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--> --> -->To understand the performance of convective-scale ensemble forecasts, it is necessary to investigate the forecast error growth (Johnson et al., 2014) of dynamic variables (horizontal wind, temperature, and water vapor mixing ratio) caused by the initial-state errors, which also helps the design of ensemble forecast systems and the explanation of warm-season convection (Melhauser and Zhang, 2012; Chen et al., 2018). For understanding the atmospheric predictability, the nature of the underlying dominant scale-interactions needs to be assessed (Bierdel et al., 2017). Studies of error growth dynamics at the convective scale have shown that moist instability and latent heat release are the predominant mechanisms for promoting rapid nonlinear error growth (Zhang et al., 2003, 2006, 2007; Tan et al., 2004; Hohenegger et al., 2006; Hohenegger and Sch?r, 2007), thereby explaining atmospheric predictability at the mesoscale. In convective-scale forecasts, small-scale and small-amplitude initial errors were found to grow quickly in the convective area, transfer to larger scales (upscale) and subsequently contaminate larger-scale forecasts (Tan et al., 2004; Zhang et al., 2007; Selz and Craig, 2015); this theory is summarized as the three-stage model for atmospheric error growth (Fig. 1). Sun and Zhang (2016) further highlighted butterfly effects (i.e., upscale growth of small-scale initial errors) on both intrinsic and practical limits at the meso and convective scales. On the other hand, larger-scale initial errors generally lead to greater forecast divergence of heavy-rainfall systems (Bei and Zhang, 2007) and dominate forecast accuracy (Sun and Zhang, 2016). The importance of large-scale errors is also confirmed by many other studies (Durran and Gingrich, 2014; Durran and Weyn, 2016; Weyn and Durran, 2017). Considering the presence of flow-dependent multiscale perturbations derived from ensemble data assimilation, Johnson and Wang (2016) found that the small-scale perturbations could influence precipitation forecasts after an approximate 5-h lead time. Overall, the scale interaction in terms of error growth within convective-scale forecasts requires further examination (Bierdel et al., 2017; Bachmann et al., 2019, 2020; Selz et al., 2019; Zhang, 2019).
Figure1. Summary of the three-stage conceptual error growth model [Reprinted from (Zhang et al., 2007)].The initial errors affect the precipitation forecast error, which in turn influences the error growth through moisture dynamics (Bei and Zhang, 2014; Klasa et al., 2019; Zhang, 2019). Nielsen and Schumacher (2016) identified the need to investigate the evolution of precipitation uncertainty due to forecast error growth. To this end, a spectral metric to assess the spatial predictability of precipitation was proposed; namely, the decorrelation scale (Surcel et al., 2015). Bachmann et al. (2019, 2020) employed the decorrelation scale to measure the predictability limit with respect to the effects of terrain and radar data assimilation. Wu et al. (2020) used the power ratio of the decorrelation scale to quantify the scale-dependent impact of initial errors on the precipitation forecast in a warm-sector heavy rainfall event. In short, the metric provides a quantitative way to establish and measure the relationship between the error growth and precipitation forecast.
The error growth of dynamic variable and precipitation forecast uncertainties measures the predictability of convective events and depends on the strengths of large-scale forcing, which is generally categorized into strong- and weak-forcing regimes (Done et al., 2006; Keil and Craig, 2011; Done et al., 2012; Flack et al., 2016; Keil et al., 2019). General results show that for events dominated by strong large-scale forcing, the large-scale initial errors play a more important role than that controlled by local instabilities (Johnson et al., 2014; Surcel et al., 2015). In a more recent study, Selz et al. (2019) highlighted the importance of identifying the up- or downscale impact of associated errors across scales on mesoscale flow for different convective regimes.
Although convective-scale predictability is believed to be highly regime-dependent, there have been few related studies on it, particularly for different convective regimes in the YHRB (Luo and Chen, 2015; Chen et al., 2018). Many studies have revealed the complexity of warm-season extreme rainfall events in this area (Zhao et al., 2007; Sun and Zhang, 2012; Zhang and Zhang, 2012; Fu et al., 2013; Luo et al., 2013; Zhang et al., 2019), and it is therefore necessary to investigate the error growth (defined as ensemble variability) dynamics and associated impact on precipitation forecasts (precipitation uncertainties) in both strong- and weak-forcing regimes for the YHRB. Furthermore, as our ultimate goal is to construct an optimal convective-scale ensemble forecast system for the YHRB region, the sensitivity of errors at different scales is also of interest (Weyn and Durran, 2019), especially for the realistic flow-dependent errors derived from an ensemble data assimilation system (Johnson and Wang, 2016; Flora et al., 2018). What is the evolution mechanism of errors at different scales within convective-scale forecasts, and how do they interact with each other to regulate the spread performance and further impact the predictability of convective weather? Although these issues are of great worth to ensemble design, few studies have focused on them.
