Observations and modeling simulations are presented that illustrate the importance of a density contrast and the upstream response to the time dependence of stratified flow over the Knight Inlet sill. Repeated sections of velocity and density show that the flow during ebb and flood tides is quite different: a large lee wave develops early in flood tide, whereas lee-wave growth is suppressed until the second half of ebb tide. There is a large upstream response that displaces as much water as accumulates in the lee wave, one that is large enough to also block the flow at a depth roughly consistent with simple dynamics. There is a large density contrast between the seaward and landward sides of the sill, and a “salty pool” of water is found in the seaward basin that is not found landward. The interface with this salty pool demarks the point of flow separation during ebb, initially suppressing the lee wave and then acting as its lower boundary. A simple two-dimensional numerical model of the inlet was used to explore the important factors governing the flow. A base simulation that included the landward–seaward asymmetry of the sill shape, but not the density difference, yielded a response that was almost symmetric with a large lee wave forming early during both flood and ebb tide. The simulation behaves more like the observations when a salty pool of water is added seaward of the sill. This salty pool induces flow separation in the model and suppresses growth of the lee wave until late in ebb. This effect is termed “density-forced” flow separation, a modification of “postwave” flow separation that allows for a density gradient across an obstacle.
This paper discusses the evolution of the stratified tidal flow over a sill in Knight Inlet, British Columbia, Canada. The problem of stratified flows over topography is a many-faceted one and arises in many geophysical situations. Tidal flows oscillate over sills and banks in the coastal ocean, creating internal lee waves that are subsequently released as tidal bores and solitons (Apel et al. 1985). Stratification interacts with underwater sills and contractions to limit the exchange of properties between adjacent basins via internal hydraulic controls (Stommel and Farmer 1953), a process that is sometimes modified by the tide (e.g., Wesson and Gregg, 1994). Stratified effects are also important in creating lee waves in nonoscillating flows; for instance, large downslope windstorms in the lee of mountain ranges are large sporadic lee waves that cause great damage to structures downwind of the mountains and exert enhanced drag on the mean atmospheric flow (Klemp and Lilly 1978). Nash and Moum (2001) observe similar lee waves where coastal currents interact with underwater banks and show that they are an important source of mixing and drag on the mean flow.
Recently, Knight Inlet has received much attention because of the Knight Inlet Sill Flows Experiment (Farmer and Armi 1999a; Klymak and Gregg 2001). Farmer and Armi (1999a) presented one of the first quantitative observations of a growing lee-wave response and therefore the first comprehensive dataset with which to test numerical models of the process (e.g., Peltier and Clark 1979; Durran 1986). In their observations they found that during ebb tide a large lee wave forms, but only during the second half of the ebb tide. Once fully formed, the lee wave has a fast jet along the topography (analagous to the downslope windstorms) bounded above by a weakly stratified, very slow wedge of fluid. They also note that there is a flow separation in the lee of the sill that they attribute to bottom boundary layer separation and that the separation drops with time. Two modeling efforts have attempted to duplicate these observations, and while both were able to achieve large lee waves, the lee waves formed too early in the tide in comparison with the observations (Cummins 2000; Afanasyev and Peltier 2001). Both studies cited the failure of their models to simulate the observed flow separation as the reason for this difference in the timing.
This paper expands on the work of Farmer and Armi (1999a) by covering other important aspects of the flow, including the whole tidal cycle, the upstream response, and the consequences of a sharp density contrast across the sill. These factors all play a role in the development of the flow but are often neglected when formulating numerical models. Models of stratified flow over isolated bodies, both in the atmosphere and the ocean, usually use the stratification from upstream of the body after the lee wave has developed (Peltier and Clark 1979; Cummins 2000); in Knight Inlet, the upstream stratification at peak flow is considerably different from the stratification before the wave develops. These numerical models also use uniform stratification across the sill, while we demonstrate below that there is a striking density contrast.
First, we compare flood and ebb tides. The tidal cycle in Knight Inlet is strong enough that the flow reverses at all depths and lee wave develops in the lee of the sill during both flood and ebb tides. A similar situation occurs in other tidally driven flows in the ocean, such as the well-studied one over Stellwagen Bank (Chereskin 1983). As we show below, in Knight Inlet the timing of the growth of the flood and ebb waves is markedly different.
