a. Experiment

From 19 August to 1 September 1996 we obtained water property and velocity data on the New England shelf on board the R/V Seward Johnson near the 70-m isobath south of Nantucket in a region known as the “mud patch” for its fine sediments (Fig. 1). The vessel steamed slowly east and west (to prevent the profiler from moving under the ship) in a box 1.5 km long (along shelf) and 1 km wide (across shelf) centered near 40°30′N, 70°30′W. On 1 September we were forced into port by Hurricane Edouard. We returned on 4 September for several more days of work. Posthurricane data are presented in MGa. Our date convention is that noon of 1 January is yearday 0.5.

b. Profiler measurements

Temperature and salinity measurements were made with SeaBird sensors mounted on the Modular Microstructure Profiler (MMP), a loosely tethered, free-falling instrument ballasted to fall at 0.5 m s⁻¹. Generally profiling in 12-h shifts at night, we completed a full water column profile approximately every four minutes during peak operating efficiency, resulting in 1469 profiles before the hurricane. By monitoring data from a small acoustic altimeter attached to the MMP, we were able to safely profile to within a few meters of the bottom. Though MMP quantities are measured as a function of pressure, all quantities are plotted versus depth for comparison with acoustic Doppler current profiler data, which produces an average error of less than 1%.

c. Velocity

Two shipboard broadband ADCPs operated continuously throughout the cruise and provided velocity data in 4-m vertical bins between 12 and 56 m (150 kHz), and 1-m bins between 8 and 32 m (600 kHz), though the higher noise of the 600-kHz instrument precluded reliable resolution of scales below 2–3 m. The ADCP sampling applies an effective 8-m Bartlett filter to the 150-kHz data and an effective 2-m filter to the 600-kHz data; hence only every other depth bin is truly independent. An 8-m Bartlett filter was applied to the 600-kHz data to mimic this effect and give the same resolution for all data. In our analysis we use composite velocity data, which combined both ADCPs at 4-m intervals to span the broadest range of the water column (velocities were averaged in the depth range of instrument overlap). Baroclinic velocity was computed by removing the depth-mean of each velocity component. Shear was calculated by first-differencing composite velocity over 4-m intervals. There is a loss in shear variance due to both the first-differencing process and the Bartlett filter response. Though specific phase information of the lost shear cannot be regained, we estimated that average 4-m shear variance would be 1.5 times the variance we measured based on spectra of observed shear and knowledge of the filter transfer functions. This is smaller than the factor of 2.11 used by Gregg (1989), most likely because of the lower vertical mode nature of this internal-wave field. Additional variance loss due to beam separation was also possible (Alford and Pinkel 2000). None of the variance loss estimates were used to correct the data.

a. Water properties and barotropic flow

Water on the shelf was strongly stratified, with a thermocline typically between 15 and 20 m (Fig. 2). Changes in stratification reflected divergences in vertical wave velocities and horizontal advection of local gradients, the later identifiable through T/S variability (MGa; Gardner et al. 2001). Density changes can be used to calculate a perturbation pressure that is later used to estimate energy fluxes. For hydrostatic waves, perturbation reduced pressure (dimensional pressure divided by an average density, ρ₀) is given by (Kunze et al. 2002)

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where ρ′ is the density perturbation from a slowly evolving mean density profile. For calculation of the low-passed density profile, our data were supplemented with a time series of CTD data generously provided by W. Gardner, Texas A&M University. The second term within the brackets ensures that perturbation pressure is baroclinic (has a zero depth average). Pressure perturbations were dominated by a low-mode structure (Fig. 3). The high-frequency thermocline depressions associated with solibores are visible as sharp perturbation pressure events during, for example, yeardays 235, 241, and 242.

Barotropic (depth averaged) flow primarily consisted of a nearly circular mixed tide (0.11 ± 0.06 m s⁻¹) superimposed on a comparably strong along-shelf flow (0.11 ± 0.04 m s⁻¹), where the range given is the standard deviation. The combination at times produced large variations in barotropic energy (Fig. 4a). Observations of tidal and subtidal flow were consistent with previous studies in this area (Voorhis et al. 1976; Chapman and Beardsley 1989; Sundermeyer and Ledwell 2001).

b. Internal wave observations

The baroclinic wavefield consisted of evolving low-frequency, low-mode waves, punctuated by episodic groups of high-frequency solibores. Both phenomena can be observed in time series of baroclinic velocity, depth-averaged baroclinic energy, and shear variance (Fig. 4). Baroclinic energy and shear variance are visually dominated by a series of energetic but short-lived spikes (Figs. 4c, d). Most of these events appear as pulses of northward velocity at 16 m (Fig. 4b). There is a large body of theoretical and observational work describing such high-frequency, nonlinear motions as solitons or solibores; they are considered further in section 3d.

