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1.7 Sinking
The simulation of a continuous release of a sinking pollutant is based on the sinking plume equations developed by Koh and Chang (1973). As the jet descends, water is entrained and particles of mass are stripped away. The initial density of the jet is therefore equal to ρo, the density of the contaminant. As the jet descends and entrains water, it increases in cross section, and the jet density approaches that of the surrounding water. In the presence of a density gradient or thermocline, the jet may reach an equilibrium level in the water column. Otherwise, the jet will eventually arrive at the sea floor. The equations governing this behavior, written with the coordinate s along the jet centerline, are as follows (Koh and Chang, 1973):
Conservation of Mass Flux (Fm)
dFm / ds = Eρw12 (3-20)
Conservation of Momentum (M)
Entrainment Rate (E)
E = 2Rjπ (α1Urel + α2Usinγsinθ2) 14 (3-22)
In Equations 3-9 through 3-11,
Fm = mass flux along the jet centerline s (kg/sec)
E = entrainment rate (m2/sec)
M = momentum per unit length (kg m/sec2)
Fb = buoyancy force per unit length (kg/sec2)
Fd = drag force per unit length (kg/sec2)
U = ambient water velocity (m/sec)
Urel = velocity difference between jet and ambient
Rj = jet radius (m)
α1, α2 = entrainment coefficients
ρw = ambient water density (kg/m3)
γ = angle between jet trajectory and ambient current
θ2 = angle between jet trajectory and vertical axis
j = vertical unit vector
The coefficient α1 is set to 0.081; α2 is set to 0.353 (Koh and Chang, 1973). The buoyant force, Fb, and drag force, Fd, both per unit length along the jet, are
Fb = πRj2g(ρ - ρw) 15 (3-23)
and
Fd = ρwCdRj(Usinγ)2 16 (3-24)
in which
g = gravitational acceleration (9.81 m2/sec)
ρ = density of descending jet (kg/m3)
Cd = drag coefficient (1.3).
The rate of mass loss from the jet to the water column is computed as
dm/dt = Ah(Co - C) 17 (3-25)
In Equation 3-25, A is jet surface area (m2), C is the pollutant concentration (kg/m3) within the jet, and Co is the concentration in the surrounding water. The mass transfer coefficient h is computed according to an empirical engineering relationship (Holman, 1981):
where
Deff = effective diffusivity (assumed as 5x10-6 m2/sec)
Rj = jet radius (m)
Re = Reynolds Number = 2URj/v
Sc = Schmidt Number = Deff/v
v = kinematic viscosity (m2/sec)
1.8 Volatilization from the Water Column
The procedure outlined by Lyman et al., (1982) is followed here:
(1) Compute Henry's Law constant (H):
H = Pvp/(S/MW) (3-27)
Pvp = vapor pressure (atm)
S = solubility (mg/l)
MW = molecular weight (gm/mole)
(2) For H < 3 x 10-7, volatilization can be neglected.
(3) For H > 3 x 10-7,
compute nondimensional Henry's Law constant H':
H' = H/R T (3-28)
R = gas constant (atm - m 3/mole -  ) T = temperature ( )
(4) Compute the liquid-phase exchange coefficient (K5):
(5) Compute the gas-phase exchange coefficient (K6):
(6) Compute overall mass transfer coefficient (K7):
K7 = (H'K5K6)/(H'K6 + K5) 21 (3-31)
The coefficients K5, K6, and K7 are in cm/hr.
The actual mass transfer rate from the water column to the atmosphere is then
dm/dt = K7m/d 22 (3-32)
in which m is the amount of pollutant mass, assumed distributed evenly over the depth d. The volatilization depth for dissolved substances is limited to the maximum of one half the wave height, or a diffusive depth d,
where
Dz =vertical diffusivity (m2/sec)
_t =model timestep (sec)
1.9 Horizontal and Vertical Transport
Transport currents in the physical fates submodel include mean flow and tides. Horizontal shears in the current field contribute to dispersion of contaminants in the modeled water column. Additional dispersion is parameterized through diffusivity parameters, Dx and Dy, calculated as a function of the characteristic length scale of the problem:
Dx = Dy = 0.01L1.13 24 (3-34)
which gives Dxy in cm2/sec, for the length scale L in cm (Okubo, 1971, 1974).
