Hydraulics and mixing in the Hudson River estuary: A numerical model study of tidal variations during neap tide conditions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003jc001954, 2004 Hydraulics and mixing in the Hudson River estuary: A numerical model study of tidal variations during neap tide conditions Petter Stenström Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden Received 19 May 2003; revised 16 February 2004; accepted 27 February 2004; published 21 April [1] Three-dimensional numerical modeling is performed to study intratidal and alongchannel variability in stratification and mixing in the Hudson River estuary. The modeled fields show good agreement with observations, both qualitatively and quantitatively. Estuarine circulation dominates the mean fields, and intratidal variability is dominated by tidal straining that acts to strengthen the stratification during ebb and weaken it during flood. Mixing is mainly confined to a bottom layer during flood but occurs higher up in the water column during ebb. Mixing across the halocline shows marked along-channel variability due to bathymetric effects. During ebb, mixing occurs preferentially at an abrupt channel expansion seaward of a channel constriction at the George Washington Bridge, as predicted by Chant and Wilson [2000]. The salt flux across the halocline in this region, averaged over ebb, exceeds kg m 2 s 1, a factor of 3 greater than the along-channel average. Increased residence time of tracers should be expected in this region due to the strong mixing but also due to observed secondary circulation [Chant and Wilson, 1997]. Mixing across the halocline during flood is small, except for early flood, before the well-mixed bottom layer is developed. Mixing is then localized to the landward slope of sills. INDEX TERMS: 4235 Oceanography: General: Estuarine processes; 4255 Oceanography: General: Numerical modeling; 4568 Oceanography: Physical: Turbulence, diffusion, and mixing processes; KEYWORDS: estuarine circulation, stratification, turbulent mixing, salt flux, tide, numerical modeling Citation: Stenström, P. (2004), Hydraulics and mixing in the Hudson River estuary: A numerical model study of tidal variations during neap tide conditions, J. Geophys. Res., 109,, doi: /2003jc Introduction [2] The temporal variation of the stratification in estuaries reflects the competition between mean advective processes that act to increase the stratification and stabilize the water column, and turbulent mixing that acts in the opposite direction. The mean advection, the so-called estuarine circulation, is due to the longitudinal density gradient that is the main characteristic of an estuary, with lighter water at the head and denser water seaward. This longitudinal density gradient, together with the surface setup at the head due to the freshwater discharge, drives a circulation where lighter water is carried seaward on top of a return current of heavier seawater. Tidal forcing interacts with this circulation in various ways. During ebb when the tidal wave withdraws seaward, the bed shear acts together with the estuarine circulation to promote stratification; with weak mixing, maximum stratification is reached near low water. Conversely, during flood, the bed shear counteracts the estuarine circulation to reduce stratification, which reaches a minimum near high water. This process is known as tidal straining of the density field [Simpson et al., 1990]. The tide also supplies mixing energy so that topography-induced turbulence (tidal stirring), shear instabilities and breaking Copyright 2004 by the American Geophysical Union /04/2003JC001954$09.00 internal waves all contribute to vertical exchange of momentum and tracers. Further, Nepf and Geyer [1996] used the term overstraining to denote conditions at the end of flood under which tidal straining may overturn parts of the density field, producing regions of hydrostatic instability and subsequent convective mixing. [3] The interaction between the estuarine circulation and turbulent mixing is nonlinear in the sense that the stable stratification produced by the advection of lighter fluid over heavier in the estuarine circulation inhibits turbulent mixing, while at the same time turbulent mixing diffuses the momentum of the estuarine circulation vertically, thereby reducing the horizontal advection. Owing to this feedback, small changes in the turbulent mixing energy supplied by the tide or other sources may lead to large changes in the circulation [Jay and Smith, 1990]. Particularly when the velocities associated with the river discharge, tidal velocities and density-driven shear velocities are all of the same magnitude, a strong interaction and a strong variability over a tidal cycle should be expected. [4] From an ecological perspective, the balance between advective and turbulent processes is important because the estuarine circulation is the most efficient mechanism for flushing river-borne pollutants out to the open sea; the greatest horizontal mass flux occurs at maximum stratification and minimum turbulent mixing [Nunes Vaz et al., 1989]. Turbulent mixing increases the tracer residence time 1of12

