On the Fraction of Internal Tide Energy Dissipated Near Topography
Louis C. St. Laurent Department of Oceanography, Florida State University, Tallahassee, Florida
Jonathan D. Nash College of Oceanic and Atmospheric Science, Oregon State University, Corvallis, Oregon
Abstract. Internal tides have been implicated as a major source of me- chanical energy for mixing in the ocean interior. Indeed, microstructure and tracer measurements have indicated that enhanced turbulence levels occur near topography where internal tides are generated. However, the details of the energy budget and the mechanisms by which energy is transferred from the internal tide to turbulence have been uncertain. It now seems that the energy levels associated with locally enhanced mixing near topography may constitute only a small fraction of the available internal tide energy flux. In this study, the generation, radiation, and energy dissipation of deep ocean internal tides are examined. Properties of the internal tide at the Mid-Atlantic Ridge and Hawaiian Ridge are considered. It is found that most internal tide energy is generated as low modes. The Richardson number of the generated internal tide typically exceeds unity for these motions, so direct shear instability of the generated waves is not the dominant energy transfer mechanism. It also seems that wave-wave interactions are ineffective at transferring energy from the large wavelengths that dominate the energy flux. Instead, it seems that much of the generated energy radiates away from the generation site in low mode waves. These low modes must dissipate somewhere in the ocean, though this likely occurs over O(1000 km) distances as the waves propagate. Additionally, a small fraction of energy flux is generated as high mode waves. These have slower group speeds, higher shear, and faster wave-wave interaction rates than the low modes, and so are likely to dissipate as turbulence close to the topography of the generation site. Preliminary estimates suggest that turbulent dissipation near sites of generation accounts for 30% or less of the total generated energy flux. Based on these estimates, a parameterization of the mixing sustained by high mode internal tide generation is proposed.
1. Introduction 1998; Egbert and Ray, 2000; 2001; Jayne and St. Lau- rent, 2001) have implicated the internal tides as a major Breaking internal waves are the main cause of di- source of mechanical energy for mixing, providing up to apycnal mixing in the ocean’s interior. They may be 1 TW of power to the deep ocean. While some studies generated by wind at the sea surface or by flow over suggest this amount of energy is sufficient for powering the topography of the seafloor. While mean flows and the mixing that closes the meridional overturning cell eddy activity in the deep ocean are characterized by of the ocean’s thermohaline circulation (Webb and Sug- −1 O(1) mm s currents, deep barotropic tidal currents inohara, 2001), other studies suggest that internal tides −1 are O(10) mm s and may be an important source provide only half of the needed energy (Munk and Wun- of internal waves. Recent studies (Munk and Wunsch,
45 46 ST. LAURENT AND NASH
sch, 1998). Wind forcing may also supply a significant enhanced turbulence at Cobb Seamount in the north- fraction of the energy input to the internal wavefield east Pacific to a semidiurnal internal tide. They found through the generation of near-inertial swell (Alford, that the largest turbulence levels occurred along a beam 2001,2003a). This highlights the need for studies to of internal tide energy emanating from the seamount examine the relative contributions from each of these rim. Kunze and Toole (1997) describe microstructure sources to internal waves and mixing. observations from Fieberling Seamount in the north- Egbert and Ray (2000, 2001) have identified a number east Pacific. They found the largest mixing levels at of deep ocean regions where barotropic tidal energy is the summit of the seamount where diurnal internal tide likely being transferred to internal tides. These regions energy was trapped. Internal tides are also produced are generally associated with three types of topogra- along the continental margins of the ocean basins, and phy: (i) oceanic islands, (ii) oceanic trenches, and (iii) evidence of internal tide driven mixing at Monterey mid-ocean ridges. Oceanic islands such as Hawaii, and Canyon off California was reported by Lien and Gregg oceanic trenches such as those in the west Pacific Ocean, (2001) and Kunze et al. (2002a). There, enhanced tur- are steep topographic features with typical slopes (s; bulence levels were found along the ray path of a semi- rise over run) of 0.1 to 0.3 (Seibold and Berger, 1996). diurnal internal tide beam extending out a distance of Egbert and Ray (2000, 2001) identify the oceanic islands 4 km away from the topography. Althaus et al. (2003) of Micronesia and Melanesia as sites accounting for over present fine structure observations of internal tides at 100 GW of internal tide production, while Hawaii