Long-term change of the atmospheric energy cycles and weather disturbances

Abstract

Weather disturbances are the manifestation of mean atmospheric energy cascading into eddies, thus identifying atmospheric energy structure is of fundamental importance to understand the weather variability in a changing climate. The question is whether our observational data can lead to a consistent diagnosis on the energy conversion characteristics. Here we investigate the atmospheric energy cascades by a simple framework of Lorenz energy cycle, and analyze the energy distribution in mean and eddy fields as forms of potential and kinetic energy. It is found that even the widely utilized independent reanalysis datasets, NCEP-DOE AMIP-II Reanalysis (NCEP2) and ERA-Interim (ERA-INT), draw different conclusions on the change of weather variability measured by eddy-related kinetic energy. NCEP2 shows an increased mean-to-eddy energy conversion and enhanced eddy activity due to efficient baroclinic energy cascade, but ERA-INT shows relatively constant energy cascading structure between the 1980s and the 2000s. The source of discrepancy mainly originates from the uncertainties in hydrological variables in the mid-troposphere. Therefore, much efforts should be made to improve mid-tropospheric observations for more reliable diagnosis of the weather disturbances as a consequence of man-made greenhouse effect.

1 Introduction

Weather disturbances accompany air motion, and the air motion is primarily driven by the energy imbalance of the earth’s atmosphere. In this sense, how the weather disturbances or extremes react to changing climate (Fischer and Knutti 2015; Rahmstorf and Coumou 2011) requires an accurate diagnosis of the atmospheric energy structure, since weather disturbances occur as mean atmospheric energy cascades into smaller eddies. The atmospheric energy can be decomposed into a simple energetics framework of the Lorenz energy cycle (Lorenz 1955). The merit of this framework is to systematically illustrate the atmospheric energy conversions between the available potential energy and the kinetic energy, and between the mean field and the eddy field (see Fig. 1). Since the pioneering work of Lorenz (1955), however, few investigated the atmospheric energetics. Oort (1964) and Peixóto and Oort (1974) estimated hemispheric and global energy cascading structure, whose works had been updated with recent observational and modeling data by several independent researches (e.g., Li et al. 2007; Marques et al. 2009, 2011; Murakami et al. 2010). Yet, most of the previous studies focused on the long-term climatological mean state of energetics, but not on the time-dependent changes.

Fig. 1

Schematic diagram of Lorenz energy cycle. P and K’s indicate available potential energy and kinetic energy, respectively, and C’s and their corresponding arrows indicate energy conversion between the energy terms. Numbers are long-term annual mean energy (10⁵ J m⁻²) and conversion (W m⁻²) estimated from NCEP2/ERA-INT, where red and blue colors imply significant increase and decrease from 1980–1994 to 2000–2014, respectively. G, T, D, and, B are generation, transport, dissipation, and boundary flux, respectively

The current situation of a warming climate may be favorable for the increase in total atmospheric energy. On the other hand, it is known that the available potential energy could decrease due to the decreased meridional gradient of surface air temperature (SAT) (Holland and Bitz 2003; Lee 2014), and that the general circulation slow down (Held and Soden 2006; Kjellsson 2014; Vecchi and Soden 2007), even if the total atmospheric energy increases. Specifically, the global mean SAT in the last 15 years (2000–2014) was about 0.4 K higher than that of the previous period (1980–1994) (Dai et al. 2015; Yeo et al. 2016), even under the recent warming hiatus (Kosaka and Xie 2013; Trenberth and Fasullo 2013; Watanabe et al. 2014). As the total potential (${E}_{P}$) and internal $({E}_{I})$ energy of the atmosphere depends on the air temperature (Lorenz 1955) (i.e., ${{E}_{P}}+{{E}_{I}}=-\frac{{{c}_{p}}}{g}\underset{{{p}_{0}}}{\overset{{}}{\mathop \int }}\,T~dp$), the warming SAT implies an increase in the total $({E}_{P}+{E}_{I})$ energy of the troposphere. However, due to strong hydrostatic and geostrophic constraints, not all of the energy can be converted into kinetic energy, and the available potential energy that can induce air motion is just a tiny fraction (<1%) of the total energy. This available portion of atmospheric energy can remain as the mean available energy, but also can be converted into eddy kinetic energy via baroclinic processes (disturbances grown by extracting potential energy from the mean state) and barotropic processes (disturbances grown by extracting kinetic energy from the mean state). The resulting eddy kinetic energy is a measure of weather variability, and the weather disturbances are the manifestation of mean energy cascading into eddies. However, the tendency of weather variability remains equivocal.

