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The hourly, 0.25o × 0.25o ERA5 reanalysis data from the European Centre for Medium-Range Weather Forecasts (Hersbach et al., 2020), available on 37 vertical levels, is used in this study to calculate energy budgets and to conduct empirical orthogonal function (EOF) (Weare and Newell, 1977) analysis (which is used to determine the dominant mode of eddy flow during this event). A 30-min, 8-km precipitation dataset, produced by using the US Climate Prediction Center morphing technique (CMORPH), is employed to investigate precipitation variation and to conduct the wavelet analysis (Erlebacher et al., 1996). The CMORPH data is verified via comparison with gauge observations to provide a credible estimate of precipitation over South China (Shen et al., 2010).
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This study uses two types of energies: available potential energy (APE) and kinetic energy (KE), which are effective measurements for the thermodynamical and dynamical conditions of the real atmosphere, respectively. Since torrential rainfall is directly caused by eddy flow, which is embedded in a favorable background environment (Markowski and Richardson, 2010; Fu et al., 2015, 2016), we utilize an energy analysis method that can reflect the variations of both eddy flow and its background environment (Murakami, 2011). The budget equations are as follows:
where the overbar represents an N-day temporal average. A sample basic variable S (e.g., zonal wind u, meridional wind v, vertical velocity ω, specific volume α, temperature T, potential temperature θ, geopotential Φ) can be decomposed as
$S=\bar{S}+S{'}$ , where$\bar{S}$ denotes the background environment (contains signals with periods >N days) and$ S\text{'} $ represents eddy flow (contains signals with periods ≤N days). AM and KM denote APE and KE of the background environment, respectively; AT and KT represent APE and KE of eddy flow, respectively. AI and KI are APE and KE of the interaction flow, respectively, which are developed to link eddy flow to its background environment (Murakami, 2011; Fu et al., 2016, 2018). The physical meaning of each term is shown in Table 1.Term Physical meaning Term Physical meaning AM Background-environment’s available potential energy $G\left({A}_{\mathrm{M} }\right)$&$G(\overline{ {A}_{\rm{T} } })$ Tendencies of AM and AT due to diabatic production/extinction, respectively KM Background-environment’s kinetic energy Conversion term $ C\left(X,Y\right) $ A positive/negative value means energy X/Y is converted into energy Y/X AT Eddy-flow’s available potential energy $ B\left({A}_{\mathrm{M}}\right) $ & $ B\left({K}_{\mathrm{M}}\right) $ 3-dimensional transport of AM and KM by background environment, respectively KT Eddy-flow’s kinetic energy $B(\overline{{A}_{\rm{T}} })$ 3-dimensional transport of AT by both eddy flow and background environment AI Interaction-flow’s available potential energy $B(\overline{ {K}_{\rm{T} } })$ 3-dimensional transport of KT by both eddy flow and background environment KI Interaction-flow’s kinetic energy $ R\left({A}_{\mathrm{M}}\right) $ Effects of vertical heat-transport by both eddy flow and background environment $ F\left({A}_{\mathrm{I}}\right) $ Transport of AI by eddy flow AM→AI→AT Downscale energy cascade of available potential energy $ F\left({K}_{\mathrm{I}}\right) $ Transport of KI by eddy flow AT→AI→AM Upscale energy cascade of available potential energy $ D\left({K}_{\mathrm{M}}\right) $ Dissipation of KM due to friction KM→KI→KT Downscale energy cascade of kinetic energy $D(\overline{ {K}_{\rm{T} } })$ Dissipation of KT due to friction KT→KI→KM Upscale energy cascade of kinetic energy Energy path Conversion/transport that links different energies Baroclinic energy conversion The energy conversion between available potential energy and kinetic energy Table 1. Terminology used in this study, where symbols “→” and “←” denote the directions of energy conversion.
