Inertia-gravity Waves Research Articles

Numerical solution of the boundary value problem for inertia-gravity internal waves

This paper presents a numerical calculation of the boundary value problem for the equation of free internal inertia-gravity waves in an unbounded basin of constant depth in the Boussinesq approximation and the presence of a two-dimensional vertically inhomogeneous flow. The boundary value problem for the vertical velocity amplitude has complex coefficients and is solved both numerically and by perturbation theory. Using the example of calculating the decrement of attenuation of internal waves and momentum wave flows, it is shown that the exact numerical calculation gives significantly better estimates in comparison with the perturbation method. In particular, at minimum divergence in the dispersion curves for both calculation methods, the imaginary part of the wave frequency, interpreted as the decrement of attenuation, can differ by two or three orders of magnitude. Vertical wave momentum fluxes are comparable to turbulent ones and may exceed them, with results obtained by the numerical method being almost an order of magnitude smaller than those calculated by the perturbation theory method.

Submesoscale Eddies Detected by SWOT and Moored Observations in the Northwestern Pacific

AbstractThe Surface Water and Ocean Topography (SWOT) mission provides a good opportunity to study fine‐scale processes in the global ocean but whether it can detect balanced submesoscale eddies is uncertain due to the “contamination” by unbalanced inertial gravity waves. Here, based on concurrent observations from SWOT and a mooring array in the northwestern Pacific, we successfully captured two submesoscale cyclonic eddies with negative sea level anomalies (SLAs) in spring 2023. We find that the SLA amplitude and equivalent radius of the first (second) eddy are 2.5 cm and 16.0 km (2.0 cm and 18.8 km), respectively. For both eddies, their vertical scales are around 150 m and their horizontal velocities and Rossby numbers exceed 15.0 cm/s and 0.4, respectively. Further analysis suggests that similar submesoscale eddies can commonly occur in the northwestern Pacific and that SWOT is capable to detect larger submesoscale eddies with scales greater than ∼10 km.

A performance study of horizontally explicit vertically implicit (HEVI) time-integrators for non-hydrostatic atmospheric models

We conduct a thorough study of different forms of horizontally explicit and vertically implicit (HEVI) time-integration strategies for the compressible Euler equations on spherical domains typical of nonhydrostatic global atmospheric applications. We compare the computational time and complexity of two nonlinear variants (NHEVI-GMRES and NHEVI-LU) and a linear variant (LHEVI). We report on the performance of these three variants for a number of additive Runge-Kutta methods ranging in order of accuracy from second through fifth, and confirm the expected order of accuracy of the HEVI methods for each time-integrator. To gauge the maximum usable time-step of each HEVI method, we run simulations of a nonhydrostatic baroclinic instability for 100 days and then use this time-step to compare the time-to-solution of each method. The results show that NHEVI-LU is 3x faster than NHEVI-GMRES, and LHEVI is 8x faster than NHEVI-GMRES, for the idealized cases tested. The baroclinic instability and inertia-gravity wave simulations indicate that the optimal choice of HEVI method is LHEVI. For the fastest time-to-solution, the second and third order methods are best although the better accuracy of the high-order methods (particularly the fourth order method) should be considered.In the future, we will report on whether these results hold for more complex problems using, e.g., real atmospheric data and/or a higher model top typical of space weather applications.

A super-convergent quasi-discontinuous Galerkin method for approximating inertia-gravity and Rossby waves in geophysical flows

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P1DG and non conforming P1NC Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P1NC discrete frequencies compared to the P1DG ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P1DG and P1NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P1NC approach.

The Influence of Topography on the Diurnal Rainfall Propagation in the Bay of Bengal

Abstract A significant diurnal offshore propagation of rainfall is observed extending from the eastern coast of India to the central Bay of Bengal. This study focuses on understanding the influence of topography over the Indian subcontinent on this rainfall propagation through a series of semi-idealized mesoscale numerical simulations. These simulations with varying topography highlight the crucial role of inertia–gravity waves, driven by diurnal mountain–land–sea thermal contrast between India and the Bay of Bengal, in initiating and promoting the offshore propagation of convective systems in the Bay. These waves’ phase speed of around 14.8 m s−1 aligns well with the speed of diurnal rainfall propagation. Even after eliminating the impact of Indian topography, the offshore propagating signal persists, suggesting a secondary rather than dominant effect of terrain on offshore rainfall propagation. Furthermore, the topography affects the depth of diurnal heating within the land’s boundary layer, which thus influences the amplitude, phase, and speed of the inertia–gravity waves. Specifically, the presence of higher mountains along the coastal area drives faster waves by increasing heating depth, resulting in faster rainfall propagation. Significance Statement This study advances our comprehension of the fundamental driver behind diurnal offshore rainfall propagation and the manner in which coastal terrain influences the rainfall pattern. We demonstrate that the diurnal offshore propagation of rainfall is closely related to inertia–gravity waves generated by thermal contrasts, and we successfully distinguish these waves in our study. Furthermore, our findings indicate that elevated coastal topography contributes to a greater heating depth near the coastline, which plays a crucial role in driving faster gravity waves and, consequently, leading to faster rainfall propagation. These outcomes provide a deeper insight into the mechanism governing offshore rainfall propagation and underscore the impact of real-world topography.

