Editorial Type:
Article
Article Type:
Research Article

Spectral Budget of Rotational and Divergent Kinetic Energy in Global Analyses

Zongheng Li aCollege of Meteorology and Oceanography, National University of Defense Technology, Changsha, China

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Jun Peng aCollege of Meteorology and Oceanography, National University of Defense Technology, Changsha, China

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https://orcid.org/0000-0003-4585-036X
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Lifeng Zhang aCollege of Meteorology and Oceanography, National University of Defense Technology, Changsha, China

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Online Publication:
23 Feb 2023
Print Publication:
01 Mar 2023
DOI:
https://doi.org/10.1175/JAS-D-21-0332.1
Page(s):
813–831
Free access

Abstract

To study the multiscale interactions between rotational and divergent components of atmospheric motion, a new formulation of spectral budget of rotational kinetic energy (RKE) and divergent kinetic energy (DKE) based on the primitive equations in the pressure coordinate is derived, with four main characteristics: 1) horizontal kinetic energy (HKE) spectral transfer is exactly divided into spectral transfer of RKE and DKE, 2) the exact spectral conversion term between DKE and RKE is constructed, 3) the Coriolis term is considered, and 4) both the baroclinic conversion from available potential energy (APE) and the vertical flux of HKE act only on DKE. With this new formulation, outputs from ERA5 global reanalysis are investigated. At planetary scales, HKE spectral transfer, mainly attributed to β effect, is dominated by downscale DKE transfer. At synoptic scales, it is dominated by an upscale transfer of RKE energized by conversion of DKE mainly due to the Coriolis effect. The ultimate source of DKE in the upper troposphere is conversion of APE, while in the stratosphere it is the vertical flux. At mesoscales, the spectral transfers of RKE and DKE are both downscale, and conversion from RKE to DKE exists at sub-800-km scales in the upper troposphere, which is mainly attributed to the contribution from relative vorticity. At different heights, the intersection scales of RKE and DKE spectra are affected by the scales of positive peaks of the local spectral conversion from DKE to RKE around total wavenumber 10.

Significance Statement

The purpose of this study is to explore more physical insights on the dynamics underlying the atmospheric energy spectra from the perspective of rotational and divergent components of motion. We derive a new formulation of the spectral rotational and divergent kinetic energy budget in the pressure coordinate for the global atmosphere, with application to ERA5 global reanalysis. Our results reveal the differences of spectral energy budget between rotational and divergent motions at different heights and scales. This new formulation provides a good tool for revealing the multiscale cascade and interaction between atmospheric rotational and divergent motions. Future work should investigate these dynamical processes with higher-resolution simulations and datasets.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Jun Peng, pengjun@nudt.edu.cn; Lifeng Zhang, zhanglif_qxxy@sina.cn

Abstract

To study the multiscale interactions between rotational and divergent components of atmospheric motion, a new formulation of spectral budget of rotational kinetic energy (RKE) and divergent kinetic energy (DKE) based on the primitive equations in the pressure coordinate is derived, with four main characteristics: 1) horizontal kinetic energy (HKE) spectral transfer is exactly divided into spectral transfer of RKE and DKE, 2) the exact spectral conversion term between DKE and RKE is constructed, 3) the Coriolis term is considered, and 4) both the baroclinic conversion from available potential energy (APE) and the vertical flux of HKE act only on DKE. With this new formulation, outputs from ERA5 global reanalysis are investigated. At planetary scales, HKE spectral transfer, mainly attributed to β effect, is dominated by downscale DKE transfer. At synoptic scales, it is dominated by an upscale transfer of RKE energized by conversion of DKE mainly due to the Coriolis effect. The ultimate source of DKE in the upper troposphere is conversion of APE, while in the stratosphere it is the vertical flux. At mesoscales, the spectral transfers of RKE and DKE are both downscale, and conversion from RKE to DKE exists at sub-800-km scales in the upper troposphere, which is mainly attributed to the contribution from relative vorticity. At different heights, the intersection scales of RKE and DKE spectra are affected by the scales of positive peaks of the local spectral conversion from DKE to RKE around total wavenumber 10.

Significance Statement

The purpose of this study is to explore more physical insights on the dynamics underlying the atmospheric energy spectra from the perspective of rotational and divergent components of motion. We derive a new formulation of the spectral rotational and divergent kinetic energy budget in the pressure coordinate for the global atmosphere, with application to ERA5 global reanalysis. Our results reveal the differences of spectral energy budget between rotational and divergent motions at different heights and scales. This new formulation provides a good tool for revealing the multiscale cascade and interaction between atmospheric rotational and divergent motions. Future work should investigate these dynamical processes with higher-resolution simulations and datasets.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Jun Peng, pengjun@nudt.edu.cn; Lifeng Zhang, zhanglif_qxxy@sina.cn
Keywords: Dynamics; Energy transport; Energy budget/balance; Spectral analysis/models/distribution

1. Introduction

The kinetic energy (KE) spectra of the atmosphere in the upper troposphere and lower stratosphere, obtained from observations such as horizontal wind and temperature measurements collected by commercial aircraft flights mainly at midlatitudes, show a distinct spectral transition from a −3 slope to a −5/3 slope near 500 km (e.g., Nastrom and Gage 1985). Up to now, the dynamical mechanisms producing atmospheric KE spectra remain controversial. The −3 region is commonly thought to be the result of an enstrophy cascade (Kraichnan 1967; Charney 1971). However, some studies also argued that an enstrophy cascade may not be necessary for the −3 region (O’Gorman and Schneider 2007), and even that the −3 region may be entirely the result of analysis errors (Lovejoy et al. 2009). While some early work suggested that −5/3 region might be the result of an inverse energy cascade driven by convection (Gage 1979; Lilly 1983), the energy cascade now is generally thought to be downscale (e.g., Cho and Lindborg 2001; Lindborg 2007; Waite and Snyder 2009; Callies et al. 2014). Various mechanisms have been proposed for such a downscale cascade, including quasigeostrophic dynamics near the tropopause (Tulloch and Smith 2006, 2009; Asselin et al. 2016a,b), inertia–gravity waves (IGWs; Dewan 1979; Bühler et al. 2014; Callies et al. 2014; Herbert et al. 2016; Kafiabad et al. 2019), and stratified turbulence (Lindborg 2006; Riley and Lindborg 2008). In addition to the mechanisms above, rotating stratified turbulence, in which the effect of rotation cannot be ignored, is also considered possible mechanism (Waite and Bartello 2006; Bartello 2010; Kafiabad and Bartello 2018). Kafiabad et al. (2019) have recently argued that the shallower slope of observed spectra is also close to −2, which can be explained with scattering of IGWs by geostrophic background flow. Given the complexity of the energy cascade, there may be several different mechanisms that play a dominant role at different scales, as well as at different heights. For example, Rodda and Harlander (2020) have suggested that two separate phenomena can contribute to the mesoscale spectrum: at the large-scale end of the mesoscale, IGWs may be more important and at the small-scale end more nonlinear phenomenon such as rotating stratified turbulence may be important. Wang and Bühler (2020) have suggested that waves dominate a mesoscale spectrum in the lower stratosphere, while in the upper troposphere the vortices are just as important as, or possibly more important than, waves. Among various proposed theories, the decomposition of the energy spectra into its rotational and divergent modes is one of the fundamental issues in the research of atmospheric energy spectra (e.g., Lindborg 2007; Lindborg and Brethouwer 2007; Bühler et al. 2014; Lindborg 2015; Li and Lindborg 2018). The relative importance and interaction of these two components at different scales are crucial to understand the dynamics underlying atmospheric energy spectra (Deusebio et al. 2013).

In addition to the often studied spectral transition from −3 to −5/3 near 500 km, KE spectra in high-resolution general circulation models and global reanalysis also exhibit a shallowing at larger scales. For example, the KE spectra derived from global high-resolution simulations present not only a mesoscale shallowing but also an obvious spectral transition around 2000 km in the stratosphere (Skamarock et al. 2014). Burgess et al. (2013) also found that the KE spectra exhibit a distinct transition between steep and shallow spectral ranges above 250 hPa based on an operational analysis, and the occurrence and variation of this spectral transition is related to the relative magnitude of the rotational and divergent components.

The rotational and divergent components of the horizontal kinetic energy (HKE) spectra, i.e., the rotational kinetic energy (RKE) and divergent kinetic energy (DKE) spectra, respectively, can be obtained by the Helmholtz decomposition of the horizontal velocity field (Koshyk and Hamilton 2001). Just like the two sides of a coin, this decomposition has both advantages and disadvantages compared with other advanced methods such as normal mode decomposition (Bartello 1995; Waite 2020), wave–vortex decomposition (Callies et al. 2014; Bühler et al. 2017; Wang and Bühler 2020), or high-order balanced–unbalanced decomposition (Kafiabad and Bartello 2016, 2018). On the strength side, the Helmholtz decomposition can be easily and straightforwardly applied to both model outputs and observation data. It is also capable of handling relatively complicated geometries and variable Coriolis parameter (e.g., Hamilton et al. 2008; Augier and Lindborg 2013; Waite and Snyder 2009, 2013; Peng et al. 2014a,b; Menchaca and Durran 2019). Furthermore, it can be applied to particular layers, suitable for the study of atmospheric properties that change significantly with altitude (e.g., Waite and Snyder 2009, 2013; Blažica et al. 2013; Peng et al. 2014a,b; Wang et al. 2018). On the weakness side, it is not as accurate as the other methods mentioned above in identifying the wave and vortex contributions. For example, the wave–vortex decomposition proposed by Callies et al. (2014) is based on the Helmholtz decomposition, but further considers the energy equipartition property of hydrostatic IGWs. The spectra of geostrophic and ageostrophic energies by normal mode decomposition will intersect at a different scale than that of RKE and DKE (Ambacher and Waite 2020). The main reasons are as follows: the linear IGWs can have both rotational and divergent part (Bartello 1995), therefore the divergence component does not accurately represent IGWs; furthermore, the Rossby waves have some divergent motion at large scales and low latitudes. In addition, Helmholtz decomposition, as a linear decomposition, ignores the divergent part that is slaved to the balanced rotational part due to nonlinearities (e.g., Warn et al. 1995; Kafiabad and Bartello 2016, 2018). Although the focus of present paper is limited to the rotational and divergent components of the KE spectra, it also needs to keep the weaknesses of the Helmholtz decomposition in mind in the interpretation of the related results.