ac- Mendocino Escarpment. In addition to finding evidence counts for 20 GW. They also identify mid-ocean ridge for strong mixing near the topography, they find en- topography in the Atlantic and Indian Oceans as each hanced mixing at sites where beams of internal tide en- accounting for over 100 GW of internal tide production. ergy refract beneath the sea surface. These topographic features have slopes that vary over Evidence of internal tide driven mixing from the a wide range of spatial scales. truly deep ocean was found during the Brazil Basin Tracer Release Experiment (BBTRE; Polzin et al., a. Observations of internal tides and mixing 1997; Ledwell et al., 2000; St. Laurent et al., 2001). This study was conducted near the Mid-Atlantic Ridge. The contribution of internal tides to oceanic ve- St. Laurent et al. (2001) present vertically integrated locity and temperature records has long been recog- dissipation data that are modulated over the spring- nized. Wunsch (1975) and Hendershott (1981) present neap tidal cycle with a small lag of about a day. reviews of earlier work. Renewed interest in internal Maximum levels of dissipation, vertically integrated to tides came with observations of sea surface elevation by 2000 m above the bottom, reach 3 mW m−2. These the Topex/Poseidon altimeter. These data have been observations span a network of fracture zones west of used to constrain hydrodynamic models of the tides, the Mid-Atlantic Ridge, with topographic relief varying and hence produce accurate estimates of open-ocean by up to 1 km between the crests and floors of abyssal barotropic tides (Egbert et al., 1994; Egbert, 1997). A canyons. The dominant topography of the fracture zone multi-year record of satellite observations is now avail- system has a slope less than 0.1 as the canyons are gen- able, and the semidiurnal tides (M2 and S2)canbe erally 30 to 50 km wide. Elevated turbulent dissipa- dealiased from the record. Ray and Mitchum (1996) tion rates were found along all regions of the fracture used along-track Topex/Poseidon records to examine zone topography, but turbulence levels appeared most the surface manifestation of internal tides generated enhanced over the slopes. Above all classes of topogra- along the Hawaiian Islands. They found that both first phy, turbulence levels decrease to background levels at and second baroclinic modes of the semidiurnal inter- heights above bottom greater than about 1000 m. nal tides were present in the data, and that the sig- Observations of deep, tidally driven mixing have also nal of internal-tide propagation could be tracked up to been made as part of the Hawaii Ocean Mixing Exper- 1000 km away from the Hawaiian Ridge. In a further iment (HOME; Pinkel et al., 2000). Both near- and analysis of the altimetry data, Ray and Mitchum (1997) far-field observational components were conducted, and estimate that 15 GW of semidiurnal internal tide en- fine- and microstructure measurements allowed for es- ergy radiates away from the Hawaiian Ridge in the first timates of tidal energy flux and turbulent dissipation. baroclinic mode. Preliminary HOME observations to be reported here Microstructure observations have provided direct ob- were presented by Kunze et al. (2002b). servations of mixing driven by internal tides at steep Figure 1 shows turbulent dissipation profiles from topography (Lueck and Mudge, 1997; Kunze and Toole, several sites where internal tides are believed to sup- 1997; Lien and Gregg, 2001; Kunze et al., 2002a,b; Al- port mixing. Average profiles from sites of internal tide thaus et al., 2003). Lueck and Mudge (1997) attributed INTERNAL TIDE ENERGY DISSIPATED NEAR TOPOGRAPHY 47
generation include the Mid-Atlantic Ridge of the Brazil Basin (Polzin et al., 1997; Ledwell et al., 2000; St. Lau- rent et al., 2001), and the 3000-m isobath of the Hawai- ian Ridge (Kunze et al., 2002b). The latter was aver- aged from 16 stations spanning roughly ∼1000 km along the Hawaiian Ridge. These stations were occupied over a 3-week period during October 2000 and capture en- ergy flux radiated from both strong and weak sites of internal tide generation along the ridge. Profiles from the Oregon Slope (Moum et al., 2002) and the Virginia Slope (Nash et al., 2003) are also shown, as turbulence at these sites is likely supported by dissipating low mode internal tides. At all sites, the dissipation rates are max- imum along the topography, and decay away from the topography with height. Enhanced turbulence levels are found to extend up to 1000 m from the bottom. The energy flux carried by the internal tide can radi- ate as propagating internal waves, and these waves are subject to a collection of processes that will eventually lead to dissipation. Shear instability, wave-wave inter- actions, and topographic scattering all act to influence the rate of dissipation and control whether the inter- nal tide dissipates near the generation site or far away. Understanding the physical cascade that allows energy in the internal tide to power turbulence is one goal of ocean mixing research.
