1. Introduction
On a global scale, turbulent mixing helps to maintain the ocean’s overturning circulation by mixing cold, dense abyssal water with lighter water above it (Munk and Wunsch 1998). Turbulent mixing creates lateral density gradients that drive circulations, and it redistributes nutrients important for biological productivity. It is also a sink for energy from the ocean circulation and tides and thus is important to quantify so as to construct accurate energy budgets. The location and magnitude of mixing have been shown to significantly affect the circulation in global climate models (Jayne 2009; Melet et al. 2013), and the vertical divergence of the resultant buoyancy flux can profoundly affect water-mass modification and mass flux (Ferrari et al. 2016; McDougall and Ferrari 2017; de Lavergne et al. 2017). Although mixing can have important effects on the larger-scale circulation, it occurs on much smaller scales than most models resolve and so must be parameterized. Quantifying and understanding the processes leading to turbulence is essential to develop accurate parameterizations.
Breaking internal lee waves are one important source of deep mixing. They have previously been observed in shallow coastal and fjord environments (Nash and Moum 2001; Klymak and Gregg 2004; Nakamura et al. 2010), but not in the deep ocean until recently. Deep turbulent events phase locked to the tidal forcing were observed during the Hawaii Ocean Mixing Experiment (HOME; Levine and Boyd 2006; Aucan et al. 2006; Klymak et al. 2008). Subsequent modeling showed that these events were due to breaking lee waves generated by tidal flow over the ridge and swept over the ridge when the flow reverses (Legg and Klymak 2008; Klymak et al. 2010b). Under these circumstances, dissipation is expected to scale as
A significant amount of work lately has dealt with 3D effects (Pinkel et al. 2012; Warner and MacCready 2014; MacKinnon et al. 2017; Puthan et al. 2021), which allow some flow to go around instead of over obstacles, in addition to creating asymmetries in tidal flow patterns that complicate simple lee-wave interpretations and create additional modeling challenges. Specifically, detailed model/observation comparisons for such 3D tidal situations are rare.
At Luzon Strait, strong barotropic tidal flow over two ridges generates some of the world’s largest internal tides (Alford et al. 2011b, 2015). In contrast to the Hawaiian Ridge, the internal wave field in Luzon Strait is complicated by the 3D bathymetry, frequent intrusions of the Kuroshio, and strong mesoscale variability. While a large fraction of the energy lost from the barotropic tide at Luzon Strait radiates away as low-mode internal tides, a significant amount of dissipation occurs locally in the form of breaking lee waves (Alford et al. 2011b; Buijsman et al. 2012, 2014). Observations made in Luzon Strait in 2010 as part of the Internal Waves in Straits Experiment (IWISE) revealed extremely large density overturns and turbulence levels on the ridge slopes (Alford et al. 2011b). Buijsman et al. (2012) used a 2D model to confirm the formation and breaking of lee waves on the western ridge at 20.6°N, also showing that dissipation at this location was increased during semidiurnal forcing due to a resonance between the two ridges.
In the summer of 2011, a mooring array was deployed in Luzon Strait as part of the IWISE experiment, with the goal of observing these processes over multiple spring–neap tidal cycles and different mesoscale states. Here we present observations from one mooring (S9) deployed in southeast Luzon Strait (Fig. 1). The western ridge is deeper at this latitude, so resonance effects are not expected. However, the topography at this location is more three-dimensional, in contrast to the northern portion of the Strait where lee-wave formation and dissipation may be approximated by parameterizations based on flow over two-dimensional ridges. In this paper we combine these data with a high-resolution, realistic 3D MITgcm simulation to examine the roles of three-dimensional topography and low-frequency flow in setting dissipation rate and explore the ability of the model to reproduce the observed flow and turbulence.
