Given that atmospheric movement is 3D, a numerical model should contain a 3D description of such motion. Owing to limitations in computing capability, one-dimensional (z) and two-dimensional (x, z) models are often used in the early stages of numerical model development. However, the reduced dimensionality causes a series of distortions that make it difficult to correctly consider the wind field, and thus misrepresents the convergence of horizontal airflow. This is particularly true for one-dimensional models. The two-dimensional models can only describe horizontal movement in the positive and negative directions, and cannot describe the horizontal rotation of flow, thus limiting the energy exchange between the ambient wind and the convective circulation. Therefore, to reasonably describe various phenomena of the atmosphere, a numerical model should take into account the 3D structure of atmospheric movement. Since atmospheric phenomena are usually 3D, as mentioned above, it is useful to consider as much of the atmospheric compressibility as possible in cloud-scale convective events, and it is necessary to develop a 3D cloud-scale model for compressible atmosphere, in which the equations should be simplified as little as possible (or little transformation with assumptions, except for improving some functions), and should contain three-phase hydrometeors.
The atmospheric equations can be expressed by using Eulerian or Lagrangian coordinate. But which one performs better? The representation of natural clouds should involve a group of hydrometeor particles growing as they move under certain dynamic and thermodynamic conditions. The movement of these particles is determined by both the dynamic field and the particle movement characteristics such as terminal velocity, mass, shape, and surface roughness; and their growth is determined by water vapor, hydrometeor interactions, and ambient thermal conditions. Although particles occur in groups, the movement of each particle is controlled by the physical properties of the particle itself. The movement of a large particle is mainly affected by its own attributes and the environmental field, but small particles are also affected by the movement of large particles. However, the distance of interaction between large and small particles is difficult to obtain through computational physics and experiments. Generally, it is assumed that the movement of a single particle is not constrained by surrounding particles of the same type. Although the hydrometeor field is a discontinuous field, it is not a dynamic rupture field. It does not block the interactions of cloud macro fields with microphysical ones. To achieve numerical model results that are as close as possible to the physical picture of the real atmospheric movement, the following options are available: the macroscopic field of cloud can be described by the Eulerian method; the background field of cloud, which can be treated as a continuous field, can be described by the semi-Lagrangian method; and the precipitation field can be dealt with via the total Lagrangian method (Xu and Duan, 2001; Xu, 2012).
Since AWM models require a detailed understanding of the structure and evolution of cloud systems, one direct approach is to improve the temporal and spatial resolution of models. First, enabling clouds to be resolved is necessary, but to achieve this, what resolution is needed? Importantly, the resolution of radar observations should be matched, which is currently about 200 m. Therefore, model resolution should be comparable to this level.
There are several advantages to increasing model resolution, including: (1) a reduction in calculation errors; (2) good representation of small-scale information; (3) accommodating and describing the excitation of short waves; and (4) expanding model simulation capabilities for multi-wave (components of a spatial wave group at a given time) and varying-wave (component variation of a wave group at different times) moments. However, meso- and small-scale movements are usually 3D, non-hydrostatic, and nonlinear, which is difficult to control during model integration. Therefore, improving model resolution is not only a single technical strategy; it also involves several relevant scientific and technological issues.
During AWM operations, it is necessary to obtain cloud structure and its evolution in detail from model outputs. Therefore, the use of an explicit cloud microphysics scheme is required to directly describe the cloud–precipitation microphysical processes at play. The study by Zhang et al. (1988) showed that a cloud microphysics (explicit) scheme has advantages for stratiform cloud precipitation, while a convection parameterization (implicit) scheme is beneficial for convective precipitation. A direct (explicit) description of cloud–precipitation processes is suitable for stratiform cloud with a simple dynamic structure, while an indirect (implicit) parameterization scheme is beneficial for convective cloud with complex dynamic structures under coarse-resolution conditions. Recent studies have shown that the description of small-scale movement is poor at high resolutions (grid distance of less than 2 km), implying that some flaws may exist when explicitly describing subgrid-scale movement over a fine grid. This is contrary to the original intention of developing explicit schemes, meaning that it is important to resolve why explicit schemes have advantages for large-scale and uniform cloud systems but weaker ability than implicit schemes when describing medium- and small-scale and non-uniform cloud systems.
