Observations of breaking internal tides on the Oregon continental slope during a 40-day deployment of 5 moorings along 43°12′N are presented. Remotely generated internal tides shoal onto the slope, steepen, break, and form turbulent bores that propagate upslope independently of the internal tide. A high-resolution snapshot of a single bore is captured from lowered acoustic Doppler current profilers (LADCP)/CTD profiles in a 25-h time series at 1200 m. The bore is cold, salty, over 100 m tall, and has a turbulent head where instantaneous dissipation rates are enhanced (ε > 10−6 W kg−1) and sediment is resuspended. At the two deepest slope moorings (1452 and 1780 m), similar borelike phenomena are observed in near-bottom high-resolution temperature time series. Mean dissipation rates and diapycnal diffusivities increase by a factor of 2 when bores are present ( W kg−1 and m s−1) and observed internal tides are energetic enough to drive these enhanced dissipation rates. Globally, the authors estimate an average of 1.3 kW m−1 of internal tide energy flux is directed onto continental slopes. On the Oregon slope, internal tide fluxes are smaller, suggesting that it is a relatively weak internal tide sink. Mixing associated with the breaking of internal tides is therefore likely to be larger on other continental slopes.
Using satellite altimetry and moored arrays, internal tides have been observed radiating thousands of kilometers from their generation sites (Ray and Mitchum 1997; Cummins et al. 2001; Ray and Cartwright 2001; Alford 2003; Zhao and Alford 2009). Low-mode internal tides become incoherent with the surface tide as they propagate away from where they are generated, becoming increasingly difficult to track (Rainville and Pinkel 2006). As a result, it is unknown where the bulk of internal tide energy dissipates. One possibility is that internal tides break over rough bathymetry or oceanic boundaries, such as continental slopes. This is supported by recent observations of internal tides shoaling on continental slopes (Nash et al. 2004; Klymak et al. 2008; Martini et al. 2011; Kelly et al. 2012), enhanced turbulence due to internal wave scattering from rough topography (Johnson et al. 1994; St. Laurent and Schmitt 1999; Ledwell et al. 2000; Moum et al. 2002), and enhanced turbulence in canyons (Carter and Gregg 2002; Kunze et al. 2012; Thurnherr et al. 2005). This paper follows on the Nash et al. (2007) and Martini et al. (2011) observations, reporting on the mechanism by which mixing is enhanced over the Oregon continental slope. Specifically, dissipation is associated with strongly nonlinear internal borelike features. Importantly, the bores appear to arise from the steepening of the incident internal tide rather than the local conversion of energy from barotropic to baroclinic motions.
The term “internal bore” has been used to describe a variety of oceanic phenomena, all of which share common features. Typically, bores are described as high-frequency wave trains that propagate horizontally and have a shocklike front followed by a turbulent head, analogous to swash or whitewater at a beach (Henyey and Hoering 1997). Vertical transport is primarily due to mixing, and there is a net stratification change after the passage of a bore. High-resolution numerical simulations by Venayagamoorthy and Fringer (2005) show that bores form when internal tides break on continental slopes. As internal tides move onshore, isopycnals become steeper and steeper, leading to an unstable density profile that overturns and breaks. The turbulent fluid then continues upslope as an internal bore.
Upslope bores may be one of several important mechanisms by which exchange occurs between deep and shallow oceans. As they propagate upslope, turbulent bores transport water-mixing heat, sediment, and nutrients across isopycnals. In addition, bores may attract organisms that feed on resuspended nutrients or other uncovered organisms. On the Conch Reef in the Florida Keys, large predatory fish and schools of reef fish orient themselves along similar upslope-moving bore fronts and convergence zones, most likely taking advantage of the increased nutrients (Leichter et al. 1996).
Internal bores have been observed at several deep sites with steep bathymetry. In the Scotland–Faroe Channel, two distinct upslope-propagating solibores [similar to both a turbulent bore and a nonlinear solitary wave (Henyey and Hoering 1997)] occurred four days apart (Hosegood and van Haren 2004). These solibores were nonperiodic, solitary events that were most likely a response to larger-scale atmospheric forcing. Near the crest of Kaena Ridge, Klymak et al. (2008) observed internal bores phase locked to the semidiurnal barotropic tide. Numerical simulations verify that these bores formed during conversion from a barotropic to baroclinic tide on the steep slopes of the ridge (Legg and Klymak 2008). Periodic, upslope bores have also been observed in both lowered acoustic Doppler current profiler (LADCP) time series and 10-day moored temperature records at two locations on the Mid-Atlantic Ridge (Dale et al. 2010). The Mid-Atlantic Ridge bores have semidiurnal periodicity and occur where the slope is supercritical to the M2 tide, but it is unclear whether they are barotropically or baroclinically forced. Baroclinically forced upslope bores have been reproduced in numerical simulations (Legg and Adcroft 2003) where shoaling mode-1 internal tides break on near-critical continental slopes.
Martini et al. (2011) observed that the internal tide on the Oregon continental slope has two major components: a remotely generated internal tide that shoals onshore and a locally generated internal tide that propagates offshore. The remotely generated internal tide is initially mode 1 and obliquely incident, that is, propagating northward nearly parallel to the Oregon coast (Martini et al. 2011; Kelly et al. 2012). Upon impinging on the slope, the incident internal tide is reflected to higher wavenumbers, turning the wave shoreward. The mode-1 internal tide offshore of the Oregon slope is relatively weak (0.5 kW m−1) compared to other sites of local tidal generation such as the Hawaiian Ridge (Lee et al. 2006) or Luzon Strait (Alford et al. 2011). Theoretical (Thorpe 1999) and laboratory (Dunkerton et al. 1998) studies have shown that obliquely incident internal tides also form upslope fronts or bores, produced by the nonlinear superposition of incident and reflected waves (Thorpe 1992). The formation of bores may be a mechanism by which shoaling internal tides generated in the deep ocean are dissipated on the Oregon slope.
