To better identify more details and local characteristics of the convective-scale precipitation, a high-resolution numerical simulation is necessary to resolve the target region. WRF (weather research and forecasting, V3.7.1) model is configured with two-nested domains with resolutions of 18 km (505×290) and 3 km (655×547). The GFS (global forecast system) reanalysis data with 0.5° resolution and 6-h interval are used as the initial and boundary fields. In this study, we use Yonsei University PBL scheme (YSU). Other physical parameterizations include the Lin scheme, RRTM (rapid radiative transfer model) longwave radiation and Dudhia shortwave radiation schemes,^[12] and the Noah LSM.^[13,14] The Grell–Devenyi ensemble cumulus scheme^[15] is used for the outer domain. The simulation is integrated from 18 UTC 7 July 2013 and lasts for 61 h with the first 6 h as the spin-up time. The model outputs with 10 min interval from the inner D02 domain are utilized to make subsequent analysis. The hourly precipitation data at a 0.1° horizontal resolution observed by automatic weather stations (AWSs) in China merging with the climate prediction center morphing technique (CMORPH) satellite data^[16,17] are used to illuminate the observed precipitation pattern during the event.

To explore the distribution modality of the local total EA related to severe weather system development, the TE budget equations both in constant density fluid and in variable density fluid^[2] are revisited and discussed again.

2.2.1. The local TE budget equation in a variable density fluid

From the momentum equation in the inviscid and compressible fluid

where ρ and p denote density and pressure, v represents the three-dimensional velocity vector, and Φ is the potential for any conservative force per unit mass (e.g., gz for a uniform gravitational field, with symbols g and z representing the gravitational acceleration and geopotential).

By defining the kinetic energy density (or energy per unit volume) K = ρυ²/2, and taking dot product with υ, equation (1) becomes an equation indicating the change rate of K, yielding

For the internal energy (I = C_vT, with C_v and T denoting the specific heat of dry air at constant volume and the temperature, respectively) and potential energy (Φ) evolutions at an adiabatic atmosphere, they obey

respectively. Combining Eqs. (2)–(4), we obtain

By utilizing the mass continuity equation, equation (5) is rewritten as

where E = ρ(υ²/2 + I + Φ) is the total energy (TE, or total energy density, i.e., the TE per unit volume of the fluid).

Equation (6) is the TE equation in the unforced, inviscid and adiabatic, compressible fluid. Note that, the material derivative of E herein is expanded into local tendency and advection, with the latter integrated into the flux form. One important point is that, the flux form in Eq. (6) includes not only the total energy density flux (υE) but also an additional term (υp). That is to say, as the work is done by the fluid against the pressure force, the energy transfer occurs.

It can be seen that the local change of TE in Eq. (6) is balanced by the flux divergence of (E + p). Thus, a general conservation equation (6) is given, producing a regular and neat form.

2.2.2. Simplify in the constant density fluid

For a constant density (ρ₀) fluid, the momentum equation (1) is simplified as

where ϕ = p/ρ₀, the viscosity is also omitted. The mass continuity equation becomes ∇ · υ = 0.

For the advective term on the left-hand side of Eq. (1), using the identity below,

We have

where ω ≡ ∇ × υ and B = (ϕ + Φ + υ²/2) are vorticity and Bernoulli function for constant density fluid, respectively.

Similar to the operation in variable density flow, taking dot product with υ and including a constant density ρ₀ to yield

Obviously, ρ₀υ · (ω × υ) equals to zero, thus the kinetic energy (with K = ρ₀υ²/2), equation (2) becomes

Considering Φ is time-independent, therefore,

or simplified as where E = K + ρ₀Φ is noted as the TE in constant density flow.

By comparison, equations (13) and (6) are completely identically in form, in which the local budget of total energy is determined by the flux divergence of (E + p) in both constant and variable density fluids.

