1. Introduction
Breaking internal waves are the main cause of diapycnal mixing in the ocean. They may be generated by wind at the sea surface or by flow over the topography of the seafloor. While mean flows and eddy activity in the deep ocean are characterized by O(1 mm s−1) currents, barotropic tidal currents in the deep ocean are O(10 mm s−1) and may be an important source of internal waves. Munk and Wunsch (1998) review the evidence suggesting that tides do about 3.7 TW of work on the global ocean, with 2.5 TW contributed by the semidiurnal lunar tides. Egbert and Ray (2000, 2001) have examined least squares fits of models for the global barotropic tide to TOPEX/Poseidon altimetry data, and they interpret model residuals in terms of “tidal dissipation.” In Egbert and Ray's studies, dissipation refers to any mechanisms that transfer energy away from the barotropic tide: they are not able to distinguish whether barotropic energy is lost to baroclinic waves (the internal tide) or to bottom friction. Egbert and Ray (2000, 2001) estimate that up to 1 TW of power is lost from barotropic tides in the deep ocean. Frictional dissipation can be estimated from the relation ρcd|u|u2, where cd = 0.0025 is the drag coefficient. For typical open-ocean tidal speeds, u ≃ 0.03 m s−1, the frictional dissipation is less than 0.1 mW m−2. Globally, this accounts for less than 30 GW of barotropic tidal power loss. Thus, nearly all barotropic power conversion in the deep ocean must occur as internal tides.
Egbert and Ray (2000, 2001) identify a number of deep ocean regions where barotropic tidal energy is likely being transferred to internal tides. These regions are generally associated with three types of topography: (i) oceanic islands, (ii) oceanic trenches, and (iii) midocean ridges. Oceanic islands such as Hawaii, and oceanic trenches such as those in the west Pacific Ocean, are steep topographic features with typical slope s (rise over run) from 0.1 to 0.3 (Seibold and Berger 1996). Egbert and Ray (2000, 2001) identify the oceanic islands of Micronesia and Melanesia as sites accounting for over 100 GW of internal tide production, while Hawaii accounts for 20 GW. They also identify midocean ridge topography in the Atlantic and Indian Oceans as each accounting for over 100 GW of internal tide production. These topographic features have slopes that vary over a wide range of spatial scales.
a. Observations of internal tides and mixing
The contribution of internal tides to oceanic velocity and temperature records has long been recognized. Wunsch (1975) and Hendershott (1981) present reviews of earlier work. Renewed interest in internal tides came with observations of sea surface elevation by the TOPEX/Poseidon altimeter. These data have been used to constrain hydrodynamic models of the tides, and hence produce accurate estimates of open-ocean barotropic tides (Egbert et al. 1994; Egbert 1997). A multiyear record of observations is now available, and the semidiurnal tides (M2 and S2) can be dealiased from the record. Ray and Mitchum (1996) used along-track TOPEX/Poseidon records to examine the surface manifestation of internal tides generated along the Hawaiian Islands. They found that both first and second baroclinic modes of the semidiurnal internal tides were present in the data and that the signal of internal tide propagation could be tracked up to 1000 km away from the Hawaiian Ridge. In a further analysis of the altimetry data, Ray and Mitchum (1997) estimate that 15 GW of semidiurnal internal tide energy radiates away from the Hawaiian Ridge in the first baroclinic mode. TOPEX/Poseidon altimetry records have also been used by Cummins et al. (2001) to show that internal tides generated at the Aleutian Ridge can be tracked over 1000 km into the central Pacific, though the total internal tide energy flux is only about 2 GW.
Microstructure observations have provided direct observations of mixing driven by internal tides at steep topography (Lueck and Mudge 1997; Kunze and Toole 1997; Lien and Gregg 2001). Lueck and Mudge (1997) attributed enhanced turbulence at Cobb Seamount in the northeast Pacific to a semidiurnal internal tide. They found that the largest turbulence levels occurred along a beam of internal tide energy emanating from the seamount rim. Kunze and Toole (1997) describe microstructure observations from Fieberling Seamount in the northeast Pacific. They found the largest mixing levels at the summit of the seamount where diurnal internal tide energy was trapped. Internal tides are also produced along the continental margins of the ocean basins, and evidence of internal-tide-driven mixing at Monterey Canyon off California was reported by Lien and Gregg (2001). They found enhanced turbulence levels along the ray path of a semidiurnal internal tide beam extending out to a distance of 4 km away from the topography.
Evidence of internal-tide-driven mixing from the truly deep ocean was found during the Brazil Basin Tracer Release Experiment (Polzin et al. 1997; Ledwell et al. 2000). This study was conducted near the Mid-Atlantic Ridge. St. Laurent et al. (2001) present vertically integrated dissipation data that are modulated over the spring–neap tidal cycle with a small lag of about a day. Maximum levels of dissipation, vertically integrated to 2000 m above the bottom, reach 3 mW m−2. These observations span a network of fracture zones west of the Mid-Atlantic Ridge, with topographic relief varying by up to 1 km between the crests and floors of abyssal canyons. The dominant topography of the fracture zone system has a slope less than 0.1 as the canyons are generally 30–50 km wide. Elevated turbulent dissipation rates were found along all regions of the fracture zone topography, but turbulence levels appeared most enhanced over the slopes. Above all classes of topography, turbulence levels decrease to background levels at heights above bottom greater than about 1000 m.
