Seawater circulation in sediments driven by interactions between seabed topography and fluid density

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /wrcr.20121, 2013 Seawater circulation in sediments driven by interactions between seabed topography and fluid density L. F. Konikow, 1 M. Akhavan, 2 C. D. Langevin, 3 H. A. Michael, 4 and A. H. Sawyer 4,5 Received 21 March 2012; revised 24 January 2013; accepted 29 January 2013; published 12 March [1] Measurements of submarine groundwater discharge (SGD) in coastal areas often show that the saltwater discharge component is substantially greater than the freshwater discharge. Several mechanisms have been proposed to explain these high saltwater discharge values, including saltwater circulation driven by wave and tidal pumping, wave and tidal setup in intertidal areas, currents over bedforms, and density gradients resulting from mixing along the freshwater-saltwater interface. In this study, a new mechanism for saltwater circulation and discharge is proposed and evaluated. The process results from interaction between bedform topography and buoyancy forces, even without flow or current over the bedform. In this mechanism, an inverted salinity (and density) profile in the presence of both a bedform on the seafloor and an upward flow of fresher groundwater from depth induces a downward flow of saline pore water under the troughs and upward flow under the adjacent crest of the bedform. The magnitude and occurrence of the mechanism were tested using numerical methods. The results indicate that this mechanism could drive seawater circulation under a limited range of conditions and contribute 20% 30% of local SGD when and where the process is operative. Bedform shape, hydraulic conductivity, hydraulic head, and salinity at depth in the porous media, aquifer thickness, effective porosity, and hydrodynamic dispersion are among the factors that control the occurrence and magnitude of the circulation of seawater by this mechanism. Citation: Konikow, L. F., M. Akhavan, C. D. Langevin, H. A. Michael, and A. H. Sawyer (2013), Seawater circulation in sediments driven by interactions between seabed topography and fluid density, Water Resour. Res., 49, , doi: /wrcr Introduction 1 National Research Program, U.S. Geological Survey, 431 National Center Reston, Virginia, USA. 2 Department of Civil & Environmental Engineering, University of Delaware, Newark, Delaware, USA. 3 Office of Groundwater, U.S. Geological Survey, 411 National Center Reston, Virginia, USA. 4 Department of Geological Sciences, University of Delaware, Newark, Delaware, USA. 5 Now at Department of Earth and Environmental Sciences, University of Kentucky, Lexington, Kentucky, USA. Corresponding author: L. F. Konikow U.S. Geological Survey, 431 National Center, Reston, VA 20192, USA. (lkonikow@usgs.gov) American Geophysical Union. All Rights Reserved /13/ /wrcr [2] Submarine groundwater discharge (SGD), which includes the discharge into coastal marine waters of terrestrially derived groundwater and recirculated seawater [see Burnett et al., 2003; Moore, 2010; Bratton, 2010], is an important pathway for transport of nutrients and contaminants to coastal waters [e.g., Burnett et al., 2006; Slomp and Cappellen, 2004; Johannes, 1980; Andersen et al., 2007]. SGD consists of both terrestrial fresh groundwater and seawater exchanges. Direct measurements of SGD near coastlines often indicate a large saline water discharge that may be substantially greater than the freshwater discharge [Taniguchi et al., 2002; Michael et al., 2005; Santos et al., 2009; Swarzenski et al., 2009]. However, sufficient seawater inflow to balance this outflow has not been widely observed [Michael et al., 2005]. [3] Several mechanisms have been proposed to drive the circulation of seawater into and out of the sea bed [Huettel and Webster, 2001; Santos et al., 2012] and explain the very large proportion of saline SGD relative to fresh SGD that is commonly observed [e.g., Michael et al., 2003; Martin et al., 2007; Santos et al., 2009]. These mechanisms include dispersion-induced density gradients (and mixing) along the freshwater-saltwater interface [Cooper, 1959], seasonal changes in the upland water-table elevation [Michael et al., 2005], tide- and wave-induced hydraulic gradients in the nearshore zone [Robinson et al., 2007; Xin et al., 2011], tidal pumping [Taniguchi, 2002; Li et al., 2009], wave-induced pressure variations [Riedl et al., 1972; Shum, 1992; Shum and Sundby, 1996; King et al., 2009; Cardenas and Jiang, 2011], bedform migration [Elliott and Brooks, 1997; Precht et al., 2004; Pilditch and Miller, 2006], and current-induced pressure variations as a result of flow over bedforms (a Bernoulli effect) [Thibodeaux and Boyle, 1987; Elliott and Brooks, 1997; Cardenas and Wilson, 2007]. But it is not clear that these mechanisms are sufficient to explain what some consider to be anomalously large saline SGD relative to the fresh SGD derived directly from hydraulic gradients on land. Bokuniewicz [1992] states, A mechanism must be sought to explain the penetration of salt to depths of decimeters in the sediment against upward advection of fresh water. Abarca et al. [2007] demonstrated the importance of 1386

2 Figure 1. Schematic cross section showing the relation between (a) a classical coastal aquifer flow system in which terrestrially derived freshwater pumps seawater into and through the aquifer and (b) the conceptual model for the seawater circulation mechanism proposed in this paper. In the classical flow system, pressure increases with depth along the seawater boundary (Figure 1a). In the proposed mechanism, the pressure increases laterally along the seawater boundary due to topographic variation of the sea floor. considering the topography of aquifer boundaries in analyzing seawater intrusion problems, but the effect of smallscale local topographic relief on seawater circulation in SGD has not been considered. In studies of hyporheic flow, laboratory experiments and numerical simulations show that fluid density variations can affect flow and solute transport across planar or rippled beds [Boano et al., 2009; Jin et al., 2011]. [4] The process of convective seawater circulation has been studied extensively in the context of saltwater intrusion, beginning with the early work of Cooper [1959] and Cooper et al. [1964]. The density gradient along the dispersed interface between fresh and saline groundwater has been shown to drive seawater circulation in coastal aquifers. As fresh groundwater flows toward the coast, salt disperses into (and is entrained by) the freshwater, requiring seawater recharge to replace the discharging salt. A diagram