J. Fluid Mec. (24), vol. 55, pp. 45 443. c 24 Cambridge University Press DOI:.7/S2224576 Printed in te United Kingdom 45 Mixing and entrainment in ydraulically driven stratified sill flows By MORTEN HOLTEGAARD NIELSEN, LARRY PRATT AND KARL HELFRICH Woods Hole Oceanograpic Institution, MS 2, 36 Woods Hole Road, Woods Hole, MA 2543, USA (Received 6 Marc 23 and in revised form 2 May 24) Te investigation involves te ydraulic beaviour of a dense layer of fluid flowing over an obstacle and subject to entrainment of mass and momentum from a dynamically inactive (but possibly moving) overlying fluid. An approac based on te use of reduced gravity, sallow-water teory wit a cross-interface entrainment velocity is compared wit numerical simulations based on a model wit continuously varying stratification and velocity. Te locations of critical flow (ydraulic control) in te continuous model are estimated by observing te direction of propagation of small-amplitude long-wave disturbances introduced into te flow field. Altoug some of te trends predicted by te sallow-water model are observed in te continuous model, te agreement between te interface profiles and te position of critical flow is quantitatively poor. A reformulation of te equations governing te continuous flow suggests tat te reduced gravity model systematically underestimates inertia and overestimates buoyancy. Tese differences are quantified by sape coefficients tat measure te vertical non-uniformities of te density and orizontal velocity tat arise, in part, by incomplete mixing of entrained mass and momentum over te lowerlayer dept. Under conditions of self-similarity (as in Wood s similarity solution) te sape coefficients are constant and te formulation determines a new criterion for and location of critical flow. Tis location generally lies upstream of te critical section predicted by te reduced-gravity model. Self-similarity is not observed in te numerically generated flow, but te observed critical section continues to lie upstream of te location predicted by te reduced gravity model. Te factors influencing tis result are explored.. Introduction Te concept of ydraulic control plays an enormous role in understanding flow troug a constriction and te influence it as on te basin circulation at bot ends. Hydraulic teory was originally developed for engineering purposes, but wit Stommel & Farmer s (953) study of estuary flow and Long s (954) towing experiments, tis penomenon gained te attention of oceanograpers and meteorologists. An appealing tougt in current geopysical researc is tat ydraulic control may exist on te boundaries between te deep basins and marginal seas of te world s oceans and so be applied in te study of global termoaline circulation and climate variability. Tus, from measurements of stratification in te Norwegian Sea, Hansen, Present address: Institute for Hydrobiology and Fiseries Science, University of Hamburg, Olbersweg 24, 22767 Hamburg, Germany. Autor to wom correspondence sould be addressed: lpratt@woi.edu.
46 M. H. Nielsen, L. Pratt and K. Helfric Turrell & Østerus (2) ave made suggestions about long-term canges of te dense flow of Arctic water into te Nort Atlantic. Assumptions often made in ydraulic teories include te neglect of friction and mixing and tat te water column consists of a number of layers, eac aving uniform density and velocity. Important results ave been obtained in tis way, including te significance of te composite Froude number and te concept of multiple controls (e.g. Farmer & Armi 986). Some elementary effects of friction ave been discussed by Pratt (986) wo sowed tat simple quadratic bottom drag forces te flow towards criticality and displaces te control section downstream of its usual location (suc as a sill or narrows). In naturally occurring flows, mixing can also play an important role. Prominent examples are te excange flow troug te Strait of Gibraltar, in wic an intermediate layer is formed by mixing between te inflow and outflow (Bray, Ocoa & Kinder 995), and te Denmark Strait overflow, wic is diluted as it descends into te Nort Atlantic (e.g. Käse & Osclies 2; Girton, Sanford & Käse 2). Tese examples sow ow mixing works to alter vertical gradients and induce canges in te flow properties in te along-cannel direction. In an attempt to capture te essence of tese effects wile retaining te simplicity of layer models, some investigators introduce a cross-interface entrainment velocity tat carries mass and momentum from one layer to te next. Wile quite common in studies of general circulation (e.g. Pedlosky 996, cap. 3), tis formulation as not been widely used in ydraulic teory. Gerdes, Garrett & Farmer (22) ave discussed te consequences of an entrainment velocity for a single active layer. Fluid is allowed to enter te layer across its upper interface and it is assumed tat te anomalous density and momentum are instantly mixed over te tickness of te layer. Te active layer tus remains vertically omogeneous in density and orizontal velocity, altoug bot quantities are allowed to vary in te along-cannel direction. For te case in wic te upper layer is motionless, it is sown tat entrainment acts in a similar way to bottom friction, forcing te flow toward criticality and sifting te control point downstream from a sill or narrows. (Tese tendencies may cange, owever, if te upper layer is in motion.) In reality, te mixing process associated wit entrainment may not be complete. Fluid may be mixed over part of te water column, but not necessarily all te way to te bottom or instantaneously. In te Bab al Mandab, wic is tougt to contain significant mixing over its lengt, tere exists smootly varying density and velocity over te water column and no distinct interface (Murray & Jons 997; Pratt et al. 999, 2). Suc observations call into question te use of layer models. Te purpose of tis work is to evaluate te performance of a layer (slab) model in te presence of entrainment and incomplete vertical mixing. We will concentrate on ydraulically driven flows, wic differ from tose broader flows normally associated wit general ocean circulation in aving muc stronger inertial effects and more intense mixing. Some insigt into te questions raised above is provided by Garrett & Gerdes (23) wo looked into te ydraulics of a omogeneous two-dimensional flow wit vertical sear (and no entrainment). In te absence of friction, it is sown tat critical flow occurs at te sill of an obstacle, as wit a slab model. Te critical condition itself can be expressed as an integral over te layer dept d: +d g dz =. u2