The energy flux carried by the internal tide can radiate as propagating internal waves, and these waves are subject to a collection of processes that will eventually lead to dissipation. Shear instability, wave–wave interactions, and topographic scattering all act to influence the rate of dissipation and control whether the internal tide dissipates near the generation site or far away.
In this paper we will review several issues relevant to the role of internal tides in mixing the deep ocean. In particular, we consider the following questions. What is the shape of the energy flux spectrum? Is the generated internal tide stable to shear instability? How efficient are wave–wave interactions at modifying the internal-tide spectrum? What causes the internal tides to dissipate? Are internal tides likely to dissipate near their generation sites or far away? In section 2, we discuss some linear theory for the internal tide and present a calculation of internal tide energy flux. The nondimensional parameters that define the wave response to flow over topography are discussed, and the energy-flux spectrum is described. The effects of finite depth and variable stratification are discussed in section 2a. Calculations of energy flux at midocean ridge topography are presented in section 2b. Energy flux production along steeper topography is considered in section 2c. In section 3, we discuss issues related to the radiation and dissipation of internal tides. We consider the stability of the internal tide to shear (section 3a), and the influence of wave–wave interactions (section 3b). The influence of topographic scattering on the internal tide spectrum is considered in section 3c. Finally, some implications for mixing are discussed in section 4.
2. Internal tide energy flux
Internal tides are produced in stratified regions where barotropic tidal currents flow over topography. Internal tides radiate throughout the oceans with frequencies identical to those of the barotropic tides (M2, S2, O1, K1, etc.) and their harmonics, provided the frequency ω falls in the range f < ω < N. Poleward of the inertial latitudes where ω = f, internal tides are trapped over topography. Thus, while internal tides of diurnal frequency can only propagate freely equatorward of about ±30° latitude, semidiurnal tides are freely propagating throughout most of the ocean, as ω
The tidal excursion parameter serves to classify two regimes of interest. In one regime, the tidal excursion is less than the scale of the bathymetry. The waves of this “internal tide” regime radiate mainly at the fundamental frequency of the tide in both the up- and downstream directions. In the other regime, the tidal excursion exceeds the length scale of the bathymetry. This is referred to as the “quasi-steady lee wave” regime, with waves that propagate only in the upstream direction with frequencies −kU(t) and phase speeds −U(t). Higher harmonic internal tides manifest themselves mainly as quasi-steady lee waves.
The steepness parameter, s/α, is also used to distinguish two regimes. Topography is referred to as “supercritical” when its slope is greater than the slope of a tidal beam (s/α > 1), while “subcritical” topography has slope less steep than a tidal beam (s/α < 1). Using (1) with ω = 1.4 × 10−4 s−1, N = 1 × 10−3 s−1, and f = 1 × 10−4 s−1, α ≃ 0.10 provides a reference value for the slope of an M2 tidal beam in the deep ocean. It follows that many oceanic island and trench features are supercritical, while large-scale features of midocean ridge topography are subcritical.
The equations for supercritical topography are reviewed by Baines (1982), and solutions for specific topographic geometries can be computed using ray tracing methods. Solutions for ridge, slope, and shelf topography have been discussed by Rattray (1960), Baines (1982), and Craig (1987). A simplified model for internal tide generation at a topographic step was proposed by Stigebrandt (1980) and applied to deep ocean topographies by Sjöberg and Stigebrandt (1992). However, the equations for arbitrary supercritical topography h(x) are difficult to solve due to the necessity of a nonlinear bottom boundary condition, w(h) = U·∇h + u·∇h, where U is the barotropic current vector and (u, w) are the lateral and vertical components of the baroclinic velocity. The baroclinic wave response to barotropic tidal flow over arbitrary steep topography has instead been studied using numerical simulations. In particular, numerical studies have been carried out for internal tides at the Hawaiian Ridge (Holloway and Merrifield 1999; Merrifield et al. 2001) and the Aleutian Ridge (Cummins et al. 2001). Simulations of internal tide generation at steep idealized topographies are presented by Li (2002, manuscript submitted to J. Mar. Res.) and Khatiwala (2002, manuscript submitted to Deep-Sea Res.) The properties of quasi-steady lee waves have also been studied numerically for atmospheric (Lott and Teitelbaum 1993) and oceanographic (Nakamura et al. 2000) cases.
Models for internal tide generation by subcritical topography have been developed by Cox and Sandstrom (1962), Baines (1973), Bell (1975a,b), Hibiya (1986), and Llewellyn Smith and Young (2002). These models apply a linearized bottom boundary condition w(−H) = U·∇h. Two approximations are made in this linearization: (i) the neglect of the term u·∇h involving the lateral current vector u of the internal tide and (ii) the use of a constant depth z = −H in place of z = h(x) in applying the boundary condition. Both of these approximations are justified when the slope parameter satisfies s/α ≪ 1. With application of the linear boundary condition, solutions for arbitrary bathymetry can be obtained by superposition.
a. Equivalent modes and finite depth
The integration in (2) is performed over a continuous spectrum of wavenumbers. In Bell's theory, the internal waves radiated away from topography are assumed to never reach the upper ocean, where the waves would reflect at the upper surface. This assumption is justified if the internal waves are dissipated rapidly after generation. For waves that are not rapidly dissipated, the neglect of upper-ocean reflection is justified only in an ocean of infinite depth.
