a. Turbulence and mixing

While we anticipated strong internal lee wave generation in response to subinertial flows over the 200-m high topographic undulations (∼3-km alongslope wavelength), the near-bottom mean flow during the period when XCP surveys were conducted (27 May onward; u < 0.05 m s⁻¹) was too weak to generate waves (MacCready and Pawlak 2001; Thorpe 2001). Instead, we find the current and density structure consistent with along-isobath flow around rather than over the undulations. Internal lee waves were not observed judging by the absence of topographic horizontal scales in the overlying water column. Lee waves may have been generated prior to 27 May (u ∼ 0.10 m s⁻¹), but these were not captured by our sampling.

Despite weak alongslope currents and the absence of lee waves, intense mixing was observed—primarily above regions with semidiurnally near- and subcritical bottom slopes at depths greater than 1000 m, as opposed to over the supercritical upper slope (Fig. 2). Numerical simulations by Legg (2004a,b) indicate that neither the cross- nor along-isobath barotropic tidal forcing is able to generate a high-wavenumber response where we observed elevated mixing. Rather, this region is in a “shadow zone” for locally generated internal tides (Legg 2004a), which tend to propagate above and along a semidiurnal characteristic emanating from the shelf break. A tide generated at the local shelf break would interact with the bottom more than 30 km from the shelf break where mixing is weak (Fig. 2).

b. Background variability

Strong northerlies (10–15 May) prior to the mooring deployments produced a 1-m rise in sea level along the coast, as evident in the time series of winds and sea surface elevation (Fig. 6). Following this storm, subinertial currents (estimated by convolving each time series with a 43.5-h Hanning window at each depth level) were southward at 15–20 cm s⁻¹ for ∼1 week, subsiding to ∼5 cm s⁻¹ for the remainder of the observation period (Fig. 7). The strong southward flow extends to within a few hundred meters of the bottom. In addition to these synoptic-time-scale changes, higher frequencies (∼2-day period) are observed, with phase propagating upward from the bottom. Barotropic currents were generally alongslope and dominated by the subinertial flow. Reversals in the along-isobath current were observed in the bottom 100 m at the moorings (located in a gully). While suggestive of a topographic eddy, these features were too weak (<2 cm s⁻¹) to be resolved by the spatial XCP surveys, and so we are not able to interpret them as persistent features.

Tidal forcing is predominantly semidiurnal, and the surface tide amplitudes agree well with modeled barotropic results (TPXO.5, Egbert 1997). Spring tides peaked on 25 May when amplitudes exceeded those of neap tides by ∼50%. Barotropic tidal ellipse amplitudes (computed with TPXO.5, not shown) range from 1 cm s⁻¹ at the 1000-m isobath to 4 cm s⁻¹ at 200 m; variance ellipse major axes were aligned across isobaths with cross-isobath currents exceeding along-isobath by a factor of 2.

c. Energy flux 〈u′p′〉

Estimates of the baroclinic energy flux are used to diagnose the dynamics of the internal wave field through energy budgets. The energy flux F_E = 〈u′p′〉 is the covariance of the wave-induced pressure p′ and velocity u′. By definition, p′ and u′ are baroclinic fluctuations, and therefore have vanishing temporal and depth averages.

The perturbation pressure was determined following Kunze et al. (2002a). The density anomaly is estimated as ρ′(z, t) = ρ(z, t) − ρ(z), where ρ(z, t) is the instantaneous density and ρ(z) is the time-mean vertical density profile, averaged over at least one wave period (>12.4 h). Alternatively, ρ′(z, t) may be defined in terms of the vertical displacement of an isopycnal ξ(z, t) so that ρ′(z, t) = (ρ/g)N²ξ(z, t). The pressure anomaly p′(z, t) is calculated from the density anomaly using the hydrostatic equation,

View Expanded

Although the surface pressure p_surf(t) is not measured, it can be inferred from the baroclinicity condition that the depth-averaged pressure perturbation must vanish:

