Sediment and Contaminant Transport in Surface Waters - Chapter 5 doc

175
5
Hydrodynamic Modeling
When considering the currents in surface waters as diverse as rivers, lakes, estu-
aries, and nearshore areas of the oceans, it is evident that large variations exist
in the length and time scales describing these currents. Signicant length scales
vary from the vertical dimensions of the microstructure of stratied ows (as lit-
tle as a few centimeters) to the size of the basin (up to several hundred kilometers),
and time scales vary from a few seconds to many years. Although in principle
the equations of uid dynamics can describe the motions that include all these
length and time scales, practical difculties prohibit the use of the full equations
of motion for problems involving any but the smallest length and time scales.
Because of this, considerable effort and ingenuity have been expended to approxi-
mate these equations to obtain simpler equations and methods of solution.
The result is that many different numerical models of currents in surface
waters currently exist. The primary difference between these models is usually
the different length and time scales that the investigator believes is signicant
for the specic problem. For example, if the details of the ow in the vertical
are not thought of as signicant, one can use a two-dimensional, vertically inte-
grated model; this may be either steady state or time dependent, depending on
whether the time variation is considered signicant. To investigate ows where
vertical stratication due to temperature and/or salinity gradients is signicant
but horizontal ows in a transverse direction are not, a two-dimensional, hori-
zontally integrated model is relatively simple and may be useful. More complex
three-dimensional, time-dependent models may be necessary when ow elds
vary signicantly in all three directions and with time. In whatever numerical
model chosen, for reasons of accuracy and stability, the grid sizes in both space
and time must be smaller than the smallest space and time scales that are thought
to be signicant.
Fluid mechanics is a fascinating and diverse science; texts on the subject
abound and should be consulted for descriptions of the fundamental processes

and its many and diverse applications. Numerous texts also exist that emphasize
civil engineering applications such as river ooding and control, the design of
control structures, and the modeling of plumes from power plants or dredging
operations. The present chapter is rather brief and does not discuss these subjects;
its purpose is simply to give an overview of hydrodynamic modeling as applied
to the transport of sediments and contaminants in surface waters. Most examples
are rather elementary and are meant to illustrate interesting and signicant fea-
tures of ows that affect sediment and contaminant transport. A few more com-
plex examples are given to illustrate the present state of the art in hydrodynamic
modeling. In the following chapters, additional applications of hydrodynamic
© 2009 by Taylor & Francis Group, LLC
176 Sediment and Contaminant Transport in Surface Waters
models are described as a complement to the analyses of specic sediment and
contaminant transport problems. However, for a thorough understanding of the
rich and fascinating eld of uid mechanics in surface waters, additional articles,
texts, and conference proceedings should be consulted (e.g., Sorensen, 1978; Mei,
1983; Martin and McCutcheon, 1999; Spaulding, 2006).
In the present chapter, the basic three-dimensional, time-dependent conserva-
tion equations and boundary conditions that govern uid transport are presented
rst. This is followed by brief discussions of eddy coefcients, the bottom shear
stress due to currents and wave action, the surface stress due to winds, sigma
coordinates, and the stability of the numerical difference equations. By integra-
tion of the three-dimensional equations over the water depth, simpler and more
computationally efcient, vertically integrated (or vertically averaged) models
result and are the topic of Section 5.2. These reduced models are comparatively
easy to analyze and require little computer time, but they do not give details of
the vertical variation of the ow. For some problems of sediment and contami-
nant transport, this detail is not necessary and a vertically integrated model is
adequate and gives comparable results to those from a three-dimensional model.
Two-dimensional, horizontally averaged models also have been developed and

are useful for a qualitative understanding of the ow and for preliminary trans-
port studies where vertical variations of the ow are signicant but where ow
in one horizontal direction can be neglected (e.g., a thermally stratied lake or
a salinity stratied estuary). The basic equations and an example of this type of
model are presented in Section 5.3. Three-dimensional, time-dependent models
and applications of these models are described in Section 5.4. In the modeling of
sediment transport, an important parameter for erosion is the bottom shear stress;
this stress is due not only to currents but also to wave action. A simple model of
wave action and an application to Lake Erie are described in Section 5.5.
5.1 GENERAL CONSIDERATIONS IN THE
MODELING OF CURRENTS
5.1.1 B
ASIC EQUATIONS AND BOUNDARY CONDITIONS
The basic equations used in the modeling of currents are the usual hydrodynamic
equations for conservation of mass, momentum, and energy, plus an equation of
state. In waters with variable salinity, a conservation equation for salinity is also
needed. In sufciently general form for almost all modeling of surface waters,
these equations are the following.
Mass conservation:r
t
t

t
t

t
t

t
t


RRRR
t
u
x
v
y
w
z
0
(5.1)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 177
Conservation of momentum in the x-direction:r
t
t

t
t

t
t

t
t

t
t

t

t
t
t
u
t
u
u
x
v
u
y
w
u
z
fv
p
xx
A
u
x
H
1
R
¤¤
¦
¥
³
µ
´


t
t
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´
y
A
u
yz
A
u
z
Hv
(5.2)

