stage,GRdominated the rapid enhancement ofKR,which was conducive to the rapid enhancement of ETR.During the maintenance stage,GRand HFRjointly maintained a high value ofKR,which was conducive to the maintenance of ETR.

It can be seen from the above that ETR was mainly concentrated in the rapid increase stage and the maintenance stage.Therefore,we further discuss the physical meaning of the main contributing terms during the rapid increase stage and the maintenance stage.In the rapid increase stage,the geopotential height is distributed with a “lower in the north and higher in the south”pattern,and the rotational wind is an anticyclonic southwesterly wind.When the meridional rotational wind (uR) crosses the isobar from south to north,the pressure gradient force does positive work.Therefore,the joint action ofuRand the meridional geopotential gradient controls the conversion of APE toKRthrough the barotropic process,leading to the rapid enhancement ofKRduring this stage (Fig.7a).During the maintenance stage,the meridional potential gradient and the rotational wind continue to increase,which leads to the enhancement of the conversion of APE toKRthrough the barotropic process (Fig.7b).At the same time,the value ofKin the southwest of the torrential-rain area is relatively small,while the value in the northeast of the torrential-rain area is relatively large (Fig.7c).Therefore,the anticyclonic rotational wind transportsKfrom the southwest to the northeast of the torrential-rain area,consumingKR.

Fig.7.Horizontal distribution of the (a–c) rotational wind (vector arrows;units: m s-1),(a,b) geopotential height(shaded;units: gpm),and (c) K (shaded;units: J m-2) at 200 hPa in the (a) rapid increase stage and (b,c)maintenance stage.

In conclusion,during the ETR process,the conversion of kinetic energy from the sub-grid to grid scale made the ETR develop slowly.The APE was converted intoKRthrough a barotropic process,leading to a sharp enhancement ofKR,which was conducive to the sharp enhancement of ETR.KRwas consumed owing to the outward transportation ofKin the rotational wind direction,which basically offset theKRproduced by the barotropic process.Therefore,the high value ofKRwas basically maintained,which was conducive to the maintenance of ETR.Furthermore,the conversion of APE toKRthrough the barotropic process depended on the joint action of the meridional rotational wind and the meridional potential gradient.

4.Discussion and conclusions

Based on ERA5 reanalysis data and hourly precipitation data from meteorological stations in China,the ETR event that occurred in Henan Province,China,18–21 July 2021,was investigated from the perspective of kinetic energy.According to the regional hourly precipitation intensity variability in the torrential-rain area,the ETR process was divided into an initial stage,a rapid increase stage,and a maintenance stage.The spatial and temporal distribution ofKand its relationship with hourly rainfall intensity was discussed,and the mechanism of ETR enhancement and maintenance was diagnosed using the rotational wind kinetic energy equation.The study addresses the lack of mechanistic research on the 21·7 ETR from a kinetic energetic viewpoint,and provides a reference for the forecasting and early warning of torrential rainstorms.The main conclusions are:

(1) During the 21·7 ETR process,Kand its variation were largest at 200 hPa,andKmainly derived fromKR.

(2) The evolution ofKRat 200 hPa was most closely related to the hourly rainfall intensity during the ETR process,and the correlation coefficient was as high as 0.9.In particular,the peak value ofKRwas 8 h ahead of the hourly rainfall intensity,which has a certain significance for indicating the development and maintenance of ETR.

(3) The conversion of kinetic energy from the sub-grid to grid scale made the ETR develop slowly during the initial stage.The APE was converted intoKRthrough the barotropic process,leading to a sharp enhancement ofKR,which was conducive to the sharp enhancement of ETR during the rapid increase stage.Furthermore,the conversion of APE toKRdepended on the joint action of the meridional rotational wind and the meridional potential gradient.During the maintenance stage,KRwas consumed owing to the outward transportation ofKin the rotational wind direction,which basically offset theKRproduced by the barotropic process.Therefore,the high value ofKRwas basically maintained,which was conducive to the maintenance of ETR.

From the perspective ofK,our research diagnoses the 21·7 ETR process and further improves the level of understanding regarding the enhancement and maintenance mechanism of this event.The intensification of the horizontal gradient of geopotential height caused by a cold low may result in the increase ofKRin the upper troposphere,which indicates enhanced upper-level anticyclonic circulation in the torrential-rain area (Cai et al.,2022).

However,many scientific issues worth exploring remain.For example,the divergent flow was formed to the right of the jet entrance,which strengthened the local ascending motion and induced a lower-level vortex,causing the rain peak during the ETR (Fu et al.,2022).The interaction of the lower-level,middle-level,and upper-levelKRandKD-may be the reason why the peak ofKRwas found to be 8 h ahead of the rainfall intensity during the ETR.Therefore,the balance ofKand the conversion betweenKDandKRat different levels during the ETR process are deserving of further study.

Acknowledgements.This study was jointly supported by the National Natural Science Foundation of China (Grant Nos.42275013,42030611 and 42175008) and the Open Grants of the State Key Laboratory of Severe Weather (Grant No.2021LASWB17).