Solute dispersion in the coastal boundary layer of southern Lake Michigan

Authors

  • Pramod Thupaki,

    1. Department of Civil and Environmental Engineering, Michigan State University, East Lansing, Michigan, USA
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  • Mantha S. Phanikumar,

    Corresponding author
    1. Department of Civil and Environmental Engineering, Michigan State University, East Lansing, Michigan, USA
    • Corresponding author: M. S. Phanikumar, Department of Civil and Environmental Engineering, 1449 Engineering Research Court, Room A130, Michigan State University, East Lansing, MI 48824, USA. (phani@msu.edu)

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  • Richard L. Whitman

    1. United States Geological Survey, Great Lakes Science Center, Porter, Indiana, USA
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Abstract

[1] We evaluate a three-dimensional, nested-grid nearshore model of Lake Michigan for its ability to describe key aspects of hydrodynamics and solute transport using data from a field study conducted in summer 2008. Velocity comparisons with observations from five bottom-mounted ADCPs at different depths in the coastal boundary layer (CBL) show that the numerical model was able to simulate currents and flow reversals accurately within the inertial boundary layer, however model accuracy reduced close to the shoreline. Power spectra of observed and simulated velocity time series at different locations showed that the hydrodynamic model was able to describe the energy contained in the inertial scales, but over-predicted turbulent dissipation rates. As a result, model-predicted values of energy in the smaller, dissipation scales were lower compared to values calculated from observations in the CBL. Differences between energy contained in the observed and simulated velocity spectra increased as the shoreline is approached. Observations showed that vertical variations in the alongshore and cross-shore velocities were dominated by inertial waves. Inaccuracies in representing energy dissipation rates and processes not explicitly described in the hydrodynamic model (e.g., anisotropy, waves) could potentially contribute to errors in describing transport in the CBL. Measurements from a continuous dye release experiment from a riverine outfall were described using a nearshore model with a mean horizontal dispersion coefficient of 5.6 m2/s. Improved representations of physical processes (such as turbulence, internal waves and wave-current interactions) can be expected to provide better descriptions of solute transport in the CBL.

1 Introduction

[2] Addressing science questions aimed at effective management of coastal resources often calls for the ability to accurately describe the dispersion of material in the nearshore region. Questions involving beach closures and the fate and transport of pathogenic microorganisms such as bacteria, viruses and protozoan parasites are intimately linked to solute dispersion. Recreational beaches in southern Lake Michigan are impacted by microbial contamination entering the nearshore environment via discharges from riverine outfalls in addition to contributions from non-point sources. One of the major sources of contamination at river sites is discharge from combined sewer overflows (CSOs). Beaches are closed to the public whenever levels of fecal indicator bacteria (FIB) such as Escherichia coli exceed local standards, with significant economic and health-risk tradeoffs associated with beach closures [Rabinovici et al., 2004; Dorfman and Rosselot, 2010]. Traditional methods of beach management are observation-based and require 24 hours to run the assays during which time conditions at the beaches can change. Hence, well-tested hydrodynamic and transport models, with the potential to make short-term forecasts, are attractive tools for beach management.

[3] A number of biotic and abiotic factors influence the transport, inactivation (loss per unit time) and survival of FIB in the nearshore environment. Key processes include bacterial inactivation due to sunlight (direct DNA damage) with turbid waters contributing to increased survival by limiting sunlight penetration, removal of bacteria from the water column due to sedimentation after attachment to suspended particles, increased loss of bacteria at elevated temperatures (due to damage of bacterial cell components as well as increased predation risk), and dependence on pH, nutrient content and dissolved oxygen, among other factors [Hipsey et al., 2008]. Although the relative importance of these factors can change depending on the geographical setting, beach orientation and beach type (open versus embayed), nearshore hydrodynamic and wave processes play a key role in controlling overall FIB fate and transport [Ge et al., 2012a, 2012b]. A budget analysis including physical and biological processes affecting FIB fate and transport indicated that dilution of FIB due to mixing is a key process that dominates overall transport [Thupaki et al., 2010]. Since peak concentrations of bacteria are determined by the combined action of dilution and other loss processes, dilution rates should be accurately represented to improve model performance. Therefore, the ability to describe transport of a conservative tracer accurately is a necessary first step to modeling bacterial fate and transport.

[4] Small-scale features dominate the mixing and transport of riverine plumes in the nearshore coastal boundary layer. Analysis of river plume mixing dynamics shows that the spatial domain can be separated into: (a) the near-field region where mixing is dominated by buoyancy and momentum of the plume and (b) the far-field region where plume dynamics is primarily controlled by turbulent diffusion. Based on field observations of the Grand River plume in Lake Michigan, [Nekouee, 2010] classified surface buoyant plumes into two major categories - shore-attached plumes, which occur when along-shore currents are strong, and unattached plumes. Unattached plumes were further classified into five sub-categories based on a Richardson number (Fischer et al., 1979) – radial spreading, offshore spreading, side deflecting, diffuse offshore spreading and diffuse shore impacting plumes.

