Tidal interaction of stratified flow with a sill

Transcription

1 Deep-Sea Research, Vol. 27A, pp. 239 to /80/ $02.00/0 "~" Pergamon Press Ltd Printed in Great Britain Tidal interaction of stratified flow with a sill in Knight Inlet DAVID M. FARMER* and J. DUNGAN SMITH t (Received 22 May 1979; revision received 23 October 1979; accepted 16 October 1979) Abstract--Observations of tidally driven flow of stratified water over a sill in Knight Inlet, British Columbia, have revealed a broad variety of different interactions depending upon the degree of stratification and the strength of tidal forcing. The flows are described on the basis of a Froude number dependence relating the barotropic tidal velocity to the long internal wave speed over the sill crest. For flows critical or supercritical with respect to all internal modes, an internal hydraulic jump occurs. However, typical summer conditions result in flow that is subcritical with respect to the lowest mode but supercritical with respect to higher modes; the interaction in this case results in a massive mode 2 lee wave or jump accompanied by instabilities up to 50 m high. Early in the tidal cycle the boundary layer separates from the sill and apparently suppresses lee wave formation, but this effect is inhibited at higher speeds. A further, unexpected finding is the generation of nonlinear mode 1 internal wavetrains upstream of the sill during strong tides. While these waves are similar in appearance to those predicted by LEE and BEARDSLEY (Journal of Geophysical Research, 79, , 1974), there are important differences in the relative location and timing of their formation. The observations reveal the way in which lee waves and jumps relax as the tidal flow slackens, subsequently evolving into travelling internal bores or surges. The observations are compared with recent laboratory studies. INTRODUCTION WE HAVE recently acquired observations of a broad variety of phenomena associated with the advection of stratified water across a sill by tides. Although the study of stratified flow over an obstacle has a long history, it is only recently that detailed experimental studies of such flows have been carried out in an oceanographic context. In particular, two investigations, one in Massachusetts Bay (HAuRY, BR1SCOE and ORR, 1979) and our own in Knight Inlet have focused on the interaction between tidal flow and bottom topography and the structure of the surges that are generated. In this preliminary account we discuss the principal types of response that we have observed and relate some of our observations to the results of recent laboratory experiments. Knight Inlet is a 102-km long 0ord on the British Columbia coast (Fig. 1). It has a maximum depth of 500 m, an average width of about 3 km, and two sills, the innermost of which is 74 km from the head and 63 m deep. As a consequence of this geometry and of the quite large tides (range of 3 to 5 m), tidal currents over the inner sill can exceed 0.8 m s- t. The summer density structure in Knight Inlet is typical of many 0ords on the British Columbia coast, having a sharp pycnocline at 5 to 10 m along much of its length, separating the relatively undiluted river water from more weakly stratified and more saline fluid beneath. The strength and distribution of the stratification play a crucial role in determining the dynamics of flow over the sill. Figure 2 shows representative examples of density profiles in August and November, together with the first three eigenfunctions calculated for long internal waves. The * Institute of Ocean Sciences, Patricia Bay, Box 6000, Sidney, B.C., Canada. t Department of Oceanography, University of Washington, Seattle, Washington, U.S.A. 239

2 240 DAVID M. FARMER and J. DUNGAN SMITH COLUMBIA "~ " linaklini m x\ R. PACIFIC m~kn/ght INLET ~ Fr anklin R. CA.AOA 5/ +5,., "-...~ Q- --U. S.A. ~ 126= 20~ W Hoeya Head sill --200m :,!v?.." ~.~'~ rj~ :~' Depth Profile ~... ~.~:?..: ~' Fig. I. Map of Knight Inlet, showing location of inner sill, near Hoeya Head, responsible for the generation of internal waves and jumps. Most of the fresh water enters the fjord from the Klinak ini and Franklin rivers at its head. eigenfunctions were calculated for waves having the M E tidal period, using a shooting method with integration steps at 0.5-m intervals. It is apparent that in summer the strong surface stratification acts as a lid, taking little part in mode 2 oscillations associated with large vertical movements at depth. In contrast the mode 1 oscillations involve maximum vertical excursions relatively close to the surface. Not surprisingly, below the top few meters, a mode 2 eigenfunction for the summer is very similar to a mode 1 eigenfunction for November. We might therefore expect a large disturbance at depth, such as flow over a sill, to excite mode 2 oscillations in the summer and mode 1 oscillations in winter, and we have frequently found this to be the case. The much weaker stratification in winter is associated with reduced river discharge (Fig. 3A). There is also a progressive weakening of the stratification towards the inlet mouth associated with the mixing of fresh and salt water and the gravitational circulation that this mixing induces. The basic mechanism of tidal interaction with the sill in Knight Inlet was first determined in August 1977 (SMITH and FARMER, 1977). Strong currents during, say, an ebb tide, can produce critical conditions over the sill, with a return to subcritical flow via an internal hydraulic jump or large amplitude lee wave behind the sill. As the tide slackens, the disturbance moves slowly back over the sill and travels upstream, evolving as an internal surge or bore. HAURY et al. (1979) obtained evidence of this type of response in Massachusetts Bay and MAXWORTHY (1979) modelled it in a three-dimensional laboratory

3 Tidal interaction of stratified flow with a sill in Knight Inlet 241 O 2,----I SIGMA -T I O 'T,,-C I " AUG 15,1977 / / ~ /~.' /f. I: I : i.. /.: /... / / / (S # SIGMA-T I 0 I -~-~ i: r ~....., //) 240 /,./" NOV 22, Fig. 2. (Left) Density profiles, together with the corresponding eigenfunctions (Right) for the first three internal modes of 12.5-h period observed a few kilometers up-inlet of the sill in August and November. Note the broken density scale in the upper figure required to show the much greater vertical density gradient in summer. The corresponding internal wave speeds for the first three modes were 0.98, 0.41, and 0.33 m s -1 in August and 0.58, 0.36, and 0.21 m s -I for November. experiment. More detailed observations have led us to modify our earlier ideas on the mechanisms for generation of wave trains and this issue is taken up in connection with Maxworthy's laboratory experiments in the Discussion. As we shall show, at least two quite different surge generating mechanisms have been observed in Knight Inlet, the precise response depending upon the stratification and the strength of the tidal current.

