Synthesis of benthic flux components in the Patos Lagoon coastal zone, Rio Grande do Sul, Brazil

Authors


Corresponding author: J. N. King, U. S. Geological Survey Florida Water Science Center, 7500 S.W. 36th St., Davie, FL 33314, USA. (jking@usgs.gov)

Abstract

[1] The primary objective of this work is to synthesize components of benthic flux in the Patos Lagoon coastal zone, Rio Grande do Sul, Brazil. Specifically, the component of benthic discharge flux forced by the terrestrial hydraulic gradient is 0.8 m3 d−1; components of benthic discharge and recharge flux associated with the groundwater tidal prism are both 2.1 m3 d−1; components of benthic discharge and recharge flux forced by surface-gravity wave setup are both 6.3 m3 d−1; the component of benthic discharge flux that transports radium-228 is 350 m3 d−1; and components of benthic discharge and recharge flux forced by surface-gravity waves propagating over a porous medium are both 1400 m3 d−1. (All models are normalized per meter shoreline.) Benthic flux is a function of components forced by individual mechanisms and nonlinear interactions that exist between components. Constructive and destructive interference may enhance or diminish the contribution of benthic flux components. It may not be possible to model benthic flux by summing component magnitudes. Geochemical tracer techniques may not accurately model benthic discharge flux or submarine groundwater discharge (SGD). A conceptual model provides a framework on which to quantitatively characterize benthic discharge flux and SGD with a multifaceted approach.

1. Introduction

[2] Numerous investigators advocate a multifaceted approach to characterize submarine groundwater discharge (SGD). Burnett et al. [2006] conduct SGD field investigations in five distinctly different hydrogeologic settings on five continents. They conclude that SGD is ubiquitous in the coastal zone, and that challenges of heterogeneity, scale, and multiple forcing mechanisms necessitate a variety of modeling techniques. The call for multifaceted characterization is also due to interpretation challenges associated with individual methods, and synthesis challenges related to the use of different methods at the same location.

[3] A conceptual model is applied in the present work to the Patos Lagoon coastal zone, Rio Grande do Sul, Brazil (Figure 1), where Windom et al. [2006] model the component of benthic discharge flux that transports 228Ra, and where analytical methods are employed to model components of benthic flux forced by the terrestrial hydraulic gradient; surface-gravity wave setup; surface gravity waves propagating over a plane, rigid, porous bed; and a component of benthic flux associated with the groundwater tidal prism. The conceptual model honorsBurnett et al. [2006]'s call for multifaceted characterization and addresses interpretation and synthesis challenges associated with the current SGD tool set. The conceptual model is geophysical, because the model relies on physics in natural systems. The term “geophysical” refers here to the study of the Earth with physics and not the niche-specific use of observational techniques, which employ a specific class of instruments.

Figure 1.

Patos Lagoon coastal zone, Rio Grande do Sul, Brazil. (a) Windom et al.'s [2006] study area is bounded by the shoreline, and the yellow polyline. (b) Elevation detail along the sandy barrier that separates the Patos Lagoon from the Atlantic Ocean in Rio Grande do Sul, Brazil. Elevation data are from National Geophysical Data Center [2010]. Transect C is detailed by Windom et al. [2006].

[4] An SGD is “any and all flow of water on continental margins from the seabed to the coastal ocean, regardless of fluid composition or driving force” [Burnett et al., 2003]. The SGD occurs over a broad range of spatial scales, from centimeter and smaller to kilometer and larger. Burnett et al. [2003]explain that thermal and density gradients, osmotic pressures, and bed-current interaction force SGD at smaller scales, and oceanic processes such as tide force SGD at larger scales. It also occurs over a range of temporal scales, from second and smaller to annual and larger. SGD is strictly a marine process; groundwater discharge to lakes and freshwater rivers is not SGD. The definition of SGD is independent of any specific measurement technique. For example, SGD is not exclusively the flux required to force activity balance for some radionuclide, geochemical tracer. SGD is routinely characterized in the literature by three key elements: a wide array of driving forces, variable chemical composition, and unlimited spatial and temporal scales.

