1. Introduction
The intricate interplay between solar insolation, atmospheric circulation patterns, and the redistribution of energy shapes our daily weather, regional climate, and ecosystems. Understanding of the fundamental processes that govern the transformation and redistribution of energy within the atmosphere is crucial for unraveling the complex dynamics of weather and climate systems and their response to external forcings. A seminal work that has remarkably contributed to this field is the introduction of the globally defined theory of available potential energy (APE) (Lorenz 1955b). Physically, the APE represents the maximum of potential energy available for reversible conversions into kinetic energy (KE) (Tailleux 2013). This approach, exemplified by the widely accepted four-box Lorenz energy cycle (LEC), provides valuable insights into the relationships between eddy KE and eddy APE with respect to zonal-mean KE and APE. In most models, the LEC is dominated by the energy conversion pathways pertaining to baroclinic unstable eddies associated with storm-track dynamics (Oort and Peixoto 1976; Li et al. 2007). Presumably, the LEC is likely modulated by exchanges unrelated to storm-track dynamics, but their relative importance is difficult to quantify in a globally defined APE framework. To understand these, a local view of APE, not restricted to any specific type of “mean” and capable of describing the local energy transformations across various spatial and temporal scales, is a priori needed, but has remained uncommon. As the mean atmospheric circulation is far from zonal, how to interpret the LEC and its responses to climate change is far from straightforward in the global APE framework. In this paper, we show that investigating the issue from a local APE viewpoint is useful to get detailed insights into the nature of the LEC. The LEC and globally defined APE theory have played a pivotal role in assessing the global energy budget using reanalysis datasets (Li et al. 2007; Marques et al. 2010; Kim and Kim 2013; Pan et al. 2017; Ma et al. 2021), historical model simulations (Boer and Lambert 2008; Lembo et al. 2019), and climate model projections (Ahbe and Caldeira 2017; Michaelides 2021; Kanno and Iwasaki 2022). However, the global character of APE and its focus on the zonal-mean circulation are increasingly recognized as important limitations of the framework, owing to the difficulties in linking the volume-integrated energy pathways to regional ones. Attempts to adopt a regional perspective on APE estimation have been made (e.g., Oort and Peixoto 1976; Li et al. 2007; Ahbe and Caldeira 2017), but few studies have used a rigorously derived local APE framework.
These conceptual difficulties and limitations associated with the global APE framework prompted the quest for local available energetics. The possibility to construct Lorenz APE theory from a local principle was first established by Holliday and McIntyre (1981), Andrews (1981), and Shepherd (1993); see also Tailleux (2013, 2018) for a review and most recent formulation. In contrast to the global APE theory, the local APE theory defines the total APE of a system as the volume integral of a positive definite APE density. Physically, the APE density of a fluid parcel represents the path-independent work against buoyancy forces (defined relative to a Lorenz reference state density field) needed to move a fluid parcel from its reference position to its actual position. Because the APE density represents a fundamental property of a fluid parcel, it is physically meaningful to speak of its advection, diffusion, conversion with kinetic energy, or fluxes through regional boundaries, which is not possible with the sign indefinite integrand of the global APE theory.
Importantly, the local APE framework is not restricted to any specific type of mean. By defining the concept of an “eddy” relative to a temporal mean, ensemble means, or other types of mean, the local theory provides a versatile and adaptable framework for systematically investigating regional energy conversions not necessarily restricted to storm-track dynamics in many new and different ways. The usefulness of the approach was first demonstrated by Kucharski and Thorpe (2000) and then by Novak and Tailleux (2018), which provided a three-dimensional representation of local APE density and its budget terms using the ERA-Interim reanalysis dataset. The globally integrated local APE can be easily computed using pressure-level datasets with higher accuracy than the traditional calculation of Lorenz APE on isobaric surfaces under the quasigeostrophic approximation. One key insight revealed by the local APE framework that had remained inaccessible to the global APE framework is the importance of the advection of APE in the storm-track regions (Novak and Tailleux 2018) or in the core region of tropical cyclones (Harris et al. 2022).
In this study, we aim to present a comprehensive analysis of the local APE framework, focusing on its interactions with kinetic energy terms, which inherently possess a local nature. In fact, the multiscale nature of atmospheric eddies (e.g., Faranda et al. 2018; Lovejoy 2019; Franzke et al. 2020) implies that all scales reproduce the same features of energy conversions within eddies and that energy is transferred upscale and downscale along a known kinetic energy cascade.