[5] A detailed classification of the coastal boundary layer following [Boyce, 1974; Rao and Schwab, 2007] divides the coastal zone into: (1) a nearshore region that includes the surf and swash zones (2) a frictional boundary layer (FBL) where bottom and lateral friction effects dominate (3) an inertial boundary layer (IBL) where large-scale oscillations due to inertial effects adjust to the presence of the shoreline and (4) the offshore region that represents the open lake environment. Together the IBL and the FBL are referred to as the coastal boundary layer (CBL) [Csanady, 1972] and represent the region where flow adjusts to the lateral boundary and transitions from the inertial- to the frictional-scale features. Using current measurements [Murthy and Dunbar, 1981] found that the width of the frictional boundary layer (defined as the distance from the shore to the point of peak kinetic energy) is about 2 km and width of the inertial boundary layer (defined as distance to the point where inertial scales begin to dominate flow) is about 9 km in Lake Huron.

[6] Several studies have investigated the dynamics of river plumes. [Jones et al., 2007] examined mixing of buoyant surface plumes with ambient waters in idealized flows. Using a combination of dye-studies, ADCP measurements, and surface drifters, mixing due to turbulent diffusion was calculated for a number of different sites. A detailed review of methods used to calculate turbulence in the nearshore is provided by [Burchard et al., 2008]. [Spydell et al., 2009; Brown et al., 2009], and [Alosairi et al., 2011] have examined the effect of shear due to wave-generated alongshore and rip currents, on dispersion in the nearshore. Breaking waves in the surf zone play an important role in wave-dominated environments as quantified by [Pearson et al., 2009]; however, relatively few studies attempted to quantify the effects of waves on nearshore solute transport, particularly in the context of the Great Lakes. Earlier studies [e.g., Bowen and Inman, 1974] suggested that mixing across the surf zone is much larger than in the alongshore direction and proportional to (H2/T) where H is the wave height and T is the wave period. In relatively low wave-energy environments such as the Great Lakes (e.g., the mean wave height in Southern Lake Michigan during summer 2008 was 0.2m), the effects of waves and wave-current interactions on solute transport may not be as important as in a marine environment, the focus of several previous studies [Svendsen and Putrevu, 1994; Bowen and Inman, 1974; Longuet-Higgins, 1970].

[7] Shear-augmented diffusion is an important mechanism that can explain enhanced mixing rates in the nearshore and was first observed in the Great Lakes by [Csanady, 1966]. Due to the combined effects of vertical or lateral shear and turbulence, a dye patch in the nearshore region can spread at a much faster rate compared to a situation in which mixing is only due to eddy-diffusion driven by turbulence. In shallow, nearshore areas of the Great Lakes where circulation is primarily wind-driven, the velocity gradients responsible for producing shear are larger in the vertical compared to those in the horizontal plane. This shear-augmented diffusion was investigated by [Ojo et al., 2006a, 2006b] and quantified for the Corpus Christi Bay, Texas using ADCP measurements. They found that shear-augmented diffusion can dominate turbulent mixing in shallow estuarine waters producing effective mixing rates that are 10 to 20 times higher than estimates based on turbulence alone. [Chen et al., 2009] identified the interfacial stress term as an important factor affecting lateral spreading of the plume in the near-field. [Hetland, 2005] identified that vertical mixing is greatest in the near-field and wind-driven mixing is most important just beyond the near-field.

[8] The horizontal turbulent diffusion coefficient (K) for oceans and lakes is related to the characteristic length scale of the flow. [Okubo, 1971] and [Murthy, 1976] used oceanographic data to arrive at a power-law relation (1) which estimates net turbulent diffusion in the open ocean, away from boundary effects.

display math(1)

[9] Here, the coefficients a and b include the effects of several factors such as wind speed, direction and fetch, surface heat fluxes, vertical stratification, current and wave fields, and Δ is the characteristic length-scale. Assuming a homogeneous, isotropic turbulent flow results in the value of b=4/3. The so-called “4/3 power-law” has a theoretical basis following the Kolmogorov hypothesis and the work of Batchelor [Batchelor, 1950]. However, homogeneous, isotropic turbulence is an idealization rarely found in nature and data reported from field studies produce a value less than 4/3 for b. For example, Borthwick [Borthwick, 1980] obtained b = 1.12 in the surface layer of the Swansea Bay. Equation ((1)) was found to describe diffusion (away from any shoreline) in small lakes as well for length scales ranging from 10 m to greater than 100 m. For example, Lawrence et al. [Lawrence et al., 1995] were able to fit their tracer data for a small lake in Vancouver, British Columbia using the coefficients a = 3.2 × 10–4 and b = 1.1. In their study to simulate contaminant transport at multiple scales in the Scheldt River and estuary [de Brauwere et al., 2011] have used a scale dependent diffusion coefficient of K = 0.03∆1.15.