One possible and important reason is that explicit schemes are used to describe the clouds directly. Although it is possible to describe the details of cloud processes with an explicit scheme, it is the dynamical processes that support the occurrence, development, and decay of the cloud, and therefore constitute the key factor. When a dynamical framework is appropriate, the general structure of a cloud would become realistic. Only under these conditions does a detailed description make sense. For a dynamical framework for large-scale and uniform clouds, the average airflow in the grid area is close to that of the grid itself, and thus the cloud structure supported by the average airflow is similar to that of real cloud. Therefore, when the cloud structure is generally suitable, describing the fine-scale characterization with an explicit scheme works well. As far medium- and small-scale non-uniform clouds, considerably large differences exist between the average airflow and the real one. As a result, the average airflow over the grids is significantly different to the actual airflow. The difference might be between convective clouds and stratiform clouds, or even between cloud and cloudless conditions. If the model cannot capture the basic characteristics of no-cloud or cloud area, the detailed explicit scheme is meaningless.
Regarding the resolution of models, currently, the subgrid-scale processes are still fairly complex and important, such as the subgrid cloud and its interaction with the relevant scale, the subgrid gravity-wave drag, turbulent diffusion, and so on. Therefore, an explicit cloud microphysical scheme needs to be placed in coordination with subgrid physical processes to describe these physical processes reasonably.
Next, we use an example to illustrate this issue. As shown in Fig. 1a, at the beginning, the average ascending velocity in the grid is zero. This means that there is no dynamical process for cloud formation, and the condition should be no-cloud. However, the average being equal to zero does not mean that there is no ascending movement in the grid. In fact, it can be an average result over a distribution with multiple vertical movements. Since there is indeed ascending movement, there should be cloud formed (Fig. 1). In the same way, if the grid has some uniform ascending movement, stratiform cloud will form. Depending on the intensity of non-uniform ascending movement, mixed cloud or convective cloud should be generated (Fig. 2). Therefore, correctly describing the cloud field depends primarily on a suitably represented airflow field. That is, the simulated airflow field should has the same pattern as that of the actual airflow field.
Figure 1. Potential patterns of vertical motion and cloud distribution under the condition that the gridbox-averaged vertical velocity is 0 m s–1: (a) no cloud; (b) one cloud cell; (c) two cloud cells; and (d) three cloud cells.
Figure 2. Potential patterns of vertical motion and cloud types under the condition that the gridbox-averaged vertical velocity is greater than 0 m s–1: (a) stratiform cloud; (b) stratiform and cumulus mixed cloud; and (c) cumulus (convective) cloud.
Subgrid movement is contained in implicit schemes, which do not simply estimate the formation of cloud only by ascending movement. In fact, a variety of convective parameterization schemes have taken into account the effects of subgrid movement on clouds and precipitation, as well as the influence of subgrid vertical transfer on temperature and humidity structure. In view of this point, although the cumulus convective parameterization is described in an implicit, indirect, and approximate manner, its representation of physical processes is relatively complete. On the other hand, despite explicit cloud microphysical schemes describing cloud processes in detail, they still need to be based on a reasonable and adaptive dynamic framework. Otherwise, fake phenomena might appear, with fine-scale cloud microphysics described but important physical mechanisms missed. Therefore, mixed-description schemes have been introduced—that is, both explicit and implicit descriptions are applied to a scheme (Tiedtke, 1993; Tan et al., 2013). However, this approach should heed the possibility of “repetition.” That is, a cloud process might be considered in both the explicit and implicit schemes.
Experimental studies using hybrid schemes have also shown that the importance of implicit cumulus convection parameterization is significantly reduced when the horizontal resolution of a model is below 5 km (Zhang et al., 1988; Yu and Lee, 2010). Therefore, some researchers have suggested that cumulus convection parameterization schemes should be avoided as much as possible at horizontal resolutions below 5 km, and all cloud–precipitation microphysical processes should be described by the explicit scheme (Zhang, 1998).
From the above analysis, it can be concluded that when the difference between the average grid movement of a model and the actual movement is very small, the model provides a movement style that is close to that of the real atmosphere. This indicates that the dynamic framework of the model is correct and reliable. In such a case, a good explicit cloud–precipitation scheme may achieve decent results (both macro- and micro-fields are accurate). When the macroscopic dynamical field is inappropriate, even though the implicit convective parameterization schemes enable approximate representation of subgrid movement, they are still not capable to produce the specific subgrid movement. This requires increasing the model resolution as much as possible to modify the subgrid movement into a grid-distinguishable movement, and to reduce the effects of subgrid movement.
