Key words: model, hydrodynamics, sediment transport, resuspension, ... on atmospheric friction and shear stress in shallow water and seagrass areas are ...
Hydrobiologia 444: 1–23, 2001. E.P.H. Best & J.W. Barko (eds), Modelling Sediment Resuspension, Water Quality and Submersed Aquatic Vegetation. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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Hydrodynamic and sediment transport modeling with emphasis on shallow-water, vegetated areas (lakes, reservoirs, estuaries and lagoons) Allen M. Teeter1 , Billy H. Johnson1, Charlie Berger1 , Guus Stelling2 , Norman W. Scheffner1 , Marcelo H. Garcia3 & T.M. Parchure1 1 U.S.
Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, U.S.A. 2 Delft University of Technology, Department of Fluid Mechanics, Delft, GA 2600, The Netherlands 3 University of Illinois at Urbana-Champaign, Department of Civil Engineering, Urbana, IL 61801, U.S.A. Key words: model, hydrodynamics, sediment transport, resuspension, submersed aquatic vegetation
Abstract Modeling capabilities for shallow, vegetated, systems are reviewed to assess hydrodynamic, wind and wave, submersed plant friction, and sediment transport aspects. Typically, ecosystems with submersed aquatic vegetation are relatively shallow, physically stable and of moderate hydrodynamic energy. Wind-waves are often important to sediment resuspension. These are open systems that receive flows of material and energy to various degrees around their boundaries. Bed shear-stress, erosion, light extinction and submersed aquatic vegetation influence each other. Therefore, it is difficult to uncouple these components in model systems. Spatial changes in temperature, salinity, dissolved and particulate material depend on hydrodynamics. Water motions range from wind-wave scales on the small end, which might be important to erosion, to sub-tidal or seasonal scales on the large end, which are generally important to flushing. Seagrass modifies waves and, therefore, affects the relationships among the non-dimensional scaling parameters commonly used in wave analysis. Seagrass shelters the bed, often causing aggradation and changes in grain size, while increasing total resistance to flow. Hydrodynamic friction can not be well characterized by a single-parameter equation in seagrass beds, and models need appropriate enhancement when applied to these systems. Presently, modeling is limited by computational power, which is, however, improving. Other limitations include information on seagrass effects expressed in frictional resistance to currents, bed-sheltering, and wave damping in very shallow water under conditions of both normal and high bed roughness. Moreover, quantitative information on atmospheric friction and shear stress in shallow water and seagrass areas are needed. So far, various empirical equations have been used with wind or wave forcing to describe resuspension in shallow water. Although these equations have been reasonably successful in predicting suspended sediment concentrations, they require sitespecific data. More detailed laboratory and field measurements are needed to improve the resuspension equations and model formulation pertaining to seagrass beds.
Introduction Hydrodynamic and sediment transport models can be used as parts of integrated models, along with water quality and submersed vegetation models, to address potential ecosystem responses to some event, change or condition of interest. Submersed aquatic vegetation is sensitive to underwater light conditions, flow and wave action, physical bed-sediment characteristics,
and nutrient availability in water column and sediment (Best et al., 2001). Recent advances in computational fluid dynamics and sediment transport model formulations enable computation of flows and sediment fluxes in three-dimensions (3D). However, spatial and temporal scales, and model processes must be carefully selected for a particular study to insure that the model will solve the problem at hand and that computations can be carried out in the practical sense.
2 Table 1. Growth limiting TSS concentration in reation to water depth in Laguna Madre, Texas Depth (m)
Growth Limiting TSS Concentration (mg l−1 )
1 2 3 4
18 7 3 1
Understanding the integrated effects resulting from some ecosystem change involves many areas of research, including flow modeling, turbulent mixing, transport processes, sediment transport, nutrient dynamics, etc. This paper focuses on hydrodynamics, shallow-water waves, macrophyte interactions with hydrodynamics and fine-sediment resuspension and transport. Typical physical ecosystem characteristics Typically, ecosystems with submersed aquatic vegetation are shallow and of moderate hydrodynamic energy. In this paper, most examples pertain to shallow coastal systems where the submersed vegetation is composed of seagrass. Laguna Madre, Texas and Florida Bay, Florida, U.S.A., are typical for near-shore coastal areas. They both are micro-tidal, and can have hyper-saline conditions due to low freshwater inflow, evaporation and sluggish flushing. Both the Laguna Madre and Florida Bay have surface areas of about 1500 km2 , and each is for more than 75% covered by seagrass. Mean depth for Laguna Madre is 0.8 m to mean tide level. Florida Bay is also shallow with a depth range of 1–4 m. Wind-generated waves are important to sediment resuspension in both systems. The depth range for submersed vegetation in coastal waters can be limited by light conditions. The total suspended sediment concentration (TSS) limiting growth can be estimated in a simplistic way by assuming that submersed vegetation requires 20% of incident light for long-term survival and by assuming a relationship between the water-column extinction coefficient for photosynthetically active radiation (PAR). Based upon the TSS–PAR relationship developed by Burd & Dunton (2000) for Laguna Madre, we demonstrated that even low levels of resuspension and TSS can limit the depth range for seagrass.