Topological Signature of Stratospheric Poincaré-Gravity Waves

Abstract The rotation of Earth breaks time-reversal and reflection symmetries in an opposite sense north and south of the equator, leading to a topological origin for certain atmospheric and oceanic equatorial waves. Away from the equator, the rotating shallow-water and stably stratified primitive equations exhibit Poincaré inertia–gravity waves that have nontrivial topology as evidenced by their strict superinertial time scale and a phase singularity in frequency–wavevector space. This nontrivial topology then predicts, via the principle of bulk-interface correspondence, the existence of two equatorial waves along the equatorial interface, the Kelvin and Yanai waves. To directly test the nontrivial topology of Poincaré-gravity waves in observations, we examine ERA5 data and study cross correlations between the wind velocity and geopotential height of the midlatitude stratosphere at the 50 hPa height. We find the predicted vortex and antivortex in the relative phase of the geopotential height and velocity at the high frequencies of the waves. By contrast, lower-frequency planetary waves are found to have trivial topology also as expected from theory. These results demonstrate a new way to understand stratospheric waves and provide a new qualitative tool to investigate waves in other components of the climate system.

The high-frequency and rare events barriers to neural closures of atmospheric dynamics

Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.

The Relationship between Convectively Coupled Waves and the East Pacific ITCZ

Abstract Longstanding climate model biases in tropical precipitation exist over the east Pacific (EP) Ocean, especially during boreal winter and spring when models have excessive Southern Hemisphere (SH) precipitation near the intertropical convergence zone (ITCZ). In this study, we document the impact of convectively coupled waves (CCWs) on EP precipitation and the ITCZ using observations and reanalyses. We focus on the months when SH precipitation peaks in observations: February–April (FMA). CCWs explain 93% of total precipitation variance in the SH, nearly double the percent (48%) of the NH during FMA. However, we note that these percentages are inflated as they inevitably include the background variance. We further investigate the three leading high-frequency wave bands: mixed Rossby–gravity waves and tropical depression–type disturbances (MRG–TD type), Kelvin waves, and n = 0 eastward inertia–gravity waves (IG0). Compared to their warm pool counterparts, these three CCWs have a more zonally elongated and meridionally narrower precipitation structure with circulations that resemble past observational studies and/or shallow water theory. We quantify the contribution of all CCWs to four different daily ITCZ “states”: Northern Hemisphere (NH) (nITCZ), SH (sITCZ), double (dITCZ), and equatorial (eITCZ) using a new precipitation-based ITCZ-state algorithm. We find that the percent of total precipitation variance explained by each of the CCWs is heightened for sITCZs and eITCZs and diminished for nITCZs. Last, we find that nITCZs are most prevalent weeks after strong CCW activity happens in the NH, whereas CCWs and sITCZs peak simultaneously in the SH. Significance Statement Convectively coupled atmospheric waves (CCWs) are a critical feature of tropical weather and are an important source of precipitation near the region of highest precipitation on Earth called the intertropical convergence zone (ITCZ). Given three decades of climate model biases in CCWs and ITCZ precipitation over the east Pacific (EP) Ocean during spring, few studies have examined the relationship between CCWs and the springtime EP ITCZ. We explored the CCWs and EP ITCZ relationship through calculations of the percent of precipitation that comes from CCWs. A significant portion of the tropical precipitation is associated with CCWs during spring. CCWs are even more impactful when the ITCZ is in the SH or on the equator, which are both problematic in climate models.