2 Figure 1. Hudson River estuary. Dots show the conductivity-temperature-depth (CTD) stations. The George Washington Bridge (GWB), Battery Park (BP), and Riverdale (RD) are also shown. in the estuary. Following an extensive field program in the Hudson River estuary (Figure 1) in 1993, a number of different studies have been made to determine the relative importance of estuarine hydrodynamic processes under moderate to strong stratification, over a tidal period and over a neap to spring tide transition. Nepf and Geyer [1996] focused on near-bed stratification over a tidal cycle using detailed velocity and salinity transects in a subarea south of the George Washington Bridge. They found tidal straining to dominate the variability in that maximum and minimum near-bed stratification occurred during late ebb and late flood respectively, with a well-mixed bottom layer developing during flood. During ebb, vertical mixing and stretching of the density field acted to reduce stratification, thereby blurring the effect of tidal straining. The rapid growth of the well-mixed bottom layer during flood was attributed to overstraining. Overall, regions of active mixing were traced through Richardson number estimates. A clear flood-ebb asymmetry was also observed here. Regions of active mixing were confined to the bottom well-mixed layer during flood. During ebb, significant stratification was found within active mixing regions and these regions were sometimes disconnected from the bed indicating that internal shear instabilities contributed to mixing. [5] Chant and Wilson [1997] studied secondary circulation based on cross-channel measurements of salinity and velocity in the vicinity of a headland at the George Washington Bridge (Figure 1). They found that under highly stratified conditions, the secondary circulation produced clear cross-channel density gradients. They also observed a tidally driven eddy develop seaward of the headland after maximum ebb and found evidence of upwelling within the core of this eddy. The mean flow field was thus not solely the result of estuarine circulation, but also of tidally induced secondary motion. [6] Ullman and Wilson [1998] fitted a depth-integrated numerical model to ADCP data to determine the spatially varying bottom drag coefficient. In general they found the values for the drag coefficient to be lower than values reported from previous studies, which they attributed to the comparatively strong stratification in the Hudson River estuary; a stable density gradient suppresses vertical exchange of momentum and thus reduces the drag coefficient. [7] Chant and Wilson [2000] studied the influence of bathymetry on the along-channel density and velocity fields using data from the ten measurement stations shown in Figure 1. Estimates of two-layer Froude numbers indicated that the flow was appreciably supercritical during ebb and mainly subcritical during flood. They further observed large-amplitude halocline excursions associated with topographic features preferentially during ebb. The halocline descended in the water column in the vicinity of channel constrictions and ascended in channel expansions. Expressing halocline slope as a function of channel geometry and flow state as described by the two-layer Froude number, they showed that this behavior was consistent with inviscid hydraulics. From Richardson number estimates they suggested that vertical mixing was enhanced immediately downstream of constrictions. [8] Peters [1997] and Peters and Bockhorst [2000, 2001] analyzed microstructure observations from cruises in the Hudson River estuary in summer 1994 and in summer and fall A relatively straight and uniform part of the channel just seaward of station 1 in the 1993 cruises (compare Figure 1) was sampled. Variations in turbulent dissipation rates [Peters, 1997; Peters and Bockhorst, 2000] and salt flux [Peters, 1997; Peters and Bockhorst, 2001], over a semidiurnal tidal cycle and over a neap to spring tide transition, were described. In summary, the most intense mixing was found to occur during flood on neap tides and during ebb on spring tides. The mixing during flood on neap tides occurred mainly within the bottom layer, and mixing across the halocline was weak. Peters [2003] analyzed microstructure observations from more recent cruises in May 2001, with the aim to study if the enhancement of mixing downstream of channel constrictions, predicted by Chant and Wilson [2000], is a significant feature. He analyzed data from a stretch of the river around the constriction at George Washington Bridge and found that averaged over tidal cycles, mixing across the halocline was strong everywhere with enhancement by a factor 2 3 just seaward of the constriction. [9] In the present paper, three-dimensional numerical modeling is performed to study intratidal and along-channel variability in stratification and mixing using the nonhydrostatic MITgcm [Marshall et al., 1997]. The simulations complement the observational studies described above and in particular allow the along-channel variation of vertical salt flux to be studied over a longer stretch of the river. Numerical simulation also avoids bias to certain times in the tidal cycle, which often is difficult to avoid in purely 2of12