[10] The homogenous and isotropic turbulent field assumptions, made in the Okubo relation (1) are a reasonable description of mixing in open waters. However, the transport of tracers in the CBL is affected by the presence of the lateral (shoreline), top (free surface) and bottom (lake bed) boundaries. While large-scale features dominate lake-wide circulation, small-scale features dominate circulation close to the shore. Correctly describing the mixing processes within the CBL is important since many transport processes of relevance to coastal communities primarily occur in this region. [Ojo et al., 2006b] have shown that the presence of shear structure is responsible for significantly enhancing mixing of a tracer in shallow estuarine conditions. When diffusion is controlled by shear, the exponent b in equation ((1)) is known to approach the limiting value of 1.0, while the upper limit of 4/3 represents diffusion in the inertial sub-range [Murthy, 1976].

[11] Describing mixing processes in a numerical model of the CBL can be a challenge due to the range of scales involved. While circulation is primarily wind-driven, vertical mixing across the thermocline is affected by thermal stratification, and internal waves. Cross-shelf thermal variability due to Poincare waves and a strongly tilted thermocline reported by [Troy et al. [2012]; Wong et al. [2012] also affect nearshore circulation and transport. From the point of FIB, a transport model should have the ability to describe two key aspects of the river plumes (both related to hydrodynamics) accurately: (a) advective transport including flow reversals and the frequency of flow reversals and (b) the nature and strength of mixing near the shoreline (which control the size and shape of the river plume). We examine both aspects in the present paper using field observations and 3D hydrodynamic and solute transport modeling.

[12] Large-scale circulation in Lake Michigan has been successfully described using eddy-resolving hydrodynamic models with 2 km grid resolution [Beletsky and Schwab, 2001, 2008; Beletsky et al., 1999; Beletsky et al., 2006]. Several studies have examined the problem of describing multi-scale mixing in the nearshore [de Brauwere et al., 2011; Hetland, 2005; Li and Hodgins, 2004; Nekouee, 2010]. Vertical mixing is popularly implemented using the κ − ε or the Mellor-Yamada (MY) 2.5 level [Mellor and Yamada, 1982] turbulence closure model. A review of algebraic closure models has been provided in [Burchard and Bolding, 2001; Burchard et al., 2008]. Horizontal mixing is implemented using the simpler Smagorinsky model, which has the advantage of being computationally stable, robust, and simple to implement. However, the model introduces significant damping [Moin and Kim, 1982] and improvements such as the dynamic Smagorinsky model and shear-improved Smagorinsky model have been used in the past [Leveque et al., 2007].

[13] In this study, we focus on mixing of a river plume in a low-wave energy lake environment characterized by flow reversals with a circulation that is primarily wind-driven. Riverine outfalls usually have beaches on either side (i.e., both upstream and downstream sides relative to the prevailing current direction). Which beach sites are impacted by contaminated water at any given instant of time depends on the direction of plume travel; therefore flow reversals are an extremely important aspect of FIB transport. One of the objectives of the paper is to assess the ability of a nested grid nearshore model based on the well-documented Smagorinsky and MY formulations to accurately describe flow reversals in the nearshore. We use velocity measurements from several ADCPs and data from a dye release experiment to evaluate the ability of these formulations to describe mixing of a conservative tracer within the CBL.

2 Materials and Methods

2.1 Site Description

[14] The Ogden Dunes beaches are located near Portage, Indiana in Southern Lake Michigan (Figure 1) and frequently experience degraded water quality due to contamination from the nearby outfall of Burn Ditch (USGS Gage # 04095090). Field experiments were conducted between May 2008 and September 2008 to study mixing characteristics and solute transport in the CBL near the outfall. The data from these field experiments form the basis for evaluating the numerical models presented in this paper.

Figure 1.

(a) Map of southern Lake Michigan showing important sites in the field study conducted during summer 2008 (b) Finite-difference mesh (2 km resolution) of Lake Michigan used to compute lake-wide circulation. Mesh for the nearshore model (100 m resolution) is shown in red. (c) Part of the nested nearshore grid used to compute nearshore circulation and transport showing the bathymetry. The dye release location is marked with a red x in the channel in (a).

[15] Five bottom-mounted Acoustic Doppler Current Profilers (ADCPs) were deployed in an upward looking configuration. Figure 1 shows the locations of the ADCPs and other important sites in the study area and details of all deployments are given in Table 1. The RDI-Monitor and RDI-BBADCP were deployed in the IBL, while the RDI-Sentinel was deployed in the FBL. Additional data were collected in the FBL for a one-week period at locations N1 and N2 using two Nortek Aquadopp current profilers. All deployment locations were chosen based on earlier numerical studies [Liu et al., 2006; Thupaki et al., 2010]. Instruments were programmed so that hydrodynamic measurements had a standard deviation of around 0.1 cm/sec [Teledyne, 2006].