We utilize the fifth major global reanalysis produced by ECMWF (ERA5; Hersbach et al. 2020) to examine the climatology and long-term changes in energetic terms, offering a holistic depiction of the atmospheric energy cycle within this local energetic framework. Building upon the work of Novak and Tailleux (2018), our study not only presents a complete cycle of the local energetic framework but also investigates its observed changes over the last few decades. By providing a benchmark for understanding large-scale circulation changes through the local APE framework, this work will contribute to the advancement of atmospheric energetic research. The remainder of this paper is organized as follows: Section 2 provides a brief introduction to the data and methodology employed; section 3 documents the climatology of energetic terms and their budget components; section 4 discusses the observed long-term changes in local energetics; and finally, section 5 concludes with a summary and further discussion.
2. Data and methods
a. Data
To analyze the energy cycle, we use the daily ERA5 dataset covering the period 1979–2021 at a horizontal resolution of 1°, with 20 vertical pressure levels up to 50 hPa (Hersbach et al. 2020). We also tried more vertical levels extending up to 10 hPa. However, the results are similar, suggesting that the 20 levels we considered are enough to get robust results. The principal atmospheric fields used include horizontal and vertical wind, temperature, and geopotential. Energetic terms are first computed using daily data and then aggregated into seasonal means. We focus on the boreal winter (December–February), when the jet stream and eddies are stronger in the Northern Hemisphere.
b. Definition of low-frequency and high-frequency APE and KE densities
c. Detailed expressions of the local energetic framework
In this part, the budget equations for various forms of energy are expressed mainly according to Novak and Tailleux (2018) and Novak (2016), where the detailed derivation can be found. However, we made some reformulations in the calculations of some terms of the APE budget including the eddy advection term in the high-frequency APE budget and diabatic heating term. The eddy advection term in Eq. (29) in Novak and Tailleux (2018) is calculated as a divergence term
The physical meanings of the terms of Eq. (5) from left to right are as follows: local temporal tendency of low-frequency APE indicating how low-frequency APE will change, mean advection of low-frequency APE which horizontally transports APE, conversions from low-frequency KE to APE, conversions from high-frequency to low-frequency APEs, and diabatic heating terms which generate low-frequency APE. The interpretations of the terms of Eq. (6) are as follows: local temporal tendency of high-frequency APE indicating how high-frequency APE will change, mean advection of high-frequency APE and eddy advection of total APE which both describe the horizontal transport of APE, conversions from high-frequency KE to APE, conversions from low-frequency to high-frequency APEs, and diabatic heat terms which describe the generation of high-frequency APE.
The physical meanings of the terms of Eq. (9) from left to right are as follows: local temporal tendency of low-frequency KE, mean advection of low-frequency KE, conversions from high-frequency to low-frequency KEs, horizontal divergence of the high-frequency momentum flux, mean advection of low-frequency geopotential anomaly with respect to the reference state, conversions from low-frequency APE to KE, the friction flux, and terms associated with the spherical geometry. The interpretations of the terms of Eq. (10) are as follows: local temporal tendency of high-frequency KE, mean advection of high-frequency KE, eddy advection of high-frequency KE, conversions from low-frequency to high-frequency KEs, low-frequency eddy advection of high-frequency geopotential anomaly with respect to the reference state, conversions from high-frequency APE to KE, the friction flux, and terms associated with spherical geometry.
d. Long-term changes
The statistical significance levels of long-term changes in the energetic terms and their budget between two different time epochs (i.e., 2007–21 and 1980–94) are estimated by the two-sided Student’s t test. A 10-day Lanczos (Duchon 1979) filter is implemented to the original atmospheric fields to eventually partition the APE and KE terms into low and high frequencies, thereby enabling us to investigate interactions between synoptic-scale (i.e., high-frequency) waves and Rossby (i.e., low-frequency) waves associated with synoptic-scale weather systems and the slowly varying circulation, respectively.
