The physical fates submodel uses a variable timestep to resolve transient toxic effects in the water column, and to compute efficiently long-term concentration changes in the sediments. The model computes a reference timestep, Δt, based on the Eulerian (fixed) grid size established on the seafloor and the (time-variable) maximum water column transport velocity　Umax: Δt = (ΔxΔy)^1/2/ (2 Umax)　(3-48)

 where Δx and Δy are the grid cell dimensions in the x and y directions. In shallow water, the timestep may be limited by the vertical mixing velocity, in which case an imbedded timestep

is used in the advection computations.

 The initial timestep is then set equal to a fraction of the reference value, and allowed to increase with time to the reference value. A small initial value is necessary to allow resolution of evaporation processes in the case of a floating substance, and to resolve the formation of a momentum/density plume in the case of a substance which sinks.

 Thereafter, the timestep is equal to the time-variable reference value, until all water column concentrations are below a specified threshold value, and all contaminants in the water column have been advected outside the fixed grid boundaries or settled to the bottom sediments. Then the model advances on daily, monthly and seasonal timesteps until all sediment and shoreline concentrations are also below the threshold value.

　

 The maximum number of particles which can be represented in the model is fixed. If the total number at a given time is at or near the maximum, and additional particles are needed to allow continuous input of contaminant, the model performs a compression of the particle arrays. This compression is based on the identification of geometrically "nearest-neighbors", and the combining of their attributes: mass, time-since-release, x", y", and z-locations. A new particle is created with mass equal to the sum of the masses of the two nearest-neighbor particles, and location and time-since-release are computed based on linear-weighting of the existing values based on the mass of each particle. This process is continued until sufficient "free space" is created in the arrays to allow the program to proceed. This results in a relatively uniform spatial distribution of particles as the program proceeds.

　

A　area

C　concentration

Ca, Cdis　adsorbed and dissolved contaminant concentrations

Cd　drag coefficient

Co　initial concentration

Cs　minimum concentration at sediment-water interface or saturation

Css　suspended particulate matter concentration

Cw　ambient concentration

C3, C4　constants

C*　entrainment coefficient

d　depth

di　mean particle diameter in size class i

do　droplet size distribution

D　slick diameter, turbulent dispersion coefficient

Da, Db　surface dispersion rates

Dbio　sediment bioturbation rate

Deff　effective diffusivity

Ds　spillet diameter

Dx, Dy, Dh　horizontal dispersion coefficients

Dz, Dv　vertical dispersion coefficient

e　energy dissipation rate

E　water entrainment rate of sinking spill

f　Coriolis parameter

Fb　buoyant force

Fd　drag force, densemetric Froude number

Fevap　fraction evaporated

Fm　mass flux

Fwc　fraction water-in-oil

g　gravitational acceleration

h　mass transfer coefficient

hi　mixing depth, ice thickness

ho　initial entrainment depth

H　Henry's Law constant

H'　nondimensional Henry's Law constant

Hb　breaking wave height

k　contaminant decay rate

K　molecular diffusivity, ice roughness amplification factor

Koc　adsorbed/dissolved partition coefficient

K1, K2, K3, K4　surface slick related coefficients

K5, K6, K7　water column volatilization coefficients

L　length scale, shoreline length

m　mass

mi　mass of oil on shoreline segment i

M　momentum per unit length, spillet mass

MW　molecular weight

Pvp　vapor pressure

Q　total pollutant mass per unit area

Qdi　entrainment rate per unit surface area

ri　removal rate, resurfacing rate

rj　process rates

R　gas constant

R^*　uniformly distributed random variate [-1 ≦ R^* ≦ 1]

Rj　radius of convective jet

Re　Reynolds number

s　coordinate along jet centerline

st　surface tension

Sb　beach slope

Sc　Schmidt number

S　solubility, fraction of sea surface covered by oil

t　time

T　temperature

Tw　wave period

u　component of current vector in x direction

U　ambient current speed

Umax　maximum water column transport velocity

Uoil　oil drift speed

Uth　threshold velocity

Uw　water speed beneath ice

v　component of current vector in y direction

vh, vz　horizontal and vertical currents

V　net horizontal current velocity

　advective transport vector

Vs　particle settling velocity, oil storage volume

Vm　spill volume

Vmax　maximum amount of oil deposited on shore

Vdiff　vertical diffusive velocity

wi　terminal size velocity

W　wind velocity

Wb　beach width

Wf　fall velocity for denser-than-water substance

Wi　deposition width for shoreline i

Wr　river width

Ws　threshold wind speed for breaking waves

W10　wind speed at 10 m height

z　coordinate down into sediments

Zb　intertidal area height

α, α2　entrainment coefficients

γ　angle between jet trajectory and ambient current

δ　slick thickness

δi　holding capacity for shoreline type i

Δd　particle diameter interval

Δt　time step

Δm　mass of oil removed from shoreline segment i

Δx　grid cell dimension in x direction

Δy　grid cell dimension in y direction

θ2　angle between jet trajectory and vertical axis

η　latitude

μ, μo　dynamic viscosity

ρ, ρw　density

σ　standard deviation

τ　stress

υ, υw　kinematic viscosity (m/r)

ψ　stream function

Ω　angular speed of earth's rotation