1. Introduction
Floods caused by heavy precipitation along the mei-yu front in the Yangtze River valley (YRV) are a major natural hazard over eastern China. Despite numerous studies in the past few decades on the formation and development mechanism of mei-yu front heavy precipitation (see Gao et al. 2018; Ding 2019 for reviews), the skill of its operational quantitative forecasts remains low. Thus, it is essential to study the predictability of this type of event. However, compared with the extensive level of mechanism research, studies on the predictability are lacking and those that have been conducted suggest a strong sensitivity to the errors in the initial conditions (Bei and Zhang 2007; Luo and Chen 2015; Zhuang et al. 2020).
Research on atmospheric predictability can be traced back to the pioneering work of Thompson (1957). Under the assumption of a perfect model, this study investigated how the initial error of the observation affected the results of the numerical weather prediction. Subsequently, Lorenz (1963) used the famous Lorenz model to first reveal the chaotic nature of the atmosphere. Then, Lorenz (1969) studied the nonlinear multiscale interaction of the atmospheric motions, demonstrating that the smallest-scale error can influence the largest-scale motion through the progression processes, although its direct effect is apparently very small, i.e., the famous “butterfly effect.” Lorenz also suggested that the error growth rate accelerates when the scale decreases, i.e., the small-scale errors grow faster than the larger ones. Based on this, Lorenz proposed the intrinsic predictability limit for many typical scales of the atmosphere. Although the model upon which the theory was based was not very proper, this theoretical study of limited predictability remains widely understood and accepted to this day. Observations have shown that the atmospheric kinetic energy spectrum follows a k−3 power law (horizontal wavenumber k) at scales larger than ∼400 km, with a transition to k−5/3 at scales of less than 400 km (Nastrom and Gage 1985; Lindborg 1999). Based on this, Rotunno and Snyder (2008) applied the generalized Lorenz model to both the two-dimensional vorticity equation (with a k−3 spectrum) and the surface quasigeostrophic equation (with a k−5/3 spectrum) to demonstrate the differences between the flows with unlimited and limited predictability. Their distinction, as indicated by the authors, was mainly dependent on the kinetic energy spectrum of the basic flow rather than the dynamics that govern the error growth.
As multiscale characteristics, in the presence of moist convection, the rapid upscale spreading of small-scale initial errors leading to limited meso- and larger-scale predictability has been verified by numerical models (Zhang et al. 2002, 2003, 2006, 2019; Melhauser and Zhang 2012; Sun and Zhang 2016) and gained wide acceptance in the meteorology community. The error-growth dynamics and the limited intrinsic predictability of convective precipitation are also consistent with the findings about storm scale predictability (Park 1999; Walser and Schär 2004; Wapler et al. 2015), characterized by stronger nonlinearity and faster error growth (Hohenegger and Schär 2007a; Yano et al. 2018). Furthermore, based on idealized moist baroclinic waves, Zhang et al. (2007) proposed a three-stage conceptual model through exploring the mesoscale error growth dynamics. In the initial stage, the small-scale errors grow rapidly, driven by convective instability and moist processes, and then quickly saturate. In the stage 2, the errors start to transform to larger-scale balanced motions. In the third stage, the errors continue to grow by the baroclinic instability with a slower rate. In a real case study, Selz and Craig (2015) demonstrated the efficiency of this model in explaining the atmospheric predictability. In these studies, the central focus was on the main statistical characteristics of error growth based on small-scale, small-amplitude homogenous isotropic ensembles.
Besides, the relative importance of the large- and small-scale initial errors on the atmospheric predictability is still under debate. Actually, the concept of downscale error cascade can be also found in Lorenz (1969), although only one paragraph was devoted to the large-scale case and the main emphasis was on the growth of small-scale errors. More generally, Lorenz suggested that the impact of initial errors with the same magnitude is insensitive to their scales, i.e., his experiment A and B. Some recent studies have demonstrated that the downscale error cascade of large-scale errors tends to dominate the upscale error growth of small-scale errors (Bei and Zhang 2007; Durran and Gingrich 2014; Durran and Weyn 2016). However, few studies have investigated what kind of spatial structure of the initial error could result in the largest forecast uncertainty under the current observational error accuracy.
Investigating what kind of initial error develops relatively quickly and subsequently leads to the largest forecast uncertainty is an important aspect in atmospheric predictability studies. To this end, Mu et al. (2003) proposed the conditional nonlinear optimal perturbation (CNOP) method to examine the specific initial error that causes the greatest forecast uncertainty, under the given initial constraint condition. To date, the CNOP method has been widely used in predictability research related to the weather, climate and oceans (e.g., Yu et al. 2009, 2012; Wang et al. 2013a,b; Dai et al. 2016; Liu et al. 2018). However, it has not yet been applied in the predictability of meso- and convective-scale weather systems such as heavy precipitation.
In the present study, the impact of the optimally growing initial errors measured by the CNOP method on the mesoscale predictability of typical mei-yu front heavy precipitation events over eastern China is explored. The specific spatial structure and the growth of the optimally growing initial error are of particular interest in this study, which is hereafter organized as follows: an overview of the typical mei-yu front heavy precipitation events employed in this study is given in section 2. Section 3 introduces the model, including its configuration, simulation performance and experimental design, along with the methods used in this study, i.e., the definition of the CNOP and the related analysis methods. Section 4 presents the results, followed finally by a brief summary and some further discussion in section 5.
