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Modeling of residence time in the East Scott Creek Estuary, South Carolina, USA
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Journal of Hydro-environment Research 2 (2008) 99e108 www.elsevier.com/locate/jher
Modeling of residence time in the East Scott Creek Estuary, South Carolina, USA Feleke Arega*, Straud Armstrong, A.W. Badr South Carolina Department of Natural Resources, Land, Water and Conservation Division, Columbia, SC, USA Received 14 February 2008; revised 3 July 2008; accepted 22 July 2008
Abstract A numerical modeling study was carried out to compute average residence time in a tide-dominated East Scott Creek Estuary, South Carolina. The East Scott Creek estuary is a long system of meandering tidal creeks and salt marsh between Edisto Island and the Edisto Beach barrier island, in South Carolina. A coupled hydrodynamic and solute transport model was developed. The flow and solute transport models were based on depth-integrated conservation equations. The equations were discretized by using the total variation diminishing (TVD) finite-volume method. The numerical model predictions were verified against a set of field-measured hydrodynamic data, with the model-predicted water elevations and velocities in good agreement with the field measurements. A remnant function method has been used to quantify the transport mechanism for a dissolved substance in a spatially varying situation with multiple sources, using a high-resolution mass-preserving hydrodynamic and mass-transport model. The spatially varying average residence times for a tide-dominated environment were investigated through a series of numerical experiments using a passive dissolved and conservative tracer as a surrogate. The result indicated that the average residence time varies with the tidal amplitude. The average residence time for the whole estuary for spring-tide condition was found to be about 22 h. The corresponding average residence time for a tracer placed at the head of the estuary was about 170 h. These findings provide useful information for understanding the transport process in the East Scott Creek Estuary that can be used to assess the impact of coastal development in and around the estuary. Ó 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. Keywords: Hydrodynamics; Residence time; The remnant function; Tracer; Scott Creek
1. Introduction The health of an estuarine ecosystem is governed by its physical, chemical, and biological processes. One physical process that affects the health and water quality of the system and indicates its susceptibility to impairment is the rate of water exchange between the estuary and the open sea. Various time scales, residence time, turnover time, age, and flushing time have been associated with the exchange of water and its
* Corresponding author. South Carolina Department of Natural Resources, Land, Water and Conservation Division, PO Box 167, Columbia, SC 29202, USA. Tel.: þ1 803 734 0073; fax: þ1 803 734 9200. E-mail address: [email protected] (F. Arega).
constituents between an estuary and the open ocean. The determination of these exchange-time scales is particularly important when conducting environmental impact assessments for coastal developments or in restoration of tidal hydrology or coastal wetland restoration and development. Generally, it is important to know the time scale required for a pollutant discharged into a water body to be transported out of the system under normal hydrological conditions and the time elapsed within the estuary since the pollutant entered the system. For example, to assess the impact of manmade closure or disruption of flow to natural tidal waterways on the assimilative or reception capacity of the water body, it is customary to evaluate pre- and post-closure tidal-exchangetime scales. Such comparisons could also assist with developing new remedial solutions. The different time scales
1570-6443/$ - see front matter Ó 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jher.2008.07.003
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associated with the exchange processes depend on the tidal exchange between the system and the open seadwhich is a complicated function of the freshwater runoff, tidal range, topography and bathymetry, density stratification, and wind. The East Scott Creek estuary used to be part of the Scott Creek estuary and is a system of tidal flats, salt marsh, and tidal creeks between Edisto Island and the Edisto Beach barrier island, South Carolina. An earthen causeway for South Carolina Highway 174 divides the Scott Creek Estuary into two nearly equal-length tidal creeks, East Scott Creek that drains east into the Atlantic Ocean through Jeremy Inlet and West Scott Creek that drains west into Big Bay Creek, a tidal tributary near the mouth of the South Edisto River (see Fig. 1). The disruption of the natural flow through Scott Creek, associated with resort development on Edisto Beach, is believed to have caused accelerated sedimentation and other water quality problems. This paper presents an application of a coupled high-resolution hydrodynamic and solute transport model to the East Scott Creek Estuary to compute the distribution of the average residence time of a passive dissolved and conservative matter as part of an ongoing study to restore tidal hydrology in the whole Scott Creek Estuary. The spatially varying average residence time in East Scott Creek Estuary was evaluated through a series of numerical tests. In the following, a brief description of the project site will be given and followed by the numerical-model description, calibration and verification, and definition of residence time and its computation.
2. Site description The East Scott Creek Estuary and its associated tidal flats occupy approximately 1 km2 of inundated area during a spring high-tide condition. The length of the creek from the causeway to the mouth at Jeremy Inlet is about 4.2 km. An overview of the site is illustrated in Fig. 1, along with a contour map of the bed elevation of the creek in Fig. 2. The creek is well mixed and strongly influenced by semidiurnal tides. Average and spring tidal ranges are 1.85 and 2.6 m, respectively. The creek area near the causeway undergoes wetting and drying during normal tides. Nearly the whole flood plain is flooded by tides of amplitude greater than 1 m. The tidal form number was computed as the ratio of the main diurnal and semidiurnal tide component amplitudes, (K1 þ O1)/(M2 þ S2) Defant (1961). It was found to be 0.21 indicating the semidiurnal nature of tides at Edisto Creek. The creek slopes at about 1 * 105 inland from Jeremy Inlet. Salinity measurements, made near the causeway (head of estuary) during tidal flood and ebb, varied by 1e2 psu for ocean salinity of 33.2 psu, which indicates minor freshwater inflow to the East Scott Creek Estuary. 2.1. Residence time scale and its computations The time history of a pollutant once it is released to a given estuary or coastal water body, is important for evaluating its ecological impacts. In estuarine and coastal water bodies, oscillatory tidal mixing processes can interact with channel
Fig. 1. Location map of East Scott Creek.
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
Fig. 2. Bed elevation map of East Scott Creek.
geometry, causing localized variability in divergence and dispersion of pollutants. Hence, it is important to know the time required for the pollutant to leave the system, and this can vary depending on different external forcing functions. More specifically, the time required for it to leave the system through the open boundary (i.e., residence time of the substance) is an important time scale for assessing the environmental impacts of pollution release or any impairment. The residence time is often used as a measure for representing the time scale of the physical process within the estuary and its open boundaries. Reviewing literature, there are several concepts of exchange-time scales, such as age, residence time, transit time and turnover time that have been used to study water exchange and transport between estuaries and their open seas. It seems there are no uniquely agreed definitions for these different time scales or agreed standard methods for their determination (Monsen et al., 2002). In many cases, the underlying concept and computational steps have been based on an idealized circumstance that is constrained by critical assumptions, but the validation of those assumptions has not always been considered when applied to a specific real-world problem (Monsen et al., 2002). Generally, there are several varying approaches that have been used by researchers in estimating the different time scales of transport in estuaries (e.g., Choi and Lee, 2004; Hilton et al., 1998; Yuan et al., 2007). These include use of dye studies, empirical formulas, and the recent use of advanced numerical methods. In this work, we review the fundamental concept of residence time as a transport time scale in coastal environments as laid out or defined in the works of Zimmerman (1976), Dronkers and Zimmerman (1982), Bolin and Rode (1973) and Takeoka (1984) and its application to tidal-dominated environment. Bolin and Rode (1973) summarized previous results and introduced a more rigorous definition for the time scales of ‘age’, ‘transit time’, and ‘turnover time’. Zimmerman (1976) introduced a new ‘residence time’, which was different from that defined by Bolin and Rodhe (1973) and that deals with moving individual particles for a spatially varying situation.
