Performance of a Global Spectral Model with Dry Air-Mass and Total Air-Mass Conserving Dynamical Cores: A Case Study of the July 2021 Henan Extreme Rainfall Event

Mass conservation is one of the fundamental laws that should be ensured in the design of the dynamical core for a numerical model. The mass conservation in atmospheric models is generally represented in two aspects. One is related to the equation form describing the air mass, such as the mass continuity equation and its derivative formulas (Peng et al., 2020; Zhang et al., 2020). The other is related to the numerical discretization formulation for solving the governing equations, such as a semi-Lagrangian advection scheme (Su et al., 2013; Wong et al., 2013). The present paper is an extension of the work of Peng et al. (2020) and provides real case studies with the full physical process parameterizations to examine the mass conservation problem at the level of the continuous set of equations. This mass conservation problem, which refers to the nonconservation of dry air mass at meteorological scales, is found in some global and climate models, such as the Integrated Forecast System (IFS) at ECMWF (Wedi et al., 2015) and the Community Atmosphere Model (CAM) at NCAR (Neale et al., 2012). That is, the assumed total air-mass conservation (TMC) in the models prescribes that there exists no obvious source or sink of total air mass in an air parcel, although many physical processes in the real atmosphere act as sources or sinks of total water. If there is a source or sink of total water, the total air mass is compensated by a false increase or decrease in dry air mass. Under the circumstances of the nonconservation of dry air mass, on the one hand, the equation set of the mass continuity would not include the water source or sink term, resulting in an incorrect description of the change in moist air in the model; on the other hand, the falsely compensated source of dry air mass would excite local divergence/convergence forcing, resulting in false gravity wave perturbation. As a consequence, the numerical model would further diverge from the real case.

The effects of mass variations in the physical processes acting in local mass continuity laws of moist air have long been recognized (Gu and Qian, 1990; Wacker and Herbert, 2003; Lackmann and Yablonsky, 2004; Wacker et al., 2006). These effects, called the mass source or sink effects, are generally attributed to the mass transport of condensates by evaporation and precipitation at the earth’s surface. Earlier studies revealed that for heavily precipitating systems or heavy rainfall events [> 25 mm day⁻¹ by Gu and Qian (1990) and > 250 mm day⁻¹ by Lackmann and Yablonsky (2004)], the precipitation mass sink effect should be considered in the moist air density equation. However, since the moist pressure is not only a direct measurement of total air mass, but also one of the components conventionally observed by surface stations, the TMC is favored to support the moist pressure operating as one of the prognostic variables in the numerical model. Meanwhile, the mass source or sink effects are relatively less influential at low resolutions, as the mass changes due to evaporation and precipitation are more likely to cancel each other out at the coarse grid spacing. Both convenient implementation and the insignificance of the negative impacts with TMC at low resolutions usually support the use of TMC in previously developed numerical models. Moreover, mass conservation is represented at the level of the continuous sets of equations. Modifying the continuity equation of the numerical model with TMC for the dry air-mass to be conserved has an impact on all the other equations. Therefore, it is not a simple process to update the existing model with TMC to a model with dry air-mass conservation (DMC). To avoid these intricacies, even in state-of-the-art numerical weather prediction (NWP) models, either the TMC is adopted or the total air mass is falsely balanced by changing the dry air mass. For example, in ECMWF IFS before cycle 46R1, a spectral mass fixer was adopted for the total air mass every 24 hours to correct numerical error (Malardel et al., 2019). From the meteorological point of view, however, this is obviously inconsistent with the fact that there exists no evident source (sink) of dry air, leading to the so-called nonconservation problem of dry air mass. On the other hand, with the finer resolution of numerical models (Shen et al., 2020; Wedi et al., 2020), the nonconservation problem of local mass becomes more obvious (Malardel and Bechtold, 2019), and the mass effect of condensates on the dynamics becomes more significant (Bacmeister et al., 2012). Thus, the challenges faced by the traditional TMC assumption would then be more serious.

Several researchers and weather research centers have attempted in recent years to solve the nonconservation problem of dry air mass in the dynamical core of the global NWP model. Lauritzen et al. (2018) developed the new NCAR version of the spectral element dynamical core as part of the Community Earth System Model (CESM 2.0). In this version, both the dry air-mass vertical coordinate and the DMC-deduced continuity equation are introduced. Almost at the same time, Peng et al. (2019) designed a modified nonhydrostatic moist global spectral dynamical core using a dry air-mass vertical coordinate, expressed the mass continuity equation in terms of the dry air density, and derived the governing equations with a new temperature variable. In addition, Malardel et al. (2019) focused on the nonconservation of dry air mass in the operational ECMWF IFS equations and proposed a new coupling between the continuity equation and physics via the total water tendencies from physics parameterizations to formally conserve dry air mass instead of total air mass. That is, the continuity equation remains an equation for total air mass, but the additional mass source or sink term is added to the continuity equation and the diagnostic forms of both surface pressure and pressure vertical velocity. Among the aforementioned three methods, Lauritzen et al. (2018) and Peng et al. (2019) adopted a similar scheme, but Lauritzen et al. (2018) ignored the source or sink terms in deriving the formula of moist pressure vertical velocity, which is a diagnostic variable in hydrostatic dynamical cores. The method proposed by Malardel et al. (2019) is restricted to the total air-mass continuity equation and ignores corresponding modifications of other prognostic variables to minimize the changes needed for the current IFS. Moreover, the model state related to the physics mass tendencies cannot be updated at the present time step, and compensating dry air fluxes still occur within each time step. To develop a dry air-mass conserving hydrostatic dynamical core for a global NWP spectral model, Peng et al. (2020) adopted the same method for the nonhydrostatic to the hydrostatic version, which not only obeys the inherent mass conservation of dry air but also obtains a rigorous derivation of the formula used to compute the full hydrostatic pressure vertical velocity. Simulations of an idealized tropical cyclone in the new hydrostatic version illustrate the obvious improvement of the new dynamical core in extreme rainfall events. Zhang et al. (2020) developed a multiscale dynamical model with moist dynamics and parameterized physics in dry air-mass vertical coordinates and established a general physics–dynamic coupling workflow.