Second, we discuss the response upstream of the sill. “Upstream influence” is a term that arises in the calculation of the flow over an obstacle. In an infinite depth flow with an upstream flow speed (U), a background stratification (N), and an obstacle height (h) small enough so that Nh/U ≲ 0.75, Long's solution allows us to calculate the flow over the obstacle (Long 1955). Observations show that for larger obstacles (Nh/U ≳ 0.75) the stratification and flow velocity are altered upstream in what is called upstream influence (Baines 1977). This happens when some fluid in the flow does not have enough kinetic energy to initially make it over the sill. Dense water is decelerated and accumulates behind the obstacle, exchanging kinetic for potential energy, altering the stratification and velocity arbitrarily far upstream. To predict the steady-state flow over large obstacles, an iterative approach can be employed that successively modifies the upstream condition until it can accommodate the large obstacle and still conserve mass and energy (Baines 1988). However, numerical models show that this process takes a long time to evolve (Lamb 1994; Pierrehumbert and Wyman 1985). The presence of an upstream influence complicates the choice of initial conditions in numerical models since what is needed is the stratification that would exist in the absence of the tide. In this paper, we show that the flow field is continually evolving upstream of the Knight Inlet sill during both tides and is therefore at least partially responsible for the continued evolution of the lee wave.
Third, we discuss the considerable density gradient across the sill and its influence on the growth of the lee waves. Knight Inlet is an estuary with a freshwater source at its head (the Klinaklini River) and an outlet to the ocean. The sill blocks the introduction of dense ocean water into the inner basin of the fjord. This dense fluid has been noted in previous surveys of the fjord (see Stacey et al. 1995, their Fig. 10), but has not been commented on in the recent discussions of the sill flow. We will show below that it plays an important role in both the flow separation observed during ebb tide and the difference in the timing of the lee-wave response between ebb and flood.
This paper presents a full tidal cycle of velocity and density data taken near the Knight Inlet sill and also some numerical modeling runs meant to demonstrate the importance of the upstream influence and the density contrast. We start by discussing Knight Inlet and our collection methods (section 2). Then we use average density profiles to show that there is a density contrast across the sill and to deduce some hydrodynamic properties of the flow a priori (section 3). In section 4 we present the full tidal cycle of data (with the caveat that the data are really a composite of two tidal cycles owing to data-collection limitations). We quantify the upstream response and the densest water coming over the sill and then show data details of the dense salty pool of water. Section 5 presents numerical modeling results that show that not including the salty pool in a numerical model of Knight Inlet garners a symmetric lee-wave response on either phase of the tide, whereas adding a dense pool suppresses lee-wave growth until after the lee wave drops below the sill crest. Section 6 summarizes our findings and their implications to understanding lee-wave growth.
2. The Knight Inlet experiment
The Knight Inlet experiment took place from 17 August to 14 September 1995 in Knight Inlet, British Columbia (Fig. 1). Three research vessels, the CSS Vector, the R/V Miller, and the CSS Bazan Bay, occupied the sill in concert in an attempt to observe the sill flows and the subsequent release of lee waves into propagating undular waves. All three vessels were equipped with acoustic Doppler current profilers to collect velocity, CTDs to map density versus depth (by measuring conductivity, temperature, and pressure), and GPS systems to track their positions. The boats were also equipped with high-frequency echo sounders, an invaluable tool for observing internal flow features by measuring acoustic backscatter from biology and density fine structure (Seim et al. 1995).
These vessels occupied a number of lines over the Knight Inlet sill (Fig. 2), some of which have been previously described. In particular, Klymak and Gregg (2001) show that the flow within the lee waves has a substantial three-dimensional component, likely originating from stretched vertical vortices shed from the headlands that bracket the sill. The evolution of the alongchannel flow during ebb tide was described by Farmer and Armi (1999a), while turbulence measurements in the lee of the sill are described in Klymak and Gregg (2002).
Most of the data in this paper were collected on the R/V Miller using the Shallow Water Integrated Mapping System (SWIMS). SWIMS is a CTD package designed to sample in clear water while being towed at less than 2 knots, making both up and down casts useful. In Knight Inlet, horizontal profile spacing varied with the water depth from less than 50 to near 100 m. For presentation purposes, the data were smoothed vertically by 1 m and then treated as vertical casts instead of angled casts. This is usually adequate except when sampling nearly vertical features, where the data presentation should be treated with caution. Our current profiler was an RDI 150-kHz broadband ADCP. Data from the ADCP were sparser than possible because of a data dropout problem; smoothing achieves an approximately 50-m horizontal resolution.
