Editorial Type:
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Article Type:
Research Article

Decomposition of Vertical Velocity and Its Zonal Wavenumber Kinetic Energy Spectra in the Hydrostatic Atmosphere

Nedjeljka Žagar aMeteorological Institute, Universität Hamburg, Hamburg, Germany

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https://orcid.org/0000-0002-7256-5073
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Valentino Neduhal aMeteorological Institute, Universität Hamburg, Hamburg, Germany

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Sergiy Vasylkevych aMeteorological Institute, Universität Hamburg, Hamburg, Germany

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Žiga Zaplotnik bFaculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
cEuropean Centre for Medium-Range Weather Forecasts, Bonn, Germany

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Hiroshi L. Tanaka dCenter for Computational Sciences, Tsukuba University, Tsukuba, Japan

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Online Publication:
22 Nov 2023
Print Publication:
01 Nov 2023
DOI:
https://doi.org/10.1175/JAS-D-23-0090.1
Page(s):
2747–2767
Free access

Abstract

The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions (HKE). One challenge that has limited progress in describing the VKE spectrum is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. New theoretical developments show that for every zonal wavenumber k, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory predicts a VKE ∝ k−5 and a VKE ∝ k1/3 power law for the Rossby and IG waves, assuming a k−3 and a k−5/3 power law for the Rossby and IG HKE spectra, respectively. The Kelvin and mixed Rossby–gravity wave VKE spectra are predicted to follow k−1 and k−5 power laws, respectively. The VKE spectra for ERA5 data from August 2018 show that the Rossby VKE spectra approximately follow the predicted a k−5 power law. The expected k1/3 power law for the gravity wave VKE spectrum is found only in the SH midlatitude stratosphere for k ≈ 10–60. The inertial range IG VKE spectra in the tropical and midlatitude troposphere reflect a mixture of ageostrophic and convection-coupled dynamics and have slopes between −1 and −1/3, likely associated with too steep IG HKE spectra. The forcing by quasigeostrophic ageostrophic motions is seen as an IG VKE peak at synoptic scales in the SH upper troposphere, which gradually moves to planetary scales in the stratosphere.

Significance Statement

The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions. One challenge is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. It is shown that for every zonal wavenumber, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory is successfully applied to the ERA5 data. It leads the way for a more accurate computation of momentum fluxes.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Nedjeljka Žagar, nedjeljka.zagar@uni-hamburg.de

Abstract

The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions (HKE). One challenge that has limited progress in describing the VKE spectrum is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. New theoretical developments show that for every zonal wavenumber k, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory predicts a VKE ∝ k−5 and a VKE ∝ k1/3 power law for the Rossby and IG waves, assuming a k−3 and a k−5/3 power law for the Rossby and IG HKE spectra, respectively. The Kelvin and mixed Rossby–gravity wave VKE spectra are predicted to follow k−1 and k−5 power laws, respectively. The VKE spectra for ERA5 data from August 2018 show that the Rossby VKE spectra approximately follow the predicted a k−5 power law. The expected k1/3 power law for the gravity wave VKE spectrum is found only in the SH midlatitude stratosphere for k ≈ 10–60. The inertial range IG VKE spectra in the tropical and midlatitude troposphere reflect a mixture of ageostrophic and convection-coupled dynamics and have slopes between −1 and −1/3, likely associated with too steep IG HKE spectra. The forcing by quasigeostrophic ageostrophic motions is seen as an IG VKE peak at synoptic scales in the SH upper troposphere, which gradually moves to planetary scales in the stratosphere.

Significance Statement

The spectrum of kinetic energy of vertical motions (VKE) is less well understood compared to the kinetic energy spectrum of horizontal motions. One challenge is a lack of a unified approach to the decomposition of vertical velocities associated with the Rossby motions and inertia–gravity (IG) wave flows. This paper presents such a unified approach using a linear Rossby–IG vertical velocity normal-mode decomposition appropriate for a spherical, hydrostatic atmosphere. It is shown that for every zonal wavenumber, the limit VKE is proportional to the total mechanical energy and to the square of the frequency of the normal mode. The theory is successfully applied to the ERA5 data. It leads the way for a more accurate computation of momentum fluxes.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Nedjeljka Žagar, nedjeljka.zagar@uni-hamburg.de
Keywords: Inertia-gravity waves; Rossby waves; Vertical motion; Kinetic energy; Filtering techniques; Spectral analysis/models/distribution

1. Introduction

The vertical velocity w, w = dz/dt, is a key variable associated with the conversion of the available potential energy to kinetic energy in the global atmosphere. Vertical motions define clouds and precipitation associated with day-to-day weather, the vertical energy propagation by internal gravity waves, and transport of trace constituents. The vertical velocity is needed to represent wind stresses in the horizontal momentum equations in order to compute vertical momentum fluxes from unresolved motions (e.g., Liu 2019). However, w is not an observed quantity of the global observing system. Sporadic observations of vertical velocity make evident the missing variance by the models, at least locally (e.g., Dörnbrack et al. 2018).

Dynamical cores of several global numerical weather prediction (NWP) models, which are used to produce the reanalyses, are hydrostatic meaning that the vertical velocity is a diagnostic quantity usually derived from the mass continuity equation. In climate research, diagnostic of vertical velocity and the computation of the global energy cycle are commonly carried out in the system with the pressure vertical coordinate, requiring the pressure vertical velocity ω. For example, the model of the European Centre for Medium-Range Weather Forecasts (ECMWF) computes ω at the hybrid sigma–pressure level η as
ω(η)=(dpdt)η=−∫0η∇η⋅(VH∂p∂η)dη+VH⋅∇ηp,
where VH = (u, υ) and ∇η are the horizontal velocity and 2D gradient operator on the η surface, respectively, and p is the η-level pressure (Simmons and Burridge 1981). The widely used pressure-level ω(p) is obtained by linearly interpolating model-level ω(η). In the pressure system,
ω(p)=−∫pTp∇p⋅Vdp,
and pT is the pressure at the top of the model atmosphere, usually pT = 0.

The vertical velocity computed by Eq. (1) or Eq. (2) contains signatures of both the Rossby and gravity wave dynamics, of resolved and parameterized physical processes as well as numerical effects. The separation of the governing dynamics and various processes is difficult, especially in the tropics where a frequency gap between the gravity and Rossby dynamical regimes, which is present in the middle latitudes, disappears. But, even in the extratropics the two regimes coexist at a large part of subsynoptic scales where the divergent-dominated dynamics increasingly projects on the inertia–gravity waves (e.g., Žagar et al. 2017), requiring a somewhat arbitrary cutoff scale for the computation of gravity wave momentum fluxes.

We derive herein a new method for linear decomposition of the vertical velocity and associated kinetic energy spectra for the two main dynamical regimes in the hydrostatic atmosphere. The method provides zonal wavenumber kinetic energy spectra of vertical motions as a function of latitude and altitude (pressure level) and is demonstrated using the ERA5 data (Hersbach et al. 2020). It leads the way to a more accurate computation of vertical momentum fluxes in the spherical atmosphere.

Traditionally, the vertical velocity associated with the Rossby wave and inertia–gravity (IG; or gravity) wave dynamics has been studied by computing each of the dynamical components independently of each other. For example, in the middle latitudes, synoptic-scale vertical velocities coupled with ageostrophic motions in baroclinic Rossby waves can be estimated using the quasigeostrophic omega equation (e.g., Hoskins et al. 1978; Stepanyuk et al. 2017). At mesoscales, vertical motions are largely due to internal gravity waves generated by processes such as interaction of the flow with orography, surface and boundary layer processes, tropospheric moist convection, frontogenesis, imbalances of synoptic jets and wave–wave interactions (e.g., Fritts and Nastrom 1992; Fritts and Alexander 2003). Associated vertical velocities can then be quantified using the polarization equations from linear wave theory on the f plane (e.g., Nappo 2002).

A more common way to diagnose internal gravity waves in hydrostatic numerical simulations is using information derived from the divergence field (e.g., Dörnbrack et al. 2018). While suitable for extratropical mesoscale processes, this approach is less informative in the tropics, where the large-scale divergence field includes a mix of dynamical modes, i.e., the Kelvin and mixed Rossby–gravity (MRG) waves on top of a spectrum of inertia–gravity waves. Not only are vertical velocities for each of these modes poorly known, their horizontal velocities are also just as reliable as the methods that are used to filter the waves from the observed or simulated circulation (Knippertz et al. 2022). The Kelvin and MRG waves, together with IG waves, constitute the non-Rossby part of the discrete, linear mode spectrum for the stratified, rotating atmosphere bounded at the top, and their horizontal wavenumber spectra have been analyzed in several studies in recent years (Žagar et al. 2009b; Stephan et al. 2021; Žagar et al. 2022).

The analytical solutions of the Rossby and non-Rossby modes on the sphere, i.e., eigensolutions of the linearized primitive equations, are known as the normal-mode functions (NMF; e.g., Kasahara 2020). As we show in this paper, the normal-mode framework provides also vertical velocities associated with the Rossby and non-Rossby waves in the spherical atmosphere. The involved hydrostatic framework is considered suitable even for model simulations at horizontal resolutions as high as a few kilometers (e.g., Craig and Selz 2018; Dueben et al. 2020).

The decomposition of both horizontal and vertical motions in terms of the Rossby and non-Rossby modes within the same framework provides a consistent comparison of the zonal wavenumber horizontal and vertical energy spectra for the two regimes. The former has been extensively studied including a transition from the −3 power law to −5/3 power law at scales between 1000 and 500 km (e.g., Nastrom and Gage 1985; Rodda and Harlander 2020, and references therein). In contrast, the spectrum of kinetic energy of vertical motions, i.e., the vertical kinetic energy (VKE) spectrum, is poorly known. Observations are few and their spatial coverage is limited (e.g., Bacmeister et al. 1996; Schumann 2019; Dörnbrack et al. 2022). Aircraft observations reveal a local maximum in the VKE at scales between 10 and 100 km, depending on the regional forcing (Dörnbrack et al. 2022, their Fig. B2).