Conservation of momentum in the y-direction:r
t
t

t
t

t
t

t
t

t
t

t
t
t
t
v
t
u
v
x
v
v
y
w
v

z
fu
p
yx
A
v
x
H
1
R
¤¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

t
t

t
t
¤
¦
¥
³
µ
´
y
A
v
yz
A
v
z
Hv
(5.3)
Conservation of momentum in the z-direction:r
t
t

t
t

t
t

t
t


t
t

t
t
t
t
¤
w
t
u
w
x
v
w
y
w
w
z
p
z
g
x
A
w
x
H
1
R
¦¦

¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´
y
A
w
yz

A
w
z
Hv
(5.4)
Energy conservation:r
t
t

t
t

t
t

t
t

t
t
t
t
¤
¦
¥
³
µ
´

t

t
T
t
u
T
x
v
T
y
w
T
zx
K
T
xy
K
H HHvH
T
yz
K
T
z
S
t
t
¤
¦
¥
³
µ

´

t
t
t
t
¤
¦
¥
³
µ
´

(5.5)
Salinity conservation:r
t
t

t
t

t
t

t
t

t
t
t

t
¤
¦
¥
³
µ
´

t
t
S
t
u
S
x
v
S
y
w
S
zx
K
S
xy
K
H HHv
S
yz
K
S

z
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´
(5.6)
An equation of state:r
S = S(p,S,T) (5.7)
where u, v, and w are uid velocities in the x-, y-, and z-directions, respectively
(z is positive upward); t is time; f is the Coriolis parameter, which is assumed
constant; p is the pressure; S is the density; A
H
is the horizontal eddy viscosity
and A
v

is the vertical eddy viscosity; K
H
is the horizontal eddy conductivity and
K
v
is the vertical eddy conductivity; g is the acceleration due to gravity; T is the
temperature; S is salinity; and S
H
is a heat source term. In this chapter, for sim-
© 2009 by Taylor & Francis Group, LLC
178 Sediment and Contaminant Transport in Surface Waters
plicity and following convention, the symbol S denotes the density of water; in
other chapters, S denotes the bulk density of the sediment, whereas S
w
denotes
the density of water.
Several approximations are implicit in these equations. These are (1) eddy
coefcients are used to account for the turbulent diffusion of momentum, energy,
and salinity; and (2) the kinetic energy of the uid is small in comparison with
the internal energy (which is proportional to the temperature) of the uid so that
energy transport (Equation 5.5) is dominant and can be described by the transport
of internal energy (temperature) alone.
However, these equations are more general, and hence more complex and
computer intensive, than usually necessary for most surface water dynamics. In
most applications, they can be further simplied by making the following approx-
imations: (1) vertical velocities are small in comparison with horizontal veloci-
ties so that a hydrostatic approximation is valid, and (2) variations in density are
small and can be neglected except in the buoyancy term in the vertical momen-
tum equation (the Boussinesque approximation). With these approximations, the
above equations reduce to the following:

t
t

t
t

t
t

u
x
v
y
w
z
0 (5.8)
t
t

t
t

t
t

t
t

t
t


t
t
t
t
u
t
u
x
uv
y
uw
z
fv
p
xx
A
u
r
H
2
1
R
xx
y
A
u
yz
A
u

z
Hv
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

(5.9)
t
t

t
t

t
t

t
t

t
t

t
t
t
t
v
t
uv
x
v
y
wv
z
fu
p

yx
A
v
r
H
2
1
R
xx
y
A
v
yz
A
v
z
Hv
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦

¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´
(5.10)
t
t

p
z
gR
(5.11)
t
t

t
t

t

t

t
t

t
t
t
t
¤
¦
¥
³
µ
´

t
t
T
t
uT
x
vT
y
wT
zx
K
T
xy
K

H HHvH
T
yz
K
T
z
S
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

(5.12)
t
t


t
t

t
t

t
t

t
t
t
t
¤
¦
¥
³
µ
´

t
t
S
t
uS
x
vS
y
wS

zx
K
S
xy
K
H HHv
S
yz
K
S
z
t
t
¤
¦
¥
³
µ
´

t
t
t
t
¤
¦
¥
³
µ
´

(5.13)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 179
S = S(S,T) (5.14)
where S
r
is a reference density.
The appropriate boundary conditions depend on the particular problem to be
solved. At the free surface, z = I(x,y,t), usual conditions include (1) the specica-
tion of a shear stress due to the wind,
RTRTA
u
z
A
v
z
vx
w
vy
w
t
t

t
t
, (5.15)
where T
x
w
and T

y
w
are the specied wind stresses in the x- and y-directions, respec-
tively; (2) a kinematic condition on the free surface,
t
t

t
t

t
t

HHH
t
u
x
v
y
w0
(5.16)
(3) the pressure is continuous across the water-air interface and therefore the uid
pressure at the surface equals the local atmospheric pressure p
a
,
p(x,y,I,t)=p
a
(5.17)
and (4) a specication of the heat ux at the surface,
qK

