Application and Characteristic Analysis of the Moist Singular Vector in GRAPES-GEPS

Ensemble prediction is a probabilistic forecast method employed to solve the uncertainty problem of a single numerical forecast. That uncertainty is derived from errors in the initial conditions and from the model itself (Buizza and Montani, 1999). The initial perturbation approach was the first proposed perturbation method and provided a solid scientific research foundation for global ensemble prediction systems. The use of a singular vector (SV) in the atmosphere was first introduced by Lorenz (1965) for the predictability problem and was subsequently improved upon in many studies, leading to approaches such as the ensemble initial perturbation scheme (Buizza and Palmer, 1995; Molteni et al., 1996; Hakim, 2000; Leutbecher et al., 2008) and targeted observation method (Palmer et al., 1998; Buizza and Montani, 1999; Buizza et al., 2007; Yamaguchi et al., 2009). The SV is based on a tangent linear model (TLM) and its adjoint model, reflects the fastest-growing direction in phase space under certain constraints, and can represent the instability of atmospheric baroclinicity at mid-to-high latitudes (Buizza et al., 1993; Hoskins et al., 2000). The mathematical theory and the physical significance of the SV are clear, and thus it has become one of the main initial perturbation methods in some global weather forecast centers—for example, the European Centre for Medium-Range Weather Forecasts (ECMWF), the Japan Meteorological Agency, and Australia’s Bureau of Meteorology. The SV calculated using a moist physical process in the TLM is called the moist SV, whereas the vector calculated using a dry physical process is called the dry SV (Ehrendorfer et al., 1999; Diaconescu and Laprise, 2012).

At present, the linearized physical process used to calculate the SV in the Global/Regional Assimilation and Prediction System—Global Ensemble Prediction System (GRAPES-GEPS) includes only vertical diffusion and a subgrid-scale terrain parameterization scheme, called the linearized planetary boundary layer (PBL) scheme (Li and Liu, 2019), which is a typical dry physical process. Mahfouf (1999) proposed that adding linearized physical processes to the TLM to calculate SVs could effectively improve the spatial SV structure (Liu et al., 2019). Therefore, adding a moist physical process to the TLM and studying the structure and characteristics of the moist SV will be helpful to construct a more reasonable initial field of the ensemble forecast.

Linearization parameterizes the physical processes in the TLM and normalizes them according to the tangent linear equation so that the TLM will have more physical information. The linearized physical processes in the ECMWF global forecast system include vertical diffusion, surface drag, gravity wave drag, longwave radiation, large-scale condensation, and deep cumulus convection (Mahfouf, 1999; Diaconescu and Laprise, 2012). The latter two are usually called moist physical processes. The reason for not using linearized moist physics in a relatively early stage may be due to the difficulties of constructing a linear approximation of moist physical processes. However, the interaction of diabatic processes (moist physics) and adiabatic processes (dry physics) is much more important in the atmosphere. To date, the TLM has been developed, and many studies have added moist physics to SVs. Although linear moist physical processes may be simple and therefore cannot fully describe the adiabatic process, these processes contain the basic principles of moist physics. Buizza (1994) emphasized the importance of adding vertical diffusion and surface drag to the TLM, pointing out that the addition of dry physical processes can effectively suppress the rapid growth of the shallow structure near the ground, which can be dispersed rapidly in the nonlinear model. Barkmeijer et al. (2001) and Puri et al. (2001) studied moist SVs in the target areas of tropical cyclones (TCs) using the ECMWF TLM and found that moist SVs are more sensitive than dry SVs to the area of the TC. Kim and Jung (2009) added large-scale precipitation to the MM5 TLM and attempted to apply the moist norm to obtain moist SVs. They found that moist physics could produce more small structures, but the moist norm was much too sensitive. Zadra et al. (2004) calculated the subtropical SV in winter using dry physical processes and moist physical processes (stratospheric cloud precipitation and convective precipitation, respectively). Their study showed that stratospheric cloud precipitation in moist physical processes has a significant effect, while the effect of convection precipitation is not obvious, and moist physical processes enhance the transmission of SV energy to the jet stream. Coutinho et al. (2004) pointed out that the parameterization of large-scale condensation plays a major role in the calculation of the SV in subtropical regions, but that the parameterization of the remaining processes (gravity wave drag, longwave radiation and deep cumulus convection) has little effect on the calculated SV in subtropical regions. Moist SVs occur in baroclinically unstable areas and are also affected by water vapor. Hoskins and Coutinho (2005) studied the role of moist SVs in high-impact weather prediction and proposed that the addition of linearized large-scale condensation to the TLM has a significant effect on improving the short-term prediction of extreme events.

