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1 Stratified Flow over Topography: Bifurcation Fronts and Transition to the Uncontrolled State Author(s): Laurence Armi and David Farmer Source: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 458, No (Mar. 8, 2002), pp Published by: The Royal Society Stable URL: Accessed: :33 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings: Mathematical, Physical and Engineering Sciences.

2 If THE ROYAL /rspa B SOCI ETY Stratified flow over topography: bifurcation fronts and transition to the uncontrolled state BY LAURENCE ARMI1 AND DAVID FARMER2t 1Scripps Institution of Oceanography, La Jolla, CA , USA 2Institute 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 t Present address: Graduate School of Oceanography, University of Rhode Island, Narragansett, RI , USA. 458, ? 2002 The Royal Society

3 514 L. Armi and D. Farmer 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

4 Stratified flow over topography 515 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

5 516 L. Armi and D. Farmer w 51? ? 50o30' Figure 1. Bottom: chart of Knight Inlet, British Columbia. Middle inset: bathymetry in the neighbourhood of the sill with depth given in metres. The straight line running across the sill corresponds to the ship track followed in observations described in the text. Inset above: sketch showing measurement approach. The CSS Vector (37 m length shown here greatly enlarged) supports an acoustic Doppler profiler, echo-sounder, towed conductivity-temperature-depth (CTD) sensor array and towed profiling CTD. The sketch is superimposed on the acoustic echo-sounder image shown subsequently in colour in figure 10b, as though viewed in elevation from the North, with the ebb tidal current.flowing from left to right. Small inset (top): plan view of vessel showing acoustic Doppler beams and towed instrumentation.

6 Stratified flow over topography 517 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 lob. (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

7 518 L. Armi and D. Farmer 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 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

8 Stratified flow over topography 519 (a)...: ::: A'. Shear flow instabilities due to upstream influence %A Weakly Subcritical stratified ---flow- *.- layer (b).. :." ~ '.. '' 7. (b)=. - - Flow separation Upstream shape -- due to sill control a (c) Still controlled state Control downstream - at bifurcation front (d)a., :.-... ::'::"::. Uncontrolled state (. A 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

9 520 L. Armi and D. Farmer (a) =+.50.,I: E (b) 0 w a)6 " Aug O1 >15.14, - <0.2 mns [ <0., 90-1 o2 <0.( o0.6 <01. I I 0 *~*1.2 +ms7 100, AO -600 E E ~~~~~~~ U distance (in) U w Figure 4. Acoustic images through an ebb-tide sequence with strong forcing. Doppler velocity vectors, coded as to magnitude, are shown for the vertical and downstream components at the same aspect ratio. Inset: time of measurement relative to tidal phase.

10 Stratified flow over topography (c) = " ~ E (d) 0 I w oe I " E distance (in) W Figure 4. (Cont.) For description see opposite.

11 522 L. Armi and D. Farmer E (f) 4UU w i SS n80 ~30 Aug > <0.2ms 80-., <0. '5-4' ^^ '-0.4 * <0. _ ^^^^^ _ -?p0.6 <0. o9-0r2- -*?? ^ 0.8 <1. -*??7 ' -, --* 1.0 < ms 100 I E distance (m) W Figure 4. (Cont.) For description see p I

12 0 0I 0 20-,- 40- " E o distance (in) Figure 5. Expanded version of bifurcations corresponding to figure 4b, c, e shown her illustrating the change in position and shape of the bifurcation with

13 524 L. Armi and D. Farmer ; I..,. I i! *e *...: ' T I ^ a:20 -O E 3v0 1 2 E distance (m) Figure 6. (a) The density field observations of the same sequence as figure 4. Horizontal arrows indicate the magnitude of the non-dimensional barotropic forcing, Uo. Marker points show the location of bifurcation for each traverse. the bifurcation will be determined by the control parameters. By analogy with the discussion of strongly forced flows in Farmer & Armi (1986), when the bifurcation advances downstream of the crest, the Froude number at the bifurcation is unity, but the flow is not controlled in the usual sense since the bifurcation is free to move, depending on the strength of the forcing. In the more weakly forced flow discussed by Farmer & Armi (1999a), the internal hydraulic control always remains at the crest since the bifurcation is at all times upstream of this point. W

