Tidal circulation and buoyancy effects in the St. Lawrence Estuary

Transcription

1 Atmosphere-Ocean ISSN: (Print) (Online) Journal homepage: Tidal circulation and buoyancy effects in the St. Lawrence Estuary François J. Saucier & Joël Chassé To cite this article: François J. Saucier & Joël Chassé (2000) Tidal circulation and buoyancy effects in the St. Lawrence Estuary, Atmosphere-Ocean, 38:4, , DOI: / To link to this article: Published online: 21 Nov Submit your article to this journal Article views: 425 View related articles Citing articles: 40 View citing articles Full Terms & Conditions of access and use can be found at

2 Tidal Circulation and Buoyancy Effects in the St. Lawrence Estuary François J. Saucier* and Joël Chassé Division of Ocean Sciences, Maurice Lamontagne Institute Department of Fisheries and Oceans 850, Route de la Mer Mont-Joli, QC G5H 3Z4 [Original manuscript received 23 November 1998; in revised form 4 April 2000] abstract The action of tides on density-driven circulation, internal gravity waves, and mixing was investigated in the St. Lawrence Estuary between Rimouski and Québec City. Timevarying fields of water level, currents and density were computed under typical summer conditions using a three-dimensional hydrostatic coastal ocean model that incorporates a second order turbulence closure submodel. These results are compared with current meter records and other observations. The model and the observations reveal buoyancy effects produced by tidal forcing. The semi-diurnal tide raises the isopycnals over the sills at the head of the Laurentian Trough and English Bank, producing internal tides radiating seaward. Relatively dense intermediate waters rise from below 75-m depth to the near surface over the sills, setting up gravity currents on the inner slopes. Internal hydraulic controls develop over the outer sills; during flood, surface flow separation occurs at the entrances of the Saguenay Fjord and the upper estuary west of Ilet Rouge Bank. Early during ebb flow (restratification), the surface layer deepens to encompass the tops of the sills. As the ebb current intensifies, the model predicts the formation of seaward internal jumps over the outer sills, which were confirmed from acoustic reflection observations. As the internal Froude number increases further, flow separation migrates up to sill height. As a result of these transitions, internal bores emanate from the head region one to two hours before low water. We find that the mixing of oceanic and surface waters near the sills is driven by the vertical shear produced during ebb in the channel south of Ilet Rouge, the shear produced in the bottom gravity flood currents, and, to a lesser extent, the processes over the sills. résumé L action des marées sur l écoulement associé à la flottabilité, les ondes de gravité internes et le mélange a été étudiée pour l estuaire du Saint-Laurent entre Rimouski et la ville de Québec. Les variations du niveau d eau, des courants et de la densité ont été calculées dans des conditions estivales typiques en utilisant un modèle hydrostatique tridimensionnel pour les eaux côtières incorporant un sous-modèle de fermeture turbulente au second ordre. *Corresponding author: saucierf@dfo-mpo.gc.ca ATMOSPHERE-OCEAN 38 (4) 2000, /2000/ $1.25/0 Canadian Meteorological and Oceanographic Society

3 506 / François J. Saucier and Joël Chassé Ces résultats ont été comparés avec des enregistrements de courantomètres et d autres observations. Le modèle et les observations nous révèlent des effets de flottabilité produits par la marée. La marée semi-diurne élève les isopycnes au-dessus des seuils à la tête du chenal Laurentien et du banc des Anglais, produisant des marées internes qui se propagent vers l océan. Des eaux intermédiaires relativement denses, se trouvant sous 75 m de profondeur, remontent prés de la surface au-dessus des seuils, produisant des courants de gravité sur les pentes internes. Des contrôles hydrauliques internes se développent au-dessus des seuils extérieurs; durant le flot, une séparation de l écoulement de surface a lieu à l entrée du fjord du Saguenay et dans l estuaire moyen, à l ouest des bancs de l Île Rouge. Au début du jusant, la couche de surface épaissit pour recouvrir les seuils. Lorsque le courant du jusant s intensifie, le modèle prédit la formation de sauts internes à l aval des seuils extérieurs. Ceci a été confirmé par des observations de réflexion acoustique. Comme le nombre de Froude interne augmente, la séparation de l écoulement atteint la hauteur du seuil. Par suite de ces transitions, des mascarets internes émanent de la région une ou deux heures avant l étale de la marée. Nous trouvons que le mélange des eaux de surface et océaniques près des seuils est causé par le cisaillement vertical produit durant le jusant dans le chenal au sud des bancs de I Île Rouge, par le cisaillement produit dans les courants de densité de flot et, d une façon moins importante, par les processus au-dessus des seuils. 1Introduction Late Holocene glacial events have forged the present shape of the Estuary and Gulf of St. Lawrence, built around the major valley called the Laurentian Trough, which extends 1500 km from the continental margin to an abrupt end near Tadoussac (D Anglejan, 1990). The upper estuary, about 180 km long, has many fjord-like features, such as relatively deep basins separated by shallow underwater sills and banks controlled by glacial erosion (Fig. 1). Its waters consist of a mixture of freshwater descending from the St. Lawrence River (mean flow of m 3 s 1, and close to m 3 s 1 during the spring freshet), and cold intermediate and deeper oceanic waters moving in over the sills from the Laurentian Trough (e.g., Neu, 1970). The Saguenay River, with an average flow rate of m 3 s 1, and other smaller tributaries enhance estuarine circulation near the outer sills (Sa, Sb and Sc on Fig. 1). The upstream reach of salt water is found on the shallow banks near Île d Orléans, where it is well mixed with the St. Lawrence River by strong tidal flows, albeit in relatively small amounts. Stratification and density then increase steadily at all depths seaward from Île-aux-Coudres, and markedly near Ilet Rouge. Mixed tides propagating from the Atlantic Ocean range from 6 m at the outer sills to over 7 m at Île d Orléans, and produce currents over 2 m s 1 throughout. They dissipate over 3 GW of energy while progressing over the outer sills and into the upper estuary (Forrester, 1972; Reid, 1977; Godin, 1979). In addition to energy dissipation along the bottom, tides, through their interaction with the topography, also redistribute buoyancy and give rise to strong gravitational effects such as internal waves, fronts and jets (Neu, 1970; Forrester, 1974; Ingram, 1976, 1978, 1985; Reid, 1977; Muir, 1979; DeGuise, 1977; Mertz and Gratton, 1990; Ingram and El-Sabh, 1990; Galbraith, 1992; Seibert et al., 1978). These distributed effects govern the turbulent mixing of freshwater with waters from the intermediate and deep layers of the Gulf

4 Tides and Buoyancy Effects in the St. Lawrence Estuary / 507 Fig. 1 Bathymetric chart of the St. Lawrence Estuary between Les Escoumins and Québec City. The chart encompasses the head of the Laurentian Trough, ending near Tadoussac, and the saline part of the upper estuary region upstream to Québec City. The maximum depths with respect to mean sea level available over the main sills, labelled Sa to Se, are: Sa: 67 m; Sb: 35 m; Sc: 20 m; Sd: 32 m; Se (English Bank): 32 m. The positions of tide gauge stations and current meters are shown. Bold-italic numbers indicate that density (conductivity and temperature) was recorded. Stations 39 and 40 are about 5 km upstream from station 35 in the Saguenay Fjord and are not shown. The volume V and area A of the water basins between Trois-Rivières and Pointe des Monts are, respectively (at mean sea level): Trois-Rivières to Québec City, A m 2, V m 3 ; Québec to Tadoussac, A m 2, V m 3 ; Tadoussac to Pointe des Monts (lower estuary), A m 2, V m 3 ; Saguenay Fjord, A m 2, V m 3. Note that the map scale differs in the two panels. of St. Lawrence, resulting in a nutrient-rich surface layer continuously moving through the lower estuary (e.g., Neu, 1970; Steven, 1974; Reid, 1977). No theoretical explanation of these scales of motion can ignore the strong three-dimensional nature of the basin, the gravitational effects associated with stratification, the rotational effects, and the non-linearity or time dependencies. In this paper, we make use of current meter data and the predictions of a three-dimensional numerical estuarine model to examine this tidal machine. Huntsman (1923) first suggested that tidal energy drives the estuarine mixing in