[5] Benthic flux math formula is a rate of flow across the bed of a water body, per unit area of bed (defined within the SGD context by King et al. [2009]). The units of benthic flux are a function of fluxed property. For example, the units for benthic volume flux are [L3 T−1 L−2] = [L T−1], where [L] is a length dimension and [T] is a time dimension; the units for benthic mass flux are [M T−1 L−2], where [M] is a mass dimension. Benthic flux is independent of water body, driving force, chemical composition, and spatial and temporal scale. Benthic flux occurs in oceans, estuaries, rivers, lakes, lagoons, ponds, and all other locations where flow crosses the permeable bed of a body of water. Benthic chemical fluxes contribute to the chemical composition of water bodies. Benthic flux is a vector quantity. Benthic flux has bidirectional benthic discharge flux math formula and benthic recharge flux math formula components. (Vector notation is not employed in the present work where a model details magnitude only.) SGD is a benthic water discharge flux to coastal regions of a marine water body. The present work primarily employs the more general term “benthic flux,” except in some statements, where reference is made to other investigators' explicit use of the term “SGD.” Where benthic discharge flux is to the coastal region of a marine water body, the terms “benthic discharge flux” and “SGD” are equivalent.

2. Geophysical Conceptual Model

[6] Scientists employ conceptual models to constrain problems or pose tractable solutions. Conceptual models are abstract representations of more complex prototypes. For example, the popular geochemical conceptual model for SGD [Moore, 1996] models the flux of a geochemical tracer to the ocean with an activity balance. Some investigators employ the geochemical tracer as a surrogate for other constituents, such as nutrients or metals. The universal validity of this surrogacy is not well established in the SGD literature.

[7] The present manuscript employs a geophysical conceptual model to characterize benthic flux and SGD. This conceptual model is rooted in fundamental laws of physics. Force interaction and nonlinearity are elements of the conceptual model. It should be possible to draw similar, mutually supporting conclusions from both geophysical and geochemical conceptual models.

[8] The following equation symbolically represents the geophysical conceptual model:

display math

where math formula is (1) a function math formula of components i forced by unique mechanisms and (2) a function math formula of the nonlinear interaction between any given component i and all other components and nonlinear combinations of components j that force math formula at the location. King [2012] describes a preliminary version of the geophysical conceptual model.

[9] Both surface-water and groundwater processes generate benthic gradients that force a benthic flux of water and constituents between surface waters and the porous medium that constitutes the bed of the water body. Individual forcing mechanisms drive benthic flux. For example, a terrestrial hydraulic gradient can force fresh terrestrial groundwater toward fresh surface-water bodies, such as rivers and lakes. The groundwater and surface water have the same density in this example. The interaction of forcing mechanisms also drives benthic flux. For example, where a terrestrial hydraulic gradient forces fresh groundwater toward the ocean, the terrestrial hydraulic gradient interacts with a buoyancy force due to the density variation that exists between fresher groundwater beneath land and more saline groundwater beneath the ocean [Cooper et al., 1964]. Groundwater and surface water have different densities in this example. Dispersion also exists within the density transition zone in this example, where fresh and saline groundwater mix. Terrestrial hydraulic gradient, buoyancy, and dispersion interact to force benthic flux.

[10] The geophysical conceptual model is based on fundamental laws of physics. Newton's second law of motion requires that the sum of forces on a body is balanced by the body's time rate of change of momentum, which is equivalent to the product of the mass of the body and the acceleration of the body. The well-known Navier-Stokes equations are an application of Newton's second law to fluids. The Navier-Stokes equations relate fluid velocity, pressure, density, and viscosity. The equations are a statement of the conservation of momentum, in which the sum of the unsteady acceleration of the fluid and the convective acceleration of the fluid is equivalent to the sum of the divergence of stress within the fluid and other body forces acting on the fluid. The divergence of stress is the sum of pressure gradient and viscosity terms. The geophysical conceptual model requires that the acceleration of fluid through a differential volume that spans the benthic interface explain the sum of all forces that contribute to the divergence of stress on the differential volume and all other body forces acting on the differential volume. The Navier-Stokes equations govern benthic flux and SGD. Consideration of all forces is a key element of Newton's second law, the Navier-Stokes equations, and the geophysical conceptual model. Explicit consideration of all forces acting on the benthic interface is especially important within the context of the existing SGD literature, which does not universally honor this concept.

[11] With periodic forcing, benthic flux may transition between recharge and discharge. Where periodic forcing exists, benthic flux components may interact in a constructive or destructive manner, such that the component of benthic recharge flux forced by one process cancels or diminishes the component of benthic discharge flux forced by another process.