A Linear Theory for Periodic Convectively Forced Gravity Waves near a Coastline

Abstract This study presents a simple 2D linear analytical model aimed at investigating gravity waves forced by temporally periodic convection near a coastline. This investigation encompasses two distinct convective heating scenarios: deep convective heating and stratiform heating/cooling. Our model explores the intricate behavior of gravity waves in proximity to a time-dependent convective source and examines their propagation characteristics across diverse atmospheric conditions. Close to the convective source, gravity waves demonstrate nearly horizontal propagation with vertically aligned phase lines. The velocity of their propagation primarily depends on the vertical scale of the convective heating. The presence of a tropopause further extends their horizontal reach through partial wave ducting between the surface and the tropopause. However, the horizontal scale of the convective heating also plays a crucial role in determining the horizontal wavelength, and consequently, affecting the horizontal propagation speed of the gravity waves. If the heating horizontal scale is small compared to the horizontal scale of free waves at the forcing frequency, the heating vertical scale determines the vertical wavelength and thus the horizontal wavelength. However, if the heating horizontal scale is large, the horizontal wavelength determined by the heating vertical scale has little energy, so that the horizontal wavelength is mainly determined by the heating horizontal scale. Moreover, longer periods of convective heating and stronger background winds contribute to an increased downstream propagation distance of the gravity waves away from the source. Additionally, inertia-gravity waves generated by diurnal convection can propagate horizontally over greater distances at a higher latitude but become confined or trapped at latitudes exceeding 30°.

The Rotational and Divergent Kinetic Energy Spectra of Geostrophic Vortices and Inertia‐Gravity Waves

AbstractThe rotational and divergent kinetic energy (RKE and DKE) spectra of geostrophic vortices (Rossby waves; RWs) and inertia‐gravity waves (IGWs) in the global atmosphere are investigated with ERA5 reanalysis. The separation of RWs and IGWs in physical space is based on the normal‐mode decomposition, and the Helmholtz decomposition produces their RKE and DKE spectra at different layers, with a focus on spherical wavenumbers 10 ≤ l ≤ 100. In the upper troposphere and the middle and lower stratosphere, the RKE spectra of the total mode closely resemble the horizontal kinetic energy (HKE) spectra of RWs over most wavenumbers; the DKE spectra of the total mode are more comparable to the DKE spectra rather than the HKE spectra of IGWs, although their slopes are similar. The HKE of RWs is dominated by its rotational component, accounting for more than 80% of the HKE in most ranges of concern. Although the HKE of IGWs is dominated by its divergent component, its rotation component is also significant, with an average percentage exceeding 25% in all three vertical layers analyzed. Care must be taken when employing divergence as a proxy of IGWs, as the DKE may underestimate the HKE of IGWs. With the increase of altitude and the decrease of scale, the contribution of the divergent component increases in the horizontal circulation of both RWs and IGWs.

Developmental mechanism of the SWCV nonlinear inertial waves

Based on the Boussinosq Approximation formulas in the symmetric, cylindric coordinates, the nonlinear effects of the internal inertial gravity waves on the meso-micro scale convective activities of Southwest China Vortex system are analyzed by the multi-scale and perturbation approximation methods. The results obtain two main conclusions: (1) When the atmospheric stratification is stable, the inertial gravity waves for Southwest China Vortex may also develop into finite-amplitude wave packet with a solitonic characteristics of large amplitude and short duration, forming a long-narrow wave band which activates and organizes severe convective activities like high wind and thunderstorm of Southwest China Vortex. When the atmospheric stratification is unstable, the inertial gravity waves exhibits attenuating oscillation characteristics and develops into finite-amplitude wave packet with large amplitude and fast speed, forming queue-type wave band that activates and organizes extreme weather like persistent heavy rainfall of Southwest China Vortex. (2) Under different atmospheric stratifications, the effects of thermal forcing on the inertial gravity waves for Southwest China Vortex are different. In the stable atmospheric stratification, the thermal forcing mainly intensifies the inertial gravity waves and has no significant effect on its duration. And in the unstable atmospheric stratification, the thermal forcing not only strengthens its growth but also obviously extends its duration. The research has revealed the some nonlinear characteristics of the internal inertial gravity waves for Southwest China Vortex, and improved the theoretical understanding about the critical role of the internal inertial gravity waves dynamic processes and its influence mechanism on the meso-micro scale severe convection weather for Southwest China Vortex.