Typical median total suspended material (TSM) concentrations in and near seagrass beds in the Laguna Madre are 18 mg l−1 over a water column depth of 1 m. At bare areas in the Laguna Madre, however, annual median TSM values of 150 mg l−1 have been reported for a 2.0-m water column (Brown & Kraus, 1997). The difference in TSM concentration levels between bare and vegetated areas in the Laguna Madre is striking, and has been attributed partly to high winds over this area. At the same time, mean current magnitudes at a 2.1 m deep, non-vegetated, site were 7 cm s−1 , while they were 4.5 cm s−1 at a 1.3-m-deep Thalassia testudinum (turtlegrass) vegetated area. Mean significant wave heights for these sites were 0.14 m for the bare and 0.04 m for the vegetated site (Teeter, 2000a). In Florida Bay, tripton (TSM minus algal biomass) levels range from 8 to 30 mg l−1 (Phlips et al., 1995). Tripton contributes 54–92% to light attenuation within the water column, with chlorophyll-containing particles the next most important contributor. The average station median-TSM value was 18.2 mg l−1 for 16 stations within Florida Bay, with a mean station depth of 2.0 m (with an SD of 0.6 m), and no apparent significant relationship between TSM and depth observed over a 24-month period. These coastal systems are not dynamic with respect to sediment transport under typical, non-storm conditions, as indicated by the relatively low TSS values above. Since most frequently occurring suspended sediment and underwater light levels are most important for growth of submersed vegetation, an integrated sediment transport modeling study could aim at simulations of long-term suspended sediment concentrations in relation to wind and wave action. Concepts of scales and scale interactions Energy inputs into systems predominated by submersed vegetation originate from instantaneous wind speed, diurnal fluctuations in wind speed and irradiance, climatic fluctuations, and variations in season and catchment hydrology. In tidal waters, strong inputs can occur at both tidal and sub-tidal periods. Purging and flushing rates depend on larger time scales, while shear stresses depend on smaller time scales (although riverine shear stresses may depend on seasonal fluctuations in river discharge). Geometry is important to instantaneous and residual flows. Small-scale geometric features, such as a channel or bar, can have important large-scale effects on residual flows.
3 Ideally, all scales of space and time are included in a model but this is not the case today. In practice, averaging must be done to eliminate some ranges of spatial and temporal scales due to current limits on computer memory and computational speed. Averaging leads to closure assumptions. Closure assumptions require skill, experience, and often a-priori knowledge on the part of the modeler. This a-priori knowledge must increase as the averaging interval is increased, and the averaging interval increases as the intended prediction period is extended. In general, it can be argued that a model formulation must fulfill the following requirements, given the model purpose: 1. The model must contain relevant physical processes and geometrical/topographical detail. This requires accurate flows and fluxes, possibly in 3D, and, to accomplish this, turbulence or turbulent mixing, possibly the transport of salt and heat, and sediments. 2. A stable numerical solution must be obtained within a reasonable amount of computational effort. This entails compromising on the flow and closure equations, as well as on spatial resolution. 3. The solution must be sufficiently accurate. This requires the use of numerical methods that are more accurate than the boundary and geometry data. To fulfill these modeling requirements, the modeler has to focus on the most important questions and associated processes, and make correct judgements with the information at hand. The modeler must often compromise between what is desirable and what is possible. Simulation will only provide additional information beyond statistic description as long as the simulation errors do not exceed variations in the quantities being simulated. This limit will depend on the types of processes, errors in initial state specification, and the quality of the simulation method. Integration of model processes Bed shear stress, sediment resuspension or erosion, light extinction and submersed vegetation influence each other through positive or negative feedbacks (James & Barko, 1994). Bed shear stresses depend on wind-waves (usually), tides, freshwater inflows, water depth, wind fetch lengths and on the presence of submersed vegetation. Momentum transfer from winds to waves depends on wave steepness and is reduced by the presence of submersed vegetation.
