Beach Boundary Layer: A Framework for Addressing Recreational

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Environ. Sci. Technol. 2010, 44, 8804–8813

Beach Boundary Layer: A Framework for Addressing Recreational Water Quality Impairment at Enclosed Beaches S T A N L E Y B . G R A N T * ,†,‡,§ A N D B R E T T F . S A N D E R S ‡ Department of Chemical Engineering and Materials Science, Department of Civil and Environmental Engineering, Henry Samueli School of Engineering, University of California, Irvine, California 92697, United States, and Department of Civil and Environmental Engineering, School of Engineering, University of Melbourne, Victoria 3010, Australia

Received May 21, 2010. Revised manuscript received September 9, 2010. Accepted September 16, 2010.

Nearshore waters in bays, harbors, and estuaries are frequently contaminated with human pathogens and fecal indicatorbacteria.Trackingdownandmitigatingthiscontamination is complicated by the many point and nonpoint sources of fecal pollution that can degrade water quality along the shore. From a survey of the published literature, we propose a conceptual and mathematical framework, the “beach boundary layer model”, for understanding and quantifying the relative impact of beach-side and bay-side sources of fecal pollution on nearshore water quality. In the model, bacterial concentration in ankle depth water Cankle [bacteria L-3] depends on the flux m′′ [bacteria L-2 T-1] of fecal bacteria from beach-side sources (bather shedding, bird and dog feces, tidal washing of sediments, decaying vegetation, runoff from small drains, and shallow groundwater discharge), a cross-shore mass transfer velocity k [L T-1] that accounts for the physics of nearshore transport and mixing, and a background concentration Cbay [bacteria L-3] attributable to bay-side sources of pollution that impact water quality over large regions (sewage outfalls, creeks and rivers): Cankle ) m′′/k + Cbay. We demonstrate the utility of the model for identifying risk factors and pollution sources likely to impact shoreline water quality, and evaluate the model’s underlying assumptions using computational fluid dynamic simulations of flow, turbulence, and mass transport in a trapezoidal channel.

Introduction Beaches located inside coastal embayments, or “enclosed beaches”, are sheltered from the large waves and strong currents of open coastlines and, for this reason, are popular destinations for recreational bathers, particularly families with young children. Unfortunately, these coastal sites also have among the highest rates of beach closures and health advisories (1, 2). To our knowledge there are no literature reviews of the physical and biological processes that control nearshore water quality at enclosed beaches, despite the explosion of papers recently published on the topic. This * Corresponding author phone: (949) 824-8277; fax: (949) 8242541; e-mail: [email protected]. † Department of Chemical Engineering and Materials Science. ‡ Department of Civil and Environmental Engineering. § University of Melbourne. 8804

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critical review is intended to fill that gap, and provide a new conceptual framework (the “beach boundary layer model”) for predicting the impacts of beach-side and bay-side pollution on nearshore water quality. For the purposes of this review, we define “nearshore” as the shallow (ankle to waist depth) water along the shoreline where water samples are collected for routine monitoring of recreational beach water quality. We further divide sources of microbial pollution into those located in, or inland, of the nearshore (“beachside”) or offshore of the nearshore (“bay-side”). The review does not encompass predictive regression models of beach water quality (3), although the findings of regression studies are discussed to the extent they shed light on relevant physical and/or biological processes. While the review focuses primarily on fecal indicator bacteria (FIB)sthe water quality index used by public health officials to determine if a beach is safe for swimming and recreationsmany of the results and conclusions are transferable to the viruses, bacteria, and protozoa that cause recreational waterborne illness. The review begins by rationalizing the division of embayments into nearshore and bay regions, and the division of fecal inputs into beach-side and bay-side sources. This is followed by the derivation, testing, and application of the beach boundary layer model. Spatiotemporal Scales: Bay versus Nearshore. Whitman and co-workers (4) argue that water quality impairment of shoreline waters should be evaluated from the perspective of a “beachshed”, that includes all nonpoint and point sources of FIB and pathogens that potentially affect shoreline water quality, coupled with the biological and physical processes that amplify or diminish their concentrations. A beachshed for a FIB impaired coastal embayment is illustrated schematically in Figure 1. We conceptually divide the embayment into two regions: (1) bay and (2) nearshore. This division is motivated by at least four considerations. First, most routine water quality monitoring occurs in waters very close to shorestypically in ankle to waist depth watersbut water quality can vary significantly over a few meters in the crossshore direction. For example, Boehm et al. (5) report that total coliform concentrations in the nearshore waters at Avalon Beach, Catalina Island, decreased >100 fold from ankle (∼50 cm) to 10 m depth. Second, most humans recreate very close to shore, particularly young children who are among the most vulnerable to recreational waterborne illness (6-8). Third, some sources of fecal pollution (e.g., point sources of treated or untreated sewage) affect water quality 10.1021/es101732m