of this classical convective seawater circulation process (Figure 1a), patterned after the Henry [1964] problem, provides an analogous conceptual basis for the proposed mechanism that can drive seawater circulation as a component of SGD in coastal zones. Because the discharge of seawater dispersed into the mixing zone results from its entrainment by advected fresh groundwater, the resulting flux of seawater in SGD should be substantially less than that of the fresh SGD that is driving it. [5] A proposed new mechanism for driving convective seawater circulation is based on a 90 rotation (Figure1b) of the classical conceptual model shown in Figure 1a. Although the classical model requires a lateral inflow of freshwater, our proposed mechanism requires an upward flow of fresh or brackish groundwater. A second difference between the classical model and our proposed mechanism is the cause of the pressure increase along the seawater boundary. In the classical model, pressure increases with depth along the seawater boundary. In our proposed model, topographic relief on the sea bed causes a lateral pressure variation along the seawater boundary. [6] It is well known that the sea bed has bathymetric (or topographic) relief over a range of scales [e.g., see Barnard et al., 2006]. Resistivity surveys, offshore groundwater sampling, and coring also prove the presence of fresh and brackish water beneath saline surface water bodies [Manheim et al., 2004; Swarzenski et al., 2006]. [7] Several previously described small-scale (bedformscale) processes that can drive shallow seawater circulation are illustrated in Figures 2a 2c and reviewed by Santos et al. [2012]. The mechanism being evaluated in this study is illustrated in Figure 2d. This type of seawater circulation is expected to be active in the two zones highlighted by red boxes in Figure 2e, which presents a conceptual schematic cross section through a coastal aquifer environment that contains a lower permeability confining layer. These two zones of possible occurrence correspond closely with the nearshore scale and embayment scale of Bratton [2010]. Occurrence requires bedform topography (Figure 2d) and fresh groundwater at depth with a sufficiently high hydraulic head to produce a potential for upwelling and discharge of fresher groundwater as SGD (Figure 2e). The topography is critical because the depth of water varies across the top surface of the bedform, and consequently, the equivalent freshwater head at the seabed surface is greater at the trough than at the crest of the bedform. However, if the freshwater head in the underlying aquifer is too 1387

3 Figure 2. Schematic cross section of coastal aquifer environment illustrating selected local-scale processes that can drive seawater circulation through shallow sediments. (a) Currrent pumping through bedforms. (b) Pore water turnover in migrating bedforms. (c) Wave pumping over either a flat or rippled bed. (d) Density-driven exchange in rippled bedforms (this study). (e) Larger-scale view showing coastal area underlain by aquifers (stipled pattern) and confining layer (brown dashed pattern). Seawater is represented by darker blue color. Seawater circulation driven by topography-density interaction may occur where fresh groundwater discharges to saline surface water, as in areas indicated with red boxes in Figure 2e. high relative to sea level (and relative to the equivalent freshwater heads across the top of the bedform), as might be expected very close to the shoreline, upwelling freshwater may overwhelm any tendency for negative buoyancy to drive seawater into the underlying sediments. At the other end of the spectrum, if the freshwater head is relatively low, as might be expected further offshore, or if the hydraulic conductance between the aquifer and the overlying seabed is low, as might be expected where a confining layer is thickest, then upward flux of freshwater can be eliminated or limited to a slow diffusive flux, and the density-topography mechanism would not yield any seawater circulation. Although the presence of a shallow confining layer enables fresh groundwater to migrate further offshore, it is not a necessary condition for offshore upwelling of fresher groundwater. [8] The objectives of this study were to demonstrate whether small topographic relief on the seafloor, coupled with an inverted pore-water salinity profile, can generate a seawater circulation flow system in the upper parts of a subsea sediment profile and to evaluate primary process controls. The approach is to use variable-density groundwater flow and transport modeling to evaluate the feasibility of the process being operative and the magnitude of circulation for a range of plausible conditions. 2. Numerical Model [9] In this study, the SEAWAT computer program was used to simulate the effects of the proposed mechanism on saltwater circulation. SEAWAT is designed to simulate variable-density groundwater flow and transport [Guo and Langevin, 2002; Langevin et al., 2003; Langevin and Guo, 2006]. The latest version of SEAWAT (Version 4) was used for the simulations reported here [Thorne et al., 2006; Langevin et al., 2007]. SEAWAT has been benchmarked with a wide variety of test problems and is routinely used to model subsurface freshwater and saltwater mixing processes Governing Equations [10] SEAWAT simulates variable-density groundwater flow by solving the following governing equation: r 0 K 0 rh 0 þ 0 rz ¼ S þ ; (1) 0 where is fluid density [ML 3 ]; m is dynamic viscosity [ML 1 T 1 ]; m 0 is dynamic viscosity [ML 1 T 1 ] of the reference fluid; K 0 is the hydraulic conductivity tensor of material saturated with the reference fluid [LT 1 ]; h 0 is the hydraulic head [L] measured in terms of the reference fluid; 0 is the density [ML 3 ] of the reference fluid; z is elevation [L]; S s,0 is specific storage [L 1 ], defined as the volume of water released from storage per unit volume per unit decline of h 0 ; t is time [T]; is porosity. [11] Freshwater with a density of kg/m 3 is the reference fluid. Fluid density is calculated in SEAWAT using 1388

