Low-mode Internal Tides Research Articles

Enhanced generation of internal tides under global warming

A primary driver of deep-ocean mixing is breaking of internal tides generated via interactions of barotropic tides with topography. It is important to understand how the energy conversion from barotropic to internal tides responds to global warming. Here we address this question by applying a linear model of internal tide generation to coupled global climate model simulations under a high carbon emission scenario. The energy conversion to high-mode internal tides is projected to rise by about 8% by the end of the 21st century, whereas the energy conversion to low-mode internal tides remains nearly unchanged. The intensified near-bottom stratification under global warming increases energy conversion into both low and high-mode internal tides. In contrast, the intensified depth-averaged stratification reduces the modal horizontal wavenumber of internal tides, leading to increased (decreased) energy conversion into high (low)- mode internal tides. Our findings imply stronger mixing over rough topography under global warming, which should be properly parameterized in climate models for more accurate projections of future climate changes.

Reflection and Scattering of Low-Mode Internal Tides on the Continental Slope of the South China Sea

Abstract It is well known that strong low-mode internal tides generated in Luzon Strait propagate westward to impinge continental slopes in the northeastern South China Sea (SCS). The reflection and scattering of these internal tides including diurnal and semidiurnal components on the slopes are quantitatively investigated using two sets of mooring data and a linear internal tide model with realistic topography and stratification. Flux reflections computed from mooring data collected on the continental slopes are consistent with the linear model. Based on the results of the observations and simulations, a map of low-mode internal tide reflection and scattering coefficients along the continental margin in the northeastern SCS is revealed. On average, diurnal internal tides lose 38% of their energy to high modes (≥mode 4) that are assumed to dissipate on the slopes, transmit 28% onto the continental shelf, and reflect 31% back to the deep ocean. On the contrary, most of the semidiurnal energy (89%) transmits onto the continental shelf, and only 11% is scattered to high modes (7%) and reflected back to the deep ocean (4%). For diurnal internal tides, a large fraction of energy that is scattered to high modes and reflected back to the deep sea can be attributed to the critical–supercritical slopes, while the weak reflection for the semidiurnal energy is due to the subcritical slopes. These quantitative descriptions for evolutions of low-mode internal tides incident to the slopes provide an energy budget map on the continental slopes in the northeastern SCS.

Enhanced Dissipation of Internal Tides in a Mesoscale Baroclinic Eddy

Abstract The dissipation of low-mode internal tides as they propagate through mesoscale baroclinic eddies is examined using a series of numerical simulations, complemented by three-dimensional ray tracing calculations. The incident mode-1 internal tide is refracted into convergent energy beams, resulting in a zone of reduced energy flux in the lee of the eddy. The dissipation of internal tides is significantly enhanced in the upper water column within strongly baroclinic (anticyclonic) eddies, exhibiting a spatially asymmetric pattern, due to trapped high-mode internal tides. Where the eddy velocity opposes the internal tide propagation velocity, high-mode waves can be trapped within the eddy, whereas high modes can freely propagate away from regions where eddy and internal wave velocities are in the same direction. The trapped high modes with large vertical shear are then dissipated, with the asymmetric distribution of trapping leading to the asymmetric distribution of dissipation. Three-dimensional ray tracing solutions further illustrate the importance of the baroclinic current for wave trapping. Similar enhancement of dissipation is also found for a baroclinic cyclonic eddy. However, a barotropic eddy is incapable of facilitating robust high modes and thus cannot generate significant dissipation of internal tides, despite its strong velocities. Both energy transfer from low to high modes in the baroclinic eddy structure and trapping of those high modes by the eddy velocity field are therefore necessary to produce internal wave dissipation, a conclusion confirmed by examining the sensitivity of the internal tide dissipation to eddy radius, vorticity, and vertical scale. Significance Statement The oceanic tides drive underwater waves at the tidal frequency known as internal tides. When these waves break, or dissipate, they can lead to mixing of oceanic heat and salt which impacts the ocean circulation and climate. Accurate climate predictions require computer models that correctly represent the distribution of this mixing. Here we explore how an oceanic eddy, a swirling vortex of order 100–400 km across, can locally enhance the dissipation of oceanic internal tides. We find that strong ocean eddies can be hotspots for internal tide dissipation, for both clockwise and anticlockwise rotating vortices, and surface-enhanced eddies are most effective at internal tide dissipation. These results can improve climate model representations of tidally driven mixing, leading to more credible future predictions.

