1. Introduction
Oceanic internal lee waves and their associated breaking and energy dissipation are thought to play a considerable role in the energy and momentum budgets of the mean and mesoscale ocean circulation. Numerical model studies show that turbulent mixing generated by breaking lee waves, in addition to internal tides and near-inertial waves, are an important driver of the global overturning circulation and may account for up to one-third of the internal wave-driven water mass transformation in the deep ocean (Nikurashin and Ferrari 2013). Estimates for the global energy flux into lee waves vary between 0.2 TW (Nikurashin and Ferrari 2011) and up to almost 0.5 TW (Scott et al. 2011). Even though this is less than the 0.7–1.3 TW going globally into internal tides (Egbert and Ray 2000; Munk and Wunsch 1998; Nycander 2005; Garrett and Kunze 2007; Falahat et al. 2014), numerical model studies show that lee waves can have a significant impact on the ocean stratification (Melet et al. 2014). Lee waves seem to play a considerable role in the energy budget of mesoscale eddies, extracting as much as 0.12 TW from the eddy field in the Southern Ocean (Yang et al. 2018). Furthermore, where in the water column lee waves break and dissipate their energy—whether this occurs in the bottom layer or if energy propagates upward—matters for the overturning circulation (Melet et al. 2014). The fraction of lee-wave energy ultimately available for turbulent mixing depends on the presence of sheared background flows, which can, through conservation of wave action, lead to an appreciable amount of lee-wave energy feeding back into the mean flow (Kunze and Lien 2019).
Lee waves may also play an important role for the momentum balance of the global mean and mesoscale flow field. The interaction between flow and bathymetry, and subsequent lee-wave generation, is often expressed as a drag on the flow and therefore termed (topographic) form drag. In this framework, the radiation of waves is a pathway for energy and momentum in the horizontal and vertical. Only where the waves break is energy dissipated and momentum deposited (e.g., Bell 1975b; Durran 2003). It is the asymmetry of lee waves generated by quasi-steady flow, as opposed to purely tidal lee waves, that allows for net momentum transports. This redistribution of horizontal momentum in the vertical has been shown to have effects on the large-scale ocean circulation (Naveira Garabato et al. 2013; Trossman et al. 2016). In the presence of background flows, tidal lee waves may also lead to net momentum fluxes (Shakespeare and Hogg 2019).
Observations of oceanic lee waves generated by quasi-steady or low-frequency flow are sparse. As they are arrested over bottom topography, these waves have zero frequency in a reference frame fixed to topography and thus in theory do not imprint a direct signature on moored time series. Cusack et al. (2017) observed lee waves over and behind a bathymetric feature in Drake Passage using Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats and found vertical energy and momentum fluxes associated with these waves. Similarly, Meyer et al. (2016) observed lee waves using EM-APEX floats near the Kerguelen Plateau in the Southern Ocean. Using measurements from an underway CTD, Johnston et al. (2019) observed lee waves over a submarine ridge at the southern end of Palau. Eakin et al. (2011) observed vertically propagating lee waves in seismic reflection data over bathymetric features in the Caribbean Sea. A study of internal waves generated by mesoscale eddies by Clément et al. (2016) finds correlation between eddy decay and higher-frequency energy content in moored velocity records. We are unaware of other direct low-frequency lee-wave observations in the ocean and one has to revert to atmospheric observations of mountain waves for further observational studies. Tidal motions play only a subordinate role in the atmosphere, making the observation of lee waves easier than in oceanographic environments where the generation and propagation of tidal internal waves masks lee waves generated by the low-frequency flow.
A number of observational studies exist on internal tidal waves generated at steep, supercritical topography (Klymak et al. 2008; Legg and Klymak 2008; Nakamura et al. 2010; Pinkel et al. 2012; Alford et al. 2014). These studies show highly nonlinear, high-mode internal waves that lose part of their energy through breaking near the generation sites during certain phases of the tide. In addition to shear-driven mixing processes, these waves exhibit convective breaking mechanisms in transient features associated with critical flow conditions. Depending on the tidal excursion length scale in relation to the topographic length scale, a tidal wave may reach a quasi-steady state during certain phases of the tide (Musgrave et al. 2016), thus bridging the gap between tidal oscillatory waves and steady lee waves.
Here we present direct observations of quasi-steady internal lee waves that are generated both by tidal and lower-frequency flow over a tall submarine ridge. Observations taken at both spring and neap tide allow us to contrast conditions with dominating tidal flow and dominating mean flow. In the following, we describe the study region and observation techniques in section 2, discuss lee-wave parameters and present results in section 3, and discuss implications of the results in section 4.
2. Experimental details
a. Study region
The measurements were carried out over a tall submarine ridge north of the island group Palau in the tropical western North Pacific (Fig. 1). The northern end of Palau generally lies within the westward-flowing surface-intensified North Equatorial Current (NEC) with eastward-flowing undercurrents beneath it (Schönau and Rudnick 2015; Qiu et al. 2015; Zeiden et al. 2019). During the time of the survey at the onset of La Niña conditions following a relatively strong El Niño event, the large-scale currents deviated from this general picture and the eastward-flowing North Equatorial Counter Current (NECC) meandered around the northern tip of Palau, leading to general eastward surface flow.
Overview of the study region based on a global bathymetry product (Smith and Sandwell 1997) and multibeam measurements during the field campaign. (left) Bathymetry of the western Pacific. (center) Bathymetry in the vicinity of the island group Palau. Bathymetry contours are shown every 1000 m. (right) Bathymetry of the ridge north of Velasco Reef with mooring location and towyo survey line. Thick contours show bathymetry at 1000 m intervals, and the thin contours at 100 m intervals.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
The very shallow, island-like Velasco Reef at the northern end of Palau drops to about 1000 m depth within a horizontal distance of only a few hundred meters and extends as a long ridge around this depth for several kilometers northward (Fig. 1c). The ridge is mostly two-dimensional with variations in ridge height on the order of 100 m. The bottom depth drops to about 3000 m within just a few kilometers from the ridge crest on the eastern side and, less steep, to about 2000 m on the western side (Fig. 2).
Ridge-normal bathymetry as measured with the shipboard multibeam system (gray) and two Gaussian fits to bathymetry based on varying topography height h0 and constant ridge width W = 3 km (see text for details). The towyo sampling pattern is shown with gray lines and the location of the mooring projected onto the towyo line is marked. Also shown are characteristic wave slopes at semidiurnal (M2) and diurnal (K1) frequencies calculated with a mean stratification profile from (1).