5 508 / François J. Saucier and Joël Chassé the St. Lawrence Estuary. Neu (1970) and Reid (1977) suggested that runoff is the other primary control. Higher runoff increases the hydraulic head of the river (by about 30 cm at Québec City during spring) and the water s stability. In this paper, the exchanges of momentum or other quantities with the atmosphere are ignored. In effect, the rate of work by the wind, following Farmer and Freeland (1983) is Z V d τ w da (1) E w A where τ w ρc d U A 2 is the wind stress, ρ A 1.27 kg m 3 is air density, C d is the dimensionless drag coefficient (say ), U A is the RMS wind speed, and V d is the surface drift velocity (on the order of U A following Farmer and Freeland, 1983). Using a relatively high U A 10 m s 1 over the estuary (area A given in Fig. 1), we find E w <10 8 J s 1, about 5% of the energy dissipated by tides in the estuary. If it is further assumed that the discharge rates and the properties of the oceanic source waters vary slowly compared to tides. The mean circulation and the density field must reach a near steady state, wherein the buoyancy input from the rivers is balanced by the input of oceanic waters through tidal mixing. This state must hold over the large neap to spring modulation in the amplitude of the semidiurnal tide. Clearly, over days to seasons, changing external conditions (e.g., runoff, storms, remote events in the Gulf of St. Lawrence) force the density, the circulation, and the mixing intensity to continuously readjust. This adjustment may be achieved over approximately one week for changes in runoff, that is the approximate residence time of freshwater estimated considering: the volume of the upper estuary (see Fig. 1); the mean conditions for the discharge rate; the mean density of the upper estuary, say ρ < 1022 to 1023 kg m 3 ; and the density of the intermediate source waters passing over the sills, ρ s < 1025 kg m 3. The adjustment is achieved over similar timescales for changes in deep water properties at the mouth. Thus, while the estuary may never reach steady state, the same tidally driven processes operate at various scales and intensities to bring the system toward such a state. This steady state, when examined under typical summer conditions in the St. Lawrence Estuary, is the focus of this paper. In the following section, we present the model description, and its calibration and validation using tide gauge and current-meter measurements. Section 3 describes the general three-dimensional circulation over semi-diurnal tidal periods. Further validation of the model predictions takes place as it reveals the tidal pumping of intermediate waters over the outer sills, the generation of internal tides and fronts, and the development of bottom density currents within the inner basins (as documented by Drainville (1968), Therriault and Lacroix (1975), and Seibert et al. (1978) for the inner slopes of the Saguenay Fjord). In particular, Section 4 is devoted to comparisons between the model predictions for internal tides and the results of previous studies. The model predicts the internal tides radiating into the lower estuary from the outer sills (observed by Forrester, 1974), and also equally large ones within the upper estuary and generated on English Bank. From published tidal currents

6 Tides and Buoyancy Effects in the St. Lawrence Estuary / 509 (>3 m s 1 ) and vertical density changes (Brunt-Väisälä frequency N < 10 2 s 1 ) over the outer sills (Fig. 1) bounding the head region (e.g., Reid, 1977; Seibert et al., 1978), one notes the potential but yet unnoticed importance of internal hydraulics (e.g., Long, 1953, 1954, 1955; Farmer and Smith, 1980). Converging surface fronts occur at tidal periods over these sills (Ingram, 1976, 1985), at the mouth of the Saguenay Fjord (Drainville, 1968), and near English Bank (sill Se on Fig. 1). In Section 5, the model provides a demonstration of the control cycles over the sills. In particular, the model predicts how large internal bores (e.g., seen in Simard (1985) and Galbraith (1992)) may emanate from the outer sills during ebb flow. Finally, Section 6 turns to a field that is presently rather difficult to predict with numerical models, that of turbulent buoyancy fluxes. The large tidal oscillations of the isopycnals observed at the head of the Laurentian Trough near Tadoussac (e.g., Hachey et al., 1956) have suggested to Steven (1974) and others (e.g., Ingram, 1983; Wang et al., 1991; Galbraith, 1992) that it may be the most intense mixing region in the St. Lawrence Estuary. Clearly, internal tides and strong flows over the sills locally produce significant mixing (Forrester, 1974; Therriault and Lacroix, 1976; Reid, 1977; Ingram, 1983; Gratton et al., 1988; Galbraith, 1992). But the model below suggests that the shear produced by the ebb flow of low salinity surface waters over oceanic waters near Île Verte, and that associated with bottom gravity currents in the North Channel, contribute more to the mixing in the St. Lawrence Estuary. 2 Numerical modelling and validation A hydrostatic solution to the mass, momentum and density conservation equations in the Boussinesq approximation is reached from the Cartesian formulation initially put forward to model oceanographic processes in the North Sea by Backhaus (1983, 1985). This model was thereafter modified to reproduce the strong and complex currents in the waters between Vancouver Island and the mainland by Stronach et al. (1993). The model was also applied at various stages of development to the St. Lawrence Estuary and Gulf (J. Stronach, unpublished manuscript; Chassé, 1994; Government of Canada, 1997; Saucier et al., 1999). Here a new application of this model is presented, modified to improve horizontal and vertical mixing, resolution, and to include the geopotential height of the St. Lawrence River. Simulations of tides and the associated density changes are sought for periods of one to two months, allowing the examination of neap to spring changes, under no atmospheric stresses, typical summer runoff values, and constant far field density conditions. The model description, including initial and boundary conditions, and its calibration and validation, follow. a Domain The model domain extends from Rimouski to the upper limits of tidal influence near Trois-Rivières and at the head of the Saguenay Fjord. This domain is described by a rectangular grid on a Lambert equal-area projection assuming uniform rotation. The

7 510 / François J. Saucier and Joël Chassé Table 1 Model parameters. Variable Value Description x 400 m Horizontal grid resolution ( cells) z i 5, 10, 15, 20, 25, 30, 35, 45, 55, Base of model layers 65, 75, 90, 105, 125, 150, 175, 200, 250, 300, 350 m θ N Mean latitude t 60 s Time step Q i Trois-Rivières: m 3 s 1 River runoffs June July 1994 Saguenay: m 3 s 1 Net Upper Estuary: 700 m 3 s 1 γ 0.4 Coefficient of horizontal diffusivity in Smagorinsky scheme n s m 1/3 Manning coefficient for bottom drag ε m Allowed error in successive over-relaxation solution α 0.5 Weighting of implicit solution η(m 2 ), φ(m 2 ) m, (Eastern Standard Time) Water level tidal amplitude and phase η(s 2 ), φ(s 2 ) m, at the corner node near Rimouski for η(n 2 ), φ(n 2 ) m, five main constituents (the amplitude η(k 1 ), φ(k 1 ) m, of the other constituents is less than η(o 1 ), φ(o 1 ) m, m) grid resolution is 400 m in the horizontal, allowing a description of the main channels and sills (Fig. 1), and limited by the available computing power. The bases of 20 model layers are defined in Table 1, along with other model parameters. The thickness of the bottom layer is locally adjusted to meet the observed depth. The topography, the mean sea level (Z 0 ) (added to charted depths), and the hydraulic head were linearly interpolated on this grid using the complete database of soundings, charts and other geodetic levelling and tide gauge records from the Canadian Hydrographic Service (Fig. 1 and Table 2). The minimum depth in the model (the thickness of the surface layer) is set to 5 m, greater than the maximum water level oscillation amplitudes that are expected in order to maintain a finite layer thickness and precision. Shallower depths were redistributed to conserve mass at mean sea level. b Governing Equations Mass conservation: Momentum conservation: u t + u? =u fv + 1 ρ P x u x + v + w 0 (2) y z u A H u A H u A VM 0 (3) x x x y z z

8 Tides and Buoyancy Effects in the St. Lawrence Estuary / 511 Table 2 Positions of tide gauge stations (see also Fig. 1). The number stn identifies the stations in Table 3. The number under CHS is the Canadian Hydrographic Service station number. The length of the series (in days) is the record length used to determine the mean sea level (Z 0 ), the gravitational potential, and the observed water level tidal constituents. C. T. is the centre date for the records that were analyzed. Z 0 is the distance above the chart datum Z c, which is raised over the Great Lakes Reference Datum 85 (approximating the geoid, International Great Lakes Datum 1985, 1992) by the distance Z c GLRD, and the hydraulic head is given in the last column as Z 0 1 Z c GLRD. These values were interpolated over the axes of the St. Lawrence and Saguenay estuaries, and were kept constant in the transverse directions. The x- derivative of the hydraulic head was added to the barotropic pressure terms in eqs. (3) and (4). Length C.T. Z 0 Z c -GLRD Head Station stn CHS Lat. (8N) Long. (Days) mm yy (m) (m) (m) Rimouski Bic Les Escoumins Trois-Pistoles Ile Verte Tadoussac Cacouna Rivière-du-Loup Saint-Siméon Pte-aux-Orignaux Saint-Irénée Cap-aux-Oies St-Joseph-Rive St-Jean-Port-Joli Île aux Grues Berthier-sur-Mer St-Francois I.O Ste-Anne-Beaupré St-Laurent (I.O.) Québec (Lauzon) Port-Alfred Chicoutimi Montmorency Neuville Portneuf Grondines Cap-à-la-Roche Brickyard Batiscan Trois-Rivières Port-St-François v t + u? =v + fu + 1 ρ P y v A H v A H v A VM 0 (4) x x y y z z Density conservation: σ t t + u? =σ t x A H σ t x σ t A H A V σ y y z σ t z 0 (5)