Research on observational knowledge, mechanical analysis and forecasting methods of extreme rainfall events have always been key points in the field of atmospheric science (Luo et al., 2020). These points are also carefully taken into account by forecasters in operational workflows, such as the Beijing extreme rainfall of 21 July 2012 (Chen et al., 2013; Zhang et al., 2013), the extreme rainfall event of 7 May 2017 over the coastal city of Guangzhou (Yin et al., 2020), the warm-sector torrential rainfall event in Beijing area during 15–16 July 2018 (Lei et al., 2020), and the record-breaking heavy rainfall event over southern China in August 2018 (Zeng et al., 2020). Beginning from 17 July 2021, Henan Province encountered heavy precipitating processes, and the maximum 24-h rainfall amount from 0800 BT (Beijing Time) 20 to 0800 BT 21 July reached 624.1 mm, which was much greater than the entire accumulated rainfall amount of 509.5 mm in 2019. Moreover, the maximum hourly rainfall amount of 201.9 mm was greater than that in the “75.8” Henan extreme rainfall event (198.5 mm h⁻¹; Li et al., 2015). As a consequence, direct and indirect disasters induced by extreme rainfall events have attracted worldwide attention. From the perspective of quantitative forecasting, the current operational numerical prediction and the traditional experiences with heavy rainfall events have difficulty explaining this extreme rainfall event. Therefore, predicting such extreme rainfall events requires further understanding of their initiation mechanisms, ongoing development, and structural characteristics. This extreme rainfall amount and the long-lasting concentrated areas correspond to a high local precipitation mass sink from the global perspective. It is worthwhile to study the effect of the assumption of mass conservation on the predictions of this extreme rainfall event.

The main goal of this study is to compare the results from global numerical models based on DMC and TMC, and then attempt to reveal the potential advantage of the dynamical core with DMC in predicting the July 2021 (“21.7”) Henan extreme rainfall event. This manuscript is organized as follows. The numerical model and datasets are described in Section 2. A comparison of features of dynamical cores with DMC and TMC is given in Section 3. The analyses of observed precipitation and convective clouds on 20 and 21 July are presented in Section 4. Then, a comparison of the predictions of the extreme rainfall by the ECMWF IFS and the Yin-He Glo-bal Spectral Model (YHGSM) is performed, followed by a comparison of the predictions from the YHGSM with DMC and TMC dynamical cores in Section 5. Section 6 provides an analysis of the possible feedback mechanism for the positive effects of precipitation forecast in the model with a DMC dynamical core. Finally, Section 7 gives the conclusions and discussion.

The data used in this study include real-time observations and predictions from global NWP models. For the former, hourly surface observations, which include rainfall, 2-m temperature, and 10-m horizontal wind, and the digital infrared radiation (IR) images of Fengyun-2 series Geostationary Meteorological Satellite G (FY-2G) are provided by the National Meteorological Information Centre (NMIC, https://data.cma.cn/data/cdcdetail/dataCode/A.0012.0001.html) and the National Satellite Meteorological Centre (NSMC, http://www.nsmc.org.cn/nsmc/cn/home/index.html) of the China Meteorological Administration (CMA), respectively. For the latter, two kinds of predictions are forecasted by global NWP models of 1) the ECMWF IFS, the operational data of which were received from NMIC; and 2) the YHGSM, which was developed by the National University of Defense Technology.

The ECMWF IFS outputs in this manuscript are the products of Cy47r2. This version adopts the hydrostatic moist global dynamical core using a total air-mass vertical coordinate and the TMC (Malardel et al., 2019). The physical processes therein include radiative transfer, turbulent mixing, convection, cloud microphysics, surface exchange, subgrid-scale orographic drag and nonorographic gravity wave drag, with at least one physical scheme representing each of the processes. For details, refer to the IFS documentation (ECMWF Cy47r1; ECMWF, 2020). Meanwhile, the IFS Cy47r2 adopts a horizontal grid spacing of approximately 9 km with a cubic octahedral reduced Gaussian grid truncated at the spectral truncation N = 1279 and sets the model top to 0.01 hPa with 137 vertical levels. It runs 10-day predictions twice per day at 0800 and 2000 BT. YHGSM adopts a hydrostatic moist global spectral dynamical core with a dry air-mass vertical coordinate (Wu et al., 2011; Peng et al., 2020). The mass continuity equation is expressed in terms of dry air density under the DMC law, and the governing equations are reformulated with a new temperature variable. Meanwhile, a two-time-level semi-implicit semi-Lagrange scheme for time integration, a spherical harmonic spectrum expansion based on a trigonometric truncation for the horizontal direction, and a finite element discretization for the vertical direction are used (Yang J. H. et al., 2015, 2017; Yin et al., 2018, 2021). The resolution of YHGSM in this study is a linear reduced Gaussian grid truncated at N = 1279, corresponding to a horizontal grid spacing of approximately 15.67 km. Similarly, YHGSM runs the 10-day predictions twice at 0800 and 2000 BT per day, with model sets of 137 vertical levels and a time step of 600 s. The model physics schemes include the new Tiedtke cumulus parameterization scheme (Zhang and Wang, 2017), the WRF Single-Moment 5-class (WSM5) scheme (Hong et al., 2004) modified with an additional prognostic equation of cloud fraction (Forbes and Tompkins, 2011; Forbes et al., 2011), the rapid radiative transfer model for the general circulation model (Mlawer et al., 1997; Iacono et al., 2008) on both shortwave and longwave radiative flux calculations, and the Yin-He land surface scheme.

The initial conditions of YHGSM and IFS are produced by their own data assimilation systems. The initial condition used in this study is produced by the ensemble-variational four-dimensional data assimilation system (Zhang et al., 2012, 2022; Liu et al., 2016), which includes an ensemble data assimilation cycle and a deterministic four-dimensional variational data assimilation cycle. The observations, including the conventional data on the Global Telecommunication System, Global Positioning System Radio Occultation, radiance data from the infrared and microwave sounder instruments (e.g., the Advanced Microwave Sounding Unit-A temperature-sounding channels and Microwave Humidity Sensors humidity-sounding channels mounted on MetOp and NOAA series satellites and the second generation of the MicroWave Humidity Sounder onboard the FY-3C and FY-3D satellites), etc., are assimilated in the data assimilation system (Zhu et al., 2014; Ma et al., 2022). Detailed information is provided in Zhang et al. (2022).

To quantitatively evaluate the precipitation forecast, a feature-based quality measure called SAL (Wernli et al., 2008) is used in this study. SAL considers aspects of the structure (S), amplitude (A), and location (L) of the precipitation field in a prespecified domain. The amplitude component A shows the normalized difference of the domain-averaged precipitation values of the model predictions and the observations. The location component L represents a combination of the normalized distance and the average distance; the normalized distance is calculated between the gravity centers of the predicted and observed precipitation fields, while the average distance is calculated between the gravity centers of the total precipitation fields and the individual precipitation objects. The structure component S hints at the normalized difference in the scaled volume of the precipitation objects. The closer the value of any component is to zero, the higher the possibility of a perfect forecast is. The SAL is applied to the rainfall prediction effects on both 20 and 21 July with a prespecified domain of 30°–40°N, 105°–125°E. The 95th default percentile of all grid point values larger than 0.1 mm is used as the threshold to define precipitation objects in this study.