To deal with the above concerns, we design several ensembles of convective-scale simulations using an Observation System Simulation Experiment (OSSE) setup to investigate the regime-dependent error growth for 13 warm-season convective events. The objectives of this study are: (1) to explore the error growth characteristics within 12-h forecast range in both strong- and weak-forcing regimes over the YHRB and identify possible differences between the two subsets; (2) to understand the impact of error growth on precipitation forecasts across spatial scales; and (3) to investigate the sensitivity of lateral boundary and realistic flow-dependent initial condition (IC) errors at different scales. The results of this investigation provide insights into understanding the impacts of realistic flow-dependent perturbations implemented in convective-scale ensemble forecasts, particularly those related to under-dispersion issues in current convective-scale ensemble forecasts (Tennant, 2015; Sun and Zhang, 2016), and offer further suggestions for the improvement of an optimal convective-scale ensemble forecast for the YHRB.
The rest of this paper is organized as follows: The model and ensemble design approaches are described in section 2, along with details of analysis methods and the cases examined. Results from synthetic analysis and typical case studies are presented in sections 3 and 4. Section 5 provides a summary and discussion.
2.1. Model configuration
The Advanced Research core of the Weather Research and Forecasting (WRF) model, version 3.7.1 (Skamarock et al., 2008), is utilized as the numerical model. A one-way nested framework is designed with 180 × 180 and 258 × 258 horizontal grid points for 18- and 3-km grid spacing, respectively (Fig. 2). There are 41 terrain-following hydrostatic-pressure vertical levels and a model top of 10 hPa in both outer and inner domains.
Figure2. (a) Model domain configuration: the black square indicates the outer domain (18-km resolution); the red square indicates the inner domain (3-km resolution); black dots show the distribution of simulated sounding datasets; blue circles show locations of the seven radar sites. (b) Terrain height within the inner domain, the Dabie Mountains (31°N, 116°E), Huang Mountains (30°N, 117.5°E), and Mu-fu Mountains (29°N, 115°E).The ICs and lateral boundary conditions (LBCs) are derived from the U.S. National Centers for Environmental Prediction Global Forecast System (GFS) operational analysis data. The physical parameterization schemes include the WSM6 microphysics scheme (Hong and Lim, 2006), Yonsei University boundary layer scheme (Hong et al., 2006), RRTM longwave radiation scheme (Mlawer et al., 1997), and Goddard shortwave radiation scheme (Chou and Suarez, 1999). Additionally, the Grell-3 cumulus scheme (Grell and Dévényi, 2002) is used in the inner domain.
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2.2. OSSE framework and ensemble generation
Following Zhuang et al. (2020), an ensemble data assimilation and forecast system (Wang et al., 2013) is constructed (Fig. 3) to generate flow-dependent IC and LBC perturbations. The Ensemble Square Root Filter approach proposed by Whitaker and Hamill (2002) is employed as the data assimilation method. For the present study, the system is modified to run in an OSSE setup to remove model uncertainties (Yussouf and Stensrud, 2012; Gasperoni et al., 2013; Johnson and Wang, 2016; Madaus and Hakim, 2017).
Figure3. Schematic of the ensemble data assimilation configuration. The outer-domain ensemble data assimilation is initialized at 0000 or 1200 UTC for each case, while the inner domain ensemble data assimilation is initialized 21 h after the outer domain initialization using the outer-nest ensemble for both the initial and boundary conditions. “Obs” indicates observations.To create a “true” atmospheric state, we first perform a 36-h nature run initialized from the GFS analysis 24 h before the analysis time for each case with a 3-km resolution covering the outer domain (Fig. 2a). The mesoscale sounding datasets are then generated at randomly chosen locations within China’s mainland (Fig. 2a) and interpolated from the “true” run with observation errors of 2.5 m s?1, 1.2 K, and 0.005 kg kg?1 for wind velocity, temperature, and water vapor, respectively (Snook et al., 2015). For the inner domain, the “true” run is interpolated to seven real CINRAD-SA radars (Zhu and Zhu, 2004) (Fig. 2b) with observation errors of 2 m s?1 and 5 dBZ for radial velocity and reflectivity (Johnson et al., 2015).