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The residence time for each material element was defined as the time taken for the element to reach the outlet. He applied these concepts to the Dutch Wadden Sea. Following the definition given by Zimmerman (1976), Takeoka (1984) established relationships between the residence time and other time scales and thus complemented the concept of residence time. Zimmerman’s residence time is a complement of the age, which is defined as the remainder of the lifetime of a particle considered (Takeoka, 1984). Takeoka (1984) introduced general properties of residence time in a coastal sea and derived the average residence times for an ideal one-dimensional channel based on solving the advection-diffusion equation. He introduced the so-called remnant function to define the residence time. Consider a water body that exchanges materials with other reservoirs where a certain amount of mass was introduced to a water body and defining the amount of material in a water body at t ¼ 0 as Ro and the amount of the material that still remains in the water body at time t as R(t); then R(t) is the amount of material whose residence time is larger than t. From these definitions the residence time distribution function can then be defined as: 1 dRðtÞ : u¼ Ro dt
ð1Þ
It can further be assumed that as t goes to infinitely large, R(t) approaches zero, then with the average residence time based on the first moment of u(t), tr can be calculated as: ZN tr ¼ tuðtÞdt:
ð2Þ
0
Integrating the above equation gives: ZN ZN RðtÞ tr ¼ dt ¼ cðtÞdt Ro 0
ð3Þ
0
where cðtÞ ¼ RðtÞ=Ro is called the remnant function (Takeoka, 1984). Since the remnant function is defined for an individual parcel of the material considered, it can be directly applied to calculate the residence time for a pollutant that is discharged into a water body at a particular location and time. The remnant function can be obtained by integrating temporal model-predicted pollutant concentration distributions over the model domain. 2.1.1. Upper limit of integration In Eq. (3), theoretically the integration should proceed to the time when the residual mass reaches zero. This may take an infinitely long time and is impractical for many applications. Hence, Eq. (3) needs a proper upper limit of integration. In this application a methodology proposed by Yuan et al. (2007) was used. For each simulation the model was run until the relative error of the cumulative average residence time
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n (tn err) for the whole estuary was less than tcr , where terr is defined as follows: ðnþ1ÞT trnT tr n ¼ ð4Þ terr ðnþ1ÞT tr
where the spatially varying t(x, y) and t0(x, y) can be estimated in similar fashion with and without return-flow, respectively. The value r(x, y) is an estimate of the effect that the tracer return-flow exerts on the local residence time whereas the spatial average of r is an estimate of the return-flow factor term r in the tidal prism method in Eq. (5) for the whole water body.
where T is the period of a tidal cycle. In this study tcr was set to be 0.001.
2.2. Hydrodynamic model
2.1.2. Return-flow factor is a factor that accounts for the fraction of solute that leaves the estuary during ebb and returns to the estuary at rising tide. The return-flow has a significant effect on the increase of the residence time, and it depends on three important factors (Sanford et al., 1992). These are the phase of the tidal flow in the connecting channel relative to the flow along the coast, the amount of mixing that occurs once the water is outside the embayment, and the strength of the inlet flow relative to the strength of the coastal current. There is no simple method to quantify the exact return-flow factor. Here we follow a method that was used by Cucco and Umgiesser (2006) to study the residence time distribution in Venice Lagoon. Recalling the classical tidal-prism method, the average residence time tav of a small and well mixed embayment is given by TVav tav ¼ ð1 rÞP
TVav P
ð5Þ
t0 : ð1 rÞ
ð7Þ
Therefore, it is possible to estimate the return-flow factor by computing the two residence times tav and t0. Eq. (7) is independent of the tidal prism P, the tidal period T, and the basin average volume Vav. In order to estimate the return-flow factor for each forcing scenario, two simulations are carried out. One in which all of the tracer mass that exits the creek is set to zero and therefore no tracer can reenter the creek. This allows the computation of the value of t0. Another simulation where fraction of solute that leaves the estuary during ebb and returns to the estuary at rising tide that allows computation of tav. Since residence time is space-varying, this method enables us to compute not only an overall value for r but also the spatially variable return-flow factor r(x, y), given by rðx; yÞ ¼
U
where U ¼ (h hu hv)T and 0
1 hu 1 F ¼ @ hu2 þ gh2 hTxx A; 2 huv hTxy 1 hv C B G ¼ @ huv hTyx A; 1 hv2 þ gh2 hTyy 2 0
0
ð10Þ
1
vzb tbx C B C B gh vx ro C S¼B B C @ vzb tby A ; gh vy ro
ð6Þ
Combining the two Eqs. (5) and (6) we have tav ¼
U
0
where T is the average tidal period, Vav the basin average volume, P the tidal prism or intertidal volume, and (1 r) is the fraction of new water entering the basin during flooding. The term r is the return-flow factor. Its value varies between 0 and 1. If we define t0 as the residence time for r ¼ 0 for the case with no return-flow, Eq. (5) gives t0 ¼
The depth-integrated continuity and momentum shallowwater equations are the basis of the flow model. Under the assumption that the fluid pressure is hydrostatic in the vertical, these equations can be written for a spatial domain U with a boundary vU in integral form as: Z I Z v UdUþ ðFdy GdxÞ ¼ SdU; ð9Þ vt vU
tðx; yÞ t0 ðx; yÞ tðx; yÞ
where u and v are, respectively, the depth-averaged velocities in the x and y directions. The terms tbx and tby account for bed resistance in the x and y directions, respectively. ro is the fluid density. zb is the bed elevation above a horizontal datum plane, and g is the gravitational constant, and Ti,j (with i, j ¼ x, y) is the depth-averaged turbulence shear stress acting in the idirection on a plane that is perpendicular to the j-direction. In this application a simple eddy viscosity model (Rodi, 1984) is used, and the bed stresses are computed in terms of a drag law. 2.3. Solute transport model Dissolved scalars are modeled by solving depth-averaged transport equations that account for advection, turbulent diffusion, dispersion, and source- and sink-terms. These equations appear as Z I Z v FQ dy GQ dx ¼ SQ dU; QdUþ ð11Þ vt vU U
ð8Þ
where Q ¼ (hc)T and
U
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
FQ ¼huch Exx þExy ; GQ ¼hvch Eyx þEyy ; SQ ¼s;
ð12Þ
where c corresponds to the depth-averaged concentration of the dissolved scalar, s is a general source/sink term. Exx, Exy, Eyx, and Eyy are elements of the dispersion tensor. The dispersion tensor accounts for longitudinal dispersion (Elder, 1959) and transverse mixing (Ward, 1974) and is computed locally, depending on the orientation of the currents (Arega and Sanders, 2004). 3. Numerical method The numerical solution of the flow model was obtained by using a finite- volume scheme that closely follows the development presented by Bradford and Katopodes (1999) and used by Bradford and Sanders (2002) and Arega and Sanders (2004). Key features of the model include the monotone upstream scheme for conservation laws (MUSCL) (Van Leer, 1979) approach for a piecewise linear description of the spatial variability of the solution, Hancock’s predictor-corrector method of time stepping, and Roe’s (1981) method to solve Riemann problems and compute mass and momentum fluxes. Generally, the method involves three steps: the predictor, the flux computation, and the corrector. The scheme is designed on an unstructured curvilinear computational grid consisting of quadrilateral cells. The purpose of the predictor step is to set the stage for flux calculations at the half-time level. This makes the corrector step, second order, accurate in time. In the predictor step, the primitive differential form of the governing equations was solved from the base-time level, forward in time, and at the one-half time level. Gradients of variables in local coordinates are computed by using a Double Minmod slope limiter to preserve the monotonicity of the solution near discontinuities (Sweby, 1984). To obtain second order spatial accuracy, predicted values of h, u, and v to the left and right of each cell face were then linearly reconstructed (based on cell average values and limited gradients) using the MUSCL (Van Leer, 1979). In the flux calculation step, the exchange of mass and momentum were computed for each cell face using the half time (predictor) solution. Fluxes were computed at the midpoint of each cell face and in the direction of the cell face normal vector. The solution, on either side of each cell face, defines a Riemann problem that is solved with Roe’s (1981) Godunov-type upwind scheme. After the fluxes are computed, the corrector step follows by solving the integral form of the governing equations, from the base time level forward one full time level based on the fluxes computed at the half-time level. A detailed description of this method can be found in Bradford and Katopodes (1999) and in Arega and Sanders (2004).
