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In numerical weather prediction and atmospheric modelling in general, the question of predictability has received considerable attention since the work of Thompson (1957) and Lorenz (1963, 1969). The investigation of the potential for growth of small errors superimposed on certain atmospheric flows is the prime subject of atmospheric predictability studies (Zhang etal., 2003; Tan et al., 2004), as well as studies of atmospheric instability (Molteni and Palmer, 1993). If a system is particularly unstable, small errors become amplified in model forecasts and severely limit the accuracy of the prediction of the flow. Mu et al. (2004) considered the term ‘predictability’ as the analysis of factors and mechanisms that yield the uncertainties of forecast results, and exploration of methods and approaches to reduce these uncertainties.
Predictability studies can be performed by the impact method or optimal perturbation method. In the first method, the value of an input parameter is changed by a small amount, and a new model solution is evaluated for this new parameter value and then compared to the control solution (Thompson, 1957; Zhang et al., 2002, 2003). This method is applied successfully in studies for which the number of input parameters is small.
The linear singular vector (LSV) is one of the classic optimal perturbation methods whose evolutions are well described by the tangent linear version of the nonlinear model (Lorenz, 1965; Molteni and Palmer, 1993; Ehrendorfer and Errico, 1995; Ehrendorfer et al., 1999; Frederiksen, 2000). This method is based on the assumption of the validity of the tangent linear model. However, Zhang et al. (2002) found that error growth rates at small scales depend on the difference amplitude, suggesting that nonlinearity is important. Errico et al. (2002) declared that the evolution of the perturbations with structures consistent in size and shape with initial condition uncertainty can be substantially nonlinear, even after only a single day's growth. Adjoint technique fails to reproduce fully the evolution of forecast differences from the convective scale to larger scale that is present in their experiments (Zhang et al., 2003). The maximum possible error growth over a finite time interval in which fully nonlinear error evolution is considered may be assessed by solving a nonlinear optimization problem (Ehrendorfer et al., 1999).
Mu and Duan (2003) proposed a new concept called conditional nonlinear optimal perturbation (CNOP), which can be obtained numerically by solving a nonlinear optimization problem. Mu and Zhang (2006) pointed out that, when the initial perturbation is small enough and the evolution time is not too long, CNOPs are very similar to LSVs. On the other hand, when the tangent linear model fails to approximate the original nonlinear model because of the effects of nonlinearity, there exists a remarkable difference between CNOPs and LSVs. Jiang et al. (2008) revealed that CNOPs, similar to LSVs, have the property of norm dependence. Afterwards, Rivière et al. (2008) explored the behaviour of CNOPs in a baroclinic unstable flow, and pointed out that CNOPs essentially differ from LSVs in the presence of a positive zonal-mean shear at initial time and in a broader meridional extension. Furthermore, Rivière et al. (2009) described a new nonlinear technique to compute the sensitivity of synoptic perturbation growth to environmental moisture, which considered the impact of moisture on the growth of CNOPs. Till now, the CNOP method has been widely used in many atmospheric and oceanic research fields (Duan et al., 2004; Mu and Jiang, 2008; Mu et al., 2009).
As a midlatitude synoptic system, cut-off low-pressure is frequently observed on the south side of the west-wind belt in the Northern Hemisphere. Hsieh (1949) pointed out that the conditions for the formation of such a cold vortex over North America were found to be a rapid intrusion of cold air into the upper west-wind belt. Chen and Chou (1994) conducted an investigation of 60 cold vortices over the western North Pacific during the warm seasons of 1982–1987. One of the interesting findings of that study was the existence of a northwest-type and/or a south-type jet streak for the majority of cold lows. The northwest-type jet mainly formed in the northwest sector of the vortex and then propagated downstream and dissipated in its southwest sector, whereas the south-type jet usually formed in the southwest and southeast of the vortex and then propagated downstream and dissipated in its northeast or northwest sector. Zhao and Sun (2007) investigated cut-off low-pressure systems during the period June to August 1998 leading up to the record flood in Northeast Asia and pointed out that baroclinicity played an important role in the initiation of cut-off low-pressure systems. Hu et al. (2010) investigated the seasonal climatology of cut-off low-pressure systems and associated precipitation patterns over northeast China during 1979–2005. It seems that most work on cut-off low-pressure systems focused on the climatological and synoptic characteristics of these systems (Zheng, 1992; Sun etal., 2002; Wu et al., 2010). However, the forecasting of the formation of cold vortices and their associated precipitation is still an important and difficult issue for operational forecasting researchers. On the basis of these considerations, it is necessary to explore the impact of initial errors on the evolution of cut-off low-pressure systems over northeast China so as to better understand the forecast error sources and the related initial error evolution mechanisms.