Microstructure observations of three-layered mixing on the shelf of the northwestern South China Sea

This study presents microstructure observations on the northwestern South China Sea (nwSCS) shelf, where energetic turbulence drove a high dissipation rate on the order of 10−5 W kg−1 in May 2021. Together with the hydrography and current observations, the results show a typical three-layered structure of dissipation on the shelf and reveal different mixing regimes for different layers. Below the mixed layer, mixing over the shallow shelf displays periodically varying characteristics, indicating low-mode internal tides. Strong mixing occurred in the stably stratified layer, reaching an approximate magnitude of 1 × 10−5 W kg−1. Our analysis suggests that the strong mixing in this layer is attributed to the shear behavior of the low-mode internal waves because the observed dissipation rate increases with increasing vertical shear and stratification as described by the MacKinnon and Gregg parameterization. In the bottom boundary layer, mixing by tidal shear is dependent on the Richardson number and trends with velocity and vertical shear. In addition, our analysis also reveals that mixing continuously contributed to downward heat transport to the deep layers. This study can improve our understanding of the vertical structure of fine-scale mixing and its regimes, and their effects on the shelf of the nwSCS.

Characterization of Internal Tide Incoherence: Eulerian versus Lagrangian Perspectives

AbstractThe Lagrangian and Eulerian surface current signatures of a low-mode internal tide propagating through a turbulent balanced flow are compared in idealized numerical simulations. Lagrangian and Eulerian total (i.e., coherent plus incoherent) tidal amplitudes are found to be similar. Compared to Eulerian diagnostics, the Lagrangian tidal signal is more incoherent with comparable or smaller incoherence time scales and larger incoherent amplitudes. The larger level of incoherence in Lagrangian data is proposed to result from the deformation of an Eulerian internal tide signal induced by drifter displacements. Based on the latter hypothesis, a theoretical model successfully predicts Lagrangian autocovariances by relating Lagrangian and Eulerian autocovariances and the properties of the internal tides and jet. These results have implications for the separation of balanced flow and internal tides signals in the sea level data collected by the future Surface Water and Ocean Topography (SWOT) satellite mission.

Surprises in Physical Oceanography: Contributions from Ocean Acoustic Tomography

The author has employed ocean acoustic tomography over the past 30 years to examine problems in physical oceanography, often with surprising results. Tomography offers an accurate measurement of current or sound speed (temperature) averaged over 100s to 1000s of kilometers. It has been used to test the fundamental equation for the speed of sound in seawater and to show that the normal state of the ocean has smooth, stable, well-behaved characteristics of acoustic propagation. This latter property has been challenging for ocean models to reproduce. Tomography measures barotropic tidal currents with remarkable accuracy, and it was used to discover that low-mode internal tides radiate far into the ocean’s interior, while retaining a surprising coherence. Tomographic measurements of large-scale temperature are complementary to point measurements by hydrography; the present observing system, relying principally on altimetry and Argo floats, has little skill in predicting the available tomographic data. The addition of tomographic data to the system would substantially reduce the uncertainty of ocean state estimates. The information content of integral acoustic observations is best exploited as a constraint on ocean models by data assimilation. The applications of tomography for measuring large-scale barotropic current, relative vorticity, and temperature have been under-exploited. Dedicated research programs, supporting novel acoustical oceanography, associated instrumentation development, and multidisciplinary research would further the applications of this observational approach.