Mixing and entrainment in ydraulically driven stratified sill flows 47 It can be sown from tis formula tat te dept average velocity ū is (gd) /2,so tat te flow at te control section appears by traditional measure to be supercritical. Te effects of bottom drag and internal friction are discussed by Garrett (24) wo argues tat te critical condition is uncanged, but tat te position of te control section is sifted to a point downstream of te sill. Bottom drag forces te control section downstream wereas internal friction does te opposite. In te limit of large viscosity, te internal flow is rendered slab-like and te control-section position is completely determined by bottom drag. In tis case, te control section is predicted to lie were te bottom slope equals te negative of te quadratic drag coefficient, also te result obtained from a pure slab model (Pratt 986). For weak internal friction te flow becomes strongly seared and te effect of internal dissipation on te position of te control section becomes as strong as tat of bottom drag. In tis case, te control section is forced back upstream to near te sill. We will revisit tis last effect in te discussion of our results. To gain insigt into te utility of te layer model wit entrainment, we compare predictions based on tis approac to numerical simulations based on a model wit continuous stratification. We first review ( 2 and 3) te results of Gerdes et al. (22) and sow some examples constructed by applying teir teory wit te entrainment parameterization of Ellison & Turner (959). We next present a series of numerical results from te continuous model sowing non-ydrostatic continuously stratified excange flow over an obstacle ( 4). Tese flows are set up troug a lock-excange experiment, configured to produce an overflow across te sill wit a relatively inactive reverse flow above. (An independent barotropic flow can be added to increase te strengt of te upper-layer flow.) Non-ydrostatic effects are allowed in order to study variations in te aspect ratio of te model, but most runs involve small ratios and are nearly ydrostatic. Te critical section for te overflow is estimated (following Hogg, Ivey & Winters 2a) by introducing small-amplitude waves into te fluid at different locations and determining te point at wic upstream propagation is cut off. We ten attempt to fit te overflow to a layer representation by defining an upper interface tat bounds te overflowing fluid. Te sape of tis interface and te location of te critical section are compared to te results obtained by integrating te equations of Gerdes et al. (22) beginning wit common upstream conditions. Altoug predicted trends are found in te continuous model, substantial quantitative disagreement is found ( 5) between te two solutions in terms of te sape of te interface and te position of te control section. We attempt to isolate te source of te disagreement between te continuous and layer model by reformulating te sallow-water equations, taking into account vertical variations of velocity and density witin te active layer ( 6). Te departure from te ordinary sallow-water equations is contained in tree sape parameters α, β and γ tat depend on te vertical profiles of orizontal velocity and density and tat vary wit te along-cannel coordinate. Tese parameters appear as coefficients in te sallow-water equations for te vertically averaged properties of te active layer. It can be proved tat α and (for ydrostatically stable stratification) tat β, and tese in turn imply tat te flow tends to be more inertial tan wat would be indicated by te vertical mean velocity and density. Sallow-water teory is recovered wen α = β = γ =, but it is sown tat significant departures from tis condition occur in te numerical simulations and in some geopysical applications ( 7). In oter words, te effects of vertically varying density and orizontal velocity lead to important canges in te budgets for momentum and mass tat make te flow more inertial tan would be expected from ordinary sallow-water teory. In addition,
48 M. H. Nielsen, L. Pratt and K. Helfric u ρ = ρ u 2 (x) d(x) ρ = ρ 2 (x) z (x) x Figure. Definition sketc sowing a deep omogeneous layer flowing beneat an inactive upper layer. it is sown tat under conditions of constant α, β and γ, te critical condition is F 2 = β/α, were F is te Froude number based on te dept-averaged flow. Te section were tis condition occurs lies upstream of te location at wic F 2 =. Constant α, β and γ occur under conditions of self-similarity, wic is not found in te simulations, but can occur under te conditions described by Wood (968). As it turns out, te location predicted by tis criterion lies quite close to te critical section identified in te full numerical simulations ( 6). Te findings of Garrett (24) suggest tat agreement could be fortuitous, a consequence of internal frictional effects not present in te slab model. 2. Governing equations for te layer system Te Gerdes et al. (22) formulation applies to a ydrostatic Boussinesq flow in a deep layer tat is fed by entrainment from an overlying dynamically inactive upper layer (figure ). Te flow is confined to a cannel wit bottom elevation z = (x) and widt tat is assumed ere to be constant. Te upper layer (layer ) may ave a non-zero velocity u, but tis velocity is spatially uniform and unaffected by canges in te lower layer. Te entrainment process is represented by an (positive downwards) entrainment velocity w e tat carries mass and momentum across te interface separating te layers. We consider cases of entrainment into te lower layer (w e > ), but not detrainment from te lower layer (w e < ). Momentum and mass carried across te interface are assumed to mix instantaneously over te dept of te lower layer, so tat te lower-layer density ρ 2 and orizontal velocity u 2 remain vertically uniform. Te lower-layer density will, owever, vary orizontally, as will te value of reduced gravity g (x)=g(ρ 2 (x) ρ )/ρ. Under tese conditions, te evolution of te lower-layer velocity, tickness (d), and Froude number (F = u 2 / g d) are given by (Gerdes et al. 22) du 2 dx = F 2 ( F 2 )d d(d) dx = F 2 ( F 2 ) {w e ( + { 2 w e u 2 2F u ) + u 2 d 2 u 2 F 2 dx ) d F 2 dx ( 4F 2 + u 2u 2 }, (2.) }, (2.2)
Mixing and entrainment in ydraulically driven stratified sill flows 49 3 2 z 3 2 2 3 x Figure 2. Solutions to (2.) (2.3) in terms of interface elevation for flow wit u =, w e =, u 2 d = and wit various upstream depts. Subcritical (supercritical) solutions are indicated by solid (dased) curves. Te topograpy consists of te single Gaussian saped obstacle sown as te lowest curve. All quantities are non-dimensional. and { ( df 2 dx = 3F 2 we F 2 + ( F 2 )d u 2 2 u ) F 2 + d }. (2.3) u 2 dx Let s represent te obstacle eigt, L te obstacle lengt, and g te value of g at some upstream section x = x. Ten te above set of dimensional equations may be considered non-dimensional if te following replacements are made: x x/l, g g, (d,) (d,)/ s, (u,u 2 ) (u,u 2 )/(g s ) /2 ; w e w e L/(g 3 s) /2. Te non-dimensional version will be adopted ereinafter and terefore te obstacle eigt and lengt and te upstream value of g will all be considered unity. To illustrate some of te properties of solutions to te above, we will consider te flow over a simple obstacle (figure ). Te values u 2 (x )=u,d(x )=d,andf(x )=F are specified at an upstream location x = x and (2.) (2.3) are integrated to find te downstream solution. If tis is done for te case w e =, a set of standard ydraulic solutions are obtained (figure 2). Te values of u 2 d,g ( = ) and te Bernoulli function u 2 2 /2+g d +g are conserved for eac solution. For te particular family of solutions sown, te volume flow rate per unit widt u d as been set to unity and d is varied. For d > 2.4, te solutions are completely subcritical, meaning tat u 2 is everywere less tan te speed ( g d) of long gravity waves. As sown in figure 2, te interface elevation dips down and ten rises as te fluid passes te obstacle. A family of supercritical solutions (u 2 > g d) exists for d <.5 and tese experience a rise in interface elevation as te obstacle is passed. Also present are te ydraulically controlled solution (d =2.4...), wic is subcritical upstream and supercritical downstream of te obstacle, and its (unstable) mirror image for d =.5. A more complete review of te properties of tese solutions appears in Baines (995). Wen entrainment or bottom drag is present in te slab model, it is natural to ask weter u 2 (g d) /2 remains te wave speed for wic ydraulic properties are judged.
