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

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 proportional to k (-5) and a VKE proportional to 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 approximate to 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.