We then generate the initial ensemble in the outer domain, by adding analysis perturbations from the first 30-member European Centre for Medium-Range Weather Forecasts global ensemble prediction products (Hagedorn et al., 2008; Hagedorn et al., 2012) to temperature, horizontal wind, and water vapor mixing ratio in the ICs and LBCs. After that, the mesoscale sounding data assimilation for the outer domain is initialized at 0000 or 1200 UTC (0008 or 2000 LST) for each case (Table 1) over a 24-h period with 3-h intervals. For the inner domain, the initial ensemble is obtained by downscaling the analysis states at the seventh assimilation cycle in the outer domain using the WRF “ndown” tool (Daniels et al., 2016). The radar data (including reflectivity and radial velocity) are then assimilated for 3-h with 10-min intervals in the inner domain, while the outer domain provides larger-scale LBC perturbations every 15-min during the assimilation cycles.
| Number | Start time | Subset |
| 1 | 0000 UTC 23 June 2013 | Weak forcing |
| 2 | 0000 UTC 5 July 2013 | Weak forcing |
| 3 | 1200 UTC 6 July 2013 | Strong forcing |
| 4 | 0000 UTC 7 July 2013 | Weak forcing |
| 5 | 0000 UTC 21 July 2013 | Weak forcing |
| 6 | 0000 UTC 22 July 2013 | Weak forcing |
| 7 | 0000 UTC 1 June 2014 | Strong forcing |
| 8 | 0000 UTC 15 June 2014 | Strong forcing |
| 9 | 0000 UTC 25 June 2014 | Strong forcing |
| 10 | 0000 UTC 26 June 2014 | Weak forcing |
| 11 | 1200 UTC 1 July 2014 | Strong forcing |
| 12 | 0000 UTC 12 July 2014 | Weak forcing |
| 13 | 1200 UTC 24 July 2014 | Strong forcing |
Table1. Summary of cases.
The horizontal and vertical covariance localization radii for the outer domain are 120 km and 6 km, and the corresponding values in the inner domain are 20 and 5 km (Wang et al., 2013; Snook et al., 2015), respectively. To maintain spread within the assimilation cycles, a multiplicative covariance inflation of 1.15 (Anderson and Anderson, 1999; Tong and Xue, 2005) and a relaxation inflation of 0.5 to prior ensemble variability (Zhang et al., 2004) are applied to the outer and inner domains.
The control experiment (CTRL) uses the first 20 members of the ensemble analysis in the inner domain and the LBC perturbations are provided by the outer domain ensemble forecasts and perform 12-h convective-scale ensemble forecasts (Fig. 3). Several sensitivity experiments (IC_MULTI, IC_SMALL, IC_TRANS, and IC_LARGE) are conducted to further clarify the scale-based sensitivity of errors. These experiments employ the same LBCs from the ensemble mean analysis of the outer domain (no LBC perturbations) and differ in initial perturbations that are constructed in three steps: (i) subtract the ensemble mean from each analysis state to obtain intermediate perturbations; (ii) scale each perturbed variable by retaining information at determined spatial scales; and (iii) add the scaled perturbations back to the ensemble mean (Table 2). Specifically, IC_MULTI uses the same IC perturbations as CTRL with LBC perturbations excluded to investigate the sensitivity of larger-scale LBC errors. By retaining the small-scale components of the flow-dependent IC perturbations, the up-amplitude and upscale processes of small-scale initial errors and the associated impact on precipitation can be isolated and examined via IC_SMALL. By retaining the larger-scale components of the IC perturbations, the evolution and the associated impact of larger-scale initial errors on precipitation can be isolated and examined via IC_LARGE. The definition of scale range is specified in section 2.3.
| Experiment | IC perturbations | LBC perturbations |
| CTRL | Flow-dependent_3 km | Flow-dependent_18 km |
| IC_MULTI | Flow-dependent_3 km | None |
| IC_SMALL | Filtered information < 36 km from Flow-dependent_3 km | None |
| IC_TRANS | Filtered in formation > 36 km and < 120 km from Flow-dependent_3 km | None |
| IC_LARGE | Filtered information > 120 km from Flow-dependent_3 km | None |
Table2. Summary of IC perturbations and LBC perturbations used in the ensemble experiments.
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2.3. Representation of scale-dependent error growth
The forecast error is quantified using the difference total energy (DTE; Zhang et al., 2007):where
To measure the scale-dependent error growth, we classify the spatial scale into three scale ranges (Zhuang et al., 2020): small scale (36 km ≥ wavelength); transition scale (120 km ≥ wavelength > 36 km); and larger scale (wavelength > 120 km). To obtain variable fields at different scales, the discrete cosine transform (DCT) ( Denis et al., 2002; Surcel et al., 2015; Wu et al., 2020) method is used. Compared to traditional Fourier transform methods, DCT is able to avoid discontinuity problems at domain boundaries.
We also introduce a vertical function (Nielsen and Schumacher, 2016) to calculate the two-dimensional root-mean vertically integrated DTE (RMDTE). To compare the error growth between different convective regimes, the normalized root-mean vertically integrated DTE (NRMDTE) (Nielsen and Schumacher, 2016; Klasa et al., 2019) at different scales is calculated to eliminate the variability independent of the magnitude of the total background flow (Appendix A).