For understanding the dynamics behind atmospheric energy spectra, not only the relative magnitude of the rotational and divergent components is important, but also the multiscale nonlinear interactions between these two components. As the atmosphere is a multiscale rotating stratified fluid, it is natural to ask the following questions: What would the interactions between the rotational modes be like, what would the interactions between the divergent modes be like, and to what extent these two types of interactions contribute to the total horizontal kinetic energy cascade? Indeed, some efforts have been made on these issues in previous studies. However, shortcomings of the used formulations of spectral energy budget seriously limit the results. Most of these studies either analyzed spectral energy budget within the framework of purely horizontal and nondivergent flow, resulting in only the interactions between the rotational modes being considered (e.g., Fjørtoft 1953; Boer and Shepherd 1983; Shepherd 1987; Trenberth and Solomon 1993; Straus and Ditlevsen 1999; Burgess et al. 2013), or simply computed the rotational component of the nonlinear KE spectral flux only with the rotational component of the velocity, resulting in the introduction of so-called complementary flux produced by interactions involving the divergent component of the velocity (e.g., Augier and Lindborg 2013). However, Helmholtz-decomposed KE spectra from observational datasets show that the divergent component actually dominates over the rotational component in the lower stratosphere (Callies et al. 2016; Li and Lindborg 2018), and even in the upper troposphere where the rotational component slightly dominates over the divergent component (Cho et al. 1999; Lindborg 2007), the level of the divergent component is still not negligible. That is to say, the divergent and rotational components of the atmospheric flows are on the same order of magnitude in the mesoscale. The divergent velocity may also contribute to the interactions between the rotational modes, which implies that the part of the interactions involving the divergent component of the velocity may belong to the latter. Therefore, a complete picture of the contributions of rotational and divergent components to the atmospheric energy spectra can be understood only through deriving the corresponding spectral budget from the primitive equation.

In general, the main purpose of the present paper is first to develop a rigorous formulation for the spectral energy budget of the divergent and rotational components and then to apply the novel formula to the state-of-the-art global reanalysis data. With the increased resolution and improved model, the global reanalysis now can resolve the atmospheric state across many scales reasonably well (Jewtoukoff et al. 2015; Watanabe et al. 2015; Gupta et al. 2021) and therefore provides a useful basis for studying energy spectra in a relatively realistic atmosphere. In doing so, our purpose is to gain more physical insights on the contributions of rotational and divergent components at various scales and at different heights, thereby deepening our understanding of the relationship between energy spectra and spectral budget. Moreover, it also provides additional insights on the multiscale energetics of the global atmosphere from the perspective of the divergent and rotational kinetic energy.

The remainder of the manuscript is arranged as follows. In section 2, the derivation of spectral HKE budget equation is reviewed, and the derivation of spectral budget equations of RKE and DKE is described. In section 3, the results based on global reanalysis data are presented and compared with previous studies. Subsequently, the characteristics of spectral RKE and DKE budget of global atmosphere obtained by the new formula are also presented. In section 4, we summarize and discuss our findings.

2. Methodology

a. Spectral budget equation of horizontal kinetic energy

At first, we make a review on the derivation of the spectral budget equation of HKE by Augier and Lindborg (2013). The horizontal momentum equation in the pressure coordinates, where variables are functions of time t, longitude λ, latitude φ, and pressure p, can be written as
∂tu=−(u⋅∇u+ω∂pu)−∇Φ−f(φ) ez×u+Du,
where f(φ) is the Coriolis parameter varying only with latitude φ, i.e., f(φ) = f0 sinφ, u = (u, υ) is the horizontal velocity vector, ω = Dtp is the pressure vertical velocity with Dt being the material derivative, ∇ is the horizontal gradient operator, ez is the vertical (upward) unit vector, and Du denotes the dissipation of u. The hydrostatic equation is ∂pΦ = −α = −RT/p, where Φ is the geopotential and α is the volume per unit mass. The main advantage of the p coordinates is that the mass continuity equation can be reduced to a diagnostic relationship: ∇3 ⋅ V = 0, where ∇3 = (∇, ∂p) is the three-dimensional gradient operator and V = (u, ω) is the three-dimensional velocity. The main disadvantage of the pressure coordinate is that the lower boundary of the atmosphere becomes very complicated because terrain passes through the isobaric surface. This can be overcome with the method developed by Boer (1982). Here, our focus is on the upper troposphere and stratosphere, where the influence of the topography is relatively small. For clarity, the spectral energy budget is derived for the pure pressure levels, and all quantities are calculated without regard to topography.
HKE per unit mass (EK) is given as
EK=(1/2 ) u⋅u.
Taking the dot product between u and Eq. (1) and reorganizing the different terms, we obtain the budget equation for HKE as follows:
∂tEK=−∇⋅EKu−∂p(ωEK)−∇⋅(uΦ)−∂p(ω Φ)−ω α−u⋅(ζez×u)−u⋅(fez×u)+u⋅Du.
In the above equation, the vertical vorticity is defined as ζ = roth(u) = ∇ × u, and the horizontal divergence is defined as δ = divh(u) = ∇ ⋅ u. The horizontal advection of the horizontal velocity is computed using the relation identity used in Augier and Lindborg (2013), i.e.,
u⋅∇u=∇EK+ζez×u.
Obviously, u ⋅ (ζez × u) and u · (fez × u) are equal to zero. This indicates that these two terms have no contribution to the budget of HKE, but in spectral budget analysis, their corresponding terms have effects on the transfer of HKE between different scales.
The spectral budget equation of kinetic energy has been derived by Augier and Lindborg (2013) based on spherical harmonic decomposition. Each scalar function defined on the sphere can be expanded in terms of spherical harmonic functions Ylm (Boer 1983), where l is the total wavenumber and m is the zonal wavenumber. The basis functions are Ylm = Plmeimλ with Plm Legendre polynomials. For spherical harmonics, the basis functions are eigenfunctions of the horizontal Laplacian operator on the sphere, i.e., ∇2Ylm=−l(l+1)Ylm/re2, where re is Earth’s radius. The streamfunction, truncated at total spherical harmonic wavenumber N, is expanded as
ψ(λ,ϕ,p)=∑0≤l≤N∑−l≤m≤lψlm(p)Ylm(λ,ϕ),
where ψlm are the corresponding spherical harmonic coefficients and the other variables are expanded in the same fashion. Following the definition in Augier and Lindborg (2013), the mean over a pressure level of the product of two functions can be written as
⟨ab⟩=∑l≥0∑−l≤m≤l(a,b)lm,
with
(a,b)lm≡R{alm*blm},
where R denotes the real part and * denotes the complex conjugate. For two horizontal vector fields a and b, the scalar product can be written as
⟨a⋅b⟩=∑l≥0∑−l≤m≤l(a,b)lm,
with (a,b)lm=R{alm*⋅blm}. Note that only the total wavenumbers larger than zero (i.e., l ≥ 1) is considered in the present work. Considering that spherical harmonic functions Ylm are the eigenfunctions of the horizontal Laplace operator, the vector product is calculated using the following formula:
(a,b)lm=re2l(l+1)R{roth(a)lm*roth(b)lm+divh(a)lm*divh(b)lm}.
Thus, the HKE spectrum can be defined as (Koshyk and Hamilton 2001)
EKlm=(u,u)lm2=re22l(l+1)(|ζlm|2+|δlm|2),
where ζlm and δlm are the spherical harmonic coefficients of vertical vorticity and horizontal divergence, respectively.
Following Augier and Lindborg (2013), the spectral HKE budget is derived by substituting Eq. (1) into the time differentiation of Eq. (10); that is, ∂tEKlm=(u,∂tu)lm. Reorganizing the different terms, the spectral HKE budget on each level can be written as
∂tEKlm=CAP→Klm+TKlm+∂pFK↑lm+Llm+DKlm,
where TKlm is the spectral transfer term due to nonlinear interactions, Llm is the spectral transfer term arising from the Coriolis term, FK↑lm is the total HKE vertical flux, defined as the sum of the pressure flux and the turbulent HKE flux, CAP→Klm is the conversion term from APE to HKE, and DKlm is the diffusion term. Each of the terms corresponds to a term in Eq. (3).

b. Spectral budget equations of rotational and divergent kinetic energy

Horizontal velocity u can be divided into rotational velocity uR and divergent velocity uD by Helmholtz decomposition:
u=uR+uD,
with
uR=∇×(ψ ez),uD=∇χ,
where ψ and χ are the spherical streamfunction and velocity potential, respectively. Substituting Eq. (12) into Eq. (2) yields
EK=ER+ED+uR⋅uD,
where ED = (1/2)uD ⋅ uD and ER = (1/2)uR ⋅ uR, corresponding to the DKE and RKE per unit mass, respectively. uR ⋅ uD is not necessarily zero everywhere in physical space, but the mean over a pressure level of this term in the global region should be equal to zero.

The tendency equations for DKE and RKE in open systems were first shown in Buechler and Fuelberg (1986). For detailed expressions of these equations, please see appendix A. According to the equations in appendix A, the global energy cycle is for available potential energy (APE), generated by differential heating, to first affect DKE and then RKE, with DKE serving as a type of catalyst (Chen and Wiin-Nielsen 1976). In addition to giving more physical insights mentioned above for the variability of kinetic energy, this kind of decomposition also provides a guideline for the following spectral budget equations of RKE and DKE.