The large-scale part of the VKE spectrum is available only from numerical model simulations. High-resolution nonhydrostatic simulations suggest a nearly flat spectrum of the VKE in the zonal wavenumber domain (e.g., Terasaki et al. 2009; Müller et al. 2018) whereas the spherical harmonics decomposition (global horizontal wavenumber) reveals two maxima, one at synoptic scales and the other near the effective resolutions of the models and dependent on convection modeling (e.g., Skamarock et al. 2014; Polichtchouk et al. 2022; Morfa and Stephan 2023). The interpretation of the global horizontal wavenumber VKE spectra is challenging not least because the spherical harmonics decomposition does not distinguish latitudinal variation in the VKE spectra. A new analytical derivation presented in this paper couples the VKE spectra with dynamics of horizontal motions. This is made possible by a new approach to the computation of the zonal wavenumber spectra of the kinetic energy of horizontal velocities [i.e., horizontal kinetic energy (HKE) spectra] in the NMF framework. The application of the new developments to ERA5 data exposes significant latitudinal variations, i.e., anisotropy of both the HKE and VKE spectra.

The paper consists of four sections, including the introduction that serves as the first section. Section 2 derives, for the first time, the regime-dependent kinetic energy spectra of vertical motions in the pressure coordinate system within a multimodal framework and alongside it, the latitudinal HKE spectra. The validation and application of the new framework to the ECMWF ERA5 data are presented in section 3. Conclusions and outlook are given in section 4.

2. Regime-dependent vertical motions and their kinetic energy spectra

We first derive equations for vertical motions associated with the Rossby and non-Rossby modes in the hydrostatic atmosphere using the normal-mode framework. This is followed by the derivation of the kinetic energy spectrum of vertical velocity and discussion of spectral slopes. Finally, we derive the zonal wavenumber kinetic energy spectrum of horizontal velocity in modal framework building upon the existing, global three-dimensional energy spectra from the MODES software (Žagar et al. 2015) and associated theory that is reviewed in Kasahara (2020) and Tanaka and Žagar (2020).

a. Normal-mode decomposition of the global horizontal motions

We start with the adiabatic, hydrostatic equations in the pressure coordinate system, linearized about a motionless basic state on a rotating Earth with the globally average vertical temperature profile To(p):
∂u∂t−2Ωsin⁡φυ+1acos⁡φ∂Φ∂λ=0,
∂υ∂t+2Ωsin⁡φu+1a∂Φ∂φ=0,
∂∂t(pR∂Φ∂p)+(κTop−dTodp)ω=0,
∂ω∂p=−∇⋅V=−1acos⁡φ[∂u∂λ+∂∂φ(υcos⁡φ)].
Here u, υ, and ω are eastward, northward, and vertical velocity components in pressure coordinates, and Φ = gh is geopotential, h being the geopotential height. The independent variables are longitude λ, latitude φ, pressure p, and time t. The constants are the gravity g, Earth’s radius a, and rotation rate Ω, the gas constant for dry air R, and the specific heat at constant pressure Cp, with κ = R/Cp. The bottom boundary condition at p = ps corresponds to no mass flux through the surface:
∂∂t(∂Φ∂p+pγoRToΦ)=0atp=ps,
where γo = κTo/p − dTo/dp is evaluated at p = ps. The top boundary condition Φω → 0 as p → 0 guarantees energy conservation since it requires that Φ is bounded for all t (Cohn and Dee 1989).
Taking the derivative with respect to pressure of Eq. (3c) and using Eq. (3d) to express ∂ω/∂p, we get the equation
∂∂t∂∂p(1Γo∂Φ∂p)−∇⋅V=0,
where V = (u, υ), and the static stability parameter Γo is defined as
Γo=Rp(κTop−dTodp)=Rpγo.
Equations (3a), (3b), and (5) form a system that is solved seeking solutions of the form
(u, υ, Φ)T(λ,φ,p,t)=(u′, υ′, Φ′)T(λ,φ,t)×G(p).
The separation of horizontal and vertical dependencies leads to a set of horizontal equations and the vertical structure equation
ddp(1ΓodGdp)+1gDG=0
with the boundary conditions
dGdp+pΓoRToG=0atp=ps,withTo=Ts,and1ΓoGdGdp→0asp→0.
Given boundary conditions (9) and realistic stability profiles Γo(p), solutions of Eq. (8) exist only for a discrete set of positive eigenvalues D, and the corresponding eigenfunctions G(p) are orthogonal in the sense that
1ps∫0psGi(p)Gj(p)dp=δij,
where δij = 1 if i = j and δij = 0 otherwise. Details about analytical and numerical solutions of Eq. (8) can be found, for example, in Staniforth et al. (1985) and Cohn and Dee (1989).
The horizontal structure equations correspond to the global rotating shallow-water equations for wind (u′, υ′) and height, h′ = Φ′/g, perturbations. The corresponding nondimensional variables u˜, υ˜, and h˜ are obtained using the mean fluid depth D and the rotation rate as
u=u˜gD,υ=υ˜gD ,h=h˜D ,t=t˜/(2Ω).
The nondimensional linear shallow-water equations on the sphere, also known as the Laplace tidal equation without forcing (Longuet-Higgins 1968), are
∂u˜∂t˜−υ˜sin⁡φ+γcos⁡φ∂h˜∂λ=0,
∂υ˜∂t˜+u˜sin⁡φ+γ∂h˜∂φ=0,
∂h˜∂t˜+γ ∇˜⋅V˜=0,
where V˜=(u˜,υ˜) and ∇˜=a∇ are the nondimensional horizontal velocity and “del” operator, respectively. The nondimensional parameter γ is defined as
γ=gD2aΩ=ϵ−1/2,
where ϵ = 4a2Ω2/(gD) is the Lamb’s parameter, and D is the separation constant, an eigenvalue of (8), called “equivalent depth” (Taylor 1936). It couples Eqs. (8) and (12) and corresponds to the mean depth of the linear shallow-water equations. In nondimensional Eq. (12), the transform (11) reduces the number of physical parameters to the single dimensionless parameter γ that also absorbs D.
The equation set (12) is a linear system and its solution W can be expressed as a linear combination of waves Wnk propagating along the latitude circles with nondimensional frequencies ν˜ (frequency ν scaled by 2Ω) and with meridional structures defined by the Hough functions Θnk(φ) as
Wnk(λ,φ,t)=(u˜,υ˜,h˜)T(λ,φ,t˜)=Θnk(φ)eikλe−iν˜nkt˜,
where Hnk(λ,φ)=Θnk(φ) eikλ is a single Hough harmonic defined by the Hough vector function Θnk(φ) with the meridional modal index n, for a given γ (i.e., D) and a zonal wavenumber k.
The computation of the Hough functions Θnk(φ) and associated eigenfrequencies ν˜ can be carried out by expanding the eigensolutions of (12) in terms of the spherical vector harmonics that are given in terms of the associated Legendre polynomials Pn′k, where integer index n′ is the degree of the polynomial (Swarztrauber and Kasahara 1985). For every k and D, the solutions Θnk(φ) exist for a range of modal indices n ≥ 0. For each n, the vector Θnk(φ) consists of the meridional structure functions for (u˜,υ˜,h˜), U, V, and Z, respectively:
Θnk(φ)=[Unk(φ), iVnk(φ),Znk(φ)]T.
The factor i, (i=−1) in front of V accounts for the phase shift of π/2 of V with respect to U. The orthogonality condition of the Hough vector functions Θnk is written as
∫−11Θnk⋅(Θn′k)* dμ=∫−11(UnkUn′k+VnkVn′k+ZnkZn′k)dμ=δnn′,
and similar for the orthogonality of the Hough harmonics for each D:
12π∫02π∫−11Hnk⋅(Hn′k′)*dμdλ=δnn′δkk′,
where μ = sinφ.

For k > 0, the eigensolutions are organized in two distinct groups depending on their frequencies: predominantly divergent inertia–gravity modes that propagate eastward or westward (EIG and WIG modes, respectively), and westward-propagating rotational waves of the Rossby–Haurwitz type. Another subscript, which would denote the three wave species of the normal modes (Rossby, EIG, and WIG waves), does not appear explicitly as it is absorbed within the meridional modal index n. The fastest eastward mode, which corresponds to the n = 0 EIG solution on the sphere, is the Kelvin wave, whereas the fastest westward-propagating rotational mode is the MRG wave (n = 0 Rossby mode in Swarztrauber and Kasahara 1985). For k = 0, all rotational modes have zero frequency while the eastward-propagating and westward-propagating inertia–gravity modes have frequencies of the opposite sign and the same magnitude that are assigned to the eastward and westward propagations, respectively.

b. Discrete three-dimensional solutions

For a stably stratified atmosphere represented in terms of M layers, there are M eigenvalues D of (8), D1 > D2 > … > Dm > … > DM. For every Dm, a number of discrete wave solutions of (12) in terms of Hough harmonics is defined by the maximal number of waves along a latitude circle and the meridional truncations of the Hough functions Θnk(φ) for the Rossby and IG modes.