The focus of this study is how to use SVs to produce initial values more reasonably in GRAPES-GEPS. Liu et al. (2017) developed TLMs and adjoint models based on the global nonlinear model of GRAPES. Linearized physical processes include vertical diffusion, subgrid-scale terrain parameterization, deep cumulus convection, and large-scale condensation. However, the calculation of the SV in GRAPES-GEPS does not consider moist physical processes. It is known from previous studies that linearized large-scale condensation has an important influence on the SV, so some experiments in which large-scale condensation was added to the TLM of GRAPES-GEPS were carried out in this study. Based on these experiments, this paper analyzes the structural changes in SVs after adding moist physical processes from the perspectives of the energy norm, energy spectrum analysis, vertical structure, and spatial profile, and the correctness and rationality of the moist SV are verified. Besides, the ensemble forecast of the moist SVs is evaluated.

For an initial state vector X, if ${{M}}$ is used to represent the nonlinear prediction model, the nonlinear equation can be expressed as (Buizza and Palmer, 1995):

Here () means ${{M}}$ is a function of ${X}$. Then, a small disturbance ${X'}$ is superimposed on the state vector. Over a finite period, the small disturbance can be represented by a linear approximation:

where ${{M}_{\rm{l}}}$ is a linear approximation of the nonlinear model ${{M}}$, which can be expressed as:

in which ${{L}}(t,{t_0})$ is the forward tangent linear operator, ${{X}}'({t_0})$ is the perturbation at the initial time, and ${{X}}'(t)$ is the evolved perturbation after linear integration from the initial time ${t_0}$ to the evolved time $t$. The superscript ' is omitted from the following equations, and the state vectors discussed hereafter are all perturbations.

The solution of the SV can be achieved with the largest ratio of the evolved perturbed vector to the initial perturbed vector:

where $\left[, \right]$ denotes the Euler product and ${{E}}$ is the transform operator. The state vector ${{X}}$ of the physical space needs to be transformed into the dimensionless vector ${{\tilde X}}$ in Euler space using a transform operator ${{E}}$:

Then, Eq. (4) can be written as:

Equation (5) can be converted into a singular value decomposition problem:

where $\lambda $ is the eigenvalues of the matrix ${{{({{E}}\cdot{{L}}}}\cdot{{{E}}^{ - 1}}{{)}}^{\rm{T}}}\cdot{{({{E}}\cdot{{L}}}}\cdot{{{E}}^{ - 1}}{{)}}$. The singular value of ${{{{E}}\cdot{{L}}}}\cdot{{{E}}^{ - 1}}$ is $\sqrt \lambda $.

Many studies have shown that using the energy norm to calculate SVs can obtain mesoscale baroclinic perturbations (Hoskins and Coutinho, 2005; Diaconescu and Laprise, 2012; Liu et al., 2013).

The input and output variables of the GRAPES TLM and adjoint models in the process of solving the GRAPES-SV are the perturbed wind speeds $(u',v')$, the TL vectors of the perturbed Exner pressure ($\prod ' = \prod - {\prod _{\rm{r}}}$), and the perturbed potential temperature $(\theta ' = \theta - {\theta _{\rm{r}}})$. Based on these perturbed variables, the energy norm formula can be expressed as (Liu et al., 2013):

where ${\rho _{\rm{r}}}$ is the reference density, ${c_p}$ is the constant-pressure specific heat, ${T_{\rm{r}}}$ is the reference temperature, ${\theta _{\rm{r}}}$ is the reference potential temperature, ${\prod _{\rm{r}}}$ is the reference dimensionless air pressure, and $\varphi $ is the latitude of the spherical coordinate.