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Defining a general ridge width is complicated owing to the asymmetry of the ridge-normal bathymetry. Following Klymak et al. (2010), the Gaussian shape of the ridge may be described by h = h0 exp(−x2/W2) with the topography height over the ocean floor h0 and the ridge width W. A ridge width W = 3 km results in Gaussian shapes that describe the cross-ridge bathymetry relatively well when using h0 = 1000 m on the western side and h0 = 2500 m on the eastern side (Fig. 2). The topographic aspect ratio α0 = h0/W thus varies between 0.3 on the western side and 0.8 on the eastern side.
b. Observations
All data were collected within the framework of the Office of Naval Research-sponsored research project Flow Encountering Abrupt Topography (FLEAT). The program was designed to study various aspects of flow–topography interactions, including their impact on energy and momentum balance of larger-scale flows. Various moored, shipboard, and autonomous measurement systems were deployed during the course of the experiment. Here, we focus on shipboard measurements during June 2016 and observations from one mooring that was deployed for 10 months in the vicinity of the ridge.
1) Shipboard observations
Shipboard measurements were collected during cruise RR1607 on R/V Roger Revelle between 2 and 24 June 2016. The survey over the ridge was designed to contrast conditions between spring and neap tide with the anticipation of low-frequency-flow-generated lee waves being less masked by internal tide generation over the ridge during neap tide. The ridge was therefore surveyed during two periods from 7 to 9 June at spring tide and again on 13 and 14 June during neap tide conditions as predicted by a global tidal model (TPXO; Egbert and Erofeeva 2002).
(i) Towyos
Towed profiling observations (towyos) were carried out using a Seabird 9/11-plus CTD on a standard rosette. The instrument package was towed at a speed of about 0.8 kt (1 kt ≈ 0.51 m s−1) back and forth along a line crossing the ridge in zonal direction at 8.55°N. At this latitude, the ridge crest is located at a depth of 943 m and thereby slightly shallower than to the north and south. While towing, the instrument package was initially cycled at vertical speeds of 1 m s−1 between the surface and 40 m above the seafloor, reaching to depths between 900 and 1500 m depending on position along the ridge. This resulted in a sawtooth-like sampling pattern with a horizontal resolution of about 500 m near the ridge crest and about 1000 m in deeper water. After a few ridge crossings, the upper turnaround depth was adjusted to about 500 m water depth to increase the horizontal resolution between individual profiles to about 300 m near the ridge crest and about 500 m in deeper water. The survey line was also slightly shortened to reduce the time needed for one ridge crossing from about 6 h during the initial occupation to about 3 h or a quarter of the semidiurnal tidal phase space. This sampling pattern was kept for the remainder of both the spring and the neap tide survey. The resulting trajectory of the instrument package is shown in Fig. 2.
During the neap tide survey, one RBR Solo thermistor sampling at 1 Hz and one Seabird SBE37 Microcat, recording temperature, conductivity, and pressure at 6 s sampling interval, were clamped on the hydrowire to provide measurements of the upper 500 m of the ocean. The pressure record from the slower SBE37 was used to depth-grid temperature measured by the RBR Solo with a faster response time. Temperature and conductivity were then used to calculate density. Resulting density profiles have much larger error bars than using data measured by the 9/11-plus system and are only used to estimate the influence of upper ocean stratification on bottom-pressure perturbations (appendix B).
A constant offset was applied to the ship’s GPS position to account for the horizontal distance between ship and instrument package due to drag on the hydrowire. Matching multibeam bathymetry with altimeter readings from the instrument package, we determined an optimal offset of 250 m.
Turbulent dissipation rate ε was estimated from CTD data based on Thorpe scales (Thorpe 1977; Dillon 1982). Details of the Thorpe scale method are shown in appendix A.
(ii) LADCP
Two lowered acoustic Doppler current profilers (LADCP) were attached to the rosette. Both instruments were Teledyne RD Instruments (TRDI) Workhorse types operating at 300 kHz and 8 m bin size. The LADCPs provided measurements of shear that extended beyond the shipboard sonar system’s (HDSS, see below) reach and into the bottom layer masked by sidelobe contamination in the HDSS data. Unfortunately, bottom tracking on the LADCPs failed and we have relatively low confidence in absolute LADCP velocity profiles outside the reach of HDSS data due to missing reference absolute velocities near the bottom. Measurements of vertical velocity shear are not affected.
(iii) HDSS
Shipboard velocity measurements were obtained using the Revelle’s Hydrographic Doppler Sonar System (HDSS) consisting of a 50 kHz and a 140 kHz sonar system (Pinkel 2012). Data from the two systems were combined and reach to about 1000 m water depth at a vertical resolution of 15 m. While this puts velocities at ridge-crest level into reach of the measurement system, bottom reflection of the sidelobe contamination renders the near-bottom 10% unusable. Individual velocity profiles were averaged to a temporal resolution of 1 min.
2) Moored observations
An array of subsurface moorings was deployed near the northern end of Velasco Reef for ten months from June 2016 through April 2017. From this array, mooring F3 was deployed near the ridge at a water depth of 1666 m (Figs. 1, 2) and was equipped with a combination of 75 and 300 kHz TRDI ADCPs, providing velocity measurements over nearly the whole water column. Velocity was measured with a burst of 10 pings every 10 min that were subsequently averaged to one value every 10 min. In addition to velocity, temperature and conductivity was measured at various depths along the mooring using RBR Solo thermistors and Seabird SBE37 CTDs.
3. Results
a. Flow conditions
Moored and shipboard velocity observations near the ridge reveal a complex flow pattern with a combination of low-frequency flow divided by various shear layers, tides, and a tidal fortnightly modulation. As the ridge is oriented in a north–south direction, zonal flow components correspond to ridge-normal flow in the following. The low-frequency ocean currents in the vicinity of the ridge were strongly layered with large vertical shear between the layers during the time of the experiment. The zonal velocity component from the moored observations (Figs. 3a and 4a) shows this banded structure with time-mean eastward flow in the upper 150 m, westward flow between 200 and 300 m, eastward flow between 350 and 700 m and westward flow in the range around the ridge crest from 800 to 1000 m. The time-mean meridional velocity component exhibits a similarly banded structure at smaller current amplitudes (Fig. 4b). HDSS observations, recorded during the two cross-ridge surveys, closely match the velocity pattern seen in the mooring data (Figs. 3b and 4). Mean eastward flow in the surface layer reflects the atypical situation of the NECC meandering around the northern end of Palau as discussed above. Mean flow near the ridge-crest level was mostly westward and northward at about 0.05 m s−1 in each component. Average conditions deviate between mooring and HDSS at the depth of the ridge crest; as the HDSS observations were taken closer to the ridge (cf. Fig. 2) we consider these a more accurate estimate of the flow directly interacting with topography. The 36-h low-pass-filtered moored velocity time series in Fig. 3c shows that the banded structure lasted throughout the time of the surveys. The cause of the highly banded structure remains an open research question on which we do not further elaborate here. We only note that the strongly sheared layers show up in other observations near Velasco Reef (MacKinnon et al. 2019) but the number of layers is reduced in the farfield (Zeiden et al. 2019).