View Expanded

For internal waves over a slope, we must also consider the contribution to ρ′ from the barotropic tide's cross-isobath velocity (Baines 1982), which produces baroclinic vertical displacements. For no normal flow at the bottom (defined by z = −sx), a barotropic tidal velocity u_bt = u^o_bt cos(ω_M2t) induces a vertical velocity w_bt = −u_bts at z = −H and a vertical velocity w_bt(z) = u_bts(z/ H) throughout the water column. The associated vertical displacement ξ_bt(t) = ∫^t₀ w_bt(t*) dt* is

ξ_btξ^o_btzHω_M2t(3)

where ξ^o_bt = u^o_bts/ω_M2 is the maximum vertical displacement. For bottom slopes and cross-isobath barotropic tidal velocities typical of the Virginia slope (s ∼ 0.1 and u^o_bt ∼ 1 cm s⁻¹), ξ^o_bt ∼ 7 m. In comparison with the observed 20–100-m internal tide vertical displacements, ξ_bt is small. We nevertheless remove the barotropic-induced baroclinic pressure anomaly, p^′_bt = ∫⁰_z ρN²ξ_bt(ẑ, t) dẑ at each station prior to computing internal wave energetics. For this calculation, the M₂ across-isobath barotropic transport q_bt is spatially uniform and is used to compute u_bt as q_bt/H; our XCP observations and the TPX0.5 model both yield consistent estimates (q^o_bt ≃ 11 m² s⁻¹).

The perturbation velocity is defined as

View Expanded

where u(z, t) is the instantaneous velocity, u(z) is the time-mean of that velocity, and u_o(t) determined by requiring baroclinicity:

View Expanded

Full-depth profiles of pressure or velocity were not available from the moored profilers, which did not sample the upper 80 m of the water column. It was therefore necessary to extrapolate u and p to the surface to determine u_o(t) and p_o(t). This was done by fitting the barotropic and first baroclinic mode to the available data. To estimate the error introduced by excluding the upper 80 m of data, we subsample the XCP/XCTD profiles, perform a mode fit to extrapolate the data, and then compare the computed energy flux with that calculated using the original full-depth profile. We find that the depth-integrated energy flux computed from partial-depth data underestimates |∫⁰_−H 〈u′p′〉 dz| by 0.12 kW m⁻¹ on average and produces a 0.20 kW m⁻¹ rms error. MP-derived estimates of 〈u′p′〉 should be regarded with caution. Flux estimates from the XCP/XCTD surveys are fully resolved in space, but lack temporal coverage. We thus combine both measurements to evaluate the internal wave climate.

Temporal averages [p∗(z), u∗(z)] for the XCP/XCTD data were the means over four occupations at each station, generally spanning ∼15 h. These were used to determine the perturbation quantities and energy flux. The superinertial baroclinic fluctuations shown in Figs. 8a, 8d, and 8g were used as perturbation quantities to compute energy fluxes for the moored profiler data.

Profiles of perturbation velocity and isopycnal displacement are also used to estimate the depth-averaged baroclinic horizontal kinetic energy density HKE = 〈u^′2 + υ^′2〉/2 and available potential energy density APE = N²〈ξ²〉/2. For a single propagating wave, HKE:APE is an intrinsic property of the wave, equal to (ω² + f²)/ (ω² − f²) = 2.14 for M₂ at 36.5°N.

d. High-frequency variability

The subinertial variability (Fig. 7) was removed from the MP data to produce baroclinic records with near-inertial and higher frequency content. High-frequency time series, produced by modal decomposition and temporal filtering of the baroclinic velocity fields, are presented in Fig. 8. We decompose the vertical structure using WKBJ-stretched flat-bottom internal wave modes. While dynamically inappropriate for internal wave propagation on a slope, flat-bottom modes are useful for separating low and high vertical wavenumber contributions. In contrast, slope modes (Wunsch 1969) are dynamically relevant, but are ill-defined for near-critical bottom slopes. In addition, a single slope mode will represent a variety of physical wavenumbers and contain both shoreward and offshore energy propagation over a supercritical bottom, obscuring interpretation of the modal decomposition. In contrast, decomposition into flat-bottom modes isolates the energetics associated with a particular wavenumber without including the dynamical response of that wavenumber to the bottom within the mode description.