Identification of stratospheric disturbance information in China based on the round-trip intelligent sounding system

Abstract. Assessing the role of physical processes in the stratosphere under climate change has been one of the hottest topics over the past few decades. However, due to the limitations of detection techniques, stratospheric disturbance information from in situ observations is still relatively scarce. The round-trip intelligent sounding system (RTISS) is a new detection technology, developed in recent years, that can capture atmospheric fine-structure information about the troposphere and stratosphere via three-stage (rising, flat-floating, and falling) detection. Based on the structure function and singular measure relationships, we quantify stratospheric small-scale gravity waves (SGWs) over China, using the Hurst and intermittency parameters, and discuss their relationship with inertia-gravity waves (IGWs). The results show that the enhancement of SGWs in the stratosphere is accompanied by weakening of the IGWs below, which is related to the Kelvin–Helmholtz instability (KHI), and is conducive to the transport of ozone to higher altitudes from lower stratosphere. The parameter space (H1, C1) shows sufficient potential in the analysis of stratospheric disturbances and their role in material transport and energy transfer.

Diurnally Varying Ekman Layer in a Rossby Wave

Abstract Weak but persistent synoptic-scale ascent may play a role in the initiation or maintenance of nocturnal convection over the central United States. An analytical model is used to explore the nocturnal low-level jets (NLLJ) and ascent that develop in an idealized diurnally varying frictional (Ekman) boundary layer in a neutrally stratified barotropic environment when the flow aloft is a zonally propagating Rossby wave. Steady-periodic solutions are obtained of the linearized Reynolds-averaged Boussinesq-approximated equations of motion on a beta plane with an eddy viscosity that is specified to increase abruptly at sunrise and decrease abruptly at sunset. Rayleigh damping terms are used to parameterize momentum loss due to radiation of inertia–gravity waves. The model-predicted vertical velocity is (approximately) proportional to the wavenumber and wave amplitude. There are two main modes of ascent in midlatitudes, an afternoon mode and a nocturnal mode. The latter arises as a gentle but persistent surge induced by the decrease of turbulence at sunset, the same mechanism that triggers inertial oscillations in the Blackadar theory of NLLJs. If the Rayleigh damping terms are omitted, the boundary layer depth becomes infinite at three critical latitudes, and the vertical velocity becomes infinite far above the ground at two of those latitudes. With the damping terms retained, the solution is well behaved. Peak daytime ascent in the model occurs progressively later in the afternoon at more southern locations (in the Northern Hemisphere) until the first (most northern) critical latitude is reached; south of that latitude the nocturnal mode is dominant.

Accuracy and stability analysis of horizontal discretizations used in unstructured grid ocean models

One important tool at our disposal to evaluate the robustness of Global Circulation Models (GCMs) is to understand the horizontal discretization of the dynamical core under a shallow water approximation. Here, we evaluate the accuracy and stability of different methods used in, or adequate for, unstructured ocean models considering shallow water models. Our results show that the schemes have different accuracy capabilities, with the A- (NICAM) and B-grid (FeSOM 2.0) schemes providing at least 1st order accuracy in most operators and time integrated variables, while the two C-grid (ICON and MPAS) schemes display more difficulty in adequately approximating the horizontal dynamics. Moreover, the theory of the inertia-gravity wave representation on regular grids can be extended for our unstructured based schemes, where from least to most accurate we have: A-, B, and C-grid, respectively. Considering only C-grid schemes, the MPAS scheme has shown a more accurate representation of inertia-gravity waves than ICON. In terms of stability, we see that both A- and C-grid MPAS scheme display the best stability properties, but the A-grid scheme relies on artificial diffusion, while the C-grid scheme does not. Alongside, the B-grid and C-grid ICON schemes are within the least stable. Finally, in an effort to understand the effects of potential instabilities in ICON, we note that the full 3D model without a filtering term does not destabilize as it is integrated in time. However, spurious oscillations are responsible for decreasing the kinetic energy of the oceanic currents. Furthermore, an additional decrease of the currents’ turbulent kinetic energy is also observed, creating a spurious mixing, which also plays a role in the strength decrease of these oceanic currents.

Numerical Solution of the Boundary Value Problem for Internal Inertia-Gravity Waves

Numerical Solution of the Boundary Value Problem for Internal Inertia-Gravity Waves

Evolution of weak, homogeneous turbulence with rotation and stratification

This article concerns the time-evolution, spectral structure and scaling of weak turbulence subject to rotation and stable stratification. The flow is expressed as a combination of particular solutions, referred to as modes, of the linearised governing equations without viscosity or diffusion. Modes are of two types: oscillatory ones which represent inertial-gravity waves and time-independent ones that express a non-propagating (NP) component of the flow. The presence of the NP component, which plays an active role in the dynamics apart from in the case of pure rotation, renders wave-turbulence analysis problematic because the NP mode is non-dispersive. Equations are derived for the time evolution of the modal amplitudes, evolution which is due to nonlinearity and visco-diffusion. Subsequent analysis assumes that one or other (or both) of the Rossby and Froude numbers is small (weak turbulence). Given this assumption, the NP component is found to evolve independently of the wave one and a numerical scheme, similar to, though significantly different from classical direct numerical simulation, is used to determine its time evolution. The treatment of the wave component assumes its amplitude is large compared with the NP one, otherwise there are seemingly intractable difficulties of closure in the analysis. Given this further assumption, the wave component decouples from the NP one. Evolution equations for the wave spectra are derived using wave-turbulence analysis and are integrated numerically. As might be expected, these equations indicate that nonlinear coupling of wave modes is dominated by resonances. Results are given for both the NP and wave components.

