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Improved Diurnal Cycle of Precipitation on Land in a Global Non-Hydrostatic Model Using a Revised NSAS Deep Convective Scheme


doi:  10.1007/s00376-023-3121-7

  • In relatively coarse-resolution atmospheric models, cumulus parameterization helps account for the effect of subgrid-scale convection, which produces supplemental rainfall to the grid-scale precipitation and impacts the diurnal cycle of precipitation. In this study, the diurnal cycle of precipitation was studied using the new simplified Arakawa-Schubert scheme in a global non-hydrostatic atmospheric model, i.e., the Yin-Yang-grid Unified Model for the Atmosphere. Two new diagnostic closures and a convective trigger function were suggested to emphasize the job of the cloud work function corresponding to the free tropospheric large-scale forcing. Numerical results of the 0.25-degree model in 3-month batched real-case simulations revealed an improvement in the diurnal precipitation variation by using a revised trigger function with an enhanced dynamical constraint on the convective initiation and a suitable threshold of the trigger. By reducing the occurrence of convection during peak solar radiation hours, the revised scheme was shown to be effective in delaying the appearance of early-afternoon rainfall peaks over most land areas and accentuating the nocturnal peaks that were wrongly concealed by the more substantial afternoon peak. In addition, the revised scheme enhanced the simulation capability of the precipitation probability density function, such as increasing the extremely low- and high-intensity precipitation events and decreasing small and moderate rainfall events, which contributed to the reduction of precipitation bias over mid-latitude and tropical land areas.
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  • Figure 1.  Area-averaged diurnal cycle of mean total precipitation (mm h−1) in July 2021 over the six regions of (a) the Sichuan basin, (b) southern China, (c) tropical Africa, (d) the Indochina Peninsula, (e) the eastern United States, and (f) tropical South America using surface rain gauge data (black solid), GPM (black long dashed), CMORPH (black short dashed), and the simulations in CLOS1_CLI (red solid), CLOS1_PBL (green solid), CLOS2_ADV (blue solid), and CLOS1_CLI_ADVTRI (purple solid).

    Figure 2.  The (a, c, e, g) composite diurnal phase (LST) and (b, d, f, h) amplitude (mm d−1) of total precipitation during summertime (JJA 2021) for the northern hemisphere in (e, f) CTL, and (g, h) EXP in comparison with (a, b) CMORPH and (c, d) GPM data. In panels (a), (c), (e), and (g), the grids with minimal amplitude (<0.1 mm d−1) and a slight amplitude (0.1–1 mm d−1) are masked out and shown in low saturation colors, respectively. The six black rectangles in (a) correspond to the selected regions in Fig. 1.

    Figure 3.  Composite diurnal phase of precipitation based on (a) surface rain gauge observation, (b) CMORPH retrieval, (c) GPM analysis, (d) CTL, and (e) EXP simulations in China.

    Figure 4.  Area-averaged diurnal cycle of precipitation amount (mm d−1, black solid), intensity (mm d−1, red solid), and frequency (%, blue solid) for (a) rain gauge observations, (b) CTL, and (c) EXP simulations, and the probability density functions (%, shaded) of mean precipitation rate by (d) rain gauge observation, (e) CTL, and (f) EXP in Southern China JJA 2021. In panels (d)–(f), the precipitation rates (1–250 mm d−1) are binned with logarithmic sizes.

    Figure 5.  The diurnal variations of area-averaged precipitation amount over the (b) coastal region, (c) coast–inland transition region, and (d) inland region of Southern China during JJA 2021. The geographic extent of the three subregions is given in (a).

    Figure 6.  Area-averaged diurnal cycle of precipitation amount (mm d−1, black solid), intensity (mm d−1, red solid), and frequency (%, blue solid) of (a) rain gauge observations, (b) CTL, and (c) EXP, and the probability density functions (%, shaded) of the mean precipitation rate of rain gauge in the (d) observations, (e) CTL, and (f) EXP in the Sichuan basin for JJA 2021.

    Figure 7.  The 3-hourly distribution of (a) CAPE (shaded) and 900-hPa horizontal winds (barbs with a full flag of 4 m s–1, green dots filled area of wind speed ≥ 4 m s–1), (b) the diurnal component of CAPE generation rate due to large-scale advective process (shaded) and 900-hPa horizontal winds (barbs with a full flag of 2 m s–1) in EXP during JJA 2021. The diurnal component is estimated as the deviation of the deviation from its daily mean. The gray shading indicates elevations ≥ 1000 m. Line AB denotes the position of the cross-section of Fig. 8.

    Figure 8.  The vertical cross-section of wind vectors of ERA5 (left) and EXP simulation (right) in the lower troposphere along AB in Fig. 7a in the Sichuan Basin. The wind vectors represent the projection of the three-dimensional wind (u, v, 10ω). The vertical component ω is amplified 10 times because it is much smaller than the horizontal ones. The shading indicates the vertical wind speed ω.

    Figure 9.  Composite diurnal phase of precipitation (mm d−1) of (a) CMORPH, (b) CTL, and (c) EXP simulations over tropical South America, and a Hovmöller diagram of hourly precipitation (mm d−1) of (d) CMORPH, and (e) CTL and (f) EXP simulations over the region (1°–6°N; 75°–65°W), which is denoted by the rectangle in (a). In (a)–(c), the grids with low amplitude (< 0.1 mm d−1) and slightly low amplitude (0.1–1 mm d−1) are masked out and shown in low saturation colors, respectively.

    Figure 10.  The probability density functions of precipitation rates over land areas of China (top) and tropics (bottom) during JJA 2021 for CMORPH (gray dot), GPM (black dot), CTL (red dot), and EXP (blue dot). A logarithmic horizontal axis (left) is used for regular precipitation rates of 0.1–250 mm d−1, and a linear horizontal axis (right) is employed to show the wide frequency variation among precipitation rates 0–200 mm d−1.

    Figure 11.  The averaged (left) ETS and (right) BIAS scores of daily precipitation for the first 3-day simulations during JJA 2021 over land areas of China (top) and tropics (bottom) in CTL (red) and EXP (blue) against the surface rain gauge observations in China and GPM in tropics.