14 Stratified flow over topography 525 E distance (m) W Figure 6. (Cont.) Note the density field at is repeated at the bottom of (a) and the top of (b). The bifurcations corresponding to parts (b), (c) and (e) of figure 4 are shown combined in figure 5 with an aspect ratio of 1:1. Each image acquired at successive times is shown as the bifurcation moves downstream. The first image at shows the bifurcation 100 m upstream of the sill crest (0 m); as shown later, the flow is

15 526 L. Armi and D. Farmer controlled at the crest so that the bifurcation itself is subcritical. The shape of the bifurcation is determined by the requirement of control at the crest and matching of upstream conditions. At the bifurcation has moved ca. 10 m downstream of the crest, the interface is very steep and unstable. By the bifurcation is 100 m downstream of the crest, representing the strongest response during the tide. As discussed later, this extreme response occurs towards the end of the ebb tide even when the current has slackened somewhat. These results are summarized in figure 6, where we show the density structure obtained from the towed instruments displayed as a function of time, together with the non-dimensional barotropic forcing discussed in? 3. Note that the spacing of the sensors introduces a step-like quantization in some of the density contours (e.g. at 14.45), but the timing of frontal passage is clearly evident. The barotropic forcing described in the next section is derived from the depth-averaged current within the active layer, measured over the sill crest. Near the boundaries in the vicinity of the sill, enhanced bottom friction and form drag reduce the flow speed, creating a phase lag between the near-shore current and that in the centre of the channel. For this reason our calculations are all based on the measured velocities rather than transports derived from the change in tidal height. 3. Analysis The establishment of the stratified flow (figure 3a-c) is discussed extensively in Farmer & Armi (1999a). A consequence of the bifurcation is that it transforms a continuously stratified flow into one that behaves as a layered flow; in the present case, just a single layer is dynamically active, except in the final relaxation stage, when a two-layer exchange takes place. A limiting example of two-layer exchange flow discussed by Farmer & Armi (1986) is the arrested wedge in which the interface intersects a boundary or capping layer, the position of which is determined by the strength of the topography and non-dimensional forcing (figure 7). The topographically controlled front at the tip of this wedge differs in a fundamental way from a gravity current. Topography need play no role in gravity currents (i.e. Von Karman 1940; Benjamin 1968). On the other hand, the position and shape of hydraulic fronts are determined by the topography. Moreover, the hydraulic front conserves energy and is not dissipative; in contrast, gravity currents conserve momentum and not energy. Because of the pressure forces acting on the topography beneath hydraulic fronts, they may constitute an important component of atmospheric drag and determine the structure of the severe downslope wind. While our approach is based on layered flows, we note that at least for the case of flow through a contraction, a continuum of solutions has been found with bifurcation fronts for continuous stratification (Armi & Williams 1993) and we expect that a similar generalization applies to sill flows. The special stratified solution of Smith (1985), for example, is likely to be one of a continuum of solutions similar to those shown in figure 7 for the two-layer system. Our analysis follows the quasi-steady analysis given by Farmer & Armi (1986); the validity of the quasi-steady analysis will be explored later in this section. We normalize all depths with respect to the total depth of the flow at the crest (Yl + Y2)0. The subscript '0' refers to properties above the sill crest and subscripts '1' and '2' refer to the upper and lower layers, respectively. All velocities are made non-

