(i) surface layer — 
Ö(pu;h)/öt + 0 (pu;ujh)/Oxj + 0(p<u'ju';>h)/0x,; + P + (p € u';u'4 > Ib - p«u'u4» la) + 26; j,k®;jPurh = 0 (1) 
Oph/0t + 0 pu;h/öx; -puz(a) = 0 (2) 
06h/0t + 0 Ou;h/Ox; + hô «0'u;»/O0x, , «0'vi» |y. <B'u'; >|, -0u3(a) = source + sink terms (3) 
(ii) subsurface layers — 
O(pu;h)/ôt + 0 (puju;h)/0x; + (p «u'ju'5; » h/Ox; + Pj * (p<u'iu'3>|p - P<u';u'3 > |a) + (puju3(b)-pu;u3(a)) + 
2 ej j, ko jPuyh =0 (4) 
dph/ot + 0 pu;h/öx; + (pu3(b)-pu3(a)) = 0 (S) 
06h/0t + à Ou:h/Ox; + hd <B'u'; >/0x; 4 «0'u'5»4g «0t,» [4 (©u3(b)-0u3(a)) = source + sink terms (6) 
  
Horizontal diffusion is based on constant eddy diffusivity 
scaled in proportion to the grid size. The dependent variables 
for dissolved/suspended substances include salinity (S), 
temperature (T) and sediment concentration (C,). The 
density p is determined from an equation of state. 
The required boundary conditions—water levels, open- 
boundary velocities and water properties, river runoff, and 
winds—are obtained from existing observing networks or 
from historical fields. Wind stress p«u';u'4» Ib) is 
calculated at each time step using the 10-m local wind and the 
drag coefficient dependence given by Large and Pond (1981). 
Initial conditions for p are specified from climatological data. 
The circulation models are prognostic in all dependent 
variables, and are spun-up from rest (u;=0) until transients 
are removed from the system and it is in dynamic balance 
with the applied forces. At that point the model is suitable for 
assimilating observed surface fields from the satellite and 
ground wave radar sensors. 
The governing equations are solved using semi-implicit finite 
difference methods on uniformly spaced Cartesian grids 
(Stronach et al., 1993). The shelf grid spacing is 5 km, while 
the grid spacing for the Strait of Georgia is 1 km. The 
number of layers and the layer thickness can be varied to 
resolve the important oceanic and effluent dispersion 
processes. Between 14 and 20 layers are used in the present 
applications. 
OIL SPILL MODEL 
A generalized oil spill model (SPILLSIM) has been developed 
to predict the trajectory and weathering properties of a range 
of petroleum products using the surface currents calculated 
with the circulation models. The oil spill model incorporates 
transport and diffusion, surface spreading, evaporation, 
vertical dispersion and emulsification using the Mackay et al. 
(1980) analytical approach. Mackay's approach treats the oil 
in two components, a "thick" portion of the slick and a "thin" 
portion. The spill is divided into a time-series of discrete 
parcels. Each parcel is initialized with a volume and area 
associated with the thick and thin slick fractions; the change 
in the volume and area of the slick fractions with time is then 
calculated as a function of the weathering processes. 
428 
Horizontal dispersion of the oil is simulated by dividing each 
parcel into several thousand particles. The oil volume 
associated with the parcel at any time is divided equally 
amongst the particles in order to account for weathering, 
shoreline, recovery and other boundary volume losses. The 
particles are advected by the current u;, and diffused using a 
random walk model. Mathematically, each particle of oil is 
defined by a position vector R(t) with location at time t given 
by: 
t 
( 
R(t) = J (uj(x;,t) + u;") dt (7) 
0 
where u;(x;,t) is the total surface current vector derived from 
the circulation models. Diffusion is produced by the turbulent 
part of the flow field u;' arising from subgrid scale motions. 
This integral is solved in a sequence of N discrete time steps 
with increment At, for M particles comprising each oil parcel. 
The result is M position vectors 
N 
R,(NAt) = 2 (uj(x;,nAt)At + saxrg); /=1,2,...,M (8) 
n=l 
For a particle that moves a distance which is uniformly 
distributed in the range (0,s) in a time step At, the probability 
of the particle being at location s at time t satisfies the 
diffusion equation with apparent diffusivity D. The random 
walk step size is given by s=(6DAt)1/ . ‘This procedure 
constitutes a Monte Carlo method (Bauer, 1958) where the 
diffusive effects of u;' are simulated by a random walk 
analogy consisting of M trials each with N random 
displacements having magnitudes between -s and s. ry isa 
random number in the range (-1,1). 
Details of the weathering calculations are given in Hodgins et 
al. (1991) and the reference cited previously. In addition to 
weathering, oil absorption by the shoreline is modelled up to 
a limit based on substrate and the incident wave energy. 
The outcome of the calculation is a set of particle positions R; 
for each parcel comprising the spill, with attributes giving the 
Intemational Archives of Photogrammetry and Remote Sensing. Vol. XXXII, Part 7, Budapest, 1998 
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