Effects of dispersion on zero and fast mode spectra in three-dimensional rotating stratified flows

Utilizing an eigenfunction decomposition, we study the scaling of energy spectra in the zero and fast modes of a three-dimensional (3D) rotating stratified fluid as a function of dispersivity, i.e. of ε = f/N : where f,N are the Coriolis parameter and Brunt-Vaisala frequency. Throughout we work in a unit aspect ratio domain and set these parameters such that the Froude and Rossby numbers are comparable and < < O(1). In fact it is the added possbility of interscale energy transfer by means of fast-fast-fast (near-)resonant interactions in the presence of dispersivity that motivates this inquiry. Starting from the well understood reference state of ε=1, employing a random large scale forcing, we see that the fast modes always transfer energy to small scales. But their scaling steepens from k^-1 to k^-5/3 for k_f < k < k_d (where k_f,k_d are the forcing and dissipation scales) as ε moves away from unity. On the other hand, the zero modes follow a k^-3 scaling for k_f < k < k_ε and a k^-5/3 form for k _ε < k < k_d with k_ε = k_d when ε differs from unity --- i.e. in the non-dispersive case, instead of being dissipated a fraction of the energy deposited at small scales by the fast modes appears to be transferred upwards into the zero modes. Further, in all cases the zero mode energy for all k < k_f grows indicating a clear inverse transfer of energy. Curiously, we notice an unequal partitioning of energy between the different modes at k=k_f --- specifically, the zero modes always have a greater amount of energy at the forcing scale. Ofcourse, this naturally leads to a steep-shallow transition in the total energy spectrum.