Stratified flow over topography: bifurcation fronts and ...

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10.1098/rspa.2001.0887

Stratified flow over topography: bifurcation fronts and transition to the uncontrolled state By Laurence A r m i1 a n d David Farmer2 † 1

2

Scripps Institution of Oceanography, La Jolla, CA 92093-0225, USA Institute of Ocean Sciences, Sidney, British Columbia, Canada V8L 4B2

Received 15 January 2001; accepted 16 July 2001; published online 25 January 2002

A distinguishing feature of controlled stratified flows over topography is the formation of a wedge of partly mixed fluid downstream of a bifurcation or plunge point. We describe observations acquired over a sill in a coastal inlet under progressively increased tidal forcing. This wedge of partly mixed fluid is displaced downstream as the flow undergoes a continuous transition from control over the obstacle crest to an uncontrolled state. The effects of changing barotropic forcing and relative density difference between the plunging flow and partly mixed layer above combine to determine the fluid dynamical response. The relative density difference in turn is determined by the prior history of the flow as well as small-scale mixing. In general it decreases with time as denser fluid is entrained into the intermediate layer, thus increasing the effective forcing. For sufficiently strong tidal velocities and small relative density difference, the wedge of partly mixed fluid is displaced downstream of the crest and topographic control is lost. Such flows occur naturally in the ocean over sills and ridges, and in the atmosphere as severe downslope winds. Keywords: stratified flow; topography; fronts; hydraulic control

1. Introduction Stratified flow over topography presents challenging fluid dynamical problems with far reaching implications for circulation in the atmosphere and ocean. The classical problem of stratified atmospheric flow over a mountain has been the subject of intense analysis for over half a century (cf. Queney 1948), motivated by its role in accounting for drag on the larger-scale circulation as well as hazards due to severe downslope winds and clear air turbulence (Lilly 1978; Gill 1982; Baines 1995). Less well studied, but of oceanographic relevance, is the flow of stratified water over topographic features on the continental shelf and inshore areas where mixing and selective exchange exert their influence on coastal circulation (Farmer & Smith 1980; Nash & Moum 2001). A distinguishing feature of continuously stratified flow over topography is the formation of a bifurcation enclosing partly mixed fluid. Detailed observations of the establishment of such topographic flows have been described by Farmer & Armi (1999a). Small-scale shear instabilities are responsible for the initial phase of mixing † Present address: Graduate School of Oceanography, University of Rhode Island, Narragansett, RI 02882-1197, USA. c 2002 The Royal Society

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above the obstacle crest. This mixing leads to a slowly moving, weakly stratified layer that progressively increases in volume with time and is bounded beneath by a density step. This step is a necessary consequence of entrainment and shear instability along the bounding streamline and is a source of the mixed fluid: a striking example of small-scale processes contributing to the larger-scale response. As pointed out by Baines (1995, p. 259), it is important to distinguish between real flows and theoretical or numerical ones where the lower boundary is assumed to be a streamline. This artificial restriction excludes separation effects widely observed in the laboratory and natural environment (Scorer 1955; Hunt & Snyder 1980; Huppert & Britter 1982; Farmer & Smith 1980; Farmer & Armi 1999a). With few exceptions, such as Sykes’s (1978) triple-deck calculations, numerical models have not included boundary-layer separation. By forcing streamlines to follow the topography, numerical models generate a large amplitude internal wave which we do not observe. This was pointed out in our first paper (Farmer & Armi 1999a, cf. fig. 9), which subsequently led to the controversy and exchange of views presented by Afanasyev & Peltier (2001a), Farmer & Armi (2001) and Afanasyev & Peltier (2001b). Cummins (2000) has also drawn attention to the consequences of excluding boundary-layer separation, which can lead to misleading interpretation of numerical solutions in the critical phase of flow establishment. Without flow separation, establishment of the high drag state is initiated with overturning of a vertically propagating internal wave launched over the topography. Within a short time, much shorter than actually observed, amplification of this wave leads to breaking and development of the wedge of stagnant fluid. Cummins’s (2000) results from a numerical case with a modified leeside topography that simulated boundary-layer separation greatly improved comparison with the observations of Farmer & Armi (1999a) by suppression of overturning motion and the ensuing mixing that otherwise leads to a rapid formation of the wedge of fluid and the high drag state. As in our study of flow establishment (Farmer & Armi 1999a), we take advantage of the opportunities provided by stratified flow over a sill in Knight Inlet, British Columbia. Coastal inlets of this type have the benefit of predictable tidal currents, variability in stratification and well-charted bathymetry. Our earlier study took advantage of weaker tidal forcing; the topic addressed here only occurs when the tidal current is strong enough to force the bifurcation downstream of the sill. These bifurcations can undergo transitions through a sequence of states and may evolve in such a way that topographic control is lost. Moreover, a subtle link is established between the mechanism of entrainment and the larger-scale response including shape and position of the bifurcation. The significance of these findings lies both in their implication for strongly forced stratified flow over topography and as a demonstration of the way in which the larger-scale response can be sensitively dependent on small-scale mixing. Recent laboratory results (Pawlak & Armi 1998, 2000) illustrate the special character of shear instability in steep downslope flows; the enhanced entrainment efficiency of the instability helps to account for the changing density of the trapped fluid, thus providing the link between small-scale processes and the larger-scale response. Stratified flows in the natural environment usually possess a continuous upstream density distribution. However, a consequence of the topographic response is that the continuously stratified flow evolves into an equivalent layered behaviour with trapped fluid isolating the downslope flow from that above. This isolation justifies Proc. R. Soc. Lond. A (2002)