 2010 American Chemical Society

Published on Web 10/15/2010

FIGURE 1. A “beachshed” for fecal indicator bacteria at an enclosed beach. of the entire bay, while others (e.g., a small storm drain) have localized impacts on nearshore water quality. Finally, different physical processes drive water circulation in the bay and nearshore regions, as described next. Bay-Wide Circulation. At the bay-scale, water circulation is driven by tides, density gradients, and winds. Tides exchange water between the bay and offshore, promote chaotic mixing, and drive a residual circulation that varies horizontally, provides nutrients and oxygen, and transports and dilutes pollutants (9-11). Baroclinic circulation originates from the within-bay mixing of water parcels with different densities. When a river discharges to an ocean embayment, for example, dense ocean water is drawn into the bay, mixes with the fresh water, and the product is returned to the ocean. In low mixing environments, the recirculation is vertically stratified and in high mixing environments the recirculation is laterally stratified (11). Winds exert a stress on the water surface that can enhance the exchange of water between the bay and offshore, generate surface water waves, and drive within-bay circulation (11). Nearshore Circulation. Mixing and circulation in the nearshore region is affected by “down-scaling” of the bayscale processes described above (tides, density gradients, and winds) (12) and physical processes confined to the shoreline (13). Breaking waves are an example of the latter. The cascade of momentum and energy from breaking waves generates along-shore and cross-shore currents that often dominate the nearshore transport of contaminants and sediments, particularly on open (unsheltered) coastlines (13-16). In sheltered embayments, waves are typically smaller and human activities can play an important role. For example, water currents along the City of Seattle waterfront in Elliot Bay are dominated by boat traffic; in particular, ferries that continuously apply thrust while docked in the harbor (17). Boat wakes generate turbulence capable of resuspending sediments, and presumably particle-associated bacteria, at the shoreline (18). Indeed, the use of mechanical flow devices to artificially “enhance” circulation has been marketed as a way to improve beach water quality, although questions persist about the efficacy, practicality, and ecological impact of such an approach (2). Generally speaking, nearshore pollutant transport in embayments occurs by advection and turbulent mixing (19, 20). Sources of Microbial Pollution: Bay-Side versus BeachSide. Sources of microbial pollution in nearshore waters can be divided into two categories: (1) bay-side sources that impact water quality over large regions of the bay, and (2) beach-side sources that have localized and transient impacts. Because these two source categories operate over different

spatial and temporal scales, from a modeling perspective their effect on ankle depth water quality can be treated separately. In particular, we assume that bay-side sources contribute a background concentration Cbay that varies with location in the bay (e.g., higher at shoreline sites closer to a point source discharge of sewage) and varies slowly over time (e.g., higher during periods of increased point source loading or reduced tidal mixing). In contrast, beach-side sources are characterized by a shoreline loading rate m′ (units of bacteria per time per length of beach) the magnitude of which varies by beach, and over tidal and longer time scales at any particular beach. The contribution of beach-side and bay-side pollution to the concentration Cankle of FIB in ankle depth water is assumed to be additive, viz: Cankle ) f(m′) + Cbay

(1)