9 512 / François J. Saucier and Joël Chassé Hydrostatic equation: P z ρg (6) with ρ ρ 0 1 σ t (in MKS units); u (u,v,w) is the velocity along the horizontal axes x and y (see Fig. 1), and vertical axis z (positive upward), f is the Coriolis parameter, P is the pressure, ρ is the density, ρ 0 is the reference density (i.e., 10 3 kg m 3 ), g is the gravitational acceleration, A H is the horizontal eddy viscosity and diffusivity for momentum and density, and A VM, A Vσ are the vertical eddy viscosity and diffusivity, respectively. c Solution Method As in Backhaus (1985), the external mode is treated implicitly, and no major restriction on time step occurs other than for the accuracy in amplitude and phase of surface gravity waves. The solution for the external mode is done by means of successive over-relaxation applied to the continuity equation. The divergence at a given time step is combined with the divergence at the next time step, the latter based on future water levels, which makes the scheme implicit. The continuity equation is finally solved as an elliptic form for the water levels at the advanced time. The stability and accuracy of the numerical model for propagating long external gravity waves is discussed in Stronach et al. (1993). The advective terms in the density equation (5) are formulated explicitly, but this does not affect the stability (e.g., Pohlmann, 1996). The horizontal derivatives use a vector upstream scheme, using the method of characteristics, and the vertical derivative term uses a component upstream second-order scheme similar to the one-dimensional Lax-Wendroff scheme (see Stronach et al., 1993). The vertical eddy diffusion terms are solved implicitly using a Crank-Nicholson solution. d Horizontal Diffusivity The horizontal viscosity and diffusivity are described following Smagorinsky (1963) and Blumberg and Mellor (1987), A H γ x 2 " u 2 2 v u v + +0Ø5 + x y y x 2 # 1 2 (7) with γ 0.4 and x 400 m. To this physically based diffusion the numerical scheme effectively adds numerical dispersion in the x-y plane, produced from the explicit solution to the advective part of equations (3) to (5) using the vectorupstream scheme (e.g., Stronach et al., 1993). The corresponding diffusion coefficient can be estimated according to Roache (1972) to be of the order 0.5 u x. With u < 1m s 1, the numerical dispersion is of the order of 200 m 2 s 1.

10 Tides and Buoyancy Effects in the St. Lawrence Estuary / 513 e Vertical Diffusion We follow Mellor and Yamada (1974, 1982), with improvements from Kantha and Clayson (1994), in the implementation of a level 2 turbulence closure model. The vertical eddy viscosity and diffusion coefficients are A VM lqs M A V σ lqsσ (8) where l is the turbulent length macroscale, q 2 is twice the turbulent kinetic energy, and S M and Sσ are stability factors in mixing coefcients dened from (uwù vw) lqs M u z Ù v z (9) σ t σ t w lqsσ Ø (10) z The stability factors, S M and Sσ, are dependent upon the ux Richardson number, derived from neutral turbulent ow data S M A 1 A 2 Sσ 3A 2 γ 1 (γ 1 + γ 2 )R f 1 R f (11) B 1 (γ 1 C 1 ) [B 1 (γ 1 C 1 )+6A 1 +3A 2 (1 C 2 )]R f B 1 γ 1 [B 1 (γ 1 + γ 2 ) 3A 1 ]R f where, γ 1 ; (1/3) (2A 1 /B 1 ), γ 2 (B 2 /B 1 )(1 C 3 ) 1 (6A 1 /B 1 ), (A 1, A 2, B 1, B 2, C 1, C 2, C 3 ) (0.92, 0.74, 16.6, 10.1, 0.08, 0.7, 0.2), and R f is the flux Richardson number, the ratio of destruction rate of turbulent kinetic energy by buoyancy to its production rate by shear, related to the gradient Richardson number Ri Sσ (12) where: Rf S σ SM Ri (13) Ri g σ t Û z ρ[( uû z) 2 +( vû z) 2 ] (14) retaining the negative root such that R f 0 when Ri 0. With this model, mixing is suppressed when R f > The turbulent kinetic energy is determined from the super-equilibrium approximation, that is an immediate and local balance between turbulent energy production (by shear or buoyancy) and dissipation " u 2 # 2 v gl 2 2 σt 1 S M l 2 q 2 z + z + Sσ q 2 z B 1 Ø (15)

11 514 / François J. Saucier and Joël Chassé No universal formulation exists for specifying the turbulent length scale l (see Davies et al., 1997, for a review). Mellor and Yamada (1982) and Kantha and Clayson (1994) use a prognostic equation for q 2 l. Here, for the lack of a more general formulation and the general inadequacy of data to suggest better approximations, we propose to make use of Blackadar (1962) and Mellor and Durbin (1975) to extract a basic scale in terms of an integral of the turbulence energy with f Hydraulic Head The departure of the mean sea level from the geoid produces a static pressure gradient that must be added to the dynamic barotropic pressure gradient terms in the momentum equations. The Great Lakes Reference Datum is defined as an Earth s equipotential surface merging with the mean sea level at Rimouski, near the north- κz l(z) (16) 1+κzÛl 0 0 Z jzjqdz l 0 α H 0 Z qdz (17) H where κ is the von Kármán constant, H is depth, and α is a constant set to α 0.01 (see Mellor and Yamada, 1974). Within stable stratified shear flows, the vertical extent of instabilities is limited by stratification over the Ozmidov length scale (e.g., Rohr et al., 1988; Kantha and Clayson, 1994). We take note of this limit (Kantha and Clayson, 1994) Nl < 0Ø53 (18) q but the present simulations show that l from (16) is always near but less than the Ozmidov scale. At mid-depth, the computed length scale is between 0.1 m and 3 m, in general agreement with recent measurements in an estuary by Peters (1997). Further investigations on mixing length scale models (e.g., Davies et al., 1995, 1997) require new velocity and density profiles from the St. Lawrence Estuary. The inclusion of other functional forms for l(z) to cope with the effect of stratification was not deemed necessary at this stage. The numerical dispersion in the vertical, of the order 0.5 w z, is of the same order as the coefficients solved from (8), and adds to the inaccuracy of buoyancy and momentum fluxes. In particular, it is found that the modelled pycnocline is generally smeared when compared to observations, which affect simulated internal waves and turbulent quantities, which are dependent upon stratification.

12 Tides and Buoyancy Effects in the St. Lawrence Estuary / 515 eastern model boundary (Fig. 1). Recall that the depth interpolated from the charts are with respect to the chart datum, and that the mean sea level is added to this depth in order to model tidal oscillations. For example, at Trois-Rivières, near the southwestern end of the domain, the chart datum is raised nearly 3 m above the geoid, and the mean sea level Z 0 is raised one metre over the chart datum (Table 2). Thus, with respect to the geoid, the mean sea level is about 4 m higher at Trois-Rivières than at Rimouski. The most significant drop in mean sea level with respect to the geoid occurs upstream from Québec City and in the Saguenay River, while the values downstream are smaller and sometimes suspicious (e.g., Ile Verte or Saint-Josephde-la-Rive). Thus only the derived static pressure gradients estimated upstream of Québec City and Tadoussac were included in the model. g Boundary Conditions 1 friction The flow normal to any solid boundary is set to zero. A free-slip boundary condition is applied to the side walls. The bottom stress is a quadratic function of bottom velocity of the Manning-Chezy type (e.g., Chow, 1959), τ b ρ 0g ju C 2 b ju b (19) z where g is the gravitational acceleration, u b is the bottom layer velocity, and C z is the Chezy coefficient related to a Manning s n, in metric units (Table 1) C z H 1Û6 2 rivers The discharge rates from twenty tributaries (with discharge rates greater than 10 2 m 3 s 1 ) were specified by a change in surface elevation and dilution in landbounding cells, at each time step, so that they accommodate the incremental momentum and buoyancy fluxes. The summer discharge rates into sub-regions of the estuary are given in Table 1. At the upstream boundary near Trois-Rivières, the observed hourly water level and the mean daily flow rate, estimated from the sum of the upstream sources, were linearly interpolated to the model time steps and were used for the simulations reported here. 3 open boundary The amplitudes and phases of the 15 main water level tidal constituents are specified along the open boundary near Rimouski in the lower estuary. We make use of the tide gauge observations at the eastern grid corner (Rimouski), and, in order to describe the changes in the constituents along the open boundary, we make use of another implementation of the model for the Gulf of St. Lawrence (J. Stronach, personal communication). This model, with 5-km lateral resolution, and forced with 15 observed water level tidal constituents at Cabot and Belle-Isle Straits, produced a n (20)