3. Climatology of the atmospheric energetics
a. Climatology of APE and KE terms
Figure 1 shows the spatial distribution of the boreal winter climatology of APE and KE and their interannual standard deviations. The APE and KE terms are further partitioned into low-frequency (periods of more than 10 days) and high-frequency (periods of less than 10 days) parts, which are denoted by subscripts L and H, respectively. Along with their budget terms, the energetic interactions between the transient disturbances on synoptic scales and low-frequency Rossby waves (i.e., L = low-frequency waves + zonal mean flow) can be explicitly evaluated. The centers of PL are mostly concentrated over the high-latitude region north of 60°N with maximum amplitude in the upper troposphere at 400 hPa (Figs. 1a,e). Interestingly, this vertical distribution contrasts with that of the classical Lorenz energy cycle, which shows a maximum of the zonal-mean APE near the surface in polar regions (Peixóto and Oort 1974; Li et al. 2007; Kim and Kim 2013). It is mainly because the local APE is defined as the integrated buoyancy force for each air parcel between the actual level and the reference level of pressure. The reference pressure level should satisfy
The spatial distribution of the vertically integrated (1000–50 hPa; mass-weighted) climatological (a) low-frequency APE PL (shading), (b) high-frequency APE PH, (c) high-frequency APE KH, and (d) low-frequency KE KL in the boreal winter (December–February) during 1980–2021. (e)–(h) As in (a)–(d), but for zonal average over the globe. The values in parentheses represent the global mean of energetic terms. The contours denote the annual standard deviation, and the units of these energetic terms are in 105 J m−2.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
In the distribution of KH, there are two maxima over the North Pacific and North Atlantic regions in the Northern Hemisphere matching up with locations of the storm tracks at around 300 hPa (Figs. 1c,g), where strong transient disturbances occur. Note that the Pacific KH center is less collocated with the PH maximum than that over the Atlantic (Fig. 1c). There are two centers of standard deviation residing on the northward and southward sides of the KH peaks (Fig. 1g), indicating a meridional oscillation of storm tracks on the interannual time scale. Upstream and equatorward of the KH maxima, KL peaks over the northwestern Pacific and the northwestern Atlantic, showing the location of the jet stream (Figs. 1d,h). Differently from the Pacific jet, a clear separation between the subtropical and eddy-driven jets is seen over the Atlantic (Fig. 1d), identifying the split jet structure. The standard deviations are shifted northward and to the downstream area of the KL maxima, overlapping with the storm-track centers. Momentum converges there and feeds back onto the low-frequency circulation which sustains the jet. In the latitude–pressure cross-sectional plot, two standard deviation centers are displayed on the northern and southern sides of the KL peaks over the Southern Hemisphere but not over the Northern Hemisphere (Fig. 1h) where the jet may vary more in terms of intensity on interannual time scales during the boreal winter. This splitting feature of the jet over the Southern Hemisphere is also visible in KH (Fig. 1g). Note that KL is dominated by the subtropical jet that could be due to the fact that KL captures the dynamics of the waveguide of the subtropical jet. The magnitude of KE in transient disturbances is about one-sixth of that of the low-frequency circulation on global average.
b. Climatology of low-frequency APE budget terms
Figure 2 shows the distribution of the budget terms for PL according to Eq. (5). Note that we show terms with a relatively small magnitude in supplementary figures (i.e., local tendency in Fig. S2). The negligible role of the residual term, estimated as the differences between the local tendency term and the sum of the terms on the right-hand side of Eq. (5), shows that the PL budget is well closed, and moreover, the residue is mostly situated in the stratosphere.
The distribution of climatological budget terms for low-frequency APE (W m−2). Vertically integrated (1000–50 hPa; mass-weighted) (a) mean advection (shading), (b) conversion from high-frequency to low-frequency APEs C(PH, PL), (c) conversion from low-frequency KE to APE C(KL, PL), and (d) generation of low-frequency APE G(PL) in the boreal winter (December–February) during 1980–2021. (e)–(h) As in (a)–(d), but for zonal averages over the globe. The values in parentheses represent the global mean of conversion terms. The contours denote the annual standard deviation.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
Large anomalies of the mean advection mainly appear in the midlatitudes, where there is concurrent occurrence of strong winds and a large PL gradient in the Northern Hemisphere (Fig. 2a). There are two salient positive anomalies of mean advection terms: One is over eastern China extending eastward toward the Pacific across Japan; and the other is over the eastern United States and the northwestern Atlantic. We further decompose the mean advection into different components in meridional, zonal, and vertical directions (Fig. S3). Interestingly, a generally strong compensation is revealed between the meridional and zonal components, with relatively small contributions coming from the vertical component. The positive anomalies over land are mainly related to meridional advection on the western side of the PL peaks associated with northerly cold air intrusion, while those over the downstream ocean are mostly associated with zonal advection on the PL peaks by the westerlies, consistent with cold air outbreaks. Near the poles, positive anomalies in the lower atmosphere are mostly attributed to the vertical component due to the extremely cold continents, such as Greenland and Antarctica (Figs. 2a,e). The visible downward motion there would transport PL from the mid- to lower tropospheres along the gradient. Theoretically, the advection term should integrate to zero globally under the assumption of mass conservation, in which case the volume-integrated budget equations would reduce to those of the classical LEC budget. This assumption is the backbone of the classic LEC budget, which uses globally integrated terms. The terms associated with spherical geometry in the kinetic energy budget are either canceled out or often neglected because of their small magnitude (Oort 1964). Then, the sum of APE and KE will be conserved under frictionless and adiabatic flow. The nonzero advection part mainly comes from the vertical component (Figs. S3e,h) and resembles positive anomalies of wind convergence, mostly the low-frequency component (Figs. S4c,d). Note that the vertical velocity may not be reliable over Greenland and Antarctica due to some data artifacts and our method of using coarse vertical integrals in stable boundary layers over complex terrain. The nonzero values of convergence mean that the continuity equation is not satisfied, and thus, the mass and energy are not totally conserved. This is a common problem in the reanalysis due to data assimilation (Hersbach et al. 2020). Although methods have been proposed to correct mass and energy conservation issues in reanalysis products (Trenberth et al. 2009), implementing these in practice represents a major undertaking that we did not attempt in this study. For this reason, the globally integrated values estimated using reanalysis may not be reliable and should therefore be interpreted with caution. Although such a limitation has remained largely unnoticed so far, it should be noted that all published studies of the classic LEC based on uncorrected reanalysis products may be affected. A key hypothesis of this study is that local energy conversions are only affected at the second order by mass and energy conservation errors in reanalysis products and therefore more reliable. The standard deviations mainly peak in the polar regions in the mid–upper troposphere, which likely depends on the PL values (Fig. 2a).