4 242 DAVID M. FARMER and J. DUNGAN SMITH RIVER DISCHARGE ~ 50( 0,,,, L J,,,, I,, i,,,,,,, ~ JULY I.O ~ , 5, 6, 7, 8, 9,10,11,12 i i. 2, 5,4, 5, 6, 7, 8~ 9~-i0~1-~ Fig. 3. (A) River discharge derived from observed stage-discharge heights, of the Klinaklini River at approximately monthly intervals. (B) Wave speeds computed for the first two internal modes from density profiles obtained near the sill, as indicated in the text. Horizontal dashed lines, corresponding to sectionally averaged maximum tidal velocities over the sill during a spring and neap tide in July 1978, are included for comparison with the calculated wave speeds. Our observations were made with profiling current meters and CTDs (conductivitytemperature-depth recorders), with moored current meters and thermistor chains, with recording pressure gauges, and with high frequency (100 to 200 khz) echo soundings. Ship location was obtained using microwave positioning equipment, the data being recorded on magnetic tape. The patterns of internal waves generated near the sill also have been observed with air photos and with ship's radar. The acoustic images, in particular, provide a wealth of detail showing the structure of the flow over the sill and of the resulting surges and intrusions. Indeed, the images have proved so useful that it would scarcely have been possible to describe the structure of these dynamic and rapidly changing processes without their help. The source of acoustic reflection remains uncertain, although biological reflectors, microstructure, and turbulence have been suggested (ORR and HEss, 1978). In Knight Inlet at least some of the reflectors display a diurnal movement common to the diel migration of certain zooplankters. Perhaps the most surprising result has been the great variety of responses that has been observed. It appears that this variety is a direct consequence of variations in the strength of tidal forcing and of the stratification. Specific examples of some of the tidal interactions that we have observed have also been seen or been inferred from observations in other locations such as Massachusetts Bay, and it is of course probable that some types of sill flows occurring elsewhere do not occur in Knight Inlet. Nevertheless, the range of responses observed in Knight Inlet, covering a broad variety of oceanographic conditions, may serve as a prototype for many similar flows over obstacles in coastal waters.

5 Tidal interaction of stratified flow with a sill in Knight Inlet 243 CLASSIFICATION OF SILL FLOWS To place the broad variety of observed internal responses in perspective, it is useful to determine a suitable parameterization of the conditions associated with each response. Relevant physical conditions might include stratification, the tidally induced current, the sill shape and size, wind stress, and gravitationally driven flow. The wind stress is highly variable in both space and time, but it does not appear to play a dominant role in the sill dynamics. Wind induced effects, except very close to the surface relative to the pycnocline, are invariably small in comparison to the tidally induced exchanges. In contrast, strong winds may have a significant influence on the circulation away from the sill. Likewise the gravitational flow, though intimately linked with and dependent upon the sill induced dynamics, does not in itself produce current speeds comparable to the tidal flow. The sill shape plays an important role and its asymmetry appears to account for much of the observed asymmetry between flood and ebb response. There has been relatively little theoretical investigation of time-dependent stratified flows over large obstacles of the type observed in Knight Inlet. Some perspective may be gained from the models of BELL (1975) and LEE (1972) applicable to oscillatory flow (U = Uo cos ogt) over relatively small obstacles. Two limiting cases of Bell's theory are provided by the quasi-static limit for which oj/n <~ 1 and the acoustic limit for which ~o >> Uo/L, where 09 is the M2 tidal frequency, N is the Brunt V~iis~il~i frequency and L a characteristic length scale, such as the e-folding half-width of the obstacle. Taking a depth averaged N of 0.01 to 0.05 s -1 and L = 500 m, we find ~/N = to and Uo/ogL = 7.1. This places the Knight Inlet sill flow in the quasi-static limit. A difficulty associated with the analysis of the present data arises from the great size of the obstacle, which inevitably introduces a response of large amplitude. The problem of steady flow of linearly stratified fluid over large obstacles has been solved by LONG (1955) and others. Unfortunately, the applicability of Long's model to Knight Inlet appears questionable, both on account of the restrictions to steady state and linear stratification and also because of uncertainty as to the validity of the hypothesis of zero upstream influence (BAINES, 1977). A quasi-static response leads to the generation of lee waves or hydraulic transitions. Hydraulic control must be expected if critical conditions occur over the sill crest. SAMBUCO and WHITEHEAD (1976) have shown that for a two-layer fluid, rotation effects will be significant whenf212/2g'ho is of order one or larger, wherefis the Coriolis parameter, I is an integral length scale of the obstacle, g' = gap/p the reduced gravity, and ho the fluid depth over the crest. In Knight Inlet the flow is never strictly 2-layer, which makes an application of these results uncertain. However, if we consider a broad range of density differences and interface depths that might approximate the variety of conditions encountered, we find that f212/2g'ho is of order 10-3, suggesting that Coriolis effects on control will probably be negligible. This is to be expected, because the time taken for a fluid particle to move over the upstream part of the sill is a relatively small part of a pendulum day. Hydraulic control is most appropriately examined in terms of a densimetric Froude number. Because the density structure is continuous, we express the Froude number Fi as Fi= O/ci, i= 1,2,..., (1) where U is the maximum sectionally averaged tidal current and ci is the speed of a long internal wave of mode i at the crest of the obstacle. Thus there will be a separate densimetric Froude number corresponding to each mode. In practice we have numerically evaluated the phase speeds for observed density profiles in the neighborhood of the sill,