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2.4. Representation of spatial precipitation uncertainty
To evaluate spatial precipitation uncertainties, the decorrelation scale method proposed by Surcel et al. (2015) is applied. As described in that study, complete decorrelation of ensemble forecasts can be considered as a lack of precipitation predictability by the ensemble at a given scale λ0. For the scale λ ≤ λ0, there is no precipitation predictability. For the scale λ ≥ λ0, the precipitation fields of ensemble members are correlated, indicative of some predictability. The power ratio is used to quantitatively assess the precipitation uncertainty (Wu et al., 2020):where
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2.5. Case selection
Thirteen mei-yu-season heavy-rainfall events (June and July 2013–2014) (Liu et al., 2012; Sun and Zhang, 2012; Luo and Chen, 2015) over the YHRB are selected, ranging from local self-organized convective events to synoptically driven mei-yu-front events. Based on the convective adjustment timescale
Figure 4a shows the evolution of
Figure4. The (a) convective-adjustment time scale 
Figure5. (a, b) Ensemble-mean 850-hPa wind vector (units: m s?1) and equivalent potential temperature (blue solid contours lines, 344–348 K, 2 K interval, indicating the location of the mei-yu front) averaged over cases for the (a) strong- and (b) weak-forcing regime calculated from the outer domain ensemble. The red boxes indicate the inner domain. (c, d) Precipitation frequency computed from the observed hourly precipitation exceeding 0.5 mm h?1 for the (c) strong- and (d) weak-forcing regime. The black triangles indicate the mountains shown in Fig. 2.2
3.1. Spatiotemporal characteristics of total error growth
Figure 6 shows the temporal evolution of total NRMDTEs for both strong-forcing (Fig. 6a) and weak-forcing (Fig. 6e) regimes. In the strong-forcing regime, the mean total NRMDTE manifests a moderate evolution, while in the weak-forcing regime it shows a larger variation with a maximum in the afternoon (4–8 h), the variation of which is in accordance with the daytime phase diurnal peak of convection in the YHRB (Sun and Zhang, 2012).
Figure6. Time series of the 12-h NRMDTE of CTRL at different scales for (a–d) strong- and (e–h) weak-forcing cases: (a, e) total NRMDTE; (b, f) small-scale NRMDTE; (c, g) transition-scale NRMDTE; (d, h) large-scale NRMDTE. The thick black curves represent the NRMDTEs averaged over cases of each subset while the dashed gray curves represent the NRMDTEs for each case. S1–S3 indicates the error growth stages for each subset.Figures 7 and 8 give the total NRMDTEs and ensemble mean precipitation (time–longitude and time–latitude diagrams) averaged over cases in each subset. The events under strong forcing exhibit a strong easterly progression (Fig. 7a, Fig. 8a), while a relatively stationary feature for both variables appears in the weak-forcing regime (Fig. 7e and Fig. 8e). These results indicate that forecast errors can be transported to a broad area covered by the influence of a strong frontal system during the strong-forcing regime, but the errors are often restricted to a smaller area in the weak-forcing regime. In addition, the NRMDTE (Fig. 8e) increases with forecast lead time toward lower latitudes in the weak-forcing regime. It is speculated that this southward motion is associated with the small-scale convection (e.g., isolated storms, squall lines, or multicell storms) triggered by local factors in the southern region. Overall, these results indicate that forecast errors often closely follow the precipitation system.
Figure7. Hovm?ller (meridionally averaged mean NRMDTE at different scales, time–longitude) diagrams for the NRMDTE of CTRL averaged over cases for the (a–d) strong- and (e–h) weak-forcing regime: (a, e) total NRMDTE; (b, f) small-scale NRMDTE; (c, g) transition-scale NRMDTE; (d, h) larger-scale NRMDTE. Black contours (0.2–1.6 mm, contoured every 0.2 mm) indicate the ensemble mean 0.5 h accumulated precipitation and the shading denotes regions with precipitation exceeding 0.6 mm.