(inset) Map showing entire Luzon Strait bathymetry and mooring location (yellow). (main panel) Detail map of mooring location (yellow star) and bathymetry contours. The black line shows the position of the model transect discussed in the text. Black circles and line show depth-mean observed velocity integrated over 1 day from yearday 209 to 210, during the fourth spring tide. Cyan shows the mean observed velocity in the lower 500 m integrated over the same period.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
2. Data and methods
a. Mooring
Mooring S9 (19°20.147′N, 121°1.839′E), one element of a larger array deployed to measure energy flux, dissipation and structure of the internal tides generated at the complicated two-ridge system of Luzon Strait (Alford et al. 2011b; Pickering et al. 2015), was designed to measure full-water-column velocity and density on a slope on the eastern ridge in the southern Luzon Strait (Fig. 1) at a depth of 2286 m. The bathymetry is supercritical with respect to the dominant diurnal frequency (Alford et al. 2011b). The mooring consisted of two stacked McLane moored profilers (MP), sampling depths of 330–1260 and 1300–2214 m, respectively. The lower MP did not sample the lower 75 m because of the mooring’s having been deployed somewhat deeper than targeted. To avoid excessive mooring knockdown due to the very strong mean velocities in the region associated with the Kuroshio and its meanders as well as the strong internal tides, the mooring was designed with a large subsurface float at 300 m (below the strongest velocities) to keep the lower part of the mooring as taut as possible for optimal McLane profiler performance. Each MP profiled at 0.25 m s−1, resulting in a profile approximately every 1.5 h. The MPs have an effective vertical resolution of 10 m for velocity and 2 m for density (Alford 2010). Near-surface velocity was measured by an upward-looking long-ranger deployed at 300 m and a thermistor chain (T chain) extending upward to a smaller float near 30-m depth. Velocity, isopycnal displacement, and turbulence measured from the mooring are shown in Fig. 2.
Time series of data from mooring S9. (a) Barotropic velocity from observations (black) and model (blue). Four periods centered on spring tides highlighted in later sections are shown with colored bars at the top. (b) Depth-integrated ϵ from observations (black) and model (blue). (c) Zonal velocity at S9, with isopycnals contoured every 400 m. (d) The log10 of turbulent dissipation rate ϵ.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
b. Model
The MITgcm (Marshall et al. 1997) was used in a 3D configuration, similar to that described in Buijsman et al. (2014) but with higher horizontal and vertical resolution. The model uses realistic topography merged from high-resolution gridded multibeam data with a resolution of ∼300 m and 30-arc-s resolution global topography/bathymetry grid (SRTM30_PLUS) data from the Smith and Sandwell database with a resolution of ∼1 km (Smith and Sandwell 1997). It was run with 150 vertical levels and 250-m horizontal resolution (telescoped to boundaries). A horizontally uniform, depth-varying stratification is used, derived from temperature and salinity data collected in between the ridges (Alford et al. 2011b). The tidal forcing at the east, west, north, and south model boundaries is composed of barotropic velocities (Fig. 2a, red) extracted from TPXO7.2 (Egbert and Erofeeva 2002), which agree well with depth-average velocity measurements (black). Dissipation is computed in the model from Thorpe sorting using the scheme of Klymak and Legg (2010). The model was run for yeardays 207–214, spanning the fourth spring tide observed in the mooring measurements.
c. Turbulence
For the observational and model data, overturns are computed following Thorpe (1977), who proposed that the size of overturns in potential density profiles could be related to the size of turbulent eddies and the turbulent dissipation rate. The distance each parcel of water must be moved vertically is defined as the Thorpe displacement. The Thorpe-scale LT is defined as the rms Thorpe displacement over each overturning region. The Thorpe scale is assumed to be proportional to the Ozmidov scale LO (Ozmidov 1965), the maximum overturning length scale in a stratified fluid; this relationship has been found experimentally to be LO = 0.8LT (Dillon 1982). Turbulent dissipation rate is then computed as
Spurious overturns are identified and rejected using several standard tests, including statistical run-length tests (Galbraith and Kelley 1996) and instrument noise levels (Ferron et al. 1998). Thorpe scales computed from profiles of potential temperature are nearly identical to those computed from density, as also found by Alford et al. (2011b) and Mater et al. (2015), owing to the dominance of temperature in setting density in the region.
Two measurement effects introduce possible high biases into the dissipation rate estimates. First, while a preponderance of evidence supports the approximate equivalency of appropriately averaged Thorpe and Ozmidov scales (Dillon 1982; Moum 1996; Mashayek et al. 2021), recent work suggests a possible high bias of time-mean dissipation rate by a factor of 2 or 3 estimated from Thorpe scales where large overturns arise from convective instability (Mater et al. 2015; Scotti 2015). Because microstructure was not directly measured, we have no means of confirming such a bias in our data; any bias if present does not affect our primary conclusions about the causes, phasing, and mesoscale dependence of the turbulence. Importantly, such biases are expected to equally impact observed and modeled dissipations, which both employ Thorpe scales.
The MP data are also subject to a positive (high) bias of up to a factor of 3 arising from the relatively slow profiling speed in comparison with internal tide vertical velocities (see the appendix). In essence, when the MP is sampling an internal tide with vertical velocities comparable to the profiling speed, density inversions are measured that are both larger and smaller than their actual extent depending on tidal phase, causing a high bias resulting from the nonlinearity of the calculation.