3 Figure 2. Prismatic channel geometry with depth equal to the (top) maximum cross-channel depth over the entire rectangular cross section and with (middle) width adjusted to preserve the true (bottom) crosssectional area. Note that H in equations (7a) and (7b) is the height of the bottom above some datum, i.e., H = const. D. The vertical dashed line in the lower panel marks the location of the George Washington Bridge. In the text the terms constriction, contraction, and sill are used, referring to minimums in cross-sectional area, width, and depth, respectively. observational studies. The results are compared to the overall behavior described by Nepf and Geyer [1996] and Chant and Wilson [2000]. Modeled vertical salt fluxes are compared to the estimates of Nepf and Geyer [1996], Peters [1997, 2003], and Peters and Bockhorst [2001]. The forcing is chosen to represent a semidiurnal tidal cycle during neap tide conditions. Results are presented for an 8 km stretch of the river with comparatively large changes in cross-sectional area. [10] Below, the measurements are first briefly described in section 2, followed by a description of the numerical model and the numerical simulations in section 3. In section 4, the modeled fields are analyzed with regard to hydraulics and mixing. The results are then discussed in section Data [11] A series of surveys was made in the Hudson River estuary over a 5 day period (29 April 3 May) in 1993 with an acoustic Doppler current profiler (ADCP) and a conductivity-temperature-depth (CTD) instrument. Chant and Wilson [1997, 2000] give details of these surveys and of the instruments used. A 10 km stretch in the vicinity of the George Washington Bridge was sampled (Figure 1). This stretch is characterized by large variations in depth and cross-sectional area (Figure 2). Longitudinal transects were run on the first and the last days of the period, and lateral transects during the 3 middle days. River discharge was high over the period, about 1500 m 3 /s on the first day and decreasing to about 850 m 3 /s on the last day. Semidiurnal tidal transport increased from about ±8000 m 3 /s to about ±14000 m 3 /s. In general, strong stratification was observed, although the density gradient decreased by nearly a factor 2 over the measurement period [Ullman and Wilson, 1998]. Here, only data from 29 April, representing neap tide conditions, will be used, and the relevant numbers are thus 1500 m 3 /s and ±8000 m 3 /s for the river discharge and the tidal transport respectively. 3. Numerical Simulations [12] Numerical simulations were performed with the MITgcm [Marshall et al., 1997]. This model solves the three-dimensional, nonhydrostatic, Navier-Stokes equations over a rectangular grid using a finite-volume formulation. Topography is represented by partial steps [Adcroft, 1997], i.e., cells may be partially filled at solid boundaries to allow a close fit. In the present simulations a quadratic grid with 37.5 m horizontal resolution was linearly interpolated from depth data sampled at 150 m intervals. To isolate the effects on the flow of variations in channel cross section, the depth data were resampled to a straight, prismatic channel with a rectangular cross section. The depth was set to the maximum cross-channel depth, and 3of12

4 Table 1. Parameters for the Numerical Simulations Parameter Value Unit Description g 9.81 m/s 2 gravitational acceleration r kg/m 3 reference density C S 0.2 Smagorinsky constant K mh, K rh 4 m 2 /s horizontal viscosity and diffusivity coefficients Dx, Dy, Dz 37.5, 37.5, 0.2 m grid spacing n x, n y, n z 720, 30, 104 number of grid points Dt 1.0 s time step T hours forcing period U R 0.1 m/s river flow U m/s forcing amplitude the width was adjusted to preserve the true cross-sectional area (Figure 2) Subgrid Parameterization [13] Given a flux-gradient form for turbulent correlations [e.g., Rodi, 1987] and further assuming that the timedependent, advective and diffusive terms can be neglected [Peters and Bockhorst, 2001], the balance for turbulent kinetic energy can be written e ¼ P þ B ¼ K @z 2! þ K r g ; that is, a steady local balance is assumed between shear production, P, buoyancy production, B, and viscous dissipation, e. The coefficients of eddy viscosity and eddy diffusivity are denoted K m and K r. Following Smagorinsky [1963], the eddy viscosity coefficient is expressed as ð1þ K m ¼ ðc S DÞ 4=3 e 1=3 ; ð2þ where C S is a constant and D is the nominal grid spacing. Inserting equation (1) into equation (2) gives s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K m ¼ ðc S DÞ 2 2 þ K r r A scheme was implemented in the model that calculates the (vertical) eddy coefficients from equation (3) locally at every time step. Following Winters and Seim [2000], K m = K r was assumed, i.e., a turbulent Prandtl number of unity. When the square root argument in equation (3) is negative, K m and K r are set to the background values 10 4 m 2 /s and 10 6 m 2 /s respectively. The larger value for K m is needed for numerical stability. The horizontal eddy coefficients were held constant. The values for the Smagorinsky constant, C S, and for the horizontal eddy coefficients are given in Table 1, together with other parameters for the simulations. The vertical grid resolution was set to 0.2 m, which is in the lower part of the range of turbulent overturning scales found during ebb by Peters [1997] and Peters and Bockhorst [2000]. During flood, the overturning scales are larger. Most of these overturning scales are thus explicitly simulated Boundary Conditions [14] At the upper boundary the model has an implicit free surface formulation and at solid boundaries a no-slip K m condition was used. At the lateral open boundaries (note that these are located at Battery Park and Riverdale (compare