
Boreal winter (January) and summer (July) global mean energy components are calculated to compare the performance of CRAI and ERA5 datasets. The results are shown in Fig. 2. Note that the energy terms named with initial letters of A or K (i.e., those in the thick solid black boxes in Fig. 2) are called reservoir terms, the terms starting with letter C are called conversion terms, and those starting with letter G (D) are called generation (dissipation) terms. Figure 2 shows that the energy reservoir terms computed from CRAI maintain high consistency with those from ERA5, in both summer and winter. As is known, AZ decomposes into a stationary wave ASE and a transient wave ATE; compared with the transient wave, the stationary wave cannot be ignored, since ASE equates to 60%–70% of the magnitude of ATE. The nonlinear conversion of CATE is an important term in the energy analysis as it directs energy from stationary to transient forms, meaning that the damping of stationary temperature via the horizontal sensible heat transient fluxes can play an important role in the general circulation (Zhao and Zhang, 2014).
Figure 2. Comparison of energy components between CRAI and ERA5 (brackets) in January (blue) and July (red). Various energy terms (boxes) and conversions are in unit of 10^{5} J m^{–2} and W m^{–2}, respectively. The dotted, dotdashed, and dashed frames denote the planetaryscale, barotropic, and baroclinic processes, respectively.
KZ is approximate to the sum of KSE and KTE (i.e., there is almost no net global energy conversion during barotropic processes). Simultaneously, KSE is only 1/3 the magnitude of KTE, which is consistent with the previous analysis (Ulbrich and Speth, 1991).
In general, global energy components from these two reanalyses are very similar. However, the main difference comes from conversions during the baroclinic process (CES and CAS), especially in July. At the same time, we can find that generation and dissipation terms of the two reanalysis datasets show differences that are slightly larger than those in reservoir terms. In addition, we can see that almost all of the energy reservoir terms calculated by CRAI are larger than those calculated by ERA5. Meanwhile, most conversions show similar magnitude. Regarding the CZ component that is used to show the conversion between vertical wind and temperature, the descriptive ability of CRAI for Hadley cell features is relatively weaker than that of ERA5.
By comparing the globally averaged energy components, as depicted in Fig. 2, magnitude of the difference between CRAI and ERA5 in reservoir and conversion terms is larger in July than in January. Therefore, we focus on the features of energy components in summer. Figure 3 shows the mean AZ in July of 2010–16 calculated by CRAI and ERA5 as well as their differences. General patterns of energy components obtained by the two reanalysis datasets are similar, where the maximum energies are present near high latitudes in the Southern Hemisphere, while the maximum biases also exist there, suggesting that larger temperature perturbations exist in the high latitudes, and the meridional temperature gradient is larger in CRAI than in ERA5.
Figure 3. Crosssections of the mean AZ (J m^{–2} Pa^{–1}) for July of 2010–16 from (a) ERA5, (b) CRAI, and (c) the difference between them.
Compared with AZ, the difference in KZ is much smaller (Fig. 4). Due to the existence of jet streams in the lower and upper atmosphere, the maximum KZ values computed by the two reanalysis datasets are both located over 30°S at 200 hPa. KZ of CRAI is slightly larger than that of ERA5 around 30°S, 30°N, and 60°S at the upper model levels. The difference is mainly reflected in the strength of the jet streams, but not the locations of the jet streams.
Figure 4. As in Fig. 3, but for the zonal kinetic energy (KZ; J m^{–2} Pa^{–1}).
For the barotropic process, the general patterns of KSE and KTE components of KZ calculated by the two reanalysis datasets show similarities in their maximum magnitudes in the Northern and Southern Hemispheres (Fig. 5). As described before, KTE is approximately three times larger than KSE. With regard to KSE, the maximum difference occurs at 150 hPa around the equator and over 0°–30°N, where the location corresponds to a largevalue area of KSE in both reanalysis datasets. This is due to the presence of a stronger jet stream computed by CRAI, which strengthens the energy reservoir. The distribution of zonal mean transient wave KTE presents a different pattern compared with that of KSE. A northward maximum KTE appears in the Southern Hemisphere midlatitudes, while the maximum KTE in the Northern Hemisphere stretches southward from the north pole to lowlatitude regions in the upper troposphere (300 hPa). The maximum KTE difference is observed over the tropics in the stratosphere, which reflects a weaker jet flow in CRAI.
