When using diagnostic quantities to detect meso- or small-scale precipitation, the most anticipated result is that the diagnostic quantities can grasp all the direct precipitation-producing information within the whole atmosphere. However, due to the strong "noisiness" of the atmosphere, and also some influence from large-scale systems (such as upper-level jets, high-latitude and mid-level troughs etc.), these quantities also show strong anomalies outside of the precipitation region (Ran and Li, 2014). To avoid this problem, an objective analysis technique from (Barnes, 1964) is firstly applied to the model output to filter the large-scale information that may not have direct links with the MCS precipitation. This method, used for this scale separation, can be expressed by the following formulae (Barnes, 1973; Maddox, 1980): \begin{eqnarray} \label{eq22} F_{\rm b}(i,j)&=&\frac{\sum\limits_{n=j-N}^{j+N}{\sum\limits_{m=i-N}^{i+N}{F(m,n)W(m,n)}}}{\sum\limits_{k=1}^K{W(m,n)}} ;\ \ (22)\\ \label{eq23} W(i,j)&=&\exp(-r_{m,n}^2/4a) ;\ \ (23)\\ \label{eq24} F_{\rm o}(i,j)&=&F(i,j)-F_{\rm b}(i,j) ;\ \ (24) \end{eqnarray} where (i,j) is the grid point analyzed by the method, F b is the field after the objective analysis that will be filtered from the original field, F is the original field, and F o is the filtered field that will be used to carry out the diagnosis. N is the radius of influence denoted by the grid number, (m,n) is the latitude and longitude grid points within the radius of influence, rm,n is the distance from (m,n) to the analyzed point (i,j), and a is a constant. The wavelength response function of this method is \begin{equation} \label{eq25} R(\lambda,a)=\exp(-4\pi^2a/\lambda^2) ,\ \ (25) \end{equation} where Λ is the wavelength. Equation (26) shows that the horizontal scale of the field (F b) that we can reduce from the total field is mostly determined by the value of a. In this paper, we adopt a=200, and the corresponding relationship between the response function and the wavelength can be seen in Fig. 3a. According to Fig. 3a, after objective analysis, most large-scale information with wavelength approximately greater than 200 km is contained in F b. This means the field F o F o includes all the meso-γ-scale information and a large part of the meso-β-scale information, which is suitable for our research on the diagnosis of high-resolution small-scale precipitation in an MCS. The technique is firstly applied to the basic variables, including horizontal wind (u,v), potential temperature (θ), GPT (θ*) and specific humidity (q v), and then the analyzed fields u o, v o and θ* o are used to calculated the PD to detect the precipitation area, while other variables (such as q vo) may also be analyzed to help understand the MCS' inner dynamics and thermodynamics.
The relations of the GPV (calculated from the original field), PD (calculated from the original field), and PD o (calculated from the filtered field) with heavy precipitation are compared by calculating the spatial correlation coefficients between these three quantities and the simulated 15-min accumulated precipitation. The term "spatial" refers to the whole area of the model domain [approximately (31°-39°N, 110°-120°E). In addition, the three quantities are integrated in the vertical direction, which can be represented by \(\langle|\ |\rangle=\int_z=0.5\;\rm^z=9.75\;\rm|\ |dz\). This operator is used to obtain a one-level field of the diagnostic variable to correspond to the precipitation data, and also to include all the important information at different levels that may be relevant to the precipitation. Regardless of the spin-up time interval (approximately 0200-0500 UTC), one can see in Fig. 3b that \(\langle|\rm PD|\rangle\) (red line) shows an evidently higher correlation with precipitation than \(\langle|\rm GPV|\rangle\) (blue line), which gives further validation to the application of deformation in precipitation diagnosis, as well as vorticity. On the other hand, after filtering the large-scale information, the correlation between \(\langle|\rm PD_\rm o|\rangle\) and precipitation increases significantly compared with that between \(\langle|\rm PD|\rangle\) and precipitation, with the highest correlation coefficient reaching 0.7. This gives a sound reason for carrying out the objective analysis before using the new quantity describing the small-scale precipitation in the MCS.