where f is a functional relationship derived later. In this section we survey bay-side pollution sources that contribute to the term Cbay, and beach-side pollution sources that contribute to the term m′. Bay-Side Pollution Sources. Bay-side contributions to the term Cbay include both point- and nonpoint sources of fecal pollution. Large point sourcesssuch as rivers and partially treated or untreated sewage effluentscan have bay-wide impacts on water quality via at least two pathways: (a) direct discharge to the bay, and/or (b) discharge outside of the bay (e.g., in the coastal ocean), followed by import into the bay by tides and currents. In general, the mixing of river or wastewater effluent into receiving waters occurs in three stages, namely: (1) near-field mixing which is dominated by the buoyancy and/or momentum flux of the effluent at the point of discharge; (2) far-field mixing which is dominated by ambient turbulence; and (3) midfield mixing which is transitional between near-field and far-field (21-23). Farfield plume dispersion can be modeled using either Eulerian formulations (gradient diffusion models) or Lagrangian formulations (random walk particle tracking techniques) (21). Examples of both Eulerian and Lagrangian plume tracking tools are now available online for some regions. One example is the Great Lakes Forecasting System (24), which includes a gradient diffusion model for simulating impacts on beach water quality from the Grand River plume in Lake Michigan (25). Another is the Lagrangian particle tracking tool maintained by the Southern California Coastal Ocean Observing System that predicts, based on hourly high frequency radar measurements of surface currents in the coastal Pacific Ocean, the trajectory of the Tijuana River Estuary plume near San Diego, California (26). The latter modeling tool was VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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recently used to determine the relative impact of both river and wastewater outfall plumes on coastal beach water quality along the California/Mexico border (27). Nonpoint sources of bay-side pollution include vessel waste discharges (28-30) and diffusive, advective, and/or erosive release of FIB from sediments (31-35). Federal law in the U.S. prevents the discharge of raw (untreated) sewage in no discharge zones, including the Great Lakes, all navigable waters of the U.S., and within 3 miles of the coasts (30). However, based on random surveys of recreational boaters it appears that illegal dumping of sewage is commonplace in some areas of the U.S., due to a paucity of pump-out facilities, a lack of appreciation of the potential environmental impacts, and the limited range of many recreational boats (29). In the U.S., less stringent laws and regulations apply to the dumping of untreated gray water in coastal areas (e.g., from galleys, baths, showers, and dishwaters) and a recent U.S. EPA assessment of wastes generated from cruise ships reported average fecal coliform concentrations in untreated gray water from 3 to 40 × 106 bacteria/100 mL, or “one to three orders of magnitude greater than typical fecal coliform concentrations in untreated domestic wastewater” (30). In practice, estimates of Cbay for a particular beach can be determined empirically, by measuring pollutant concentration in deeper waters offshore, or predicted by the Lagrangian or Eulerian plume tracking models described above. At many enclosed beaches, the background concentration will be a minor contributor to nearshore water quality, and therefore Cbay ≈ 0. Beach-Side Pollution Sources. The contribution of beachside pollution to the term m′ includes bather shedding, bird or animal feces, beach sediments, decaying marine vegetation (e.g., associated with wracklines), runoff from small drains, and submarine groundwater discharge (SGD): m′ ) m′bather + m′feces + m′seds + m′wrack + m′drains + m′SGD (2) Simple expressions for each of these loading rates are proposed in Table 1; literature sources for key parameter values are also shown in the table. Present understanding of beach-side sources is summarized below. Elmir et al (36) estimate that the average marine bather sheds 300 000 enterococci bacteria in the first 15 min of immersion. Less enterococci bacteria are shed with subsequent immersions, and sand attached to the skin of bathers does not appear to be the source of FIB, consistent with a previous study that concluded most FIB released by bathers is associated with fecal matter (37). The deposition of dog and bird droppings on foreshore sediments is frequently implicated as a source of FIB in nearshore waters (35, 38-40); however, nearshore FIB concentrations are not always correlated with bird abundance, even in tidal ponds specifically managed as bird habitat (41). Beach sands can harbor FIB (42), genetic markers for human fecal contamination (enterococci bacteria with the human esp gene and human specific HF183 Bacteroidales) and human pathogens (Vibrio vulnificus, and Cryptosporidium) (43, 44). Potential sources of FIB in beach sands include human and animal feces, runoff, spilled sewage, and/or the growth of environmentally adapted FIB strains (44-48). FIB present in foreshore sediment can transfer into the water column with the rising tide; that is, the tides “wash” the sand free of bacteria (42). Shoreline vegetationsincluding the macroalga Cladophora in freshwaters (49, 50) and seaweed in marine waters (35, 51)sis another potential source of FIB. Indeed, the tidal washing of FIB-laden foreshore sediments and decaying seaweed in wracklines may explain why nearshore water quality at many marine beaches in California is worse during spring tides when the daily maximum tide range is large (52). 8806