Table 1. Details of ADCP Deployments and Sampling Locations
InstrumentLocationCoordinatesPing rate (Ensemble interval)Depth (m)
600 kHz TRDI-MonitorM41.71059 N, 87.20996 W1Hz (5 min)18.3
600 kHz RDI-BBADCPB41.69717 N, 87.10078 W0.1Hz (15 min)17.6
1200 kHz TRDI SentinelS41.63813 N, 87.18539 W1Hz (10 min)9.6
2000 kHz Nortek AquadoppN141.66677 N, 87.06297 W1Hz (12 min)4.1
2000 kHz Nortek AquadoppN241.63315 N, 87.18839 W1Hz (12 min)5.2
Ogden Dunes 1OD141.6279 N, 87.1966 W--
Ogden Dunes 2OD241.6299 N, 87.1875 W--
Ogden Dunes 3OD341.6298 N, 87.1831 W--

[16] A continuous-release dye study was conducted on June 24, 2008 using Rhodamine WT as a tracer, which was released into the Burns Ditch outfall (shown in Figure 1) at a constant rate. Concentrations of the dye entering the lake were measured using a Turner Designs Self-Contained Underwater Fluorescence Apparatus (SCUFA) unit moored at the mouth of the outfall. Plume evolution was tracked by taking multiple transects using a towed SCUFA unit and a hand-held GPS with sub-meter accuracy (Leica GS20 Professional Data Mapper), on a small motorboat. The procedure was repeated to provide snapshots of the plume at two different instants of time separated by approximately 3 hours in time. Tracer breakthrough data at the beaches were obtained by taking water samples in knee-deep water, every hour, close to the outfall. Vertical temperature profile data were collected using Nexsens Micro-T temperature loggers attached, at 1 m interval, to a steel cable with a buoy at one end to keep it vertical. The cable was anchored close to location S.

2.2 Numerical Modeling

[17] Circulation in Lake Michigan was modeled using the hydrodynamic equations in their primitive form derived from the Reynolds-averaged form of the Navier-Stokes equations for mass and momentum transport. More details about the equations are available in [Vallis, 2006]. The equations were solved using a modified version of the Princeton Ocean Model [Blumberg and Mellor, 1987; Mellor, 1998] that was successfully used to model large-scale circulation in Lake Michigan by [Beletsky et al., 1999; Beletsky and Schwab, 2001; Beletsky et al., 2006]. The numerical model was adapted to be executed in a nested-grid configuration, so as to resolve small-scale features. Tracer transport was modeled using the unsteady, three-dimensional advection diffusion equation ((2)).

display math(2)

[18] Here AH and KV, which are the eddy diffusivity coefficients in the horizontal and vertical directions, are related to the eddy viscosity values AM and KM (eddy viscosity in the horizontal and vertical directions) in the momentum equations via the turbulent Prandtl number Pr = AM / AH. Using the Smagorinsky formulation [Smagorinsky, 1963; Pope, 2000] for eddy viscosity in the horizontal directions AM is defined as:

display math(3)

[19] In equation ((3)), mixing in the horizontal directions (x, y) is assumed to be equal, i.e. anisotropy is neglected. However, computed eddy-viscosity depends on the scale of the unresolved sub-grid scale processes (Δx, Δy). The non-dimensional number (α) has a value around 0.1. Mixing in the vertical is implemented using the Mellor-Yamada turbulence closure model [Mellor and Yamada, 1982]. The horizontal directions are discretized using an Arakawa-C grid and the vertical is discretized using σ-levels that follow the contours of the bathymetry. The momentum equations are solved using a leapfrog method for the internal mode and the second-order accurate Smolarkiewicz scheme is used to solve for advection. Wind stress at the water surface is the prime driver of circulation in Lake Michigan. This is calculated based on wind speed and direction recorded at meteorological stations located around Lake Michigan [Liu and Schwab, 1987]. Quality controlled wind measurements are made available by the National Climatic Data Center (NCDC) and the National Data Buoy Center (NDBC) on their websites. The observed wind datasets were interpolated to the computational grid using a nearest neighbor method [Schwab and Morton, 1984]. Seasonal and diurnal changes in temperature of the water column depend on heat flux due to solar insolation and ambient air temperature. These fluxes are calculated [McCormick and Meadows, 1988] based on meteorological observations recorded at weather monitoring stations around Lake Michigan.