2.2.3. Difference of local energy budget equation between variable and constant density fluids

2.2.3.1 Difference in form

What differences are there between two expressions? Except that variable density ρ utilized in Eq. (6) is specified as constant density ρ₀ in Eq. (13), the total energy formula E itself utilized in two equations is different in nature. The internal energy vanishes in Eq. (13). Why does this happen? Reverting back to the initial deriving process in constant density fluid, ∇ · υ = 0 is applied to the mass continuity equation. Therefore, as the connective term between kinetic energy equation (2) and internal energy equation (3), ∇ · υ = 0 leads to the internal energy change in Eq. (3) disappearing, which indicates that both kinetic energy equation (2) and potential energy equation (4) are sufficient to completely determine the evolution of a system. Just due to this important point, the internal energy tendency is excluded in Eq. (13) for constant density fluid.

2.2.3.2 Difference of the integral of energy in a closed domain

More traditionally, given a spatial domain with rigid boundary condition and without mass flux, it is usually the integral operator but not the local tendency of energy is carried out to investigate the rate of change of TE within the closed volume. From Eqs. (6) and (13), the divergence flux term vanishes by adopting integral over an enclosed volume with rigid boundaries, implying that the TE is conserved for whatever constant or variable density fluids. However, for energy components within the closed domain, energy exchange occurs among kinetic, internal and potential parts in variable density flow, while the case is not the same for the constant density atmosphere. This is because the integral of kinetic energy and gravitational potential energy in constant density fluid are written as

According to the divergence theorem, the kinetic energy conservation is satisfied within the closed volume. Note that is a constant and not impacted by a redistribution of flow. Therefore, no exchange presents between kinetic energy and potential energy within a closed constant-density domain, and the kinetic energy itself remains conserved.

2.2.3.3 Difference of usage for time-varying case

The derived TE equations (6) and (13) play roles in two completely different kinds of environmental atmospheres. Generally, the former should have wider application regimes because of its less restriction. However, the latter is simpler to use under some suitable approximation and assumption. For example, the Boussinesq equations are enough to derive TE budget in an incompressible fluid. At this time, the internal energy is divorced from the other components of energy, therefore this term is absent in Eq. (13). Thus, the simplified version of local TE equation (13) exerts its maximal advantage under Boussinesq approximation. In the subsequent analysis, more general usage for time-varying case in variable density fluid will be illustrated as an example.

To investigate the local total EA for a real severe weather event, a heavy precipitation that occurred in western Sichuan near the transition zone between Sichuan Basin and the TP from 8 to 10 July 2013 is generally described. Due to the complex local geography and the special geological structure over mountainous areas, mudslides broke out accompanying the rainstorm, causing casualties and economic losses. Due to the strong local characteristics, high-resolution numerical simulation is performed to resolve the finer-scale processes.

The circulation backgrounds and synoptic situations are analyzed in Ref. [18]. The precipitation occurred in the western Sichuan basin to the east edge of TP (Fig. 1(a)). The simulated precipitation has similar pattern and magnitude of order with the observation (Fig. 1(b)), except that a little stronger simulated precipitation presents north of 32° N. More local details of precipitations are resolved in Fig. 1(b).

Fig. 1.

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Fig. 1. (a) Observed and (b) simulated total rainfall amount (mm) over western Sichuan Basin from 00 UTC 8 to 06 UTC 10 July 2013. The terrain height above 3 km is outlined; the evolution of area-averaged (c) 10 min precipitation intensity (mm 10 min−1) and (d) vertical motion intensity at 7.58 km level (m·s−1) from 00 UTC 8 to 06 UTC 10 July 2013. Arrows represent the valley/peak times (T1, T3/T2, T4) of precipitation and convection evolutions.