6 244 DAVID M. FARMER and J. DUNGAN SMITH down to but not exceeding, sill depth. For each successive monthly sequence of profiles, data from four stations within l0 km of the sill were used and an average value found for both the first and the second mode eigen values cl, c2, for a wave of period 12.5 h. The calculations showed that phase speeds at the four stations varied by about 10~o for a given CTD sequence; occasionally a single station would yield phase speeds very different from those of neighboring stations and for these rare occurrences, presumably associated with the presence of large internal waves, the values were excluded from the average. Our approach was intended to provide a representative stratification parameter for an unsteady flow subject to significant perturbations near the sill. The choice of a procedure for determining ci is necessarily arbitrary. However, it should be noted that this Froude number differs from that normally used, in which the density structure over the full depth is taken into account. Our choice is governed by the recognition that upstream blocking must occur (see section on Time dependent effects) and the behaviour of the flow over the crest will depend principally upon the density structure down to the sill depth. As the phase and amplitude of tidal elevation only change by 3~o along the inlet, the sectionally averaged current U is well approximated by the time derivative of observed tidal height ~(t): s d~ U - A dt' (2) where S is the surface area up-inlet of the sill and A the cross-section over the sill. The respective values for Knight Inlet are S = m 2 and A = 7.5 l04 m E giving a ratio of S/A = The parameter F i provides a basis for classifying sill flows if the response is hydraulically controlled. While the analogy with the hydraulics of steady homogeneous or layered flows is obviously imperfect the data from Knight Inlet suggest that calculations of F i provide a reasonably consistent basis for an initial classification of observed responses. Figure 3 (B) shows calculated long wave speeds for the two lowest internal modes derived from density profiles obtained at monthly intervals over the period 1977 and 1978, together with values of maximum ebb tide currents during a spring and neap tide in July Phase speeds for the lowest mode reach maximum values in the summer, when the river discharge is maximum. The phase speeds for second mode waves follow a generally similar curve but with much reduced range. Evidently within the range of maximum tidal flows several possibilities exists with respect to criticality over the crest. For example, in mid-winter even the smallest tides will produce flows critical with respect to all internal modes (F1, F 2... > 1), while in mid-summer most flows will be subcritical with respect to the first mode but may be supercritical with respect to the second (F 2 > 1 > F~). In spring and autumn, flows critical with respect to the lowest mode may also occur (F1 = 1). We shall present examples for each of these conditions using acoustic images to demonstrate the structure of the response. OBSERVATIONS OF SILL FLOWS The broad variety of flows observed over the sill in Knight Inlet appears to fit into two general classes; those involving the formation of internal lee wave trains and those associated with a single large isopycnal depression behind the sill. In the latter case, the response is similar to the internal hydraulic jumps familiar from laboratory observations of layered flow (LONG, 1954; ARMI, 1975), and we shall use the term hydraulic jump to describe them. However, an internal hydraulic jump might also be considered as a large

7 Tidal interaction of stratified flow with a sill in Knight Inlet 245 breaking lee wave and the relationship between these two qualitatively different types of response is taken up later. Other important phenomena that do not necessarily fit into this classification but occur concurrently with jumps or lee waves include the separation of the boundary layer behind the sill crest and also the formation of internal bores or surges that leave the sill ; discussion of these is deferred to the following section. Hydraulic jumps, mode 1 The simplest example occurs with a strong tide during a period of weak stratification when the flow is supercritical with respect to all internal modes. Unfortunately the weak stratification in winter is also associated with much weaker acoustic images. In Fig. 4 we present two images made in April 1978, one from Knight Inlet and a second, somewhat clearer example, from Boundary Pass. In both examples reflectors near the surface upstream of the crest plunge downward in the lee of the sill. In the Boundary Pass example the reflectors reach a depth of about 100 m. Calculating the densimetric Froude numbers for the Knight Inlet example yields F1 = 1.93, F2 = No current measurements have been made in Boundary Pass, but estimated speeds are thought to be substantially in excess of the 0.1 m s- 1 mode 1 long internal wave speed deduced from density profiles. Thus both of these flows should be critical with respect to all internal modes. We identify the observed distribution of acoustic reflectors with distortion of the flow field occurring as the deeper fluid from 10 m down to sill depth in each case accelerated down the lee face before returning to subcritical conditions via an hydraulic jump. The return to subcritical flow appears to produce a more diffuse and hence less readily visible acoustic reflection. By analogy with the hydraulic behavior of simple layered flows we would expect the hydraulic jump to involve significant energy dissipation, a contention supported visually by the contorted character of the reflectors in this region. Hydrualic jumps, mode 2 A quite different response occurs with moderate to strong tides during a period of strong stratification in summer. In Knight Inlet this typically leads to tidal flow over the sill that is subcritical with respect to mode 1 waves. In Fig. 5 we present an example from 23 July 1978 during maximum ebb flow when the sectionally averaged tidal current was 0.80 m s- 1. The corresponding Froude numbers are F 1 = 0.96 and F 2 = At this time the flow was approximately critical or slightly subcritical with respect to the first mode. Acoustic reflectors at the pycnocline show a gradual deepening over the sill, followed by an abrupt shoaling just upstream of the crest. Subsequently they descended irregularly to over 120 m. Just downstream of the crest a somewhat diffuse band of reflectors near the surface soon coalesced to form a pycnocline reflection similar to, although not so sharp as, that seen upstream of the crest. CTD profiles observed just downstream of the sill crest confirm that significant stratification persists near the surface. Evidently the reflectors associated with the pycnocline upstream of the crest, which consist of a sequence of discrete reflecting layers, split; the deeper reflectors plunge downwards and the near-surface reflectors, after initial spreading, re-form as a slightly weaker pycnocline. Large instabilities, up to 50 m high, can be seen on the deepening interface. Approximately 500 m downstream of the sill crest the deep reflectors start to return to the surface, although they become sufficiently diffuse that

