a. Time dependence

The time-dependent flow over the Knight Inlet sill is quite complicated and has been discussed in detail by Farmer and Armi (1999a), Armi and Farmer (2002), and Klymak and Gregg (2003). An example of data from a weak ebb tide, a strong ebb tide, and a strong flood tide are shown in Fig. 4. During each tide a nonlinear lee wave formed downstream of the sill. The response was analogous to hydraulic control of a two-layer fluid, described in detail by Baines (1984,1995: see Fig. 5.28 and p. 296). Under a wide range of conditions, water upstream of the obstacle does not have enough energy to surmount the obstacle, and so the lower layer is partially blocked. Water accumulates in the dense layer lifting the interface upstream, while there is a divergence downstream dropping the interface there. A nonsteady situation persists near the sill until enough potential energy is stored in the lower layer to drive all of the lower layer over the obstacle at a fast enough rate. Once in steady state the flow at the obstacle crest is hydraulically controlled in a manner analogous to water passing over a weir, and the two layers meet a criticality criterion of G² = u²₁/g′h₁ + u²₂/g′h₂ = 1 (Armi 1986). The lift of the interface upstream and the drop downstream propagate away from the obstacle as waves, raising the energy of the water upstream and lowering it downstream. Water flowing over the obstacle must dissipate energy for the upstream and downstream conditions to match. In a two-layer (or one layer) fluid much of this dissipation takes place through an abrupt hydraulic jump.

Many features analogous to the two-layer case can be seen in Fig. 4 and in Klymak and Gregg (2003). The isopycnals have been raised upstream and depressed downstream of the sill and form waves that radiate away from the sill. There is a plunging flow over the sill and an abrupt rebound of isopycnals to the background state. There are also differences with the simple two-layer analogy. Farmer and Armi (1999a) and Armi and Farmer (2002) show that a critical Froude number can be found near the sill crest along the strong shear interface between the plunging flow and the flow aloft. However, the stratified interface does not follow a simple hydraulic solution (Farmer and Armi 1999a) as isopycnals peel off from the interface successively. Instead of a localized hydraulic jump, there is a large region in the lee of the sill where the flow adjusts to the downstream state (Fig. 4i); we will show below that this region dissipates considerable energy. There are also other ways to dissipate the energy required of the hydraulic response. As we will see below, and was also pointed out by Farmer and Armi (1999a), the flow develops shear instabilities that are quite vigorous. There is also the possibility of strong bottom friction (Nash and Moum 2001) and lee vortices (Klymak and Gregg 2001; MacCready and Pawlak 2001). These sinks of energy will all be quantified in section 4 and demonstrated by example below.

b. Weak ebb tide

Our first example shows the response over the sill during a weak ebb tide (Fig. 5). There were two shear regions in this flow. The first shear region was along the bottom of the lee wave starting near 10-m depth 1 km landward of the sill crest and extending downstream; it dropped, crossing isopycnals until −0.5 km (Fig. 5a). This shear interface was populated with flow instabilities evident in the strong backscatter that began near the surface and then became braided as the shear interface deepened over the sill crest (Fig. 5b). These billows had dissipation rates approaching ε = 10⁻⁴ W kg⁻¹ (Fig. 5c).

A second turbulent region was observed in the flow separation near the sill crest. The turbulence began at the sill crest and extended west to −0.75 km at about 80-m depth. This interface was not detected clearly by the echo sounder (Fig. 5b), presumably because the stratification was much weaker at these depths. Careful examination of the echo-sounder data showed signs of breaking waves (see Fig. 6 for larger images); previous studies in Knight Inlet have shown large instabilities near this location (Farmer and Smith 1980; Farmer and Armi 1999a; Klymak and Gregg 2003). The region was populated by overturns, but the implied turbulence, 3 × 10⁻⁶ W kg⁻¹, is weaker than in the wave shear layer (Fig. 5c). The water beneath the separation was stably stratified and not turbulent.

There was little turbulence observed in the rest of the water column. Upstream of the sill crest the water was quiescent, as it was downstream below the wave shear layer. The turbulence in the wave shear layer was localized to the shear layer, dropping off quickly above and below. Sampling with SWIMS did not reach closer than 5 m from the bottom, and so turbulent overturns in the bottom boundary layer may have been missed.