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the use of an appropriate layered analysis, as was first shown by Wood (1968) and Smith (1985). In the present study, the limiting single-layer theory for two-layer exchange flows, first introduced in a pair of papers by Armi & Farmer (1986) and Farmer & Armi (1986), provides the starting point for analysis of the position and shape of the resulting bifurcation. Largier (1992) and Stephens & Imberger (1997) have illustrated the application of this theory to the position of the plunge point in tidally forced two-layer estuarine flows. Here we explore the implications of this approach to downslope continuously stratified flows and compare the results with new observations in an oceanographic environment.

2. Observations The observations discussed here were acquired in Knight Inlet, a 120 km long fjord in British Columbia with a sill across which tidal forcing generates the flows of interest (figure 1). Run-off from the Franklin and Klinaklini rivers provides a strong near-surface stratification which is maximum during the summer. The tidal currents are ca. 1 m s−1 across the 60 m deep sill crest. This combination of bathymetry and forcing results in a Reynolds number of 108 . A singular advantage of this environment relative to studies of the equivalent problem in atmospheric flow over mountains is that several ship traverses of the evolving flow can be made within a given halftidal period with the measurements simultaneously spanning all depths of interest. In contrast, aircraft operations are necessarily limited to a few flights and the scale of atmospheric flows is much larger, making a detailed section difficult to complete (Lilly 1978), especially given the constraint that the measurements are generally limited on any given traverse to the flight path and altitude. A detailed explanation of the measurement approach is included in Farmer & Armi (1999a), and we limit the present discussion to a brief summary. The observations were acquired from the CSS Vector as she slowly traversed the sill. Our primary focus was on ebb tides, when the current was directed towards the west, since in contrast to the opposing flood tides the ebb flow is relatively two-dimensional in the centre of the channel permitting a much simplified analysis. Figure 1 (upper) shows the deployment scheme at 10:1 aspect ratio, with the plan view of the ship somewhat enlarged, and at 1:1 aspect ratio in the inset. Up to eight internally recording conductivity–temperature sensors were suspended from the bow for acquisition of continuous density measurements. A 150 kHz broadband acoustic Doppler current profiler (ADCP) was used to measure the current field. Backscatter acoustic images were acquired with a 120 kHz echo-sounder, an example of which is shown as background to figure 1 (upper). For many traverses we acquired density structure using continuous profiles of conductivity and temperature as a function of depth (CTD). All instruments were referenced to a common time base and navigation was achieved with the differential global positioning system (DGPS). A chart of Knight Inlet is also shown with an enlarged inset providing bathymetry in the neighbourhood of the sill. Traverses were carried out along the central section as indicated. In order to comply with the convention to always have the flow from left to right, all sections are illustrated as though viewed from the north. Aerial photographs were acquired from a floatplane at an altitude of 300 m. An example looking up-inlet towards the East (figure 2) shows the CSS Vector about to traverse the surface expression of the bifurcation plunge line, beyond which can Proc. R. Soc. Lond. A (2002)