Near-Surface Reflection and Nonlinear Effects of Low-Mode Internal Tides on a Continental Slope

AbstractTwo sets of mooring data were collected at two sites (MA and MB) along a cross-slope section on the northeastern continental slope in the South China Sea (SCS). These data are used to investigate evolution and energy decay of low-mode semidiurnal (M2) internal tides on a subcritical slope with respect to M2. At the deep portion of the slope (~1250 m; MA), the M2 internal tides show upward energy propagation, while vertically standing M2 internal tides are often observed at shallow MB (~845 m). A two-dimensional linear internal tide model with realistic topography and stratification reproduces the observations, suggesting that low-mode M2 internal tides incident on subcritical slopes evolve into vertically propagating internal waves due to topographic scattering, propagate upward to the boundary, and reflect from the sea surface. The reflection point largely depends on the phase between the modal components of the incoming flux. In the near-surface reflection region, two kinds of nonlinear effects are observed to decay energy of the incoming internal tides. One is the resonant parametric subharmonic instability which transfers M2 internal tides to diurnal subharmonics M1 (=M2/2), but the instability is found to mainly depend on the incident waves. The other one is the nonresonant wave–wave interaction, producing two higher-harmonic M4 (=2M2) rays with opposite vertical propagation. A strong westward mean flow is observed in the interacting region, with amplitude comparable to that of the incident waves. This mean flow also appears to be generated by the nonlinear reflection of the M2 internal tides.

The Drag on the Barotropic Tide due to the Generation of Baroclinic Motion

AbstractThe interaction of a barotropic flow with topography generates baroclinic motion that exerts a stress on the barotropic flow. Here, explicit solutions are calculated for the spatial-mean flow (i.e., the barotropic tide) resulting from a spatially uniform but time-varying body force (i.e., astronomical forcing) acting over rough topography. This approach of prescribing the force contrasts with that of previous authors who have prescribed the barotropic flow. It is found that the topographic stress, and thus the impact on the spatial-mean flow, depend on the nature of the baroclinic motion that is generated. Two types of stress are identified: (i) a “wave drag” force associated with propagating wave motion, which extracts energy from the spatial-mean flow, and (ii) a topographic “spring” force associated with standing motion at the seafloor, including bottom-trapped internal tides and propagating low-mode internal tides, which significantly damps the time-mean kinetic energy of the spatial-mean flow but extracts no energy in the time-mean. The topographic spring force is shown to be analogous to the force exerted by a mechanical spring in a forced-dissipative harmonic oscillator. Expressions for the topographic stresses appropriate for implementation as baroclinic drag parameterizations in global models are presented.

Toward global maps of internal tide energy sinks

Internal tides power much of the observed small-scale turbulence in the ocean interior. To represent mixing induced by this turbulence in ocean climate models, the cascade of internal tide energy to dissipation scales must be understood and mapped. Here, we present a framework for estimating the geography of internal tide energy sinks. The mapping relies on the following ingredients: (i) a global observational climatology of stratification; (ii) maps of the generation of M2, S2 and K1 internal tides decomposed into vertical normal modes; (iii) simplified representations of the dissipation of low-mode internal tides due to wave-wave interactions, scattering by small-scale topography, interaction with critical slopes and shoaling; (iv) Lagrangian tracking of low-mode energy beams through observed stratification, including refraction and reflection. We thus obtain a global map of the column-integrated energy dissipation for each of the four considered dissipative processes, each of the three tidal constituents and each of the first five modes. Modes ≥6 are inferred to dissipate within the local water column at the employed half-degree horizontal resolution. Combining all processes, modes and constituents, we construct a map of the total internal tide energy dissipation, which compares well with observational inferences of internal wave energy dissipation. This result suggests that tides largely shape observed spatial contrasts of dissipation, and that the framework has potential in improving understanding and modelling of ocean mixing. However, sensitivity to poorly constrained parameters and simplifying assumptions entering the parameterized energy sinks calls for additional investigation. The attenuation of low-mode internal tides by wave-wave interactions needs particular attention.