3. Results
a. Oceanographic overview and context
Barotropic (depth averaged) velocity shows mixed but diurnally dominated tides (Fig. 2a), as well as a weaker low-frequency flow that will be discussed below. Following tradition in the region’s literature, we define flood and ebb as toward the South China Sea (west) and Pacific (east), respectively. The moored data span about 50 days, encompassing four spring–neap cycles. Velocity, displacement (Fig. 2c), and dissipation rate (Fig. 2b,d) are strongly enhanced during each spring, which are labeled in color at the top of Fig. 2a.
Lateral barotropic tidal displacements obtained by integrating the observed depth-average velocities during the fourth spring tide (Fig. 2a) are about 5 km (Fig. 1, black) and oriented perpendicular to the fan. Meanwhile, associated lateral displacements from near-bottom velocities (cyan), where tidal flows are stronger as well as showing a low-frequency flow, are larger. These near-bottom flows show a clear hysteresis as near-bottom ebb-tide flows are steered northward by the fan, while flood-tide flows are directed more westward. Because it is the near-bottom flows that generate the lee-wave response, these differences between the depth-mean and near-bottom flows prove to be important. In terms of the excursion parameter Ltides/Lbump, depth-mean and near-bottom flows give values of ∼1 and 2, respectively.
In contrast to the geometry and dynamics to the north, the S9 mooring is south of the main Kuroshio path, such that low-frequency flows are due to eddies generated by the Kuroshio meanders rather than a strong northward velocity (Pickering et al. 2015, Fig. 4). Depth-averaged low-frequency flow (Fig. 3a, thick line) is about 0.1 m s−1, veering slowly from northeastward to northwestward and finally to nearly westward (Fig. 3b) and decreasing gradually with time from the first to the fourth spring tide. Near-bottom flow (thin line) is generally in the same direction (Fig. 3b), but significantly stronger and showing a spring–neap cycle suggesting that some of the flow is rectified; that is, the hysteresis of the tidal flow gives rise to a perceived low-frequency flow at the mooring location. These effects are also seen in the mean profiles of low-passed U, V, and |U| (Figs. 4b–d), which show enhancement near the bottom of the low-frequency flow.
Three-day low-passed (a) current speed and (b) direction for barotropic currents (thick lines) and currents below 1700 m (thin lines).
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
Time-mean depth profiles computed over each spring-tide period of (a) N2, (b) low-passed zonal velocity, (c) low-pass-filtered meridional velocity, and (d) low-pass-filtered velocity magnitude. Colors refer to individual spring tides as indicated in Fig. 2.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
Near-bottom flow decreases in time over the four spring tides (Fig. 3a), as does depth-integrated dissipation rate (Fig. 2b). The hypothesis that the stronger near-bottom flow during the earlier spring tides contributes to the stronger dissipation will be examined below.
Stratification (Fig. 4a) decreases with depth as typical for this region, with no noticeable dependence on time.
b. Internal tide and lee-wave response
Velocity (Fig. 2c, colors) reflects some of the low-frequency features above but is dominated by the strong and complicated internal tide response, documented in Alford et al. (2011b), Buijsman et al. (2012) and Pickering et al. (2015). Isopycnal displacements (gray) are tidally dominated, with values exceeding 300 m near 1800-m depth during spring tides.
Turbulent dissipation (Fig. 2d) is strongest deeper in the water column and extends to about 700 m off of the bottom, with a large spring–neap cycle. Dissipation integrated from the bottom to 1400-m depth (Fig. 2b) varies by an order of magnitude between spring and neap, with the peak value decreasing by a factor of 2 from the first to the fourth spring tide.
Observed time-averaged dissipation (Fig. 5, black) increases toward the bottom, reaching a maximum of nearly 10−6 W kg−1 in the bottom 200 m. This profile is similar to those observed at the Hawaiian Ridge (Klymak et al. 2006) and elsewhere in Luzon Strait (Alford et al. 2011b), with dissipation increasing from middepth to the bottom. Both the decay scale and the magnitude of dissipation rate vary by about a factor of 2 among the four individual spring tides, which are colored corresponding to the bars at the top of Fig. 2. Both the magnitude and vertical extent of dissipation and diffusivity are larger during the first spring tide, and decrease with each subsequent spring tide, as better seen when plotted on a linear scale (Fig. 6). Under the assumption of constant mixing efficiency ∼0.2 (Osborn 1980), turbulent diffusivity Kρ ≡ 0.2ϵN2 (Fig. 5b) exceeds 10−1 m2 s−1, potentially subject to the biases mentioned in the previous section.