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TABLE 1. Parameterizations for Six Beach-Side Sources of FIBa line source for bather shedding ˙′ m′bather ) SN S, number of FIB shed per bather (bacteria bather-1) S ) 300,000 ENT bacteria bather-1 ref 36 ˙ ′, number of bathers wading into water every hour per N meter of beach (bathers h-1 m-1)

line source for fecal droppings m′feces ) W′fecesCfeces/TF W′feces, weight of feces deposited on foreshore sediments during low tide per meter of beach (g m-1) Cfeces, concentration of FIB in feces (bacteria g-1)

Cfeces )

{

4 × 107ENT g-1(dog) 3 × 105ENT g-1(bird) ref 39

TF, time over which a single flood tide occurs (ca., 6 h) line source for sediment washing m′seds ) W′sedsCseds/TF Cseds, concentration of FIB in foreshore sediment (bacteria (dry weight sediment)-1) W′seds, weight of sand washed free of bacteria by the rising tide per meter of beach (g m-1) W′seds ) Fbdwηr/sin (tan -1s) ≈ 350 kg m-1 Fb ≈ 1.7 g cm3, bulk density of dry beach sand dw ≈ 5 cm, depth of sediment washed by the rising tide ηr ≈ 1 m, tide range s ≈ 0.25, beach slope

line source for wrack washing m′wrack ) W′wrackCwrack/TF W′wrack, weight of wrack deposited on foreshore sediments per meter of beach (g m-1) Cwrack, concentration of FIB in wrack (bacteria g-1) Cwrack ≈ 4 × 104 ENT g-1 [ref 99]

line source for runoff from periodically spaced drains m′drains ) QdrainsCdrains/Ldrains Cdrains, concentration of FIB in runoff from drains (bacteria (volume)-1) Qdrains, daily average volumetric flow rate from drains (volume (time)-1) Qdrains ≈ 200 liters per day [unpublished data for small drains in Newport Bay] Ldrains, spacing of drains along the shore (distance) Ldrains ≈ 10 m [unpublished data for small drain spacing in Newport Bay]

line source for shallow groundwater discharge m′SGD ) Q′SGDCSGD Q′SGD, volumetric discharge of shallow groundwater per meter of shoreline (volume (time)-1 (length)-1) CSGD, concentration of FIB in SGD (bacteria (volume)-1) Q′SGD ≈ 3 L min-1 m-1 [estimated for Avalon Bay, ref 76] a

ENT, enterococci bacteria.

Dry and wet-weather runoff from small drains contributes terrestrial flux of FIB and other contaminants to nearshore waters (53, 54). During dry weather, nonpoint source runoff originates from activities in the watershed that generate excess water, such as over irrigation and car washing. In arid

urban settings such as southern California, runoff that does not evaporate or infiltrate, flows through a network of gutters, curbs, and underground pipes (collectively referred to as the storm sewer, see Figure 1) to the beach, where it typically discharges without treatment through small drains distributed around the perimeter of the bay. Dry and wet weather runoff from drains is a documented source of FIB and molecular markers of human fecal pollution at recreational beaches (53-56), and bather illness rates appear to increase with proximity to drains discharging dry weather runoff (57). In general, the volume of runoff flowing out of any single storm drain during dry weather is small and intermittent. Nevertheless, dry weather runoff from drains can potentially affect nearshore water quality because of the high concentration of FIB carried in runoff and the location of drain outlets close to the waterline where water depths are shallow and dilution is minimal (15, 58, 59). The recreational water quality impacts posed by wet and dry weather urban runoff, and modeling tools available for planning and mitigation, are reviewed elsewhere (60, 61). The coastal discharge of shallow groundwater can also contribute to the terrestrial flux of FIB and other fecal indicators to nearshore waters, particularly if the shallow groundwater is contaminated with sewage from poorly functioning onsite disposal treatment systems or leaking sewage collection infrastructure (62-67). Deriving the Beach Boundary Layer Model. In this section we derive the “beach boundary layer model”, which in effect is an explicit expression for the functional relationship f that appears in eq 1. Assuming that the bacterial removal follows first-order kinetics (elaborated further below), the balance between FIB advection, turbulent mixing, and removal in any differential element in the water column can be expressed mathematically as follows (68):

[ ]

[ ] [ ]

∂ ∂C ∂C ∂C ∂C ∂C ∂ ∂C ε + ε + +U +V +W ) ∂t ∂x ∂y ∂z ∂x x ∂x ∂y y ∂y ∂ ∂C ε - krC ∂z z ∂z