3. Flow parameters
The sill geometry is irregular, with a sharp seaward face, a slightly sloping plateau, and then a sharp landward face. The sill is very three-dimensional, especially on the landward side where there is a notch in the topography that steers the flood tide flow (Klymak and Gregg 2001). The seaward basin is about 150 m deep, and the sill about 60 m deep, and so the seaward sill height is hS ≈ 90 m (Fig. 3). The landward basin is much deeper, reaching 450 m deep in places, and so we can say that the landward sill height is hL > 250 m. The aspect ratio of the seaward side of the sill is AS ≈ 0.25, while the landward side is less steep with AL ≈ 0.17 through the deepest path of the channel (thalweg).
Tidal currents oscillate back and forth over this sill, driven by the rising tidal heights in the open ocean. The barotropic velocity through a section can be estimated from the rate of change of the surface height, ζ, as U ≈ (S/A) dζ/dt, where S is the surface area of the inlet up inlet of the section and A is the sectional area. The distance from the sill to the head of the inlet (100 km) is short in comparison with the barotropic wavelength (∼2400 km for a basin 300 m deep), and so the approximation is valid. Heights are given by a tide gauge 3.3 km east of the sill crest.
There is a difference in the average density of the seaward and landward basins. We compiled two sets of density profiles, one from west of −1 km, and the other from between 2.5 and 4 km. The data were averaged isopycnally to determine the average depth of isopycnals. The results, with bootstrap error bars, are presented in Fig. 4. There is a striking density gradient across the sill at all depths, with heavier water in the seaward basin. Below the sill crest there is water in the seaward basin that is denser than the water to 180-m depth in the landward basin. We will refer to this dense water as the “salty pool” hereinafter.
These mean observations raise a number of expectations. We expect that there will be an exchange flow across the sill driven by the density difference. The R/V Vector made a continuous survey that came near the sill crest often enough to estimate the net flux over a tidal cycle. The flow over the sill crest is largely barotropic (Fig. 5), but the water is more dense during flood than ebb tide. On average, there is a net 3 m2 s−1 inflow of water denser than σθ = 24.125 kg m−3 into the landward basin, balanced by a return flow out of the basin at lighter densities. The almost barotropic flow during both tides indicates that the control of the exchange is strongly affected by the tide.
The value of (Nh/U) over the sill is very large. In Knight Inlet, if we take N at the sill crest and U as the maximum barotropic tidal velocity in each basin, we get NShS/US ≈ 4.5 for the seaward basin, and NLhL/UL ≈ 25 for the landward. This parameterization may be a little naive, since the stratification is not constant in Knight Inlet and drops off so strongly below the sill crest, so we can also scale the depth coordinate by the stratification in a Wentzel–Kramers–Brillouin (WKB)-type scaling, which will tend to reduce the apparent height of the sill:
With this scaling, we still get Nh/U ≈ 4 seaward and Nh/U ≈ 13 landward of the sill. In strict terms, the cutoff value of Nh/U depends on the ratio of the obstacle height to the water depth h/D and is different than the value of 0.75. Again we are hampered by a lack of theory for nonuniform stratifications. However, if we use the WKB scaling we get h/D ≈ 0.2 in the seaward basin and h/D ≈ 0.35 in the landward basin. For his “semi-infinite” experiments, Baines (1995) used a value of ≈0.3 (see Fig. 17 below). Therefore the high value of Nh/U puts Knight Inlet sill flow well into the range at which we would expect wave breaking and upstream influence to be important, an expectation that is borne out by the time-dependent observations below.
4. Time-dependent response
Our data were collected while the R/V Miller crossed over the sill during flood and ebb tide (Fig. 6). It took 30–45 min to complete each transect over the sill, with most runs extending 1.3 km seaward and 2.5 km landward of the crest. We did not have continuous use of the boat for 12 hours, and so the picture in Fig. 6 was made using two days of data, with the break during ebb tide (between 6f and 6g). The two ebb tides are qualitatively similar and had similar barotropic tidal forcing but were taken seven days apart on either side of the fortnightly tidal cycle. The results compare well with the continuous series presented by Farmer and Armi (1999a), and the flow at the two high tides is very similar (Figs. 6a,b,q), justifying this composite. Our CTD failed for part of Fig. 6g.