From the evolution of the area-averaged hourly rainfall intensity (Fig. 1(c)), two unambiguous peaks present at 06 UTC on 8 and at 09 UTC on 9 July 2013. Both p.m. peaks accompany twice as large amplitude growths of the vertical motion (Fig. 1(d)). The variation of rainfall intensity is basically consistent with the swing of vertical velocity intensity from 00 UTC 8 to 06 UTC 10 July 2013 (Figs. 1(c) and 1(d)), demonstrating their close correlation. Thus, the initiation and movement of convection may be a key for identifying the convective-scale precipitation development. Based on the simulation and preliminary analyses for the heavy precipitation process with warm vortex,^[18] the local total EA related to the convection development and precipitation growth are analyzed by utilizing Eq. (6) from the viewpoint of energy budget.

Similar diurnal cycles for the intensity evolutions of precipitation and vertical motion emerge on 8 and 9 July 2013. Correspondingly, what anomalies do the energy distributions take on during the evolutions of convection and precipitation evolutions? Figures 2(a)–2(d) show the vertical distributions of TE perturbation from background separation along the strong convection centers, representing the total EA at four typical times (T₁–T₄ shown in Fig. 1(d)).

Fig. 2.

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	Fig. 2. The vertical distributions of total energy perturbation from background separation along the strong convection centers (31.2° N herein), representing the total energy anomaly (contour, kg·m−1·s−2). The vertical motion is superposed with shaded (m·s−1) (a) at valley 1 time; (b) at peak 1 time; (c) at valley 2 time; (d) at peak 2 time.

Note that the extraction of perturbation refers to Refs. [19] and [20], in which the background field of TE is not simple time average or spatial mean as usual, but defined as to vary both with space and time. In an easy-to-understand mathematical language, for the TE, E(x,y,z,t) as an instance in this study, its spatial mean over a fixed area (e.g., domain 2 herein) is and its time average over a time window (the whole simulation period except for the first 6 h as spin-up time) is . Thus, , the time average of , is just a function of z. If ΔE is defined as , the deviation from its time average can account for the gradual increase/decrease of the areal-mean TE with time as the system moves. Then the background and perturbation are computed as and . Therefore, E₀ varies with both location and time, containing the averaged spatial pattern plus the change in its mean value with time. The perturbation TE sufficiently embodies the energy anomaly within the simulated domain during the overall period of severe weather process, and also considering the variation of background field with system evolution.

Before system development, there is vast convergence of sinking motions (shaded), except for a weak updraft between 103° E and 104° E below 5 km level (Fig. 2(a)). The positive EA (contour) with the value >100 kg·m⁻¹·s⁻² dominates east of 103° E in middle-lower troposphere at time T₁ (or noted as Valley1 time, below are the same), implying local energy accumulation. Above 14 km level, negative EA spreads widely from west to east.

At time T₂ (or peak 1, Fig. 2(b)), a pair of strong convection cells (>7 m/s) stretch upwards, with two centers at (7 km, 103.5° E) and (11 km, 103.2° E), respectively. The weak sinking motion is located at both flanks of the strong updrafts. The precipitation occurs between two updrafts. The negative energy perturbation less than 300 kg·m⁻¹·s⁻² dominates the regions between 103°–104° E, superposing strong convection, which means local energy consumption to feed the convection growth. Different from the case at time T₁, upper-level positive TE anomaly, rolling out a long way off the center of strong convection, is coupled with the lower-level negative TE anomaly.

On 9th July 2013, another diurnal cycle with one wave peak and one valley of system evolution (Figs. 1(c) and 1(d)) occurs. Correspondingly, a similar positive-negative coupled EA presents at time T₃ (or valley 2, Fig. 2(c)), similar to that at valley 1 time except with a little weak intensity, signifying the re-accumulation of local energy. At time T₄ (or peak 2, Fig. 2(d)), the left-side convection in Fig. 2(b) disappears and is replaced by a wide region of sinking motion (Fig. 2(d)). The leading-side convection and related precipitation propagating to 103.7° E, overlaid by negative EA, is weakened and stretched upwards to mid-higher level. The distribution modality also presents lower-level negative and upper-level positive EA above convection and precipitation, with similar pattern but smaller strength relative to the case at peak 1 (Fig. 2(b)).