Figure 1. The coupling of wind, waves, seagrass and bed-surface shear stresses under the influences of seagrass roughness, wave steepness and resuspension. Resuspension affects seagrass through its influence on light extinction within the water column over longer time scales than the other interactions.
Erosion depends on total water column shear stress and the fraction of this shear stress reaching the bed (100% in non-vegetated areas) and, therefore, on the sheltering effects of seagrass. Submersed vegetation reduces local resuspension due to wind-generated waves (Ward et al., 1984; James & Barko, 1994; Hamilton & Mitchell, 1996). Sediment resuspension affects seagrass areas mainly through its impact on water clarity and light penetration. Light extinction depends on total suspended sediment concentrations, particle size and/or flocculated state of the particles, adsorption, water chemistry as discussed by Van Duin et al. (2001). Submersed vegetation depends on light and nutrient availability, temperature and physical stability. The interactions between wind, wave, seagrass, and bed shear stresses, and the influences of wave steepness, seagrass roughness, and resuspension are shown schematically in Figure 1. The feedback from resuspension to seagrass roughness, shown in Figure 1, operates over a much longer time scale than the other interactions shown. Non-vegetated areas are prone to resuspension that can decrease water clarity and may prevent seagrass establishment or cause further seagrass decline (Onuf, 1994). On the other hand, seagrass beds slow water movement, damp waves, trap and hold sediments (Fonseca, 1996). Seagrass reduces shear stress at the sediment bed to lower levels than would occur on a non-vegetated bottom. At the same time, it greatly increases total resistance to flow and wave damping, absorbs shear stress and shelters the sediment beds. Previous studies have documented the effects of submersed vegetation on total flow friction and on
4 wind-wave damping (see Madsen et al., 2001). Thus, there are feedbacks between the presence of seagrass, water clarity and the establishment of new seagrass. Model components can be linked together or coupled in various ways to reflect ecosystem functioning. Direct coupling of hydrodynamic and sedimenttransport models admits density effects associated with transported water-borne constituents (dissolved and particulate) into the momentum equations, and bed elevation changes into the mass continuity equation. Hydrodynamic and sediment transport models can be operated un-coupled if suspended concentrations are low, and bed elevation changes insignificant. Sediment transport can be un-coupled from submersed vegetation provided changes in the distribution and density of the latter are small or otherwise unimportant to the problem at hand. Layout of the paper to follow In this paper, the various ecosystem components and couplings presented in the last sub-section are described as they pertain to hydrodynamic and sediment transport modeling in shallow systems dominated by submersed vegetation. The two coastal systems described earlier will be used as examples for seagrass dominated systems to illustrate physical characteristics, modeling requirements and approaches. Hydrodynamic modeling is described in the next section using examples from three models. Wind-generated shear stress and wave modeling is described for shallow water. Plant-related frictional characteristics affecting hydrodynamics and wind-generated waves are described next. Sediment transport modeling is described in the last section with emphasis on model formulations.
Hydrodynamic modeling methods Geometric flexibility, high-dimensional representation and computational efficiency are desirable attributes of a numerical hydrodynamic model. Domains to be modelled usually have irregular bathymetric and boundary features which are most accurately represented using unstructured grids or meshes. Numerical simulations are worthless unless the model is an accurate representation of the system being modeled. Flow variations are best calculated using a threedimensional (3D) model, although depth-averaged models may be appropriate for near-homogeneous
systems. Hydrodynamic model simulations used for sediment transport or water quality computations have long simulation times, often demanding on computer resources. Because of limits in the ability to perform massive computations, compromises are usually made to insure that a given model can be operated in an economical manner for the problem for which it was developed. Some tradeoffs are built into models which can be traced to the developer’s background in fluid dynamics and/or the flow problems he/she worked on during development. As a result, many approaches for the casting of equations, fitting of geometry and solving equations are used in numerical models. For any specific model domain and ecosystem, other tradeoffs may be required in the model spatial resolution, state variables, temporal resolution, and simulation duration. The features of three models are given as examples in the sub-sections that follow. CH3D (Johnson et al., 1993) is a finite-difference model that uses a curvilinear grid with boundary-fitted coordinates. The RMA10 model (King, 1993) uses finite elements and a Galerkin variant of the weighted residual solution method. ADCIRC (Luettich et al., 1992) is a model which uses a generalized wave-continuity equation and finite elements. CH3D and RMA10 have been applied in 3D and 2D modes. RMA10 also has the capability for 1D and 2D-laterally averaged modes. Three-dimensional modeling Over the past decade, many researchers have developed and applied 3D hydrodynamic models. Since 3D models are state-of-the-art in environmental hydrodynamic modeling, they deserve special attention. One of the first decisions to be made in a model study may be whether or not a 3D model is essential. Other options will be described later. To expedite 3D computations, free-surface model formulations normally assume the vertical pressure distribution to be hydrostatic. The full 3D momentum equations are thereby reduced to a set of two momentum equations and an integrated continuity equation. The governing equations of fluid motion can be cast differently, e.g. the primitive or wave equation forms. For the spatial integration of the governing equations, either the finite difference, finite volume or the finite element approaches can be traced to the concept of the method of weighted residuals. For model time integration, finite differences are generally