Ehrendorfer et al. (1999) studied the perturbation growth characteristics of SVs under moist physics and proposed that the use of moist energy norms may lead to the unclear growth of disturbances. Hoskins and Coutinho (2005) pointed out that the SV with perturbed water vapor is similar to that without perturbed water vapor, so the dry energy norm is a better choice for calculating SVs. ECMWF uses the total energy norm (i.e., the dry energy norm without water vapor) for calculating SVs. To make a clean comparison between the moist TLM and dry TLM, this paper also uses the energy norm without water vapor.

The linearized large-scale condensation process developed by Liu et al. (2019) is the large-scale cloud and precipitation scheme developed by Tompkins and Janisková (2004). This scheme describes the relationship between moist processes and clouds. The governing equation for humidity is:

where $C$ is the condensation or evaporation process of clouds on a large scale, ${E_{{\rm{prec}}}}$ is the change in water vapor by the evaporation of precipitation, and ${D_{{\rm{conv}}}}$ is the source term of water vapor for the convection outflow.

The governing equation for temperature is:

where $n$ and $m$ signify the freezing of rainwater and the melting of snow, respectively, $L_{\rm{a}}$ is the latent heat coefficient of evaporation/sublimation, and ${L_{\rm{a,f}}}$ is the latent heat coefficient of freezing and melting.

To avoid the abnormal growth of some false perturbations in the TLM, some constraints and conventions are derived from ECMWF (2017). It can be seen from the above formulae that the calculation of SVs by large-scale condensation is mainly reflected in the temperature and humidity. However, since the SV adopts the dry energy norm, the effect of large-scale condensation is mainly reflected in the temperature term.

The initial conditions of the control forecast in the ensemble prediction system are derived from the GRAPES 4DVAR method (Zhang et al, 2019) with a horizontal resolution of 0.5° and 60 vertical layers (The parameters of SV is showed in Table 1). After 60 iterations with the Lanczos algorithm, approximately 30 SVs can be produced in GRAPES-GEPS. These 30 SVs can be combined with random (Gaussian) linear combinations to form 30 initial perturbations. In this study, the evolved SVs were not involved in the initial perturbation, and the model perturbation was shut down. The complete test period ranged from 1200 UTC 1 May to 1200 UTC 5 May 2019, and from 1200 UTC 16 May to 1200 UTC 20 May 2019, spanning a total of 10 cases, the evolved time of the initial perturbation was 48 h, the ensemble forecast time was 240 h, and the forecast interval was 24 h. During this period, the atmospheric circulation in the Northern Hemisphere exhibits the characteristics of summer. In addition, the monsoon begins, meaning precipitation is abundant in South China.

In the original SV calculation process, only the linearized boundary layer scheme, including the terrain parameterization scheme and vertical diffusion, is used. In this study, large-scale condensation was added to calculate the SV, and the result was compared with that of the original SV scheme. The experimental setup is shown in Table 2.

Statistical analysis was performed on the results of the 10 test cases. Figure 1 shows the distribution of the 10-case average energy norm of the SV at the initial time and the difference between the moist SV and dry SV. As shown in Fig. 1c1 and c2, in the Northern Hemisphere, the proportion of internal energy (red) in the moist SV energy is larger than that of the kinetic energy (blue), indicating that the moist physical process mainly affects the temperature term of the SV. The large-scale condensation process mainly affects precipitation through the temperature term, which is consistent with the above. The growth of the energy at the initial time in the Southern Hemisphere has similar characteristics, but the growth rate is not as large as that in the Northern Hemisphere. This may be due to different seasonal characteristics; at the initial time, the Southern Hemisphere is in winter, and the precipitation characteristics are not obvious. Therefore, the influence of large-scale condensation on the SV structure is reflected mainly in the temperature and further affects precipitation by releasing the latent heat of condensation.