To accurately diagnose weather disturbances under warming climate, a close inspection on the atmospheric energy structure of the ongoing climate change is therefore worthwhile. Here we analyze the atmospheric energy distribution and conversion under Lorenz energy cycle frameworks, and compare the two independent estimates from widely utilized datasets in climate sciences seeking for the origin of their possible discrepancies.

2 Data and methods

2.1 Data and variables

The primary dataset analyzed in this study is the NCEP-DOE AMIP-II Reanalysis (Kanamitsu et al. 2002) (NCEP2, in 2.5° $\times$ 2.5° horizontal resolution, provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their web site http://www.esrl.noaa.gov/psd/). Daily averages of horizontal winds (u, v), pressure velocity $(\omega )$, temperature $(T)$, geopotential height ($\Phi$), relative humidity (RH), and surface pressure (${p}_{s}$) were analyzed. The 1° $\times$ 1° daily ERA-Interim (Dee et al. 2011) (ERA-INT) was also utilized to confirm the energetics and its change in NCEP2. We compared the 15-year periods of 1980–1994 and 2000–2014 to assess the atmospheric energy structures before and after the late-1990s global climate shift (e.g., Bond et al. 2003). The datasets were divided into the 15-year subsets of the total period to secure robustness, however, it coincided well with the long-term change point (especially for NCEP2). We also assessed decadal periods (1980s and 2000s) or first and last halves, but the results basically remain unchanged.

2.2 Definition of mean state and eddy

We decomposed the variables into slowly varying basic states, denoted by bars ($\bar{X}$), and the deviations from them, denoted by primes ($X^{\prime}$). The basic state was defined as a monthly mean of daily averages. Defining the basic state in such a way has advantage over using fixed climatology, in that it enables one to assess long-term change as well as interannual variation of the mean energy. The disadvantage is that one cannot evaluate the sub-monthly scale fluctuations. After calculating Lorenz energy terms, monthly mean was taken so that the eddy terms were dropped in the final time-averaged version, i.e., $\underset{{{t}_{1}}}{\overset{{{t}_{2}}}{\mathop \int }}\,X'~dt=0$. We tested different definitions for the mean state, e.g., period-specific fixed climatology or 11-year running climatology, but the main conclusions on the eddy energy structures were unchanged. The governing equations, time-averaged energy, conversion, generation, dissipation, and boundary flux terms are given in the Appendixes 1, 2, 3, 4. More detailed derivation of the energy cascading structure can also be found in Murakami (2011).

3 Results

3.1 Globally averaged atmospheric energetics

The resulting energy distribution and cascade can be summarized as the following Lorenz energy cycle schematics (Fig. 1). The numbers in boxes indicate P _M (mean available potential energy), P _E (eddy available potential energy), K _E (eddy kinetic energy), and K _M (mean kinetic energy), counter-clockwise from the top-left corner, respectively. The arrows indicate energy conversion (C), generation (G), dissipation (D), transport (T), and boundary flux (B). The long-term estimates (1979–2014, 10⁵ J m⁻² for the energy terms and W m⁻² for conversion terms) of NCEP2/ERA-INT are indicated as numbers, and are colored if the differences between the 1980–1994 period and the 2000–2014 period are significant at a 95% confidence level (red and blue indicate increase and decrease, respectively).

