We investigate the dynamics of inertia-gravity wave modes in 3D rotating stratified fluids. We start by deriving a reduced PDE, the GGG model, consisting of only wave-mode interactions. In principle, comparing this model to the full rotating Boussinesq system allows us to gauge the importance of wave-vortical-wave vs. wave-wave-wave interactions in determining the transfer and distribution of wave-mode energy. As in many atmosphere-ocean phenomena we work in a skewed aspect ratio domain (H/L) with Fr = Ro < 1 such that Bu = 1. Our focus is on the equilibration of wave-mode energy and its spectral scaling under the influence of random large-scale forcing. As anticipated, when forcing is applied to all modes, with Fr=Ro ~ 0.05 and H/L=1/5, the wave-mode energy equilibrates and its spectrum scales as a power-law. For the same parameters, when forcing is restricted to only wave modes, the energy fails to equilibrate in both the full system as well as the GGG subsystem at resolutions we can achieve. This cleary demonstrates the importance of the vortical mode in determining the wave-mode energy distribution. Proceeding to the second set of simulations, i.e. for the larger Fr=Ro ~ 0.1 in a less skewed aspect ratio domain with H/L=1/3, we observe that the energy of the GGG subsystem equilibrates. Further, the full system with forcing restricted to wave modes also equilbrates and both yield identical energy spectra. Thus it is clear that the wave-wave-wave interactions play a role in the overall dynamics at moderate Ro, Fr and aspect ratios. Apart from theoretical concerns, these results highlight the difficulty in properly resolving wave-mode interactions when simulating realistic geophysical phenomena.
Deep Dive into Nonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows.
We investigate the dynamics of inertia-gravity wave modes in 3D rotating stratified fluids. We start by deriving a reduced PDE, the GGG model, consisting of only wave-mode interactions. In principle, comparing this model to the full rotating Boussinesq system allows us to gauge the importance of wave-vortical-wave vs. wave-wave-wave interactions in determining the transfer and distribution of wave-mode energy. As in many atmosphere-ocean phenomena we work in a skewed aspect ratio domain (H/L) with Fr = Ro < 1 such that Bu = 1. Our focus is on the equilibration of wave-mode energy and its spectral scaling under the influence of random large-scale forcing. As anticipated, when forcing is applied to all modes, with Fr=Ro ~ 0.05 and H/L=1/5, the wave-mode energy equilibrates and its spectrum scales as a power-law. For the same parameters, when forcing is restricted to only wave modes, the energy fails to equilibrate in both the full system as well as the GGG subsystem at resolutions we can
Inertia gravity (IG) waves, resulting from the rotating and stratified nature of geophysical fluids, play an important role in atmosphere-ocean dynamics [1]. A broad overview of their properties with relevance to the ocean can be found in Garrett & Munk [2], and a recent review focussing on the observational characterization of the oceanic wave-field can be found in Polzin and Lvov [3].
The general dynamics of these waves are reviewed by Sommeria & Staquet [4]. Further, Wunsch & Ferrari [5] place these waves in context when considering the general circulation of the ocean, while Fritts & Alexander [6] review their influence on diverse phenomena in the middle atmosphere.
In their influential papers, Garrett & Munk [2], [7] pointed out the importance of a deeper understanding of the dynamics of the IG wave modes. In this regard, one avenue of progress has been to focus upon the subset of resonant interactions among these waves [8]. In general, the dispersive nature of the linear waves results in reduced nonlinear transfer between modes by dispersive phase scrambling. However, resonant interactions are special nonlinear interactions for which phase scrambling is absent, and nonlinear transfer remains strong. Thus resonant interactions are critical for the dynamics when they are present. See for example [9] for a broad introduction to nonlinear dispersive equations, [10] for statistical theories of the so-called weak turbulence, [11], [12] for geophysical perspectives and [13], [14] for the roles of resonances and near-resonances in the specific case of the 3D rotating Boussinesq system. [8] distinguished between three classes of resonant interactions that lead to induced diffusion, elastic scattering and parametric sub-harmonic instability. The picture put forth was, for a large-scale initial data: (i) elastic scattering leads to 3D isotropy of an initial condition that is only horizontally isotropic, (ii) sub-harmonic instability moves energy downscale, (iii) induced diffusion tends to force the high wavenumber spectrum towards equilibrium. A detailed account of these resonances can be found in the extensive review by Muller et al. [15]; their review also describes approaches that account for more than resonant interactions, but involve other approximations (such as the direct interaction approximation). Further, the use of resonances to derive kinetic equations under the weak turbulence paradigm has been an active area of work -for an overview see Lvov et al. [16].
In particular recent work has shown a correspondence between observed IG energy spectra and particular steady state solutions of the relevant kinetic equations [17], [18].
Unfortunately, as was pointed out by McComas & Bretherton [8], resonances do not account for the bulk of interactions among IG waves. A formal statement regarding the sparsity of these resonances can be found in Babin et al. [19]. This is all the more an important issue when Ro, F r are small (implying strong rotation and stratification) but far from zero, i.e. 0
Ro, F r < 1 (where Ro, F r are the Rossby and Froude numbers respectively). Indeed, having non-zero F r and/or Ro opens the door for near-resonances to enter the dynamics and these interactions can, both quantitatively and qualitatively, change the behavior of the system [14]. This naturally motivates the construction of a model (called the GGG model) that includes all possible wave-mode interactions. Quite interestingly, as the two-dimensional (2D) stratified Boussinesq system only supports wave modes, this new 3D system can be looked upon as an extension of the full 2D
stratified problem [20], [21], [22].
So far, our discussion has been restricted to considering wave modes in isolation. In reality rotating and stratified fluids support an additional vortical mode of motion [23], and in fact in geophysically relevant limits, interactions among vortical modes lead to the celebrated quasigeostrophic (QG) equations [13], [19], [24]. Indeed, one of the many contributions by Prof. Majda (in collaboration with Prof. P. Embid) in the field of geophysical fluid dynamics concerns a formal statement on the emergence of QG dynamics, from the governing rotating Boussinesq equations, in the limit Ro ∼ F r = → 0 while holding Bu ∼ 1 [25], [19]. In addition to clarifying the nature of the QG equations and constructing a general framework for averaging over fast-modes in geophysical systems [26], their work also predicted the emergence of a new regime, the so-called vertically sheared horizontal flows (VSHF) when Ro ∼ 1 while F r = → 0 [25], [27]. Since then, numerical work in the appropriate parameter regime has confirmed the emergence of VSHF modes when considering a rotating Boussinesq fluid under random forcing [24], [28], [29], [30].
As it happens, the vortical mode plays an important role in the redistribution of wave-mode energy by means of the so-called wave-vortical-wave interactions. In fact, this is thought to be th
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