4 the following linear equation of state that relates fluid density to solute concentration: ¼ 0 þ 0:7143ðCÞ; (2) where C is the concentration of the density-controlling solute [ML 3 ]. The effect of viscosity dependence on solute concentration was not considered as part of this investigation by setting 0 = equal to one. [12] The solute in this case represents the dissolved salts commonly found in seawater, which is assumed to have a total dissolved salts concentration of 35.0 parts per thousand (ppt). Salt transport is represented as being conservative (nonreactive) as is common for these types of flow process studies. Transport of a conservative solute species is represented in SEAWAT (through the integrated MT3DMS routines) by solving the following form of the transport ð Þ h ¼r m þ a q i rc rðqcþ; (3) where D m is the molecular diffusion coefficient [L 2 T 1 ]; a is the dispersivity tensor [L]; q is specific discharge [LT 1 ] Numerical Methods [13] Finite-difference approximations of flow and transport equations were solved iteratively in SEAWAT using preconditioned conjugate gradient methods. Flow convergence was achieved when the head and residual mass flux criteria met the specified threshold of m and kg/d, respectively, or smaller (depending on the specific case tested values as small as were sometimes required to achieve a stable symmetrical solution). Transport convergence was achieved when subsequent changes in the relative concentration difference did not exceed or smaller. Upstream weighting and a finely discretized model grid were used to minimize oscillations and numerical dispersion. Explicit coupling with a one time step lag of the flow and solute-transport equations was used for solving the coupled system of equations. [14] The configuration of the model grid and boundary conditions was intentionally designed to create symmetric flow and solute-transport patterns. For some problems, however, asymmetry was encountered in model results. An analysis indicated that round off error in the solution was the cause of the asymmetry. Problems with asymmetry were eliminated by compiling SEAWAT such that all real variables are represented with quadruple precision. The simulations were then run using convergence tolerances that were reduced by 5 or more orders of magnitude. 3. Numerical Experiments [15] The feasibility of this topography-density process contributing saltwater to SGD is evaluated through a series of numerical experiments designed to demonstrate its existence and assess the factors controlling its magnitude. A test problem was designed to represent idealized but realistic conditions encountered in nearshore coastal environments. The experiment is designed to assess whether seawater circulation occurs in the absence of other recognized driving forces. We do not attempt to maximize fluxes by adjusting parameters, boundary conditions, or dimensions. [16] We assume that the seabed topography is a symmetrical series of repeating sinusoidal waveforms extending away from the shoreline and is underlain by groundwater having a lower salinity and lower density than seawater present above the seabed. The waveforms are also assumed to be infinite in length and locally under a constant depth of seawater; we do not consider other bedform shapes that would be affected by the mechanism proposed here nor larger-scale trends in the depth of seawater. The underlying fresher water is assumed to have a hydraulic head slightly greater than sea level. Terrestrial recharge would be the cause for the higher head, but we do not explicitly simulate groundwater flow from a recharge area. For our purpose of investigating the proposed circulation mechanism, we assign a head at depth so that there is a potential for upward flow. We assume for our preliminary feasibility analyses that hydrostatic conditions exist in the sea. That is, there are no currents, tides, or waves in the surficial saltwater body, and that SGD will not affect the salinity in the surficial seawater. Furthermore, we assume that the effects of transient flow and changes in fluid storage are negligible in the porous media. Because of the assumed boundary conditions, symmetry allows us to simulate just a single representative waveform Test Problem [17] A schematic of the discretized model domain for the numerical experiments is shown in Figure 3. The model represents a cross section of single, symmetrical, sinusoidal bedform having a local crest-to-trough relief 0.2 m on the seafloor and a wavelength of 0.5 m. The seafloor lies m below sea level. The model was discretized into 25 columns, 1 row, and 101 layers. The deepest layer was reserved for representing the boundary conditions associated with the underlying aquifer. The horizontal spacing between nodes was 0.02 m, and the vertical spacing ranged from m below the troughs to m under the crest of the bedform. Groundwater flow and solute transport parallel to the shoreline was assumed to be negligible and, therefore, a two-dimensional model in the vertical plane normal to the shoreline was applied. [18] Because the waveform is assumed to be one of a repeating sequence, the resulting symmetry provides a basis for assuming that both vertical boundaries coincide with streamlines, and therefore can be represented as an impermeable no-flow boundary so that no fluid or solute flux can cross the vertical side boundaries of the model domain. A specified (constant) head boundary condition was assigned to the top layer of the model to represent the overlying open seawater. The equivalent freshwater head on this upper boundary is calculated in SEAWAT using the height of the overlying column of seawater. Inflow along this boundary is assumed to enter with a concentration of 35.0 ppt, whereas outflow from any cell in the top layer leaves with the simulated concentration in the cell. Because the depth of the top layer below sea level varies with position, 1389

5 Figure 3. Cross section of model domain, showing dimensions, boundary conditions, and numerical grid. the equivalent freshwater head (a function of gh) is slightly greater in the trough than at the crest of the waveform. [19] A general-head boundary condition (similar to a specified-head condition; see Guo and Langevin, 2002) was assigned to the bottom layer of the model because this type of boundary (as opposed to a specified-flux condition) would allow the heads at the base of the model domain to self-adjust in response to changing conditions in the rest of the model domain and the flux into the aquifer across the base to similarly self-adjust (or even go to zero if conditions warranted). This bottom boundary allows groundwater flow into or out of the model domain depending on the head gradient at the boundary. The hydraulic head on this bottom boundary was set at 0.09 m, slightly above sea level, to simulate an upward driving force for the underlying fresher water. [20] Any inflow on this lower boundary was assumed to enter the domain with a concentration of 10.0 ppt. The initial salt concentration in the model was 10 ppt. Additional key model parameters are listed in Table 1. If there were no topographic relief on the seafloor and the sea bed was uniformly at a depth of 3 m below sea level, then the properties and boundary conditions of this