(3)

The variables in eq 3 represent the concentration of FIB in the water column (C), time (t), spatial coordinates in the alongshore (x), cross-shore (y), and vertical (z) directions, current velocities in the alongshore (U), cross-shore (V), and vertical (W) directions, turbulent diffusion coefficients in the alongshore (εx), cross-shore (εy), and vertical (εz) directions, and a first-order removal rate constant (kr). A number of nonconservative and ecological processes can affect nearshore FIB concentrations, including (1) sequestration and differential survival in sediment (69-73) (2) bacterial mortality by exposure to sunlight (74, 75), reactive oxygen species (76, 77), and predacious protozoa (78-80), (3) selection and growth (46, 81-83), and (4) resuscitation/ recovery of injured (viable but nonculturable) cells (84). In eq 3, all of these nonconservative and ecological processes are lumped into a single first-order decay term (last term in the equation)san approach that will be inadequate in many settings. For example, if FIB mortality is dominated by protistan grazing, as appears to be the case in both fresh and marine waters (78-80), then the first-order rate “constant” kr may depend on the abundance and clearance rates of dominant predators. A simple first-order decay term is also inadequate for modeling light-induced bacterial mortality, which varies diurnally and with depth through the water column (76, 85), and the turbulent sedimentation and resuspension of particle-associated FIB (86). Rather than attempt to address these shortcomings by significantly increasing the complexity of eq 3 (e.g., see ref 86), here we adopt the opposite tact by setting the loss term to zero, in effect neglecting all nonconservative and ecological processes. By focusing exclusively on the transport aspect of the problem, we can elucidate the role that physical processes

play in regulating nearshore water quality in settings where ecological and nonconservative processes are less important. The opposite approachsassuming FIB concentrations are controlled only by nonconservative and ecological processess is also interesting, and no doubt real systems exhibit some mixture of transport and nonconservative/ecological control. It is also interesting to note that nonconservative and ecological processes may be less important in situations where FIB and/or pathogens are monitored using cultureindependent methods, such as quantitative polymerase chain reaction (qPCR). Measurements of genome copy number by qPCR may be insensitive to the aggregation state of cells in a sample, their association with particles, or their distribution between culturable and nonculturable statessall factors that can affect measurements of FIB by conventional culturedependent methods (87, 88). After dropping the bacterial removal term, eq 3 can be further simplified by invoking a set of transport-related assumptions that may apply in the nearshore region of coastal embayments: (1) turbulent diffusion dominates over advection in the cross-shore direction (|VC| , |εy ∂C/∂y|); (2) advection dominates over turbulent diffusion in the longshore direction (|UC| . |εx ∂C/∂x|); and (3) FIB concentrations quickly reach quasi-steady-state (∂C/∂t ≈ 0). Assumptions (1) and (2) imply that nearshore currents follow the shoreline topography. Assumption (3) implies that nearshore concentrations of FIB quickly achieve quasi-steady-state in response to changes in pollutant loading and transport rates. Invoking these assumptions and depth-averaging the remaining terms in eq 3 (where depth-averaged pollutant j we arrive at a classic “boundary concentration is denoted C) layer” balance between alongshore advection and cross-shore turbulent diffusion, where Z(y) represents the water depth at cross-shore distance y: U(y)Z(y)

[

j j ∂C ∂C ∂ ε (y)Z(y) ) ∂x ∂y y ∂y

]

(4)

In formulating eq 4 we allow for the possibility that alongshore velocity, water depth, and turbulent diffusivity all vary with cross-shore position y (see schematic at the top of Figure 2). The emphasis on alongshore advection and cross-shore diffusion is consistent with detailed FIB budgets carried out for recreational beaches in both Lake Michigan (68) and Huntington Beach in southern California (15, 89). Exact Solution of the Beach Boundary Layer Equation. The boundary layer equation above is standard in many engineering and scientific textbooks (90). What is unique to this application is the prismatic geometry of the boundary and an alongshore velocity and cross-shore eddy diffusivity that vary with cross-shore position. Despite these complications, we show in the Supporting Information that eq 4 can be solved exactly given the following parametrizations for the cross-shore variation in water depth, along-shore velocity, and cross-shore turbulent diffusion: Z(y) ) z0√y/yo

(5a)

U(y) ) u0√y/yo

(5b)