Figure 1), i.e., far from the region of interest around George Washington Bridge), a modification of the condition proposed by Khatiwala [2003] was used. This is a two-part condition that decomposes the total velocity at the boundary into a depth-independent (vertically averaged) part and a perturbation part and treats them separately. The perturbation velocity is radiated out of the domain using a variant of the Orlanski [1976] implementation of the classical Sommerfeld radiation condition. The depth-independent part, U, is also radiated, using a simple integrated condition [cf. Nycander and Döös, 2003; Stenström, 2003]: U c h D ¼ 0; where h is the sea level perturbation and D is depth. The phase velocity p was set to the speed of a long gravity wave, i.e., c =± ffiffiffiffiffiffi gd, with the negative (positive) sign at the left (right) boundary. To force a net flow through the domain, a forcing term was added to the right hand side of equation (4) at the seaward boundary: ð4þ F U ¼ U R þ U 0 sin 2p t ; ð5þ T where U R is the river discharge, U 0 is the amplitude of the tidal wave, t is time, and T is the period. [15] Data were available neither for the seaward boundary at Battery Park nor for the landward boundary at Riverdale. Instead a uniform time-dependent velocity was forced at the seaward boundary, consistent with the measured time-dependent net flow due the tide and the river discharge. Salinity profiles at the boundaries were adjusted iteratively by simple trial and error to fit data at CTD stations in the interior. Figure 3 shows a comparison between modeled and measured profiles at CTD stations 1, 7 and 10, i.e., at the boundaries of the central reach that was in focus during the field program and at the headland at the George Washington Bridge. While the fit could be improved, this was not considered vital for the purpose of the present study; of most importance for an assessment of the dynamics is that the magnitude of the different flow components, the river discharge and the barotropic tide, and the salinity range agree well with the measured conditions. 4. Results 4.1. Temporal Variation in Stratification [16] The modeled salinity field at four points over the tidal cycle is shown in Figure 4. Consistent with the measurements, the stratification is nearly linear from maximum ebb to late ebb. During flood a well-mixed bottom layer propagates landward through the domain and lifts the salinity field to create a sharper halocline above. For closer assessment of the temporal variation in stratification, the advection-diffusion equation for salt is vertically differentiated, following Nepf and @x @z : ð6þ 4of12

5 Figure 3. Comparison between measured and modeled profiles at (top) maximum ebb and (bottom) maximum flood for CTD stations (left) 1, (middle) 7, and (right) 10. Solid lines show streamwise velocity, and dashed lines show salinity. Thin lines show measured profiles, and bold lines show modeled profiles. Positive velocities are up-estuary. On the right-hand side, the first term represents straining, the second stretching, and the third diffusion of the salinity field. Cross-channel terms are neglected and the alongchannel salinity gradient and velocity are considered uniform. To facilitate comparison to Nepf and Geyer [1996], we focus on the near-bed region, defined as the bottom 40% of the local depth. Vertical derivatives were simply estimated as the difference between the top and bottom of this region and the horizontal derivative as the central difference over 60 grid points (2250 m). The terms of equation (6) are plotted in Figure 5. Consistent with Nepf and Geyer [1996], the temporal variation is dominated by tidal straining. Straining promotes stratification during ebb and diminishes stratification during flood. There is an asymmetry so that the tidally averaged straining is negative, representing the straining induced by the mean estuarine circulation. The stretching term, which represents the vertical stretching of the salinity field due to differential vertical advection, counteracts straining during ebb and diminishes stratification. The magnitudes of these terms very closely resemble those calculated from data by Nepf and Geyer [1996] Two-Layer Model: Analysis of Halocline Behavior [17] To analyze the hydraulics of the flow we shall follow Chant and Wilson [2000] and derive an expression for the along-channel slope of the interface in an inviscid two-layer flow (noting that an inviscid, energy-conserving description will only capture some aspects of a continuously stratified, viscous flow, and that any quantitative statements are uncertain). Starting from the Bernoulli equations for an inviscid two-layer flow in a channel, u 2 1 2g þ H þ h 1 þ h 2 ¼ C 1 u 2 2 2g þ H þ h 2 þ r 1 r 2 h 1 ¼ C 2 ; ð7aþ ð7bþ where C i are constants, H is the bottom elevation above some datum, and r i, h i, and u i are the density, thickness, and velocity of layer i, respectively, subtracting equation (7b) from equation (7a) and using the continuity equation, q i = u i Bh i, to replace the layer velocities with layer discharges, we get 1 2g q 2 1 B 2 h 2 1 q2 2 B 2 h 2 þ r 2 r 1 h 1 ¼ C; ð8þ 2 r 2 where B is the (vertically uniform) channel width. Differentiating with respect to the along-channel direction, 5of12