42 M. H. Nielsen, L. Pratt and K. Helfric Altoug u =(g d) /2 would appear, on te basis of (2.) (2.3), to be te correct critical condition, it as not yet been sown tat tis corresponds to te presence of stationary waves. Moreover, Garrett & Gerdes (23) argue tat te introduction of non-conservative processes in a sallow-water slab model gives rise to a family of dispersive long waves, some of wic ave upstream group velocity exceeding u (g d) /2. Tis result would appear to cloud te traditional idea tat upstream propagation cannot occur at a control section. Altoug teir result is based on a model wit quadratic bottom drag, te same issues are raised wen entrainment is present. Tese issues can be laid to rest by considering te full time-dependent equations for a sallow-layer subject to quadratic bottom drag and entrainment (Appendix A). Teir caracteristic form is [ t + ( u 2 ± (g d) /2) ] (u2 ± 2(g d) /2) ( ) /2 [ d x g [ = g d dx C u 2 u 2 u u 2 d + w e d d t + ( u 2 ± (g d) /2) ] g x ( ) ] + g g /2 ± (2.4) 2u 2 d and [ ] t + u 2 g = g w e x d, (2.5) were C d is te bottom drag coefficient. Tere are tree caracteristic speeds: u 2 ± (g d) /2 and u 2, and a sketc of te corresponding caracteristic curves for a ypotetical steady flow over an obstacle appears in figure 3. Tese curves represent te pats tat small-amplitude perturbations of te steady flow would take in te (x,t)-plane. Te dark solid curves correspond to signals wit speed u 2 (g d) /2, wile te dased and faint solid curves correspond to speeds u 2 +(g d) /2,andu 2.Te latter two ave positive tilt everywere, corresponding to downstream propagation of information. It is assumed tat u 2 =(g d) /2 at a location sligtly downstream of te sill and tis is represented by a vertical solid curve in te (x,t)-plane. Upstream propagation is possible only to te left of tis line. Te above formulation is made possible because bottom drag and entrainment do not involve x- ort-derivatives of te dependent variables. Under tese conditions, information propagates along caracteristic curves tat are defined in te same way as if te flow were conservative. Te effect of non-conservation is tat te signals temselves are modified as tey propagate. Even so, te solution at any location upstream of te sill is influenced only by information tat exists upstream of te sill. For example, te solution at location B (figure 3) is influenced only by initial conditions lying along AC. It would appear ten tat u 2 ± (g d) /2 and u 2 are te only signals speeds relevant to te ydraulics of te flow and tat apparent dispersive beaviour as identified by Garrett & Gerdes (23) is really a manifestation of te alteration of information along te signal pats by entrainment, bottom topograpy or, in teir case, bottom friction. A related principle tat can guide te selection of wavelengt relevant to ydraulic criticality is tat of non-dispersion. Wen forced locally, a stationary non-dispersive wave is unable to transport energy away from te site of forcing, resulting in resonance. Tis process is evident in te rigt-and sides of (2.) (2.3), wic indicate resonant excitation wen te speed u 2 (g d) /2 of a non-dispersive wave becomes zero. Resonant growt is avoided only wen te sum of te forcing terms (te numerators)
Mixing and entrainment in ydraulically driven stratified sill flows 42 t B C A Critical section x u(y) m Figure 3. Te lower frame sows a ydraulically controlled flow in wic entrainment and/or bottom drag ave caused te critical section to be located downstream of te sill. Te upper frame sows te caracteristic curves for te steady flow, also te pats in te (x,t)-plane along wic small amplitude disturbances to te steady flow would travel. Te caracteristic speeds are given by u 2 (g d) /2 (tick, solid curves) u 2 +(g d) /2 (dased) and u 2 (tin, solid). adduptozero: ( w e 3 u 2 2 u ) = d (F 2 =), (2.6) u 2 dx wic follows from any of (2.) (2.3). If w e =, ten te only momentum source is te orizontal component of te bottom pressure, ere d/dx. Critical flow must occur were tis slope is zero, at te sill in figure. Entrainment provides a source of mass and of momentum (if u u 2 ) to te lower layer. If u is non-positive, critical flow must occur on te downslope d/dx< of te obstacle. Critical flow can occur on te upslope d/dx> of te obstacle if te upper-layer velocity is positive and sufficiently strong (u > 3u 2 /2). If te magnitude of te left-and side of (2.6) is everywere larger tan te maximum bottom slope, ten critical flow will be completely expunged from te problem and no ydraulic transitions will occur. (Hogg et al. (2a) ave sown tat ydraulic beaviour is expunged from a viscous, stratified, excange flow for sufficiently small values of te parameter g 5 s/ν 2 L 2 were