Time series of zonal (ridge-normal) velocity for the time around the lee-wave experiment. (a) Moored ADCP observations (mooring location in Fig. 1). (b) Shipboard sonar (HDSS) observations during the two surveys. Note how data cuts out at the bottom as the ship crosses back and forth across the ridge. (c) The 36-h low-pass-filtered moored time series. (d) Bandpass-filtered moored time series with cutoff periods of 11 and 25 h encompassing semidiurnal and diurnal tidal time scales. (e) Barotropic tidal zonal velocity prediction from a model (TPXO; Egbert and Erofeeva 2002), depth-averaged zonal velocities from mooring and HDSS, and results of tidal harmonic analysis of the depth-averaged moored time series. The survey-mean velocity was subtracted from the depth-averaged HDSS time series. (f) Zonal velocity above the ridge-crest level, calculated as depth mean between 750 and 950 m from moored and HDSS observations. (g) The 4-h low-pass-filtered HDSS time series from (f) indicating ridge-normal flow strength in the vicinity of the ridge crest.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Mean current conditions during the spring and neap tide surveys from the shipboard Hydrographic Doppler Sonar System (HDSS) and the moored velocity measurements. (top) Zonal (ridge-normal) current component u. (bottom) Meridional current component υ. The ridge-crest depth level is shown with horizontal dashed lines.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Tidal variability of the flow was dominated by a semidiurnal and, to lesser extent, diurnal signal that was modulated by a clear spring–neap cycle. Figure 3d shows the moored ridge-normal velocity time series bandpass-filtered around the semidiurnal and diurnal tidal frequencies. Spring–neap modulation is apparent with stronger tidal velocities around 7 June and weaker tidal velocities around 14 June. This is reflected in the barotropic or depth-averaged moored zonal velocity record showing tidal amplitudes of about 0.1 m s−1 during spring tide and reduced tidal modulation during neap tide (Fig. 3e). The phase of the fortnightly cycle as predicted by a tidal model (TPXO; Egbert and Erofeeva 2002) matches relatively well with the spring–neap cycle observed here. However, amplitudes of the TPXO-predicted tide are reduced by about a factor of 2 when compared to the observations and the phase is offset by about 10°. We attribute this mismatch to the coarse resolution of the TPXO model of only 1/6° where local bathymetry is not resolved sufficiently. For example, bottom depth in the TPXO model at the mooring site is 2754 m compared to multibeam-measured depth of 1666 m.
Velocities around the ridge-crest level, calculated as a depth-mean over the range of 750–950 m, show higher tidal amplitudes than the depth-averaged velocities (Fig. 3f). Harmonic analysis of the moored velocity time series at ridge-crest level reveals peak tidal amplitudes of about 0.15 m s−1 during spring tide and only 0.05 m s−1 during neap tide. The semidiurnal tidal constituents M2 and S2 alone account for 55% and 20% of the tidal energy at the ridge-crest level. Tidal currents are mostly ridge normal with tidal ellipse aspect ratios of 4 and 8 for the M2 and S2 constituents, respectively.
The 4-h low-pass-filtered HDSS ridge-crest-level velocity time series in Fig. 3g sums up the flow conditions expected to be important for lee-wave generation. During spring tide, tidal motion back and forth across the ridge dominates. The westward mean flow is superimposed. The ratio of average semidiurnal (tidal) energy to overall kinetic energy in the flow above the ridge crest is 0.58 in the spring tide case and only about 0.02 for the neap tide survey. In contrast, the westward mean flow dominates during neap tide with a ratio of lower-frequency flow energy to overall flow energy at 0.65. This ratio drops to 0.05 during spring tide. The moored time series does not show a clear mean westward flow in the range of 750–950 m during the neap tide survey (cf. Fig. 3f); however, as noted above, the HDSS data recorded closer to the ridge are expected to better reflect current conditions in the direct vicinity of the topography.
b. Lee-wave parameter space
Both tides and low-frequency flows generate internal lee waves as they interact with bottom topography. While the lee-wave response at any given time is dictated by the total velocity (sum of tides and mean flow), we first consider these in isolation. In the following we discuss a number of important parameters for each type of lee wave before we present observations of waves above the ridge.
In shallow-water flow, the water depth D above the obstacle is an important parameter in setting the wave response (e.g., Farmer and Smith 1980). Here, D is not expected to play an important role in the response to topography as U0/(ND) ≈ 0.02 ≪ 1. For systems with small U0/(ND), we do not expect waves to interact with the surface; we thus treat the ocean as infinitely deep (Mayer and Fringer 2017).
In his seminal work on the generation of tidal internal waves under linear conditions, Bell (1975a) shows that for small tidal excursion parameter ξ, the internal wave response is mainly at the fundamental forcing frequency. For increasing ξ water particles are advected farther across the whole width of the ridge and waves at higher harmonics of the fundamental frequency are excited. Musgrave et al. (2016) show how under moderate conditions the time-dependent superposition of this multifrequency response generates a transient lee wave for
The vertical group speed of a nonlinear wave scales approximately as cgzαefUC (Klymak et al. 2010). This translates to vertical propagation of
Quasi-steady lee waves and tidal lee waves react very differently when encountering background flows or shear layers. For tidal waves near the acoustic limit (Bell 1975b), the presence of a steady background flow means that the ray pattern radiating away from the ridge will not be symmetric but tilted owing to Doppler shifting. For quasi-steady lee waves, shear layers can reduce the wave frequency to the local inertial frequency. In these critical layers, phase propagation relative to the background current stalls and hence no upward energy propagation is possible (Bell 1975b; Booker and Bretherton 1967). Quasi-steady lee waves dissipate their energy in these critical layers. However, as the first shear layer is located about 300 m above the ridge crest it takes about 3 h for the lee wave to reach the critical layer at the vertical group speed cgz estimated above. As this reaches tidal time scales where flow conditions above the ridge cannot be approximated as steady anymore, it is unclear whether much of the nonlinear lee-wave response reaches the critical layer and dissipates its energy there. We know of no observations of this.
From the above considerations, we expect a combination of tidal flow and mean flow at the ridge crest to form quasi-steady lee waves within relatively short time scales (faster than half a tidal cycle). During spring tide when oscillatory velocities are strong, lee waves can form on both sides of the ridge. The mean flow will bias the resulting lee waves by reducing the ridge crest velocity during eastward tidal flow and increasing it during westward flow. During neap tide when the mean flow dominates and inhibits oscillatory flow across the ridge, lee waves will be generated only on one side of the ridge.
c. Lee-wave observations
The measurements reveal the presence of breaking internal waves in the vicinity of the ridge. In the following we highlight one towyo section across the ridge in detail before we show all sections from both surveys. The example section shown in Fig. 5 was conducted at the beginning of the spring tide survey. The ship crossed the ridge in the westward direction during a phase of westward flow above the ridge crest. Isopycnals over the ridge are vertically displaced by up to 80 m relative to a survey-mean density profile. The dissipation rate of turbulent kinetic energy ε from Thorpe scales is highly elevated downstream of the ridge near topography. Elevated levels of ε are also found along the backward (relative to the flow direction) sloping band of wavy features, for example, clearly visible at x = 0.5 km around 750 m depth.