obtain second order spatial accuracy. The advective and diffusive fluxes at each cell face are evaluated separately and then summed at each cell face. The corrector solution is then updated for the next iteration. Generally, the numerical schemes for flow and scalar transport have previously been tested in benchmark problems and have performed well. Bradford and Sanders (2002) presented applications involving a partial dam break problem as well as a tsunami run-up problem. Zhou et al. (2001) presented applications involving transcritical flow over a channel mound and other problems involving hydraulic jumps and bores. In both cases the flow was well resolved without spurious oscillations, and predictions compared well with either exact solutions or measurements. Generally, the scheme conserves fluid mass to numerical precision and preserves monotonicity and stationarity. Gross et al. (1999) made comparative studies of scalar-transport schemes and concluded that TVD schemes are ideal for resolving contact discontinuities without artificially diffusing the front or adding spurious oscillations. The performance of the scheme to model a real-world problem (flow and solute transport) in tidal wetland can be seen in Arega and Sanders (2003, 2004). 4. Model calibration and validation East Scott Creek was discretized into 6450 grid cells. The lateral grid size, within the creek, varies between 1.2 and 2.8 m. The bathymetric data appearing in Fig. 2 were obtained from several sources and then compiled into a digital elevation map (DEM) that serves as the computational grid. The flow equations (continuity and momentum) were forced by a verified Ocean tide just offshore of the creek at Edisto Beach. The hydrodynamic model was applied to characterize circulation and mixing in East Scott Creek. The main parameter of calibration was the Manning coefficient. A constant Manning coefficient of 0.02 was used. The model was numerically integrated with a time step of 0.1 s and was calibrated by using water-surface elevation data collected during Dec 2004 through Jan 2005. Water surface elevations referenced to
3.1. Solute transport Solute transport predictions are made by using the same computational grid, time levels, and predictor-corrector scheme as used for the hydrodynamic predictions. Predicted values of c are linearly reconstructed by using MUSCL to
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Fig. 3. Surface elevations and water current monitoring stations.
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NAVD88 were sampled by using Druck KPS1 5 m pressure transducers at five locations (E1, E2, E3, E4, and E5). Fig. 3 shows the location of the monitoring stations for water-surface elevations and acoustic Doppler current meters used for model calibration and verification. Using tide levels at Edisto Beach, the model was forced for the period of Dec 12, 2004, to January 10, 2005, and comparisons between the measured and predicted water levels were made. Fig. 4 presents the comparisons between the measured and predicted water levels at the monitoring stations E1 and E4. There was strong agreement between the observed and predicted water-surface elevations. Quantitative model performance in surface-water propagation was evaluated by using different statistical tests to compare observed and computed time series data. Root mean square (RMS) values of 7.0 and 8.0 cm were computed for E1 and E4, respectively. Computed mean errors (ME) were 0.3 and 0.00 cm for stations E1 and E4, respectively. The negative root mean errors showed that the model over-predicted the values. Generally, the agreement was satisfactory and the errors were less than or comparable to previously reported values for similar estuarine model studies. In MayeJune 2006, water-surface elevations and water currents were measured at stations AE and S. The model performance was verified with these data. Fig. 5 shows comparisons of model-predicted surface elevations and currents against measured data for stations AE and S (see Fig. 3 for locations), respectively. The left column shows comparisons of respective water-surface elevations, and the right column shows comparisons of the currents. There was strong agreement between the observed and predicted watersurface elevations and currents. Overall, the model was able to capture the tidal propagation reasonably well. The comparisons were sufficiently accurate to justify the use of the model for transport time scales studies.
In dry-weather conditions, there is no noticeable freshwater input to East Scott Creek. Practically, there is no longitudinal salinity gradient between the mouth and the head. For the sake of completeness in calibrating the mass-transport model, we set the model with constant ocean salinity. The model reproduced a constant salinity without any longitudinal gradient, and this is in agreement with the data we have. 5. Residence time computation The calibrated and verified hydrodynamic model was applied to investigate the average residence time scales for East Scott Creek. The average residence time can be used to quantify the transport process and estimate how long it takes for the substance to leave the estuary. First, in East Scott Creek one of the concerns is how long it will take for a dissolved substance (pollutant) discharged into the head of the estuary to leave the system through Jeremy Inlet. The head of East Scott Creek is near the causeway where pollutant discharges could occur owing to intense vehicle traffic. Second, the average residence time of the whole creek is of great interest for different water quality management issues. In order to compute the average residence time at the head of the creek, transport of a passive tracer without decay was simulated. To compute the mean residence time of particles released at the head of the creek, a unit concentration of tracer was released at the center of all grid cells at the head of the estuary at high tide. The model was forced for different tidal amplitudes. Three tidal amplitudes were tested; a harmonic tide of 1.3, 1.0 and 0.5 m amplitudes and a period of 12 h were used. The 1.3 m amplitude is a typical spring-tide condition in the creek. For each tidal run, two simulations were carried out. First, assuming that all materials leaving the creek through Jeremy Inlet during the ebb period will not be returned in the
Fig. 4. Comparisons of model-predicted and observed water surface elevations.
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
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Fig. 5. Comparison of predicted and observed water-surface elevation and current for station S (top row) and station AE (bottom row).
next flooding period. Second, we assumed that the fraction of materials that left the creek during the ebb period could be returned to the creek in the subsequent flood period. We assumed the boundary condition during the flooding period to be equal to the concentration at the boundary cells. In the absence of any observed data on the fraction of materials returned during the flood period, this assumption seems reasonable. Furthermore, this could be an optimum condition for this ebb-dominated shallow system with very small net intertidal storage where the flushing is dominated by internal circulation. In the first scenario, zero-tracer concentration was used at the open boundary in Jeremy Inlet on the assumption that no tracer that left the estuary through the open boundary can return. For each tidal setup, the model was initially run for 4 days to obtain a dynamic-equilibrium condition in the flow field which was used as the initial condition for hot start. In total, six model runs were done that correspond to the three different tidal amplitudes (with and without return). The model was then run until the criterion set in Eq. (4) was satisfied. Similarly, to compute the mean residence time of the whole creek, a unit concentration of tracer was released at the center
of each grid cell (throughout the whole creek domain) at high tide and a total of six model runs were done that correspond to three different tidal amplitudes with and without return. The model was then run until the criteria set in Eq. (4) satisfied. The average residence times were calculated by using Eq. (3). Return-flow factors were computed by using Eq. (8). Table 1 shows computed average residence times and return-flow factors for the tracer released near the headway. Similarly, Table 2 shows computed average residence times and return-flow factors for the whole creek. Fig. 6 shows a plot of cumulative residence time (left panel) and remnant function variations (right panel) for East Scott Creek for masses released throughout the estuary at tidal amplitudes of 0.5 m (top panel), 1.0 m (middle panel) and 1.3 m (bottom panel) with and without return-flow. Similarly, Fig. 7 shows plot of cumulative residence time and remnant function variations for East Scott Creek for masses released at the head of the estuary for various tidal amplitudes with and without return-flow. Return-flow factors of 0.03, 0.02, and 0.02 were computed for materials released at the head of the creek for tidal amplitudes of 0.5, 1.0 and 1.3 m, respectively. On the other hand, return-flow factors of 0.15, 0.07, and 0.10 were computed for the whole creek for tidal amplitudes of 0.5, 1.0
Table 1 Average residence times and return-flow factors for masses released near the headway
Table 2 Average residence times and return-flow factors for the whole creek system
Average residences times in hours
Average residence times in hours
Tide amplitude
tavg
to
r
Tide amplitude
tavg
to
r
h ¼ 0.5 m h ¼ 1.0 m h ¼ 1.3 m
252 190 170
244 187 167
0.03 0.02 0.02
h ¼ 0.5 m h ¼ 1.0 m h ¼ 1.3 m
86 35 22
73 33 20
0.15 0.07 0.10
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
106
Hours
100
1 η=0.5 m
50
0
0.5
0
200
400
600
800
1000
Hours
80
0
0
200
400
600
800
1000
0
200
400
600
800
1000
0
200
1
60
η=1.0 m
40
0.5
20 0
0
200
400
600
800
1000
80
Hours
return no return
0 1
60
η=1.3 m
40
0.5
20 0
0
200
400
600
0
800
Time(Hour)
400
600
800
Time(Hour)
Fig. 6. Computed cumulative average residence time (left panel) and relative residual variations (right panel) for the East Scott Creek Estuary (whole estuary) as a function of tidal amplitude.