The most prominent feature of the internal tidal cycle is the lee wave that forms downstream of the sill crest during both ebb and flood tides. During ebb, the wave starts to form 2.5 h into the tide with a small wedge between σθ = 21 kg m−3 and σθ = 23 kg m−3 (Fig. 6f). The lee wave grows as the ebb tide strengthens, first with a steepening leading face and a thickening of the layer between σθ = 21 kg m−3 and σθ = 23 kg m−3 (Fig. 6h), and then with a thickening of deeper isopycnals between σθ = 23 kg m−3 and σθ = 24 kg m−3 (Fig. 6i), before abruptly collapsing at slack tide. A lee wave forms on the landward side of the sill shortly thereafter during flood tide (Fig. 6l), but this time it is first apparent in deeper isopycnals (σθ = 24 kg m−3). The flood lee wave grows and encompasses shallower isopycnals, reaching its largest size at peak flood tide (Fig. 6n), after which it collapses and propagates seaward during slack water (Fig. 6q).
a. Upstream response
The first aspect of the flow worth noting is the upstream response during both flood and ebb tides. The progression can be seen in Fig. 6. At slack tide (Fig. 6b) isopycnals are all near their average depth in the landward basin. As the tide progresses all isopycnals deeper than σθ = 17 kg m−3 are raised above their average depth as far landward as our measurements extend. A slowly moving wave crest can be seen along the σθ = 24 kg m−3 isopycnal progressing upward and landward with time. This disturbance continues until slack tide (Fig. 6i) at which time it starts to relax before rapidly collapsing (Fig. 6j). The upstream disturbance also has a clear velocity signal, with deep water slowed and the mid water column accelerated. The sequence is repeated for flood tide (Figs. 6l–q), with all isopycnals below σθ = 17 kg m−3 raised in the seaward basin before abruptly collapsing at slack tide.
We quantify the upstream response by considering the excursion of the σθ = 24 kg m−3 isopycnal upstream of the sill during ebb and during flood. This isopycnal is near the mode-1 zero-crossing, and so its excursion could be likened to the interface displacement in a two-layer flow. During ebb tide the landward side of the sill is the upstream side. Two kilometers landward there is a 20-m excursion of the σθ = 24 kg m−3 isopycnal (Fig. 7) continuing to build until late in the ebb tide, when it suddenly collapses during the transition to flood tide. During slack tide the isopycnal rapidly drops over 40 m, and during flood it starts a slow climb back up to its mean depth. The response at 2 km is larger than the response far upstream because of its proximity to the sill crest; stations farther landward show smaller excursions of 10 m. Some of this difference is because the upstream disturbance has not fully evolved during the tide as compared with the steady-state case; some is because the response over the sill is local.
During flood tide, the seaward basin is upstream of the sill. A station at −0.8 km shows elevations of the σθ = 24 kg m−3 isopycnal between 15 and 20 m (Fig. 7, seaward). As during ebb tide, the upstream wave during flood tide shows an asymmetry with the forcing, getting higher earlier in the flood tide, and then staying that way until just before the low water when it drops abruptly.
The velocity field is also altered upstream during both flood and ebb tides. Water at mid depths is accelerated relative to the barotropic velocity, while deeper water is decelerated. As an example, consider any point upstream of 1 km in Fig. 6h; the velocity profile is almost stagnant below 70 m and reaches peak velocity near depth 20 m. The shear field is superimposed on a barotropic acceleration caused by the constriction (both lateral and vertical) of the inlet near the sill.
The upstream response is strong enough that there is blocking of the densest water in the flow (e.g., “partial blocking”: Farmer and Denton 1985). The densest water at the sill crest changes with time as plotted in Fig. 8. During low tide the density of the water at the sill crest is about σθ = 24.2 kg m−3. This density increases until peak ebb tide when it reaches between σθ = 24.6 kg m−3 and σθ = 24.8 kg m−3. It stays high until just the end of flood tide when the water at the sill crest starts to get lighter again. The drop in density continues well into ebb, particularly during spring tides when the densest water does not reach σθ = 24.2 kg m−3 again until peak ebb tide.
The depth from which water is withdrawn can be estimated by comparing the maximum density at the sill crest with the background densities from the upstream basin, seaward for flood and landward for ebb (bottom panel, Fig. 8). The interpolation is meaningless when there is an exchange flow over the sill, but becomes valid past peak ebb. For both tides, water is withdrawn from as deep as 125 m, more than 60 m below the sill crest, and the deepest withdrawal is just after peak tide in both cases. The maximum depth of withdrawal corresponds quite well to the steady-state depth of withdrawal, given by Dmax = Umax/N (de Young and Pond 1988). For Dmax = 65 m and Umax = 0.7 m s−1 at the sill crest, we calculate N = 0.01 s−1, which is the stratification near the sill crest for both basins. However, the nonsinusoidal response of the densest water means that this simple relationship does not apply instantaneously.