The long-term averaged energy terms, P _M , P _E , K _E , and K _M , are balancing at around 39/41, 2.5/3.3, 4.0/4.1, and 6.8/6.6 × 10⁵ J m⁻² (NCEP2/ERA-INT), respectively. Their changes from 1980–1994 to 2000–2014 periods are −0.6*/−0.5*, +0.6*/0.0, +0.2*/0.0, and −0.1*/0.1 × 10⁵ J m⁻², respectively, where asterisks indicate significant differences at 95% confidence level. The eddy energy estimates of ERA-INT are generally larger than those of NCEP2 partly owing to its higher spatial resolution. The mean available potential energy in NCEP2/ERA-INT is converted to eddy potential energy [C(P _M , P _E )] at a rate of 1.5/2.1 W m⁻², to eddy kinetic energy [C(P _E , K _E )] at 1.3/2.2 W m⁻². In global sense, the eddy kinetic energy reinforces mean circulation [C(K _E , K _M )] at 0.3/0.4 W m⁻², but the conversion direction of mean available potential energy and mean kinetic energy [C(P _M , K _M )] do not agree with each other between the datasets (−4.8/0.8 W m⁻²). Our results generally agree with the following previous studies: Marques et al. (2009) estimated P _M , P _E , K _E , and K _M as 40.2/41.6 (NCEP2/ERA-40), 4.31/4.76, 6.08/6.48, and 8.02/8.19 J m⁻² for their 1979–2001 analysis period, respectively. Kim and Kim (2013) estimated 40.45, 2.77, 4.62, and 7.81 J m⁻² from NCEP2 for 1979–2008 period, respectively. Above two estimates and current analysis slightly differ in the detailed formulation, analysis period, definition of mean, horizontal resolution, and/or vertical extent of analysis. Nonetheless, all the energy terms and their conversion are robustly estimated and lie within the same order of ranges.

3.2 Long-term changes in atmospheric energetics

The interannual variation of atmospheric energy shows some meaningful long-term changes between the two periods and their discrepancy between the datasets. Figure 2 shows the time series of the annual mean global energy cascading structure within the troposphere (mass-weighted integrated from the surface to 200 hPa), in terms of the available potential energy (P’s, left) and kinetic energy (K’s, right) that are stored in the mean field ( _M ’s, top) and as a form of eddy ( _E ’s, bottom). The two datasets, NCEP2 and ERA-INT, agree well on their interannual variability of all forms of energy terms. The mean energy tends to increase after major El Niño events, e.g., 1983 and 1998, while the eddy energy enhances during the El Niño developing years, e.g., 1982 and 1997. Global mean energy is higher during boreal winter compared to boreal summer, possibility due to the asymmetric configuration of land and ocean between the hemispheres. In general, both datasets share common interannual variability and comparable fluctuation ranges.

Fig. 2

The time series of the globally averaged annual mean energy terms estimated from (a–d) NCEP2 and (e–h) ERA-INT. Straight orizontal lines indicate significant change of means (at a 95% confidence level) from the 1980–1994 period to the 2000–2014 period. Black lines represent annual mean, while blue and red lines show boreal winter (DJF) and boreal summer (JJA) values, respectively

However, it is intriguing that the two datasets exhibit substantial discrepancy in long-term change of the energy cascading structure. The NCEP2 results (Fig. 2a–d) illustrate that, from the 1980–1994 period to the 2000–2014 period, all energy forms underwent meaningful changes. In the recent period, the mean available energy that can potentially be converted into kinetic energy has reduced as the colder Polar Regions warm faster than the warmer tropical regions. A direct consequence of this less available energy is a weaker mean kinetic motion. On the other hand, the ERA-INT results (Fig. 2e, f) show less reduction of the mean available energy and no notable change of mean kinetic motion. Moreover, unlike NCEP2, the eddy energies, P _E and K _E , between the two periods are not significantly changed in the ERA-INT (Fig. 2g, h).