and 1.3 m, respectively. Generally, they are small. This is due to the relatively deeper water depths (larger volume), in the expanded area, beyond Jeremy inlet into the ocean, that made the tracer concentration of water leaving the estuary too small. They do indicate spatial and tidal variabilities, however. The return-flow factor variability with tidal amplitudes for pollutant released at the headway is minimal compared to the
1
Hours
300
return no return
η=0.5 m
200
0.5 100 0
0
200
400
600
800 1000 1200 1400
Hours
0
0
200
400
600
800 1000 1200 1400
1
200 η=1.0 m
0.5
100
0
0
200
400
600
800
1000
0
0
200
0
200
400
600
800
1000
400
600
800
1000
1
200
Hours
variability for mass released throughout the whole creek. On the other hand, the variability of return-flow factor with tidal amplitude, for mass released throughout the creek, could partly be explained by the fact that during-spring-tide condition the large volume of water in the creek diluted the mass, and the concentration reaching the boundary cells in ebbing water is somewhat reduced. A maximum return-flow factor of
η=1.3 m 0.5
100
0
0
200
400
600
Time(Hour)
800
1000
0
Time(Hour)
Fig. 7. Computed cumulative average residence time (left panel) and relative residual variations (right panel) for masses released at the head of the East Scott Creek Estuary as a function of tidal amplitude.
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
0.15 for the neap-tide condition for the whole creek was computed. This means that about 15% of the creek water that has been ejected during ebb returns to the creek during the subsequent flood event. A plot of the remnant functions indicated that most of the flushing out of the tracer occurred during the first tidal period, with the remaining mass being mixed with the net non-tidal storage and taking quite long to leave the system. Because the net non-tidal storage is quite small, the impact of ebb and flood tide conditions on the remnant function is not well defined. This is particularly true for the no-return conditions. For return conditions, there is a rise and fall with small amplitude (the rise due to return mass brought in during flooding); however, for the neap tide and the whole creek condition in Fig. 6, a relatively high amplitude of remnant function during the return condition was computed in the first few days. This is due to the relatively high concentration situation in the ebbing water that was used as boundary condition. Generally, in this shallow- and ebbdominated environment, the relative contribution of the returnflow is small. Computations indicate that the residence times for the whole creek are 22, 35 and 86 h for tidal amplitudes of 1.3, 1.0 and 0.5 m, respectively. Similarly, the residence times for a conservative substance released at the head of the creek are 170, 190 and 252 h, respectively. The results showed that the tide range had a significant impact on the residence time of the creek. In the absence of any freshwater from the upstream boundary, pollutants released at or near this headway took a longer period to leave the system. For the tested tidal amplitudes, the oscillatory flooding and ebbing processes caused the mass to be trapped near the headway within a 1.5 km reach for an extended period before reaching a downstream area (1.5 km downstream) where they could have been ejected in one tide cycle. While the mass released at the headway is leaving during the ebb period, the new flood from the ocean pushes it back for several tides cycles before reaching 1.5 km downstream of the headway. That indicates that particles near the headway have very small net tidal excursion. This contributes to higher residence time for masses released at the headway. 5.1. Conclusion A high-resolution depth-integrated two-dimensional hydrodynamic and transport model was developed for the East Scott Creek Estuary. The model was calibrated and verified with measured water-surface levels and currents. Using the verified model, a series of numerical modeling experiments were made to compute mean residence time of the East Scott Creek Estuary. A remnant function method has been employed to quantify the transport mechanism for a dissolved substance in a spatially varying situation with multiple sources. This was accomplished by using a high-resolution mass-preserving hydrodynamic and mass-transport model. The average residence time of a conservative tracer for the whole of the East Scott Creek Estuary is relatively short, 22 h for the spring-tide condition. The average residence time for a conservative tracer
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released at the head of the estuary is quite long, 7 days for the spring-tide condition. The application demonstrates that unlike other simple and grossly simplified methods that compute bulk-exchange-time scales, the use of remnant function with high-resolution model provides the spatial distribution of transport characteristics where both the advective and diffusive processes are taken into account. The importance of the return-flow from the ocean into the creek is shown. The returnflow factors were computed and found to be spatially varying and dependent on tidal amplitude; however they are small. The maximum return-flow factor computed was 0.15, indicating that a maximum 15% of the creek water that has been ejected to the ocean returns to the creek during the next flood event. Generally, the model results suggest that using the conservative-tracer method in a two-dimensional modeling configuration provides a quantitative measure of the mean long-term transport timescale for a tidally-dominated shallow-water estuary. Acknowledgments The authors are thankful to Brenda Hockensmith, Joe Gellici, Andy Wachob and Scott Harder for their valuable help during field surveys. We are thankful to Prof. Richard Styles of The University of South Carolina for his help in data collection and processing. This study in part was supported by U.S. Department of Commerce, National Oceanic and Atmospheric Administration (DOC/NOAA) via a grant (grant # NA03NOS4630167) to David Whitaker of the South Carolina Department of Natural Resources Marine Resources Division. Disclaimer: Any statements, opinions, findings, conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the Federal sponsors, or the South Carolina Department of Natural Resources and no official endorsement should be inferred. References Arega, F., Sanders, B.F., 2003. Modeling circulation and mixing in tidal wetlands of the Santa Ana River. Estuarine and Coastal Modeling 145, 46. Arega, F., Sanders, B.F., 2004. Dispersion model for tidal wetlands, Journal of Hydraulic Engineering. ASCE 130 (8), 739e754. Bolin, B., Rodhe, H., 1973. A note on the concepts of age distribution and transit time in natural reservoirs. Tellus 25, 58e63. Bradford, S.F., Katopodes, N.D., 1999. Hydrodynamics of turbid underflows Part I: formulation and numerical analysis. Journal of Hydraulic Engineering, ASCE 125 (10), 1006e1015. Choi, K.W., Lee, J.H.W., 2004. Numerical determination of flushing time for stratified water bodies. Journal of Marine Systems 50, 263e281. Cucco, A., Umgiesser, G., 2006. Modeling the Venice Lagoon residence time. Ecological Modelling 193, 34e51. Defant, A., 1961. Physical Oceanography, Vol. 2. Pergamon Press. Elder, J.W., 1959. The dispersion of marked fluid in turbulent shear flow. Journal of Fluids Mechanics, Cambridge, England. 5, 544-560. Gross, E.S., Koseff, J.R., Monismith, S.G., 1999. Evaluation of advective schemes for estuarine salinity simulations. Journal of Hydraulic Engineering 125 (1), 32e46. Hilton, A.B.C., McGillivary, D.L., Adams, E.E., 1998. Residence time of freshwater in Boston’s inner harbor, Journal of Waterway, Port. Coastal, and Ocean Engineering 124 (2), 82e89.
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Monsen, N.E., Cloern, J.E., Lucas, L.V., Monismith, S.G., 2002. A comment on the use of flushing time, residence time, and age as transport time scales. Limnology and Oceanography 47 (5), 1545e1553. Roe, P.L., 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43, 357e372. Sanford, L., Boicourt, W., Rives, S., 1992. Model for estimating tidal flushing of small embayments. Journal of Waterway, Port. Coastal, and Ocean Engineering 118 (6), 913e935. Sweby, P.K., 1984. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM (Soc. Ind. Appl. Math.) Journal of Numerical Analysis 21, 995. Takeoka, H., 1984. Fundamental concepts of exchange and transport time scales in a coastal sea. Continental Shelf Research 3 (3), 322e326.
Van Leer, B., 1979. Towards the ultimate conservative difference scheme. V: a second order sequel to Godunov’s Method. Journal of Computational Physics 32, 101e136. Ward, P., 1974. Transverse dispersion in oscillatory channel flow. Journal of the Hydraulics Division 100 (HY6), 755e770. Yuan, D., Lin, B., Falconer, R.A., 2007. A modeling study of residence time in a macro-tidal estuary, Estuarine. Coastal and Shelf Science 71, 401e411. Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M., 2001. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics 168, 1e25. Zimmerman, J.T.F., 1976. Mixing and flushing of tidal embayment in the Western Dutch Wadden Sea, Part I: distribution of salinity and calculation of mixing time scales. Netherlands Journal of Sea Research 10, 149e191.