Scale interactions between eddy flow and its background environment can occur directly through energy cascades. The energy cascade terms of the APE are
$ C\left({A}_{\mathrm{M}},{A}_{\mathrm{I}}\right) $ and$ C\left(\overline{{A}_{\mathrm{T}}},{A}_{\mathrm{I}}\right) $ , which have a relation:$C\left({A}_{\mathrm{M}},{A}_{\mathrm{I}}\right)+C\left(\overline{{A}_{\mathrm{T}}},{A}_{\mathrm{I}}\right)= F\left({A}_{\mathrm{I}}\right)$ , where$ F\left({A}_{\mathrm{I}}\right) $ represents transport of AI by eddy flow. When$ C\left({A}_{\mathrm{M}},{A}_{\mathrm{I}}\right) > 0 $ and$ C\left(\overline{{A}_{\mathrm{T}}},{A}_{\mathrm{I}}\right) < 0 $ , AM→AI→AT, i.e., a downscale APE cascade occurs (Table 1). If$ C\left({A}_{\mathrm{M}},{A}_{\mathrm{I}}\right) < 0 $ and$ C\left(\overline{{A}_{\mathrm{T}}},{A}_{\mathrm{I}}\right) > 0 $ , AT→AI→AM, i.e., an upscale APE cascade occurs. Otherwise, no energy cascade occurs. The energy cascade terms of KE are$C({K}_{\mathrm{M}},{K}_{\mathrm{I}})= $ $\overline{u}\text{div}\overline{{u}'{\boldsymbol{u}}'} +\overline{v}\text{div}\overline{{v}'{\boldsymbol{u}}'}-\mathrm{tan}\phi (\overline{u}\overline{{u}'{v}'}-\overline{v}\overline{{u}'{u}'})/a$ and$C(\overline{{K}_{\mathrm{T}}},{K}_{\mathrm{I}})= \overline{{u}'{\boldsymbol{u}}'}\cdot \text{grad}\overline{u}+\overline{{v}'{\boldsymbol{u}}'}\cdot \text{grad}\overline{v}+\mathrm{tan}\phi (\overline{u}\overline{{u}'{v}'}-\overline{v}\overline{{u}'{u}'})/a$ , where div(·) is the divergence operator;$ \boldsymbol{u} $ is the three-dimensional wind vector ($ \boldsymbol{u}=\stackrel{̄}{\boldsymbol{u}}+\boldsymbol{u}\mathbf' $ ); φ is latitude; grad ( ) is the gradient operator; and a is the Earth’s radius. Their relation is$ C\left({K}_{\mathrm{M}},{K}_{\mathrm{I}}\right)+C\left(\overline{{K}_{\mathrm{T}}},{K}_{\mathrm{I}}\right)=F\left({K}_{\mathrm{I}}\right) $ , where$ F\left({K}_{\mathrm{I}}\right) $ represents transport of KI via eddy flow (Table 1). A downscale KE cascade (KM→KI→KT) occurs when$ C\left({K}_{\mathrm{M}},{K}_{\mathrm{I}}\right) > 0 $ and$ C\left(\overline{{K}_{\mathrm{T}}},{K}_{\mathrm{I}}\right) < 0 $ , and an upscale KE cascade (KT→KI→KM) appears when$ C\left({K}_{\mathrm{M}},{K}_{\mathrm{I}}\right) < 0 $ and$ C\left(\overline{{K}_{\mathrm{T}}},{K}_{\mathrm{I}}\right) > 0 $ . Otherwise, no KE cascade occurs. Term$ B\left(\overline{{K}_{\mathrm{T}}}\right) $ can be decomposed into two parts. The first part (FP) is$\text{div}\left(\overline{\boldsymbol{u}}(\overline{u'^{2}}+\overline{v'^{2}})/2\right)$ , which denotes three-dimensional transport of KT by the background environment. It can be regarded as a type of scale interaction between eddy flow and its background environment via transport. The second part (SP) is$\text{div}\left[\overline{\left(\right(u'^{2}+v'^{2})/2+\varPhi{'})\boldsymbol{u}{'}}\right]$ , which represents three-dimensional transport of KT and$ \Phi \text{'} $ by eddy flow. Detailed physical significances, expressions, and spheres of application for Eqs. (1)–(4) are provided by Murakami (2011) and Fu et al. (2016). -
A Morlet wavelet-based analysis (Erlebacher et al., 1996) is applied to the 30-min, key-region averaged CMORPH precipitation (using data from 1 April to 1 October). The result indicates that the torrential-rain-producing eddy flow was mainly governed by a quasi-daily signal (Figs. 4a and b). This is consistent with the findings of Jiang et al. (2017), Chen et al. (2017), Chen et al. (2018a), and Wu et al. (2020a), as they all confirmed the notable diurnal variations of heavy precipitation over South China. As Figs. 4a and b show, the background environment of this event was jointly dominated by a quasi-weekly oscillation (QWO; over 95% confidence level), a quasi-biweekly oscillation (QBO; over 90% confidence), an intraseasonal oscillation (ISO; over 90% confidence level), and oscillations with longer periods (these signals cannot be identified by using a half-year dataset). A schematic illustration of the wavelet analysis is shown in Fig. 4c.