Figure 5. Crosssections of the mean (a, b, c) KSE (J m^{–2} Pa^{–1}) and (d, e, f) KTE (J m^{–2} Pa^{–1}) waves for July of 2010–16 from (a, d) ERA5, (b, e) CRAI, and (c, f) the difference between them.
As mentioned in Section 2.1, CKTE is the conversion term between KSE and KTE, and this nonlinear conversion directs energy from KTE to KSE (Fig. 2), which means that KE transfers from transient waves to stationary waves, resulting in strengthening of KSE. This always corresponds to a local jet maximum and plays a balancing role between the energy reservoir terms. Ulbrich and Speth (1991) proposed that the maximum CKTE is in the same region of the maximum jet stream (i.e., a positive maximum CKTE indicates a region with a local minimum KSE and maximum KTE). This phenomenon can be interpreted as a physical forcing mechanism that brings the atmosphere to a more balanced state. From the difference in the conversion terms computed by the two reanalysis datasets, we can see that a high consistency is maintained, and the main difference comes from high latitudes in the Southern Hemisphere (Fig. 6).
Figure 6. Crosssections of the mean conversion term CKTE (10^{–6} W m^{–2} Pa^{–1}) for July of 2010–16 from (a) ERA5, (b) CRAI, and (c) the difference between them.
By further analyzing the detailed process associated with baroclinic conversions, the globally averaged values of stationary and transient waves maintain good consistency (Fig. 2). To comprehensively investigate the detailed performance of baroclinic conversions, the zonal mean ASE and ATE are displayed in Fig. 7. It is shown that the largest ASE values are located at 300 hPa near 30°N and over the Antarctic below 850 hPa in both CRAI and ERA5. A dominant difference in ASE also occurs in these areas. From a conceptual perspective, ASE characterizes systematic differences in temperature; it is clear that the differences between the low latitudes in the Northern Hemisphere and Antarctic region can be largely caused by the influence of complex terrain. Compared with ASE, the maximum of transient wave energy term ATE in the Northern Hemisphere is further poleward; the zonal mean ATE distribution has a structure similar to that of KTE (Figs. 5d, e), and it forms an approximately symmetrical structure in the Northern and Southern Hemispheres within the low–mid troposphere, while a weak difference is shown near the Antarctic (Figs. 7d, e).
Figure 7. Crosssections of (a, b, c) ASE (J m^{–2} Pa^{–1}) and (d, e, f) ATE (J m^{–2} Pa^{–1}) for July of 2010–16 from (a, d) ERA5, (b, e) CRAI, and (c, f) the difference between them.
It is easy to see that from the globally averaged energy components shown in Fig. 2, the main difference in the baroclinic process comes from the conversion terms such as CES. CES can quantitatively describe the process of warm air rising and cold air sinking as a transfer process of APE to KE. The distribution characteristics of CES described by the two reanalysis datasets are consistent (Fig. 8), and the difference between the two datasets is highly similar to the case of ASE (Fig. 7c). Considering the topographical features at low–mid latitudes and near the South Pole as well as the different patterns for other energy components that have analyzed (figures omitted), it is not difficult to conclude that differences in geographic distribution of topography in the two datasets possibly dominates the differences in the distribution of CES.
Figure 8. Crosssections of the mean conversion term CES (10^{–6} W m^{–2} Pa^{–1}) for July of 2010–16 from (a) ERA5, (b) CRAI, and (c) the difference between them.