In this article, an optimization problem related to the maximum uncertainties caused by initial errors during the formation of a cold vortex is formulated and a CNOP method is adopted to explore the above problem. The organization of this article is as follows. In section 2, the method of conditional nonlinear optimal perturbation is introduced. The dry total energy and moisture energy norms are illustrated. Section 3 contains a brief description of the model used and the cold vortices studied, one 2006 case and one 2007 case. Linear singular vector and conditional nonlinear optimal perturbation are compared in section 4. Furthermore, the properties of CNOP are illustrated. Section 5 explores the linkage of CNOP signals with the background field characteristics. The article is concluded with a summary and their possible implications on mesoscale predictability and targeting observations.
2. Method: LSV and CNOP
In this subsection, we briefly introduce the conditional nonlinear optimal perturbation (CNOP) method used in this article. Assuming that for fixed time T > 0 and initial condition X|t=0 = X0 with X corresponding to the state vector in phase space, the propagator M is well–defined; X(T) = MT(X0) is the solution of the nonlinear model at time T. Perturbations x0 to the initial condition X0 result in deviations from the original trajectory so that the system follows a new trajectory X̃(T) = MT(X0 + x0) = X(T) + x(T). Then, the nonlinear evolution of x0 is defined as x(T) = MT(X0 + x0) − MT(X0).
For chosen norm ||•||, CNOP is the initial perturbation , which makes the objective function J(x0) acquire its maximum value under the initial constraint condition ||x0||2 ≤ δ,
δ is a predefined positive constant representing the magnitude of the initial uncertainty. The local projection operator P is employed in the objective function J(x0), and the P value inside (outside) the verification region is 1 (0) (Buizza and Palmer, 1995).
The spectral projected gradient 2 (SPG2) optimization algorithm (Birgin et al., 2000) can be employed to calculate the minimum value of a function subject to initial constraint conditions when the objective function and its gradient to the initial condition are provided in advance. The great strength of the SPG2 method is its ability to solve problems with higher dimensions. For the problem to be solved here, approximately 50 iterations are generally regarded as the stopping condition for convergence. Generally speaking, a numerical algorithm only yields a local minimal point. To obtain a CNOP, we have tried several different starting perturbations in order to ensure that the value of the objective function of the CNOP is the largest. A detailed introduction to CNOP calculations can be found in the work of Jiang etal. (2008).
Similarly, the linear singular vector, denoted by LSV, is obtained by maximizing a modified version of the objective function J(x0), which is acquired by replacing the nonlinear evolution of the perturbation with its tangent linear evolution. The optimization algorithm employed is still the SPG2 method with the same constraint condition but a smaller value of δ. Due to the linear characteristics of the LSV, multiplying an LSV by a constant yields another LSV. Thus, we can compare the CNOP and LSV with the same initial norms.
2.2. Energy norms
Energy norms have been widely employed in previous studies of singular vectors (SVs) and CNOPs (e.g. Buizza et al., 1993; Buizza, 1994; Ehrendorfer et al., 1999; Mu and Zhang, 2006; Rivière et al., 2008, 2009). Ehrendorfer and Errico (1995) computed singular vectors in total energy norm to investigate the mesoscale predictability for two explosive cyclogenesis events. Kleist and Morgan (2005) used the dry energy-weighted forecast errors as response function to assess the influence of initial condition uncertainties on a snowstorm over the eastern United States. Therefore, in our studies for the chosen cold vortices, the dry total energy (DTE) norm is still tried first in the definitions of the LSV and CNOP, which is the perturbation kinetic plus potential energy about a state with spatially invariant reference temperature (Tr = 270 K) and surface pressure (pr = 1000 hPa). The dry total energy employed both in the initial constraint condition and the objective function to measure the magnitude of the state vector is based on a discrete form of the energy integral (Ehrendorfer and Errico, 1995; Mu et al., 2009):
where cp and Ra are the specific heat at constant pressure and gas constant of dry air, respectively (with numerical values of 1005.7 J kg−1 K−1 and 287.04 J kg−1 K−1, respectively), and the integration extends over the full horizontal domain D and vertical direction σ. The terms u′, v′, T′ and are the perturbed zonal and meridional horizontal wind components, temperature and surface pressure, respectively.