[12] Force interaction is a key component to the geophysical conceptual model. Benthic flux components may interact both linearly and nonlinearly. For example, consider a hypothetical location where benthic discharge flux is forced by two and only two mechanisms. If benthic discharge flux at this hypothetical location is equivalent to the sum of the two components, then benthic discharge flux is linear. If benthic discharge flux is equivalent, for example, to the sum of the first component and the square root of the second component, then benthic discharge flux is nonlinear. The presence of the first component of benthic discharge flux diminishes the contribution of the second component in the hypothetical nonlinear system.

[13] The findings of Xin et al. [2010] are specific examples of nonlinear interactions. Xin et al. [2010] quantify components of benthic discharge flux forced by the following interactions: (1) terrestrial hydraulic gradient, buoyancy, and dispersion (all associated with the coastal hydrogeology conceptual model described by Cooper et al. [1964]); (2) tide, terrestrial hydraulic gradient, buoyancy, and dispersion; (3) wave setup, terrestrial hydraulic gradient, buoyancy, and dispersion; and (4) tide, wave setup, terrestrial hydraulic gradient, buoyancy, and dispersion. Xin et al. [2010] use a numerical model to calculate a 2.30 m3 d−1 component of benthic discharge flux integrated over a hypothetical offshore area 1 m in the alongshore direction by 50 m in the offshore direction. For this base case, terrestrial hydraulic gradient, buoyancy, dispersion, and interactive combinations of these processes force benthic discharge flux to the hypothetical domain. Combinations include the following: (1) terrestrial hydraulic gradient and buoyancy, (2) terrestrial hydraulic gradient and dispersion, (3) buoyancy and dispersion, and (4) terrestrial hydraulic gradient, buoyancy, and dispersion. This calculation excludes all other forcing mechanisms to the hypothetical domain. They add tide, wave setup, and both tide and wave setup to the base case to model 5.54, 5.28, and 7.42 m3 d−1 components of benthic discharge flux to the offshore area. Although Xin et al. [2010] do not explicitly describe nonlinearity in the hypothetical domain, nonlinearity clearly exists.

[14] Additional processes not considered by Xin et al. [2010] likely force benthic flux in a natural system. For example, Xin et al. [2010] do not include other components of benthic discharge flux forced by surface gravity waves, such as the component forced by surface gravity waves propagating over a plane, rigid, porous bed [Riedl et al., 1972; King et al., 2009]; the component forced by surface gravity waves interacting with bed forms [Shum, 1992; Shum and Sundby, 1996]; or the component forced by surface gravity wave runup on a sloped beach face. (Grantham [1953] and Hughes [2004] discuss the general concept of wave runup.)

[15] Other investigators limit geophysical conceptual models of benthic discharge flux to an explicit set of components. For example, in a highly cited work, Li et al. [1999] propose a geophysical conceptual model for benthic discharge flux to Moore's [1996] SAB study area. Li et al. [1999] propose that all benthic discharge flux to the study area is exclusively equivalent to the sum of components of benthic discharge flux forced by tide, wave setup, and terrestrial hydraulic gradient. Li et al. [1999]do not include other processes that force water across the bed, identify as a model limitation the absence of other processes from the linear, three-component sum, acknowledge nonlinear process interactions, or incorporate the constructive or destructive interference of transient, oscillatory process.

[16] Numerous investigators identified other processes that force water across the bed prior to Li et al. [1999]. For example, Huettel et al. [1996] showed with laboratory observations that the interaction of currents with bedforms force benthic flux; Riedl et al. [1972] showed with analytical modeling and application to Moore's [1996] SAB study area that the interaction of propagating waves with a plane, rigid, porous bed force benthic discharge flux. Li et al. [1999]do not include the current-bedform interaction ofHuettel et al. [1996]or the wave-bed interaction ofRiedl et al. [1972] in their geophysical conceptual model.

3. Application to the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazil

[17] The geophysical conceptual model is applied to transect C [Windom et al., 2006] of the Patos Lagoon coastal zone, Rio Grande do Sul, Brazil. Windom et al. [2006] use the geochemical tracer 228Ra (half life, 5.75 years) to model an 8.5 × 107 m3 d−1 component math formula of benthic discharge flux that transports 228Ra to a 15,600 km2 study area, which extends in the positive y direction from shore to a point 65 km offshore (Figure 1). The 8.5 × 107 m3 d−1 component is normalized to 350 m3 d−1 m−1 shoreline for application to transect C. Windom et al. [2006] assume that a single 20.5 Bq m−3end-member228Ra activity describes all benthic discharge flux to the 15,600 km2 study area. Windom et al. [2006] do not detail the sensitivity of the single endmember assumption on the model of math formula. They do show that 228Ra activity varies in the offshore along the 65 km transect, from 2.5 to 4.2 Bq m−3 at the shoreline, to 0.8 to 3.0 Bq m−3, approximately 60 km offshore; and that 228Ra activity varies in beach groundwater, from 6.8 to 42.7 Bq m−3.