test problem would generate a uniform steady upward flux (specific discharge, q) of m/s (0.19 cm/d) Results [21] The equilibrium flow field and salinity distribution calculated using SEAWAT (Figure 4) indicates that circulation of seawater as a component of SGD occurs under the conditions of the numerical experiment. The basic pattern (Figure 4b) includes influx of descending seawater at the troughs of the bedform and outflow of upwelling mixed groundwaters beneath the crest of the bedform. This basic flow pattern is consistent with fluxes through rippled bedforms, as observed in laboratory experiments for similar systems (but with waves or currents in the surficial waters) [e.g., Webb and Theodor, 1972; Huettel et al., 1996; Precht and Huettel, 2004], which showed pore water upwelling and release at or close to the crests of rippled bedforms. [22] The calculated volumetric seawater flux is m 3 /s (equivalent to q¼ m/s), which can be compared to an upward volumetric flux rate of m 3 /s flowing into the system across the bottom boundary of the model domain (q ¼ m/s (0.17 cm/d) over the basal surface area of 0.10 m 2 and an average interstitial velocity of m/s (0.68 cm/d) when considering the effective porosity). The total discharge on the top boundary (i.e., the calculated SGD) is m 3 /s (q ¼ m/s). Thus, 24.8% of the SGD is derived from shallow seawater circulation, which infiltrates the seabed across nearly 50% of the surface area of the model domain. From the surface areas where seawater infiltration occurs (see Figure 4b), the minimum, maximum, and average travel (residence) times are 0.23, 0.89, and 0.50 years, respectively (based on forward tracking of five uniformly spaced particles on the top face of each of the 12 cells where infiltration occurs). [23] The depth of seawater circulation is determined on the basis of where the salinity equals 17.5 ppt (which thereby includes 70% of the range of salinities between the upper and lower limits in the model). In this hypothetical test case, the saltwater front penetrates 0.77 m below the seabed in the trough of the bedform. In a normalized (or nondimensional) reference, the depth of penetration is 3.9 times greater than the height of the crest of the bedform above the trough elevation, and the fraction of seawater penetration of the aquifer thickness is [24] The flux along the upper and lower boundaries of the domain varies spatially (Figure 5). In the top layer, flux consists of inflow along the outermost six columns on each side and outflow (SGD) for the middle 13 nodes (Figure 5a). Because the SGD occurs only through the 13 central cells along the top boundary, the average specific discharge where SGD occurs is m/s (0.44 cm/d), but ranges from to m/s. The salinity of the SGD, as reflected in the concentrations of the 13 discharging cells along the middle part of the top boundary, varies from 11.5 to 29.6 ppt over a distance of 0.13 m from the crest down each side of the waveform. The fluxweighted mean salinity of the SGD is 15.8 ppt. The inflow across the bottom boundary (Figure 5b) is also affected by the geometry of the domain, and the inflow is slightly greater at the edges than in the center of the grid. Table 1. Model Parameters Parameter Value Density of seawater, s (kg/m 3 ) 1025 Density of freshwater, f (kg/m 3 ) 1000 Longitudinal dispersivity, L (m) 0.02 Transverse dispersivity, T (m) Porosity, (dimensionless) 0.25 Effective molecular diffusion coefficient, D m (m 2 /s) Aquifer thickness, b (m) Horizontal hydraulic conductivity, K H (m/s) Vertical hydraulic conductivity, K V (m/s) Initial concentration (ppt)

6 KONIKOW ET AL.: SEAWATER CIRCULATION IN SGD Figure 4. Calculated (a) equilibrium salinity distribution and (b) flow path lines with travel times showing seawater circulation in SGD around seabed waveform for test case. Horizontal exaggeration ¼ 5x Sensitivity Analysis [25] Because the results are expected to depend strongly on the particular dimensions and properties of the base case discussed above, we also evaluated the sensitivity of seawater circulation to variations in selected model parameters. Results were assessed by comparing the calculated values for key indicators of the system behavior, including the volumetric saltwater inflow across the top boundary, the flux ratio (defined as the ratio of saltwater flux coming in from the top boundary to the SGD), and the normalized depth of penetration, which is the ratio of the depth of penetration of the saltwater front (assumed to be indicated by a salinity of 17.5 ppt) relative to the crest height (a measure of bedform shape, which is 0.20 m in the base case) Hydraulic Conductivity [26] We varied the hydraulic conductivity between Kx ¼ and m/s in all cells, except those on the constant-head boundary. In evaluating these changes, the anisotropy ratio (Kx /Kz) was kept constant at a value of 10 and other model parameters were unchanged. Figure 5. (a) Variation of boundary flux and salinity with lateral position in the grid. (b) The vertical scale is expanded to more clearly show the inflow variability across the bottom boundary. Positive values of flux indicate inflow; negative values reflect outflow. 1391

7 Figure 6. Effect of varying hydraulic conductivity on (a) calculated depth of penetration of the seawater front (red curve and left axis) and the rate of seawater inflow on top boundary (blue curve and right axis), and (b) normalized measures of penetration depth (relative to crest height) (red curve) and seawater circulation (relative to SGD) (blue curve). Vertical anisotropy (K x /K z ) kept constant at a ratio of 10. Stable circulation of seawater occurs only to the left of the dashed vertical line. Figure 7. Sensitivity of selected model results to variability in assumed value of effective porosity. Colors indicate which vertical axis applies to each curve. Stable circulation of seawater occurs only to the right of the dashed vertical line. The results (Figure 6) indicate that over a range of more than 2 orders of magnitude in K x, the magnitude of saltwater circulation changed almost linearly with K x (Figure 6a), although the ratio of seawater inflow (q SW ) to total outflow (or SGD) on the top boundary (q SGD ) only varied within a relatively narrow range between 0.24 and 0.32 (Figure 6b). Over this same range in K x, increasing hydraulic conductivity coincided with a small increase in the depth of penetration of the seawater front. However, a small additional increase in the value of K x beyond m/s caused the salinity distribution to pass a critical point (at a depth of penetration of about 5 times greater than the crest height), where the underlying equivalent freshwater head