We hypothesize that borelike features are the primary driver of mixing on the slope and the primary sink of the onshore-propagating tide. Owing to sparse sampling, we are unable to identify whether the features observed on the Oregon slope are bores, solibores, or some other nonlinear phenomena. However, these features are qualitatively similar to the bores described in Cacchione and Wunsch (1974), Pineda (1991), Slinn and Riley (1996), Legg and Adcroft (2003), Klymak et al. (2008), Legg and Klymak (2008), and Dale et al. (2010) and are therefore referred to as bores hereafter.
Instrumentation and methods are discussed in section 2. A high-resolution case study of a bore from an 18-h LADCP depth–time series is presented in section 3. Longer (40 day) and more detailed observations from time series at three moored locations enable descriptions of spatial and temporal variability, construction of composites, and identification of forcing mechanisms (section 3). Lastly, conditions for wave breaking and bore formation are discussed and comparisons with global estimates are made (section 4).
1) Moored array
A zonal line of five moorings at 43.21°N were deployed across the Oregon continental slope from the 500-m to 3000-m isobath for 40 days in October 2005 (Fig. 1). More details are given in Martini et al. (2011). On each mooring, a McLane Moored Profiler (MMP) made full-depth profiles of pressure, temperature, conductivity, and horizontal velocity every ~1.5 h, marginally resolving individual bores but providing detailed information about their vertical structure at discrete times. Within 20 m of the bottom (mab), below the MMP profiling range, high-resolution time series from Sea-Bird SBE-37 Microcats (temperature and salinity) and SBE-39 T-loggers (temperature), sampling every 60 and 30 s, respectively, are used to identify the bores, which are not always profiled by the MMPs.
2) LADCP/CTD Time series
To better resolve spatial variability on the slope, full-depth 15–24-h time CTD series of salinity, temperature, and pressure profiles were obtained at eight stations using the Sea-Bird SBE-911plus CTD on the R/V Wecoma. A pair of up/downlooking 300-kHz RDI LADCPs mounted to the CTD cage measured full-column velocity every three hours, and velocities were verified against the shipboard ADCP following Visbeck (2002). Between each full-depth profile the instrument package was cycled up and down in the bottom 500 m (~30 min per miniprofile) to maximize temporal resolution near the bottom. The LADCP/CTD package contained an Optical Backscatter Sensor (OBS) to measure suspended sediment in Formazin Turbidity Units (FTU).
b. Internal tides
The semidiurnal baroclinic internal tide is extracted from MMP and LADCP horizontal velocity and density depth–time series using methods described in Martini et al. (2011). Briefly, windowed, 3.5-day sliding semidiurnal harmonic fits are computed at each depth to obtain semidiurnal velocities and vertical displacements. Vertical displacement, ζ(z), is the vertical distance an isopycnal is displaced from its mean depth. Vertical displacements are then depth integrated to find the pressure perturbation
where ρ0 is the depth-mean density, and the depth-mean of the pressure perturbation, which are subtracted in (1) to satisfy the baroclinicity condition (Kunze et al. 2002; Kelly et al. 2010). Similarly, the depth-mean or barotropic velocity UBT is subtracted from each velocity profile u(z) to obtain the baroclinic velocity u′(z). The product of the baroclinic pressure perturbation and velocity averaged over a semidiurnal period gives the semidiurnal energy flux . For a single, freely propagating internal wave, energy flux is parallel to the group velocity of a wave, pointing in the direction of wave energy propagation.
c. Dissipation rate and diffusivity estimates
Following Alford et al. (2006), MMP profiles are used to obtain dissipation rates from density overturns (Dillon 1982). Density profiles are first sorted to produce stable profiles and Thorpe overturn lengths. The Thorpe length is the vertical distance each parcel must be moved to form the stable profile (Thorpe 1977). Averaging Thorpe lengths over each overturn produces the Thorpe scale, LT. Dissipation rates are then given by
where Ni is instantaneous stratification over the overturn (Dillon 1982). Diapycnal diffusivity is then obtained using
Dissipation rate time series are also estimated from T-loggers/Microcat pairs at the base of each mooring. These instruments were closer to the bottom and have a much higher temporal resolution than the MMPs. This method is described in more detail in the appendix. Briefly, overturn size is estimated from temperature inversions between the two instruments. Thorpe displacements are found from the overturns, where in which Tupper is the upper temperature, Tlower the lower temperature, and is the 2-h smoothed vertical temperature gradient. Overturns are only calculated from observed inversions, Tupper < Tlower. At the base of a mooring the nearly linear relation between temperature and salinity is used to obtain the instantaneous stratification of each overturn. Observed overturn size is defined here to be the Thorpe scale, then used to calculate dissipation rates using (2). The size of the overturn is not a “true” Thorpe scale as it is only an instantaneous measure of vertical water parcel displacement and not the mean over the overturn. However, dissipation rates computed from Microcat/T-logger pair time series compare well to those from MMPs, indicating that, although they are less accurate, they can be used to produce a time series that resolves abrupt changes in dissipation rate. Hereafter, dissipation rates obtained using this method are referred to as Microcat dissipation rates.
a. Internal bore case study
In Nash et al. (2007), a turbulent hotspot with semidiurnal periodicity was observed at LADCP station 4.3 on the Oregon slope (Figs. 2a–f), but details were not discussed. From the observations described here, we infer that this turbulent hotspot is caused by the passage of upslope-propagating bores. The observations at LADCP station 4.3 are discussed in this context.