However, many studies have indicated that increasing the model resolution—especially up to around 1 km—does not definitively improve simulation results (Schwartz et al., 2014; Seiki et al., 2015). The reason for this is that there are problems in the existing dynamic framework design and numerical computation method, which cannot correctly describe the movement episodes with a spatial resolution of less than 2 km. This is possible because the equations are nonlinear, and accurate numerical solution is difficult to obtain. Therefore, a suitable numerical integration scheme, which has some conservation requirements such as mass conservation and energy conservation, is required. For weather and climate models, such a requirement may be satisfied with a resolution of around 2 km. For AWM models, however, there may be other requirements at a resolution of less than 2 km, such as whether the components of these conservative quantities are appropriate. Momentum conservation means that the momentum of all scale movements is constant. However, in the case when the momentum of large-scale movement is marginally large, and that of small-scale movement is marginally small, or vice versa, the total momentum might still be conserved. In such a case, the composition of the momentum component of the movement is different, or the characteristics of the movement or the dominant scale have been changed; that is, the nature of the movement has been changed, which might result in a distortion of movement. Consequently, “fidelity” is required in a numerical model.
In terms of model resolution, aside from considering the coordination between model dynamic processes and cloud microphysical processes, the effects of model resolution on cloud microphysical and parameterization schemes should also be considered. For instance, the atmosphere may be saturated with a relative humidity of 70% for a horizontal grid distance of 10 km, and the part greater than 70% can be condensed into clouds. However, saturation can only take place under relative humidity up to 100% in the case of the resolution being around 1 km. In addition, the effect of turbulence on cloud microphysical processes should also be considered at resolutions of around 1 km or higher.
In the early stages of numerical model development, the first author of this paper has noted that choosing a differential scheme with short waves attenuated as small as possible, and paying attention to controlling the effects of the waves with wave length of doublet grid distance, play a slight positive role in reducing computational errors when the existing differential schemes are used. However, the idea of “fidelity” had not been proposed for numerical computation at that time.
In fact, a better simulation of wind field relies on a better dynamical framework of numerical models. Generally, it seems difficult to improve the dynamical framework, and only slight benefits can be obtained by improving the dynamical framework. This might be an excuse for shrinking back from difficulties given by the developers of large- and medium-scale models. As for the dynamic requirements of AWM models, regardless of the benefits being large or small in improving the dynamical framework, it is worthwhile improving the dynamical framework when facing such difficulties. At the very least, it can be seen that only a set of complete equations is not enough in terms of the dynamical framework, and a conservative numerical computational scheme for differential calculation of the equations is required too. Zhong (1992a) and Zhong et al. (2002) proposed a constructive principle for spatial discretization in which the idea and method for all-energy conservation are contained. This method deserves attention, and it might be worth trying to apply it to the improvement of the dynamical framework in AWM models.
A conservative numerical computation scheme for spatial discretization was employed in the fidelity model developed by Zhong (1992b). This new method, proposed in the 1990s but received scarce attention, is based on the characteristics of energy conversion in the actual atmosphere. In view of energy balance, the energy conversion and constraint mechanism among the internal energy, gravity energy, internal energy, and latent heating, as well as the energy distribution mechanism between these elements, are reconstructed, so as to maintain the energy and mass characteristics of the element. Compared to the existing dynamic framework of nonhydrostatic models, the newly proposed scheme can effectively suppress the “noise” due to false sources and sinks caused by the damage of energy balance after discretization. Although the “conservation” scheme keeps the shortwaves of atmosphere, the development of false noise may introduce computational problems leading to instability (Fig. 3). Figure 3 compares the analytic solution and the general numerical solution. It can be seen that there is considerable noise in the conventional numerical method. These false and small noises not only distort the physical picture, but also cause problems of calculation instability.
Figure 3. Comparison of (a) a conservative scheme with (b) a conventional scheme (Qing Zhong, personal communication).