Journal of Hydro-environment Research 2 (2008) 99e108 www.elsevier.com/locate/jher
Modeling of residence time in the East Scott Creek Estuary, South Carolina, USA Feleke Arega*, Straud Armstrong, A.W. Badr South Carolina Department of Natural Resources, Land, Water and Conservation Division, Columbia, SC, USA Received 14 February 2008; revised 3 July 2008; accepted 22 July 2008
Abstract A numerical modeling study was carried out to compute average residence time in a tide-dominated East Scott Creek Estuary, South Carolina. The East Scott Creek estuary is a long system of meandering tidal creeks and salt marsh between Edisto Island and the Edisto Beach barrier island, in South Carolina. A coupled hydrodynamic and solute transport model was developed. The flow and solute transport models were based on depth-integrated conservation equations. The equations were discretized by using the total variation diminishing (TVD) finite-volume method. The numerical model predictions were verified against a set of field-measured hydrodynamic data, with the model-predicted water elevations and velocities in good agreement with the field measurements. A remnant function method has been used to quantify the transport mechanism for a dissolved substance in a spatially varying situation with multiple sources, using a high-resolution mass-preserving hydrodynamic and mass-transport model. The spatially varying average residence times for a tide-dominated environment were investigated through a series of numerical experiments using a passive dissolved and conservative tracer as a surrogate. The result indicated that the average residence time varies with the tidal amplitude. The average residence time for the whole estuary for spring-tide condition was found to be about 22 h. The corresponding average residence time for a tracer placed at the head of the estuary was about 170 h. These findings provide useful information for understanding the transport process in the East Scott Creek Estuary that can be used to assess the impact of coastal development in and around the estuary. Ó 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. Keywords: Hydrodynamics; Residence time; The remnant function; Tracer; Scott Creek
1. Introduction The health of an estuarine ecosystem is governed by its physical, chemical, and biological processes. One physical process that affects the health and water quality of the system and indicates its susceptibility to impairment is the rate of water exchange between the estuary and the open sea. Various time scales, residence time, turnover time, age, and flushing time have been associated with the exchange of water and its
* Corresponding author. South Carolina Department of Natural Resources, Land, Water and Conservation Division, PO Box 167, Columbia, SC 29202, USA. Tel.: þ1 803 734 0073; fax: þ1 803 734 9200. E-mail address: [email protected] (F. Arega).
constituents between an estuary and the open ocean. The determination of these exchange-time scales is particularly important when conducting environmental impact assessments for coastal developments or in restoration of tidal hydrology or coastal wetland restoration and development. Generally, it is important to know the time scale required for a pollutant discharged into a water body to be transported out of the system under normal hydrological conditions and the time elapsed within the estuary since the pollutant entered the system. For example, to assess the impact of manmade closure or disruption of flow to natural tidal waterways on the assimilative or reception capacity of the water body, it is customary to evaluate pre- and post-closure tidal-exchangetime scales. Such comparisons could also assist with developing new remedial solutions. The different time scales
1570-6443/$ - see front matter Ó 2008 International Association for Hydraulic Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jher.2008.07.003
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associated with the exchange processes depend on the tidal exchange between the system and the open seadwhich is a complicated function of the freshwater runoff, tidal range, topography and bathymetry, density stratification, and wind. The East Scott Creek estuary used to be part of the Scott Creek estuary and is a system of tidal flats, salt marsh, and tidal creeks between Edisto Island and the Edisto Beach barrier island, South Carolina. An earthen causeway for South Carolina Highway 174 divides the Scott Creek Estuary into two nearly equal-length tidal creeks, East Scott Creek that drains east into the Atlantic Ocean through Jeremy Inlet and West Scott Creek that drains west into Big Bay Creek, a tidal tributary near the mouth of the South Edisto River (see Fig. 1). The disruption of the natural flow through Scott Creek, associated with resort development on Edisto Beach, is believed to have caused accelerated sedimentation and other water quality problems. This paper presents an application of a coupled high-resolution hydrodynamic and solute transport model to the East Scott Creek Estuary to compute the distribution of the average residence time of a passive dissolved and conservative matter as part of an ongoing study to restore tidal hydrology in the whole Scott Creek Estuary. The spatially varying average residence time in East Scott Creek Estuary was evaluated through a series of numerical tests. In the following, a brief description of the project site will be given and followed by the numerical-model description, calibration and verification, and definition of residence time and its computation.
2. Site description The East Scott Creek Estuary and its associated tidal flats occupy approximately 1 km2 of inundated area during a spring high-tide condition. The length of the creek from the causeway to the mouth at Jeremy Inlet is about 4.2 km. An overview of the site is illustrated in Fig. 1, along with a contour map of the bed elevation of the creek in Fig. 2. The creek is well mixed and strongly influenced by semidiurnal tides. Average and spring tidal ranges are 1.85 and 2.6 m, respectively. The creek area near the causeway undergoes wetting and drying during normal tides. Nearly the whole flood plain is flooded by tides of amplitude greater than 1 m. The tidal form number was computed as the ratio of the main diurnal and semidiurnal tide component amplitudes, (K1 þ O1)/(M2 þ S2) Defant (1961). It was found to be 0.21 indicating the semidiurnal nature of tides at Edisto Creek. The creek slopes at about 1 * 105 inland from Jeremy Inlet. Salinity measurements, made near the causeway (head of estuary) during tidal flood and ebb, varied by 1e2 psu for ocean salinity of 33.2 psu, which indicates minor freshwater inflow to the East Scott Creek Estuary. 2.1. Residence time scale and its computations The time history of a pollutant once it is released to a given estuary or coastal water body, is important for evaluating its ecological impacts. In estuarine and coastal water bodies, oscillatory tidal mixing processes can interact with channel
Fig. 1. Location map of East Scott Creek.
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
Fig. 2. Bed elevation map of East Scott Creek.
geometry, causing localized variability in divergence and dispersion of pollutants. Hence, it is important to know the time required for the pollutant to leave the system, and this can vary depending on different external forcing functions. More specifically, the time required for it to leave the system through the open boundary (i.e., residence time of the substance) is an important time scale for assessing the environmental impacts of pollution release or any impairment. The residence time is often used as a measure for representing the time scale of the physical process within the estuary and its open boundaries. Reviewing literature, there are several concepts of exchange-time scales, such as age, residence time, transit time and turnover time that have been used to study water exchange and transport between estuaries and their open seas. It seems there are no uniquely agreed definitions for these different time scales or agreed standard methods for their determination (Monsen et al., 2002). In many cases, the underlying concept and computational steps have been based on an idealized circumstance that is constrained by critical assumptions, but the validation of those assumptions has not always been considered when applied to a specific real-world problem (Monsen et al., 2002). Generally, there are several varying approaches that have been used by researchers in estimating the different time scales of transport in estuaries (e.g., Choi and Lee, 2004; Hilton et al., 1998; Yuan et al., 2007). These include use of dye studies, empirical formulas, and the recent use of advanced numerical methods. In this work, we review the fundamental concept of residence time as a transport time scale in coastal environments as laid out or defined in the works of Zimmerman (1976), Dronkers and Zimmerman (1982), Bolin and Rode (1973) and Takeoka (1984) and its application to tidal-dominated environment. Bolin and Rode (1973) summarized previous results and introduced a more rigorous definition for the time scales of ‘age’, ‘transit time’, and ‘turnover time’. Zimmerman (1976) introduced a new ‘residence time’, which was different from that defined by Bolin and Rodhe (1973) and that deals with moving individual particles for a spatially varying situation.