Depth profiles of time-averaged (left) ϵ and (right) turbulent diffusivity Kρ. The S9 profile for the entire record is shown in black. Blue is the MITgcm profile. Other colors are profiles for individual spring tides.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
Depth profile of time-averaged ϵ averaged over each spring tide period, plotted on a linear scale.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
Model dissipation, shown later, is slightly stronger than observations near the bottom, but decays more rapidly, giving a model depth-integrated dissipation rate (Fig. 2b, blue) approximately 1.5 times as great as observed. As noted above, observations are potentially biased high by up to a factor of 3, giving rise to the possibility that the model overestimates the turbulence by up to a factor of 4.5. On the other hand, the model might underrepresent roughness on smaller scales than resolved (≈300 m), which could give rise to additional small-scale lee-wave formation and breaking (e.g., Nikurashin and Ferrari 2010; Hibiya et al. 2017; Hibiya 2022) that would be captured in the observations but not the model. Notwithstanding these areas for improvement, the MITgcm model with the Klymak and Legg (2010) closure predicts observed magnitude and depth decay of the turbulence fairly well, as was also found by Alford et al. (2014).
Zooming in to a representative period (Fig. 7), the model reproduces the magnitude and timing of velocity, displacements and dissipation reasonably well, though there are important differences. Broadly, near-bottom velocity (Figs. 7b,c, colors) and displacements (black) in the model and observations show good agreement in both magnitude and phasing. Modeled (Fig. 7d) and observed (Fig. 7e) dissipation rates are both enhanced toward the bottom and show comparable tidal modulation.
Observed and model data from mooring location for a representative 3-day period. (a) Depth-mean zonal velocity from observations (black) and model (blue). Also shown are (b) model and (d) observed zonal velocity and (d) model and (e) observed ϵ; A, B, and C indicate times of spatial transects shown in Figs. 9–11. Gray lines in (c) and (e) show the sampling pattern of the MP.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
As demonstrated in Buijsman et al. (2012), the ridge at this location is strongly supercritical to both semidiurnal and diurnal tides and thus conducive to the formation of high mode lee waves (Klymak et al. 2010b). Both the observations and the model are indeed consistent with a breaking lee wave formed on downslope (eastward) flow and swept past the mooring when the flow slackens, as seen at Kaena Ridge, Hawaii (Klymak et al. 2008, 2010a; Alford et al. 2014). Specifically, near-bottom flow during ebb tide (upslope flow) lifts isopycnals in both observations and the model, generating a lee wave east of the mooring. Immediately after the flow switches from ebb to flood, isopycnals drop sharply (e.g., yearday 209.4) before rebounding as the lee wave sweeps past the mooring. Dissipation rate in both observations and model (Figs. 7d,e) shows phase propagation, with strong dissipation seen first between 1600 and 1800 m before increasing at 2000–2200 m.
The reasonable agreement in vertical structure, magnitude, and phase between observations and model encourages the use of the model to obtain further inferences on the structure of the lee-wave response. Cross sections of the MITgcm model aligned in the direction of dominant barotropic flow (240°T, where T indicates true) and passing through the mooring location (Fig. 1; black line) are shown at three times, A, B, C, which are indicated in Fig. 7.
During flood tide (Fig. 8; time A), lee wave of approximately 600-m vertical wavelength forms on the steep western side of the ridge. Because the slope is so steep, the wave is somewhat under-resolved in the model (250-m horizontal resolution). Nonetheless, the ≈600-m vertical wavelength and its slope close to that of the bottom are consistent with predictions from theory (Klymak et al. 2010b), as demonstrated in the next section.
Spatial section at time A (Fig. 7) of a(top) long-transect velocity (toward 60°T), and (bottom) ϵ from the MITgcm model during ebb tide. In the top panel, contours of log10ϵ > 5.8 are shown in white. Isopycnals are contoured in black in both panels. The dashed line indicates the location of S9.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
During ebb tide, the wave forms on the more gently sloped side of the ridge (Figs. 9 and 10; times B and C), and also shows the expected vertical wavelength of about 600 m and a slope close to the that of the seafloor. Before the flow switches from ebb to flood (Fig. 9), the model shows a typical lee-wave response, with a strong, bottom-intensified northeastward jet (>0.5 m s−1) over the ridge crest. Isopycnals are depressed and then rebound, with strong dissipation just northeast of the mooring location. A lee wave is seen forming with isopycnals pushed downward east of the mooring, forming a trough near x = 9 km before rebounding sharply at x = 10 km. Turbulence is largest just above this sloping jet, near 2000-m depth and between x = 6 and 10 km. Just before this, the turbulence occurred nearly entirely east (upslope) of the mooring (dashed). At this time, the sloping feature has just begun to propagate to the left as the flow slackens.