Here we define a bore to be a turbulent “slug” of water that propagates upslope, similar to swash at a beach. Two bores are observed during the LADCP time series at yearday 265.4 and 265.15, hereafter referred to as “Bore A” and “Bore B,” respectively. Bore A is considered the prototypical bore, used as the model by which other bores are identified in the mooring time series. The leading edge of Bore A, or the “bore front,” is first observed at yearday ~265.4, followed by the body of the bore with onshore velocities due to upslope propagation, increased dissipation rates due to high turbulence within the bore, and decreased temperature and increased salinity due to the upslope transport of water. The height and duration of each bore (~80 m and >2.5 h for Bore A) is determined by examining these properties.
At yearday ~265.4, the front of Bore A first reaches the LADCP (black arrows). It is characterized by cross-isobath (zonal) velocity (Fig. 2a) abruptly switching from downslope (blue) to upslope (red) and along-isobath (meridional) velocity switching from south (blue, Fig. 2b) to north (red). At the same time, isopycnals lift sharply (black lines) as the 80-m thick dense, cool (Fig. 2c), and salty (Fig. 2d) bore moves up the slope. After the bore front passes (after yearday 265.4), dissipation rates are elevated (ε > 10−7 W m−3) above background levels (Fig. 2e) and the water becomes more turbid (Fig. 2f). Immediately before the bore front passes the LADCP, isopycnals plunge downward.
The vertical structure of Bore A can be seen in the OBS data (Fig. 2f), where suspended material represents a passive tracer advected with larger-scale water movement. At the bore front, after yearday 265.4, particulates are carried upward. Particulates remain at the top of Bore A in a relatively well-mixed 20 m tall layer, while immediately below in the core (at the 1032.1 kg m3 isopycnal), turbidity decreases. Turbidity within the upper layer increases before the front passes at yearday 265.3, but is not locally suspended bottom material because it is detached from the seafloor.
Another bore, Bore B, is observed ~6 h before Bore A (Fig. 2, yearday 265.15). Both Bore A and Bore B have similar characteristics: velocity rotates onshore (Fig. 2a), isopycnals uplift, temperature drops (Fig. 2c), and sediment is resuspended (Fig. 2f). However, dissipation rates are nearly two orders of magnitude lower in Bore B and appear to be more bottom trapped than Bore A. At the start of the LADCP time series, nearly 12 h before Bore A, a similar feature is observed, suggesting bores are episodic, occurring at semidiurnal intervals. Bore B may therefore be a secondary front associated with this feature.
b. Moored time series
The LADCP time series only captured a single snapshot of a bore. To understand temporal variability and produce a composite picture of an “average” bore at other stations, the longer MMP records are utilized. Because individual MMP profiles are almost 3-h apart near the bottom, they temporally underresolve the bores, easily missing the abrupt velocity and temperature jumps used to identify bores (example shown in Fig. 3). However, bores (red triangles, Fig. 3a) and associated temperature drops (black and gray lines, Fig. 3) can be observed in near-bottom Microcat and T-logger data, which sample every 60 and 30 s, respectively. Similar to observations at LADCP station 4.3, as the bore front passes the mooring at MP3, near-bottom zonal velocity switches from downslope to upslope (red line, Fig. 3c) and isopycnals uplift (black lines, Fig. 3d). Bores arise roughly every ~12 h, suggesting a link to the semidiurnal tide. Overturns and enhanced dissipation rates (squares, Fig.3b) generally occur immediately after the passage of the bore front. Occasionally, MMPs profile through the 150-m tall head (first, third, and fourth bores, red triangles), resulting in high dissipation rates. More often, MMPs miss the head (second bore, open red triangle), leading to an underestimate of bore size and dissipation rates.
Bore passages are identified from 60-s Microcat data using dT/dt, the change in temperature over time. At all moorings, a bore front is identified when the temperature drops abruptly, such that ∂T/∂t falls below an empirically predefined threshold, <−0.002°C min−1. Temperature is first smoothed with a 30-min running boxcar to remove dT/dt spikes from overturns (which have smaller temperature changes but over very short time scales, <10 min). The threshold criteria is most stringent at the deep moorings where the vertical temperature gradient is smallest.
Using the dT/dt criteria, many bores are detected at both deep MP3 and MP4 slope moorings (Figs. 4a,c). At the shallowest slope moorings, MP5 and MP6, neither bores nor overturns are observed (in spite of greater stratification), consistent with smaller dissipation rates derived from density overturns in the MMP data. At MP3 bores occur intermittently, while at MP4 bores occur nearly every tidal cycle. At both sites bores are equally likely to occur during spring and neap tides. Overturns cluster immediately after the bore front passage (time t = 0 h), ranging in size from 0–20 m. Dissipation rates ε (Figs. 4b,d) generally increase during the passage of the bore front, suggesting that near-bottom turbulence is intensified by the bores.
c. Bore composites
Profiles from the MMPs only provide glimpses of the bore structure at moorings MP3 and MP4 owing to their coarse temporal resolution (Fig. 5). To create composite pictures of the bores at each mooring (Figs. 6 and 7), each bore in the temperature time series (colored lines, Fig. 5a) is aligned with a previously chosen “reference” bore [black line, (a,b) in both figures]. Then, each MMP profile is referenced and placed appropriately in a 12.42-h window centered at the nearest front (c,d). The referenced profiles are averaged in 15-min and 10-m bins to form the composite (Fig. 6). Before compositing, velocities are high passed with a 2-day cutoff to remove aliasing of low frequency variability.