T
z
HT T
va

t
t
R () (5.18)
where q is the energy ux, H is a surface heat transfer coefcient, and T
a
is the
air temperature. At the bottom, the conditions are (1) those of no uid motion
or a specication of shear stress in terms of either integrated mass ux or near-
bottom velocity and (2) a specication of temperature or a specication of heat
ux. Variations in these boundary conditions and additional boundary conditions
are discussed in the following sections.
5.1.2 EDDY COEFFICIENTS
The numerical grid sizes for a problem usually are determined from consider-
ations of the physical detail desired and computer limitations. Once the grid size
is chosen, it is implicitly assumed that all physical processes smaller than this can
either be neglected or approximately described by turbulent uctuations. Turbu-
lent uctuations usually manifest themselves in an apparent increase in the vis-
cous stresses of the basic ow. These additional stresses are known as Reynolds
stresses. The total stress is the sum of the Reynolds stress and the usual molecular
viscous stress. In turbulent ow, the latter is comparatively small and therefore can
be neglected in most cases. Analogous to the coefcients of molecular viscosity,
© 2009 by Taylor & Francis Group, LLC
180 Sediment and Contaminant Transport in Surface Waters
an eddy viscosity coefcient can be introduced (as has been done in the equations
above) so that the shear stress is proportional to a velocity gradient. Similarly, an

eddy diffusion coefcient can be introduced so that the heat and salinity uxes are
proportional to temperature and salinity gradients, respectively. In turbulent ow,
these coefcients are not properties of the uid as in laminar ow but depend on
the ow itself, that is, on the processes generating the turbulence.
The determination of these turbulent eddy coefcients is a signicant prob-
lem in hydrodynamic modeling. Two approaches to this problem are (1) the use
of semi-empirical algebraic equations to relate the eddy coefcients to the local
ow conditions and (2) the use of turbulence theory to relate the eddy coefcients
to the turbulence kinetic energy and a turbulence length scale by means of trans-
port equations for these quantities. Both approaches depend on laboratory and
eld measurements to quantify parameters that appear in the analyses. However,
the use of turbulence theory, although more complex, is also more general and
requires less adjustment of parameters for a specic problem.
In the rst approach, semi-empirical algebraic relations are used that relate the
eddy coefcients to processes (which generate turbulence) and to density changes
(which reduce turbulence). Because the scale and intensity of the vertical and hori-
zontal components of turbulence are generally quite different, it is convenient to
consider these effects separately, as has been done in the equations presented above.
The vertical eddy viscosity, A
v
, and vertical eddy diffusivity, K
v
, should in general
vary throughout the system. Some of the more important generating processes of
this vertical turbulence and causes for its variation include (1) the direct action of
the wind stress and heat ux on the water surface, (2) the presence of vertical shear
in currents due to horizontal pressure gradients, (3) the presence of internal waves,
and (4) the effect of bottom irregularities and friction due to currents and waves.
If the density of the water increases with depth, stability effects will reduce
the intensity of the turbulence. These effects depend on the Richardson number,

dened by
Ri
g
z
u
z

t
t
t
t
¤
¦
¥
³
µ
´
R
R
2
(5.19)
where u is the mean horizontal velocity. Various empirical equations have been
developed that relate the eddy viscosity coefcient and the Richardson number. A
typical relation is that developed by Munk and Anderson (1948):
AA Ri
vv


0
12

110()
/
(5.20)
where A
v0
is the value of A
v
in a nonstratied ow. Values for A
v0
are generally on
the order of 1 to 50 cm
2
/s.
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 181
In a nonstratied ow, it is believed that the eddy diffusivity is approximately
equal to the eddy viscosity. However, for a stratied ow, the mechanisms of
turbulent transfer of momentum and heat are somewhat different, and this leads
to different dependencies of these coefcients on the Richardson number. For
example, a semi-empirical relation (Munk and Anderson, 1948) similar to Equa-
tion 5.20 suggests that
K
v
=K
v0
(1 + 3.33 Ri)
−3/2
(5.21)
where K
v0

is the value of the vertical eddy diffusivity in a nonstratied ow. Many
additional relationships for A
v
and K
v
similar to the above two equations have
been proposed and used.
Horizontal viscosity coefcients are generally much greater than the vertical
coefcients. It is found from experiments that the values of the horizontal vis-
cosity coefcient increase with the scale  of the turbulent eddies. An empirical
relation of this type is
Aa
H
E
13 43//
C (5.22)
where a is a constant and F is the rate of energy dissipation (e.g., see Okubo, 1971;
Csanady, 1973). Observations indicate values of 10
4
to 10
5
cm
2
/s for A
H
for the
overall circulation in the Great Lakes (Hamblin, 1971), with smaller values indi-
cated in the nearshore regions. In numerical models, A
H
and K