The horizontal wavenumber spectra of the RKE and DKE on each level, referred to as the RKE spectrum and DKE spectrum, respectively, are defined as
ERlm=(uR,uR)lm2=re22l(l+1)(ζ,ζ)lm,
and
EDlm=(uD,uD)lm2=re22l(l+1)(δ,δ)lm.
Now, we begin to derive the spectral budget equations of RKE and DKE by substituting Eq. (1) into the time partial derivative of Eqs. (15) and (16); that is,
∂tERlm=(uR,∂tuR)lm=(uR,∂tu)lm−(uR,∂tuD)lm,
and
∂tEDlm=(uD,∂tuD)lm=(uD,∂tu)lm−(uD,∂tuR)lm,
respectively. Note that (uR, ∂tuD)lm and (uD, ∂tuR)lm are equal to zero by applying the identical Eq. (9). Then, the Eq. (18) becomes as follows:
∂tEDlm=−(uD,u⋅∇u+ω ∂pu)lm−(uD,∇Φ)lm−[uD,f(φ)ez×u]lm+(uD,Du)lm.
In what follows, we show the derivations in detail using DKE as an example. First, the HKE turbulent vertical flux is separated from the three-dimensional advection term of Eq. (19), as follows:
−(uD,u⋅∇u+ω ∂pu)lm=−(uD,u⋅∇u)lm−(uD,ω ∂pu)lm/2−(uD,ω ∂pu)lm/2+∂p(u,ω u)lm/2−∂p(u,ω u)lm/2=−[(uD,u⋅∇u)lm−(u,u ∂pω)lm/2]+[(∂pu,ω uD)lm/2−(uD,ω ∂pu)lm/2]+[(∂pu,ω uR)lm/2+(uR,ω ∂pu)lm/2]−∂p(u,ω u)lm/2,
where the two terms in the second set of square brackets represent the effects of horizontal velocity vertical shear and vertical motion on the spectral transfer of DKE; the two terms in the third square bracket correspond to the conversion terms B and C, and the separated vertical flux term corresponds to the term VF in appendix A.
Using the identical Eq. (4) and the mass continuity equation, the two terms in the first set of square brackets in Eq. (20) can be rewritten as
−[(uD,u⋅∇u)lm−(u,u ∂pω)lm/2]=−[(uD,∇Ek)lm+(u,u δ)lm/2]−(uD,ζez×u)lm,
where the first two terms in the square brackets represent the spectral transfer due to nonlinear advection and the last term is the corresponding vorticity-related term, which can be further separated into a spectral transfer term of DKE and a conversion term between RKE and DKE as follows:
−(uD,ζez×u)lm=−(uD,ζez×u)lm/2−(uD,ζez×u)lm/2=−[(uD,ζez×u)lm/2+(u,ζez×uD)lm/2]+[(uR,ζez×uD)lm/2−(uD,ζez×uR)lm/2],
where the two terms in the first set of square brackets represent the rotational effect due to (relative) vorticity on the spectral transfer of DKE; the two terms in the second set of square brackets correspond to the conversion term Az in appendix A.
Second, the term related to the horizontal divergence of the geopotential is decomposed into vertical flux and baroclinic conversion by using the mass continuity equation, the hydrostatic equation, and the identical Eq. (9) together:
−(uD,∇Φ)lm=−re2l(l+1)R{δlm*(∇2Φ)lm}=R{δlm*Φlm}=(δ,Φ)lm=−(∂pω,Φ)lm=−∂p(ω,Φ)lm−(ω,α)lm.
Therefore, we obtain the same conversion term as well as the vertical flux terms as those in Augier and Lindborg (2013).
Third, the Coriolis term is also further decomposed into energy transfer and conversion in the same way as the vorticity-related term:
−(uD,f ez×u)lm=−[(uD,f ez×u)lm/2+(u,f ez×uD)lm/2]+[(uR,f ez×uD)lm/2−(uD,f ez×uR)lm/2],
where the two terms in the first set of square brackets represent the rotational effect due to the Coriolis force on the spectral transfer of DKE; the two terms in the second set of square brackets correspond to the conversion term Af in appendix A.
Combining spectral transfer-related terms and conversion-related terms, the spectral DKE budget equation can be written as
∂tEDlm=CAP→Dlm+∂pFD↑lm+TDlm−CD→Rlm+DDlm.
Similarly, the spectral RKE budget equation can be derived as
∂tERlm=TRlm+CD→Rlm+DRlm.
In Eqs. (25) and (26), TRlm and TDlm are spectral transfer terms representing the transfer of RKE and DKE between wavenumbers. DRlm and DDlm are spectral diffusion terms, respectively. There are two terms—which are essentially the same as the corresponding terms in Eq. (11)—that only appear in the spectral budget equation of DKE; namely, FD↑lm is the spectral vertical flux of DKE, which is equal to the total HKE vertical flux FK↑lm, and CAP→Dlm is the spectral conversion term from APE to DKE. This implies that only DKE can be transported vertically, and direct conversion can only occur between APE and DKE. CD→Rlm represents the spectral conversion term from DKE to RKE, according to the fact that it appears in the two equations with the same magnitude and opposite signs. Moreover, this term does involve the interaction of the two modes and corresponds to conversion terms in physical space very well (Buechler and Fuelberg 1986). More specifically, the spectral conversion term can be divided into three components, corresponding to contributions from vertical motion, Coriolis effect, and relative vorticity, respectively. Thus, the spectral RKE and DKE budget equations are consistent with the form of the RKE and DKE budget equations in physical space. The detailed expressions of these terms are as follows:
TDlm=−[(uD,∇EK)lm+(u,u δ)lm/2]+[(∂pu,ω uD)lm−(uD,ω ∂pu)lm]/2−[(uD,f ez×u)lm+(u,f ez×uD)lm]/2−[(uD,ζ ez×u)lm+(u,ζ ez×uD)lm]/2,
TRlm=[(∂pu,ω uR)lm−(uR,ω ∂pu)lm]/2−[(uR,f ez×u)lm+(u,f ez×uR)lm]/2−[(uR,ζ ez×u)lm+(u,ζ ez×uR)lm]/2,
FD↑lm=−[(ω, Φ)lm+(u,ω u)lm/2],
CAP→Dlm=−(ω,α)lm,
CD→Rlm=−[(∂pu,ω uR)lm+(uR,ω ∂pu)lm]/2+[(uD,f ez×uR)lm−(uR,f ez×uD)lm]/2+[(uD,ζ ez×uR)lm−(uR,ζ ez×uD)lm]/2,
DDlm=(uD,Du)lm, and DRlm=(uR,Du)lm.
The reason why we do not ignore the Coriolis term as in previous studies is that it is closely related to the conversion of DKE and RKE [term Af in Eqs. (A1) and (A2), appendix A] and sometimes even acts as a major contributor (Chen and Wiin-Nielsen 1976). According to our derivation, the linear transfer term [Eq. (B2), appendix B] should be decomposed in the same fashion as the vorticity-related nonlinear spectral transfer when switching from the spectral budget of HKE to the spectral budget of DKE and RKE [Eq. (24)]. As a result, the Coriolis effect is included in the spectral fluxes and conversion terms [Eqs. (27), (28), and (31)], which implies that it acts as both transferring energy between different scales and converting energy between different types. By doing so, the total spectral transfer term of HKE (TTKlm), defined as the sum of nonlinear and linear spectral transfer terms, can be exactly divided into spectral DKE and RKE transfer terms, i.e.,
TTKlm=TKlm+Llm=TDlm+TRlm.

c. Flux form of the spectral budget equations of rotational and divergent kinetic energy

Given the two-dimensional spherical harmonic spectrum, the total wavenumber spectrum can be further obtained by summing over all corresponding zonal wavenumbers. For example, the total wavenumber spectrum of HKE at some pressure level is defined as
EK[l]=∑|m|≤lEKlm.
The total wavenumber spectra of RKE and DKE are defined in a similar way.
It is convenient to investigate transfers of RKE and DKE between scales by constructing the corresponding spectral fluxes (Augier and Lindborg 2013; Peng et al. 2015a,b), which are defined as
ΠR[l]=∑n≥l∑|m|≤nTRnm and ΠD[l]=∑n≥l∑|m|≤nTDnm,
respectively. The other spectral fluxes are defined in the corresponding way, such as the nonlinear and linear fluxes in Eq. (11):
ΠN[l]=∑n≥l∑|m|≤nTKnm and ΠL[l]=∑n≥l∑|m|≤nLnm.
Then, from Eq. (33), we can obtain
ΠTT[l]=ΠN[l]+ΠL[l]=ΠD[l]+ΠR[l].
Similarly, we can also calculate the corresponding cumulations of the spectral conversion terms and the other terms in Eqs. (25) and (26) by summing over all the spherical harmonics with total wavenumber greater than or equal to l, which are as followed:
EK,R,D[l]=∑n≥l∑|m|≤nEK,R,Dnm,
CD→R[l]=∑n≥l∑|m|≤nCD→Rnm,
CAP→D[l]=∑n≥l∑|m|≤nCAP→Dnm,
FD↑[l]=∑n≥l∑|m|≤nFD↑nm,
DR,D[l]=∑n≥l∑|m|≤nDR,Dnm.
Furthermore, to consider that the density strongly varies with atmospheric height, with the formulation in Augier and Lindborg (2013), the vertically integrated KE spectrum over a pressure range from pb at the lowest level to pt at the top level is obtained via
EK[l]ptpb=∫ptpbdpg∑−l≤m≤lEKlm.
The vertically integrated nonlinear and linear spectral fluxes of kinetic energy is defined as
ΠN[l]ptpb=∫ptpbdpg∑n≥l∑|m|≤nTKnm and ΠL[l]ptpb=∫ptpbdpg∑n≥l∑|m|≤nLnm.
The vertically integrated form of other cumulative terms [Eqs. (38)–(42)] are defined in the corresponding way. From Eq. (37), we can further obtain
ΠTT[l]ptpb=ΠN[l]ptpb+ΠL[l]ptpb=ΠD[l]ptpb+ΠR[l]ptpb.
When Eqs. (25) and (26) are vertically integrated and summed as in Eq. (44) over all the spherical harmonics with total wavenumber greater than or equal to l, we obtain
∂tED[l]ptpb=ΠD[l]ptpb+CAP→D[l]ptpb−CD→R[l]ptpb+FD↑[l](pb)−FD↑[l](pt)+DD[l]ptpb,
∂tER=ΠR[l]ptpb+CD→R[l]ptpb+DR[l]ptpb,
which are the final flux form of the spectral budget equations of DKE and RKE used in what follows.