5 employed. Friction treatment, vertical discretization and mixing-coefficient specification can be handled differently in models. Some models have capabilities to wet and dry areas as water level changes. Some solve additional equations such as wave-form momentum or turbulence transport equations. The basis for governing flow equations are the 3D Navier–Stokes equations. With hydrostatic and Boussinesq assumptions, and with an eddy-viscosity closure on the Reynolds stress terms, they are expressed for RMA10 in two momentum equations and an integrated continuity equation as: ρ
∂P Du − · σx + − x = 0; Dt ∂x
(1)
ρ
∂P Dv − · σy + − y = 0; Dt ∂y
(2)
∂h ∂ζ ∂a ∂ζ + uζ − ua + vζ − ∂t ∂x ∂x ∂y ζ ∂v ∂a ∂u + + va dz = 0; ∂y ∂x ∂y a where: σx
Exx = Exy Exz
∂u ∂x ∂u ∂y ∂u ∂z
Eyx ; σy = Eyy Eyz
∂v ∂x ∂v ∂y ∂v ∂z
(3)
.
(4)
D/Dt = ∂/∂t + u(∂/∂x) + v (∂/∂y) + w(∂/∂z), u,v,w = x,y,z velocity components, t is time, σx ,σy = x,y internal stress terms, E-terms are eddy viscosities, P is pressure, ρ is density, is the gradient operator, h is the depth, uζ , vζ = x,y velocity components at the water surface, ζ is the water surface elevation, ua , va = x,y velocity just above the bed, and a is the bed elevation. x , y = x,y combined Coriolis, bed-friction, and wind forces:
x = ρv −
x = −ρv −
ρgua (u2a + va2 )1/2 + ψ Ua2 cos(); C2 + va2 )1/2 C2
ρgua (u2a
+ ψ Ua2 sin().
= 2 ω sin(φ), ω is the rate of angular rotation of the earth, φ is the local latitude, g is the gravitational acceleration, C is a Chezy or Manning friction formulation, = ρ a Cd set using Wu (1980), ρ a is the
atmospheric density, Cd is the atmospheric drag coefficient, Ua is the standard-height wind speed, and is the wind direction counterclockwise from easterly. ADCIRC uses a wave-form of the governing equations derived by taking the time differential of the primitive continuity equation, and the y- and xderivatives of the x- and y-momentum equations. By substituting the latter into the time-differentiated continuity equation, and adding the primitive continuity multiplied by a constant τ 0 (Luettich et al., 1992): ∂ uvh ∂ζ ∂ 2ζ u2 h + − ∂ + (5) + τ −∂ 0 2 ∂t ∂t ∂x ∂x ∂y
∂ p vh − h + g(ζ − αη) ∂x ρ + (MDBS)x + τ0 uh uvh v2 h ∂ −∂ + ∂ − uh − h + ∂y ∂x ∂y
∂ p + g(ζ − αη) + (MDBS)y + τ0 vh = 0 , ∂y ρ where α is the effective earth elasticity factor, η is the the Newtonian equilibrium tidal potential, and MDBS is the combined momentum diffusion, momentum dispersion, baroclinic forcing, and stress terms. This model formulation was developed to eliminate spurious numerical modes from the finite element solution, possesses accurate dispersion relations for long waves, and was selected for use in the ADCIRC model (Luettich et al., 1992). A model domain including the east coast of the USA, Gulf of Mexico, and Caribbean Sea was developed, verified to tidal stations and used to predict the primary diurnal and semi-diurnal tidal constituents over a 30 000-node mesh (Westerink et al., 1993). These results have been used to drive smaller, near-shore models. The ADCIRC model has also been used extensively to predict storm surges, and these results have been entered into storm databases. Mixing and stratification Density stratification due to temperature, dissolved substances and suspended material can be critical to vertical mixing, vertical velocity profiles and transport. If stratification is important, the vertical dimension may need to be modeled. A k – ∈ turbulence closure model is employed to compute vertical eddy viscosity and diffusivity in CH3D (where k is turbulent kinetic energy, and ∈
6 is the dissipation rate for turbulent kinetic energy). The scheme solves local equations but does not transport turbulence quantities. Since advection generally dominates horizontal transport in estuaries and bays, a simpler algebraic closure is used in the horizontal plane. Such a closure would not be appropriate for problems in which small-scale features are to be resolved and accurately computed. A third-order spatial accuracy scheme based on Leonard’s (1979) QUICKEST is used in CH3D and has proven to work well in partially stratified water bodies such as Chesapeake Bay. An algebraic vertical viscosity and diffusivity scheme (Mellor & Yamada, 1982; level 2.5) is used in RMA10. A horizontal turbulence closure scheme (Smagorinsky, 1963; see review by Speziale, 1998) is used to dynamically set eddy viscosity terms in Equation 4 and horizontal diffusion coefficients in transport equations to be described later. Layer- and area-averaged models Models in two- or one-dimension (2D and 1D) are formulated by integrating the 3D equations of motion over appropriate dimension(s). In doing so, 1D and 2D models lose the capability to predict variations in the missing dimension(s) and some assumptions must be introduced. However, the 2D depth-averaged approach may be justified and well suited for shallow, near-homogeneous systems such as those considered here.