Bore composites at MP3 and MP4 are broadly similar to the bore observed at LADCP 4.3 but differ from each other in detail (Figs. 6 and 7). At the deeper MP3 mooring, bores occur less frequently (37 realizations), but are larger (>200-m tall), faster (u > 0.1 m s−1), and have higher dissipation rates. At the shallower mooring, MP4, bores occur more frequently (51 realizations), but are smaller (<100-m tall), slower (u < 0.1 m s−1) and have lower dissipation rates. Bores at MP3 have a strong northward velocity, which bores at MP4 lack. Composite dissipation rates are elevated 1–2 h before the passage of the bore front, but are likely biased owing to fewer profiles with extremely elevated dissipation rates (Figs. 5c,d). This is verified by comparing mean dissipation rates from the temperature time series (Figs. 4b,d) and dissipation rates at LADCP station 4.3 (Fig. 2), neither of which are elevated 1–2 h before the passage of the bore front. Vertical displacements of isotherms, ζ, are calculated from the temperature time series using ζ = (T − Tref)/(∂T/∂z)CTD, where T is the temperature, Tref the reference temperature (here the start of the bore window), and (∂T/∂z)CTD the mean vertical temperature gradient estimated from CTD profiles (Figs. 6–7b).
The mean velocity in each bore is estimated by averaging the high-passed velocities from the MMP profiles over the 100m thickness and 0–4 h duration after the front’s passage for each bore (Fig. 8). Because of the sparse temporal sampling of the MMPs (~3 h per profile), the first profile pair which passes through the bore often occurs more than 1 h after the passage of the front. Therefore, we do not always measure the bottom velocity after the passage of the bore front. Most of the averages only contained one up/down profile pair; therefore the average shown is the first profile taken after the passage of the front. The calculation was also done using only profiles occurring within 2 h after the passage of the front, and the estimated velocities were nearly identical, but with fewer realizations. Mean velocities within each bore are predominantly to the northeast at MP3 (Figs. 8a,c, black) and east-southeast at MP4 (Figs. 8b,c, gray), consistent with a internal tide that turns onshore as it shoals—suggesting upslope propagation. In the 12.42-h semidiurnal period immediately following each bore front the net flow is upslope (not shown), suggesting fluid is transported upslope by the bores.
d. Dissipation rates and mixing
Within 300 mab at moorings MP3 and MP4, mean dissipation rates ( W kg−1) and diffusivities ( m2 s−1) are always elevated above background levels [ W kg−1, m2 s−1, Gregg (1989)]. When bores are present, both dissipation rates and diffusivities become even larger (Figs. 4 and 9). When bores pass MP4, dissipation rates and diffusivities are enhanced in the bottom 250 m, the thickness of the composite bore, dropping to nonbore levels above 1200 m. Similarly, dissipation rates and diffusivities are enhanced by a factor of 2 in the bottom 300 m when bores are observed at MP3. With or without bores, dissipation rates and diffusivities are always larger at the deeper MP3 than the shallower MP4. Dissipation rates may remain elevated near the bottom when bores are not present due to weaker turbulence caused by downslope flow following each bore (van Haren and Goustiaux 2010).
The semidiurnal periodicity of the bores suggests that there are three possible processes by which they are formed: 1) conversion of the barotropic tide to a locally generated internal tide, for example, Legg and Klymak (2008); 2) locally-generated internal tides reflecting or scattering from the slope; or 3) shoaling low-mode internal tides breaking on the slope (Slinn and Riley 1996; Dunkerton et al. 1998; Thorpe 1999; Legg and Adcroft 2003). Here, we will determine which process most likely forms bores by comparing dissipation (which is associated with the bores) to tidal forcing, then determining to which signal bores are predominantly phase-locked.
1) Dissipation rate
Dissipation rates are expected to rise and fall with changes in the magnitude of the forcing, either the barotropic, local baroclinic, or remote baroclinic tide. To determine whether barotropic or baroclinic tides cause enhanced turbulence associated with bores, correlations between the semidiurnal tidal velocities within 300 mab (the height of the largest bore) to dissipation rates are computed (Table 1). To examine whether enhanced turbulence is associated with local or remote baroclinic tides, correlations with energy fluxes within 200 mab are also computed. Onshore energy fluxes are associated with remotely generated internal tides that shoal onto the slope, while offshore fluxes are associated with locally generated internal tides (Martini et al. 2011; Kelly et al. 2012).
Correlations between the semidiurnal quantities and dissipation rates ε are computed as
where and are the mean values from MMP data in the bottom 200 m, such that W kg−1 and W kg. A logarithm is applied to reduce biasing from large dissipation rates, which are several decades larger than the mean. To find the direction of energy flux that corresponds to the largest dissipations, MMP dissipation rates are sorted with respect to cross- and along-slope semidiurnal energy flux, Fx and Fy, respectively, and averaged over 0.2 W m−2 bins (Fig. 10).
At MP3 dissipation rates are significantly correlated to onshore and northward baroclinic energy fluxes (Table 1). The lack of correlation with barotropic velocities and offshore fluxes suggests increased dissipation is not caused by either the barotropic tide or locally generated internal tides. Dissipation rates are largest and most overturns occur when flux is onshore and northward (Fig. 10a). Therefore, turbulence at MP3 appears to be associated with remotely-generated internal tides.
