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 ǫ = f/N , where f is the Coriolis parameter and N is the 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. Working in regimes characterized by moderate Burger numbers, i.e. Bu = 1/ǫ < 1 or Bu ≥ 1, our results indicate profound change in the character of vortical and wave mode interactions with respect to Bu = 1. Previous analytical work concerning the qualitatively different nature of these interactions has been in limiting conditions of rotation or stratification domination (i.e. when Bu ≪ 1 or Bu ≫ 1 respectively). As with the 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. In fact, in a sense the wave modes behave in an asymmetric manner about ǫ = 1. 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 plays an energetically smaller role in the overall dynamics. Combining the vortical and wave modes, 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. There is no such transition when ǫ > 1 and the wave modes dominate the total energy for all k > kf .