Testing the consistency of the resonant wave interaction approximation with simulated dynamics of idealized 2D internal wave fields

TL;DR

This study interrogates the consistency of the resonant wave interaction approximation (RWIA) for internal waves by performing idealized 2D, non-rotating DNS forced at low wavenumbers and examining the resulting energy transfers. By analyzing the energy equation in wave amplitudes and testing three standard RWIA derivations, the authors show that many energy transfers arise from non-resonant interactions and that the assumed slow-time evolution of wave amplitudes or energies is not satisfied in the simulations. They find self-consistency issues with the conventional RWIA derivations and demonstrate that resonance conditions do not reliably predict the dominant energy transfers in their DNS. The results suggest caution in applying RWIA to real ocean internal waves in this simplified framework and motivate further testing in more realistic 3D, rotating, and stratified settings with near-resonant broadening considerations.

Abstract

Nonlinear interaction and breaking of internal ocean waves are responsible for much of the interior ocean mixing, affecting ocean carbon storage and the global overturning circulation. These interactions may affect the observed Garrett-Munk wave energy spectrum, in addition to the recently explored interaction of waves with ocean eddies. According to the resonant wave interaction approximation, that is commonly used to derive the kinetic equation for the energy spectrum, the dominant interactions are between wave triads whose wavevectors satisfy $\mathbf{k}=\mathbf{p}+\mathbf{q}$, and whose frequencies satisfy $ω_{\mathbf{k}}=|ω_{\mathbf{p}}\pmω_{\mathbf{q}}|$. In order to test the validity of the resonant wave interaction approximation, we examine several analytical derivations of the theory. The assumptions underlying each derivation are tested using idealized direct 2D numerical simulations, representing near-observed energy levels of the oceanic internal wave field. We show that the slow-amplitude assumptions underlying the derivations are inconsistent with the simulated dynamics in this particular set of simulations. In addition, most of the triads satisfying the resonant conditions do not contribute significantly to nonlinear wave energy transfer in our simulations, while some interactions that are dominant in nonlinear energy transfers do not satisfy the resonance conditions. We also point to possible self-consistency issues with some derivations found in the literature.

Figures (18)

• Figure 1: Characterizing the energy in the direct numerical simulations: (a) Time series of standardized (mean removed, divided by std, with both specified in the figure) total energy per unit volume (J/m$^3$) for 200 days for the three runs. Energy spectra as a function of the horizontal wavenumber, $P(k_x)$, are depicted in panel (b), and energy spectra as a function of the vertical wavenumber, $P(k_z)$, are depicted in panel (c). The forced wave vectors are those with $k_x/(2\pi/L_x)\le2$ and $k_z/(2\pi/L_z)\le2$ except for $k_x=0$ or $k_z=0$.
• Figure 2: (a--c) Time-average of the interaction term ($\mathsf{M}_{abc}\left(\mathbf{k,p,k-p}\right) A_{b,\mathbf{p}}A_{c,\mathbf{k-p}}A_{a,\mathbf{k}}^{\ast}$ in eq. \ref{['eq:energy-equation']}) for nondimensional wavevectors $\mathbf{k}=(40,4)$, $(24,12)$, and $(12,24)$, for the medium forcing amplitude run, normalized (for each $\mathbf{k}$) by its maximum value. The color range corresponds to a part of the range of values plotted to include the weakest 10% of the interactions, which account for about 50% of the total energy transfers, as shown by Supplementary Fig. \ref{['fig:interaction-CDF']}. The positive values (red) correspond to gain and the negative ones (blue) to loss of energy by wavevector $\mathbf{k}$. (d) A zoom into the region of $\mathbf{p}\sim\mathbf{k}$ for panel (c). The dashed and solid, blue, red, and black lines represent the different resonance curves, see the text for details. The forced wave vectors are those with $k_x/(2\pi/L_x)\le2$ and $k_z/(2\pi/L_z)\le2$ except for $k_x=0$ or $k_z=0$.
• Figure 3: Testing the slow amplitude assumption, requiring $V_1\gg1$, where $V_1$ is the ratio of the time scale of the slow amplitude and the linear wave period (eq. \ref{['eq:V1']}). Color shading shows $V_1$ and the black contours show the spectral energy $\log_{10}E(\mathbf{k})$ with contour lines drawn from $-8$ to 2 with a contour interval of 1. (a) weak, (b) intermediate, and (c) strong forcing. The contour lines show that there is significant energy in wavenumbers that do not satisfy the slowness condition (red shading), spanning most resolved wavenumbers.
• Figure 4: Time series of the fast and slow amplitudes of three particular wavevectors in the weakly forced simulation. These panels demonstrate that the slow amplitude (blue) varies on a time scale not much longer than that of the linear wave period, as also quantified in Fig. \ref{['fig:V1']}.
• Figure 5: Testing whether the three-product of wave amplitudes varies slowly, as quantified by criterion $V_2$, eq. \ref{['eq:V2']}, for the weak forcing case, for the wavevectors denoted in the titles of each panel.