To identify the sources for the difference (bias) in energy components between the two reanalysis dataset, we examine the performance of each forecast variable separately (Fig. 9). The bias in wind field is concentrated near the equator; where zonal wind bias is located in the upper layer (similar to KTE) while meridional wind bias occurs from the surface to upper levels. Meanwhile, there is certain bias near the Antarctic. The temperature bias is mainly concentrated in the lower troposphere, where there is an obviously negative bias at high latitudes in the Southern Hemisphere, while the largest positive deviation appears in the Northern Hemisphere near 30°N, and the overall distribution is similar to that of ASE (Fig. 7c). In terms of specific humidity, we can see that CRAI represents a certain dry pattern at low–mid latitudes in the Northern Hemisphere compared to ERA5; the large difference is mostly concentrated near 30°N, which is similar to that for temperature. For the vertical velocity field, the ascending motion in CRAI near the equator is obviously stronger than that in ERA5. In contrast, low latitudes in the Northern Hemisphere correspondingly show strong sinking motion. Near 30°N, the lowlevel sinking motion is slightly larger, while the mid–upper troposphere is dominated by strong upward air motion. At the same time, it can be found that the vertical velocity bias is concentrated in the Northern Hemisphere, while only some excessive sinking motion occurs in the lower layer at high latitudes of the Southern Hemisphere.
Figure 9. The zonal mean difference between ERA5 and CRAI for (a) zonal wind (m s^{–1}), (b) meridional wind (m s^{–1}), (c) temperature (K), (d) specific humidity (g kg^{–1}), and (e) vertical speed (omega; Pa s^{–1}) for July of 2010–16. Panel (f) shows the topography of CRAI.
CES is calculated by using the formula A11 in Appendix. In A11, ω represents the vertical pvelocity, T_{v} represents the virtual temperature, R represents the gas constant, p represents the pressure, g represents the gravity of earth, and * represents the deviation from zonal mean. Apparently, CES is related to the vertical pvelocity and virtual temperature, and the latter can be calculated with the temperature and specific humidity. Therefore, by comparing the pvelocity, temperature, and humidity, it is found that there is clearly a reason for the large difference in the energy components at midlatitudes and near the Antarctic. According to these common features, it is not difficult to find that the large differences should be related to the complex terrains at the corresponding locations (Fig. 9f).

In the above section, the mean state of global energy cycle is investigated. Here, the longterm monthly mean characteristics of energy reservoir and conversion terms are summarized, together with the corresponding correlation between ERA5 and CRAI during 2010–16 (Table 1). In Table 1, the monthly mean values are calculated by averaging the global energy components over the 7yr period, and confidence intervals with a 99% confidence level are used to estimate the distribution range. It is shown that the energy reservoir and conversion terms are mainly within this significant interval. To further illustrate the monthly evolution of energy reservoirs and conversions, standard deviation is calculated (Table 1) to represent the dispersion characteristics. Table 1 clearly shows that energy reservoirs computed by the two reanalysis datasets maintain an obvious consistency, with some correlation coefficients of above 99%. The spreading of AZ calculated by CRAI is higher than that by ERA5; however, the subdivided components (ASE and ATE) are basically similar.