εy(y) ) ε0√y/yo

(5c)

where zo,uo,ε0 represent the water depth, along-shore velocity, and eddy diffusivity all specified at cross-shore distance y ) y0. Because εy and Z have the same y dependence, our model assumes that turbulent diffusion scales linearly with water depth, εy ∼ Z1, consistent with Kolmogorov-Prandtl turbulence closure with water depth as the turbulent length scale (91). After combining eqs 4 and 5, performing a similarity transformation, and applying two boundary conditions VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 2. The exact solution assumes that water depth, along-shore velocity, and turbulent diffusivity all increase with cross-shore distance (top schematic). Cross-section depth profiles (top row) and predicted pollutant concentration fields (bottom row) corresponding to the CFD (left column) and exact (right column) solutions. j Cjbay far offshore and a finite pollutant loading rate m′ at (Cf the shoreline) the following exact solution to eq 4 is obtained: j (x, y) ) C

[ ]

yom′ -y2uo j bay Ei +C 2εozo 4εox

(6)

where the exponential integral function is defined as follows. Ei(x) ) -

∫

∞

-x

e-tdt t

(7)

Comparison of Exact Solution and Numerical Simulations. The exact solution derived above was obtained after considerable simplification of eq 3, not only relative to ecological and nonconservative processes neglected (see discussion above), but also relative to the limited set of physical processes assumed to dominate nearshore transport. To determine if the latter “transport-related” assumptions are realistic, we compared the exact solution to a computational fluid dynamics (CFD) simulation of coupled Reynolds- and depthaveraged (2-dimensional) fluid mass, momentum, turbulence, and pollutant transport equations using a Godunovtype finite volume scheme with k-eps turbulence closure (92-94). Save the steady-state assumption, the CFD simulations employed none of the assumptions or parametrizations used to derive the exact solution. Thus, a comparison of concentration fields predicted from the CFD simulations and the exact solution should constitute a fair test of the assumptions employed to derive the latter. The CFD solution was implemented over a long straight channel (2000 m long by 100 m wide) with a line source of pollution located along one bank (m′ ) 100 bacteria m-1 s-1 for x > 0) (Figure 2, left panel). The upstream and downstream boundary conditions consisted of a constant inlet flow rate (1 m3 m-1 s-1) and 8808

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constant outlet water depth (6 m at the centerline), respectively, and the channel cross-section was trapezoidal with a constant side slope (s ) 0.25) out to a depth of 6 m (20 m from the bank). Details of the CFD simulations, and the approach used to assign parameter values for the exact solution’s velocity and eddy diffusivity fields, are presented in the Supporting Information. The pollutant concentration field predicted for one CFD simulation (for a grid dimension of 2 × 1 m ) length by width) is shown in Figure 2 (lower left panel). The highest pollutant concentrations are confined to a thin region close to the shore, consistent with the idea that beach-side pollutant loading leads to the development of a concentration boundary layer, or “beach boundary layer”. The concentration boundary layer is characterized by a sharp concentration front that moves bayward with increasing alongshore distance x. At the furthest downstream location, x ) 2000 m, the numerical simulation illustrated in Figure 2 predicts that pollutant concentrations decrease >100 000 fold over a crossshore distance of 4 m), but do not reliably predict the ankle-depth pollutant concentration, most likely due to numerical diffusion and the concentration singularity at y ) 0. Nearshore Pollutant Concentration. Estimates for the impact of specific pollution sources on ankle depth water quality can be obtained by setting y ) yankle in eq 6 (note the overbar, representing depth-averaged quantities, has been dropped for notational simplicity): Cankle )

m′ + Cbay zok

k)

2ε0 yBBL

yBBL ) -y0Ei[-l/x] ≈ y0log[0.562x/l] l)

2 u0 yankle 4ε0

Predicting the Impact of Beach-Side Pollution. Assuming that FIB concentration in ankle depth water is dominated by beach-side pollution (i.e., Cankle > >Cbay), eq 8a can be rewritten as follows:

(8a)

Cankle m′ ) Ccriterion m′criterion

(10a)

(8b)

m′criterion ) zokCcriterion

(10b)

(8c)

(8d)