These differences between NCEP2 and ERA-INT are more clearly identified by comparing Figs. 3 and 4 (∆ = the 2000–2014 mean minus the 1980–1994 mean). Among the four energy components and their conversion efficiencies in NCEP2, the increase of eddy kinetic energy (K _E ) is most pronounced (Fig. 3a–d). For all seasons, the eddy-related kinetic energy was much larger in the 2000–2014 period than it was in the 1980–1994 period in NCEP2 (Fig. 3d), but such enhancement was not apparent in ERA-INT (Fig. 4d). The increase in K _E is in line with the systematic increase in the conversion of eddy potential energy into eddy kinetic energy [C(P _E ,K _E )] in NCEP2 (Fig. 3g). In detail, the C(P _E , K _E ) change is concentrated in the southern mid-latitudes (60°S–30°S) throughout the year, the northern subtropics (~20°N) around September, and the northern mid-latitudes (30°N–60°N) in winter only. Most of the changes described above are, however, unclear in ERA-INT, except for the small increase in the mid-latitude eddy conversion characteristics (Fig. 4g). In the next section, we will investigate this strengthening of C(P _E , K _E ) with greater details. Readers may also find in Figs. 3e, f, h and 4e, f, h that the temporal changes in energy conversions other than C(P _E , K _E ) are not distinctively different between NCEP2 and ERA-INT.

Fig. 3

Hovmöller diagrams of the time difference (the 2000–2014 period minus 1980–1994 period) in the longitudinal averages of the vertically integrated (a–d) energy and (e–h) conversion terms estimated from NCEP2. Significant changes at a 95% confidence level are shaded with diagonal lines

Fig. 4

Same as Fig. 3, but for ERA-INT estimates

3.3 Long-term change in baroclinic conversion and data discrepancy

As our definition of eddies generally cover synoptic-scale ones, the baroclinic eddy conversion and its resulting eddy kinetic energy is indicative of the weather disturbances. We first elaborate on the southern hemispheric mid-latitudes in NCEP2 where the change is the largest and compare the result with ERA-INT estimates. The vertical structure of the eddy conversion reveals that the maximum conversion occurs at mid-latitude mid-troposphere (~ 500 hPa), and its long-term change is the largest just above this level, i.e., around 400-hPa mid-troposphere (Fig. 5). There is also a smaller, but significant, increase at the northern mid-latitude mid-troposphere and at the lower-level southern mid-latitudes, while the opposite can be observed over the tropics. In general, the conversion of available potential energy into eddy kinetic energy, C(P _E , K _E ), increases in the mid-latitudes in NCEP2.

Fig. 5

Zonally averaged annual mean C(P _E , K _E ) (contours) and its difference between the two periods (shades, 2000–2014 minus 1980–1994). Diagonal lines indicate the significant differences at 95% confidence level

To explain why C(P _E , K _E ) has increased greatly, Fig. 6 shows the covariance histogram of 400-hPa eddy vertical motion (${\omega }^{{\prime}}\propto -{w}^{{\prime}}$) and eddy temperature ($T^{\prime}\propto {\alpha }^{{\prime}}$) fields in NCEP2 (refer to the definitions in Appendix). The variabilities of both $\omega ^{\prime}$ and $T^{\prime}$ increased in the 2000–2014 period, but their expansions were not isotropic and preferred a direction that further increased their coherency (i.e., at −45°-angle diagonal direction in the covariance plot). That is, the air parcels favor rising motion where the temperature is warmer and favor sinking motion where it is colder. To be more specific, the sinking motion becomes more coherent with colder air, while the upward motion is realized by vigorous convection accompanied by a strong latent heat release. In the covariance histogram, it can be inferred from the increased probability centered at the small positive $\omega ^{\prime}$-negative $T^{\prime}$, and large negative $\omega ^{\prime}$-positive $T^{\prime}$ directions.