3. Diagnostic analysis based on ERA5

a. Presentation of the data and tool

The data analyzed herein are horizontal wind fields, vertical velocity, and geopotential from the fifth generation of global reanalysis (ERA5) of the European Centre for Medium-Range Weather Forecasts (ECMWF) (Hersbach et al. 2020), available from the Climate Data Store (CDS) at https://cds.climate.copernicus.eu/. The grid is regular latitude–longitude, with a horizontal resolution of 0.25° × 0.25°. As mentioned above, the global reanalysis data at such a horizontal resolution can resolve the atmospheric state across many scales reasonably well, although it is not enough to analyze the whole mesoscale. In fact, global reanalysis has been used to research atmospheric energy spectra for many years. For example, the quasigeostrophic characteristics of the atmosphere, including upscale energy flux and downscale enstrophy flux, are already shown by spectral analysis with the first-generation global reanalysis data (Boer and Shepherd 1983) or a T799 ECMWF operational analysis (Burgess et al. 2013); the global energy spectra of the Rossby waves and IGWs have been computed by the Hough harmonics based on the ECMWF interim reanalysis (Žagar et al. 2017). As pointed out by Malardel and Wedi (2016), the magnitude and scale of the nonlinear interactions will to some extent depend on the choices made for the model dynamical cores as well as the physical parameterizations, which should be kept in mind when understanding the diagnose results from the global reanalysis.

The variables used in this paper are obtained at 0000, 0600, 1200, and 1800 UTC in January 2018 from 500 to 10 hPa. One month of global data sufficiently characterizes synoptic-scale and mesoscale spectral properties (Trenberth and Solomon 1993; Straus and Ditlevsen 1999; Burgess et al. 2013). In this paper, all calculation results presented are 1-month averages and the seasonal variations are not considered for with only one month of data. We adopt spherical harmonic transform as the spectral analysis method in this paper. The software package used here for performing the spherical harmonic transform is the windspharm library, which is a Python package maintained by Dawson (2016).

To facilitate the discussion that follows, the three spatial scales are roughly divided, the planetary scale (larger than 5700 km, l < 7), the synoptic scale (between 5700 and 2000 km, 7 ≤ l < 20), and the mesoscale (smaller than 2000 km, l ≥ 20) according to the relation between the total wavenumber and the wavelength in spherical harmonics [wavelength (km) ≃ 40 000/l]. Here, the mesoscale range refers to the definition by Orlanski (1975).

b. Horizontal kinetic energy spectra

Figure 1a shows HKE spectra at different pressure levels from the middle troposphere (500 hPa) to the middle stratosphere (10 hPa) based on Eq. (34). At each level, the spectrum of HKE decreases sharply when the total wavenumber exceeds a certain value, which is usually considered a result of dissipation effects (Burgess et al. 2013). Graphically, we conservatively estimate that the dissipation range begins at l = 100. Thus, in what follows the spectral slope is calculated in the range of l = 10–100 corresponding to wavelengths between 4000 and 400 km. At 500 and 250 hPa, the spectral slope is approximately constant throughout the entire range l = 10–100. Above 100 hPa, however, the spectral slope shows an obvious change from a steeper slope to a shallower slope. Between 250 and 100 hPa, the transition appears and moves to larger scale with altitude (shown in Fig. 6).

Fig. 1.
Fig. 1.

(a) HKE spectra at selected pressure levels spanning the middle troposphere (500 hPa) to the middle stratosphere (10 hPa) are presented in different colors. (b) HKE spectra and the fitting spectra at 250 hPa (blue line) and 100 hPa (green line) are presented. Lines with slopes of −3 and −5/3 are added to both panels for comparison.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

To quantitatively compare the difference between 250 and 100 hPa height spectral lines, we use the same least squares method described in Burgess et al. (2013) to fit the energy spectra between l = 10–100 and find the changepoint of the spectral slope in this range. The fitting results at 250 and 100 hPa based on the ERA5 data are shown in Fig. 1b. At 250 hPa, the approximate −3 spectral slope extends into the smaller end of mesoscales corresponding to wavelengths of 400 km or less. This slope is characteristic of two-dimensional turbulent or geostrophic turbulent enstrophy cascading (Kraichnan 1967; Charney 1971). The energy spectra at 500 hPa are shallower than those at 250 hPa. This behavior of the power law is similar to earlier results (Boer and Shepherd 1983; Burgess et al. 2013). At 100 hPa, the slope of the spectrum changes dramatically around l = 20 from −3.9 to −2.1, and the shallower part likewise extends into the smaller end of mesoscales corresponding to wavelengths of 400 km or less. Compared to the proposed −5/3, the shallower slope is closer to −2. A possible explanation for the −2 spectral slope was provided by Kafiabad et al. (2019), who attribute it to result of the IGWs scattering.

c. Divergent and rotational kinetic energy spectra

To facilitate comparison with previous results, we choose the two layers 400–250 hPa and 250–10 hPa to represent the upper troposphere and the stratosphere, respectively, which are generally consistent with the chosen height ranges in Augier and Lindborg (2013). In what follows, unless otherwise specified, the results presented are vertically integrated ones over these two layers.

Figure 2 presents the HKE spectra and their two component spectra (RKE and DKE) vertically integrated over the upper troposphere (Fig. 2a) and stratosphere (Fig. 2b) based on Eq. (43). In addition, two-segment and single-segment linear fitting are performed for HKE spectra and their two component spectra in the range of l = 10–100. It highlights the importance of the relative magnitude between the divergent and rotational components to the change of HKE spectral slope. In the upper troposphere (Fig. 2a), the HKE spectrum has no obvious transition, the slope of the DKE spectrum is −1.71, and the slope of the RKE spectrum is −2.90. The rotational component is at least one order of magnitude larger than the divergent component except within the dissipation range. In the stratosphere (Fig. 2b), the slope of the HKE spectrum changes slightly around l = 27 from −3.24 to −2.58, while the slope of the DKE and RKE spectrum is −1.69 and −3.27 at l = 10–100, respectively. The divergent component intersects the rotational component at l ≃ 79 corresponding to a wavelength of approximately 500 km within mesoscales. Our results obviously show that the DKE spectrum has a slope close to −5/3 in both the upper troposphere and the stratosphere while the RKE spectral slope is close to −3, which has been also shown in many previous studies (e.g., Burgess et al. 2013; Skamarock et al. 2014). The different slopes of these two components are also similar with spectral slopes of the balanced and unbalanced modes (Kafiabad and Bartello 2018), since the rotational mode roughly corresponds to the balanced mode, and the divergent mode to the unbalanced mode.

Fig. 2.
Fig. 2.

HKE spectrum (black) and its divergent (red) and rotational (blue) components integrated over two layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere, with lines of slope −3 and −5/3 added to both panels for comparison. The corresponding linear fitting results are represented by dashed lines. Other details are as in Fig. 1.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

Compared with the observed spectra obtained by Li and Lindborg (2018), the DKE spectrum in this paper never shows a much steeper slope at wavelengths larger than around 800 km. However, the steeper slope of DKE obtained by Li and Lindborg (2018) at these scales should be unreal. Actually, the method used by them is originally proposed in Lindborg (2015). As pointed out by Bierdel et al. (2016), the steeper slope of DKE retrieved by this method should be related to a bad signal-to-noise ratio, i.e., the big error of DKE is caused by small errors in RKE at large scales that contaminate the divergent component. Another major difference is that the shallowing of the rotational spectrum is missing in our Fig. 2. However, the shallowing of the rotational spectrum in Li and Lindborg (2018) exists at wavelengths smaller than 200 km in the upper troposphere and smaller than 500 km in the lower stratosphere. According to Lindborg and Brethouwer (2007), the shallowing of the rotational spectrum is an indication of the weakly rotating stratified turbulence, as the rotational and divergent modes closely couple due to the strong nonlinear interactions. Thus, the steeper rotational spectrum obtained here around 500 km indicates that the stratospheric flow in ERA5 at corresponding scales shows more wavelike behavior than stratified turbulence, which is in agreement with experimental result in Rodda and Harlander (2020), while at wavelengths smaller than 200 km in the upper troposphere, due to the limited effective resolution of ERA5, the steeper spectrum should be mainly attributed to the effect of dissipation.

d. Nonlinear and linear spectral fluxes of horizontal kinetic energy

Energy transfer between different scales can be usually investigated by constructing the nonlinear spectral flux. Different from the method based on the barotropic vorticity equation (Boer and Shepherd 1983; Burgess et al. 2013; formula not shown), the framework taken in Eq. (11) developed by Augier and Lindborg (2013) does not restrict consideration to only the rotational component of the flow. Thus, it provides a complete representation of the nonlinear spectral flux and allows us to investigate the energy transfer in three-dimensional hydrostatic primitive atmosphere (Augier and Lindborg 2013).