Component ERA5 CRAI Correlation Mean Standard deviation Mean Standard deviation AZ (10^{5} J m^{–2}) 56.75 ± 0.55 2.56 57.82 ± 0.56 2.62 0.992 KZ (10^{5} J m^{–2}) 7.29 ± 0.13 0.61 7.22 ± 0.11 0.50 0.999 ASE (10^{5} J m^{–2}) 3.15 ± 0.12 0.56 3.34 ± 0.12 0.56 0.993 KSE (10^{5} J m^{–2}) 1.77 ± 0.10 0.46 1.77 ± 0.10 0.46 0.999 ATE (10^{5} J m^{–2}) 4.15 ± 0.06 0.30 4.11 ± 0.06 0.29 0.997 KTE (10^{5} J m^{–2}) 5.16 ± 0.05 0.24 5.11 ± 0.05 0.24 0.998 CZ (W m^{–2}) 1.10 ± 0.06 0.28 1.17 ± 0.06 0.28 0.942 CAS (W m^{–2}) 0.42 ± 0.04 0.20 0.36 ± 0.04 0.20 0.989 CKS (W m^{–2}) –0.15 ± 0.01 0.07 –0.14 ± 0.01 0.06 0.993 CAT (W m^{–2}) 1.62 ± 0.04 0.18 1.62 ± 0.04 0.19 0.986 CKT (W m^{–2}) –0.45 ± 0.01 0.07 –0.45 ± 0.01 0.06 0.994 CES (W m^{–2}) 0.54 ± 0.03 0.14 0.71 ± 0.03 0.13 0.915 CET (W m^{–2}) 1.76 ± 0.02 0.11 1.74 ± 0.02 0.10 0.985 CATE (W m^{–2}) –0.41 ± 0.02 0.10 –0.39 ± 0.02 0.11 0.993 CKTE (W m^{–2}) –0.04 ± 0.01 0.06 –0.06 ± 0.01 0.07 0.995 GZ (W m^{–2}) 3.14 ± 0.06 0.28 3.15 ± 0.07 0.32 0.916 GSE (W m^{–2}) 0.53 ± 0.04 0.18 0.74 ± 0.04 0.21 0.976 GTE (W m^{–2}) –0.27 ± 0.02 0.09 –0.27 ± 0.02 0.09 0.907 DZ (W m^{–2}) 1.70 ± 0.05 0.23 1.76 ± 0.05 0.24 0.914 DSE (W m^{–2}) 0.36 ± 0.02 0.10 0.52 ± 0.02 0.10 0.822 DTE (W m^{–2}) 1.34 ± 0.01 0.06 1.35 ± 0.01 0.07 0.957 Table 1. Monthly mean values and corresponding correlations of the globally averaged energy components for ERA5 and CRAI in 2010–16
On the contrary, the standard deviation of KZ in CRAI is smaller than that in ERA5, while the subdivided terms (KSE and KTE) maintain a high level of agreement. The main difference lies in the conversion terms, where the correlation coefficient for CES is only 91.5%, while that for CZ is 94.2%. There is no obvious difference in the spreading characteristics (standard deviation). It seems difficult to find the main source of difference between the two reanalysis datasets. The large difference in the monthly mean values of CES (0.54 W m^{–2} for ERA5 and 0.71 W m^{–2} for CRAI) leads to subsequent evolution difference. The globally averaged energy generation and dissipation terms are based on energy reservoirs and conversions by budget equations. Obviously, in the equations with large differences in conversion terms, the correlation between generation and dissipation terms is weak. Therefore, temporal differences in energy reservoir and conversion terms are not very important in diagnosing the generation and dissipation rates.
In regard to the annual cycle of global energy reservoir terms, Fig. 10 displays the annual evolution of these energy components averaged in 2010–16. It is clear that these energy terms shows apparent periodic variation characteristics. Compared to APE, KE terms are more consistent with each other. Magnitudes of AZ and ASE in CRAI are obviously higher than those in ERA5, and the largest difference appears in summer (2.7% larger than ERA5 for AZ in August and 10.1% larger than ERA5 for ASE in July). However, the evolution characteristics are similar between CRAI and ERA5.
Figure 10. Monthly mean (2010–16) time series of the globally averaged (a) AZ, (b) KZ, (c) ASE, (d) KSE, (e) ATE, and (f) KTE from ERA5 (dashed line) and CRAI (solid line). The unit of each variable is 10^{5} J m^{–2}.
Different from energy reservoir terms, energy conversion terms display some discrepancies, especially for conversions from ASE to KSE (CES; Fig. 11). Clearly, CZ and CET maintain relatively similar evolution features, while CES, which represents the conversion from ASE to KSE, presents certain differences. The variation between boreal spring and summer is obviously larger than that between boreal autumn and winter. The largest CES difference appears in July and CRAI is about 37.8% larger than ERA5. As previously analyzed, the eddy conversion term from APE to KE is essentially caused by variations in the temperature and vertical velocity in the latitudinal circulation. Therefore, positive biases in the temperature and vertical velocity at 30°N and near the surface over the Antarctic play an important role in this corresponding difference.
Figure 11. Monthly mean (2010–16) time series of the globally averaged (a) CZ, (b) CES, and (c) CET from ERA5 (dashed line) and CRAI (solid line). The unit of each variable is W m^{–2}.
By further analyzing the other conversion terms in baroclinic and barotropic processes (Fig. 12), it is inferred that conversion terms between the mean and eddy potential energy are produced by the eddy transport of heat and temperature gradient. Moreover, the eddy transport of momentum and gradient in the mean angular rotation creates conversion rates between the mean and eddy KE. It can be concluded that both the KE and potential energy conversion terms present high degrees of consistency based on the two reanalysis datasets, and CRAI products can effectively depict seasonal evolution characteristics of the temporal evolution in various energy components.
