Vortical and Wave Modes in 3D Rotating Stratified Flows: Random Large Scale Forcing

TL;DR

This study investigates the energy dynamics of vortical and wave modes in 3D rotating stratified flows under large-scale forcing, revealing how their interactions and spectra change with the parameter =f/N, especially around =1.

Contribution

It provides a detailed spectral analysis of vortical and wave modes in rotating stratified flows, highlighting the asymmetric behavior around =1 and the impact of  on energy saturation and transfer.

Findings

Wave energy saturates quickly for <1, with spectra steepening as  decreases.

Wave modes dominate energy for >1, with no saturation observed.

Vortical modes exhibit a k^{-3} spectrum for =1 and smaller, with inverse energy transfer.

Abstract

Utilizing an eigenfunction decomposition, we study the growth and spectra of energy in the vortical and wave modes of a 3D rotating stratified fluid as a function of $\epsilon = f/N$. Working in regimes characterized by moderate Burger numbers, i.e. $Bu = 1/\epsilon^2 < 1$ or $Bu \ge 1$, our results indicate profound change in the character of vortical and wave mode interactions with respect to $Bu = 1$. As with the reference state of $\epsilon=1$, for $\epsilon < 1$ 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 $\epsilon$ decreases: we see a shift from $k^{-1}$ to $k^{-5/3}$ scaling for $k_f < k < k_d$ (where $k_f$ and $k_d$ are the forcing and dissipation scales, respectively). On the other hand, when $\epsilon > 1$ 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 $\epsilon = 1$. With regard to the vortical modes, for $\epsilon \le 1$, the signatures of 3D quasigeostrophy are clearly evident. Specifically, we see a $k^{-3}$ scaling for $k_f < k < k_d$ and, in accord with an inverse transfer of energy, the vortical mode energy never saturates but rather increases for all $k < k_f$. In contrast, for $\epsilon > 1$ 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.