Fig. 6

Covariance histogram (contours) and its difference (shades) of 400 hPa eddy vertical velocity (${\omega}^{\prime}$) and eddy air temperature (${T}^{\prime}$) averaged over the southern hemisphere mid-latitude (60°S–30°S). Thin blue (1980–1994) and thin red (2000–2014) lines indicate individual annual deviation from their long-term (1979–2014) mean, and the thick lines indicate decadal averages

The covariance between the vertical motion and temperature fluctuation can result from thermodynamic processes that incorporates latent heat release. The mean relative humidity’s vertical structure reveals a significant increase of moisture contents in the mid- to upper-troposphere (Fig. 7a, b). With the extended—but not necessarily stronger—direct cell at the tropics which accompanies the latitudinal expansion of the Hadley cell (Kang and Lu 2012; Lu et al. 2007) and the higher eddy potential energy, the mid-tropospheric moisture supply increases at the mid-latitude (Fig. 7c). This moistening of the mid-troposphere is also consistent with the increased eddy potential energy and the increased pole-ward transport of moisture. With higher relative humidity and moisture contents, the condensation induced by the anomalous upward motion increases the air temperature, which in turn enhances the convective motion. As a result, higher relative humidity in the mid-troposphere favors a larger latent heat release, thus strengthening the coherency between the eddy vertical motion and the temperature fluctuation. On the contrary, in ERA-INT, the change of eddy vertical motion and eddy temperature covariance is nearly isotropic (Fig. 8), and the baroclinic eddy conversion do not strengthen. This is because the vertical profile of moisture is different from NCEP2, and its change is opposite that the mid-latitude mid-troposphere is drying in ERA-INT instead of moistening (Fig. 9).

Fig. 7

(a) Vertical profiles of relative humidity (RH) and (b) pressure velocity during the 1980–1994 period (blue) and the 2000–2014 period (red). Thick solid lines indicate the decadal mean values. (c) Zonal mean of annual mean potential temperature (dotted contours) and zonal wind (m s⁻¹, red shadings), superimposed onto the change in meridional moisture transport (g kg⁻¹ m s⁻¹, stream lines). The vertical component of moisture transport is arbitrarily scaled to 100 times its original value

Fig. 8

Same as Fig. 6, but for ERA-INT

Fig. 9

Same as Fig. 7, but for ERA-INT

4 Conclusion and discussion

We looked into the atmospheric energy distribution and conversion in the two independent but widely utilized datasets in climate sciences, NCEP2 and ERA-INT, putting emphasis on the baroclinic eddy conversion [C(P _E , K _E )] processes. Through the Lorentz energy cycle, mean general circulation was adequately represented by the mean kinetic energy of atmosphere. Baroclinic processes were described as the conversion of mean available potential energy into eddy kinetic energy, and barotropic processes as the conversion of mean kinetic energy into eddy kinetic energy (Murakami 2011).

Available potential energy is converted into kinetic energy when warm air rises and cold air sinks (positive w-T covariance). This is equivalent to exhausting available energy to generate air motion. Conversely, when air sinks in warmer regions and rises in colder regions (negative w-T covariance), a buildup of potential energy at the expense of kinetic energy occurs. In this manner, for the conversion of mean available potential energy to eddy kinetic energy, the positive covariance is favorable. In the mid-latitude indirect circulation region, where most of the mean-to-eddy energy conversion is established by baroclinic eddies, the dominant means of the energy conversion process are synoptic-scale disturbances associated with baroclinic instability (Oort 1983). These eddy disturbances are the dominant ways of redistributing atmospheric energy from the tropics to higher latitudes. At the same time, the mean-to-eddy energy conversion processes are subject to the mean climate change. A quantitative assessment of atmospheric energy distribution and its long-term change thus will provide a useful insight into the future of weather disturbances under the changing climate.