16 - Stratified flow over topography 527 Uo =0-4^ moderate >.- h < U0 < (2/3)3/2 stagnant> UO= (2/3)3/2 - > \flui intermediate (2/3)3/2 <U<l 1 - ~ :~... UO= 1 strong Uo>l Figure 7. Sketch showing barotropically forced two-layer flow over a sill (adapted from Farmer & Armi 1986). dimensional with respect to the product of reduced gravity and the total depth at the sill: (g'(yl + Y2)o)1/2. The present analysis makes the simplification that the flow may be approximated by a single moving layer with depth Y2 and reduced gravity g' characterizing the stratification. An important feature of our subsequent analysis is the variability of g' in space and time due to mixing and entrainment across the interface. Continuity can be used to define the non-dimensional barotropic forcing: Uo - ul0y0 + u2oy20. (3.1) Here Uo is a non-dimensional flow rate, consistent with that introduced by Armi & Farmer (1986) and Farmer & Armi (1986); later q is used for the dimensional equivalent. Although a continuum of solutions exists, it is useful to consider two different classes. First, when the front is upstream of the crest, control at the sill crest can be written in non-dimensional form as,2 F22o= u20 = 1. (3.2) Y20

17 528 L. Armi and D. Farmer From (3.1) and (3.2) the non-dimensional depth at the sill crest is expressed as Y20 = U2/3 (3.3) The above solutions occur when the flow is strong enough to arrest the upper layer (ul = 0) and prevent an exchange flow, yet not so strong that equation (3.3) can still be satisfied at the sill crest, i.e. Uo < 1. Subcritical conditions then exist at all points upstream of the crest. Under stronger barotropic forcing the second class of solutions occurs and the bifurcation cannot be sustained upstream of the crest. It then adjusts to a position downstream of the crest. Critical conditions, 2 F2f = U2 = 1, (3.4) Y2f will then exist at the location of the bifurcation where the calculation now includes the active lower layer and the density step that separates it from the trapped intermediate layer. The subscript 'f' now refers to the properties at the location of the bifurcation front. The depth at this location is then derived from (3.4) and the continuity equation, and is Uo U2fY2f, (3.5) Y2f = U2/3. (3.6) Equations (3.3) and (3.6) are plotted in figure 8. These examples correspond to the intermediate and strongly forced cases discussed by Farmer & Armi (1986). Once the bifurcation has been forced downstream of the crest, equation (3.6) applies. The critical condition is the same, but it now occurs at the bifurcation rather than over the sill crest. Figure 8 also shows values and times derived from the observations of 30 August Returning to figure 5, not only is the position of the bifurcation front shown moving across the sill crest with increasing barotropic forcing, but the shape of the front is also seen to change. The controlled case at has a gentle slope upstream of the control at the crest, while the plunge lines at and are steep with critical conditions at the nose of the bifurcation front. The bifurcation needs to be understood in terms of its position relative to the hydraulic state. Suppose the bifurcation is downstream of the crest. As pointed out above, the Froude number is unity at the location of the bifurcation. The critical Froude number at the bifurcation does not represent a hydraulic control in the usual sense, but rather identifies the junction of two flows, one without the reduced gravity interface upstream and one with a reduced gravity interface that is supercritical immediately downstream of the bifurcation. Modification of the flow speed or density difference is accommodated by an adjustment in the position of the critical junction. The bifurcation itself does not 'control' the flow and exerts no upstream influence. We refer to this class of flows as the uncontrolled state. If the barotropic current increases, the flow is momentarily supercritical at, and just downstream of, the bifurcation and its position can no longer be sustained. It retreats downstream, where the flow is slower (equation (3.5)), until it again represents the point at which the Froude number is unity. If the barotropic flow decreases, the bifurcation is at a subcritical location and advances upstream to a position where the flow is faster. It does so

18 Stratified flow over topography Y2f 1.4- I -1 strong Y moderate moderate /3)3/2 /1507 intermediate Figure 8. Non-dimensional depths associated with the bifurcation front. To the right of Uo = 1, the normalized depth shown is the depth beneath the nose of the front y2f; for Uo < 1, the depth y20 is the height of the interface above the sill crest when the flow is controlled at the crest. The moderate, intermediate and strong flow regions correspond to the nomenclature illustrated in figure 7. Times when the flow has relaxed to an average flow are shown in italics. UO until it again reaches the point at which the Froude number is unity. In this way the bifurcation constantly adjusts its location to match the forcing. If, however, the forcing drops to a level at which the bifurcation can advance upstream of the crest, control is re-established at the crest and the bifurcation lies in subcritical flow where it represents the upstream influence of the hydraulic control at the crest. This occurs in the final stage of the flow evolution, when the combination of decreasing current and reduced stratification leads to Uo < 1; at this point the front has retreated upstream of the crest. When the tidal current decreases still further, such that U0 < (2)3/2, an exchange flow begins. The precise moment at which this occurs cannot be deduced from our data, but has evidently occurred by As is evident from figure 8, Uo is an excellent predictor of the frontal location, but our use of the non-dimensional forcing Uo masks the important physical process associated with entrainment and resulting change in the density difference. As the ebb flow progresses, entrainment into the intermediate layer leads to a reduced density

