Stratified Flow Separation in the Lee of the Knight Inlet Sill

Transcription

1 Stratified Flow Separation in the Lee of the Knight Inlet Sill Jody M. Klymak 1 and Michael C. Gregg Applied Physics Laboratory and School of Oceanography, University of Washington, Seattle Abstract. Stratified flows observed in nature and the laboratory often separate from the lee of topography, creating a stagnant low-flow pool of water beneath the swifter flow aloft. Existing theories and numerical models of stratified flow over topography do not exhibit this behavior because it is usually assumed that the topography is a streamline of the flow. This produces predictions of flow velocities near the topography that are often far in excess of those observed. We present observations of the flow separation of tidal flow over the Knight Inlet B.C. sill during ebb tides. Previously, it has been thought that the flow separation in Knight Inlet arose from boundary layer separation at the sharp sill crest, a process which is quite complicated to model since it depends on the accurate simulation of boundary layer processes. We show, however, that the mechanism mediating the flow separation in Knight Inlet is a dense pool of water trapped seaward of the sill that the lighter fluid simply passes over. Using a dense pool of water in a numerical model, we can simulate the development of the flow quite well without resorting to complicated boundary-layer physics. 1. Introduction Introducing an obstacle into a stratified flow can produce internal disturbances, which under a wide range of conditions consist of a strong downslope flow on the downwind side of the obstacle. In the atmosphere, the downslope flow is sometimes manifested as violent windstorms in the lee of mountain ranges [Lilly, 1978]. In the ocean, downslope flows have been observed over sills in tidal channels [Farmer and Smith, 19] and over banks on continental shelves [Nash and Moum, 2]. Sometimes the strong downslope current predicted by theory is prevented by flow separation in the lee of the obstacle. Flow separation can occur from a sharp obstacle in a homogenous fluid when flow in the bottom boundary layer is reversed by the pressure gradient. The situation is more complicated when the flow is stratified, and appears to be governed either by boundary layer separation or by stratification depending on the strength of the stratification [Baines, 1995]. Laboratory flows show both mechanisms at work; in the first case, the flow separates from the obstacle crest and flows over a 1 Applied Physics Laboratory, 113 NE 4th St, Seattle WA, jklymak@apl.washington.edu turbulent well-mixed region (Fig. 1a). In the second case, a lee-wave forms, which then encounters dense water and rebounds (Fig. 1b). Figure 1. Sketches of three different flow-separation scenarios, slightly modified from Baines [1995, Fig. 5.8] to include the layer of blocked dense water. a) Boundary layer separation in a homogenous or weakly-stratified fluid, note the overturning underneath the separation. b) Flow separation due to a uniform stratification. Note the dense water blocked upstream of the obstacle and trapped downstream. c) Flow separation due to an unequal stratification as we are proposing for Knight Inlet. Flow separation has also been observed in the ocean. Farmer and Smith [19] present acous- a b c 1