Tidal Dissipation in Stably Stratified and Semiconvective Regions of Rotating Giant Planets: Incorporating Coriolis Forces

We study how stably stratified or semiconvective layers alter tidal dissipation rates associated with the generation of inertial, gravito-inertial, interfacial, and surface gravity waves in rotating giant planets. We explore scenarios in which stable (nonconvective) layers contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. Our model is an idealized spherical Boussinesq system incorporating Coriolis forces to study effects of stable stratification and semiconvective layers on tidal dissipation. Our detailed numerical calculations consider realistic tidal forcing and compute the resulting viscous and thermal dissipation rates. The presence of an extended stably stratified fluid core significantly enhances tidal wave excitation of both inertial waves (due to rotation) in the convective envelope and gravito-inertial waves in the dilute core. We show that a sufficiently strongly stratified fluid core enhances inertial wave dissipation in a convective envelope much like a solid core does. We demonstrate that efficient tidal dissipation rates (and associated tidal quality factors )—sufficient to explain the observed migration rates of Saturn's moons—are predicted at the frequencies of the orbiting moons due to the excitation of inertial or gravito-inertial waves in our models with stable layers (without requiring resonance locking). Stable layers could also be important for tidal evolution of hot and warm Jupiters and hot Neptunes, providing efficient tidal circularization rates. Future work should study more sophisticated planetary models that also account for magnetism and differential rotation, as well as the interaction of inertial waves with turbulent convection.

Probing signals of atmospheric gravity waves excited by the July 29, 2021 MW8.2 Alaska earthquake

It is commonly believed that the atmosphere is decoupled from the solid Earth. Thus, it is difficult for the seismic wave energy inside the Earth to propagate into the atmosphere, and atmospheric pressure wave signals excited by earthquakes are unlikely to exist in atmospheric observations. An increasing number of studies have shown that earthquakes, volcanoes, and tsunamis can perturb the Earth's atmosphere due to various coupling effects. However, the observations mainly focus on acoustic waves with periods of less than 10 min and inertial gravity waves with periods of greater than 1 h. There are almost no clear observations of gravity waves that coincide with observations of low-frequency signals of the Earth's free oscillation frequency band within 1 h. This paper investigates atmospheric gravity wave signals within 1 h of surface-atmosphere observations using the periodogram method based on seismometer and microbarometer observations from the global seismic network before and after the July 29, 2021 MW8.2 Alaska earthquake in the United States. The numerical results show that the atmospheric gravity wave signals with frequencies similar to those of the Earth's free oscillations 0S2 and 0T2 can be detected in the microbarometer observations. The results confirm the existence of atmospheric gravity waves, indicating that the atmosphere and the solid Earth are not decoupled within this frequency band and that seismic wave energy excited by earthquakes can propagate from the interior of the Earth to the atmosphere and enhance the atmospheric gravity wave signals within 1 h.

Application of balloon-borne high precision GPS for inertia gravity wave characterization

Atmospheric gravity waves have conventionally been studied and characterized using temperature and wind data from radiosondes. In this study we outline and adapt several previously established wave-detection methods for use with solely high-resolution position and velocity measurements from student-constructed GPS units. We consider both vertical and horizontal profiles, utilizing Fourier, hodograph, and polarization analyses, which primarily target low-frequency inertia gravity waves. Using eleven balloon flights over central Montana and New Mexico conducted during 2021–2023 summers by the Balloon Outreach Research Exploration and Landscape Imaging Systems (BOREALIS) program, we show the validity of our procedures in successfully locating inertia gravity waves and finding wave parameters consistent with those of existing studies. A total of eleven waves were found in seven of the flights, with the same wave likely showing up more than once in two flights. The detected waves generally had intrinsic frequencies of one to two times the Coriolis frequency, spanned vertical wavelengths between one and two kilometers, and had periods of between 10 and 15 h.

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

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.