The Horizontal Spectrum of Vertical Velocities near the Tropopause from Global to Gravity Wave Scales

Vertical motions are fundamental for atmospheric dynamics. Compared to horizontal motions, the horizontal spectrum of vertical velocity w is less well known. Here, w spectra are related to spectra of horizontal motions in the free atmosphere near the tropopause from global to gravity wave scales. At large scales, w is related to vertically averaged horizontal divergent motions by continuity. At small scales, the velocity energy spectra reach anisotropy as in stably stratified turbulence. Combining these limits approximates the w spectrum from global to small scales. The w spectrum is flat at large scales when the divergent spectrum shows a -2 slope, reaches a maximum at mesoscales after transition to -5/3 slopes, and then approaches a fraction of horizontal kinetic energy. The ratio of vertical kinetic energy to potential energy increases quadratically with wavenumber at large scales. It exceeds unity at small scales in stratified turbulence. Global and regional simulations and two recent aircraft measurement field campaigns support these relationships within 30% deviations. Energy exchange between horizontal and vertical motions may contribute to slope changes in the spectra. The model allows for checking measurement validity. Isotropy at large and small scales varies between the datasets. The fraction of divergent energy is 40%-70% in the measurements, with higher values in the stratosphere. Spectra above the tropopause are often steeper over mountains than over oceans, partly with two -5/3 subranges. A total of 80% of w variance near the tropopause occurs at scales between about 0.5 and 80 km.