2. Internal tide energy flux
Several nondimensional parameters are needed to model the physical regime of internal tide generation. One parameter, kU0/ω, measures the ratio of the tidal excursion length scale U0/ω to the length scale of −1 the topography k . This parameter is discussed by Figure 1. Average profiles of turbulent dissipation Bell (1975) and others, and distinguishes a wave re- from several sites where internal tides support mixing. sponse dominated by the fundamental tidal frequency Oregon Slope data (Moum et al., 2002; shown with 95% (kU0/ω < 1) from a lee-wave response involving higher confidence intervals) and Virginia Slope data (Nash et tidal harmonics (kU0/ω > 1). A second parameter, al., 2003) show mixing supported by the dissipation of δ = h0/H, measures the ratio of the topographic am- low mode internal tides. The 95% confidence inter- plitude h0 to the total depth H. A third parame- vals for data from Brazil Basin fracture-zone valleys, ter, s/α, measures the ratio of the maximum topo- crests, and slopes are shown as blue, green, and red graphic slope s = |∇h| to the ray slope given by shaded bands, respectively, as described by Ledwell et 1/2 α = ω2 − f 2 / N 2 − ω2 . This parameter also al. (2000). Dissipation at the 3000-m isobath of the distinguishes two regimes. In the case of s/α < 1, the Hawaiian Ridge was derived from data described in the topographic slopes are less steep than the radiated tidal text. beam, and internal wave generation is termed subcrit- ical. In the case of s/α > 1, the topographic slopes exceed the steepness of the radiated beam and the in- ternal wave generation is termed supercritical. The crit- ical generation condition is met when the radiated tidal beam is aligned with the slope of the topography. The subcritical generation of internal tides was first considered by Cox and Sandstrom (1962), Baines (1973), 48 ST. LAURENT AND NASH
and Bell (1975). These studies examined subcritical Models for generation at abrupt depth discontinu- topography in the limit of δ 1ands/α 1, for ities (s/α = ∞) have also been formulated. Rattray which the bottom boundary condition can be linearized (1960) considered the first-mode baroclinic response to w(−H)=U ·∇h. In this case, the internal tide to barotropic flow over shelf topography. Stigebrandt generation problem can be solved for topography of ar- (1980) also considered a topographic step and exam- bitrary shape by Fourier decomposition. Later studies ined the full spectrum of baroclinic modes. In addition have examined the initial transient wave response, fi- to re-examining internal tides at a step, St. Laurent et nite depth effects, depth-varying stratification, and spa- al. (2003) consider generation at a knife-edge ridge, a tial variations in topography and tidal forcing (Hibiya, top hat-ridge, and a top-hat trench. They argue that 1986; Khatiwala, 2003; Li, 2003; Llewellyn Smith and the knife-edge serves as a simple model for calculating Young, 2002; St. Laurent and Garrett, 2002). Of cen- internal tide energy flux from tall oceanic ridges. tral interest in all studies is the production of baro- Here, we present two internal tide energy flux esti- clinic energy as the barotropic tide responds to changes mates. For the case of internal tide generation at mid- in topography. In the limit of small tidal excursion ocean ridge sites, the dominant energy flux is generated (kU0/ω 1), the conversion rate is equal to the en- by bathymetry that is subcritical (s/α < 1). St. Lau- E ergy flux ( f ) away from the topography. Llewellyn rent and Garrett (2002) used a linearized model to ex- Smith and Young (2002) show that the energy flux in amine the energy flux spectrum for the semidiurnal in- the limits of s/α 1andkU0/ω 1is ternal tide at a Mid-Atlantic Ridge site (Fig. 2a). The ∞ energy flux spectrum is clearly “red,” with modes 1-10 −2 ∗ Ef = E0H knh˜(kn)h˜ (kn)∆k, (1) accounting for 60% of the total energy flux. This can n=1 be contrasted to an estimate for supercritical topogra- phy (s/α > 1). St. Laurent et al. (2003) considered k n k αnπ/H where n = ∆ = denote the horizontal energy flux generated by tidal flow over a knife-edge h˜ wavenumbers of the resonant modes and is the Fourier ridge (Fig. 2b). In this calculation, the ridge height transform of the topography. Here, was taken as 3/4 of the water depth (δ =0.75), with a −1 2 2 2 2 1/2 deep water barotropic current amplitude of 0.015 m s . 1 (N − ω )(ω − f ) E ρ U 2H2 St. Laurent et al. (2003) argue that the knife-edge 0 = π ω 0 (2) 2 is a good model for general tall ridges in the ocean. is a convenient metric for the energy flux amplitude, Here, the knife-edge calculation is compared to an ob- with the rest of (1) being nondimensional. St. Laurent servational based estimate of internal tide energy flux and Garrett (2002) have discussed the representation of from the Hawaiian Ridge (Fig. 2b). The observationally (1) using an integral over a continuous spectrum rather based estimate was calculated using the measured per- than a summation over discrete modes. turbation pressure and velocity profiles from the 3000- Internal tide generation at topography of finite steep- m isobath. These measured properties were projected ness was considered by Baines (1982). In that study, ray onto the first fourteen flat-bottom internal wave modes. tracing methods were developed to permit the full range The same dimension values for buoyancy frequency −1 −5 of δ and s/α. The integral equations