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The residence time for each material element was defined as the time taken for the element to reach the outlet. He applied these concepts to the Dutch Wadden Sea. Following the definition given by Zimmerman (1976), Takeoka (1984) established relationships between the residence time and other time scales and thus complemented the concept of residence time. Zimmerman’s residence time is a complement of the age, which is defined as the remainder of the lifetime of a particle considered (Takeoka, 1984). Takeoka (1984) introduced general properties of residence time in a coastal sea and derived the average residence times for an ideal one-dimensional channel based on solving the advection-diffusion equation. He introduced the so-called remnant function to define the residence time. Consider a water body that exchanges materials with other reservoirs where a certain amount of mass was introduced to a water body and defining the amount of material in a water body at t ¼ 0 as Ro and the amount of the material that still remains in the water body at time t as R(t); then R(t) is the amount of material whose residence time is larger than t. From these definitions the residence time distribution function can then be defined as: 1 dRðtÞ : u¼ Ro dt
ð1Þ
It can further be assumed that as t goes to infinitely large, R(t) approaches zero, then with the average residence time based on the first moment of u(t), tr can be calculated as: ZN tr ¼ tuðtÞdt:
ð2Þ
0
Integrating the above equation gives: ZN ZN RðtÞ tr ¼ dt ¼ cðtÞdt Ro 0
ð3Þ
0
where cðtÞ ¼ RðtÞ=Ro is called the remnant function (Takeoka, 1984). Since the remnant function is defined for an individual parcel of the material considered, it can be directly applied to calculate the residence time for a pollutant that is discharged into a water body at a particular location and time. The remnant function can be obtained by integrating temporal model-predicted pollutant concentration distributions over the model domain. 2.1.1. Upper limit of integration In Eq. (3), theoretically the integration should proceed to the time when the residual mass reaches zero. This may take an infinitely long time and is impractical for many applications. Hence, Eq. (3) needs a proper upper limit of integration. In this application a methodology proposed by Yuan et al. (2007) was used. For each simulation the model was run until the relative error of the cumulative average residence time
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n (tn err) for the whole estuary was less than tcr , where terr is defined as follows: ðnþ1ÞT trnT tr n ¼ ð4Þ terr ðnþ1ÞT tr
where the spatially varying t(x, y) and t0(x, y) can be estimated in similar fashion with and without return-flow, respectively. The value r(x, y) is an estimate of the effect that the tracer return-flow exerts on the local residence time whereas the spatial average of r is an estimate of the return-flow factor term r in the tidal prism method in Eq. (5) for the whole water body.
where T is the period of a tidal cycle. In this study tcr was set to be 0.001.
2.2. Hydrodynamic model
2.1.2. Return-flow factor is a factor that accounts for the fraction of solute that leaves the estuary during ebb and returns to the estuary at rising tide. The return-flow has a significant effect on the increase of the residence time, and it depends on three important factors (Sanford et al., 1992). These are the phase of the tidal flow in the connecting channel relative to the flow along the coast, the amount of mixing that occurs once the water is outside the embayment, and the strength of the inlet flow relative to the strength of the coastal current. There is no simple method to quantify the exact return-flow factor. Here we follow a method that was used by Cucco and Umgiesser (2006) to study the residence time distribution in Venice Lagoon. Recalling the classical tidal-prism method, the average residence time tav of a small and well mixed embayment is given by TVav tav ¼ ð1 rÞP
TVav P
ð5Þ
t0 : ð1 rÞ
ð7Þ
Therefore, it is possible to estimate the return-flow factor by computing the two residence times tav and t0. Eq. (7) is independent of the tidal prism P, the tidal period T, and the basin average volume Vav. In order to estimate the return-flow factor for each forcing scenario, two simulations are carried out. One in which all of the tracer mass that exits the creek is set to zero and therefore no tracer can reenter the creek. This allows the computation of the value of t0. Another simulation where fraction of solute that leaves the estuary during ebb and returns to the estuary at rising tide that allows computation of tav. Since residence time is space-varying, this method enables us to compute not only an overall value for r but also the spatially variable return-flow factor r(x, y), given by rðx; yÞ ¼
U
where U ¼ (h hu hv)T and 0
1 hu 1 F ¼ @ hu2 þ gh2 hTxx A; 2 huv hTxy 1 hv C B G ¼ @ huv hTyx A; 1 hv2 þ gh2 hTyy 2 0
0
ð10Þ
1
vzb tbx C B C B gh vx ro C S¼B B C @ vzb tby A ; gh vy ro
ð6Þ
Combining the two Eqs. (5) and (6) we have tav ¼
U
0
where T is the average tidal period, Vav the basin average volume, P the tidal prism or intertidal volume, and (1 r) is the fraction of new water entering the basin during flooding. The term r is the return-flow factor. Its value varies between 0 and 1. If we define t0 as the residence time for r ¼ 0 for the case with no return-flow, Eq. (5) gives t0 ¼
The depth-integrated continuity and momentum shallowwater equations are the basis of the flow model. Under the assumption that the fluid pressure is hydrostatic in the vertical, these equations can be written for a spatial domain U with a boundary vU in integral form as: Z I Z v UdUþ ðFdy GdxÞ ¼ SdU; ð9Þ vt vU
tðx; yÞ t0 ðx; yÞ tðx; yÞ
where u and v are, respectively, the depth-averaged velocities in the x and y directions. The terms tbx and tby account for bed resistance in the x and y directions, respectively. ro is the fluid density. zb is the bed elevation above a horizontal datum plane, and g is the gravitational constant, and Ti,j (with i, j ¼ x, y) is the depth-averaged turbulence shear stress acting in the idirection on a plane that is perpendicular to the j-direction. In this application a simple eddy viscosity model (Rodi, 1984) is used, and the bed stresses are computed in terms of a drag law. 2.3. Solute transport model Dissolved scalars are modeled by solving depth-averaged transport equations that account for advection, turbulent diffusion, dispersion, and source- and sink-terms. These equations appear as Z I Z v FQ dy GQ dx ¼ SQ dU; QdUþ ð11Þ vt vU U
ð8Þ
where Q ¼ (hc)T and
U
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FQ ¼huch Exx þExy ; GQ ¼hvch Eyx þEyy ; SQ ¼s;
ð12Þ
where c corresponds to the depth-averaged concentration of the dissolved scalar, s is a general source/sink term. Exx, Exy, Eyx, and Eyy are elements of the dispersion tensor. The dispersion tensor accounts for longitudinal dispersion (Elder, 1959) and transverse mixing (Ward, 1974) and is computed locally, depending on the orientation of the currents (Arega and Sanders, 2004). 3. Numerical method The numerical solution of the flow model was obtained by using a finite- volume scheme that closely follows the development presented by Bradford and Katopodes (1999) and used by Bradford and Sanders (2002) and Arega and Sanders (2004). Key features of the model include the monotone upstream scheme for conservation laws (MUSCL) (Van Leer, 1979) approach for a piecewise linear description of the spatial variability of the solution, Hancock’s predictor-corrector method of time stepping, and Roe’s (1981) method to solve Riemann problems and compute mass and momentum fluxes. Generally, the method involves three steps: the predictor, the flux computation, and the corrector. The scheme is designed on an unstructured curvilinear computational grid consisting of quadrilateral cells. The purpose of the predictor step is to set the stage for flux calculations at the half-time level. This makes the corrector step, second order, accurate in time. In the predictor step, the primitive differential form of the governing equations was solved from the base-time level, forward in time, and at the one-half time level. Gradients of variables in local coordinates are computed by using a Double Minmod slope limiter to preserve the monotonicity of the solution near discontinuities (Sweby, 1984). To obtain second order spatial accuracy, predicted values of h, u, and v to the left and right of each cell face were then linearly reconstructed (based on cell average values and limited gradients) using the MUSCL (Van Leer, 1979). In the flux calculation step, the exchange of mass and momentum were computed for each cell face using the half time (predictor) solution. Fluxes were computed at the midpoint of each cell face and in the direction of the cell face normal vector. The solution, on either side of each cell face, defines a Riemann problem that is solved with Roe’s (1981) Godunov-type upwind scheme. After the fluxes are computed, the corrector step follows by solving the integral form of the governing equations, from the base time level forward one full time level based on the fluxes computed at the half-time level. A detailed description of this method can be found in Bradford and Katopodes (1999) and in Arega and Sanders (2004).