The pattern of interactions between PH and PL [i.e.,
The climatological global-mean value of C(KL, PL) is −1.58 W m−2 (Fig. 2c) in contrast to previous estimates with a minor magnitude and inconclusive sign using the classic LEC due to the canceling effect between the Hadley and Ferrell cells (Li et al. 2007). This highlights the important role of low-frequency waves in transforming PL to KL, which is not considered in the zonal-mean component of LEC. Furthermore, the pattern of C(KL, PL) mostly reflects the overturning of the thermally direct circulation, while the Ferrel cell plays a dominant role in the LEC (Li et al. 2007). Another possible reason partly accounting for the differences could be large negative anomalies near the surface around the South Pole (Fig. 2g), which are mainly attributed to the strong downward motion there associated with high and complex topography (Fig. S7f) or because the separation into eddy and mean components is different. The conversions from KL to PL are determined by the product of
The C(KL, PL) and diabatic generation terms are the dominant contribution terms in the PL budget (Figs. 2c,d). The opposite sign between them indicates that the major part of the PL generation term is mostly consumed by conversion into low-frequency kinetic energy by C(KL, PL), particularly in the tropics. The generation term is mainly contributed by
c. Climatology of high-frequency APE budget terms
As discussed above, the low-frequency APE is the major source for high-frequency APE through eddy mixing across strong temperature gradients at 300 hPa (Figs. 3c,h). The conversions from low-frequency to high-frequency APEs C(PL, PH) and from high-frequency APE to KE C(KH, PH) constitute the two most important terms for the PH budget analysis and largely compensate each other (Figs. 3c,d,h,i), which is in agreement with the classic LEC framework. The C(PL, PH) and C(KH, PH) terms are maximized poleward of and at the location of the storm-track entrance regions, respectively. The centers of C(KH, PH) are shifted downward at about 400 hPa below those of C(PL, PH) in the vertical dimension (Figs. 3h,i). This vertical mismatch of centers between two conversions is mainly transformed by the eddy advection term of Eq. (6) (i.e.,
The distribution of climatological budget terms for high-frequency APE (W m−2). Vertically integrated (1000–50 hPa; mass-weighted) (a) mean advection (shading), (b) eddy advection of high-frequency total APE, (c) conversion from low-frequency to high-frequency APEs C(PL, PH), (d) conversion from high-frequency KE to APE C(KH, PH), and (e) generation of high-frequency APE G(PH) in the boreal winter (December–February) during 1980–2021. (f)–(j) As in (a)–(e), but for zonal averages over the globe. The values in parentheses represent the global mean of conversion terms. The contours denote the annual standard deviation.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
d. Climatology of high-frequency KE budget terms
The midtropospheric KH between 900 and 300 hPa is mainly converted from PH over the upstream part of the storm-track regions in the Northern Hemisphere as discussed in the previous section (Figs. 4b,g). Most of KH is consumed by
The distribution of climatological budget terms for high-frequency KE (W m−2). Vertically integrated (1000–50 hPa; mass-weighted) (a) mean advection (shading), (b) conversion from high-frequency APE to KE C(PH, KH), (c) low-frequency eddy advection of high-frequency geopotential anomaly with respect to reference state
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
e. Climatology of low-frequency KE budget terms
Analogously to the KH budget, C(PL, KL) is mostly compensated by
The distribution of climatological budget terms for low-frequency KE (W m−2). Vertically integrated (1000–50 hPa; mass-weighted) (a) mean advection, (b) conversion from low-frequency APE to KE C(PL, KL), (c) mean advection of low-frequency geopotential anomaly with respect to reference state
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
4. Long-term changes in the atmospheric energetics
In the previous section, we have identified the three-dimensional spatial distribution of climatological features of atmospheric energetics. Now, we proceed to examine their long-term changes, which can help to provide a new perspective for understanding climate change. Figure 6 displays the long-term changes in different energetic forms and associated budget terms averaged over 0°–360°, 60°S–60°N during 1980–2021. The polar region is excluded because of the nonconservation issue of mass and energy there. There is a significant increasing trend of PH and KH, suggesting intensified eddy and storm activities, which is consistent with previous studies on historical changes in eddy energetics using reanalysis (Pan et al. 2017; Ma et al. 2021). These prominent increases in PH and KH well correspond to the significant increasing trend of the conversion terms, C(PL, PH) and C(PH, KH), respectively. This finding indicates an increase in baroclinic instability, which mainly occurs in the midlatitudes at the synoptic scale and is associated with the conversion of PL to PH and subsequently to KH (Lembo et al. 2019). The low-frequency energetic forms also show increases but do not reveal any significant long-term changes. It is noteworthy that climate model studies tend to present a suppressed energy cycle in response to a warming scenario (Lucarini et al. 2010; Michaelides 2021; Kanno and Iwasaki 2022). This controversy between observed historical and projected future changes may be because the model cannot replicate the observed atmospheric energy cycles, mostly too vigorous in model simulations as suggested by Boer and Lambert (2008). Revealing the exact reason why there are differences between historical and future changes in energy cycles is out of the scope of our current study. Our work overall agrees with the previous literature using reanalysis, and it may be useful to investigate this controversy from a local perspective in the future. Next, the characteristics of long-term changes in energetics will be explored from a regional view over the boreal winter hemisphere.