Sample section across the ridge from the spring tide survey. Towyo data are depth averaged into 25 m size bins for better visualization. The ship moved in a westward direction as indicated by the arrow in (b). (a) Vertical isopycnal displacement ζ (color along instrument package trajectory) and potential density anomaly σ0 (black contours). The thick contour is the σ0 = 27.25 isopycnal. Density contours are repeated in the following panels. (b) Zonal (cross-ridge) velocity u from HDSS and LADCP. LADCP velocity is shown in the bottom layer where sidelobe interference inhibits shipboard sonar measurements. Characteristic wave slopes are shown for the semidiurnal tidal frequency (dotted), intrinsic lee-wave frequency at 0.05 m s−1 flow speed (dash–dotted), and lee-wave frequency at 0.15 m s−1 flow speed (dashed). (c) Dissipation rate of turbulent kinetic energy ε. Only bins with log10(ε) > −9.5 W kg−1 are shown. (d) Inverse Richardson number (Ri−1). Only bins with Ri−1 > 0.25 are shown.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
The vertical wavelength of a high-mode stationary wave implied by the velocity scale is λ0 = 2πUc/N (Klymak et al. 2010). For peak ridge crest velocities Uc ≈ 0.15 m s−1 the expected vertical wavelength λ0 is approximately 350 m. Due to the background shear layers it is difficult to make out the velocity signature of the waves. In their model runs, Klymak et al. (2010) find vertical displacements of about 0.25λ0. Vertical displacements of up to about 80 m observed here also scale as approximately 0.25λ0. For comparison, Alford et al. (2014) found vertical wavelengths λ0 ≈ 300 m for high-mode oscillatory lee waves over Kaena Ridge near Hawaii.
The characteristic slope α of an internal wave can be calculated following (1) based on the wave frequency ω. Characteristics for the semidiurnal tide
Cross-ridge towyo sections during spring tide. Time progresses from left to right. (top) potential density σθ (black contours) and vertical displacement ζ (colors). The thick contour is the σ0 = 27.25 isopycnal. Circles and wedges show the time occupied in M2 phase space by each ridge crossing. (second row) Zonal (u) and (third row) meridional (υ) velocities as measured by the shipboard HDSS. Arrows in the second row indicate the driving direction of the ship. (fourth row) Turbulent dissipation rate (ε) estimates from density inversions. (bottom) Inverse Richardson numbers (Ri−1). Black contours in all rows repeat the density field from the top row. Dashed lines in the top row in column 5 show characteristic wave slopes with increasing steepness for waves at semidiurnal periods and periods set by the topographic length scale of the ridge (see Fig. 5b and text for details).
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
All cross-ridge sections from both the spring and neap tide surveys are shown in Figs. 6 and 7, respectively. They show similar features discussed for the single section above. Isopycnals exhibit wave-like fluctuations over the ridge in many of the sections. Vertical displacements of isopycnals are
Cross-ridge towyo sections during neap tide. See Fig. 6 for details.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Turbulent dissipation rates ε reach up to 10−6 W kg−1 and are thus elevated by several orders of magnitude over oceanic background levels. Enhanced levels of ε near topography are found to be shifting with the tidal flow back and forth across the ridge for the spring tide survey. For example, columns 1, 4, and 12 in Fig. 6 show enhanced ε on the western side of the ridge during westward flow. For the neap tide survey, overall levels of ε are weaker and high values are mostly found on the western side of the ridge. As for the example section, inverse Richardson numbers reach beyond 2 for some of the regions of elevated ε, but not all, indicating the presence of both shear-driven and convectively driven turbulence processes.
d. Survey-average conditions
We highlight the differences between spring and neap tide survey by calculating time-mean of flow and turbulent dissipation rate. The left panels in Fig. 8 show the time mean over each survey with values bin-averaged on a 500 m width × 50 m depth grid. The ridge-normal flow during both surveys is very similar with westward flow near the ridge crest and eastward flow aloft. The time-mean turbulence pattern for the spring tide survey is relatively symmetric across the ridge and thereby indicative of breaking internal tidal waves generating this pattern. This is as expected for tidal velocities dominating over the mean flow. The value of ε is slightly higher on the western side, possibly indicating that the superposition of the westward mean flow on top of the tidal flow leads to swifter flows and a stronger hydraulic response on the western side. In contrast, the survey-average turbulent dissipation rate pattern for the neap tide is highly skewed toward the western side of the ridge. We conclude that a steady lee wave, set up by the lower-frequency flow across the ridge, breaks and generates the asymmetric pattern in turbulent dissipation. Individual ridge crossings show that the tide, even though weaker than at spring tide, still modulates this time mean pattern (Fig. 7). However, as the tide is not strong enough to overcome the westward flow during neap tide, we mostly see elevated turbulent dissipation rates associated with breaking internal lee waves on the western side. This is a central result of our study.
Selective averages. (top) Zonal HDSS velocity, (middle) turbulent dissipation rate ε, and (bottom) vertical isopycnal displacement ζ. (left two columns) Time-mean conditions during spring and neap tide. (center two columns) Time-mean conditions during the spring tide survey separated into eastward and westward flows as determined from HDSS velocities at ridge-crest level (cf. Fig. 3g). (right four columns) Observations from the spring tide survey split even further into four phases of the tide as indicated by the bottom time series panel. Here, eastward and westward peak tidal flows are defined by absolute zonal ridge-crest-level HDSS velocities above 0.04 m s−1.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Absolute values of turbulent dissipation are higher during spring tide than during neap tide near the ridge and higher up in the water column, consistent with stronger internal tide generation, local breaking over the ridge and upward radiation where some of the waves also locally dissipate.
Viewed in terms of height above bottom (Fig. 9), time averages of turbulent dissipation rate ε follow a familiar pattern of decreasing turbulence away from the bottom into the interior ocean (e.g., Polzin et al. 1997; Waterhouse et al. 2014). The lower 200 m above bottom exhibit an order of magnitude difference in turbulent dissipation on either side of the ridge during the neap tide survey. At 400 m above bottom and beyond, ε is of comparable size between eastern and western sides with an order-of-magnitude difference between spring and neap tides. This may indicate that low-frequency flow-generated lee waves cannot pass through critical layers set up by the strong shear layers, only becoming apparent during neap tide when no tidal internal waves are breaking and causing higher ε aloft.