Figure 1.  Schematic diagram of the SV energy norm (units: J m⁻³) at the initial time. The left-hand column is the Northern Hemisphere and the right column is the Southern Hemisphere. Blue represents the kinetic energy (kic). Red and green are the internal energy containing the temperature term (thp) and the pressure term (pip), respectively. (a1, a2) DRY-SV; (b1, b2) MOIST-SV; (c1, c2) MOIST-SV minus DRY-SV.

It can be seen from Fig. 2 that at the evolved time (48 h), the proportion of the energy norm of the moist SV is similar to that of the dry SV, while the energy norm of the moist SV is nearly twice that of the dry SV. From the differences between the moist SV and dry SV (Figs. 2c1 and c2), the increase in kinetic energy (blue) in the evolved moist SV is more significant than that in the evolved dry SV, and the internal energy of temperature also increases. The reason is that the SVs are defined as the fastest-growing perturbation at the evolved time. Therefore, the evolved SVs represent the growth in the perturbation energy of the initial SV. There is a significant increase in the internal energy of the moist SV after tangent linear integration, which may be due to the release of the latent heat of condensation. Compared to the evolved dry SV, the addition of the linearized large-scale condensation process increases not only the internal energy but also the kinetic energy of the moist SV. This is due to the interaction among the physical processes that produces feedback between the variables; that is, the growth of internal energy promotes the growth of kinetic energy.

Figure 2.  As in Fig. 1 but for the energy norm (units: J m⁻³) at the evolved time (48 h).

By averaging the energy norm of the 30 SVs in the vertical direction, the energy norm vertical profile in Fig. 3 can be obtained. The peak initial energy norm of the dry SV in the Northern Hemisphere is found approximately in the layers 28−30 (the middle layer of the troposphere), while the peak distribution of the initial energy norm of the moist SV is approximately in layers 20−25. Compared to the total energy norm of the dry SV, the total energy norm of the lower layer in the moist SV is increased. The perturbation energy norm of the evolved dry SV features two peaks: one transmitting upward and one transmitting downward. The upward peak is related to the jet, which is consistent with previous research (Coutinho et al., 2004; Li and Liu, 2019). The evolved moist SV maintains this feature, and it can be seen that the energy norm of the valley (layers 15−20, corresponding to 850−700 hPa) between the two peaks also increases, and this increase is mainly reflected in the internal energy. The Southern Hemisphere also has similar characteristics. However, the two-peak structure at the evolved time in the Southern Hemisphere is less obvious, and the downward energy norm peak is weaker than the upward energy norm peak. In general, the addition of the linearized large-scale condensation process maintains the basic characteristics of the energy distribution of the dry SV and simultaneously increases the perturbation energy norm in the lower layer of the troposphere. This tends to stimulate the instability of the middle and lower layers.

Figure 3.  Vertical distribution of the energy norm (units: J m⁻³). The left-hand column is the Northern Hemisphere and the right column is the Southern Hemisphere. (a1, a2) DRY-SV; (b1, b2) MOIST-SV. The black line is the initial time (magnified 10 times) and the red line is the evolved time. The solid line is the total energy (TE) norm and the dashed line is the kinetic energy (KE) norm.