6 Figure 4. Modeled salinity contours. x (positive up-estuary), and again using the continuity equation to make the layer velocities explicit variables, gives u2 1 h 1 u2 þ u2 2 h 2 þ u2 þ r 2 r 1 r ¼ 0: ð9þ Substituting the layer Froude number F i 2 = u i 2 /(g 0 h i ) and multiplying by g/g 0, equation (9) can be written as F 2 2 F2 þ Bg u2 1 u2 1 ¼ 0: ð10þ 2 /@x = (@H/@x 1 /@x), we get an expression for the along-channel rate of change of upper layer thickness (indirectly slope of F2 2 þ Bg u2 1 u2 2 ð1 G 2 : ð11þ Þ [18] The subdivision of the simulated (continuous) flow into two discrete layers for estimation of layer velocities and Froude numbers can be made in several ways. First, isotachs and isohalines do not coincide when mixing is active, and therefore a choice has to be made whether to base the subdivision on the velocity field or the salinity field. Second, the halocline is not sharp over the entire tidal cycle; rather, the stratification is almost linear during ebb. It was chosen here to use the 9 ppt isohaline to approximate the interface. This is a somewhat arbitrary choice, but the estimation of Froude numbers was found not to be very sensitive to the choice of subdivision. The reduced gravity, g 0, was calculated at every point along the channel from the difference between the average densities of the layers at that point. The Froude numbers are plotted in Figure 6. [19] The isohalines make large vertical excursions primarily at maximum ebb and maximum flood (compare Figure 4). At maximum ebb, the isohalines drop sharply as the flow enters the lateral contractions just downstream of the two sills (compare Figure 2). Returning to equation (11) and Figure 2, we note tend to have the same sign. Therefore when the surface layer velocities exceed the bottom layer velocities, as is the case during ebb, the two terms are of the same sign. If the flow is supercritical (G 2 > 1), then when the channel deepens (@H/@x <0) and contracts (@B/@x < 0), the two terms will both act to increase the thickness of the surface layer. Conversely, when the channel shoals and expands, the two terms will act to decrease the thickness of the surface layer. During flood, bottom layer velocities exceed or equal surface layer velocities and the two terms will be of opposite signs and thus act in conflict on the surface layer. However, the second term will in general be small since the difference between surface and bottom layer velocities is small during flood, and the first term is thus expected to dominate. This 6of12

7 Figure 5. The terms in the advection-diffusion salt equation (6). The curves represent time variation (dash-dotted line), straining (solid line), stretching (dashed line), and a residual (dotted line), which includes mixing. A negative value for any term means that stratification is promoted. The smooth solid curve shows time in tidal cycle, with the positive part corresponding to flood. The plot is for the same location as studied by Nepf and Geyer [1996] and should be compared to Figure 6 in that paper. is confirmed by the isohaline drop over the sills during flood (Figure 4) where the flow appears to be marginally supercritical Shear Instability: Identification of Potential Mixing Regions [20] In steady stratified flow, stability can be measured in terms of the gradient Richardson number Ri ¼ N 2 ðdu=dzþ 2 ¼ g dr=dz r 0 ðdu=dzþ 2 ; ð12þ where z is the vertical coordinate. When Ri is less than a certain critical value, Ri c, vertical mixing may occur through shear instabilities. According to linear stability theory Ri c should be equal to However, as shown by Geyer and Smith [1987], if Ri is defined with mean quantities, active mixing is likely to occur for values greater than 0.25 since internal waves give locally enhanced shear that may trigger instabilities. Nepf and Geyer [1996] used Ri c = 0.40 to identify regions of active mixing in the Hudson River estuary, but noted that at least during flood, Ri gradients were so sharp that the exact value of Ri c did not matter. [21] Ri estimates (Figure 7) show that at maximum flood, mixing is potentially active throughout the bottom layer and that above this layer, the flow is by a wide margin stable due to strong stratification. This is consistent with the findings by Nepf and Geyer [1996] who also showed that the region of active mixing roughly coincided with the region below the velocity maximum; above this maximum, the vertical shear changes sign so that straining acts to strengthen the stratification, producing a sharp lid to the region of active mixing. Also consistent with the findings by Nepf and Geyer [1996], there is significant stratification within regions of low Ri during ebb, and patches with Ri < 0.25 are disconnected from the bottom, indicating that shear instabilities in the interior contribute to mixing in addition to topographically induced turbulence (tidal stirring). Particularly between 17 km and 18 km from Battery Park, where the isohalines ascend in the water column as the ebb flow exits the topographic constriction at 18 km, there is potentially strong mixing. Chant and Wilson [2000] also found low Ri in this region due to enhanced vertical shear Vertical Salt Flux [22] The (vertical) eddy coefficients were calculated explicitly by the numerical model from local values of the resolved shear and stratification (Figure 8). The vertical salt flux can then be estimated [Peters, 1997]: J S ¼ ; ð13þ where the factor 10 3 converts from practical salinity to concentration units. 7of12