Time-mean turbulent dissipation rate ε in height above bottom for the spring and neap tide surveys on both sides of the ridge.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Averaging the spring tide survey into different phases of the tide shows how tidal waves set up and break on either side of the ridge. The center panels in Fig. 8 show zonal velocity, turbulent dissipation rate, and vertical isopycnal displacement averaged into eastward and westward flows as indicated in the low-pass-filtered HDSS time series in Fig. 3g. The data are averaged on the same 500 m × 50 m grid as the overall survey means. Time-mean velocities show how the flow over the ridge crest changes direction. Turbulent dissipation rate is biased toward the west during westward flow and, although not as clear, stronger on the eastern side during eastward flow. Time-mean displacement shows generally elevated isopycnals on the upstream side of the ridge.
Further separation of the spring tide survey into peak tidal flows and slack tide is shown in the four right panels in Fig. 8. The bottom panel shows the separation into four tidal phases using the 8-h low-pass-filtered HDSS ridge-crest level ridge-normal velocity. Peak tidal flows are defined as ridge-normal flow magnitude above 0.04 m s−1. During slack tides, regions of higher turbulent dissipation rate are concentrated around the ridge crest with a slight bias toward the downstream side relative to the preceding tidal flow. During peak tidal flow, the pattern is also biased toward the downstream side but less concentrated around the ridge crest. Displacement patterns during peak tidal flow exhibit the upstream lifting and downstream suppression seen in the center panels. Displacement during slack tide does not show a clear pattern.
e. Form drag
As CTD measurements were only obtained to within a safe distance from the bottom, density data are missing in the lower 20–60 m above bottom. In this bottom layer we assume a constant density profile as given by the deepest density estimate in each CTD profile. Pressure anomalies obtained by integrating ρ′ from 480 m depth to the bottom are shown in Fig. 10. For most profiles, no density data exist at shallower depths and we have to restrict our analysis to the deeper part of the water column, assuming that variations in the density field higher up in the water column do not exert a leading order influence on the pressure field and hence form drag at depth. A comparison with bottom pressure calculated from full-depth density profiles shows that this introduces an uncertainty in individual bottom-pressure anomaly estimates of up to 100 N m−2. The uncertainty in form drag estimates introduced by only integrating density in the deeper part of the water column is about 1.5 × 104 N m−1 (appendix B). In addition, we are missing the external component of the bottom-pressure anomaly caused by variations in sea surface height (see, e.g., Warner et al. 2013). Nonhydrostatic pressure perturbations, also missing in pB as estimated here, have been found to play only a subordinate role for nonlinear internal waves (Warner et al. 2013; Moum and Smyth 2006).
Pressure anomaly (colors) as estimated from depth-integrated density anomalies, isopycnals σ0 (contours), and form drag Df (N m−1) printed in white on top of bathymetry. Section-average velocity above ridge-crest-level UC is shown with white arrows. (top) Ridge crossings during the spring tide survey. (bottom) Neap tide survey results.
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
To obtain meaningful values for the form drag, horizontal integration of bottom pressure in (7) has to start and end at the same bottom depth. We integrate over the part of the ridge shallower than 1250 m as for most of the ridge crossings the profiles do not reach deeper on both sides of the ridge. Form drag calculated this way for each ridge crossing is shown in Fig. 10 and plotted in Fig. 11; absolute values of form drag are in the range of (0.01–4) × 104 N m−1. For reference, Warner et al. (2013) find maximum form drag of 1 × 104 N m−1 during peak flood tidal currents of 0.2 m s−1 at a topographic feature in Puget Sound. Other observational studies compute form drag per unit area. When averaged over the horizontal integration distance of dx = 2.7 km in (7), form drag of 1 × 104 N m−1 translates into about 3.7 N m−2. In comparison, over a 20 m tall bank on the Oregon shelf, Nash and Moum (2001) find form drag ranging from 0.5 to 1.8 N m−2 at flow speeds between 0.08 and 0.35 m s−1. At the southern end of Palau the form drag associated with lee-wave generation over a steep submarine ridge was estimated to 1 N m−2 (Johnston et al. 2019).
Form drag Df from each ridge crossing in relation to section-average HDSS ridge-crest velocity UC. Results are marked by spring (blue) and neap (pink) tides. Where available, form drag estimates calculated from full-depth density profiles are shown in connection with their partial-depth estimates (orange; see appendix B for details). Error bars show one standard deviation for ridge-crest-level velocity UC and 1.5 × 104 N m−1 for form drag DF based on the comparison with full-depth profiles (appendix B). A bluff body parameterization for form drag (teal) with varying drag coefficient CD and a form drag parameterization for high-mode nonlinear waves (purple; Klymak et al. 2010) are shown for ridge-height parameter hm = 300 (solid) and 600 m (dashed).
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
The survey-average form drag is 3.6 × 103 N m−1 for the spring tide survey and 3.0 × 103 N m−1 for the neap tide survey. However, these averages should be interpreted with caution as we likely do not have high enough resolution of the changing form drag over time to calculate a meaningful temporal average that would eliminate the effects of the tide.
Form drag approximately scales with the cross-ridge velocity at the ridge-crest level (Fig. 11). It acts both in easterly and westerly direction, generally against the flow at ridge-crest level. Westward flow (negative velocity in Fig. 11) leads to positive form drag, that is, a force opposing the flow, and vice versa for eastward flow. There are however a few exceptions during neap tide where flow, albeit at low amplitude, and form drag have the same sign.
Kinetic energy of the flow lost to form drag is calculated as UCDf, where UC is the velocity of the flow interacting with the bathymetry. Here, we calculate UC as before as an average of HDSS ridge-normal velocity over the layer between 750 and 950 m depth. We note that UC is the superposition of tidal and mean flow. For a two-dimensional ridge and purely barotropic tidal flow, the power loss due to form drag would equal the tidal barotropic to baroclinic conversion term (Warner et al. 2013). For oscillatory flow, form drag can have a component termed inertial drag in Warner and MacCready (2009) and Warner et al. (2013) that is in quadrature with velocity and does no net work on the flow. In Warner et al. (2013) the inertial drag is associated with the sea surface tilt driving tidal barotropic flow. As we do not include any sea surface variability in the pB estimates, we do not expect inertial drag to play a large role. Survey-average power loss is 1450 and 200 W m−1 for the spring and neap tide survey, respectively. Dividing by the horizontal integration distance dx = 2.7 km used in the form drag calculation, this translates to power losses per unit area of 538 and 76 mW m−2 for spring and neap tide. For reference, in their study in Puget Sound mentioned above, Warner et al. (2013) find a tidally averaged power loss to form drag of 200 mW m−2. Peak power losses during strong tidal currents were 1.7 W m−2 in Warner et al. (2013), compared to peak power losses of 1.2 W m−2 during strong westward flow at spring tide found here.
