MODELING OF ESTUARINE WATER QUALITY 719
g acceleration of gravity
K frictional resistance coefficient
t time
b mean channel width
A cross-sectional area of the channel.The terms on the right hand side of the equation of
motion are, in sequence, the rate of momentum change by
mass transfer, the frictional resistance (with the absolute
value sign to assure that the resistance always opposes the
direction of flow), and the potential difference between the
ends of the channel element. In the continuity equation the
right hand side represents the change in storage over the
channel length per unit channel width. To minimize com-
putation, the equation of motion is applied to the channel
elements and the continuity equation to the junctions.
Both equations are rendered into partial difference form
and solved for each channel element and junction, using a
modified Runge-Kutta procedure. The results comprise the
predicted channel velocities, flows, and cross-sectional areas
and the predicted water surface elevations at each junction
for each time interval. These data are then input to the water
quality component of the model. The equations are put into
finite difference form and solved to give the concentration of
the substance at each junction.
Ward and Espey (1971) and Masch and Brandes (1971)
describe a segmented hydrodynamic and water quality model
which has been applied to Texas estuaries. Each segment is
a square one nautical mile on each side, and the estuary is
divided into these segments. Hydrodynamic transport across
segment boundaries is represented much as the equations
given above and occurs in response to forcing flows from
river inflow at the head of the estuary and tidal exchange
at the lower end. The model is able to simulate water stage
change within each segment and flows between segments
with change in tides, and the averages of the flows are used
in conjunction with the water quality portion of the model to
forecast concentrations of conservative and nonconservative
constituents.
A third type of two-dimensional model is that of
Leendertse (1970) who developed a water-quality simula-
tion model for well-mixed estuaries and coastal seas (i.e.,
no stratification) and applied it in Jamaica Bay, New York.
Leendertse and Gritton, 1971, have extended the model
to include the transport of several dissolved waste con-
stituents in the water, including any interactions among
them. The changing tide level influences the location of
the land-water boundaries in the shallow areas of coastal
waters. To simulate this process, procedures were devel-
oped in the model to allow for time-dependent boundary
changes. Large amounts of numerical data are generated
by the computer program developed from the simulation
model. To assist the investigator in extracting important
and meaningful results from these data, machine-made
drawings were used to graphically present the results of
the computation.
The basic mass-balance equation for 2-dimensional trans-
port of waste constituents in a well mixed estuary (uniformconcentration in the vertical directions) is given in Leendertse
(1970) as:
tHP
xHUP
yHVPxHDP
xyHDP
xyy() ( ) ( )⎛
⎝⎜⎞
⎠⎟−⎛
⎝⎜⎞
⎠⎟⎟HSA 0where
P integrated average over the vertical of the waste
constituents mass concentration
U and V vertically averaged fluid velocity (compo-
nents in the x (eastward) and y (northward) directions
respectively)
S A source function
D x and D y dispersion coefficients
H instantaneous depth at a point.The generalized mass-balance equation for n constitu-
ents is written in matrix notation as
tHP
xHUP
yHVP
xHDP
xyHDP
yxy() ( ) ( )⎛
⎝⎜⎞
⎠⎟⎛
⎝⎜⎞
⎠⎟⎟[]KHP HS Dwhere
P mass-concentration vector with n elements
[ K ] reaction matrix (^) S source and sink vector.
The reaction matrix [ K ] in its most general form can give
rise to a non-linear transport equation. This occurs because
the individual elements of the matrix can be defined as func-
tions of their own concentration, or that of other constitu-
ents or both. Since the elements of [ K ] are multiplied by the
elements of the concentration vector, such non-linear terms
imply kinetics of an order higher than first.
Point sources, such as occur at the location of sewage
discharges into the estuary, are simulated by adding delta-
function source terms to the source vector.
For two-dimensional flow in a well-mixed estuary, verti-
cal integration of the momentum and continuity equations
yields the following basic equations for the flow model
U
t
U
U
x
V
u
y
fV g
x
g
UU V
CH H
V
t
U
V
x
s
z
r
t
()^2212 /
2
1
0
x
V
V
y
fU g
y
g
VU V
CH H y
s
z
r
t
()^2212 /
2
1
= 0
C013_006_r03.indd 719C013_006_r03.indd 719 11/18/2005 12:49:01 PM11/18/2005 12:49:01 PM