    Table 1.  Numerical Experiments

    Experimental Name Trigger of Deep Convection Closure of Deep Convection
    CLOS1_CLI(CTL) The default trigger functions:
    (1) the distance between the CSL and LFC of the parcel without entrainment < 120–180 hPa; (2) the difference between the LFC with and without sub-cloud entrainment < 25 hPa; (3) the convective inhibition defined by the integral of negative buoyancy from the CSL to LFC > −80 –−120 m2s−2.
    $ {M}_{\mathrm{B}\mathrm{E}}=-\dfrac{A-\alpha \left(w\right){A}_{\mathrm{c}}}{\tau }\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $, adjusting the CWF to the climatology in $ \tau $.
    CLOS1_PBL $ {M}_{\mathrm{B}\mathrm{E}}=-\dfrac{A-{A}_{\mathrm{B}\mathrm{L}}}{\tau }\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $ , adjusting the CWF to the boundary layer contribution in $ \tau $.
    CLOS2_ADV $ {M}_{\mathrm{B}\mathrm{E}}=-{\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}}\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $ , using the $ {\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ to drive deep convection.
    CLOS1_CLI_ADVTRI Besides the trigger functions described above, a constraint of $ {\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} > 110\;\mathrm{ }\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{h}}^{-1} $ is added. Same as CLOS1_CLI
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Manuscript received: 13 June 2023
Manuscript revised: 24 October 2023
Manuscript accepted: 30 November 2023
通讯作者: 陈斌, bchen63@163.com
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Improved Diurnal Cycle of Precipitation on Land in a Global Non-Hydrostatic Model Using a Revised NSAS Deep Convective Scheme

    Corresponding author: Xindong PENG, pengxd@cma.gov.cn
  • 1. State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, Beijing 100081, China
  • 2. University of Chinese Academy of Sciences, Beijing 100049, China
  • 3. CMA Earth System modeling and Prediction Centre, Beijing 100081, China
  • 4. 2035 Future Laboratory, PIESAT Information Technology Co Ltd., Beijing 100195, China

Abstract: In relatively coarse-resolution atmospheric models, cumulus parameterization helps account for the effect of subgrid-scale convection, which produces supplemental rainfall to the grid-scale precipitation and impacts the diurnal cycle of precipitation. In this study, the diurnal cycle of precipitation was studied using the new simplified Arakawa-Schubert scheme in a global non-hydrostatic atmospheric model, i.e., the Yin-Yang-grid Unified Model for the Atmosphere. Two new diagnostic closures and a convective trigger function were suggested to emphasize the job of the cloud work function corresponding to the free tropospheric large-scale forcing. Numerical results of the 0.25-degree model in 3-month batched real-case simulations revealed an improvement in the diurnal precipitation variation by using a revised trigger function with an enhanced dynamical constraint on the convective initiation and a suitable threshold of the trigger. By reducing the occurrence of convection during peak solar radiation hours, the revised scheme was shown to be effective in delaying the appearance of early-afternoon rainfall peaks over most land areas and accentuating the nocturnal peaks that were wrongly concealed by the more substantial afternoon peak. In addition, the revised scheme enhanced the simulation capability of the precipitation probability density function, such as increasing the extremely low- and high-intensity precipitation events and decreasing small and moderate rainfall events, which contributed to the reduction of precipitation bias over mid-latitude and tropical land areas.

    • In atmospheric general circulation models (GCMs) with relatively coarse resolution, the statistical effects of sub-grid convection, just as other subgrid-scale processes, need to be described with a physical parameterization scheme. The use of cumulus convective parameterization in the numerical model can be illustrated with a cycle. The occurrence of and overall intensity of cumulus convective activity are controlled by large-scale processes, and in turn, the adjustment of cumulus convection to the ambient atmosphere is also fed back into the grid-scale atmosphere (Arakawa, 2004). This lies at the core of the cumulus convective parameterization scheme and is referred to as the closure assumption condition. Several convective schemes are based on the classical “quasi-equilibrium” theory of large-scale and convective processes introduced by Arakawa and Schubert (1974) (AS hereafter). It assumes that when convection is adjusted fast enough, the rate of atmospheric unstable energy consumed by convective processes can be comparable to that produced by other non-convective processes, such as advection, radiation, and turbulent diffusion in the atmospheric boundary layer.

      The accuracy of the AS quasi-equilibrium theory has been well verified with observational data and increasing application of cloud-resolving models. It is deduced that the variability of convective available potential energy (CAPE) is a result of the temperature variation of both the air parcel and its ambient surroundings, which is related to the thermal properties of the atmospheric boundary layer and free tropospheric atmosphere, respectively (Emanuel et al., 1994; Zhang, 2002). Observations over tropical and mid-latitude land areas further reveal that net CAPE variability is primarily controlled by temperature and humidity changes in the boundary layer rather than in the free troposphere, indicating that the AS quasi-equilibrium assumption is not well satisfied under rapidly varying surface forcing conditions (Zhang, 2002; Donner and Phillips, 2003). Results of cloud-resolving models also show that the temporal evolution of convection remains broadly consistent with the large-scale forcing at long timescales and that the convective response significantly lags behind the large-scale forcing at short timescales (Jones and Randall, 2011; Davies et al., 2013).

      The diurnal cycle of precipitation is a key factor in assessing physical processes and numerical models due to its significance in simulating local rainfall. There are still profound difficulties in reproducing the observed diurnal cycle of precipitation with GCMs, although the mean state of precipitation is simulated quite reasonably. The observed rainfall peak during the late afternoon and early evening over land is incorrectly simulated at noon or early afternoon by numerical models, concurrent with when the strongest surface heat and moisture fluxes appear (Dai and Trenberth, 2004; Lee et al., 2007; Covey et al., 2016; Christopoulos and Schneider, 2021). Stronger diurnal amplitudes of precipitation, characterized by considerable regional variability, are shown over land areas rather than oceans in the warm season. Local atmospheric instability associated with diurnal solar heating over the land surface and regional mesoscale circulations forced by heating differences are considered the vital causes for the appearance of rainfall peaks over lands in the afternoon (Yang and Smith, 2006; Dai et al., 2007; Zhou et al., 2008). The occurrence of nocturnal precipitation over land areas may be related to nocturnal radiative cooling or the downstream propagation of mountain-induced convection, such as the nocturnal rainfall in the central United States and Sichuan Basin of China, resulting from complex interactions related to topographic circulations, the nocturnal low-level jet, and well-organized mesoscale convective systems (MCSs) (Yang and Smith, 2006; Carbone and Tuttle, 2008; Zhang et al., 2019). Representation of nonequilibrium convection caused by rapid variations in boundary layer forcing using the equilibrium closure assumption may be a major cause of model bias in simulating the diurnal precipitation phase over land areas. The approach of linking convective processes with the free tropospheric large-scale forcing while decoupling it from the boundary layer forcing shows a significant improvement in parameterized results of cumulus convection (Xie and Zhang, 2000; Zhang, 2002, 2003).

      Observations by Donner and Phillips (2003) and Zhang (2002) hint that the AS closure can be further refined as a quasi-equilibrium state of the tropospheric large-scale processes and convective activities. Zhang (2002, 2003) optimized the closure assumption of the Zhang-McFarlane scheme in the National Center for Atmospheric Research (NCAR) Community Climate Model (CCM3) single-column model so that deep convection is entirely controlled by free tropospheric large-scale processes rather than boundary layer processes. Rainfall simulations with the new scheme show a much better agreement with the observations for both the timing of convective activity and the rainfall amounts, which corrects the original misleading assumption that the evolution of convection broadly follows the boundary layer forcing and surface fluxes. Bechtold et al. (2014) proposed a new diagnostic closure assumption to represent both equilibrium convection and nonequilibrium convection, which is completed by subtracting the CAPE production due to boundary layer forcing, the latter defined at different boundary-layer timescales over land and ocean for convective depletion. The results of the long- and short-term European Centre for Medium Weather Forecasts (ECMWF) Integrated Forecasting System (IFS) runs show a 4-5-h effective delay of appearance of the afternoon rainfall peak over land, even though the simulation of nocturnal precipitation is still tricky.