Figure8. As in Fig. 7 but for zonally averaged NRMDTE averaged over cases in each subset.2
3.2. Spatiotemporal characteristics of error growth at different scales
We further analyze error growth at and across spatial scales. In general, the trends of NRMDTEs for different cases are generally consistent with each other for each subset (gray lines in Fig. 6), confirming our classification of cases. Similar to Zhang et al. (2007), we conclude the scale-dependent error growth into three stages for the strong- and weak-forcing subsets, respectively. In the strong-forcing regime, a “stepwise” increase with increasing spatial scale that is similar to the three-stage conceptual model proposed in Zhang et al. (2007) is observed. Stage 1 corresponds to rapid up-amplitude growth and saturation that is defined as the point at which the NRMDTE reaches its maximum value (Zhang, 2019) of small-scale errors around the first 2 h (Figs. 6b, 7b and 8b). This result indicates the rapid predictability loss at the relevant scales (Lorenz, 1969; Judt, 2018). At Stage 2, transition-scale NRMDTE remains increasing and slightly spread beyond the convective zones (Figs. 7c and 8c), which is attributable to an upscale transfer of small-scale errors. During Stage 3, transition-scale NRMDTE saturates at around 6 h while larger-scale NRMDTE remains a slight increase (Figs. 6d, 7d and 8d), which may gradually influence mesoscale predictability from the initial higher-latitude areas where the mei-yu front formed, demonstrating an interaction between the convective and larger scales (Nielsen and Schumacher, 2016).In the weak-forcing regime, small-scale errors (Figs. 6f, 7f and 8f) experience a rapid up-amplitude growth similar to that under strong forcing during the first 2 h (Stage 1) but remain a steeper and longer increase during 2–8 h (Stage 2), demonstrating lower predictability than that under strong forcing (Keil et al., 2014; Zhuang et al., 2019). Transition-scale errors (Figs. 6g, 7g and 8g) experience an evolution similar to but slightly smoother than that of small-scale errors during the same period. Such evolutions of the NRMDTEs for the small and transition scales are likely driven by the convection diurnal peak, in which convective events usually strengthen during the afternoon due to solar heating. After 8 h (Stage 3), small- and transition-scale NRMDTEs both exhibit reduction, albeit with a slight increase in larger-scale NRMDTE (Figs. 6h, 7h and 8h), reflecting the relationship between the moist convection and error growth is weaker at the larger scale.
Overall, the above findings confirm that upscale error growth (Zhang et al., 2007; Selz and Craig, 2015) is evident over the YHRB, but more obvious for the strong-forcing regime. Meanwhile, for the weak-forcing regime, the error growth is largely modulated by the convection afternoon peak (Nielsen and Schumacher, 2016; Klasa et al., 2019; Wu et al., 2020). This result highlights a strong regime-dependent feature. In addition, we also find the peak of forecast errors in the weak-forcing regime is most prominent for the small and transition scales. As the diurnal cycle of convection can be regarded as “external forcing” from a quasi-stationary terrain or solar forcing, the peak (or other big fluctuations) of error growth for the weak-forcing regime should be regarded as added value superimposed on the basic error growth emerging from the strong moist convection (Nielsen and Schumacher, 2016) and explain the scale-dependent error growth.
Note that larger-scale NRMDTEs are slightly damped during the first few hours in both regimes (Figs. 6d and h). This may be due to the poor representation of synoptic- and subsynoptic-scale information using small-domain convective-scale ensemble forecasts. A similar phenomenon can also be found in previous studies (Bei and Zhang, 2007; Zhang, 2019). A comparison of the NRMDTEs with those from a larger-domain experiment (not shown) reveals that the relatively small domains and the terrain along the southern boundary do not influence the 12-h evolution trend of forecast error. However, the terrain at the boundary does have a slight negative impact on the up-amplitude error growth at the small-scale, particularly for the terrain-forcing events under weak forcing, which requires further evaluation beyond the scope of this study.
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3.3. Impact of error growth on precipitation
A novelty of this study is a systematic examination of the impact of error growth on the precipitation forecast. Previous studies have revealed a strong correlation between error growth and precipitation (Johnson et al., 2014; Flack et al., 2018), but they did not consider this issue in terms of spatial scales.Figure 9 shows the power ratio [Eq. (2), hereafter referred to as R for brevity] averaged over cases within each subset at different lead times, which is used to quantitatively assess the precipitation uncertainty. In Stage 1 (0.5 h, black curves) for both strong- and weak-forcing regimes, the rapid increase of the flow-dependent small-scale errors (Figs. 6b and f) leads to increased R originated from small scales. The distribution of R decreases monotonically with increasing scale, revealing an approximately positive relationship between error growth and precipitation uncertainties in the small scale.
Figure9. Temporal evolution of the power ratio (R) for 0.5 h accumulated precipitation calculated from CTRL averaged over cases for the (a) strong- and (b) weak-forcing regime. The red reference line represents total loss of precipitation predictability. The dashed lines correspond to Stage 3 of error growth.A divergence occurs between the strong- and weak-forcing regimes at the start of Stage 2 (2 h, blue curves). For the strong-forcing regime, R features fluctuations on scales around 20 km, with the distribution peak shifting to the larger scales until approximately 6 h (the end of Stage 2), implying a loss of predictability at around 60 km (Fig. 9a). This indicates the impact of upscale error growth on precipitation at the meso-β-scale (Fig. 9a) and can be explained by the rapid displacement of individual cells driven by large-scale forced ascent (Done et al., 2006; Zhang et al., 2007; Liu and Tan, 2009; Flack et al., 2018) around the mei-yu front. After 6 h (Stage 3), when the NRMDTEs generally reach saturations at both the small and transition scales (Figs. 6b and c), R continues to increase as a result of the up-amplitude growth of larger-scale errors (Fig. 6d) with the distribution peak remaining unchanged.