For NWP models, the formulations of dynamical cores depend on the choice of DMC or TMC law. The derivation of the continuity equation and its related flux equations or diagnostic formulas differ in the representations of terms on the right-hand side (RHS) of the equations. The physical considerations implied are different treatments on the mass changes of the moist part of the atmosphere. Both models used in this study are globally hydrostatic spectral models with mass-based vertical coordinates. Figure 1 shows a comparison of part of the control equations and diagnostic formulas between dynamical cores with DMC and TMC.

Figure 1.  Comparison of some of the equations and diagnostic formulas in different dynamical cores with DMC and TMC.

From the physical perspective, an air parcel is presumed to be a composite of dry air, water species (such as water vapor, cloud water, cloud ice, rainwater, and precipitating snow/graupel/hail), and an ensemble of other gases of negligible concentration. The dry air mass inherently obeys no sources or sinks in air parcels at meteorological scales, and any change in the water species mass should correspond to the same variation in total air mass (Fig. 1a). However, the numerical models are generally implemented under the initial assumption of TMC inside an air parcel. When the water species exchanges between the atmosphere and the land or sea occur in the physical package, the sources or sinks of total water in the physical step have to be artificially compensated by dry air mass (Fig. 1b), leading to the so-called nonconservation of dry air mass. Here, the total water is the sum of all the prognostic water species, and its sources or sinks are generally related to the water species exchanges through evaporation and precipitation and the corresponding subgrid transport in the physical package.

With regard to the formulations, although the expressions of the continuity equations with TMC and DMC are similar other than their use of moist air or dry air density, respectively, the control equations of the components of the atmosphere differ in dynamical cores. At the starting point of TMC, the continuity equation of total air mass is a control equation in the dynamical core, while there is no equation for dry air. Thus, the variation in the dry air mass is deduced from the difference between the total air mass and total water mass (the box marked with red dots in Fig. 1b). As the sources or sinks of total water are not considered in the continuity equation of total mass, the representing term ${S_w} = 1/{\rho _t}( - \partial {J_w}/ $$ \partial z + \partial {F_p}/\partial z) $—where $ {F_p} $ is the precipitation flux with a positive downward trend and $ {J_w} = \overline {{{\rho '}_w}w'} $ is the subgrid vertical flux of total water computed by the parameterizations—falsely exists on the RHS of the dry air-mass continuity equation. The mass changes of dry air and total water show an anti-correlation. However, from the starting point of DMC, the continuity equation of dry air is an inherent control equation in the dynamical core, and the continuity equation of total air mass is induced by a combination of those of dry air and total water (the box marked with blue dots in Fig. 1a). Then, the term representing the sources or sinks of total water exists in the RHS of the total air-mass continuity equation; the mass variations of total air and total water show a naturally positive correlation.

Due to the direct correlations to the continuity equation in the hydrostatic NWP model, the expressions of $ \partial {\pi _{\rm s}/\partial t}$, the pressure vertical velocity $ \omega $, and the $ \eta $-coordinate vertical velocity $ \dot{\eta} $ in the dynamical core with DMC differ from those of TMC in a similar way. The mass effect of total water $ {S_w} $ should add to the RHS of the three expressions in the dynamical core with DMC (Fig. 1a). Therefore, when there is a sink of total water throughout the vertical column ($\displaystyle\int_{{\it{\pi}} = 0}^{\pi = {\pi _{\rm s}}} {{S_w}} {\rm d} \pi < 0$), the tendency equation of hydrostatic pressure at surface $ \partial {\pi _{\rm s}/\partial t}$ in the dynamical core with DMC would experience this further decrease. This will lead to a lower surface pressure in the models with DMC than that with TMC at the next time step. The pressure vertical velocity $ \omega $, which is an essential variable in the NWP model but needs to be diagnosed in the hydrostatic dynamical core, expresses a similar correlation. The source or sink of total water due to evaporation or precipitation adds a nonvanishing vertical velocity component into the pressure vertical velocity. When the water sink occurs in the vertical column ($\displaystyle\int_{{\it{\pi}} = 0}^{\it{\pi}} {{S_w}} {\rm d}{\it{\pi}} < 0$), the induced decrease in pressure vertical velocity implies ($\displaystyle\int_{\pi = 0}^{\pi = {\pi _{\rm s}}} {{S_w}} {\rm d} \pi < 0$) a stronger vertical motion in the dynamical core with DMC than that with TMC. It is worth noting that, since the mass transfer of total water across the earth’s surface represents the net outcome of precipitation and evaporation effects, these effects of total water in $ \partial {\pi _{\rm s}}/\partial t $ and $ \omega $ would vanish in the special situation that precipitation and evaporation balance exactly.

Regarding the nature of DMC and TMC, another choice faced in NWP models with mass-based vertical coordinates is the dry air mass or total air mass the model is based on. To strictly conserve the dry air mass, a dry air-mass vertical coordinate is recommended, which means the dry hydrostatic pressure is introduced as one of the prognostic variables of the NWP model (Fig. 1c). Since in models such as both the IFS and the YHGSM, the coupling between the continuity equation and the physical parameterizations is performed in grid-point space after the computation of the explicit dynamics, the dry air-mass vertical coordinate favors the consistent physics–dynamics coupling (Lauritzen et al., 2018; Peng et al., 2020). As in the dry air-mass vertical coordinate, the coordinate surfaces remain constant throughout the physics updates. While in the total air-mass vertical coordinate, the effect of water vapor is implicitly included in the definition of the (moist) pressure level. If there is a loss of water vapor mass during the physics updates, the moist pressure levels cannot stay constant and thus the coordinate surfaces where the returned values from the physics actually differ from those where the explicit dynamics was computed. This inconsistency will lead to some inconvenience for the physics–dynamics coupling (Zhang et al., 2020). In addition, as mentioned in Lauritzen et al. (2017, 2018), the dry air-mass vertical coordinate is a convenient convention for the total energy conservation and the consistent coupling between the spectral element and the conservative semi-Lagrangian scheme. More importantly, the expressed terms in the dry air-mass vertical coordinate may provide more meaningful and clearer information, such as the diagnostic moist pressure vertical velocity $ \omega $ (Fig. 1c; Peng et al., 2020). However, the dynamical core with DMC and total air-mass vertical coordinate is also adopted in some of the NWP models to minimize the changes to the current model with TMC (Malardel et al., 2019).