The primary influence of finite depth on the internal tide energy flux is that the largest scale of internal wave response is limited to that of the mode-1 wave. This suggests that using k0 = k1 − ½δk as the lower limit of integration in (2) provides the first-order finite depth correction to Bell's theory. The magnitude of this correction to the energy flux estimate depends on the shape of the bathymetric spectrum ϕ(k), with the correction being largest for spectra that are red at low wavenumbers. There is additional inaccuracy in (2) that arises when doing a continuous integration over wavenumbers instead of a discrete sum over modes. In practice, this amounts to only a small error in the integrated estimate of energy flux.
In the calculations that follow, we will integrate (2) from k0 to ∞ and refer to the wavenumbers given by (4) as &ldquo=uivalent modes.” This term signifies that the internal-tide response at a given wavenumber kj may never occur as an actual baroclinic mode. This is particularly true for the internal tides generated at high wavenumbers, which may dissipate before ever reaching the upper ocean. However, by referring to equivalent modes, we may easily refer to a given wavenumber portion of the internal-tide spectrum.
b. Energy flux estimates for midocean ridge topography
1) Topography in one dimension
Here, we consider internal tides generated at midocean ridge topography. We anticipate that the dominant wavelengths of midocean ridge topography will have subcritical slopes, justifying the use of subcritical theory. We will use Bell's (1975a,b) theory to estimate the energy flux according to (2), using k0 = k1 − ½δk as the lower limit of integration.
It has long been established that the roughness of midocean ridge topography varies according to the spreading rates of the seafloor (Small and Sandwell 1992). In general, the roughness and steepness of ridge topography are inversely proportional to the spreading rate of the seafloor. To examine the likely range of energy flux production at midocean ridges, we consider topography from two regions. An example of fast-spreading seafloor topography is taken from a region of the East Pacific Rise (EPR) near (17°S, 115°W), just south of the Garrett Fracture Zone. An example of slow-spreading seafloor topography is taken from a region of the Mid-Atlantic Ridge (MAR) near (27°S, 14°W), just north of the Rio Grande Fracture Zone. These two regions are useful, as they represent extreme cases. Multibeam bathymetry data for these regions are available from the RIDGE Multibeam Synthesis Project, and maps showing the data are presented in Fig. 2. Sections of multibeam data have 200-m resolution in the lateral directions.
Bathymetric spectra from these regions were computed along several sections of the multibeam data (Fig. 3). The spectra have been normalized such that the integrated spectrum gives the mean square height of topography,
The length scale corresponding to the amplitude of the tidal excursion u0/ω is an additional important parameter, requiring estimates of the tidal velocities. Barotropic tidal velocity data for the EPR and MAR sites considered here were produced using the TPXO.3 global tidal model (Egbert et al. 1994; Egbert 1997) for the year between 1 January 2000 and 1 January 2001. Records of barotropic tidal currents (U, V) were produced at 2° Mercator intervals spanning the bathymetric sections. The M2 components of these records were extracted, and tidal ellipses were computed using standard methods (e.g., Godin 1972). The spatially averaged M2 tidal ellipses for each site are shown in Fig. 4. Current amplitudes along the semimajor and semiminor axes of the ellipses were projected onto the FZ and XFZ directions to give estimates of u0 for use in the energy flux calculations. Current amplitudes along the EPR bathymetry are (0.010, 0.008) m s−1 for the (FZ, XFZ) directions, respectively. The corresponding current amplitudes for the MAR bathymetry are (0.031, 0.026) m s−1. Tidal excursion amplitudes in the XFZ directions are 53 m and 185 m for the EPR and MAR sites, respectively.
Topographic slopes on wavelengths greater than 1 km are subcritical. Therefore, we assume that theory for subcritical topography is applicable, and we use the spectra of XFZ bathymetry shown in Fig. 3 with expression (2) to calculate energy flux spectra. The fine spatial resolution of these one-dimensional sections allow us to examine the spectrum of energy flux to very large equivalent modes.
Energy flux estimates were made using frequency parameters ω = 1.4 × 10−4 s−1 for the M2 tide, f = (−4.3 × 10−5, −7.3 × 10−5) s−1, and Nb = (6 × 10−4, 1 × 10−3) s−1, for the EPR and MAR sites respectively. These bottom values of buoyancy frequency were taken from the data in Fig. 5, not from the exponential fits to the profiles. Tidal current amplitudes of 0.008 m s−1 (EPR) and 0.026 m s−1 (MAR) were used to calculate the XFZ spectra, which are shown in Fig. 6. Estimated equivalent mode numbers from (6) are given along the upper axes. Spectra from both EPR and MAR sites are “blue” with a dependence that is roughly k3/2 at low wavenumbers and “red” with a dependence that is roughly k−2–k−3 for the high wavenumbers. The spectral peaks occur near mode 8 in the EPR data, and near mode 5 in the MAR data.