For the weak-forcing regime, with the consistent, up-amplitude growth of small- and transition-scale NRMDTEs (Figs. 6f and g) at Stage 2 (2–8 h), R retains a monotonic distribution with the peak centered at the grid scales (Fig. 9b), implying a thorough predictability loss at the meso-γ-scale as a result of the localized error growth (Fig. 7e). After 8 h, the small- and transition-scale NRMDTEs become saturated (Figs. 6f and g) and R displays increase at all scales, particularly at the larger scales, corresponding to the subsequent upscale error growth.
Overall, these results demonstrate a strong correlation between error growth and precipitation in a scale-dependent manner. Specifically, the up-amplitude error growth increases the magnitude of precipitation uncertainty while the upscale error transfer before saturation at a given scale influences its spatial distribution. These results also demonstrate how multiscale initial perturbations impact precipitation at different scales, particularly under various large-scale forcings.
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4.1. Role of larger-scale lateral boundary errors
In convective-scale ensemble forecasts, LBC perturbations generally exert larger-scale errors to the inner domain from LBCs in a one-way nested configuration (Nutter et al., 2004) and dominate the ensemble variability after the first 6–12 h (Vié et al., 2011; Kühnlein et al., 2014). Figure 10 shows a comparison of CTRL (black curves) and IC_MULTI (gray curves), which reveals that the LBC errors exert impacts on the error growth at the transition and larger scales for both subsets. Such impacts can also be seen from the comparison between CTRL and IC_MULTI in terms of a spatial view as shown in Fig. 8 and Fig. 11. In addition, the relative difference of total NRMDTE between IC_MULTI and CTRL shows a quasi-linear increase with time and reaches 30% at 12 h, reflecting the results from Zhang (2019), which is primarily attributable to the larger-scale component (Figs.10d, h and Fig. 11d, h).
Figure10. Time series of 12-h mean NRMDTE at different scales of CTRL (black curves), IC_MULTI (gray curves), IC_SMALL (blue curves), IC_TRANS (red curves), and IC_LARGE (yellow curves) averaged over cases for the (a–d) strong- and (e–h) weak-forcing regime: (a, e) total NRMDTE; (b, f) small-scale NRMDTE; (c, g) transition-scale NRMDTE; (d, h) larger-scale NRMDTE. S1–S3 indicate the error growth stages for each subset.
Figure11. Zonally averaged Hovm?ller (time–latitude) diagrams for the NRMDTE of IC_MULTI averaged over cases for the (a–d) strong-forcing and (e–h) weak-forcing regime: (a, e) total NRMDTE; (b, f) small-scale NRMDTE; (c, g) transition-scale NRMDTE; (d, h) larger-scale NRMDTE. Black contours (0.2–1.6 mm, contoured every 0.2 mm) indicate the corresponding ensemble mean 0.5 h accumulated precipitation and the shading denotes regions with precipitation exceeding 0.6 mm.Regarding the impact of LBC errors on precipitation uncertainty, we find a clear decay of R during the strong-forcing regime (Fig. 12a) after 4 h. This finding demonstrates a practical predictability limit from LBC errors, as R decays in both amplitude and spatial scale of the distribution peak in IC_MULTI after 4 h, revealing under-dispersion issues compared with CTRL and the need to improve LBC perturbations for convective-scale ensemble forecasts. This feature can also be identified from 4–6 h in Fig. 10c, as the transition-scale NRMDTEs for IC_TRANS, IC_LARGE, and IC_MULTI all experience decays, while larger-scale NRMDTEs cease increasing afterward (Fig. 10d). Accordingly, this time period can also be regarded as a bifurcation point at which transition-scale errors can either reach a theoretical limit or continue to grow upscale to larger scale and impact precipitation at the meso-β-scale. Comparably, though R in IC_MULTI also exhibits a decay relative to CTRL under the weak-forcing regime (Fig. 12b), it nevertheless continues to increase with time. In short, these results support the hypothesis that error growth and its associated impact on precipitation can be constrained by the large-scale control imposed by the LBCs (Nielsen and Schumacher, 2016; Klasa et al., 2019), particularly for the strong-forcing regime.