422 M. H. Nielsen, L. Pratt and K. Helfric ν is te vertical eddy vicosity. If we interpret g s as a scale for velocity an ν/ s as a scale for w e ten te condition (2.6) leads to a similar conclusion.) 3. Examples of solutions wit Ellison & Turner entrainment A standard parameterization for te entrainment velocity is tat due to Ellison & Turner (959, ereinafter referred to as ET). In te present non-dimensional units, w e is given by u u 2 L ( ).8.Ri (R i <.8), w e = s +5R i (3.) (R i.8), were ( ) 2 R i = F 2 u. (3.2) u 2 If te upper layer is motionless (u =) ten F = Ri 2 and te requirement R i <.8 means tat entrainment only occurs for supercritical flows. Use of te ET expression for w e in (2.6) leads to a more specific constraint on te position of a critical section: d dx = u ( 3 u 2 2 u ) [ ] L.8(u /u 2 ) 2. (R u 2 s (u /u 2 ) 2 i <.8), +5. (3.3) (R i.8). According to (3.2) (wit F = ), te condition R i.8 is satisfied if u /u 2 > 2.2 or if u /u 2 <.2. In te first case, te upper layer flows in te same direction and at a greater speed tan te lower-layer critical flow, and te critical section lies on te upslope of te obstacle. In te second case, te upper-layer velocity is negative and te critical section lies on te downslope of te obstacle. It is interesting to observe ow te conservative solutions of figure 2 are altered wen ET entrainment is introduced. To tis end we ave calculated a family of solutions by fixing te upstream (x = 3) values of u 2 d and of g at unity and varying te upstream value of d, as before. Te value of L/ s is fixed at 5 and solutions are obtained by integrating te dimensionless versions of (2.) (2.3) from x = 3 in te direction of positive x. Wen te upper layer is motionless (u =) te subcritical (solid curves) solutions remain uncanged from te previous case, as sown by te curves of interface elevation (figure 4a) and Froude number (figure 4b). On te oter and, te supercritical (dased) solutions suc as te one wit d( 3) =.5 experience rapid increases in dept owing to entrainment. Critical flow at te sill is obtained wen te upstream flow is subcritical and as value d( 3) = 2.4 or wen te upstream flow is supercritical and as value d( 3) =.26. In eac case, te subcritical and supercritical brances of te solution tat occur downstream of te critical section are sown. Te appropriate coice of downstream solution is te one tat allows te fluid to pass smootly troug te critical section. For example, we would follow te subcritical (solid) curve beginning at d( 3) = 2.4 and continue on to te supercritical (dased) branc downstream of te sill. Te subcritical downstream branc as an upstream continuation wit values u 2 d and g different from tose used to generate te family of curves in figure 3(a). Tis solution is unstable, as is te supercritical-to-subcritical solution wit d( 3) =.26.
Mixing and entrainment in ydraulically driven stratified sill flows 423 (a) 4 3 z 2 3 2 2 3 (b) 4 3 F 2 3 2 2 3 x Figure 4. (a) Same as figure 2 except tat w e is given by (3.) wit L/ s =5.(b) Te Froude numbers F =(g d) /2 corresponding to te solutions sown in (a). Intersections between different solution curves in figure 4(a) do not carry te same significance as would be te case in a conservative system. In te latter, intersections imply te existence of two solutions wit te same dept and fluxes, but different interface slopes. Suc beaviour is indicative of critical flow since it implies tat stationary disturbances can exist at te section in question. An example is te intersection point corresponding to critical sill flow in figure 2. For te (nonconservative) solutions sown in figure 4(a), an intersection implies only tat te depts of te two solutions are equal, not necessarily te fluxes or values of g. For example, te intersection between te dased curves near x =.9 involves two solutions wit identical depts but different Froude numbers (as sown in te figure 4b). As pointed out by Gerdes et al. (22), entrainment tat occurs in te presence of non-positive u tends to pus te flows towards criticality. Tis property can be
424 M. H. Nielsen, L. Pratt and K. Helfric (a) 7 6 5 z 4 3 2 (b) 3 2 2 3 3 2 F 3 2 2 3 x Figure 5. As figure 4 except tat u =. seen by inspection of (2.3), wic sows tat F 2 / x < (>) wen F 2 > (<), not accounting for te influence of topograpy and provided u /u 2. Tis beaviour is confirmed by te Froude number beaviour of te solutions (figure 4b). Te previous case as u =, so entrainment occurs only wen te flow is supercritical. An important consequence is tat critical flow can occur only at te sill. We now consider two cases wit finite upper velocity, u = (figure 5) and u =3 (figure 6). Inspection of figure 5(a) sows tat critical transitions occur downstream of te sill as predicted by (3.3). As in te previous case, entrainment tends to pus te solutions towards a critical state (figure 5b) and, in te case of some of te supercritical curves, tis results in te formation of an infinite interface slope corresponding to a ydraulic jump. Jumps are represented in te figure by vertical terminations of te dased curves. Te case of u > (figure 6) is less straigtforward because it is more difficult to generalize te tendency for entrainment to pus te flow towards or away from criticality. According to (2.3), te tendency is to force te flow away from te critical
Mixing and entrainment in ydraulically driven stratified sill flows 425 (a) 5 4 z 3 2 (b) 3 2 2 3 3 2 F 3 2 2 3 x Figure 6. As figure 4 except tat u =3. state if u is sufficiently large tat ( u /u 2 )F 2 < 2. Also, (2.6) sows tat displacement of te critical section upstream requires u /u 2 > 3/2. Te solutions sown in figure 6 were computed wit a strong upper layer velocity (u 2 = 3) resulting in satisfaction of bot conditions. Te traditional, ydraulically controlled solution begins wit upstream dept =.5 and remains subcritical until x =.2 were a transition to supercritical flow occurs (figure 6a). As it turns out, tere is anoter control section rigt at te upstream boundary x = 3 were te dept is =. Sub- and supercritical solution curves branc from tis point, but only te supercritical branc is continuous across te entire obstacle. Te Froude number curves of figure 6(b) sow a tendency for solutions to move away from criticality, except were topograpy may reverse tis trend.