Figure 3 presents the nonlinear spectral fluxes ΠN[l]ptpb (black solid line) vertically integrated over two different layers corresponding approximately to the upper troposphere (Fig. 3a) and stratosphere (Fig. 3b) based on Eq. (44). By construction, the fluxes are equal to zero at l = lmax and l = 0; thus, they represent energy transfer only between scales. Note that, negative values of ΠN[l]ptpb represent upscale energy transfer and positive values of ΠN[l]ptpb represent downscale energy transfer. In the upper troposphere (Fig. 3a), the ΠN[l]ptpb(black solid line) decreases to reach a valley between l = 4 and l = 9 with a minimum of approximately −0.33 W m−2, increases to 0 at l ≃ 18, and continues to increase to reach a maximum before falling to zero at the largest wavenumbers. This indicates that the nonlinear HKE spectral transfer is upscale at wavelengths larger than 2200 km and downscale smaller than this scale. In the stratosphere (Fig. 3b), ΠN[l]ptpb reaches a minimum equal to −0.53 W m−2 at l = 4, increases to 0 at l ≃ 15, then increases to reach a maximum and before falling to zero at the largest wavenumbers. This indicates that there is an upscale transfer of HKE larger than wavelengths of approximately 2700 km and a downscale transfer of HKE smaller than this scale.

Fig. 3.
Fig. 3.

Nonlinear spectral flux of HKE (ΠN[l]ptpb; black solid line); nonlinear spectral flux of HKE based on the barotropic vorticity equation (Πrot[l]ptpb; blue dashed line); the complementary flux (ΠN[l]ptpb−Πrot[l]ptpb; red dashed line) (unit: W m−2) integrated over layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

Figure 3 also presents the nonlinear spectral flux computed only with rotational flow, referred to below as Πrot[l]ptpb(blue dashed line), which is equivalent to the spectral flux computed under the framework based on the barotropic vorticity equation (Boer and Shepherd 1983; Burgess et al. 2013). The most significant characteristic is that Πrot[l]ptpb is negative at all scales in both the upper troposphere and stratosphere, indicating an upscale energy transfer in both layers, consistent with previous studies (Boer and Shepherd 1983; Burgess et al. 2013). Unsurprisingly, there exists differences between Πrot[l]ptpb and ΠN[l]ptpb in each layer, especially in the upper troposphere. In the upper troposphere (Fig. 3a), Πrot[l]ptpb is very different from ΠN[l]ptpb at l ≥ 4. The Πrot[l]ptpb reaches a minimum equal to −0.57 W m−2 at l = 10, much smaller than that of ΠN[l]ptpb, and cannot reproduce the downscale characteristics of ΠN[l]ptpb at wavelengths smaller than 2200 km. In the stratosphere, Πrot[l]ptpb is very close to ΠN[l]ptpb at l < 13, but different from ΠN[l]ptpb at smaller scales. More specifically, it is also unable to present downscale energy transfer at mesoscales, which is similar to the upper troposphere (Fig. 3a).

In some earlier studies (e.g., Kitamura and Matsuda 2006; Augier and Lindborg 2013), the barotropic nonlinear flux is interpreted as the RKE spectral flux, without considering the effects of vertical wind shear and vertical velocity on RKE. The complementary flux, ΠN[l]ptpb−Πrot[l]ptpb (red dashed line), is thought to be associated with the energy transfer by the divergent part of velocity. However, the way these two fluxes represent the spectral fluxes of RKE and DKE, respectively, is not very rigorous. As shown in Eq. (28), the divergent velocity may also contribute to the interactions between the rotational modes, which implies that part of the interactions involving the divergent component of the velocity may belong to the latter. It is natural to ask what the exact spectral fluxes of RKE and DKE should be like, which is just the initial motivation for us to derive the novel formulation for the spectral budget of RKE and DKE shown in section 2a.

In addition to the nonlinear spectral flux, the linear spectral flux also contributes to energy transfer between different scales. Figure 4 presents the linear (red dashed line) and nonlinear (black dashed line) spectral fluxes of HKE and the sum of both (black solid line) integrated over two different layers corresponding approximately to the upper troposphere and the stratosphere based on Eq. (44). Note that the vertical axes in the two subfigures are not the same. In the upper troposphere (Fig. 4a), the cumulative Coriolis term (ΠL[l]ptpb) increases from zero to a peak equal to 0.36 W m−2 at l = 4, and passes through zero line at l = 7, indicating a downscale energy transfer at l < 7. In the stratosphere (Fig. 4b), the ΠL[l]ptpb also indicates a downscale transfer at planetary scales, with a maximum equal to 1.74 W m−2. The planetary-scale downscale spectral transfer is consistent with previous computations (Lambert 1987; Koshyk and Hamilton 2001; Augier and Lindborg 2013), and should be caused by the presence of a nonnegligible background vorticity gradient (i.e., β effect) (Lambert 1987). This interpretation is consistent with the following facts: in the f-plane case, where f is constant and a Fourier decomposition is used, the Coriolis term at each wavenumber is zero (Koshyk and Hamilton 2001; Augier and Lindborg 2013); however, in the sphere case, where f varies with latitudes (i.e., β effect) and a spherical harmonic decomposition is used, the Coriolis term can exchange RKE and DKE between adjacent wavenumbers (Lambert 1987), although it does not involve rotational–rotational (or divergence–divergence) interactions (Augier and Lindborg 2013).

Fig. 4.
Fig. 4.

Nonlinear spectral flux of (black dashed line) HKE and linear spectral flux of (red dashed line) HKE due to the Coriolis term, and the sum of the two (black solid line) integrated over layers approximately to (a) the upper troposphere and (b) the stratosphere. Other details are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

Although ΠL[l]ptpb (red dashed line) is negligible at l > 10 for both layers, its amplitude at planetary scales is larger than that of the nonlinear spectral flux (black dashed line), especially in the stratosphere. As a result, total HKE flux (ΠTT[l]ptpb; black solid line) shows a downscale transfer around l = 5. This is reminiscent of the two-dimensional ideal model results in Rhines (1975), which shows that in the presence of β effect, the cascade produces a field of waves without loss of energy, and energy stops transferring upscale at a certain one. The direct consequence of the upscale transfer of the nonlinear spectral flux is the emergence of large circular eddies. Due to the presence of β effect, as a restoring force, these eddies tend to form waves, such as planetary waves. In other words, the existence of the β effect prevents the transfer of energy to the large-scale end of planetary scales (Read et al. 2018).

e. Spectral rotational and divergent kinetic energy budget

As the extension of Augier and Lindborg (2013), detailed RKE and DKE spectral budget is created in order to gain more physical insights into the role of the rotational and divergent components of motion.

Figure 5 shows the cumulative spectral transfers of total HKE (ΠTT[l]ptpb; black solid line), DKE (ΠD[l]ptpb; red solid line), and RKE (ΠR[l]ptpb; blue solid line), net vertical flux [ΔFD↑[l]=FD↑[l](pb)−FD↑[l](pt); magenta dashed line], conversion from APE to HKE (CAP→D[l]ptpb; green dashed line), and conversion from DKE to RKE (CD→R[l]ptpb; cyan dashed line) integrated over two different layers corresponding approximately to the upper troposphere (Fig. 5a) and the stratosphere (Fig. 5b) in Eqs. (46) and (47).

Fig. 5.
Fig. 5.

Spectral fluxes of HKE (black solid line), DKE (red solid line) and RKE (blue solid line); cumulative conversion (cyan dashed line) from DKE to RKE and conversion (green dashed line) from APE to HKE; and cumulative net vertical flux (magenta dashed line) in (a) the upper troposphere and (b) the stratosphere. Cumulative conversion (cyan dashed line) from DKE to RKE and its three components related with vertical motion (black dashed line), Coriolis effect (red dashed line), and relative vorticity (blue dashed line) in (c) the upper troposphere and (d) the stratosphere. Other details are as in Fig. 3. The inset is an expanded view of the mesoscale subrange (l ≥ 20).

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

In the upper troposphere (Fig. 5a), the total HKE spectral flux ΠTT[l]ptpb (black solid line; same in Fig. 4a) presents a fluctuating shape, suggesting the scale range of upscale or downscale cascade according to its negative or positive. It reaches a maximum equal to 0.09 W m−2 at l = 5, decreases to a minimum equal to −0.42 W m−2 at l = 8 and increases again to a maximum equal to 0.20 W m−2 at l ≃ 34 before decreasing to zero.

At planetary scales, the total HKE spectral flux (black solid line) is dominated by the DKE spectral flux ΠD[l]ptpb (red solid line). ΠD[l]ptpb increases at approximately l < 5 and decreases between l = 5 and l = 8, indicating a transfer of DKE from wavelengths on the order of 10 000 km to wavelengths between 8000 and 5000 km. At l < 5, cumulative conversion CAP→D[l]ptpb (green dashed line) strongly decreases, indicating a conversion from APE to DKE of approximately 0.82 W m−2 (CAP→D[1]ptpb−CAP→D[5]ptpb). At these scales, the cumulative net vertical flux ΔFD↑[l] (magenta dashed line) increases by 0.71 W m−2 and tends to remove the DKE. The combined effect of the above two still makes DKE enhanced. Between l = 5 and l = 8, the decrease of ΠD[l]ptpb is accompanied by a decrease of the RKE spectral flux ΠR[l]ptpb (blue solid line), as well as increases of the CD→R[l]ptpb (cyan dashed line) and CAP→D[l]ptpb in the same range. The increase of CD→R[l]ptpb from 0.39 to 0.59 W m−2 between l = 5 and l = 8 corresponds to a conversion from RKE to DKE of approximately 0.20 W m−2, close to the conversion from DKE to APE of 0.17 W m−2 in the same range. This indicates that the RKE is converted to DKE, and a portion of DKE is further converted to APE in this range.