The integrated energy flux of these XFZ spectra are roughly 0.2 and 4.3 mW m−2 for the EPR and MAR sites respectively. Integrated energy flux levels for the FZ sections were roughly 0.1 mW m−2 (EPR) and 5.7 mW m−2 (MAR). Bathymetric variance levels for different sections at a given site exhibit 50% variations, even between adjacent sections. Also, the relative orientation of the tidal ellipse to a given section influences the energy flux estimate significantly because of the dependence of (2) on
2) Topography in two dimensions
To evaluate (10), a two-dimensional power spectrum must be defined from bathymetry. The multibeam data previously discussed are inadequate for this purpose because of their nonuniform coverage. Gridded bathymetry data from satellite altimetry and ship sounding records have been prepared by Smith and Sandwell (1997, version 8.2; SS97 hereafter). These data provide full spatial coverage, though at a coarse resolution of roughly 3 km. However, given the red spectrum of internal tide energy flux, the SS97 data are adequate for resolving the spatial scales accounting for most of the energy.
As a means of assessing the quality of spectra computed from the SS97 data for the MAR site (Fig. 2b), we have computed spectra along the same bathymetric sections used with the multibeam data. The SS97 spectra are shown along with multibeam spectra in Fig. 7. The SS97 spectra have been wavenumber averaged with a 3-point running boxcar, and standard error confidence bands are shown. The SS97 data are noisier and more statistically uncertain, as they contain significantly fewer degrees of freedom than the multibeam spectra. However, the spectra clearly are comparable at the standard error confidence level, and the integrated variance levels out to the SS97 Nyquist wavenumber (k = 2π/6 km) are within 25%. We note that SS97 data were constrained by the same multibeam data used in this study and in this comparison. Thus, these multibeam sections have strongly influenced the quality of the SS97 spectral estimates for this region.
An estimate of internal tide energy flux from (10) was made for a region of MAR topography between 22° and 32°S and 8° and 20°W (Fig. 8). The region was divided into four sections, and records of barotropic tidal currents were generated using the TPXO.3 model. The M2 current ellipses were computed in each quadrant (Fig. 8). The semimajor and semiminor axes of each tidal ellipse were used to define a local reference frame, and the SS97 data were regridded to these tidal ellipse coordinates. Power spectra of bathymetry were computed using the tidal ellipse coordinates for each of the four quadrants, and these were used to produce an average spectrum ϕ(k, l). An energy flux estimate was made using (10) with (ue, υe) = (0.028, 0.021) m s−1, the average semimajor and semiminor current amplitudes of the tidal ellipses shown in Fig. 8. Stratification data from Levitus et al. (1994) and Levitus and Boyer (1994) were used to estimate an average value of Nb = 8.7 × 10−4 s−1 for the region. Equivalent modes were identified by substituting the total wavenumber Kj = (k2 + l2)1/2 into (6), with N0 and N1 parameters taken from the fit in Fig. 5b. The average spectrum of energy flux is shown in Fig. 9. We find Ef = 2.9 ± 1.3 mW m−2 in the first 25 modes, with the first 10 modes accounting for Ef = 2.2 ± 1 mW m−2. The total area of the region is 1.3 × 1012 m2, such that 3.8 ± 1.7 GW of internal tide production into equivalent modes 1–25 occurs on average in this region. For comparison, estimates of internal tide production at the Hawaiian Ridge range from 9 to 20 GW (Ray and Mitchum 1997; Egbert and Ray 2000; Merrifield et al. 2001).
Our energy flux estimates along bathymetric sections suggest that equivalent modes 1–25 likely account for 80% of the total vertical energy flux (Fig. 6). Thus, an additional 1 mW m−2 may be generated as high equivalent modes. With this addition to our two-dimensional spectral estimate, 4 ± 1 mW m−2 is our best estimate of energy flux for this MAR site.
c. Energy flux production at steeper topography
We have presented the perturbation expansion of energy flux (26) as a means of assessing the change in efficiency of power production as topographic slopes of increasing steepness are considered. Higher-order corrections are clearly necessary to accurately assess the true change in energy flux for topography with finite steepness. However, we find the first correction is a rather small adjustment to the linear solution. We therefore suggest that the energy flux prediction of subcritical theory provides a useful estimate of internal tide power for general topographic slopes with s < α. This conclusion is supported by the work of Balmforth et al. (2002), who have extended the series in (26) using MAPLE. They find a convergent series for the case of sinusoidal topography with maximum steepness satisfying the critical condition s/α = 1, predicting a 56% increase in the energy flux production over the linear solution. Balmforth et al. also consider the case of a Gaussian seamount, finding only a 14% increase in energy flux over the linear prediction. The numerical results of (2002, manuscript submitted to J. Mar. Res.) indicate a 70% increase over the linear prediction of energy flux for critical sinusoidal topography. For supercritical slopes, Li finds that a reduction in the buoyancy stratification inhibits the generation of internal tides, with a significant decrease in energy flux, though this result is sensitive to the model's dissipation scheme. Khatiwala (2002, manuscript submitted to Deep-Sea Res.) also finds reduced energy-flux production at supercritical topography. Further research on supercritical generation is needed.