Figure 4. Wavelet analysis of the target region-averaged precipitation by using Morlet wavelet-based analysis. Panel (a) shows the wavelet power spectrum (shading), where grey dots mark the regions exceeding the 95% confidence level, and the two black dashed lines show the 72-h time window of this event. The thick black solid curved line indicates the cone influence outside of which the edge effect become important. Panel (b) illustrates the 72-h mean wavelet power spectrum during the event (thick red line), where purple and green dashed lines outline the notable signals in the background environment, and the thick grey solid line shows the position of the 3-day running mean (which separates the original time series into the eddy flow and its background environment). Panel (c) is a schematic illustration of the wavelet analysis results, where blue bold characters show the typical signals. BE = Background environment; QDS = Quasi-daily signal; QWO = Quasi-weekly oscillation; QBO = Quasi-biweekly oscillation; ISO = Intraseasonal oscillation.
The northern and southern sections of the KR experienced completely different torrential rainfall; one experienced frontal precipitation, and the other experienced warm-sector precipitation. Nevertheless, in terms of AM and KM, the northern and southern sections of the KR exhibited similar features (e.g., vertical distribution, peaks) to those of the whole KR (Figs. 5a and b). This means that the frontal and warm-sector precipitation shared a similar background environment. For the torrential-rainfall-related eddy flow, in the middle and upper troposphere, AT and KT within the northern and southern sections exhibited similar features to those of the whole KR (Figs. 5c and d). However, in the lower troposphere, the KT values within the two sections notably differed from each other (Fig. 5d). This means that the essential differences between frontal and warm-sector rainfall mainly lay in the dynamical features in the lower troposphere. Therefore, a detailed budget on KT is effective to clarify the fundamental differences between frontal and warm-sector torrential rainfall.
Figure 5. Panel (a) illustrates the horizontally averaged AM (J kg–1) within the whole key region (black line) and its southern (blue line) and northern sections (red line), respectively. Panel (b) is the same as (a), but for KM. Panel (c) is the same as (a), but for AT. Panel (d) is the same as (a), but for KT. The thick grey dashed lines divide the vertical levels into the upper (200–450 hPa), middle (450–700 hPa), and lower (700–950 hPa) layers. The ordinate represents pressure (hPa).
As mentioned above, for the middle and upper troposphere, energy features within the KR can effectively represent those within the northern and southern sections. In contrast, for the lower troposphere, the northern and southern sections should be analyzed separately. In order to show the energy budgets at various vertical layers, three equal-weight layers are defined: the upper layer (200–450 hPa), the middle layer (450–700 hPa), and the lower layer (700–950 hPa). Using ERA5 reanalysis data, each term in Eqs. (1)–(4) is first calculated at every grid point and then averaged horizontally within the KR, the northern section, and the southern section, respectively. After that, the corresponding results are integrated vertically in the upper, middle, and lower layers, respectively, to represent the overall energy features within a specified region at a selected layer.