At synoptic scales (Fig. 5a), the total HKE spectral flux (black solid line) is mainly dominated by the RKE spectral flux (blue solid line) at 7 ≤ l < 18. The ΠD[l]ptpb (red solid line) is larger than zero at l > 8, indicating a downscale transfer of DKE at wavelengths smaller than 5000 km. The ΠR[l]ptpb increases abruptly between l = 8 and l ≃ 34, associated with a sharply decrease in CD→R[l]ptpb (cyan dashed line) of approximately 0.59 W m−2 and a strong decrease in CAP→D[l]ptpb (green dashed line) of approximately 0.80 W m−2. This indicates that the direct forcing feeding the RKE at these scales is mainly from a significant conversion of DKE, which ultimately comes from a conversion of APE due to baroclinic instability. Further analysis on the three components of CD→R[l]ptpb as shown in Eq. (31) demonstrates that such significant conversion of DKE to RKE is dominated by the Coriolis related contribution (Fig. 5c). In addition, ΠR[l]ptpb is almost zero at l ≃ 23 corresponding to the wavelength approximately 1700 km, indicating there is a strong upscale transfer of the RKE at l < 23 and a downscale transfer at smaller scales. This means a portion of RKE at synoptic scales is transferred upscale toward the planetary scale and another portion of RKE downscale toward the mesoscale.

At mesoscales (Fig. 5a), the ΠR[l]ptpb (blue solid line) decreases after reaching a plateau at 0.10 W m−2, and the ΠD[l]ptpb (red solid line) is decreasing and of the same magnitude at l > 34. This shows that both RKE and DKE transfer toward smaller scales. At these scales, the net vertical flux ΔFD↑[l] (magenta dashed line) increases and tends to remove the DKE. This shows that the upper-tropospheric mesoscale atmosphere is not directly forced by upward-propagating gravity waves, consistent with previous studies (Koshyk and Hamilton 2001; Augier and Lindborg 2013) despite some different details. It can also be seen that, although not remarkably, the CD→R[l]ptpb (cyan dashed line) turns to increase at wavelengths smaller than 800 km (local conversion is negative), showing that the conversion is from RKE to DKE in this range, which is actually governed by the contribution from relative vorticity (Fig. 5c).

Figures 5b and 5d present the same quantities as Figs. 5a and 5c, but integrated over the stratosphere. The spectral fluxes and cumulative conversions are very different from the terms integrated over the upper troposphere.

At planetary scales (l < 7; Fig. 5b), the total HKE flux (black solid line) is still dominated by the downscale DKE flux (red solid line) but has a larger peak equal to 1.21 W m−2. Significantly, the RKE flux (blue solid line) is downscale, which is quite different from the upper troposphere. More specifically, at l ≤ 3, a stronger increase in the RKE flux is accompanied a stronger decrease in the CD→R[l]ptpb (cyan dashed line) from 0.58 to 0.04 W m−2, corresponding to a conversion from DKE to RKE of approximately 0.54 W m−2. At these scales, the curve of CAP→D[l]ptpb (green dashed line) goes down and up, indicating a conversion of 0.3 W m−2 from APE to DKE. Between l = 3 and l = 8, a stronger decrease in the RKE flux is accompanied a stronger increase in the CD→R[l]ptpb, corresponding to a conversion from RKE to DKE of approximately 0.91 W m−2. At l ≤ 8, the curve of ΔFD↑[l] (magenta dashed line), which goes down at l ≤ 5 and up at 5 < l ≤ 9, shows that the net vertical flux of DKE tends to inject DKE and then remove it.

At synoptic scales (Fig. 5b), the HKE flux (black solid line) is obviously upscale and dominated by the RKE flux (blue solid line) at 7 ≤ l < 18. Between l = 8 and l ≃ 34, the ΠR[l]ptpb is significantly increasing and the CD→R[l]ptpb (cyan dashed line) is decreasing, indicating there is a large amount of RKE converted from DKE. The significant conversion from DKE to RKE is also dominated by the Coriolis related contribution (Fig. 5d), as the case in the upper troposphere. Meanwhile, the decreasing ΔFD↑[l] (magenta dashed line) from 1.26 to 0.05 W m−2 tends to enhance the DKE by 1.21 W m−2. Remarkably, CAP→D[l]ptpb (green dashed line) is opposite to the case in the upper troposphere, indicating that APE is not the source for the DKE at these scales.

At mesoscales (Fig. 5b), the transfers of both RKE (blue solid line) and DKE (red solid line) are downscale, but the RKE flux is much larger than that of DKE. The CD→R[l]ptpb (cyan dashed line) is close to zero, indicating a very weak conversion between DKE and RKE. This is because different components in CD→R[l]ptpb cancel each other, which is quite different from the case in the upper troposphere. The CAP→D[l]ptpb (green dashed line) is still increasing toward the largest wavenumber, indicating that APE is not the source for the DKE. In contrast to the upper troposphere, the positive ΔFD↑[l] (magenta dashed line) is always decreasing in the stratosphere, indicating the DKE spectrum in this range is mainly forced by an upward energy flux.

The DKE and RKE spectral fluxes presented here (Figs. 5a,b) are to some extent similar to balance breakdown scenario that was described in Kafiabad and Bartello (2018, see their Fig. 2). First, the DKE spectral flux at l > 7 is characterized by an upscale transfer regime followed by a downscale transfer regime, which is basically consistent with their unbalanced energy transfer spectra. The upscale transfer of DKE might be due to the upscale transfer of RKE because DKE is slaved to RKE at small Rossby number. Second, the RKE spectral transfer in the stratosphere is downscale at l ≤ 6 and upscale at 6 < l ≤ 20, and upscale at l ≤ 23 in the upper troposphere. Very similarly, their balanced transfer spectra are downscale at l ≤ 5 and upscale at 5 < l ≤ 20. In addition, there are also two significant differences we would like to highlight. One occurs at l ≤ 7(8) in the upper troposphere (stratosphere), where the DKE spectral transfer is downscale largely due to the β effect. However, such feature does not appear on their unbalanced energy transfer spectra. This is most likely because the numerical experiments in Kafiabad and Bartello (2018) did not consider spatially varying f. The other occurs at the wavenumbers larger than about 20, where the RKE spectral transfer is downscale. This feature is not obvious on their balanced transfer spectra at smaller scales. This may be because part of RKE should belong to unbalanced mode when Rossby number is no longer small (Bartello 1995).

Furthermore, balanced motion implies that DKE is much less than RKE, which holds for both the small Froude number balance possible at mesoscales and the small Rossby number balance relevant to synoptic scales (Saujani and Shepherd 2006); in contrast, unbalanced motion implies that DKE is larger than or comparable to RKE. Thus, it seems that rotational and divergent modes can be approximated to represent balanced and unbalanced motion better at synoptic scales than at mesoscales.

To further investigate the variational characteristics of spectral budget with height, not limited to the division of the stratosphere and the upper troposphere, the vertical structures of different fluxes and cumulative terms calculated based on Eqs. (35), (37), and (39)–(41) are shown in Fig. 6.

Fig. 6.
Fig. 6.

Spectral (a) HKE flux, (b) DKE flux, (c) RKE flux, (d) cumulative conversion from DKE to RKE, (e) cumulative conversion from APE to HKE, and (f) cumulative vertical flux divergence (colored; unit: 10−4 m2 s−3) at l ≥ 10. The contour interval is 0.4 × 10−4 m2 s−3 and the thick black contour represents the value of zero.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

Figures 6a–c show the vertical structures of spectral fluxes, highlighting the concentrations of downscale and upscale fluxes around the tropopause. They also show that the direction of the HKE spectral transfer is sensitive to the height, and whether the rotational flux or divergent flux dominates at different heights is also different. Below approximately 125 hPa, the vertical structure of the total HKE spectral flux has two concentrations of positive flux (ΠTT[l] > 0) with two peaks (approximately 300 and 125 hPa) indicating strong downscale transfer of HKE, and an obvious concentration of negative flux (ΠTT[l] < 0) indicating strong upscale transfer of HKE (Fig. 6a). Compared with the DKE spectral flux which presents a bit larger region of downscale transfer than the HKE spectral flux (Fig. 6b), the features of HKE spectral flux resemble more closely those of RKE spectral flux (Fig. 6c). Approximately between 125 and 70 hPa, the HKE transfer is weakly upscale (Fig. 6a). This feature comes from both the upscale DKE spectral transfer (Fig. 6b) and the upscale RKE spectral transfer (Fig. 6c). It is possible that the part of divergence component is slaved to the rotational component. (e.g., Warn et al. 1995; Kafiabad and Bartello 2016, 2018). Above approximately 70 hPa, RKE spectral flux dominates the HKE spectral flux again. HKE and RKE spectral transfers present the upscale/downscale alternate structure in the vertical, with an upscale transfer at about 50 hPa, a relatively strong downscale transfer at approximately 30 hPa and then an upscale transfer above 20 hPa (Figs. 6a,c). DKE presents downscale transfer above 50 hPa (Fig. 6b).

The variation characteristics of CD→R[l]ptpb roughly correspond to the characteristics of spectral fluxes of HKE and RKE (Fig. 6d). In the upper troposphere, CD→R[l]ptpb generally decreases with wavenumbers (∂CD→R[l]/∂l<0), indicating that the conversion is mainly from DKE to RKE, except for wavelengths less than approximately 800 km (l > 50) near 300 hPa. Note that, at approximately 100 and 50 hPa, the conversion is mainly from DKE to RKE, both accompanied with the upscale transfer of HKE there. From Fig. 6e, it is shown that the trend of CAP→D[l]ptpb decreases with increasing wavenumbers below 250 hPa and above 20 hPa, which indicates that the conversion from APE to DKE is dominant in the corresponding height, while between 250 and 20 hPa, the situation is the opposite. At mesoscales, between 400 and 250 hPa (i.e., the upper troposphere), ∂pFD↑[l] increases as wavenumbers, indicating a removal of DKE by the vertical flux, while below 400 hPa and above 250 hPa, ∂pFD↑[l] decreases as wavenumbers, indicating an injection of DKE by the vertical flux (Fig. 6f). Further calculation of the vertical flux at the bottom and top of the upper troposphere shows that it is downward at 400 hPa and upward at 250 hPa (not shown). The wavenumber ranges of the removal of DKE by the vertical flux at these scales nearly coincide with those of the conversion from RKE to DKE, which implies that there are IGWs excited between 400 and 250 hPa and then they carry energy upward and downward. At larger scales, as the height rises, ∂pFD↑[l] alternately increases and decreases with wavenumbers indicating that the vertical flux in this region represents the upward-propagating waves.