Figure 12. Monthly mean (2010–16) time series of the globally averaged (a) CAS, (b) CKS, (c) CAT, (d) CKT, (e) CATE, and (f) CKTE from ERA5 (dashed line) and CRAI (solid line). The unit of each variable is W m^{–2}.
The generation of potential energy via the radiative heating and the dissipation of KE via the friction can be barely measured directly. They can only be calculated by the balance of energy reservoir terms. Figure 13 shows the annual evolution of globally averaged generation and dissipation terms based on CRAI and ERA5. Obviously, the largest difference comes from the generation and dissipation of stationary waves (i.e., the systematic difference of reanalysis datasets); in terms of GSE, the largest difference exists in summer and the CRAI value is larger than the ERA5 value by about 34.7% in June. For the planetaryscale and transient waves, the two reanalysis datasets maintain a high degree of consistency, which are now given more attention in climatological research. The CRAI dataset seems to have a good application prospect.

AZ is the zonal available potential energy
$$\hspace{136pt}{\rm{AZ}} = \dfrac{\gamma }{{2{\rm{g}}}}{([\overline T ]  \{ \overline T \})^2}. \tag{A1}$$ (A1) ASE is the stationary eddy available potential energy
$$\hspace{156pt} {\rm{ASE}} = \dfrac{\gamma }{{2{\rm{g}}}}[{\overline {{T^{*}}} ^2}]. \tag{A2}$$ (A2) ATE is the transient eddy available potential energy
$$ \hspace{157pt} {\rm{ATE}} = \dfrac{\gamma }{{2{\rm{g}}}}[\overline {{T^{'2}}} ]. \tag{A3}$$ (A3) KZ is the zonal kinetic energy
$$ \hspace{136pt} {\rm{KZ}} = \dfrac{1}{{2{\rm{g}}}}({[{\overline u} ]^2} + {[{\overline v} ]^2}). \tag{A4}$$ (A4) KSE is the stationary eddy kinetic energy
$$ \hspace{135pt} {\rm{KSE}} = \dfrac{1}{{2{\rm{g}}}}[{\overline {{u^{*}}} ^2} + {\overline {{v^{*}}} ^2}]. \tag{A5}$$ (A5) KTE is the transient eddy kinetic energy
$$\hspace{137pt} {\rm{KTE}} = \dfrac{1}{{2{\rm{g}}}}[\overline {{u^{'2}}} + \overline {{v^{'2}}} ]. \tag{A6}$$ (A6) KE is the eddy kinetic energy
$$\hspace{24pt} \hspace{126pt} {\rm KE} = {\rm KSE} + {\rm KTE.} \tag{A7}$$ (A7) CAS is the conversion from AZ to ASE
$$ \begin{aligned} \hspace{48pt}\, {\rm{CAS}} = &  \dfrac{\gamma }{{\rm{g}}}([{\overline v} ^{*}\overline T ^{*}]\dfrac{{\partial [\overline T ]}}{{r\partial \varphi }} + [\overline \omega ^{*}\overline T ^{*}] \\ & \cdot \left(\dfrac{\partial }{{\partial p}}([\overline T ]  \{ \overline T \})  \dfrac{R}{{p{c_p}}}([\overline T ]  \{ \overline T \}))\right).\end{aligned} \;\;\;\;\;\;\;\;\;\, \tag{A8}$$ (A8) CAT is the conversion from AZ to ATE
$$ \hspace{54pt} \begin{aligned}{\rm{CAT}} = &  \dfrac{\gamma }{{\rm{g}}}([\overline {v'{T^{'}}} ]\dfrac{{\partial [\overline T ]}}{{r\partial \varphi }} + [\overline {\omega '{T^{'}}} ] \\ & \cdot \left(\dfrac{\partial }{{\partial p}} ([\overline T ]  \{ \overline T \})  \dfrac{R}{{p{c_p}}}([\overline T ]  \{ \overline T \}))\right).\end{aligned} \; \tag{A9}$$ (A9) CZ is the conversion from AZ to KZ