To compute a surface mixing depth condition for this submodel, the U.S. Army Corps of Engineers Shore Protection Manual (CERC, 1984) equations for deep and shallow water wave forecasting are used to estimate wave height based on wind speed, fetch, and duration. The CERC (1984) equations give the significant wave height Hs, and one half of this crest-to-trough value is used as the initial surface mixing depth for the model. For floating substances, entrained particles are injected into this surface layer. As mass is entrained from the surface slick, new particles are created in this surface layer.
A particle-based random walk algorithm (Reed, 1980) is used to simulate both horizontal and vertical dispersion in the water column. Particles diffuse with velocities vi,
where the subscript i = 1,2,3 corresponds to the horizontal and vertical directions x,y,z, Di is associated diffusivity, Δt is the submodel timestep, and R* is a random variate uniformly distributed over the interval -1.0≦R*≦1.0. The value of Dz is set equal to 0.0001 m2/sec, a typical oceanic value (Kullenberg, 1982). For the information of the user, the model prints out the maximum mixing depth as a function of time. The concentrations of contaminant averaged over sub-intervals of this depth are then used to compute an exposure field in the water column.
1.10 Bottom Contamination
Contaminants in the water column are carried to the sea floor primarily by adsorption to suspended particulates, and subsequent settling. The ratio of adsorbed (Ca) to dissolved (Cdis) concentrations is computed from standard equilibrium partitioning theory as
Ca/Cdis = KocCss 26 (3-36)
Here Koc is a dimensionless partition coefficient, and Css is the concentration of suspended particulate matter in the water column expressed as mass of particulate per mass of water. The model uses a default value of total suspended solids of 10 mg/l (Kullenberg, 1982). These values can be changed by the user. Diffusion spreads mass vertically and horizontally in the water column. In addition, the adsorbed fraction of the total mass, Ca/(Ca + Cdis), settles through the water at the settling rate Vs. For deposition purposes, the model assumes that the concentration of suspended matter near the sediments is a factor of 10 greater than in the overlying water column (Kullenberg, 1982).
The model assumes that the duration of the release will be short (i.e., days) compared to sediment diffusion times (i.e., years). Then the diffusion equation with a decay term,
can be solved for a single loading, Q, of pollutant to the sediment. The solution to Equation (3-37) is
where
Q = total pollutant mass per unit area (mt/m2)
Dbio = sediment bioturbation rate (m2/day)
t = time (days)
z = depth (positive down) into the sediments (m)
k = decay rate (per day)
Contaminants which sink directly to the sediments may be returned to the water column by the process of dissolution. The dissolution mass transfer rate is formulated as
dm/dt = hAc(Cs-Cw) 29 (3-39)
(Thibodeaux 1977, 1979), where h is the mass transfer coefficient (m/day), Ac is the area (m2) contaminated at concentration Cs, and Cw is the ambient (background) concentration of the contaminant. Cs is the minimum of the contaminant concentration at the sediment - water interface and the saturation concentration. The mass transfer coefficient h is computed from
h = 0.36(VL/υ)0.8(υ/Dv)0.33Dv/L 30 (3-40)
where V = the horizontal water velocity (m/sec), L = an effective length (m; Ac = L2, assuming a square area), Dv = the vertical diffusion coefficient at the bottom, and v = the kinematic viscosity of water (Thibodeaux, 1977, 1979).
Contaminant concentrations in sediment are distributed between adsorbed and dissolved states by linear partitioning, as in the water column. The particulate-to-interstitial water ratio, however, is taken to be 0.45 (CERC, 1984). The ratio of adsorbed to dissolved contaminant is then computed from Equation 3-36.
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