720 MODELING OF ESTUARINE WATER QUALITY
z
txHU
y() ()0HVwhere
f Coriolis parameter
g Acceleration of gravity
C Chezy coefficient
txs Component of the wind stress in the x directiontys
Component of the wind stress in the y direction
r Water density
z water level elevation relative to the reference
plane.The wind stress components are given bytur wtur wxsysaaWW22sincoswhere
u wind stress coefficient ≈ 0.0026r (^) a atmospheric density
W wind velocity
w angle between the wind direction and the y axis.
In the finite difference approximations of these equations,
the discrete values of the variables are described on a space-
staggered grid. The position and time coordinates (x, y, t)
are represented on the finite grid by (j ∆ x, k ∆ y, n ∆ t), for j, k,
n 0, 1/2, 1, 3/2, K.
Water levels and pollutant concentrations are computed
at integer values of j and k (x and y directions). Water depths,
obtained from a field survey, are given at half-integer values
of j and k. The velocity component U (x directed) is com-
puted at half integer values of j and integer values of k, and
the velocity component V (y directed) is computed at integer
values of j and half-integer values of k.
The set of finite different equations used to approxi-
mate the momentum and mass-balance equations are then
presented at two adjacent time levels, n and (n 1/2).
Numerical computation of the reaction matrix terms in the
mass-balance equations is accomplished by a sequential
use of forward and backward information. If M constitu-
ents are transported, then for constituent i(1 i M), in
the first operation at time level n (going from t to t 1/2t),
information is used in the reaction matrix terms on the level
(t ∆ t) for all constituents for which the sequence num-
bers m is smaller than i. Information at the level t is used
for which m i. In this step, the constituents are computed
is ascending order, from 1 to M.
In the second operation, at time level n 1/2 (going
from t 1/2 ∆t to t ∆ t), the constituents are computed in
descending order, M to 1. Information on the level t 1/2t
is used for all constituents whose m i, and information on
the level t 1/2t is used for all constituents whose m i.
This procedure centers the reaction matrix information of the
mass-balance equations within the time interval t to t ∆ t.
The reaction matrix terms which involve the ith constituent
itself are taken centered over each half time step.
The sequential use of finite-difference approximations
for the continuity equations at n and n 1/2 results in
alternating forward and backward differences. This means
that over a full time step the terms are either central in time
or averaged over the time interval. In the first operation at
time level n (going from t to t 1/2t), the momentum and
continuity equations are solved first for the water levels
and x-directed velocities at time level n 1/2. The infor-
mation generated is then used in the mass balance equa-
tions to obtain the constituent concentrations at time level
n 1/2.
The results of this first operation are then used at time level
n 1/2 to determine the unknowns in the second half timestep,
going from t 1/2t to t ∆ t. Again, the momentum and con-
tinuity equations are solved first, but this time the water levels
and y-directed velocities at time level n 1 are obtained. This
new information is then used in the mass balance equation to
obtain pollutant concentrations at time level n 1.
This procedure is then repeated for each succeeding full
time step. The model can be used to investigate the influence
of wind on low and circulation in the area covered, together
with its effect on water levels and distribution of pollutants.
This was the first time that real wind effects were investi-
gated in detail.
The need for three dimensional models has been rec-
ognized for salt wedge type and moderately stratified
estuaries, and three dimensional mathematical models
of real estuaries have been developed. Leedertse and Liu
(1975) developed a three dimensional code for water
movements, salinity, and temperature which was applied
to San Francisco and Chesapeake Bays and later to the
Bering Sea, Chukchi Sea, the Beaufort Sea, and the Gulf
of Alaska (Liu and Leendertse 1987). Other three dimen-
sional models include that of Oey (1985) who modeled the
Hudson-Raritan estuary.
SOURCES AND SINKS
In addition to the hydraulic regime of an estuary, the other
factors which have great influence on the water quality of
estuaries are the sources and sinks of the materials. The cir-
culation patterns and water movement in estuaries will dic-
tate the distribution of fresh and salt water in the estuary.
Superimposed on this distribution is another pattern made up
of materials introduced by sources and lost to sinks.
In all these equations, the source and sink terms become
zero for conservative substances. For non-conservative sub-
stances, reactions that take place may usually be represented
by first-order kinetics, i.e., the rate of reaction is propor-
tional to the concentration of the material. In some cases the
reaction term defines the fundamental reaction mechanism,
whereas in other uses it is an empirical approximation to the
phenomenon.
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