The grid data of GRAPES can be converted to spectral coefficients, and the variables can be expanded according to spherical harmonic functions. Then, the energy spectrum can be written as a function related only to the total number of spherical waves. The results of an energy spectrum analysis at different heights are shown in Fig. 4. From this figure, at 500 hPa (Figs. 4a1 and a2), the energy spectrum peak of the dry SV at the initial time is at approximately 15 wavelengths, while the evolution time is at 12−13 wavelengths; that is, the propagation energy is increasing (Coutinho et al., 2004; Li and Liu, 2019). This increase in the propagation energy is a feature that distinguishes the subtropical SV from the Lyapunov vector. The moist SV also maintains this feature, and at the initial time, the energy spectrum peak of the moist SV is at approximately 20 wavelengths, and the scale is smaller than that of the dry SV. At the evolved time, the energy spectrum of the moist SV exhibits increasing propagation energy. Compared to that of the dry SV, the growth in the total energy of the moist SV is reflected at 15−40 wavelengths; that is, the energy growth caused by the moist SV manifests as a relatively small-scale weather system. At 850 hPa, the moist SV energy is more concentrated at relatively small-scale wavelengths, and the energy peak of the moist SV is significantly higher than that of the dry SV at the evolved time. The release of latent heat of condensation leads to an energy increase at the small and medium scales; therefore, the SV under the linearized large-scale condensation process is more unstable and can grow rapidly, which is beneficial for describing the changing characteristics of a weather system.

Figure 4.  Energy spectrum analysis at different levels (units: ${10^{ - 7}}$ K²). The solid line is DRY-SV, the dashed line is MOIST-SV, the black line is the initial time (magnified 50 times), and the red line is the evolved time. The left-hand column is the Northern Hemisphere and the right-hand column is the Southern Hemisphere. (a1, a2) 500 hPa; (b1, b2) 850 hPa.

Singular values can reflect the growth rate of the perturbation over the time interval, and the 10-case average of the singular values is demonstrated in Fig. 5. It can be seen that, in both hemispheres, the singular value of moist SVs is larger than that of dry SVs, which means the moist SVs contain more uncertainty information. As the number of SV steps increases, the growth rate decreases and the first few SVs grow larger.

Figure 5.  The 10-case average of the singular values. Red lines represent dry SVs and blue lines represent moist SVs. Dashed and solid lines are for the Northern Hemisphere and the Southern Hemisphere, respectively.

Figure 6 shows the distribution of the SVs at 500 hPa on different days. The dry SV02 on 1 May (Fig. 6a) is distributed in the 150°E ridge area of the midlatitudes, which is also an area with obvious baroclinic features, while the moist SV02 on the same day (Fig. 6b) not only covers the ridge area but also substantially disturbs the area behind the East Asian trough, indicating that the moist SV can generate more disturbances than the dry SV in this baroclinically unstable region. The moist SV02 is more compact, meaning that the amplitude of its perturbation is greater than that of the dry SV02. It should be noted that the positive or negative sign does not affect the characteristics of SVs. The differences of sign can be eliminated when constructing the initial perturbation. Figures 6c and d show another example of the same sign, where the dry SV03 on 2 May (Fig. 6c) is located in the unstable area in the mid-to-high latitude region, while the moist SV03 (Fig. 6d) on that day is more widely distributed, extending to the trough around 70°E, and has a larger perturbation.

Figure 6.  Distributions of the initial (a, b) SV02 on 1 May and (c, d) SV03 on 2 May at 500 hPa, for the (a, c) dry SV and (b, d) moist SV. Shading and arrows indicate the potential temperature (unit: K) and the wind field (units: m s⁻¹) of the SV, respectively (with an amplification factor of ${10^3}$). The contour lines are the 500-hPa geopotential height of the control member at 1200 UTC on that day.

At the evolved time (48 h), the growth of the dry (Fig. 7a) and moist (Fig. 7b) SV02 on 1 May is consistent with the atmospheric circulation, while the moist SV02 can cover the entire low-pressure circulation region in 150°E−180°, and the small- and medium-scale perturbations have also developed. The dry (Fig. 7c) and moist (Fig. 7d) SV03 on 2 May exhibit a similar character, and the growth rate in the moist SV03 is larger than that of the dry SV03.

Figure 7.  As in Fig. 6 but for the evolved time (48 h). The contour lines are the 48-h forecast of the control member starting at 1200 UTC on that day.

The evolution of the SVs is synchronous with that of the atmospheric circulation, which is flow-dependent. The moist SVs maintain the basic characteristics of the dry SVs but are wider than the dry SVs, can contain more small- and medium-scale information, and have more energy. To analyze the spatial characteristics of these SVs, the vertical structure of the disturbance will be analyzed further.