From above analyses, a clear image of local total EA is portrayed. At the first valley time, local TE experiences heap up to prepare for the future system development. Then the peak time local energy is consumed to feed the convection growth. Till another valley time of the second diurnal cycle, local energy goes through piling up again. It is stored to supply for the convection re-development of the second time. Undoubtedly, the local energy undergoes great expenditure once again. This is the image of local energy anomaly below the height where updraft could touch upwards. Continuing to extend upwards, an opposite-phase EA spreads above 14 km level, just like a wide cloud anvil in appearance (Fig. 2(b)). Therefore, the local total EA presents lower-level negative and upper-level positive modality at peak time while it transforms into an opposite pattern at valley time, with consistent variation regular in intensity with system evolution.

Since the local TE shows some anomaly and exhibits a meaningful spatial image during the evolution of severe weather, the possible internal mechanism responsible for the local EA variation is necessary to be revealed further by utilizing Eq. (6). To avoid the inaccuracy introduced by time derivative, the flux divergence term in Eq. (6) is computed based on high-frequency model output, 10-minutes intervals herein, to give attribution to local TE change during the severe weather event.

Reverting back to Eq. (6), if the divergence flux term is moved to the right-hand side of this formula, the backhander minus sign is attached thus it becomes −∇ · [υ(E + p)]. This new form is composed of four terms, −υ · ∇E, −υ · ∇p, −E∇ · υ, −p∇ · υ, denoting E and p advections effects for the former two terms and three-dimensional divergences of E and p for the latter two. It is the coaction among them that produces local total EA. Since similar diurnal cycles for the system evolution on 8 and 9 July 2013, the cases at valley 1 (left column, Figs. 3(a), 3(c), 3(e), 3(g), and 3(i)) and peak 1 (Figs. 3(b), 3(d), 3(f), 3(h), and 3(j)) times are used to make analyses as examples. At valley 1 time, positive advections (shaded in Figs. 3(a) and 3(c)) and the alternatively distributed convergence/divergence of E and p (shaded in Figs. 3(e) and 3(g)) clearly present near 103.5° E in the lower troposphere. Their interaction leads to weak positive EA (Fig. 3(i)), implying energy reservation before the system development. At peak 1 time, positive advections of E and p (contour in Figs. 3(b) and 3(d), with solid line representing positive advection) and divergence of E and p (contour, dashed line denoting divergence) overlaid the strong updraft (shaded). Thus, advection and divergence effects have clear task assignments, respectively. One is in charge of supplying energy for the local convection growth, the other expends local energy. After offsetting each other between positive advections and divergences of E and p because of their opposite signs, the remainder parts account for the graphic distribution of local total EA shown in Fig. 3(j). There are scattered negative energy divergence fluxes within updrafts at peak 1 time. At the following time (i.e., 10 min later, Fig. 3(l)), the left-side branch of convective cells is weakened and the convective systems propagate eastwards, there are always fragmented positive divergence fluxes of energy dispersed at the leading side of system propagation.

Fig. 3.

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Fig. 3. The zonal-vertical cross sections of energy divergence flux and its components in Eq. (6) along 31.2° N at valley 1 time (shaded, kg·m−1·s−2, (a), (c), (e), (g), and (i)) and peak 1 time (contour, kg·m−1·s−2, superposed with vertical motion, m·s−1, (b), (d), (f), (h), (j)). (a) and (b) −υ · ∇E; (c) and (d) −υ · ∇p; (e) and (f) −E∇ · υ; (g) and (h) −p∇ · υ; (i) and (j) −∇ · [υ(E + p)]. (k) The three-dimensional divergence (shaded, 10−4 s−1), temperature perturbation (contour, K), and wind vector (m·s−1). (l) The same as that in panel (j), except at 10 min-later time.