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Figure 2. (a) Aerial photograph from 300 m altitude looking east along Knight Inlet, showing the CSS Vector approaching the sill (0157 GMT 28 August 1995) just before collecting the acoustic echo-sounder image shown in figures 1 and 10b. (b) Bathymetry and shoreline oriented as in the photo with the viewing position, ship and direction of ebb flow shown.

be seen a series of lines corresponding to internal solitary waves which have been discussed previously (Farmer & Armi 1999b). The surface expression that renders these internal responses visible from the air primarily arises from modulation of Proc. R. Soc. Lond. A (2002)


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the short gravity-capillary waves, which in turn modify the sky reflection. The twodimensional character of the ebb flow is evident in the straightness across the sill of both the plunge line visible about five ship lengths (200 m) ahead of the ship and the internal solitary waves about ten ship lengths further ahead. The streamline bifurcation and trapping of mixed fluid over a sill is illustrated schematically in figure 3. Farmer & Armi (1999a) discuss the way in which a streamline bifurcation occurs over the sill crest as shear instability creates mixed fluid (figure 3a). The bifurcating streamlines enclose a continuously increasing volume which extends downstream (figure 3b). Only when sufficient fluid has been entrained into the wedge of mixed fluid is the pressure gradient on the leeside of the sill favourable. Boundary-layer separation no longer occurs and the downslope flow is established (figure 3c). Increased flow over the sill can then push the bifurcation further downstream to create the strongly forced case (figure 3d ) which is of primary interest in the present context. In order to illustrate the essential features of strongly forced flow we examine a sequence of traverses across the sill. Figure 4 shows a sequence of sections acquired at successive intervals during a spring ebb tide on 30 August 1995. The sequence begins (figure 4a) at a time intermediate between that represented in figure 3b,c. For each section we show the downstream and vertical components of flow in vector form, superimposed on an acoustic backscatter image. The density structure is presented subsequently in figure 6. A bifurcation in the streamlines (figure 4a) has formed and the bottom boundary layer separates from the sill crest creating a downstream jet between 30 and 70 m. Over the sill crest the interface descends to two-thirds of its height above the crest far upstream of the sill, consistent with the behaviour of a single-layer hydraulically controlled reduced gravity flow. Twenty-two minutes later (figure 4b) the bifurcation has been displaced downstream by the increasing strength of the tidal flow. The interface slope has increased and the point at which the boundary layer separates has slightly advanced down the lee face of the sill. As the volume of mixed fluid downstream of the bifurcation increases, the separation point deepens. Forty minutes later (figure 4c), the bifurcation has been forced downstream and is immediately above the crest. The downslope flow separates some distance downstream of the sill crest, where the depth is ca. 90 m, and is followed by an internal hydraulic jump. As discussed subsequently, the displacement of the bifurcation downstream occurs as a result of both the strong tidal current and the reduction in density difference between the trapped fluid and downslope flow. Approximately twenty-five minutes later (figure 4d ) the bifurcation appears downstream of the crest. For one further measurement (figure 4e) the bifurcation advances still further downstream, even though the tidal current has slightly decreased. Thereafter, the bifurcation retreats upstream. Figure 4f shows the relaxation of this flow as the barotropic forcing declines. The slackening tide allows a bi-directional exchange to take place, with the weakly stratified upper layer moving east above the residual deeper flow towards the west. During the exchange and relaxation, the flow remains controlled over the sill crest with a 20 m thick lower layer. The flow sequence discussed above may be contrasted with the establishment of moderately forced flows described previously (i.e. Peltier & Clark 1979; Smith 1985; Farmer & Armi 1999a, fig. 7d ; Cummins 2000), in which the location at which the Proc. R. Soc. Lond. A (2002)



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Figure 3. Schematic summarizing the formation and evolution of a bifurcation front as discussed in the text.

flow bifurcates is well upstream of the sill crest. As long as the bifurcation remains upstream of the crest, the internal hydraulic control remains at the crest. The flow between the bifurcation and the control is subcritical and hence the position of Proc. R. Soc. Lond. A (2002)


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