Time series of different energetic (kJ m−2) and associated conversion (mW m−2) terms in winter averaged over 0°–360°, 60°S–60°N during 1980–2021. (a) PL, (b) PH, (c) KH, (d) KL, (e) C(PL, PH), (f) C(PH, KH), (g) C(KH, KL), and (h) C(KL, PL). The polar regions are excluded because mass and energy are not conserved there in the ERA5 reanalysis. The straight line indicates the linear trend. Slope and significance levels are denoted by s and p values, respectively, in the upper-left corner of each panel. The red shading denotes the 95% confidence interval of the fitted line.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
a. Long-term changes in APE and KE terms
Figure 7 shows the long-term changes in various energetic forms between two 15-yr time periods (i.e., 2007–21 and 1980–94). We also checked the changes between two 20-yr time periods, which are highly consistent with those presented in the following analysis. There is a prominent decrease in the low-frequency APE over the Arctic (Fig. 7a). This is expected as the temperature over the Arctic increases much faster than that over other regions, a phenomenon referred to as “Arctic amplification” (Serreze et al. 2009). The increases in temperature reduce the departure from the isobaric mean over the Arctic. As a result, the magnitude of differences in pressure between actual state and low-frequency state becomes smaller, characterized by positive anomalies (Fig. S14) in contrast to the negative climatology (Fig. S1). The decreases in differences in
The long-term changes (104 J m−2) in energetic terms in the boreal winter (December–February) during 1980–2021. (a) Low-frequency APE, (b) high-frequency APE, (c) high-frequency KE, and (d) low-frequency KE. Black dots mark the grid points where the changes are at the 0.05 significance level according to the Student’s t test. (e)–(h) As in (a)–(d), but for zonal averages over the globe.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
b. Low-frequency APE budget changes
The long-term changes in the budget terms are examined to delineate how the contribution of different conversion terms varies between the two time periods. The long-term changes in PL mean advection feature an increased band from northeastern China to Japan and the adjacent ocean in conjunction with its climatological extrema (Figs. 8a and 2a). The latitude–pressure cross section of changes shows a dipole in the mid–upper troposphere between 500 and 300 hPa, with negative values between 50°N and the North Pole and positive values between 40° and 50°N (Fig. 8b). This pattern is generally consistent with the climatological mean advection, indicating a stronger meridional advection from the polar region toward the midlatitudes. Albeit significant, the changes over the tropics tend to be fragmentized. There is a dipolar pattern of zonal-mean changes in C(PH, PL) over the high latitudes between 600 and 400 hPa (Fig. 8f). This dipole resides on the northern and southern sides of its climatological negative center, suggesting a southward displacement of the conversion extrema. Interestingly, further checking the distribution of C(PH, PL) dipole changes is characterized by a contrasting response over the two storm tracks, with decreases from northeastern China to Japan and increases from the northeastern United States to the Atlantic (Fig. 8b). These conversion anomalies serve as a compensation effect to the mean advection term (Figs. 8a,b). For the C(KL, PL) term, the changes mainly appear in the tropical and polar regions, particularly in the former (Figs. 8c,g). There are negative anomalies over the eastern equatorial Pacific and the Maritime Continent extending southeastwardly (Fig. 8c), accompanied by anomalous upward motion of warm air converting PL to KL. Between these negative anomalies, positive anomalies are seen over the western and southeastern Pacific associated with anomalous downward motion of warm air transforming KL into PL. This dipolar pattern of anomalies over the western and central Pacific agrees with a strengthening and westward shift of the Pacific Walker circulation in recent decades (Sohn and Park 2010; Takahashi and Watanabe 2016), which probably arises from the internal climate variability (Bordbar et al. 2017; Chung et al. 2019) such as the warming of the Atlantic (McGregor et al. 2014) and phases of the interdecadal Pacific oscillation (England et al. 2014). The zonal-mean long-term changes are also characterized by a dipole within the northern tropics with decreases and increases over the northern and southern parts, respectively (Fig. 8g), in favor of a northward shift of the ascending branch of the Hadley circulation. To the north, the positive anomalies denote a strengthening of the descending branch of the Hadley cell due to stronger conversions into PL from KL. Changes in the PL generation term exhibit a reversed dipole as compared to C(KL, PL) over the tropics, compensating with each other (Figs. 8d,h). The increases and decreases in the generation term are mainly related to the changes in latent heat release associated with the potential temperature changes. Although the contribution from different budget balances may vary between the two periods, their offsetting effects result in minor changes in the tendency of PL (Figs. S15a,d).