formulation, such as that of Smagorinsky (1963) mentioned earlier, may be required. It should be noted that accurate, spatially-resolved bathymetric data are important in general. “Geometry is everything” is a good rule of thumb for hydrodynamic modeling. Langmuir circulation cells are ubiquitous in shallow aquatic systems, consisting of wind-aligned roll vortices which trap floatable materials in windrows. These long spiraling circulations have alternating rotations. Plant litter often marks the downwelling convergences between cells. In shallow water these circulations reach the bottom and respond rapidly to wind changes. Downwelling current speeds can be of the same order as the mean flows (cm s−1 ; Leibovich, 1983), and could dominate over- to under-canopy mass exchanges. Langmuir circulations are produced by interactions between wave orbital motions and the background shear-flow (Faller, 1969). Cross-cell dimensions are only about three times the water depth so circulation cells are too small to be resolved in hydrodynamic models (and would require special model formulations). Dispersion within Langmuir cells is many times higher than the background turbulent values (Faller & Auer, 1988). Reproducing very small-scale eddies is not generally considered necessary when modeling large bays and estuaries for the purpose of providing flow fields to water quality models. However, getting the vertical turbulence right and reproducing the residual circulation are extremely important for water quality modeling.
Wind-driven circulation Since wind-generated shear-stress acts on the water surface, modeling is best done with a 3D model. Standard depth-averaged models use the vector of the mean flow to compute shear-stress, and generally poorly reproduce wind-driven circulation in shallow water. Some special 2D depth-averaged model formulations for bed stress have been developed to improve the ability of these models to simulate wind-driven circulation especially in shallow water (Davies, 1988; Hearn & Hunter, 1988; Hunter & Hearn, 1989). Small-scale features Relatively small man-made structures including channels, depth features, flow features such as gyres, and/or recirculations require appropriate spatial resolution to be resolved in model grids and to be accurately computed. An adequate horizontal mixing
Winds and wind-generated waves in shallow water Wind-generated waves produce shear stresses important to resuspension in shallow water and vegetated areas, and are often included in sediment transport modeling. However, some resuspension models have successfully used wind alone without calculation of wave characteristics directly. Aalderink et al. (1985) compared four models, two using maximum near-bed wave orbital velocity and two using wind speed. TSM data for a 1 m deep lake, collected hourly for 2 weeks were used in the model evaluation. The two models using wind alone (with and without wind thresholds) better matched the observed TSM than did the two resuspension models using wave-induced flows. All models compared used simultaneous erosion and deposition and a background concentration that was
7 not subject to deposition. These assumptions will be discussed later in the section on sediment transport. Pejrup (1986) points out that where wave heights and depths change appreciably, wind speed (being relatively constant over an area) may correlate better to TSM concentrations than wave height measured at a point. Analysis of time-series TSM from a microtidal estuary indicated that wind alone, regardless of direction, had the best correlation to TSM levels (Pejrup, 1986). Arfi et al. (1993) tested an expression relating wind speed and water column buoyancy to calculate thresholds for resuspension and obtained results that were similar in magnitude to wave-based threshold estimators. Although wave characteristics are critical, wave shear stress and the overall balance of momentum input from the atmosphere are critical to resuspension in large shallow lagoons and estuaries. These shallow water bodies respond to winds at small spatial scales (for example, depth-limited wind-generated waves, Langmuir circulation cells and buoyant eddy overturning). Resuspension model studies of shallow systems have used wave measurements or results from wave models driven by winds to provide wave parameters for bottom shear stress calculations, and have been reasonably successful simulating TSM levels (Luettich et al., 1990; Hawley & Lesht, 1992; Sheng et al., 1992; Lick et al., 1994). Near-bed wave orbital velocity depends on wave height and wave period and is the critical parameter