There is also a low-frequency, clockwise wave apparent in baroclinic velocity, especially near yeardays 231–232, 236–239, and 242 (Fig. 4b). The flow is nearly circularly polarized (equal northward and eastward magnitude); hence there is not a strong signal at the wave period in baroclinic energy or shear variance (Figs. 4c, d). Instead, the low-frequency variability in energy and shear variance represents the timescales over which wave energy waxes and wanes. In the remainder of this section attention is restricted to low-passed (below 0.17 cph) motions, which overall make up 84% of baroclinic kinetic energy and 72% of shear variance.

Low-frequency waves had large-scale but variable vertical structures. Figure 2 shows depth–time maps of each component of low-passed baroclinic velocity and the associated shear variance. The vertical structure of velocity suggests low-mode internal waves. On yeardays 231–233 and 240–243, there appeared to be a first-mode, clockwise-polarized, standing wave, with flow above the thermocline (z ≤ 20 m) in the opposite direction of subthermocline flow. At other times, yeardays 237–238, higher modes were clearly present. Midcolumn shear variance was on average concentrated near the thermocline (depth 10–25 m), consistent with low-mode internal wave structure (Fig. 2). Evolving vertical wave structure at times elevated shear variance below the thermocline (e.g., yearday 238–239).

c. Internal wave statistics

The distribution of energy between vertical modes significantly fluctuated, which has important implications for the magnitude and location of shear variance, as well as for internal-wave-interaction-based models of turbulence production. A linear wave field can be represented as a superposition of internal waves of distinct frequency (ω) and vertical mode number (j), which together determine the magnitude of horizontal wavenumber (e.g., Munk 1981; Levine 2002). The vertical structure of each mode is governed by (Gill 1982; Thorpe 1998; Levine 2002)

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where c_j is the separation constant (eigenvalue), and waves are assumed to be hydrostatic. Vertical velocity and vertical displacement associated with each mode are proportional to Ψ_j, while the horizontal velocity is proportional to dΨ_j/dz.

We begin by considering the average frequency content of observed waves. Baroclinic energy and shear display a GM-like spectral slope for a midrange of frequencies, but diverge at both the high and low ends of the spectrum (Fig. 5). There are distinct and significant peaks near inertial and semidiurnal (M₂) frequencies. At the other end, there is an elevated shoulder of energy and shear above 0.5 cph that Levine attributes to high-frequency solitons (section 3d).

The main focus here is on the distribution of energy amongst vertical modes, H_j. There are two motivations for this focus, both ultimately related to internal-wave-generated turbulent mixing. First, higher modes contribute a relatively larger amount of shear. Hence, for a given level of wave energy, the distribution of relative modal energy controls the magnitude and location of shear variance and Richardson number (A12), which in turn determine the likelihood of shear instabilities to occur (Miles 1961; Thorpe 1978). Second, models of mode distribution are explicitly used by recent parameterizations of the turbulent dissipation rate (e.g., MGa; Gregg 1989; Polzin et al. 1995). Such models combine observations of low-mode waves with an assumed steady spectral shape to extrapolate the rate of spectral energy transfer to smaller scales (and hence to mixing) through wave–wave interactions.

In practice, the vertical mode shapes presented here are calculated from numerical solution of (2) using the average stratification profile (Fig. 6). Velocity data were fit to the barotropic and first five baroclinic modes using a least squares minimization. For comparison, we also calculate empirical orthogonal functions (EOFs), those coherent structures that maximize variance between measured velocity at different depths (Fig. 6, gray). The strong similarity between theoretical modes and EOFs is encouraging.

A mode-fit velocity profile was created at each time by combining the projected amplitude and vertical structure of the first five baroclinic modes. Examples of mode-fit eastward and northward velocity during two sample time periods are shown in Fig. 7. The modal fit captures the dominant low-mode signal well, though it does not reproduce the small-scale fluctuations that are a relatively more important part of shear variance. Overall, the mode fit captures 90% of the baroclinic energy and 72% of the shear variance.

On average, low-frequency wave energy was weighted more strongly toward low modes than predicted by the GM H_j distribution, even using j∗ = 1 as suggested by Levine (2002) (Fig. 8; appendix). The flatter shape of the average shear modal distribution (Fig. 8b) reflects the opposing tendencies of the two components of shear: energy is concentrated in low modes while higher modes contribute more shear (A12).