An extreme rainfall event hit Henan Province from 0800 BT 17 to 0800 BT 22 July 2021. The persistent rainfall amount exceeded 50 or 100 mm day⁻¹ and spread from the Taihang Mountains to the Nanyang Basin, covering most of the province. In particular, some areas encountered 24-h rainfall amounts over 250 mm, and 19 national meteorological stations logged record maximum daily rainfall amounts (figure omitted).

Throughout the evolution of the extreme event, persistent episodes of 24-h rainfall amounts over 100 mm and even over 250 mm occurred in the middle-northern part of Henan Province on both 20 and 21 July 2021 (see Fig. 2); the rainfall amount of this extreme event represented a record high. Specifically, from 0800 BT 20 to 0800 BT 21 July 2021, heavy precipitation with a 24-h rainfall amount larger than 100 mm hit the middle-west of Henan, and the representative station for Zhengzhou recorded a maximum 24-h rainfall amount of 624.1 mm (Fig. 2a). In particular, the hourly rainfall rate of 201.9 mm h⁻¹ at Zhengzhou during the period of 1600–1700 BT (Fig. 2c), which was even greater than the highest rate of 198.5 mm h⁻¹ recorded during the “75.8” extreme rainfall event of Henan in 1975, broke the record of the hourly rainfall rate for national inland meteorological stations. On 21 July 2021, the coverage of the rainfall amount exceeding 50 mm showed a long south–north band throughout nearly all of central Henan and western Shandong Provinces. The heavy precipitation over 100 mm moved northward, with a dominant distribution from the north of Henan Province to the west of Shandong Province (Fig. 2b). Furthermore, there were also areas receiving 24-h rainfall amounts over 250 mm in northern Henan Province. The representative station of Huixian recorded a maximum 24-h rainfall amount of 447.1 mm. Meanwhile, the maximum hourly rainfall amount was 74.5 mm, appearing in the same period as that from 1600 to 1700 BT 20 July 2021 (Fig. 2d).

Figure 2.  Distributions of 24-h accumulated rainfall (mm) from national meteorological stations: (a) 0800 BT 20 to 0800 BT 21 July, (b) 0800 BT 21 to 0800 BT 22 July; and observed hourly rainfall of (c) Zhengzhou (ZZ) from 0800 BT 20 to 0800 BT 21 July, (d) Huixian (HX) at 0800 BT 21–22 July 2021.

Black body temperature (TBB) from satellite observations usually acts as an indicator for quantitatively representing the location, area, and evolution of mesoscale convective systems (MCSs) (Maddox, 1980; Zeng et al., 2020). An examination of satellite images can indicate the development of MCSs causing extreme rainfall. According to the definition of MCSs by TBB features, a meso-$ \beta $ or larger-scale MCS is generally defined if three conditions are satisfied. First, a contiguous cold cloud region with an infrared brightness temperature $ \leqslant - 52^\circ{\rm C} $ is identified; second, the area of the identified cloud region should be larger than 30,000 km²; and third, the duration time of this cloud region should exceed 3 h (Jirak et al., 2003; Yang X. R. et al., 2015).

The satellite images from FY-2G show that the convective clouds located at high levels in Henan and its vicinities were very active during 20–21 July 2021, showing a total of 4 meso-$ \beta $ or larger-scale MCSs with the persistent back-building of a new convection from the prior MCS (Table 1; Fig. 3). From the cloud top height indicated by the minimum infrared brightness temperature, the second (from 1800 BT 20 to 0900 BT 21 July) and the third (from 1400 BT 21 to 0100 BT 22 July) MCSs illustrated stronger convective processes (Table 1), with both of their minimum infrared temperatures at initiation and maturation stages lower than −70°C (Figs. 3g, i, j). Meanwhile, both the second and the third MCSs exhibited larger cloud shields at their maturation stages than the other two, as the maximum areas of the four MCSs were 6.36 × 10⁴, 9.21 × 10⁴, 12.13 × 10⁴, and 4.80 × 10⁴ km², respectively (Table 1). Thus, both the cloud top height and area of the cloud shield showed stronger updrafts for the second and third MCSs, which showed a higher uprushing cloud top. When focused on the shapes of convective clouds during the evolution of MCSs, the eccentricities of the cold cloud area (the short axis compared to the long axis) at the maturation stages of the four MCSs were 0.49, 0.66, 0.70, and 0.50 (Table 1), respectively. According to the definition of MCS shape by infrared brightness temperature (Maddox, 1980; Jirak et al., 2003; Yang X. R. et al., 2015), the third MCS satisfied the criterion of the mesoscale convective complex (MCC), while the other three belonged to elongated MCSs. The four MCSs showed relatively stationary movements throughout the evolution of their life cycles. The first two MCSs experienced their whole life cycles from the initiation stage to the decay stage in the center-north and center-east of Henan Province (Figs. 3a–c, 3f–h), and the third initiated in the center of Henan Province with slightly eastward movement in the developing stage and a subsequently northward move in the dissipating stage (Figs. 3i–k). The fourth initiated at the northern border of Henan Province and showed a slightly southward movement (Figs. 3l–n). As a consequence, with the coeffects of favorable synoptic situations and topography (Yin et al., 2022), these four relatively stationary MCSs exhibited a persistent back-building of the new MCS from the end of the prior MCS (Fig. 3), which led to the new extreme rainfall events in Henan during 20–21 July 2021.

Figure 3.  Infrared brightness temperature (°C) at the initiation, maturation, and decay stages of the 4 MCSs from 0400 BT 20 to 0800 BT 22 July 2021 and satellite images from FY-2G at 1600 and 1700 BT 20 July 2021.

Of particular note is the extreme rainfall episode during 1600–1700 BT 20 July. Zhengzhou station recorded a new record of hourly rainfall amount of 201.9 mm. As seen from the infrared satellite images (Figs. 3d, e), the new convective cloud became more concentrated and rapidly penetrated higher from 1500 to 1700 BT. The minimum infrared brightness temperatures of −64.15, −67.15, and −68.15°C hint at a persistent strengthening (Table 1). With the merging of the cold cloud shield from the south, the convective cloud formed the second MCS at 1800 BT 20 July. Although the areas of convective clouds at 1600 and 1700 BT could not satisfy the currently existing criterion of meso-$ \beta $ MCS from the geostationary satellite perspective, the uprushing cloud tower indicated by infrared brightness temperatures and the large areas of cloud shields support the classification of the convective clouds as a developing meso-$ \gamma $ MCS. This meso-$ \gamma $ MCS has been confirmed by detailed surveys on composite radar reflectivity (Yin et al., 2022).