Further onshore at MP4 (8.5 km from MP3), dissipation rates are strongly correlated with onshore energy fluxes and zonal baroclinic velocity amplitudes (Table 1). When flux is onshore, dissipation rates are strongest (Fig. 10), suggesting dissipation rates increase when the internal tide is remotely generated as at MP3. Rarer but larger overturns also occur when flux is to the southwest. Southwestward flux is likely generated at a bump to the north of MP4 (Martini et al. 2011; Kelly et al. 2012), suggesting these larger overturns are caused by locally generated internal tides. Thus, turbulence at MP4 appears to be primarily driven by remotely generated internal tides, but at times is also driven by a combination of remotely and locally generated internal tides.
2) Tidal phasing
Bores are expected to be phase locked to their forcing. To determine whether bores are phase locked to the barotropic or baroclinic tide, we translate the bores into semidiurnal tidal reference frames. The advantage of the tidal reference frame is that we can observe at which point in each tidal cycle the bore front passes the mooring, that is, phase locking.
The tidal reference frame (trf) is a semidiurnal window (12.42 h), centered at the time (defined as ttrf = 0) when the instantaneous semidiurnal velocity is zero, changing from negative to positive (∂u/∂t > 0). The harmonic fits used to find the semidiurnal velocity cannot resolve M2 and S2, so the tidal reference frame also includes phase modulation by the S2 tide. In the tidal reference frame, a bore front is observed at ttrf = tbore − tu=0, where tbore is the time the bore front is observed, and tu=0 is the time when the semidiurnal velocity is zero and ∂u/∂t > 0. If ttrf < 0 (ttrf > 0), the bore front passes the mooring before (after) the velocity changes direction. To translate into the trf, we use barotropic, baroclinic, and total velocities that are closest to the bottom: 61 mab and 27 mab at MP3 and MP4, respectively.
Phase locking is determined by calculating a bore probability density function (PDF) with 2-h bins in each of the three tidal reference frames: barotropic, baroclinic, and total (barotropic + baroclinic) (Fig. 11). The distribution of bore observations varies in the barotropic, baroclinic, and total tidal reference frames (Fig. 11).
At the 1800-m isobath (MP3), bores are most likely to occur when the total zonal velocity (Fig. 11c), which is predominantly baroclinic, switches from positive to negative (down- to upslope) at time t = 0 h. Bores at MP3 are also phase locked to the barotropic tide (Fig. 11b), in both the cross- and along-slope directions. Barotropic (Fig. 11a) and baroclinic (Fig. 11b) PDFs are similar in magnitude, but their peaks occur at different times in the tidal reference frame. Bores are observed at similar times in the total and baroclinic reference frames, which is expected because the total velocity is predominantly baroclinic.
At 1500-m (MP4) bores are most likely to occur immediately after zonal velocities (barotropic, baroclinic, and total) all switch from offshore to onshore (Figs. 11d,e,f). Bores are more strongly phase locked to the across-slope baroclinic velocity (Fig. 11e) than the barotropic velocity (Fig. 11d) as its PDF is taller and narrower.
At both moorings bores are most phase locked to the total velocity, suggesting that both barotropic and baroclinic tides control when bore formation occurs. Strong correlations between onshore energy fluxes and dissipation rates suggest that the bores are primarily driven by shoaling internal tides. But, the interplay of temporally varying remote and local tides may also modulate their strength and phase, causing phase locking to both barotropic and baroclinic velocities.
a. Internal tides
We have discussed several features that suggest bores are primarily driven by shoaling internal tides.
We envision bores form when remotely generated internal tides shoal onto the slope and break, losing energy through the production of turbulence as they propagate upslope. By examining a steady internal tide energy budget, we will determine whether the shoaling internal tide is energetic enough to drive the observed turbulence.
The energy budget balances the influx of tidal energy from remote sources with observed turbulence. If the bulk of the shoaling internal tide is completely dissipated in the bottom 500 m of the water column by internal bores, the energy budget, following Kunze et al. (2002, 1995); Katsumata (2006) and Carter et al. (2008), is
where F is the incident depth-integrated energy flux, ρ the density, and ε the dissipation rate. Internal tide energy is assumed to be in steady state (dE/dt = 0), meaning that the remotely generated internal tide does not grow or decay. Local generation may be an additional source of tidal energy on the slope, but it is not included here. Nonlinear fluxes [the nonlinear advection fE = uE + up where E is the total energy (Moum et al. 2007)], are more than an order of magnitude smaller than internal tide fluxes so are also not included.
In this simple energy budget, we assume the internal tide shoals, breaks, and forms bores—causing the upslope energy convergence. Dissipation rates are estimated from the mean values at MP3 and MP4 within 500 mab when bores are observed (from Fig. 9, W kg−1). Since bores are not observed at MP5, we also assume that the majority of the shoaling internal tide breaks before the 1077-m isobath, though some of the incident internal tide might also reflect offshore.