A vertical section was constructed through the large-value area of potential temperature of SV03 on 2 May at 500 hPa. As shown in Fig. 8a2, the dry SV tilts westward with height at the initial time, which is a basic feature of the midlatitude baroclinic atmosphere. At the evolved time (48 h, Fig. 8b2), the energy exhibits large growth, and with the transmission of energy upward and downward, the typical baroclinic structure gradually turns into a barotropic structure.

Figure 8.  Horizonal distribution and vertical section of the dry SV03 potential temperature (unit: K) at 500 hPa (with an amplification factor of ${10^3}$) on 2 May. The left-hand column is the horizontal distribution and the right-hand column is the vertical profile along the black line in the left-hand column. (a1, a2) At the initial time (00 h); (b1, b2) at the evolved time (48 h).

Figure 9 shows that the moist SV03 on 2 May maintains the basic characteristics of the dry SV regarding both its horizontal structure and its vertical structure; that is, the moist SV can generate the perturbation in the baroclinically unstable region in the midlatitudes, but the disturbance range caused by the moist SV is wider, the scale is smaller, and the amplitude is larger, which shows that the moist SV contains more medium- and small-scale information.

Figure 9.  As in Fig. 8 but for the first moist SV03 on 2 May.

Figure 10 shows the 48-h forecast beginning at 1200 UTC 2 May 2019. Figure 10a shows the 24-h accumulated (3−4 May from 1200 UTC) convective rain, and Fig. 10b the non-convective rain (refers to large-scale precipitation). The rain in Fig. 10b is located mainly to the east of the Sea of Japan while covering the 850-hPa low-pressure circulation center. From the evolution of the two dry SVs (Figs. 10c and e), the perturbation is distributed within the transverse trough region of the 850-hPa geopotential height field. In addition to this transverse trough region, the two evolved moist SVs (Figs. 10d and f) are also distributed in the low-pressure circulation and its trough extension. The perturbation of this area is more consistent with the location of the large-scale condensation process, indicating that the moist SVs can produce perturbations related to large-scale condensation and precipitation, which is not a characteristic of the dry SVs.

Figure 10.  Distribution of the 48-h forecast precipitation of the dry (moist) SVs beginning at 1200 UTC 2 May 2019 for the (a) 48-h convective precipitation (rainc) (unit: mm) and 850-hPa geopotential height field (units: gpm) and (b) 48-h large-scale precipitation (rainnc) (unit: mm) and 850-hPa temperature field (unit: K). Panels (c, e) show the 48-h evolution of the first and second dry SVs at 850 hPa. Panels (d, f) are the same as (c, e) but for the moist SV. The shading, arrows and contours indicate the potential temperature (unit: K), wind field (units: m s⁻¹) and pressure of the SV, respectively (with an amplification factor of ${10^3}$).

The target area of the SVs is divided into the Northern and Southern hemispheres. Therefore, a calculation scheme that separates the Northern Hemisphere and the Southern Hemisphere is also adopted for evaluation of the meteorological field. The scoring area is 20°−80°N in the Northern Hemisphere and 20°−80°S in the Southern Hemisphere. The results of the ensemble prediction from the 10 cases show little difference in the scores between the Northern and Southern hemispheres. Taking into account both the root-mean-square error (RMSE) and the spread, Figs. 11a-d give the ratio of the spread to the RMSE (consistency), and Figs. 11e and f directly plot the spread and RMSE in the early stage of the forecast.

Figure 11.  (a−d) Ratio of the spread to the RMSE (consistency) of the ensemble forecast: (a, c) Northern Hemisphere; (b, d) Southern Hemisphere (red line is for the dry SV and blue line for the moist SV); (a, b) zonal winds (u) at 850 hPa (units: m s⁻¹); (c, d) zonal winds at 10 m (units: m s⁻¹). Panels (e, f) plot the spread and RMSE directly in the black box of (a, c) from the forecast time of 0 h.