$$ \hspace{55pt} {\rm{CZ}} =  \dfrac{R}{{p{\rm{g}}}}([\overline \omega ]  \{ \overline \omega \})([\overline {{T_v}} ]  \{ \overline {{T_v}} \}). \quad\quad\;\;\;\;\, \tag{A10}$$ (A10) CES is the conversion from ASE to KSE
$$ \hspace{124pt} {\rm{CES}} =  \dfrac{R}{{p{\rm{g}}}}[\overline \omega ^{*}  \overline {{T_v}} ^{*}]. \;\;\; \tag{A11}$$ (A11) CET is the conversion from ATE to KTE
$$\hspace{140pt}\!{\rm{CET}} =  \dfrac{R}{{p{\rm{g}}}}[\overline {\omega '{T_v}'} ] . \tag{A12}$$ (A12) CKS is the conversion from KZ to KSE
$$ \begin{aligned} {\rm{CKS}} = & \! \dfrac{1}{{\rm{g}}}([{\overline u} ^{*}{\overline v} ^{*}]\dfrac{{\partial [{\overline u} ]}}{{r\partial \varphi }} \!+\! \dfrac{{t{\rm{g}}\varphi }}{r}[{\overline u} ^{*}{\overline v} ^{*}][{\overline u} ] + [{\overline v} ^{*}{\overline v} ^{*}]\dfrac{{\partial [{\overline v} ]}}{{r\partial \varphi }} \\ &  \dfrac{{t{\rm{g}}\varphi }}{r}[{\overline u} ^{*}{\overline u} ^{*}][{\overline v} ] + [\overline \omega ^{*}{\overline u} ^{*}]\dfrac{{\partial [{\overline u} ]}}{{\partial p}} \\ & + \dfrac{{\partial [{\overline v} ]}}{{\partial p}}[\overline \omega ^{*}{\overline v} ^{*}]). \end{aligned} \tag{A13}$$ (A13) CKT is the conversion from KZ to KSE
$$ \begin{aligned} \!\!\!\!\! {\rm{CKT}} = &  \dfrac{1}{{\rm{g}}}([\overline {u'v'} ]\dfrac{{\partial [{\overline u} ]}}{{r\partial \varphi }} \\ & + \dfrac{{t{\rm{g}}\varphi }}{r}[\overline {u'v'} ][{\overline u} ] + [\overline {v'v'} ]\dfrac{{\partial [{\overline v} ]}}{{r\partial \varphi }}  \dfrac{{t{\rm{g}}\varphi }}{r}[\overline {u'u'} ][{\overline v} ] \\ & + [\overline {\omega 'u'} ]\dfrac{{\partial [{\overline u} ]}}{{\partial p}} + \dfrac{{\partial [{\overline v} ]}}{{\partial p}}[\overline {\omega 'v'} ]). \end{aligned} \tag{A14}$$ (A14) ${C_{{\rm{ATE}}  {\rm{ASE}}}}$ (CATE) is the conversion from ATE to ASE$$\begin{aligned} \!\!\!\!\! {C_{{\rm{ATE}}  {\rm{ASE}}}} = & \frac{\gamma }{{\rm{g}}}\left( {{{\overline {u'T'} }^*}\frac{1}{{r\cos \varphi }}\frac{{\partial {{\overline T}^*}}}{{\partial \lambda }}} \right.\\ & \left. { + \;{{\overline {v'T'} }^*}\frac{1}{{r\cos \varphi }}\frac{{\partial {{\overline T}^*}}}{{r\partial \varphi }}} \right). \quad \quad \quad \quad \quad \quad\;\; \end{aligned} \tag{A15} $$ (A15) ${C_{{\rm{KTE}}  {\rm{KSE}}}}$ (CKTE) is the conversion from KTE to KSE$$ \! \begin{aligned} {C_{{\rm{KTE}}  {\rm{KSE}}}} = & \frac{1}{{\rm{g}}}\left( {{{\overline {u'u'} }^*}\frac{1}{{r\cos \varphi }}\frac{{\partial {{\bar u}^*}}}{{\partial \lambda }} + {{\overline {u'v'} }^*}\frac{{\partial {{\bar u}^*}}}{{r\partial \varphi }}} \right.\\ & + {\overline {u'v'} ^*}{{\bar u}^*}\frac{{t{\rm{g}}\varphi }}{r} + {\overline {v'v'} ^*}\frac{{\partial {{\bar v}^*}}}{{r\partial \varphi }}  {[\overline {u'u'} ]^*}{{\bar v}^*}\frac{{t{\rm{g}}\varphi }}{r}\\ & \left. { + \;{{\overline {u'v'} }^*}\frac{1}{{r\cos \varphi }}\frac{{\partial {{\bar v}^*}}}{{\partial \lambda }}} \right). \end{aligned} \;\; \;\; \;\; \tag{A16}$$ (A16) GZ, GSE, GTE, DZ, DSE, and DTE are the generation of AZ, ASE, ATE, KZ, KSE, and KTE, respectively.