To check the performances of NWP models in predicting the extreme rainfall event, the predictions of 24-h precipitation on 20 and 21 July from the IFS and the YHGSM are compared. Furthermore, predictions at lead times of 84, 60, and 36 h, i.e., predictions for 20 July starting at 2000 BT 17, 18, and 19 July and predictions for 21 July starting on 18, 19, and 20 July, respectively, are chosen to analyze the accuracy and consistency of predictions related to the extreme rainfall event.

Figure 4 presents the results of 24-h accumulated precipitation on 20 July 2021 from the IFS and the YHGSM. Compared to the observation in Fig. 2a, both models basically predicted the occurrences of 24-h rainfall amounts less than 100 mm, but predictions from both models showed northwestern shifts or southwestern shifts of rainfall centers at lead times of 84, 60, and 36 h. Moreover, the maximum amounts of rainfall centers predicted by both models were much smaller than the observations for all three lead times (Fig. 4). To some extent, precipitation forecasts for 20 July from both models failed and were not expected. Nevertheless, a further comparison of predictions from the YHGSM (Figs. 4a, c, e) and the IFS (Figs. 4b, d, f) at the same lead time showed obvious differences in the location, shape of rainfall distribution and the maximum 24-h rainfall amount, especially at longer lead times (e.g., 60 and 84 h) and for heavier precipitation (e.g., exceeding 100 or 250 mm day⁻¹). As far as the location and shape were concerned, the south–north orientation bands with rainfall amounts over 50 mm predicted by the YHGSM remained relatively stable in the center-west of Henan, with the rainfall centers occurring on the western border for all three lead times (Figs. 4a, c, e). However, the spatial distributions of heavy rainfall predicted by the IFS showed a change from a southwest–northeast to a south–north orientation, and the rainfall centers changed from the southwest to the western border of Henan as the lead time decreases (Figs. 4b, d, f). With regard to the maximum rainfall amounts, both models exhibited a smaller value change from 84- to 60-h lead times but a larger value variation from those of 60–36 h. As a comparison, the YHGSM predicted a much larger 24-h rainfall amount than the IFS at each lead time, showing a better consistency with the largest predicted value of 491.6 mm starting at 2000 BT 17 July 2021 (Fig. 4e).

Figure 4.  Accumulated precipitation (mm) from 0800 BT 20 to 0800 BT 21 July 2021 predicted by (a, c, e) the YHGSM and (b, d, f) the IFS starting at 2000 BT 19, 18, and 17 July 2021. The pink line is the border of Henan Province.

Regarding the rainfall episode from 0800 BT 21 to 0800 BT 22 July 2021 (Fig. 5), the spatial distributions of 24-h accumulated precipitation predicted by the IFS and YHGSM were much closer to the observations (Fig. 2b) than the comparisons on 20 July 2021. In particular, the predictions started at 2000 BT 20 July 2021 (Figs. 5a, b). Although there were also northwestward migrations of spatial distributions for rainfall amounts over 250 mm, the south–north heavy rainfall bands from the northwest of Henan Province to the north in Hebei and Shanxi Provinces predicted by both models were very close to the observations (Fig. 2b), with the maximum 24-h rainfall amounts of 471.6 mm (YHGSM) and 534.7 mm (IFS) compared to the observed 447.1 mm. However, when comparing predictions between the YHGSM and the IFS at the three lead times, the performances of the YHGSM (Figs. 5a, c, e) were also better than those of the ECMWF IFS (Figs. 5b, d, f) on both spatial distributions and rainfall amounts. The YHGSM exhibited stable south–north bands of heavy rainfall in northwestern and northern Henan at all three lead times, while the heavy rainfall distributions predicted by the IFS showed northward adjustment as the lead time decreases. Moreover, changes in daily rainfall amount predicted by the YHGSM and the IFS differed largely. The heavy rainfall areas with 24-h accumulated values exceeding 100 and 250 mm were stably predicted by the YHGSM at lead times of 84, 60, and 36 h (Figs. 5a, c, e). While the daily rainfall amount predicted by the IFS becomes increasingly heavier as the lead time decreases, with a predicted amount of no more than 100 mm at the 84-h lead time (Fig. 5f) but exceeding 250 mm at the 36-h lead time (Fig. 5b). In addition, the maximum 24-h rainfall amounts predicted by the YHGSM (393.0, 512.6, and 471.6 mm at lead times of 84, 60, and 36 h) were closer to the observations (447.1 mm) than those predicted by the IFS (93.3, 174.5, and 534.7 mm), indicating better accuracy and consistency.

Figure 5.  Accumulated precipitation from 0800 BT 21 to 0800 BT 22 July 2021 predicted by (a, c, e) the YHGSM and (b, d, f) the IFS starting at 2000 BT 20, 19, and 18 July 2021, respectively.

To quantitatively evaluate the performances of both models, the SAL technique is applied to the predictions for 20 and 21 July 2021. The SAL values are presented in Table 2. Among the six predictions for these two days, almost the whole S and A values of the YHGSM and the IFS are negative other than those of predictions on 21 July at the 36-h lead time. This result hints at the overall underestimation of precipitation in both models, which is also shown in the rainfall distributions of Fig. 4 and Fig. 5. However, the relatively small L values indicate that the predicted rainfall locations are relatively reliable. As stated by Gong (2010), the A and L values are more indicative functions than the S value. A further analysis of the A and L values from both models is conducted to compare the performances of both models on this extreme rainfall event. Corresponding to the differences between those shown in Fig. 4 and Fig. 5, both the A and L values of the predictions on 20 and 21 July from the YHGSM are closer to zero than those from the IFS, other than the predictions on 21 July at the 36-h lead time. The five negative A values of both models show the underestimations of precipitation, but the larger A values from the YHGSM hint at the stronger precipitation forecasts than those from the IFS, which means that the former are closer to the observed values. For the predictions on 21 July, the A values from both the YHGSM and the IFS represent the overestimations of precipitation at the 36-h lead time, and the larger A value from the YHGSM shows a stronger overestimation. This may be attributed to the larger area of rainfall amount over 100 mm predicted by the YHGSM, as the maximum 24-h rainfall amount predicted by the YHGSM is smaller than that of the IFS (Figs. 5a, b). On the other hand, as far as the L values are concerned (Table 2), the L values from the YHGSM are smaller than those from the IFS, again indicating the consistently better predictions from the YHGSM.