      The method of emphasizing the large-scale processes in convective triggers is often used to suppress frequent, weak daytime convection caused by a CAPE-based convective parameterization (Bechtold et al., 2004; Lee et al., 2008). For example, a trigger threshold is defined as a function of grid-scale vertical velocity at the cloud base or the mean relative humidity at sub-cloud layers in some convective schemes (Pan and Wu, 1995; Han et al., 2020). Xie et al. (2019) substantially improved the modeling of the diurnal variations of terrestrial precipitation in the Department of Energy's Exascale Earth System Model (E3SM) Atmosphere Model version 1 (EAMv1) by using an additional convective trigger function. Specifically, they introduced the dynamical CAPE generation rate (dCAPE) due to large-scale processes (Xie and Zhang, 2000) and unrestricted convective launch level (ULL) (Wang et al., 2015) to modulate moist convective initiation. This dynamical constraint of the dCAPE trigger suppresses the frequent release of CAPE associated with weak convective activity and allows CAPE to accumulate for delayed and intense convection (Xie and Zhang, 2000; Xie et al., 2004). The ULL trigger helps to capture the nocturnal elevated convection above the boundary inversion (Wang et al., 2015). Following the methods of Xie et al. (2019), Cui et al. (2021) obtained profoundly improved simulations of the diurnal phase of summer precipitation over land with the NCAR Community Atmosphere Model version 5 (CAM5) model, especially regarding the downstream propagation of convective systems, using a further optimized threshold of the dCAPE trigger and an environmental entrainment rate based on observations. Many other studies also found evidence of dCAPE and ULL triggers to improve the precipitation and its diurnal variation modeling in climate simulations, such as Wang et al. (2020) and Li et al. (2023).

      The role of boundary layer forcing in regulating convection is overemphasized in previous deep convective parameterizations because of the use of CAPE-based trigger or closure assumption, which is the leading cause of the early appearance of rainfall peaks over land areas in GCMs (Xie and Zhang, 2000; Zhang, 2002, 2003; Lee et al., 2007). This problem can be alleviated by modifying the closure assumption or trigger function with a tight link of convective activity and the CAPE production rate due to tropospheric large-scale forcing. The goal of the current study is to examine the impact of these revisions in the new simplified Arakawa–Schubert (NSAS hereafter) convective scheme on the precipitation simulated with the Yin-Yang-grid Unified Model for the Atmosphere (YUNMA), with a particular emphasis on the diurnal cycle.

      The remainder of this paper is organized as follows. The numerical model and physical parameterization schemes, including the target convective parameterization, are given in section 2. Section 3 shows the numerical results of the sensitivity experiments using two new diagnostic closures and a convective trigger function that are all linked to the free tropospheric large-scale forcing. In addition, the effects of the revised convective trigger on the simulations of the diurnal cycle, the probability density distribution, and skill scores of the precipitation are analyzed. Finally, a summary and discussion appear in section 4.

    2.   Numerical Model and Physical Schemes
    • A global non-hydrostatic atmospheric model, the YUNMA model (Li et al., 2015; Li and Peng, 2018; Chen et al., 2023), is used in this study. It is built on the quasi-uniform spherical Yin-Yang grid, a redevelopment of the China Meteorological Administration Global Forecasting System (CMA-GFS), which is known as the global/regional assimilation and prediction system (GRAPES) (Chen et al., 2008). Local refinement is possible with a multi-level nesting technique. Because of the complete representation of inertial forcing in the YUNMA dynamical core, the model coordinate can be rotated arbitrarily for the convenience of specific computational domain selection. More details can be found in Chen et al. (2023).

      The physical package coupled in the YUNMA model includes the longwave and shortwave radiation schemes of the rapid radiation transfer model for general circulation models (RRTMG) (Iacono et al., 2008), the atmospheric boundary layer approach of the medium-range forecast (MRF) scheme (Hong and Pan, 1996) with surface-layer parameterization of Monin–Obukhov similarity theory (Beljaars, 1995), the new simplified Arakawa-Schubert (NSAS) cumulus convective parameterization scheme (Arakawa and Schubert, 1974; Pan and Wu, 1995; Han and Pan, 2011), and a double-moment mixed-phase cloud microphysics scheme (Liu et al., 2003) with an explicit cloud-cover prognostic scheme (Ma et al., 2018), and the common land-surface model (CoLM) (Dai et al., 2003).

    • The version of the NSAS deep convective scheme was developed by Han and Pan (2011) and originated from Pan and Wu (1995). This scheme describes a convective cloud with only one updraft and a saturated downdraft employing a bulk mass-flux approach, simplified by Grell (1993). The intensity of the triggered convection, i.e., the mass flux, is estimated based on the “quasi-equilibrium” closure assumption defined by the cloud work function (CWF) (Arakawa and Schubert, 1974). The CWF is a measure of the total buoyancy used to represent the efficiency of atmospheric kinetic energy generation, being different from CAPE with sufficient consideration of entrainment mixing of the updraft and environment. The convective consumption rate of the CWF (hereinafter denoted with A in the equations) can be expressed in two ways. First, the CWF is adjusted to a reference state A0 after a time scale $ \tau $ of convective adjustment. Second, the convection is suppressed with a reduced CWF based on the environmental thermodynamic changes as a result of cumulus-induced subsidence. A closure is then formulated, and the cloud-base mass flux ($ {M}_{\mathrm{B}\mathrm{E}} $) can be obtained diagnostically from the closure. The default closure assumption in the NSAS (Han and Pan, 2011) suggests that the CWF is adjusted to a reference climatology derived from the observations (marked as CLOS1_CLI) and the $ {M}_{\mathrm{B}\mathrm{E}} $ is formulated as:

      where $ {A}_{\mathrm{c}} $ is the climatology of CWF and $ \alpha \left(w\right) $ a vertical velocity-related factor. $ {A}^{{'}} $ is a CWF at the updated thermodynamic state modified by an arbitrary cloud-base mass flux, $ {M}_{\mathrm{B}\mathrm{E}}^{{'}}\delta t $. Compared with the closure that consumes the total CWF (A0 = 0) within $ \tau $, the default closure in the NSAS is more realistic because the CWF is adjusted to its climate state, and the estimated $ {M}_{\mathrm{B}\mathrm{E}} $ is related to the large-scale vertical velocity at the cloud base.

      The current NSAS version in the YUNMA model has introduced many other published improvements which include: (a) increasing the relative humidity (RH)-dependent term of updraft entrainment rate by a factor of 10 to weaken the strength of convection in a dry environment (Bechtold et al., 2014; Han et al., 2017), (b) modifying the constant detrainment rate to be the sum of the organized term and turbulent term (Bechtold et al., 2008, 2014; De Rooy et al., 2013), (c) considering the temperature dependency of precipitation auto-conversion parameter (Han et al., 2016), (d) revising the threshold of the convective trigger to be proportional to the RH averaged over the sub-cloud layers (Han et al., 2020), and (e) using the convective inhibition calculated with the integral of the negative buoyancy from the convection starting level (CSL) to the level of free convection (LFC) as additional trigger condition (Han et al., 2017). However, the simulated precipitation still shows a large bias in the diurnal cycle. The application of the NSAS calls for additional tuning in the triggering and closure assumptions for a better precipitation simulation.