Through the choice of the dry total energy norm, the moisture perturbation field is set to zero at the initial time. However, the perturbation x(T) may carry moisture with the time evolution. The moisture energy for the perturbed mixing ratio contribution is still discussed in the later numerical results, which can be defined as described by Ehrendorfer et al. (1999),
where L is the latent heat of condensation per unit mass (with a numerical value of 2.5104 × 106 J kg−1), and q′ is the perturbed mixing ratio.
3. Description of the model and synoptic situations
The model used here is the non-hydrostatic fifth-generation mesoscale model (MM5) developed by the Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR). The tangent linear model and its adjoint version are also available. The physics include a dry convective adjustment scheme, grid-resolvable large-scale precipitation, the Kuo cumulus parametrization scheme and the high-resolution Blackadar planetary boundary-layer (PBL) scheme.
The horizontal resolution is 90 km, with 57 (55) grid points in the north–south (east–west) direction, and the vertical range is divided into 11 sigma levels, with the top pressure at 100 hPa. The model domains are shown in the following figures, which change with cases.
The first simulation is initialized at 1200 UTC 20 July 2006 with the National Centers for Environment Prediction (NCEP) Final (FNL) operational global analysis data, which is available on 1.0 × 1.0 degree grids. From the analysis, we can find that at the initial time, there exists a depression over northeast China. After 24 hours, this depression weakens and disappears (not shown). At the same time, a cold tongue to its north extends southward and arrives at northeastern China. At 1200 UTC on 22nd, this cold tongue is cut off and a cold vortex matures over this area. Comparatively, the simulated cold vortex is warmer than the observed one (as shown in Figure 1). The second simulation is initialized at 1200 UTC 9 July 2007 with NCEP FNL operational global analysis data reanalysed with conventional observations (radiosondes and surface reports) by the Cressman objective analysis scheme. From the analysis, it is found that with time the depression to the west side of the blocking high extends southeast, and is cut off by the warm air from the south side of the ridge. After 48 hours, a cold vortex located to the south of a blocking high has formed, which comprises a typical dipole blocking pattern. Comparatively, the modelled deepening of the cold vortex is stronger than the observed one, which is shown in Figure 2. Certainly, the 2007 case is warmer than the 2006 case. The above two simulation deficiencies may be partly due to the low horizontal and vertical resolutions of the model and to the use of a simple description of moist physical process, which we consider as acceptable for our studies because we focus on the synoptic-scale systems. The relatively large grid spacing chosen here leads to a large domain size, and thus makes the CNOP computation tractable. Detailed investigations of mesoscale structures associated with cold vortices are outside the scope of the present study. In order to explore the effect of initial perturbations on the cold vortex system, the local projection operator P is employed in the cold vortex area, as illustrated in Figures 1(c) and 2(c).
4. Properties of LSV and CNOP
For the above two cold vortex cases, the first linear singular vectors (LSVs) and conditional nonlinear optimal perturbations (CNOPs) are computed with an optimization time interval of 48 h. The initial constraint condition δ is defined as 0.39 J kg−1, so that the magnitude of the initial perturbations is within the analysis errors. The following numerical experiments show that CNOPs are on the boundaries of the initial constraint. Hence, the norms of the LSVs and CNOPs are both equal to δ.
4.1. The 2006 case
Figure 3 presents the temperature and wind vectors of the CNOP and LSV for this cold vortex at the 0.55 σ-level at 1200 UTC 20 July 2006. It is observed that the CNOP shows strong resemblance to the LSV, both of which are heavily localized in the north of the mature cold vortex. However, the primary perturbation regions of the CNOP are farther to the north than those of the LSV. The temperature perturbation field presents a warm–cold–warm pattern in which the cold centre is the strongest. The strong cold perturbation centre corresponds geographically to a northerly wind perturbation, and the warm centre corresponds to a southerly wind perturbation, which mainly forms an anticyclone vortex in the west and a weak cyclonic vortex to the east. The maximum amplitudes of the horizontal wind and temperature of the CNOP at this level are 0.83 m s−1 and 1.22 K, respectively. Similarly, the surface pressure and temperature of the CNOP are shown in Figure 4. The maximum value of the ps perturbations located in the north of the cold vortex is 0.73 hPa, and the minimum value lies in the area of the mature cold vortex. Additionally, the positive surface pressure perturbations usually correspond geographically to the negative temperature perturbations and vice versa. That is to say, perturbations in the wind, temperature and surface pressure fields were related to each other, a finding which matches the thermal wind balance qualitatively.