[18] This application of the geophysical conceptual model relies on both Windom et al.'s [2006] 350 m3 d−1 m−1 model of the component math formula of benthic discharge flux that transports 228Ra to transect C, and on analytical solutions to governing equations, which model components of benthic discharge flux. As with all models, the abstraction process involves compromises, such as the representation of heterogeneous and transient processes with homogeneous and steady-state assumptions.

3.1. Terrestrial Hydraulic Gradient Within the Water Table

[19] Bokuniewicz [1992] employs a Fourier cosine transform to solve the Richards Equation for the integrated component math formula of benthic discharge flux forced by a terrestrial hydraulic gradient from an unconfined, surficial hydrogeologic unit with an elevated groundwater table at a ground watershed divide

display math

where math formula is dimensionless benthic discharge flux forced by the terrestrial hydraulic gradient c/s, K is hydraulic conductivity, b is alongshore width, and d is depth of the surficial hydrogeologic unit. Bokuniewicz [1992] shows that the solution bounds observations by Bokuniewicz and Zeitlin [1980] made with seepage meters in the Great South Bay, New York.

[20] The analysis makes assumptions for K and d (Table 1). The hydrogeologic unit through which the terrestrial hydraulic gradient acts is shallow soil and beach material of high permeability. The integrated component math formula of benthic discharge flux forced by the terrestrial hydraulic gradient is 0.8 m3 d−1 m−1. This component is integrated over a 1 m wide area that extends from shore to a point offshore at which the terrestrial hydraulic gradient ceases to force math formula. Windom et al. [2006] suggest that the 1.2 × 106 km2 Mercosul aquifer may discharge to the Atlantic Ocean in this region of South America. The analytical solution for the component math formula does not describe the component of benthic discharge flux forced by the regional aquifer. The model represents 0.2% of Windom et al.'s [2006] component math formula of benthic discharge flux that transports 228Ra.

Table 1. Model Inputs and Outputs for Application of Bokuniewicz's [1992] Analytical Solution for the Component math formulaof Benthic Discharge Flux Forced by a Terrestrial Hydraulic Gradient From an Unconfined, Surficial Hydrogeologic Unit With an Elevated Groundwater Table at a Ground-Watershed Divide(2) Applied to Windom et al.'s [2006] Transect C in the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazila
ParameterLower BoundModelUpper BoundUnitsSource
  • a

    Model sensitivity is represented with lower and upper bounds.

Inputs
s10,0008,0006,000mFigure 1
d102040mAssumed for surficial hydrogeologic unit
c121518mFigure 1
K5 × 10−55 × 10−45 × 10−3m s−1Assumed for surficial hydrogeologic unit
math formula1.2 × 10−31.9 × 10−33.0 × 10−3  
math formula1,000400150  
 
Outputs
math formula4.79.720mEquation (2)
math formula0.020.830m3 d−1 m−1Equation (2)
math formula0.01%0.2%7%  math formula from Windom et al. [2006]

3.2. Groundwater Tidal Prism

[21] King et al. [2010] define the groundwater tidal prism P as the water volume that inundates a porous medium, forced by one tidal cycle. King et al. [2010] employ Nielsen's [1990]solution to the well-known, governing Boussinesq equation to quantify the groundwater tidal prism

display math

where n is porosity, A is tidal amplitude, math formula is the wave number for the propagating groundwater wave, and math formula is the dimensionless groundwater tidal prism. King et al. [2010] detail assumptions and limitations, and show that the solution compares favorably with both laboratory data and qualitative observations by numerous investigators of tidally forced benthic discharge flux.

[22] The analysis makes assumptions for n, K, d, and slope of the beach face sb (Table 2). The integrated component math formula of benthic discharge flux associated with the groundwater tidal prism is 2.1 m3 d−1 m−1. This component is integrated over a 1 m wide area that extends from shore to a point offshore at which math formula ceases. Models of components math formula and math formula employ d and Kof the same magnitude because both processes act on the same shore-proximate hydrogeologic region. The model represents 0.6% ofWindom et al.'s [2006] component math formula of benthic discharge flux that transports 228Ra. Mass conservation requires equivalence between components math formula of benthic discharge flux and math formula of benthic recharge flux, both associated with the groundwater tidal prism.