13 516 / François J. Saucier and Joël Chassé Fig. 2 Initial σ t condition along a cross section from Québec City to Rimouski. This condition is made laterally uniform. It represents the observed profiles described by Forrester (1974) and temperature-conductivity profiles observed during summer 1975 in the upper estuary. month-long simulation used to establish the spatial derivative of each constituent amplitude and phase along the eastern boundary. During inflow, a zero gradient condition is applied to all velocity components on the eastern open boundary. Density is specified from the mean summer profile derived by Forrester (1974). During outflow (u > 0), the radiation condition is specified to ξ u, v, while zero-gradient is specified on density. ξ t ξ + u 0 (21) x h Initial Conditions and Experimental Setup The initial density condition (Fig. 2) was made laterally uniform and smoothly varying along the estuary and the Saguenay Fjord. In the lower estuary, the mean summer profile derived by Forrester (1974) is specified and extrapolated to gradually merge, near the outer sills, with temperature and salinity profiles observed throughout the estuary and Saguenay Fjord during the summer of We assume that those are typical of other summer observations (e.g., Neu, 1970). Runoff and water levels at Trois-Rivières during the months of June and July 1994 were close to climatological values ( m 3 s 1 for mean runoff) and are also specified as typical summer values. The model spins up from rest and the boundary forcing ramps up for two days. Density and mixing come to a near steady state, modulated by tides, over an e-folding time of about one week, representing the residence time of

14 Tides and Buoyancy Effects in the St. Lawrence Estuary / 517 freshwater and the relatively small discrepancy between the initial condition and the mean equilibrium condition. The solution may become slightly denser (mixing generally too high), or fresher (mixing generally too low) than the initial or observed conditions given small model parameter changes (e.g., topography, friction, eddy diffusivity and viscosity, and numerics). The tidal analyses following use the results from the last 30 days of a 40-day long simulation in this context. i Calibration The calibration was performed by adjusting the Manning coefficient n (see Table 1) through a few simulations to reproduce accurate water level tidal amplitudes and phases over the model domain (derived from Foreman, 1977). Table 3 shows the phase and amplitude differences found between the five main observed and modelled constituents at 20 tide-gauge stations downstream from Québec City (Dept. of Fisheries and Oceans, Ottawa). Upstream from Québec City, because the effect of bottom stress and varying runoff prevent accurate harmonic representations of the tide (e.g., Godin, unpublished manuscript), the model s hourly results were directly compared with the observations. The modelled tidal phases, amplitudes and waveforms compared well with the observations (not shown). We refer the reader to further analyses of tidal propagation found in Godin (1979) and El-Sabh and Murty (1990). j Validation Against Current Meters After calibration, the model results were compared with 51 current meter records, each of a few days to one year duration, acquired between 1973 and Figure 1 shows the current meters positions and Table 4 shows their depths. One density recording was rejected, and no records were available for depths over 50 m. The data and the model results were synthesized in tidal harmonic constituent forms (Foreman, 1977). Tables 5 to 9 show the comparison between observed and modelled amplitudes and phases for current components u, v and density anomaly σ t (computed from temperature and conductivity at sea level pressure using Unesco, 1983). Figure 3a shows the observed hourly u current component against the u component synthesized from model-derived tidal constituents. We first note that over 90% of the observed current component hourly variance can be reproduced from a synthesis of the tidal harmonics estimated from the data, and over 80% from a synthesis of the model-derived harmonics (over the records lengths). Meteorological events are barely noticed in all current records, as shown in Fig. 3a, but the density changes over periods of a few days may be sporadically as large as that for the M 2 tide (not shown). Thus in Fig. 3b we compare the harmonic predictions derived from both the modelled and observed densities, with the mean values removed, illustrating the results from Tables 5 and 9. The semi-diurnal tide clearly dominates the current and density variability, with amplitudes generally exceeding 0.5 to 1 m s 1 and 0.5 to 1 kg m 3 for the axial cur-

15 518 / François J. Saucier and Joël Chassé Fig. 3 Comparison with current meter records. The blue lines show observations and the red lines show the model results. The horizontal time axis spans the record length to a maximum set to the first 28 days, but actual records may be longer (see Table 4). Each tick mark represents one day. (a) Harmonic synthesis from the modelled velocity along the axis of the estuary (using a 40-day simulation) compared with hourly unfiltered observations. See Tables 5 to 9 for the comparison between the observed and the predicted amplitudes and phases of the main constituents. (b) Har-

16 Tides and Buoyancy Effects in the St. Lawrence Estuary / 519 Fig. 3 (Continued) monic synthesis from the model compared with the harmonic synthesis from the observed hourly density (mean Z 0 removed from both series for comparisons). See Tables 5, 7 and 9 for the comparison between modelled and observed density tidal constituents M 2, Z 0 and MS f.

17 520 / François J. Saucier and Joël Chassé Fig. 3 (Concluded)

18 Table 3. Comparison between the model and the observations for the water level tidal constituents M 2, S 2, N 2, K 1 and O 1 for stations seaward from Québec City. Station number (stn) refers to Table 2. The amplitude of the tide is given by η (10 2 m) and its phase by φ(8est). The superscript o represents observed values and η and φ are the modelled minus the observed values. M 2 S 2 N 2 K 1 O 1 Stn. η o η φ o φ η η φ o φ η o η φ o φ η o η φ o φ η o η φ o φ Mean Tides and Buoyancy Effects in the St. Lawrence Estuary / 521

19 522 / François J. Saucier and Joël Chassé Table 4. Positions and sampling intervals for current meters. Figure 1 shows the positions of the instruments. The last column indicates whether or not temperature and salinity were observed (bold numbers in Fig. 1). The sampling interval varies from 10 to 60 minutes, but all data were subsampled at hourly intervals. Start date Latitude Longitude Depth Length Stn. (8N) (8W) (m) yy mm dd (Days) σ t no yes yes yes yes no yes yes no no no no yes no no no no no yes yes yes yes yes yes yes yes yes yes yes yes no yes yes yes yes yes yes yes yes yes no yes yes yes yes yes yes yes yes yes yes

20 Tides and Buoyancy Effects in the St. Lawrence Estuary / 523 Table 5. Comparison between observations and model results for the tidal current and density constituent M 2 (see Table 4 for station locations). The amplitudes of the current components and density are given by (u,v) (10 2 ms 1 ) and σ t, respectively; their phase φ is with respect to Eastern Standard Time. The superscript o represents observed values, and v and φ are the predicted minus the observed values. The model error is estimated in the last row from (1/N) i ( u i 2 /u i 2 ) for amplitudes where N is the number of observations. For the phases, the error is simply expressed as the mean difference between the observations and the model. Stn. u o u φ o φ u v o v p φ v o φ v σ t o σ t φ σ o φ σ Skill

21 524 / François J. Saucier and Joël Chassé Table 6. Axial tidal current constituents S 2, N 2, K 1 and O 1 (u in 10 2 m s 1, φ in 8est; see Table 5 for notation). S 2 N 2 K 1 O 1 Stn. u o u φ o φ u u o u φ o φ u u o u φ o φ u u o u φ o φ u

22 Tides and Buoyancy Effects in the St. Lawrence Estuary / 525 Table 7. Mean current and density over the record lengths for observed series, and over one month for the model (see Table 5 for notation). Stn. u o u v o v σ t o σ t

