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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 ∝ k 1/3 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 k 1/3 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.

Worldwide tsunamis driven by atmospheric waves—or planetary meteotsunami waves—are extremely rare events. They mostly occur during supervolcano explosions or asteroid impacts capable to generate atmospheric acoustic-gravity waves including the Lamb waves that can circle the globe multiple times. Recently, such ocean waves have been globally recorded after the Hunga Tonga–Hunga Ha‘apai volcano eruption on 15 January 2022, but did not pose any serious danger to the coastal communities. However, this study highlights that the mostly ignored destructive potential of planetary meteotsunami waves can be compared to the well-studied tsunami hazards. In practice, several process-oriented numerical experiments are designed to force a global ocean model with the realistic atmospheric response to the Hunga Tonga–Hunga Ha‘apai event rescaled in speed and amplitude. These simulations demonstrate that the meteotsunami surges can be higher than 1 m (and up to 10 m) along more than 7% of the world coastlines. Planetary meteotsunami waves thus have the potential to cause serious coastal damages and even human casualties during volcanic explosions or asteroid impacts either releasing intense acoustic energy or producing internal atmospheric gravity waves triggering the deep-ocean Proudman resonance at a speed of ∼212 m s −1 . Based on records of catastrophic events in Earth’s history, both scenarios are found to be realistic, and consequently, the global meteotsunami hazards should now be properly assessed to prepare for the next big volcanic eruption or asteroid impact even occurring inland.

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L 1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic L p theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.

The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.

Equatorial Kelvin waves can be affected by subtropical Rossby wave dynamics. Previous research has demonstrated the Kelvin wave growth in response to subtropical forcing and the resonant growth due to eddy momentum flux convergence. However, the relative importance of the wave-mean flow and wave-wave interactions for the Kelvin wave growth compared to the direct wave excitation by the external forcing has not been made clear. This study demonstrates the resonant Kelvin wave excitation by interactions of subtropical Rossby waves and the mean flow using a spherical shallow-water model. The use of Hough harmonics as basis functions makes Rossby and Kelvin waves prognostic variables of the model and allows the quantification of terms contributing to their tendencies in physical and wave space. The simulations show that Kelvin waves are resonantly excited by interactions of Rossby waves and the balanced zonal mean flow in the subtropics, provided the Rossby and Kelvin wave frequencies, which are modified by the mean flow, match. The resonance mechanism is substantiated by analytical expressions. The Kelvin wave tendencies are caused by velocity and depth tendencies: The velocity tendencies due to the meridional advection of zonal mean velocity can be outweighed by the zonal advection of Rossby wave velocity or by the depth tendencies due to Rossby wave divergence. Identifying the resonant excitation mechanism in data should contribute to the quantification of Kelvin wave variability originating in the subtropics.

The equatorial mixed Rossby-gravity wave (MRGW) is an important contributor to tropical variability. Its excitation mechanism capable of explaining the observed MRGW variance peak at synoptic scales remains elusive. This study investigates wave-mean flow interactions as a generation process for the MRGWs using the barotropic version of the global Transient Inertia-Gravity And Rossby wave dynamics model (TIGAR), which employs Hough harmonics as the basis of spectral expansion, thereby representing MRGWs as prognostic variables. High accuracy numerical simulations manifest that interactions between waves emanating from a tropical heat source and zonal mean jets in the subtropics generate MRGWs with the variance spectra resembling the one observed in the tropical troposphere. Quantification of spectral tendencies associated with the MRGW energy growth underscores the significance of wave-mean flow interactions in comparison to excitation mechanisms driven by external forcing and wave-wave interactions. The MRGW growth and amplitude depend on the asymmetry in the zonal mean flow that may explain not only seasonal variability but also differences between the troposphere and the middle atmosphere.

In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler-Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler-Poincar\\'{e} equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler-Boussinesq-\\(\\alpha\\) and primitive equations-\\(\\alpha\\) models, however feature a different regularizing second order operator.

Generalized Lagrangian mean theories are used to analyze the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste (J. Fluid Mech., 2018) who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphism or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian center of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-\\(\\alpha\\) equations arise as Lagrangian averaged Euler equations when using the \\(L^2\\)-geodesic mean on the volume preserving diffeomorphism group of a manifold without boundaries, imposing a `Taylor hypothesis', which states that first order fluctuations are transported as a vector field by the mean flow, and assuming that fluctuations are statistically isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. These results generalize an earlier observation by Oliver (Proc. R. Soc. A, 2017) to manifolds in geometrically fully intrinsic terms.

We report the existence of a new regime for domain wall motion in uniaxial and near-uniaxial ferromagnetic nanowires, characterised by applied magnetic fields sufficiently strong that one of the domains becomes unstable. There appears a new stable solution of the Landau-Lifshitz-Gilbert equation, describing a nonplanar domain wall moving with constant velocity and precessing with constant frequency. Even in the presence of thermal noise, the new solution can propagate for distances on the order of 500 times the field-free domain wall width before fluctuations in the unstable domain become appreciable.