Using the orthogonality conditions (10) and (17), the forward discrete transform consists of the vertical and horizontal projection of the data vector Xj(λ, φ, pj) = (u, υ, h)T at time t on jth pressure level onto the vertical structures and the Hough harmonics,
Xm(λ,φ)=Sm−1∑j=1JXjGm(pj)
and
χnk(m)=12π∫02π∫−11Xm⋅[Θnk(φ;m)]*e−ikλdμdλ,
respectively. Here, Sm is a nondimensionalization matrix that removes physical dimensions of wind and geopotential height after the vertical projection. The Sm matrix is a 3 × 3 diagonal matrix with elements gDm, gDm, and Dm. The number of vertical levels with data is J, and the number of vertical modes M cannot exceed J, M ≤ J. Referring to the previous section, Wm≈SmXm.
The computation of the Hough expansion coefficients χnk(m) is carried out using the fast Fourier transform along the latitude circles and the Gaussian quadrature for the integration in the meridional direction. The inverse pair is
Xm(λ,φ)=∑n=1R∑k=−KKχnk(m)Θnk(φ;m)eikλ,
and
Xj(λ,φ,pj)=∑m=1MGm(pj)SmXm(λ,φ) .
In Eq. (20), the parameter R denotes the total number of meridional modes which combines all of the Rossby and non-Rossby modes. The implementation of Eqs. (18)–(21) in the MODES software package (Žagar et al. 2015) defines R = RE + RW + RR = RnR + RN and applies the same truncations for the three wave types, i.e., with RN = RE = RW. The first mode in the list of Rossby modes is the MRG mode as the fastest westward-propagating balanced mode. The EIG modes start with the Kelvin wave and continue with the mode that corresponds to the so-called eastward MRG wave in the equatorial β-plane framework (Matsuno 1966). By using a common index n for the three wave species in (20) and later on, we avoid adding the fourth summation in the equations that would go over wave species. For physical interpretation, n goes from 1 to RN − 1 for Rossby modes, from 0 to RE − 1 for EIG modes, and from 0 to RW − 1 for the WIG modes. Since the Kelvin wave is n = 0 EIG mode and the MRG wave is the n = 0 Rossby mode, tropical modes are discussed as non-Rossby or IG modes depending on whether they include or not, respectively, the Kelvin and MRG waves. In the extratropics, using IG or non-Rossby modes makes no difference since the two special waves are equatorially trapped. Solutions for the case of k = 0 are not unique and MODES applies the so-called K modes derived by Kasahara (1978).

c. Frequency relationships for the Rossby and gravity waves

The analytical expressions for the frequencies of the eastward or westward inertia–gravity waves and Rossby waves on the sphere exist only for a limiting case when γ → ∞ (i.e., a large D or a small ϵ), and it was obtained already by Hough (1898) and Haurwitz (1940). Known as the dispersion relationship for the Rossby–Haurwitz waves, it reads as
νR=−2Ωkn′(n′+1),
where n′ = k + n, the degree of the Legendre polynomial. Here, n—the Hough function index—starts from 0. In the literature, n′ and n are usually used without special distinction as k + n is the horizontal wavenumber on the sphere, equivalent to the global wavenumber in the spherical harmonics expansion. The dimensional frequencies for the IG waves on the sphere are for a small ϵ approximated by
νG2=gDa2n′(n′+1).
The nondimensional frequency ν˜ is ν˜ =ν/(2Ω). The mathematical derivation of Eqs. (22) and (23) is available in Swarztrauber and Kasahara (1985).
The limit dispersion relationships on the sphere can be compared with the expressions for the equatorial β plane,
νR=−βkk2+(2n+1)β/candνG2=c2k2+βc(2n+1),
where c=gD and n stands for the meridional mode index of the Hermite polynomial which builds eigensolutions for u, υ, and h on the equatorial β plane. For the midlatitude Rossby waves, the linearized barotropic potential vorticity equation gives the dispersion relationship in the form
νR=−βkk2+l2+f2/c2,
whereas for the inertia–gravity waves we have νG2=c2k2+c2l2+f2, where l is the planar meridional wavenumber. The derivation of (24) and (25) is available in textbooks (e.g., Gill 1982). The planar horizontal wavenumber (k2 + l2)1/2 is in spherical geometry replaced by k + n. Although valid for the barotropic fluid in the absence of the background wind, Eqs. (22)–(25) provide theoretical understanding and guide interpretation of observations. More generally, the baroclinic atmosphere is represented in the previous subsection as a superposition of barotropic models with varying D.
On the sphere or on the equatorial β plane isotropy is lost meaning that the scaling of the barotropic wave frequency in terms of k and n is not the same. In all three cases (the sphere, the f plane, and the equatorial β plane), the Rossby wave frequency scaling with respect to the zonal wavenumber is
νR∝k−1.
Scaling of νR with n is different on the sphere and the equatorial β plane, with Eqs. (22) and (24) giving
νR∝n−2andνR∝n−1,
respectively. For large-scale waves on the equatorial β plane, applying k → 0 to Eq. (24) gives νR ≈ −ck/(2n + 1), implying νR ∝ κ0. Here, κ represents a wavenumber in either zonal or meridional direction because large-scale tropical variability is associated with the few lowest n (e.g., Žagar et al. 2017).
The gravity wave frequencies scale with the zonal wavenumber as
νG∝k
in all three cases (the f plane, the equatorial β plane, and the sphere), whereas the scaling with respect to n is
νG∝nandνG∝n1/2
on the sphere and the equatorial β plane, respectively. For the large-scale equatorial IG waves, accounting for the smallness of both k and n gives νG ∝ κ1/2. For the Kelvin and MRG waves, the scalings are
νK∝kandνM∝k−1,
respectively. The MRG dispersion relations are discussed in Paldor et al. (2018), who showed that the dispersion relation for the MRG waves on the sphere when ϵ → 0 is νM = −ϵ/(k + 1), as originally derived by Longuet-Higgins (1968). The dispersion relationship for the Kelvin waves on the sphere was derived by Boyd and Zhou (2008) and it reads νK=k/ϵ−1/(2k)+1/(2ϵ)(1+ϵ/k2)1/2, which can be approximated by νK≈k/ϵ.

We shall make use of these scaling laws in the derivation of the scaling laws for the vertical kinetic energy in the next subsection and in the discussion of the results.

With respect to Eq. (22), it is necessary to address the question to what extent the dispersion relationships on the sphere, derived for the case D → ∞, apply to frequencies associated with Ds ranging between a few meters and 10 km, as computed for the atmosphere with a finite depth. This question is addressed in appendix A, where we demonstrate that the dispersion relationship (23) for large D is an excellent approximation for numerically computed IG frequencies for many equivalent depths and meridional modes. On the other hand, the relationship (22) provides a good approximation for nearly all Rossby frequencies of the first vertical mode (D1 ≈ 10 km), for the mesoscale range of wavenumbers with D > 200 m and for the waves with large k and equivalent depths D > 10 m. Furthermore, we show that the frequencies of all Rossby modes can be well approximated by a more elaborate formula, structurally similar to the β-plane relationship (25), implying similar scaling and asymptotic behavior.

d. Modal decomposition of the vertical velocity

For a single vertical mode m (i.e., a single D), the Hough harmonics expansion (14) for height h˜ is
h˜(λ,φ,t˜)=∑n=1R∑k=−KKχnkZnk(φ)eikλe−iν˜nkt˜,
with m dropped for simplicity. Taking time derivative of (31) for every wave component (n, k) we get
∂∂t˜h˜nk=−iν˜nkZnk(φ)eikλe−iν˜nkt˜=−iν˜nkh˜h.
Using (32) in (12c) gives a nondimensional horizontal wind divergence for a single wave component
−γ∇˜⋅V˜nk=−iν˜nkh˜nk,
whereas the divergence in physical space is a sum over all wave components:
γ∇˜⋅V˜=∑n=1R∑k=−KKiν˜nkχnkZnk(φ)eikλ.
By multiplying (33) by 2Ω, the nondimensional frequency ν˜nk on the right-hand side of the equation becomes dimensional frequency νnk and the nondimensional divergence ∇˜⋅V˜ on the left-hand side becomes the dimensional divergence ∇ · V. This is equivalent to multiplying the result of the right-hand side of (33) by a gDm/γ for every m. Equation (33) provides a novel way for the computation of divergence decomposed into Rossby and gravity modes. Its application to the reanalysis data focusing on the regime quantification of tropical divergence will be the subject of a subsequent manuscript.
We proceed with the dimensional form of (33) in the computation of the pressure vertical velocity (2). Using the vertical decomposition (21) for the two wind components, we can express divergence as
∇⋅V=∑m=1M(∇⋅V)mGm(p),
and use it in (2):
ω(λ,φ,p)=−∑m=1M∫0pGm(p′)(∇⋅V)mdp′.
Using ∇ ⋅ V from (33) multiplied by 2Ω, a single component of the vertical velocity associated with the Rossby or non-Rossby mode, as defined by the value of n, is
ωnk(λ,φ,p;m)=−∫0piνnk(m)χnk(m)Znk(φ;m)eikλGm(p′)dp′.
The complete signal ω in physical space is obtained by a summation of all components:
ω(λ,φ,p)=∑m=1M∑n=1R∑k=−KKωnk(λ,φ,p;m).
On the other hand, along a single latitude circle φ at a given pressure level p, ω can be represented using the Fourier series expansion in terms of K waves as
ω(λ,φ,p)=∑k=−KKω^k(φ,p)eikλ.
Comparing this expression with Eqs. (36) and (37), we see that the complex expansion coefficient ω^k(φ,p) is given by
ω^k(φ,p)=−i∑m=1M∑n=1Rνnk(m)χnk(m)Znk(φ;m)∫0pGm(p′)dp′.
In this equation, the vertical structure functions G(p) and the Hough functions for the geopotential height field Z(φ) project the spectral space signal νnk(m)χnk(m) on the latitude φ and pressure level p in a mode-selective way. Given (φ, p), modal components ω^k are additive. The summation of the meridional modes associated with the Rossby and IG (or non-Rossby) modes in Eq. (39) provides ωR or ωG, respectively, and total ω is their sum, ω = ωR + ωG. Further decomposition of ω for any species means limiting the range of meridional indices in Eq. (39) to a subset of Rossby or non-Rossby modes or any single mode.