derived by Baines (N =0.004 s ) and Coriolis frequency (5.5 × 10 −1 (1982) are difficult to apply to arbitrary topography, s ) were used for both the knife-edge and observa- and Baines did not examine the sensitivity of the con- tional estimates; wave modes for the knife-edge ridge version rate to the steepness of the topography. Using were calculated for a water depth of 3000 m. As shown an approach similar to Baines, Craig (1987) specifically in the figure, both the knife-edge and observationally considered a continental slope of finite steepness, but based spectra are red. Furthermore, the spectra show only considered s/α ≤ 2. Taking a different approach, excellent correspondence in modal content. Modes 1-10 St. Laurent and Garrett (2002) examined the first order account for more than 95% of both the modeled and correction to linear theory estimates of energy flux for observed energy flux; in fact, modes 1 and 2 account sinusoidal topography of subcritical steepness (s/α < for 88% of the baroclinic energy flux in both the obser- 1 2 vationally based and model estimate. The generation 1), finding that Ef = Ef(linear) 1+ 4 (s/α) + ··· . Full series expansions for the full corrections for sinu- of low mode internal tides is clearly favored by both soidal and gaussian topographies were derived by Balm- rough mid-ocean ridge topography and abrupt Hawaii- forth et al. (2002) for increasing steepness up to the like ridges alike. critical condition (s/α = 1). They find increased lev- The shear (uz) of the generated internal tides is also 2 2 els of conversion at critical slopes, though the increase of interest, as the Richardson number (Ri = N /uz ) varies from only 14% for the gaussian ridge to 56% for is a crucial stability parameter for baroclinic flow. St. the sinusoidal case. Laurent and Garrett (2002) examined the shear of in- INTERNAL TIDE ENERGY DISSIPATED NEAR TOPOGRAPHY 49 log Energy Flux
−2 −1 4.3 mW m 10 kW m Integral
Figure 2. (a) Energy flux spectrum for subcritical internal tide generation at a Mid-Atlantic Ridge site. Multibeam bathymetry data were used to calculate the spectrum, and the standard error confidence band is shown. The spectrum has units of (mW m−2)/(rad m−1), and the net integrated energy flux is 4.3 mW m−2. (b) Energy flux for supercritical internal tide generation at a knife-edge ridge, assumed to be a simple representation of the energy flux spectrum from the Hawaiian Ridge. The spectrum has units of (kW m−1)/(rad m−1), and the net integrated energy flux is 10 kW m−2. In both panels, the normalized cumulative integral of energy flux is shown. In b, an observationally based estimate of the energy flux spectrum at the 3000-m isobath of the Hawaiian Ridge is shown (confidence band in blue shading) .
mode number mode number 1 2 3 5 10 30 100 300 1 2 3 5 10 30 100 300 8 10 a. MAR b. Hawaii
6 10
4 10
2 Richardson Number 10
0 10
10−3 10−2 10−1 10−3 10−2 10−1 vertical wavenumber m (rad m−1) vertical wavenumber m (rad m−1)
Figure 3. The Richardson number estimated by integrating the cumulative shear variance of increasing modes for internal tides generated at (a) a mid-ocean ridge and (b) a knife-edge ridge. In both panels, the first 300 baroclinic modes are shown. A Richardson number of unity is indicated for reference. 50 ST. LAURENT AND NASH
3. Internal tide radiation and decay
Given the dominance of low mode generation, much of the internal tide’s energy flux will radiate away from the generation region. Direct evidence for radiated low modes has been documented at a number of in- 3
[km] 10 ε ternal tide generations sites. Ray and Mitchum (1996) L have examined the sea-surface elevation signature of low mode semidiurnal tides generated at Hawaii. They find that a coherent signal of radiation extending over 1000 km to the north and south of the ridge. Ray
decay length and Mitchum (1997) estimate that the mode-1 radia- 1.0 2 10 tion from Hawaii accounts for 15 GW of energy flux. 2.0 This compares to 20 GW as estimated by Egbert and Ray (2000) for the total barotropic tidal power loss at 0 1 10 10 Hawaii. In an approach similar to Ray and Mitchum, integrated energy flux ∫ 0
−1 sociated with the largest energy fluxes. In other words, 10 104 the integrated dissipation is fractionally less important mode number at sites of strong energy flux. We postulate the pres- 1 0 bottom reflection time 2 ence of dissipative high modes is only weakly related to 10 5 3 10 the amount of low mode energy flux generation, so that 5 dissipation occurs in higher proportion at sites of weak 10
generation. 1 10 5 106
In the following we consider the processes that influ- time (days) 3 time (sec) ence the evolution of the radiated internal tide spectrum 2 and contribute to its decay. These include wave-wave 2 10 wave interaction time 7 interactions and bottom scattering. The efficiency of 10 1 these processes will determine whether the internal tide