In what follows, we try to associate the movement of the intersection points of RKE and DKE spectra and the spectral conversion from DKE to RKE, since this conversion acts as the only source for RKE. The corresponding results are presented in Fig. 7. Figure 7a shows at 10 ≤ l ≤ 100, the changepoints of the HKE spectral slope (red dots) obviously existing in the 225–10 hPa region and the intersection points of the RKE and DKE spectra (black dots) existing in the 175–10 hPa region. The minimum wavenumber of the intersection points is l = 26 at 50 hPa, while that of changepoints is l = 19 at 70 hPa. In the upper troposphere (approximately between 250 and 400 hPa), the RKE spectra are always larger than the DKE spectra, without intersection points and obvious changepoints. In the stratosphere (approximately between 225 and 10 hPa), the changepoints of HKE spectra first move obviously to larger scales and then slightly change to smaller scales, while the intersection points appear from 175 hPa and move in a similar way. As noted earlier, the intersection point reflects a rough transition from RKE-dominated scales to DKE-dominated scales and it is the relatively larger contribution from the DKE spectra that shallows the HKE spectra at corresponding scales. Figure 7b shows the DKE spectra and RKE spectra between 225 and 70 hPa. In this region, the change of DKE spectra with increasing altitude is relatively small compared to that of RKE spectra and the movement of the intersection points toward larger scales mainly results from a preferential decrease in synoptic-scale RKE (Fig. 7b), which is consistent with the result in Burgess et al. (2013).

Fig. 7.
Fig. 7.

(a) Spectral conversion from DKE to RKE (colored; unit: 10−4 m2 s−3). Red dots represent the changepoint wavenumber of the spectral slope at given levels described in section 3b. Black dots represent the intersection wavenumber of the rotational and divergent spectra at given levels. The contour interval is 0.2 × 10−4 m2 s−3, and the zero contour is omitted. (b) RKE spectra (solid lines) and DKE spectra (dashed lines) at selected pressure levels. Lines with slopes of −3 and −5/3 are added to (b) for comparison.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

Also shown in Fig. 7a is the local spectral conversion from DKE to RKE (CD→R[l]). It is clearly shown that the intersection-point wavenumbers are roughly parallel to the scales of positive peaks of the local spectral conversion from DKE to RKE around total wavenumber 10. It seems that such peak of this local conversion roughly determines the scales where the RKE spectra begin to decline dramatically from nearly the same level of the RKE spectra at large scales, as shown in Fig. 7b. In addition, one can also see from Fig. 7a that the spectral conversion presents many variations at planetary and synoptic scales, which is mainly attributed to the Coriolis effect.

4. Summary and discussion

A rigorous formulation of the spectral RKE and DKE budget in the hydrostatic framework [Eqs. (25) and (26)] has been presented and applied to study the ERA5 for the first time. The main features of the formulation include the following: 1) the Coriolis effect is taken into account in both spectral energy transfer and conversion between RKE and DKE, 2) the baroclinic conversion term associated with APE and the vertical flux term of HKE act only on DKE, 3) the HKE spectral transfer between different spherical harmonics is exactly divided into the spectral transfer of RKE and DKE, and 4) the exact spectral conversion term between DKE and RKE is constructed to ensure that it is equal in magnitude and opposite in sign in their respective budget equations. Compared with previous formulations, the respective contribution of rotational and divergent components in kinetic energy spectral budget can be revealed.

With this new formulation, the energy spectra and spectral budget of RKE and DKE based on ERA5 are quantitatively presented. Note that in this study the mesoscale refers to scales shorter than 2000 km, and ERA5 can only resolve part of the mesoscale motions. The energy spectra based on ERA5 show that the HKE spectrum in the upper troposphere decreases at a relatively fixed slope from the synoptic scale to the dissipation range, while the HKE spectrum in the stratosphere presents an obvious spectral transition characteristic, i.e., from a steeper slope at synoptic scales to a shallower slope at mesoscales. The spectra of RKE and DKE are further obtained by Helmholtz decomposition. The rotational component dominates the HKE spectrum at synoptic scales and mesoscales in the upper troposphere, while the divergent component is comparable to or even dominates the rotational component within mesoscales in the stratosphere. In both the upper troposphere and stratosphere, the slope of the DKE spectrum is close to −5/3, and the slope of the RKE spectrum is close to −3. The relationship between the intersection points of the RKE and DKE spectra as well as the changepoints of the spectral slope of the HKE spectra is analyzed above 225 hPa, and the corresponding results imply that the transition of the slope of the HKE spectrum is related to the dominant mode at different scales.

The spectral budget is depicted schematically in Fig. 8 and summarized as follows. In the upper troposphere, the HKE spectral transfer is downscale at wavelengths larger than approximately 6700 km, dominated by downscale DKE transfer, while the RKE at these scales is forcing by an upscale transfer from synoptic scales. Between approximately 6700 and 2200 km, the HKE spectral transfer is dominated by an upscale transfer of RKE, which is energized by the conversion of DKE mainly due to Coriolis effect. And the ultimate source of DKE comes from the conversion of APE due to baroclinic instability. Smaller than approximately 2200 km, the HKE spectral transfer becomes downscale again, own to the fact that the spectral transfers of RKE and DKE are overall both downscale and of the same magnitude. Interestingly, the conversion is from RKE to DKE at wavelengths smaller than 800 km due to relative vorticity. In addition, at almost all scales, the net vertical flux tends to remove DKE.

Fig. 8.
Fig. 8.

Schematic of spectral DKE and RKE budget for global atmosphere. Note that ΔFD↑ represents the contribution of the net vertical flux, not the vertical flux itself.

Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-21-0332.1

In the stratosphere, the HKE spectral transfer is also dominated by the downscale DKE transfer at wavelengths larger than 5700 km. Between approximately 5700 and 2200 km, the HKE spectral transfer is dominated by an upscale transfer of RKE, which is energized by the conversion of DKE. However, the ultimate source of DKE comes from the net vertical flux. Smaller than approximately 2200 km, the HKE spectral transfer becomes downscale again and is actually dominated by a downscale transfer of RKE. In contrast to the upper troposphere, the DKE tends to convert into APE, and is positively forced by the net vertical flux at nearly all the scales except between approximately 8000 and 5000 km. In both the upper troposphere and stratosphere, the conversion between DKE and RKE is weak at mesoscales and significantly enhanced at synoptic scales. The DKE and RKE spectral fluxes are to some extent similar to balance breakdown scenario that was described in Kafiabad and Bartello (2018). One of the significant differences occurs at l ≤ 7–8, where the DKE spectral transfer is downscale while such feature does not appear on their unbalanced energy transfer spectra. This is most likely because their numerical simulations did not consider spatially varying f. Another difference occurs at wavenumbers larger than about 20, where the RKE spectral transfer is downscale while this feature is not obvious on their balanced transfer spectra. This may be because part of RKE should belong to unbalanced mode when Rossby number is no longer small (Bartello 1995).

The vertical structure of each term could provide more details about the spectral budget of DKE and RKE. The upscale and downscale HKE transfers present wavelike structures with increasing height at synoptic scales and mesoscales. In addition, an upscale DKE transfer significantly appears near 100 hPa, while an upscale RKE transfer appears near 10 hPa. In both the upper troposphere and the stratosphere, the spectral conversion term is mainly from DKE to RKE, unlike the conversion between APE and HKE, which differs significantly in direction at different heights. The spectral conversion from RKE to DKE occurs at wavelengths less than approximately 800 km (l > 50) near 300 hPa where RKE is larger than DKE. The HKE spectral slope and the location of the spectral slope transition at 10 ≤ l ≤ 100 change with the height. The positive peak of the local conversion from DKE to RKE around total wavenumber 10 roughly determines the scales where the RKE spectra begin to decline dramatically, and therefore affects the transition of the HKE spectra which depend on the intersection scales of RKE and DKE spectra.

These results show that our spectral RKE and DKE budget formulation is a convenient tool to investigate the issue of the atmospheric dynamics. In particular, we have shown that the linear flux related to Coriolis effect accounts for the downscale energy transfer at planetary scales, which should be divided into two parts acting on the RKE and DKE transfers, respectively. The RKE spectral flux presents a strong downscale transfer at mesoscales. It is obviously different from the flux computed only with the rotational part of the velocity field, since the latter one accounts only for the upscale KE flux (e.g., Boer and Shepherd 1983; Augier and Lindborg 2013).

However, caution and more work are required before generalizing these results calculated from the reanalysis, especially at mesoscales. First, the horizontal resolution of current reanalysis data is not sufficient to resolve the entire mesoscale range. As a result, the dynamics underlying the spectral transition from −3 to −5/3 around 500 km has not been well investigated in current work. This issue can be further explored by applying our new formulation to global higher-resolution atmospheric simulations. Second, aspects of the reanalysis not well constrained by measurement are largely influenced by some key factors of the forecast model, such as resolution level and physical parameterizations. Vertical resolution is known to determine the accuracy of vertical discretization (e.g., representing gravity waves in the vertical direction) and the effect of vertical mixing, and will therefore influence energy spectra and spectral budget (Brune and Becker 2013; Augier and Lindborg 2013). Higher horizontal resolution means more realistic variance of topography and other surface fields, and the reduction of explicit hyperdiffusion, thus tends to produce much shallower slope of the spectra at mesoscales (Malardel and Wedi 2016). In addition, spectral energy budget analysis from a hierarchy of models with reduced complexity also show that surface drag and momentum vertical mixing as the key processes for influencing the transfer of energy in a stratified atmosphere (Malardel and Wedi 2016). The effects of these factors on the DKE and RKE spectral budget require further investigation.

