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We have derived the limit spectra of the kinetic energy of vertical motions in the hydrostatic atmosphere and applied it to reanalysis data. This project has been ongoing since 2018, and it is since 2020 funded by the DFG within the TRR181 “Energy transfers in the atmosphere and ocean” project.
Within the project, we have extended the linear normal-mode function (NMF) theory to the unified computation of vertical velocities associated with the Rossby waves and inertia-gravity (IG) waves, and their zonal wavenumber kinetic energy spectra (i.e. vertical kinetic energy, or VKE, spectra). In the same framework, we have developed 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). The decomposition of both horizontal and vertical motions in terms of the Rossby and gravity waves within the same framework provides a consistent comparison of the zonal wavenumber horizontal and vertical energy spectra for the two regimes.
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 (Table 1).
The latitude-by-latitude and level-by-level HKE and VKE spectra applied to the ERA5 reanalysis data expose major latitudinal and seasonal variations i.e., anisotropy of both the horizontal and vertical energy distribution. The ERA5 data does not have an adequate resolution or representation of gravity waves. Therefore, the theoretical k1/3 power law for the IG VKE spectrum is found only in the 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 quasi-geostrophic ageostrophic motions is seen as an IG VKE peak at synoptic scales in the upper troposphere which gradually moves to planetary scales in the stratosphere.
The regime decomposition of vertical velocity paves the way for the decomposition of the vertical momentum fluxes due to the Rossby waves, IG waves, the Kelvin and mixed Rossby-gravity waves across scales that are currently being completed. For more details on this, contact Valentino Neduhal.
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 (E) as a function of the zonal wavenumber k. The wavenumber index k is used for the large scales tropical flows where both k and meridional mode index n are small.
Figure 1: (a)-(b) The horizontal and (c)-(d) vertical kinetic energy of (a), (c) Rossby and (b), (d) non-Rossby motions in the latitude belt 30°S-60°S in different layers, as indicated in the legend. Spectra are averaged for the ERA5 data in August 2018. Note that non-Rossby modes consist only of the IG modes in the extratropics. Adapted from https://doi.org/10.1175/JAS-D-23-0090.1
Žagar, N., V. Neduhal, S. Vasylkevych, Ž. Zaplotnik, and H. L. Tanaka, 2023: Decomposition of Vertical Velocity and Its Zonal Wavenumber Kinetic Energy Spectra in the Hydrostatic Atmosphere. J. Atmos. Sci., 80, 2747–2767, https://doi.org/10.1175/JAS-D-23-0090.1.