Scattering of Low-Mode Internal Tides at a Continental Shelf

AbstractA series of laboratory experiments are performed to investigate the scattering of low-mode internal tides at a continental shelf by varying the criticality parameter and normalized topographic height independently. A wide-range synchronized particle image velocimetery (PIV) measures the velocity fields of the internal tides. Beams radiate from both the shelf break and the bottom of the slope, indicating that energy transfers from low modes to higher modes, which is verified by the modal decomposition. Energy is also transferred to higher harmonics, whose amplitude is less than a quarter of that of the first harmonic. The fraction of energy transmitted onshore and dissipated on the topography is determined by both the criticality parameter and the normalized topographic height, while the fraction of energy reflected offshore is dependent only on the criticality parameter. Mean flow with a shear structure induced by internal tides is observed along the continental slope, with horizontal velocity generally half of the amplitude of the incident waves. A net onshore transport along the slope is caused by the onshore current with larger thickness. The strength of the mean flow is dependent on both the criticality parameter and the normalized topographic height, and a linear relationship between the energy of the mean flow and the vertical shear of internal tides is revealed.

Generation of second-mode internal solitary waves during winter in the northern South China Sea

Field measurements of second-mode internal solitary waves (mode-2 ISWs) during the winter on the upper continental slope of the northern South China Sea were reported in Yang et al. (2009), but their generation mechanism remains elusive. We investigated this issue with a multi-modal evolution model and theoretical analysis, which suggest that the observed mode-2 ISWs were generated by a shoaling mode-2 semidiurnal internal tide (IT) that emanated from the Luzon Strait. The results show that two groups of mode-2 ISWs usually appear within one semidiurnal tidal period, successively riding on expanded and subsequently compressed pycnoclines. The number of wave groups largely depends on the amplitudes of the ITs; that is, a larger IT produces larger mode-2 ISWs. Furthermore, intermodal coupling dominates the evolution of a mode-1 IT, highlighting the importance of considering mode scattering in the propagation of low-mode ITs.

Scattering of low-mode internal tides at different shaped continental shelves

A series of numerical simulations have been performed to investigate the scattering of low-mode internal tides on different shaped continental shelves, and topographies with a linear slope, Gaussian slope and sinusoidal slope for a wide range of slope steepness are used in the model. The results indicate that 3 beams radiate from the abrupt shelf break, and 2 beams radiate from the gentle shelf beak during the scattering process. The greatest amount of energy can be reflected by the linear slope, whereas the least energy can be transmitted onto the shelf, and the results are contrary to those for the Gaussian slope. Modal decomposition are applied and the results indicate that the transmitted wave is dominated by mode 1, whereas the energy fluxes of higher modes increase with critical parameters and then decrease. For the reflected waves, the energy flux of the higher mode is obviously influenced by the topography. More energy in high modes can be reflected off-shore by topography with an abrupt junction point, and thus, the magnitude of vertical shear is enhanced in front of the slope. In addition, the region where the shear is influenced by the reflection process is within a wavelength of the incident wave.