2. Overview of the typical mei-yu front heavy precipitation events
During the 1998, 2016, and 2020 mei-yu seasons, frequent heavy precipitation events occurred over the YRV with devastating socioeconomic impacts (Zhao et al. 1998; Bi et al. 2017; Liu and Ding 2020; Zhang et al. 2020). In this study, these three heavy precipitation events were chosen for analysis, i.e., 1200 UTC 20 July–1200 UTC 21 July 1998 (case1), 0000 UTC 2 July–0000 UTC 3 July 2016 (case2), and 0000 UTC 6 July–0000 UTC 7 July 2020 (case3). They all occurred in a typical mei-yu synoptic environment, i.e., the distributions of the western North Pacific subtropical high (WNPSH) and the mei-yu front control the general precipitation areas, the midlevel troughs and moisture supply through low-level jets (LLJs) could provide favorable mesoscale conditions, for heavy-precipitation production (Liu et al. 2008; Sampe and Xie 2010; Ding et al. 2020). Figure 1 show the respective spatial distributions of the 24-h accumulated precipitation for the three cases, in which the main rainbands are all located over the YRV and cover a wide area. More specifically, the rainband in case1 (Fig. 1a) has a west–east axis. The heaviest precipitation is located in southern Hubei Province, with a record-breaking accumulated precipitation amount of 286 mm at Wuhan station. A southwest–northeast-oriented rainband with a width of ∼250 km can be seen in case2 (Fig. 1b), in which the heaviest precipitation is located in southern Anhui Province, with the maximum accumulated precipitation amount reaching up to 280 mm. The rain belt in case3 (Fig. 1c) also orients from southwest to northeast, with a width of ∼200 km, and the heavy precipitation center (>100 mm) is located in southern Anhui and Hubei and the junction of Hubei and Hunan provinces, with the heaviest amount being 387 mm.
Observed 24-h accumulated precipitation (mm) for (a) case1, (b) case2, and (c) case3. The observation of case1 was obtained by interpolation of rain gauge data using the Cressman method, capturing a reasonable spatial distribution but with a loss of high values. In addition, the merged rain gauge–satellite gridded hourly precipitation dataset for China was used for case2 and case3, due to this dataset starting from 2008 (Pan et al. 2012; Shen et al. 2014).
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
Figure 2 shows the observational evolutions of the large-scale synoptic circulations every 12 h for the three cases, which are quite similar and exhibit typical circulation characteristics of the mei-yu season. Taking case2 as an example (Figs. 2d–f), there is a slow-moving shortwave trough at 500 hPa throughout the event, which transports cold and dry air to the YRV. Meanwhile, the southeast of China is mainly controlled by the WNPSH. At 850 hPa, there is a relatively strong southwesterly LLJ along the northwest flank of the WNPSH, transporting warm and moist air to the YRV. The mei-yu front locates at a region with large meridional gradients of equivalent potential temperature (θe) and horizontal wind shear (Ding and Chan 2005), orienting from southwest to northeast. Under this situation, the cold and dry air (low θe) will combine with the warm and moist one (high θe, above 345 K), which is beneficial to form and develop the mesoscale convective systems (MCSs) in the mei-yu front through the sustained moisture supply. Compared with the circulation situation in case2, the WNPSH in case1 retreats to the coastal areas of South China and the mei-yu front is oriented from west to east. In case3, the WNPSH has an obvious westward extension and northward jump, with warmer and moister air transporting to the YRV than case2.
Large-scale synoptic circulation from the ERA5 reanalysis dataset at (a) 1200 UTC 20 Jul, (b) 0000 UTC 21 Jul, and (c) 1200 UTC 21 Jul 1998; (d) 0000 UTC 2 Jul, (e) 1200 UTC 2 Jul, and (f) 0000 UTC 3 Jul 2016; and (g) 0000 UTC 6 Jul, (h) 1200 UTC 6 Jul, and (i) 0000 UTC 7 Jul 2020. Contours are the 500-hPa geopotential height (gpm); colored shading is the 850-hPa equivalent potential temperature (θe; K); and wind barbs are the 850-hPa wind (m s−1). A reference barb is 5 m s−1. The green lines represent the geographic map.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
3. Model and methods
a. Model and experimental design
The Weather Research and Forecasting (WRF) Model, version 4.0 (Skamarock et al. 2019a) was used in this study. The model integration domain covered most of the YRV (23.0°–36.7°N, 104.3°–126.1°E) at a grid spacing of 3 km. In the vertical direction, a hybrid vertical coordinate system was used, which was constituted by 50 levels [including 9 in the planetary boundary layer (PBL) below 850 hPa] with the top at 30 hPa. The highest vertical resolution is 52 m, and the average of all levels is 470 m, with a finer one of 150 m in the PBL. As suggested by Skamarock et al. (2019b), this vertical resolution is slightly coarser than horizontal resolution. The following physical parameterization schemes were used: the WRF single-moment 6-class microphysics scheme (Hong and Lim 2006), the Noah land surface model (Tewari et al. 2004), the Rapid Radiative Transfer Model for Global Climate Model longwave and shortwave radiation scheme (Iacono et al. 2008), and the Yonsei University planetary boundary layer scheme (Hong et al. 2006). Note that the choice of the single-moment microphysics scheme was mainly due to the limited computational resources and memory despite of the inferiority of the double-moment microphysics schemes (Igel et al. 2015). The cumulus parameterization was turned off. The model’s initial and boundary conditions were derived from the ERA5 reanalysis dataset with a ∼31-km resolution, and the boundary conditions updated every 3 h. It was integrated for 24 h with an 18-s time step. Based on the above settings, a simulation with no spinup was performed for each case, i.e., initializing at 1200 UTC 20 July 1998, 0000 UTC 2 July 2016, and 0000 UTC 6 July 2020, respectively. These simulations are referred to as the control runs (“CNTL runs”).