H
are often chosen
as the minimum values required for numerical stability (see below).
Implicit in the use of algebraic equations for the eddy coefcients is the
assumption that the production and dissipation of turbulent mixing are in local
equilibrium (Bedford, 1985). However, in many cases, this is not an accurate
assumption and the transport of turbulence must be considered. For this purpose,
equations have been developed that describe the transport of turbulent kinetic
energy, k, and viscous dissipation, F. Alternately, because the viscous dissipation
is proportional to k
3/4
/, where  is a turbulence length scale, a transport equation
for  can be used instead of a transport equation for F. In either case, the resulting
model for turbulence is generally known as a k-F (turbulence closure) model.
An example of a k-F model that is widely used is the following (Mellor and
Yamada, 1982; Galperin et al., 1988; Blumberg et al., 1992). In this model, the
vertical mixing coefcients are given by
AA KK
vvM vvH
 
ˆ
,
ˆ
UU
(5.23)
ˆ
,
ˆ
AqS KqS
vMvH

CC (5.24)
where q
2
/2 is the turbulent kinetic energy, S
M
and S
H
are stability functions dened
by solutions to algebraic equations given by Mellor and Yamada (1982) as modi-
ed by Galperin et al. (1988), and V
M
and V
H
are constants. The stability functions
© 2009 by Taylor & Francis Group, LLC
182 Sediment and Contaminant Transport in Surface Waters
account for the reduced and enhanced vertical mixing in stable and unstable verti-
cally density-stratied systems in a manner similar to Equations 5.20 and 5.21.
The variables q
2
and are determined from the following transport (or con-
servation) equations:
t
t

t
t

t
t

t
t

t
t
t
t
q
t
uq
x
vq
y
wq
zz
K
q
z
q
22 2 2 2
() ()( )
ĐĐ
â
ă
ả
á
ã

t

t
Ô
Ư
Ơ

à


t
t
Ô
Ư
Ơ

à

Đ
â
ă
ă
ả
á
ã
ã
2
22
A
u
z
v

z
v
22
2
3
1
g
K
z
q
B
F
o
vq
R
Rt
t

C
(5.25)
t
t

t
t

t
t

t

t

t
t
() ( ) ( ) ( )q
t
uq
x
vq
y
wq
zz
K
222 2
CCC C
qq
v
q
z
EA
u
z
v
z
t
t
Đ
â
ă
ả

á
ã

t
t
Ô
Ư
Ơ

à


t
t
Ô
Ư
Ơ

à
()
2
1
2
C
C

Đ
â
ă
ă

ả
á
ã
ã

t
t
ê
ô

ơ

ạ


ằ


2
3
1
g
K
z
q
B
F
o
v
R

R
W

C
(5.26)
where K
q
=0.2q, the eddy diffusion coefcient for turbulent kinetic energy; F
q
and F

represent horizontal diffusion of the turbulent kinetic energy and turbulence
length scale and are parameterized in a manner analogous to Equation 5.22;

W
is a
wall proximity function dened as

CWK1
2
2
EL(/ ); (L)
1
=(I z)
1
+(h+z)
1
;
L is the von Karman constant; h is the water depth; I is the free surface elevation;
and E

1
, E
2
, and B
1
are empirical constants set in the closure model.
The above and similar k-F models have been used extensively, and the coef-
cients appearing in them have been determined from laboratory experiments as well
as from comparison of results of the numerical models with eld measurements.
Applications of these models are described in the following sections and chapters.
5.1.3 BOTTOM SHEAR STRESS
5.1.3.1 Effects of Currents
A shear stress is produced at the sediment-water interface due to physical interac-
tions between the owing water and the bottom sediments. This stress depends
in a nonlinear manner on the ow velocity and the bottom roughness. It has been
investigated extensively and quantied approximately by means of laboratory
experiments, eld tests, and model calibrations. Results for the shear stress are
usually reported as
ẩ = Sc
f
qq (5.27)
â 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 183
where c
f
is a coefcient of friction and is dimensionless; q is a near-bed, or refer-
ence, velocity; and a bold symbol denotes a vector (i.e., the shear stress is a vector
aligned with the ow velocity). The components of the stress can be written as
U
x