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The upper-tropospheric divergent wind field (Fig. 3a), middle-tropospheric warm advection (Fig. 3b), and lower-tropospheric southwest monsoonal wind (Fig. 3c) provided a favorable background environment② for the torrential rainfall. As Fig. 6a shows, upper-tropospheric KM was maintained through the baroclinic conversion [i.e., C(AM, KM)] of AM to KM, which sustained the anticyclonic divergent wind field. As seen from Fig. 6b, AM and KM in the middle layer were mainly sustained via three-dimensional transport of AM [i.e., B(AM)] and baroclinic conversion, respectively. The former maintained the temperature gradient (temperature decreased towards the southeast), and the latter sustained the westerly wind. These features contributed to the maintenance of middle-tropospheric warm advection. KM in the lower troposphere was sustained via baroclinic conversion of AM to KM (not shown), which maintained the strong southwest monsoonal wind.
Figure 6. Vertical integral of the key-region averaged budget terms of Eqs. (1)–(4) (blue values, units: W m–2) among the upper layer (200–450 hPa) (a) and the middle layer (450–700 hPa) (b). Blue percentages (within small blue boxes) indicate the contributions of the most favorable factors for the maintenance of the background environment (i.e., AM and KM). Green percentages (within small green boxes) indicate the contributions of the most favorable factors for the maintenance of the eddy flow (i.e., AT and KT). Green arrows show net-import transport, blue arrows show net-export transport, purple arrows show conversions between different types of energies, dark red arrows show diabatic production/extinction of APE, curved arrows show the effects of the dissipation terms, and the red dashed curved arrows outline the energy-cascade-related paths that provided energy to KT. The dominant factor for the maintenance/dissipation of each energy is shaded with light purple/blue. The purple arrows may be shaded in two colors because the conversions that they represent are dominant factors for two types of energies. UL = upper layer; ML = middle layer; BE = background environment; IF = interaction flow; EF = eddy flow; BCEC = baroclinic energy conversion, where “+” and “–” represent the release and production of APE, respectively. DSEC/UPEC = downscale/upscale energy cascade.
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In the lower layer, the energy paths that supported KT within the northern and southern sections notably differed from each other (cf., Figs. 7a and b). For the northern section, its dominant energy path was the baroclinic energy conversion of AT to KT [i.e.,
$ C(\overline{{A}_{\mathrm{T}}},\overline{{K}_{\mathrm{T}}}) $ ], which accounted for ~80% of the energy supply for KT (Fig. 7a). This conversion occurred mainly due to a direct thermal cycling (purple ellipse in Fig. 2a) associated with the front in the northern section, during which relatively warm air ascended along the tilted front while relatively cold air descended. This resulted in a net lowering of the air-column’s barycenter and an increase in its wind speed. The remaining 20% energy supply for KT was contributed by three-dimensional transport of KT [i.e.,$ B (\overline{{K}_{\mathrm{T}}} ) $ ] (Table 2). Of this, the transport by the southwesterly wind through the northern section’s southern boundary made the largest contribution (not shown). Decomposing this transport into the FP and SP shows that the transport by the background environment (i.e., FP) made a larger contribution (~58%) than eddy flow (i.e., SP).Figure 7. The same as Fig. 6, but for the northern section of the lower layer (950–700 hPa) (a) and southern section of the lower layer (b), where the red percentages (within small red boxes) indicate the contributions of the second most favorable factors that sustained KT. NS = northern section; SS = southern section.
