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GRAPES-REPS, which was developed by the China Meteorological Administration, has 15 members (one control member) and serves the region of China (15°–55°N, 70°–140°E). This system adopts terrain-following coordinates with a 15-km horizontal resolution of 502 × 330 grids and has 51 vertical levels. It runs twice a day (at 0000 and 1200 UTC) out to 72 h of forecast at 6-h intervals. The initial condition uncertainty is represented by a multi-scale blending method based on the Ensemble Transform Kalman Filter approach (Zhang et al., 2015), and the lateral boundary conditions and initial conditions are provided by the different members of the GRAPES global EPS, which is also running operationally at the CMA. The model uncertainty is represented by applying a multiphysics scheme and an SPPT scheme (Wang et al., 2018). In GRAPES-REPS, the multiphysics approach is applied through a combination of two boundary parametrization schemes (MRF and YSU) and four convective cumulus parameterization schemes (New Kain–Fritsch, Old Kain–Fritsch, Betts–Miller–Janjic, and Simplified–Arakawa–Schubert). The domain and topography for the model simulation is shown in Fig. 1, and the regional domain is the same as the verification plots.
We applied the SPP, SPPT, and SKEB schemes to develop and evaluate different combinations of multiple stochastic physics schemes. Five stochastic experiments and one control (CTL) experiment were conducted in GRAPES-REPS (Table 1) for a summer month (1–30 June 2015) over China. Different combinations of stochastic physics schemes represent different aspects of the model uncertainties.
Experiment Initial condition Physical perturbation method CTL Downscaling None SPP Downscaling SPP SPP_SPPT Downscaling SPP, SPPT SPP_SKEB Downscaling SPP, SKEB SPPT_SKEB Downscaling SPPT, SKEB SPP_SPPT_SKEB Downscaling SPP, SPPT, SKEB Table 1. Experiments conducted in this study.
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We selected the 18 key parameters that may have a significant impact on precipitation from the Kain–Fritsch convection (Kain and Fritsch, 1990; Kain, 2004), Medium-Range Forecast Planetary Boundary Layer (MRF-PBL) (Hong and Pan, 1996) and WRF Single-Moment 6-class (WSM6) microphysics (Hong and Lim, 2006) and Monin–Obukhov (Beljaars, 1995) surface layer parameterization schemes. Descriptions and ranges of the parameters selected are presented in Table 2.
Parameter Scheme Default Range Description BRCR MRF 0.5 [0.25, 0.75] Critical Richardson number PFAC MRF 2.0 [1, 3] Profile shape exponent for calculating the momentum diffusivity coefficient KARMAN MRF 0.4 [0.38, 0.42] Von-Kármán constant CFAC MRF 7.8 [3.9, 15.6] Coefficient for Prandtl number XKA Monin–Obukhov 2.4×10−5 [1.2×10−5, 5.0×10−5] The multiplier for heat/moisture exchange coefficient CZO Monin–Obukhov 0.0156 [0.01, 0.026] The Charnock parameter EACRC WSM6 1.0 [0.6, 1] Snow/cloud-water collection efficiency DENG WSM6 500 [300, 700] Density of graupel (kg m−3) XNCR WSM6 3.0×108 [1.0×107, 1.0×109] Maritime cloud content (%) N0R WSM6 8.0×106 [5.0×106, 1.2×107] The intercept parameter of rain (m−4) PEAUT WSM6 0.55 [0.35, 0.85] Collection efficiency for cloud water converting to rain (g/g.s) DIMAX WSM6 5.0×10−4 [3.0×10−4, 8.0×10−4] The limited maximum value for the cloud-ice diameter (m) PD Kain–Fritsch 1.0 [0.5, 2] The multiplier for downdraft mass flux rate PE Kain–Fritsch 1.0 [0.5, 2] The multiplier for entrainment mass flux rate PH Kain–Fritsch 150 [50, 350] The starting height of downdraft above updraft
source layer (hPa)TIMEC Kain–Fritsch 2400 [1800, 3600] Average consumption time of convective available
potential energy (s)W0 Kain–Fritsch 0.75 [0.04, 0.1] The threshold vertical velocity in the trigger function (m s−1) TKEMAX Kain–Fritsch 5.0 [3, 12] The maximum TKE value in sub-cloud layer (m2 s−2) Table 2. The parameters selected in this paper. The identifier of parameters and the scheme that the parameters are in are presented in the first and the second column, respectively. The default values of the parameters are given in the third column, and the fourth column indicates the realistic empirical ranges of the parameters. Finally, definitions of the parameters are provided in the last column.