The first four stretched baroclinic modes were fit to WKBJ-stretched velocity data u′, which was bandpass filtered between 0.7 and 1.3 ω_M2, where ω_M2 represents the M₂ tidal frequency (12.4 h). We thus obtain a low-mode representation of the data:

View Expanded

where u_n is the modal amplitude associated with the mode-n vertical structure function, Z_n(z). The high vertical wavenumber representation is simply the difference between the total and low-mode velocity, each bandpassed between 0.7 and 1.3 ω_M2.

The pressure perturbation is predominantly first mode and of semidiurnal frequency, so only the raw p′ is shown (p′ has a redder vertical wavenumber spectrum than u′ or ξ′ by a factor of k⁻²_z since pressure is related to the vertical integral of vertical displacement). As expected, the velocity and vertical displacement fluctuations have a much broader modal content—high-wavenumber velocity fluctuations are of similar magnitude to the low modes and are intensified near the bottom.

A clear transition in the character of the flow was observed on 29 May, when both velocity fluctuations and pressure perturbations increased dramatically. At the same time, the along-isobath barotropic current became negligible. These changes are also evident in frequency spectra of 10-m shear (Fig. 9). Prior to 29 May, shear was dominantly near- and subinertial. Afterward, the observed shear was dominantly semidiurnal. In the following, we focus on data from the final week of the deployment (28 May–4 June 1998), because this was the time when XCP surveys (performed 27–29 May) captured the spatial structure of the strong semidiurnal shear (and energy flux).

Intensified semidiurnal shear became evident in the lower 100–200 sm (300–500 m) during 30 May–3 June (Fig. 9b). During that period, motions with 250–300-m vertical wavelengths, 5–10 cm s⁻¹ amplitudes, and 0.6– 0.7 cm s⁻¹ upward phase velocity dominate the high-wavenumber time series (Figs. 8c,f). These waves are the source of the high shear shown in Fig. 5. We interpret these as high-k_z waves of semidiurnal and higher intrinsic frequencies with downward energy propagation. We believe the source for these waves to be inshore of the moorings due to near-critical reflection of an onshore-propagating low-mode internal tide, consistent with the semidiurnal characteristic shown in Fig. 5. Cross-isobath trends in the semidiurnal phase of the mode-1 amplitudes of u, υ, and p, as computed through harmonic analysis at each of the moorings (not shown) support this hypothesis. Low-mode phase at the offshore mooring leads that at the onshore mooring by ∼30° in u, υ, and ∼10° in p, suggesting onshore propagation. Harmonic analysis of the XCP records indicates similar phase shifts.

e. Time dependence

Significant temporal variability was observed in the depth-averaged energetics of the internal wave field (Fig. 10). Although near-inertial and higher-frequency variability is included in these estimates, the dominant contribution to the energy flux is from semidiurnal fluctuations. Most notable is the dramatic increase in the along-isobath energy-flux 〈υ′p′〉 on 29 May. The onset of 1 kW m⁻¹ northward energy flux coincides with a relaxation of the ∼10–15 cm s⁻¹ barotropic southward currents (Fig. 7c), an increase in perturbation potential energy density (as a result of semidiurnal vertical displacements; Figs. 8f and 10b), and a shift in frequency content of 10-m shear from sub- and near-inertial to semidiurnal (Fig. 9).

A decrease in HKE:APE from its semidiurnal value (∼2) was observed along with the increase in energy flux after 29 May. We will show in section 3b that reduced HKE:APE near the boundary is a signature of a horizontally standing mode; in the limit of reflection from a vertical wall, HKE/APE ∼ 0 near the wall, an artifact of having significant vertical displacements yet no normal flow at the boundary. This suggests that horizontally standing modes in the cross-isobath direction may dominate the energetics during periods of high energy flux. This interpretation will be borne out by later analysis.