23 526 / François J. Saucier and Joël Chassé Table 8. Comparison between observations and the model for the tidal current constituent M 4 (u o > 0.10 m s 1 ). Stn. u o u φ o φ u v o v φ v o φ v rent and the density, respectively (Table 5 and Fig. 3). Table 5 shows good agreement between the observed and modelled M 2 amplitudes and phases. As shown in Table 5, the model reproduces the observed density M 2 amplitude within an error of 25% for 24 instruments out of 39. Table 6 shows the relative agreement between other main observed and modelled semi-diurnal and diurnal along-channel current harmonics. Table 7 shows that the density and the circulation have established a steady state that compares relatively well with the observations. Table 8 shows that the quarter-diurnal current harmonic, generated from non-linear terms in the momentum and density equations have amplitudes of the same order as diurnal components, and are also well reproduced by the model. Finally, Fig. 3b underlines the importance of fortnightly modulations in the density field, associated with large changes in mixing intensity (Section 6). Diagnostics show that some important differences in velocity and density signals between the model and the observations are quite sensitive to the vertical or horizontal sampling positions in the model. This is particularly true for MS f phases, shifting by about 1808 across the pycnocline near 10- to 30-m depth. For model-derived harmonic amplitudes in the upper estuary, relatively small changes in the horizontal sampling position (less than 500 m) can also produce differences over 50% in the amplitude of the current or the density change harmonics. Older instrument positions may not be this precise. Large discrepancies are found for current meters nos. 9 and 30. Record no. 9 shows complex currents and is suspicious when compared to nearby upstream and downstream measurements. The station no. 30 is near the surface and better current comparisons are obtained for instruments 29 and 31 that were deployed deeper on the same mooring. The density changes at the shallow (10 m) stations 28 and 30 are also not well reproduced and the semi-diurnal component is nearly 1808 out of phase. This is suspicious considering the overall tidal circulation, but the current shear is high and complex in this region of strong density effects, as shown below. Finally, while the currents are rather well reproduced for stations 45, 47 and 48, the model underestimates the M 2 density amplitude.

24 Tides and Buoyancy Effects in the St. Lawrence Estuary / 527 Table 9. Comparison between observations and the model for the tidal current and density constituent MS f (u in 10 2 m s 1 ; φ in 8est; σ t in kg m 3 ). The transverse current component v is omitted. The column labelled -z shows the instrument depth. Stn. -z (m) u o u φ o φ u σ t o σ t φ σ o φ σ Tidal circulation The irregularities in the mixed tides, changes in initial conditions, the runoff, and offshore conditions, mainly affect the magnitude of the modelled semi-diurnal changes in the circulation and density fields. Qualitatively, one sees little difference from one semi-diurnal cycle to the next. Figures 4 to 7 show hourly modelled currents and densities over a semi-diurnal summer cycle of average amplitude, for various depth intervals, extracted from a simulation. Because of the lack of space, the reader is referred to Government of Canada (1997) and Saucier et al. (1999), where charts of the surface circulation are presented for the whole domain. a Tidal Flood Figures 4a and 5a show that the ebb-flowing surface waters from the Saguenay Fjord and the North Channel first come to stop against emerging denser waters near sill Sb at LW 1 2 hrs (where LW stands for Low Water). Surface flow reversal takes place

25 528 / François J. Saucier and Joël Chassé only one to two hours later in the South Channel. Mariners have noted that when the flood begins, the surface waters from the lower estuary move southeasterly from the head region and then south around Ilet Rouge (Dominion of Canada, 1936; also noted by Reid, 1977; Ingram, 1985, and from surface buoys trajectories). Near LW 1 2 hrs, intermediate waters are pushed up along the lower estuary north shore and move over sill Sc against the outflowing Saguenay surface waters (Fig. 5). A surface front forms between the surfacing intermediate waters and the surface waters. A segment of this frontal structure then rotates clockwise at the mouth of the Saguenay Fjord between LW 12 and LW 1 3 hrs, until the dense waters find the paths into the outer fjord basin (Fig. 4). Strong surface convergence begins along the front. Near LW 1 3 hrs to LW 1 4 hrs, the dense waters invade sill Sb and, in turn, produce a surface front and initiate subduction into the North Channel, in just the same way as over sill Sc. The current and density changes across the fronts increase, and the fronts maintain stationary positions over the lee sides of the sills until high water. The surface signature of the two major convergence fronts, near sills Sb and Sc, is visible in aerial photographs taken at high tide (Fig. 8). The positions of the fronts in the aerial photographs are within 1 km of the modelled fronts. The surface density contrasts extending over Ilet Rouge bank at high tide were previously described by Ingram (1976, 1985) as sharp features (smaller than 2 m in width), with density changes across by 2 to 5 kg m 3. The same change occurs in the model over approximately one cell width, and its magnitude decreases by a factor of two from spring to neap tides. In the next section, we show that these fronts are associated with internal hydraulic jumps (e.g., Farmer et al., 1995; Farmer and Armi, 1986). Other fronts are modelled at various stages of the tide near Batture aux Alouettes, Île aux Lièvres, La Malbaie and Île-aux-Coudres (e.g., Figs 5a and 6, see observations by Greisman and Ingram, 1977; El-Sabh and Murty, 1990). Figure 6 shows the evolution of density in the area of rapid shoaling near Île-aux-Coudres. Although density variability is mainly controlled by salinity changes in the St. Lawrence (e.g., Galbraith, 1992) temperature and salinity variations are sufficiently well correlated that Figs 5a and 6 compare with spatial structures observed at various stages of the tide in summer satellite thermal infrared images (Lacroix, 1987; Thibault 1995; Lavoie et al., 1996). Upward displacements of intermediate waters over the sills, by over 75 m, occur within three hours (see the vertical velocities in Fig. 7). The dense waters passing the sills (σ t > 24 kg m 3 ) set up gravity currents on the landward slopes of sills Sa, Sb, Sc and Se. The surface waters of the St. Lawrence and the Saguenay stop their seaward displacement and remain nearly still near Île aux Lièvres and in the Saguenay Fjord, only slowly backing up during large flood tides. b Ebb flow Near HW 1 1 hr (where HW stands for High Water) upwelling comes to stop on the outer slopes while bottom density currents still go strong over the inner slopes (Figs 4c and 7). Between HW 1 2 hrs to HW 1 3 hrs, intense downwelling, with vertical

26 Tides and Buoyancy Effects in the St. Lawrence Estuary / 529 Fig. 4 Hourly horizontal currents modelled over a mean summer semi-diurnal cycle (tidal range of 3 m) between Les Escoumins and Saint-Siméon (the tidal reference port is Rimouski). (a) Horizontal depth-averaged currents between the surface and 20-m depth; (b) same as (a) between 20 and 50 m; (c) same as (a) between 50 and 100 m. The same model results for surface currents over the whole model domain can be found in Government of Canada (1997). velocities greater than m s 1, takes place through the water column over the outer slope (Fig. 7). This is associated with the continuity of the flow over the rapidly increasing depth, and the negative buoyancy of the portion of oceanic waters previously brought up in the upper estuary and now returning. Surface waters that were backed up in the Saguenay Fjord and near Île aux Lièvres drain to restratify the head region. They maintain a relatively sharp leading edge as they move over the sills Sb and Sc (Fig. 5a); the Saguenay waters first, followed by the St. Lawrence waters. The flow over sills Sb and Sc rapidly becomes supercritical and produces large internal deformation on the lee sides. More attention is focused on the physics over the outer sills in the following sections. The pattern of surface ebb flow (Fig. 4a) is in agreement with observations presented elsewhere (Ingram, 1976; Reid, 1977; Seibert et al., 1978; Government of Canada, 1939). The main outflow region of the estuary forms a surface jet from the

27 530 / François J. Saucier and Joël Chassé Fig. 4 (Continued) North Channel, with velocities exceeding 3 m s 1 during spring tides, that branches around Ilet Rouge. The southern branch carries over twice as much surface water than does the branch north of Ilet Rouge. The northern branch, flowing over sill Sb, is intensified by the Saguenay waters and rejoins the southern branch on the southern bank. There, as is observed and modelled, the flow only reverses with large flood tides. c Tidal Streams and Buoyancy Fluxes over the Sills The monthly average tidal flood stream (that is the mean transport during flood), summed over sills Sa, Sb and Sc, is about m 3 s 1 (Table 10). This is somewhat smaller than the value of m 3 s 1 computed by Farquharson (unpublished manuscript) over his section A18 using observations and other parameter estimates. Since the tide is nearly standing in the Saguenay Fjord (see Tables 2 and 3, tide gauges 5 and 19), the mean tidal stream over sill Sc should be m 3 s 1 (the Saguenay tidal prism may be estimated from its area, m 2, times the average tidal range, 3.9 m), less than 2% higher than the mean modelled stream (computed from Table 10). Over a modelled tidal cycle of average range, twice as much oceanic water is

28 Tides and Buoyancy Effects in the St. Lawrence Estuary / 531 Fig. 4 (Concluded) forced over the southern bank (sill Sa) than over sill Sb (Table 10), and the tidal flood stream is 50% higher during spring tides than during neap tides over both sills. The buoyancy fluxes over the sills are sensitive to blocking (e.g., Reid, 1977). From Bernouilli s equation and the Brunt-Väisälä frequency, given by N 2 g ρ 0 the height a water parcel can reach near the sill before its kinetic energy is completely transformed into potential energy is approximately πun 1 (Farmer and Denton, 1985). Given u > 0.2 m s 1 and N > 10 2 s 1 near sill depth for a spring tide, we find h > 60 m. This value suggests that the buoyancy fluxes over the sills are quite sensitive to the mean stratification as well as to the tidal range. Examining the effect of tidal range, Table 10 shows the modelled buoyancy fluxes, B s, over the three outer sills (including Sc), averaged over monthly and also 14-day periods centred on neap and spring tides, computed from ZZ B s g (σ t (z) σ t )u? ndγ (23) Γ σ t z (22)

29 532 / François J. Saucier and Joël Chassé Fig. 5 Density anomaly σ t field at hourly intervals over a mean semi-diurnal cycle between Les Escoumins and Saint-Siméon (same cycle as in Fig. 4). (a) Surface; (b) Bottom.