[20] Circulation near the shore is characterized by small-scale features, but it is also influenced by large-scale, lake-wide circulation. Large-scale circulation in the hydrodynamic model was resolved using a lake-wide grid with a uniform horizontal resolution of 2 km and 20 σ-levels in the vertical direction. The small-scale features near the shore were resolved using a nested hydrodynamic model with a horizontal resolution of 100 m and 20 σ-levels in the vertical. Results from the lake-wide hydrodynamic model were interpolated to provide boundary conditions for the nested model. The lake-wide and nearshore computational grids used by the hydrodynamic and transport models are shown in Figures 1b and 1c. Energy and mass exchange between the large-scale and nearshore models were described as being “one-way interaction”, by ignoring the contributions of nearshore hydrodynamics to lake-wide circulation. Open boundaries of the nested grid domain were modeled using an upstream advection boundary condition for tracers (temperature, salinity, dye) and a radiation boundary condition for velocity variables. Surface elevation (hydrostatic pressure) was assigned at the boundary based on interpolated values from the lake-wide model and the radiation boundary condition is used to ensure reduced spurious reflections at the boundary.

3 Results

[21] A comparison of vertically-averaged currents based on observations and the nearshore hydrodynamic model is shown in Figure 2 for different locations within the CBL. Although the lake-wide model with a 2 km grid resolution (Figure 1b) was able to resolve the large-scale features of circulation (comparison not shown), the nested grid model with a 100 m grid resolution provided a better description of the currents close to the shore. Bottom friction, wind stress at the top surface and, internal waves produce a dynamically changing vertical velocity structure at the location M which is in the IBL. The vertical variability in alongshore and cross-shore velocity is less prominent at location N2 which is in the FBL. Contour plots of observed and simulated alongshore and cross-shore velocity profiles at locations M and N2 are shown in Figure 3. A summary of model performance metrics in the form of RMSE values based on the vertically-averaged currents has been shown in Table 2. The vertical-average values of alongshore and cross-shore currents were calculated by interpolating simulation results to observational z-levels and computing the mean. The alongshore and cross-shore components of the velocity were calculated by approximating the coastline to be at an angle of 30o to the East-West direction. Velocity observations from the surface bins were removed and a high-frequency filter was used to reduce noise in observed velocities.

Figure 2.

Comparison of observed (red lines) and simulated (black lines) depth-averaged velocities at different locations within the coastal boundary layer.

Figure 3.

Comparison of observed and simulated velocity profiles for the alongshore (u) and cross-shore (v) velocities at two different locations in the coastal boundary layer (a) Location M (b) Location N2.

Table 2. Summary of RMSE Values for Current Speeds at Different Locations in the CBL
LocationRMSE (m/s)
M0.031
S0.037
N10.048
N20.042

[22] Comparison between simulated and observed temperature at location S is presented in Figure 4 in which the box plots denote the variability in temperature within the water column. The measurement uncertainty for temperature (+/- 1°C) is not shown. Advective transport in the CBL in southern Lake Michigan is characterized by a strong alongshore current that changes direction frequently. The ability of the numerical model to predict the reversals in flow direction is shown in Figure 5 by comparing the observed and model-predicted times of flow-reversal. Clearly, the nested-grid model with a 100 m resolution was able to capture flow reversals (both timing and their frequency) accurately. The slight mismatch in the observed and simulated flow-reversal times at the location B is attributed to measurement error (the BBADCP was an older ADCP model that was used to make measurements at this location).

Figure 4.

Comparison of observed and simulated variability in temperatures at location S. The box plots show the variability in the vertical direction. Uncertainty in the measurement was ±1°C (not shown in the figure).

Figure 5.

Comparison of observed and simulated times of flow reversals (expressed in Julian days of the year 2008) in the alongshore direction.

[23] Model performance at different scales of motion was analyzed by comparing the power spectral density (PSD) based on observations and model results. Figure 6 shows the PSD computed using Welch's method [Press et al., 2007] for observations and simulation at the locations M, S and N1. The energy peak at inertial frequency (approximately 18 hr for the latitude of Lake Michigan) was accurately predicted by the model. Observations seem to show a second peak around 0.5 cycles per hour (CPH) in the energy spectrum which was not simulated by the numerical model. Least-squares power-law fits for the observed and simulated velocity power-spectra between 10 CPH and 0.1 CPH are also shown in Figure 6. Slopes of the power spectra are over-predicted by the numerical model and energy contained in the smaller scales (higher frequencies) was not accurately described by the hydrodynamic model, at both offshore and nearshore locations. This supports our earlier observation that the hydrodynamic model did not reproduce small-scale features in the water column as accurately. This is similar to the experience of others who have used the Smagorinsky-MY2.5 turbulence closure schemes to simulate nearshore hydrodynamics [Nekouee, 2010].

Figure 6.

Comparison between observed (red lines) and simulated (black lines) power spectral densities (PSDs) at different locations within the coastal boundary layer. The dashed lines represent curve-fits to the data in the frequency range 10 CPH to 0.1 CPH. (a, b) Location M (c,d) Location S (e,f) Location N1.