In the following we briefly explain the motivation for selecting the above parameters. The parameters and their ranges were determined based on the literature (e.g., ECMWF IFS Documentation Part IV: P HYSICAL PROCESSES (CY25R1), 2003; Reynolds et al., 2011; Baker et al., 2014; Johannesson et al., 2014; Di et al., 2015; Mccabe et al., 2016), and following the parameter sensitivity analysis work of Di et al. (2015), Baker et al. (2014) and Johannesson et al. (2014) and consultations with GRAPES physics parameterization experts (J. CHEN, G. XU, Q. LIU, personal communication, 2017).
In the boundary layer parameterization scheme, boundary layer height is defined as the level where the bulk Richardson number reaches the critical value of Richardson number (BRCR) (ECMWF IFS Documentation Part IV: P HYSICAL PROCESSES (CY25R1), 1–174). Furthermore, Hong and Pan (1996) found that convective precipitation is particularly sensitive to the critical Richardson number (BRCR). In addition, the profile shape exponent for calculating the momentum diffusivity coefficient (PFAC) is highly sensitive in the simulation of precipitation because it directly affects the development of convection in the boundary layer (Di et al., 2015). The Von-Kármán constant (KARMAN), which is a constant of the logarithmic wind profile in the surface layer, and the CFAC, which is a coefficient for the Prandtl number at the top of the surface layer, were both closely related to characteristics of momentum, heat, and mass transfer (Reynolds et al., 2011; Di et al., 2015). These parameters were therefore selected for the boundary layer parameterization scheme.
In the surface layer parameterization scheme, Zhang and Anthes (1982) found that the structure of the PBL is highly sensitive to the roughness length. The roughness length is therefore indirectly perturbed through the Charnock parameter (CZO), which determines the magnitude of the wind-speed-dependent roughness length over the oceans and was also selected in the random parameter scheme (Baker et al., 2014). The multiplier for the heat/moisture exchange coefficient (XKA) is both sensitive and significant, as the XKA value predominantly reveals the strength of the flux exchange (Di et al., 2015). The parameters CZO and XKA were therefore selected in the surface layer parameterization scheme.
In the convection parameterization scheme, the most significant and sensitive parameters are the downdraft and entrainment mass flux rates, which represent mixing of the cloud with the environment (Kain, 2004). Yang et al. (2012) and Di et al. (2015) indicated that PD and PE, which are the multipliers for the downdraft and entrainment mass flux rates, respectively, are closely related to the downdraft and entrainment mass flux rates and physically affect the convective process. The starting height of the downdraft above the updraft source layer (PH), which controls the structure of the downdraft, has a significant effect on the convection process (Di et al., 2015). In addition, the mean consumption time of the convective available potential energy (TIMEC) efficiently controls the development of convection and has a considerable impact on convective precipitation (Johannesson et al., 2014; Di et al., 2015). The threshold vertical velocity (W0) in the trigger function is highly sensitive (Kain, 2004; Li et al., 2008) and can be stochastically perturbed for ensemble forecasts (Bright and Mullen, 2002). The intensity of updraft mass flux at the updraft source layer is assumed to be a function of turbulent kinetic energy (TKE) for shallow convection and the maximum turbulent kinetic energy (TKEMAX) has been proven to be significant for convective precipitation (Yang et al., 2012; Di et al., 2015). The parameters PD, PE, PH, TIMEC, W0, and TKEMAX were therefore selected in the convection parameterization scheme.