The above analyses reveal that the IFS and YHGSM could basically predict the occurrences of rainfall amount smaller than 100 mm on both 20 and 21 July 2021 from 84- to 36-h lead times. Especially for the predictions on 21 July 2021 at the 36-h lead time, the IFS and YHGSM predict the maximum 24-h rainfall amounts of 534.7 and 471.7 mm, respectively, at nearly the northwestern locations compared to the corresponding observed amount of 447.1 mm. However, other than the prediction from the YHGSM at the 84-h lead time, both models show northwestern offsets of rainfall centers and obviously weak 24-h rainfall amounts on 20 July 2021. Interestingly, when comparing performances between the IFS and YHGSM on these two days, the YHGSM predicts more similar distributions of heavy rainfall and closer rainfall amounts to the observations than the IFS, providing particularly better stability and consistency predictions. Such performance is consistent with the result of Peng et al. (2020), who found that the idealized tropical cyclone (TC) test with the DMC dynamical core showed more intense TC-like storms and more compact rainbands than the original IFS-like dynamical core at longer lead times. Moreover, earlier studies (Lackmann and Yablonsky, 2004; Wacker et al., 2006) also illustrated that, considering the precipitation mass sink in the local mass continuity law of moist air is important for model simulations of extreme weather situations, such as polar cold air outbreaks, tropical cyclones, and other heavily precipitating systems. It is natural to ask to what extent the differences between dynamical cores with DMC and TMC account for the differences between the YHGSM and the IFS.

As mentioned in Section 5.1, the YHGSM shows better performances of precipitation forecasts in terms of both consistency and daily rainfall amount than the IFS on 20 and 21 July 2021. To validate the possible effects of dynamical cores with DMC and TMC on predicting the extreme rainfall event, this section compares the predictions from the YHGSM with DMC and TMC.

Figures 6 and 7 present the predicted 24-h accumulated precipitation on 20 and 21 July 2021 by the YHGSM with DMC and TMC at lead times of 84, 60, and 36 h. The differences between the rainfall distributions in the DMC and TMC frameworks were not very obvious for the weak precipitation. Most of the spatial locations and shapes of the outlines with rainfall amounts of 1, 10, 25, and even 50 mm were basically similar in both frameworks. They showed northwestward migrations compared to the observation on 20 July 2021 (Fig. 6) and a slightly northward migration to the observation on 21 July 2021 (Fig. 7), as analyzed in Section 5.1.

Figure 6.  As in Fig. 4, but for accumulated precipitation predicted by the YHGSM with DMC and TMC.

Figure 7.  As in Fig. 6, but for predictions on 21 July 2021.

However, there were obvious differences in heavier precipitation between the predictions from the YHGSM with DMC and TMC, especially on the daily rainfall amount and the stability (Figs. 6, 7). For example, the rainfall areas over 100 mm predicted by the YHGSM with DMC were larger than those with TMC at all three lead times, and the distributions of these heavy rainfall areas were more stable in the former than in the latter. Furthermore, rainfall areas exceeding 250 mm could almost be predicted by the YHGSM with DMC at all lead times, while there were only two lead times with 24-h rainfall amounts over 250 mm predicted by the model with TMC (Figs. 6f, 7b). In addition, the maximum 24-h rainfall amounts predicted on 20 July 2021 were 260.4, 191.3, and 491.6 mm by the YHGSM with DMC (Figs. 6a, c, e) and 220.2, 210.2, and 280.5 mm by the YHGSM with TMC (Figs. 6b, d, f) at lead times of 84, 60, and 36 h, respectively. Those on 21 July 2021 were 471.6, 512.6, and 393.0 mm in the former (Figs. 7a, c, e) and 441.8, 209.4, and 143.6 mm in the latter (Figs. 7b, d, f). Compared to the observed 624.1 and 447.1 mm on 20 and 21 July 2021, respectively, the maximum 24-h rainfall amounts predicted by the YHGSM with DMC were more precise and more stable than those with TMC.

From the quantitative perspective, the predictions from the YHGSM with DMC are closer to the observations, since the majority of A and L values of applying SAL to the YHGSM with DMC are closer to zero than those with TMC (Table 2). That is, although both the DMC and TMC dynamical cores show underestimations and offsets of precipitation forecasts, the YHGSM with DMC predicts relatively larger rainfall distributions that are closer to the observations. In addition, the relatively smaller variations in the A and L values hint at more stable precipitation forecasts.

Consequently, the overall performance on two days supports the larger daily rainfall amounts and relatively more accurate predictions by the YHGSM with DMC.

The results in Section 5 show that predictions are relatively closer to the observations in terms of both stability and accuracy with the DMC dynamical core. To understand the reason for this improved performance, this section analyzes the possible mechanism for the positive effect of the DMC dynamical core. To utilize the maximum difference in predictions between the two dynamical cores but with a prediction that is closer to the observation, predictions from the YHGSM with DMC and TMC on 21 July 2021 starting at 2000 BT 18 July 2021 are chosen.

Figure 8 presents the observed and predicted 6-h rainfall distributions from 0200 BT 21 to 0800 BT 22 July by the YHGSM with DMC and TMC. The observations (Figs. 8a, d, g, j, m) showed that persistent heavy rainfall in the five episodes were concentrated mainly in central to northwestern Henan Province and northern Shanxi and Shandong Provinces. Other than the period between 0200 and 0800 BT 21 July 2021, extreme rainfall areas exceeding 100 mm (6 h)⁻¹ occurred in northwestern Henan. The predicted 6-h rainfall from the YHGSM with DMC and TMC showed similar distributions, with the slightly northwestward migrations relative to the observations. However, the predicted areas with rainfall amounts exceeding 50 mm from the YHGSM with DMC (Figs. 8b, e, h, k, n) were larger than those with TMC (Figs. 8c, f, i, l, o). Moreover, the maximum 6-h rainfall amounts were larger and closer to the observations in the former. That is, there was an area with a rainfall amount of over 100 mm in each 6-h period in the former, while the latter showed the maximum 6-h rainfall amount only exceeding 50 mm (Fig. 8c), especially the last two episodes, in which the largest values were even less than 50 mm (Figs. 8j, l).

Figure 8.  Predicted precipitation (mm) every 6 hours from 0200 BT 21 to 0800 BT 22 July 2021 by the YHGSM with DMC and TMC, and the corresponding observations.

Theoretical analyses in Section 3 reveal that the term representing sources or sinks of total water should be added to the tendency equation of moist hydrostatic pressure $ \partial {\pi _s}/\partial t $ and the diagnostic moist pressure vertical velocity $ \omega $ in the DMC dynamical core. This means that if there is a high local loss of total water, the last terms of the RHS of both expressions are less than 0, illustrating the further decrease in both $ \partial {\pi _s}/\partial t $ and $ \omega $. These findings suggest that the mass effect of total water loss would decrease the surface pressure and strengthen the vertical motion. Figure 9 displays the mean sea level pressure at 0800, 1400, 2000 BT 21, and 0200 BT 22 July predicted by the YHGSM with DMC, TMC, and the differences between them. The surface pressure predicted by the YHGSM with DMC was basically smaller than that with TMC in Henan Province and its vicinity. The maximum difference reached a large value of 3 hPa. In particular, northwestern Henan, where the predicted heavy rainfall correspondingly occurred, showed a large local difference in surface pressure. Especially for the difference in surface pressure at 2000 BT 21 July and 0200 BT 22 July, there were areas of −1.5 to −2 hPa for the differences in surface pressure between the two dynamical cores over northwestern Henan Province.