    • Similar to Arakawa and Schubert (1974), the closure based on the quasi-equilibrium of CWF changes between convective and large-scale processes in the free troposphere (Zhang, 2002, 2003; Donner and Phillips, 2003) can be formulated as:

      with large-scale advection, radiation, and convection being denoted by the subscript ADV, RA, and CU, respectively. The estimation of deep convection is then dependent on the mean horizontal and vertical advection of temperature and humidity rather than the boundary layer processes controlled by surface solar heating if the radiation processes are ignored. Therefore, we test three revisions of the NSAS scheme to show the impacts on the precipitation simulation.

      The first diagnostic closure (noted as CLOS1_PBL) refers to the method of Bechtold et al. (2014), which adjusts the CWF to the boundary layer contribution ($ {A}_{\mathrm{B}\mathrm{L}} $) within the $ \tau $. In this way, the kinetic energy generated by the boundary layer forcing is not used to drive deep convection. The formulation of cloud-base mass flux becomes:

      The $ {A}_{\mathrm{B}\mathrm{L}} $ is defined as:

      and

      where $ {\tau }_{\mathrm{B}\mathrm{L}} $ denotes the boundary-layer time scale and is defined to be equal to the convective turnover time over land in Bechtold et al. (2014); $ T $ and $ {q}_{v} $ are the current temperature and specific humidity profiles, respectively; $ {\left.\Delta T\right|}_{\mathrm{B}\mathrm{L}} $, $ {\left.\Delta {q}_{v}\right|}_{\mathrm{B}\mathrm{L}} $ denotes the changed temperature and water vapor due to advection, radiation, and turbulent processes inside the boundary layer during the physical time step $ \Delta t $ of the convective scheme.

      The second diagnostic closure (noted as CLOS2_ADV) estimates the cloud-base mass flux using the CWF generation rate determined by large-scale advective forcing $ {\left(c{\partial A}/{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ (Zhang, 2002, 2003) which can be formulated as:

      and

      where $ {\left.\Delta T\right|}_{\mathrm{A}\mathrm{D}\mathrm{V}} $, $ {\left.\Delta {q}_{v}\right|}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ denotes the updated temperature and specific humidity due to large-scale advection during the physical time step $ \Delta t $ of the convective scheme. The closure described in Eq. (6) uses a different diagnostic method from CLOS1_CLI in Eq. (1) and CLOS1_PBL in Eq. (3) with no need to introduce the convective adjustment time. The modifications in CLOS1_PBL and CLOS2_ADV both largely associate convective processes with large-scale advective forcing.

      The $ {\left( {\partial A}/{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ term can also be used as a dynamical constraint for convection initiation (Xie and Zhang, 2000; Xie et al., 2019; Cui et al., 2021; Li et al., 2023). A third test is the revision of the trigger function (marked as CLOS1_CLI_ADVTRI) in the NSAS deep convective scheme by enhancing the convective triggering threshold without any modification to the default closure. Compared with those in CLOS1_PBL and CLOS2_ADV, the convection with the revised trigger function is easier to artificially modulate based on the observations by tuning the threshold of the trigger function. An appropriate trigger threshold is given as:

      in the YUNMA model with a 0.25° horizontal resolution and 200-s time step.

    • The simulation using the default trigger and closure in the NSAS is referred to as CLOS1_CLI (CTL). Three sensitivity experiments, CLOS1_PBL, CLOS2_ADV, and CLOS1_CLI_ADVTRI (Table 1), which use two revised diagnostic closures and one modified trigger function, respectively, to explicitly link the deep convection to the free-tropospheric large-scale forcing based on the CWF, serve to illustrate the effect of large-scale forcing on subgrid-scale convection over land. The sensitivity experiments are expected to delay the early appearance of the maximum rainfall in CTL by suppressing the convection during the early afternoon, primarily driven by the surface flux forcing in the boundary layer. The decreased convective consumption of CWF generated by boundary layer forcing in the early afternoon allows for the accumulation of CWF for a longer time until stronger convection is triggered in the late afternoon.

      Experimental Name Trigger of Deep Convection Closure of Deep Convection
      CLOS1_CLI(CTL) The default trigger functions:
      (1) the distance between the CSL and LFC of the parcel without entrainment < 120–180 hPa; (2) the difference between the LFC with and without sub-cloud entrainment < 25 hPa; (3) the convective inhibition defined by the integral of negative buoyancy from the CSL to LFC > −80 –−120 m2s−2.
      $ {M}_{\mathrm{B}\mathrm{E}}=-\dfrac{A-\alpha \left(w\right){A}_{\mathrm{c}}}{\tau }\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $, adjusting the CWF to the climatology in $ \tau $.
      CLOS1_PBL $ {M}_{\mathrm{B}\mathrm{E}}=-\dfrac{A-{A}_{\mathrm{B}\mathrm{L}}}{\tau }\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $ , adjusting the CWF to the boundary layer contribution in $ \tau $.
      CLOS2_ADV $ {M}_{\mathrm{B}\mathrm{E}}=-{\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}}\dfrac{{M}_{\mathrm{B}\mathrm{E}}^{{{'}}}\delta t}{{A}^{{{'}}}-A} $ , using the $ {\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ to drive deep convection.
      CLOS1_CLI_ADVTRI Besides the trigger functions described above, a constraint of $ {\left(\dfrac{\partial A}{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} > 110\;\mathrm{ }\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{h}}^{-1} $ is added. Same as CLOS1_CLI

      Table 1.  Numerical Experiments

    • In this study, the global configuration of the YUNMA model on the standard Yin-Yang grid is used to test the revised NSAS scheme. A horizontal resolution of 0.25° and 60 layers in the vertical direction with the model top at 36 km are specified in the YUNMA model. An upward-stretched layer specification is used with the highest vertical resolution of 20 m near the surface. A 200-s time step is utilized in the integration.

      All CTL and sensitivity experiments were integrated for 96 h, which were initiated at 0000 UTC every three days from 1 July 2021 to 31 July 2021 for a one-month batched simulation. The simulated precipitation was first checked to find the most effective closure/trigger revision in the sense of the rainfall diurnal cycle. The best sensitivity experiment was referred to as EXP, and an extension of the batched EXP simulation was carried out from 1 June 2021 to 31 August 2021 in comparison with CTL for detailed analysis. The prognostic quantities were output hourly on pressure levels for analysis.

