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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 33
The Role of Upstream Waves and a Downstream Density Pool in the Growth of Lee
Waves: Stratified Flow over the Knight Inlet Sill
JODY M. KLYMAK*
AND
MICHAEL C. GREGG
Applied Physics Laboratory and School of Oceanography, University of Washington, Seattle, Washington
(Manuscript received 2 May 2001, in final form 3 December 2002)
ABSTRACT
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 ‘‘densityforced’’ flow separation, a modification of ‘‘postwave’’ flow separation that allows for a density gradient across
an obstacle.
1. Introduction
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
* 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.
E-mail: [email protected]
q 2003 American Meteorological Society
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
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KLYMAK AND GREGG
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
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FIG. 1. Knight Inlet: a long sinuous fjord 300 km north of Vancouver on the British Columbia coast. The study site is a sill halfway
along the length. The inlet opens into Queen Charlotte Strait, and
eventually the Pacific Ocean, and has a glacier-fed river at its head.
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 leewave 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,
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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
FIG. 2. The 60-m-deep sill separates an ocean basin 175 m deep
from a landward basin 400 m deep. The topography is three-dimensional with headlands on either side of the channel that extend underwater. The seaward face of the sill is abrupt and less three-dimensional than the landward face, which has a deep, southwardtending notch. The data presented in this paper were mostly collected
in east–west lines made in the center of the channel, near the dashed
line. The deepest part of the channel is indicated with a dot–dashed
line labeled thalweg. The origin is at the sill crest: 50.6758N,
126.018W.
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
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 h S ø 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 h L . 250 m.
The aspect ratio of the seaward side of the sill is A S ø
0.25, while the landward side is less steep with A L ø
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, z, as U
ø (S/A) dz/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 21 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 m 2 s 21 inflow of
water denser than s u 5 24.125 kg m 23 into the landward
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KLYMAK AND GREGG
FIG. 3. Stratification on either side of the sill and the profile of the sill. The right- and leftmost plots are of the mean
buoyancy frequency in each basin (see Fig. 4 for how the averages were formed).
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
N S h S /U S ø 4.5 for the seaward basin, and N L h L /U L ø
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:
z WKB (z) 5
E
0
z
N(z9)
dz9.
Nsill
(1)
With this scaling, we still get Nh/U ø 4 seaward and
Nh/U ø 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 ø 0.2 in the seaward
basin and h/D ø 0.35 in the landward basin. For his
‘‘semi-infinite’’ experiments, Baines (1995) used a value
of ø0.3 (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
FIG. 4. The average density profiles (s u ) seaward (x , 21 km)
and landward (2.5 , x , 4 km) of the sill. Standard deviations are
derived by using a bootstrap method. The inset figure shows the full
density range from about s u 5 7 kg m 23 to s u 5 25 kg m 23 . The
depth of the averages is limited by the depth of our CTD data.
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
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FIG. 5. (a) Time series of net flux as measured by CSS Vector, derived as a vertical sum of data
in (b). The smoother thick line is the volume flux estimated from the tide gauge scaled to match the
two-dimensional flux. (b) Raw velocity and density time series at the sill crest. Note the dense water
intrusions at flood. (c) Net flux as a function of isopycnals. Positive is a net landward flux.
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 s u 5 21 kg m 23 and s u 5 23 kg m 23 (Fig.
6f). The lee wave grows as the ebb tide strengthens,
first with a steepening leading face and a thickening of
the layer between s u 5 21 kg m 23 and s u 5 23 kg m 23
(Fig. 6h), and then with a thickening of deeper isopycnals between s u 5 23 kg m 23 and s u 5 24 kg m 23 (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 (s u 5 24 kg m 23 ). 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 s u 5 17 kg m 23 are raised above their
average depth as far landward as our measurements extend. A slowly moving wave crest can be seen along
the s u 5 24 kg m 23 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 s u 5 17 kg m 23 raised in the seaward basin before
abruptly collapsing at slack tide.
We quantify the upstream response by considering
the excursion of the s u 5 24 kg m 23 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 twolayer 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 s u 5 24 kg m 23 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 20.8 km shows elevations of the
s u 5 24 kg m 23 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
JULY 2003
KLYMAK AND GREGG
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 s u 5 24.2 kg m 23 . This density increases until
peak ebb tide when it reaches between s u 5 24.6 kg
m 23 and s u 5 24.8 kg m 23 . 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 s u 5 24.2 kg m 23 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 5 Umax /N (de Young and Pond
1988). For Dmax 5 65 m and Umax 5 0.7 m s 21 at the
sill crest, we calculate N 5 0.01 s 21 , 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
E
1`
D5
2`
P
]h
dx,
]x
(2)
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 5 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
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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 23
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
20.5 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 s u 5 24.25 kg
m 23 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 s u 5 24.25 kg
m 23 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
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FIG. 6. Data taken from the Knight Inlet sill during ebb tide: (a)–(f ) from 2025 to 2354 UTC 25 Aug
1995, and (g)–(k) 1443 UTC 31 Aug to 0049 UTC 1 Sep. The four darkest isopycnals correspond to s u
5 17, 23, 24, and 24.25 kg m 23 . The average position of these four isopycnals (see Fig. 4) is plotted with
thick gray lines. Thin isopycnals are chosen from the density every 5 m in the background seaward
stratification (Fig. 3). At the top of each plot, the time at each end point in the transect is shown in hours
after high tide, followed by the tidal flux (10 3 m 3 s 21) (calculated from the tide gauge; Freeland and Farmer
1980).