for resuspension of bed sediments. Short-period oscillatory currents forced by wind-generated waves are more effective at developing bed shear stress than the same current magnitudes forced by tides due to boundary layer effects (Luettich et al., 1990). Wave models have not been specifically developed for vegetated areas, but waves can be predicted using a number of different methods. In shallow water areas, waves ‘feel’ the bottom when wave length exceeds twice the depth, and the resulting bottom stress can resuspend sediments and dissipate the waves. Moreover, wave growth in very shallow vegetated areas appears to be limited by depth, bottom friction and fetch. Whatever wave modeling approach is employed, a model requirement for an enclosed shallow water system should be that the total bed force over the model domain is less than the total atmospheric force. This cannot be assured unless atmospheric shear stress is calculated and compared to that from waves. Some amount of wave shear stress is normally expended where waves break at the shoreline The interest here,
however, is the wave shear stress over the greater area of the sediment bed, and therefore shoreline processes will not be considered. The remainder of this section will examine momentum transfer and the overall shear stress balance between waves, currents and the atmosphere. Atmospheric shear stress The coupling of winds to waves and currents starts with momentum transfer at the air–water interface. Atmospheric shear stress (τa , Pa) is calculated using wind speed Ua , in m s−1 , adjusted if necessary to a reference height of 10 m (CERC, 1984): τa = ρa Cd Ua2 ,
(6)
where ρ a is the air density (about 1.225 kg m−3 ), and Cd is the atmospheric drag coefficient appropriate for wind referenced to 10 m height. The atmospheric drag coefficient can be derived by assuming a logarithmic velocity profile and neutral atmospheric stability as: 2 κ Cd = , (7) ln (10/zo ) where κ is the von Karman constant (0.4), and zo is the surface roughness coefficient, m. Hsu (1974) developed an expression to relate zo to wave steepness (significant wave height Hs over wave length Lw ) and atmospheric friction velocity U∗a : zo =
Hs Hs 2 U∗a = U 2 ]deep water , gLw 2πC 2 ∗a
(8)
where U∗a is the atmospheric shear velocity, and C is the wave celerity. By substituting analytical expressions for fully-developed oceanic wave conditions and setting the ratio of C to U∗a equal to 29 , Hsu (1988) proposed: 2 0.4 Cd = . (9) 14.56 − 2 ln Ua Various linear expressions have been proposed relating Cd to Ua . For example, for oceanic conditions and neutral atmospheric stability, Wu (1980) proposed: Cd =(0.8 + 0.065 Ua ) × 10−3 , while for Lake Washington, USA, Atakturk & Katsaros (1999) found: Cd = (0.87 + 0.078 Ua ) × 10−3 . For shallow water with depth-limited waves, Cd may be less than for open ocean conditions. For 1.3– 2.1-m-deep locations in the Laguna Madre, Teeter
8 (2001a) used wind and wave measurements and Equation 8 to develop the empirical expression: Cd = (0.73 + c1 Ua ) × 10−3 , Ua > 2 m s−1 (10) for shallow water where the coefficient c1 varies with depth, fetch or wind direction, and with the presence of seagrass. The value of c1 ranged from 0.067 to 0.086 for four stations. By substitution of a depthlimited expression for Hs , and an expression relating C/U∗a to Ua into Equations 7 and 8, the following expression has been proposed for shallow water (Teeter, 2001a): Cd =
0.4 16.11 − 0.5 ln (h) − 2.48 ln (Ua )
2 (11)
and found to agree with Cd values determined from wind and wave data in the Laguna Madre. Equation 11 predicts smaller Cd values than Equation 9 when h< 22.2 Ua −0.96 . For example, when h 1, bl = a) , E(gs > 1) = E(gs = 1) S(gs = 1, bl = a) gs>1 E(gs = 1) > 0 and τb > τce (gs > 1),
(32)
where S(gs,bl) is the grain-class sediment mass per unit area in a bed layer. Erosion thresholds for silt fractions are taken to be independent of their bed layer location. Erodibility is also linked to the structure of the bed (Dixit, 1982). Models often use a layered bed structure (Ariathurai et al., 1977; Teisson, 1991; Hamm et al., 1997). For the type III model formulation, a layered bed algorithm was developed with variable silt concentrations by layers, depending on initial conditions, and on erosional and depositional history. A fully-settled near-surface concentration distribution with respect to the cohesive fraction is assumed. After deposition occurs, hindered-settling rate is calculated by bed layer, and material is transported vertically downward in the bed using class-aggregated transport parameters, until the specified density distribution is achieved. The mass conservation equation for bed layer consolidation is: Wh (bl)S(gs, bl) dS(gs, bl) =− + dt hs (bl) Wh (bl − 1)S(gs, bl − 1) , hs (bl) > hso (bl), hs (bl − 1) (33) where hs (bl) is the bed layer thickness, hso (bl) is the specified fully-settled thickness, and the hindered settling rate is computed from an empirical hinderedsettling function. Hindered-settling consolidation is inhibited by deposition or erosion greater than 0.01 g/m2 /s. In the bed, volumes of grain classes are