      The NCEP final analysis data (FNL) was used to establish the initial conditions for the YUNMA model in this study. The CMORPH_V0.x precipitation data provided by the Climate Prediction Center, the GPM IMERG Late Precipitation product of NASA, and the merged hourly surface rain-gauge observations in China and CMORPH data, provided by CMA, served as observational facts for analysis. The CMORPH_V0.x data have a horizontal 0.25° resolution ranging between 60°S and 60°N in 3-hour intervals. The GPM data, a combination of microwave infrared estimation and rain gauges, are of 0.1° spatial and 30-minute temporal resolutions. For the convenience of verification, the CMORPH_V0.x precipitation products are linearly interpolated with an hourly temporal resolution, and the GPM and CMA merged precipitation are remapped onto the 0.25° grid with the exact spatial resolution as model outputs. The ECMWF ERA5 monthly averaged reanalysis data of 0.25° resolution are used to examine the large-scale circulation. The method of diurnal phase and amplitude analysis of precipitation can be found in Sakaeda et al. (2017) and Dias et al. (2018), which is a Fourier transformation of the 24-h time series of precipitation at a fixed grid.

    3.   Numerical Results
    • The averaged diurnal cycle of total precipitation in July 2021 over six selected regions is presented in Fig. 1 using the default (CTL) and three revised trigger/closure schemes described above against the observations. The selected areas include two mid-latitude continental regions, i.e., the Sichuan Basin (27°–32°N, 100°–107°E) and the eastern United States (30°–45°N, 100°–80°W), and several tropical regions, including southern China (23°–26°N, 110°–117°E), the Indochina Peninsula (10°–20°N, 100°–110°E), tropical Africa (0°–15°N, 10°–30°E), and tropical South America (7.5°S–0°, 75°–65°W).

      Figure 1.  Area-averaged diurnal cycle of mean total precipitation (mm h−1) in July 2021 over the six regions of (a) the Sichuan basin, (b) southern China, (c) tropical Africa, (d) the Indochina Peninsula, (e) the eastern United States, and (f) tropical South America using surface rain gauge data (black solid), GPM (black long dashed), CMORPH (black short dashed), and the simulations in CLOS1_CLI (red solid), CLOS1_PBL (green solid), CLOS2_ADV (blue solid), and CLOS1_CLI_ADVTRI (purple solid).

      Figure 1 indicates that the errors of the diurnal cycle of terrestrial precipitation reflected in many GCMs also exist in the YUNMA model coupled with the default NSAS deep convective scheme (CLOS1_CLI (CTL)), with an early appearance of a late-afternoon rainfall maximum and even a failure to simulate the nocturnal peaks. The afternoon peaks of precipitation over the Indochina Peninsula and tropical Africa in the control run occur about 4 h earlier than observations, and the nocturnal rainfall peak observed at 0200 local solar time (LST) in the Sichuan basin is incorrectly presented as an afternoon peak at 1200 LST. Except for the eastern United States, most areas peak around 1200–1500 LST, corresponding to the peak time of the sensible heat and latent heat fluxes.

      The peak time simulated in CLOS1_PBL and CLOS1_CLI (CTL) are similar over most areas, but the amplitude in CLOS1_PBL is more extensive, especially in tropical continents where convective precipitation is predominant. In CLOS1_CLI (CTL), the deducted CWF climatology is mainly derived from the boundary layer contribution, which illustrates the reason for the similar peak times in CLOS1_CLI (CTL) and CLOS1_PBL. Compared with the control run, the simulated peak time of precipitation is around 2000 LST in the CLOS2_ADV run, which indicates the linkage of large-scale advection and convective processes being effective in delaying the terrestrial precipitation peak time.

      The simulation of nocturnal precipitation maximum in the Sichuan basin displays a noticeable improvement in the CLOS2_ADV run, with a larger amplitude and an early-evening peak against those in CLOS1–CLI (CTL) run. We can also notice that the peak time over some areas, such as Southern China, the Indochina Peninsula, and tropical South America, featured by the afternoon peak at 1500–1800 LST in observations, are over-delayed to around 2000 LST in CLOS2_ADV. In CLOS1_PBL and CLOS2_ADV, the deep convective processes are both essentially driven by the unstable kinetic energy produced by the free-tropospheric large-scale forcing, but the estimated diurnal cycle of precipitation is significantly different. Compared with CLOS1_PBL, the unstable kinetic energy consumption within a time step due to deep convection being diagnosed is much smaller in CLOS2_ADV. A delayed appearance of the peak time of precipitation is therefore observed in the CLOS2_ADV run.

      Among all the sensitivity experiments, the CLOS1_CLI_ADVTRI run agrees best with observations over most areas, except for the Sichuan basin. It is suggested that the revised trigger in CLOS1_CLI_ADVTRI effectively improves the diurnal cycle simulation over land, in which the convection initiation is constrained with the free tropospheric large-scale advective forcing. The diurnal cycle in the CLOS1_CLI_ADVTRI run is easily adjusted to fit the observations by repeatedly trying different thresholds in Eq. (8). Different convective triggering thresholds manifested two aspects in the precipitation simulations. First, the triggering of early-afternoon thermal convection activity controlled by the boundary layer forcing can be greatly suppressed. The unstable energy frequently consumed by convective processes is therefore reduced. Convective instability remains until much later in the day, and a delayed peak appears. Second, a reduced ratio of convective to total precipitation is realized with an increase in the threshold, because some of the originally triggered deep convection were inhibited. The setting of the threshold should consider its impact on precipitation over different regions because of these sensitivities. For example, a small threshold is appropriate in Southern China, where an afternoon peak in precipitation is dominant, while a large threshold works well in the Sichuan Basin, where precipitation peaks at nighttime. A fixed trigger threshold is used here for simplicity, although it does have limitations. The CLOS1_CLI_ADVTRI run with a threshold of 110 J kg−1 h−1, selected from a series of sensitivity experiments with different thresholds, is demonstrated to be conducive to reducing the obvious simulation bias of the diurnal cycle of precipitation in CTL.

    • Based on the fact that the CLOS1_CLI_ADVTRI run achieves the best diurnal cycle simulation of precipitation in comparison with the observations, we further investigate the geographical variation of the rainfall diurnal cycle in tropical and mid-latitude regions in the northern hemisphere (15°S–45°N) with the CLOS1_CLI_ADVTRI (EXP) in comparison with the default NSAS scheme CLOS1_CLI (CTL). We extend the sampling of precipitation simulation to three months (June to August 2021) for confident simulation of diurnal cycle variation. Figure 2 shows the composite diurnal phase and amplitude of the total precipitation during summertime in the northern hemisphere (June-July-August, JJA) in 2021 for EXP and CTL. The diagnostic methods of Sakaeda et al. (2017) and Dias et al. (2018) are used to estimate the phase and amplitude of diurnal precipitation variation. The CTL produces peak time of precipitation ahead of the CMORPH and GPM observations over most areas of the tropical and mid-latitude lands (Figs. 2a, c, e). In contrast, EXP significantly delays the terrestrial diurnal phase over tropical Africa, tropical South America, subtropical regions such as South Asia, Southeast Asia, and southern China, and the mid-latitude regions of the eastern United States. The number of grids with peak time at noon is also reduced (Fig. 2g). Even though there are still biases between EXP and the observations, a clear improvement is observed in comparison with CTL. Minor amplitude enhancements are observed over land areas of tropical Africa and tropical South America (Figs. 2b, d, f, h).