b. Flood–ebb asymmetry
A second aspect of the flow worth noting is the flood–ebb asymmetry of the lee-wave response. The asymmetry can be seen in Fig. 6, with a delayed and smaller lee wave during ebb as compared with flood tide. A way to quantify the internal response is to consider the time dependence of the form drag on an obstacle. The formal definition of form drag is
where P is the pressure along the obstacle and h is the obstacle height (Baines 1995). This number reflects the net drag over the obstacle that results from the asymmetry of the pressure field from one side to the other. (In the language of shallow-water hydraulics: sub- and supercritical flows over symmetric obstacles have drags of F = 0 because even though the interface is deflected, it does so symmetrically over the obstacle; for a transcritical flow |F| > 0). This formal definition of form drag is very difficult to use with our data because we do not know the surface pressure gradient and because our coverage of the density field varies horizontally and in the depth of individual casts. We reformulated the form drag by estimating the baroclinic pressure at the depth of the sill (60 m) and multiplying it by the slope of the sill. The integration was carried out between −3 and 3 km. The resulting number is not the drag on the flow exerted by the sill, but it does gives a single measure of the asymmetry of the isopycnals over the sill. Calculating this “pseudodrag” parameter for all the passes made over the sill using SWIMS yields the time history shown in Fig. 9. We also perform this calculation in the numerical model below and demonstrate that it a good measure of wave growth.
The timing of the drag response in Fig. 9 is quite regular through both neap and spring tides. At high water, the response starts near zero and slowly becomes more negative. During neap tides (light “x”s), it reaches −0.5 by peak ebb and then levels out until just before low tide, when it drops back to zero. During spring tides (dark “x”s), the response continues to strengthen (more negative) past peak ebb forcing, and then starts to drop back toward zero, though it is still negative at low tide. In both cases, the ebb response is asymmetric with the forcing, with stronger drags later in the tide. The limited data indicate that flood tide response is more symmetric, rising earlier in the tide and reaching its highest drag just a little past peak flood tide. However, the response is still asymmetric, ramping up to the maximum value rather than growing sinusoidally.
There is a strong density gradient across the sill, which means that during ebb tide the water passing seaward over the sill impinges on a pool of denser water. The dense pool of water has the ability to suppress the growth of lee waves, and so its time-dependent behavior is of considerable interest. For the purposes of discussion we will consider water denser than σθ = 24.25 kg m−3 to be the salty pool, corresponding to the densest water passing seaward over the sill during ebb tide (Fig. 8).
During flood tide the dense pool is lifted over the sill crest and intrudes into the landward basin as a density current (Fig. 6o). We did not follow the fate of this density current. As the tide reverses the dense inflow is slowed and eventually stopped as the salty pool is pushed down and seaward (Figs. 6a,b). An hour into ebb tide (Fig. 6d), the situation is as pictured in Fig. 10. There is a point 1.2 km east of the sill crest where the ADCP shows arrested flow below σθ = 24.25 kg m−3 with strong seaward flowing velocity above. The shear between these two distinct layers creates shear instabilities visible in the echo sounder. The shear layer originates from the sill at 1.2 km (marked “separation point” in Fig. 10).
The situation is very reminiscent of the behavior of a salt wedge in a river (Geyer and Smith 1987). As with a salt wedge, stronger tidal forcing pushes the salty pool farther seaward (Figs. 6d–f). The shear layer at the top of the salty pool can also be seen in Fig. 7i of Farmer and Armi (1999a) originating 0.8 km landward of the sill crest. Finally, at peak tide the pool is pushed below the sill crest, shown in detail in Fig. 11. The seaward-flowing water lighter than σθ = 24.25 kg m−3 encounters dense water west of the sill and separates from the topography. The salty pool itself is quiescent and well stratified. The shear line between the salty pool and light water has developed shear instabilities.
This separation point continues to drop with time as the tide progresses for two reasons: First, there is a divergence in the dense pool at the sill with strong seaward velocities, but no source of water from landward, causing the interface to drop. Second, once it has dropped below the sill crest, the water above develops a downward momentum that deflects the interface. This causes a rebound on the interface that can be seen in Figs. 6f–i and is usually about 10 m in amplitude. The size of this rebound is approximately related to the stratification and the vertical velocity by h ≈ w/N = 0.25 m s−1/0.02 s−1 = 12 m.
In summary, the salty pool
is denser than the water passing over the sill during ebb tide,
is not pushed back to the sill crest until peak ebb forcing (3.1 h into the ebb tide), and
is not turbulent below its upper interface.
We will discuss the importance of these properties below, but first we consider a pair of numerical simulations.