e. Zonal wavenumber kinetic energy spectrum of vertical motions

The zonal wavenumber spectrum of the VKE is
EVk(φ,p)=(1−δk0)|ω^k|2,
where δk0 = 1/2 for k = 0 and 0 otherwise (with ω^k presented only for positive k). Using the Parseval theorem, the VKE per unit mass integrated around a latitude circle, EV, is equal to the sum of its components in all zonal wavenumbers plus the zonal mean state:
EV(φ,p)=1L∫0Lω22dx=∑k=−KK|ω^k|22=∑k=0KEVk,
where L = 2πa cosφ is the circumference at latitude φ and dx = a cosφdλ. Limiting the summation (41) to the Rossby and non-Rossby (or IG) modes gives EVR and EVG, respectively.
It is useful at this point to recall the zonal wavenumber spectrum of horizontal motions (Žagar et al. 2015); for every k, the total mechanical energy, a sum of the available potential energy and kinetic energy, is given by
Ik=(1−δk0)∑m=1M∑n=1RgDm|χnk(m)|2,
with energy in a single wave mode (k, n, m) given by
Ink(m)=12gDm|χnk(m)|2,i.e.,Ink(m)∝|χnk(m)|2.
In contrast to EVk, which is defined for a pressure level and a latitude circle, Ik is global. The components Ink(m) are additive because of 3D orthogonality of the expansion functions Gm(p) and Hnk(λ,φ;m) [Eqs. (10) and (17)]. In contrast, the components of |ω^k|2 are not additive. This is because the Hough functions for the geopotential height Znk(φ;m) are not orthogonal but are a part of the L2 norm [Eq. (16)]. Nevertheless, we can involve only a subset of modes in (39) in order to study the VKE spectra of the Rossby and IG modes, EVR and EVG, respectively. Effects of the multiplicative terms on our VKE spectra are found to be small (not shown).
Based on Eqs. (39) and (40), we make the following ansatz for a single component of the VKE spectrum Ek:
Enk(m)∝|νnk(m)χnk(m)|2∝[νnk(m)]2Ink(m).
Besides energy of horizontal motions, (44) accounts for modal frequencies that couple the VKE spectra with the horizontal wavenumber through the dispersion relations in section 2c. The ansatz (44) does not account for the role of the geopotential height Hough function Znk(φ;m) with the mathematical justification provided in appendix B including Fig. B1. Physically, it is meaningful to consider the role of latitude-dependent Hough function to be one of filtering the latitudinal signal of the global coefficients in (39), similar to what G(p) performs for the vertical domain.
Given the ansatz (44) and the power laws −3 and −5/3 associated with the energies IRk and IGk of the Rossby and IG modes, respectively, the power laws for limit spectra of the VKE can be argued to be
EVR∝κ−5andEVG∝κ1/3
for the Rossby and gravity modes, respectively. This is summarized in Table 1. The same power laws −5 and 1/3 for the EVR and EVG, respectively, are obtained from the analytical dispersion relations for waves in a barotropic fluid on the β plane discussed above.
Table 1.

Scaling laws for the frequency, the total mechanical energy I (sum of the kinetic energy of horizontal motions and available potential energy), and the vertical kinetic energy (KE) E as a function of the zonal wavenumber k. The wavenumber index κ is used for the large-scale tropical flows where both k and meridional mode index n are small.
Table 1.

The −5 power law for VKE is associated with the geostrophic wind divergence which is due to the β term and is proportional to υgβ/f. On the other hand, ageostrophic motions, a part of quasigeostrophic turbulence, which is characterized by the −3 power law, project on the IG modes. This can be compared with a common assumption that the VKE is proportional to the square of the horizontal wind divergence that leads to a −1 power law for EVR after assuming a −3 power law for the HKE spectrum. The difference between the −1 slope derived from the square of divergence and the power law EVR∝k−5 arise from the latter involving the assumption of the VKE proportional to the square of the wave frequency whereas the former has VKE proportional to the square of the horizontal wavenumber. Strong ageostrophic motions are intrinsically related to the baroclinic Rossby wave dynamics as will be seen in the horizontal and vertical IG spectra at synoptic forcing scales. The VKE spectra of midlatitude IG modes are thus a mixture of ageostrophic circulation, inertia–gravity waves, internal gravity waves due to various sources, and coherent structure across scales that are not waves and may even be numerical artifacts. Appendix B provides a consistent mathematical discussion of the VKE limit spectra from divergence and the mechanical energy.

The large-scale tropical flows and the two special tropical wave solutions are given a separate treatment in Table 1. For scales with k < 7, Žagar et al. (2017) showed that global energy spectra of horizontal motions associated with non-Rossby modes approximately follow a −1 slope. Involving the frequency scaling discussed in the previous subsection and a nearly white spectrum of the planetary-scale horizontal circulation in tropics suggests a somewhat steeper EVR than EVG spectra at the largest tropical scales. The Kelvin wave spectra were discussed in Žagar et al. (2022) and shown to have slopes between −3 and −5/3 at synoptic and subsynoptic scales and a nearly flat spectrum at planetary scales in the stratosphere. This is related to the Kelvin wave being a mixture of nondivergent and irrotational flow depending on the zonal wavenumber. In Table 1 we use a steeper spectrum of −3 as the limit spectrum that gives a power law of k−1 for EVK. The MRG energy spectra are characterized by a maximum at synoptic scales and they approximately follow a k−3 power law for wavenumbers k > 10 (Žagar et al. 2009b). The spherical MRG wave frequency is inversely proportional to k−1 (Paldor et al. 2018) suggesting a −5 power law for the MRG VKE spectrum EVM. For k = 0, all frequencies of the Rossby modes are zero, and therefore, the VKE is zero.

f. Latitude- and altitude-dependent kinetic energy spectra of horizontal motions

Similar to the expansion of the height field in terms of the Hough functions Z, we can expand the two components of the horizontal velocity at a single time instant as
u(λ,φ,p)=∑m=1MgDm∑n=1R∑k=−KKχnkUnk(φ;m)eikλGm(p),
υ(λ,φ,p)=∑m=1MgDm∑n=1R∑k=−KKχnkiVnk(φ;m)eikλGm(p),
and define the complex expansion coefficients u^k(φ,p) and υ^k(φ,p) for the zonal and meridional velocity components along the latitude circle as
u^k(φ,p)=∑m=1MgDm∑n=1Rχnk(m)Unk(φ;m)Gm(p)withu(λ,φ,p)=∑k=−KKu^k(φ,p)eikλ,
υ^k(φ,p)=∑m=1MgDm∑n=1Riχnk(m)Vnk(φ;m)Gm(p)withυ(λ,φ,p)=∑k=−KKυ^k(φ,p)eikλ.
Then, the horizontal kinetic energy per unit mass around a latitude circle, EH, can be computed as
EH(φ,p)=∑k=0KEHk(φ,p)=∑k=0K(1−δk0)(|u^k|2+|υ^k|2).
In contrast to Ik, which involves total energy summation in the vertical and meridional directions for every k, EHk is the kinetic energy for a single vertical pressure level and a latitude circle, just like the VKE EVk. It also shares its disadvantage of having components of |uk|2 and |υk|2 not being additive, although this does not significantly affect the value of the HKE decomposition for our purpose. The two HKE components are obtained by keeping the subset of the values of the meridional mode index n corresponding to the Rossby and non-Rossby modes in Eq. (47), which gives EHR and EHG, respectively. As for the VKE, we shall denote the spectra of the non-Rossby (or IG) horizontal kinetic energy with the letter G. The deviations of EHR and EHG from the expected average slopes of −3 and −5/3 are helpful in the interpretation of the VKE spectra. A final remark on the new zonal wavenumber HKE spectra is that their computation requires the coefficients χnk(m) of the 3D NMF expansion. A rather limited use of the NMF decomposition explains why such 1D spectra have not been studied before.

3. Decomposition of the vertical velocity and kinetic energy in the ECMWF system

a. MODES implementation and validation

Referring to Žagar et al. (2015), the computation of ω within the MODES package required the implementation of the pressure vertical coordinate (Kasahara 1984; Tanaka 1985). The default version of MODES is in the terrain-following σ coordinate, but the evaluation of vertical velocity in the σ system involves the term dependent on the surface pressure which is cumbersome to use. MODES is thus complemented by an option for the pressure system.

The computation of ω in MODES is based on the following steps. First, we compute the meridionally dependent part of Eq. (39), ∑n=1Rνnk(m)χnk(m)Znk(φ;m) for every k and m. This is followed by the integration of the vertical part which is solved by the finite-difference method, and the summation over all m. This results in a set of Fourier expansion coefficients for the total ω field which are then transformed to physical space by the inverse Fourier transform. By partitioning the summation over n, one can filter contributions to ω from different Rossby and IG modes.

If the vertical structure functions Gm(p) are constructed analytically then Eq. (36) involves the evaluation of derivatives ∂Gm(p)/∂p as shown by Tanaka and Yatagai (2000). They compared the spherical and Hough harmonics expansions for the computation of ω and showed that the Hough harmonics expansion provides a suitable representation of the pressure vertical velocity.

An example of our retrieval of ω(φ, p) using MODES is shown in Fig. 1 in comparison with the ω field retrieved from the ECMWF ERA5 archive and decomposed along the same latitude using the 1D FFT in Python. The two fields are in very good agreement but not identical for several reasons. There are differences between the model-level pressure at points along the latitude and a constant pressure level used in the MODES expansion. Other factors causing the differences, especially in the lower troposphere and in the polar regions are the vertical and meridional truncations, respectively. For the vertical truncation, M < J always due to a rapid reduction in the values of equivalent depth (Žagar et al. 2009a). More smoothed fields of ω from MODES are expected also due to the truncated input fields. Nevertheless, the reconstruction of the ω field by the new method is very good and certainly sufficiently accurate for the intended decomposition, as confirmed by statistics of root-mean-square differences between the two fields (not shown).