Figure 4 shows the horizontal distribution of the 15-min accumulated precipitation and PD (also, a vertical integration of PD absolute values; PD here, and in the following parts, indicates the variable calculated from F o fields) during the banding convective stage and the cluster convective stage of the MCS. As shown in Fig. 4a, at 1000 UTC, two main precipitation regions exist in East China: one located in approximately (35°-36.5°N,116°-119°E), which shows a cluster-shaped pattern; and the other stretching from approximately (34°N, 117°E) westwards to (32°N,112°E), which shows a band-shaped pattern. Correspondingly, strong anomalies of PD in Fig. 4b also present a cluster-band-united distributional pattern. At 1200 UTC (Figs. 4c and d), the band-shaped precipitation has a weakening trend, especially in Anhui, where the precipitation almost disappears. Meanwhile, the cluster-shaped precipitation evolves into two parallel bands (indicated by the red straight line), with high values of PD also organized into two corresponding bands. At 1400 UTC (Figs. 4e and f), the band-shaped precipitation pattern disappears. A cluster-shaped pattern of precipitation remains , and it shows an intensifying trend. Corresponding to this, PD also shows a similar evolution. At 1600 UTC (Figs. 4g and h), with the eastward movement of the MCS system, the latitudinal scale of the precipitation region decreases evidently, and PD also presents a trend to mainly develop longitudinally.
From the above analysis, it can be concluded that the newly derived parameter PD describes the strong rainfall within the MCS well, thus showing great potential to be used in detecting the evolution of small-scale precipitation operationally.
To explain the close correlation between PD and heavy precipitation in the MCS physically, Fig. 5 shows the distribution of PD in the whole atmosphere along the "EPC" location (35.3°N) during the passage of a precipitating cell. From Figs. 5a-d, the most evident distinction between the precipitating and non-precipitating atmosphere is that the PD shows strong anomalies in the whole precipitating troposphere. This means that this quantity can grasp the typical structures of the precipitating atmosphere, as distinguished from the non-precipitating atmosphere. To see these typical structures, PD is written as follows by incorporating Eqs. (8) and (9) into Eq. (21): \begin{eqnarray} \label{eq26} D^2&=&\left(\frac{\partial {v}_{\rm h}}{\partial z}\right)^2\nabla_{\rm h}^2\theta^\ast+E^2\left(\frac{\partial\theta^\ast}{\partial z}\right)^2+ \Bigg[2\left(-\frac{\partial v}{\partial z}\frac{\partial \theta^\ast}{\partial x}-\frac{\partial u}{\partial z} \frac{\partial \theta^\ast}{\partial y}\right)E_{\rm sh}\frac{\partial \theta^\ast}{\partial z}+2\left(-\frac{\partial u}{\partial z}\frac{\partial \theta^\ast}{\partial x}+ \frac{\partial v}{\partial z}\frac{\partial \theta^\ast}{\partial y}\right)E_{\rm st}\frac{\partial \theta^\ast}{\partial z}\Bigg], \ \ (26)\end{eqnarray} where E2=E st2+E sh2 is the total deformation. Equation (27) shows that PD is a complete reflection of vertical wind shear, moist baroclinicity, flow deformation and convective stability of the atmosphere. By analyzing these elements, including the thermodynamic element denoted by GPT and the dynamic element such as deformation and vertical wind shear, physical processes associated with heavy precipitation that are able to be reflected by PD can thus be discussed.
4.2.1. Thermodynamic processes
According to its definition, GPT is the potential temperature multiplied by the condensation latent heat release function, which can be written as \begin{equation} \label{eq27} \theta^\ast=\theta\eta , \ \ (27)\end{equation} where \(\eta=\exp[(Lq_\rm vs/c_p T)(q_\rm v/q_\rm vs)]^k\) is the condensation latent heat release function. The most significant characteristic of GPT is the involvement of the weight coefficient (q v/q vs)k into η. This means that the amount of water vapor that can condense will depend on the current humidity of the atmosphere. In an absolutely dry atmosphere with a weight coefficient of (q v/q vs)k=0, GPT reduces to potential temperature and no moisture condenses. In a saturated atmosphere with a weight coefficient of (q v/q vs)k=1, all the moisture can be condensed and GPT thus becomes equivalent potential temperature. Accordingly, in a neither entirely dry nor entirely saturated atmosphere, GPT assumes only part of the moisture in an air parcel can be condensed by (q v/q vs)k« 1, which theoretically conforms more to the real atmosphere (Mason, 1971). In this paper, as shown above, all the fields are given an objective scale-separation analysis before they are applied, which means the GPT we use for the PD calculation is actually \begin{eqnarray} \label{eq28} \theta_{\rm o}^\ast&=&\theta^\ast-\theta_{\rm b}^\ast=\theta\eta-\theta_{\rm b}\eta_{\rm b}=(\theta_{\rm o}+\theta_{\rm b})(\eta_{\rm o}+\eta_{\rm b})-\theta_{\rm b}\eta_{\rm b}\theta_{\rm b}\eta_{\rm o}+\theta_{\rm o}\eta_{\rm b}+\theta_{\rm o}\eta_{\rm o} , (28)\end{eqnarray} where θ* and η are the GPT and condensation latent heat function evaluated by the model ouput, θ* b and η b are the analyzed large-scale GPT and condensation latent heat function obtained by Eq. (23), and θ o and η o (θ o=θ-θ b, η o=η-η b) represent the small-scale information obtained by filtering θ* b and η b from the original field. From Eq. (29), it can be seen that, after scale separation, θ* o is composed of three parts: the coupling of large-scale potential temperature and the small-scale condensation latent heat function (denoted by θ o1*=θ bη o); the coupling of small-scale potential temperature and the large-scale condensation latent heat function (denoted by θ o2*=θ oη b); and the coupling of small-scale potential temperature and the small-scale condensation latent heat function (denoted by θ o3*=θ oη o).
