Effects of Dispersion on Vortical and Wave Modes in 3D Rotating Stratified Flows: Random Large Scale Forcing

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical (geostrophic) and wave (ageostrophic) modes of a three-dimensional (3D) rotating stratified fluid as a function of dispersion, i.e. of ǫ = f/N , where f,N are the Coriolis parameter and Brunt-Vaisala frequency. Throughout we employ a random large scale forcing in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are roughly comparable and much less than unity. This inquiry is motivated by recent analytical work on the change in character of vortical-wave mode dynamics with dispersion, and especially the asymmetry about ǫ = 1. As with the non-dispersive reference state of ǫ = 1, for ǫ < 1 the wave mode energy saturates quite quickly and the ensuing forward cascade continues to act as an efficient means of dissipating ageostrophic energy. Further, these saturated spectra steepen as ǫ decreases: we see a shift from k to k−5/3 scaling for kf < k < kd (where kf and kd are the forcing and dissipation scales, respectively). On the other hand, when ǫ > 1 the wave mode energy never saturates and comes to dominate the total energy in the system. With regard to the vortical modes, for ǫ ≤ 1, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a k scaling for kf < k < kd and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all k < kf . In contrast, for ǫ > 1 and increasing, the vortical modes contain a progressively smaller fraction of the total energy indicating that the 3D quasigeostrophic subsystem is disappearing. The total energy for k > kf and ǫ ≤ 1 shows a transition as k increases wherein the vortical modes contain a large portion of the energy at large scales, while the wave modes dominate at smaller scales. Indeed, there is no such transition when ǫ > 1 and the wave modes dominate the total energy for all k > kf .