426 M. H. Nielsen, L. Pratt and K. Helfric 4. Solutions based on a numerical model wit continuous stratification To wat extent do te solutions of te previous section reproduce te caracteristics of continuously stratified deep overflows wen te vertical mixing of entrained fluid is incomplete? We seek a limited answer to tis question by performing a set of numerical experiments based on te incompressible Boussinesq equations in two dimensions: U +(U )U = p g + (K m U), (4.) t ρ ρ g + U g = (K s g ), (4.2) t and U =, (4.3) in wic U is te velocity vector, ρ is te reference density, and g and p are te perturbation quantities of te density and te pressure, respectively, all variables according to te non-dimensionalization introduced in 2. K m and K s are te diffusion coefficients for momentum and density, respectively. Te numerical sceme used to solve tese equations is te second-order projection metod described by Bell et al. (989a,b) and Bell & Marcus (992). Essentially, te projection is to estimate U at te new time step n + using te pressure gradient calculated from te preceding alf time step n /2, i.e. te pressure gradient is taken as a source term in (4.). Tis estimate for U will be divergent, and so te non-divergent part, obtained as te curl of U, is removed. Tis part is ten used to find te perturbation pressure gradient at te new alf time step n +/2. Te diffusion coefficients K m and K s are calculated according to Smagorinsky (963). Te Smagorinsky constant, wic is used to calculate K m from te computed velocity and density gradients, is taken to be.7 (Winters & Seim 2). Also following Winters & Seim, te turbulent Prandtl number is taken as unity, and so K m = K s. Te model domain is bounded by rigid impermeable walls at te top and bottom and as open boundaries at te ends. Te top is orizontal, and te bottom includes a topograpic obstacle described by ( ( ) ) 2 L = exp x, (4.4) s in wic is scaled by s,x by L, and were s /L is te obstacle aspect ratio. At te top and te bottom we operate wit a slip velocity, in accordance wit te neglect of surface drag in (2.) (2.3). Te computational domain consists of an ortogonal curvilinear grid following te topograpy. It is created from te model domain by te metod described by Ives & Zacarias (987). Tis metod conformally maps grid points designated on te boundaries of te model domain (in Cartesian coordinates) onto a rectangle. Ten, te interior points of te grid are calculated using a Poisson solver tecnique. For te present purpose, te grid points on te vertical boundaries are distributed so tat a ig density of grid points is obtained in te lower part of te pysical domain, wic is were te active layer of fluid is found. Along te top and bottom boundary, te grid points are distributed wit constant spacing. We ave used different computational domains, all wit 5 grid points in te vertical and grid points in te orizontal, but wit different lengts and eigts
Mixing and entrainment in ydraulically driven stratified sill flows 427 of te obstacle. In every case, te total lengt of te domain was times or more te lengt of te obstacle. Te eigt of te obstacle was eiter.5 or. of te total eigt of te domain. Wit lower-layer depts up to twice te obstacle eigt, we ave kept te lower layer approximately decoupled from te upper layer. Te initial conditions for all our model runs consist of a two-layer stratification trougout te model domain. To te left of te obstacle, te lower-layer dept is set to twice te obstacle eigt, and to te rigt te lower layer is kept sligtly below te crest of te obstacle. Furter, bot fluids are stagnant initially, and so tese experiments resemble a dam-break problem. In case of inflow troug te open boundaries (resulting from adjustments witin te model domain or a barotropic flow troug it), te densities assigned initially at tese places are used. Sponge layers are also applied at bot ends in order to damp reflections from te open boundaries. Finally, we control te barotropic flow component by prescribing te value of te streamfunction on te upper boundary. All model results presented are of flows tat ave developed to steady state and are smoot wit no turbulence. In times immediately after initiation of te flow, owever, overturning Kelvin Helmoltz billows are present and resolved. Tese billows are subsequently dissipated by te mixing sceme, leaving te smoot steady flow. Te coice of te non-ydrostatic model and te mixing sceme was made partly for convenience, but also to maintain a connection wit te modelling of Hogg, Winters & Ivey (2b). However, te coice of model is not crucial. Any model tat produces incomplete mixing of density and momentum troug te lower layer and from wic te vertical entrainment can be determined is adequate to test te central ideas. Weter te model produces more (or less) mixing tan would occur in a laboratory or oceanograpic flow is of secondary importance. As it turns out, te redistribution of momentum and velocity as a result of mixing produced in te model is qualitatively similar to te distributions observed in te Romance fracture zone overflow ( 7), but tis is not by design. Te model is allowed to run rougly until te disturbances from te initial dam break ave reaced te ends of te domain. At tis time, te flow becomes nearly steady in te vicinity of te obstacle. Tree different obstacle aspect ratios are used ( s /L =.4, 3.54 2 and.77 2 ) and te upper-layer velocity is varied witin eac case by prescribing te barotropic flow. Te main restriction on te barotropic component is tat it sould not be so large as to make te upper layer dynamically active. We impose tis restriction by requiring tat te upper-layer Froude number must remain muc smaller tan te lowerlayer Froude number. By performing a suite of experiments for eac s /L, we map out a parameter space in wic te general caracter of te final flow can be categorized. One quantity of particular interest is te location of a critical section. Consider an example of te model output for a run wit an aspect ratio s /L =.4, a domain eigt of 2 s and wit zero barotropic flow (figure 7). Te overall pattern consists of a positive lower level flow tat spills over te obstacle. Te overflow underlies a reverse flow wit a velocity u tat is everywere muc less in magnitude tan u 2. Entrainment occurs were te lower layer accelerates and flows over te obstacle, as sown in te pattern of subducted streamlines (figure 7c). Te effect of te entrainment on te distributions of density and velocity can be seen in figures 7(a) and7(b), respectively. For te density, te interfacial area between te