[24] Circulation in the CBL is anisotropic as a result of interactions between large-scale circulation and the shoreline. Flow is predominantly in the alongshore direction as shown in Figures 2 and 3. Figure 7 shows the anisotropy of currents in the flow by presenting the ratio of two characteristic velocities math formula in the alongshore and cross-shore directions respectively. The characteristic velocity u* in the alongshore direction is defined as math image, math formula, math formula is the vertical mean of velocity, math formula where h denotes the water depth. A similar velocity v*is defined in the cross-shore direction as well. The mean (time-averaged) values of the characteristic velocities in the alongshore and cross-shore directions are presented in Table 3. Although time-averaged values of simulated and observed characteristic velocities are similar, significant differences are noted in their ratios (anisotropy) shown in Figure 7. The mismatch in anisotropy between observations and the simulation can be expected to produce important differences in comparisons of plume shapes and solute breakthrough (time series) data.

Figure 7.

Observed (red) and simulated (black) ratios of characteristic velocities in the alongshore (u *) and cross-shore (v *) directions as a measure of anisotropy at (a) location M, (b) location S, (c) location N1, and (d) location N2.

Table 3. Observed (Simulated) Characteristic Velocities in the Alongshore and Cross-Shore Directions
LocationMSN1N2
Alongshore characteristic velocity, u* (m/s)0.03(0.05)0.02(0.03)0.02(0.02)0.03(0.01)
Cross-shore characteristic velocity, v* (m/s)0.03(0.05)0.02(0.02)0.02(0.01)0.02(0.01)

[25] Data from the Rhodamine (RhWT) dye release experiment conducted at the Burns Ditch outfall are compared with results from the dye-transport model in Figure 8. Due to several complicating factors including boat traffic, loss of fluorescence due to sunlight, lateral and longitudinal dispersion and retention of tracer mass within the river dead zones, there was uncertainty in the actual dye concentrations that entered the lake at the mouth of the outfall. Assuming an uncertainty of (±1σ) in the values of the RhWT concentrations at the mouth of the outfall, uncertainty in the simulated concentrations at the Ogden Dunes beaches are included in the comparisons presented in Figure 8 in the form of error bars. It is common practice to calibrate dispersion coefficients in the model using field data. The dispersion values were adjusted to fit observations by varying the horizontal eddy diffusion coefficient AH in equation ((2)). A mean horizontal dispersion coefficient of 5.6 m2/s was used to simulate the observed RhWT breakthrough at Ogden Dunes beaches, shown in Figure 8. The sampling station OD3, being closest to the outfall, was most accurately predicted by the model. Moving further from the outfall reduces the accuracy of the numerical model. Because of the complex processes that are responsible for mixing and transport of contaminants in the nearshore, it is useful to examine the spatial extent of the plume as well as the time series at the individual beaches. Although lack of sufficient sampling points prevented a more detailed quantitative comparison of the spatial extent of the plume, the comparisons between model results and synoptic measurements of the plume shown in Figure 9 indicate that a reasonable agreement was obtained close to the outfall. Due to the uncertainty of RhWT concentrations entering the lake, spatial comparisons of the dye plumes were shown for minimum, average and maximum loadings entering the lake. The difference between observed and simulated concentrations was greater as one moved away from the outfall. However, differences were within the uncertainty of plume concentrations at the outfall as shown by the time series comparisons presented in Figure 8.

Figure 8.

Comparisons between simulated (lines) and observed (symbols) concentrations of Rhodamine WT at the Ogden Dunes beach locations OD1,OD2 and OD3 respectively.

Figure 9.

Comparison of simulated and observed Rhodamine WT concentrations (ppb) at the Burns Ditch outfall for two synoptic measurements within the plume. Due to uncertainty in the concentrations of the dye entering the lake, simulations are shown for minimum, average and maximum loadings.

4 Discussion

[26] Our results show that alongshore and cross-shore velocities close to the shoreline could be resolved using a 100 m nested grid with open boundary conditions computed from the results of the lake-wide model. Velocity comparisons with observations, presented in Figures 2 and 3 show that the numerical model is able to simulate the hydrodynamics better in deeper waters. As shown in Table 2, model accuracy at observation points located in the IBL (locations M, B) show a significantly lower error relative to comparisons for points within the FBL (locations S, N1, N2). However, flow reversals due to changes in wind-direction and large-scale circulation, are accurately simulated by the numerical model (Figure 5) at all locations and this result has important implications for modeling FIB transport in the coastal boundary layer.