19 530 L. Armi and D. Farmer 2.0- * 0 (a) 1.0- * * O (b) 40- * C" E/ e I-- E cl 0 l l l l I l l ?* (c) 40- * 0 I- 20- n I I I I Figure 9. (a) Relative density difference across the interface at the nose of the bifurcation front when it is strongly forced, or at the sill crest for intermediate flows. (b) Tidal transport at the sill. (c) Height of interface over the sill crest for intermediate flows (below dashed line), or at the bifurcation for the strongly forced uncontrolled state (above dashed line). time difference at the bifurcation. Thus the leading edge of the bifurcation occurs at a deeper isopycnal in the upstream flow and is more sensitive to changes in barotropic forcing. This sensitivity is further enhanced by the progressively smaller slope near the crest of the topography. In the above discussion the changing density step is incorporated in the normalization of the barotropic forcing used to determine the bifurcation depth, so that the density effect itself is not explicitly illustrated in figure 8. As the bifurcation is

20 Stratified flow over topography 531 progressively forced towards and over the sill crest by the tidal current, small-scale instability leads to further entrainment from the lower to the intermediate layer. This entrainment has the effect of progressively reducing the interfacial density step, as well as the density step at the bifurcation itself. But we have already seen from the previous discussion that the position is fixed by the requirement of critical Froude number at the bifurcation. Thus any change in the density step must also alter the bifurcation position. In figure 9a, the density step is shown as a function of time and is seen to drop to one half of its initial value in a little over 1 h. We also show (figure 9b) the transport per unit breadth q and in figure 9c the actual depth Y20 of the interface over the sill crest for the controlled case and the depth Y2f at the bifurcation for uncontrolled cases. The effect of the decreasing density step is to increase the effective forcing. For example the bifurcation passes downstream over the sill crest when the transport q = 42 m2 s-1, but retreats upstream over the crest when the transport is only 27 m2 s-1, or about 0.64 as much. The normalized velocity is Uo = 1 in each case. We now test the assumption implicit in the quasi-steady analysis, in which variability of the forcing is taken to occur over time-scales which are long relative to that required for the flow to adjust to the presence of the topography. Subject to this assumption, the speed at which the bifurcation is displaced downstream by the increasing current is OXf _ayf/ot y_ yf Uo /dh Ot dh/dx Uo Ot / dx' where the position of the front Xf, and the height of the topography, h, are referenced with respect to the location of the bifurcation. We evaluate the speed at which the front moves in the neighbourhood of the crest under changing conditions of forcing for the subcritical and supercritical cases respectively. When the front is upstream of the crest, we can relate the conditions at the sill to the conditions at the intersection, using the Bernoulli equation: 2o20 + Y20 = U2f + Y2f - h. (3.8) Since Y2f - h = 1, continuity at the front implies that the depth of the intersection is defined by Y2f = Uo(3U/3-2)-1/2 (3.9) Application of L'H6pital's rule to (3.7) and (3.9) gives a frontal speed of Oxf 4 OUo /02h Ot 3 Or a~q~t~ Ox2' x2'~ a~~t ~(3.10) For the supercritical case, when the front is just downstream of the crest, the frontal speed is Xf 2-1/3 0Uo dh1 at 3 at dx(3.11) This result is finite near the crest for the subcritical case, but sensitive to the precise shape of the topography. For the Knight Inlet sill the dimensionless curvature is ca The time rate of change of the barotropic forcing is 10-4 s-1, which implies a translation speed of 6 m s-1. This exceeds the internal wave speed (ca. 1 m s-1) and thus the quasi-steady assumption must break down shortly before the bifurcation