The 24-h accumulated precipitation distributions simulated by the model are shown in Fig. 3. Compared to the observations (Fig. 1), the spatial distributions of the rainbands are reproduced. The west–east rainband in case1 and southwest–northeast ones in case2 and case3 are all well simulated. Moreover, the heavy precipitation centers, i.e., southern Hubei Province and Chongqing City in case1, southern Anhui Province in case2, are captured as well, although their intensities are all overestimated. Also, the precipitation is slightly underestimated over the regions in the junction of Anhui and Hubei provinces and moderately overestimated over the regions in the junction of Jiangxi and Hunan provinces in case2. In case3, the simulated heavy precipitation centers located in southern Hubei and the junction of Hubei and Hunan provinces are about 100 km south of those observed. And the heavy precipitation centers located in southern Anhui province is slightly underestimated. To further demonstrate the model’s capability of reproducing the heavy precipitation processes, the simulated temporal evolution of hourly precipitation is also compared, which is comparable with the observation as well (not shown). Note that the overestimation of precipitation with convection-permitting resolutions is also found in other studies (e.g., Weisman et al. 2008; Schwartz et al. 2009, 2010, 2014; Li et al. 2019). Moreover, threat scores (TS; Gilbert 1884) computed from the standard 2 × 2 contingency table (Hamill 1999) are also used to quantitatively verify the model performance. In this study, TS refers to the ratio of observations that are correctly simulated. The values of TS range from 0 to 1, and the higher the better. The TS scores for 24-h accumulated precipitation of case2 (case3) with thresholds of 0.1, 10, 25, and 50 mm are 0.69 (0.65), 0.57 (0.64), 0.47 (0.50), and 0.33 (0.30), respectively. This is comparable to a recent study by Zhou et al. (2021). Thus, we have sufficient confidence in the model’s capability of reproducing the heavy precipitation events.
Simulated 24-h accumulated precipitation (mm) in the CNTL runs for (a) case1, (b) case2, and (c) case3.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
Generally, although there are some inevitable biases in the simulated location and intensity of the heavy precipitation centers, the model successfully depicts the main features of the three mei-yu front heavy precipitation events. Thus, this model framework can be used to study the impact of the initial error growth on their mesoscale predictability. In addition, for each case, a summary of experiments conducted in this study is listed in Table 1.
A summary of experiments conducted in this study.
b. Methods
1) CNOP method and its applications
(i) CNOP method
According to the definition, the CNOP is the global maximum of J(x0) in phase space. Mathematically, its existence and uniqueness cannot be generally proven. However, physically, when specific atmospheric and oceanic problems are concerned, the CNOP is considered to be existed. Meanwhile, its uniqueness is usually validated by different initial guess values. The crucial step to the CNOP method is how to obtain CNOP numerically. There is no general optimization algorithm to obtain the global maximum value in a nonlinear complex optimization problem with high dimensions. Thus tentatively, different initial guess values are often used to calculate the CNOP. Besides, the convergence (i.e., how many iteration steps are needed, how to stop the iteration according to the objective function or gradient change) should be comprehensively considered according to specific physical problems and computing resources. It is also worthwhile to mention that in addition to the global maximum of J(x0), there probably exists local maximum in some cases, which is called local CNOP (Mu and Zhang 2006) and could possess clear physical meanings. For example, some studies (e.g., Duan et al. 2004; Mu et al. 2014b) showed that the CNOP (local CNOP) represented the optimal precursor that most likely evolved into the El Niño (La Niña) event. Dai et al. (2016) demonstrated that the CNOP (local CNOP) yielded the optimal precursor could develop into the positive (negative) North Atlantic Oscillation. In terms of initial errors, the local CNOP could lead to different prediction errors when considering the problems for the error growth (Mu et al. 2007; Duan et al. 2009; Mu et al. 2009; Wang et al. 2013a; Ma et al. 2022).
(ii) Settings for computing CNOP
(iii) PSO algorithm
In this study, the particle swarm optimization (PSO) algorithm (Kennedy and Eberhart 1995) was used to compute the CNOP. In addition to the general characters of intelligent optimization algorithms, i.e., do not need to compute the gradients of the complicated system and have good performance in solving the optimization problem [see Wang et al. (2020) for details], PSO algorithm also has the advantages of simple coding, less uncertain parameters, and better computational efficiency, i.e., faster convergence speed than genetic algorithm (Hassan et al. 2005; B. Mu et al. 2015; Zhang et al. 2017). And it has been applied to calculate the CNOP in the simple model (Zheng et al. 2017), moderate complexity model (M. Mu et al. 2015), and complex models (Yang et al. 2020). Thus, in this study, a nonlinear optimization system that combined the high-resolution WRF Model and the PSO algorithm was first built to obtain the CNOP.