More important than the particular shape of average mode distribution, the relative strengths of each mode changed significantly and independently during the experimental period. Figure 9a represents the changing relative modal amplitudes by a time series of stacked histograms of low-frequency energy in each mode. Both the total energy, E(t), and the relative contribution of each mode, H_j = H_j(t), changed significantly throughout the cruise (A8). The formal correlation between the amplitudes of different modes was quite low, especially between the lowest and highest measured modes (Table 1). Note that spectral models like GM assume there is no correlation between the phases of individual waves, but they do assume a prescribed relationship between the evolving average energy (amplitude) of low and high modes.

The observation of nonsteady mode distribution is one of the major conclusions of this paper. A thorough analysis of the reasons for spectral variability is beyond the scope of this work, but we can offer two tentative hypotheses. First, variations in the background stratification and mean shear may funnel internal-wave forcing energy into different modes. For example, MacKinnon and Gregg (2003b, manuscript submitted to J. Phys. Oceanogr.) argue that changes in stratification and a significant nonlinear bottom drag may alter the modal content of wind-generated near-inertial waves. Additionally, internal tide generation at the shelf edge may be sensitive to the meanders of a strong shelfbreak jet. Second, low-frequency waves have on average a different modal energy distribution than high-frequency waves. In particular, high-mode waves do not have strong peaks at low frequencies (Fig. 10). Changes in the total energy of low-frequency waves hence produce changes in vertical mode content.

Changes in relative modal amplitude decouple the energy and shear content of the wavefield. For example, during yeardays 237–239, there was a relatively large amount of shear from higher modes, while the total energy level was low (Fig. 9b). Changing modal amplitude also affects the vertical distribution of shear relative to stratification, and consequently the Richardson number. For illustration, we return to the two profiles of Fig. 7. The energy and shear modal distributions for these times are shown in Figs. 11a,b. During the first time period (Fig. 11, left panels), shear was comparable to stratification in the thermocline (12–28 m), but dropped drastically below the thermocline (25–40 m). During the second time period (right panels), there was more energy and shear in high modes, specifically in mode 5. The result was a shear profile that was comparatively larger below the thermocline, occasionally rising high enough above stratification to potentially produce shear instabilities. The existence of a nonsteady spectrum also violates a crucial assumption behind wave–wave interaction based models of turbulence production. A revised dissipation rate parameterization that accounts for observed spectral properties is presented in MGa.

To be fair, spectral models like that of Garrett–Munk were never intended to represent low-frequency internal waves, especially the internal tide. Such models are successfully used in estimating turbulence in the open ocean because the majority of shear variance comes from the continuum wavenumber range of the spectrum. However, on the continental shelf low-frequency waves are strong enough to independently produce order-1 Richardson numbers.

d. Solibores

One of the most interesting dynamical features observed on the summer shelf was the episodic passage of strongly nonlinear, high-frequency solibores. Energetic nonlinear internal solitary waves have been observed at a variety of locations around the world (Ostrovsky and Stepanyants 1989; Apel et al. 1995). In most cases, they are believed to be generated by interactions between tidal flow and steep topography. In a fluid with a strong pycnocline separating two comparatively well-mixed layers, theory and experiment predict a dominant mode-1 structure in baroclinic velocity, with an upper-layer flow in the direction of wave propagation (onshore for shelf waves), a bottom-layer flow in the opposite direction, and large (usually downward) isopycnal displacements centered in the thermocline (Apel et al. 1995).

We observed 15 solibores during our fortnight of observation, 5 of which occurred during microstructure profiling periods (Fig. 4). Though observed wave arrivals were consistent with a semidiurnal generation mechanism, there was huge variation in wave strength and properties from day to day. Solibores were picked out of the velocity record by hand based on a simple ad hoc criterion: events with depth-averaged baroclinic energies more than 2 × 10⁻² J kg⁻¹ above a 30-min running mean were classified as solibores. This criterion did not capture all bursts of high-frequency energy near wave crests, but did capture the major events. The duration of solibore events defined using this criteria ranged from 20 min to 4 h, with little apparent correlation between solibore duration and strength. The depth-averaged, event-averaged baroclinic horizontal kinetic energy ranged from 2.5 × 10⁻³ to 2.5 × 10⁻² J kg⁻¹. All 15 events consisted of strong shoreward flow in the upper water column and shelfward flow below, qualitatively consistent with two-layer soliton theory.