Table 2. Model Inputs and Outputs for Application of King et al.'s [2010] Analytical Solution for the Component math formula of Benthic Discharge Flux Associated With the Groundwater Tidal Prism (3) Applied to Windom et al.'s [2006] Transect C in the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazila
ParameterLower BoundModelUpper BoundUnitsSource
  • a

    Model sensitivity is represented with lower and upper bounds.

  • b

    R. D. Ray, Topex/poseidon: Revealing hidden tidal energy, 2011, http://svs.gsfc.nasa.gov/stories/topex/tides.html.

Inputs
n0.350.400.45 Assumed
math formula12.4212.4212.42h−1Defined
σ0.50.50.5h−1 math formula
d152025mAssumed
K5 × 10−55 × 10−45 × 10−3m s−1Assumed
λt1.8 × 10−15.3 × 10−21.6 × 10−2m−1 math formula
math formula0.050.100.15mRay (2011)b
sb0.070.050.03 Assumed
ϵ0.130.110.08  math formula
 
Outputs
math formula2.8262.8302.829 King et al. [2010]
P0.141.16.0m3 m−1Equation (3)
math formula0.262.112m3 d−1 m−1P/T
math formula0.08%0.6%3%  math formula from Windom et al. [2006]

3.3. Surface Gravity Wave Setup

[23] As a breaking wave propagates from the break point toward shore, the mean water surface increases, or sets up. A setdown, or decrease in the mean water surface, occurs as waves approach the break point from deep water. Li et al. [1999] employ Longuet-Higgins [1983] to quantify the integrated component math formula of benthic discharge flux forced by the wave setup

display math

integrated from shore to a point offshore at which the setup no longer forces the flux, where math formula is the slope of the setup water surface, math formula is the distance from the break point to shore, Kb is the breaking parameter, and Hb is the breaking wave height. Longuet-Higgins [1983] describes laboratory experiments that qualitatively validate the component math formula.

[24] The analysis makes assumptions for sb, Hb, and K (Table 3). The integrated component math formula of benthic discharge flux forced by surface gravity wave setup is 6.3 m3 d−1 m−1. This component is integrated over a 1 m wide area that extends from shore to a point offshore at which wave setup ceases to force math formula. Models of components math formula, math formula, and math formula employ Kof the same magnitude because these processes act on the same shore-proximate hydrogeologic region. The model represents 1.8% ofWindom et al.'s [2006] component math formula of benthic discharge flux that transports 228Ra. Mass conservation requires equivalence between components math formula of benthic discharge flux and math formula of benthic recharge flux, both forced by surface gravity wave setup.

Table 3. Model Inputs and Outputs for Application of Longuet-Higgins's [1983] Analytical Solution for the Component math formula of Benthic Discharge Flux Forced by Surface Gravity Wave Setup (4) Applied to Windom et al.'s [2006] Transect C in the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazila
ParameterLower BoundModelUpper boundUnitsSource
  • a

    Model sensitivity is represented with lower and upper bounds.

Inputs
Kb0.780.780.78 Canonical
sb0.030.050.07 Assumed
sw0.00600.00900.013  math formula
Hb0.250.500.75mAssumed
Lsz13.115.716.9m math formula
K5 × 10−55 × 10−45 × 10−3m s−1Assumed
 
Outputs
math formula0.326.395m3 d−1 m−1Equation (4)
math formula0.090%1.8%27%  math formula from Windom et al. [2006]

3.4. Surface Gravity Waves Propagating Over a Plane, Rigid, Porous Bed

[25] Surface gravity waves propagating over a plane, rigid, porous bed, force a component of benthic flux math formula. Riedl et al. [1972] and King et al. [2009] develop analytical solutions for the component magnitude math formula by employing a solution by Reid and Kajiura [1957] for wave damping. The positive, discharge phase of the math formula sinusoid generates a benthic discharge flux averaged over the wave period T [King et al., 2009]

display math

The amplitude of math formula is math formula, where a is wave amplitude, g is gravitational acceleration, k is permeability of the porous medium, and ν is the kinematic viscosity of water. The complex wave number math formula has real math formula and imaginary math formula components, where math formula is the wave radial frequency, his mean surface-water depth, math formula is a dimensionless model parameter, and math formula is the thickness of the porous medium. King et al. [2009] detail assumptions and limitations, and show that the solution compares favorably with laboratory data.