Figure12. Temporal evolution of the power ratio (R) for 0.5 h accumulated precipitation of (a, b) IC_MULTI, (c, d) IC_SMALL and (e, f) IC_LARGE averaged over cases for the (a, c, e) strong- and (b, d, f) weak-forcing regime. The red reference line represents total loss of precipitation predictability. The dashed lines indicate Stage 3 when larger-scale errors become dominant.2
4.2. Role of small-scale initial errors
Small-scale initial errors can grow up-amplitude and upscale predominated by moist convection (Hohenegger et al., 2006; Zhang et al., 2007; Selz and Craig, 2015). Figure 13 shows that the large NRMDTE values at the transition scale (Figs. 13c and g) and larger scale (Figs. 13d and h) appear later than those of the small scale (Figs. 13b and f) for IC_SMALL, indicative of the upscale transfer. In addition, the total NRMDTE of IC_SMALL (blue curves in Figs. 10a and e) is substantially lower than that of IC_MULTI (gray curves in Figs. 10a and e), with reduced magnitude of mean small-scale NRMDTE at Stage 1 (Figs. 10b and f) and subsequent stages through a weaker upscale transfer. However, for both convective regimes, we find the evolution trend [i.e., the stepwise feature under strong forcing (Figs. 10b–d) and the afternoon maximum under weak forcing (Figs. 10f–h)] of IC_SMALL are similar to those of IC_MULTI. This suggests that the magnitude of the forecast error is mainly dominated by large-scale flow but moist convection determines the trend (Nielsen and Schumacher, 2016).
Figure13. As in Fig. 11 but for IC_SMALL.Regarding precipitation uncertainties, initial small-scale errors produce R within the meso-γ-scale and gradually extend to larger scales (Figs. 12c and d). Specifically, in the strong-forcing regime, the distribution peak of R does not shift to a meso-β-scale within 2–4 h (Fig. 12c) compared with IC_MULTI (Fig. 12a) and is followed by a clear decay after 4 h. Inconsistent with the strong-forcing regime, R exhibits a slight increase between 2 and 8 h (blue to yellow curves in Fig. 12d) and remains large afterward as a result of consistent up-amplitude growth of small-scale NRMDTE (Fig. 10f) in the weak-forcing regime. In summary, initial small-scale errors can gradually grow up-amplitude and upscale, causing a dispersion within ensemble members during the weak-forcing regime, reflecting greater sensitivity than in the strong-forcing regime (Weyn and Durran, 2019).
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4.3. Role of larger-scale initial errors
Flow-dependent larger-scale initial errors generally correspond to uncertainties of the larger-scale systems (e.g., mei-yu front). The total NRMDTE in IC_LARGE shows similar temporal (Figs. 10a and e) and spatial (Figs. 14a and e) features with IC_MULTI (Figs. 10a, e and Figs. 11a, e) after 4 h. A further comparison of NRMDTEs between IC_LARGE and IC_MULTI at different scales reveals that, even in the absence of small-scale errors in the initial state, the small-scale component is still rapidly generated (Figs. 10b and f; Figs. 14b and f) and then transferred to the larger scale (Figs. 10d and h; Figs. 14d and h), corresponding to the downscale cascade as proposed by Durran and Gingrich (2014). Additionally, we can also see the total NRMDTE in IC_LARGE is obviously larger than that of IC_SMALL for both regimes (Figs. 10a and e). These results highlight that larger-scale initial errors generally control the error growth after 4 h while the errors at smaller scales are only effective at early lead times.
Figure14. As in Fig. 11 but for IC_LARGE.Figures 12 e–h display the precipitation uncertainties in IC_LARGE. At 0.5 h, R in IC_LARGE is lower than that in IC_SMALL, revealing that larger-scale initial errors do not immediately impact the precipitation. At later times, R in IC_LARGE (Figs. 12e and f) is found to be similar to that of IC_MULTI (Figs. 12a and b) for both subsets, indicating that the limit of practical predictability of precipitation can be extended by reducing the initial errors at the larger scale and emphasizing the necessity and importance of larger-scale perturbations in ensemble design (Johnson and Wang, 2016; Surcel et al., 2016). Specifically, R in the strong-forcing regime displays a clear change in distribution during 2–4 h (blue and cyan curves in Fig. 12e) at scales < 40 km and decays afterward. By contrast, R keeps a consistent increase during the entire forecast range with a monotonical distribution in the weak-forcing regime, which relates to the local instability under unstable environments (i.e., higher CAPE) that stabilizes the spatial error propagation and destroys mesoscale predictability eventually.
Additionally, the scale-based experiments shown in Fig. 10 can also be regarded as “reduced” experiments (Nielsen and Schumacher, 2016; Zhang et al., 2016; Flora et al., 2018) that can be used to detect practical versus intrinsic predictability by examining whether the ratio for NRMDTE evolution of a “reduced” ensemble over IC_MULTI remains constant or rapidly converges to one. In this study covering the YHRB, a converging tendency within ensemble experiments in both strong- and weak-forcing regimes is found (Fig. 10) at the small and transition scales, suggesting intrinsic predictability limits at relevant scales even within a 12-h forecast range. This result corroborates the conclusion by Nielsen and Schumacher (2016) that the magnitude of error growth is governed by the interaction of moist convection with the large-scale flow.