In order to gain a full description of the vertical structure of the CNOP, the vertical cross-sections of the temperature and horizontal meridional wind perturbations through the centre (along 56°N) are shown in Figure 5. It is observed that the CNOP has a deep structure that extends throughout the entire troposphere. Moreover, the CNOP exhibits a distinct baroclinic structure, and it can be observed that there is a westward tilt with increasing height, reaching maximum temperature values of 1.22 K at σ = 0.55 and maximum values in the meridional wind field of 0.88 m s−1 at σ = 0.65. The maximum amplitudes of the horizontal meridional wind and temperature are in the mid-level troposphere, as can be seen in Figure 6.
To further observe the impact of nonlinearity on the perturbation evolutions, the temperature and wind vectors of the evolved CNOP and negative CNOP at 1200 UTC 22 July 2006 are shown in Figure 7. Evidently, these two patterns are not symmetrical, an observation that reveals the role of nonlinearity in our system. What is more, as can be observed by comparing Figures 3(a) and 7, there is an increase in the spatial extension of the structures. Similarly, the temperature perturbation field of the evolved CNOP presents a warm–cold–warm pattern in which the cold centre is still the strongest. The warm centre corresponds geographically to an anticyclone vortex and the cold centre corresponds to a cyclone vortex. What is more, by comparing Figures 1(c) and 7(a), we find that the effect of the CNOP on the mature cold vortex is to shift the vortex northward and produce much colder temperatures. In addition, Figure 8 illustrates that at the optimization time the initially tilted structures become quasi-barotropic. That is to say, the baroclinic mechanism plays a part role in the evolutions of these perturbations. Also, the meridional wind perturbations propagate upward and their maximum values are concentrated in the upper troposphere. The temperature perturbations propagate upward and downward, and are vertically distributed across the entire troposphere.
In addition, we calculate the growth rate of CNOPs and LSVs. The growth rate is defined as the ratios of the maxima in Eq. (1) by the value of the initial constraint parameter δ. Numerical results show that the growth rate of CNOP is 380, whereas that of LSV is 342. It is evident that the nonlinear evolution of CNOP over the projection area is larger than that of LSV. Figure 9 presents the vertical profiles of the individual terms in the energy norms of the CNOP at the initial and final times. Note that values at the initial time are multiplied by a factor of 50 in order to better observe their vertical distribution. The initial energy norm of CNOP is defined over the whole domain, whereas the evolved energy norm of the CNOP at the optimization time is defined in the projection area. The figure is labelled with K for the kinetic energy contribution, P for the potential energy contributions, which consist of the contributions of the temperature and surface pressure terms included at the lowest level, and Q for the mixing ratio contribution. At the initial time, the potential energy is concentrated in the middle troposphere and the kinetic energy term is almost homogeneous in the whole troposphere. At the optimization time, the kinetic energy dominates the contribution at the upper levels. The potential energy and the q′ contributions are both very small. At the lower levels, the largest contribution still comes from kinetic energy, followed by the q′ contribution and then the potential energy. The above results illustrate that the perturbation growth during this cold vortex is effectively related to the upper dynamic process and the lower moisture process.
The above analysis reveals the importance of moist physical processes. In order to further observe the evolution of q′, the distributions of q′ at σ = 0.975 after 6 and 48 hours are shown in Figure 10. We find that the mixing ratio perturbations also appear first in the north of the cold vortex, subsequently propagating all around, and that the positive mixing ratio perturbations are in geographic conformity with the positive temperature perturbations and vice versa. This correspondence can be explained by the evaporation and condensation processes. Thus, it appears that all the variables of the initial and evolved CNOPs are related to each other, a finding that matches the above expected balances.