The relationship between the expected energy loss UCDf and the measured energy loss, calculated as the integral over the dissipation rate of turbulent kinetic energy ε below 500 m depth and over each ridge crossing
Expected form drag power loss UCDf and observed levels of turbulent dissipation rate integrated around the ridge for each cross-ridge section. Markers are colored by ridge-crest-level velocity UC and divided by shape into spring tide (circles) and neap tide (squares) surveys. The solid gray line indicates a 1:1 relationship, the dark dashed line a least squares fit to data points from the spring tide survey (slope = 0.1), and the dark dash–dotted line a fit to the neap tide survey results (slope = 0.15).
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
4. Summary and concluding remarks
We observed internal waves over a tall ridge under a combination of tidal and low-frequency flow. In the parameter space set by a topographic Froude number, the waves formed within relatively short times compared to tidal scales. The resulting waves resembled quasi-steady topographic lee waves and were highly dissipative on the lee side of the ridge. Average conditions during spring and neap tides revealed a contrasting pattern in the resulting lee waves. During spring tide, when the tidal flow dominated, breaking waves were observed on either side of the ridge. During neap tide, when the low-frequency westward flow at the ridge crest dominated over the tidal flow, breaking lee waves formed mostly on the western side of the ridge. We conclude that the observed lee waves are generated by a combination of tidal flow and mean flow. As the wave generation consumes energy and can carry momentum, it provides a sink term for both the low-frequency ocean circulation and the tides.
A few studies find similar patterns in the deep ocean. Musgrave et al. (2017) find that the superposition of a subinertial, topographically trapped diurnal tide and radiating semidiurnal tide leads to modulation of the tidal lee-wave response in a channel on the Mendocino Ridge. When the currents from the two tidal components interact constructively, a tidal lee wave forms and strong turbulent dissipation rates are observed in the lee of the topography. When interacting deconstructively, no tidal lee wave forms. The results presented here may be interpreted analogously: during neap tide, when tidal currents are weakened and do not dominate over the mean flow anymore, the constructive superposition with the mean flow leads to tidal lee-wave generation on the western side while no lee waves are observed on the eastern side. Dale and Inall (2015) observe an asymmetry in tidal lee waves and associated turbulence at a saddle amid steep and complex Mid-Atlantic Ridge topography. They speculate that the asymmetry could be due to background flow, a similar situation as presented here.
A recent theoretical study by Shakespeare and Hogg (2019) examines the influence of a mean flow on momentum fluxes associated with tidal-wave generation. Bell (1975b) already noted that for a mean background flow generating lee waves, the presence of tides can increase momentum fluxes by several tens of percent but did not elaborate further. Shakespeare and Hogg (2019) explore this phenonemon in more detail and show how the bottom background flow Doppler shifts the internal tide at generation and leads to a net momentum flux directed against the bottom flow. Unfortunately, form drag estimates presented here are too coarse to explore this phenonemon in further detail.
The observations show form drag on the order of 104 N m−1, acting both on tidal flow and mean flow. The form drag scales approximately linearly with ridge-crest velocity and is captured relatively well by the Klymak et al. (2010) parameterization when adjusting the topographic height parameter. A comparison of expected loss due to form drag with observed energy loss in the vicinity of the ridge suggests that between 85% (neap tide) and 90% (spring tide) of the energy lost to form drag radiates away and dissipates elsewhere. A rough estimate of about 1 kW m−1 for the M2 internal tide energy flux on the eastern side of the ridge supports this conclusion. A recent study by Zheng and Nikurashin (2019) shows, based on idealized numerical simulations and linear theory, that for the Drake Passage region about 30%–40% of lee waves generated by low-frequency flow dissipate locally near topography while the remaining part of the waves decay downstream of the topographic feature.
Mixing and drag induced by the ridge influence the low-frequency flow. A motivating question for this study is the general importance of submarine ridges for the energy and momentum budget of low-frequency flows. We do not have an estimate for the form drag averaged over a tidal cycle reliable enough to estimate the stress on the mean flow separated from that on the total flow. The kinetic energy of a 0.05 m s−1 amplitude flow that extends over 500 m in the vertical is about 7 × 103 J m−2. Power loss due to form drag was estimated to about 50 mW m−2 for the neap tide survey. Assuming that during neap tide the low-frequency flow contributes the majority to the lee-wave formation on the western side of the ridge, the (0.05 m s−1 × 500 m) flow would be consumed within about 3.5 h at this power loss. Since there are still tidal currents contributing at neap tide, this estimate is likely too fast, but it shows that the ridge exerts a strong influence on the flow around its ridge-crest level.
Acknowledgments
This work was funded by the Office of Naval Research through the Departmental Research Initiative Flow Encountering Abrupt Topography (FLEAT) under Grants N00014-15-1-2592, N00014-15-1-2264, N00014-15-1-2302, and N00014-16-1-3070. We thank captain Murline and his crew on the R/V Roger Revelle for their outstanding support. All members of the science team greatly contributed to the successful measurements. We are grateful to Pat and Lori Colin from the Coral Reef Research Foundation in Palau for their help with logistics and outreach activities onshore. We appreciate valuable comments from two anonymous reviewers that helped to improve this study.
APPENDIX A
Turbulence Estimates
Spurious overturns resulting from noise in the density measurements are excluded using a method developed by Ferron et al. (1998). Before sorting the density profile into its stable state, an intermediate profile is constructed that takes the noise in the measurements into account by discretizing the measurements at the accuracy level of the density profile (1 × 10−3 kg m−3).
APPENDIX B
Bottom-Pressure Estimates and Form Drag Error
Density anomalies ρ′ are integrated from 480 m depth to near the bottom to infer bottom-pressure perturbations p′ used in the form drag calculations. Constraints in profiling timing, as aliasing of the internal tide in the cross-ridge sections was to be avoided, led to only partial coverage of the water column with the shipboard CTD for most of the survey. However, full depth CTD profiles were recorded for the first two ridge crossings and allow for an error estimate of bottom pressure calculated from only the deeper part of the density profile (Fig. B1). Full-depth density profiles constructed from shipboard CTD measurements at depth and RBR Solo and SBE37 Microcat data from higher up in the water column provide additional error estimates during the neap tide survey. Absolute values of bottom-pressure anomalies differ on average by 20 N m−2 between the two methods. We note that 65% of the values are within 40 N m−2 of each other and only 3% differ by more than 100 N m−2. The sign of the bottom-pressure anomalies is the same for both methods.
Bottom-pressure anomaly
Citation: Journal of Physical Oceanography 50, 5; 10.1175/JPO-D-19-0257.1
Form drag from the integration of (7) with pB calculated from full-depth density profiles differs from the partial-depth integrated profiles by 0.6 × 104 N m−1 on average (see yellow markers in Fig. 11 for individual values). The maximum difference occurred during the first ridge crossing of the neap survey at 1.4 × 104 N m−1. We assign a conservative uncertainty of 1.5 × 104 N m−1 to the form drag estimates, thereby spanning all observed differences.