Internal wave resonant triads in finite-depth non-uniform stratifications

We present a theoretical study of nonlinear effects that result from modal interactions in internal waves in a non-uniformly stratified finite-depth fluid with background rotation. A linear wave field containing modes $m$ and $n$ (of horizontal wavenumbers $k_{m}$ and $k_{n}$) at a fixed frequency $\unicode[STIX]{x1D714}$ results in two different terms in the steady-state weakly nonlinear solution: (i) a superharmonic wave of frequency $2\unicode[STIX]{x1D714}$, horizontal wavenumber $k_{m}+k_{n}$ and a vertical structure $\bar{h}_{mn}(z)$ and (ii) a time-independent term (Eulerian mean flow) with horizontal wavenumber $k_{m}-k_{n}$. For some $(m,n)$, $\bar{h}_{mn}(z)$ is infinitely large along specific curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, where $N_{0}$ and $f$ are the deep ocean stratification and the Coriolis frequency, respectively; these curves are referred to as divergence curves in the rest of this paper. In uniform stratifications, a unique divergence curve occurs on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for those $(m,n\neq m)$ that satisfy $(m/3)<n<(3m)$. In the presence of a pycnocline (whose strength is quantified by the maximum stratification $N_{max}$), divergence curves occur for several more modal interactions than those for a uniform stratification; furthermore, a given $(m,n)$ interaction can result in multiple divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane for a fixed $N_{max}/N_{0}$. Nearby high-mode interactions in a uniform stratification and any modal interaction in a non-uniform stratification with a sufficiently strong pycnocline are shown to result in near-horizontal divergence curves around $f/\unicode[STIX]{x1D714}\approx 1$, thus implying that strong nonlinear effects often occur as a result of interaction within triads containing two different modes at the near-inertial frequency. Notably, self-interaction of certain modes in a non-uniform stratification results in one or more divergence curves on the $(\unicode[STIX]{x1D714}/N_{0},f/\unicode[STIX]{x1D714})$ plane, thus suggesting that even arbitrarily small-amplitude individual modes cannot remain linear in a non-uniform stratification. We show that internal wave resonant triads containing modes $m$ and $n$ at frequency $\unicode[STIX]{x1D714}$ occur along the divergence curves, and their existence is guaranteed upon the satisfaction of two different criteria: (i) the horizontal component of the standard triadic resonance criterion $\boldsymbol{k}_{1}+\boldsymbol{k}_{2}+\boldsymbol{k}_{3}=0$ and (ii) a non-orthogonality criterion. For uniform stratifications, criterion (ii) reduces to the vertical component of the standard triadic resonance criterion. For non-uniform stratifications, criterion (ii) seems to be always satisfied whenever criterion (i) is satisfied, thus significantly increasing the number of modal interactions that result in strong nonlinear effects irrespective of the wave amplitudes. We then adapt our theoretical framework to identify resonant triads and hence provide insights into the generation of higher harmonics in two different oceanic scenarios: (i) low-mode internal tide propagating over small- or large-scale topography and (ii) an internal wave beam incident on a pycnocline in the upper ocean, for which our results are in qualitative agreement with the numerical study of Diamessis et al. (Dynam. Atmos. Oceans., vol. 66, 2014, pp. 110–137).

Low-Mode Internal Tide Propagation in a Turbulent Eddy Field

AbstractUnderstanding and predicting how internal tides distort and lose coherence as they propagate through the ocean has been identified as a key issue for interpreting data from the upcoming wide-swath altimeter mission Surface Water and Ocean Topography (SWOT). This study addresses the issue through the analysis of numerical experiments where a low-mode internal tide propagates through a quasigeostrophic turbulent jet. Equations of motion linearized around the slower turbulent field are projected onto vertical modes and assumed to describe the dynamics of the low-mode internal tide propagation. Diagnostics of the terms responsible for the interaction between the wave and the slow circulation are computed from the numerical outputs. The large-scale change of stratification, on top of eddies and jet meanders, contributes significantly to these interaction terms, which is shown to be consistent with an independent scaling analysis. The sensitivity of interaction terms to a degradation of the slow field spatial and temporal resolution indicates that present-day observing systems (Argo network, altimetry) may lack the spatial resolution necessary to correctly predict internal tide evolution. The upcoming SWOT satellite mission may improve upon this situation. The number of vertical modes required to properly estimate interaction terms is discussed. These results advocate development of a simplified model based on solving a modest number of the linearized equations subject to a prescribed mesoscale field and internal tide sources.

Reflection of Linear Internal Tides from Realistic Topography: The Tasman Continental Slope