Comparing eqs 1 and 8a we conclude that, in our version of the beach boundary layer model, the functional relationship in eq 1 is given by f(m′) ) m′/z0k. The cross-shore mass transfer velocity k [units of L T-1] encapsulates the physics of nearshore transport and mixing, and is proportional to the reference eddy diffusivity ε0 and inversely proportional to yBBL, where the latter is an estimate for the width of the beach boundary layer. The approximate expression for yBBL in eq 8c is valid provided that x > 10l. Solving for the flux of FIB out of ankle depth water, m′′ ) m′/z0, eq 8a can be rearranged into a form that is mathematically identical to the film theory expression for interfacial mass transfer (90): m′′ ) k(Cankle - Cbay)

FIGURE 4. Predictions of the contribution of beach-side pollution to exceedences of the U.S. EPA criterion for enterococci bacteria (104 bacteria (100 mL)-1) at marine recreational beaches.

(9)

These results have significant implications for the influence of beach-side pollution sources on FIB concentrations in ankle depth waters, as explored in the next section.

where Ccriterion is a water quality criterion and m′criterion is the beach-side pollution loading rate above which an exceedence of the criterion occurs. For the set of parameters values in Table S1 (Supporting Information), x ) 1000 m, and using the EPA recommended criterion for enterococci bacteria (ENT) at marine bathing beaches (104 bacteria per 100 mL), we obtain the following model parameter values: yBBL ) 0.7 m, k ) 3 × 10-4 m/s, and m′criterion ) 3 × 105 bacteria m-1 h-1. Given this criterion loading rate, and the line source expressions listed in Table 1, plots of eq 10a are presented for seven different beach-side pollution sources in Figure 4. While all pollution sources can trigger criterion exceedence (i.e., Cankle/Ccriterion > 1), some sources are clearly more problematic than others. Consistent with the findings of Wright et al. (39), dog feces are a potent contributor of ENTsas little as 1 g of dog feces deposited at low tide per meter of beach is sufficient to cause a water quality exceedence during the following rising tide (red line, Figure 4). By comparison, an average of 30 and 80 g of dry feces are produced by small and large dogs, respectively, during a single fecal event (39). For all other sources, the model predicts that the concentration of ENT in ankle depth water is 10 times greater than the EPA criterion whenever one or more of the following conditions are met: (1) ENT concentrations VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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in foreshore sediments exceed 500 bacteria/10 g (green line in Figure 4); (2) more than 10 bathers wade into the water per hour per meter of beach (dark blue line); (3) ENT concentration in shallow groundwater exceeds 2000 bacteria/ 100 mL (light blue line); (4) more than 70 g of bird droppings are washed by the tide per meter of beach (brown line); (5) more than 500 g of decaying vegetation are washed by the tide per meter of beach (pink line); and (6) ENT concentration in runoff exceeds 500 000 bacteria/100 mL in small drains spaced 10 m apart (black line). It should be noted that the relatively low impact predicted for drains assumes a flow rate of 200 L day-1 (see Table 1), which is typical for dry weather runoff volumes from small (104-105 m2) urban catchments in Newport Bay in southern California (unpublished data), but will be on the low side for many other beach settings. At beaches where the average flow rate of runoff from drains exceeds 200 L day-1, the black line in Figure 4 will shift to the left. Indeed, Dorsey (59) evaluated water quality improvement projects at 17 beach sites in California, and found that dry weather diversionssin which dry weather runoff from storm sewer drains is diverted from the beach to the sanitary sewer system for treatmentswere the most likely to improve beach water quality. The location of curves in Figure 4 depends on the value chosen for the cross-shore mass transfer velocity. Intriguingly, the value adopted here (k ) 3 × 10-4 m/s) is within the range of k values (10-5 to 0.1 m/s) reported for interfacial mass transfer in turbulent flows (95), potentially linking the problem of beach water quality to the well-developed literature on turbulence-driven interfacial flux (95). However, it is unclear at this stage how environmental factors unique to beaches (e.g., generation of nearshore turbulence by waves and currents, the prismatic geometry of the interface, and bed roughness, to name a few) and to FIB and pathogens (e.g., their origin within the nearshore wedge, size, diffusivity, and potential particle association) influence the value of k. Therefore, the mass transfer velocity value used to generate the curves in Figure 4 should be viewed as illustrative, and by no means definitive. One quirk of our exact solution is the semi-infinite nature of the line source geometry, which starts at fixed point along the beach, at x ) 0. A less general result can be obtained by j forcing ∂C/∂x ) 0 in eq 4, whereupon the predicted crossshore mass transfer velocity becomes k)