8 Figure 6. Froude numbers: G 2 (solid line), F 2 1 (dashed line), and F 2 2 (dotted line) averaged over (top) flood and (bottom) ebb. The ebb values are larger because the river flow and the tidal flow are then in the same direction, giving larger velocities. [23] Figure 9 shows modeled profiles of vertical salt flux, averaged over flood and ebb respectively, for the same location in the vicinity of station 1 (compare Figure 1) where Peters and Bockhorst [2001] estimated salt flux from microstructure data (compare Figure 7a in their paper) and where Nepf and Geyer [1996] estimated salt flux based on a Munk-Anderson parameterization of K r and observed salinity gradients (compare Figure 10 in their paper). Consistent with their estimates, the maximum values are found close to the bottom where vertical shear is strong. The maximum modeled values are in the same order of magnitude as the estimates of Peters and Bockhorst [2001], but Figure 7. Richardson number estimates. The linear stability criterion is Ri > of12

9 Figure 8. Modeled vertical eddy diffusivity. The background value is 10 6 m 2 /s. one order of magnitude smaller than the estimates of Nepf and Geyer [1996]. [24] In Figure 10, the along-channel variation of the vertical salt flux, averaged over flood and ebb respectively, is plotted. During ebb, the maximum flux is found just seaward of the constriction at George Washington Bridge, where the channel expands abruptly. The average vertical salt flux during ebb in this region is approximately kg Figure 9. Modeled profiles of vertical salt flux at the location studied by Nepf and Geyer [1996] and Peters and Bockhorst [2001], in the vicinity of station 1. The curves show the modeled fluxes averaged over ebb (dashed line) and over flood (solid line). 9of12

10 Figure 10. Along-channel variation in vertical salt flux equation (13), averaged over (top) ebb and (bottom) flood and for the 8 and 10 ppt isohalines. The dashed lines show the along-channel averages. m 2 s 1, corresponding to an increase by a factor 3 as compared to the along-channel average. No increase in salt flux is seen downstream of the seaward constriction, presumably because the expansion after this constriction is less abrupt. During flood, regions with moderately enhanced vertical fluxes are found on the landward side of the two sills. The sills are more important during flood since lower layer velocities then exceed surface layer velocities. The lower layer, which is sensitive to depth variations, then controls the dynamics. Conversely, during ebb, surface layer velocities are greater and the dynamics are controlled by the surface layer, which feels only width variations. 5. Discussion [25] A nonhydrostatic three-dimensional numerical model was set up to study intratidal and along-channel variability in stratification and mixing during neap tide conditions in the Hudson River estuary. Three-dimensional modeling of a full tidal cycle has to the author s knowledge not been tried before for this estuary. The modeled fields showed good agreement with data, although the fit could most likely be improved by more systematic data assimilation and with better information about velocity and salinity distribution at the upstream and downstream boundaries. The simulations confirmed many of the results from earlier, purely observational studies in the Hudson River estuary and in addition allowed the along-channel variation in vertical salt flux to be explored over the tidal cycle. Numerical modeling has the advantage that the results cover the dynamics without bias to particular sampling times in the tidal cycle. Numerical modeling also allows the relative magnitude of different forcing components to be systematically varied and the estuary response to be studied. The latter was however beyond the scope of the present paper. The results are discussed below, followed by a discussion of a few important issues that need to be further explored in order to increase the quantitative confidence in numerical results Intratidal Variation [26] The simulated fields were found to be clearly dominated by estuarine circulation, with maintained stratification over the entire tidal cycle. Tidal straining was found to dominate the intratidal variability in that maximum and minimum near-bed stratification occurred during late ebb and late flood respectively. Regions with potentially active mixing, identified by low Ri, were confined to the well-mixed bottom layer during maximum and late flood, whereas during ebb and early flood, patches of low Ri and potentially active mixing were found also in the stratified interior of the fluid. These patches indicate that shear instabilities in the interior contributed to mixing in addition to topography-induced turbulence. [27] The expression for the vertical eddy coefficients, equation (3), can be rewritten in terms of Ri: p K m;r ¼ ðc S DÞ 2 T ffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Ri; T ¼ ð@v=@zþ 2 þ ð@v=@zþ 2 : ð14þ When Ri > 1, the square root argument is negative and the viscosity is set to the background viscosity. It is clear from equation (14) and from Figures 7 and 8 that Ri and K m,r follow each other closely; low Ri corresponds to high K m,r and vice versa. The expression for the vertical salt flux, J S, contains the product of K r and the salinity gradient, but the variability of K r was much larger, so increased vertical salt flux was found only in areas of high K r and low Ri. Vertical salt flux above the bottom layer was hence significant only 10 of 12