8 246 DAVID M. FARMER and J. DUNGAN SMITH precise movements are hard to determine. The overall length of the response is of the same order as the length of the sill (1 to 2 km). At lower Froude numbers, spreading of the near-surface reflectors becomes less pronounced and irregularly spaced waves appear downstream of the crest. Figure 6 was made during an ebb flow with F1 = 0.86, F 2 = This image also shows a secondary spreading of reflectors (marked 'b' on the figure). This secondary spreading is a common feature and is especially apparent during flood tides. Figure 7 shows evidence of three stationary features over the sill during a flood tide. The first of these, marked 'a' in the figure, looks quite like a solitary wave of the type observed in laboratory experiments (DAVIS and AcRIvos, 1967). The second wave 'b' resides in the wake of the first, where the isopycnals, corresponding to the layers of acoustic reflectors, are already spread further apart. In the lee of the second feature, the deeper reflectors plunge down with signs of large scale instabilities. The final feature 'c' appears to be confined to the pycnocline. To summarize, we can say that with flows for which 1 > F 1 > 0.8, F 2 > 2, the intense pycnocline near the surface splits at some point close to the crest of the sill. Water below the pycnocline plunges down the lee slope of the sill in an accelerating and hence progressively thinner flow, the upper interface of which may be unstable, leading to the growth of large (50-m) rolling vortices. By analogy with the theory of layered hydraulics, we identify this accelerating flow as being supercritical. It continues to deepen and accelerate until the momentum exchange is sufficiently great that it can no longer remain supercritical, at which point the interface rises via an hydraulic jump. The interface becomes more diffuse although it does not immediately rise to its former level. In contrast, the sharp pycnocline rises abruptly over the sill crest, suffering further deformation through increased separation of isopycnals or by the generation of short wavelength and irregular oscillations (Fig. 6), before reforming downstream of the sill. The initial divergence causes a significant reduction in density stratification; the reduced stratification supports a second and even a third stationary wave-like feature further downstream. Lee waves Well-defined trains of lee waves have been observed during a moderate ebb tide in strongly stratified summer conditions and also during similar tides in the more weakly stratified conditions of November. In both summer and winter observations, separation of the flow behind the sill was clearly visible, but the structure of the lee waves and the relationship between the separation point and the waves was quite different. The summer series began 2 h after high water slack during a 3.2-m ebb tide on 30 July A dense band of acoustic reflectors associated with the boundary layer on top of the sill provides a graphic indication of flow separation. Initially there were no lee waves and the separation point is close to the sill crest. During the latter part of the ebb, lee waves form and the separation point moves down the crest. Figure 8 shows the situation 2 h before low water slack. The respective Froude numbers at this time are F 1 = 0.5, F 2 = 1.3. The fresh surface layer, appearing as a thick black line along the upper boundary of the image, appears to take little part in the flow. However, there is a distinct thinning of the surface layer over the first wave, indicating that it is of second mode. This would be consistent with the effects of the large density difference at the pycnocline which ensures that beneath the pycnocline the eigenfunction for second mode waves is similar to that of a first mode wave when the pycnocline is taken as the upper boundary. It is interesting to

9 O. 20 ~40 T ~60 80 O0 I I I I I t ' I I I I m 0 I 2 km Fig. 4. Acoustic images of first mode response in April 1978 in Boundary Pass, B.C. (Left) and in Knight Inlet (Right). The tidally driven flow is from right to left. In each example, flow is supercritical with respect to all internal modes and acoustic scatterers near the surface plunge down below the crest level on the lee side of the sill. The crescent-shaped band (a) is a multiple reflection of the sea bed. 2O ~40 -r" la] I00" I 0 I I I I I 500 I Fig. 5. Acoustic image made during a strong ebb tide on 23 July 1978 (F1 = 0.96, F 2 = 2.41). Flow is from right to left. Note the splitting of near-surface reflectors over the sill crest and the large instabilities on the plunging interface over the lee slope of the sill. Small variations in the horizontal scale arise from variations in vessel speed. Corrections to the horizontal scale for this and other figures have been made using microwave vessel positioning data. [jktcing p. 246]

10 l,j. :i::!... I I I I I I I 0 I m Fig. 6. The flow is similar to that in Fig. 5 except that the current is less strong (F~ = 0.86, F2 = 2.09). Note the second stationary wave (b) and the irregularly spaced short waves on the pycnocline. 20- ~ I- o-601 IJ.I a 80" I I I I I I m Fig. 7. Multiple stationary waves over the sill during a flood tide on 24 July The gentler slope on the up-inlet face of the sill allows the development of a more elongated response. F~ = 0.93, F 2 = Horizontal lines associated with the pycnocline diverge at the first wave (a). The weaker stratification in the wake of (a) supports a second, larger wave at (b). Following (bt the spreading reflectors can be seen plunging down the lee face of the sill, contorted by instabilities. The final wave (c) is confined to the pycnocline. O IO IO00m Fig. 8. Mode 2 lee waves on an ebb tide during strongly stratified summer conditions, 0323PST, 20 July The dark band emanating from part-way down the sill is associated with the separated boundary layer. F~ = 0.5, F z = 1.33.

11 @ b C d O'L _ I I 20- ~.,,~. ~..~ 'q ~4ov I " ~60- c", " ~ i - ii " IO0-i" " "r 4!. ~-, ". ~ ~'1 I I I I ' I I m Fig. 9. The development of a mode 1 lee wave trair~over the sill in November 1978 and its subsequent evolution as an internal undular bore. Image (a) was made during max. ebb. As the ebb flow slackened, more waves formed. Eventually the leading wave passed over the sill crest (d) forming the leading edge of a train of large amplitude internal waves (e) that travels up the inlet.

12 0 t r,, 30-~ t I0 I I minutes 0-1- I- ~2o D 30? : 4.- :,Y< ~':,,-.- 4,~.k i i i i i i 0 I minutes Fig. 11. Images of a mode 1 internal undular bore (a) and a mode 2 bore (b), generated in November 1978 and July 1978, respectively. The vertical scale is similar in each case, but the time scale is greatly compressed in the upper figure to allow for the much slower passage of the first mode bore past the stationary vessel.

13 I 0 I I000 I 2000 m I.... "~, ] 0 0 I m 20 4o "r I--- ~o I00 I I I I 0 I000 : m Fig. 12. Showing the relaxation of a massive mode 2 internal lee wave or hydraulic jump and subsequent evolution of a mode 2 wave train. As the ebb tide slackens the large instabilities contorting the plunging interface on the lee of the sill appear to coalesce into waves that then travel back up the inlet.

14 t~ : "~40-1- cl w / 80 I00 - q --::- T -.. ;2_[i;/ " I / I o i i I F ; ;,o Fig. 13. Three images made during and after a strong ebb tide on 20 July 1978 showing the generation of a nonlinear mode 1 internal wavetrain, upstream of the sill crest. The upper two images have the same horizontal and vertical scales as the lower one but are only separated by 60 min. A well-defined wave train (A) was formed close to the time of max. ebb. The lower image, made 3.5 h later, shows the isopycnal deformation (D) associated with the flood tide (D). The waves initially at (A) have moved up channel to (B) and are followed by a long wave of depression (B~). (E} is a multiple reflection of the bottom.