Figure 6 shows the vertical cross sections of θ* o and its three components at 1400 UTC when a precipitation cell is passing "EPC". The cell is profiled by the 0.5 g kg-1 contour of total hydrometers' specific mixing ratios, and the precipitation caused by it is denoted by the gray bars. As shown in Fig. 6a, within the convective cell, evident horizontal variations of GPT are found. In the mid-lower troposphere below 6 km, "EPC" (118.43°E) is located between a positive GPT anomaly area and a negative GTP anomaly area, which makes a strong horizontal gradient of GPT over "EPC". By comparing Figs. 6b-d to Fig. 6a, it can be seen that θ o1*, which is the coupling of large-scale potential temperature and the small-scale condensation latent heat function, is the leading factor determining θ o*. Since the large-scale potential temperature (θ b, figure omitted) mainly shows a vertical variation, it actually acts as a weight function that makes the low-level information more dominant. The small-scale condensation latent heat function (η o) thus becomes the main factor that influences θ o*. As shown in Fig. 6e, the configuration of positive and negative values of η o is highly in accordance with those of θ* o in Fig. 6a. According to the definition of η o, its distribution is actually related to the moisture content of the atmosphere that can be reflected by the weight function (q v/q vs)k (k is 45 here). As shown in Fig. 6f, evident small-scale relative humidity anomalies appear around the "EPC" location at 118.43°E. The positive anomaly is in the front (east) part of the cell, where warm moist inflow ascends and brings the moisture upwards, indicated by the streamlines in Fig. 6f. The negative humidity anomaly is at the back (west) part of the cell, where the rear-inflow downward motions prevail. Correspondingly, negative anomalies of the condensation latent heat function in Fig. 7e show downward motions, while positive anomalies have upward motions. This means that PD includes the small-scale moisture variation caused by the convective cell inner motions through the condensation latent heat function.
4.2.2. Dynamic processes
With θ o*≈θ bη o, the PSD and PRD can be respectively written as \begin{eqnarray} \label{eq29} S&=&(\nabla\times {v}_{\rm so})\cdot\nabla(\theta_{\rm b}\eta_{\rm o})=\eta_{\rm o}(\nabla\times {v}_{\rm so})\cdot\nabla\theta_{\rm b} +\theta_{\rm b}(\nabla\times {v}_{\rm so})\cdot\nabla\eta_{\rm o}\approx&\theta_{\rm b}(\nabla\times {v}_{\rm so})\cdot\nabla\eta_{\rm o} \ \ (29)\end{eqnarray} and \begin{eqnarray} \label{eq30} R&=&(\nabla\times {v}_{\rm ro})\cdot\nabla(\theta_{\rm b}\eta_{\rm o})=\eta_{\rm o}(\nabla\times {v}_{\rm ro})\cdot\nabla\theta_{\rm b}+ \theta_{\rm b}(\nabla\times {v}_{\rm ro})\cdot\nabla\eta_{\rm o}\approx&\theta_{\rm b}(\nabla\times {v}_{\rm ro})\cdot\nabla\eta_{\rm o} ,\ \ (30) \end{eqnarray} where v ro=(-u o,v o,0) and v so=(v o,u o,0) are transformers of the wind vector v ho=(u o,v o,0). In Eqs. (30)-(31), the term that relates the three-dimensional gradient of the large-scale potential temperature θ b is neglected because of its much smaller magnitude than the other part (figure omitted). With Eqs. (30)-(31), PD can be written as \begin{eqnarray} \label{eq31} D^2&=&[\theta_{\rm b}(\nabla\times {v}_{\rm so})\cdot\nabla\eta_{\rm o}]^2+[\theta_{\rm b}(\nabla\times {v}_{\rm ro})\cdot\nabla\eta_{\rm o}]^2\nonumber\\[1mm] &=&\theta_{\rm b}^2\left[\left(\frac{\partial u_{\rm o}}{\partial z}\right)^2+\left(\frac{\partial v_{\rm o}}{\partial z}\right)^2\right] \left[\left(\frac{\partial\eta_{\rm o}}{\partial x}\right)^2\!+\!\left(\frac{\partial\eta_{\rm o}}{\partial y}\right)^2\right]\!+\! \theta_{\rm b}^2E_{\rm o}^2\left(\frac{\partial\eta_{\rm o}}{\partial z}\right)^2\!