Fig. 1.
Fig. 1.

Comparison of the pressure vertical velocity ω from ERA5 (red lines) and ω reconstructed by MODES (blue lines) at pressure level 75.2 hPa, along 60°S at 1200 UTC 11 Aug 2018. ERA5 ω is decomposed using the 1D FFT in Python. (a) Power spectra; (b) ω(λ).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The horizontal and vertical structures of regime-decomposed ω are illustrated in Figs. 2 and 3. Figure 2 shows vertical motions illustrative of internal gravity wave above the Antarctic Peninsula that projects onto the westward-propagating inertia–gravity modes. Note that the large-scale part of the IG modes, associated with the gradient wind balance and orography, is filtered out by removing k < 7. The Rossby wave component, ωR, has large scales and two orders of magnitude smaller amplitude. The upward and downward motions are found superimposed on the front and rear side of the large-scale wave in the horizontal circulation, respectively, as expected from the geostrophic wind divergence, ∇ ⋅ Vg = −υgβ/f. The vertical cross section of the retrieved ω over the Antarctic Peninsula in Fig. 3 shows that data corroborate expectations from the decomposition as in Fig. 1. A part of the standing signal over the topography projects on both eastward and westward modes whereas the propagating wave is made of the WIG modes. We do not intend here a detailed examination of relatively high-resolution features over orography, as it would require decomposition at multiple time steps during the day. Features of ω in the lower troposphere are expected to somewhat deviate from ERA5 due to the vertical and horizontal truncation and lower resolution of MODES, similar to Fig. 1.

Fig. 2.
Fig. 2.

Pressure vertical velocity ω at level near 75 hPa at 1200 UTC 11 Aug 2018. (a) ω from ERA5 data, (b) ω derived by MODES that is further split into (c) Rossby modes (ROT), (d) inertia–gravity (IG) modes, (e) eastward IG (EIG), and (f) westward IG (WIG). The IG modes in (d)–(f) are filtered for k > 7. Small-scale features of EIG and WIG modes sum up to zero over the Andes between 20° and 35°S. The contours in (c) are the horizontal wind speed (in m s−1). The Rossby mode ω is multiplied by a factor of 100.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

Fig. 3.
Fig. 3.

As in Fig. 2, but the vertical cross section along 70°S across the Antarctic peninsula.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The ω decomposition is motivated by the need to quantify vertical momentum fluxes associated with equatorial non-Rossby waves. This requires decomposition of both horizontal and vertical velocities. Having decomposed ω, quantification of the vertical momentum can be performed and it is a subject of follow-on papers. Here we discuss the decomposition of the VKE spectra into the Rossby and non-Rossby modes. The VKE spectra are analyzed for August 2018 using ERA5 data (Hersbach et al. 2020) once per day at 1200 UTC. To remain close to the levels at which dynamical fields were computed, the pressure levels are assigned as the average pressure of the 137 model levels from the definition of the hybrid sigma–pressure coordinate. The horizontal grid is the regular Gaussian grid with 1280 points along the latitude circle with 320 circles between the equator and pole. The numerical truncations are K = 350 zonal wavenumbers, and R = 600 meridional modes including equal numbers of the Rossby and both types of inertia–gravity modes (RR = RE = RW = 200). The number of vertical modes M = 60 suffices to represent most of variance in the lower troposphere while providing an accurate representation of the middle atmosphere.

By definition of the Rossby modes in the linear normal-mode function decomposition, ageostrophic motions in the extratropical troposphere will project on the inertia–gravity modes. In addition, the gradient wind balance in the polar stratosphere will partly project on the IG modes. How are these properties presented in the HKE and VKE spectra at different latitudes and pressure levels? The HKE and VKE is computed along every latitude circle and averaged in bins of 10° latitude width for every level over all days for August 2018. Vertical averaging is carried out following the identification of gross properties of the spectra as discussed in the next section.

b. Horizontal kinetic energy spectra for August 2018

The horizontal kinetic energy spectra EH, EHR and EHG computed by Eq. (47) are shown in Fig. 4. Compared to the 3D total energy spectrum (Žagar et al. 2017), the spectra in Fig. 4 provide details of anisotropy of the spherical flow. There is evidence of more energetic planetary-scale Rossby waves in the midlatitude stratosphere in the Southern Hemisphere (SH) than in the Northern Hemisphere (NH) which has summer (Fig. 4e2 versus Fig. 4a2). The Rossby wave spectra in the summer hemisphere rapidly attenuate with altitude by Charney–Drazin filtering (Charney and Drazin 1961). We do not show the HKE spectra for the high-latitude belts 60°–80° as they are similar to the midlatitude belts except for a significantly steeper drop in energies because of a smaller zonal scale of the same k. The VKE maxima at synoptic scales in extratropics, especially in the SH (winter), are present in both EHR and EHG spectra due to baroclinic Rossby wave dynamics (Figs. 4d,e). The EHG maximum at k = 5–10 is the manifestation of ageostrophic dynamics in extratropical winter (SH), whereas in the tropics it is due to the MRG waves.

Fig. 4.
Fig. 4.

The horizontal kinetic energy (HKE) spectra averaged over latitude belts and altitude bands for the ERA5 data in August 2018. Dashed lines show theoretical slopes with power laws −3, −5/3, −1, and 0, with 0 denoting the white spectrum. Dotted lines are empirical fit for average energy spectrum between 200 and 400 hPa for zonal wavenumbers k = 20–80. (left) Total HKE; (center) the Rossby part, EHR; (right) the non-Rossby part, EHG. The y axis for the left and center columns is the same.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The Rossby and non-Rossby spectra in Fig. 4 show the extent to which the ERA5 data on average deviate from the theoretical power laws for the inertial range in different latitude belts and altitudes. In particular, Fig. 4 is informative about differences between the upper troposphere and the stratospheric levels which on average have large weight in the 3D spectra due to associated equivalent depths [Eq. (42)]. For this reason, and because of the presence of the available potential energy in the 3D spectrum Ik, the non-Rossby total 3D energy spectrum for August 2018 (not shown) does not show a prominent energy maximum at synoptic scales, although such maxima are clearly seen in daily energy spectra of the ECMWF model analyses and forecasts on the MODES web page.1

In addition to lines of the theoretical power laws (dashed lines), Fig. 4 contains empirical fits (as dotted lines) of the spectra averaged over model levels between 200 and 400 hPa for k = 20–80. Even though individual samples may largely deviate from the theoretical power laws (Dörnbrack et al. 2022), it is informative to compare slopes of average spectra with theoretical expectations. Overall, the HKE spectra in the upper troposphere suggest an insufficient Rossby wave variance in the extratropical SH and a lack of the non-Rossby wave variance at all latitudes in ERA5 in August 2018. On the other hand, the upper troposphere in the tropics and in the subtropical summer hemisphere (Figs. 4b,c) appear more balanced, especially the deep tropics. This is seen as a shallower than −3 slope of the HKE spectra at 20 < k < 200 in Fig. 4c2. On the other hand, the steepest non-Rossby spectrum is found in the subtropical upper troposphere (Fig. 4b3). At all latitudes, stratospheric subsynoptic EHG spectra are shallower than the tropospheric spectra, especially in winter extratropics (Fig. 4e3). In fact, stratospheric IG spectra follow the −5/3 power law at smaller synoptic scale in SH extratropics and their amplitudes exceed that of the Rossby spectra at fairly large scales, a property found in several previous studies using the Helmholtz decomposition (e.g., Koshyk and Hamilton 2001). There are other properties of the EH spectra in Fig. 4 that we leave for future studies since such spectra are here presented for the first time.

c. Vertical kinetic energy spectra for August 2018

Now we discuss the VKE spectra averaged over the same latitude belts as the HKE spectra but for a few more vertical layers to show gradual changes in vertical velocities in ERA5 data. Many physical processes and numerical effects in the model are likely to manifest in the structure of the VKE spectra in the tropics and the middle and high latitudes as discussed in what follows.

1) Extratropics

The VKE spectra for middle latitudes are shown in Fig. 5 and can be compared with the high-latitude spectra in Fig. 6 and subtropical belts in Fig. 7. The first notable feature is the redness of the Rossby spectra, EVR, compared to the spectra of the IG modes, EVG, as expected. The Rossby VKE approximately follows the expected −5 power law from scales from about k = 7 to about k = 150 (Figs. 5a,b, 6,a,b, 7a,b). A slope steeper than −5 for k > 120 in Fig. 5 is associated with the horizontal spectra EHR in Fig. 4 stronger deviating from a −3 power law. A much steeper drop in VKE at small scales in the polar regions in Fig. 6 compared to Fig. 7 reflects a much higher resolution for the same wavenumber in the high latitudes compared to midlatitudes. Planetary-scale EVR spectra in the upper troposphere are more flat in the high than in the middle latitudes, similar to what was found for the EHR spectra. Note that the comparison with the HKE spectra is only qualitative since the HKE spectra are in J kg−1 (or m2 s−2) whereas the VKE spectra are in Pa2 s−2.

Fig. 5.
Fig. 5.