A particularly energetic event was observed on yearday 235. Northward velocity over a 24-h period including this time shows two packets of high-frequency energy (Fig. 12). Both packets were superimposed on the gradually varying internal tide. After the passage of each packet, local velocities, isopycnal displacements, and potential energy reflected a different phase of the internal tide (approximately one-quarter of a tidal period had elapsed), which provides some justification for the use of “solibore” rather than “soliton” (Henyey and Hoering 1997). Velocity and shear contours of the larger, second packet are shown in the lower half of Fig. 12.

Profiles taken immediately before, and averaged during, the first three solibore troughs are shown in Fig. 13. Each pulse in the solibore packet brought strong on (off) shore velocities above (below) the thermocline and a large downward isopycnal displacement. The depth of the velocity zero-crossing (and maximum isopycnal displacements) during this and other observed solibores was 2–7 m lower than that of the linear first baroclinic mode, consistent with the fully nonlinear numerical simulations of Vlasenko et al. (2000). The large, quasi–two-layer flow in each wave trough produced a 2- to 10-fold increase in shear variance. Within wave troughs, the inverse Richardson number was pushed above the threshold for shear instability due to both large strains above the depth of peak displacement (15–25 m), and strong shear in the troughs (25–35 m; Fig. 13). Unlike the case of propagating internal waves, the depth ranges of peak shear and peak strain in these mode-1 standing waves are not collocated, resulting in two regions of supercritical Richardson number (Fig. 13e)

a. Methods

The change in energy at a given location (neglecting turbulent decay) can be written as the divergence of an energy flux,

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For a propagating internal wave, energy-flux can be written two different ways, dubbed F₁ and F₂,

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where E represents total (kinetic plus potential) baroclinic energy density, c_g is the group velocity, and u′, p′ are perturbation wave velocity and reduced pressure. All quantities are averaged over the water-column depth and should be averaged in time over a wave period (Kunze et al. 2002). The second term in each equation represents the advection of wave energy by wave velocity, and is neglected in the limit of linear waves. Both methods are used here to compute energy fluxes and will subsequently be referred to as method one and method two. For the first method, F₁, energy density and group velocity are calculated based on ADCP-measured velocity variance and vertical mode shape, using the dispersion relation. For the second method, F₂, perturbation (baroclinic) velocity estimates from the ADCP are combined with perturbation pressure estimates from the microstructure profiler. With either method, the difficult task remains of deciding what period to average over in a wave field where near-inertial, semidiurnal, and high-frequency waves all contribute substantial energy (Fig. 5). The first method capitalizes on the excellent temporal resolution of the ADCP to bandpass velocity, and hence to separate flux from different frequency waves. However, when simply looking at energy densities, flux direction cannot be determined and, in fact, propagating waves cannot be distinguished from horizontal standing waves, which have no associated energy flux. Thus, calculations using this method represent an upper bound. On the other hand, the second method robustly determines energy-flux magnitude and direction, but is limited by the noncontinuous (12 hours on, 12 hours off) nature of MMP sampling.

b. Flux results

For the first method, F₁, energy density of each mode is multiplied by the appropriate group speed for that mode. For a given mode, the total energy (horizontal and vertical kinetic energy plus potential energy) is related to the measured horizontal kinetic energy by a frequency-dependent correction function,

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where Ψ_j, c_j are defined in (2). Group speed, c_g = ∂ω/∂k, is a function of frequency and mode number (Gill 1982). Energy flux is calculated for each vertical mode as a function of frequency by multiplying the rotary frequency spectra of each modal amplitude and group velocity and integrating over the internal wave band. The total energy flux for each mode is given in Table 2a. Fluxes from near-inertial (1–1.6 cpd) and semidiurnal (1.7–2.5 cpd) bandpassed waves are shown in Tables 2b,c. The overwhelming majority of flux was due to first-mode waves; the dramatic decrease in energy flux with increasing mode number reflected both the declining energy (Fig. 8) and decreasing group speed of higher modes. Half the total flux was from the semidiurnal band. Though there was comparable energy in near-inertial and semidiurnal waves (Fig. 5), the lower group velocity of near-inertial waves resulted in a order of magnitude lower energy flux.