[26] The component math formula is integrated in the offshore direction over a 1 m wide area that extends from shore 65 km to the offshore boundary of Windom et al.'s [2006] study area. Integration over the transect is key to the magnitude of the estimate. The component math formula loses contact with the bed approximately 20 km offshore, where the amplitude of benthic flux approaches zero (Figure 2e) and the depth is approximately 30 m (Figure 2a). The amplitude is less than 10−5 m s−1 at points more than 500 m offshore (Figure 2f), where the depth is approximately 5 m (Figure 2b). The amplitude of math formula is a maximum of 2 × 10−5 m s−1 at a point less than 20 m offshore, where the depth is less than 1 m (Figures 2f and 2b).

Figure 2.

For the 5 s period, 0.5 m amplitude surface gravity wave detailed in Table 4: (a and b) Mean water depth h, in meters; (c and d) dimensionless depth math formula; wave amplitude a, in meters; and wave number math formula, in inverse meters; and (e and f) benthic flux amplitude α, in meters; all versus distance offshore along transect C of Windom et al. [2006]. (a), (c), and (e) detail shore to a point 65 km offshore; (b), (d), and (f) detail shore to a point 500 m offshore. Bathymetric data are from Windom et al. [2006]. The alignment of transect C is shown in Figure 1.

[27] The analysis makes assumptions for T, ao, and k (Table 4). The integrated component math formula of benthic discharge flux forced by surface gravity waves propagating over a plane, rigid, porous bed is 1400 m3 d−1 m−1. The damping and shoaling of the propagating 5 s period, 0.5 m amplitude surface gravity wave causes offshore variability (Figure 2). The majority of math formula occurs close to shore (Figures 2e and 2f). Specifically, 99% of math formulato the 65 km-wide transect occurs between shore and a point 15 km from shore; 73% occurs between shore and a point 5 km from shore. When the Patos Lagoon coastal zone is calm, with no surface gravity waves, math formula may diminish considerably or cease to exist. The model represents 400% of Windom et al.'s [2006] component math formula of benthic discharge flux that transports 228Ra. The ratio of the component math formula to Windom et al.'s [2006] component math formula, both integrated over the same study area, is similar to ratios for other studies in the SAB (Table 5). Mass conservation requires equivalence between components math formula of benthic discharge flux and math formula of benthic recharge flux, both forced by surface gravity waves propagating over a plane, rigid, porous bed.

Table 4. Model Inputs and Outputs for Application of King et al.'s [2009] Analytical Solution for the Component math formula of Benthic Discharge Flux Forced by a Surface Gravity Waves Propagating Over a Plane, Rigid, Porous Bed (5) Applied to Windom et al.'s [2006] Transect C in the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazila
ParameterLower BoundModelUpper BoundUnitsSource
  • a

    Model sensitivity is represented with lower and upper bounds.

Inputs
g9.8069.8069.806m s−2Canonical
ρ1,03010301030kg m−3Canonical
ν1.17 × 10−61.17 × 10−61.17 × 10−6m2 s−1Canonical
T456sAssumed
ao0.40.50.6mAssumed
k1.5 × 10−112.0 × 10−112.5 × 10−11m2Assumed
σ1.61.31.0s−1 math formula
Kb0.780.780.78 Canonical
 
Outputs
math formula50014002800m3 d−1 m−1Equation (5)
math formula140%400%800%  math formula from Windom et al. [2006]
Table 5. Comparison of Ratios of the Component of Benthic Discharge Flux Forced by Surface Gravity Waves on a Plane, Rigid, Porous Bed ( math formula) to the Component of Benthic Discharge Flux That Transports 226Ra or 228Ra ( math formula or math formula)a
StudyStudy AreaLength (km)Width (km)Area (km2)(m3 d−1 m−1) math formula 
math formula math formula math formula
  • a

    Components are integrated over designated study areas in the South Atlantic Bight or Patos Lagoon coastal zone, and are normalized per meter shoreline (m−1) for comparison.