Kinetic energy spectra of vertical motions (VKE) per unit mass in the midlatitude belt between (a),(c) 30° and 60°S and (b),(d) 30° and 60°N, averaged for August 2018. (a),(b) Rossby mode VKE and (c),(d) IG modes VKE. The bottom spectrum (thick blue line) in each panel is the average over levels between 10 and 30 hPa, and the spectra above belong to layers lower in the atmosphere as defined in the legend. Black lines show power laws discussed in section 2.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for the latitude belt (a),(c) 60°–80°S and (b),(d) 60°–80°N.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for the latitude belt (a),(c) 10°–30°S and (b),(d) 10°–30°N.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The IG VKE spectra vary more distinctly than the Rossby spectra not only vertically but also latitudinally (summer versus winter of August 2018). First, note a remarkable maximum in the VKE spectra in Fig. 5c at k = 3 in the SH midlatitude stratosphere shifting to synoptic scales k = 7–8 in the troposphere where baroclinic Rossby waves dynamics creates intense ageostrophic circulation in the winter and Rossby waves penetrate the stratosphere. The maximum is present also in the SH high-latitude lower stratosphere (Fig. 6c) and SH subtropical troposphere (Fig. 7c), but absent in the high-latitude troposphere (Figs. 6c,d) and in the subtropical stratosphere (Fig. 7c), because Rossby wave activity is either weak or waves do not propagate vertically. Similar synoptic-scale peaks in VKE spectra are seen in spherical harmonics decomposition of simulated vertical velocity by kilometer-scale models, both hydrostatic (Polichtchouk et al. 2022) and nonhydrostatic (Skamarock et al. 2014; Morfa and Stephan 2023) including a shift to larger scales in the stratosphere.

The tropospheric IG VKE spectra have slopes between −1 and −1/3 for a range of synoptic and subsynoptic scales 10 < k < 100 in both hemispheres. In agreement with more shallow EHG extratropical spectra in NH compared to SH in Fig. 4, the EVG spectrum in Figs. 6d and 7d (NH, summer) is somewhat shallower than in Figs. 6c and 7c (SH, winter). The difference may be attributable to convective activity in boreal summer projecting onto IG modes. The difference between the spectral slopes of EHG and theoretically assumed −5/3 explains much of the difference between EVG at scales 10 < k < 100 and expected limit spectra with 1/3 slopes.

Finally, we find the expected positive slope of the VKE spectra associated with gravity waves [Eq. (45)] in the SH stratosphere in midlatitudes (Fig. 5c). This “ideal” gravity wave VKE spectrum with a slope close to 1/3 starts at about k = 10 and extends toward k = 60. For comparison, the EHG spectrum in the stratosphere in Fig. 4j is close to −5/3. Moving downward from the 10 hPa level, positive VKE slopes narrow in k and disappear near 200 hPa. In contrast, the IG modes in the NH stratosphere (a summer period) have flat VKE spectra and quickly drop off.

2) Tropics

The presentation of the tropical VKE spectra in Fig. 8 is somewhat different from the extratropical case because of the presence of the two special equatorial modes, the Kelvin and MRG wave. The non-Rossby spectra (the sum of the IG, Kelvin and MRG modes, Fig. 8a) are almost indistinguishable from the total spectra (not shown) but also from the IG spectra (Fig. 8c). This is because the VKE spectra for the Kelvin and MRG waves (Fig. 9) have one or two orders of magnitude less VKE than the IG modes at every k.

Fig. 8.
Fig. 8.

As in Fig. 5, but for the tropical belt between 10°S and 10°N. (a) non-Rossby (IG, MRG, and Kelvin) modes, (b) Rossby modes, (c) IG modes, (d) EIG, and WIG modes.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

Fig. 9.
Fig. 9.

As in Fig. 8, but for the (a) Kelvin waves and (b) MRG waves.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The Rossby spectra, EVR, in the tropical belt within 10° away from the equator (Fig. 8b) appear rather similar to their extratropical counterparts. The most significant difference is more variance at scales shorter than a k ≈ 100 compared to the extratropics, especially in the lower stratosphere, as also seen in the horizontal spectra in Fig. 4. At planetary scales, tropical EVR are steeper compared to their extratropical counterparts. Changes in EVR vertically are similar to that in the extratropical summer hemisphere.

The EVG spectra in the tropics are significantly shallower compared to the extratropics; their slope is close or somewhat steeper than −1/3 (Fig. 8c). At planetary scales, the EVG spectrum is nearly white in the troposphere and develops a slightly negative slope in the stratosphere in agreement with EHG and Table 1. The non-Rossby and IG VKE spectra maintain their shapes throughout the upper troposphere and stratosphere, until the wavenumber of about k = 125 in the stratosphere and up to about k = 250 in the midtroposphere. The overall whiteness of the IG spectra reflects the fact that tropical convection, which generates variance in vertical velocity, has scales from the planetary scale (i.e., ITCZ) to the small individual convective cells not resolved by the analysis grid (Bergman and Salby 1994). The slopes between −1/3 and −2/3 reflect a lack of variance in tropical convective activity in ERA5. If the slope of −2.39 estimated for EHG in the upper troposphere (Fig. 4c3) is used instead of −5/3 in Table 1, the EVG spectra should be around −0.4 which is similar to what is shown in Fig. 8. Overall, ERA5 shows no increase in VKE at the mesoscale range of wavenumbers in agreement with tropical spectra of vertical velocity in the ECMWF with parameterized deep convection in Polichtchouk et al. (2022, their Fig. 2d).

In the stratosphere and within the tropical tropopause layer, the westward IG modes dominate at planetary scales k = 1–2 (spectra drawn with dashed lines in Fig. 8d) and are significantly smaller at other wavenumbers. This can be related to the quasi-biennial oscillation (QBO) in August 2018 being in its easterly phase and the horizontal eigenstructures derived with respect to the state of rest. The role of the phase of the QBO on the VKE spectra and the vertical momentum fluxes is a subject of future work.

The VKE spectra for the Kelvin wave and the MRG wave are shown in Fig. 9. For the Kelvin wave, the VKE increases going from 10 to 300 hPa, and there is little change of VKE below this level. Overall, the Kelvin wave VKE distribution is similar to that of IG modes except that amplitudes are smaller and the spectra at subsynoptic scales are steeper and noisy. On the other hand, the MRG VKE spectra are similar to those for the Rossby modes. Both Kelvin and MRG spectra approximately follow the predicted sloped of −1 and −5, respectively, over a range of scales between k = 10 and k = 100. Even if the VKEs for the IG modes and the Kelvin waves were of similar amplitudes, a difference in the slope alone accounts for a significant difference in vertical velocity and thus momentum fluxes. This can be illustrated by a single zonal wavenumber, for example, k = 20. The ratio between VKEs for power laws −1 (Kelvin wave) and −1/3 (IG modes) gives about 14% of EVK compared to EVG at k = 20. This implies that the observed Kelvin wave vertical velocity is about one-third of that in the IG modes. At planetary and large synoptic scales, the Kelvin VKE exceeds the MRG VKE in the upper troposphere that may be associated with the phase of the QBO. This and other aspects of the Kelvin and MRG vertical velocities will be discussed in a separate study on the quantification of the vertical momentum fluxes driving the QBO using the complete ERA5 dataset.

4. Conclusions and outlook

We derived expressions for vertical velocity associated with the Rossby–Haurwitz waves and inertia–gravity (IG), Kelvin, and mixed Rossby–gravity waves for a hydrostatic atmosphere. The applied linear decomposition, based on the normal-mode functions, projects ageostrophic motions on the inertia–gravity modes. As a result, the IG modes contain both high-frequency signals and low-frequency components that result from the balance between the linear and nonlinear terms of the prognostic equations (e.g., Errico 1984; Warn and Menard 1986; Ko et al. 1989; Tribbia 2020). A further differentiation between the linear and nonlinear balance, which would decompose IG modes into ageostrophic component coupled with the quasigeostrophic dynamics and gravity waves, is beyond the scope of the present work, and will be the subject of future work.

The new decomposition method provides the vertical velocity along latitude circles at pressure levels. It is implemented in the MODES software (Žagar et al. 2015) and applied to the ERA5 data. The spectra of the kinetic energy of vertical velocity (VKE) are computed for August 2018 and discussed in relation with the KE spectra of the horizontal motions (HKE spectra) for the same latitude belts. The intended application to longer datasets (e.g., the complete ERA5 period) will provide further insight and quantify vertical velocities and associated momentum fluxes due to the Kelvin, MRG and IG modes in driving low-frequency variability such as the QBO. As an example, our method enables decomposition of the vertical momentum fluxes in the gray zone of the zonal wavenumbers k ≈ 7–40 where unbalanced dynamics gradually grows comparable and eventually exceeds in amplitude balanced dynamics.

The analytically derived expression coupling the limit VKE spectrum, EV, and the total mechanical energy (kinetic plus available potential energy) spectrum, TE, for every modal component, states that EV ∝ ν2TE, where ν is the normal mode frequency. Invoking frequency relationships for the Rossby and inertia–gravity modes and the k−3 and k−5/3 power laws for the EH within the inertial range, the VKE spectra of the Rossby–Haurwitz and inertia–gravity waves follow k−5 and k1/3 power laws, respectively. At the planetary scales in the tropics, which constitute much of the divergent circulation, both Rossby and IG VKE spectra are nearly white. The Kelvin and mixed Rossby–gravity waves are expected to follow k−1 and k−5 power laws, respectively. However, the linear IG modes include also ageostrophic circulation, which is coupled with the divergence and expected to follow a k−1 power law, and the level to which the internal gravity waves make the observed k−5/3 power law of the horizontal kinetic energy depends on the latitude (season), altitude and, maybe most of all, model characteristics.