obtain second order spatial accuracy. The advective and diffusive fluxes at each cell face are evaluated separately and then summed at each cell face. The corrector solution is then updated for the next iteration. Generally, the numerical schemes for flow and scalar transport have previously been tested in benchmark problems and have performed well. Bradford and Sanders (2002) presented applications involving a partial dam break problem as well as a tsunami run-up problem. Zhou et al. (2001) presented applications involving transcritical flow over a channel mound and other problems involving hydraulic jumps and bores. In both cases the flow was well resolved without spurious oscillations, and predictions compared well with either exact solutions or measurements. Generally, the scheme conserves fluid mass to numerical precision and preserves monotonicity and stationarity. Gross et al. (1999) made comparative studies of scalar-transport schemes and concluded that TVD schemes are ideal for resolving contact discontinuities without artificially diffusing the front or adding spurious oscillations. The performance of the scheme to model a real-world problem (flow and solute transport) in tidal wetland can be seen in Arega and Sanders (2003, 2004). 4. Model calibration and validation East Scott Creek was discretized into 6450 grid cells. The lateral grid size, within the creek, varies between 1.2 and 2.8 m. The bathymetric data appearing in Fig. 2 were obtained from several sources and then compiled into a digital elevation map (DEM) that serves as the computational grid. The flow equations (continuity and momentum) were forced by a verified Ocean tide just offshore of the creek at Edisto Beach. The hydrodynamic model was applied to characterize circulation and mixing in East Scott Creek. The main parameter of calibration was the Manning coefficient. A constant Manning coefficient of 0.02 was used. The model was numerically integrated with a time step of 0.1 s and was calibrated by using water-surface elevation data collected during Dec 2004 through Jan 2005. Water surface elevations referenced to
3.1. Solute transport Solute transport predictions are made by using the same computational grid, time levels, and predictor-corrector scheme as used for the hydrodynamic predictions. Predicted values of c are linearly reconstructed by using MUSCL to
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Fig. 3. Surface elevations and water current monitoring stations.
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NAVD88 were sampled by using Druck KPS1 5 m pressure transducers at five locations (E1, E2, E3, E4, and E5). Fig. 3 shows the location of the monitoring stations for water-surface elevations and acoustic Doppler current meters used for model calibration and verification. Using tide levels at Edisto Beach, the model was forced for the period of Dec 12, 2004, to January 10, 2005, and comparisons between the measured and predicted water levels were made. Fig. 4 presents the comparisons between the measured and predicted water levels at the monitoring stations E1 and E4. There was strong agreement between the observed and predicted water-surface elevations. Quantitative model performance in surface-water propagation was evaluated by using different statistical tests to compare observed and computed time series data. Root mean square (RMS) values of 7.0 and 8.0 cm were computed for E1 and E4, respectively. Computed mean errors (ME) were 0.3 and 0.00 cm for stations E1 and E4, respectively. The negative root mean errors showed that the model over-predicted the values. Generally, the agreement was satisfactory and the errors were less than or comparable to previously reported values for similar estuarine model studies. In MayeJune 2006, water-surface elevations and water currents were measured at stations AE and S. The model performance was verified with these data. Fig. 5 shows comparisons of model-predicted surface elevations and currents against measured data for stations AE and S (see Fig. 3 for locations), respectively. The left column shows comparisons of respective water-surface elevations, and the right column shows comparisons of the currents. There was strong agreement between the observed and predicted watersurface elevations and currents. Overall, the model was able to capture the tidal propagation reasonably well. The comparisons were sufficiently accurate to justify the use of the model for transport time scales studies.
In dry-weather conditions, there is no noticeable freshwater input to East Scott Creek. Practically, there is no longitudinal salinity gradient between the mouth and the head. For the sake of completeness in calibrating the mass-transport model, we set the model with constant ocean salinity. The model reproduced a constant salinity without any longitudinal gradient, and this is in agreement with the data we have. 5. Residence time computation The calibrated and verified hydrodynamic model was applied to investigate the average residence time scales for East Scott Creek. The average residence time can be used to quantify the transport process and estimate how long it takes for the substance to leave the estuary. First, in East Scott Creek one of the concerns is how long it will take for a dissolved substance (pollutant) discharged into the head of the estuary to leave the system through Jeremy Inlet. The head of East Scott Creek is near the causeway where pollutant discharges could occur owing to intense vehicle traffic. Second, the average residence time of the whole creek is of great interest for different water quality management issues. In order to compute the average residence time at the head of the creek, transport of a passive tracer without decay was simulated. To compute the mean residence time of particles released at the head of the creek, a unit concentration of tracer was released at the center of all grid cells at the head of the estuary at high tide. The model was forced for different tidal amplitudes. Three tidal amplitudes were tested; a harmonic tide of 1.3, 1.0 and 0.5 m amplitudes and a period of 12 h were used. The 1.3 m amplitude is a typical spring-tide condition in the creek. For each tidal run, two simulations were carried out. First, assuming that all materials leaving the creek through Jeremy Inlet during the ebb period will not be returned in the
Fig. 4. Comparisons of model-predicted and observed water surface elevations.
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Fig. 5. Comparison of predicted and observed water-surface elevation and current for station S (top row) and station AE (bottom row).
next flooding period. Second, we assumed that the fraction of materials that left the creek during the ebb period could be returned to the creek in the subsequent flood period. We assumed the boundary condition during the flooding period to be equal to the concentration at the boundary cells. In the absence of any observed data on the fraction of materials returned during the flood period, this assumption seems reasonable. Furthermore, this could be an optimum condition for this ebb-dominated shallow system with very small net intertidal storage where the flushing is dominated by internal circulation. In the first scenario, zero-tracer concentration was used at the open boundary in Jeremy Inlet on the assumption that no tracer that left the estuary through the open boundary can return. For each tidal setup, the model was initially run for 4 days to obtain a dynamic-equilibrium condition in the flow field which was used as the initial condition for hot start. In total, six model runs were done that correspond to the three different tidal amplitudes (with and without return). The model was then run until the criterion set in Eq. (4) was satisfied. Similarly, to compute the mean residence time of the whole creek, a unit concentration of tracer was released at the center
of each grid cell (throughout the whole creek domain) at high tide and a total of six model runs were done that correspond to three different tidal amplitudes with and without return. The model was then run until the criteria set in Eq. (4) satisfied. The average residence times were calculated by using Eq. (3). Return-flow factors were computed by using Eq. (8). Table 1 shows computed average residence times and return-flow factors for the tracer released near the headway. Similarly, Table 2 shows computed average residence times and return-flow factors for the whole creek. Fig. 6 shows a plot of cumulative residence time (left panel) and remnant function variations (right panel) for East Scott Creek for masses released throughout the estuary at tidal amplitudes of 0.5 m (top panel), 1.0 m (middle panel) and 1.3 m (bottom panel) with and without return-flow. Similarly, Fig. 7 shows plot of cumulative residence time and remnant function variations for East Scott Creek for masses released at the head of the estuary for various tidal amplitudes with and without return-flow. Return-flow factors of 0.03, 0.02, and 0.02 were computed for materials released at the head of the creek for tidal amplitudes of 0.5, 1.0 and 1.3 m, respectively. On the other hand, return-flow factors of 0.15, 0.07, and 0.10 were computed for the whole creek for tidal amplitudes of 0.5, 1.0
Table 1 Average residence times and return-flow factors for masses released near the headway
Table 2 Average residence times and return-flow factors for the whole creek system
Average residences times in hours
Average residence times in hours
Tide amplitude
tavg
to
r
Tide amplitude
tavg
to
r
h ¼ 0.5 m h ¼ 1.0 m h ¼ 1.3 m
252 190 170
244 187 167
0.03 0.02 0.02
h ¼ 0.5 m h ¼ 1.0 m h ¼ 1.3 m
86 35 22
73 33 20
0.15 0.07 0.10
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
106
Hours
100
1 η=0.5 m
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0
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80
0
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40
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20 0
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0 1
60
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40
0.5
20 0
0
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0
800
Time(Hour)
400
600
800
Time(Hour)
Fig. 6. Computed cumulative average residence time (left panel) and relative residual variations (right panel) for the East Scott Creek Estuary (whole estuary) as a function of tidal amplitude.
and 1.3 m, respectively. Generally, they are small. This is due to the relatively deeper water depths (larger volume), in the expanded area, beyond Jeremy inlet into the ocean, that made the tracer concentration of water leaving the estuary too small. They do indicate spatial and tidal variabilities, however. The return-flow factor variability with tidal amplitudes for pollutant released at the headway is minimal compared to the
1
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300
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0
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Hours
variability for mass released throughout the whole creek. On the other hand, the variability of return-flow factor with tidal amplitude, for mass released throughout the creek, could partly be explained by the fact that during-spring-tide condition the large volume of water in the creek diluted the mass, and the concentration reaching the boundary cells in ebbing water is somewhat reduced. A maximum return-flow factor of
η=1.3 m 0.5
100
0
0
200
400
600
Time(Hour)
800
1000
0
Time(Hour)
Fig. 7. Computed cumulative average residence time (left panel) and relative residual variations (right panel) for masses released at the head of the East Scott Creek Estuary as a function of tidal amplitude.