15 Tidal interaction of stratified flow with a sill in Knight Inlet 247 note that the separation point is forward of the first lee wave trough and the resulting shear layer does not appear to take part in the lee wave modulation, although it becomes progressively more diffuse with distance from the sill. Lee waves with quite different properties were observed in November Figure 9(b) and (c) represent this type of response. The first mode Froude number was 1.7, suggesting supercritical conditions over the crest, which were nevertheless able to support a train of large amplitude lee waves. The waves in this case, as expected, are of first mode, but the trough of the leading wave is now much closer to the crest and lies directly above the separation point. In,contrast to the summer example, the shear layer rises abruptly behind the separation point. The wavelength of lee waves in the examples of Figs. 8 and 9 is about 250 m for each case, in contrast to the larger scale response in Fig. 5. TIME-DEPENDENT EFFECTS Although flow over the sill in Knight Inlet is 'quasi-steady' in the sense indicated in the section on Classification, the time dependence of the tidal forcing is essential to the generation of the internal disturbances observed away from the sill. We identify three distinct types of time-dependent response: (1) internal tides, (2) travelling internal surges arising from the advance of lee waves or jumps over the sill against the slackening tide, and (3) stationary wave trains formed upstream of the sill crest that subsequently evolve into travelling surges. Internal tides While we are not yet in a position to describe the details of the generation of an internal tide, we can say that we have observed the M 2 component in Knight Inlet and found that typically it has an amplitude of about 20 m at a depth of 60 m. Time series CTD profiles in the summer show that the vertical excursion of isopycnals continues to increase with depth well below 30 m, suggesting that the internal tide is of second mode (see Fig. 2). Various models have been developed to explain the generation of internal tides over a sill (for example, STIGEBRANDT, 1976 and BLACKFORD, 1978); however none is strictly applicable to the Froude numbers and stratification encountered in Knight Inlet. While it seems inevitable that the large disturbances generated in the lee of the sill must put significant energy into the tidal component, there is also another mechanism that needs to be considered. If the density gradient at depths greater than the sill crest exceeds a critical value, there will be insufficient kinetic energy available to raise the fluid over the crest, resulting in upstream blocking. If the gradient is constant, blocking can only be avoided if the velocity satisfies the criterion: U 2 ~ NEh 2. (3) Below the sill crest, N ~ 3.3 x 10 -a s-1 so that for a sill height of say 450 m, the velocity required to prevent blocking would be 1.5 m s-1. This value far exceeds tidal velocities, even over the sill crest itself, so that some blocking must always occur. Because the blocking is controlled by the oscillating flow, it will be manifested as a periodic rise in isopycnals on the upstream side of the sill (up-inlet on an ebb tide, down-inlet on a flood) that propagates upstream and has the same period as the tide. The extent to which such an internal tide would be coupled to the other tidal phenomena described below is not yet clear.

16 248 DAVID M. FARMER and J. DUNGAN SMITH Collapse of lee waves and jumps LEE (1972) and BELL (1975) considered the problem of waves generated by an obstacle in a time-varying current. While LEE (1972) and LEE and BEARDSLEY'S (1974) analyses were motivated by the Massachusetts Bay observations, they have been the source of recent discussion germane to the Knight Inlet data and warrant examination. LEE (1972) suggested a classification of sill flows beyond the quasi-steady-acoustic limits, including blocked flows applicable to low Froude numbers and mixing flows for high Froude numbers. LEE and BEARDSLEY (1974) ascribed HALPERN'S (1971) Massachusetts Bay surge observations to blocking effects ; they supposed that as the flood tide slackens, only fluid close to or above sill depth would move over the sill, forming a warm front that would evolve as an undular surge downstream of the sill due to nonlinear effects. LEE (1972) also examined, both theoretically and with a laboratory model, the quasisteady lee wave problem, which he did not consider directly applicable to the Massachusetts Bay example. He showed that if the obstacle was not so large as to reflect the waves, lee waves formed downstream would advance back over the obstacle as the flow slackened. BELL (1975) interpreted Lee's model results in terms of a more general theory in which the higher frequency waves are seen as harmonics generated by interaction with the corresponding spatial derivatives defining the obstacle shape. MAXWORTHY (1979) argued that a lee wave model as opposed to Lee and Beardsley's 'warm-front' model is applicable to the Massachusetts Bay observations. Maxworthy's laboratory results appear to be consistent with Lee's laboratory and theoretical study of quasi-steady lee waves, although Maxworthy observed a single large depression in the lee of his three-dimensional obstacle and ascribed the subsequent evolution of a wave train to the interaction of nonlinear steepening and dispersion as well as the collapse of a mixed region. As Lee used no flow visualization techniques we do not know what lee wave structure occurred in his model, but the distinction between BELL'S (1975) interpretation of Lee's results on the one hand and Maxworthy's model on the other appears to lie in the emphasis given to the generation of short waves by interaction of the flow with the obstacle as opposed to the formation of short waves due to interaction between nonlinear and dispersive effects. A possible re-interpretation of Maxworthys' model results, motivated by our recent Knight Inlet observations, is given in the Discussion. In Knight Inlet we have seen a well-defined train of lee waves travel back over the sill to become an undular bore upstream of the crest, and we have also observed a single large lee wave or jump collapse to form an undular bore, although as we have discussed, boundary layer separation plays an important role in our interpretation of the mechanism. We have not observed phenomena corresponding to the partial blocking model of LEE (1972) and LEE and BEARDSLEY (1974), but we have observed the formation of waves on the shallow pycnocline upstream of the sill. This latter phenomenon does not appear to have been described previously. The advance of a lee wave train over the sill in November 1978 is shown in Fig. 9. Fifteen acoustic images were made during the critical stage of formation and advance, allowing the process to be tracked in some detail. In Fig. 10 we show the positions of acoustically observed wave troughs for the first six waves found by relating continuously monitored microwave positioning data to time marks made on the acoustic images. In the same figure we have also plotted the sectionally averaged tidal current over the sill, found from central differences of 15-rain tidal height observations made 6 km west of the sill. The leading wave forms soon after maximum ebb is reached and remains stationary for