2 : 2 tic images of ebb tide flow over the Knight Inlet B.C sill that clearly show flow separation. They attribute the flow separation to boundary layer separation and offer observations of flow-separation in a two-layer tank experiment as argument. More recently, Farmer and Armi [1999] observed the same flow separation and point out the role it plays in delaying the onset of strong downslope currents. Numerical models of stratified flow over topography that have constant stratification and include no boundary layer physics are unable to produce the observed flow dynamics, as recently demonstrated by detailed comparisons between the Knight Inlet flow with a numerical model by Cummins [2]. A similar discrepancy between the experiments of Baines [1995] and a numerical study by Lamb [1994] was found. We present density and velocity data from ebb and flood tide over the Knight Inlet sill. We observe that there is a strong density contrast across the sill, and that this density contrast argues for a modification of the density-induced flow separation mechanism; instead of allowing a lee-wave to develop, the higher stratification suppresses the lee wave (Fig. 1c). We present two runs of a numerical model, one without this density contrast that behaves much like the model by Cummins [2], and a second, which includes a dense pool, that behaves much more like the data. The conclusion is that the physics of the flow in Knight Inlet can be represented by choosing the appropriate initial stratification. 2. Observations The geophysical flow we study is over a sill in Knight Inlet, 3 km north of Vancouver B.C. The fjord is divided into a 1 m deep seaward basin and a 35 m deep landward basin by a 65 m deep sill. The water in the inlet is stratified by the continuous mixing of glacier run-off from the Kliniklani River and ocean water from Johnston Strait to the west. We observed flow properties over the sill from mid August until early September, just after the peak runoff season. Consequently, there is a sharp surface stratification about 2 m thick that caps a more gentle stratification at depth. A barotropic tide with a 1.5 m amplitude drives an oscillating semi-diurnal tide over the sill, producing barotropic velocities at the sill crest of.65 m s sill depth seaward landward σ θ / kg m -3 Figure 2. The average density profiles (σ θ ) seaward and landward of the sill. Standard deviations are derived via a bootstrap method. The inset figure shows the full density range from about σ θ = 7 kg m 3 to σ θ = 25 kg m 3. The data presented here was taken from the R/V Miller as she steamed east west across the sill. Density was measured using a Sea-Bird CTD mounted on the weighted tow-body SWIMS, calculating density from temperature and conductivity. Simultaneous profiles of velocity were made using an RDI 15 khz broad-band acoustic Doppler current profiler. Smoothed and gridded onto a 5 m horizontal and 1 m vertical grid, we produced density and velocity maps of the flow over the sill. This relatively coarse grid was supplemented by qualitative information sampled every second from a highgain 12 khz echo-sounder that scatters off inhomogeneities in the water column. Part of our argument is that there is a stratification contrast across the sill. The water seaward of the sill is denser than the water landward, and the stratification is higher (Fig. 2). For instance, on the landward side we measured the density at 1 m depth to be about σ θ =.25 kg m 3, whereas on the seaward side that same density was, on average, at 65 m depth. The density contrast is evident at all depths. Of course individual profiles vary greatly with the tides, with internal wave displacements in the deep water of well over 4 m.

3 : a.125 b WEST - EAST/ m WEST - EAST/ m WEST - EAST/ m WEST-EAST / m 2 23 e 23 f 23 g U / ms -1 c U / ms -1 seaward -5 - landward WEST - EAST/ m WEST - EAST/ m WEST - EAST/ m WEST-EAST / m Figure 3. Data taken from the Knight Inlet sill during ebb and tide. a) and b) were collected August 25, 1995, 2:25 until 23:54 GMT. c) g) on August 31, 14:43. The four darkest isopycnals correspond to σ θ = 16,21,23, kg m 3. A thin isopycnal is marked every σ θ =.25 kg m 3 below σ θ = 23 kg m 3. d) and h) show the ship track, and have a colorbar for reference to the velocity data in the other plots. In c) d) dark velocities are seaward (towards the west), and in e) h) dark velocities are landward (towards the east) Time Dependence: Ebb and Flood tide Curiously, the strongest internal response in the lee of the sill during ebb tide lags the barotropic forcing by the tide [Farmer and Armi, 1999]. The sequence of events is presented here. Just after slack tide the response is dominated by an arrested wedge flow. Beneath σ θ =.125 kg m 3 the isopcynals are tilted down landward, but the flow is held in place by the barotropic pressure gradient, which is driving the flow aloft seaward (Fig. 3a). By peak ebb tide there is a weak asymmetric response to the sill, but not a very dramatic one (Fig. 3b). There has been a lifting of the isopycnals upstream of the sill, indicating an upstream disturbance has passed, and a general tilting of the isopycnals to seaward. Past the sill crest the water lighter than σ θ = 23 kg m 3 has thickened and decelerated, creating a small wedge of fluid. Water between σ θ = 23kg m 3 and σ θ =.125kg m 3 on the other hand has thinned and accelerated. The water beneath σ θ =.125 kg m 3 is almost stagnant. Though we are at peak tide, there is little evidence of a strong downslope flow. Only near the end of ebb tide does a strong asymmetric response develop (Fig. 3c). This response is characterized by a dense layer that penetrates to almost m depth, and a much wider intermediate wedge layer that widens rapidly in the lee of the sill. Qualitatively, this response is quite similar to those observed in the lee of mountain ranges [i.e. Lilly, 1978], and in numerical models [i.e. Peltier and Clark, 1979]. However, even when the downslope flow is strongest, the flow still separates at m depth. In contrast to the delayed internal response during ebb tide, a downslope flow sets up quite quickly during flood tide (Fig. 3e g). By peak flood tide the isopycnals have been compressed upstream of the sill, and a large and deep asymmetric response has formed in the lee of the sill. The dense water plunges with velocities greater than 1m s 1 to depths below our ability to measure it, and intermediate water forms a very thick wedge. The flow stays the same for almost three hours during flood tide, and only relaxes as the barotropic forcing does.