dissipates near the generation region, or far away. 3 10 108 3 4 5 10 20 30 a. Wave-wave interactions latitude The transfer of energy between wave modes due to wave-wave interactions has been the focus of a number Figure 5. The time scales for wave-wave (PSI) interac- of studies, and a review of this subject was given by tions between the M2 internal tide and the internal wave M¨uller et al. (1986). Olbers and Pomphrey (1981) and continuum for modes 1, 2, 3, 5, and 10, adapted from Hirst (1991, 1996) specifically examined the wave-wave Olbers (1983). The Parametric Subharmonic Instability ◦ interactions between the internal tide and the internal- interaction for the M2 tide occurs equatorward of ±30 wave continuum, and estimated the time scale for the latitude. Bottom reflection times for the various modes energy transfer away from modes 1-10 to higher modes. were estimated for an exponential stratification over a They considered the Parametric Subharmonic Instabil- depth of 4000 m. ity (PSI) resonant interaction, which transfers tidal en- ω ergy with frequency to higher vertical wavenumbers z/b curves for τg were computed using N = N0e with with frequency ω/2. Thus, the PSI mechanism is an ef- −1 (b, N0) =(1300 m, 0.005 s ), and the results are shown fective mechanism for dissipating the M2 internal tides ◦ in Figure 5. Low modes have the fastest group speeds, at latitudes less than about 30 . Estimates of the dis- with τg ≤ 12 hrs for mode 1 and τg ≥ 4daysfor sipation time scale for the M2 internal tides by PSI are given in Figure 5 (adapted from Olbers, 1983). The mode 10. Thus, low modes will easily complete one round trip between the bottom and surface before ex- transfer time scale is O(100) days for mode 1, consis- periencing the effects of wave-wave interactions. For tent with the observed weak spatial decay of low mode j ≥ internal tides generated at the Hawaiian Ridge (Ray modes 10, energy transfer to higher modes will oc- cur over a time scale comparable to, or faster than, the and Mitchum, 1996) and the Aleutian Ridge (Cummins bottom reflection time. Thus, wave-wave interactions et al., 2001). The PSI transfer time scale decreases to may be the dominant mechanism of spectral evolution O(1) day for mode 10. Hirst (1996) discusses conditions for high mode components of the internal tide, while where strong tidal forcing can result in reduced transfer time scales, even for low mode internal tides. bottom reflection may be significant for low modes. Other studies have considered the additional reso- b. Bottom reflection and topographic nant interactions occurring in the general internal wave continuum over the range of frequencies between f scattering and N. The “elastic scattering” and “induced diffu- The interaction of internal waves with bottom bathym- sion” interactions may influence internal tides at all etry is often discussed in terms of wave reflection from latitudes. In all cases, wave interaction time scales sloping topography and wave scattering from rough for low modes with tidal frequencies typically exceed topography. Particular attention has been given to re- O(10) days M¨uller et al., 1986; Hirst, 1991). flection from planar slopes (e.g., Eriksen, 1982). Reflec- It is useful to compare the wave interaction time tion from slopes with nonplanar geometries has been scale to the time for a wave group to travel from bot- considered by Gilbert and Garrett (1989) and M¨uller tom to surface and back again, defined according to and Liu (2000a,b). Scattering of waves from random τ c−1dz, g =2 g where the integral is over the ocean rough topography was considered by M¨uller and Xu depth and cg is the vertical component of the group (1992), while Thorpe (2001) considered scattering from speed. The latitude and mode number dependent rough topography with an underlying slope. The latter 52 ST. LAURENT AND NASH
studies are relevant to our discussion of low mode inter- the the Hawaiian Ridge can also be used to estimate nal tides from arbitrary topography in the deep ocean. the local dissipation fraction. These estimates imply As discussed by M¨uller and Xu (1992), both re- that at least 75% of the generated energy flux is radi- flection and scattering redistribute the energy flux in ated to the far field in mode 1, leaving q ≤ 0.25 as the wavenumber space. M¨uller and Xu (1992) find that fraction that dissipates near the ridge. Two considera- scattering is more efficient than reflection at redistrib- tions make this preliminary estimate uncertain. First, uting energy flux to higher modes, particularly at tidal Ray and Mitchum’s 15 GW only accounts for the energy and near inertial frequencies. However, M¨uller and Xu flux in the lowest mode. Direct observations (Figure 2b (1992) find less than 10% of the incoming radiation is and Kunze et al., 2002b), indicate that modes 2 and redistributed during a scattering event at typical deep higher carry an appreciable fraction of the energy flux. ocean topography. Most of the incoming radiation un- Second, some of the 20 GW of barotropic power loss at dergoes pure reflection as from a flat bottom, such that Hawaii may be dissipated by frictional drag along the the wavenumber content of the spectrum is mostly pre- shallow areas of the ridge. Further analysis of HOME served. St. Laurent and Garrett (2002) examined the observations will establish a more certain estimate. scattering of the mode-1 internal tide through a “second Althaus et al. (2003) have also estimated the lo- generation” calculation. They find mode-1 scattering cal dissipation fraction using their measurements of occurs with an efficiency of about 10%, comparable to finestructure and microstructure from the Mendocino the efficiencies estimated by M¨uller and Xu (1992). Escarpment site. They used the method of Kunze et al. (2002a) to estimate the baroclinic energy flux vp of 4. Dissipation at generation sites radiated internal tides. It is possible to rewrite (4) as Sites of strong internal tide generation can both radi- ρ dz q = . (5) ate low mode internal waves that propagate