Velocity variance in the cross-isobath direction exceeds that of the along-isobath direction by ∼25% (Figs. 8 and 10b). The northward energy flux (Fig. 10a) is therefore not due to a single propagating internal wave, for which the ratio of along-to cross-isobath velocity variance should be ω²/f² ∼ 2.78. Furthermore, the squared coherence,

View Expanded

is very low: coh²_υp ∼ 0.05 before 29 May and ∼0.16 afterward (with a maximum ∼0.33 on 29–30 May). Thus, less than one-third of the velocity/pressure variance is associated with the net along-isobath energy flux on 29–30 May (Figs. 10 and 11). In the cross-isobath direction, coh²_up = 0.015 throughout. These results suggest that the measured flux is not the product of a single propagating wave, for which coh² ∼ 1. For reference, a coherence of 0.03 is significant at the 95% confidence level, assuming independence of 10-m data.

f. Energetics

Depth-integrated energy fluxes calculated from the XCP/XCTD surveys are generally along-isobath and northward (Fig. 11). There is also a suggestion of cross-isobath convergence—in waters deeper than 1000 m, the flux is slightly upslope, while in shallow water, the flux is either along-isobath or downslope.

We illustrate this convergence in more detail with cross sections of energy flux (Fig. 12). The energy flux is decomposed by projecting the perturbation velocity (u′, υ′) and pressure (p′) onto the flat-bottom vertical-mode structure functions and computing the energy flux from those projections (6). This allows separation of low modes (which may propagate long distances) from high wavenumbers (which are presumed to be generated and dissipated locally because of their slow group velocities and susceptibility to nonlinearity). Cross sections of energy flux and inferred diffusivity are shown along the ridge crest (Figs. 12a–c) and in the gully (Figs. 12d–f). Vertical averages of the energy flux are shown above each plot (Figs. 12a′, b′, d′, e′).

Mode 1 is propagating onshore and its energy flux is converging (Figs. 12a,d). The high-wavenumber signal is dominated by a divergent beam of strong offshore energy flux near the bottom below 900 m (1100 sm; Figs. 12b,e). Such a beam may be formed through near-critical reflection from a supercritical linear slope (Eriksen 1982, 1985). The partitioning of energy flux may also be interpreted in terms of supercritical slope modes (2D modes in a supercritical wedge Wunsch 1969), for which a low-k onshore energy flux is balanced by an intense high-k offshore flux near the bottom.

In both of our ridge–crest and gully sections, the near-bottom beam (Figs. 12b,e) originates near the 900-m isobath and coincides with the near-bottom layer of elevated diffusivity (Figs. 12c,f). At the shoremost station, the high-wavenumber wave field is weak; inferred diffusivities are small (K_ρ ∼ 10⁻⁵ m² s⁻¹) and not bottom-intensified despite the similarly rough bottom. The two slope environments (ridge and gully) appear indistinguishable in terms of energy-flux convergence and diffusivity.

Ridge and gully sections each indicate low-mode convergence of similar magnitude to the high-wavenumber divergence in the lower water column. The average low-mode convergence [ridge: (−6 ± 2) × 10⁻⁸ W kg⁻¹, gully: (−4 ± 2) × 10⁻⁸ W kg⁻¹] exceeds the high-wavenumber divergence [ridge: (3 ± 3) × 10⁻⁸ W kg⁻¹, gully: (2 ± 3) × 10⁻⁸ W kg⁻¹], consistent with there being another sink for the converging cross-isobath energy flux, such as local turbulent dissipation ϵ or the divergence of along-isobath energy flux.

To determine the most significant contributions in the internal wave energy budget, we consider the evolution of depth-averaged baroclinic energy E:

View Expanded

where G is the baroclinic generation by the barotropic tide. The internal tide energy density increased during the XCP surveys and cannot be ignored. From Fig. 10, the rate of change of baroclinic perturbation energy density was at most ∂E/∂t = 1 (J m⁻³)/day = 1 × 10⁻⁸ W kg⁻¹.