-\nonumber\\[1mm] &&2\left(\frac{\partial v_{\rm o}}{\partial x}+\frac{\partial u_{\rm o}}{\partial y}\right)\left(\frac{\partial v_{\rm o}}{\partial z} \frac{\partial\eta_{\rm o}}{\partial x}+\frac{\partial u_{\rm o}}{\partial z}\frac{\partial\eta_{\rm o}}{\partial y}\right) \frac{\partial\eta_{\rm o}}{\partial z}\theta_{\rm b}^2-\nonumber\\[1mm] &&2\left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)\left(\frac{\partial u}{\partial z} \frac{\partial \eta }{\partial x}-\frac{\partial v}{\partial z}\frac{\partial \eta }{\partial y}\right)\frac{\partial \eta }{\partial z}\theta _{\rm b}^2\nonumber\\[1mm] &=&D_1^2+D_2^2 +D_3^2, (31)\end{eqnarray} where \begin{eqnarray} \label{eq32} D_1^2&=&\theta_{\rm b}^2\left[\left(\frac{\partial u_{\rm o}}{\partial z}\right)^2+\left(\frac{\partial v_{\rm o}}{\partial z}\right)^2\right]\left[ \left(\frac{\partial\eta_{\rm o}}{\partial x}\right)^2+\left(\frac{\partial\eta_{\rm o}}{\partial y}\right)^2\right] ,\ \ (32)\\[1mm] \label{eq33} D_2^2&=&\theta_{\rm b}^2 E_{\rm o}^2\left(\frac{\partial \eta_{\rm o}}{\partial z}\right)^2 ,\ \ (33)\\[-2mm]\nonumber \end{eqnarray} and \begin{eqnarray} \label{eq34} D_3^2&=&-2\left(\frac{\partial v_{\rm o}}{\partial x}+\frac{\partial u_{\rm o}}{\partial y}\right)\left(\frac{\partial v_{\rm o}}{\partial z} \frac{\partial\eta_{\rm o}}{\partial x}+\frac{\partial u_{\rm o}}{\partial z}\frac{\partial \eta_{\rm o}}{\partial y}\right) \frac{\partial\eta_{\rm o}}{\partial z}\theta _{\rm b}^2-\nonumber\\ &&2\left(\frac{\partial u_{\rm o}}{\partial x}-\frac{\partial v_{\rm o}}{\partial y}\right) \left(\frac{\partial u_{\rm o}}{\partial z}\frac{\partial \eta_{\rm o}}{\partial x}-\frac{\partial v_{\rm o}}{\partial z} \frac{\partial \eta_{\rm o}}{\partial y}\right)\frac{\partial \eta_{\rm o}}{\partial z}\theta_{\rm b}^2. \ \ (34)\end{eqnarray}
In Eq. (32), PD squared (i.e., PD2) contains several terms that respectively couple vertical wind shear, the horizontal and vertical gradient of the condensation latent heat function, and deformation, comprehensively. Figure 7 shows the vertical distributions of the magnitude of PD, PD2 and the three components of PD at 1400 UTC 17 August 2009, along the same section as in Fig. 5. As in Fig. 7a, within the convective cell enclosed by the red line, strong precipitation occurs, with the 15-min maximum accumulated rainfall amount reaching 60 mm (118.43°E, also the "EPC" location). Over the precipitation region, PD exhibits large positive anomalies below the height of 6 km, which further verifies its ability in diagnosing strong precipitation. Strong centers of PD are mainly oriented in the boundary layer below the height of 2 km (within approximately 118.05°E, 118.5°E and 118.81°E, respectively), with the largest center (approximately 118.5°E) almost collocated with the precipitation center (118.43°E, also the "EPC" location). On the other hand, it seems that PD in the strong precipitation area develops much higher than that in the weak precipitation area, which is likely due to the deep convection in the strong precipitation area. As in Fig. 7a, the convective cell over the "EPC" area reaches up to a height of 14 km and positive anomalies of PD extend up to 6 km. This is also a factor that contributes to the large vertically integrated absolute PD values in the strong precipitation area (Fig. 4f). Comparing Figs. 7a and b, we can see that PD and PD2 show very similar distributions, which then gives us a reasonable basis upon which we can use PD2 to analyze the physical processes contained within PD. According to Figs. 7c-f, the three components of PD2 basically have comparable magnitude over the precipitation area, which implies an equal importance of containing vertical wind shear, a horizontal and vertical gradient of the condensation latent heat function, and deformation, in precipitation diagnosis. The process associated with the condensation latent heat function has been explained in the previous section. In this section, we focus on the dynamic processes associated with vertical wind shear and deformation.