= Sc
f
qu (5.28)
U
y
= Sc
f
qv (5.29)
where q=(u
2
+v
2
)
1/2
.
The coefcient of friction depends on the grain size of the sediment bed; sedi-
ment bedforms such as mounds, ripples, and dunes; and any biota present. Typical
values for c
f
are between 0.002 and 0.005. In the absence of site-specic informa-
tion, a value of 0.003 often is chosen for smooth, cohesive sediment beds. A more
accurate value for c
f
can be determined as an average over the entire sediment bed
from calibration of modeled to measured currents when the latter are available.
From turbulent ow theory and assuming a logarithmic velocity prole, it can
be shown that the coefcient of friction can be determined from
c
h
z

f

¤
¦
¥
³
µ
´
K
2
0
2
2
ln
(5.30)
where L is von Karman’s constant (0.41), z
0
is the effective bottom roughness,
and h is a reference distance or depth. For three-dimensional models or two-
dimensional, horizontally integrated models, h is the thickness of the lowest layer
of the numerical grid; for vertically integrated models, h is the water depth. From
this formula and by assuming a value for z
0
, c
f
can be calculated and will vary
locally as a function of depth and z
0
. However, the difculty with this formula is
that an effective value for z

0
is difcult to determine accurately. This is often done
by model calibration.
For a more accurate determination of z
0
and c
f
, eld measurements of veloc-
ity proles can be used (e.g., Cheng et al., 1999; Sea Engineering, 2004). As a
specic example, consider the measurements and analyses made by the USGS
and Sea Engineering, Inc., for the Fox River (Sea Engineering, 2004). For fully
developed turbulent ow in a river, the near-bottom velocity prole is given by
u
u
n
z
z

*
K
C
0
(5.31)
where u
*
=(U/S)
1/2
. This is known as the universal logarithmic velocity prole, or
law of the wall (e.g., Schlichting, 1955). It can be rewritten as
© 2009 by Taylor & Francis Group, LLC

184 Sediment and Contaminant Transport in Surface Waters
CCnz
u
unz
K
*
0
(5.32)
Once u(z) is obtained from eld measurements and n z is plotted as a function of
u, then the z intercept of the line given by the above equation is z
0
and the slope
of the line is L/u
*
.
To determine u(z), more than 100 vertical proles of current velocities were
measured at various locations in the Fox during four separate ow events. From
this, z
0
and u
*
were determined. Because U = Su
*
2
, the coefcient of friction then
can be calculated from Equation 5.27 as
c
U
f
avg


T
R
2
(5.33)
where U
avg
is the vertically averaged velocity. For each vertical prole, c
f
as deter-
mined in this way is shown in Figure 5.1. Also shown is c
f
as determined from
the law of the wall (Equation 5.30) with z
0
equal to 0.2 cm, the average from all
the measurements. There is a reasonable t to the data except for shallow waters,
where the law of the wall under-predicts the values of c
f
compared with measured
data. This discrepancy may be due to the larger irregularities (bottom roughness)
in the local bathymetry near shore; this would increase z
0
and hence c
f
. Because
of this discrepancy, a semi-empirical equation given by
c
h
f



0 004
119 128
.
. exp( . )
(5.34)
was chosen to approximate the data (Figure 5.1) and was later used in hydrody-
namic modeling of the Fox River.
0.025
0.020
0.015
0.010
0.005
0
0 1 2 3
Eq. 5.30
Eq. 5.34
Data
4 5 6 7 8
Depth (m)
C
f
9
FIGURE 5.1 Bottom shear stress coefcient of friction, c
f
, as a function of depth from
measurements, from the law of the wall with z
0
= 0.2 cm, and from Equation 5.34. (Source:

From Sea Engineering, 2004.)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 185
In general, the total shear stress acting on the sediment bed is considered
to consist of a shear stress due to skin friction and a shear stress due to form
drag. The former is due to viscous forces acting tangential to the surface, whereas
the latter is due to the normal stresses (mostly pressure) acting on the surface.
Because form drag is caused by normal forces, it is not considered to contribute
signicantly to the shear stress causing sediment erosion. Equation 5.30 implicitly
includes the effect of form drag when z
0
is an effective bottom roughness; this
usually is determined as part of the hydrodynamic model calibration. Explicit
equations for the form drag due to sand dunes in rivers have been developed
(Einstein and Barbarossa, 1952; Engelund and Hansen, 1967; Wright and Parker,
2004); for sandy sediments, the bottom shear stress then can be explicitly sepa-
rated into a frictional shear stress and a shear stress due to form drag and calcu-
lated in this way.
5.1.3.2 Effects of Waves and Currents
In general, bottom shear stresses in surface waters are generated by a combina-
tion of waves and currents. A highly nonlinear relationship exists between the two
processes. Grant and Madsen (1979) have developed a detailed description of these
processes. They assumed that the water column near the sediment bed can be sepa-
rated into a wave boundary layer and a current boundary layer; the large scale of
the current boundary layer makes the velocity gradient associated with it smaller
than the velocity gradient for the wave boundary layer. Because of this, even if the
magnitudes of the wave orbital velocity and the current velocity are the same, the
shear stresses due to the waves will be greater than those due to the currents.
Soulsby et al. (1993) reviewed much of the experimental and theoretical work
on wave and current interactions. The model chosen for presentation here (Christ-

offersen and Jonsson, 1985) was developed from the original Grant and Madsen
(1979) work and was shown to reproduce experimental data well. In this model,
the total bottom shear stress, È, can be divided into the periodic wave component,
È
w
, and the steady current component, È
s
:
È = È
w
+ È
s
(5.35)
where the current and wave make an arbitrary angle, R, with each other. The shear
stress due to a current was represented as
TR
sf
cu
1
2
2
(5.36)
whereas the shear stress due to waves was written as
TR
wwbm
cu
1
2
2
(5.37)