The long-term changes in the budget of low-frequency APE budget (W m−2) in the boreal winter (December–February) between 2007–21 and 1980–94. (a) Mean advection (shading), (b) conversion from high-frequency to low-frequency APEs C(PH, PL), (c) conversion from low-frequency KE to APE C(KL, PL), and (d) generation of low-frequency APE G(PL) in the boreal winter (December–February) during 1980–2021. (e)–(h) As in (a)–(d), but for zonal averages over the globe. Black dots mark the grid points where the differences are at the 0.05 significance level according to the two-sided Student’s t test. The contours denote the long-term climatology during 1980–2021.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
c. High-frequency APE budget changes
Most of the PH budget term changes appear in eddy advection, C(PL, PH), and C(KH, PH), while relatively marginal differences are shown in the mean advection and the generation terms. Note that the differences in C(PL, PH) play an important role in the PH budget changes, but not for the PL budget changes. This is expected as C(PL, PH) is the main source of PH (Fig. 3). The spatial distribution of integrated eddy advection changes displays a wave-like pattern with alternating positive and negative anomalies within the same latitude band (Fig. 9b). The latitude–pressure cross section of eddy advection changes is characterized by a meridional dipole structure in the midtroposphere with positive anomalies over the North Pole and negative anomalies in the midlatitudes (Fig. 9g), suggesting a northward transport of PH by high-frequency eddies. The positive anomalies of PH over the polar regions are mainly transferred into PL (Fig. 9h), with a secondary contribution from C(KH, PH) (Fig. 9i). The negative counterpart in the midlatitudes is mostly replenished by PL (Fig. 9h).
The long-term changes in the budget of high-frequency APE budget (W m−2) in the boreal winter (December–February) between 2007–21 and 1980–94. (a) Mean advection (shading), (b) eddy advection, (c) conversion from low-frequency to high-frequency APEs C(PL, PH), (d) conversion from high-frequency KE to APE C(KH, PH), and (e) generation of high-frequency APE G(PH). (f)–(j) As in (a)–(e), but for zonal averages over the globe. The contours denote the long-term climatology during 1980–2021.
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
d. High-frequency KE budget changes
The major changes in the KH budget come from C(PH, KH), the low-frequency eddy advection of high-frequency geopotential anomaly
The long-term changes in the budget of high-frequency KE budget (W m−2) in the boreal winter (December–February) between 2007–21 and 1980–94. (a) Mean advection (shading), (b) conversion from high-frequency APE to KE C(PH, KH), (c) low-frequency eddy advection of high-frequency geopotential anomaly with respect to reference state
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
e. Low-frequency KE budget changes
The major changes in the KL budget are the mean advection, C(PL, KL), and the mean advection of the low-frequency geopotential anomaly, while C(KH, KL), the convergence of momentum fluxes, and dissipation play a relatively minor role (Fig. 11). The changes in mean advection display alternating positive and negative anomalies in the midlatitudes, particularly over the Pacific (Fig. 11a). A prominent zonal dipolar pattern of anomalies is present with increases and decreases over the western and eastern Pacific, respectively. This pattern opposes the climatological mean advection with their centers shifted eastward, suggesting a weakening of the mean advection. However, a reversed dipole is displayed in the geopotential anomaly advection changes, compensating the mean advection impact (Fig. 11c). Over the tropics and poles, stronger anomalies are found in the C(PL, KL) and the geopotential anomaly advection term, offsetting with each other (Figs. 11b,c,h,i). These counteracting effects, among different terms, result in small changes in the local tendency (Figs. S16a,b).