From energy budget (5), we find observed mixing rates within the bores can dissipate as much as 0.23 kW m−1 of the incident internal tide before MP5. This assumes that the shoaling internal tide propagates 26 km onshore, the distance from the base of the slope to MP5, and dissipation rates are uniform below 1077 m and at a maximum, that is, when bores are observed ( W m−2). Observations at the mooring 100 km offshore suggest that the mode-1 internal tide has an average northward magnitude of 0.5 kW m−1 throughout the 45-day sampling period, but it is unclear what fraction of this northbound internal tide shoals onto the slope. The rough energy budget presented here suggests the northbound internal tide is energetic enough to maintain observed dissipation rates if it breaks on the slope.
b. Phase speeds
In studies by Thorpe (1999) and Dunkerton et al. (1998), bores formed by shoaling internal tides move upslope at the phase speed of the incident internal tide. Using the phase differences between observations at MP3 and MP4, we can compare the internal tide phase speed to the upslope speed of the bores. We first compute the internal tide phase speed, then compare it to three separate estimates of bore speed.
From phase lags between semidiurnal fits to zonal velocities at MP3 and MP4, the mean upslope internal tide phase speed within 100 mab is m s−1. Lag correlations between temperature time series produce a similar phase speed, m s−1 (Fig. 12). These are smaller than the theoretical mode-1 phase speeds, m s−1 at MP3 and m s−1 at MP4, but similar to the mode-3 phase speeds, m s−1 at MP3 and m s−1 at MP4, which is expected because the internal tide is likely to be composed of higher modes on the slope (Kelly et al. 2012). Reduced cross-slope phase speeds may also be due to the internal tide propagating obliquely to the mooring line, in this case to the northeast.
We first estimate mean bore speeds at MP3 and MP4 from the composites, averaging upslope velocities from 0–4 h after the bore front and within 100 mab (Figs. 6–7). Average upslope velocities are 0.04 and 0.02 m s−1 at MP3 and MP4, respectively, significantly smaller than theoretical or observed internal tide phase speeds.
Nonlinear self-advection of the bores due to density differences may also drive the bores upslope. For comparison, we calculate the gravity current speeds, where is the reduced gravity, H the height of the bore, and the background density (here, the time-mean density profile at each station). From MP3 to MP4, bore thickness decreases from approximately 200 and 100 m, and therefore average gravity current speeds decrease from 0.11 to 0.07 m s−1. Gravity current speeds are similar to mean speeds from the composites, although slightly larger. But, both are still smaller than theoretical or observed internal tide speeds.
Within the same tidal cycle, 66% of bores at MP3 are followed by a bore at MP4, but we are unable to determine conclusively whether individual bores propagate from MP3 to the shallower MP4 owing to a lack of observations between the two moorings. However, were a bore able to propagate between MP3 and MP4, we can estimate the speed it would travel at using lag times between the two moorings (Fig. 13). The estimated mean bore speed between MP3 and MP4 is 0.4 m s−1; larger than the two previous estimates, yet still smaller than the speed of the internal tide. However, 90% of bores at MP3 are followed by a bore at MP4 in the following tidal cycle (+12.42 h), suggesting that, if the bores propagate between the two moorings, it may take them longer than one tidal cycle to do so. This corresponds to an average upslope propagation speed of 0.14 m s−1 (not shown), similar to the estimated reduced gravity speeds.
The three estimates presented here give a wide range of possible bore speeds. The lag analysis gives an upper bound on the bore speed, 0.4 m s−1. This is still substantially lower than the phase speed of the internal tide, suggesting that, once formed, bores propagate slower than, and independently from, the shoaling internal tide.
c. Wave breaking
Whether remotely incident internal tides are dissipated locally, transmitted onshore, or reflected offshore from sloping bathymetry is determined by slope criticality s/α: the ratio of the bathymetric slope to the internal tide ray-path slope , where ω is the wave frequency, f the inertial frequency, and N the buoyancy frequency. When internal waves propagate perpendicular to isobaths, that is, normal incidence, subcritical slopes (s/α < 1) transmit them onshore, while supercritical slopes (s/α > 1) reflect them back into the deep ocean. For example, on the Virginia continental slope, internal tides shoal directly onto the supercritical slope and are reflected offshore to high wavenumber (Nash et al. 2004). On Fieberling Guyot in the North Pacific, internal waves are reflected to very high wavenumbers and amplitudes over the portion of the slope that is near critical (s/α ≈ 1) with respect to their frequency, making them more susceptible to breaking and leading to enhanced mixing (Eriksen 1998).
On the Oregon slope, bores are only observed where the slope is supercritical with respect to the semidiurnal internal tide (Fig. 14a). This could be the result of 1) the Oregon slope being mostly supercritical, 2) biased observations as eight of the nine sampling sites are over supercritical slopes, or 3) wave breaking and bore formation occurs only on supercritical slopes. Here, we will show that, while bores are only observed on supercritical slopes, further analysis suggests wave breaking, and therefore bore formation most likely occurs on near-critical slopes.
The observed internal tide phase speed, m s−1, is significantly less than the mode-1 phase speeds, m s−1 at MP3 and m s−1. It is more comparable to the mode-3 phase speeds, m s−1 at MP3 and m s−1 at MP4, suggesting that shoaling internal tides, which are predominantly mode 1 (Martini et al. 2011), are scattered to higher wavenumber on the slope (Eriksen 1982, 1998; Kelly et al. 2012). When internal tides are reflected from a slope, 1) horizontal and vertical wavenumbers increase, causing phase velocity to decrease, and 2) wave amplitudes increase, causing horizontal velocities to increase. Particle velocities can therefore become greater than the phase velocity, leading to wave breaking.
To determine whether wave breaking occurs, we use the Froude number Fr, the ratio of the particle speed U0 to the phase velocity cp. Wave breaking should occur when the Froude number of the reflected wave is supercritical, . As expected for propagating waves that have not broken, Froude numbers of both incident mode-1 internal tide and the observed propagating internal tide on the slope are less than one (semidiurnal particle velocities are at most 0.08 m s−1). However, because there are substantial portions of the slope that are near critical, we expect that there are regions where shoaling internal tides are reflected to higher modes and amplitudes, becoming more susceptible to wave breaking.