Riedl et al. [1972]South Atlantic Bight1160100117,000  37003.9Riedl et al. [1972]/Moore [2010b]
Moore [1996]South Atlantic Bight32020640094    
King [2008]South Atlantic Bight320206400  22002.3King [2008]/Moore [2010b]
Moore [2010b]South Atlantic Bight60010060,0009201000   
King [2012]South Atlantic Bight: 20   21002.2King [2012]/Moore [2010b]
Tubbs Inlet Transect        
Windom et al. [2006]Table 4Patos Lagoon coastal zone2406515,600 350   
Patos Lagoon coastal zone: 65   14004.0Table 4/Windom et al. [2006]
Windom et al.'s [2006] transect C        

[28] Numerous investigators find that a component of benthic flux forced by surface-gravity waves is ecologically significant. For example,Almroth-Rosell et al. [2012] show that waves, wind, and currents force benthic flux—which affects benthic oxygen consumption—in a Scottish sea loch. Zhang et al. [2011]develop a numerical model of benthic nutrient uptake in coral reef communities forced by surface-gravity waves and bed-current interaction, and successfully apply the model to the Ningaloo Reef in western Australia.Precht and Huettel [2003a]characterize the ecological implications of benthic flux forced by surface gravity waves, and state that wave-forced benthic regions of the coastal zone are biocatalytic filters. Additional investigators also describe the ecological significance of benthic flux forced by both waves and bed-current interaction, including but not limited toNoffke et al. [2012], Santos et al. [2012], Precht and Huettel [2003b], Marinelli et al. [1998], Shum and Sundby [1996], Huettel and Gust [1992], Riedl et al. [1972], and Riedl and Machan [1972].

4. Discussion

[29] Numerous investigators characterize benthic discharge flux with a geochemical conceptual model math formula, where math formula is a flux of some tracer χ forced by some process i into or out of a control volume, and math formula is the change in χ within the control volume [e.g., Ellins et al., 1990; Moore, 1996; Cable et al., 1996; Charette et al., 2001; Burnett and Dulaiova, 2003; Paytan et al., 2006; Windom et al., 2006; Santos et al., 2008; Dimova et al., 2011; Smith and Swarzenski, 2012]. The geochemical conceptual model is fundamentally a mass or activity conservation statement. After summing all other tracer fluxes into or out of the control volume, the component of benthic discharge flux that transports the tracer explains any tracer deficiency within the control volume. Typically, the control volume is a reach or section of a water body, such as a river, lagoon, estuary, or the continental shelf. Investigators use a number of tracers, including radionuclides (228Ra, 226Ra, 224Ra, 223Ra, and 222Rn) from the Th and U decay series. Specific tracer fluxes required to model benthic discharge flux are either directly measured, or estimated; are a function of the tracer; and include advection and mixing of the tracer in surface-water, diffusion of the tracer, evasion of the tracer across the air-water interface, tracer production, and tracer decay.

[30] The geochemical tracer technique may not describe all benthic discharge flux to a water body [King, 2008; King et al., 2009; Moore, 2010a; Michael et al., 2011; Santos et al., 2012]. The geochemical tracer technique characterizes the component of benthic discharge flux that transports the tracer. Obviously, it is not possible to model a flux with a tracer that is not present within the flux. Where a component of benthic discharge flux exhibits the same tracer concentration or activity as the associated surface water body, it is not possible—with the tracer—to model the component. Restated, the geochemical conceptual model does not account for components of benthic discharge flux that do not enrich the control volume with tracer [King, 2008; King et al., 2009]. Moore [2010a] acknowledges that waves force SGD, and that the geochemical conceptual model does not describe SGD forced by waves. Michael et al. [2011] and Santos et al. [2012]both explain that models of SGD with radionuclide geochemical tracers do not describe benthic flux forced by short timescale or small spatial scale processes such as waves or bed-current interaction because the residence time within porous media is not sufficient to accumulate measurable tracer activity.Michael et al. [2011] show that full and accurate knowledge of the distribution of radium activity within a water body does not yield accurate predictions of SGD because heterogeneity in radium activity is not coincident with heterogeneity in SGD and because radium activity is not proportional to SGD.

[31] The geophysical and geochemical conceptual models are not in conflict. Components enriched in 228Ra with respect to surface waters may describe a portion of Windom et al.'s [2006] 350 m3 d−1 m−1 component math formula of benthic discharge flux that transports 228Ra. For example, employing a timescale argument, because the component math formula may transport fluids enriched in 228Ra, a portion of the 6.3 m3 d−1 m−1 component math formula may explain a portion of the 350 m3 d−1 m−1 component math formula. The 0.8 m3 d−1 m−1 component math formula also transports fluids enriched in 228Ra and may describe a portion of the 350 m3 d−1 m−1 component math formula. Neither the 1400 m3 d−1 m−1 component math formula nor the 2.1 m3 d−1 m−1 component math formula describe any portion of the component math formula because math formula and math formula are not sufficiently enriched in 228Ra with respect to surface waters.