To elucidate the relative contribution of several components to total divergence flux of energy, Table 1 quantifies their intensity variations at valley and peak times as illustrations, respectively. From Table 1, two advection terms take positive values within updrafts (in bold font) before and during the simulated period. As for two divergence terms, the sporadic positive and negative signals are alternatively and irregularly presented at valley times, but keep strong negative divergence at peak times within convective regions (Fig. 3), with −E∇ · υ having the largest contribution to −∇ · [υ(E + p)]. To a large extent, the divergence term of E and p are determined in modality by the three-dimensional divergence (shaded in Fig. 3(k)), which closely depends on the patterns of related thermal (contour, showing the temperature perturbation) and dynamic (wind vector) meteorological elements. It should also be noted that these components are larger in intensities at peak 1 time than that at peak 2 time, which is consistent with the system evolution (Figs. 1(c) and 1(d)). Relative to the valley time, two peak times accompany with more energy consumption and supply, producing more EA.

Table 1.

Table 1.

Table 1. The intensity variations (kg·m−1·s−2) and the relative contribution of several components to total divergence flux of energy in Eq. (6) at valley and peak times (shown in Fig. 1(d)) as illustrations, respectively. The characteristic maximal/minimal values within updrafts are denoted in bold font. . Valley 1 Peak 1 Valley 2 Peak 2 −v · ∇E −10−12 −20−120 −5−6 −2−70 −v · ∇p −5−6 −10−60 −3−3.5 −3−24 −E∇ · v −25−25 −240−60 −25−20 −140−80 −p∇ · v −10−8 −90−20 −6−8 −50−30 −∇ · [v(E + p)] −40−30 −300−150 −30−25 −180−90

Table 1. The intensity variations (kg·m−1·s−2) and the relative contribution of several components to total divergence flux of energy in Eq. (6) at valley and peak times (shown in Fig. 1(d)) as illustrations, respectively. The characteristic maximal/minimal values within updrafts are denoted in bold font. .

From above analyses, the irregularly positive–negative alternative signals of energy flux divergence bring more complexity for operational prediction. To simplify, the method of taking the absolute value of energy flux divergence is adopted herein. Thus, EA is given in a generalized sense whatever positive anomaly or negative one should be it, termed as absolute EA. After the absolute-value operator on energy flux divergence, it is convenient to routine application for forecasters to track the system evolution by visualizing the absolute EA. Figure 4 examines the horizontal distributions of absolute EA (Figs. 4(a) and 4(b)), along with the associated precipitation and convection (Figs. 4(c) and 4(d)) at valley 1 and peak 1 times. From Figs. 4(a) and 4(c), precipitation is absent south of 32° N, but the energy begins gathering in the future rainy region. As the precipitation increases and convection bursts (Fig. 4(d)), there are clear EA signals near the rain belt, which indicates that the local energy activity is frequent and the supply/consumption is very active.

Fig. 4.

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	Fig. 4. (a) and (b) The horizontal distributions of absolute total energy anomaly (shaded, kg·m−1·s−2), (c)–(d) along with the associated precipitation (contour, mm) and vertical velocity (with the value > 1 m·s−1 shaded) at valley 1 ((a) and (c)) and peak 1 ((b) and (d)) times.

Fig. 5.

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	Fig. 5. Evolutions of domain-averaged (a) positive/negative total energy anomaly (kg·m−1·s−2) and (b) its absolute value (kg·m−1·s−2) at different height levels.

Figure 5 shows the evolutions of both total EA and absolute EA. After region-averaged operation, the intensity of (absolute) energy anomaly (Figs. 5(a) and 5(b)) is weakened relative to that at peak time (Figs. 3(j) and 4(b)), but the two diurnal cycles with the first stronger (> 60 kg·m⁻¹·s⁻²) and the second weaker (about 45 kg·m⁻¹·s⁻²) are still unambiguously identified. In Fig. 5(a), positive and negative energy anomalies are roughly equivalent in magnitude but with a little stronger positive/negative value at valley/peak time by a subtler survey. The absolute EA (Fig. 5(b)) is more succinctly indicative of the system evolution (Figs. 1(c) and 1(d)), whatever in graphic visualization or from concise concept itself.