21 532 L. Armi and D. Farmer (a) (i) E distance (m) i i i i i w v d (ii) 0 - I 10 - upste downstrea <0.2 ms < <0.6 - ~.-0.6 < <1.0 i ms r- I I I I I I I I density (at) E distance (m) W Figure 10. (a) (i) Acoustic images at 10:1 aspect ratio of ebb flow over the sill, showing transition to the strongly forced case. (ii) Corresponding 1:1 aspect ratio images of the leading edge of the bifurcation. Velocity vectors are included along with upstream (blue) and downstream (red) density profiles.

22 Stratified flow over topography 533 (b) (i) E -800 ra I distance (m) ,,,, 200! 400 w Au: (ii) 80- II III 20 - ' 30 - a downstream r-- l density (Yt) E distance (m) W Figu re 10. (b) For description see opposite.

23 534 L. Armi and D. Farmer passes over the crest. In the absence of a fully time-dependent theory we anticipate that the front is advected downstream at the speed of the tidal current since there is insufficient time for it to adjust to the quasi-steady solution. The quasi-steady solution breaks down only in this limited time of ca. 200 s and over a distance of ca. 200 m upstream of the crest. When the barotropic forcing increases, the height of the interface will rise above the sill crest. Upstream adjustment causes a wave of elevation to propagate upstream along the interface. This adjustment takes some time to reach the front, during which the increased flow advects the front downstream. The net effect is to cause a steepening of the front beyond that specified in the quasi-steady solution and this can be seen in figure 10a. This figure also illustrates the presence of internal solitary waves, which are further discussed by Farmer & Armi (1999b). Notice that the slope of the interface changes from ca. 1:10 in the period (figure l0aii) to ca. 1:1 in the period (figure l0bii). The shear layer changes from a quasi-stable entrainment interface with only small (ca. 1-2 m) instabilities having intermittent entrainment at low entrainment rates of order 0.2 (Farmer & Armi 1999a; Ellison & Turner 1959), to large entrainment rates at 1:1 slope as the mixing regime turns into a mixing layer for which the stratification is primarily important as a mechanism for accelerating the lower layer. The shape of the interface may be found from eqn (5.7) in Farmer & Armi (1999a), a result that is readily demonstrated by calculation from the observed depths. The slope of the interface near the bifurcation is particularly sensitive to the bottom slope when the Froude number is near unity. The aerial photograph shown in figure 2 was taken at 01.57, just after the CSS Vector completed the sill traverse shown in figure 10a and before the traverse shown in figure 10b. The bifurcation front is located at the beginning of the smoother water approximately five ship lengths (200 m) ahead of the ship's bow. 4. Concluding remarks The observational techniques now available allow highly resolved measurements of stratified flow over topography at Reynolds numbers beyond those readily accessible in numerical models and laboratory experiments. The process discussed by Farmer & Armi (1999a), in which a continuously stratified fluid passing over topography changes to an equivalent layered flow through a streamline bifurcation enclosing a weakly stratified intermediate layer, is here explored for the strongly forced regime in which control is lost at the sill crest. The observations illustrate aspects fundamental to the description of both moderate and strongly forced flows. As the forcing changes, the response passes through a continuum of controlled states similar to those described by Farmer & Armi (1986) for two-layer flows. The observed continuum falls into two classes: first, the moderately forced case discussed by Farmer & Armi (1999a) and second, the strongly forced case in which the bifurcation moves downstream of the crest. In our observations the quasi-steady approximation is generally valid. Therefore, any point in this continuum of controlled states represents a steady solution. If the density of the trapped intermediate layer and the strength of the barotropic forcing are known, the position and shape of the bifurcation may be calculated from an inviscid, nonlinear, steady analysis.