© 2009 by Taylor & Francis Group, LLC
186 Sediment and Contaminant Transport in Surface Waters
where c
w
is the wave friction factor and u
bm
is the bottom wave orbital velocity. By
developing a model for the nonlinear interaction of waves and currents, Christof-
fersen and Jonsson (1985) were able to express the total bottom shear stress as
TR
1
2
2
cum
wbm
(5.38)
where
m 12
2
SSQcos( ) (5.39)
and
S
cu
cu
f
wbm
2
2
(5.40)
The current friction factor is calculated from

2 1 30 1
c
h
ek
k
k
fN
A
N

KK
ln ln (5.41)
with the apparent roughness, k
A
, dened as
k
kk km
A
N
w
N
w
N

¤
¦
¥
³
µ
´

30
DKD
B
S
exp (5.42)
and where E
w
, the wave boundary layer thickness, is calculated from
DP
B
w
N
k
J 045
2
. (5.43)
The existence of the wave boundary layer causes the currents to experience a
larger bed shear than they would in a condition with no waves. The roughness that
is felt by the currents is therefore larger than the Nikuradse roughness, k
N
. This is
reected through the apparent roughness, k
A
.
The wave friction factor is given by
c
mJ
w

2B

(5.44)
where C is 0.747 and J is given by
J
u
k
fm
NA

W
(5.45)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 187
where X is the absolute wave frequency and u
fm
is the wave friction velocity
dened as
u
mc
u
fm
w
bm

2
(5.46)
These methods have been applied successfully in modeling studies (e.g., Chroneer
et al., 1996) and have been applied to numerous eld studies. Drake and Cacchione
(1992) provide an example of the application of these and similar techniques to calcu-
lating wave and current generated shear stresses in the eld from velocity proles.
The bottom boundary layer and shear stress are also inuenced by suspended

sediment stratication. A general model that includes wave-current interaction
and stratication due to suspended sediment has been developed by Glenn and
Grant (1987) and improved by Styles and Glenn (2000).
5.1.4 WIND STRESS
In the modeling of the wind-driven circulation in surface waters, it is necessary to
know the horizontal shear stress imposed as a boundary condition at the surface.
This stress, U
w
, is caused by the interaction of the turbulent air and water. The
relation of this stress to the wind speed is difcult to determine from theoretical
considerations, and its value is usually based on semi-empirical formulas and on
observations. Similar to the shear stress at the sediment-water interface, a com-
mon relation assumed between these quantities is
TT
w
ad a
n
a
CW

R
1
W (5.47)
where C
d
is a drag coefcient, S
a
is the density of the air, W
a
is the wind velocity

at 10 m above the water surface, and n is an empirically determined exponent not
necessarily an integer.
Wilson (1960) has analyzed data from many different sources and has given
a best t to the data. For W
a
in units of centimeters per second (cm/s), S
a
in units
of grams per cubic centimeter (g/cm
3
), and U
w
in units of dynes per square centi-
meter (dynes/cm
2
), Wilson suggests a value of n = 2 and C
d
= 0.00237 for strong
winds and 0.00166 for light winds. From a comparison of eld data and results of
numerical models, Simons (1974) suggests values of n = 2 and C
d
= 0.003.
An additional problem in determining the wind stress is that W
a
in the above
formula is the wind velocity at the specied location over the water. Unfortunately,
wind velocities are generally measured on shore and not at the desired over-the-
water location. For large bodies of water, over-the-water winds tend to be higher
than the land values by as much as a factor of 1.5. For rivers and smaller lakes,
over-the-water winds tend to be smaller than those measured at land locations far

from the body of water. For these smaller bodies of water, a wind-shelter coef-
cient has been proposed (Cole and Buchak, 1995) that has a range between 0 and
1, depending on the shape and size of the water body.
© 2009 by Taylor & Francis Group, LLC
188 Sediment and Contaminant Transport in Surface Waters
5.1.5 SIGMA COORDINATES
In a Cartesian grid, the vertical distances between grid points depend only on z
and not on x or y. In a variable-depth basin, the number of grid points between the
sediment-water and air-water interfaces is therefore variable. Because of this, in a
basin with large depth variations, it is difcult to obtain adequate resolution of the
ow in the vertical in both the deep and shallow parts of the basin. This problem
is minimized by the use of a sigma coordinate in the vertical. The dimensionless
sigma coordinate is dened as
S
H



hz
h
0
0
(5.48)
where z is the physical vertical coordinate measured in the upward direction
from the undisturbed water surface, h
0
(x, y) is the depth of the undisturbed water,
I(x, y, t) is the displacement of the water surface above z = 0, and 0 < T <1. The
advantage of the sigma coordinate is that it divides the water column into the same
number of layers in the vertical independent of the water depth, h = h