3. Discussion
a. Shear instability
Observations by Lueck and Mudge (1997) and Lien and Gregg (2001) suggest that higher levels of shear and turbulence occur at sites of internal-tide generation by supercritical topography. At these sites, observed levels of dissipation may be the result of shear instability. However, this local dissipation may represent only a small fraction of the radiated internal-tide energy. For example, recent observations at the Mendocino Escarpment and at Hawaiian Ridge suggest that dissipation near the topography is about 10% of the energy flux radiated away as low-mode internal tides (E. Kunze 2002, personal communication).
b. Wave–wave interactions
The transfer of energy between wave modes due to wave–wave interactions has been the subject of a number of studies (e.g., McComas and Müller 1981a,b; Henyey et al. 1986; Hirst 1991, 1996), and a review of this subject was given by Müller et al. (1986). Olbers and Pomphrey (1981) and Hirst (1996) specifically examined the wave–wave interactions between the internal tide and the internal wave continuum and estimated the timescale for the energy transfer away from modes 1–10 to higher modes. They considered the Parametric Subharmonic Instability (PSI) resonant interaction, which transfers tidal energy with frequency ω to larger vertical wavenumbers with frequency ω/2. Thus, the PSI mechanism is an effective mechanism for dissipating the M2 internal tides at latitudes less than about 30°. Estimates of the dissipation timescale for the M2 internal tides by PSI are given in Fig. 11 (adapted from Olbers 1983). The transfer timescale is O(100) days for mode 1, consistent with the observed slow southward decay of low-mode internal tides generated at the Hawaiian Ridge (Ray and Mitchum 1996) and Aleutian Ridge (Cummins et al. 2001). The PSI transfer timescale decreases to O(1) day for mode 10. Hirst (1996) also discusses conditions where strong tidal forcing can result in reduced transfer timescales for low-mode internal tides.
Most other studies have considered the additional resonant interactions occurring in the general internal wave continuum over the range of frequencies between f and N. The “elastic scattering” and “induced diffusion” interactions may influence internal tides at all latitudes. In all cases, wave interaction timescales for low modes with tidal frequencies typically exceed O(10) days (Müller et al. 1986; Hirst 1991).
c. Bottom reflection and topographic scattering
The interaction of internal waves with bottom bathymetry is often discussed in terms of wave reflection from sloping topography and wave scattering from rough topography. Particular attention has been given to reflection from planar slopes (e.g., Eriksen 1982, 1985; Garrett and Gilbert 1988; Thorpe 1987, 1997, 1999). Reflection from slopes with nonplanar geometries has been considered by Gilbert and Garrett (1989) and Müller and Liu (2000a,b). Scattering of waves from random rough topography was considered by Müller and Xu (1992), and Thorpe (2001) considered scattering from rough topography with an underlying slope. For the midocean ridge topography discussed in section 2b, the latter studies are clearly most relevant.
As discussed by Müller and Xu (1992), both reflection and scattering redistribute energy flux in wavenumber space. Müller and Xu (1992) find that scattering is more efficient than reflection at redistributing energy flux to higher modes, particularly at tidal and near-inertial frequencies. However, Müller and Xu (1992) find less than 10% of the incoming radiation is redistributed during a scattering event at typical deep-ocean topography. Most of the incoming radiation undergoes pure reflection as from a flat bottom, such that the wavenumber content of the spectrum is mostly preserved.
Here, we consider the similarity between internal-tide generation and the scattering of low-mode internal tides. In this simple calculation, we exploit the fact that the wave response to a mode-1 current flowing over small topographic wavelengths is much the same as the wave response to a barotropic current. We will examine the scattering of the mode-1 internal tide through a “second generation” calculation, where the amplitude of the mode-1 current is used in place of the amplitude of the barotropic current in (2). This is meant to provide a crude estimate of the energy flux scattered from mode 1 into higher modes. The rms current for the mode-1 internal tide at the MAR site was found to be u = 0.0013 m s−1 (Table 1), corresponding to a current amplitude of
4. Conclusions
Most internal-tide-driven mixing must be supported by high equivalent modes that have more shear, making them more likely to dissipate locally by shear instability. This is particularly true of internal tides generated at critical slopes, such as those found along supercritical topography. Wave–wave interactions are also more efficient for higher modes, providing another mechanism for converting internal tide energy flux into turbulence. The “Parametric Subharmonic Instability” interaction is particularly effective since it specifically moves energy to higher equivalent modes. However, the PSI mechanism for tidal frequencies is only effective at low latitudes. The effectiveness of other classes of wave–wave interactions at transferring wave energy to higher modes is unclear, though some ideas are presented by Polzin (1999).