As in Fig. 8, but for time B.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
As in Fig. 8, but for time C.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
By time B (Fig. 10), the flow has slackened sufficiently that the lee wave is able to sweep past the mooring under a combination of its own westward propagation and the increasingly westward flow. The western (shallower) edge of the spatially sloped feature arrives at the mooring first, giving the phase propagation observed in Figs. 7d and 7e. A weaker lee wave is seen forming on the west side of the ridge now, with a trough near x = 4 km, just west of the mooring site. Isopycnals at S9 are now stretched vertically (high strain), and turbulence is strongest below 2000 m and between x = 4 and 10 km.
Both observed and modeled dissipation rate have a second peak during each diurnal period, later in flood tide (Fig. 7). These peaks are associated with the lee wave formed on the western side of the ridge, which can just be seen forming in Fig. 10 near km 4. The asymmetry of the ridge and also the weaker flow cause the dissipation response of this peak to be weaker and more bottom-intensified. The position of the mooring near the top of the ridge allows the turbulence from both lee waves to be seen as they sweep past it.
The asymmetry induced by the 3D topography as well as the near-bottom enhancement of the tidal flow are clear in Fig. 11. Near the end of ebb tide (Fig. 11a), barotropic velocity (blue arrows) is toward the northeast. However, the flow averaged over the bottom 300 m (white) is much stronger and steered northward by the topography. The asymmetry in the tidal flows at the mooring site gives rise to the perceived stronger low-frequency flows at spring tide due to tidal rectification.
Modeled depth-integrated dissipation (colors), BT velocity (blue arrows), and velocity averaged over the bottom 300 m (white arrows) at times (a) B and (b) C. (A is similar to C and so is not plotted). The magenta arrow shows near-bottom velocity measured at S9; the black line shows the transect plotted in Figs. 8–10.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
The largest dissipation during this period is east of S9, as was also seen in the model cross section. At time C (Fig. 11b), near-bottom velocity is strong and westward over the shallower areas of the slope. Dissipation is largest over the steep slopes running south of S9, associated with the lee wave formed on flood tide.
c. Lee-wave structure and scaling
According to this scaling, lee waves should then have a vertical scale of λz ≈ 2πUo/N (Klymak et al. 2010b). Using the measured values given above gives a vertical scale of λz ≈ 600 m, which is close to that observed.
As noted above, the ridge is asymmetric, with estimated slope α of 0.1 and 0.25 on the eastern and western sides of the ridge, respectively. Since lee-wave frequencies must be high enough for the wave to form before the tide has changed appreciably, α > 2ωtide, a condition that is easily satisfied at our ridge, especially for the dominant diurnal tide (Buijsman et al. 2012, Fig. 1).
The observed wave on ebb tide closely matches the bottom slope (Figs. 9 and 10), another prediction of Klymak et al. (2010b). On the flood tide (Fig. 8) the slope is slightly harder to discern as it is likely somewhat underresolved, but it does not appear inconsistent with the bottom slope. Corresponding horizontal wavelengths are 5 and 2 km for the ebb and flood tides, respectively. Corresponding respective lee-wave frequencies are then 0.1N and 0.25N, or 3 and 7 times the diurnal frequency, for the ebb and flood, respectively. Therefore, the waves are consistent with the lee-wave dispersion relation. Their slope is 4 and 10 times that of the diurnal internal tide, respectively, thus ruling our that they are simply long internal tides. Long internal tides do also exist strongly in the region but on much longer horizontal scales (Alford et al. 2011a; Buijsman et al. 2012), as predicted by Hibiya (1986).
d. Scaling with barotropic velocity
Depth-integrated turbulent dissipation rate ϵ plotted vs barotropic velocity magnitude from S9 for each spring tide period (colors) and MITgcm model (blue), both averaged with a moving 1-day window. Solid lines show fits with Eq. (2), with the best-fit slope m indicated in the legend. The black solid line is the fit over all times; the dashed line shows
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
The scaling with UBT is broadly consistent with the cubic dependence expected from Klymak (2010b), but the factor of 2 between the first spring and the last spring appears as variability. Both dissipation rate and low-frequency flow are strongest during the first spring tide, decreasing gradually in time over each spring, suggesting that inclusion of the low-frequency near-bottom flow should improve the scaling. However, simply adding the near-bottom flow from Fig. 4 does not improve the fits. That is, shifting the curves to the right by the observed low-frequency-flow vectors would be far too large a correction. Since the rectified spring-tide flows are not entirely cross ridge (Fig. 3), it is possible that using the smaller cross-ridge component might yield better results. However, the three dimensionality of the flow complicates such calculations. Hence, low-frequency flow appears to introduce a factor-of-2 variability into the dissipation rate estimates that is not captured by the simple
4. Discussion
A novel aspect of this dataset is the long time series covering several spring–neap tidal cycles and different mesoscale states, allowing investigation of how tidally forced processes can be influenced on subtidal time scales. The model (which is only forced with the tides) and the data both show general consistency with a U3 scaling expected for breaking lee waves, but an additional modulation by a factor of approximately 2 is seen, suggesting that near-bottom total flows, rather than barotropic tidal flows, set the dissipation.