17 4. \ \. \ Tidal interaction of stratified flow with a sill in Knight Inlet 249 -I " E 0- SILL CREST I - - o ~ " \ \ 500- I000- /m/ / 6- / \ T "\ E.5. \% 3 B \ \ 17oo n8'oo l~oo 2boo 21oo hr NOV. 18, 1978 Fig. 10. (A) Position of the troughs of the lee waves shown in Fig. 9, plotted against time, to show their formation and movement over the sill. The vertical scale corresponds to distance upstream ( + ) or downstream (-) of the sill crest. (B) Sectionally averaged tidal velocity over the sill crest, found by taking central differences of tidal height measured a few kilometers west of the sill. Note that lee waves do not start to form until close to max. ebb flow. an hour and a half before travelling back over the sill. As the tide slackens more waves form and advance over the sill crest. As each wave forms there is a period of rapid growth in the lee of the sill. When the wave travels over the crest its amplitude drops and then increases again on the upstream side. For example, the 18-m isopycnal of the leading wave in Fig. 10 grows to over 31 m in 54 min, but as the wave passes over the sill crest the amplitude drops to an observed minimum of 8.2 m before returning to about 15 m. A striking feature of these observations is the continuous generation of successive lee waves to replace those that have advanced over the sill. Successive waves travel more rapidly over the sill crest, although their wave speed, corrected for the changing tidal velocity, actually decreases slightly with each wave. The first three waves in Fig. 10, having maximum amplitudes just before advancing upstream of 32, 20, and 17.5 m, have wave speeds over the crest of 0.81, 0.72, and 0.69ms -1, respectively. Thus the largest wave leads the train and has the highest phase speed. The first wave escapes shortly after the ebb begins to slacken, with succeeding smaller waves following at about the same speed with respect to the crest, but at progressively lower speeds calculated with respect to the tidal current. This basically two-dimensional pattern is complicated by additional waves formed near the shore. Air photographs show that these spread radially from the two points at which the sill meets the channel walls. These latter waves seem to be smaller than the waves running transversely across the inlet, but some time after the main waves have passed over the sill crest the waves generated at the sides reach the centre and may form an interference

18 250 DAVID M. FARMER and J. DUNGAN SMITH pattern that complicates interpretation of acoustic runs along the inlet axis. Eventually, however, the main wave train emerges as a sequence of plane waves. There may also be some enhancement of the amplitude as the waves move into the slightly narrower section of the inlet. Figure ll(a) shows an example of a mode 1 bore also from November The wave train that travels away from the sill in Fig. ll(a) is associated with a net deepening of the surface layer as would be expected for an internal bore travelling on a shallow interface. It is similar to that observed during a period of weak stratification by FARMER and SMITH (1977). The contrasting example of the collapse of a single large mode 2 wave or hydraulic jump is shown in Fig. 12 and a mode 2 bore is shown in Fig. ll(b). As the tide slackens the lee wave collapses to form the second mode bore that travels back over the sill. In general we have found that first mode bores are formed by first mode lee waves or jumps and second mode bores arise from the relaxation of second mode waves. Wave trains formed upstream of the sill crest During strong ebb tides in summer a quite different wave generating mechanism has been observed. Nonlinear waves form near the leading edge of the pycnocline depression well upstream of the sill crest. This pycnocline depression is similar to that seen in Figs 5 and 6 and can be interpreted as the baroclinic adjustment necessary to compensate for the pressure drop as the deeper fluid accelerates over the sill. Figure 13 shows a sequence of images during a strong tide when these waves were being formed. The Froude numbers at maximum ebb were F~ = 1.10, F 2 = In this example the time of generation can be determined to within a 30-min period between successive acoustic images. The tidal flow deduced from tidal height measurements was at maximum ebb as the waves formed. As the tide slackened the pycnocline depression together with the train of nonlinear internal waves travelled back upstream, dispersing as it receded from the sill. By taking observations from two vessels, one anchored over the sill while the second made acoustic traverses, we have established that the waves are formed in the leading (i.e. upstream) portion of the pycnocline depression. The generation of these waves is qualitatively consistent with the prediction of LEE and BEARDSLEY'S (1974) nonlinear dispersive model in which the leading edge of a depression wave on a shallow pycnocline steepens, eventually balancing dispersion to produce a train of solitary waves. While a correct comparison with theory would require allowance for the effect of variable bottom topography and shear, it appears paradoxically that Lee and Beardsley's analysis may indeed be applicable to the generation of nonlinear waves near a sill but on the upstream rather than the downstream side. DISCUSSION We have shown that stratified flow over the sill in Knight Inlet results in a broad range of possible responses. Despite our rather arbitrary choice of technique for determining phase speeds for calculation of an internal Froude number, this parameter appears to provide a consistent basis for classifying the response. Of particular interest is the fact that second mode responses tend to occur in summer and first mode responses in winter. This must have important implications for the mean gravitational circulation. While the size of the sill in Knight Inlet limits the applicability of linear theory, it is interesting to make some qualitative comparisons. LEE (1972) carried out his quasi-static lee wave analysis for a uniform stratification N in a channel of depth H; while the