It appears that local energy conversion conservation is required at the grid point (i, j, k), rather than only in the full computational domain (Fig. 4). This idea is quite reasonable. The local energy conversion and conservation (fidelity) in a grid means the absence of obvious distortion in each grid, and thus it is possible to retain conservation in all fields in the whole domain.
Figure 4. Flowchart of a conservative scheme for local energy transfer at the grid (i, j, k) (Qing Zhong, personal communication).
As for the fidelity model discretization design, it is proposed that the local energy conversion should be satisfied at the grid point (i, j, k) as follows:
where K is kinetic energy, Φ is the gravity potential, T is temperature, F is the external force, V is wind speed, Q is specific humidity, cυ is specific heat at constant volume, cp is specific heat at constant pressure, and ρ is air density.
Guaranteeing the stability of the calculation and maintaining the energy characteristics in a grid enable the accurate description of small-scale components and multi-scale structure. It also helps to avoid the weakness of component variability due to excessive numerical smoothing and additional dissipation caused by artificial treatments. The preliminary tests of Zhong (personal communication) show that the stability of discrete integration was improved because of the improvement in the local energy conversion and conservation in a grid, with an effective resolution of 3–4 grids. In this way, the new scheme makes the discrete integration stable through satisfying the necessary and sufficient condition. On the other hand, it is noted that the effective resolution of the WRF model is 5–7 grids, the MM5 model is 8–9 grids, and the ECMWF model is 10 grids. Therefore, this approach should be taken into account in further research and experiments by AWM numerical model developers.
Atmospheric movement alternates between quiet and drastic change, and secondary small-scale movement is often accompanied by transitions between these states. However, small-scale movement is difficult to deal with in numerical models, and has been removed because it seems that small-scale movement makes no sense for weather or dynamical processes as viewed in the past. But is this really the case? In reality, it does not seem so. Next, we provide an example to explain this issue.
Figure 5 is an example of a closed streamline, with its possible distorted patterns. Figure 5a shows that when the air rises due to buoyancy, there is a “closed convection circulation,” which is a new component on the move-ment scale and is a small-scale movement. If the component is eliminated as noise in the calculation, the “embryonic convection circulation” initializing convections might be choked. Even though it is not eliminated, the movement of the closed convection might be replaced by open convective airflow or wavy patterns due to unconsciously handling of the small-scale movement or its distortion (Figs. 5b, c). From the structural characteristics of the flow patterns, it is seen that there are three emergent flow patterns that are essentially different in nature. For example, differences exist in their maintenance and development mechanism, as well as advantageous conditions. Besides, the range for unstable energy collection, the evolution of the final state, and the accompanying weather phenomena, are significantly different. If this were true in natural clouds, it would be a highly meaningful dynamics topic for numerical model development.
In addition to improving the dynamic framework, existing explicit microphysical schemes also need to be verified and improved. As far as high-resolution model results are concerned, there are three possible reasons for simulation errors. First, the model dynamical processes fail to match the cloud microphysical processes. Second, the parameter values in a cloud microphysical scheme mismatch the model resolution. Third, even though the coupled cloud microphysical scheme is complete, there are several steps to link dynamical processes with cloud microphysical processes. Therefore, are the rules employed in a model the same as those in natural clouds? How much of an influence does inconsistency in the model have? It is necessary to develop natural cloud–precipitation processes by combining field observations, radar measurements, satellite retrievals, and other data, and then compare these with numerical simulations for investigating the differences between simulations and observations and propose new methods to match dynami-cal processes with cloud microphysical processes. Additionally, model results and associated diagnostic information may be used to predict the evolution of cloud microphysical processes.
One important question is how to evaluate existing cloud microphysical schemes. Comparing the modeled precipitation with ground observations of precipitation is the most simple and common method. However, this method only understands the terminal state of the cloud microphysical processes, and is unable to investigate the evolution of cloud and precipitation processes. The radar and satellite observations make up the defect in terms of temporal and spatial resolution of ground observations of precipitation to some extent. Therefore, AWM numeri-cal models require more in-depth explorations and applications of the observations from radar, satellite, conventional ground instruments, and aircraft data, than those of weather and climate models.