We designed an OSSE-based ensemble data assimilation and forecast system to produce realistic flow-dependent IC/LBC perturbations and initialize convective-scale ensemble forecast within a 12 h forecast range (CTRL), applied the convective adjustment timescale (Done et al., 2006) to differentiate convective regimes controlled by different large-scale forcings, compared spatiotemporal error growth based on NRMDTE at different scales across convective regimes, and applied the power ratio (R) of the decorrelation scale to measure the precipitation uncertainty in terms of spatial scale. The sensitivities of LBC and IC errors at different scales were further assessed using several scale-based ensemble sensitivity experiments (IC_MULTI, IC_SMALL, IC_TRANS, and IC_LARGE).
The investigation of CTRL revealed that the error growth dynamics are highly regime-dependent, leading to varying precipitation uncertainties within the 12-h forecast range. In general, the weak-forcing events (characterized by weaker large-scale advection and higher CAPE) are less predictable, a conclusion consistent with those of previous work (Done et al., 2012; Keil et al., 2014; Klasa et al., 2019). Specifically, the forecast error under strong forcing exhibits a stepwise feature similar to that seen in Zhang et al. (2007), while the error growth under weak forcing exhibits larger variability with a peak that corresponds to the solar-forced daytime peak phase for mei-yu-season convection over the YHRB (Sun and Zhang, 2012). The corresponding assessment of precipitation uncertainties shows a strong relationship between precipitation and error growth: the up-amplitude growth of NRMDTE determines the magnitude of R at a given scale, while the upscale growth of NRMDTE before saturation influences the distribution of R across scales during the same period.
A comparison of CTRL and IC_MULTI reveals that LBC errors impose a quasi-linear increase in the magnitude of NRMDTE in both regimes, indicating that the magnitude of the forecast error is primarily driven by the large-scale flow. Specifically, in the absence of LBC errors, the associated R under strong forcing shows a consistent decay within 4 h, revealing the importance of LBC perturbation in the ensemble design, especially for the strongly forced mei-yu frontal events.
The evaluations of IC_SMALL and IC_LARGE demonstrate that scale interaction, including upscale transfer of small-scale errors and downscale cascade of larger-scale errors, is evident in both convective regimes. Generally, small-scale IC errors are effective during the first 4 h and cause precipitation uncertainties at the meso-γ-scale, while the larger-scale IC errors are responsible for the magnitude of forecast error during the subsequent forecast range and produce precipitation uncertainties at the meso-β-scale. This result indicates that the limit of practical predictability can apparently be extended by reducing the initial errors at larger-scale (Sun and Zhang, 2016) and highlights the relative importance of initial perturbations at larger scales in ensemble design (Johnson et al., 2014; Surcel et al., 2017). Thus, the scale-blending method (Caron, 2013; Wang et al., 2014) can be employed in convective-scale ensemble forecast systems to introduce larger-scale perturbations produced by a coarser resolution ensemble forecast system that better sample the uncertainties of synoptic weather systems. We also find that small-scale IC errors are more sensitive in the weak-forcing regime than that in the strong-forcing regime. To improve the ensemble design, the model resolution can be enhanced to allow more error growth at small scales when local instability plays more important roles.
This study focuses on the practical predictability within convective-scale ensemble forecasts of warm-season convective events in the YHRB. The influence of intrinsic predictability is also determined as all the forecast errors for all the scale-based sensitivity experiments exhibit convergence tendency. These findings provide insights into convective-scale predictability, particularly the regime-dependent features, which has not been demonstrated previously. However, our final goal is to build an operational convective-scale ensemble forecast applicable to the YHRB, a task for which the domain size used in this study may not be sufficiently large. A discussion on the optimal configuration for building a convective-scale ensemble forecast remains the subject of future consideration.
Acknowledgements. The authors would also like to thank Samuel J. CHILDS from Colorado State University for language editing. This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC1502103), the National Natural Science Foundation of China (Grant Nos. 41430427 and 41705035), the China Scholarship Council, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX17_0876).
We used the difference total energy (DTE) to represent energy differences between ensemble members. A vertical function (Nielsen and Schumacher, 2016) is introduced to calculate the vertically integrated root-mean DTE (RMDTE):
where
As RMDTE values can vary by convective event, we used a method proposed in Nielsen and Schumacher (2016) to compare different cases by calculating their associated normalized RMDTEs (NRMDTEs) as the ratio of RMDTE to total mean kinetic energy (TMKE). The TMKE and its integrated form for a specific region are given as
The temperature term is excluded from Eq. (A3) to allow the TMKE to vary with the convective situation in question. Thus, the NRMDTE can be calculated as
The growth of errors at different scales can be assessed by computing NRMDTE over time, which, according to Lorenz (1969), can also be used to represent predictability: as the region-averaged NRMDTE gradually increases, predictability decreases.
The convective adjustment timescale is defined as the rate at which CAPE is removed by diabatic heating associated with precipitation (Done et al., 2006):
where