During MMP profiling periods we computed an alternate estimate of energy flux, F₂, using perturbation pressure measurements from the microstructure profiler (Fig. 3). Velocity and pressure perturbation were both projected onto vertical modes to calculate a mode-by-mode estimate of energy flux (Table 2e). This method allowed resolution of eastward and northward components of energy flux separately, giving flux direction as well as magnitude (Table 2). The majority of energy flux was propagating onshore, roughly perpendicular to isobaths. On the whole, the two methods of calculating energy flux agree remarkably well, suggesting that the overwhelming majority of observed energy was from propagating (as opposed to horizontally standing) internal waves.

c. Solibore energy fluxes

Both methods of calculating energy flux allow estimates of energy flux during solibores, although using the second method we only have observations of 5 of 15 total events. For both methods, solibore energy fluxes were calculated using high-passed (above 4 cpd) velocity averaged over each solibore period (section 3d) and depth-integrated. The nonlinear terms from (5) and (6) were also included for these large-amplitude waves. The calculation of perturbation pressure (1) assumes the hydrostatic approximation, which may not be appropriate for high-frequency solibores; energy fluxes calculated here may be underestimates. It is assumed that solibores propagate at the group speed of high-frequency first-mode waves, 0.45 m s⁻¹, plus a correction term for their nonlinear amplitude (Apel et al. 1995). For the range of solibores observed (10–20-m peak vertical displacements), nonlinearity increased the estimated group speed to 0.48–0.52 m s⁻¹. Comparison of shipboard, mooring and satellite data results in group speed estimates of 0.4–0.6 m s⁻¹ (Zheng et al. 1993; Pringle and Brink 1999; Colosi et al. 2001).

Though only present for 10% of the time, cruise-averaged solibore fluxes were a fifth of the strength of the semidiurnal internal tide flux (Table 2d). Energy fluxes averaged over the duration of particular solibore events were larger and highly variable in magnitude (Fig. 14); there was a reasonable agreement in flux magnitude calculated using the two methods for most solibore events. The lower cruise-averaged energy flux using the second method (Table 2g) reflects the random subsample of events observed during profiling periods—the largest events are included but not the majority of midsize events (Fig. 14: yeardays 233, 234.5, 235.5, and 243.5). There was a range of around 40° of flux direction. Synthetic aperture radar (SAR) images taken during and before the experiment reveal surface signatures of solibore packets propagating shoreward in different directions, suggestive of several different generation sites along the shelfbreak (D. Thompson 2001, personal communication).

d. Implications of energy fluxes

Observations of energy fluxes give an upper bound on baroclinic energy available to dissipate farther upshelf. For example, if the total observed energy flux of 130 W m⁻¹ were evenly dissipated over 100 km (roughly the distance to the coast) in an ocean that was on average 40 m deep, the average dissipation rate would be 3 × 10⁻⁸ W kg⁻¹, comparable to that observed at the CMO site (MGa). For comparison, Holloway et al. (2001) studied the propagation of the M₂ internal tide onto the North West Australian shelf and observed onshore energy fluxes that decayed with distance, ranging from 1 W m⁻¹ at the 300-m isobath to 0.5 W m⁻¹ at 124 m to negligible flux at 65 m.

Alternately, a rough estimate of characteristic distances traveled by different types of waves can be calculated by dividing energy flux by a typical rate of turbulent energy dissipation. MGa observe an average dissipation rate of 10⁻⁸ W kg⁻¹ away from the surface and bottom frictional boundary layers. In a steadily dissipating shelf of constant (70 m) depth, the first-mode internal tide could propagate 90 km, while near-inertial waves and mode-5 waves could propagate only 4.7 and 2.1 km, respectively (Table 2). This result has important implications for modeling internal-wave energy and shear on a shelf; while the internal tide is a shelfwide phenomenon, shear-containing near-inertial waves and higher-mode waves, both of which may be important for turbulence generation (MGa), were probably generated locally. However, this simple calculation ignores huge variations in turbulence strength, which was in turn systematically linked to internal wave strength (MGa). Furthermore, dissipation may not be draining energy equally from all modes. If anything, turbulence is expected to drain energy most efficiently from higher modes (Polzin et al. 1995).

Dissipation may also be an important component of solibore energetics. Solibores experiencing the average dissipation rate could propagate hundreds of kilometers without being significantly drained of energy (Table 2). However, the appropriate dissipation rate to use for these waves varies between solibore events and, in general, is significantly higher than the average dissipation rate used here (MGa). The dissipation rate in the strongest solibores was large enough to significantly erode their energy over one day, which was also the length of time the waves observed to propagate onshore before disappearing near the 40-m isobath (MGa; D. Thompson and J. Barth 2001, personal communication). On the other hand, Henyey and Hoering (1997) argue that, in a borelike wave, steady dissipation can be balanced by a constant energy supply to the wave from net changes in isopycnal locations and potential energy, which in our case could be from the larger-scale internal tide.