In the microphysics parameterization scheme, the properties of the scheme are sensitive to the size distribution of ice particles and therefore the intercept parameter (N0R), which directly influences the distribution of the entire range of drop sizes in the exponential distribution of raindrop size, was selected (Hacker et al., 2011; Di et al., 2015). Jiang et al. (2010) and Baker et al. (2014) confirmed the significance for precipitation of the collection efficiency for the conversion of cloud water to rain (PEAUT) and the limited maximum value for the diameter of cloud ice (DIMAX) because these parameters affect the conversion of clouds to rain. Three other uncertain parameters—the snow/cloud water collection efficiency (EACRC), the density of graupel (DENG) and the maritime cloud content (XNCR)—are also selected based on the results of Johannesson et al. (2014) and the suggestions of experts. First, the snow-/cloud-water collection efficiency (EACRC), which represents the ratio of cloud coagulation, and the coagulation between large and small cloud droplets can convert cloud droplets into precipitation, so there is a direct impact on precipitation; second, the density of graupel (DENG), which greatly influences the precipitation efficiency and thereby represents a part of the uncertainty in the heavy rainfall process, is also perturbed; and third, the maritime cloud content (XNCR), which is a multiplier for the automatic conversion rate, is the direct factor of influence in the transformation of cloud water to rain water and thereby has an important effect on precipitation. The parameters N0R, PEAUT, DIMAX, EACRC, DENG, and XNCR were therefore selected in the microphysics scheme.
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The parameters in the SPP scheme are perturbed stochastically and vary within a reasonable range set within the realistic empirically tuned value of each parameter. Temporal and spatial correlations are obtained through a first-order auto-regressive process (Li et al., 2008; Berner et al., 2009). A lognormal distribution (Ollinaho et al., 2017) is used to describe the distribution of the perturbed parameters:
The perturbed and unperturbed parameters are referred to as ξj and
${\hat \xi _j}$ (j = 1, 18), respectively; μ is the mean of the random field and set to μ = 0 for all parameters; and σ is the specified standard deviation of the random field and set to σ = 0.8 for all parameters based on the sensitivity experiments (Xu et al., 2019). The random field${\psi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ , which approximately follows a Gaussian distribution, as in Li et al. (2008), is applied:where the variables λj, ϕj and tj are longitude, latitude and time, respectively. The
${s_j}\left({{\varphi _{\dot J}},\mu } \right)$ is a stretching function. The three-dimensional random field${\varphi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ is defined as:where
${Y_{l,m}}\left({{\lambda _j},{\phi _j}} \right)$ represents the spherical harmonics.${\alpha _{l,m}}\left({{t_j}} \right)$ is the spectral coefficient of the time-related random fields, with l and m equal to the total horizontal and zonal wave numbers, respectively. The evolution of${\alpha _{l,m}}\left({{t_j}} \right)$ is obtained by the first order Markov chain:where
${R_{l,m}}\left({{t_j}} \right)$ follows a Gaussian distribution with a mean of zero and a variance of one. τ and L are the temporal and spatial correlation scales of the random field, respectively. A stretching function${s_j}\left({{\varphi _{\dot J}},\mu } \right)$ is applied to${\varphi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ to obtain the random field${\psi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ in Eq. (2), which ensures that the random field lies within specified bounds (e.g., ψmax and ψmin): The probability density function distribution of the random field
${\psi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ is wider after stretching and no longer perfectly Gaussian. β is set to a constant of −1.27 and ψmax and ψminrepresent the upper and lower boundaries of the random field ${\psi _j}\left({{\lambda _j},{\phi _j},{t_j}} \right)$ , respectively.Note that the different parameters have independent stochastic patterns because they use different random seeds to generate the random number used to initiate the Markov process. The perturbation pattern