The wave–wave interaction timescale for equivalent mode 10 is roughly equal to the timescale over which the mode-10 tidal beam completes a round-trip between the bottom and surface. Over this characteristic timescale, the mode-10 tidal beam radiates roughly 100 km from the generation site. Higher equivalent modes will be increasingly influenced by the wave–wave interactions over shorter time- and distance scales. We suggest that for a given internal-tide-generation event, waves with equivalent modes less than 10 are likely to radiate away, with equivalent modes greater than 10 dissipating locally. For the MAR topography considered in section 2b, this suggests that roughly one-third of the generated power, about 1–2 mW m−2, may dissipate as turbulence at this site. At other sites, the amount of energy flux available in equivalent modes greater than 10 will clearly depend on the nature of the topography, and the nature of the barotropic tides. As shown in section 2b, the energy flux production at site along the EPR is only 0.2 mW m−1.
Low equivalent modes that radiate away will be influenced by topographic scattering. Energy scattered to higher modes is more likely to dissipate, so topographic scattering provides a mechanism for driving mixing away from an internal-tide-generation site. However, the efficiency of topographic scattering is low, with roughly 10% of low-mode energy flux being transferred to higher modes. It is possible that while the scattering efficiency for general topographies is low, certain topographic sites may favor enhanced scattering. For example, Gilbert and Garrett (1989) and Müller and Liu (2000b) find convex topographies are more efficient than linear or concave topographies at scattering energy to high modes. Thus, in addition to the internal tides generated at such sites, convex topographies may be favored sites for the dissipation of incoming internal-tide energy. At such sites, the scattered waves could act as a catalyst for dissipating the locally generated waves since elevated shear levels in the scattered waves will contribute to a lower overall Richardson number.
Given the inefficient nature of topographic scattering at general topographies, it is likely that low-mode internal tides radiate much of their energy through multiple bottom reflections over O(1000 km) distances. Low modes must dissipate somewhere, but the nature of low-mode dissipation is particularly unclear. Perhaps low modes are preferentially dissipated in regions of enhanced sheared currents, such as the equatorial undercurrents or the Antarctic Circumpolar Current. Perhaps low modes propagate onto the continental shelves where they dissipate in shallow waters. Finally, perhaps low modes are dissipated throughout in the interior of the deep ocean under the slow evolution of wave–wave interactions. In this scenario, much of the energy in low-mode internal tides would be uniformly dispersed through the oceanic interior. The dissipation of this energy would contribute to background levels of turbulence everywhere. The fate of low-mode internal tides, however, remains as a primary question.
Acknowledgments
We thank Eric D'Asaro, Eric Kunze, and Bill Young for carefully reviewing the manuscript. We also thank Gary Egbert, Ming Li, Stefan Llewellyn Smith, Kurt Polzin, and John Toole for helpful discussions about this work. Multibeam bathymetry data were obtained from the RIDGE Multibeam Synthesis Project. Comments and suggestions from two anonymous reviewers also improved this presentation. The authors were supported by the U.S. Office of Naval Research and the Natural Science and Engineering Research Council of Canada.
REFERENCES
Baines, P. G., 1973: The generation of internal tides by flat-bump topography. Deep-Sea Res., 20 , 179–205.
Baines, P. G., . 1982: On internal tide generation models. Deep-Sea Res., 29 , 307–338.
Balmforth, N. J., G. R. Ierley, and W. R. Young, 2002: Tidal conversion by subcritical topography. J. Phys. Oceanogr., 32 , 2900–2914.
Bell, T. H., 1975a: Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 67 , 705–722.
Bell, T. H., . 1975b: Topographically generated internal waves in the open ocean. J. Geophys. Res., 80 , 320–327.
Cox, C. S., and H. Sandstrom, 1962: Coupling of surface and internal waves in water of variable depth. J. Oceanogr. Soc. Japan, 20th Anniversary Volume, 499–513.
Craig, P. D., 1987: Solutions for internal tide generation over coastal topography. J. Mar. Res., 45 , 83–105.
Cummins, P. F., J. Y. Cherniawski, and M. G. G. Foreman, 2001: North Pacific internal tides from the Aleutian Ridge: Altimeter observations and modelling. J. Mar. Res., 59 , 167–191.
Egbert, G. D., 1997: Tidal data inversion: Interpolation and inference. Progress in Oceanography, Vol. 40, Pergamon, 81–108.
Egbert, G. D., and R. D. Ray, 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature, 405 , 775–778.
Egbert, G. D., . 2001: Estimates of M2 tidal energy dissipation from TOPEX/POSEIDON altimeter data. J. Geophys. Res., 106 , 22475–22502.
Egbert, G. D., A. F. Bennett, and M. G. G. Foreman, 1994: TOPEX/POSEIDON tides estimated using a global inverse model. J. Geophys. Res., 99 , 24821–24852.
Eriksen, C., 1982: Observations of internal wave reflection off sloping bottoms. J. Geophys. Res., 87 , 525–538.
Eriksen, C., . 1985: Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr., 15 , 1145–1156.
Garrett, C., and D. Gilbert, 1988: Estimates of vertical mixing by internal waves reflected off sloping topography. Small-Scale Turbulence and Mixing in the Ocean. J. C. J. Nihoul and B. M. Janard, Eds., Elsevier Scientific, 405–424.
Gilbert, D., and C. Garrett, 1989: Implications for ocean mixing of internal waves scattering off irregular topography. J. Phys. Oceanogr., 19 , 1716–1729.
Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.
Godin, G., 1972: The Analysis of Tides. University of Toronto Press, 264 pp.