While it is sensible that the lee-wave response should depend on the total near-bottom flow rather than the depth-averaged tidal flow (Klymak et al. 2010a), the scaling is not simple, motivating further work on the topic. The short-term message for modeling tidal dissipation is that much of the variability is in fact captured with cubic tidal models, which is attractive because such schemes can be run “offline” in GCMs. However, capturing the last factor of 2 or so requires a fuller understanding of the interplays between the low-frequency and tidal near-bottom flows, which is complicated owing to three dimensionality, tidal rectification, and variable flow paths.
A significant finding of our work is the agreement between the observations and a 3D realistically forced model, as also found by Alford et al. (2014), at least for the phasing, depth structure, and approximate magnitude of velocity, displacement, and dissipation. Important differences in vertical structure remain unaccounted for. Nonetheless, the skill of such models suggests their continued usefulness in both planning experimental efforts and also exploring improved parameterizations.
5. Conclusions
We have presented nearly full-depth temperature, salinity, and velocity measurements from a mooring deployed on a supercritical slope in Luzon Strait for 50 days. Turbulent dissipation is several orders of magnitude greater than typical open ocean values and exhibits a spring–neap cycle, varying by an order of magnitude between spring and neap. Dissipation also varies by approximately a factor of 2 for similar barotropic forcing during different spring tides, suggesting that dissipation scales with near-bottom total flow (tides plus low frequency), rather than the barotropic tidal flow alone. However, the exact scaling was not clear from our analyses.
Modeled velocity, displacement, and dissipation from a high-resolution 3D MITgcm model simulation agree well with the observations and are used to place the mooring observations in context. Model transects confirm that turbulence in the bottom 1000 m is largely due to breaking lee waves formed during tidal flow over the slope. The 3D topography influences the tidal flow over the slope, resulting in spatially variable patterns of flow and dissipation that would not be captured in simple 2D models or parameterizations.
Acknowledgments.
This work arose from the Ph.D. thesis and postdoctoral work of Dr. Andy Pickering. The other authors believe that he should be first author. However, Dr. Pickering has indicated in writing that for personal reasons he prefers that the paper be published without his being an author. We respect his wishes and deeply thank and acknowledge him for his hard work, talent, and many contributions to the IWISE project. This work was funded by the U.S. Office of Naval Research under Grants N00014-09-1-0219 (author Alford), N00014-09-1-0281 (author Nash), and ONRDC32025354 (author Buijsman). We thank the captain and crew of the R/V Revelle and R/V OR1 for their skill and hard work to acquire these measurements. We also thank John Mickett, Eric Boget, Luc Rainville, Zoe Parsons, Paul Aguilar, Tom Peacock, Hayley Dosser, Ke-shien Fu, and Chung-wei Lu for their work in deploying the moorings and staffing the deployment and recovery cruises. We thank three anonymous reviewers for their comments and suggestions, which greatly improved the paper. We also thank Zoë Geller Alford for useful comments and a careful proof of the final submitted paper.
Data availability statement.
Observational data and model output used for the plots and analyses in this paper are available upon request.
APPENDIX
Biases in Turbulent Dissipation Rate Estimated via Thorpe Scales from Profiling Instruments
a. Objectives
In applying the Thorpe-scale method to compute the dissipation rate ϵ, an implicit assumption is made that profiles are obtained instantaneously, and/or that the true profile of temperature or density did not change during the time it took for a profile. The validity of this assumption and its effect on the estimated turbulence have not been previously investigated [although it was pointed out by Thorpe (1977)]. For profiles that were obtained relatively quickly, one might assume that this assumption is well founded. However, for data obtained from a relatively slow-crawling profiler in a rapidly changing flow, this assumption may be violated because the density structure and turbulence changes and/or is advected by the lower-frequency background flow during the time taken to obtain a profile. We anticipate that vertical advection during a profile may decrease or increase both the apparent overturn size and background stratification measured by the instrument, and in some cases can introduce apparent overturns in a profile that is actually stable. This appendix sets bounds on the uncertainty and bias of dissipation estimates made in highly energetic flows such as those presented here. Some of the findings are analogous to biases introduced by sloping transects made by gliders, as Smyth and Thorpe (2012) found by sampling a numerically simulated flows as gliders would.