REFERENCES
Alford, M. H., M. C. Gregg, and M. A. Merrifield, 2006: Structure, propagation, and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii. J. Phys. Oceanogr., 36, 997–1018, https://doi.org/10.1175/JPO2877.1.
Alford, M. H., J. B. Girton, G. Voet, G. S. Carter, J. B. Mickett, and J. M. Klymak, 2013: Turbulent mixing and hydraulic control of abyssal water in the Samoan Passage. Geophys. Res. Lett., 40, 4668–4674, https://doi.org/10.1002/grl.50684.
Alford, M. H., J. M. Klymak, and G. S. Carter, 2014: Breaking internal lee waves at Kaena Ridge, Hawaii. Geophys. Res. Lett., 41, 906–912, https://doi.org/10.1002/2013GL059070.
Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 498 pp.
Bell, T. H., 1975a: Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 67, 705–722, https://doi.org/10.1017/S0022112075000560.
Bell, T. H., 1975b: Topographically generated internal waves in the open ocean. J. Geophys. Res., 80, 320–327, https://doi.org/10.1029/JC080i003p00320.
Booker, J. R., and F. P. Bretherton, 1967: The critical layer for internal gravity waves in a shear flow. J. Fluid Mech., 27, 513–539, https://doi.org/10.1017/S0022112067000515.
Clément, L., E. Frajka-Williams, K. L. Sheen, J. A. Brearley, and A. C. N. Garabato, 2016: Generation of internal waves by eddies impinging on the western boundary of the North Atlantic. J. Phys. Oceanogr., 46, 1067–1079, https://doi.org/10.1175/JPO-D-14-0241.1.
Cusack, J. M., A. C. Naveira Garabato, D. A. Smeed, and J. B. Girton, 2017: Observation of a large lee wave in the Drake Passage. J. Phys. Oceanogr., 47, 793–810, https://doi.org/10.1175/JPO-D-16-0153.1.
Dale, A. C., and M. E. Inall, 2015: Tidal mixing processes amid small-scale, deep-ocean topography. Geophys. Res. Lett., 42, 484–491, https://doi.org/10.1002/2014GL062755.
Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov length scales. J. Geophys. Res., 87, 9601–9613, https://doi.org/10.1029/JC087iC12p09601.
Durran, D. R., 2003: Lee waves and mountain waves. Encyclopedia of Atmospheric Sciences, J. R. Holton, J. Pyle, and J. A. Curry, Eds., Academic Press, 1161–1169.
Eakin, D., W. S. Holbrook, and I. Fer, 2011: Seismic reflection imaging of large-amplitude lee waves in the Caribbean Sea. Geophys. Res. Lett., 38, L21601, https://doi.org/10.1029/2011GL049157.
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, https://doi.org/10.1038/35015531.
Egbert, G. D., and S. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183–204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.
Falahat, S., J. Nycander, F. Roquet, and M. Zarroug, 2014: Global calculation of tidal energy conversion into vertical normal Modes. J. Phys. Oceanogr., 44, 3225–3244, https://doi.org/10.1175/JPO-D-14-0002.1.
Farmer, D. M., and J. D. Smith, 1980: Tidal interaction of stratified flow with a sill in Knight Inlet. Deep-Sea Res., 27A, 239–254, https://doi.org/10.1016/0198-0149(80)90015-1.
Ferron, B. H., H. Mercier, K. Speer, A. Gargett, and K. Polzin, 1998: Mixing in the romanche fracture zone. J. Phys. Oceanogr., 28, 1929–1945, https://doi.org/10.1175/1520-0485(1998)028<1929:MITRFZ>2.0.CO;2.
Garrett, C., and E. Kunze, 2007: Internal tide generation in the Deep Ocean. Annu. Rev. Fluid Mech., 39, 57–87, https://doi.org/10.1146/annurev.fluid.39.050905.110227.
Howard, L. N., 1961: Note on a paper of John W. Miles. J. Fluid Mech., 10, 509–512, https://doi.org/10.1017/S0022112061000317.
Johnston, T. S., and Coauthors, 2019: Energy and momentum lost to wake eddies and lee waves generated by the north equatorial current and tidal flows at Peleliu, Palau. Oceanography, 32, 110–125, https://doi.org/10.5670/oceanog.2019.417.
Klymak, J. M., and M. C. Gregg, 2004: Tidally generated turbulence over the Knight Inlet sill. J. Phys. Oceanogr., 34, 1135–1151, https://doi.org/10.1175/1520-0485(2004)034<1135:TGTOTK>2.0.CO;2.
Klymak, J. M., R. Pinkel, and L. Rainville, 2008: Direct breaking of the internal tide near topography: Kaena Ridge, Hawaii. J. Phys. Oceanogr., 38, 380–399, https://doi.org/10.1175/2007JPO3728.1.
Klymak, J. M., S. Legg, and R. Pinkel, 2010: High-mode stationary waves in stratified flow over large obstacles. J. Fluid Mech., 644, 321–336, https://doi.org/10.1017/S0022112009992503.
Kunze, E., and R.-C. Lien, 2019: Energy sinks for lee waves in shear flow. J. Phys. Oceanogr., 49, 2851–2865, https://doi.org/10.1175/JPO-D-19-0052.1.
Legg, S., and J. Klymak, 2008: Internal hydraulic jumps and overturning generated by tidal flow over a tall steep ridge. J. Phys. Oceanogr., 38, 1949–1964, https://doi.org/10.1175/2008JPO3777.1.
MacKinnon, J. A., M. H. Alford, G. Voet, K. Zeiden, T. S. Johnston, M. Siegelman, S. Merrifield, and M. Merrifield, 2019: Eddy wake generation from broadband currents near Palau. J. Geophys. Res. Oceans, 124, 4891–4903, https://doi.org/10.1029/2019JC014945.
Mayer, F. T., and O. B. Fringer, 2017: An unambiguous definition of the Froude number for lee waves in the deep ocean. J. Fluid Mech., 831, 1–9, https://doi.org/10.1017/JFM.2017.701.
Melet, A., R. Hallberg, S. Legg, and M. Nikurashin, 2014: Sensitivity of the ocean state to lee wave–driven mixing. J. Phys. Oceanogr., 44, 900–921, https://doi.org/10.1175/JPO-D-13-072.1.
Meyer, A., K. L. Polzin, B. M. Sloyan, and H. E. Phillips, 2016: Internal waves and mixing near the Kerguelen Plateau. J. Phys. Oceanogr., 46, 417–437, https://doi.org/10.1175/JPO-D-15-0055.1.