These data are in agreement with those collected in AMP drops through very similar flows the previous day. Figure 6a presents four drops through a widening wave. Drops 15950–15952 show that the turbulence was localized to the shear layer beneath the lee wave, starting at about 17-m depth at 0.4 km and then dropping to 25-m depth by 0 km. The turbulence reaches ε = 10⁻⁴ W kg⁻¹ in this interface, is very low below the interface (ε = 10⁻⁸ W kg⁻¹), and is moderate above the interface (ε = 10⁻⁶ W kg⁻¹). Data in drop 15951 indicate a lower dissipation rate in the wave shear layer farther upstream. Turbulence decreased downstream; at drop 15953 the dissipation at 30 m was only ε = 10⁻⁶ W kg⁻¹. High turbulence was measured at 60–80 m in drop 15953 (−0.2 km) as the instrument traveled through the flow-separation region, but decreased again below. The echo-sounder image indicated billows along the flow-separation interface.

In a pass over the sill made 30 min later (Fig. 6b) the shear interface had deepened. The interface also moved to denser isopycnals (not shown). The turbulence along this interface was again quite high (ε = 10⁻⁴ W kg⁻¹) but low above and below. Figure 6 emphasizes the time dependence of the changes in the flow.

c. Strong ebb tide

The beginning of a strong ebb tide looked much like a weak ebb tide. As the tide progressed, however, the response grew stronger (Fig. 7). A wave shear interface with the billows remained evident, but, as it approached the sill crest, it plunged more sharply. This interface rebounded at about 100-m depth and demarks an undular wave in the lee of the sill.

The wave shear layer was populated by shear instabilities that were larger with depth (Fig. 7b). The dissipation rate of these overturns was ε = 10⁻⁴ W kg⁻¹. The large undulation on the seaward side of the lee wave also has a dissipation rate above ε = 10⁻⁴ W kg⁻¹. This feature was not an overturn, but rather a very steep wave that appeared to lose energy soon downstream of the first undulation. As such, we will treat the height of the undulation as the vertical scale instead of the proper Thorpe scale in Eq. (1). The turbulence associated with this undulation extended up in the water column to 25-m depth but dropped quickly farther seaward, reaching levels consistently below ε = 10⁻⁶ W kg⁻¹ by −0.5 km.

Again, the SWIMS data agree well with the direct turbulence measurements from AMP (Fig. 8). Upstream and downstream of the undulation on the seaward side of the lee wave the turbulence was low except in the shear interface near 15-m depth (drops 15769 and 15770). In the undular wave (drop 15771) high dissipation was measured almost to the top of the water column; values were slightly above ε = 10⁻⁴ W kg⁻¹. The turbulence measured in the next drop (15772) at −0.6 km was still elevated but not as high, generally around ε = 10⁻⁵ W kg⁻¹, and drop 15773, just seaward, was below ε = 10⁻⁶ W kg⁻¹ (not shown).

d. Flood tide

Flood tide was moderately strong and did not differ as much from neap to spring (Fig. 9). A large lee wave formed early in the tide with the characteristic elevated isopycnals upstream, plunging flow over the sill crest, and a rebound to downstream conditions by 1.5 km east of the sill (Figs. 4g–i). The turbulence appeared mostly when isopycnals in the lee wave expanded, which they did at progressively deeper depths downstream. The strongest overturns occurred where the distance between isopycnals widened in the lee of the sill and individual isopycnals became vertical (Fig. 9c). Some of the turbulence was perhaps also caused by shear instability; in particular, the turbulence between 1 and 1.5 km was associated with billows in the echo-sounder image between 25- and 55-m depth. During flood it was more difficult to separate the two processes definitively, partly because the flow was more three-dimensional than during ebb and our sampling sliced the flow at an awkward angle (see below). The dissipation was low over the plateau of the sill until 1 km when it increased. By 2 km it had returned to a low level.