Figure 9.  Predicted mean sea level pressure (SLP; hPa) at 0800, 1400, 2000 BT 21, and 0200 BT 22 July by the YHGSM with (a, d, g, j) DMC and (b, e, h, k) TMC, and (c, f, i, l) corresponding differences between the two.

Figure 10 shows the similar distributions of moist pressure vertical velocity at 850 hPa at the same time as in Fig. 9. The figure shows that the pressure vertical velocities from both dynamical cores showed similar areas of values less than 0 over northwestern Henan Province, where the upward vertical motions correspond well with the rainfall areas. However, the values of pressure vertical velocity in the dynamical core with DMC were smaller than those with TMC, which indicates that the upward vertical motions were stronger in the former, corresponding well with heavier rainfall centers. For example, the minimum vertical velocities (locations labeled with the symbol ×) in the former were consistently smaller than those in the latter. The corresponding values of the minimum vertical velocities were −6.81, −5.19, −6.14, and −7.16 Pa s⁻¹ in the DMC dynamical core but −5.53, −5.0, −3.6, and −2.41 Pa s⁻¹ in the TMC at 0800, 1400, 2000 BT 21, and 0200 BT 22 July, respectively. The differences between them were in the range of −0.17 and −4.75 Pa s⁻¹, with a maximum of −4.75 Pa s⁻¹ occurred at 0200 BT 22 July. Especially for the vertical velocities at 2000 BT 21 and 0200 BT 22 July, the areas of minimum vertical velocities are found at the nearby locations of heavy rainfall centers.

Figure 10.  As in Fig. 9, but for pressure vertical velocity (Pa s⁻¹).

The cross-section (Fig. 11) of pressure vertical velocity along the longitudes of minimum values in Fig. 10 further illustrated the differences between the two types of dynamical cores. The characteristics of vertical velocity in the DMC dynamical core showed more obvious vertical motions than those in the TMC dynamical core. The stretching height at which vertical velocity is zero was higher in the former, and the minimum values of pressure vertical velocity in the cross-section, which occurred at approximately 600 hPa, were much smaller in the former than those in the latter at 1400, 2000 BT 21, and 0200 BT 22 July (Figs. 11c, e, g). Both horizontal and vertical distributions of pressure vertical velocity indicated stronger vertical motion in the DMC dynamical core, which was more favorable to the development of convection.

Figure 11.  Latitude–height cross-sections of vertical velocity at the longitude marked by the symbol × in Fig. 10 at 0800, 1400, and 2000 BT 21, and 0200 BT 22 July.

The above analyses show that the decreased mean sea level pressure and the smaller pressure vertical velocity during the heavy rainfall episodes correspond well with the theoretical analyses in the DMC dynamical core. On the other hand, the decreased sea level pressure and pressure vertical velocity, which are attributed to the superimposed mass effect of total water loss during the rainfall episode, contribute to the further development of the precipitating system. Hence, the precipitation is much heavier in the DMC dynamical core for the next episode. This is analogous to the cycle of the secondary circulation, and it is most obviously shown by the variables at 2000 BT 21 July 2022 (Figs. 8h, i, 9i, 10i) and the subsequent 0200 BT 22 July 2022 (Figs. 8k, l, 9l, 10l). However, the question of how the decreased surface pressure favors vertical motion remains.

The studies showed that surface converging lines favor the triggering and strengthening of convection in environments of high temperature, high moisture, and convective instability (Lei et al., 2020; Zeng et al., 2020; Yin et al., 2022). Figures 12a, 12b, and 12c show the 10-m wind distributions predicted by the YHGSM with DMC and TMC, and the streamline of the 10-m wind difference between them at 0800 BT 21 July. The speeds of the southeast wind in the east and northwest wind in the northwest of Henan Province from the prediction by the YHGSM with DMC (Fig. 12a) were larger than those with TMC (Fig. 12b). This led to obvious wind convergence in northwestern Henan (Fig. 12c), where heavy precipitation occurred. Meanwhile, the divergence field at 850 hPa showed a similar distribution as that of the 10-m wind. The convergence zones in both frameworks appeared in northwestern and western Henan (Figs. 12d, e), and there were two convergence centers in the difference distribution (Fig. 12f), which corresponded to the convergence area of the 10-m wind streamline. In addition, south–north oriented bands with high values (larger than 60 kg m⁻²) of total precipitable water (TPW) throughout Henan Province in both frameworks showed favorable water conditions for heavy rainfall, especially the maximum over 75 kg m⁻² appeared in the center-north (Figs. 12g, h). However, there were also positive TPW differences similarly occurring in northwestern Henan and its west when the TPW from the YHGSM with DMC was considered versus the TPW with TMC. That is, the weakly positive TPW centers corresponded well with those of the convergence area from the 10-m windstream and divergence at 850 hPa, and the indicated conditions were more favorable to the extreme rainfall areas in northwestern Henan (Fig. 8).

Figure 12.  Wind field at 10 m, divergence at 850 hPa, and total precipitable water in (a, d, g) the DMC and (b, e, h) TMC dynamical cores, and (c) the streamline and speed (shaded) of 10-m wind, (f) divergence at 850 hPa, and (i) total precipitable water (TPW) in the difference between the two dynamical cores.

Figure 13 shows the corresponding distributions as Fig. 12 but at 2000 BT 21 July. Similarly, there were obvious differences in the three variables in the northwestern corner of Henan, where heavy rainfall occurred, between the two dynamical cores. The surface wind in the DMC dynamical core showed a predominantly eastward wind in the majority of Henan, but a northward wind from the YHGSM with TMC appeared in northern Henan; then, the streamline of the wind difference between the two dynamical cores showed an obvious convergence center in the northwest (Fig. 13c). Correspondingly, predictions from both dynamical cores showed almost the same 850-hPa convergence zone in northwestern Henan, but the value of the convergence center at 850 hPa in the DMC dynamical core (Fig. 13d) was much smaller than that in the TMC (Fig. 13e). This difference represented the stronger convergence in the former, as the smaller negative values of the divergence field indicated stronger convergence. Furthermore, this convergence center of the difference field (Fig. 13f) correlated well with that of the surface wind streamline (Fig. 13c). In addition, the TPW at 2000 BT in both dynamical cores were smaller than those at 0800 BT, but the south–north band of high values with TPW exceeding 60 kg m⁻² still existed throughout the province. Meanwhile, the positive center of the TPW difference still existed in northwestern Henan between the two dynamical cores. Collocations of the three variables’ differences between the two dynamical cores supported favorable surface, low-level and moisture conditions in the DMC framework. Besides, this favorable surface to low-level convergence may bridge the decreased surface pressure and the stronger vertical motion, which then led to the relatively larger rainfall amounts that were closer to the observations, as shown in Fig. 8.