Typically, Froude numbers are calculated from observations. However, we do not directly observe the processes that lead to wave breaking nor do we know the modal content of the internal tide immediately after reflection. Therefore, we are unable to determine the reflected phase velocity and reflected Froude numbers that would indicate wave breaking. As a result, we compute theoretical Froude numbers to try to estimate where wave breaking might occur on the slope.
Following Legg and Adcroft (2003), theoretical Froude numbers after reflection for a wave impinging normally on the slope are
where θ is the angle of the bathymetric slope with respect to the horizontal plane (s = tanθ), and β the angle of internal tide slope to the horizontal plane (α = tanβ). Superscripts i and r refer to the incident and reflected internal tides, respectively; is the velocity amplitude along the propagation direction of the reflected wave.
The analysis is then expanded to include the more realistic case of obliquely-incident internal tides. The reflected particle and phase speeds, and , now depend upon the horizontal angle of incidence φi and are calculated following Eqs. (4), (5), and (7) of Eriksen (1982). For both cases, the incident internal tide is chosen to be similar to the offshore mode-1 tide observed at yearday 269.4, having a depth-integrated energy flux of 0.53 kW m−1 and an azimuthal angle of 82° (Martini et al. 2011). We assume that the incident internal tide does not change direction until after reflection, therefore φi is kept constant across the slope.
Most supercritical Froude numbers occur where the slope is near critical with respect to the semidiurnal internal tide (Fig. 14). If bores result from breaking internal tides, we would expect to see a region of supercritical Froude number just downslope of the stations where bores are observed. In the normal case, regions of supercritical Froude numbers are widespread, occurring downslope from all stations except for MP5 (Fig. 14b). In the oblique case, supercritical Froude numbers are only found where 0.89 < s/α < 1.13, just downslope from stations where bores are observed (Fig. 14c), except L3.3. During the time when L3.3 observations were made, bores were also not observed downslope at MP3, suggesting that the internal tide may have been too weak to form bores at this time. Nonetheless, the proximity of supercritical Froude number to bore observations suggests that Froude number is an important parameter for predicting bore formation. The similar distribution of supercritical Froude numbers and bores in the oblique case suggests that bores are formed when the northbound internal tide breaks on the slope (Martini et al. 2011).
The prevalence of supercritical Froude numbers suggests that more bores and, perhaps, more mixing will occur in the normal case than in the oblique case. Similarly, theoretical studies show fronts formed by normally incident shoaling internal tides are stronger and steeper than those formed by obliquely incident internal tides (Thorpe 1999). But even in the oblique case, where supercritical Froude numbers are sparser, shoaling internal tides are still completely dissipated. In the normal case, more widespread supercritical Froude numbers on the deep slope may act as a geographic filter, confining internal tide breaking to deeper portions of the slope. Internal tide propagation direction and magnitude affects the distribution of supercritical Froude numbers and therefore wave breaking, and may contribute to the heterogeneity and intermittency of bores as well as their associated mixing.
d. Comparison with global estimates
When bores are present, diffusivity is elevated by a factor of 2 compared to periods when no bores were present. For this site on the Oregon slope, shoaling internal tides can significantly increase mixing after breaking where the slope transitions from sub to supercritical, that is, where the slope is near critical with respect to the semidiurnal internal tide. From observations at the offshore mooring, we estimate that on average 0.5 kW m−1 or 25% of the 2 kW m−1 internal tide propagating from Mendocino Escarpment (Althaus et al. 2003) propagates far enough northward to reach the Oregon slope. The remainder likely dissipates before reaching the array at 43.2°N.
Using satellite altimetry, Egbert and Ray (2001) estimate the M2 surface tide generates 0.7 TW of energy that must dissipate in the deep ocean. Kunze et al. (2006) could only account for about one-third of the surface tide loss in the deep ocean to dissipation inferred from a finescale parameterization of global World Ocean Circulation Experiment (WOCE) and LADCP/CTD data. The coarse sampling may be inadequate or the parameterization may underestimate turbulence at hotspots to properly estimate global dissipation rates, but the continental margins were undersampled and may therefore be a major, but unresolved, internal tide sink. From the Egbert and Ray (2001) values, we estimate M2 internal tides radiate an average of 1.3 kW m−1 onto continental slopes (assuming 85% of available energy radiates away from generation regions as low-mode internal tides similar to observations at the Hawaiian Ridge (Klymak et al. 2006), one-third dissipates in the abyssal ocean (Kunze et al. 2006), and the global length of continental slope is ~3 × 105 km). This is significantly larger than the internal tide offshore of our site, which has an average flux of 0.5 kW m−1, indicating that the Oregon slope is likely a relatively small internal tide sink. Nonetheless, weak shoaling internal tides can cause significant dissipation rates, and there is significant internal tide energy available for mixing on other continental slopes. Presumably, internal bores generated by breaking internal tides would be stronger at sites with greater incident flux, such as the continental slope west of Luzon Strait (Klymak et al. 2011). In any case, the observed dissipation within the bores appears sufficient to account for an appreciable fraction of the average internal tide incident on the Oregon slope. Most likely, a similar fraction of the incident 1.3 kW m−1 internal tide radiated onto continental slopes worldwide would also be dissipated, substantially increasing dissipation rates.