The net energy-flux divergence ∇ · 〈u′p′〉 is obtained with increased precision by including all vertical modes (low and high wavenumbers) as well as combining both ridge and gully transects of the depth-integrated energy flux prior to calculating its gradient (Fig. 13). Evaluated at the 1000-m isobath, there is a net convergence of cross-isobath energy flux: d〈u′p′〉/dx = (−7 ± 2) × 10⁻⁸ W kg⁻¹. The along-isobath flux-divergence is estimated by differencing the XCP- and MP-derived energy-flux, and is d〈υ′p′〉/dy = 0.3 ± 0.2 W m⁻²/10 km = (3 ± 2) × 10⁻⁸ W kg⁻¹. The net divergence is ∇ · 〈u′p′〉 = (4 ± 3) × 10⁻⁸ W kg⁻¹. In comparison, the observed dissipation rates were ϵ ∼ 0.5 × 10⁻⁸ W kg⁻¹.

We estimate the barotropic–baroclinic conversion term G from the simulations of Legg (2004a). In those simulations, an ∼ 20 W m⁻¹ internal tide is generated for a 0.01 m s⁻¹ deep baroclinic cross-isobath forcing, typical of the Virginia slope. Most of the baroclinic energy flux is generated at the shelf break, with less than 10% originating over the deeper slope. Assuming that generation occurs over 5 km of slope in 1000 m of water, the local generation is G = 2 W m⁻¹/1000 m/ 5000 m = 0.04 × 10⁻⁸ W kg⁻¹. Hence it is reasonable to neglect local generation by the barotropic tide in (8).

In summary, the cross-isobath energy-flux convergence of (7 ± 2) × 10⁻⁸ W kg⁻¹ is the source term in (8), with both along-isobath divergence and turbulent dissipation being possible sinks. The net convergence ∇ · 〈u′p′〉 = (4 ± 4) × 10⁻⁸ W kg⁻¹ is large enough to both supply the baroclinic energy density increase during 29–31 May (Fig. 10) and fuel the observed turbulence production rate of ϵ ∼ 0.5 × 10⁻⁸ W kg⁻¹. However, the uncertainties are sufficiently large that we cannot preclude other energy sources supporting the observed turbulence (Figs. 2, 3).

In any event, the high-wavenumber internal waves formed over the slope must dissipate somewhere. If the only means of attenuation were local turbulent dissipation of O(10⁻⁸ W kg⁻¹), the offshore flux of high-wavenumber energy (∼1 W m⁻²) could propagate about 100 km before dissipating and support mixing well into the ocean interior. Only 5%–10% of the O(1 kW m⁻¹) cross-isobath energy flux is dissipated locally by turbulence over the slope [i.e., the dissipation efficiency, as defined by St. Laurent et al. (2002), is 0.05–0.1]. Since much of the high-k flux is directed offshore and downward (hugging the bottom), it may dissipate or scatter during interaction with the subcritical bottom away from our site of observations.

a. Reflection from an inclined plane

Following Eriksen (1982, 1985), we consider reflection of an internal wave from a sloping bottom (z = −sx). The ratio of reflected to incident wavenumbers [k^(r)_z/k⁽ⁱ⁾_z] and amplitudes [u^(r)/u⁽ⁱ⁾] is given by the reflection ratio,

View Expanded

where α = ±(ω² − f²)/(N² − ω²) for an internal wave of frequency ω. The ratio R is unity only for limiting cases (s = 0, s = ∞, α = 0, or α = ∞). Linear theory predicts amplification when the slope of internal wave characteristics is similar to the bottom slope (α ∼ s). At α = s, R = −∞. The case of large |R| is of particular interest here, as it leads to high shear and increased likelihood of turbulence.