Medium- and small-scale weather systems are the product of macro- and micro-field interactions in the atmosphere. Consequently, there are feedbacks between dynamical processes and cloud microphysical processes and, in fact, the feedback from cloud microphysical processes to dynamical fields is stronger. In addition, such a feedback plays a major role in the direction of evolution of medium- and small-scale weather systems, and the feedback pathways vary in various ways. As for the description of cloud–precipitation processes, explicit schemes offer a variety of coherent pathways between the various links. It varies as to which link plays the major role and which downstream link it evolves into. Natural evolution in clouds is one of several possible paths, and there are significant differences in the final state among the various evolutionary pathways. Therefore, it is difficult to use a single-cloud microphysical scheme to describe all types of cloud–precipitation microphysical processes. This might be the main reason for the model uncertainties caused by cloud microphysical processes and their feedbacks.
More recently, Yin et al. (2015) pointed out that it is necessary to reveal the evolution of the dominant cloud microphysical process forming severe weather systems to improve numerical simulations of heavy rainfall occurrence and development. The development of “intelligent” cloud microphysical schemes that make transformation of cloud microphysical processes from “hard rules” to ones close to “natural transformation” is required so as to make the transformation of cloud microphysical processes follow their own natural tendency in numerical models. It is well-known that the evolution of cloud–precipitation microphysical processes is variable, with many possible pathways. However, defining the evolution of cloud–precipitation microphysical processes in numerical models is based on a few cases of observations, theoretical research, or experimental results, which greatly limits the development of cloud–precipitation in numerical models. Therefore, it is necessary to combine multi-source observations for various weather systems to analyze the evolution of cloud–precipitation processes under different macro- and microphysical conditions. Based on these results, a variety of options could then be designed in a cloud microphysical scheme. By doing so, cloud microphysical processes in a numerical model can be selected properly among the various approaches according to the cloud–precipitation microphysical properties and macroscopic conditions provided by the numerical model.
Hydrometeor particles comprise cloud droplets, raindrops, ice crystals, snow, graupel, and hail. It is difficult to consider the effects of particle phase, shape, and density by classifying them only according to their mass or scale, if not sorting the hydrometeor particles into seve-ral types. To distinguish the phase, is it only necessary to divide the particles into two types (liquid and ice particles)? The shape of liquid particles can be assumed to be quasi-spherical, and the density is equivalent to water. The shape of ice particles is too diverse, making it difficult to scrutinize. However, the conditions for dry or wet growth can be used to determine whether ice particles are dry or wet (water), and thus to diagnose the density. Can solid particles be divided into ice crystals, snow, graupel or hail according to scale, density, and other characteristics? In fact, it is not necessary to divide hydrometeor particles into ice crystals, frozen drops, snow, graupel, and hail by the artificial hard rules given by the scheme designer, since such a classification increases the tasks that involve equations describing the interactions among the ice particles in a cloud microphysical scheme, and thus hinders the natural conversion between the ice particles.
AWM numerical models require a better scheme for initial and lateral boundary conditions than those of weather and climate models. For the initial conditions, objective analysis should pay attention to the structural characteristics of medium- and small-scale systems, and data at short distances should be used to fill the grid field as far as possible to retain medium- and small-scale information. Figure 6 shows the fitting curves using data at long distances and short distances. It is indicated that the fitting with data at short distances highlights the characteristics of medium- and small-scale systems.
Figure 6. Schematic diagram of the natural distribution of data (solid line), and the fitting curves using data at long distances (dashed line) and at short distances (dot-dashed line), respectively.
Regional models involve issues related to boundary conditions. In particular, the requirements of a wide movement scale spectrum and high fidelity are needed for weather models, and thus false wave information should be prevented. Therefore, thoughtful consideration is required for the processing of boundary conditions. Our previous tests indicate that several commonly used boundary treatment methods are far from perfect, and thus the suspension boundary and the hierarchical boundary are designed for this purpose. Large-scale movement from boundary inflow may be blocked at the boundary. Therefore, the suspended boundary method was utilized (Xu and Wang, 1988). At the outflow boundary, parasi-tic short waves, wave reflection, or standing waves can often be found. Consequently, the hierarchical boundary scheme was used to “escort” false waves (Xu, 2014).
|Huanbin XU, Jinfang YIN. Key Issues in Developing Numerical Models for Artificial Weather Modification[J]. J. Meteor. Res., 2017, 31(6): 1007-1017. doi: 10.1007/s13351-017-7113-3|
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