Both the RMSE and spread were calculated with a latitude weight, and the average score of the 10 cases was calculated through the mean of the mean square error, as recommended by the World Meteorological Organization (http://epsv.kishou.go.jp/EPSv/).

The blue line in Fig. 11 represents the forecast result formed by the moist SV, and the red line is the forecast result formed by the dry SV. The low-level variables (zonal winds at 10 m) of the moist SV forecasts are better than those of the dry SV forecasts, especially in the first 72 h (Fig. 11), whereas the mid-level variables are not different in the early stage, and the high-level variables of the moist SV are slightly lower than those of the dry SV at 48−72 h and higher for the rest of the forecast (not shown). Figures 11e and f show more details of the spread and RMSE, from which it can be seen that the improvement comes from an increase in spread from 00 to 72 h. According to the results of the medium- and long-term forecasts, the results of the moist SV are also improved at 144−240 h (6−10 days).

It should be noted that there is a scaled process in which an empirical coefficient $\gamma $ is adjusted to ensure that the initial perturbation can generate sufficient ensemble spread (Li et al., 2019). The RMSE does not vary much with $\gamma $, while the spread is proportional to the empirical coefficient $\gamma $. Then, the ratio of spread/RMSE greater than 1 could be caused by $\gamma $. For comparison with the operational SV, the same scale factor is used in this paper.

Considering the structural characteristics of the energy distribution and the energy spectrum, as well as the spatial distribution of the moist SV, it can be found that the addition of physical processes with large-scale condensation can form new perturbations in the SV. The increase in energy caused by the release of latent heat of condensation is reflected mainly in the middle and lower troposphere, and the scale is small. It is inferred from the results that the perturbation lasts for a short time and is likely to be eliminated during the process of nonlinear integration. Therefore, the addition of a large-scale condensation process to the calculation of SVs can improve the prediction of short-term weather systems, which is consistent with the analysis of the predictability of high-impact weather by Hoskins and Coutinho (2005). From the perspective of mid- and long-term ensemble forecasts, it is also beneficial to add moist physical processes to the calculation of SVs.

The anomaly correlation coefficient (ACC) is a common method used in the evaluation of ensemble forecasts. After deducting the mean of the climate field, it reflects the correlation between the forecast field and the observation (analysis) field. In this section, the average ACC of 10 cases is calculated to assess the impact on large-scale circulation in both the Northern and Southern Hemisphere. The average ACC ($\overline {{\rm{ACC}}} $) is calculated as follows:

Here, $Z$ is a conversion coefficient. From Fig. 12, in the Northern Hemisphere, the average ACC of 850-hPa temperature, 500-hPa geopotential height, and 850-hPa geopotential height are better for the moist SV forecast than for the dry one, especially for the 4−10 days forecast. In the Southern Hemisphere, meanwhile, the effects are not as obvious. The increase in ACC indicates that adding moist process to SVs is beneficial to medium-term large-scale circulation.

Figure 12.  ACCs for the Northern Hemisphere (left) and Southern Hemisphere (right): (a, b) 500-hPa geopotential height (units: gpm); (c, d) 850-hPa geopotential height (units: gpm); (e, f) 850-hPa temperature (unit: K). The dashed, red and blue lines denote the control member (Ctr), dry SV forecast (DRY-EM), and moist SV forecast (MOIST-EM), respectively.

The scoring area is the whole of China, covering approximately 2400 stations. Owing to the short duration of precipitation, only the first 120-h precipitation forecast of the 10 cases is evaluated here. The precipitation scores of light rain and rainstorms are basically the same. Here, only the area under the relative operating characteristic curve (AROC) scores of moderate rain and heavy rain are given. The scores are all greater than 0.5, and the closer the score is to 1, the better.