The long-term changes in the budget of low-frequency KE budget (W m−2) in the boreal winter (December–February) between 2007–21 and 1980–94. (a) Mean advection, (b) conversion from low-frequency APE to KE C(PL, KL), (c) mean advection of low-frequency geopotential anomaly with respect to reference state
Citation: Journal of Climate 37, 23; 10.1175/JCLI-D-24-0106.1
5. Summary and discussion
In this study, the boreal winter climatology and long-term changes in various energetic forms and their associated budget terms in the Northern Hemisphere are presented using ERA5 reanalysis data in the context of a local energetic framework (Novak and Tailleux 2018). The main advantage of this local energetic framework is to provide the local information of APE and its interactions with KE as compared to the classic Lorenz energy cycle. A further decomposition of various energetic forms into low- and high-frequency components enables us to further understand the spatial variations of the conversions between APE and KE in contrast to the traditional zonal-mean and eddy partitioning.
The extrema of low-frequency APE are mainly located over the poles and the tropics in the midtroposphere, where the largest departure relative to the reference state pressure occurs. The high-frequency APE is mostly concentrated in the midlatitudes peaking downstream and to the south of the low-frequency APE centers in agreement with strong eddy activity there. Two maxima are shown vertically at 500 and 200 hPa. The centers of low-frequency KE are located over the Pacific and the Atlantic at 300 hPa, while those of high-frequency KE are situated to the northeast, corresponding to the locations of the jet streams and storm tracks, respectively. The well-collocated locations of APE, KE, and their conversions may further support the rationale of this local view of atmospheric energetics. For example, PH, KH, and C(PH, KH) all have a peak of around 400 hPa in the midlatitudes of the Northern Hemisphere. By contrast, this feature is not apparent from the zonal-mean distribution of classic Lorenz energy cycle (e.g., Li et al. 2007), which assumes that the globally integrated energetics can also be interpreted in a local sense.
Low-frequency APE is mainly generated over the tropics and the poles and consumed in the midlatitudes being converted into eddy APE, which is subsequently transformed into high-frequency KE. Both conversions are baroclinic processes, and while the former is mainly related to diabatic heating anomalies at 500 hPa associated with the thermally direct circulation, the latter is mostly guided by horizontal heat transport by transient eddies at 300 hPa. Globally, part of the high-frequency KE is converted into low-frequency KE, involving barotropic processes, and the remainder is dissipated due to friction within the planetary boundary layer. This energy pathway of low-frequency APE → high-frequency APE → high-frequency KE → low-frequency KE overall is consistent with the classic LEC four-box model (e.g., Ma et al. 2021). However, our local framework provides a much clearer picture of the compensation effects at regional scales among the different terms. For instance, the conversion from high-frequency APE to KE is largely offset by the eddy advection term of total APE. The conversion from high-frequency APE to KE is mainly counteracted by low-frequency eddy advection of high-frequency geopotential anomaly
The long-term changes in the low-frequency APE are characterized by a dipole at 400 hPa in the Northern Hemisphere, with decreases over the pole and increases over Siberia. This dipole shifts southwestward over the northern Pacific in the high-frequency APE changes at 500 hPa and in the high-frequency KE trend at 300 hPa, where their climatological extrema peak. These increases are well collocated with their centers, while decreases are to the north of the centers, suggesting a strengthening and narrowing of the synoptic eddy activities. This is in stark contrast with Franzke and Harnik (2023), which show an overall weakening of eddy fluxes during 1958–2018 based on the JRA-55 reanalysis. The discrepancies of long-term changes may be caused by the different analyzed periods. Kanno et al. (2016) revealed that the Siberia cooling is much stronger in the later period after 1979 than that starting from 1958. Another possible reason could be related to the circulation uncertainty in different reanalysis products due to a lack of observation in data assimilation, particularly over the ocean (Chemke and Polvani 2020). However, ERA5 reanalysis has been found to be the closest to observations in wind performance among different reanalysis products (Fan et al. 2021; Gualtieri 2022). For the low-frequency KE, there is an anomalous tripolar pattern of trend, with increases in the northern flank of the jet and decreases in the southern flank and to the north of the jet over the Pacific. This means a northward displacement and narrowing of the Pacific jet stream under global warming.