$\exp \left({{\psi _j}} \right)$ from a randomly chosen ensemble member and model time step is shown in Fig. 2. In totality this produces a multiplicative perturbation factor$\exp \left({{\psi _j}} \right)$ within specified bounds, which is used to generate the perturbed parameter at each point in time and space.Figure 2. The perturbation pattern
$\exp \left( {{\psi _j}} \right)$ of SPP from a randomly chosen ensemble member and model timestep.The SPP scheme in this study uses the same random field generator as the SPPT scheme, but with larger scale correlation patterns (a spatial correlation scale of L = 20 and a temporal correlation scale of τ = 12 h). We conducted sensitivity experiments on the temporal and spatial decorrelations and standard deviations in GRAPES-REPS and the best results were achieved when choosing spatial decorrelations of L = 20 (i.e., each wave spans approximately 375 km in the latitude over the verified domain), temporal decorrelations of τ = 12 h and standard deviations of σ = 0.8 (Xu et al., 2019). The results indicate that larger perturbation patterns show better skill, which is also argued by Ollinaho et al. (2017). This may be explained by the fact that if the parameter perturbations vary too quickly, then it may be difficult to observe and measure their significant impact. If the parameters vary more slowly, then the perturbations have more time to affect the critical areas of the developing weather phenomenon and therefore a larger impact is observed and measured during the period of the forecast.
The perturbed parameters are confined within strictly specified bounds (between ξmax
and ξmin) to prevent them from attaining physically unrealistic values. If a perturbed parameter falls outside this range then we simply clip it back to the extremal value. -
The SPPT scheme in GRAPE-REPS represents the structural uncertainties associated with the physics parameterizations by perturbing the net tendencies with noise correlated in space and time (Yuan et al., 2016). The net tendency term is referred to as X and
$\hat X$ denotes the perturbed net tendency:The random field
$\psi \left({\lambda,\phi,t} \right)$ was presented in Eq. (2). All the namelist parameter settings of SPPT are the same as their operational configuration. The value of the standard deviation σ of the random field$\varphi \left({\lambda,\phi,t} \right)$ is set to 0.27 and β in Eq. (5) is −1.27. The decorrelation timescale τ of the random field is set to 6 h and the horizontal truncation L is set to 24. The larger perturbation is set within the range [0.2, 1.8] with a mean value of 1.0. The stochastic perturbation parameter options for SPP, SPPT, and SKEB are presented in Table 3.SPP SPPT SKEB Temporal correlation scale τ 12 h 6 h 6 h Spatial correlation scale L 20 24 [50, 100] Standard deviation σ 0.8 0.27 0.27 Specified bounds Varying with parameters [0.2, 1.8] [−0.8, 0.8] Mean of the random field μ 0.0 1.0 0.0 Constant β −1.27 −1.27 −1.27 Table 3. Stochastic perturbation parameter options for SPP, SPPT, and SKEB.
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The SKEB scheme represents the model uncertainty derived from the dissipation of energy by stochastically perturbing the stream function. In the SKEB scheme, the fraction of the dissipated energy that acts as streamfunction forcing for the resolved-scale flow is backscattered upscale in the physical parameterization processes.
The horizontal wind is perturbed stochastically (note that temperature is not perturbed in this study) according to
where su and sv are small-scale forcing terms:
Following Shutts (2005), Fψ is given by:
The random field
$\psi \left({\lambda,\phi,t} \right)$ presented in Eq. (2) and all the realizations of the random field$\psi \left({\lambda,\phi,t} \right)$ lie within the bounds [−0.8, 0.8] with a mean of 0 and a global standard deviation of 0.27. The temporal decorrelation scale τ is set to 6 h in the operational configuration. Because forcing is mainly exerted on small scales, the spectral elements are set within the range [50, 100]. The dissipation rate$\hat D\left({\lambda,\phi,\eta,t} \right)$ is set to 0.03 and the adjustment coefficient a is set to 1.5 in the operation configuration of GRAPES-REPS. These choices are partly similar to the implementation of the Global Environmental Multiscale model at the Meteorological Service of Canada (Charron et al., 2010).