Impact of Wave Interference on the Consistency Relations of Internal Gravity Waves near the Ocean Bottom

TL;DR

This study shows that classical IGW consistency relations, derived for monochromatic plane waves, can be violated when evaluating time-series data near the ocean bottom due to interference from bottom-reflected low vertical modes that form standing waves. The authors derive a new set of bottom-aware consistency relations incorporating tan(mz) factors and validate them using a high-resolution regional MITgcm simulation, demonstrating improved agreement over deep depths when low modes are appropriately accounted. They also show that filtering out low vertical modes restores the classical relations, while single-mode analyses confirm the classical relations hold for individual modes, highlighting the importance of mode composition and bottom boundary conditions in interpreting IGW fields. The work advances the understanding of IGW energy partitioning and provides practical tools for diagnosing IGW continua in the presence of bottom reflections, with implications for ocean mixing and energy cascades.

Abstract

Consistency relations of internal gravity waves (IGWs) describe ratios of cross-spectral quantities as functions of frequency. It has been a common practice to evaluate the measured or simulated signals (e.g., time series of velocity, density, etc.) against the consistency relations, as a way to determine whether an oceanic field of interest is comprised of IGWs. One such study is carried out in Nelson et al. (JGR Oceans, 125(5), 2020, e2019JC015974), which certifies that the ocean interior field in a numerical simulation of a region southwest of Hawaii is dominated by IGWs, through evaluating the consistency relations derived from time series at a depth of 620 m. However, we find that when the same procedure is applied at greater depths (e.g., 2362 m, 3062 m, and 4987 m), a clear deviation of the simulated signal from the classical consistency relations is observed. In this paper, we identify the reason for the unexpected deviation and show that it is a general phenomenon due to interference of low vertical modes under the reflection by the ocean bottom. We further derive a new set of formulae to characterize the consistency relations of these low modes and validate these formulae using model output.

Figures (11)

• Figure 1: Bathymetry in the simulation domain (24$^\circ$N to 32$^\circ$N, 193$^\circ$E to 199$^\circ$E). Adapted from skitka2024probing.
• Figure 2: Results of (a) ${E_{vk}}/{E_{hk}}$ and (b) ${E_{hk}}/{E_{p}}$ at different depths, including $612~\text{m}$ (blue dashed line), $2362~\text{m}$ (red dashed line), $3062~\text{m}$ (magenta dashed line) and $4987~\text{m}$ (green dashed line), in comparison with Eqs. \ref{['eq:r11']} and \ref{['eq:r22']} (black solid line), respectively.
• Figure 3: Results of (a) $E^{m_c}_{vk}/E^{m_c}_{hk}$ and (b) $E^{m_c}_{hk}/E^{m_c}_{p}$ with $m_c=0$ (green dashed line), $m_c=0.002~\text{m}^{-1}$ (red dashed line), $m_c=0.004~\text{m}^{-1}$ (magenta dashed line), $m_c=0.008~\text{m}^{-1}$ (blue dashed line) for the depth of $4987~\text{m}$, in comparison with Eqs. \ref{['eq:r11']} and \ref{['eq:r22']} (black solid line), respectively.
• Figure 4: A schematic drawing of the incident (blue solid line) and reflected (red solid line) waves (left), as well as the resulting standing wave (magenta solid line, right) with nodes (magenta dots) and antinodes (magenta triangles)
• Figure 5: Results of (a) $E_{vk}/E_{hk}$ (b) $E_{hk}/E_{p}$ obtained from the model and Eqs. \ref{['eq:r1new']}-\ref{['eq:r2new']} at the heights of $3350~\text{m}$ (red dashed and solid lines), $2650~\text{m}$ (magenta dashed and solid lines) and $725~\text{m}$ (green dashed and solid lines) with $m=0.002~\text{m}^{-1}$, in comparison with Eqs. \ref{['eq:r11']} and \ref{['eq:r22']} (black solid line), respectively