5. Numerical modeling
We contrast the observed internal response with that in a numerical model of an oscillatory tide over an obstacle similar to the Knight Inlet sill. The model used was the Hallberg Isopycnal Model (Hallberg 2000), a hydrostatic nonlinear layered model in which the equations of motion are solved in isopycnal layers of fluid rather than a spatial grid. We used a horizontal grid that varied from 2-m resolution near the sill to 200 m far from the sill. Twenty-five vertical layers were used with uneven stratification and thicknesses. The model was typically run for 36 h with a tidal forcing of the flow. Internal waves were absorbed 70 km away from the sill with a simple Raleigh damping that gradually increased in strength with distance away from the sill. There is a horizontal Laplacian smoothing of thickness and velocity, and a Richardson number–dependent vertical mixing. The dynamics of the free surface could be solved in the model with either a time-splitting scheme or explicitly; both schemes were tested with no discernable change to the internal dynamics, and so the faster time-splitting scheme was used for the model runs below. The bottom boundary condition was set to free slip.
In our first simulation, the stratification on each side of the sill was the same. Despite the asymmetry of the sill shape, the response during both flood and ebb was essentially identical (Fig. 12). A large lee wave is created during both phases of the tide; first the densest isopycnals steepen along the downstream face of the wave, followed by the isopycnals above. There is a large upstream disturbance and blocking of the densest water. The lee wave collapses at slack tide, with the less-dense water propagating upstream first, followed by denser water. The results of this simulation are very similar to those made by Cummins (2000), in which the internal response was quite rapid and almost in phase with the external forcing. We evaluate this quantitatively using the form drag, this time from the precise definition since the numerical model has all the relevant information to make the calculation (Fig. 13). From it, we see that the response slightly lags the forcing, similar to the observed flood tide. For comparison, the pseudodrag (2) evaluated at 60 m is also shown. The surface elevation was removed from the model and the pressure at 60 m was calculated as described above. The result shows that the pseudodrag calculation underestimates the drag during flood tide but does an adequate job during ebb. Most important, it preserves the phase information of the internal wave signal during both phases of the tide.
In our second simulation we included a density contrast and salty pool on the seaward side of the sill. The result captures the asymmetry of the lee-wave response observed in the data (Fig. 14). Like the first simulation, a large lee wave develops early in flood tide and persists until just before slack tide. During ebb tide the lee-wave growth is delayed by the addition of the density difference. An exchange flow persists for about an hour past slack tide (1240 model time). At 1340 the dense water has been pushed to the sill crest. Lighter water above is accelerating as the isopycnals landward of the sill are lifted, but a lee wave does not form. There is a small lee wave that forms by 1440 but its vertical extent is limited by the dense water at the sill crest. The dense water is dropping, however, and the flow separates from the topography about 10 m below the sill crest. This dense pool interface continues to drop and the lee wave continues to get larger, until at the end of the ebb tide (1640) it is at its maximum extent.
The drag history of this run shows the flood–ebb asymmetry (Fig. 15). Flood tides develop in phase with the forcing, with the peak drag occurring slightly before peak flood, settling into a steady state for 2 h and then relaxing. Ebb tide develops more slowly, not reaching peak drag until over an hour past ebb. This compares well with the drag history observed in the data (Fig. 9).
There are, however, marked differences between the observed and simulated responses of the flow. The salty pool is pushed seaward faster in the numerical model then it is in the data. The separation point is below the sill crest by 1440, about an hour earlier than in the data, and the ebb lee wave grows earlier in the simulation. The simulation is also very sensitive to the choice of the stratification. If the interface of the seaward layer is made a few meters shallower, it can completely suppress the growth of the lee wave, a result similar to the one Cummins (2000) simulated using a seaward channel 70-m deep. If the interface is a few meters deeper, the salty pool is easily flushed downstream and the ebb-tide lee wave grows earlier and is larger. We suspect that these differences are a consequence of the model being two-dimensional, whereas Knight Inlet is very three-dimensional. The inlet is about 1.7 km wide at the sill crest but widens dramatically to over 2.5 km less than 1 km seaward (ignoring the additional complication of a shallow bay to the north). The widening means that more water in the salty pool must be pushed seaward in the real inlet than in the simulated one. The additional inertia may account for the slower response in the observations and mean that the flushing of the seaward layer is less sensitive to changes of thickness.