To deal with the high-dimensional (∼106) problem, the feature extraction was conducted first to reduce the dimensions based on empirical orthogonal function (EOF) decomposition. To obtain the samples, a simulation was conducted for every period from 29 June to 6 July during 2001–20 by utilizing the same model configuration as the CNTL run. Each run integrated for 24 h and saved the outputs every 6 h. Then, the samples of T were obtained by subtracting the climatology, i.e., the mean of the 20-yr model simulations. The analyses demonstrated that the cumulative explained variance of the first 50 EOF modes exceeded 90% of the total, which was sufficient to depict the main features of the particular events. Hence, these 50 modes were used to construct a feature space of the initial errors to find the CNOP. The following briefly introduces the application of the PSO algorithm:
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Initialization. Randomly generate a swarm of 20 particles with two attributes, i.e., location (L) and velocity (V). Based on Eq. (4), each particle is projected from the feature space (F) back to the original space, obtaining the initial error
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The particles are evaluated to select the particle best (PB) and the global best (GB) based on the objective function values. In the first step, every particle is the PB, and the particle with the largest
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Update the velocity and location of each particle through the following procedure:where ω is the inertia weight; c1 and c2 are the learning factors, which are both set to 2 in this study; r1 and r2 are random parameters, being 0.6 and 0.3, respectively. The settings of these parameters are mainly empirical. Under this situation, the PSO algorithm will be quickly converged to the optimal solution. k is the iteration step, whose limit is set to 30, which is sufficient in this study. The final GB is then projected back to the original space to obtain the CNOP.
2) Analysis methods
For each case, a summary of experiments is listed in Table 1. The “RP runs” were initialized with 20 sets of random errors and scaled to satisfy the same constraint of Eq. (3). The random errors (referred to as “RPs”) are sampled from spatially uncorrelated, uniformly distributed random numbers, ranging from −1 to 1 K. The role of CNOPs in the mesoscale predictability of the mei-yu front heavy precipitation events and their specific characteristics were then revealed by comparing the CNOP and RP runs. Besides, from a practical perspective, the CNOP+RP runs, in which 20 sets of random errors were added to the CNOP and scaled under the same constraint, were conducted to validate its superiority. If the objective function values of the CNOP+RP runs are all smaller than that of the CNOP, then the obtained CNOP at least is the local CNOP.
(i) Error energy metrics
(ii) Spectral analysis
In this study, the discrete Fourier transform was used to conduct the two-dimensional (2D) spectral analysis. Before the calculation, a 2D detrending method developed by Errico (1985) was applied to satisfy the requiring periodic boundary conditions of the Fourier transform. Then, for both the background and error spectrum calculations, the 2D spectral coefficients of each wavenumber (kx and ky) were obtained by separately performing the transforms in the zonal and meridional directions for each variable at every model level. They were then transformed to 1D spectra (wavenumber k) following the procedure detailed in Durran et al. (2017). The final power spectra analyzed here were summed for each variable and averaged over all model levels. In addition, for the spectral decomposition, to obtain the variable fields at different wavelength scales, the discrete cosine transform (DCT; Denis et al. 2002) method was used, which can automatically deal with the aperiodicity problem (Durran et al. 2017).
(iii) Quantitative estimation of the precipitation error at different scales
4. Results
a. CNOP and its effects on prediction
Based on the nonlinear optimization system described above, the CNOPs were obtained for the three cases, which are illustrated in Fig. 4. The results show that the CNOPs exhibit different spatial patterns for different heavy precipitation events, and their magnitudes range from −1.1 to 0.9 K, from −0.7 to 0.8 K, and from −0.3 to 0.9 K for case1, case2, and case3, respectively. For case1 and case3, the temperature perturbations for CNOPs are mainly positive, indicating these spatial patterns are conducive to the generation of convection and showing the quasi-barotropic structures. For case2, however, the CNOP presents the opposite distributions between the lower and upper levels, i.e., exhibiting a baroclinic structure characteristic. In addition, the CNOPs of case1 and case3 are more conducive to the generation of convection than that of case2. Despite their particularities, the large amplitudes of all the CNOPs at lower levels are mainly located in the rainband along the mei-yu front, orienting as west–east or southwest–northeast.
Spatial patterns of the CNOPs (K) for (a)–(f) case1, (g)–(l) case2, and (m)–(r) case3 at 200, 300, 500, 700, 850, and 900 hPa.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
To study the reliability of the above algorithm, the iterative procedure had been run for three times for case2 with different initial particles. Then, the corresponding three results for case2 were obtained, i.e., result1, result2, and result3, respectively (Fig. 5). Although there exist some differences among them, they all exhibit baroclinic structure characteristic. This implies that although these patterns are not truly unique, they do give much more optimal perturbations than choices like random chance. Besides, the evolution of the maximum objective function value (J) with the iteration step is shown in Fig. 6. Except for result1, whose iteration step is up to 30 steps, the other two are iterated for 40 steps. By the end, the result2 leads to the largest J among the three results. Thus, the result2 is selected as the CNOP for case2.