Locations Features of energy paths (supporting KT) and energy cascades Upper layer (entire key region) AM → AI → AT → KT
(100%)
Downscale APE cascade (Maintains eddy-flow’s temperature gradient)
Upscale KE cascade (Favors BE’s divergent wind filed)Middle layer (entire key region) AT → KT ←$B(\overline{ {K}_{\mathrm{T} } })$ (Westerly wind in BE)
(62%) (38%)
Upscale APE cascade (Favors BE’s temperature gradient)
Upscale KE cascade (Favors BE’s westerly wind)Lower layer (northern section) AT → KT ←$B(\overline{ {K}_{\mathrm{T} } })$ (Southwesterly wind in BE)
(80%) (20%)
Upscale APE cascade (Favors BE’s temperature gradient)
None KE cascadeLower layer (southern section) AM → KM → KI → KT ←$B (\overline{ {K}_{\mathrm{T} } } )$ (Southwesterly wind in BE)
(28%) (72%)
Downscale APE cascade (Maintains temperature gradient in eddy flow)
Downscale KE cascade (Maintains southwesterly wind in eddy flow)Table 2. Three-dimensional energy paths that sustained KE of the torrential-rainfall-producing eddy flow (KT) within different regions [symbols “→” and “←” denote the directions of energy conversion; percentage below a vector denotes its relative contribution; the wind that had dominant effect for
$B(\overline{{K}_{\mathrm{T}}})$ is shown within parentheses to the right] and their energy cascade features (their main effects are shown within parentheses). BE = Background environment.For the lower layer of the southern section, two energy paths supplied energy to the torrential-rainfall-producing eddy flow (Table 2). Three-dimensional transport of KT [i.e.,
$ B (\overline{{K}_{\mathrm{T}}} ) $ ] was dominant, which accounted for ~72% of the energy supply for KT (Fig. 7b). Of this, transport of KT by southwesterly wind through the southern boundary of the southern section contributed the largest amount. Further decomposition of this transport shows that the background environment (i.e., FP) and eddy flow (i.e., SP) accounted for ~68% and ~32%, respectively. The second energy path (~28% in contribution) consisted of two processes (red dashed line with arrowhead in Fig. 7b): (i) the baroclinic energy conversion of AM to KM and (ii) the downscale energy cascade of KE from KM to KT. There is a sharp contrast between the baroclinic energy conversion within the northern and southern sections [cf.,$ C(\overline{{A}_{\mathrm{T}}},\overline{{K}_{\mathrm{T}}}) $ in Figs. 7a and b]; as for the latter, there was a conversion of KT to AT. As seen from Fig. 2b, the southwesterly wind decelerated within the southern section, which led to convergence and lifting of the air-column’s barycenter. This enhanced the eddy flow’s APE and reduced its KE, resulting in the baroclinic energy conversion of KT to AT.In the upper layer, the energy path that supplied energy to KT within the KR contained two processes (red dashed line with arrowhead in Fig. 6a): first, AM was converted into AT through a downscale APE cascade; then AT was converted to KT via baroclinic energy conversion. This was the only energy source for the upper-tropospheric eddy flow (Table 2), whereas other terms mainly acted to reduce KT (Fig. 6a).