The data in this paper present observation of a full tidal cycle of flow over the Knight Inlet sill. The observations show a large asymmetry in the lee-wave response between flood and ebb tide, with a large lee-wave response early in flood, and a delayed, smaller-amplitude response during ebb. Because they were made over a large depth and horizontal range, the observations allowed us to observe a large upstream response and the behavior of the dense salty water trapped seaward of the sill. Both observations have important implications for understanding tidal sill flows.
a. Upstream response
The presence of an upstream response during both phases of the tide in Knight Inlet is expected from laboratory and numerical studies. The value of Nh/U greatly exceeds the critical value of ≈0.75. This upstream response has been observed qualitatively by Farmer and Smith (1980) but has not been quantified before.
The upstream response raises isopycnals upstream of the sill, increasing the stratification of the middepth water column. This compression of the mid water column upstream of the sill can be seen by comparing Fig. 16a with Fig. 16d where light gray fluid has been displaced by dark gray. This displacement is large when compared with the accumulation of water in the lee wave; between the two panels 1100 m2 of water between σθ = 23 kg m−3 and σθ = 24 kg m−3 has been displaced from the landward basin, while only 800 m2 has accumulated in the seaward basin. The sudden widening of the isopycnals in the lee wave is often characterized as a “split” (Smith 1985), or a “bifurcation” (Farmer and Armi 1999b). While an accurate description of the flow in Fig. 16d, splitting implies a sudden creation of the water in the wedge, leading to the question of where the water comes from. It has typically been ascribed to mixing, either in the breaking of the wave in the lee (Peltier and Clark 1979) or in shear instabilities along the leading edge of the wave (Farmer and Armi 1999a). These observations show that there is plenty of water upstream of the sill of the right density to form the lee wave without needing mixing. How water gets into the lee wave is still problematic. In comparing Fig. 16h and Fig. 16i, it is seen that the lee wave (between σθ = 23 kg m−3 and σθ = 24 kg m−3) suddenly widens, without an apparent contraction upstream; in fact, there is an expansion upstream as well as in the lee wave. This may indicate that at this stage mixing is the source of the water, or it may point to the importance of three-dimensional effects in the lee wave (Klymak and Gregg 2001). The point is that sufficient water already exists in the lee wave and that the split may be more accurately explained as a squeezing.
A second important effect of the upstream evolution is that it is slow to evolve so that the flow cannot be called “quasi steady.” Quasi-steady means that wave motions adjust much faster than the forcing changes so that the wave field at any given time is the same as the one predicted by assuming the flow is in steady state: the condition is the frequency of the tidal forcing ω is much less than the buoyancy frequency N (Bell 1975), which is true in Knight Inlet (ω/N ≈ 10−2). The changing upstream condition, however, means that the forcing on the lee flow changes slowly. Numerical model runs indicate that a large upstream response takes a long time to evolve, much longer than a few buoyancy periods (Pierrehumbert and Wyman 1985). This lag leads to the asymmetry of the response observed Fig. 9. We ran a version of our numerical model with an idealized topography and a Knight Inlet stratification in which the flow was impulsively started to full tidal velocities. It took over 4 hours to approach a steady state—long in comparison with the tidal forcing (Klymak 2001).
Last, the strength of the upstream changes indicates that care should be taken when initializing numerical models. Numerical models need to specify an initial stratification. In the atmosphere this has usually been done with density profiles from upstream of the obstacle at the time the lee wave is observed (Peltier and Clark 1979). Similarly, Afanasyev and Peltier (2001) and Cummins (2000) used profiles from upstream of the Knight Inlet sill during peak ebb tide after the upstream response had developed. This means that the stratification used was too high at almost all depths over the sill and has an initial depth of the σθ = 24 kg m−3 isopycnal that is 15 m too high. We have not performed numerical experiments to determine how important the difference in the stratification is to the timing of the lee-wave development. However, it seems reasonable to use the average stratification rather than one that develops after the upstream influence has passed. Even if the timing of the lee-wave growth does not change, the character of the wave will. For instance, the squeezing effect mentioned above may be much less pronounced.
b. Importance of the salty pool
We believe that the asymmetry between flood and ebb is due to the density difference between the seaward and landward basins. In the absence of the tides, the density difference would drive an exchange flow. For a very submaximal flow, such as we expect here, the two-dimensional exchange flow can be found using open-channel hydraulic theory1 to be 4.4 m2 s−1. At the sill crest the lower layer would be 12.5 m thick with a velocity of 0.35 m s−1. This velocity is comparable to those found at low slack tide (Fig. 6a). The barotropic tidal velocities at the sill crest are upward of 0.7 m s−1, and so they overcome the exchange flow, but the exchange flow cannot be ignored in the dynamics.