As in Fig. 4, but for the (a)–(f) result1, (g)–(l) result2, and (m)–(r) result3 (K) for case2.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
Evolution of the maximum objective function value (J) with the iteration step. The black, red, and blue lines represent the result1, result2, and result3, respectively.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
For a further verification of the obtained CNOPs, Fig. 7 shows the temporal evolution of the J and DPE in the CNOP, CNOP+RP and RP runs. The results suggest that these three cases are similar. In the CNOP runs, J grows rapidly in the first 8 h, and thereafter slows down. For case1, however, J grows rapidly again during 16–20 h, mainly due to the increase in precipitation. In the CNOP+RP and RP runs, a clear decay of the initial errors exists in the first 2 h, which was also found in some other studies (Zhang et al. 2006; Bei and Zhang 2007), possibly resulting from the numerical diffusion and gravity waves (Bretherton and Smolarkiewicz 1989; Hohenegger and Schär 2007b). It can also be seen that the CNOP (RP) runs exhibit the largest (smallest) J among the three types of runs throughout the 24-h simulations, implying a particular importance of the CNOP-type initial errors in the mei-yu front heavy precipitation events. Considering the evolution of the DPE, the largest (smallest) precipitation errors are generally also found in the CNOP (RP) runs throughout the 24-h simulations, thus further validating the superiority of the CNOPs obtained in this study. Thus, although the obtained CNOPs are not perfect approximations to the global optimum, they can also be used to move forward in the current analysis. Besides, the amplitudes of the forecast error in case1 (case2) are the largest (smallest), which may be related to the precipitation intensity and the spatial pattern of the CNOP.
Temporal evolution of the objective function value (J) and difference precipitation energy (DPE; mm2 h−2) for (a),(d) case1; (b),(e) case2; and (c),(f) case3. The black, blue, and yellow lines represent the CNOP, CNOP+RP, and RP runs, respectively. The dotted lines indicate the 20 groups of runs. The y axis of (a)–(c) is in logarithmic form.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
To investigate the effects of the CNOPs on the prediction of the heavy precipitation events, the area-averaged 24-h accumulated precipitation amounts are depicted in Fig. 8. In the RP runs, they oscillate around those in the CNTL runs, meaning that some RPs make the 24-h accumulated precipitation stronger, and some weaker. For the three cases, the CNOPs induce the largest 24-h accumulated precipitation among the various scenarios. And the effects of the CNOPs are to make them stronger. The spatial distributions of the 24-h accumulated precipitation errors in the CNOP runs are illustrated in Fig. 9. They are consistent with the positions of the mei-yu frontal rainbands, especially the convective rain systems, hinting at the possible role of moist convection in impacting the error growth. Furthermore, the maximum absolute values of the errors reach up to 200 mm (300 mm for case1), implying the CNOPs can result in significant 24-h accumulated precipitation errors.
Area-averaged 24-h accumulated precipitation for (a) case1, (b) case2, and (c) case3. The black and red dots denote the CNTL and CNOP runs, respectively; and the blue and yellow dots represent the 20 groups of CNOP+RP and RP runs, respectively.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
The 24-h accumulated precipitation errors in the CNOP runs for (a) case1, (b) case2, and (c) case3. In each panel, the location of maximum absolute values of the 24-h accumulated precipitation errors is denoted by the plus, and the number in the bottom-right corner denotes the maximum amount.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
b. Error growth characteristics
To better depict how the initial error grows across different spatial scales, Fig. 10 shows the evolution of the DTE spectra. In all three cases, the k−5/3 slopes of the observed background spectra are captured quite well at scales from 20 to ∼300 km. For the scales shorter than 20 km (roughly 7Δx with Δx being the grid spacing), the background spectra are strongly dampened by numerical dissipation (Skamarock 2004). For the scales from 300 to ∼600 km, the background spectra generally follow a k−3 slope. And for the larger scales, the slopes are steeper. The DTE spectra in the CNOP runs are quite similar for the three cases (Figs. 10a–c). Specifically, the initial energies of the CNOPs increase with increasing scale; in the first 6 h, the DTEs at small scales grow and saturate rapidly, whereas those at larger scales grow slowly. Among them, the growth between 3 and 6 h is much more “up-amplitude” and is actually weaker for scales shorter than 20 km, than for larger scales, i.e., at scales from 20 to ∼200 km (300 km in case1). This does not mean that the dominant errors are not growing rapidly in regions of moist convection. As discussed in Lloveras et al. (2022), this mainly arises from the use for Fourier transforms for which localized spatial distributions of regions of active convection in physical space appear broad in wavenumber space. During 6–12 h, the DTEs at larger scales start to grow rapidly, since the small-scale errors have propagated to the larger ones; and finally, after 12 h, only the large-scale DTEs continue to grow at a still slower rate. These error growth characteristics for the CNOPs generally follow the three-stage conceptual model proposed by Zhang et al. (2007). At 24 h, the DTE error energies are mainly concentrated at larger scales. Also, the error spectra at scales below ∼200 km (∼300 km for case1) reach the background spectra, indicating the signal and noise are indistinguishable. Thus, these errors have basically become saturated, and no predictive skill can be obtained by any single deterministic forecast. For comparison, Figs. 10d–f also show the DTE spectra in the RP runs, which were averaged from the 20 groups. The results suggest that the initial energies of the RPs are maximized at the smallest resolved scales, parts of which even exceed the background spectral energies. This energy imbalance at small scales will be quickly smoothed by numerical diffusion and gravity waves (Bretherton and Smolarkiewicz 1989; Hohenegger and Schär 2007b). The subsequent error development is the same as the rapid saturation and upscale growth. These spectral analyses further validate the point suggested by Lorenz (1969) that the impact of errors with the same magnitude is insensitive to their scales. However, the forecast error growth and saturation for the CNOPs are faster than those for the RPs in the first 6 h, suggesting their particular importance from the common characteristics.