Similar dissipations were found in AMP data collected during flood tide (Fig. 10). Upstream, in drops 15919–15916, the turbulence was low and confined to a thin shear layer above 20-m depth. Downstream, as the sill steepens, the turbulence was high (drop 15915), especially between 25 and 50 m. This region was populated by shear instabilities and nearly vertical isopycnals, and so, again, saying whether the turbulence was from shear or breaking of the lee wave is difficult. Farther downstream (drop 15914), the turbulence below 75 m was not clearly associated with features in the echo sounder, suggesting the importance of breaking waves. By 2.1 km (drop 15913) there was almost no turbulence in the flow.

e. Three-dimensionality of the turbulence during flood

During flood tide there was large cross-inlet variability in the dissipation, as expected from the large cross-inlet variability of the flow (Klymak and Gregg 2001). On August 26 two cross-inlet lines were repeatedly occupied, one at the sill crest and the second 1.5 km downstream (Fig. 11). There was an hourglass-shaped jet of landward-flowing water at middepth flanked by two counterflowing jets [discussed in detail by Klymak and Gregg (2001)]. At middepth, the landward jet was less than 300 m wide, creating a complicated pattern of shear in the flow. The shear lines between the landward jet and the counterflows were populated by increased backscatter showing undulations with 25-m amplitudes (i.e., at −0.4 km and 50-m depth; Fig. 11b). These undulations appear as large overturns in the SWIMS CTD data implying dissipation rates close to ε = 10⁻⁴ W kg⁻¹ centered on the landward jet (Fig. 11c). The rapid return flow on the south side of the channel also appeared turbulent, but the slow return flow on the north side was mostly quiescent.

f. Upinlet turbulence

Sampling was concentrated near the sill, but some data were collected far enough away from the sill that an average profile of turbulent dissipation can be formed. Considering all drops made 6 km or more away from the sill, the dissipation rate was modest (Fig. 12). The turbulence profile was generally as high as ε = 10⁻⁷ W kg⁻¹ in the upper water column, and ε = 10⁻⁸ W kg⁻¹ at 100-m depth. Below 100 m it was 10⁻⁹ < ε < 10⁻⁸ W kg⁻¹, though the small number of samples below 100 m may have biased this number.

A larger ensemble of profiles made upinlet were collected while following nonlinear internal wave packets. In order to not bias the average, these profiles were not included in the average presented. These packets manifested themselves as a series of rapid undulations in the near-surface pycnocline. These “solibores” had dissipations reaching ε = 10⁻⁵ W kg⁻¹ in limited patches, approximately 1000 m × 10 m (Fig. 13). The structure of this example was not dissimilar to the lee waves, with a turbulent shear layer along a bottom interface.

a. Energy lost from the barotropic tide

The energy lost from the barotropic tide can be estimated by considering the phase difference between two tide gauges (Freeland and Farmer 1980). If no energy is lost, the barotropic tide in an enclosed inlet will be a perfect standing wave. If energy is lost, the reflected wave will be weaker than the incoming wave, and the seaward tide gauge will lead the landward one. If the phase lag is small, Stacey (1984) derived that the power lost from the barotropic tide between two tide gauges can be estimated from the phase lag between them, ϕ₁, as

View Expanded

where η₀ is the height amplitude of the barotropic tide (m), ω is its frequency (rad s⁻¹), S₁ = 2.4 × 10⁸ m² is the surface area upinlet of the seaward tide gauge, and S₀ = 3.6 × 10⁷ m² is the surface area between the two tide gauges.

Here a harmonic analysis at two frequencies, semidiurnal (12.42 h) and diurnal (23.93 h), estimates the phase difference between the two tide gauges. The analysis was carried out over a 1-day sliding window. The phase lag at the semidiurnal frequency between the two tide gauges was approximately constant at ΔΦ ≈ 1.8° (Fig. 15c); the phase lag at the diurnal frequency was small. The semidiurnal amplitude varied from 0.9 to 2 m over the course of the cruise (Fig. 15b). Therefore the power lost from the semidiurnal tide according to Eq. (2) varied from 4 MW at neap tide (day 231, Fig. 15d) to 20 MW at spring (day 239).

b. Energy dissipated near the sill

c. Wave flux away from the sill

The internal wave flux from the sill is estimated using two direct methods and then compared with a linear theory. The internal waves are the manifestation of the upstream and downstream propagating disturbances that are part of the response of the flow to the sill (section 3a). To estimate the wave flux we first consider the correlation between velocity and pressure perturbations:

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where Δy is the width of the inlet at the seaward and landward sections (Kunze et al. 2002). The velocity perturbations are computed by removing the barotropic velocity expected from the tide gauge data. The pressure perturbation is calculated by assuming hydrostatic pressure

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where ρ′ = ρ − ρ₀ is the density perturbation at a given depth. Throughout, ρ₀ is calculated separately on either side of the sill for each set of SWIMS runs by averaging two density profiles bracketing the slack tide.