taken into account when converting between mass and concentration. In general, it is assumed that: hs (bl) =
ns S(gs, bl) Oc S(gs = 1, bl) + + ρs ρl
gs=1
(34) ns gs=2
Os S(gs, bl) , ρl
where Oc and Os are the ratios of clay and silt masses to water masses associated with these fractions, and ρs and ρl are the particle and fluid densities. As mass is transported vertically downward as a result of consolidation, the layer concentration of the cohesive fraction is maintained constant over time, and: hs (bl) =
S(gs = 1, bl) Oc S(gs = 1, bl) + (35) ρs ρl
is imposed. Bed concentration (mass per unit volume) is S(gs,bl) / hs (bl). If a layer is withered away by erosion, it disappears at least temporarily. The erosion surface thus descends through the bed, as the surface layer thins, then stepwise through progressively deeper layers. The effect of erosion on bed mass is evaluated as: dS(gs, bl = e) = − E(gs), (36) dt e where e is the exposed bed layer index. Deposition, on the other hand, always occurs into the first layer (bl =1), and the effect of deposition on bed mass is evaluated as: dS(gs, bl = 1) = D(gs), (37) dt d In this way, the bed structure is formed by consolidation from the top layer down. After sufficient deposition and elapsed time has occurred, the bed (in the absence of erosion or further deposition) will return to the specified fully-settled structure. A comparison between types II and III model formultations A numerical fine-grained sediment transport model was formulated by Teeter (2000b) with optionally one or more conservative grain classes to test the differences between type II and III models for deposition and resuspension. The basic model is configured as a
19 vertical suspension and a unit area of sediment bed at a point, similar to the models developed by Luettich et al. (1990) and Hawley & Lesht (1992). The depthaveraged conservation of sediment mass at a point, disregarding horizontal advection and diffusion, is: h
dc = E − D. dt
(38)
Shear stresses generated by currents were calculated using a Manning’s equation. Wave shear stresses were calculated using linear wave theory, and smooth, laminar wave friction formulations as described by Luettich et al. (1990). Wave and current components were added to estimate the total bed shear stress. The point model used a fourth-order Runge-Kutta scheme to integrate conservation equations in time using time steps of 90–120 seconds. Single- and multiple-grain-class model formulations (types II and III) were compared to each other for a series of laboratory experiments and for field data from a shallow lake where resuspension came from wind-generated waves. The only difference between the models was the number of grain classes considered. Models with up to 7 grain classes were compared to laboratory flume tests which formed steadystate suspension concentrations during deposition. Mehta & Partheniades (1975) performed annularflume deposition experiments starting at high shear stresses. Initially-suspended fine-grained cohesive sediments deposited when shear stresses were reduced, forming constant, steady-state concentrations cs that depended on the initial suspension concentrations co and the bed shear-stresses. Each experiment had a 1 g l−1 initial concentration of kaolinite. The kaolinite sediment material contained about 35% of particles coarser than 2 µm, and a maximum particle size of about 45 µm . Similar results were obtained for natural fine-grained sediments from San Francisco Bay, U.S.A., Maracaibo Bay, Venezuela and the Netherlands (Mehta & Partheniades, 1975; Verbeek et al., 1993). The type-III point model was initialized with 1 g l−1 and operated over a range of shear stresses for 1-h simulation times. The concentration cs at the end of the runs was assumed to represent steady-state. Model results were compared to data from Mehta & Partheniades (1975) and are shown in Figure 7. A type II model formulation can only simulate the cs / co distribution shown in Figure 7 as a step function where cs / co = 0 when τ b < τ cd , and cs / co = 1 when τ b ≥ τ cd .