The DTE (m2 s−2) spectra in the CNOP runs for (a) case1, (b) case2, and (c) case3; and those in the RP runs for (d) case1, (e) case2, and (f) case3. The black solid lines represent the background spectra derived from the 0- to 24-h forecast lead times’ mean spectra in the CNTL runs, the black dashed lines correspond to reference lines with slopes of k−3 and k−5/3 and the different colored lines represent the DTE spectra at 0, 1, 3, 6, 12, and 24 h, respectively. Only those wavelengths greater than 6 km are shown.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
For a further quantitative comparison of the error growth in the CNOP and RP runs, the error growth rates, which are defined and measured as the multiple of increase in the DTEs across all scales during each 6-h period, are illustrated in Fig. 11. For the RP runs, the average of the 20 groups is considered. It suggests that in the first 6 h the error growth rates in the CNOP runs are significantly larger than those in the RP runs. This is induced by the rapid growth and decay of the small-scale errors in the CNOP runs and RP runs, respectively. In addition, the error development for the CNOPs mainly manifests as a rapid increase in amplitude. During 6–12 h, the error growth for the CNOPs slows down, mainly manifesting as a scale increment. Also, the error growth rates for the CNOPs are slightly smaller than those for the RPs in case2 and case3 in this stage because the errors at many smaller scales have saturated in the CNOP runs but not for the RP runs, and the errors at larger scales grow more slowly. For the same reason, the CNOP runs show even smaller error growth rates after 12 h, which is similar to those in the RP runs. In this stage, the error development for the CNOPs mainly manifests as large-scale error structural adjustments, as suggested by Zhang et al. (2007).
Error growth rates of the DTE in the CNOP and RP runs for (a) case1, (b) case2, and (c) case3. The black and gray boxes represent the CNOP and RP runs, respectively.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
Furthermore, the precipitation error growth in the CNOP and RP runs is quantitatively compared in Fig. 12. The distribution of R(λ) at different spatial scales and forecast lead times is shown in Figs. 12a, 12d, and 12g. At the very beginning, more precipitation error energies are placed at smaller scales; subsequently, R(λ) increases rapidly at small scales, suggesting rapidly growing errors (Zhang et al. 2006, 2016); at 12 h, there is a range of scales (∼100 km for case1, ∼50 km for case2, and ∼60 km for case3) for which R(λ) reaches the threshold (0.95 ≤ R(λ) ≤ 1.05), indicating that the forecasts are totally decorrelated; and then there is a clear increase in these scales with the forecast lead time. Therefore, for precipitation fields, the error growth is also characterized by rapid growth and saturation at small scales, and then an upscale cascade is triggered. This can also be found in the RP runs (Figs. 12b,e,h). However, in the RP runs, at 12 h, R(λ) reaches the threshold at ∼30 km for case3, and it has not reached the threshold for the other two cases. Hence, the R(λ) amplitudes are smaller and the corresponding precipitation error growth is much slower than those in the CNOP runs.
The distribution of R(λ) at different spatial scales and forecast lead times in (left) the CNOP runs, (center) the RP runs, and (right) the temporal evolution of the decorrelation scale λ0 for (a)–(c) case1, (d)–(f) case2, and (g)–(i) case3. The distribution of R(λ) for the RP runs is averaged over 20 groups of runs. Different colored lines denote different forecast lead times at 1, 3, 6, 12, and 24 h. The black dashed lines correspond to the thresholds of 0.95 and 1.05. (right) The black and yellow lines denote the CNOP and RP runs, respectively. The evolution of λ0 for CNOP and RP runs has also been smoothed using a 3-h running mean.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
Figures 12c, 12f, and 12i show the temporal evolution of the decorrelation scale λ0, which is further obtained by R(λ). In the CNOP runs, λ0 increases with the forecast lead time, reaching ∼100 km in the final forecast period. Thus, this suggests a total loss of predictability of the precipitation fields in the CNOP runs below ∼100 km in the final 24 h. Moreover, it is clear that the precipitation errors induced by the CNOPs grow and saturate faster than those by the RPs.
c. Relative importance of large- and small-scale initial errors
This section explores the relative importance of the large- and small-scale initial errors based on the CNOP-type initial errors. The “CNOP-L” (“CNOP-S”) run was conducted, which was the same as the CNOP run, except for the CNOP from the scales smaller (larger) than 200 km was removed (Table 1). The error growth in the CNOP-L and CNOP-S runs was compared with that in the CNOP runs, which had the initial errors in all the scales, revealing consistent conclusions among the three cases (Fig. 13). The initial amplitudes of the DTE in the CNOP-L runs are close to those in the CNOP runs and larger than those in the CNOP-S runs (Figs. 13a–c). Subsequently, the DTE and DPE in the CNOP-L runs are similar to those in the CNOP runs and clearly larger than those in the CNOP-S runs throughout the 24-h simulations. Furthermore, the 24-h accumulated precipitation errors in the CNOP-L and CNOP-S runs (Fig. 14) were also compared with those in the CNOP runs (Fig. 9), revealing that those in the CNOP-L runs to be similar to the CNOP runs in both range and amplitude, and larger than those in the CNOP-S runs. Thus, these results suggest that the larger-scale and larger-amplitude initial errors (in terms of the DTE amplitude) in the CNOPs is dominant of the error growth. However, it is worth noting that only small-scale and small-amplitude initial errors are presented in the CNOP-S runs, which can also result in large forecast uncertainties. In addition, the error growth in the CNOP-L runs is similar to that in the CNOP runs across all scales, and clearly much faster than that in the CNOP-S runs, as is evident in the DTE spectral analyses for the CNOP-L and CNOP-S runs (Fig. 15). This implies that reducing small-scale errors in the CNOPs has little effect on the results, because the larger-scale initial errors will rapidly spread to smaller scales by downscale propagation, and then trigger an upscale energy cascade, to affect the entire error growth process. Yet, reducing large-scale errors in the CNOPs can slow down the error growth across all scales.