was no longer large enough to overcome the downward buoyancy forces. Consequently, the overall vertically upward flow field was reversed, and the entire domain filled with seawater. After this occurs, there is no longer any groundwater discharge across the top boundary, and local seawater circulation is eliminated Effective Porosity [27] Porosity affects both the pore volume used for the mass storage and pore water velocity. As a result, it influences both the advection and dispersion terms in the governing solute-transport equation. We varied the value of porosity from 0.15 to 0.65 while keeping other model parameters unchanged. The results (Figure 7) indicate that decreasing the porosity less than a critical value of about 0.17 will result in saltwater filling the entire model domain, as fluid velocity and dispersion both increase relative to cases yielding a stable shallow saltwater circulation pattern. Above the critical porosity value, increasing the porosity (and reducing the velocity) results in shallower penetration of the saltwater front but has almost no effect on the magnitude of the seawater flux through the top layer of the system Hydrodynamic Dispersion [28] The relative importance of molecular diffusion and mechanical dispersion on seawater circulation in this test problem was estimated by computing the product of longitudinal dispersivity and pore velocity (which averages about 10 7 m/s in most of the flow field), which is a major component of mechanical dispersion, and comparing it to the magnitude of the effective coefficient of molecular diffusion for chloride, which is reasonably well defined for porous media transport. The comparison shows that for the parameter values in Table 1, mechanical dispersion is approximately 10 times greater than molecular diffusion as a contributor to hydrodynamic dispersion throughout most of the flow field. However, in a few parts of the flow field, particularly within the saltwater circulation zone, the flow velocities are substantially lower than average, and diffusion is the dominant mixing process in those areas. The transverse dispersivity ( T ) is always held at a value of 0.01 times L. Additionally, because the transport equation is solved using a finite-difference method, there is some small amount of numerical dispersion inherent in the numerical solution. [29] To examine the sensitivity to mechanical dispersion, the values of longitudinal dispersivity ( L ) were varied while holding the ratio of T = L at a constant value of The results (Figure 8) show a moderate sensitivity to the dispersion process, though a stable seawater circulation pattern is observed over the entire range of values tested. The magnitude of the seawater inflow and circulation is 1392

8 Figure 8. Sensitivity of selected model results to variability in assumed value of longitudinal dispersivity coefficient. Colors indicate which vertical axis applies to each curve. relatively stable over a range of values for L from to 0.1 m, but decreases noticeably as L is increased to values greater than 0.1 m. The depth of penetration, on the other hand, is least for a longitudinal dispersivity between about 0.1 and 0.2 m. When L equals 0.0 m, the only dispersive process is molecular diffusion (and the results for L ¼ m were essentially identical to those not shown for when L ¼ 0.0 m). [30] As the magnitude of dispersion increases, mixing increases, so that for relatively larger values of dispersivity, the highest concentrations in the system are noticeably less than that of seawater (Figure 9), in contrast to the results shown in Figure 4. Furthermore, the high rate of mixing yields a salinity pattern that is subparallel to the seafloor, and field observations of such a pattern would likely not be interpreted as reflecting the presence of the same type of shallow local seawater circulation reflected in Figure 4, yet in this case the seawater circulation is still 65% of that in the base case, and the fraction of SGD derived from seawater circulation is only reduced to from in the base case. The difference in the flow field for the cases illustrated in Figures 4 and 9 is illustrated by differences in the velocity vectors for the two cases (Figure 10). In the base case (Figure 10a), vertically downward flow beneath the trough (at the outer limit of the model domain) extends into model layer 13, whereas for the high-dispersivity case (Figure 10b), the downward flow only extends into model layer 2. [31] The role of the effective coefficient of molecular diffusion was also evaluated, but to isolate the effects of diffusion in these tests, mechanical dispersion was turned off (by setting the values of the dispersivity coefficients to zero) in this series of simulations. The results of varying the diffusion coefficient (Figure 11) indicate that when diffusion (and hydrodynamic dispersion) is low, the domain will fill with seawater and no SGD will occur. When the value of D m is relatively high, seawater circulation will occur and comprise about 25% of the SGD. That circulation occurs for the base value of D m explains why circulation occurs for the lowest value of L (Figure 7). These results also demonstrate that the numerical solution and the circulation process are not controlled by numerical dispersion because the flow and transport regimes change dramatically as lower values of D m are applied. The transition from circulation to complete seawater infiltration is a sharp one, and this critical change occurs at a value of D m of approximately m 2 /s Upward Flux and Overall Hydraulic Head Gradient [32] One of the underlying assumptions in this analysis is that fresher groundwater exists beneath the 5 m thick model domain. The presence of the salinity contrast in which fluid density decreases with depth implies that the hydraulic head in that underlying groundwater must be greater than sea level. If the fluid pressure in the underlying system were lower than that equivalent to hydrostatic seawater, then the domain should fill with seawater from above. If the fluid pressure in the underlying system were substantially greater than that equivalent to hydrostatic seawater, then the overall hydraulic gradient between the top of the domain (representing the seabed) and the bottom of the domain should induce a substantial upward flux of the underlying fresher pore water that may have the effect of reducing the saltwater circulation. To evaluate the sensitivity of the numerical solution to variations in the underlying fluid pressure (or head), we varied the specified head for the underlying general head boundary while keeping the constant head at the top boundary of the model unchanged. [33] The results (Figure 12) show that for the given properties and boundary conditions of this problem, a stable steady circulation of seawater occurs when the specified head on the lower boundary is greater than about m. The domain fills with seawater if the specified head is less than about m. In this case, mixing of seawater with fresher groundwater will only occur at the base of the system, and there will be no freshwater component of SGD. Above this critical value of boundary head, the fraction of SGD derived from seawater circulation steadily decreases Figure 9. Calculated equilibrium salinity distribution for case in which L ¼ 1.0 m. Horizontal exaggeration ¼ 5x. 1393