30 Tides and Buoyancy Effects in the St. Lawrence Estuary / 533 Fig. 6 Modelled hourly surface density anomaly σ t over a mean summer semi-diurnal cycle (same solution as in Figs 4 and 5) between Saint-Siméon and Cap-aux-Oies. The reference port for high and low waters is Rimouski. where Γ is the cross sectional area, u? n is the velocity component normal to the cross section, σ t (z) is the density and σ t is a reference density that we choose in the lower estuary near sill depth (σ t 24 kg m 3 ). The outer sill s net buoyancy flux (Sa 1 Sb), over a few Ms f periods, must balance the buoyancy intake from the St. Lawrence River at the head. With σ t 24 kg m 3 and the river flow Q f m 3 s 1, we have B f J m 1 s 1 at the head which is about equal to the sum of the buoyancy fluxes over sills Sa and Sb computed in Table 10 (i.e., J m 1 s 1 ). Table 10 also shows the tidal flood stream of waters with σ t > 23 kg m 3 for spring and neap tides. During neap tides, the flood stream of these waters is small over sill Sb and 15 times less than over sill Sa. During spring tides, the net stream of dense waters over sills Sa and Sb is three times larger than during neap tides, while the stream of dense waters over sill Sb approaches 70% of that over sill Sa. Reid (1977) also documented large variations in the amount of oceanic waters passing sill Sb as a function of tidal range. As reported in Table 10, the buoyancy flux doubles during spring tides over sill Sb, and becomes larger than the flux over sill Sa. For the Saguenay Fjord, the mean buoyancy flux is ten times less than the flux into the Upper Estuary, i.e., J m 1 s 1, and it is not quite balanced at the mouth in the model (with J m 1 s 1, see Table 10). This suggests that the

31 534 / François J. Saucier and Joël Chassé Fig. 7 Hourly vertical velocity (w) at 25-m depth over the same semi-diurnal tidal cycle as in Figs 4 to 6 between Les Escoumins and Saint-Siméon. water is not being renewed in the Saguenay Fjord during the present simulation. This is expected as a result of not having dense enough waters below sill depth in the lower estuary to renew these waters (e.g., Drainville, 1968), and the extreme sensitivity of estuarine circulation to initial density conditions in the inner fjord basin. In summary, one may examine the last panel in Figs 7 and 9, which shows the mean currents over the simulation period. The addition of tides and estuarine circulation produces large subsurface flood currents and strong surface ebb currents. This circulation and the mean density field (Table 7) compare well with measurements made by Neu (1970), Reid (1977), Ingram (1979), Muir (1979) and Seibert et al. (1978). The model confirms net upstream flow along the north shore in the Lower Estuary and lateral transport to the south both upstream and downstream from Ilet Rouge. Below 100 m, near Les Escoumins, the model also suggests that recirculation occurs on the flank of Ilet Rouge bank.

32 Tides and Buoyancy Effects in the St. Lawrence Estuary / Internal tides Forrester (1974) and Therriault and Lacroix (1976) documented the propagation of an internal semi-diurnal tide in the lower estuary from the head region towards Pointe des Monts. Fitting various lateral and vertical modes, given a general density profile similar to the one established here for the Lower Estuary, to the baroclinic currents downstream from Les Escoumins, Forrester (1974) detected a Poincaré internal M 2 tide in the second vertical mode propagating seaward with an axial wavenumber equal to m 1 and an axial current amplitude of 0.26 m s 1. He found the cross-channel oscillation to be a standing wave lagging the axial component by 708, while the linear theory predicts 908 (e.g., Gill, 1982). Muir (1979), DeGuise (1977), Ingram (1978), Simard (1985), Galbraith (1992) and others (see Mertz and Gratton, 1990) documented the presence of other internal gravity wave motion in the estuary. Figure 10 shows a chart of the region where reference positions are indicated for Figs 11 to 20. We first concentrate on cross section A running along the lower Estuary from sill Sb. Figure 11 shows the modelled M 2 amplitude and phase for the water level, the barotropic and baroclinic along- and cross-channel currents, and the density. The barotropic current is approximated as the depth-averaged current, and the baroclinic current in each layer is defined as the deviation from it (consistent with Forrester, 1974, although a somewhat simpler description of barotropic currents was used there). For comparison, the axial baroclinic current and phase lag computed by Forrester (1974) are also plotted on Fig. 11. The results are in good agreement with Forrester (1974). Figure 11c shows that the baroclinic current M 2 is highest in amplitude near the surface and maximum over the sloping region between Les Escoumins and Cap de Bon-Désir, down the outer slopes (recall that Forrester s most upstream instrument was 22 km from sill Sb). Figures 11g and h show the density amplitude and phase for M 2. The amplitude is maximum near the surface at the head (2 kg m 3 ) and decreases linearly to about 0.75 kg m 3 15 km from the head. Further seaward, the amplitude at the surface decreases while at 25-m depth, it remains at about 0.25 kg m 3 toward the boundary near Rimouski. Once the channel widens and the bottom flattens out, the wave propagates seaward with a phase progression equal to o m 1. These model results confirm the findings of Forrester (1974) away from the sill for the phase, the amplitude of the baroclinic along-channel surface current (about 0.25 m s 1, see Fig. 11c), and the phase of the cross-channel velocity component (lagging the long-channel component by 908) of an internal tide generated over the outer sills and propagating seaward. As suggested by Forrester (1974), the surface transverse component v is almost purely baroclinic. The maximum baroclinic current amplitude is found near the surface. The model estimates the surface axial current amplitude to be twice the value reported by Forrester. The phases fit well away from the sloping region. Measurements from current meters in Table 5 also confirm the model results for the internal tide propagating in the lower estuary (note the model evaluation for stations 2, 3, 4, 5, 7, 8, 13, 14, 20 and 21). Our results are also in good agreement with Ther-