Figure 13.  As in Fig. 12, but at 2000 BT 21 July 2021.

The discussion in Section 3 shows that the tendency of moist hydrostatic pressure and the diagnostic moist pressure vertical velocity in the DMC dynamical core positively relate to the variation of total water mass. That is, the greater the loss of total water is, the smaller both moist hydrostatic pressure and the diagnostic moist pressure vertical velocity are. From the physical perspective, when the water species are removed to the surface via precipitation in the physical parameterizations, there is a corresponding hydrostatic pressure decrease in the overlying column. As indicated by studies considering pressure changes due to the addition or reduction of total water in tropical cyclones (Lackmann and Yablonsky, 2004; Wacker et al., 2006), pressure decreases due to this mechanism generally result in unbalanced pressure-gradient forces in the lower troposphere and may contribute to vorticity generation and moisture convergence. Regarding the heavy precipitation on 20 and 21 July 2021, a similar mechanism may account for the better performances of the DMC dynamical core, as shown in the schematic diagram of Fig. 14. That is, when the conservation of dry air mass and total water mass are assumed first (Fig. 14b), the precipitation mass sink can be considered in this DMC dynamical core. This high local loss of total water leads to an additional force for the decrease in surface pressure (∆p_s) . Then, the resulting unbalanced pressure-gradient force brings in the additional cyclonic rotation of surface wind (the yellow streamlines) and strengthens the convergence in the lower troposphere. Subsequently, this strengthened low-level convergence induces another vertical velocity component $ {\omega _1} $ [the barycentric vertical velocity in Wacker et al. (2006)], which may enhance the original vertical velocity $ \omega $, implying a stronger vertical motion $ \omega ' $. Hence, under the favorable conditions of heavily precipitating systems, the above cycle may contribute to the further development of the low-pressure system, and then result in a larger rainfall amount (the yellow rain dot in Fig. 14b), which again cycles the further development of the control system and leads to the persistence of heavy precipitation. This kind of positive feedback in the DMC dynamical core is similar to the effects of the source or sink of total water discussed by Gu and Qian (1990). They considered this feature as another positive feedback of total water other than latent heating for the development of the atmospheric system. On the other hand, when the TMC is assumed first (Fig. 14a), false compensation of the dry air mass will occur in the heavily precipitating systems. The compensation will falsely excite the local divergence/convergence forcing, leading to false perturbation of the model state, and finally provide incorrect predictions.

Figure 14.  Schematic diagram of the mass effect of total water with TMC and DMC dynamical cores.

The importance of the precipitation mass sink in heavily precipitating systems has received some attention in the literature, but the pressure changes due to sources or sinks of total water are still neglected in some of the NWP models. Based on two global models, the ECMWF IFS and the YHGSM, which adopt TMC and DMC dynamical cores, respectively, this paper first compares the predictions of two heavy rainfall episodes during the “21.7” Henan extreme rainfall event. Then, a sensitivity test on the performances of heavy precipitation forecasts is conducted in the YHGSM with TMC and DMC. Subsequently, the mass effects due to sources or sinks of total water on the dynamic core are analyzed for extreme rainfall events from both theoretical analyses and case studies.

On 20 and 21 July 2021, four meso-$ \beta $ or larger-scale MCSs occurred in Henan Province with a lasting back-building of a new MCS from the end of the prior MCS. The training style of MCSs led to maximum 24-h rainfall amounts of 624.1 and 447.1 mm on 20 and 21 July, respectively. In particular, the meso-$ \gamma $ MCSs during 1600–1700 BT 20 July produced an hourly rainfall rate of 201.9 mm, which is a record-breaking hourly rainfall rate among national meteorological stations.

The performances of the YHGSM and ECMWF IFS on the 24-h accumulated precipitation on 20 and 21 July 2021 demonstrate that both models can basically predict relatively weak precipitation less than 100 mm day⁻¹ at lead times of 84, 60, and 36 h, and some of the predictions are embedded with rainfall areas of 250 mm. However, compared to the observations, both models show northwestward migrations for 2-day predictions and predict smaller rainfall amounts on 20 July 2021. When comparing performances between the ECMWF IFS and the YHGSM on 2 days, the YHGSM predicts more similar distributions of heavy rainfall and closer daily rainfall amounts to the observations than those of the ECMWF IFS, and provides notably better stability and consistency.

Simulations by the YHGSM adopting DMC and TMC dynamical cores validates that, both rainfall distributions and daily rainfall amounts from the YHGSM adopting DMC are closer to the observations on both 20 and 21 July than those from the model adopting TMC. The former predicts larger areas of rainfall amounts exceeding 100 mm and larger values of the maximum 24-h rainfall amount.

Theoretical analyses and case studies of this extreme rainfall event show that, considering sources or sinks of total water in numerical models with DMC dynamical cores may have positive feedback for the precise prediction of condensates. The mechanism may act as follows: for extreme rainfall events, the high local loss of total water causes an additional force for the decrease in surface pressure, and then the unbalanced pressure gradient forces an additional cyclonic rotation of surface wind and strengthens the convergence in the lower troposphere. Subsequently, this strengthened low-level convergence induces another vertical velocity, and enhances the vertical motion. Under the favorable conditions of heavily precipitating systems, the above cycle contributes to the persistence of heavy precipitation.

It is worth noting that for the extreme rainfall event in this paper, the mass of water vapor lost during water phase changes is shown to be important, but the mass effect of the precipitation sink only positively contributed to the precise prediction of heavy rainfall. That is, the deterministic contributions still come from the direct dynamic and thermodynamic components, and this relatively lower magnitude mass effect plays the superimposed effect in facilitating further development, such as those illustrated from the diagnosed pressure vertical velocity in Peng et al. (2020). The questions regarding the type of weather situation for which this superimposed effect can be maximized and how to quantify this mass effect for heavily precipitating systems need further work. On the other hand, the coupling between the physical parameterizations and the continuity equation in the DMC dynamical core may provide another way to solve the physical–dynamic coupling, such as that used to couple the deep convection scheme and the dynamic via the divergence of mass flux for gray-zone resolution (Malardel and Bechtold, 2019).

We would like to thank the anonymous reviewers for their helpful comments.