9 Figure 10. Velocity vectors in upper 20% of model domain for equilibrium flow field for base case (a), as shown in Figure 4, and for high-dispersivity case (b), as shown in Figure 9. Vector base coincides with node location in grid. To improve clarity, only nodes in odd-numbered model layers are depicted. Horizontal exaggeration ¼ 5x. as the head increases (Figure 12b), although the actual seawater inflow on the top boundary is fairly constant for heads between about 0.09 and 0.14 m (Figure 12a). The seawater flux and circulation then approaches zero as the head increases to 0.21 m, where the upward flux of fresher water becomes overwhelming. Thus, the model domain fills completely with the underlying fresher groundwater, and the SGD consists entirely of that deeper groundwater Salinity and Density of Underlying Fluid [34] The salinity difference between the top and bottom boundary of the model influences both the diffusion and advection terms of the solute-transport equation and consequently affects buoyancy forces that drive seawater circulation in the porous medium. The salinity (and associated fluid density) of any groundwater flowing into the system across the lower general-head boundary of the model domain is a specified property. For the base case, a salinity of 10.0 ppt was assigned to any inflow across the bottom boundary. The effect of varying that boundary inflow concentration was evaluated over a range from 8.0 to 18.0 ppt. Figure 11. Sensitivity of selected model results to variability in assumed value of the effective coefficient of molecular diffusion. Colors indicate which vertical axis applies to each curve. Stable circulation of seawater occurs only to the right of the dashed vertical line. [35] The results (Figure 13) show that a stable seawater circulation pattern develops when the inflowing concentration has a salinity of 9.0 ppt or greater. A small reduction below 9.0 results in a situation in which the inflow has too low a density to overcome the downward buoyancy effect of the overlying salty-brackish mix of pore water, and the flow field becomes downward everywhere, and the entire model domain fills with seawater from above. As the boundary salinity is increased above 9.0 ppt, the magnitude of the seawater inflow on the upper boundary gradually decreases and the depth of penetration of the circulating seawater is reduced, primarily because the density gradient with depth has been reduced Bedform Height [36] The shape and steepness of the bedform may be additional factors that affect the potential for local shallow circulation of seawater in SGD. In our numerical experiments, we maintained the sinusoidal shape while varying the crest height. In the model, the vertical discretization was adjusted for this change while keeping the same number of model layers. The height of the bedform directly affects local topographic relief and its interaction with buoyancy forces. In keeping the bedform wavelength constant, we also thereby varied the bedform steepness (defined as the crest height divided by the wavelength). Dingler and Inman [1977] indicate a typical steepness for orbital ripples in their study area of about The bedform height (and local topographic relief) is 0.2 m in the base case, and the equivalent steepness is 0.4. We evaluated crest heights ranging from 0.0 to 1.5 m for the same basic symmetrical shape. For these simulations, the top boundary head was kept at sea level and the maximum depth of water was kept at 3.0 m, so increasing the amplitude resulted in a higher elevation (and smaller depth of water, hence lower fluid pressure) on the crest of the bedform but no change in saltwater pressure at the troughs of the bedform. [37] The results for crest heights ranging from 0.0 to 1.0 m (Figure 14) show that as the crest height is decreased to less than 0.05 m (steepness less than 0.1) and approaches 1394

10 Figure 12. Effect of varying the specified head on the lower boundary on (a) calculated depth of penetration of the seawater front (red curve) and the rate of seawater inflow on top boundary (blue curve), and (b) normalized measures of penetration depth and seawater circulation. Stable circulation of seawater occurs only to the right of the dashed vertical line. zero (i.e., a flat seafloor), the upward discharge of fresher water becomes dominating: no seawater enters the system across the top boundary, and all SGD is composed of fresher water that entered the domain by upward flow across the bottom boundary. As the crest height is increased, both the depth of penetration and the magnitude of seawater circulation continue to increase over the range of values tested. The ratio of penetration depth-to-crest height (Figure 14b) reaches a maximum value at a bed height of about 0.15 m, although as the crest height increases from 0.1 to 1.5 m, the fraction of SGD derived from shallow seawater circulation increases continually up to a maximum of about These results indicate that seawater circulation requires relatively steep bedforms: exchange occurs for steepnesses greater than 0.1, and fully developed orbital ripples have a typical steepness of 0.15 [Dingler and Inman, 1977]. 