33 536 / François J. Saucier and Joël Chassé

34 Tides and Buoyancy Effects in the St. Lawrence Estuary / 537 Fig. 8 Aerial photographs taken at high water on 14 November See Fig. 5a, near high water, for a comparison with the modelled density changes. (a) Aerial photograph of the surface separation front over sill Sb viewed from the west. The darker blue water is plunging over sill Sb at the left beneath the greenish surface waters of the upper estuary, which remain nearly still. (b) The front at the mouth of the Saguenay Fjord (sill Sc) viewed from the southeast. The intermediate Gulf waters subduct from the estuary under the Saguenay surface layer (see the wake behind the anchored buoy below the tip of the front). Vorticity is shed into the dense waters from Pointe à Granite and seen at the cusp and also in the model results; conservation of vertical relative vorticity flux through subduction produces this vortex. The flow of dense waters exceeds m 3 s 1 across the front at this time. These two fronts have been observed at the same time in the tidal cycle and with roughly the same shapes on several occasions by the first author and have been reported by navigators. riault and Lacroix (1976). Their time series for locations along cross section A (Fig. 10) will further demonstrate the fit with their observations (see Figs 4 and 5 in Therriault and Lacroix (1976); P2 corresponds to their station 63 and P3 to station 190 in our Fig. 19). Although unnoticed by Therriault and Lacroix (1976), their data suggest that the wave crest propagates to their station 190 slightly later than at their head stations 63 and 90. The data at their stations 277 and 240 show the progressive pattern with about a 1008 advance over the head stations, which is consistent with our results. The interaction of the flood tide with the topography produces an elevation of the isopycnals over the sills and within the upper estuary, associated with density changes reaching an amplitude greater than 1.5 kg m 3 over sills Sb and Se, that provides the forcing to generate internal tides. Figure 12 shows M 2 modelled co-phase and co-amplitude charts for σ t at the surface and at 25-m depth. At the surface, the density change is approximately equal to that due to the barotropic forcing. It vanishes in the lower estuary and dominates the density changes at all depths close to the sill. At 25-m depth (Figs 12 c,d), we find the internal waves propagating from the head region and from English bank (sill Se). We observe little internal wave energy scattering from sill Sb into the upper estuary, as opposed to the predictions by Blackford (1978) and Mertz and Gratton (1990). Conversely, this region is dominated by seaward-propagating internal waves. This is consistent with the results of Hibiya (1990), and the transition from supercritical to subcritical flow conditions over the sills (e.g., Farmer and Freeland, 1983). Eight out of 11 M 2 density phase differences in the North Channel, (Table 5, stations 24, 28, 29, 30, 32, 33, 34, 37, 38 and 43) confirm the presence of an internal M 2 tide propagating seaward in the upper estuary (Figs 12 c,d). Forrester (1974) and Galbraith (1992) have noted that Poincaré waves in the first and second vertical modes are evanescent owing to the channel width near the head region (as narrow as 7 km). Furthermore, only waves of mode two or higher may propagate in equilibrium with rotation seaward from Ilet Rouge bank (as the channel width becomes larger than 25 km). The internal Rossby radius for an internal wave of vertical scale m 1 < 100 m, is a < N/mf, where N < 10 2 s 1 is the Brunt-Vaïsälä

35 Table 10. Semi-diurnal tidal streams and buoyancy fluxes over the outer sills. Neap tide Spring tide Mean Stream σ t > 23 kg m 3 B s Stream σ t > 23 kg m 3 B s <B s > (10 5 m 3 s 1 ) (10 4 m 3 s 1 ) (10 6 J m 1 s 1 ) (10 5 m 3 s 1 ) (10 4 m 3 s 1 ) (10 6 J m 1 s 1 ) (10 6 J m 1 s 1 ) Sill Sa Sill Sb Subtotal Sill Sc Total / François J. Saucier and Joël Chassé

36 Tides and Buoyancy Effects in the St. Lawrence Estuary / 539 Fig. 9 Mean monthly-modelled currents (i.e., Z 0 component) between Les Escoumins and Cap-aux- Oies: (a) integrated between the surface and 20-m depth; (b) integrated between 20-m and 50-m depth; (c) integrated between 50-m and 100-m depth; (d) integrated over depths larger than 100 m. See Table 7 for the comparison with the observations.

37 540 / François J. Saucier and Joël Chassé Fig. 10 Map showing the positions of cross sections and sampling positions in Figs 11 to 20. frequency and f < 10 4 s 1 is the Coriolis parameter, is of the order of 10 4 m. Thus rotation effects become important shortly after generation, although we do not examine the theoretical aspects of this transition here. 5 Internal hydraulics The flow structures over the sills depend upon the depth, length and shape of the constriction, the stratification, and the velocity. To describe the nature of this flow, we can use the densimetric Froude number according to linear theory, F i u/c i, where c i is the phase speed of a long internal wave of vertical mode i at the sill crest (Farmer and Smith, 1980). For long sills, such as in Puget Sound (Geyer and Cannon, 1982), friction becomes important (e.g., Pratt, 1986), that is when the parameter C d L/H > 1, where C d is the drag coefficient, L is the sill s length and H its depth. Using the conservative values C d > , L > m, and H > 30 m, one finds an upper bound of C d L/H > 0.5 for sills Sb, Sc, Sd or Se, which suggests that the flow over these sills can be regarded as a potential flow. The effects of rotation may also be neglected at the length scale of these sills, following the analysis from Sambuco and Whitehead (1976) (see Farmer and Smith, 1980). Knowing that critical conditions are met for any mode if they are met for mode 1 (e.g., Farmer and Smith, 1980), we concentrate on mode 1. Let us consider the flow on vertical sections crossing sills Sb and Sc along the mean current directions (Figs 13 and 14). While we consider the two-dimensional aspects of the flow on these sections, the confluence of outflows over these two sills is highly three-dimensional and probably leads to more complicated dynamics. From Fig. 13, we calculate a depth-avergaed N > s 1 during ebb, and N > s 1 and less near the surface to s 1 at sill depth during flood. We have c 1 > N 1 H somewhere in the mid-flow away from the friction-dominated bottom layer. If we choose u as the depth-averaged velocity over the sill, u varies from 2 m s 1 to 1.8 m s 1 from ebb to flood (Fig. 13); with H 30 m over the crest, we find an internal wave speed c 1 > 0.3 m s 1 during ebb and c 1 > 1.2 m s 1 during flood so that 0 < F 1 < 6.7 during ebb and 0 < F 1 < 1.5 during flood (using surface stratification). In such ranges, complex flow transitions occur as responses to varying flow and stratification conditions, and

38 Tides and Buoyancy Effects in the St. Lawrence Estuary / 541 Fig. 11 Baroclinic M 2 tidal current and density amplitude and phase as a function of distance from sill Sb (on the axis shown as section A, Fig. 10, extending toward the northeast): (a) amplitude of surface elevation as a function of distance from the sill; (b) phase of surface elevation (8EST); (c) amplitude of axial currents at 2.5- (bold line), (regular line) and 225-m (dashed line) depth the barotropic current component amplitude and phase are shown dotted; (d) phase of the axial currents; (e) amplitude of the cross-channel currents; (f) phase of the cross-channel currents; (g) density amplitude; (h) phase of density. The asterisks show the «axial minus continuity» (or baroclinic component) and cross-channel M 2 amplitude and phase lag taken from Forrester (1974), as well as his prediction for the current phase lags at the origin (sill Sb). Note that Forrester (1974) worked in Atlantic Standard Time, and that he defined the velocity component u positive inland, as opposed to positive seaward here.

39 542 / François J. Saucier and Joël Chassé Fig. 12 Co-amplitude and co-phase (8EST) charts of density component M 2 : (a) and (b) surface; (c) and (d) 25-m depth.

40 Tides and Buoyancy Effects in the St. Lawrence Estuary / 543 Fig. 13 Density (left) and velocity (right) at two-hour intervals during a semi-diurnal cycle (same as in Fig. 4) on section B (shown in Fig. 10) from the upper to the lower estuary across sill Sb. The velocity component is tangent to the cross section and the maximum vector magnitude is 3.1 m s 1. Fig. 14 Density anomaly σ t and velocity changes at two-hour intervals during a semi-diurnal cycle (same cycle as in Fig. 4) on section C (shown in Fig. 10) from the upper to the lower estuary across the sill Sc. The velocity is tangent to the cross section and the maximum magnitude is 2.5 m s 1.

41 544 / François J. Saucier and Joël Chassé Fig. 15 Undular internal jump observed with acoustic sounding over sill Sb two hours before low water. no steady state is ever approached (Farmer and Freeland, 1983). When F 1 > 1, supercritical conditions occur, that is, the flow speed exceeds internal wave speed, thus inhibiting the propagation of wave-like motion in the upstream direction. If the flow becomes subcritical away from the sill, an internal hydraulic jump may develop. This condition is modelled in Figs 13 and 14 during both ebb (near HW 1 4 hrs on the outer slopes) and flood tides (near HW on the inner slopes), while the isopycnals are sharply depressed on the lee sides. When F 1 < 0.3, we may expect lee waves to develop, but not in the present hydrostatic model (e.g., Stacey and Zedel, 1986). However, lee waves were indeed observed from acoustic soundings in the early stage of ebb flow (Fig. 15). Farmer and Freeland (1983) discuss conditions for which these undulations, if generated before slack currents, could propagate as internal bores over the sill (e.g., Haury et al., 1979). Gravity currents on the inner slopes, associated with flow separation at the surface (see Fig. 8), are shown in Figs 13 and 14 near high water. Two to three hours after high water, the current turns to ebb and restratification takes place over the sills. Hydraulic jumps develop over the outer sills Sb and Sc at about HW 1 4 hrs. They are associated with a sharp depression of the isopycnals near the sills. They eventually break up as bottom flow separation migrates up to sill height near the time of low

42 Tides and Buoyancy Effects in the St. Lawrence Estuary / 545 Fig. 16 Left column: time series of density as a function of depth through a semi-diurnal cycle at points along cross section A (see Fig. 10 for location). The distance from sill Sb is shown. The right column shows the depth-integrated density, approximating the steady isopycnal oscillations.