4 : 4 P / MPa σ θ =.25 U / m s separation point -5 5 WEST - EAST / m Figure 4. Detail of slack tide flow (Fig. 3a). Upper plot is contoured density over seaward velocity. Thin black lines are the CTD tracks. Bottom panel is an echosounder image Flow Separation and Density Contrast P / MPa U / m s K-H billows flow separation WEST - EAST / m Figure 5. As in Fig. 4 except for peak ebb tide (Fig. 3b) The reason the flow separates during ebb tide and not during flood tide is that there is denser water seaward of the sill than landward. At the end of flood tide, the dense water is still elevated seaward of the sill, but the density current is pinched off, forming an arrested wedge. The interface between the two water masses is at σ θ =.125kg m 3 and can be seen in both the velocity and echo-sounder data (Fig. 4). The tip of the arrested wedge is about 12 m east of the sill crest and marks the interface between the dense water that is slowly being pushed back seaward and the lighter water that is flowing over it. The tip of the wedge can be thought of as a separation point, but its location leaves little doubt that the flow separation is due to the density contrast of the two fluids, and not to the shedding of boundary layer vorticity. By peak ebb tide the dense water has dropped (gradually, 25 m over 2.5 hours) until it is below the sill crest (Fig. 5). Again, there is a contrast between this dense water and the densest water passing seaward over the sill crest. The interface between the two waters is marked by a strong shear, with the dense pool of water almost stagnant, and the lighter water swiftly ebbing seaward. This shear is strong enough to drive the local Richardson number below 1/4 as required for the Kelvin-Helmholtz instability seen in the echo-sounder image as three 15 m tall billows. These instabilities are very similar to the instabilities at the flow-separation interface presented by Farmer and Smith [19], indicating that we are considering the same phenomena. The interface between the jet of ebbing water and the stagnant water below is always co-incident with the density layer, even in a flow configuration that is unlikely to be boundary layer separation (slack tide Fig. 4). This argues that density is what forces the flow separation. A second argument lies in the nature of the water below the separation interface. In the boundary layer separation cases presented by Baines [1995], the stagnant pool of water beneath the jet is very turbulent and well mixed. In contrast, the water beneath the jet shows no sign of vigorous turbulence, and remains well stratified. 3. Model We show the importance of including a dense layer of water in a numerical simulation of the flow over the Knight Inlet sill. We used the Hallberg Isopycnal Model, a layered model, in a twodimensional domain meant to simulate Knight In-

5 : 5 let. The main advantage of this isopycnal model is that it has a positive-definite advection of layer thickness scheme that allows isopycnals to intersect topography. The model was forced with a prescribed barotropic tidal force that oscillated with the tidal frequency, and it included lateral boundary conditions that artificially damped internal waves far from the domain of interest. The first run was initialized with an equal density distribution on either side of the sill (Fig. 6, top panels). The response is very similar in character to that observed during flood tide. A strong asymmetric flow develops almost immediately and continues for the duration of the tide until the barotropic forcing relaxes. Like flood tide, the asymmetric flow consists of a dense plunging jet bounded above by a low-velocity, weakly stratified wedge of fluid marked by hydraulic jumps at each density layer. In contrast, adding a dense layer of water west of the sill crest creates a situation more similar to the ebb tide flow described above (Fig. 6, bottom panels). At slack, there is a strong exchange flow characterized by the lighter water flowing seaward over the denser water. As the tide progresses, this denser water relaxes below the sill level and mediates a flow separation at about m depth. The relaxation of the dense water is faster than the relaxation observed in the data. Nonetheless, a strong downslope flow does not develop until late in the tide. The flow is somewhat asymmetric, with isopycnals compressed upstream of the sill crest and widening downstream, but the sharp plunging flow is not observed until 17: h, just before slack tide. These results can be compared and contrasted with the results reported by Cummins [2]. He used an equal stratification throughout his modeling domain, and found results that were essentially the same as our equal-stratification case (Fig. 6). He also showed that by modifying the topography of the sill so that it is only m deep seaward of the sill crest, the flow can look like the data observed. His modified sill geometry was meant to represent boundary layer separation, however, this study indicates that it could just as easily represent a denser layer of water trapped seaward of the sill. This dense layer also models the dynamics of the exchange flow during slack tide better than the modified geometry solution. 4. Summary and Discussion We have demonstrated that the observations in Knight Inlet show that the flow separation observed during ebb tide is co-incident with the interface between the densest water passing over the sill crest and a denser pool of water trapped to the seaward. As has been observed before, this flow separation suppresses the development of a strong downslope flow during ebb tide [Farmer and Armi, 1999]. A numerical model run without a dense pool of water confirms this hypothesis, and shows a strong asymmetric downslope flow in the lee of the sill, whereas a run including a dense layer is able to delay the downslope flow until later in the ebb tide. These observations argue against a flow domiated by boundary layer separation (Fig. 1a). If we calculate the Froude number and the aspect ratio of the flow, we find that for the ebbing flow Nh/U 4 and h m /A d.5, which the experiments in Baines [1995] indicate are well within the regime where boundary layer separation is not expected to occur. Our numerical experiment with no stratification across the sill and those by Cummins [2] both show flow separation at the bottom of a large lee-wave as predicted by Baines [1995] (Fig. 1b). In this work we have further argued that the strong lee-wave can be suppressed by a baroclinic pressure gradient across the sill, an effect not explicitly considered before. A sensitivity study shows that the suppression of th lee-wave was very sensitive to the thickness of the dense layer. Changing the stratification so that the dense pool is a little shallower completely suppresses the lee-wave, while making it deeper allows a lee-wave to form much earlier in the tide. Our experiments show that the timing and depth of the flow separation depend very closely on two other time dependent effects. First, the time dependence of the dense layer that forces the flow separation has a large vertical excursion. We found it difficult to model this excursion accurately, perhaps due to our method of forcing the model, or due to threedimensional effects. Secondly, the density of the densest water passing from upstream continuously increases over the course of the tide. Denser water in the overflow can penetrate through the dense pool to a greater depth. Since the stratification N is not constant with depth and varies across the sill,