O(1000) km ∇· vp dz + ρ dz distances before dissipating and generate high-wavenumber motions which dissipate locally. Here we attempt to as- Althaus et al. (2003) report that q 0.01 for the local sess the fraction of internal tide energy flux that dissi- dissipation fraction along the ridge crest of Mendocino pates as turbulence near generation regions. We define Escarpment. Dissipation occurring at sites where the the “local dissipation fraction” as, ridge-generated beam refracts from the sea surface is ρ dz not included in this estimate. q = , (4) It is clear that the local dissipation fraction may have Ef an order-of-magnitude variability (q ∼0.01–0.3). This where the column integrated dissipation in (4) is space/time likely depends on the details of the generating topog- E averaged, and f is the average energy flux per unit raphy (e.g., δ, s/α, kU0/ω), and a multitude of other area. factors. Direct measurements of turbulent dissipation rates above Mid-Atlantic Ridge topography have been de- 5. Parameterizing local mixing scribed by St. Laurent et al. (2001). These data were collected over two 4-week sample periods in a re- St. Laurent et al. (2002) have suggested a semi- gion between 12◦ − 22◦W longitude and 20◦ − 25◦S empirical operational scheme for estimating enhanced latitude above fracture zone topography in the Brazil turbulent mixing rates in regions where internal tides Basin. The average dissipation rate, vertically inte- are generated. They build on a parameterization of in- grated to a height of 2000 m above the bottom, was ternal tide energy flux used with a hydrodynamic model 1.2 ± 0.2mWm−2. If this dissipated energy is as- for the tides (Jayne and St. Laurent, 2001). Jayne and sumed to be provided by the internal tide, it can be St. Laurent (2001) estimated the energy flux as compared to estimates of generated internal tide en- 1 2 2 ergy flux. St. Laurent and Garrett (2002) considered Ef ρNbκh U . (6) theinternaltideenergyflux over the region between 2 ◦ ◦ ◦ ◦ 8 −20 W longitude and 22 −32 S latitude, and found Here ρ is the reference density of seawater, N is the −2 b 4 ± 1mWm as the spatially averaged generated buoyancy frequency along the seafloor, and (κ, h)are power. Taken together, these dissipation and genera- the wavenumber and amplitude scales for the topo- tion estimates suggest an average local dissipation frac- graphic roughness. The barotropic tidal speed U is tion of q =0.3 ± 0.1. determined through solution of the Laplace tidal equa- Comparison of the estimates of Ray and Mitchum tions. Further details of the barotropic model and the (1997) and Egbert and Ray (2000) for internal tides at tidal simulation are given by Jayne and St. Laurent INTERNAL TIDE ENERGY DISSIPATED NEAR TOPOGRAPHY 53
(2001). They find that estimates from (6) account for ζ is the vertical decay scale, and 0 is the maximum about 1 TW of energy flux into the deep ocean. This dissipation rate at the bottom, z = −H. This gives provides a constraint on the amount of energy available −( + ) for the enhancement of turbulence near sites of internal e H z /ζ F (z)= . (9) tide generation. However, as we have discussed, not all ζ(1 − e−H/ζ) of the generated energy flux contributes to local mix- ing, since some energy radiates away from the genera- St. Laurent et al. (2002) examined a number of tion sites as low mode waves. Only q · Ef is dissipated diffusivity sections estimated using (8) and (9) with locally. ζ = 500 m. We note that this choice is consistent Motivated by the dissipation and generation esti- with profiles of shown in Figure 1, but is otherwise mates from the abyssal Brazil Basin, we propose an somewhat arbitrary. Stratification estimates were de- initial estimate of the local dissipation fraction of q = rived from Levitus and Boyer (1994) and Levitus et al. 0.3 ± 0.1. We have limited information on which to as- (1994). These reveal significant spatial variations in sess the variability of this parameter for different in- tidal mixing, especially with respect to spatial varia- ternal tide generation regions, though we suspect it to tions in bottom roughness. depend on the “color” of the generated energy flux spec- As an example, a zonal section of diffusivity, aver- trum. While we expect q 0.3 to be appropriate for aged in the latitude band between 24◦Sand28◦S, is the mid-ocean ridge regions similar to the Brazil Basin, shown in Figure 6 for the South Atlantic. The Mid- it may be an overestimate for regions like the Mendo- Atlantic Ridge (15◦W), and the Walvis Ridge (5◦E) cino Escarpment (Althaus et al., 2003) or the Hawaiian are sites where elevated levels of diffusivity reach the Ridge. upper ocean. At abyssal depths, diffusivity estimates Our model for the turbulent dissipation rate follows in the Angola Basin exceed those in the Brazil Basin. as This signal is tied to differences in abyssal stratifica- (q/ρ)Ef (x, y)F (z), (7) tion, with weaker stratification occurring in the An- gola Basin. Since the internal tide energy flux is pro- E x, y where f ( ) is the time-averaged map of generated portional to N , the diffusivity scales as N −1 at these F z b b internal tide energy flux, and ( ) is a function for the abyssal depths. In Figure 6, it appears that most en- vertical structure of the dissipation, chosen to satisfy hanced levels of mixing occur within the envelope (the energy conservation within an integrated vertical col- min/max range in local bathymetry) of rough topogra- 0 F z dz . umn, −H ( ) =1 The