Mesoscale Spectra of Vertical Vorticity and Horizontal Divergence in an Idealized Baroclinic Wave Simulation

Abstract A new formulation of the spectral budget of vertical vorticity and horizontal divergence suitable for the mesoscale atmosphere on an f plane is derived. Compared to previous formulations in large-scale studies, there are three main improvements: (i) both the squared vorticity (SV; i.e., enstrophy as usual) and squared divergence (SD) spectra are taken into account, (ii) the spectral transfers of SV and SD between scales are exactly constructed under the nonlinear advection of the full horizontal velocity, and (iii) the general relationship between spectral energy and SV/SD transfers is derived. With this new formulation, the atmospheric spectra of divergent and rotational motion components are investigated through numerical simulation of idealized dry baroclinic waves. Spectral budget analysis shows that, in the present dry simulation, the upper troposphere is almost completely dominated by the downscale SV transfer at all scales, while the lower stratosphere is dominated by the downscale SV transfer at synoptic scales and by the downscale SD transfer at mesoscales. The pressure-related term is largely cancelled out by the conversion term between SV and SD at both levels, but at the small-scale end of lower-stratospheric mesoscales there exists a significant net positive forcing, accounting for the distinct spectral transition of the total spectrum there. An explicit association between spectral energy and SV/SD transfers is further made. In the upper troposphere, the downscale energy cascade is mainly governed by the downscale SV transfer, while in the lower stratosphere, it is mainly governed by the residual term related to nonuniformly distributed vertical velocity. Significance Statement The purpose of this study is to explore the dynamics underlying the atmospheric spectra of divergent and rotational motion components. The traditional analysis of enstrophy is first extended to include both squared vertical vorticity (SV) as usual and squared horizontal divergence (SD), and then a new formulation of the spectral SV and SD budget suitable for the mesoscale atmosphere is derived, with application to the dry baroclinic waves simulation. Our results clearly reveal the different physical processes governing the vorticity and divergence spectra at different heights. We also derive the general relationship between spectral energy and SV/SD transfers, which allows explicitly associating spectral energy and SV/SD fluxes and thus provides additional physical views on the mesoscale energy cascade. Further work should consider the effects of other physical processes neglected here.