From Fig. 13, in the forecast of the first 48 h, the 10-case statistical precipitation score of the moist SV is better than that of the dry SV from moderate to heavy rainfall. At 72 h, the moist SV has a higher precipitation score in moderate rain than the dry SV, while the precipitation scores in heavy rain decrease. One reason for this result may be that the linearization of large-scale condensation causes the precipitation distribution to become wider rather than narrower. Therefore, under the same water vapor conditions, the moderate rainfall in the moist SV experiment increases, while the heavy rainfall decreases.

Figure 13.  AROC scores of the precipitation ensemble forecasts. The solid line denotes moderate rain (10 mm) and the dashed line denotes heavy rain (25 mm), with red and blue representing the forecasts formed by the dry SVs and moist SVs, respectively.

Based on the GRAPES global TLM and adjoint model with the addition of a linearized large-scale condensation physical process, the moist SVs in GRAPES-GEPS were calculated in this study, and experiments based on 10 cases carried out. The structures of dry and moist SVs were analyzed from the perspectives of the energy norm, energy spectrum, and horizontal and vertical structures.

At the initial time, the proportion of internal energy in the moist SV energy is larger than that of kinetic energy, and the effect of the moist physical process on the SV manifests mainly in the temperature. At the evolved time, compared to the evolved dry SVs, the addition of the linearized large-scale condensation process increases not only the internal energy but also the kinetic energy of the evolved moist SVs. That is, the growth of internal energy promotes the growth of kinetic energy. The addition of the linearized large-scale condensation process increases the perturbation energy in the lower layer of the troposphere. This tends to stimulate the instability of the middle and lower layers.

From the results of the energy spectrum and spatial structure, the propagation energy of the dry SV is increasing (Coutinho et al., 2004; Li and Liu, 2019), and the moist SV also maintains this feature. The release of the latent heat of condensation leads to an energy increase at the small and medium scales. Therefore, after adding the linearized large-scale condensation process, the SVs are more unstable and can grow rapidly, which is beneficial for describing the changing characteristics of a weather system. The evolution of the SVs is synchronous with that of the atmospheric circulation, which is flow-dependent. At the evolved time, the energy of both SVs exhibits large growth, and as energy is transmitted upward and downward, the typical baroclinic structure gradually turns into a barotropic structure. However, the moist SVs are wider, more compact, and can contain more small- and medium-scale information than the dry SVs.

Further, the sensitivity of the moist SV to precipitation was also analyzed in this study. The locations of the two evolved moist SVs are more consistent with the large-scale condensation process than those of the two evolved dry SVs, indicating that the moist SVs will cause perturbations associated with large-scale condensation and precipitation, which is not a characteristic of the dry SVs.

Finally, some ensemble prediction experiments were carried out. The low-level variables of the moist SV forecast are better than those of the dry SV forecast, especially in the first 72 h, whereas the mid-level variables are not different in the early stage, and the high-level variables of the moist SV are slightly lower than those of the dry SV at 48−72 h but higher the rest of the time. The increase in energy caused by the release of latent heat of condensation is reflected mainly in the middle and lower troposphere, and the scale is small. Therefore, the addition of a large-scale condensation process to the calculation of SVs can improve the prediction of short-term weather systems, which is consistent with the conclusion reached by Hoskins and Coutinho (2005). The ACC indicates the moist SVs are favorable for medium-term large-scale circulation. The moist SVs respond well to short-term precipitation according to the precipitation score of the 10 cases. At 72 h, the precipitation score for moderate rain in the moist SV is higher than that in the dry SV, but the precipitation score for heavy rain is not as good as that in the dry SV. One reason for this result may be that the linearization of large-scale condensation leads to a wider rather than a narrower precipitation distribution. Therefore, under the same water vapor conditions, the moderate rainfall in the moist SV experiment increases, while that of heavy rain decreases. Overall, these results suggest that it is beneficial to add moist physical processes to the calculation of SVs.

Acknowledgements. The corresponding author appreciates the National Key R&D Program of China (Grant Nos. 2017YFC1502102 and 2017YFC1501803). This study was also supported by the GRAPES Special Project of Numerical Prediction Center of the China Meteorological Administration.