      Figure 2.  The (a, c, e, g) composite diurnal phase (LST) and (b, d, f, h) amplitude (mm d−1) of total precipitation during summertime (JJA 2021) for the northern hemisphere in (e, f) CTL, and (g, h) EXP in comparison with (a, b) CMORPH and (c, d) GPM data. In panels (a), (c), (e), and (g), the grids with minimal amplitude (<0.1 mm d−1) and a slight amplitude (0.1–1 mm d−1) are masked out and shown in low saturation colors, respectively. The six black rectangles in (a) correspond to the selected regions in Fig. 1.

      In zooming in over China, observations show mainly afternoon rainfall peaks in northeastern China and southern China which are the result of increased surface heating (Figs. 3a–c). Compared with CTL, EXP with its enhanced dynamical convective trigger of $ {\left({\partial A}/{\partial t}\right)}_{\mathrm{A}\mathrm{D}\mathrm{V}} $ significantly postpones the diurnal phase on most grids in northeastern and southern China, with the early-afternoon maximum of precipitation in CTL being replaced by the late-afternoon maximum in EXP. The similarity of the diurnal phase between the simulation and observation is significantly improved in southern China when an additional convective trigger of the enhanced CWF threshold is used. A clear diurnal-cycle transition of the early-afternoon peak at the coastline and late-afternoon maximum inland is observed, similar to Jiang et al. (2017). In the Sichuan basin, east of the Tibetan Plateau, observational precipitation is dominated by night and early-morning peaks (Figs. 3a–c) as a result of the mountain-valley wind thermodynamic circulation and moisture convergence created by the low-level jet in the nocturnal boundary layer (Zhang et al., 2019). However, the default NSAS scheme (CTL) produces early-afternoon rainfall peaks over this region (Figs. 3d and 1a), which is quite different from the daytime subsidence flows that correspond to the minimum observational precipitation in the Sichuan basin (Jin et al., 2013; Zhang et al., 2019). The EXP improves the rainfall cycle simulations with a decreased number of early-afternoon-peak grid points over the western Sichuan basin (Fig. 3e). Unfortunately, it is not satisfactory in simulating the peak precipitation which occurs at night and early morning in the eastern Sichuan basin with the revised trigger condition. This likely relates to an inadequate expression of topographic slope as well as low-level jets in the nocturnal boundary layer.

      Figure 3.  Composite diurnal phase of precipitation based on (a) surface rain gauge observation, (b) CMORPH retrieval, (c) GPM analysis, (d) CTL, and (e) EXP simulations in China.

      The area-averaged diurnal cycle of the amount, intensity, and frequency of precipitation in CTL and EXP against the surface rain gauge observations in southern China are shown in Figs. 4ac. The precipitation amount, intensity, and frequency measures are defined according to Zhou et al. (2008), in which the intensity and frequency are defined as the mean and percentage of the precipitation rate over 0.1 mm d−1, respectively, and the amount is the accumulated precipitation rates. Surface observations show peak rainfall amount and frequency at about 1600 LST and precipitation intensity peaks between 1400 and 1600 LST. The precipitation amount and intensity in CTL show their maxima at 1200 LST. Even though the frequency peaks at 1700 LST, very high rainfall frequency appears during 1000–1700 LST. The diurnal peak of all three measures is displayed at 1700 LST in EXP by introducing the enhanced dynamical constraint of large-scale advective forcing, resulting in an increasing agreement with the observations (Figs. 4ac). The EXP run tends to reduce the model biases over southern China by suppressing weak convection which occurs too frequently. Figure 4d shows the observed diurnal variation of the probability density function of precipitation. Weak precipitation (1–5 mm d−1) occurs successively during a full 24-h day and the percentage of large precipitation rates (>10 mm d−1) increases in the afternoon. The overlapped high-frequency large precipitation shows a peak of rainfall intensity (Fig. 4a) at 1600 LST. In CTL, the simulation displays a strong probability density function of precipitation at 1200–1400 LST (Fig. 4e) for large and heavy (10–50 mm d−1) rainfall events. The enhanced dynamic convective trigger in EXP suppresses convection of all strengths in a given day (Fig. 4f). A dramatic decrease in rainfall is found between 0800 and 1600 LST, corresponding to the enhanced boundary layer forcing period. Instead, a high probability density function of heavy rainfall is observed during 1200–1700 LST, which effectively delays peak precipitation (Fig. 4f).

      Figure 4.  Area-averaged diurnal cycle of precipitation amount (mm d−1, black solid), intensity (mm d−1, red solid), and frequency (%, blue solid) for (a) rain gauge observations, (b) CTL, and (c) EXP simulations, and the probability density functions (%, shaded) of mean precipitation rate by (d) rain gauge observation, (e) CTL, and (f) EXP in Southern China JJA 2021. In panels (d)–(f), the precipitation rates (1–250 mm d−1) are binned with logarithmic sizes.

      Figure 4 shows the afternoon peaks of the area-averaged amount, frequency, and intensity of precipitation for a rough area selection in Southern China. A detailed investigation of the simulated diurnal variation of precipitation over the coastal region, coast-inland transition, and inland region is also conducted. Based on the distance from the coastline, all grid points in southern China are categorized as either a coastal region, coast–inland transition region, or an inland region, as shown in Fig. 5a. However, the given coastal region in Fig. 5a, does not match the coastline due to the 0.25° coarse resolution. As shown in Jiang et al. (2017) and Chen et al. (2018), the observed precipitation features a propagating mode of peak rainfall from the coastal region toward the inland region (Figs. 5bd). During summer (JJA), the observations exhibit an early-morning sub-peak rainfall at 0700 LST and an afternoon peak at 1400 LST near the coastline. A slightly later peak appears at 1500 LST in the coast-inland transition region and a peak displays at 1600 LST in the inland zone. The model successfully captures the lag of the peak time from the coast to the inland region although biases from observations still exist (Figs. 5bd). A more pronounced reduction is noticed in the midday precipitation amount simulated by EXP, especially in the coastal and transition regions, due to the strict limitation of the convective triggering controlled by the boundary layer forcing.

      Figure 5.  The diurnal variations of area-averaged precipitation amount over the (b) coastal region, (c) coast–inland transition region, and (d) inland region of Southern China during JJA 2021. The geographic extent of the three subregions is given in (a).