[32] The present work employs analytical solutions that describe individual benthic flux components forced by singular mechanisms. These solutions do not account for nonlinear interactions that exist between individual components or for constructive or destructive interference that may exist between oscillatory components, such as interactions between components math formula and math formula. The relative magnitudes of math formula, math formula, math formula, and math formula are largely a function of the area over which forcing processes act. Components math formula, math formula, and math formula are shore proximate; math formula is active to a point approximately 20 km offshore (Figure 2e).

[33] Models detailed in the present work, including math formula, are normalized, averaged, or integrated over some time period. Strictly, models also include a math formula element. This temporal aspect of mathematics is dropped from the notation, for simplicity of presentation.

[34] Although this characterization is multifaceted (Table 6), it is not complete. Clearly, other processes may also force math formulain the Patos Lagoon coastal zone, such as buoyancy, density gradients, temperature gradients, the interaction of currents with bedforms, turbulence, bioturbation, dispersion, diffusion, sea-level rise, or terrestrial hydraulic gradients associated with confined hydrogeologic units. Analytical methods detailed in the present work cannot characterize nonlinear relationships, such as the one that may exist between math formula and math formula. Existing science lacks tools to quantitatively characterize all benthic flux components and associated nonlinear interactions.

Table 6. Synthesis of Benthic Flux Components in the Patos Lagoon Coastal Zone, Rio Grande do Sul, Brazila
ProcessOrientationParameterValue (m3 d−1 m−1)Source
  • a

    Components are time averaged over some period, such that math formula might also be included in the symbolic representation.

Terrestrial hydraulic gradientDischarge math formula0.8Table 1
Groundwater tidal prismDischarge math formula2.1Table 2
Recharge math formula2.1
Wave setupDischarge math formula6.3Table 3
Recharge math formula6.3
228Ra transportDischarge math formula350Windom et al. [2006]
Surface-gravity waves on a plane, rigid, porous bedDischarge math formula1400Table 4
Recharge math formula1400

5. Conclusions

[35] The present work details the following components of benthic flux to transect C of the Patos Lagoon coastal zone (Figure 1):

[36] The component math formula of benthic discharge flux forced by a terrestrial hydraulic gradient from the unconfined, surficial hydrogeologic unit is 0.8 m3 d−1 m−1.

[37] Components math formula of benthic discharge flux and math formula of benthic recharge flux, both associated with the groundwater tidal prism, are 2.1 m3 d−1 m−1.

[38] Components math formula of benthic discharge flux and math formula of benthic recharge flux, both forced by surface gravity wave setup, are 6.3 m3 d−1 m−1.

[39] Components math formula of benthic discharge flux and math formula of benthic recharge flux, both forced by surface gravity waves propagating over a plane, rigid, porous bed, are 1400 m3 d−1 m−1.

[40] The present work also details the following conclusions related to geochemical tracer techniques:

[41] Geochemical tracer techniques, such as Moore [1996], characterize the component of benthic discharge flux that transports the tracer. It is not possible to model with a geochemical tracer technique, a component of benthic discharge flux that does not enrich surface waters with the tracer.

[42] Geochemical tracer techniques, such as Moore [1996], that rely on the enrichment of surface waters with a tracer, do not accurately model “any and all flow of water on continental margins from the seabed to the coastal ocean, regardless of fluid composition or driving force” (Burnett et al. [2003]definition of SGD), where tracer half life exceeds the transport time scale of certain bed-proximate geophysical processes that force benthic discharge flux, such as surface gravity waves or bed-current interaction.

[43] Windom et al. [2006] detail a 350 m3 d−1 m−1 component math formula of benthic discharge flux to the Patos Lagoon coastal zone forced by math formula transport processes. This component does not describe all benthic discharge flux or SGD to the study area.

Acknowledgments

[44] Conversations with a number of leading SGD investigators at both the 2010 AGU Meeting of the Americas in Brazil and the 2010 AGU Fall Meeting, and comments from the editor, associate editor, Arturo Torres, W. Barclay Shoemaker, and four anonymous reviewers improved the manuscript. The U.S. Geological Survey, Water Mission Area, partially funded this work.

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