11 during ebb and early flood. During maximum and late flood, the strong stratification above the well-mixed bottom layer effectively shut down vertical transport. Partly in contrast to previous observational studies, Figure 10 shows a significant transport across the halocline also during flood. The main part of this transport occurred during early flood, before the bottom layer had been fully developed Along-Channel Variation [28] Chant and Wilson [2000] suggested that enhanced mixing should be found downstream of channel constrictions due to observed strong vertical shear in these regions. In the present study, an extended region of low Ri was found immediately seaward of the contraction at George Washington Bridge during maximum ebb (Figure 7). Increased vertical salt flux during ebb by a factor 3 as compared to the along-channel average was also found in this region (Figure 10). However, correspondingly low Ri and increased vertical salt flux was not found at the exit from the seaward contraction, presumably because the expansion after this contraction is less abrupt. It can hence be stated that increased mixing is found primarily at abrupt channel expansions, where the halocline is released after having descended in the water column when entering the contraction (compare Figure 4). [29] Modest increases in vertical salt flux in the vicinity of channel constrictions were found also during flood, primarily early flood, but here it was rather the sills that controlled the along-channel variability. This asymmetry in hydraulic response was explained by the fact that lowerlayer velocities exceeded surface layer velocities during flood, and that the lower layer, which is sensitive to depth variations, then controlled the dynamics. The isohalines dropped sharply over the sill crests (compare Figure 4) and increased vertical salt fluxes were found just downstream of these drops Turbulence Closure [30] A crucial part of a numerical simulation aiming at quantifying vertical fluxes is of course the parameterization of subgrid-scale turbulent fluxes. Obviously, the finer the grid resolution, the more processes are explicitly simulated by the model and the less is left to parameterization. The vertical resolution in the present simulations was set in the lower range of turbulent overturning scales observed by Peters [1997] and Peters and Bockhorst [2000]. Most of these scales were thus resolved by the model, but no formal sensitivity analysis was performed to see if refining the resolution further would affect the results. The horizontal eddy coefficients were not calculated by the closure scheme, but were given the minimum constant values needed for numerical stability. This should not be of major importance for the results though, since vertical fluxes were the focus. [31] The present subgrid turbulence scheme assumed a steady local balance for turbulent kinetic energy between shear production, buoyancy production and viscous dissipation. Time-dependent, advective and diffusive terms were thus neglected. In addition K m = K r was assumed, i.e., a turbulent Prandtl number of unity. In a stably stratified fluid, internal waves can transport momentum more efficiently than scalars and K m > K r. Conversely, in an unstable fluid, convection can transport scalars more efficiently than momentum and K m < K r. The turbulence scheme neglected also these effects. Owing to these simplifications, the quantitative results are uncertain primarily in regions of larger Ri, where turbulence is produced by fluctuating internal wave-induced shear rather than the mean shear [Peters and Bockhorst, 2001] Open Boundary Conditions [32] The lateral open boundary conditions in a numerical simulation should ideally let disturbances in the interior pass out of the domain with minimum reflection, while at the same time allow forcing of the interior. This is a major difficulty in three-dimensional numerical modeling and some spurious reflections are always generated. In the present simulations, the lateral open boundaries were moved far from the region of interest, and spurious boundary effects are believed to have a minor influence on the results. However, it is not always computationally feasible to increase the length of the domain as required, and it may also be preferred to apply the forcing closer to the region of interest if this is where forcing data is available. Further evaluation of the performance of the boundary conditions is therefore needed Secondary Circulation [33] A specific issue in the present simulations was that the channel was straightened and the cross section simplified to isolate the effects of along-channel variations in cross-sectional area on the flow. Cross-channel structure and secondary circulation, which has been shown to be significant at certain locations in the studied reach, primarily around the headland at the George Washington Bridge [Chant and Wilson, 1997], were thus ignored. In the context of the present study, the important effect of secondary circulation is to further increase the residence time of tracers in the estuary. Downstream of the headland at George Washington Bridge (seaward during ebb and landward during flood) mixing across the halocline and secondary circulation thus act together to increase the residence