parameterization for the phy characterizing the section. Moreover, the vertical turbulent diffusivity follows from the Osborn (1980) re- extent of strongest enhancement generally does not fill lation for the mechanical energy budget of turbulence, the vertical extent of the topography envelope. This is qE x, y F z consistent with observations of turbulence above rough k Γ f ( ) ( ). v ρN 2 (8) topography (e.g., Figure 2 of St. Laurent et al., 2001). Our section of diffusivity across the Brazil Basin (Fig. 6) Here, Γ is related to the mixing efficiency of turbu- may be compared to the section of Polzin et al. (1997), lence, and Γ = 0.2 is generally used (Osborn, 1980). with notice to the significant differences in color axes The diffusivity given in (8) is meant to quantify only used in both presentations. In addition, Polzin et al. the enhanced levels of mixing supported by the topo- (1997) collapsed two survey transects into one zonal sec- graphically enhanced dissipation of internal tide energy. tion and show bathymetry from only one transect; en- A background diffusivity of 0.1 × 10−4 m2 s−1 is still hanced diffusivity appearing at significant heights above assumed to be sustained in all regions, and must be this bathymetry may therefore occur much closer to the added to the estimate from (8). Contributions to the bottom. diffusivity from other sources of enhanced mixing, such This mixing scheme is a preliminary attempt to for- as turbulence in critical layers and frictional boundary mulate internal tide dissipation in terms of the topo- layers, could also be added to (8). graphically generated internal tide. It simply assumes In the estimates we present here, our selection of the that the internal tide energy flux is generated accord- vertical structure function F (z) in (7) and (8) is mo- ing to (6) and that a fraction (q =0.3) of the energy tivated by turbulence observations shown in Figure 1. flux is dissipated locally. A more physically based rep- These observations suggest that the dissipation rate has resentation would predict values of q and ζ that vary a similar structure at many sites, and can roughly be spatially and depend on the details of the bathymetry modeled as an exponential function that decays away and the tidal forcing. At this point, however, it is dif- from the topography. Specifically, we use a dissipation ficult to accurately evaluate the value of q even in a profile of the form (z)=0 exp(−(H + z)/ζ), where region where substantial fine- and microstructure ob- 54 ST. LAURENT AND NASH
0.1 0.3 1 3 10 30 −4 2 −1 k v (10 m s ) 0
1000
2000
3000
depth (m) 4000
5000 BrazilMid−Atlantic Angola Ridge 6000 Basin Basin −30o −20o −10o 0o 10o longitude
Figure 6. Estimated diffusivity section for the latitude band between 24◦ − 28◦S for the Atlantic. A background diffusivity of 0.1 × 10−4 m2 s−1 has been added to the parameterized estimates from (8). The envelope defined by deepest (shaded) and shallowest (white line) bathymetry in the 4◦ latitude band is indicated. servations have been made, so it makes little sense to estimated using the characteristic equation for a tidal introduce additional tunable parameters into this simple beam, dx/dz = α−1, such that parameterization. 0 N 2 z − ω2 1/2 This approach is meant to be a first-round effort to x ( ) dz. ∆ =2 ω2 − f 2 (10) model enhanced mixing rates in regions of internal tide −H generation. It is intended for application in OGCMs, For typical stratification in the deep ocean, this radia- particularly those used at coarse resolution to study the tion distance is ∆x 100 km. Thus, waves correspond- long time scales associated with the thermohaline cir- ing to low modes can be expected to travel O(100) km culation. A preliminary implementation of the parame- before scattering from topography. During topographic terization in such a model is discussed by Simmons et scattering, much of the low mode energy flux is pre- al. (2002), whose initial results suggest that the equi- served, suggesting that low modes can radiate through librium behavior of OGCM simulation is significantly many bottom encounters. Observations of internal tides improved by this new mixing scheme. Thus, it is hoped generated at Hawaii (Ray and Mitchum, 1996) and the that the current parameterization, though crude, may Aleutian Ridge (Cummins et al., 2001) demonstrate serve as a useful tool in studies of long term climate. that low modes can be tracked for O(1000) km away from their generation site. 6. Discussion Mixing driven by the internal tide must be sup- ported by a wavefield with sufficient amplitude and Having considered the various mechanisms that can high-wavenumber content such that the superposition act to degrade internal tides into turbulence, it is possi- of waves is susceptible to shear- or convective-instability. ble to speculate about the nature of internal tide driven At sites where critical generation occurs, elevated levels mixing in the deep ocean. Low mode waves are rela- of shear along radiated beams will support enhanced tively uninfluenced by wave-wave interactions and their turbulence. Additionally, the transfer of low mode en- Richardson numbers are large. In regions where addi- ergy to smaller scales will be enhanced in regions of tional sheared currents are absent, low modes will prop- rough or near-critical topography where wave reflec- agate away from their generation sites, carrying much tion and scattering occur. Wave-wave interactions act of the generated energy flux. The lateral distance for efficiently on higher wavenumbers, providing another one round trip between the bottom and surface can be mechanism for converting internal tides into turbulence. INTERNAL TIDE ENERGY DISSIPATED NEAR TOPOGRAPHY 55