Wind driven wave resuspension in the 600-km2, 3.2-m-deep Lake Balaton, Hungary, was observed by Luettich et al. (1990). Water samples were collected at mid-depth from a boat at a 2-m deep site. Current speed, wave height, and wave period data presented by Luettich et al. were interpolated to 0.1 h for use as boundary conditions for the point model. Model coefficients were first manually set to reasonable values. Subsequently, an automatic model-coefficient adjustment procedure was used to optimize model coefficients so that the adjustment process would not affect the model comparison. Results for one and four grainclasses are shown in Figure 8. The type II model had difficulty following depositional phases of the timeseries record, while the type III model did much better during deposition and slightly better during erosion. Sediment effects on underwater light conditions Factors affecting underwater light conditions are described by van Duin et al. (2001). In addition to TSM, flocculation affects cohesive sediments and light conditions, and is a factor that can be calculated within a sediment transport model. Particle size affects the ratio of mass to surface area, and light scattering. Fine sediments do not exist in the environment as individual particles but as flocs or particle aggregates. Gibbs & Wolanski (1992) measured the optical effects of varying floc size using two fine-grained sediments. Floc size was manipulated by changing the flow in a carousel flume. When large flocs were disrupted, backscatter intensity about doubled for the same TSM. Gibbs & Wolanski (1992) concluded that floc-state is important to the optical properties of suspensions. Flocculation can be modeled by calculating particle motions and collisions which result in floc formation or breakup. However, this approach would not be practical for an ecosystem model. Floc spectra, such as those described earlier in Equation 27, might be used along with site information to dynamically calculate floc surface area per unit volume in a sediment transport model to provide information to a light model.
Conclusions and recommendations Hydrodynamic and sediment transport models have features including appropriate physics-based equations, geometric flexibility, adequate accuracy and economical computing. Often, trade-offs are required
20
Figure 7. Comparison of type II (ns=1) and III (ns=7) models to laboratory steady-state suspensions formed over a range of shear-stresses (data from Mehta & Partheniades, 1975).
Figure 8. Comparisons of a type II (ns=1) model and a type III (ns=4) model with field TSM measured by Luettich et al. (1990).
to successfully complete a particular study with a particular numerical model. Alternate spatial discretization schemes are employed. The finite difference method applied with boundary-fitted coordinates, such as CH3D, provides a degree of geometric flexibility while retaining economical computing capabilities. For example, the application of CH3D to Chesapeake Bay produced long-term solutions of acceptable accuracy. The finite element method provides a greater amount of geometric flexibility but at some computational cost. The finite element discretization according to the Galerkin method of weighted residuals can use standard Navier-Stokes equations or wave equations. RMA10 is an example of the former and uses mixed linear and quadratic interpolation to avoid spurious modes. The RMA10 model is fully-implicit in the spatial domain and gains efficiency back by taking
long time steps. ADCIRC is governed by wave equations and uses linear elemental interpolation. ADCIRC first solves the wave equations and, subsequently original momentum equations with certain boundary conditions solved iteratively. These models have been successfully applied. Hydrodynamic models applied to seagrass vegetated areas require enhancement to accurately model flows and shear stresses, taking submersed plant deflection by currents into account. This is because hydrodynamic friction in seagrass beds can not be characterized by a single parameter as in standard friction formulations (such as those of Chezy or Manning). Moreover, the influences of bed roughness and friction on shallow water waves have not been wellstudied and need to be resolved. Reasonable wave predictions can be made for non-vegetated areas. In seagrass beds, however, models require modification accounting for high bed roughness, since existing models appear to over-estimate wave heights appreciably. Likewise, standard open-water atmospheric drag equations overestimate atmospheric shear stress in shallow water. Wave shear-stresses in seagrass areas and wave roughness equations require modeler’s attention to insure that the total bed shear stress is consistent with atmospheric inputs. To improve the description of sediment resuspension within submersed vegetation, more detailed laboratory and field measurements are needed to quantify the bed sheltering effects of submersed vegetation. Resuspension models generally follow the type I formulation including simultaneous erosion and deposition, and successfully model TSS concentrations. A drawback of these models is that parameters vary spatially and require data from multiple locations. Model comparisons indicate that the type-III multiple grain-class formulation describes erosion and deposition processes better than the single-grain type II formulation. The additional information required by the type III formulation shows less variation than the properties of the cohesive fraction, used in the type II formulation. Type I and III formulations are recommended for modeling resuspension where predicting TSS concentration is the model objective. Where significant erosion and deposition occur, or where bed material properties such as grain size and bulk density vary, the type III formulation is recommended. So far, application of sediment transport models to seagrass systems have focused largely on calculating suspended solids concentrations, which were
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