428 M. H. Nielsen, L. Pratt and K. Helfric (a) 3 2 Dept Dept (b) 3 2.2.4.6.8..2..2.3.4.5.6.7 (c) 3 Dept 2 2..5..5.5..5 2. Along-cannel distance Figure 7. Output of te non-ydrostatic model run for an obstacle aspect ratio at.4 and an upper-layer flow at almost zero. (a) Contours of density anomaly between te upper and te lower layer; (b) contoursofspeed;(c) te streamlines. All units are non-dimensional according to 2. two omogeneous layers increases gradually and ends up extending all te way to te bottom. For te velocity, as te flow approaces te obstacle, sligtly iger values are found in te upper part of te lower layer. Tis seems to cange as entrainment increases; te igest velocities gradually being found near te bottom. Note tat te lower layer is not at all omogeneous in density and velocity as entrainment takes place. In order to diagnose te model output in te frame of reference provided by (2.) (2.3), a layered structure is defined. Te interface is defined as te location of te isopycnal g =. (in te non-dimensional formulation), wic is close to te density of te upper layer, and so practically all parts of te fluid tat ave undergone mixing will be considered as belonging to te lower layer. We also attempted to define an interface based on te level of zero orizontal velocity, but te resulting contour was quite irregular. Having defined te interface, te mean velocity ū 2, te mean reduced gravity ḡ, and te Froude number F for te lower layer are calculated as follows: ū 2 = d ḡ = d +d +d u 2 dz, (4.5) g dz, (4.6)
Mixing and entrainment in ydraulically driven stratified sill flows 429 (a) u and ρ 2.5 2..5..5 + d (b).5.3 u and ρ..5 ρ u 2 w e.2. w e (c).5 u and ρ..5 β/α F 2 2..5..5.5..5 2. Along-cannel distance Figure 8. A layered interpretation of te model output sown in figure 7. (a) Te eigt of te obstacle (tick line) and te interface elevation (tin line); (b) mean velocity, ū 2 (solid line), te mean density anomaly, ρ (long dases), and te entrainment velocity, w e (sort dases use axis on te rigt); (c) te Froude number, F 2 = ū 2 2 /(ḡ d) (tick line), and β/α (solid line), defined in 6. and F 2 = ū2 2 ḡ d. (4.7) Te entrainment velocity (w e ) is calculated using conservation of volume for te lower layer and it is verified tat te result closely approximates to te vertical velocity across te interface produced by te model. Wen determining te location of te interface, noise arises in te along-cannel direction owing to te discrete representation of te density field. Tis noise is removed by applying a moving average to te variables pertaining to te layered description. Te interface elevation and mean velocity and density obtained using te above procedure wit te figure 7 model run are sown in figure 8. A good measure for te degree of mixing tat is taking place is te mean density, non-dimensionalized by its value at x = 2. and sown in te middle panel in te form ḡ. Te latter is seen to decrease by more tan 5% over te region were w e is largest (also sown figure 8b), implying a decrease by more tan a factor of 2 in te mean layer density. Te Froude number F based on te mean flow (figure 8c) reaces te value unity at x =.56. To estimate te location of te actual control point for eac model run, a number of wave excitation experiments were carried out. A small-amplitude approximately ydrostatic disturbance was introduced in different locations near te crest of te obstacle by raising te isopycnals a small amount. Te disturbance ad te sape of
43 M. H. Nielsen, L. Pratt and K. Helfric one sine wave and was fitted smootly to te steady solution. Te bottom and rigid lid pressures were adjusted to eliminate any barotropic component (wic would travel upstream at an infinite speed). Ten, te model was run forward and te resulting wave-propagation pattern was found by subtracting te undisturbed flow field. Te purpose of te experiments is to determine te section at wic small-amplitude long waves remain stationary. Te procedure is similar to tat used by Hogg et al. (2b) wo calculated te evolution of a free disturbance using linearized versions of te governing equations. Interpretation of te results of tis approac is subject to several sources of uncertainty. As noted by Hogg et al. (2b), te gravest wave modes of te flow can generally be classified as internal or vortical. Te former ave te strongest effect on stratification and are terefore of primary interest ere. Vortical modes can, in principle, propagate upstream troug a critical section for internal waves, tese waves apparently cause no alteration of te upstream stratification. Te waves we introduce involve disturbances of te stratification, but not te vorticity, tereby minimizing te generation of vortical modes. A second complication is te presence of friction and buoyancy diffusion, wic can lead to dispersion and strong damping as te wavelengt goes to infinity. Our guiding principle is tat te waves associated wit ydraulic control sould be non-dispersive, and we terefore select wavelengts sufficiently long to be ydrostatic, but sorter tan te caracteristic scale for wic damping becomes important. It is difficult to determine te precise time or space scale for any damping of te disturbance. However, following Hogg et al. (2b), te relative effects of advection versus diffusion can be estimated by comparing te ratio of te time for te wave to propagate one wavelengt to te time for momentum to diffuse vertically over te dept of te active layer t a /t b = K m /k, were te turbulent diffusion coefficient K m as been normalized by (g 3 s) /2 and te disturbance wavelengt k by s. Tey found tat teir results were not sensitive to te viscosity for t a /t d < 2. For te calculations presented below (figure ), K m =.5 3 in te neigbourood of te sill crest and te primary wavelengt of te truncated sine wave disturbance k =.77, giving t a /t d =8 4, suggesting tat damping of te disturbance is probably insignificant. Figure 9 sows te temporal evolution of te bottom velocity u(x,) resulting from disturbances introduced at x = (at te crest of te obstacle),.4,.28,.42 and.56. In eac case, te disturbance develops into two parts, one of negative velocities (relative to te steady solution) on te upstream side and one of positive velocities on te downstream side. Wen te wave is excited at x =, it appears to be able to propagate in bot te upstream and te downstream direction. Tis suggests tat te flow is subcritical at x =. Wen excited at x =.28, or downstream, propagation takes place in te downstream direction only, indicating tat te flow regime is supercritical in tis area. At x =.4, te pase speed of te part of te wave to te upstream side seems close to zero, sowing tat te control point is close to tis location. Evidently, te place of control is muc farter upstream tan were F 2 =(x =.56, as sown in figure 8). 5. Comparison between te continuous and sallow-water models How well does te layer model capture te dynamics of te deep overflows seen in te numerical simulations? We now investigate tis issue in te context of te steady