24 Stratified flow over topography 535 The forcing has both barotropic and internal components. The internal component depends on the history of entrainment from the lower to the upper layer, sometimes called the 'detrainment' (Baines 1998), and the circulation in the weakly stratified layer. As shown by Pawlak & Armi (1998), the character of the instability differs from the classical Kelvin-Helmholtz form, being dependent on acceleration of the lower layer. This acceleration depends in turn on the slope of the interface, which is determined by the larger-scale hydraulic response. In general, a steeper interfacial slope results in greater acceleration, which increases the entrainment rate and consequently decreases the density difference across the sheared interface. For a given barotropic transport, the effective forcing of the internal response is thus increased, causing a more rapid transition to the strongly forced state. A surprising result is that the effective forcing of the internal response may increase, even as the barotropic transport remains steady or decreases. There is a brief period during the transition between the moderate and strongly forced solutions during which the quasi-steady solution breaks down. The quasisteady assumption requires that the speed at which the bifurcation moves be small relative to the speed at which information on its equilibrium state can propagate upstream from the control towards the bifurcation. This information travels in the form of an interfacial adjustment which is limited by the long internal wave speed. If the time required for adjustment is long relative to the time required for significant displacement of the front by the tidal current, the shape of the interface will change from that given by the steady solution. In this unsteady transition the wedge-like structure of the intermediate layer will steepen through a combination of the accumulation of fluid advancing upstream and other nonlinear effects. While the detailed response during the unsteady part of the transition is not described by a steady non-hydrostatic model, a group of solitary internal waves is observed to form at this time (see figure 10 and Farmer & Armi 1999b). As in any geophysical environment there are three-dimensional aspects to the response, primarily arising at the sides of the channel. This will certainly be true in the present case, where the channel broadens considerably downstream of the sill. Since the flow is supercritical downstream of the crest, however, displacements of the supercritical interface take some distance to propagate along characteristics into the centre of the channel where our measurements were made. The Froude number varies with position, but is at least unity up to the hydraulic jump. The angle of these characteristics with respect to the downstream flow, that is the internal Mach angle, is 45? for a Froude number of unity. Thus characteristics propagating towards the centre from each shore will meet at a point at least b/2 downstream of the sill, where b is the channel breadth. For a breadth of 2 km applicable here, the characteristics would meet at least 1 km downstream of the crest, which is outside of any of the images analysed here. As discussed by Farmer & Armi (1999a), smallscale instability and entrainment across the unstable interface separating the deeper flow from the intermediate layer can be measured and was shown to be sufficient to account for the observed interfacial deepening. In the neighbourhood of the hydraulic jump three-dimensional effects can be expected, but we emphasize that in this paper, as in Farmer & Armi (1999a), these occur downstream of the dynamic process being discussed, for which a two-dimensional analysis is appropriate. A further confirmation of the two-dimensional character of the flow is provided by the aerial photograph shown in figure 2. The plunge line in the acoustic image

25 536 L. Armi and D. Farmer Park o/// Range ont Di.*-distance in nautical miles-- I, I I I riciu EAS^T t IDenver) EAST I 50 I Figure 11. Interpolated isotachs adapted from Lilly (1978), showing trapped mixed layer of low speed, located well downwind of the continental divide, indicative of strongly forced flow. (figure 10b) corresponds to a modulation of the surface gravity-capillary wave field which is essentially straight in the aerial photograph and aligned with the sill. In shallow and stratified coastal waters it appears that topographically trapped fronts are widespread (cf. Largier 1992; Stephens & Imberger 1997; Farmer & Denton 1985). The situation in the atmosphere is less easy to infer, due to the great difficulty of acquiring comprehensive data in mountain flows. Examination of the classical data of Lilly (1978), figure 11, illustrates features that might be interpreted as similar to the strongly forced examples. Specifically, the position of the intermediate layer is well downstream of the continental divide and the wind speeds and density difference together imply a non-dimensional barotropic forcing, Uo = 1. In this case it must be pointed out that the stratification aloft is weaker, so that in contrast to the modest vertical displacements at the upper interface in our example, quite large excursions in the tropopause might be expected. We thank the officers and crew of the CSS Vector for their assistance in the field program. We are also indebted to those who helped with different aspects of the field program including operation of the RV Gnarly, the floatplane and data analysis. We are especially grateful to Kevin Bartlett, Ward Cartier, Nancy Hurlbirt, Grace Kamitakahara-King, Geno Pawlak, Rich Pawlowicz and Les Spearing. Financial support from the US Office of Naval Research is gratefully appreciated. References Afanasyev, Y. D. & Peltier, W. R. 2001a On breaking internal waves over the sill in Knight Inlet. Proc. R. Soc. Lond. A457, Afanasyev, Y. D. & Peltier, W. R. 2001b Reply to comment on the paper: on breaking internal waves over the sill in Knight Inlet. Proc. R. Soc. Lond. A 457,