Miles, J. W., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech., 10, 496–508, https://doi.org/10.1017/S0022112061000305.
Moum, J. N., 1996: Energy-containing scales of turbulence in the ocean thermocline. J. Geophys. Res., 101, 14 095–14 109, https://doi.org/10.1029/96JC00507.
Moum, J. N., and W. Smyth, 2006: The pressure disturbance of a nonlinear internal wave train. J. Fluid Mech., 558, 153–178, https://doi.org/10.1017/S0022112006000036.
Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.
Musgrave, R. C., J. A. MacKinnon, R. Pinkel, A. F. Waterhouse, and J. Nash, 2016: Tidally driven processes leading to near-field turbulence in a channel at the crest of the Mendocino Escarpment. J. Phys. Oceanogr., 46, 1137–1155, https://doi.org/10.1175/JPO-D-15-0021.1.
Musgrave, R. C., J. A. MacKinnon, R. Pinkel, A. F. Waterhouse, J. Nash, and S. M. Kelly, 2017: The influence of subinertial internal tides on near-topographic turbulence at the Mendocino ridge: Observations and modeling. J. Phys. Oceanogr., 47, 2139–2154, https://doi.org/10.1175/JPO-D-16-0278.1.
Nakamura, T., Y. Isoda, H. Mitsudera, S. Takagi, and M. Nagasawa, 2010: Breaking of unsteady lee waves generated by diurnal tides. Geophys. Res. Lett., 37, L04602, https://doi.org/10.1029/2009GL041456.
Nash, J. D., and J. N. Moum, 2001: Internal hydraulic flows on the continental shelf: High drag states over a small bank. J. Geophys. Res., 106, 4593–4611, https://doi.org/10.1029/1999JC000183.
Naveira Garabato, A. C., A. J. G. Nurser, R. B. Scott, and J. A. Goff, 2013: The impact of small-scale topography on the dynamical balance of the ocean. J. Phys. Oceanogr., 43, 647–668, https://doi.org/10.1175/JPO-D-12-056.1.
Nikurashin, M., and R. Ferrari, 2011: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean. Geophys. Res. Lett., 38, L08610, https://doi.org/10.1029/2011GL046576.
Nikurashin, M., and R. Ferrari, 2013: Overturning circulation driven by breaking internal waves in the deep ocean. Geophys. Res. Lett., 40, 3133–3137, https://doi.org/10.1002/grl.50542.
Nycander, J., 2005: Generation of internal waves in the deep ocean by tides. J. Geophys. Res., 110, C10028, https://doi.org/10.1029/2004JC002487.
Ozmidov, R. V., 1965: On the turbulent exchange in a stably stratified ocean. Izv., Atmos. Ocean. Phys., 1, 853–860.
Pinkel, R., 2012: Velocity imprecision in finite-beamwidth shipboard Doppler sonar: A first-generation correction algorithm. J. Atmos. Oceanic Technol., 29, 1569–1580, https://doi.org/10.1175/JTECH-D-12-00041.1.
Pinkel, R., L. Rainville, and J. Klymak, 2012: Semidiurnal baroclinic wave momentum fluxes at Kaena Ridge, Hawaii. J. Phys. Oceanogr., 42, 1249–1269, https://doi.org/10.1175/JPO-D-11-0124.1.
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, https://doi.org/10.1126/science.276.5309.93.
Qiu, B., S. Chen, D. L. Rudnick, and Y. Kashino, 2015: A new paradigm for the North Pacific subthermocline low-latitude western boundary current system. J. Phys. Oceanogr., 45, 2407–2423, https://doi.org/10.1175/JPO-D-15-0035.1.
Schönau, M. C., and D. L. Rudnick, 2015: Glider observations of the north equatorial current in the western tropical Pacific. J. Geophys. Res. Oceans, 120, 3586–3605, https://doi.org/10.1002/2014JC010595.
Scott, R. B., J. A. Goff, A. C. Naveira Garabato, and A. J. G. Nurser, 2011: Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography. J. Geophys. Res., 116, C09029, https://doi.org/10.1029/2011JC007005.
Shakespeare, C. J., and A. M. Hogg, 2019: On the momentum flux of internal tides. J. Phys. Oceanogr., 49, 993–1013, https://doi.org/10.1175/JPO-D-18-0165.1.
Smith, W. H. F., and D. T. Sandwell, 1997: Global sea floor topography from satellite altimetry and ship depth soundings. Science, 277, 1956–1962, https://doi.org/10.1126/science.277.5334.1956.
Thorpe, S., 1977: Turbulence and mixing in a Scottish Loch. Philos. Trans. Roy. Soc., A286, 125–181, https://doi.org/10.1098/RSTA.1977.0112.
Trossman, D. S., B. K. Arbic, J. G. Richman, S. T. Garner, S. R. Jayne, and A. J. Wallcraft, 2016: Impact of topographic internal lee wave drag on an eddying global ocean model. Ocean Modell., 97, 109–128, https://doi.org/10.1016/j.ocemod.2015.10.013.
Voet, G., J. B. Girton, M. H. Alford, G. S. Carter, J. M. Klymak, and J. B. Mickett, 2015: Pathways, volume transport and mixing of abyssal water in the Samoan passage. J. Phys. Oceanogr., 45, 562–588, https://doi.org/10.1175/JPO-D-14-0096.1.
Warner, S. J., and P. MacCready, 2009: Dissecting the pressure field in tidal flow past a headland: When is form drag “real”? J. Phys. Oceanogr., 39, 2971–2984, https://doi.org/10.1175/2009JPO4173.1.
Warner, S. J., P. MacCready, J. N. Moum, and J. D. Nash, 2013: Measurement of tidal form drag using seafloor pressure sensors. J. Phys. Oceanogr., 43, 1150–1172, https://doi.org/10.1175/JPO-D-12-0163.1.
Waterhouse, A. F., and Coauthors, 2014: Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate. J. Phys. Oceanogr., 44, 1854–1872, https://doi.org/10.1175/JPO-D-13-0104.1.
Yang, L., M. Nikurashin, A. M. Hogg, and B. M. Sloyan, 2018: Energy loss from transient eddies due to lee wave generation in the Southern Ocean. J. Phys. Oceanogr., 48, 2867–2885, https://doi.org/10.1175/JPO-D-18-0077.1.
Zeiden, K. L., D. L. Rudnick, and J. A. MacKinnon, 2019: Glider observations of a mesoscale oceanic island wake. J. Phys. Oceanogr., 49, 2217–2235, https://doi.org/10.1175/JPO-D-18-0233.1.
Zheng, K., and M. Nikurashin, 2019: Downstream propagation and remote dissipation of internal waves in the Southern Ocean. J. Phys. Oceanogr., 49, 1873–1887, https://doi.org/10.1175/JPO-D-18-0134.1.