If a mode-1 internal tide (λ_z = 2H = 3000 m; H is the water depth) reflects to produce waves with λ_z = 300 m (as observed) then [k^(r)_z/k⁽ⁱ⁾_z] = 10, which requires s/α = 0.8 or s/α = 1.2 (9). The distribution of bottom slopes along the Virginia continental slope (Fig. 14) indicates that both of these values of the bottom slope are common near and onshore of the 1100-m isobath.

Interactions with nearly critical slope s could also directly lead to mixing through shear instability associated with Ri < 1/4. These may act locally to break down the most unstable waves, yet permit stable waves to propagate into the interior to later produce turbulence through the mechanism suggested by Gregg–Henyey (Gregg 1989). For example, consider the shoaling of a mode-1 internal tide with a 2 cm s⁻¹ WKBJ amplitude (0.4 kW m⁻¹, consistent with Fig. 13). Near the 1000-m isobath, the mean squared shear associated with such a first-mode wave is S² ∼ 10⁻¹⁰ s⁻². Given a bottom stratification N² ∼ 10⁻⁶ s⁻², a (10⁴–10⁵)-fold increase in S² would produce unstable Ri ∼ 1/4. Bottom slopes within 20% of critical could produce this intensification.

b. A ray-tracing approach

To better demonstrate that wave reflection may lead to the observed energy-flux distribution, a ray-tracing model (9) of the wave–topography interaction is used. We assume 2D bathymetry, neglecting alongshore topographic variability. Computations were performed over a 100-km cross-isobath and 1500-sm-deep constant stratification domain. To focus on the deep dynamics (i.e., near 1000 sm), we neglect the continental shelf by extending the continental slope to the surface at the shelf break. The incident wave field is a sum of rays comprising a mode-1 vertically standing wave at the offshore boundary (x = 100 km) with zonal velocity amplitudes chosen to produce a shoreward energy flux of 0.4 kW m⁻¹. Information about the wave amplitude is assumed to be transmitted along characteristics until they leave the domain. Surface reflections alter only the vertical propagation direction [k^(r)_z = −k⁽ⁱ⁾_z]. Topographic reflections follow (9) after Eriksen (1982). Subcritical bottom reflection alters vertical propagation direction and amplitude [k^(r)_z = −Rk⁽ⁱ⁾_z, k^(r)_x = Rk⁽ⁱ⁾_x, u^(r) = Ru⁽ⁱ⁾]. Supercritical reflection alters horizontal propagation direction and amplitude [k^(r)_z = Rk⁽ⁱ⁾_z, k^(r)_x = −Rk⁽ⁱ⁾_x, u^(r) = −Ru⁽ⁱ⁾]. Vertical displacements were computed by integrating vertical velocity in time; pressure anomalies were computed using (1).

Figure 16 shows the time-averaged energy flux in the shoreward, seaward, and net fields. The shore- and seaward fields are intensified near critical slopes, dramatically increasing amplitudes and wavenumbers. As a result, high-wavenumber fluxes propagating both on- and offshore are produced (Fig. 16c). In the absence of dissipation, these form beams that propagate until they leave the domain. In reality, these beams would be unstable and dissipate by turbulence. Destructive interference between on- and offshore beams would therefore be reduced (Gilbert and Garrett 1989; Legg and Adcroft 2003).

Concave continental slopes are often overlooked for their potential for strong internal-wave–topography interactions because analytic solutions (Gilbert and Garrett 1989) and observations (Gilbert 1993; Zervakis et al. 2003) suggest that they are much less efficient at intensifying shear than convex ones. However, Gilbert and Garrett's (1989) results were based on a single analytic form (parabolic) for the bottom slope. While this particular choice allowed for an exact analytic solution, it also has a unique property that the amplitudes of upslope and downslope (incident and reflected) wavefields were equal, resulting in destructive interference. From our ray-tracing solutions, only these parabolic slopes produce complete destructive interference, while more-realistic concave slopes produce complex high-wavenumber interference patterns. Recent numerical simulations in which this symmetry is not required (Legg and Adcroft 2003) find a similar intensification of high wavenumbers over both concave and convex slopes. Observations have also provided anecdotal evidence that enhanced dissipation can occur over concave slopes (Toole et al. 1997). This has important implications, because most continental slopes are concave in a WKBJ-stretched sense. They should not be overlooked as potential sites of mixing.