b. Data
To test our assumptions, we use data from a traditional mooring (with an array of sensors at fixed locations on the mooring line) deployed nearby in Luzon Strait during IWISE (Pickering et al. 2015). The mooring is not part of the dataset used in this paper but shows similar overturning scales and is also strongly tidally forced. The mooring was deployed on the eastern slope of the western ridge, where previous measurements and models indicated strong overturns and turbulent dissipation associated with breaking lee waves formed by tidal flow over the ridge (Alford et al. 2011b; Buijsman et al. 2012). Instruments included SBE-56 (Fig. A1) and RBR-1060 temperature loggers sampling at 1-s intervals and SBE-37 CTDs sampling at 6-s intervals. Our observations span the depth range between 600 and 2100 m, with an average vertical spacing between instruments of about 50 m, increasing to O(10 m) near the bottom. All measurements were corrected for mooring blowdown using pressure measured by the SBE-37s and then were interpolated onto a depth–time grid with a vertical spacing of 10 m.
Example data from T chain 3: (top) barotropic tidal velocity from TPXO, (middle) temperature, and (bottom) turbulent dissipation rate ϵ.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
c. Resampling
For this analysis the gridded mooring temperature data are interpolating onto the sawtooth depth–time path of a simulated profiling instrument traveling up and down at constant vertical speed. Each up or down segment of the path (Fig. A2) is then assigned to a “profile” at the time of the midpoint of the segment. Thorpe-scale analyses are then applied to both the resampled profile pairs and the “real” profile obtained instantaneously at the midpoint time and compared. Resampling is done for an ensemble of sampling paths with start times shifted by 2 min, in order to average over all possible phases between sampling and the flow. For this analysis, we use data from yeardays 165 to 185, a period encompassing an entire spring cycle during which all instruments were functioning. The Matlab code used to resample data along simulated profiles is available online (https://github.com/andypicke/SimProfiler).
Zoom of example data from T chain 3, Showing ϵ (colors) and isotherms (black). The white line shows an example sampling path for w = 0.25 m s−1. Dashed vertical lines indicate the midpoint time of profiles.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
d. Results
1) Effect of noninstantaneous profiles
We examine the effect of noninstantaneous profiles sampled by a profiling instrument at four different speeds. The 0.25 m s−1 case simulates sampling by a McLane MP. We find that the ensemble mean profiles have a positive bias that is inversely proportional to the profiling speed (Fig. A3). The ensemble mean profile (averaged over yeardays 165–185) is biased by approximately a factor of 2 for w = 0.25 m s−1. Time series of the depth-integrated dissipation (averaged over 1-day periods; Fig. A4) exhibit similar biases that are approximately independent of time (or the magnitude of the dissipation rate).
Time-mean profiles of dissipation rate at T chain 3 for different profiling speeds. The black line is the true profile using all data. The orange line is the mean of the resampled ensemble. Gray is all individual resample profiles. Dashed lines indicate ±1 standard deviation.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
Depth-integrated time series of dissipation rate at T chain 3 for different profiling speeds. The black line is the true profile using all data. The yellow line is the mean of the resampled ensemble. Gray is all individual resampled profiles. Dashed lines indicate ±1 standard deviation.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
(i) Histograms
Figure A5 shows histograms of overturn size for the true T-chain data as well as the resampled data for w = 0.15. Distributions of overturn size and Thorpe scale are similar, but the resampled data contain more larger and smaller values than their true values—in other words, the resampled data have a broader distribution. Histograms of N2 are both approximately lognormal, but the resampled data are shifted to larger values. Dissipation rates follow a similar pattern, with resampled data exhibiting a similar distribution shifted to larger values. Larger Thorpe scales and N2 thus both lead to larger values of ϵ, with stratification having a larger impact since ϵ ∝ N3.
Histograms of (top left) the patch size L, (top right) the rms overturn size LT, (bottom left) the patch stratification N2, and (bottom right) the patch dissipation rate ϵ for true and resampled data at T chain 3. The mean of each distribution is indicated by the dashed lines.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
(ii) CTD scenario
We also present another scenario designed to simulate continuous CTD sampling ([such as 24–36-h lowered acoustic Doppler current profiler (LADCP)/CTD stations] since it is a common measurement technique. The typical profiling speed for a shipboard CTD is 1 m s−1, ignoring the short sections of the water column near the top and bottom where it is normally slowed. We also simulate sampling on the downcast only, which is often done since CTD measurements are contaminated by the rosette’s wake on upcasts. The sampling results in an average bias close to zero (Fig. A6), but there is significant uncertainty (factor of 2) for daily averages of individual CTD profiles as compared with the full time average. Thus, we concluded that the profile speed of the CTD is high enough to significantly reduce the errors and bias from noninstantaneous profiles seen at lower speeds, but the undersampling from using only the downcasts increases the uncertainty in daily averages (which, for this particular case, contain about 20 profiles and thus have a degree of randomness depending on how they sample the tidally modulated signals characteristic of this particular site).