Acknowledgments.

We thank the editor, Prof. Peter Bartello, and three anonymous reviewers for helpful comments and suggestions on the manuscript. This research is supported by the National Natural Science Foundation of China (Grants 42275062, 41975066, and 41705037).

Data availability statement.

All the data are used during this study are from the ECMWF reanalysis dataset ERA5 hourly data on pressure levels from 1979 to the present (Hersbach et al. 2020), openly available from the Climate Data Store (CDS) at https://cds.climate.copernicus.eu/.

APPENDIX A

Terms in the Rotational and Divergent Kinetic Energy Budget

The tendency equations for DKE and RKE in open systems are given as follows:
∫∫∂tEDK︸DDKE=∫∫−uD⋅∂tuR︸INTD−[∫∫−ζ(υRuD−uRυD)︸Az+∫∫−f(υRuD−uRυD)︸Af+∫∫−ω∂pERK︸B+∫∫−ωuR⋅∂puD︸C]+∫∫−uD⋅∇Φ︸GD+∫∫−∇⋅EKuD︸HFD+∫∫−∂p(ωEK)︸VF+∫∫uD⋅Du︸DD,
∫∫∂tERK︸DRKE=∫∫−uR⋅∂tuD︸INTR+[∫∫−ζ(υRuD−uRυD)︸Az+∫∫−f(υRuD−uRυD)︸Af+∫∫−ω∂pERK︸B+∫∫−ωuR⋅∂puD︸C]+∫∫−uR⋅∇Φ︸GR+∫∫−∇⋅EKuR︸HFR+∫∫uR⋅Du︸DR.
In the above two equations, the double integral symbols (∫∫) represents the integration of an atmosphere in pressure coordinates.

The four terms enclosed by square brackets are the conversion from DKE to RKE because they appear in both equations but with opposite signs. Terms Az and Af depend on the relative orientations and magnitudes of uR and uD. The integrand of the terms Af and Az represents the work of a tangential force on the tangential (rotational) motion, which is produced by the exchange of fluid elements to satisfy the conservation of angular momentum. Term B (C) describes the vertical exchange of rotational (divergent) momentum since uD was observed to produce vertical motion, which is responsible for the exchange of momentum at different levels (Buechler and Fuelberg 1986).

Term GD (GR) represents the conversion between APE and DKE (RKE) due to cross-contour flow (Chen and Wiin-Nielsen 1976). Terms HFD and HFR denote horizontal flux divergence of total kinetic energy by uR and uD, respectively. The term VF is the vertical flux divergence of the HKE, which acts only on the DKE. Terms INTR and INTD arise from uR ⋅ uD and represent interactions between RKE and DKE due to the existence of each other. Terms DD and DR are dissipation terms representing frictional processes as well as energy transfers between resolvable and unresolvable scales of motion.

If these equations are integrated over the entire mass of the atmosphere, flux terms HFD, HFR, and VF vanish. Similarly, terms GR, INTD, and INTR can be expressed as Jacobians of streamfunction and velocity potential, also integrating to zero globally. Neglecting dissipative processes further, RKE can increase only by conversion from DKE.

APPENDIX B

Terms in the Spectral Horizontal Kinetic Energy Budget

Detailed expressions of the terms in the spectral kinetic energy budget are given as follows:
TKlm=−(u,u⋅∇u)lm−(u,δu/2)lm+[(∂pu, ωu)lm− (u,ω∂pu)lm]/2,
Llm=−(u,fez×u)lm,
FK↑lm=−[(ω, Φ)lm+ (u,ωu)lm/2],
CAP→Klm=−(ω,α)lm,
DKlm=(u,Du)lm,
−(u,u⋅∇u)lm=−(u,∇EK)lm−(u,ζez×u)lm.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) HKE spectra at selected pressure levels spanning the middle troposphere (500 hPa) to the middle stratosphere (10 hPa) are presented in different colors. (b) HKE spectra and the fitting spectra at 250 hPa (blue line) and 100 hPa (green line) are presented. Lines with slopes of −3 and −5/3 are added to both panels for comparison.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    HKE spectrum (black) and its divergent (red) and rotational (blue) components integrated over two layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere, with lines of slope −3 and −5/3 added to both panels for comparison. The corresponding linear fitting results are represented by dashed lines. Other details are as in Fig. 1.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Nonlinear spectral flux of HKE (ΠN[l]ptpb; black solid line); nonlinear spectral flux of HKE based on the barotropic vorticity equation (Πrot[l]ptpb; blue dashed line); the complementary flux (ΠN[l]ptpb−Πrot[l]ptpb; red dashed line) (unit: W m−2) integrated over layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Nonlinear spectral flux of (black dashed line) HKE and linear spectral flux of (red dashed line) HKE due to the Coriolis term, and the sum of the two (black solid line) integrated over layers approximately to (a) the upper troposphere and (b) the stratosphere. Other details are as in Fig. 3.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Spectral fluxes of HKE (black solid line), DKE (red solid line) and RKE (blue solid line); cumulative conversion (cyan dashed line) from DKE to RKE and conversion (green dashed line) from APE to HKE; and cumulative net vertical flux (magenta dashed line) in (a) the upper troposphere and (b) the stratosphere. Cumulative conversion (cyan dashed line) from DKE to RKE and its three components related with vertical motion (black dashed line), Coriolis effect (red dashed line), and relative vorticity (blue dashed line) in (c) the upper troposphere and (d) the stratosphere. Other details are as in Fig. 3. The inset is an expanded view of the mesoscale subrange (l ≥ 20).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Spectral (a) HKE flux, (b) DKE flux, (c) RKE flux, (d) cumulative conversion from DKE to RKE, (e) cumulative conversion from APE to HKE, and (f) cumulative vertical flux divergence (colored; unit: 10−4 m2 s−3) at l ≥ 10. The contour interval is 0.4 × 10−4 m2 s−3 and the thick black contour represents the value of zero.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    (a) Spectral conversion from DKE to RKE (colored; unit: 10−4 m2 s−3). Red dots represent the changepoint wavenumber of the spectral slope at given levels described in section 3b. Black dots represent the intersection wavenumber of the rotational and divergent spectra at given levels. The contour interval is 0.2 × 10−4 m2 s−3, and the zero contour is omitted. (b) RKE spectra (solid lines) and DKE spectra (dashed lines) at selected pressure levels. Lines with slopes of −3 and −5/3 are added to (b) for comparison.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Schematic of spectral DKE and RKE budget for global atmosphere. Note that ΔFD↑ represents the contribution of the net vertical flux, not the vertical flux itself.

Fig. 1.

(a) HKE spectra at selected pressure levels spanning the middle troposphere (500 hPa) to the middle stratosphere (10 hPa) are presented in different colors. (b) HKE spectra and the fitting spectra at 250 hPa (blue line) and 100 hPa (green line) are presented. Lines with slopes of −3 and −5/3 are added to both panels for comparison.
Fig. 2.

HKE spectrum (black) and its divergent (red) and rotational (blue) components integrated over two layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere, with lines of slope −3 and −5/3 added to both panels for comparison. The corresponding linear fitting results are represented by dashed lines. Other details are as in Fig. 1.
Fig. 3.

Nonlinear spectral flux of HKE (ΠN[l]ptpb; black solid line); nonlinear spectral flux of HKE based on the barotropic vorticity equation (Πrot[l]ptpb; blue dashed line); the complementary flux (ΠN[l]ptpb−Πrot[l]ptpb; red dashed line) (unit: W m−2) integrated over layers corresponding approximately to (a) the upper troposphere and (b) the stratosphere.
Fig. 4.

Nonlinear spectral flux of (black dashed line) HKE and linear spectral flux of (red dashed line) HKE due to the Coriolis term, and the sum of the two (black solid line) integrated over layers approximately to (a) the upper troposphere and (b) the stratosphere. Other details are as in Fig. 3.
Fig. 5.

Spectral fluxes of HKE (black solid line), DKE (red solid line) and RKE (blue solid line); cumulative conversion (cyan dashed line) from DKE to RKE and conversion (green dashed line) from APE to HKE; and cumulative net vertical flux (magenta dashed line) in (a) the upper troposphere and (b) the stratosphere. Cumulative conversion (cyan dashed line) from DKE to RKE and its three components related with vertical motion (black dashed line), Coriolis effect (red dashed line), and relative vorticity (blue dashed line) in (c) the upper troposphere and (d) the stratosphere. Other details are as in Fig. 3. The inset is an expanded view of the mesoscale subrange (l ≥ 20).
Fig. 6.

Spectral (a) HKE flux, (b) DKE flux, (c) RKE flux, (d) cumulative conversion from DKE to RKE, (e) cumulative conversion from APE to HKE, and (f) cumulative vertical flux divergence (colored; unit: 10−4 m2 s−3) at l ≥ 10. The contour interval is 0.4 × 10−4 m2 s−3 and the thick black contour represents the value of zero.
Fig. 7.

(a) Spectral conversion from DKE to RKE (colored; unit: 10−4 m2 s−3). Red dots represent the changepoint wavenumber of the spectral slope at given levels described in section 3b. Black dots represent the intersection wavenumber of the rotational and divergent spectra at given levels. The contour interval is 0.2 × 10−4 m2 s−3, and the zero contour is omitted. (b) RKE spectra (solid lines) and DKE spectra (dashed lines) at selected pressure levels. Lines with slopes of −3 and −5/3 are added to (b) for comparison.
Fig. 8.

Schematic of spectral DKE and RKE budget for global atmosphere. Note that ΔFD↑ represents the contribution of the net vertical flux, not the vertical flux itself.
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