43 546 / François J. Saucier and Joël Chassé Fig. 17 Density and tangential velocity on cross section D over sill Sa (see Fig. 10). The maximum velocity is 2.5 m s 1. Fig. 18 Density and tangential velocity on cross section E passing sill Se at the left. The maximum velocity is 2.5 m s 1.

44 Tides and Buoyancy Effects in the St. Lawrence Estuary / 547 water. Figure 16 shows the model results for density time series along cross section A. We can see an internal wave train radiating from the sill three to four hours after high water (not visible from the hourly sampling shown in Fig. 13. Its propagation is led by a sharp depression of isopycnals, or internal bore, with surface waters plunging to more than 25-m depth. This model predicts the observations from Galbraith (1992), and the modelled vertical velocity changes at 25-m depth in the lower estuary (Fig. 7), associated with the propagation of the bore, are quite similar to the refraction pattern photographed near Rimouski from a Landsat satellite (Simard, 1985). Hibiya (1986, 1990) has shown how internal wave generation works in a two-layer system, in the non-rotating regime, with the build-up of a series of elementary interfacial waves emanating from the sill during the accelerating phase of the tide, and the addition of another series emanating during the decelerating phase. Our results are consistent with this model. In highly sheared tidal flow, as over sill Sb, Hibiya (1990) predicts that the seaward-propagating wave amplifies while the landward one does not build up efficiently. The pattern of the seaward-propagating interfacial wave (see Fig. 6 in Hibiya, 1990) is rather similar to the one presented here. As the flood currents increase, the pycnocline rises against the sill and is sharply depressed on the other side (consistent with our jumps and density currents on the landward sides of sills Sb and Sc, Figs 13 and 14). With the turn to ebb, the wave is relaxed and the sharp depression travels over the sill (consistent with the leading edge of the surface layer travelling over the sills and the internal jumps formed seaward). 6Mixing a Shear Instabilities Estuarine mixing occurs through shear production leading to turbulence between layers of different densities. The shear is produced from the superposition of estuarine flow, baroclinic tidal currents, internal waves, and density currents. As a result, when the gradient Richardson number Ri is less than about 0.25, internal gravity waves, starting with short ones, become unstable if kept in this regime (e.g., Turner, 1973). DeGuise (1977) found high-frequency internal gravity waves near Pointe-au- Pic, with periods less than 300 s, in the interval between maximum ebb to maximum flood. Ingram (1978) also documented high frequency internal waves off Île Verte about 1.5 hours before low tide. Figures 17 and 18 show the evolution of density and velocity on cross sections D and E (located in Fig. 10). Figure 19 shows the time series of density and velocity at the model sampling points P1 to P8 (positions shown in Fig. 10). These figures show that the currents are highly baroclinic and produce enough shear for vertical turbulent mixing to take place in the lower estuary near the head region (points P1 and P2), in the South Channel (P5, Fig. 17), and in the North Channel (P6, Fig. 18). Internal oscillations occur at frequencies near N during ebb throughout the outflow region, and those are associated with occurrences of the critical state Ri < 0.25 in the mean flow. Acoustic soundings provide further evidence of Kelvin-Helmholtz (K-H) instabilities during the early ebb period in the upper estuary (Fig. 20 here and see Simard, 1985).

45 548 / François J. Saucier and Joël Chassé Fig. 19 Modelled time series in each model layer at points P1 to P8 (labelled 1 to 8 on Fig. 10) over a typical diurnal cycle. First row: surface elevation; second row: barotropic axial current; third row: baroclinic axial currents every two layers (the top layer is labelled); fourth row: density every two layers; fifth row: instantaneous (computed each minute) occurrence of Ri < 0.25 between each layer. We expect wavelengths of the order of 2πh, where h is the thickness of the shear layer, to be most unstable and first lead to K-H instabilities (Turner, 1973), but the model cannot resolve such scales. To verify that the oscillations seen in the baroclinic currents and density are not spurious numerical instabilities, we reduced the time step by factors of 2 and 4, and found that the same oscillations occur near the buoyancy period. Thus we propose that the relative dissipation among the regions of the modelled estuary and through the tidal cycle correctly reflect the dominant processes leading to the production of turbulence. To estimate the relative timing, location and intensity of mixing events through the semi-diurnal cycle, we compute the vertical eddy buoyancy flux per unit area over the water column: B eddy g 0 Z H σ t A V σ dz (24) z Figure 21 shows the hourly vertically-integrated buoyancy flux per unit area in the region near Ilet Rouge. Although sill processes produce strong dissipation, we see in

46 Tides and Buoyancy Effects in the St. Lawrence Estuary / 549 Fig. 19 (Concluded) Figs 19 and 21 that the mixing is produced in horizontal shear layers in the main outflow channel, particularly in the South Channel, where tidally renewed deep waters are scraped off by the surface outflow through shear instabilities. This is similar to mixing in a salt-wedge estuary (Geyer and Farmer, 1989). Second in importance as a source of mixing, we find the shear produced by the bottom density currents in the North Channel during flood. Bourgault et al. (1999) have compared the shear and the stability of the water column at various phases of the tide in this region of the upper estuary from (Acoustic Doppler Current Profiler (ADCP) and Conductivity Temperature Depth (CTD) profiles. It was found that although the present model predicts smeared pycnocline and velocity profiles when compared with the observation, the gradient Richardson numbers, on the other hand, compared well. Then, in order of importance, we find mixing during early ebb flow over sills Sb and Sc. The study of mixing between the western end of the North Channel, near Île-aux-Coudres, and the salt wedge near Quebec City is outside the scope of this analysis. b Fortnightly Changes Figure 22 shows the domain-integrated vertical turbulent buoyancy flux computed over 28 days. Mixing is five to ten times more important during ebb than during flood currents. Then it is shown that the net mixing over a tidal cycle is a strong function of the tidal range, fluxes more than doubling over spring tides when com-

47 550 / François J. Saucier and Joël Chassé Fig. 20 Kelvin-Helmholtz (K-H) instabilities observed acoustically during ebb flow in the upper estuary near point P6, 1102 to 1107 est 12 August 1997, nine hours after low water at Rimouski. The horizontal axis is time and the boat was drifting in the surface current while observing this series. pared to neap tides. The mean value is B eddy J s 1. This interaction between M 2 and S 2 results in large changes in density over the MS f period (14.7 days), and such changes may serve as sensitive indicators of mixing (e.g., Geyer and Cannon, 1982; Farmer and Freeland, 1983). Therriault and Lacroix (1976), Ingram (1983) and Lacroix (1987) found modulation of the cold water extent and intensity with the neap-spring tidal cycle near the head region. Taken separately, the observed and modelled phases for the MS f density constituent exhibit two modes (Table 9). And the same two modes also appear in the differences between the model and the observations. The phase of the MS f density change is either around 408 to 808 (density increasing during spring tides) or around 2508 to 3008 (density decreasing during spring tides). Fortnightly density phase differences around 1808, among instruments at different depths, are consistent with higher mixing (downward buoyancy flux) during spring tides, bringing denser waters to shallow instruments and lighter waters to deeper ones. For depths less than 20 m seaward from English Bank, the model and the observations agree for eight out of nine instruments, showing denser waters during spring tides. For the other instruments, between 20 and 40 m, the modelled and observed phases generally differ by about The phase differences among real and virtual instru-

48 Tides and Buoyancy Effects in the St. Lawrence Estuary / 551 Fig. 21 Hourly and mean (last panel) vertically-integrated vertical turbulent buoyancy flux over a semi-diurnal cycle (same cycle as in Figs 4 to 6). ments suggest that the modelled mixing depth is underestimated. Landward from English Bank, the results are poor and suggest that modelled mixed layer depth is overestimated. For 13 out of 18 instruments seaward from English Bank, both the model and the observations suggest that the surface outflow is enhanced during spring tides to a depth of about 40 m, with an amplitude of to 10 1 m s 1. For the Lower Estuary, these results are consistent with Hibiya and Leblond (1993), who model enhanced estuarine circulation from increased mixing intensity landward during spring tides. But their model also suggests decreased estuarine circulation landward from the mixing regions, while the present observations and model cannot confirm this and show mixed results. Mixing is more uniformly distributed in the Upper St. Lawrence Estuary and further work is required to examine the changes in circulation induced by higher mixing during spring tides.