In a large-scale sense, the incident solar radiation will generate slowly varying mean field (P _M ) which balances with mean general circulation (K _M ). However, the actual realization is not one-directional from mean ( _M ’s) to eddy ( _E ’s) nor from potential energy (P’s) to kinetic energy (K’s). Instead, the direction of energy conversion depends on time and space; the eddy can first react to the perturbation of generation term which eventually feeds mean circulation, or kinetic motion can transfer energy to higher latitude to rectify potential energy distribution. The energy cascading structure in Fig. 1 should thus be understood as a summary of these various interactions. Another point to notice is that our definition of eddy focuses on relatively large-scale fluctuations in time and space that is limited by the reanalysis resolution. It broadly refers to synoptic-scale disturbances, but not the micro-scale eddies that eventually dissipates due to friction. In this research, as a result, the eddy kinetic energy was rather representative of the weather variability.

The disagreement between the datasets seemed to originate from the mid-tropospheric eddy vertical motion and eddy temperature covariance, which is responsible for baroclinic eddy conversion efficiency. The coherent mid-latitudes vertical motion that is in-phase with the temperature, or the positive w-T covariance, is a manifestation of diabatic heating, thickness advection, and differential vorticity advection, of which the last two are dynamically tangled together and partially cancel out each other (Hoskins et al. 1978). In addition, the dynamically induced vertical motion has a 90° out-of-phase relationship with the temperature in a developing system (i.e., the maximum vertical motion occurs outside the maximum temperature fluctuation in a developing system). On the other hand, the diabatic heating gives an in-phase relationship between the vertical motion and the temperature (i.e., the maximum vertical motion occurs where the maximum temperature fluctuation occurs), although the magnitude is generally smaller than those of dynamic terms (Holton and Hakim 2012). In this sense, a small increase in humidity in the mid-troposphere, in which an upward motion is embedded, can stretch the vertical extent of the baroclinic system, by stronger diabatic heating through the latent heat release that encourages a stronger upward motion. This chain of processes draws energy more efficiently from the mean flow and increases eddy kinetic energy.

We have also seen that the observation of upper-level moisture is highly uncertain. It is based on satellite-based remote sensing or on balloon-based measurement, and in addition the reanalysis products are highly dependent on the assimilation system. Therefore, the vertical structure of the moisture contents and its long-term change are still under debate (Gettelman et al. 2006; Miloshevich et al. 2006). In general, the mid-tropospheric moisture content is reported to show continuous decline under the warming climate (Paltridge et al. 2009); however, it is interesting that the mid-latitude downward branch is systematically moistening in NCEP2 (cf., note that NCEP2 also shows general drying of the mid-troposphere outside the target analysis regions). At large, the estimates based on ERA-INT agree on the increasing eddy kinetic energy, but the vertical profiles of the second-order variables, including humidity, show substantially different structures, and their long-term trends significantly differ from those of NCEP2. As a result, the baroclinic energy conversion and the vertical motion-temperature anomaly relationship disagree between NCEP2 and ERA-INT. Eventually, two widely utilized climate datasets, NCEP2 and ERA-INT, disagree on the long-term change of eddy energy and weather variability under the changing climate. It is suspected, however, that the moisture vertical profile of NCEP2 is less dependable than that of ERA-INT, as NCEP2 reanalysis does not incorporate directly assimilated observation (Kanamitsu et al. 2002).

Our planet is likely to become warmer in the future (Stocker et al. 2013). The consequence of the warming to future weather variability or eddy can be subject to the cascading of mean available potential energy into eddy energy. Understanding the energy cascade under the changing climate may thus advance the long-term extremes prediction that is limited by the current earth system models. However, this study show that today’s different datasets readily give different narratives about the prospect of long-term change in weather disturbances. The origin of the discrepancy between the datasets can be related to a range of possibilities including the characteristics of the reanalysis system. Thus, the intrinsic process involved in weather variability should be carefully concluded with various observational products (Kim and Kim 2013; Marques et al. 2010). Nevertheless, our results suggest that a simple energetics analysis would provide useful metrics.