[27] The power spectral density plots based on observations and hydrodynamic models show the energy present at a particular frequency range. This is useful when assessing the performance of the numerical model in simulating different time-scales of motion. The coastal boundary layer is unique in the sense that lake-wide circulation adjusts to the presence of lateral boundary features such as riverine outfalls and natural and artificial features along the coastline. The typical length-scale of flow features is smaller close the shoreline and energy is transferred from the low-frequency inertial scales to the high-frequency viscous-scales as we move closer to the shoreline. Although the energy peak at the inertial frequency (approximately 18 hr) is captured well (Figure 6), a second peak around 0.5 CPH is not adequately simulated by the model. The second peak around 0.5 CPH may be due to lake-wide free-oscillations in transverse and longitudinal directions that have been observed in power-spectral analyses of water level measurements by [As-Salek and Schwab, 2004; Mortimer and Fee, 1976; Rao and Schwab, 1976]. The inability of the models to simulate this feature could be due to the inadequate representation of internal and surface waves by the hydrostatic equations solved by the numerical model.

[28] As proposed by Kolmogorov's hypothesis on the turbulent cascade of energy [Kolmogorov, 1941], the power spectra can be approximated by a power law relationship that is a result of the local equilibrium (at a particular scale of motion) between the rate of energy transfer and energy dissipation. Figure 6 shows the slopes of the power spectra calculated using a least-squares method. The slopes are over-predicted by the numerical model at locations in the IBL as well as FBL. While the energy spectrum is captured well in the IBL (Figures 6a, and 6b), the higher dissipation rate in the numerical model results in reduced model accuracy as one moves into the FBL (Figures 6e, and 6f). As a result, accuracy of the circulation and transport models also reduces from the IBL to the FBL.

[29] The nested-grid nearshore numerical model explicitly resolved the larger scales of motion using a grid size of 100 m while the unresolved smaller scales are approximated using turbulence closure schemes such as the Smagorinsky model. Figure 6 shows that while the model is able to simulate the larger scales (with time scales of 100 hr to 10 hr), smaller scales (with timescales less than 5 hr) are not accurately described.

[30] Wind stress at the water surface, thermal stratification, and internal waves all result in significant variation in the velocity profile. The comparisons between vertically-averaged velocities shown in Figure 2 and the RMSE values in Table 2 reveal that, the numerical model is more accurate in the IBL, and the error increases closer to the shoreline. However, the comparison between simulated and observed velocity structure in Figure 3 shows a distinct top and bottom region in the IBL, where the velocities are in opposing directions, indicative of internal waves. These features are not present in the comparisons within the FBL (Figure 3b). The directions are generally simulated accurately, while the magnitude is sometimes over-predicted, especially in the top layer.

[31] The hydrodynamic model was able to reproduce the thermal stratification; however simulated thermocline was too diffused similar to the observations made by [Beletsky et al., 2006]. The velocity structure shown in Figure 3 indicates that the inertial oscillations observed in the IBL dominate the vertical variability in alongshore and cross-shore velocities. Closer to the shore, where the epilimnion extends to the lake-bottom, the alongshore and cross-shore velocities in the vertical are nearly uniform. The power spectral density plots also show that the inertial peak has dissipated in the FBL. Inaccuracies in simulating the circulation close to the shore maybe attributed to the inability of the model to accurately simulate dissipation of energy contained within the inertial scales.

[32] The comparisons between observed and simulated values of characteristic velocity in the alongshore and cross-shore directions, presented in Figure 7, show the presence of a highly anisotropic mixing regime in the coastal boundary layer. The numerical model was able to simulate the mean values (Table 3) accurately; however, there are important differences in the peak values that can result in significant differences when simulating dye transport. Considering the inaccuracy of model results in the FBL, it is likely that numerical models with grid-sizes of math formula(10m) or smaller may be required to reproduce the variability seen close to the shore. However, directly relating grid sizes with resolved frequencies can be simplistic and misleading. While the large-scale model has a grid-size 20 times that of the nearshore model, velocity comparisons at the location M were found to be comparable. Therefore, improving the representation of the Reynolds stresses in turbulence model and using more advanced closure schemes, are expected to improve model accuracy in describing the transfer of energy from larger to the smaller scales. Since the model used a vertical grid spacing of about 0.5m, insufficient grid resolution in the vertical direction, is not seen as a reason for the model's inability to simulate observed features accurately. Reynolds stresses are apparent stresses due to fluctuations in the unfiltered velocity field. The main aim of the various turbulence closure models is to find a suitable approximation of this term, thereby closing the governing equations. The Smagorinsky closure scheme parameterizes the sub-grid scale turbulent mixing processes based on the gradient/rate-of-strain hypothesis without solving for the turbulent kinetic energy or the length and time scales as is done by more advanced turbulence closure schemes. Momentum and density effects dominate mixing and entrainment in the near-field and wind-driven mixing processes dominate far-field behavior of the plume [Hetland, 2005; Nekouee, 2010]. In the absence of stratification, the time-scale for vertical mixing in the nearshore region is much smaller than mixing in the horizontal directions. Plume behavior is therefore dominated by horizontal mixing processes and the horizontal turbulence mixing schemes become increasingly important, especially as one moves into the far-field.