Mixing and entrainment in ydraulically driven stratified sill flows 43 Time after wave excitation..5..5..5..5..5..5.5..5 2. 2.5 3. Along-cannel distance Figure 9. Te temporal development of te orizontal velocity field resulting from wave excitation experiments, waves being excited in a narrow vicinity around x =,.4,.28,.42 and.56 (upper to lower panel). Te contours are of constant u at intervals of 2.5 3,te tick line being zero, and velocities below and above zero are to te left and to te rigt of te tick line, respectively. flow sown in figure 7. Beginning at an upstream section x = x, equations (2.) (2.3) are integrated downstream using d(x ),u(x )andg (x ) equal to te vertically averaged values computed from te numerical model. Te procedure also uses w e (x) andu as computed in te numerical simulation, altoug te latter is so small as to ave negligible effect on te outcome. As sown in te depictions of interface elevation (figure ), te solution from te slab model (upper curve) remains subcritical, wereas te flow in te continuous model (middle curve) spills over te obstacle and becomes ydraulically supercritical. An arrow indicates te approximate point of criticality for te continuous flow based on te wave speed calculations of te previous section. It is natural to ask weter te upstream conditions could be adjusted sligtly in order to produce a solution to (2.) (2.3) tat more closely resembles te continuous model. We fixed te values of d(x )andg (x ), and increased te flow rate q(x )=d(x )u(x ) until te resulting solution underwent a critical transition over te obstacle. It was necessary to increase q(x ) by 5% to acieve tis end and te resulting solution (lower curve wit te dased extension in figure ) is terefore quite different from wat is observed in te model. 6. Incorporating vertically varying velocity and density into te layer formulation In order to identify te sources of te inconsistency between te layer and continuous models, it is elpful to reconsider te development of te layer formulation
432 M. H. Nielsen, L. Pratt and K. Helfric Control section for continuous model Slab model z 2..5. Slab model wit adjusted upstream conditions Continuous model.5 (x).5..5.5..5 x Figure. Comparison of te interface eigts for te layer model and continuous model wit identical upstream conditions (upper two curves). Te common upstream conditions are imposed at x =.69 and ere d =.947, u 2 =.85, F 2 =.8, u 2 d =.36 and g =. Also sown is a layer model solution in wic te upstream layer dept and reduced gravity is te same as for te upper curve, but were te upstream value of ud as been increased by 5%. Te dased portion of tis curve indicates F>. in te presence of density and velocity fields tat vary continuously in te vertical. Consider a steady continuously stratified and seared Boussinesq ydrostatic flow tat takes place beneat an inactive and omogeneous upper fluid (figure ). Te upper fluid may ave a uniform velocity u, but is oterwise inactive. We furter partition te (dimensional) velocity and density in te overflowing layer into vertical averages and departures from te average: u 2 (x,z)=ū 2 (x)+u (x,z), ρ 2 (x,z)= ρ 2 (x)+ρ (x,z). If te budgets of momentum, volume and mass are considered for a lower-layer control volume of lengt dx, anddx is taken to zero, te following conservation laws result: d dx [ dū 2 2 + +d u 2 dz + 2ḡ d 2 + g ρ +d +d z ] ρ (x,z )dz dz = ḡ d d dx + w eu, (6.) d dx [dū 2]=w e, (6.2) [ d +d ] ρ 2 dū 2 + ρ u dz = ρ w e, dx
Mixing and entrainment in ydraulically driven stratified sill flows 433 u = constant ρ = ρ = constant w e (x) u 2 (x,z) d 2 ρ 2 (x,z) dx x Figure. Definition sketc sowing control volume extending from te bottom of te cannel (eavy line) to a bounding isopycnal, across wic tere is a positive downwards entrainment velocity w e. Te widt of te control volume is dx. Te overlying fluid is assumed to ave constant density ρ and velocity u. z = were ḡ (x)=g( ρ 2 (x) ρ )/ρ. If te product of ρ and (6.2) is subtracted from te last equation, te result is a statement of conservation of buoyancy flux: [ d +d ] ( ρ 2 ρ ) dū 2 + ρ u dz =. (6.3) dx Equations (6.) (6.3) differ from tose in te continuous model only in te neglect of effects resulting from te non-uniformity of te overlying fluid, non-ydrostatic pressure and internal friction. For eac run, it as been verified tat eac equation is generally valid to a ig degree of accuracy, implying tat tese effects are quite weak in our simulations. In te absence of te primed quantities, (6.) (6.3) form te standard sallowwater equations for te dept-averaged velocity and density (te basis of (2.) (2.3)). Departures from te slab model result from vertical non-uniformities in te density and orizontal velocity. In engineering applications (e.g. Cow 959), te departure of te momentum flux from dū 2 in (6.) is sometimes regarded as a correction to te
434 M. H. Nielsen, L. Pratt and K. Helfric momentum flux of te mean flow, as measured by te Coriolis coefficient α: dū 2 2 + +d u 2 dz = αdū 2 2, (6.4) were α = +d dū 2 2 + u 2 dz dū 2 2 = +d dū 2 2 u 2 2 dz. (6.5) Coefficients measuring te departures from te sallow-water approximations of te orizontal pressure force and te density flux can be introduced in a similar way: were and were d 2 + g +d 2ḡ ρ β(x) = +d 2 ( ρ 2 ρ ) dū 2 + +d z +d z +d +d ρ (x,z )dz dz = β 2ḡ d 2, (6.6) (ρ 2 (x,z ) ρ )dz dz d 2 ( ρ 2 (x) ρ ), (6.7) ρ u dz = γ ( ρ 2 ρ )dū 2, (6.8) (ρ 2 (x,z) ρ )u(x,z)dz γ (x) =. (6.9) ( ρ 2 (x) ρ )d(x)ū 2 (x) Te coefficients α, β and γ can be regarded as sape functions for te vertical profiles of lower-layer velocity, pressure anomaly and flux of density anomaly. (Note tat our α is Cow s β.) Te equations for conservation of momentum and buoyancy flux anomaly can now be written as d ] [αdū 2 + 2 dx βḡ d 2 = ḡ d d dy + w eu (6.) and d dx [γ ḡ dū 2 ]=, (6.) and tese along wit (6.2) ave a superficial resemblance to te flux form of te sallow-water equations. Te latter are recovered wen α = β = γ = and departures terefore give an indication of te error incurred by treating te lower layer as a slab. From te final expression in (6.5), it is easily seen tat α, and it can also be sown (Appendix B) tat β, provided te stratification is stable. Vertical variations of velocity and density terefore ave te effect of enancing te inertia