      Another interesting place is the Sichuan Basin, where the precipitation is primarily governed by the topography of the Tibetan plateau. The diurnal cycle of the precipitation measures over the Sichuan basin is shown in Fig. 6. Surface rain gauge observations display a peak of precipitation frequency at 0500 LST and a valley at noon, and the rainfall amount and intensity show their peaks at midnight and valleys at noon, which is known as “night-time rainfall” in Sichuan basin. The CTL produces an out-of-phase diurnal cycle of precipitation in comparison with the observations, showing the maximum in the afternoon and the minimum in the evening (Fig. 6b). In EXP, the diurnal cycle of precipitation amount and intensity is significantly improved with peaks at midnight and valleys at 1000 LST with an apparent diurnal variation. The increased precipitation intensity in the afternoon demonstrates better consistency with the observations (Fig. 6a) in EXP than in CTL. The precipitation frequency is reduced in the afternoon in EXP with the enhanced dynamical trigger, as shown in Fig. 6c, even though the diurnal phase of precipitation is not evidently ameliorated. The enhanced threshold of large-scale forcing for convection trigger can further reduce the precipitation frequency, especially in the afternoon, which leads to a reduction of parameterized convective precipitation and the convective-to-total precipitation ratio in numerical models (not shown). It is worth noting that the enhanced dynamical trigger in a cumulus convective parameterization can postpone the convection triggering associated with atmospheric boundary-layer activity but cannot resolve all the problems related to the diurnal cycle simulation of precipitation. Improvement of dynamic and thermodynamic processes related to complex topography is essential for the realistic simulation of diurnal cycle, such as nocturnal precipitation peaks.

      Figure 6.  Area-averaged diurnal cycle of precipitation amount (mm d−1, black solid), intensity (mm d−1, red solid), and frequency (%, blue solid) of (a) rain gauge observations, (b) CTL, and (c) EXP, and the probability density functions (%, shaded) of the mean precipitation rate of rain gauge in the (d) observations, (e) CTL, and (f) EXP in the Sichuan basin for JJA 2021.

      The diurnal variations of CAPE, CAPE generation rate associated with large-scale advective processes, and the 900-hPa horizontal winds in EXP are examined over southern and southwest China (Fig. 7). There are obvious increases in CAPE over land in southern China during 1000 and 1600 LST, and a sudden decrease of CAPE is observed at 1900 LST, which are generally controlled by the surface solar heating (Fig. 7a). Within this interval, the peaks of precipitation amount and frequency are simulated in southern China at 1700 LST (Fig. 4c), which consume a lot of CAPE. In Fig. 7b, the diurnal component is defined as the deviation from the daily mean. The diurnal component of CAPE generation rate due to advective processes shows a reverse diurnal variation to that of CAPE, manifested by a positive rate during nighttime and a negative rate in the daytime (Fig. 7b), which is related to the diurnal variations of the summer monsoon in southern China (Chen et al., 2013). The intense negative diurnal component of the dynamical CAPE generation rate in daytime over southern and southwest China contributes to the decreased precipitation frequency and amount during the surface solar heating hours in EXP compared to CTL (Figs. 46). Even though the diurnal precipitation cycle is improved in EXP (Fig. 6c), large biases exist in the diurnal precipitation cycle. As shown by Chen et al. (2014), the rainfall is strongly associated with the nocturnal CAPE produced by low-level advection processes in the Sichuan basin. This suggests that it is essential to resolve those problems related to the computation of low-level advection over regions of complex topography in a vertical terrain-following coordinate. The steepness of the Tibetan plateau inhibits the accurate mid- and low-level advection computation in the Sichuan basin.

      Figure 7.  The 3-hourly distribution of (a) CAPE (shaded) and 900-hPa horizontal winds (barbs with a full flag of 4 m s–1, green dots filled area of wind speed ≥ 4 m s–1), (b) the diurnal component of CAPE generation rate due to large-scale advective process (shaded) and 900-hPa horizontal winds (barbs with a full flag of 2 m s–1) in EXP during JJA 2021. The diurnal component is estimated as the deviation of the deviation from its daily mean. The gray shading indicates elevations ≥ 1000 m. Line AB denotes the position of the cross-section of Fig. 8.

      Figure 8 shows the vertical cross-section of lower-tropospheric wind vectors in EXP over the Sichuan Basin, along AB in Fig. 7a, in comparison with the ERA5 reanalysis. Similar to observations (Figs. 8a), a strong ascending current is simulated along the slope of the Tibetan Plateau at 1300 LST (Fig. 8c). Unfortunately, an unexpectedly strong downdraft is simulated adjacent to the Tibetan Plateau. In the southeastern Sichuan basin, EXP displays weak ascent in contrast to the observational descent. At 0100 LST, the simulation reproduces the downslope flow along the southeastern flank of the Tibetan Plateau albeit much more intense. A false downdraft remains adjacent to the plateau. Strong and widespread upward motions are shown in the basin (Figs. 8b, d). The ascending current, which is favorable for triggering convection, is weakly simulated in the basin. The biases of vertical velocity and the horizontal wind in the lower troposphere are responsible for the diurnal cycle simulation of rainfall over the Sichuan Basin in EXP.

      Figure 8.  The vertical cross-section of wind vectors of ERA5 (left) and EXP simulation (right) in the lower troposphere along AB in Fig. 7a in the Sichuan Basin. The wind vectors represent the projection of the three-dimensional wind (u, v, 10ω). The vertical component ω is amplified 10 times because it is much smaller than the horizontal ones. The shading indicates the vertical wind speed ω.

      Shifting our focus to tropical South America, Figs. 9ac shows the statistically averaged diurnal cycle of precipitation during JJA 2021 for the CMORPH observations and the CTL and EXP simulations. The mitigation of the simulated bias of the early occurring precipitation maximum is observed over tropical South America. When we zoom in on the region (1°–6°N; 75°–65°W) marked with a rectangular frame in Fig. 9a, a more evident zonal transition of the peak time can be observed in EXP that is in better agreement with the observations, which implies that the westward propagation of convective systems is governed by prevailing easterly winds (Figs. 9b, c). Time-longitude Hovmöller diagram of the mean precipitation rate averaged over 1°–6°N is shown in Figs. 9df to display the propagation of convection. The observations show less intensive and slower propagating convective systems over the western region in comparison with the eastern part, where the core of convective precipitation reaches 22 mm d−1 (Fig. 9d). Compared with CTL, EXP simulates a more consistent rainfall rate and propagation with the observation, especially the relatively weak convective rainfall in the western part (Fig. 9f). The CTL, however, displays a rainfall rate of about 15 mm d−1 at noon, being more potent than the observation.

      Figure 9.  Composite diurnal phase of precipitation (mm d−1) of (a) CMORPH, (b) CTL, and (c) EXP simulations over tropical South America, and a Hovmöller diagram of hourly precipitation (mm d−1) of (d) CMORPH, and (e) CTL and (f) EXP simulations over the region (1°–6°N; 75°–65°W), which is denoted by the rectangle in (a). In (a)–(c), the grids with low amplitude (< 0.1 mm d−1) and slightly low amplitude (0.1–1 mm d−1) are masked out and shown in low saturation colors, respectively.