The derived VKE spectra for August 2018 ERA5 data approximately follow the expected −5 slope for the Rossby modes up to k ≈ 150 depending on the latitude belt, altitude, and season. The MRG wave VKE spectra are similar to the Rossby spectra. The VKE spectra associated with the IG modes reveal a synoptic-scale VKE peak in the winter hemisphere (SH August) associated with quasigeostrophic ageostrophic motions. The peak moves to planetary scales going from the upper troposphere toward the upper stratosphere due to a strong attenuation of the Rossby waves with altitude (Charney and Drazin 1961). A part of the midlatitude winter stratosphere VKE spectra between k ≈ 10 and k ≈ 60 follows the “true” gravity wave spectrum with a power law of k1/3. The spectrum becomes shallower near the tropopause and takes slopes from −1 to −2/3 near 200 hPa level and lower in the troposphere. More shallow IG spectra are found in the NH hemisphere troposphere (summer season) and in the tropics, with slopes closer to −2/3 and −1/3, respectively, for 10 < k < 200. The slopes of VKE spectra are in a qualitative agreement with the HKE spectra that are overall steeper than expected for both Rossby and inertia–gravity wave regime. An exception is the Rossby wave horizontal kinetic energy spectrum in the summer hemisphere tropics that is significantly shallower than −3. The Kelvin waves VKE spectra follow the expected −1 power law implying a significantly smaller amplitude of their vertical velocities compared to the IG modes in the tropics.

The derived 1/3 limit spectra for the VKE in hydrostatic atmosphere suggest that the VKE spectra with slopes 2/3 or even greater in kilometer-scale models (Morfa and Stephan 2023) are either due to nonhydrostatic processes or particular modeling solutions. The latter is supported by similar VKE spectra in the hydrostatic ECMWF model (Polichtchouk et al. 2022) and by an apparent lack of resolution convergence in both hydrostatic and nonhydrostatic models (Skamarock et al. 2014; Polichtchouk et al. 2022). Using the new method, the hydrostatic component of vertical velocity in nonhydrostatic models can be computed to identify nonhydrostatic effects on the VKE spectra and momentum fluxes. This additionally provides a basis for an improved strategy for evaluating kilometer-scale models, i.e., a subject for further research.

Acknowledgments.

This paper is a contribution to the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation), Project 274762653. Ž. Zaplotnik was supported by the Slovenian Research Agency (ARRS), Grant J1-9431 and Program P1-0188. We thank Andreas Dörnbrack, Inna Polichtchouk, and Frank Lunkeit for the discussions, and them and Richard Blender for reading the paper. We are very grateful for the insightful comments by four reviewers.

Data availability statement.

The ERA5 data are available from the Copernicus Programme, via https://climate.copernicus.eu/climate-reanalysis. The default version of the MODES software is available via http://modes.cen.uni-hamburg.de. Outputs of the new decomposition are available on request.

APPENDIX A

Dispersion Relationship for the Rossby and Inertia–Gravity Waves on the Sphere

Computing frequencies of Rossby and gravity waves on the sphere amounts to finding eigenvalues of a symmetric pentadiagonal matrix (Swarztrauber 1984). Solving this problem in full generality is beyond reach of currently available analytical methods. Nevertheless, useful scaling laws and asymptotics can be obtained numerically.

To this end we compute frequencies of the waves for equivalent depths representative of Earth’s atmosphere using MODES software at T170 resolution and compare them to the limit γ → ∞ frequencies (22) and (23) for the Rossby and gravity waves, respectively (Fig. A1). We observe that (22) provides an excellent approximation for IG dispersion relationship on the sphere for all scales and equivalent depths. We omitted plotting the curves for the west-propagating IG modes in the right panel of Fig. A1 as for n > 0 they are nearly identical to those of EIG with exception of sign reversal.

Fig. A1.
Fig. A1.

Dispersion curves of the (left) Rossby and (right) inertia–gravity modes on the sphere for different equivalent depths and meridional modes compared to their respective γ → ∞ limit curves (23) and (22). Dotted lines in the left panel correspond to the fitted Rossby dispersion curves (A1).

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The picture is more complicated for the Rossby modes. First, limit D → ∞ Rossby–Haurwitz (RH) frequencies (23) match those of barotropic Rossby waves (D ≈ 10 km) outside of large-scale region (n < 5, k ≤ 5). Second, these limit RH dispersion curves approximate well those of small-scale Rossby waves (κ → ∞) regardless of equivalent depth. The accuracy of approximation improves with increase of equivalent depth and decrease of horizontal scale so that for D > 200 m the match is good throughout atmospheric mesoscales.

More precise Rossby dispersion relationship than (23) can be obtained by fitting the frequencies to the ansatz
νR(k;n,D)=−ka(n,D)k2+b(n,D)k+c(n,D),
where the fit parameters a, b, c ≥ 0. The dotted lines in Fig. A1 show the fit, which is excellent for all scales and equivalent depths. Then, taking the limits k → ∞ and k → 0 yields
lim⁡k→∞ν˜R(k)k=−1/a
and
lim⁡k→0ν˜R(k)≈−k/c
for small- and planetary-scale Rossby waves, respectively. These asymptotics hold not only for the fitted, but also for the actual frequencies as the left panel of Fig. A2 demonstrates.
Fig. A2.
Fig. A2.

Dependence of nondimensional Rossby mode frequencies on the sphere on the (left) zonal and (right) meridional scales for D = 10 km, 200 m, and 10 m.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The dependence on the meridional scale is markedly different from the ones expected from both (23) and β-plane dispersion relationships with no universal limits at both ends of the spectrum (Fig. A2, right panel). For small meridional scales, the slopes fall into the [−2, −1] range, depending on zonal wavenumber and equivalent depth, with large D producing n−2 curves. The slope gets progressively shallower as equivalent depth and zonal length scale decrease. Less can be concluded for the planetary scales, where the asymptotics is the function of k and D, with nonmonotone dependence on both.

APPENDIX B

Derivation of the Limit Spectra for the Vertical Kinetic Energy

For an arbitrary set of modes S, the scale-dependent energy of horizontal motions per unit mass can be written as
ES,k(φ,p)=∑(m,n)∈SgDm|Gm(p)χnk(m)Θnk(φ;m)|2,
where we assume k > 0. Noting that Hough and vertical structure functions are bounded, we obtain
ES,k(φ,p)≤CGCΘ[∑(m,n)∈SgDm|χnk(m)|2]=CGCΘIS,
where CΘ=max⁡|Θnk(φ;m)|2 and CG=max⁡|Gm(p)|2.
Similar argument applied to the energy integrated in the latitude belt (φ1, φ2) yields
E¯S,k(p;ϕ1,ϕ2 )=∑(m,n)∈SgDm|Gm(p)χnk(m)∫ϕ1ϕ2Θnk(ϕ;m)d(sin⁡ϕ)|2≤CG[∑mgDm|∑nχnk(m)∫ϕ1 ϕ2 Θnk(ϕ;m)d(sin⁡ϕ) |2 ]≤CG{∑mgDm[∑n|χnk(m)|2 ]×[∑n|∫ϕ1 ϕ2 Θnk (ϕ;m)d(sin⁡ϕ)|2 ]}≤CG|sin⁡ϕ2−sin⁡ϕ1|{∑mgDm[∑n|χnk(m)|2 ]×[∑n∫ϕ1 ϕ2 |Θnk(ϕ;m)|2d(sin⁡ϕ)  ]}≤CG|sin⁡ϕ2−sin⁡ϕ1 |RIS,
where the second and third inequalities arise from an appropriate variant of Cauchy–Schwarz, while the last one is due to L2 orthonormality of Hough functions.
In fact, for energy spectra of reanalysis data
EG,k∝E¯G,k∝IG∝k−5/3andER,k∝E¯R,k∝IR∝k−3;
however, proving it mathematically is difficult and weaker statements (B2) and (B3) are sufficient for our purpose.
To obtain the limit spectra for VKE, we remark that for arbitrary mode m,
|∫0pGm(p′)dp′|2≤∫0p|Gm(p′)|2dp′∫0pdp′≤ps2,
where the first step follows from Hölder inequality and the last is the consequence of (10). Then, inserting (39) into (40), making use of (B5) while noting that
|Znk(φ;m)|≤|Θnk(φ;m)|
yields
ES,k(φ,p)=|∑(m,n)∈Sνnk(m)χnk(m)Znk(φ;m)∫0pGm(p′)dp′|2≤ps2∑mgDm|∑nνnk(m)χnk(m)Znk(φ;m)|2≤ps2CΘR∑mgDm[∑n|νnk(m)|2][∑n|χnk(m)|2]≤C1νS2IS,
where the second-to-last inequality again follows from Cauchy–Schwarz, νS2=∑(m,n)∈S|νnk(m)|2 and C1=ps2CΘR. Even though the derivation involves inequalities, Fig. B1 demonstrates that in the data the spectral slopes of all expressions in (B7) are not largely different.
Fig. B1.
Fig. B1.

Vertical kinetic energy (VKE) for a random date in August 2018 along latitude circle 15°N at 171 hPa level. The Rossby (denoted R) and non-Rossby (denoted nR) VKE (full lines) is compared to three other quadratic quantities involved in the derivation of the limit VKE spectra (B7). All dashed curves are scaled so that their values at the zonal wavenumber k = 1 match that of the VKE. The constant parameter c is thus different for each curve.

Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0090.1

The argument for the band-averaged VKE spectra proceeds analogously to (B3) and yields
E¯S,k(p;φ1,φ2)≤|sin⁡φ2−sin⁡φ1|Rps2νS2IS.
Now, making use of the dispersion relationships for the Rossby and gravity waves on the plane or on sphere and of the power laws IG ∝ k−5/3 and IR ∝ k−3 in (B7) and (B8), the bounds
EG,E¯G≤C2k1/3andER,E¯R≤C3k−5
arise consistent with (45), where the constants C2 and C3 dependent on proportionality constants in the power laws for IG and IR, respectively.