Mixing and entrainment in ydraulically driven stratified sill flows 435 2.5 2. α.5 γ. β.5 2..5..5.5..5 2. Along-cannel distance Figure 2. Te sape functions α, β and γ as defined by (6.5), (6.7) and (6.9), for te model output sown in figure 7. and decreasing te buoyancy, relative to te mean flow, of te lower layer. Te underestimation of te inertia term (α ) is discussed by Cow (959) and is familiar to investigators of omogeneous open-cannel flows. Te new and more subtle result is te overestimation of te buoyancy term (β ) by te slab model. Te range of te correction γ to te buoyancy flux is less restricted; owever, it can be sown tat, wen te stratification is stable and te orizontal velocity decreases monotonically downward (upward) troug te lower layer, γ<(>). Tis result is also derived in Appendix B. As sown in figure 2, te values of α(x), β(x), and γ (x) for te numerical run of figure 7 conform to tese ranges and suggest significant departures from sallow-water teory. For example, te value of α(x) becomes as ig as.99 in te supercritical part of te flow, wereas β(x) becomes as low as.55. Te underestimation of te inertia-to-buoyancy ratio is te primary reason tat te layer model solution (top curve in figure ) is more subcritical tan te full model solution. In order to solve (6.2), (6.) and (6.), under general circumstances, a closure sceme would be required to relate te sape coefficients to te mean quantities. Tis difficulty is avoided if te density and orizontal velocity are self-similar [ρ 2 and u 2 are functions of (z (x))/d(x) only], in wic case α, β and γ are constant. If tis be te case, or if α, β and γ vary gradually on te scale of te oter forcing, (6.2), (6.) and (6.) can be used to sow tat and dū 2 dx = F 2 (β α F 2 )d d(d) dx = F 2 (β α F 2 ) d F 2 dx = 3 F 2 (β α F 2 )d {w e ( α + { 2 w e ū 2 { we ū 2 β 2 F u ) + ū2 d 2 ū 2 F 2 dx ) ( β 4 F 2 + u 2ū 2 α F 2 d dx }, (6.2) } (6.3) ( α F 2 + β 2 u ) F 2 + d }, (6.4) ū 2 dx were F 2 = ū 2 /(ḡ d). Note tat tese relations are independent of γ and tat tey reduce to (2.) (2.3) wen α = β =. It is apparent tat ydraulic criticality under te assumed conditions corresponds to F 2 = β α. (6.5) Under conditions of self-similarity, critical flow occurs were te ordinary Froude number indicates subcriticality.
436 M. H. Nielsen, L. Pratt and K. Helfric In general, self-similarity is not a property of te flow field and (6.5) is invalid. For te special case of uniform density (β = ), Garrett & Gerdes (23) ave sown tat te vertical velocity profile cannot be self-similar witout a special unrealistic form for te internal dissipation. However, Wood (968) as sown tat self-similarity can occur in te presence of stratification, provided te cannel bottom remains orizontal and te flow is forced entirely by sidewall contractions. Wood s solution takes te form ρ 2 = ρ 2 (z/d 2 (y)) and u 2 = u 2 (z/d 2 (y)), and examples of corresponding flows ave been reproduced in te laboratory (Armi & Williams 993). Te precise forms of tese functions and te corresponding constant values of α, β and γ depend on te upstream conditions. In te numerical solution under discussion, wic is not self-similar, F 2 = at x =.56, wereas F 2 = β/α at x =.2 (see figure 8c). Te direct calculation of wave speed (figure 9) indicates te actual section of criticality near x =.4, were F 2 <. Positions of predicted and measured critical sections for all te numerical experiments are summarized in figure 3. Tere are tree series of runs, eac wit a particular value of s /L, and te corresponding results are displayed in different frames. Witin eac series, te upper-layer velocity u is varied. Te position of te control section in eac case, as determined by te introduction of free waves, is indicated by a orizontal line tat reflects te uncertainty in visually isolating te exact location. Te position at wic F 2 = (filled circle) lies well downstream of tis position in eac case. We ave also indicated te position at wic F 2 = β/α (crosses), and tis lies witin te range of uncertainty of te direct measurement in eac case. F 2 did not reac unity at any section in some of te experiments, and tese are indicated by te absence of a filled circle in figure 3. Tis result as implications for certain ocean overflows tat appear to be ydraulically controlled, but lack observations indicating F 2. One example is te Bab al Mandab, were measurements of te composite Froude number (te two- or tree-layer extension of F 2 ) at te main sill and narrowest section give values < (Pratt et al. 999). As u decreases, te control section moves in te downstream direction, as expected from (2.6). Note tat for an upper-layer velocity u <.5, te experiments wit te least steep topograpy (lower frame) did not become critical at any point. For te steepest topograpy (upper frame) te influence of te upper-layer velocity is less clear; in fact, te control point appears to move sligtly upstream as u is decreased below.4. Te control is located downstream of te sill (x = ) in nearly all cases. Altoug (2.6) predicts an upstream location for sufficiently large positive u, we did not ceck tis. Te required upper-layer velocity would imply a significant contribution to te composite Froude number, and so te upper layer would no longer be inactive. Our finding tat critical flow occurs near te location were F 2 = β/α seems consistent wit te fact tat vertical variations of u and ρ in te lower layer enance te inertial caracter of te flow relative to tat deduced from te vertical mean. However, te work of Garrett & Gerdes (23) suggests tat te actual situation may be more subtle. Teir study of a omogeneous free-surface sear flow, essentially te case β = γ = and α>, sows tat te critical section indicated by direct calculation of te long-wave speed lies were F 2 >, as opposed to te position of F 2 =/α < predicted for a self-similar flow. Te real critical section terefore lies downstream of te location suggested by (6.5). Garrett (24) furter argues tat te presence of bottom drag puses te critical section downstream, wereas internal friction displaces it back upstream. In te present experiments, wic contain internal friction but no bottom drag, it is possible tat te critical section lies upstream of were it