Appendices

Appendix 1: governing equations

We start from a set of primitive equations for non-divergent hydrostatic dry air:

$$\begin{gathered} \frac{{Du}}{{Dt}} - \frac{{\tan \varphi }}{a}uv - fv = - \frac{1}{{a\cos \varphi }}\frac{{\partial \Phi }}{{\partial \lambda }} + F_{\lambda } \hfill \\ \frac{{Dv}}{{Dt}} - \frac{{\tan \varphi }}{a}u^{2} + fu = - \frac{1}{a}\frac{{\partial \Phi }}{{\partial \varphi }} + F_{\varphi } \hfill \\ 0 = - \frac{{\partial \Phi }}{{\partial p}} - \alpha \hfill \\ \frac{{D\theta }}{{Dt}} = \left( {\frac{\partial }{{\partial t}} + U \cdot \nabla } \right)\theta = \left( {\frac{\partial }{{\partial t}} + \frac{u}{{a\cos \varphi }}\frac{\partial }{{\partial \lambda }} + \frac{v}{a}\frac{\partial }{{\partial \varphi }} + \omega \frac{\partial }{{\partial p}}} \right)\theta = \left( {\frac{{p_{0} }}{p}} \right)^{\kappa } \frac{Q}{{C_{p} }} \hfill \\ \nabla \cdot U = \frac{1}{{a\cos \varphi }}\left( {\frac{{\partial u}}{{\partial \lambda }} + \frac{{\partial v\cos \varphi }}{{\partial \varphi }}} \right) + \frac{{\partial \omega }}{{\partial p}} = 0 \hfill \\ \end{gathered}$$

where $\lambda$, $\varphi$, $p$, and $t$ is longitude, latitude, pressure, and time, respectively; $U=\left(u,v,w\right)$ is 3-dimensional wind velocity, $\theta$ is potential temperature ($={\left({p}_{0}/p\right)}^{\kappa }T$), $\Phi$ is geopotential height, $\alpha$ is specific volume; $a$ is the radius of the earth, $f$ is the Coriolis parameter, $F=\left({F}_{\lambda },{F}_{\varphi },0\right)$ is friction, $Q$ is diabatic heating, ${p}_{0}$ is reference pressure (1000 hPa). All other notations follow general form.

Appendix 2: time-averaged energy terms

Monthly means of available potential energy and kinetic energy are expressed as follows:

$$\begin{aligned} P_{M} &= \frac{{C_{p} }}{2}\left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)^{2} \hfill \\ P_{E} & = \frac{{C_{p} }}{2}\left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \overline{{\theta ^{{\prime 2}} }} = \frac{{C_{p} }}{2}\gamma \overline{{T^{{\prime 2}} }} \hfill \\ K_{M} & = \frac{{\bar{u}^{2} + \bar{v}^{2} }}{2} \hfill \\ K_{E} &= \frac{{\overline{{u^{{\prime 2}} }} + \overline{{v^{{\prime 2}} }} }}{2} \hfill \\ \end{aligned}$$

where $\left\langle X \right\rangle$ denotes global averages over a given pressure level and $\gamma =-\kappa /p{{\left( {{p}_{0}}/p \right)}^{\kappa }}{{\left( d\left\langle {\bar{\theta }} \right\rangle /dp \right)}^{-1}}$ is the static stability of the dry atmosphere. P and K denote available potential energy and kinetic energy, while subscripts _M and _E indicate mean and eddy energy, respectively. It can be inferred from the equation that the available potential energy is a function of temperature deviation from its surrounding, so that relatively warmer (colder) air can potentially induce air motion. Due to strong hydrostatic constraints kinetic energy is solely expressed by the horizontal winds.