AbstractThe reflection of a low-mode internal tide on the Tasman continental slope is investigated using simulations of realistic and simplified topographies. The slope is supercritical to the internal tide, which should predict a large fraction of the energy reflected. However, the response to the slope is complicated by a number of factors: the incoming beam is confined laterally, it impacts the slope at an angle, there is a roughly cylindrical rise directly offshore of the slope, and a leaky slope-mode wave is excited. These effects are isolated in simulations that simplify the topography. To separate the incident from the reflected signal, a response without the reflector is subtracted from the total response to arrive at a reflected signal. The real slope reflects approximately 65% of the mode-1 internal tide as mode 1, less than two-dimensional linear calculations predict, because of the three-dimensional concavity of the topography. It is also less than recent glider estimates, likely as a result of along-slope inhomogeneity. The inhomogeneity of the response comes from the Tasman Rise that diffracts the incoming tidal beam into two beams: one focused along beam and one diffracted to the north. Along-slope inhomogeneity is enhanced by a partially trapped, superinertial slope wave that propagates along the continental slope, locally removing energy from the deep-water internal tide and reradiating it into the deep water farther north. This wave is present even in a simplified, straight slope topography; its character can be predicted from linear resonance theory, and it represents up to 30% of the local energy budget.

Impact of Parameterized Internal Wave Drag on the Semidiurnal Energy Balance in a Global Ocean Circulation Model

AbstractThe effects of a parameterized linear internal wave drag on the semidiurnal barotropic and baroclinic energetics of a realistically forced, three-dimensional global ocean model are analyzed. Although the main purpose of the parameterization is to improve the surface tides, it also influences the internal tides. The relatively coarse resolution of the model of ~8 km only permits the generation and propagation of the first three vertical modes. Hence, this wave drag parameterization represents the energy conversion to and the subsequent breaking of the unresolved high modes. The total tidal energy input and the spatial distribution of the barotropic energy loss agree with the Ocean Topography Experiment (TOPEX)/Poseidon (TPXO) tidal inversion model. The wave drag overestimates the high-mode conversion at ocean ridges as measured against regional high-resolution models. The wave drag also damps the low-mode internal tides as they propagate away from their generation sites. Hence, it can be considered a scattering parameterization, causing more than 50% of the deep-water dissipation of the internal tides. In the near field, most of the baroclinic dissipation is attributed to viscous and numerical dissipation. The far-field decay of the simulated internal tides is in agreement with satellite altimetry and falls within the broad range of Argo-inferred dissipation rates. In the simulation, about 12% of the semidiurnal internal tide energy generated in deep water reaches the continental margins.

Climatic Impacts of Parameterized Local and Remote Tidal Mixing

Abstract Turbulent mixing driven by breaking internal tides plays a primary role in the meridional overturning and oceanic heat budget. Most current climate models explicitly parameterize only the local dissipation of internal tides at the generation sites, representing the remote dissipation of low-mode internal tides that propagate away through a uniform background diffusivity. In this study, a simple energetically consistent parameterization of the low-mode internal-tide dissipation is derived and implemented in the Geophysical Fluid Dynamics Laboratory Earth System Model with GOLD component (GFDL-ESM2G). The impact of remote and local internal-tide dissipation on the ocean state is examined using a series of simulations with the same total amount of energy input for mixing, but with different scalings of the vertical profile of dissipation with the stratification and with different idealized scenarios for the distribution of the low-mode internal-tide energy dissipation: uniformly over ocean basins, continental slopes, or continental shelves. In these idealized scenarios, the ocean state, including the meridional overturning circulation, ocean ventilation, main thermocline thickness, and ocean heat uptake, is particularly sensitive to the vertical distribution of mixing by breaking low-mode internal tides. Less sensitivity is found to the horizontal distribution of mixing, provided that distribution is in the open ocean. Mixing on coastal shelves only impacts the large-scale circulation and water mass properties where it modifies water masses originating on shelves. More complete descriptions of the distribution of the remote part of internal-tide-driven mixing, particularly in the vertical and relative to water mass formation regions, are therefore required to fully parameterize ocean turbulent mixing.

How stationary are the internal tides in a high-resolution global ocean circulation model?