4.2. The 2007 case
Figure 11 shows the temperature and wind vectors of the LSV and CNOP at the σ = 0.55 level at 1200 UTC 9 July 2007. It is found that there is a slight difference in structure and amplitude between the LSV and CNOP. Furthermore, additional perturbations of the CNOP appear in other regions. However, the position of the CNOP, which is localized in the southern area of this mature cold vortex, is quite different from its position in the 2006 case. Similarly, the perturbations in wind, temperature and surface pressure are related to each other, a finding that also matches the results expected due to the thermal wind balance. Warm–cold temperature structures with equivalent magnitudes correspond geographically to an anticyclonic vortex, whereas another cyclonic vortex is observed to the southwest. The maximum amplitudes of the horizontal wind and temperature perturbations at this level are 0.86 m s−1 and 0.95 K, respectively. The surface pressure and temperature perturbations are shown in Figure 12, which presents a strong negative surface pressure perturbation localized in the south of the cold vortex. The minimum value of the ps perturbations is −1.3 hPa. Similarly, the negative pressure perturbations correspond geographically to the positive temperature perturbations.
In order to explore the vertical distribution of CNOP, the vertical structure of the CNOP with respect to the temperature and horizontal meridional wind fields through the centre (along 36°N) is shown in Figure 13. Evidently, the CNOP presents a weak baroclinic structure throughout the entire troposphere, with maximum temperature perturbations of 1.09 K at σ = 0.75 and 0.68 K at σ = 0.25. As shown in Figure 14, the maximum values of the magnitudes of horizontal wind speed and temperature are distributed in the low–middle levels of the troposphere.
The evolved CNOP and negative CNOP over 48 hours at 1200 UTC 11 July 2007 are shown in Figure 15. It is evident that nonlinearity plays an important role in the perturbation evolutions, which makes these two patterns present asymmetric structures. As compared with Figure 11(a), there is also a spatial extension of the structure in Figure 15. For the evolved CNOP field shown at the 0.55 σ-level, the temperature perturbation field still presents a warm–cold–warm pattern; however, the western warm centre is the strongest, and corresponds to a cyclone vortex. Compared with Figure 2(c), it is clear that the effect of the CNOP on the mature cold vortex is to make the depression appear deeper but warmer. Figure 16 shows that the vertical structures appear to be barotropic. Moreover, the temperature perturbations propagate downward and are mainly concentrated in the lower troposphere. The meridional wind perturbations propagate upward and downward simultaneously, finally converging in the surface layer and middle layer.
For this case, we also calculate the growth rate of CNOPs and LSVs. Numerical results show that the growth rate of CNOP is 222, whereas that of LSV is 198. The vertical profiles of the individual terms in the energy norms of the CNOP at the initial and final times are shown in Figure 17. Note that values at initial time are also multiplied by a factor of 50. At the initial time, the potential energy is larger than the kinetic energy term in almost the whole troposphere. At the optimization time, it can be seen that in the lower troposphere the main contributions to the final norm come from kinetic and moisture energy, followed by potential energy. In the upper levels, the kinetic energy is also the main contributor, whereas the moisture contribution plays the least significant role. This finding illustrates that perturbation growth of the cold vortex during July 2007 may be mainly related to the dynamic and moisture processes at lower levels.
5. The linkage of CNOP signals to background field characteristics
The above numerical experiments investigate the properties of the CNOP for two cold vortices over northeast China. In this section, we attempt to explore the related background field characteristics in order to provide a possible explanation for the above CNOP properties. Figure 18 shows the initial and evolved potential vorticity of the simulated background field at the σ = 0.35 level for the 2006 and 2007 cases, respectively. For the 2006 case, CNOPs are mainly located over the region of high potential vorticity associated with the cold tongue. For the 2007 case, CNOPs focus on the region of high potential vorticity associated with the warm air which goes north. The results show that the primary perturbation regions of the CNOP are closely related to the source regions of potential vorticity corresponding to the mature cold vortices, an observation that indicates the evolution of CNOPs may be directly related to the possible triggering or supporting mechanisms associated with the cold vortex. Certainly, this needs to be further explored. The vertical cross-sections (height–latitude) of horizontal zonal wind and temperature advection for the initial basic state are shown in Figure 19. For the 2006 case shown in Figure 19(a), we find that there are two upper-level jet streams and the CNOP is mainly observed in the lower level of the northern branch at approximately 55°N. Furthermore, cold advection plays an important role in this region. For the 2007 case shown in Figure 19(b), it can likewise be clearly observed that there are two upper-level jet streams. However, in this case the CNOP is located in the southern branch at approximately 35°N. Moreover, this upper-level jet is a deeper system which penetrates downward to approximately 750 hPa. Warm advection permeates the entire troposphere in this region. We have already established that the kinetic contribution dominates the energy evolutions of the CNOP for both the 2006 and 2007 cases. Therefore, it seems likely that the jet system contributes more significantly to the above results. Furthermore, the CNOP for the 2006 case is located in the area of cold advection, whereas the CNOP for the 2007 case is located in the area of warm advection, an observation that may explain the different effects of the CNOP on the cold vortices in the perturbed temperature fields. It can also be said that the main region of CNOPs is closely related to the baroclinic region of the basic state. Figure 20 shows the vertical structure of equivalent potential temperature for the initial basic state for the 2006 and 2007 cases. We find that the CNOPs are both located in regions with large horizontal potential temperature gradients, features that are usually related to the frontal zones. The above analysis about the background field characteristics presents a possible explanation for the above CNOP properties.