[33] Turbulent diffusion and shear-augmented mixing dominate transport in the nearshore and adequately representing the effect of vertical shear, either by resolving the shear structure explicitly or by calibrating the mixing coefficients in the model against observed data is essential. A mean horizontal dispersion coefficient of 5.6 m2/s was needed to represent the mixing in the nearshore region for the conditions of our dye release experiment. This is due to the effect of unresolved shear structure as well as other processes such as surface and internal waves that were not described by the model. The spatial comparison between plumes in the nearshore reveals that the model underestimated the plume extent close to the shore. This could be due to wave-induced alongshore velocity (not accounted for in the current model) or the lack of anisotropy in the turbulent diffusion field in the model. Finally, an aerial image of the river plume from the Burns Ditch outfall (Figure 10) shows the complex nature of mixing and the highly irregular shape of the interface between the river and lake waters. This is in contrast with the smooth and regular plume shapes produced by the advection diffusion equation used in POM (equation ((2))). Lagrangian (particle-based) approaches are expected to describe plume shapes better; however, since the success of Lagrangian methods depends on the accuracy of the hydrodynamic fields on which they are based, it is important to improve fundamental descriptions of hydrodynamic and wave phenomena in the coastal region in order to more accurately describe FIB fate and transport. More advanced turbulence closure schemes and coupled wave-current hydrodynamic models are expected to improve the accuracy of FIB transport models in the surf zone.

Figure 10.

An aerial image of the river plume from the Burns Ditch outfall showing complex mixing patters and the sharp interface between river and lake waters (Photo courtesy of Jerry Mobley).

5 Conclusions

[34] Field observations and results from a nested-grid nearshore numerical model were used to analyze mixing and transport processes within the coastal boundary layer of southern Lake Michigan. Comparisons between model results and ADCP measurements of velocity profiles at different locations within the IBL and FBL show that the hydrodynamic model was able describe the vertically-averaged velocity time series as well as vertical velocity profiles adequately. Comparisons between the simulated and observed power-spectral densities show that the turbulence closure scheme was able to describe the transfer of turbulent kinetic energy at the inertial scale, but was unable to accurately describe the energy transfer and dissipation at smaller scales. It was also seen that errors in simulated velocities increased as we move from the IBL to the shallow waters, dominated by viscous (friction) effects. The quadratic bottom boundary layer description used in the numerical model, in conjunction with the local (wind-driven) and large-scale forcing, was able to describe the general flow structure, in the vertical including the internal waves. Direct comparisons with observations were not possible in the surface layer of the water column due to insufficient vertical resolution of the ADCP profiles. The numerical model was able to simulate the temperature of the water column in the nearshore to within observational accuracy.

[35] Inertial waves observed in the IBL were found to dominate vertical variation in alongshore and cross-shore velocity components. The observed velocity structures show a great deal of variation in the vertical, particularly in the IBL. However, in the FBL where the thermocline extends to the lake-bed, the alongshore and cross-shore velocities showed little variation in the vertical. The model's inability to simulate dissipation of internal waves accurately may have resulted in higher error closer to the shore. The other important aspect of the hydrodynamic field in the CBL, flow reversals, is accurately described by the model. Mean values of the anisotropy in the velocity fields (characterized by the ratio of the characteristic velocities in the alongshore and cross-shore directions) were predicted reasonably well, although a comparison of the time series shows considerable mismatch between observed and simulated values of anisotropy in the flow field. Small-scale, high frequency fluctuations, typical of the highly turbulent flow fields, were observed in the measured velocity time-series but the model was unable to reproduce these features perhaps due to the use of the Smagorinsky turbulence closure model which is known to produce excessive damping.

[36] In conclusion, we found that a highly variable vertical shear structure, small-scale processes and flow reversals are all important features of circulation in the coastal boundary layer that affect mixing and transport of solutes in the nearshore environment. A three-dimensional hydrodynamic model was tested using observations from five ADCPs deployed during the summer months in southern Lake Michigan. The model was able to simulate the mean velocity field and energy contained in the larger scales of motion, but significantly under-estimated the energy contained in the smaller scales. Comparisons between the observed and simulated power-spectral densities showed that turbulent energy transfer from the large, inertial-scale eddies to the smaller scales was not accurately described by the model. For a transport model with a resolution of 100m, a mean horizontal dispersion coefficient of 5.6 m2/s was required to describe data from our dye release experiment. This value is closely approximated by the scale-dependent dispersion relation (K = 0.03Δ1.15) proposed by [de Brauwere et al., 2011].

Acknowledgments

[37] This research was funded by the NOAA Center of Excellence for Great Lakes and Human Health. We thank David J. Schwab and Dmitry Beletsky (University of Michigan) for their contributions to this research. We thank Chaopeng Shen, Jie Niu and Kashi Telsang for their assistance with the 2008 field data collection.

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