The stationarity of the internal tides generated in a global eddy-resolving ocean circulation model forced by realistic atmospheric fluxes and the luni-solar gravitational potential is explored. The root mean square (RMS) variability in the M2 internal tidal amplitude is approximately 2 mm or less over most of the ocean and exceeds 2 mm in regions with larger internal tidal amplitude. The M2 RMS variability approaches the mean amplitude in weaker tidal areas such as the tropical Pacific and eastern Indian Ocean, but is smaller than the mean amplitude near generation regions. Approximately 60% of the variance in the complex M2 tidal amplitude is due to amplitude-weighted phase variations. Using the RMS tidal amplitude variations normalized by the mean tidal amplitude (normalized RMS variability (NRMS)) as a metric for stationarity, low-mode M2 internal tides with NRMS < 0.5 are stationary over 25% of the deep ocean, particularly near the generation regions. The M2 RMS variability tends to increase with increasing mean amplitude. However, the M2 NRMS variability tends to decrease with increasing mean amplitude, and regions with strong low-mode internal tides are more stationary. The internal tide beams radiating away from generation regions become less stationary with distance. Similar results are obtained for other tidal constituents with the overall stationarity of the constituent decreasing as the energy in the constituent decreases. Seasonal variations dominate the RMS variability in the Arabian Sea and near-equatorial oceans. Regions of high eddy kinetic energy are regions of higher internal tide nonstationarity.

Topographic scattering of the low-mode internal tide in the deep ocean

We investigate the role of deep-ocean topography in scattering energy from the large spatial scales of the low-mode internal tide to the smaller spatial scales of higher modes. The complete Green function method, which is not subject to the restrictions of the WKB approximation, is used for the first time to study the two-dimensional scattering of a mode-1 internal tide incident on subcritical and supercritical topography of any form in arbitrary stratifications. For an isolated Gaussian ridge in a uniform stratification, large amplitude critical topography is the most efficient at mode-1 scattering and small amplitude topography scatters with an efficiency on the order of 5–10%. In a nonuniform stratification with a pycnocline, the results are qualitatively the same as for a constant stratification, albeit with the key features shifted to larger height ratios. Having validated these results by direct comparison with the results of nonlinear numerical simulations, and in the process demonstrated that WKB results are not appropriate for reasonable ocean predictions, we proceed to use the Green function approach to quantify the role of topographic scattering for the region of the Pacific Ocean surrounding the Hawaiian Islands chain. To the south, the Line Islands ridge is found to scatter ∼40% of a mode-1 internal tide coming from the Hawaiian Ridge. To the north, realistic, small-amplitude, rough topography scatters ∼5–10% of the energy out of mode 1 for transects of length 1000–3000 km. A significant finding is that compared to large extents of small-amplitude, rough topography a single large topographic feature along the path of a mode-1 internal tide plays the dominant role in scattering the internal tide.

A Coupled Model for Laplace's Tidal Equations in a Fluid with One Horizontal Dimension and Variable Depth

AbstractTide–topography interactions dominate the transfer of tidal energy from large to small scales. At present, it is poorly understood how low-mode internal tides reflect and scatter along the continental margins. Here, the coupling equations for linear tides model (CELT) are derived to determine the independent modal solutions to Laplace's Tidal Equations (LTE) over stepwise topography in one horizontal dimension. CELT is (i) applicable to arbitrary one-dimensional topography and realistic stratification without requiring numerically expensive simulations and (ii) formulated to quantify scattering because it implicitly separates incident and reflected waves. Energy fluxes and horizontal velocities obtained using CELT are shown to converge to analytical solutions, indicating that “flat bottom” modes, which evolve according to LTE, are also relevant in describing tides over sloping topography. The theoretical framework presented can then be used to quantify simultaneous incident and reflected energy fluxes in numerical simulations and observations of tidal flows that vary in one horizontal dimension. Thus, CELT can be used to diagnose internal-tide scattering on continental slopes. Here, semidiurnal mode-1 scattering is simulated on the Australian northwest, Brazil, and Oregon continental slopes. Energy-flux divergence and directional energy fluxes computed using CELT are shown to agree with results from a finite-volume model that is significantly more numerically expensive. Last, CELT is used to examine the dynamics of two-way surface–internal-tide coupling. Semidiurnal mode-1 internal tides are found to transmit about 5% of their incident energy flux to the surface tide where they impact the continental slope. It is hypothesized that this feedback may decrease the coherence of sea surface displacement on continental shelves.