    • The percentages of binned precipitation rates are used to capture characteristics of various precipitation intensities (Gehne et al., 2016), and are statistically calculated during JJA 2021. The percentages of various precipitation rates are compared in Fig. 10 between observation and simulation in a land area of China (20°–54°N; 100°–135°E) and tropics (20°S–20°N; 180°W–180°E). The percentage, which shows the occurring frequency of different precipitation rates, illustrates a pronounced skew toward drier conditions (Figs. 10b, d), i.e., a smaller magnitude of precipitation taking place more frequently (Sardeshmukh et al., 2015). The simulation displays a similar distribution as the observation, with the occurring percentage decreasing with the intensity. Over the land areas of both China and southern America, the model simulates significantly fewer zero-rainfall events compared with the observations, which indicates an overly lenient convective trigger in the convective parameterization scheme. Both experiments exhibit underestimation of extremely weak- (<1 mm d−1, see Figs. 10a and c) and strong-rainfall-rate events (>50 mm d1 for midlatitude land and >20 mm d−1 for tropical land), and an overestimation of weak and moderate-rainfall-rate events (1–50 mm d−1 on midlatitude land and 1–20 mm d−1 on tropical land) against the surface rain gauge observations in the mid-latitude region and satellite observations in tropics. These deviations are commonly observed in GCM results (Sun et al., 2006; Stephens et al., 2010; Dias et al., 2018). Fortunately, the occurrence of all precipitation rates is clearly improved by using the enhanced dynamical trigger. The increased frequency of extremely weak and strong precipitation and decreased weak and moderate precipitation rates are shown over land areas of China and the tropics in Fig. 10.

      Figure 10.  The probability density functions of precipitation rates over land areas of China (top) and tropics (bottom) during JJA 2021 for CMORPH (gray dot), GPM (black dot), CTL (red dot), and EXP (blue dot). A logarithmic horizontal axis (left) is used for regular precipitation rates of 0.1–250 mm d−1, and a linear horizontal axis (right) is employed to show the wide frequency variation among precipitation rates 0–200 mm d−1.

    • The impact of the convective trigger on precipitation simulation is then measured with the Equitable Threat Score (ETS) and BIAS scores (Ebert et al., 2003), which are defined as:

      and

      where H is the number of correct forecasts for observed occurrences, M is the number of events forecasted not to occur but occurred, F is the number of events forecasted to occur but not occurred, Z is the number of correct forecasts for observed non-occurrences, and Hrandom is used to adjust the H associated with a random chance. Higher quality is implied through either a higher ETS score or a BIAS score being much closer to unity. Figure 11 shows the average ETS and BIAS scores of daily precipitation for the first 3-day forecasts over the land area of China and the tropics during JJA 2021. Although the ETS scores decrease slightly for the 5–25 mm d−1 levels, score boosting is observed in extremely weak (0.1–1 mm d−1) and strong (50–100 mm d−1) levels over land areas in China in EXP. A significant reduction of bias is reflected by the BIAS scores which are much closer to unity at all precipitation levels in EXP (Fig. 8b). Both the ETS and BIAS scores illustrate an effective suppression of weak convection and a promotion of intense convection by an enhanced dynamical trigger. In tropical land areas, the BIAS score also trended closer to unity in Fig. 11d. The ETS, however, displays a lower score in the tropics, which may be related to a mismatch of convective locations in simulation and observation. The location of a specific convective target is much harder to define in the tropics than that of the more systematic convection found in mid-latitude areas.

      Figure 11.  The averaged (left) ETS and (right) BIAS scores of daily precipitation for the first 3-day simulations during JJA 2021 over land areas of China (top) and tropics (bottom) in CTL (red) and EXP (blue) against the surface rain gauge observations in China and GPM in tropics.

    4.   Conclusions and discussion
    • A convective parameterization scheme is indispensable in atmospheric GCMs with relatively low resolution to account for the subgrid-scale precipitation. Trigger and closure assumptions of the cumulus parameterization schemes implicitly decide the onset and intensity of convection in the numerical model. Consequently, the initiation and intensity of the parameterized subgrid-scale precipitation are closely associated with the cumulus parameterization. An improved convective parameterization scheme serves to catch the real diurnal cycle of precipitation.

      The NSAS convective parameterization scheme, using a convective trigger function of large-scale forcing and closure of CWF quasi-equilibrium theory, is implemented into the YUNMA non-hydrostatic atmospheric model to describe subgrid-scale convective effects. The diurnal cycle of terrestrial precipitation is verified at different places with two revised closure schemes (CLOS1_PBL and CLOS2_ADV) and an enhanced trigger threshold (CLOS1_CLI_ADVTRI) in addition to the default NSAS scheme (CTL). In the CLOS1_PBL, the CWF is adjusted to the boundary-layer contribution, and the CWF is determined by only large-scale advection in the CLOS2_ADV without changing the trigger condition. In the CLOS1_CLI_ADVTRI, the trigger of convection is enhanced by directing a CWF tendency threshold of 110 J kg−1 h−1, which makes convective onset more difficult and the convection weak in comparison with the default scheme.

      Approximately the same peak time of precipitation with the revised closure in CLOS1_PBL and the default closure in CLOS1_CLI suggests that the boundary-layer contribution is an effective approach to the CWF climatology over land areas such as the Sichuan basin, tropical North Africa, and the eastern United States. However, in southern China, the Indochina peninsula, and tropical South America, more substantial precipitation in CLOS1_PBL depicts more cloud-base vapor flux than in CLOS1_CLI. The CLOS2_ADV closure assumption significantly postpones the diurnal phase of precipitation over land, but overcorrection is observed in many areas.

      The revision of the trigger function provides a practical way to modulate the convection onset based on the observations artificially. The enhanced trigger function in CLOS1_CLI_ADVTRI using the large-scale advective CWF producing rate as the dynamical constraint is demonstrated to be more effective than the two revised closure schemes in simulating the diurnal cycle of precipitation over land. The enhanced convective trigger with an increased CWF threshold improves the diurnal precipitation pattern by reducing the frequency of convective triggering during active boundary-layer turbulence periods when large surface heat fluxes appear. As a result, a more realistic diurnal phase of the simulated precipitation is achieved over southern China, the Sichuan Basin, the Indochina Peninsula, tropical Africa, and tropical South America with an enhanced dynamical trigger in comparison with the default one. Consequently, the precipitation probability density distribution is also ameliorated with an increase in extremely weak and strong precipitation events and a decrease in weak to moderate precipitation events. Further, the bias in simulated precipitation is reduced over land areas.

      The default triggers and closure assumption in the NSAS parameterization simulate a generally incorrect precipitation peak around early afternoon over land in the CLOS1_CLI run, which suggests that the onset and intensity of convection are primarily forced by the boundary-layer processes. As the illustrations in Donner and Phillips (2003) and Zhang (2002) show, convective processes are significantly related to the CAPE production rate of large-scale advection in the free troposphere. Cumulus parameterizations with a CWF can also be improved with the convective trigger of the large-scale advective process of CWF production. The 3-month batched simulations confirm the efficiency of the enhanced dynamical trigger in CLOS1_CLI_ADVTRI. Elimination of convective onset frequency helps to postpone the early appearance of peak precipitation in the YUNMA model with the NSAS scheme. In the Sichuan basin, however, convective rainfall is significantly related to low-level CAPE production (Chen et al., 2014), thus additional improvements to the closure/trigger are expected with the consideration of the advection computation influenced by the steep topography in a terrain-following coordinate system.

      Acknowledgements. This study was supported by the National Natural Science Foundation of China (Grant Nos. 42375153, 42075151).

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