Since vertical wind shear describes the vertical structure of the flow in a cell, as an example, Figs. 8a and b show the distributions of the original zonal wind and perturbation potential temperature (obtained from the model output), with the streamlines superposed along the same section as in Fig. 7. As indicated by the solid arrows in Figs. 8a and b, the same as in Fig. 7, two significant flows prevail in the convective cell: a westward ascending inflow that moves into the cell in the boundary layer and out of the cell in the upper level at the front (east side) of the cell; and an eastward descending inflow that penetrates into the cell from the 6-km mid-levels at the back (west side) of the cell and pours down to surface. Corresponding to this flow structure, vertical wind shear presents three main large-value regions in the cell, denoted by "I", "II" and "III". Region "I" is associated with wind streaks caused by the descending flow and the front ascending flow. As in Fig. 8a, a positive zonal wind center (indicated by "H") can be found at the boundary layer at approximately 118.5°E, which is likely related to be related to the strong pressure gradient induced by the cold pool. On the other hand, in the front of the cell is a strong negative zonal wind center (indicated by "L"), with a clockwise vertical circulation superposed. With strong wind centers mostly confined in the levels below 2 km, strong wind shear thus develops. Region "II" is associated with a positive wind streak at a height of about 6 km, when the mid-level eastward flow penetrates into the cell. The flow is also denoted as the rear inflow jet, which is quite important to surface high winds (Markowski and Richardson, 2010). Region "III" is related to the upper outflow. Among these three regions, as can be seen by comparing Fig. 7c and Fig. 8c, Region "I" in the boundary layer is the most dominant shear contained in PD, because of its coupling with the condensation latent heat function (Fig. 6e) and large-scale potential temperature. This means, due to vertical wind shear, PD includes the process of the interaction between cold-pool outflow and warm inflow in the boundary, which is one of the most important processes in triggering and maintaining convection systems within MCSs.
Since deformation describes the horizontal structure of the flow, Fig. 9 shows the horizontal distributions of the wind stream, the deformation tick marks, the total deformation, and its two components at z=1.25 km and at 1400 UTC in the precipitation area of focus. The "deformation tick marks" are a series of short straight lines, which are like wind vectors but without arrows. The orientation of the tick marks is parallel to the dilatation axis, and their length is the magnitude of the total deformation. As shown in Fig. 9a, the wind over the precipitation region presents a significant deformational pattern. The deformation magnitude represented by the length of the deformation tick marks (Fig. 9b) is obviously larger in the precipitation area than the surrounding non-precipitation area. It is also noted that, in Fig. 9b, in the strong precipitation area of focus (black box), the deformation tick marks are consistently south-north oriented, largely parallel to the y-axis, which further verifies the confluence pattern of the flow. West of these south-north oriented tick marks is the westerly divergent outflow formed by the downward cold air at the back of the convective cell (Figs. 8a and 9a). East of them is the easterly caused by the inflow in the front of the convective cell (Figs. 8a and 9a). As noted above, the confluence of these two flows is the basic factor that produces the heavy precipitation. The overlap of the band structure of the precipitation area and the deformation pattern of the flow shows that this rain-producing process is contained in PD by the deformation.