How white is the sky?

The energy spectrum of atmospheric horizontal motions has been extensively studied in observations and numerical simulations. Its canonical shape includes a transition from the -3 power law at synoptic scale to -5/3 power law at mesoscale. The transition is taking place at scales around 500 km that can be seen as the scale where energy associated with quasi-linear inertia-gravity waves exceeds the balanced (or Rossby wave) energy. In contrast to the horizontal spectrum, the spectrum of kinetic energy of vertical motions is poorly known since the vertical motion is not an observed quantity of the global observing system and vertical kinetic energy spectra from non-hydrostatic models are difficult to validate.

Traditionally, vertical velocities associated with the Rossby and gravity waves have been treated separately using the quasi-geostrophic omega equations and polarization relations for the stratified Boussinesq fluid in the (x,z) plane, respectively. In the tropics, the Rossby and gravity  wave regimes are difficult to separate and their frequency gap, present in the extra-tropics, is filled with the Kelvin and mixed Rossby-gravity waves. A separate treatment of the Rossby and gravity wave regimes makes it challenging to quantify energies of their vertical motions and vertical momentum fluxes. A unified treatment and wave interactions is performed by high-resolution non-hydrostatic models but their understanding requires the toolkit of theory. 

This contribution presents a unified framework for the derivation of vertical velocities of the Rossby and inertia-gravity waves and associated kinetic energy spectra. Expressions for the Rossby and gravity wave vertical velocities are derived using the normal-mode framework in the hydrostatic atmosphere that can be considered applicable up to the scale around 10 km. The derivation involves the analytical evaluation of divergence of the horizontal wind associated with the Rossby and inertia-gravity eigensolutions of the linearized primitive equations. The new framework is applied to the global analysis data of the ECMWF system. Results confirm that the tropical vertical kinetic energy spectra associated with inertia-gravity waves are on average indeed white. Deviations from the white spectrum are discussed for latitude and altitude bands.