This calculation yields the light gray traces in Figs. 16e–h and 16m–p. The direction for the seaward fluxes was to the west, but they have been plotted as positive. The flux estimate is not likely to work on the landward side of the sill during flood tides where the velocity is strongly influenced by horizontal vortices (see Fig. 11 and Klymak and Gregg 2001). On the seaward side of the sill, ∫ u′p′dz varies with the square of the transport and reaches as high as 14 MW during flood tide on day 236 (Fig. 16f).

Wave fluxes are also estimated by calculating the amplitude of the internal waves using a modal decomposition. Wave motions are decomposed into the three lowest vertical modes. The total energy flux is

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where Δy is the inlet width, c_n is the wave speed of the mode, and E_n is the energy density:

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The amplitude of each mode, ζ_n, is calculated by regressing the displacement with the standing vertical mode structures. This expression for the energy density is an approximation assuming that energy is equipartitioned between potential and kinetic energies. We tested this assumption by calculating the kinetic energy flux on the seaward side of the sill where the flow is two-dimensional and found that it agreed well with the potential energy flux, though the results are not plotted for clarity.

The modal estimate, Eq. (7) (black line in Figs. 16e– h, m–p), gives similar results to the u′p′ estimates on the seaward side of the sill, but not on the landward where Eq. (5) is likely inaccurate. The flux tends to be 15% higher on the landward side of the sill, yielding an energy flux that peaks at 17 MW on day 236.

These direct estimates are compared with the linear estimate developed by Stacey (1985) and Stigebrandt (1980). This method uses the vertical modes and assumes continuity of the barotropic velocity at the sill crest (U_s):

View Expanded

where z_s is the sill depth. This linear estimate gives the dashed curves in Figs. 16e–h and 16m–p. There is a relatively good agreement in energy flux between the methods, though the linear estimate tends to be smaller. There is also a phase difference between the direct and linear estimates (and thus the direct estimates and the barotropic velocity at the sill); the direct estimates lead the forcing on the downstream side of the sill and lag it on the upstream side.

d. Bernoulli drop near the sill

As discussed above (section 3a), the waves that radiate from the sill at each phase of the tide raise the energy upstream and lower it downstream. This means that water flowing over the sill must dissipate energy to match the upstream and downstream conditions. The dissipation of energy in a steady-state flow can be estimated from the divergence of flux of the Bernoulli function:

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where the Bernoulli function is B = ½u² + P/ρ + gz (Baines 1995, p. 13). Nash and Moum (2001) used a one-layer approximation of this equation for an energy balance over Stonewall Bank. To estimate the right-hand side of Eq. (10), we divide the flow into five layers (divided at σ_θ = 0, 21, 22, 23, 24, and 25 kg m⁻³) and calculate the Bernoulli function in each layer at the section at the top of the sill [B^(s)_n] and the section downstream [B^(d)_n]. The volume flux q_n is assumed constant in each layer between the two sections; that is, the mixing in the flow is small. Power lost is

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The constant flux assumption is necessary because the lee vortices preclude a good estimate of downstream volume flux in each layer. The surface pressure is also not known precisely, but is estimated to be strong enough to accelerate the flow so that the volume flux remains constant in the surface layer; this correction corresponds to 0.01 m, or approximately 10% of the Bernoulli drop in the deeper layers.

The resulting energy drop (black line in Figs. 16i–l) shows spring–neap modulation and a maximum energy drop similar to the observed wave flux (∼15 MW). Interestingly, neap tides (Figs. 16i and 16l) have disproportionately small Bernoulli drops.