Goff, J. A., 1991: A global and regional stochastic analysis of near-ridge abyssal hill morphology. J. Geophys. Res., 96 , 21713–21737.
Hendershott, M. C., 1981: Long waves and ocean tides. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., The MIT Press, 292–341.
Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach. J. Geophys. Res., 91 , 8487–8495.
Hibiya, T., 1986: Generation mechanism of internal waves by tidal flow over a sill. J. Geophys. Res., 91 , 7696–7708.
Hirst, E., 1991: Internal wave–wave resonance theory: Fundamentals and limitations. Dynamics of Oceanic Internal Gravity Waves: Proc. ‘Aha Huliko’ a Hawaiian Winter Workshop, P. Müller and D. Henderson, Eds., Honolulu, HI, University of Hawaii at Manoa, 211–226.
Hirst, E., . 1996: Resonant instability of internal tides. Ph. D. dissertation, University of Washington, 92 pp.
Holloway, P. E., and M. A. Merrifield, 1999: Internal tide generation by seamounts, ridges and islands. J. Geophys. Res., 104 , 25937–25951.
Kunze, E., and J. M. Toole, 1997: Tidally driven vorticity, diurnal shear, and turbulence atop Fieberling Seamount. J. Phys. Oceanogr., 27 , 2663–2693.
Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. St. Laurent, R. W. Schmitt, and J. M. Toole, 2000: Mixing over rough topography in the Brazil Basin. Nature, 403 , 179–182.
Levitus, S., and T. P. Boyer, 1994: Temperature. Vol. 4, World Ocean Atlas 1994, NOAA Atlas NESDIS 4, 117 pp.
Levitus, S., R. Burgett, and T. Boyer, 1994: Salinity. Vol. 3, World Ocean Atlas 1994, NOAA Atlas NESDIS 3, 99 pp.
Lien, R-C., and M. C. Gregg, 2001: Observations of turbulence in a tidal beam and across a coastal ridge. J. Geophys. Res., 106 , 4575–4592.
Llewellyn Smith, S. G., and W. R. Young, 2002: Conversion of the barotropic tide. J. Phys. Oceanogr., 32 , 1554–1566.
Lott, F., and H. Teitelbaum, 1993: Linear unsteady mountain waves. Tellus, 45 , 201–220.
Lueck, R. G., and T. D. Mudge, 1997: Topographically induced mixing around a shallow seamount. Science, 276 , 1831–1833.
McComas, C. H., and P. Müller, 1981a: The dynamic balance of internal waves. J. Phys. Oceanogr., 11 , 970–986.
McComas, C. H., . 1981b: Time scales of resonant interactions among oceanic internal waves. J. Phys. Oceanogr., 11 , 139–147.
Merrifield, M. A., P. E. Holloway, and T. M. Shaun Johnston, 2001: The generation of internal tides at the Hawaiian Ridge. Geophys. Res. Lett., 28 , 559–562.
Müller, P., and N. Xu, 1992: Scattering of oceanic internal waves off random bottom topography. J. Phys. Oceanogr., 22 , 474–488.
Müller, P., and X. Liu, 2000a: Scattering of internal waves at finite topography in two dimensions. Part I: Theory and case studies. J. Phys. Oceanogr., 30 , 532–549.
Müller, P., . 2000b: Scattering of internal waves at finite topography in two dimensions. Part II: Spectral calculations and boundary mixing. J. Phys. Oceanogr., 30 , 550–563.
Müller, P., G. Holloway, F. Henyey, and N. Pomphrey, 1986: Nonlinear interactions among internal gravity waves. Rev. Geophys., 24 , 493–536.
Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45 , 1977–2010.
Nakamura, T., T. Awaji, T. Hatayama, and K. Akitomo, 2000: The generation of large amplitude unsteady lee waves by subinertial K1 tidal flow: A possible vertical mixing mechanism in the Kuril Straits. J. Phys. Oceanogr., 30 , 1601–1621.
Olbers, D. J., 1983: Models of the oceanic internal wave field. Rev. Geophys., 21 , 1567–1606.
Olbers, D. J., and N. Pomphrey, 1981: Disqualifying two candidates for the energy balance of oceanic internal waves. J. Phys. Oceanogr., 11 , 1423–1425.
Polzin, K. L., 1999: A rough recipe for the energy balance of quasi-steady internal lee waves. Dynamics of Oceanic Internal Gravity Waves, II: Proc. ‘Aha Huliko’a Hawaiian Winter Workshop, P. Müller and D. Henderson, Eds., Honolulu, HI, University of Hawaii at Manoa, 117–128.
Polzin, K. L., J. M. Toole, J. R. Ledwell, and R. W. Schmitt, 1997: Spatial variability of turbulent mixing in the abyssal ocean. Science, 276 , 93–96.
Rattray, M., 1960: On the coastal generation of internal tides. Tellus, 12 , 54–61.
Ray, R., and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23 , 2101–2104.
Ray, R., . 1997: Surface manifestation of internal tides in the deep ocean: Observations from altimetry and island gauges. Progress in Oceanography, Vol. 40, Pergamon, 135–162.
Seibold, E., and W. H. Berger, 1996: The Sea Floor—An Introduction to Marine Geology. Springer-Verlag, 356 pp.