0
+ I. A
disadvantage is that the transformation can produce errors in computing the pres-
sure gradient term; these may be signicant when steep bottom slopes are present.
Another disadvantage is that the small distances between grid points in the verti-
cal that are often generated can decrease the numerical stability of the equations.
The source of the pressure gradient difculty can be understood as follows. In
T coordinates, the x-component of the pressure gradient is written as
t
t
¤
¦
¥
³
µ
´

t
t
¤
¦
¥
³
µ
´

t
t
t
t

¤
¦
¥
³
µ
´
p
x
p
xh
h
x
p
zxS
S
S
(5.49)
The potential error in calculating this gradient is due to the numerical truncation
errors that occur from the terms on the right-hand side when these are approxi-
mated by nite differences. The rst term on the right-hand side is the pressure
gradient along lines of constant T and depends on changes in bottom topogra-
phy. The second term is the pressure gradient in the vertical across lines of con-
stant T. When steep bottom slopes are present, both of these terms may be large,
comparable in magnitude, but opposite in sign. Errors then arise due to small
truncation errors in each of these terms, that is, small differences between large
numbers.
This type of error has been extensively investigated (Haney, 1991; Mellor et
al., 1994, 1998). The general conclusion from these studies is that it can be mini-
mized by the use of ne grids in both the horizontal and vertical directions. As an
example, an investigation of this type of error for the modeling of currents using

the hydrodynamic model EFDC in the Lower Duwamish Waterway (Arega and
Hayter, 2007) is summarized in Section 5.4.
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 189
5.1.6 NUMERICAL STABILITY
In the numerical modeling of currents, the basic differential equations are approx-
imated by difference equations and then solved by various numerical procedures.
These difference equations involve grid sizes in space (e.g., %x) and time (%t).
For numerical efciency, one would like to use space and time steps as large
as are consistent with the accuracy and physical detail desired. However, other
restrictions on the allowable space and time steps are dictated by the stability of
the calculation procedure. These restrictions, of course, depend on the particular
numerical scheme used but, when present, can usually be related to the physical
space and time scales of the problem.
For example, consider an explicit, forward-time, central-space scheme. Sim-
ple one-dimensional theory indicates that limits on the time step ∆t are given
approximately by the following:
1. %t<%x/(gh)
1/2
, a restriction indicating that the numerical time step must
be less than the time it takes a surface gravity wave (speed of (gh)
1/2
) to
travel the horizontal distance between two grid points %x
2. %t<%x/u, the time step must be less than the time it takes a uid particle
to be convected horizontally a distance %x
3. %t<(%z)
2
/2A
v

, the time step must be less than the time for turbulent dif-
fusion between two grid points in the vertical
4. %t<(%x)
2
/2A
H
, the same argument as (3) but applied to horizontal
diffusion
5. %t<2Q/f, the time step must be less than the inertial period due to rotation
6. %t<%x/u
i
, where u
i
is the speed of an internal wave, an argument simi-
lar to (1) above but for internal waves
As an example, consider parameters appropriate for a large lake: u = 50 cm/s,
h=20 m, %x=2 km, %z=1 m, A
v
=10 cm
2
/s, A
h
=10
5
cm
2
/s, f=10
-4
/s, and
g=980 cm

2
/s. Corresponding to the above listing, the limiting time steps are then
approximately (1) 140 s, (2) 4000 s, (3) 500 s, (4) 2 × 10
5
s, (5) 6 × 10
4
s, and (6)
4×10
5
s. The limiting time step depends on the smallest allowed time, 140 s in
this example. However, this restriction was implicitly based on a constant-depth
basin, or average depth. For a variable depth basin where the largest and smallest
depths are 20 and 1 m, respectively, and where a T-coordinate is used with 20 grid
points in the vertical (as above), the smallest %z is 0.05 m, and the limiting time
step described by (3) is now only 1.25 s instead of 500 s.
All the above rules are equivalent to stating that the numerical time step must
be less than the time it takes for a disturbance to propagate between two grid
points. Each of these restrictions can be eliminated by various numerical proce-
dures. However, each numerical procedure has its own difculties, and which pro-
cedure is most advantageous depends on the particular problem being studied.
© 2009 by Taylor & Francis Group, LLC
190 Sediment and Contaminant Transport in Surface Waters
5.2 TWO-DIMENSIONAL, VERTICALLY INTEGRATED,
TIME-DEPENDENT MODELS
5.2.1 B
ASIC EQUATIONS AND APPROXIMATIONS
Two-dimensional, vertically integrated conservation equations are obtained by
integrating the three-dimensional equations of motion (Equations 5.8 through
5.14) over the water depth from z = −h
o