Temporal evolution of the DTE (m2 s−2) and DPE (mm2 h−2) for (a),(d) case1; (b),(e) case2; and (c),(f) case3. The black, blue, and yellow lines represent the CNOP, CNOP-L, and CNOP-S runs, respectively. The y axis of (a)–(c) is in logarithmic form. In (a)–(c), the black lines are hidden by the blue lines.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
The 24-h accumulated precipitation errors in the (right) CNOP-L runs and (left) CNOP-S runs for (a),(b) case1; (c),(d) case2; and (e),(f) case3.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
As in Fig. 10, but for the (a)–(c) CNOP-L runs and (d)–(f) CNOP-S runs.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
To sum up, larger-scale and larger-amplitude initial errors generally result in larger forecast uncertainties than smaller-scale and smaller-amplitude initial errors. In practical terms, the downscale error cascade from larger-scale initial errors is more significant than the upscale error growth from smaller-scale initial errors, and consistent with recent studies (Bei and Zhang 2007; Durran and Gingrich 2014; Durran and Weyn 2016). These findings suggest that improving the initial analysis, especially at larger scales (>200 km) can improve the forecast accuracy. However, due to the limited observational accuracy, there must be some small-scale (<200 km) and small-amplitude initial errors. Nevertheless, they can induce forecast uncertainties across all scales that are comparable to those induced by larger-scale initial errors. This further implies that the predictability of the mei-yu front heavy precipitation events examined in the present study is inherently limited.
d. Error growth mechanisms
Figure 16 shows the spatial evolution of the vertically averaged DTE in the CNOP runs and the simulated precipitation in the CNTL runs. In the first 6 h (stage 1), the error growth is confined to the precipitation regions; from 6 to 12 h (stage 2), the spatial scales of the errors gradually increase and ultimately cover the entire analysis domain; and after 12 h (stage 3), the error development mainly manifests as structural adjustments at large scales, also is affected by the convection, gravity waves, and geostrophic adjustment as suggested by Zhang et al. (2007). The amplitudes of the DTEs and the precipitation rates vary synchronously in the entire process; the larger DTEs generally correspond to the precipitation areas, indicating that moist convection is the key to the growth of the CNOPs. Combining with the spatial patterns of the CNOPs, large amplitudes of all the CNOPs at lower levels are mainly located in the rainband along the mei-yu front. Hence, these results imply that the CNOPs induce significant forecast errors under the influence of moist convection.
Spatial evolution of the vertically averaged DTE (m2 s−2) in the CNOP runs valid at 3, 6, 12, 18, and 24 h for (a)–(e) case1, (f)–(j) case2, and (k)–(o) case3. The regions where the precipitation rate is >0.1 mm h−1 in the CNTL runs are gray shaded.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
To further investigate the role of moist convection in causing rapid growth of the CNOP for each case, two “Fake Dry” runs were conducted, which were identical to the standard, moist CNTL and CNOP runs except that the diabatic contribution from microphysical processes was ignored. Unsurprisingly, with latent heating turned off, convection and precipitation in the Fake Dry runs are much delayed and weaker (not shown). The subsequent DTE evolutions in the Fake Dry runs are plotted as the black dashed curves in Fig. 17. The DTEs in the Fake Dry runs are greatly reduced by an order of magnitude at 24 h compared with that of the standard, moist runs (black solid lines), especially for case1, which sees a reduction of about 1/20. This confirms the direct dependence of the growth of the CNOPs on the moist convection throughout the 24-h simulations. Furthermore, the DTE evolutions at the three characteristic scale ranges (S: small scale, wavelength < 200 km; M: intermediate scale, 200 km < wavelength < 1000 km; L: large scale, wavelength > 1000 km) in the Fake Dry runs are plotted as the colored dashed curves in Fig. 17. The DTEs in the Fake Dry runs at the S, M, and L bands are also reduced. For the L scale, the errors have been increased in the first 4 h, but later have been decreasing, except for case3 they increase again in the last few hours. Thus, this indicates the error upscale propagation is significantly reduced with decreasing moist convection. Moreover, the errors in the Fake Dry runs grow slowly and do not saturate across nearly all scales, as is evident in the DTE spectral analyses (not shown). Therefore, moist convection plays an important role in the growth of the CNOPs.
Temporal evolution of the DTE (m2 s−2) for the CNOPs at different wavelength scales [Total (black dashed lines): all scales; L (red dashed lines): wavelength > 1000 km; M (blue dashed lines): 200 km < wavelength < 1000 km; S (yellow dashed lines): wavelength < 200 km] under the moisture environments of Fake Dry runs for the three cases: (a) case1, (b) case2, and (c) case3. The solid curves indicate the corresponding DTE evolution in the standard moist environment. The y axis is in logarithmic form.
Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0330.1
5. Summary and discussion
In this study, the CNOP method was adopted to explore the impact of the optimally growing initial errors on the mesoscale predictability of typical mei-yu front heavy precipitation events in the high-resolution WRF Model. First, three typical mei-yu front heavy precipitation events occurring in a typical mei-yu synoptic environment were selected, and their main features were found to be reproduced well by the model. Then, the CNOP for each case was obtained based on a nonlinear optimization system built using the model and PSO algorithm, and its spatial pattern was analyzed. Despite different CNOPs having particular spatial structures, the large amplitudes of the CNOPs at lower levels were mainly located in the rainband along the mei-yu front. Then, the CNOPs as the optimally growing initial errors in these mei-yu front heavy precipitation events were validated. The CNOPs caused the largest forecast errors, i.e., more than the CNOP+RPs and RPs, indicating the particular importance of the CNOPs. The effects of the CNOPs were to make the 24-h accumulated precipitation stronger and cause significant 24-h accumulated precipitation errors.
The initial spectral energies of the CNOPs increased with increasing scale. This type of initial-error spectral structure was consistent with recent studies, which adopted the practical data assimilation algorithms to generate the initial states for the ensembles and near-twin experiments. In addition, the forecast error growth for the CNOPs generally followed the three-stage conceptual model proposed by Zhang et al. (2007). Besides, the precipitation error growth could also be distinguished as upscale error growth. At 24 h, the DTE and the precipitation errors induced by the CNOPs saturated at scales below ∼200 km (∼300 km for case1) and ∼100 km, respectively. This means that different types of variables have different predictability limits and no predictive skill can be obtained by any single deterministic forecast under these scales. For this reason, ensemble forecasts have to be adopted to extract the predictable signal in the form of uncertainty, to improve the forecast skill (e.g., Kain et al. 2008; Clark et al. 2009; 2018).
The specific characteristics of the CNOPs in the mesoscale predictability of these mei-yu front heavy precipitation events are further revealed by comparing the CNOP and RP runs. By quantitatively comparing the error growth rates, R(λ) and the corresponding decorrelation scale λ0 in the CNOP and RP runs, it was found that the CNOPs with specific spatial structures grew and saturated faster than the random errors in space. It is worth noting that recent studies indicated that the current convection-permitting ensemble forecast systems (CPEFSs) are characterized by insufficient ensemble spread, which are not efficient to represent the true forecast uncertainties (e.g., Romine et al. 2014; Zhang 2018). Hence, the CNOP, as the optimally growing initial error in the meso- and convective-scale processes, can be potentially better to sample the analysis errors contributing to the subsequent forecast errors, which can be effective in constructing initial ensembles for CPEFSs (Jiang and Mu 2009; Jiang et al. 2009a,b).
The relative importance of the large- and small-scale initial errors was explored based on the CNOP-type initial errors. It was found that large-scale and large-amplitude initial errors in the CNOPs were the most influential on the forecast quality. In practical terms, the downscale error cascade from larger-scale initial errors was more significant than the upscale error cascade from smaller-scale initial errors. Our study suggests that improving the initial conditions especially the larger-scale component can be potentially efficient to improve the forecast skill, such as ingesting observations using advanced data assimilation approaches and constructing adaptive observations. In addition, it is found that CNOP method is effective to guide the adaptive observations, in which the regions of large amplitudes of the CNOPs generally correspond to the sensitive areas (Mu 2013; Mu et al. 2014a; M. Mu et al. 2015). However, this improvement in forecast skill has limits, implying the mesoscale predictability of the mei-yu front heavy precipitation events is inherently limited.
The present study has also shed light on the importance of moist convection in the forecast error growth for the CNOPs. It is worth noting that three typical heavy precipitation events were examined for the impact of optimally growing initial errors on the mesoscale predictability in this study, and the errors grew with the background dynamics, especially the moist convection. Besides, it should be noted that the heavy precipitation that occurs along the mei-yu front is an end product of multiscale weather system interactions including MCSs, the mei-yu front itself, the WNPSH, shear lines, and the low- and upper-level jet. Hence, the conclusions drawn from the present study are believed to be universal to these types of mei-yu front heavy precipitation events. In addition, it is believed that the results in this study can be potentially generalized to storm scale predictability.
Due to the limited computational resources, this preliminary application of the nonlinear optimization system only examined the temperature field with limited vertical levels and three cases. For the heavy precipitation events, the initial errors of T, u, υ, and r should be considered together. It is worth stressing that the moist energy norm was chosen as the objective function in the present study. However, for heavy precipitation events, the accumulated precipitation is of more concern. As such, it would be better to consider the 24-h accumulated precipitation as the objective function, but this would bring with it more nonlinear problems. Although such a change would offer something different, however, it is anticipated that the main conclusions of this study would not change, because of the nonlinear and rapid growth of the CNOP. Of course, future work is needed to investigate all of these aspects. This study also suggests that the PSO algorithm is a potentially useful tool for obtaining the CNOP in meso- and convective-scale processes, which will therefore be used in further studies. In the future studies, it is also strongly necessary to consider the exclusive double-moment microphysics schemes and higher vertical resolution to improve the simulation of the heavy precipitation events, especially with the convection-permitting resolution.
Acknowledgments.
The research is supported by Guangdong Major Project of Basic and Applied Basic Research (Grant 2020B0301030004). The authors gratefully acknowledge the three anonymous reviewers for their relevant comments and suggestions.
Data availability statement.
The observed precipitation data can be downloaded from the China Meteorological Administration’s data portal (http://data.cma.cn). The ERA5 reanalysis dataset, which was developed through the Copernicus Climate Change Service (C3S), is the fifth-generation ECMWF atmospheric reanalysis of global climate and can be accessed via http://climate.copernicus.eu/products/climate-reanalysis.
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