In the middle layer, two energy paths maintained KT (Fig. 6b). The baroclinic energy conversion of AT to KT contributed ~62% of the total energy income for KT within the KR. This conversion was closely related to the middle-tropospheric baroclinic shortwave trough (Fig. 3b). The second energy path was the three-dimensional transport of KT [i.e.,
$ B(\overline{{K}_{\mathrm{T}}}) $ ], which accounted for ~38% of the total energy income for KT within the KR (Fig. 6b). Further analysis indicates that westerly winds in the background environment (mainly through the key region’s western boundary) dominated this transport (Table 2). -
This section discusses the contributions of different background environment signals in maintaining the eddy flow’s wind field (in terms of KT) at the lower layer. The Lanczos band-pass filtering technique (Duchon, 1979) is utilized to separate QWO, QBO, and ISO (Fig. 4c) from the background environment (using the hourly ERA5 reanalysis data during the period from 1 April to 1 October). We define the rest of the background environment after removing QWO, QBO, and ISO as the remaining background environment (RBE). As seen from Fig. 4b, the RBE mainly contained signals that had larger periods than ISO. This is confirmed in Fig. 8d, which shows that the RBE consisted of the typical situation of a summer monsoonal season (Ding, 1994; Zhao et al., 2004). Comparisons among QWO, QBO, and ISO (cf., Figs. 8a–c) indicate that QBO had the largest intensities of geopotential height, temperature, and wind filed, whereas, QWO had the smallest intensities. Overall, QWO exhibited a vortex structure around the KR, causing weak perturbations of temperature and wind (Fig. 8a). QBO exhibited a transversal trough structure around the KR (Fig. 8b), with strong northeasterly and westerly wind perturbations appearing in its northern and southern parts, respectively. Temperature perturbations were strong within the northern section, corresponding to the front in this region. ISO exhibited a ridge structure around the KR (Fig. 8c), and its associated wind and temperature perturbations were weak.
Figure 8. The 850-hPa geopotential height (black contours, units: gpm), temperature (red contours, units: K), and wind above 3 m s–1 (a full wind bar is 4 m s–1) of QWO (a), QBO (b), ISO (c), and the RBE (d). Grey shading marks the terrain above 1500 m, and the green tilted box shows the key region, with a green dashed line dividing it into the northern and southern sections.
As Table 2 shows, for the eddy flow in the lower layer of the northern section, scale interactions were not a dominant factor for its sustainment; instead, baroclinic conversion in eddy flow was the governing factor, which accounted for ~80% of its energy source. In contrast, for the eddy flow in the lower layer of the southern section, scale interactions dominated its maintenance (~77% contribution) via two processes: a downscale energy cascade of KE (28% contribution) and a three-dimensional transport of KT by the background environment wind field (i.e., FP; 49% contribution). The FP is further decomposed to compare the transport of KT by respective QWO, QBO, ISO, and RBE wind fields. It is found that the transport due to the RBE wind field (i.e., the southwest monsoonal wind) had the largest intensity (Fig. 9d), whereas, that due to the QWO wind field was weakest (Fig. 9a). Overall, in the lower layer of the southern section, the transport of KT by the RBE contributed the largest proportion at over 60% (Table 3); the contributions of QBO and ISO (Figs. 9b and c) were also notable and accounted for 21.3% and 14.6%, respectively (Table 3). In contrast, QWO showed the smallest contribution to the transport of KT (4.1%).
Figure 9. The 850-hPa KT (shading; J kg–1) and the transport of KT (vectors, units: W m kg–1) by the wind field of QWO (a), QBO (b), ISO (c), and the RBE (d). Grey shading marks the terrain above 1500 m, and the green tilted box shows the key region, with a green dashed line dividing it into the northern and southern sections.
QWO QBO ISO RBE Transport of KT 4.1% 21.3% 14.6% 60.1% C (KM, KI) 3.7% 17.9% 6.6% 71.7% C ($ \overline{{K}_{\mathrm{T}}} $, KI) 9.1% 39.3% 13.2% 38.5% Notes: QWO = Quasi-weekly oscillation; QBO = Quasi-biweekly oscillation; ISO = Intraseasonal oscillation; RBE = Remaining background environment. Table 3. Contribution (units: %) of different BE signals in the transport of KT (i.e., FP), and KE cascade processes at the lower layer of the southern section.