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 seawardflowing water lighter than s u 5 24.25 kg m 23 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 ø w/N 5 0.25
m s 21 /0.02 s 21 5 12 m.
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KLYMAK AND GREGG
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FIG. 6. (Continued ) The ocean is west of the sill, to the left in all plots. The cartoon ship is pointing
in the direction R/V Miller was cruising. The last panel (next page) shows the ship tracks over the sill
and has a colorbar for reference to the velocity data in the other plots: (a)–(f ) follow the track that runs
straight east–west, and (g)–(q) follow the track that veers slightly to the south. East–west discretization
of the tracks is where the 50-m ADCP data grid was made.
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
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 33
FIG. 6. (Continued )
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 timesplitting 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 ob-
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KLYMAK AND GREGG
FIG. 7. Time series of the depth anomaly of the s u 5 24 kg m 23
isopycnal 20.8 km seaward of the sill crest and 2 km landward.
Seaward the isopycnal’s average depth is 30 m and landward it is 51
m, taken from Fig. 4. Sinusoids are plotted with an approximate
matching amplitude. If replotted with the density average between
1.5 and 2.5 km instead of the basin average, the 2-km curve is moved
down by 5 m.
served 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.
1455
FIG. 8. (a) Densest water at the sill crest vs tidal phase. (b) Depth
from which this water comes from in the average density. Both signals
are very noisy because the densest water measured near the sill depends on the depth of the SWIMS casts relative to depth of the sill
crest; we typically came within 3 m of the bottom, but the closest
cast could be as far as 50 m up- or downstream of the sill crest.
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
FIG. 9. Time series of pseudodrag (as described in the text) vs tidal
phase for all SWIMS runs made over the sill. Note the asymmetric
internal response during ebb tide vs the symmetric one during flood.
Tidal phase is with respect to the barotropic tidal flux as measured
by the tide gauge. Dashed line is a sinusoid for reference.
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FIG. 10. Close-up of the flow over the sill one hour past high tide.
(a) Velocity field (colored) and density field (contoured, isopycnals
denser than s u 5 24 kg m 23 are contoured every s u 5 0.25 kg m 23 .
Light sawtooth lines are approximate path taken by the CTD. (b)
Echo-sounder image with bright red representing stronger acoustic
backscatter than blue.
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 threedimensional. 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.
VOLUME 33
FIG. 11. As in Fig. 10 but at peak ebb tide.
6. Discussion
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 leewave response early in flood, and a delayed, smalleramplitude 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 ø0.75. 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
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KLYMAK AND GREGG
1457
FIG. 12. A tidal cycle of a numerical model run using realistic topography and velocities and a single stratification throughout the model
domain. Note that the responses of flood and ebb tides are almost identical. Every second isopycnal layer used in the simulation has been
plotted, and the s u 5 21 kg m 23 , s u 5 23 kg m 23 , and s u 5 24 kg m 23 isopycnals have been plotted with thicker lines. The simulation
time is shown above each panel, and Q refers to the two-dimensional tidal flux (m 2 s 21 ).
FIG. 13. (a) Time series of volume flux in Fig. 12. (b) Time series
of the form drag showing an almost symmetrical response between
flood and ebb. The drag at rest has been removed from the signal.
Dashed line is pseudodrag [Eq. (2)].
the two panels 1100 m 2 of water between s u 5 23 kg
m 23 and s u 5 24 kg m 23 has been displaced from the
landward basin, while only 800 m 2 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 s u 5 23
kg m 23 and s u 5 24 kg m 23 ) 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.
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VOLUME 33
FIG. 14. Numerical model run similar to Fig. 12 except with a density contrast and a salty pool similar to that observed in the data.
Every second layer is contoured, and dark contours correspond to s u 5 21, 23, 24 kg m 23 .
FIG. 15. As in Fig. 13 but for simulation with unequal density on
either side of the sill.
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 v is
much less than the buoyancy frequency N (Bell 1975),
which is true in Knight Inlet (v/N ø 10 22 ). 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
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KLYMAK AND GREGG
1459
FIG. 16. Summary of the response during ebb tide. Isopycnals are from the data and are contoured
at su 5 17, 23, 24, 24.25 kg m 23.
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 s u 5 24 kg m 23
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 leewave 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 twodimensional exchange flow can be found using openchannel hydraulic theory1 to be 4.4 m 2 s 21 . At the sill
crest the lower layer would be 12.5 m thick with a
velocity of 0.35 m s 21 . 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 21 ,
1
This follows from single-layer hydraulics assuming that the interface at s u 5 24 kg m 23 has a g9 5 0.01 m s 22 , an open reservoir
height of 115 m, and is blocked by a sill of 90 m.
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 5 h/W ø 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
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FIG. 17. Two laboratory experiments showing (a) boundary layer
flow separation where the flow separation takes place at the sill crest,
and (b) postwave separation, where the flow separation is farther
down the sill face (Baines 1995, p. 231). Note the turbulence and
overturning below the separation point in (a).
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
VOLUME 33
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.
Acknowledgments. 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.
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