6. Summary and discussion
In this article, conditional nonlinear optimal perturbations during the development of two cold vortices are computed with a primitive mesoscale model in a dry total energy norm. CNOPs are the initial perturbations that have the largest possible nonlinear evolutions in the targeted areas over a finite time interval. Numerical results show that the CNOPs, like the first linear singular vectors, exhibit the property of localness and baroclinicity. Perturbations in the wind, temperature and surface pressure are related to each other and match the thermal wind balance qualitatively. Nonlinear effects may result in the appearance of additional structures of CNOP compared with LSV. Moreover, the effect of nonlinearity can also be revealed by examining the evolution of the CNOP and negative CNOP, which shows that the evolved CNOP and negative CNOP present evident asymmetric characteristics. In addition, CNOPs have deep baroclinic structures that extend throughout the entire troposphere. With time, the initial perturbations increase their spatial size over larger areas and become quasi-barotropic at the optimization time.
It is a fact that CNOPs are situated in regions of large baroclinicity. For the 2006 case, the CNOP is mainly located in the north of the mature cold vortex, corresponding to the northern branch of a high-level jet with cold advection. For the 2007 case, the CNOP focuses on the south of the mature cold vortex, corresponding to the southern branch of a high-level jet with warm advection. Moreover, the primary perturbation regions of the CNOP are closely related to the source regions of high potential vorticity corresponding to the cold vortices, an observation that indicates that the evolution of CNOPs may be directly related to the possible triggering or supporting mechanisms associated with the cold vortex. The effects of evolved CNOPs on the 2006 cold vortex is to make it appear to shift northward and become much colder, whereas the effect of the evolved CNOPs on the 2007 cold vortex is to make the depression appear deeper but warmer, features that may result from the role of temperature advection of the initial basic states.
In addition, though the dry total energy norm is used in the initial constraint condition and the objective function, the moisture perturbation field appears and develops early in the model's progression. Moreover, moisture energy plays a significant role in the lower troposphere during the perturbation growth for these two cases. However, kinetic energy may be the most important contributor. Therefore, exploration of the CNOP may be helpful for further understanding the evolution mechanism of the cold vortex.
Our main conclusion is that the primary perturbation regions of CNOPs for these two cases are different and that small initial perturbations can increase quickly to larger scale. These findings place severe constraints on the accuracy of forecasts of the position and intensity of cold vortices. Furthermore, the localized structure of CNOPs is helpful for identifying the data-sensitive regions in adaptive observations (Mu et al., 2009). Therefore, we can attempt to conduct targeted observations over these areas to improve the quality of the initial conditions so as to increase our ability to predict cold vortices. Certainly, the present study of the behaviour of CNOPs during the development of cold vortices is only conducted from a synoptic perspective. The associated investigation of the mesoscale predictability of such systems is a necessary next step. Furthermore, Ehrendorfer et al. (1999) and Zhang et al. (2003) have demonstrated that moist physical processes are essential contributors to the rapid development of the initial perturbations in mesoscale predictability. To gain insight into mesoscale convective systems in the cold vortex, the use of other norms accounting for moist physical processes and vertical motion should be considered.
We are indebted to Zhou Feifan, Liu Ying and Xia Rudi for their kind assistance in this work. This research is supported by the National Natural Science Foundation of China (Grant Nos. 40905023 and 40633016). The authors are grateful to the two anonymous reviewers for their valuable comments and suggestions.