The energy cascade terms of KE in the lower layer are shown in Fig. 10a. A configuration of negative C(
$ \overline{{K}_{\mathrm{T}}} $ , KI) with positive C(KM, KI) is notable in the southern section, whereas, in the northern section, negative C($ \overline{{K}_{\mathrm{T}}} $ , KI) is not obvious. This is a direct reason for why only the southern section exhibited a clear downscale KE cascade. In order to compare the contributions of different background environment signals, energy cascade terms of KE due to respective QWO, QBO, ISO, and RBE are calculated. As Figs. 10b–d show, within the southern section, C($ \overline{{K}_{\mathrm{T}}} $ , KI) generally had a larger absolute value than C(KM, KI). This means that QWO, QBO, and ISO were more important in determining C($ \overline{{K}_{\mathrm{T}}} $ , KI) (i.e., the conversion between eddy flow KE and interaction flow KE) than C(KM, KI) (i.e., the conversion between background environment KE and interaction flow KE). Positive C($ \overline{{K}_{\mathrm{T}}} $ , KI) (i.e., the conversion of KT to KI) due to the RBE (Fig. 10f) canceled out the QWO+QBO+ISO associated negative C($ \overline{{K}_{\mathrm{T}}} $ , KI) within the northern section (Fig. 10e), which is the key reason why no obvious downscale KE cascade appeared in this section.Figure 10. Terms C(KM, KI) (solid blue; 10–5 W kg–1) and C(
$ \overline{{K}_{\mathrm{T}}} $ , KI) (shading; 10–5 W kg–1) calculated by the total background environment (a), QWO (b), QBO (c), ISO (d), QWO+QBO+ISO (e), and the RBE (f). Grey shading marks the terrain above 1500 m, and the green tilted box shows the key region, with a green dashed line dividing it into the northern and southern sections.Overall, for the lower layer of the southern section, the RBE dominated term C(KM, KI), which made a contribution of 71.7% (Table 3). For term C(
$ \overline{{K}_{\mathrm{T}}} $ , KI), QBO exhibited the largest contribution (39.3%), which is 8% higher than that of the RBE. This indicates that both the monsoonal wind (i.e., the RBE) and perturbations in the monsoonal wind (i.e., QWO, QBO, and ISO) were crucial to the downscale KE cascade, whereas their relative importance was different for terms C(KM, KI) and C($ \overline{{K}_{\mathrm{T}}} $ , KI).
Term | Physical meaning | Term | Physical meaning |
AM | Background-environment’s available potential energy | $G\left({A}_{\mathrm{M} }\right)$&$G(\overline{ {A}_{\rm{T} } })$ | Tendencies of AM and AT due to diabatic production/extinction, respectively |
KM | Background-environment’s kinetic energy | Conversion term $ C\left(X,Y\right) $ | A positive/negative value means energy X/Y is converted into energy Y/X |
AT | Eddy-flow’s available potential energy | $ B\left({A}_{\mathrm{M}}\right) $ & $ B\left({K}_{\mathrm{M}}\right) $ | 3-dimensional transport of AM and KM by background environment, respectively |
KT | Eddy-flow’s kinetic energy | $B(\overline{{A}_{\rm{T}} })$ | 3-dimensional transport of AT by both eddy flow and background environment |
AI | Interaction-flow’s available potential energy | $B(\overline{ {K}_{\rm{T} } })$ | 3-dimensional transport of KT by both eddy flow and background environment |
KI | Interaction-flow’s kinetic energy | $ R\left({A}_{\mathrm{M}}\right) $ | Effects of vertical heat-transport by both eddy flow and background environment |
$ F\left({A}_{\mathrm{I}}\right) $ | Transport of AI by eddy flow | AM→AI→AT | Downscale energy cascade of available potential energy |
$ F\left({K}_{\mathrm{I}}\right) $ | Transport of KI by eddy flow | AT→AI→AM | Upscale energy cascade of available potential energy |
$ D\left({K}_{\mathrm{M}}\right) $ | Dissipation of KM due to friction | KM→KI→KT | Downscale energy cascade of kinetic energy |
$D(\overline{ {K}_{\rm{T} } })$ | Dissipation of KT due to friction | KT→KI→KM | Upscale energy cascade of kinetic energy |
Energy path | Conversion/transport that links different energies | Baroclinic energy conversion | The energy conversion between available potential energy and kinetic energy |