26 Stratified flow over topography 537 Armi, L. & Farmer, D. M Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, Armi, L. & Williams, R The hydraulics of a stratified fluid flowing through a contraction. J. Fluid Mech. 251, Baines, P. G Topographic effects in stratified flows. Cambridge University Press. Baines, P. G Downslope flows into a stratified environment-structure and detrainment. In Mixing and Dispersion in Stably Stratified Flows: Proc. 5th IMA Conf. on Stratified Flows, Dundee (ed. P. A. Davies), pp Oxford: Clarendon. Benjamin, T. B Gravity currents and related phenomena. J. Fluid Mech. 31, Cummins, P. F Stratified flow over topography: time-dependent comparisons between model solutions and observations. Dynam. Atmos. Oceans 33, Ellison, T. H. & Turner, J. S Turbulent entrainment in stratified flows. J. Fluid Mech. 6, Farmer, D. M. & Armi, L Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow. J. Fluid Mech. 164, Farmer, D. M. & Armi, L. 1999a Stratified flow over topography: the role of small-scale entrainment and mixing in flow establishment. Proc. R. Soc. Lond. A 455, Farmer, D. M. & Armi, L. 1999b The generation and trapping of solitary waves over topography. Science 283, Farmer, D. M. & Armi, L Stratified flow over topography: models versus observations. Proc. R. Soc. Lond. A 457, Farmer, D. M. & Denton, R. A Hydraulic control of flow over the sill in Observatory Inlet. J. Geophys. Res. 90, Farmer, D. M. & Smith, J. D Tidal interaction of stratified flow with a sill in Knight Inlet. Deep Sea Res. A 27, Gill, A. E Atmosphere-ocean dynamics. Academic. Hunt, J. C. R. & Snyder, W. H Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech. 96, Huppert, H. E. & Britter, R. E Separation of hydraulic flow over topography. J. Hydraul. Div. Am. Soc. Civ. Engrs 108, Largier, J. L Tidal intrusion fronts. Estuaries 15, Lilly, D. K A severe downslope windstorm and aircraft turbulence event induced by a mountain wave. J. Atmos. Sci Nash, J. D. & Moum, J. N Internal hydraulic flows on the continental shelf: high drag states over a small bank. J. Geophys. Res. 106, Pawlak, G. & Armi, L Vortex dynamics in a spatially accelerating shear layer. J. Fluid Mech. 376, Pawlak, G. & Armi, L Mixing and entrainment in developing stratified currents. J. Fluid Mech. 424, Peltier, W. R. & Clark, T. L The evolution and stability of finite-amplitude mountain waves. Part II. Surface wave drag and severe downslope windstorms. J. Atmos. Sci. 36, Queney, P The problem of airflow over mountains: a summary of theoretical studies. Bull. Am. Meteorol. Soc. 29, Scorer, R. S Theory of airflow over mountains. IV. Separation of flow from the surface. Q. J. R. Meteor. Soc. 81, Smith, R. B On severe downslope winds. J. Atmos. Sci. 42, Stephens, R. & Imberger, J Intertidal motions within Deep Basin of Swan River Estuary. J. Hydraul. Engng 123,

27 538 L. Armi and D. Farmer Sykes, R. I Stratification effects in boundary layer flow over hills. Proc. R. Soc. Lond. A 361, Von Karman, T The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, Wood, I. R Selective withdrawal from a stably stratified fluid. J. Fluid Mech. 32,