There is a density contrast at all depths, most obviously manifested in the salty pool of water in the seaward basin. At slack tide the salty pool starts as an exchange flow (Fig. 16a) that becomes cut off, similar to a salt wedge (Geyer and Smith 1987) or a box flow (Farmer and Armi 1988). The salty pool is pushed seaward as the tidal forcing increases, but does not reach the sill crest until peak ebb forcing. As long as the salty pool is over the sill crest, it is impossible for a lee wave to develop.
Past peak ebb forcing the salty pool continues to act as a lower boundary to the ebbing flow. The flow separates from the sill during ebb tide, an effect first observed by Farmer and Smith (1980), and observed again by Farmer and Armi (1999a). Both papers attributed this flow separation to bottom boundary layer separation. However, flow can separate because it encounters water denser than itself. This is discussed in detail by Baines (1995) who identifies two flow separation regimes. The first, for low Nh/U and sharp obstacles, is boundary layer separation (Fig. 17a). The other, which he terms “postwave separation,” is for large Nh/U and gentler obstacles (Fig. 17b). Based on the criteria in Baines (1995, his Fig. 5.8), Knight Inlet falls into this second regime (a = h/W ≈ 0.25 and Nh/U > 4). The observations support this: the flow separation is along a density interface Fig. 11 above a quiescent, well-stratified salty pool. If boundary layer separation was the dominant effect, then we would expect the water beneath the flow separation to be turbulent and overturning (Fig. 17a). However, the term postwave separation does not appear applicable to Knight Inlet. The different Nh/U on either side of the sill means that during ebb tide the wave is suppressed for some of the tide. We propose that “density forced” separation may be a more general name for this type of flow separation. Only as the ebb tide flushes the salty pool downstream, does the flow take on a postwave character.
Once the salty pool is pushed seaward of the sill crest it will continue to drop, even in the absence of forcing from the flow above. The seaward volume flux in the salty pool is not being matched at the sill, causing the salty-pool interface to deflect downward. As the interface drops the downslope fluid is free to drop further, again expanding the lee wave.
The numerical model demonstrates the importance of the salty pool to the flow. In the model, the salty pool acts as a lower boundary on the ebb flow. It suppresses the lee wave until past peak ebb tide. Why it is pushed seaward faster than in the data is not clear, though the widening of the inlet to the seaward seems a reasonable explanation. The numerical model is free slip and therefore does not produce a bottom boundary layer, but there is still a density-forced flow separation. This does not rule out the importance of boundary layer physics. However, we have attempted to argue here that the flow separation is always accompanied by a density interface, which is not a characteristic of boundary layer separation. A definitive answer of the relative importance of the two effects would require better measurements of the boundary layer flow and pressure fields than are available.
An analagous situation has been considered in the atmosphere. Simulations of the flow over the Rockies initialized with uniform cross-range stratification lead to a severe overprediction of downslope windstorms and too many false severe-wind warnings in Boulder, Colorado (Nance and Colman 2000). Lee et al. (1989) suggested that the overprediction is due to the presence of dense cold pools downstream of the Rockies that prevent the formation of large lee waves. They tested this hypothesis numerically by including a cold pool of air east of the Rockies and found that it suppressed the formation of lee waves. Subsequent strengthening of the upstream flow can flush the dense pool away from the mountain, allowing downslope flows to occur; however, the onset of strong downslope flows is delayed in comparison with simulations without the dense pool. This is similar to the ebb tide in Knight Inlet where the strong lee wave cannot form until the dense water is flushed seaward of the obstacle. To our knowledge, the observations presented in this paper are the first documenting the importance of a downstream dense pool in suppressing lee-wave growth.
We thank those whose expertise made the collection of these data possible: Jack Miller, Earl Krause, Steve Bayer, and the master of the R/V Miller, Eric Boget. David Farmer and Eric D'Asaro kindly provided data. The UW School of Oceanography generously supplied time on a pair of Compaq ES40 supercomputers. Robert Hallberg and David Darr gave freely of their expertise in using the numerical model HIM. We are grateful to the U.S. Office of Naval Research for financial support under budget numbers N00014-95-1-0012 and N00014-97-1-1053, and for the SECNAV/NCO Chair in Oceanography held by MG.
Current affiliation: College of Oceanic and Atmospheric Science, Oregon State University, Corvallis, Oregon
Corresponding author address: Dr. Jody M. Klymak, College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97330. Email: email@example.com
This follows from single-layer hydraulics assuming that the interface at σθ = 24 kg m−3 has a g′ = 0.01 m s−2, an open reservoir height of 115 m, and is blocked by a sill of 90 m.