Despite the many simplifying assumptions, we find a qualitative similarity of the wavenumber content and high-k energy flux in comparing Figs. 16c and 12b,e. This confirms that the idealized mechanics hypothesized in section 3a are plausible.

1) Horizontally standing modes

Since our calculations are inviscid, no net energy-flux convergence is possible, and the vertically integrated energy flux of the reflected field is approximately equal and opposite to the incident. This forms a horizontally standing mode with weak net cross-isobath energy flux (Fig. 16c). A peculiar feature of the standing wave is that the along-slope flux 〈υ′p′〉 does not vanish, but instead has a significant northward component (0.5 kW m⁻¹ within 20 km of the coast).

This artifact can be illustrated by considering the energy flux associated with a superposition of two first-mode waves, one propagating in the −x direction (the incident wave) and the other in the +x direction (the reflected wave) having vertical and zonal wavenumbers of k^o_z = π/H and k^o_x = αk^o_z ∼ 2π/(80 km). Each wave carries a zonal energy flux of ±½u_op_o cos²(k^o_zz), where u_o and p_o are the modal amplitudes of zonal velocity and pressure. They are phased to produce no normal flow across a vertical wall at x = 0. While the cross-isobath energy flux 〈u′p′〉 of the combined wavefield is zero everywhere, the correlation between pressure anomaly of the incident (reflected) wave and the transverse velocity of the reflected (incident) wave is nonzero and produces an alongshore energy flux:

View Expanded

as illustrated by Fig. 17. Maxima 〈υ′p′〉_total occur at x = (2n − 1)π/(4k^o_x) and have a magnitude that locally exceeds the spatially uniform energy flux in either the incident or reflected propagating waves. This explains the observed along-slope energy fluxes (Fig. 11). A signature of such a superposition is that the strongest baroclinic velocities are not aligned in the direction of the net energy flux as we observe (Fig. 10).

In addition, time- and depth-averaged energy densities of a horizontally standing mode have offshore structure:

View Expanded

As compared with a single propagating wave for which HKE/APE = (ω² + f²)/(ω² − f²) uniformly, the ratio HKE/APE for a horizontally standing mode varies periodically. It is 0 at x = nπλ_x/2 and ∞ at x = (2n + 1)λ_x/4 for n = 0, 1, 2, 3, …. Hence, a signature of a standing mode is decreased HKE/APE near the boundary, as observed during the later period of strong internal wave activity (Fig. 10), and enhanced HKE/APE one-quarter wavelength offshore.

c. Interactions with corrugations

Scattering of the low-mode M₂ internal tide off the cross-isobath corrugations is another possible mechanism for generating finescale internal waves that would break and produce elevated turbulence. From Thorpe (2001), one would expect the along-slope wavenumber of the reflected wave field to match that of the local bathymetry (having along-slope wavevector k_bathy). To conserve frequency, this requires

View Expanded

where α⁻¹ ≃ 30, [k⁽ⁱ⁾_x, k⁽ⁱ⁾_y, k⁽ⁱ⁾_z] and [k^(r)_x, k^(r)_y, k^(r)_z] are the wavevectors of the incident and reflected fields, and k^(r)_y = k_bathy = 2π/λ_bathy = 2 km⁻¹. Assuming k^(r)_x = 0 allows calculation of the lowest vertical wavenumber k^(r)_z (the largest wavelength) associated with M₂ interaction with the corrugations. This yields

k^(r)_zα⁻¹k_bathy⁻¹(15)

which corresponds to vertical wavelengths of 100 m and smaller. This contrasts with the 300-m vertical wavelengths that we observe to dominate the near-bottom signal. Hence, scattering off the corrugations does not appear to be responsible for the observed shears. This is further supported by our observations that neither the variability in the high-k internal wave field nor the dissipation rate was clearly tied to the corrugation length scale. However, high-wavenumber scattering from the corrugations (15) may dissipate rapidly enough that a layer of enhanced turbulence is produced without developing a phase-locked internal wave field. Our data cannot rule out this possibility.

d. Source and variability of the internal tide

Until this point, we have not specified a source for the incident, low-mode baroclinic tide nor discussed reasons for its temporal variability. In this section, we explore some possibilities.