εankle , y > yankle yanklelog(ybay /yankle) bay

(11)

where εankle is the eddy diffusivity in ankle depth water (located at cross-shore position yankle) and ybay is the cross-shore distance over which the beach boundary layer persists. Unfortunately, this simplification, while eliminating the need to specify an alongshore distance x, does not allow for the computation of ybay from known or presumed parameters; c.f., the expression for yBBL in eq 8c. Practical Implications and Future Research. The large impact of beach-side sources on nearshore water quality, illustrated above, derives from the low dilutions associated with the prismatic geometry of the water column, together with the physics of nearshore mixing and transport which is dominated by along-shore advection (e.g., from tidal flow in and out of a tidal embayment) and cross-shore turbulent diffusion (e.g., from waves and currents). These geometrical and physical transport characteristics result in the development of a concentration boundary layersor “beach boundary layer”sthat traps pollution in shallow water very close to shore, where routine water quality monitoring is carried out and most human recreational contact occurs. As a practical matter, these observations highlight the importance of controlling beach-side sources of fecal pollution, particularly 8810

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in cases where fecal-oral pathogens are present (e.g., from sewer overflow into storm sewer drains). The very sharp concentration fronts predicted by the numerical and exact solutions are consistent with the observation that FIB concentration falls off rapidly with increasing water depth in sheltered embayments. According to eq 9, when Cankle > Cbay, beach-side pollution sources are primarily responsible for FIB impairment in ankle depth waters, and the net flux of FIB is directed from onshore to offshore; that is, m′′ > 0. The sign of the cross-shore gradient reverses (Cankle > Cbay) at beaches impacted by the onshore transport of bay-side sources, and the net flux of FIB is directed from offshore to onshore; that is, m′′ < 0. Thus, the sign of the cross-shore FIB concentration gradient indicates the relative importance of beach-side and bay-side pollution. Our results have implications relative to the design of recreational beach monitoring programs. The CFD and exact solutions predict a concentration singularity at the waterline, or “triple-point”, where water, beach, and air all intersect. Clearly, real beaches will not have infinite concentrations at the triple-point, but our results suggest that FIB concentrations in “ankle depth” water are likely to be very sensitive to small variations in how and where water samples are collected. This observation may help to explain why recreational beach water quality data exhibit extreme (fractallike) variability (96)sa fact that negatively impacts the reliability of public water quality notification programs (97) and recreational waterborne illness risk and cost assessments (98). Shifting the sampling location from ankle to waist depth should yield a more representative, and stable, estimate of the “plateau” concentration within the beach boundary layer (see Figure 3). To achieve an exact solution to the governing partial differential equation (eq 3), we neglected all ecological and nonconservative processes that might affect FIB concentrations in shallow nearshore waters, and made further assumptions regarding the dominant mixing and transport processes associated with nearshore flows. While these assumptions allow for an exact solution to the governing partial differential equation, and a specification of the function f in eq 1, it remains to be seen under what conditions they are satisfied in practice. For example, our exact solution assumes that cross-shore advective exchange is negligible (compared to other processes), and hence it may be inapplicable at beaches with rip currents and/or cross-shore directed anthropogenic currents (e.g., prop wash from boats). Another concern is the uncertainty associated with shoreline loading rate and transport parameters, which could be substantial. One advantage of the exact solution derived here is that parameter uncertainty can be propagated, for example through Monte Carlo simulations, to yield uncertainty estimates for nearshore pollutant concentrationsssomething that is difficult to accomplish in the context of numerical simulations of nearshore water quality.

Acknowledgments Research described in this paper was supported by a National Science Foundation Award No. 0724806 to S.B.G., a National Science Foundation award to B.F.S. (No. 0825165), the Santa Ana Regional Water Quality Control Board, and the County of Orange. Special thanks to the Newport Bay Technical Advisory Committee, P. Holden, A. Boehm, R. Guza, and five anonymous reviewers for feedback and guidance, G. Trinidad for help with graphics, and Dr. Jack Skinner for insisting that alongshore currents affect FIB pollution in ankle depth waters.

Supporting Information Available Derivation of the exact solution, methods used for carrying out CFD calculations and calibrating the eddy diffusivity and

velocity fields for the exact solution. This material is available free of charge via the Internet at http://pubs.acs.org.

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