While attempts have been made to parameterize local mixing by internal tide generation (Jayne and Laurent 2001; St. Laurent et al. 2002), the dissipation of propagating low-mode internal tides is often not well characterized in global climate models (Simmons et al. 2004; Decloedt and Luther 2010; Friedrich et al. 2011). Predicting internal tide shoaling and reflection in global and regional simulations requires resolving small-scale scattering, large-scale scattering (Klymak et al. 2011), and bore formation in bottom boundary layers. Turbulence in global models may be underestimated if mixing produced by internal tide breaking on continental slopes is not accounted for. If continental and other near-critical slopes contain the bulk of mixing “hotspots,” it is important to properly parameterize dissipation rates at these locations. In addition, the spatial distribution of mixing over these “hotspots” is affected by the propagation direction of incident internal tides, which may have important effects on local biological and geological processes driven by their breaking.
We have presented detailed observations of 1400–1800 m deep internal bores on the Oregon continental slope. Bores appear to form after shoaling remotely generated internal tides break on near-critical slopes, then propagate upslope independently of the internal tide. Composites derived from moored profiler data show that the average bore is turbulent, 100 m thick, cold, salty, propagates upslope and resuspends sediment. At deep moorings, 1780-m isobath MP3 and 1452-m isobath MP4, bores are observed over 36% and 88% of the 40-day deployment, respectively. They are associated with increased dissipation rates and diffusivities, raising turbulence levels on the slope. At MP3 and MP4, bores are predominantly forced by the remotely generated internal tide, but their strength and phase is likely modulated by the combination of the barotropic tide and remotely and locally generated internal tides.
Linear reflection of internal tides from near-critical slopes causes 1) the transfer of energy to high modes and 2) supercritical Froude numbers, leading to wave breaking and bore formation. In numerical simulations by Kelly et al. (2012), regions where there is conversion to high modes roughly correspond to where we identify supercritical Froude numbers, but bores or their mixing are not resolved. Therefore, conversion of shoaling internal tides to high modes in numerical simulations could be interpreted as a first step in bore formation and used to identify where wave breaking is likely to occur.
Remotely-generated M2 internal tides offshore of the Oregon slope are weak ( kW m−1) compared to estimated global averages ( kW m−1), yet their breaking substantially boosts diffusivity over the slope ( m−2 s). The formation of bores by breaking internal tides appears to be an important mechanism by which internal tides are dissipated, leading to significantly enhanced mixing at oceanic boundaries. Observations at MP3 and MP4 show spatial and temporal variability in bore generation, frequency, and characteristics. This variability is dependent upon the strength and azimuthal angle of the incident tide and the local slope bathymetry. Most likely, obliquely incident internal tides are the norm rather than the exception, likely affecting the distribution of wave breaking and turbulence on continental slopes compared to strictly normal incidence.
This work was supported by the National Science Foundation Grants OCE-0350543 and OCE-0350647. We thank Dicky Allison, Eric Boget, Andrew Cookson, Richard Dewey, Eleanor Frajka-Williams, Dave Winkel, Zhongxiang Zhao, and the captain and crew of the R/V Wecoma for their technical and at-sea expertise. The authors would also like to thank two anonymous reviewers whose comments were very helpful.
Dissipation Rates from Temperature Time Series
Microcat (upper) and T-logger (lower) data are used to calculate a high-frequency dissipation rate time series within 20 mab at each mooring (Fig. 4). Dissipation rates are calculated from Thorpe scales, LT, and the instantaneous stratification, Ni. With only two instruments at fixed depths, exact values of overturn size and, therefore, Thorpe scales and dissipation rates cannot be obtained. In addition, density overturns and instantaneous stratification cannot be directly measured as the lower T-logger only records temperature. Instead, we use temperature inversions to find overturns, using the overturn size as a proxy for Thorpe scales. This method tends to overestimate Thorpe scales, as it often only captures the largest overturn size and not the average Thorpe displacement over the overturn. Dissipation rates may be underestimated when inferred Thorpe scales become comparable to the distance of the sensors from the bottom (>30 m) due to interactions with the bottom boundary layer and inhomogenous turbulence. However, this method can still be used to examine the temporal variability of near-bottom mixing.
First, Thorpe displacements are found from temperature inversions (Fig. A1). The vertical temperature gradient approximates the sorted, stably-stratified temperature profile from which Thorpe displacements are determined following Dillon (1982). Temperature is first smoothed using a 2-h running boxcar filter to retain changes in temperature gradients due to isopycnal strain from tidal and inertial heaving but remove small-scale temperature gradients from overturns. Overturn size is then calculated from the unsmoothed temperature data, where . Only overturns from temperature inversions are kept, such that Tupper < Tlower − Δzinsts(dTadiabatic/dz), where dTadiabatic/dz is the adiabatic heating of a water parcel as it is transported vertically, and Δzinsts is the vertical distance between the instruments. At both instruments, vertical displacement errors owing to temperature calibration errors (~0.001°C) are less than 0.5 m.
Next we calculate the instantaneous stratification between the upper and lower instruments. The temperature–salinity relation is fairly linear (Fig. A2a), therefore salinity at the lower T-logger is recreated using the linear fit to the T–S relation at the upper Microcat. The reconstructed salinity is smoothed over 2-h windows, and together with the smoothed temperature data, is used to calculate density and stratification between the two instruments. Using Monte Carlo simulations, the 95% confidence intervals for Microcat dissipation rates are 1 × 10±0.56 W kg−1 when stratification is computed from reconstructed salinity.