6 : : 14:4 16: 17: WEST EAST / km WEST EAST / km WEST EAST / km WEST EAST / km U / m s 1 13: 14:4 14:4 16: 17: WEST - EAST / km WEST - EAST / km WEST - EAST / km WEST - EAST / km U / m s -1 Figure 6. Model runs of ebb tide, every second density layer plotted over velocity. Dark isopycnals correspond to σ θ = 21,23, kg m 3. Upper plots were from a run with equal stratification on either side of the sill. Lower plots were with stratification contrast across the sill the parameter space necessary to explore these effects is daunting. The phenomena of dense fluid in the lee of topography mediating flow separation and suppressing lee-wave creation may be relevant to the longstanding problem of predicting the onset of destructive downslope windstorms over mountain ranges. These downslope storms are relatively rare, despite small differences in the upstream flow field. Perhaps modeling the flow with a uniform stratification (as is usually done) is not always appropriate. The air flowing over the mountain range may not be as dense as the air at ground level, suppressing the growth of the lee-wave, as observed in Knight Inlet. Perhaps only on occasions when the air at ground level has gotten lighter, or when the air coming over the mountain range is particularly dense, will severe windstorms occur. References Baines, P. G., Topographic Effects in Stratified Flows, Cambridge Univ. Press, Cummins, P. F., Stratified flow over topography: Time-dependent comparisons between model solutions and observations, accepted Dyn. Atmos. Ocean. Farmer, D. M., and L. Armi, Stratified flow over topography: The role of small scale entrainment and mixing in flow establishment, Proc. R. Soc. Lond. A, 455, , Farmer, D. M., and J. D. Smith, Tidal interaction of stratified flow with a sill in Knight Inlet, Deep- Sea Res., 27A, 239 5, 19. Lamb, K. G., Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge, J. Geophys. Res., 99, , Lilly, D. K., A severe downslope windstorm and aircraft turbulent event induced by a mountain wave, J. Atmos. Sci., 35, 59 77, Nash, J. D., and J. N. Moum, Internal hydraulic flows on the continental shelf: High drag states over a small bank, submited J. Phys. Oceanogr. Peltier, W. R., and T. L. Clark, The evolution of finite-amplitude waves. part II: Surface wave drag and severe downslope windstorms, J. Atmos. Sci., 36, , 1979.

7 : 7 J Klymak, Applied Physics Laboratory, 113 E 4th St, Seattle WA, jklymak@apl.washington.edu This preprint was prepared with AGU s LATEX macros v4, with the extension package AGU ++ by P. W. Daly, version 1.6a from 1999/5/21.