These two-dimensional estimates of the Bernoulli drop can be compared with a similar estimate made using the cross-channel sections (Klymak and Gregg 2003). We found a 16-MW drop between two cross-channel sections at peak flood tide, similar to the value in Fig. 16j. We did not exclude mixing in the three-dimensional calculation, and so the agreement indicates that our assumption of two-dimensionality does not bias the estimate unduly.

e. Energy sinks near the sill

The Bernoulli drop must be caused by dissipation or storage of energy between our two sections or by three-dimensional motions. There is very large dissipation in the shear and convective instabilities near the sill. However, when averaged, this turbulence is not large enough to account for the Bernoulli drop. Dissipation is estimated from the measurements made during the SWIMS runs using the overturn method and summing:

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where D is the depth at each SWIMS cast. The width, Δy, is the width over which the dissipation measured during the SWIMS run is representative. Assuming that the turbulence is confined to the region where there are large lee waves and not behind the headlands, we choose Δy = 1200 m. The results (dotted line, Figs. 16i–l) suggest that the water column dissipation is too small to account for the energy drop across the sill. Even for a run that passed through the highest dissipations during flood tide (Fig. 16k and Fig. 9 above) the dissipation only reaches 50% of the energy drop across the sill, peaking at 5 MW.

There are probably other sinks of energy not captured by SWIMS. First, there could be large sinks of energy near the bottom where we did not measure with AMP or SWIMS. To estimate the capacity of this energy sink, we assume a cubic dependence:

View Expanded

where k = 0.003 (Freeland and Farmer 1980). This yields an estimate (dashed–dotted lines in Figs. 16i–l) that is approximately as large as the mid water column turbulence. If there were a turbulent region 5 m thick along the bottom, then dissipation rates in it would need to reach ε ≈ 5 × 10⁻⁵ W kg⁻¹. This is a high, but not unreasonable value. Nash and Moum (2001) observed bottom boundary layer turbulence of ε ≈ 10⁻⁴ W kg⁻¹ for flow over a shallower obstacle on the continental shelf.

The direct dissipation measurements and inferred bottom dissipation remain inadequate to account for the drop in the Bernoulli flux across the sill. There is a storage of potential energy in the lee waves. When estimated from the sections it is found to be much smaller than the other terms. For instance, a 2000-m-long interface displaced at 12.5 m h⁻¹ with a density difference across it of Δρ = 1 kg m⁻³ would have a rate of change of available potential energy dPE/dt ≈ ΔyΔxgη²Δρ/Δt = 0.6 MW. This is not a small accumulation of energy, but it is accounted for eventually in either the wave radiation or in the dissipation. In fact, the release of these waves may be responsible for the phase differences observed between the linear calculation and the wave flux estimates (Fig. 16).

A final energy sink is the lee vortices shed during each tide. Vortical motions do not show up in the wave flux estimates despite containing significant energy that can advect away from the sill region or dissipate on sidewalls. During peak flood tide the vortices have a kinetic energy of 2.3 × 10⁷ J m⁻¹ (Fig. 11). Multiplying this by the distance to the sill (1.6 km) gives 3.7 × 10¹⁰ J, which, distributed over the three hours of flood, gives a rate of 3.4 MW of energy being fed into the vortices. This only accounts for energy in the vortices toward the sill; they also extend an unknown distance downstream.

A simple parameterization of the vortex strength gives a time dependence to this energy sink. For flow separation around bluff bodies, the drag coefficient can be considered to be of order unity (C_d ≈ 1). Recent numerical experiments by MacCready and Pawlak (2001) indicate that this drag coefficient is appropriate for a ridge protruding into flow along a slope. The energy lost from a flow with velocity U impinging on an obstacle with a cross-sectional area A is given by P_D = 0.5C_dρU³A. For the pair of headlands we estimate an area of 2 × 500 m × 60 m to calculate energy removed from the mean flow (dotted line, Figs. 16i–l). The energy removed from the vortices dominates the other dissipative terms, peaking at 7 MW (Fig. 16j). In comparison with the estimate of 3.4 MW above, the vortices would need to be 3.2 km long at flood tide, which is not an unreasonable length. Klymak and Gregg (2001) show that the vorticity in each vortex between the sill and the landward section is 3400 kg m⁻¹ s⁻¹. Vorticity production at each headland was estimated at a rate of 1.5 kg m⁻¹ s⁻², and so a vortex with 3400 kg m⁻¹ s⁻¹ of potential vorticity could be produced in less than 0.75 h, and one 2 times as large in 1.5 h.