F. Arega et al. / Journal of Hydro-environment Research 2 (2008) 99e108
0.15 for the neap-tide condition for the whole creek was computed. This means that about 15% of the creek water that has been ejected during ebb returns to the creek during the subsequent flood event. A plot of the remnant functions indicated that most of the flushing out of the tracer occurred during the first tidal period, with the remaining mass being mixed with the net non-tidal storage and taking quite long to leave the system. Because the net non-tidal storage is quite small, the impact of ebb and flood tide conditions on the remnant function is not well defined. This is particularly true for the no-return conditions. For return conditions, there is a rise and fall with small amplitude (the rise due to return mass brought in during flooding); however, for the neap tide and the whole creek condition in Fig. 6, a relatively high amplitude of remnant function during the return condition was computed in the first few days. This is due to the relatively high concentration situation in the ebbing water that was used as boundary condition. Generally, in this shallow- and ebbdominated environment, the relative contribution of the returnflow is small. Computations indicate that the residence times for the whole creek are 22, 35 and 86 h for tidal amplitudes of 1.3, 1.0 and 0.5 m, respectively. Similarly, the residence times for a conservative substance released at the head of the creek are 170, 190 and 252 h, respectively. The results showed that the tide range had a significant impact on the residence time of the creek. In the absence of any freshwater from the upstream boundary, pollutants released at or near this headway took a longer period to leave the system. For the tested tidal amplitudes, the oscillatory flooding and ebbing processes caused the mass to be trapped near the headway within a 1.5 km reach for an extended period before reaching a downstream area (1.5 km downstream) where they could have been ejected in one tide cycle. While the mass released at the headway is leaving during the ebb period, the new flood from the ocean pushes it back for several tides cycles before reaching 1.5 km downstream of the headway. That indicates that particles near the headway have very small net tidal excursion. This contributes to higher residence time for masses released at the headway. 5.1. Conclusion A high-resolution depth-integrated two-dimensional hydrodynamic and transport model was developed for the East Scott Creek Estuary. The model was calibrated and verified with measured water-surface levels and currents. Using the verified model, a series of numerical modeling experiments were made to compute mean residence time of the East Scott Creek Estuary. A remnant function method has been employed to quantify the transport mechanism for a dissolved substance in a spatially varying situation with multiple sources. This was accomplished by using a high-resolution mass-preserving hydrodynamic and mass-transport model. The average residence time of a conservative tracer for the whole of the East Scott Creek Estuary is relatively short, 22 h for the spring-tide condition. The average residence time for a conservative tracer
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released at the head of the estuary is quite long, 7 days for the spring-tide condition. The application demonstrates that unlike other simple and grossly simplified methods that compute bulk-exchange-time scales, the use of remnant function with high-resolution model provides the spatial distribution of transport characteristics where both the advective and diffusive processes are taken into account. The importance of the return-flow from the ocean into the creek is shown. The returnflow factors were computed and found to be spatially varying and dependent on tidal amplitude; however they are small. The maximum return-flow factor computed was 0.15, indicating that a maximum 15% of the creek water that has been ejected to the ocean returns to the creek during the next flood event. Generally, the model results suggest that using the conservative-tracer method in a two-dimensional modeling configuration provides a quantitative measure of the mean long-term transport timescale for a tidally-dominated shallow-water estuary. Acknowledgments The authors are thankful to Brenda Hockensmith, Joe Gellici, Andy Wachob and Scott Harder for their valuable help during field surveys. We are thankful to Prof. Richard Styles of The University of South Carolina for his help in data collection and processing. This study in part was supported by U.S. Department of Commerce, National Oceanic and Atmospheric Administration (DOC/NOAA) via a grant (grant # NA03NOS4630167) to David Whitaker of the South Carolina Department of Natural Resources Marine Resources Division. Disclaimer: Any statements, opinions, findings, conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the Federal sponsors, or the South Carolina Department of Natural Resources and no official endorsement should be inferred. References Arega, F., Sanders, B.F., 2003. Modeling circulation and mixing in tidal wetlands of the Santa Ana River. Estuarine and Coastal Modeling 145, 46. Arega, F., Sanders, B.F., 2004. Dispersion model for tidal wetlands, Journal of Hydraulic Engineering. ASCE 130 (8), 739e754. Bolin, B., Rodhe, H., 1973. A note on the concepts of age distribution and transit time in natural reservoirs. Tellus 25, 58e63. Bradford, S.F., Katopodes, N.D., 1999. Hydrodynamics of turbid underflows Part I: formulation and numerical analysis. Journal of Hydraulic Engineering, ASCE 125 (10), 1006e1015. Choi, K.W., Lee, J.H.W., 2004. Numerical determination of flushing time for stratified water bodies. Journal of Marine Systems 50, 263e281. Cucco, A., Umgiesser, G., 2006. Modeling the Venice Lagoon residence time. Ecological Modelling 193, 34e51. Defant, A., 1961. Physical Oceanography, Vol. 2. Pergamon Press. Elder, J.W., 1959. The dispersion of marked fluid in turbulent shear flow. Journal of Fluids Mechanics, Cambridge, England. 5, 544-560. Gross, E.S., Koseff, J.R., Monismith, S.G., 1999. Evaluation of advective schemes for estuarine salinity simulations. Journal of Hydraulic Engineering 125 (1), 32e46. Hilton, A.B.C., McGillivary, D.L., Adams, E.E., 1998. Residence time of freshwater in Boston’s inner harbor, Journal of Waterway, Port. Coastal, and Ocean Engineering 124 (2), 82e89.
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Monsen, N.E., Cloern, J.E., Lucas, L.V., Monismith, S.G., 2002. A comment on the use of flushing time, residence time, and age as transport time scales. Limnology and Oceanography 47 (5), 1545e1553. Roe, P.L., 1981. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43, 357e372. Sanford, L., Boicourt, W., Rives, S., 1992. Model for estimating tidal flushing of small embayments. Journal of Waterway, Port. Coastal, and Ocean Engineering 118 (6), 913e935. Sweby, P.K., 1984. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM (Soc. Ind. Appl. Math.) Journal of Numerical Analysis 21, 995. Takeoka, H., 1984. Fundamental concepts of exchange and transport time scales in a coastal sea. Continental Shelf Research 3 (3), 322e326.
Van Leer, B., 1979. Towards the ultimate conservative difference scheme. V: a second order sequel to Godunov’s Method. Journal of Computational Physics 32, 101e136. Ward, P., 1974. Transverse dispersion in oscillatory channel flow. Journal of the Hydraulics Division 100 (HY6), 755e770. Yuan, D., Lin, B., Falconer, R.A., 2007. A modeling study of residence time in a macro-tidal estuary, Estuarine. Coastal and Shelf Science 71, 401e411. Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M., 2001. The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics 168, 1e25. Zimmerman, J.T.F., 1976. Mixing and flushing of tidal embayment in the Western Dutch Wadden Sea, Part I: distribution of salinity and calculation of mixing time scales. Netherlands Journal of Sea Research 10, 149e191.