Appendix 3: time-averaged conversion terms

Conversion of energy C(A, B) from A to B is calculated as

$$\begin{gathered} C\left( {P_{M} ,\,K_{M} } \right) = - \bar{\omega }\bar{\alpha } ~~ ; \quad C\left( {P_{E} ,\,K_{E} } \right) = - \overline{{\omega ^{\prime}\alpha ^{\prime}}} \hfill \\ C\left( {P_{M} ,\,P_{I} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)\nabla \cdot \left( {\overline{{\theta ^{\prime}U^{\prime}}} } \right) \hfill \\ C\left( {P_{E} ,\,P_{I} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \overline{{\theta ^{\prime}U^{\prime}}} \cdot \nabla \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right) \hfill \\ C\left( {K_{M} ,\,K_{I} } \right) = \bar{u}\nabla \cdot \overline{{u^{\prime}U^{\prime}}} + \bar{v}\nabla \cdot \overline{{v^{\prime}U^{\prime}}} - \frac{{\tan \varphi }}{a}\left( {\bar{u}\overline{{u^{\prime}v^{\prime}}} - \bar{v}\overline{{u^{\prime}u^{\prime}}} } \right) \hfill \\ C\left( {K_{E} ,\,K_{I} } \right) = \overline{{u^{\prime}U^{\prime}}} \cdot \nabla \bar{u} + \overline{{v^{\prime}U^{\prime}}} \cdot \nabla \bar{v} + \frac{{\tan \varphi }}{a}\left( {\bar{u}\overline{{u^{\prime}v^{\prime}}} - \bar{v}\overline{{u^{\prime}u^{\prime}}} } \right) \hfill \\ \end{gathered}$$

The conversion of available potential energy to kinetic energy [positive C(P _M , K _M ) or C(P _E , K _E )] results when relatively warm air (positive ${\alpha }=\frac{RT}{p}$ rises (negative $\omega$). The transfer of mean energy into (from) eddy energy is split into two parts that incorporate interaction term whose appropriate time mean is zero. Here, the subscript _I stands for interaction energy that links between the mean energy and the eddy energy, but it disappears after monthly averaging. A detailed discussion of the interaction energy can be found in Murakami (2011). Regarding the calculation of energy conversion between the eddy terms, C(P _E , K _E ), the $\omega \bullet \alpha$ formula is utilized to focus on the thermodynamic aspect of energy conversion (Kim and Kim 2013).

Appendix 4: time-averaged generation, dissipation, and boundary flux terms

Generation, dissipation, and boundary fluxes are as follows. In this study, however, generation terms were not calculated explicitly, but as a residual of fluxes. As the following terms are highly sensitive to reanalysis configuration, they are not extensively discussed in this research.

$$\begin{gathered} G\left( {P_{M} } \right) = \gamma \left( {\bar{T} - \left\langle {\bar{T}} \right\rangle } \right)\left( {\bar{Q} - \left\langle {\bar{Q}} \right\rangle } \right)~;\quad G\left( {P_{E} } \right) = \gamma \overline{{T^{\prime } Q^{\prime } }} \hfill \\ D\left( {K_{M} } \right) = - \bar{U} \cdot \bar{F}~~;\quad D\left( {K_{E} } \right) = - \overline{{U^{\prime } \cdot F^{\prime } }} \hfill \\ B\left( {P_{M} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \nabla \cdot \left[ {\frac{{\left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)^{2} - \left\langle {\bar{\theta }} \right\rangle ^{2} }}{2}\bar{U}} \right] \hfill \\ B\left( {P_{E} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \left( {\frac{{\overline{{\theta ^{{\prime 2}} }} }}{2}\bar{U} + \overline{{\frac{{\theta ^{{\prime 2}} }}{2}U^{\prime } }} } \right) \hfill \\ B\left( {K_{M} } \right) = \nabla \cdot \left[ {\left( {\frac{{\overline{u} ^{2} + \overline{v} ^{2} }}{2} + \bar{\Phi }} \right)\bar{U}} \right] \hfill \\ B\left( {K_{E} } \right) = \nabla \cdot \left( {\frac{{\overline{{u^{{\prime 2}} }} + \overline{{v^{{\prime 2}} }} }}{2}\bar{U}} \right) + \nabla \cdot \left[ {\overline{{\left( {\frac{{u^{{\prime 2}} + v^{{\prime 2}} }}{2} + \Phi ^{\prime } } \right)U^{\prime } }} } \right] \hfill \\ \end{gathered}$$