A possible source for the internal tide is the local shelf break. However, numerical simulations performed by Legg (2004a) suggest that the vertically integrated energy flux from the shelf break is weak (<0.05 kW W m⁻¹) for typical barotropic cross-isobath forcing. This is in contrast to regions like the Bay of Biscay (Pingree and New 1991) where the cross-isobath barotropic forcing is strong and the internal tide forms at the shelf break. The Virginia shelf break is more typical of continental margins for which the barotropic forcing across the shelf break is weak (Baines 1982; Sjöberg and Stigebrandt 1992).

More important is that baroclinic motions from a shelf-break-generated tide do not penetrate below the downward semidiurnal characteristic originating at the shelf break. This contrasts our observations of strong semidiurnal shear in the lower water column. Legg (2004b) also performed simulations with along-slope barotropic forcing and concluded that bottom-intensified shear was generated by the corrugations, but was only of small amplitude for reasonable barotropic forcing.

As the locally generated internal tide seems incapable of generating the observed deep shears, we search for a remote source by exploring some deep-ocean data. Unfortunately, baroclinic velocity data were not collected offshore of the Virginia slope during this experiment. However, a 2-yr record of velocity and temperature at 34°N, 70°W was collected during the Long Term Upper Ocean Study (LOTUS; Tarbell et al. 1985) and is used to assess the strength of the deep-ocean internal tide and its variability. The measured barotropic velocity (Fig. 18b) exhibits more temporal variability than the TPX0.5 barotropic tide prediction (Fig. 18a). The estimated mode-1 baroclinic tide (Fig. 18c) is even more variable in time with its amplitude only loosely linked to the barotropic spring–neap cycle.

The median and mean vertically integrated energy fluxes over the 450-day period are 0.35 and 0.5 kW m⁻¹, respectively; estimates exceeding 1 kW m⁻¹ occur 6% of the time. On average, the depth-integrated energy flux is directed northeast [away from the Blake Escarpment—a possible generation site; Hendry (1977)] with 18% of the estimates directed northwest (toward the Virginia slope) with an average magnitude of 0.35 kW m⁻¹. We hypothesize that these internal tides could survive multiple surface/bottom reflections, as their more energetic counterparts do in the Pacific Ocean (Ray and Mitchum 1996; Cummins et al. 2001). Since northward-propagating internal tides encountering zonal bathymetric slopes will gradually refract shoreward and eventually propagate fully upslope (Thorpe 2001; Fig. 2), even the northeastward-propagating internal tides at LOTUS may shoal on the Virginia slope. We conclude that deep-ocean internal tides in the North Atlantic are sufficiently energetic to be a possible source for the observed energy fluxes at the Virginia slope.

Several factors may explain the abrupt appearance of the semidiurnal tide after 29 May in our observations. In addition to the fortnightly spring–neap cycle, the internal tide is modulated on subinertial time scales through interaction with temporally variable stratification and Doppler-shifting by geostrophic currents. Both of these can alter propagation over long distances and produce highly intermittent internal tides (Wunsch 1975). Currents from the nearby meandering Gulf Stream system (∼1 m s⁻¹, comparable to the mode-1 phase speed) could also significantly modulate internal tides by deflecting their ray paths. Local along-isobath currents are likely to have only a minor effect as they could account for a frequency shift Δω = k_yυ = 9 × 10⁻⁶ rad s⁻¹ of only 10% of M₂ (assuming a mode-1 horizontal wavenumber k_y ∼ 9 × 10⁻⁵ rad m⁻¹ and along-isobath current υ ∼ 0.1 m s⁻¹).