Figure 7. Spatial distribution of cross-isobath energy flux and inferred eddy diffusivity along a ridge top (top row) and through its neighboring gully (bottom row) over the continental slope of the Mid-Atlantic Bight. The energy flux computed from projections onto the mode-1 structure functions is shown in a and d; the high wavenumbers (total flux minus mode 1) are shown in b and e. The eddy diffusivity inferred from the Gregg-Henyey scaling applied to the finescale shear (c and f) is intensified near the bottom below 1000 m, approaching 10 × 10−4 m2 s−1. The low mode convergence exceeds the high-wavenumber divergence; the excess is more than sufficient to supply the observed dissipation. 56 ST. LAURENT AND NASH
The Parametric Subharmonic Instability interaction is ple bottom reflections over O(1000) km distances. Low particularly effective, since it specifically moves energy modes must dissipate somewhere, but the nature of to higher modes. However, the PSI mechanism for tidal low mode dissipation is particularly unclear. Perhaps frequencies is only effective at low latitudes. The effec- low modes are preferentially dissipated in regions of en- tiveness of other classes of wave-wave interactions at hanced sheared currents, such as the equatorial under- transferring wave energy to higher modes is unclear, currents. Perhaps low modes propagate onto the conti- though some ideas are presented by Polzin (1999). Nu- nental shelves, where they produce bores and solibores, merical studies such as those by MacKinnon and Win- or simply dissipate in shallow waters. Nash et al. (2003) ters (2003 – this volume) will further elucidate the present observations from the continental slope of the mechanisms of this wave-to-turbulence energy transfer. U.S. Mid-Atlantic Bight (Fig. 7). They suggest the The wave-wave interaction time scale for mode 10 is incoming low mode internal tide is reflected and scat- roughly equal to the time scale over which the mode tered into high wavenumbers as it encounters the near- 10 tidal beam completes a round trip between the bot- critical Virginia Slope. The resultant high wavenumbers tom and surface. Over this characteristic time scale, presumable decay locally to produce the observed ele- the mode 10 tidal beam radiates roughly 100 km from vated dissipation and mixing rates. Nash et al. (2003) the generation site. Higher modes will be increasingly suggest that many continental slopes are near-critical influenced by the wave-wave interactions over shorter with respect to a semidiurnal characteristic. Further- time and distance scales. We suggest that for a given more, it may be possible to model the elevated dissipa- internal tide generation event, waves with mode num- tion in these regions using a model for wave scattering ber less than 10 are likely to radiate away, while those from large scale bathymetry data (Smith and Sandwell, greater than 10 dissipate locally. For the MAR site dis- 1997), provided that the incoming low mode energy flux cussed in Section 2, this suggests that roughly 1/3 of the is known. Improved global estimates of the long-range generated power, about 1-2 mW m−2, may dissipate as propagation of the internal tide (i.e., Alford 2003b) may turbulence at this site. For our knife-edge model of the help to quantify these low mode waves and their even- Hawaiian Ridge, modes greater than 10 represent only tual fate. 5% of the generated power. At other sites, the amount Finally, perhaps low modes are dissipated through- of energy flux available in modes greater than 10 will out the interior of the deep ocean, under the slow evo- clearly depend on the details of the energy spectrum lution of wave-wave interactions. In this scenario, much and on the amplitude of the tidal forcing. of the energy in low mode internal tides would be uni- Low modes that radiate away will be influenced by formly dispersed through the oceanic interior. The dis- topographic scattering. Energy scattered to higher sipation of this energy would contribute to background modes is more likely to dissipate, so topographic scat- levels of turbulence everywhere. The fate of low mode tering provides a mechanism for driving mixing away internal tides, however, remains as a primary question. from an internal tide generation site. However, the efficiency of topographic scattering is low, with only Acknowledgments. Data presented in this report were a small fraction of low mode energy flux being trans- collected by a number of investigators. We thank Tom San- fered to higher modes during each scattering event. It ford, Eric Kunze and Craig Lee for their efforts in the collec- is possible that while the scattering efficiency for gen- tion and interpretation of Hawaii Ocean Mixing Experiment eral topographies is low, certain topographic sites may data presented in this report. Additionally, we are grate- ful for Figure 1 data contributed by Jim Moum and Kurt favor enhanced scattering. For example, Gilbert and Polzin. This research has been funded by grants from NSF Garrett (1989) and M¨uller and Liu (2000b) find convex and ONR. topographies are more efficient than linear or concave topographies at scattering energy to high modes. 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