4. Discussion Figure 13. Sensitivity of selected model results to variability in value of salinity (and density) of fluid entering the model domain across the lower boundary. Colors indicate which vertical axis applies to each curve. Stable circulation of seawater occurs only to the right of the dashed vertical line. [38] Seawater circulation only occurs for a limited range of most parameter values. Of the many parameters affecting circulation, the groundwater upwelling rate (or average seepage velocity) appears to be the most important, and it is primarily controlled by the head difference (as governed by the assumed head in the underlying aquifer in the model analyses) and by the hydraulic conductivity. Our results can be related to a finger velocity number presented by Gosink and Baker [1990], who analyzed salt fingering. They present the term, w, for the vertical velocity in a finger as w ¼ Dgk : (4) [39] Bokuniewicz [1992] made a similar calculation to explain shallow salinity gradients and indicated that fingering should not occur where groundwater upwelling is much greater than w. [40] For our parameters for a case without upwelling, the value of w is about m/s. For the range of upwelling flows for the various head differences represented in Figure 12 in which seawater circulation occurs, the upward velocities range from about m/s for h ¼ m on the lower boundary to about m/s for h ¼ 0.20 m on the lower boundary. Thus, seawater circulation appears to shut down when upwelling velocities are less than the value of w. In this case, the sinking dense plumes (somewhat analogous to fingers) diffuse so much salt into the interfinger areas that the increased fluid density overwhelms the potential upwelling of fresher water and flow in the entire system becomes downward, filling the domain with infiltrating seawater. At the other end of the spectrum, seawater circulation shuts down for upwelling that exceeds about 2.5 times the value of w. In this case the upward flow is so dominating that downward advection of fingers is simply precluded. [41] This investigation demonstrated the viability of a previously unrecognized seawater circulation mechanism for one set of conditions focused on a relatively small wavelength bedform. However, this seawater circulation mechanism may exist at larger scales, and with larger rates, for other conditions and types of seabed topographic variations. This basic mechanism has been demonstrated to 1395

11 Figure 14. Sensitivity of model results to variability in the topographic relief on the seafloor, as reflected by the height of the crest of the bedform above the elevation of the troughs. Curves show effect on (a) calculated depth of penetration of the seawater front and the rate of seawater inflow on top boundary and (b) normalized measures of penetration depth and seawater circulation. Colors indicate which vertical axis applies to each curve. Stable circulation of seawater occurs only to the right of the dashed vertical line. occur at a larger scale in a terrestrial analogue and with a different fluid, as the process is analogous to air flow reported in Yucca Mountain, Nevada, where wells at the crest of the mountain exhaust air at relatively high velocity during the winter [Weeks, 1987]. Yucca Mountain is underlain by a thick unsaturated zone and the ridge of the mountain rises about 250 m above the floor of the adjacent canyon. Weeks [1987] and Thorstenson et al. [1990] demonstrate that the strong winter air circulation is caused by a topographic effect in which the higher density column of cold dry air in the adjacent valley outweighs the lower density column of warm moist air inside the mountain over the same elevation range. [42] In our analysis, we assumed that over the small width and length of the section analyzed that the head at depth was constant in space. However, it is expected that at a larger scale, the head in the underlying fresher water should generally decrease with distance offshore as the head dissipates with the upward flow and discharge of the deeper fresher groundwater. Thus, one might anticipate the occurrence of a banded pattern somewhat parallel to the shoreline of saltwater circulation due to topographicdensity interaction. Close to shore, the underlying head might be so large as to preclude SGD circulation, and far offshore, the head might be so small as to preclude any upward flow of deeper groundwater, thereby also eliminating the circulation due to topographic-density interaction. [43] Based on a compilation of worldwide SGD estimates, Taniguchi et al. [2002] report that most SGD rates are below about 0.1 m/d, with a substantial number of reported rates being much less than that. In studies of two separate field sites, Swarzenski et al. [2006] andswarzenski and Izbicki [2009] report SGD rates on the order of 0.07 m/d, without identifying contributing mechanisms. These reported rates are higher than the values of SGD computed in our numerical experiments, but our experiments were designed to isolate just one of the several mechanisms that can contribute to SGD and neither boundary conditions nor parameters of the model were adjusted to achieve a maximized flux. In a coastal setting where fresh groundwater discharges through a rippled bed, all of the various mechanisms are likely to interact in nonlinear ways, depending on the strength of currents, waves, and groundwater discharge rates. In terms of cumulative long-term effects, the topography-density interaction process would tend to be steady over time while current-, wave-, and tide-dependent processes would tend to be transient, oscillatory, or intermittent, so that the cumulative long-term contributions of the former process would be greater than a simple comparison of maximal instantaneous rates of individual processes might indicate. [44] The magnitude of the density-topography mechanism described in this study can also be put into a broader perspective by comparing the exchange rate due to this process with that generated by currents flowing over bedforms with the same characteristics as the base case (see Table 1; mean water depth of 3 m, bedform wavelength of 0.5 m, height of 0.2 m, K ¼ m/s, and a fresh groundwater upwelling rate of m/s). The average seawater flux due to buoyancy forces in the base case is cm/d. A current over the bedform of 0.26 m/s would be required to drive the same amount of flux, based on Elliott and Brooks [1997, Equations 21 and 28]. This is a moderate current for a coastal setting. Sheltered estuaries can have significantly slower currents, whereas open coasts can have significantly faster currents. Moreover, fresh discharging groundwater would tend to limit exchange due to currents and waves [Boano et al., 2008; Cardenas and Wilson, 2006], but it is a requirement for buoyancy-driven exchange described in this study. [45] The calculated seawater circulation patterns (Figure 4b) in most of the simulated cases typically went to depths of about 1 m and had travel times (or ages) of less than a year. On the other hand, the upwelling deeper groundwater has travel times on the order of 2 years, but because the age of the groundwater entering the model domain may already be decades old, the apparent groundwater age in areas having shallow seawater circulation cells may vary greatly over very short horizontal and vertical distances. [46] The presence or absence of seawater circulation switching abruptly when there is a very small change in a 1396