time Conclusion and Implications for Future Studies [34] The main result of the presented numerical simulations was that mixing across the halocline showed marked along-channel variability due to bathymetric effects. During ebb, mixing occurred preferentially at abrupt channel expansions seaward of channel constrictions. During flood, mixing was instead localized to the landward slope of sills. Increased residence time of tracers should be expected primarily immediately seaward of the headland at the George Washington Bridge, due to strong vertical mixing and observed secondary circulation in this region. [35] Another important result was that the model was in fact able to reproduce observational results, including microstructure data, both with regard to the along-channel variability and the intratidal variability. Numerical modeling could thus be a useful complement to observations in future studies, in the Hudson River estuary and in other estuaries. In the present simulations, the grid resolution was high, and the results agreed well even quantitatively with observations, in spite of the fact that a rather simplistic turbulence closure scheme was used. It should be useful though in future studies to further develop and evaluate the closure 11 of 12

12 scheme [e.g., Nunes Vaz and Simpson, 1994], particularly for studies where a coarser grid resolution has to be used for computational feasibility. The extensive data sets from the Hudson River estuary, covering along-channel variability as well as intratidal and neap to spring variability, provide a good basis for such an evaluation. [36] Acknowledgments. These studies were started when I was on a 3 month visit at the Marine Sciences Research Center, State University of New York at Stony Brook, October December I would like to thank Robert E. Wilson for introducing me to this subject and for giving me access to data from the field measurements. I would also like to thank Anders Engqvist for helpful comments on the manuscript and Göran Broström for assistance when setting up the numerical experiment on the 48-processor Linux cluster that he built. Thanks also to two anonymous reviewers, who contributed to major improvements of the manuscript. References Adcroft, A. (1997), Representation of topography by shaved cells in a height coordinate ocean model, Mon. Weather Rev., 125, Chant, R. J., and R. E. Wilson (1997), Secondary circulation in a highly stratified estuary, J. Geophys. Res., 102, 23,207 23,215. Chant, R. J., and R. E. Wilson (2000), Internal hydraulics and mixing in a highly stratified estuary, J. Geophys. Res., 105, 14,215 14,222. Geyer, W. R., and J. D. Smith (1987), Shear instability in a highly stratified estuary, J. Phys. Oceanogr., 17, Jay, D. A., and J. D. Smith (1990), Residual circulation in shallow estuaries: 1. Highly stratified, narrow estuaries, J. Geophys. Res., 95, Khatiwala, S. (2003), Generation of internal tides in an ocean of finite depth: Analytical and numerical calculations, Deep Sea Res., Part I, 50, Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey (1997), A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res., 102, Nepf, H. M., and W. R. Geyer (1996), Intratidal variations in stratification andmixinginthehudsonestuary,j. Geophys. Res., 101, 12,079 12,086. Nunes Vaz, R. A., and J. H. Simpson (1994), Turbulence closure modeling of estuarine stratification, J. Geophys. Res., 99, 16,143 16,160. Nunes Vaz, R. A., G. W. Lennon, and J. R. de Silva Samarasinghe (1989), The negative role of turbulence in estuarine mass transport, Estuarine Coastal Shelf Sci., 28, Nycander, J., and K. Döös (2003), Open boundary conditions for barotropic waves, J. Geophys. Res., 108(C5), 3168, doi: /2002jc Orlanski, I. (1976), A simple boundary condition for unbounded hyperbolic flows, J. Comput. Phys., 21, Peters, H. (1997), Observations of stratified turbulent mixing in an estuary: Neap-to-spring variations during high river flow, Estuarine Coastal Shelf Sci., 45, Peters, H. (2003), Broadly distributed and locally enhanced turbulent mixing in a tidal estuary, J. Phys. Oceanogr., 33, Peters, H., and R. Bockhorst (2000), Microstructure observations of turbulent mixing in a partially mixed estuary: 1. Dissipation rate, J. Phys. Oceanogr., 30, Peters, H., and R. Bockhorst (2001), Microstructure observations of turbulent mixing in a partially mixed estuary: 2. Salt flux and stress, J. Phys. Oceanogr., 31, Rodi, W. (1987), Examples of calculation methods for flow and mixing in stratified fluids, J. Geophys. Res., 92, Simpson, J. H., J. Brown, J. P. Matthews, and G. Allen (1990), Tidal straining, density currents and stirring in the control of estuarine stratification, Estuaries, 13, Smagorinsky, J. (1963), General circulation experiments with the primitive equations, Mon. Weather Rev., 91, Stenström, P. (2003), Mixing and recirculation in two-layer exchange flows, J. Geophys. Res., 108(C8), 3256, doi: /2002jc Ullman, D. S., and R. E. Wilson (1998), Model parameter estimation from data assimilation modeling: Temporal and spatial variability of the bottom drag coefficient, J. Geophys. Res., 103, Winters, K. B., and H. E. Seim (2000), The role of dissipation and mixing in exchange flow through a contracting channel, J. Fluid Mech., 407, P. Stenström, Department of Land and Water Resources Engineering, Royal Institute of Technology, SE Stockholm, Sweden. (petters@ kth.se) 12 of 12