19 Tidal interaction of stratified flow with a sill in Knight Inlet 251 calculations can be made for arbitrary stratification, the essential features are preserved in this simpler case. The phase speed of stationary lee waves must equal the velocity of water over the sill. Wave energy propagates away from the sill at a speed CE equal to the difference between the phase speed Cp and the group velocity Cg, which is always less than the phase speed. Lee found that for linear stratification: CE = C v- Cg = U(1 -nztr2u2/n2h2). (4) As the flow increases, the wave energy propagates away from the sill more slowly, resulting in an accumulation of energy close to the area of generation. As critical conditions are approached (02-*N2H2/n2rc2), the linear theory breaks down. However, critical conditions occur for the highest modes first, although when the flow is still weak little energy will be available for accumulation at the sill. At some intermediate stage we expect sufficient energy to be available that critical conditions for a certain mode, not necessarily the lowest, will permit the generation of a massive lee wave. The results are consistent with our observations in Knight Inlet, where we observe large mode 2 lee waves or jumps in conditions that are subcritical with respect to mode 1 waves but critical or somewhat supercritical with respect to mode 2 waves. (The existence of mode 2 waves when F 2 > 1 may partly be explained by the nonlinear properties, in particular higher phase speed, of large amplitude waves and partly perhaps as a consequence of our method of calculating phase speeds used in determining Fi.) Lee's model predicts that the location of the wave energy peak moves back and forth over the sill within the range of the tidal excursion. Again we find that the most dynamically active area lies within the tidal excursion (1 to 2 km) of the Knight Inlet sill. Apart from the obviously nonlinear shape of the waves, departure from the linear model is most noticeable in the timing of lee wave formation. Theory predicts that short waves of small amplitude should form as soon as the flow begins. Instead we find that no lee waves occur early in the tidal cycle but form rapidly only when the flow has reached a certain stage. The explanation appears to lie in separation of the boundary layer at the sill crest in the early stages of the ebb tide. Separation can be seen in acoustic images and is frequently associated with large coherent structures. The structure of lee wave formation and its time-dependent interaction with boundary layer separation has been experimentally investigated using a small laboratory tow tank, and a detailed theoretical analysis of this problem forms the subject of another study (Dr T. S. MURTY, Institute of Ocean Sciences, Victoria, personal communication). For the present, some insight into the physics that might be involved can be found in BELL'S (1975) analysis of lee waves. Bell showed that successive Fourier contributions to the lee wave structure depend upon the spatial derivatives defining the obstacle shape. Thus the fundamental mode is sensitive to the obstacle's slope, the first harmonic to its curvature, and so forth. But the effect of boundary layer separation is to reduce the contribution from these derivatives, because the separated boundary layer represents a smoother boundary than would the unseparated flow. Thus we expect boundary layer separation to inhibit the growth of lee waves. This is particularly noticable during an ebb tide over the asymmetric sill of Knight Inlet, but the concept should have application to other shapes. At a certain critical point, however, separation of the flow is suppressed. This comes about through modification of the pressure field on the downstream face of the sill (BRIGHTON, 1977); further downstream separation will actually be enhanced. Suppression will first occur when the wavelength of a lee wave is comparable to the length of the sloping portion

20 252 DAVID M. FARMER and J. DUNGAN SMITH of the lee face of the obstacle. This appears to be the explanation for the lack of lee waves in the early stages of the ebb flow (Fig. 10). Once the first lee waves form, separation is controlled and additional waves can be generated as the ebb tide slackens. It is possible that at still higher Froude numbers separation may again occur near the sill crest, but it is now controlled by the boundary layer rather than the lee wave. Our conceptual model thus invokes the separation of the boundary layer in the early stages of the tide, followed by formation of lee waves that have j ust the right wavelength to suppress separation. Successive lee waves now form behind the first wave (wave energy propagates downstream). If the tidal current and stratification are appropriate, the group velocity drops to a level where significant evergy can accumulate at the sill in the form of a massive lee wave or jump. This happens when the wavelength of the wave is comparable to the length of the sill itself and is accompanied by intense mixing. This wave need not be of first mode and in Knight Inlet we have found second mode waves to dominate the response in summer. As the tide slackens the first waves advance over the crest and are replaced by new waves forming down-stream. These waves control the location of boundary layer separation, permitting their continued generation at flow speeds that at the start of the tidal cycle had suppressed them. The waves travel upstream as a packet. Because they are of large amplitude, they evolve as either turbulent, breaking waves or as solitary waves of the type modelled by nonlinear-dispersive theory. Thus far the explanation has not invoked the nonlinear mechanism used by MAXWORTHY (1979); it is worth making some simple comparison with his model results. We first note that no lee waves are formed below a certain critical Froude number Ft. It can be shown from quasi-static lee wave theory that the wavelength 2 for flow of uniform stratification in a fluid of depth H is: 2 = 2HF(1-F2) -½, F < 1, (5) where F is defined as in (1). Because Maxworthy used measured wave speeds this Froude number also corresponds to his definition. However, he did not use a uniformly stratified fluid and an accurate comparison would require use of the observed density distribution. From the graphed values in Maxworthy's Fig. 6 we find that for the three examples F c ~ 0.35, 0.42, and 0.45 corresponding to h/h ~- 0.33, 0.50, and 0.60; from Fig. 1 the obstacle height is seen to be 5 cm giving values of H , 10.0, and 8.3 cm, respectively. Using these values (5) predicts wavelengths of 11.3, 9.3, and 8.4 cm, respectively. As the sloped part of the obstacle is 10 cm long, these values seem about right for the transition from separated to separation-suppressed flow. Maxworthy also gives values of Froude numbers for breaking lee waves (F,, ~- 0.73, 0.90, and 1.1). The last, being just supercritical, is obviously beyond the range of validity of equation (5), but the first two values yield wavelengths of 32.4 and 41.3 cm, respectively, which are at least comparable to the full obstacle width of 30 cm. This result is consistent with our observations in Knight Inlet (Fig. 5), in which massive lee waves or jumps have a horizontal scale comparable to the sill width. However, the amplitude is now sufficiently large that nonlinear effects steepen the wave ; thus we might expect the horizontal scale of the strongly deformed isopycnals to be somewhat less than predicted by linear theory. The question arises as to whether the waves described by Maxworthy are generated by the disintegration of a single large disturbance as predicted by nonlinear-dispersive theory or whether they evolve as a train of lee waves successively formed behind the obstacle while the flow speed drops, as is evidently the case for Knight Inlet in Fig. 9. In Maxworthy's