12 parameter value, even though other larger changes in the values of these same parameters yield only small changes in the computed fluxes, might be considered somewhat akin to chaos theory. It is also analogous to the abrupt change in vertical velocity arising from shifts in the balance between advective and density-gradient forces in a low-permeability, variable-salinity, confining layer, which then leads to a drastically different equilibrium salinity profile [Konikow and Arevalo, 1993]. Nondimensional groupings of parameters do not appear to predict the nature of these critical transitions between the presence or absence of stable circulation of seawater in shallow sediments. We were unable to delineate a regime for stable circulation using parameters such as the Rayleigh number or a nondimensionalized groundwater upwelling rate. However, the numerical experiments and sensitivity tests clearly indicate the relative strength and influence of each controlling parameter and boundary condition. [47] The Rayleigh number is often used as an indicator in variable-density stability problems, but this parameter is not reflective of stable displacement in an active flow field (under a regional hydraulic gradient). The difficulty in determining an appropriate nondimensional framework in a study of a dense plume in an active flow field has been noted by Schincariol et al. [1994], who stated, Once a vertical flow is imposed on this system, the nondimensional parameters governing instability become much more complex....the simple Rayleigh stability criterion, developed for problems of free convection in a box, should not be generally applicable to our problem. In another study, Wilson [2005] analyzed regional-scale topographically driven and density-gradient driven groundwater discharge to the oceans and states There is no Rayleigh number threshold for convection under these circumstances,... [48] It is important to reiterate that our analyses did not examine the interaction of the topography-density effect with other recognized driving forces for seawater circulation as a contributor to SGD. In the presence of waves, currents, and tides, the topographic-density driven component, as a separate process, may be overwhelmed. There are many quiet environments at times in coastal settings where fresh groundwater is upwelling, and this mechanism may be the dominant one in such settings and times. Examples include sheltered parts of estuaries where currents are slow moving and fetch may limit the wave action; locations where water depths are many times deeper than the bedform height (where bedforms are such a relatively small feature that moderate currents don t impart strong pressure gradients over the bedforms); and in the window of quiet conditions between fair weather wave base and storm wave base along coasts. Thus, the long-term cumulative contribution of the topography-density effect may be greater than indicated by a comparison solely with maximum flux rates generated by the other processes. In a coastal zone, all of the driving forces would be acting simultaneously, and all would be contributing to the net SGD flux. 5. Conclusions [49] The numerous mechanisms that drive saltwater circulation in coastal aquifers and result in large saltwater discharge (relative to freshwater discharge) are still not fully understood or quantified analytically. Here, we propose a new process not known to have been previously described in the literature that induces saltwater circulation in the presence of topographic relief on the seafloor if fresher groundwater is present at depth. In this mechanism, the inverted salinity (and fluid density) profile in the presence of a bedform induces a downward flow of saline water under relatively lower seabed elevations and upward convective flow under the adjacent higher seabed elevations. This mechanism is similar to the process of convective seawater circulation driven by fresh groundwater flow, density gradients, and salt dispersion, as first noted by Cooper [1959] and Cooper et al. [1964] for coastal aquifers in the context of saltwater intrusion. The seawater circulation occurs because of the interaction between bedform topography and buoyancy forces and requires no surficial current or flow to drive it, although the presence of underlying fresher groundwater is required to maintain the required density gradient. The occurrence and magnitude of this process depends on the values and interactions of several factors, including the bedform length and height, hydraulic conductivity of sediments, hydraulic head and salinity at depth in the porous media, aquifer thickness, effective porosity, and hydrodynamic dispersion parameters. [50] The results of this study indicate that local shallow circulation of seawater can indeed be driven by the coupled effects of seabed topography and fluid buoyancy effects. This circulation contributes to the SGD, and for the range of parameters and boundary conditions evaluated in this study, 20% 30% of SGD consisted of circulating seawater in most cases when it occurred. This is consistent with the modeling results of Smith and Turner [2001], who calculated that up to 35% of groundwater discharge to a coastal estuary represented saltwater circulation. [51] Sensitivity tests show that when this process is operative, the seawater circulation flux and depth of penetration of the seawater generally vary in a quasi-linear fashion over a moderate range of values of the various controlling parameters. However, for most parameters, there is a limiting value at which there is a rapid collapse of this equilibrium condition and a transfer to one of two alternate stable conditions. That is, depending on the specific boundary condition values, the system will either fill completely with seawater from above or with fresher groundwater from below. [52] The simulation results show that the flux across the seabed can vary from downward to upward within a very short lateral distance in the presence of even a small amount of topographic relief on the seafloor. This relatively shallow and local circulation of seawater will cause it to mix with upward flowing deeper pore water that may have a chemical signature partly reflective of onshore fresh groundwater and partly reflective of deeper brackish or salty groundwater. This may yield complex mixing patterns and chemical compositions that are highly variable over relatively small distances making a consistent, reliable, and logical interpretation of a limited number of waterquality analyses difficult to develop and the flow field difficult to define. [53] The magnitude of the seawater circulation flux driven by this mechanism is small but can be similar to that resulting from other small-scale driving forces. Like any 1397