$Cp$ is the specific heat at constant pressure; g is the gravity of earth; p is the pressure; r is the radius of earth; R is the gas constant; T is the temperature; Tv is the virtual temperature; u is the zonal wind; v is the meridional wind;$\omega $ is the vertical pvelocity;$\varphi $ is the latitude;$\lambda $ is the lontitude;$\gamma =  \dfrac{R}{p}{(\dfrac{\partial }{{\partial p}}[\overline T ]  \dfrac{R}{{Cp}}\dfrac{{[\overline T ]}}{p})^{  1}}$ is the stability parameter.${\overline x} $ is the time mean of x;${x^{'}}$ is the deviation from time mean; {x} is the global horizontal mean; [x] is the zonal mean; x^{*} is the deviation from zonal mean.
Component  ERA5  CRAI  Correlation  
Mean  Standard deviation  Mean  Standard deviation  
AZ (10^{5} J m^{–2})  56.75 ± 0.55  2.56  57.82 ± 0.56  2.62  0.992  
KZ (10^{5} J m^{–2})  7.29 ± 0.13  0.61  7.22 ± 0.11  0.50  0.999  
ASE (10^{5} J m^{–2})  3.15 ± 0.12  0.56  3.34 ± 0.12  0.56  0.993  
KSE (10^{5} J m^{–2})  1.77 ± 0.10  0.46  1.77 ± 0.10  0.46  0.999  
ATE (10^{5} J m^{–2})  4.15 ± 0.06  0.30  4.11 ± 0.06  0.29  0.997  
KTE (10^{5} J m^{–2})  5.16 ± 0.05  0.24  5.11 ± 0.05  0.24  0.998  
CZ (W m^{–2})  1.10 ± 0.06  0.28  1.17 ± 0.06  0.28  0.942  
CAS (W m^{–2})  0.42 ± 0.04  0.20  0.36 ± 0.04  0.20  0.989  
CKS (W m^{–2})  –0.15 ± 0.01  0.07  –0.14 ± 0.01  0.06  0.993  
CAT (W m^{–2})  1.62 ± 0.04  0.18  1.62 ± 0.04  0.19  0.986  
CKT (W m^{–2})  –0.45 ± 0.01  0.07  –0.45 ± 0.01  0.06  0.994  
CES (W m^{–2})  0.54 ± 0.03  0.14  0.71 ± 0.03  0.13  0.915  
CET (W m^{–2})  1.76 ± 0.02  0.11  1.74 ± 0.02  0.10  0.985  
CATE (W m^{–2})  –0.41 ± 0.02  0.10  –0.39 ± 0.02  0.11  0.993  
CKTE (W m^{–2})  –0.04 ± 0.01  0.06  –0.06 ± 0.01  0.07  0.995  
GZ (W m^{–2})  3.14 ± 0.06  0.28  3.15 ± 0.07  0.32  0.916  
GSE (W m^{–2})  0.53 ± 0.04  0.18  0.74 ± 0.04  0.21  0.976  
GTE (W m^{–2})  –0.27 ± 0.02  0.09  –0.27 ± 0.02  0.09  0.907  
DZ (W m^{–2})  1.70 ± 0.05  0.23  1.76 ± 0.05  0.24  0.914  
DSE (W m^{–2})  0.36 ± 0.02  0.10  0.52 ± 0.02  0.10  0.822  
DTE (W m^{–2})  1.34 ± 0.01  0.06  1.35 ± 0.01  0.07  0.957 