21 Tidal interaction of stratified flow with a sill in Knight Inlet 253 Figs 5, 8, and 9, the waves do indeed form right over the obstacle. The waves are well separated right from the time of generation, as would be expected for the linear lee-wave model interpretation. However, their dispersion after generation is quite weak. The nonlinear-dispersive model quoted by Maxworthy requires that after their initial separation the waves disperse at a rate linearly proportional to their amplitude. Thus if the rate of separation is known, it should be possible to project backwards to a point at which the waves were close enough to interact. Because this point must occur after wave generation, it places limits on the possible area of generation. Projecting backwards in this way is not easy to do, as necessary data on the position of the obstacle in the tank and the timing of the photographs are not provided; additional complications arise from the influence of radial spreading on amplitude dispersion. Nevertheless, analysis of Maxworthy's photographs using marks on the tank floor as a reference frame, especially in Fig. 5, suggests that amplitude dispersion alone cannot account for their evolution from a point over the obstacle. We conclude that while the model may indeed simulate the generation of waves due to tidal flow over sills, for the specific examples of Maxworthy's Figs 5, 8, and 9, it is not necessary to appeal to nonlinear-dispersive theory to account for them. In contrast to the mode 1 wave trains in Knight Inlet, the large mode 2 bores (Fig. lib) start as a set of two or three large waves, only evolving into a sequence of several solitary waves after travelling several kilometers along the inlet. These initial waves appear to arise directly from the relaxation of large instabilities occurring on the upper interface of the supercritical layer as the tide slackens. The surges that finally result, whether of first or second mode, and whatever the detailed mechanism of their generation, are highly nonlinear and appear to have several of the features predicted by theory based on the Korteweg-De Vries model (FARMER and SMITH, 1977). The observations we have presented indicate an energetic interaction between the tidal current and the sill. Even for relatively weak flows, large waves often showing signs of instabilities are generated. As indicated above, vertical mixing certainly takes place in the lee of the sill. This can be determined indirectly, through observation of changes in the density structure. More recently small scale velocity and density measurements have been taken from a submersible in the lee of the sill during an ebb tide (GARGETT, in press) and these demonstrate a high level of turbulence compared to the relatively quiescent conditions up-inlet of the sill. In addition to mixing produced near the sill, mixing can also be produced within the energetic bores. This process is quite different from wavebreaking on the sloping sides of an inlet as discussed by STIGEBRANDT (1976) in connection with internal tides in the Oslofjord. Rather, mixing can occur as the wave is travelling well away from any physical boundaries. Evidence of shear instabilities can be seen in the acoustic images of Fig. 11. Gargett has also directly observed intense turbulence within these waves. This description of tidal interaction with the sill in Knight Inlet presents a much more complicated picture than might have been supposed. For example, previous theoretical models applied to the circulation of Knight Inlet have either represented turbulent processes through an eddy coefficient that increases with distance from the inlet head, but whose vertical structure is independent of position (WINTER, 1973), or by a laterally uniform supply of energy for mixing throughout the inlet (LONG, 1975). Our new observations suggest that two rather different models are required, depending upon the season, with a representation of mixing that allows for a maximum intensity near the sill. While a two-

22 254 DAVID M. FARMER and J. DUNGAN SMITH layer model may be appropriate for winter stratification, it appears that a three-layer representation may be necessary for an adequate description of the circulation in summer. REFERENCES ARMI L. D. (1974) The internal hydraulics of two flowing layers of different densities. Ph.D. Thesis, University of California, 147 pp. BAINES P. G. (1977) Upstream influence and Long's model in stratified flows. Journal of FluM Mechanics. 82, BELL T. H. (1975) Lee waves in stratified flows with simple harmonic time dependence. Journal of FIuM Mechanics, 67, BLACKEORD B. L. (1978) On the generation of internal waves by tidal flow over a sill--a possible nonlinear mechanism. Journal of Marine Research, 36 (3), BRIGHTON P. W. M. (1977) Boundary layer and stratified flow over obstacles. Ph.D. Thesis, University of Cambridge, England, 201 pp. DAVIS R. E. and A. ACRIVOS (1967) Solitary internal waves in deep water. Journal of Fluid Mechanics. 29, FAarV~R D. M. and J. D. SMITH (1977) Nonlinear internal waves in a fjord. In: Hydrodynamics olestuaries and [jords. J. NIHOUL, editor, Elsevier, GARGETT A. (In press) Turbulence measurements through a train of breaking internal waves in Knight Inlet, B.C. In: Fjord oceanography, Plenum Press. HALPERN D. (1971) Observations on short period internal waves in Massachusetts Bay, Journal of Marine Research, 29, 11(~132. HAURV L. R., M. G. BRISCOE and M. H. ORR (1979) Tidally generated intei'nal wave packets in Massachusetts Bay, U.S.A. : preliminary physical and biological results. Nature, 278, LEE C. Y. (1972) Long nonlinear internal waves and quasi-steady lee waves. Ph.D. Thesis, Massachusetts Institute of Technology and Woods Hole Institution, 127 pp. LEE C. Y. and R. BEARDSLEY (1974) The generation of long nonlinear internal waves in a weakly stratified shearflow. Journal of Geophysical Research, 79, LONG R. R. (1954) Some aspects of the flow of stratified fluids II. Experiments with a two-fluid system. Tellus, 6, LONG R. R. (1955) Some aspects of the flow of stratified fluids III. Continuous density gradients. Tellus, 7, LONG R. R. (1975). Circulations and density distributions in a deep, strongly stratified, two-layer estuary. Journal o[ Fluid Mechanics, 71, MAXWORTHY T. (1979) A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. Journal of Geophysical Research, 84, C1, ORR N. H. and F. R. HESS (1978) Remote acoustic monitoring of natural suspensate distributions, active suspensate resuspensions, and slope/shelf water intrusions. Journal of Geophysical Research, 83, SAMBUCO E. and J. A. WHITEHEAD (1976) Hydraulic control by a wide weir in a rotating fluid. Journal olfluid Mechanics, 73, SMITH J. D. and D. M. FARMEa (1977) Non-linear internal waves and internal hydraulic jumps in a fjord. Geo/tuiddynamical wave mathematics: research contributions. Appl. Math. Group, Univ. of Wash., Seattle. STIGEBaAYT A. (1976) Vertical diffusion driven by internal waves in a sill fjord. Journal olphysical Oceanography, 6, 48(~495. WIY~R D. F. (1973) A similarity solution for steady-state gravitational circulation in fjords. Coastal and Marine Science, 1,