Vertical kinetic energy and divergence spectra

Using (35), VKE could also be computed from the divergence spectra PS,δk. Writing
δnk(λ,φ;m)=gDm∇⋅[χnk(m)Unk(φ)eikλ,iχnk(m)Vnk(φ)eikλ]T
for the divergence in the individual Hough mode, then substituting (B10) into (35) yields
ω^S,k=−∑(m,n)∈S∫0pGm(p′)dp′gDmχnk(m)δ^nk(m).
Now the spectra for ES,k can be estimated from (B5) as
ES,k≤ps2|∑(m,n)∈Sδ^nk(m)|2=ps2PS,δk.
On the other hand, the limiting divergence power spectra is easily computed as
PS,δk≤CDk2IS,
where CD=(2/a2)max⁡({|Unk(φ;m)|2+|∂/∂ϕ[Vnk(φ;m)cos⁡φ]|2}/cos⁡φ), whereby
ER,k≤C4k−1andEG,k≤C5k1/3,
with constants C4 and C5 again dependent on the proportionality in the power law for IR and IG, respectively.

Comparing (B9) with (B14), we observe that the former approach provides a much sharper bound on the Rossby VKE spectral slope than estimating it from divergence. This is because the dispersion relation for Rossby waves involved in (B7) is for the barotropic Rossby–Haurwitz waves with divergence due to the beta term. Quasigeostrophic turbulence with IS ∝ k−3 involves ageostrophic motions that cause divergence but in linear decomposition project on the IG modes. On the other hand, both approaches yield identical limit IG spectra.

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  • Žagar, N., J. Tribbia, J. L. Anderson, and K. Raeder, 2009b: Uncertainties of estimates of inertia–gravity energy in the atmosphere. Part II: Large-scale equatorial waves. Mon. Wea. Rev., 137, 3858–3873, https://doi.org/10.1175/2009MWR2816.1; Corrigendum, 138, 2476–2477, https://doi.org/10.1175/2010MWR3256.1.

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  • Žagar, N., A. Kasahara, K. Terasaki, J. Tribbia, and H. Tanaka, 2015: Normal-mode function representation of global 3D data sets: Open-access software for the atmospheric research community. Geosci. Model Dev., 8, 1169–1195, https://doi.org/10.5194/gmd-8-1169-2015.

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  • Žagar, N., D. Jelić, M. Blaauw, and P. Bechtold, 2017: Energy spectra and inertia–gravity waves in global analyses. J. Atmos. Sci., 74, 2447–2466, https://doi.org/10.1175/JAS-D-16-0341.1.

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  • Žagar, N., F. Lunkeit, F. Sielmann, and W. Xiao, 2022: Three-dimensional structure of the equatorial Kelvin wave: Vertical structure functions, equivalent depths, and frequency and wavenumber spectra. J. Climate, 35, 2209–2230, https://doi.org/10.1175/JCLI-D-21-0342.1.

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    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Žagar, N., F. Lunkeit, F. Sielmann, and W. Xiao, 2022: Three-dimensional structure of the equatorial Kelvin wave: Vertical structure functions, equivalent depths, and frequency and wavenumber spectra. J. Climate, 35, 2209–2230, https://doi.org/10.1175/JCLI-D-21-0342.1.

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

    Comparison of the pressure vertical velocity ω from ERA5 (red lines) and ω reconstructed by MODES (blue lines) at pressure level 75.2 hPa, along 60°S at 1200 UTC 11 Aug 2018. ERA5 ω is decomposed using the 1D FFT in Python. (a) Power spectra; (b) ω(λ).

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

    Pressure vertical velocity ω at level near 75 hPa at 1200 UTC 11 Aug 2018. (a) ω from ERA5 data, (b) ω derived by MODES that is further split into (c) Rossby modes (ROT), (d) inertia–gravity (IG) modes, (e) eastward IG (EIG), and (f) westward IG (WIG). The IG modes in (d)–(f) are filtered for k > 7. Small-scale features of EIG and WIG modes sum up to zero over the Andes between 20° and 35°S. The contours in (c) are the horizontal wind speed (in m s−1). The Rossby mode ω is multiplied by a factor of 100.

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

    As in Fig. 2, but the vertical cross section along 70°S across the Antarctic peninsula.

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

    The horizontal kinetic energy (HKE) spectra averaged over latitude belts and altitude bands for the ERA5 data in August 2018. Dashed lines show theoretical slopes with power laws −3, −5/3, −1, and 0, with 0 denoting the white spectrum. Dotted lines are empirical fit for average energy spectrum between 200 and 400 hPa for zonal wavenumbers k = 20–80. (left) Total HKE; (center) the Rossby part, EHR; (right) the non-Rossby part, EHG. The y axis for the left and center columns is the same.

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

    Kinetic energy spectra of vertical motions (VKE) per unit mass in the midlatitude belt between (a),(c) 30° and 60°S and (b),(d) 30° and 60°N, averaged for August 2018. (a),(b) Rossby mode VKE and (c),(d) IG modes VKE. The bottom spectrum (thick blue line) in each panel is the average over levels between 10 and 30 hPa, and the spectra above belong to layers lower in the atmosphere as defined in the legend. Black lines show power laws discussed in section 2.

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

    As in Fig. 5, but for the latitude belt (a),(c) 60°–80°S and (b),(d) 60°–80°N.

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

    As in Fig. 5, but for the latitude belt (a),(c) 10°–30°S and (b),(d) 10°–30°N.

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

    As in Fig. 5, but for the tropical belt between 10°S and 10°N. (a) non-Rossby (IG, MRG, and Kelvin) modes, (b) Rossby modes, (c) IG modes, (d) EIG, and WIG modes.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 8, but for the (a) Kelvin waves and (b) MRG waves.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    Dispersion curves of the (left) Rossby and (right) inertia–gravity modes on the sphere for different equivalent depths and meridional modes compared to their respective γ → ∞ limit curves (23) and (22). Dotted lines in the left panel correspond to the fitted Rossby dispersion curves (A1).

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    Dependence of nondimensional Rossby mode frequencies on the sphere on the (left) zonal and (right) meridional scales for D = 10 km, 200 m, and 10 m.

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    Vertical kinetic energy (VKE) for a random date in August 2018 along latitude circle 15°N at 171 hPa level. The Rossby (denoted R) and non-Rossby (denoted nR) VKE (full lines) is compared to three other quadratic quantities involved in the derivation of the limit VKE spectra (B7). All dashed curves are scaled so that their values at the zonal wavenumber k = 1 match that of the VKE. The constant parameter c is thus different for each curve.

Fig. 1.

Comparison of the pressure vertical velocity ω from ERA5 (red lines) and ω reconstructed by MODES (blue lines) at pressure level 75.2 hPa, along 60°S at 1200 UTC 11 Aug 2018. ERA5 ω is decomposed using the 1D FFT in Python. (a) Power spectra; (b) ω(λ).
Fig. 2.

Pressure vertical velocity ω at level near 75 hPa at 1200 UTC 11 Aug 2018. (a) ω from ERA5 data, (b) ω derived by MODES that is further split into (c) Rossby modes (ROT), (d) inertia–gravity (IG) modes, (e) eastward IG (EIG), and (f) westward IG (WIG). The IG modes in (d)–(f) are filtered for k > 7. Small-scale features of EIG and WIG modes sum up to zero over the Andes between 20° and 35°S. The contours in (c) are the horizontal wind speed (in m s−1). The Rossby mode ω is multiplied by a factor of 100.
Fig. 3.

As in Fig. 2, but the vertical cross section along 70°S across the Antarctic peninsula.
Fig. 4.

The horizontal kinetic energy (HKE) spectra averaged over latitude belts and altitude bands for the ERA5 data in August 2018. Dashed lines show theoretical slopes with power laws −3, −5/3, −1, and 0, with 0 denoting the white spectrum. Dotted lines are empirical fit for average energy spectrum between 200 and 400 hPa for zonal wavenumbers k = 20–80. (left) Total HKE; (center) the Rossby part, EHR; (right) the non-Rossby part, EHG. The y axis for the left and center columns is the same.
Fig. 5.

Kinetic energy spectra of vertical motions (VKE) per unit mass in the midlatitude belt between (a),(c) 30° and 60°S and (b),(d) 30° and 60°N, averaged for August 2018. (a),(b) Rossby mode VKE and (c),(d) IG modes VKE. The bottom spectrum (thick blue line) in each panel is the average over levels between 10 and 30 hPa, and the spectra above belong to layers lower in the atmosphere as defined in the legend. Black lines show power laws discussed in section 2.
Fig. 6.

As in Fig. 5, but for the latitude belt (a),(c) 60°–80°S and (b),(d) 60°–80°N.
Fig. 7.

As in Fig. 5, but for the latitude belt (a),(c) 10°–30°S and (b),(d) 10°–30°N.
Fig. 8.

As in Fig. 5, but for the tropical belt between 10°S and 10°N. (a) non-Rossby (IG, MRG, and Kelvin) modes, (b) Rossby modes, (c) IG modes, (d) EIG, and WIG modes.
Fig. 9.

As in Fig. 8, but for the (a) Kelvin waves and (b) MRG waves.
Fig. A1.

Dispersion curves of the (left) Rossby and (right) inertia–gravity modes on the sphere for different equivalent depths and meridional modes compared to their respective γ → ∞ limit curves (23) and (22). Dotted lines in the left panel correspond to the fitted Rossby dispersion curves (A1).
Fig. A2.

Dependence of nondimensional Rossby mode frequencies on the sphere on the (left) zonal and (right) meridional scales for D = 10 km, 200 m, and 10 m.
Fig. B1.

Vertical kinetic energy (VKE) for a random date in August 2018 along latitude circle 15°N at 171 hPa level. The Rossby (denoted R) and non-Rossby (denoted nR) VKE (full lines) is compared to three other quadratic quantities involved in the derivation of the limit VKE spectra (B7). All dashed curves are scaled so that their values at the zonal wavenumber k = 1 match that of the VKE. The constant parameter c is thus different for each curve.
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