Profiles of time-mean dissipation rate ϵ for simulated sampling by a shipboard CTD profiling at 1 m s−1, using downcasts only. Gray lines are daily averaged profiles, dashed lines are ±1 standard deviation, yellow is the mean, all for the resampled data, as compared with the real profile (thick black line).
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
2) Detailed look at one turbulent cycle
To gain better understanding of what leads to the observed bias, we look at one turbulent period in detail. In Fig. A7, resampled ϵ is plotted along downward and upward sampling paths. From this figure, it appears that the bias is largest when the slope of isopycnals is approximately equal to the slope of the profiling path in depth–time space (or equivalently, the “background” vertical velocity of the fluid is equal to that of the profiler).
A detailed look at one turbulent period: (d) ϵ estimated from full T-chain data and Ε from overturns for (e) downward and (f) upward sampling paths, plotted along each path in the ensemble. The color scale for epsilon is the same in (d)–(f). (a)–(c) The time-average profile of true (black) and resampled (blue) Ε for each respective case. Note the much larger epsilon where sampling paths are nearly parallel to slope of isopycnals.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
3) Why is the bias always positive?
We imagine an instrument profiling through a density field containing turbulent overturns with the “true” size equal to what would be measured by an instantaneous profile at the midpoint of the slower profile. During the time taken to complete a profile, the background field may evolve, and overturns may be advected vertically or horizontally (we neglect horizontal advection here). If the advection is in the direction of the profiler, the measured overturn would appear larger than the true size. If advection is in the opposite direction, overturns will be compressed and appear smaller. Assuming the phase between the sampling and any vertical advection is random, the measured overturns will be symmetrically perturbed to smaller and larger values and have the same average size. However, because
Histograms of the difference between true and resampled overturn sizes for different profiling speeds at T chain 3. The mean and standard deviation are given above each plot, and red lines show normal distributions with those parameters.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
4) Identifying and correcting biased data
Can we identify errors in epsilon from the sampled data without knowledge of the true data? One method would be to identify regions where the background vertical velocity is similar to the profiler speed. However, the slow sampling may prohibit accurate estimation of vertical velocity over the time scales we are interested in. Instead, we infer a turbulent velocity wt velocity scale based on the measured Thorpe scale and stratification, wt = LTN. Data for which wt is larger than a fraction of the profiler speed wsamp are marked as biased and discarded. As an example, for data at T chain 3, we found that the criterion wt > 0.5wsamp eliminated most of the bias in the time-averaged profile of epsilon (Fig. A9).
Time-average profiles of dissipation rate at T chain 3. The black line is the true value. The blue line is for resampled data with wsamp = 0.25. The red line shows resampled data after discarding data where wt > 0.5wsamp.
Citation: Journal of Physical Oceanography 53, 1; 10.1175/JPO-D-22-0117.1
e. Summary
This appendix provides useful constraints for planning sampling by mooring or shipboard CTD. While faster sampling speeds are preferred, the profiling speed of an MP or CTD is typically fixed. Reducing the vertical range of a profiler will reduce uncertainty due to undersampling but will not alleviate biases when the fluid’s vertical velocities are similar to the profiler speed. Preliminary full-depth measurements and local model simulations could help identify specific depths to be studied, allowing measurements to be optimized for estimating turbulence.
Do the results of this study apply to other locations? The magnitude of the bias will likely vary depending on the nature of flow and turbulence at different locations, but we expect the relationship between bias and profiling speed to be similar at most locations. We also expect the positive bias to hold at most locations, as a result of the two mechanisms discussed. However, the results presented here are for a particular location and may differ at other locations with different flow regimes and turbulent processes.
In conclusion, Thorpe-scale estimates of ϵ from data measured by profiling instruments exhibit a positive bias that is inversely proportional to the vertical sampling speed. “Profiles” obtained over a finite time result 1) in uncertainty in ϵ because they often capture only a small fraction of the overturns that contribute to the sample estimate and 2) in a positive bias associated with the nonlinearity of the Thorpe estimate of dissipation rate. Errors are largest when the background vertical velocity is of similar magnitude to the profiling speed. Our results suggest that errors are minimal for typical shipboard CTD sampling at speeds of about 1 m s−1. At the slower sampling speeds typical of moored profilers, there can be a significant bias in energetic regions with large vertical displacements. Our results should be useful in estimating uncertainty in turbulent dissipation rates measured from overturns by profiling instruments, as well as in planning sampling of turbulent overturns.
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