Further analysis of changes in the energetic budget between the periods of 2007–21 and 1980–94 helps to quantify the contribution of different terms in generating the long-term changes in the energetic forms. The zonal-mean changes in C(PL,PH) show a dipolar pattern, with negative anomalies in polar regions and positive anomalies in the midlatitudes (Fig. S17e), resembling those of PL changes. This is likely connected to the stationary wave structure changing and shifting PL tongues, where the cold air outbreaks associated with conversion to eddies give rise to storm tracks, equatorward over the continents. Interestingly, inspection of spatial distribution of changes in the midlatitudes displays a contrasting difference between the Pacific and the Atlantic (Fig. S17a). The former is characterized by an intensified conversion from PL to PH over the northwestern Pacific, while the latter shows a weakening over the northwestern Atlantic. In the downstream regions, these anomalies subsequently result in more conversions to KH over the Pacific and less over the Atlantic (Fig. S17b). Both C(PL, PH) and C(PH, KH) are associated with baroclinic processes, suggesting a stronger and weaker baroclinicity over the Pacific and the Atlantic, respectively, consistent with Wang et al. (2017). Further to the east, C(KH, KL) is enhanced over the central Pacific and weakened over the Atlantic (Fig. S17c), associated with barotropic processes. These strengthening and weakening conversion processes over the Pacific and the Atlantic indicate an accelerated energy cycle in the former and a suppressed energy cycle in the latter. The changes in C(PL, KL) are mainly featured with significant differences in the deep tropical regions, characterized by positive values over the eastern equatorial Pacific and the Maritime Continent and negative anomalies in-between over the western–central Pacific (Fig. S17d). This anomaly pattern coincides with a strengthening and westward shift of the Walker circulation likely arising from the internal climate variability (Chung et al. 2019).
Furthermore, the low-frequency APE changes in the boreal midlatitudes over East Asia are predominantly driven by an enhanced mean advection and are subsequently consumed by conversion into high-frequency APE. As a result, the high-frequency APE increases in the downstream areas. However, this intensification is mainly compensated by eddy advection rather than conversion into high-frequency kinetic energy, depicting a more complete picture than the classic paradigm. Changes in the high-frequency KE are mainly induced by conversion from the high-frequency APE, but their consumption is mostly through eddy advection of geopotential anomalies with relatively small transformation into low-frequency KE. The budget term of low-frequency KE shows no clear signals contributing to its long-term changes, which may partially explain the opposite trends of jet stream changes in previous studies (Wang et al. 2017; Franzke and Harnik 2023).
In summary, this work provides a first systematic attempt to depict the climatology and long-term changes in a local energetic framework developed by Novak and Tailleux (2018). The physical understanding of this local framework is consistent with the classic Lorenz energy cycle and more importantly provides insights about the interactions between APE and KE on different spatiotemporal scales. Moreover, the budget for various energetic forms is balanced quite well, both globally and locally. We acknowledge that only one dataset (i.e., ERA5) is used in this study. The robustness of the findings needs to be verified using other reanalysis products. Further applications of this framework in the context of teleconnection patterns (Kosaka et al. 2009), storm tracks (Mbengue and Schneider 2017), extratropical cyclone activities (Gertler and O’Gorman 2019), and monsoons (Hu et al. 2022) will be studied in the future. It is also interesting to evaluate the model performance on this local energetics and project their future changes under different socioeconomic scenarios. A moist version which explicitly considers the moisture effect can also be explored by using equivalent potential temperature or virtual temperature (Lembo et al. 2019; Harris et al. 2022). While using only virtual temperature to account for moist processes is potentially a limitation, this framework might still produce an effectively closed budget enabling physical insights into moist energetics.
Acknowledgments.
The authors thank Editor Jian Lu and two anonymous reviewers whose constructive comments led to a significant improvement of the manuscript. We thank Dr. Nili Harnik for helpful discussions. This study was supported by the Institute for Basic Science (IBS), South Korea, under IBS-R028-D1. ZL is supported by the start-up funding of the Hong Kong University of Science and Technology (Guangzhou), the National Natural Science Foundation of China (42405020), and the Guangzhou Municipal Science and Technology Project for Maiden Voyage (2024A04J4523). CF is supported by the National Research Fund of Korea (NRF-2022M3K3A109708). LN is supported by Schmidt Sciences. VL acknowledges financial support by project DROMEDAR-Grant Assignment Decree No. 1388 by the Italian Ministry of University and Research (MUR) under the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender n. 1409, funded by the European Union-NextGenerationEU, and funding from the Italian Ministry of Education, University and Research (MIUR) through the JPI Oceans and JPI Climate “Next Generation Climate Science in Europe for Oceans”-ROADMAP project (D. M. 593/2016) and from the European Union’s Horizon Europe research and innovation program Grant 101081193 (OptimESM). The analysis was conducted on the IBS/ICCP supercomputer “Aleph,” 1.43 petaflop high-performance Cray XC50-LC Skylake computing system with 18,720 processor cores, 9.59 PB storage, and 43 PB tape archive space. We also acknowledge the support of KREONET.
Data availability statement.
The ERA5 reanalysis product is available online from the website https://www.ecmwf.int/en/forecasts/dataset/ecmwf-reanalysis-v5.
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