Sjöberg, B., and A. Stigebrandt, 1992: Computations of the geographical distribution of the energy flux to mixing processes via internal tides and the associated vertical circulation in the ocean. Deep-Sea Res., 39 , 269–291.
Small, C., and D. Sandwell, 1992: An analysis of ridge axis gravity roughness and spreading rate. J. Geophys. Res., 97 , 3235–3245.
Smith, W. H. F., and D. T. Sandwell, 1997: Global sea floor topography from satellite altimetry and ship depth soundings. Science, 277 , 1956–1962.
Stigebrandt, A., 1980: Some aspects of tidal interaction with Fjord Constrictions. Estuarine Coastal Mar. Sci., 11 , 151–166.
St. Laurent, L. C., J. M. Toole, and R. W. Schmitt, 2001: Buoyancy forcing by turbulence above rough topography in the abyssal Brazil Basin. J. Phys. Oceanogr., 31 , 3476–3495.
Thorpe, S. A., 1987: On the reflection of a train of finite amplitude waves from a uniform slope. J. Fluid Mech., 178 , 279–302.
Thorpe, S. A., . 1997: On the interactions of internal waves reflecting from slopes. J. Phys. Oceanogr., 27 , 2072–2078.
Thorpe, S. A., . 1999: Fronts formed by obliquely reflecting internal waves at a sloping boundary. J. Phys. Oceanogr., 29 , 29–38.
Thorpe, S. A., . 2001: Internal wave reflection and scatter from sloping rough topography. J. Phys. Oceanogr., 31 , 537–553.
Wunsch, C., 1975: Internal tides in the ocean. Rev. Geophys., 13 , 167–182.
Diagram showing the internal wave response to tidal flow over bathymetry for the slope parameter s/α and the tidal excursion parameter ku0/ω. Internal tides occur when the scale of the topography k−1 exceeds the tidal excursion scale u0/ω. In this regime, waves radiate energy in both the up- and downstream directions [as shown by the group velocity (cg) vectors]. Quasi-steady lee waves occur when the scale of tidal excursion exceeds the scale of the topography. Wave energy is radiated in the upstream direction only
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Multibeam bathymetry data from the (a) EPR and (b) MAR. Bathymetric sections were taken along fracture zones (FZ) and across fracture zones (XFZ) to calculate spectra. Boxed regions indicate the locations of several sections described in the text
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Bathymetric spectra for the (a) EPR and (b) MAR from sections shown in Fig. 2. The standard deviations of the spectral estimates are represented by the shaded bands. In each panel, the slope spectra of the XFZ bathymetry are also shown. The wavenumber of the tidal excursion scale ω/u0 is shown for each site. The range of wavenumbers associated with subcritical and critical slopes for each XFZ spectrum is indicated
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Tidal ellipses of the barotropic M2 currents at the (a) EPR and (b) MAR sites. In both cases, the current vectors turn in the clockwise direction over the M2 tidal period. The axes defined by the semimajor and semiminor currents (Ue, Ve), and the axes aligned with FZ and XFZ directions are also shown
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Profiles of buoyancy frequency N from the (a) EPR and (b) MAR sites. Approximate exponential fits to the data are shown
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Energy flux spectra from the (a) EPR and (b) MAR sites. Standard deviations of the spectral estimates are indicated by the shaded bands. For each site, the wavenumber range for critical slopes and the tidal excursion scale are indicated. The cumulative integrals of energy flux are also shown. The equivalent modes are given along the upper axes for reference
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Spectra of bottom bathymetry from (a) an FZ section and (b) an XFZ section of MAR topography. Spectra were computed from both multibeam data and the gridded SS97 data. Standard errors of the spectra are shown
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Map showing the region of MAR topography used in calculations of internal tide energy flux. Tidal ellipses of M2 currents are also shown
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Two-dimensional spectrum of energy flux in a four-quadrant wavenumber space. The wavenumber axes were taken along the semimajor (k) and semiminor (l) axes of the tidal ellipse reference frame. Circles correspond to the equivalent modes 1, 10, and 25
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Vertical wavenumber dependence of the Richardson number for internal tides at the (a) EPR and (b) MAR sites. The standard deviations of the Richardson number estimates are indicated by the shaded bands. Equivalent modes are given along the upper axes for reference
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
The timescales for wave–wave (PSI) interactions between the M2 internal tide and the internal wave continuum for modes 1, 2, 3, 5, and 10, adapted from Olbers (1983). The PSI interaction for the M2 tide occurs equatorward of ±30° lat. Bottom reflection times for the various modes were estimated for an exponential stratification over a depth of 4000 m
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Spectra of incident and scattered mode-1 energy flux from MAR bathymetry. The wavenumber range associated with critical slopes is shown
Citation: Journal of Physical Oceanography 32, 10; 10.1175/1520-0485(2002)032<2882:TROITI>2.0.CO;2
Vertical energy flux and baroclinic currents for the M2 internal tide calculated for a section of XFZ Mid-Atlantic Ridge topography. A barotropic current amplitude of 0.026 m s−1 was used in (2) and (7) for these estimates. Values for other parameters are given in the text