(x,y) to z = I(x,y,t), where h
o
is the equi-
librium water depth and I is the surface displacement from equilibrium (e.g., see
Ziegler and Lick, 1986). From Equations 5.8 through 5.11, the resulting equations
for mass and momentum conservation are
t
t

t
t

t
t

H
t
U
x
V
y
0 (5.50)
t
t

t
t

t
t


t
t
¤
¦
¥
³
µ
´
U
t
gh
x
fV A
U
x
U
y
x
w
xH
H
TT
2
2
2
2

t
t


t
t
(/) ( /)Uh
x
UV h
y
2
(5.51)
t
t

t
t

t
t

t
t
¤
¦
¥
³
µ
´
V
t
gh
y

fU A
V
x
V
y
y
w
yH
H
TT
2
2
2
2

t
t

t
t
(/)(/)UV h
x
Vh
y
2
(5.52)
where g is the acceleration due to gravity, h = h
o
+ I is the total water depth, and
U and V are vertically integrated velocities dened by

Uudz
h
o


¯
H
(5.53)
Vvdz
h
o


¯
H
(5.54)
where u and v are the velocities in the x- and y-directions, respectively, and z
is the vertical coordinate. Equations 5.50 through 5.52 are also often written in
terms of the vertically averaged velocities, dened as
U
h
udz
U
h
avg
h
o


¯

1
H
(5.55)
V
h
vdz
V
h
avg
h
o


¯
1
H
(5.56)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 191
When this is done, the resulting equations are referred to as vertically averaged
equations. Whether written in terms of vertically integrated or vertically averaged
velocities, both sets of conservation equations lead to essentially the same results.
The bottom stress is represented by È, whose components are given by
T
xf
cq
U
h
 (5.57)
T

yf
cq
V
h
 (5.58)
where q is the velocity magnitude and c
f
is a bottom shear stress coefcient of
friction, as in Equation 5.30.
Numerous vertically integrated (or vertically averaged), two-dimensional,
time-dependent hydrodynamic models have been developed. Some are publicly
available; some are not. Examples of publicly available models include TABS-2
(USACE/WES; Thomas and McAnally, 1985); HSCTM-2D (EPA CEAM; Hay-
ter et al., 1999); FESWMS-2D (USGS; Froelich, 1989); and SEDZL and SED-
ZLJ (Ziegler and Lick, 1986; Jones and Lick, 2001a) as well as two-dimensional
versions of primarily three-dimensional models (see Section 5.4). SEDZL and
SEDZLJ were primarily developed as sediment transport models, and that aspect
of the models is discussed in some detail in Chapter 6; both these models are
essentially the same as far as hydrodynamics are concerned. Results of hydro-
dynamic calculations with SEDZL and SEDZLJ are presented here to illustrate
the two-dimensional modeling of currents and for further use in Chapter 6 in the
modeling of sediment transport. To illustrate the similarities and differences in
the currents in rivers and lakes, calculated results are shown for the Lower Fox
River and Lake Erie.
5.2.2 THE LOWER FOX RIVER
Because of the extensive hydrodynamic and sediment data available for the Lower
Fox, this river was chosen as an application of SEDZLJ (Jones and Lick, 2000,
2001a). The specic area of the river considered was from the DePere Dam to
Green Bay (Figure 1.2), a distance of approximately 11 km. A bathymetric map
of this portion of the river is shown in Figure 5.2. The upstream region of the

river is fairly wide (more than 0.5 km in some regions), with shallow pools in the
nearshore regions as little as 1 m in depth. Previous dredging in the center of the
channel created a channel 7 to 8 m deep; this has partially lled up with time but,
in the upstream portion, was still up to 5 m deep in 2002.
The ow into this section of the Lower Fox is controlled primarily by the
DePere Dam. In the calculations, it was assumed that approximately 10% of any
given ow at this dam concurrently ows in through the East River; however, dur-
ing some large events, this ow can be considerably higher. Flow contributions
due to runoff and other smaller tributaries were considered negligible. Flow rates
© 2009 by Taylor & Francis Group, LLC
192 Sediment and Contaminant Transport in Surface Waters
during a typical year vary from 30 to 280 m
3
/s. The highest ow rate in the past
80 years was about 650 m
3
/s. High ow events, in general, are caused by opening
the dams or by large storms.
In the numerical calculations, a 30- by 90-m rectangular grid was generated
to discretize the river. The water depth at each node was determined from NOAA
navigational charts. A no-slip boundary condition was imposed along the shore-
line to approximate the very low ow areas in the shallow regions near the shores
of the river.
Constant ow rate calculations were rst made to determine the general
characteristics of the hydrodynamics of the river under various ow conditions.
Calculations also were made for variable ow (ood) conditions. Results for a
constant ow rate over the DePere Dam of 280 m
3
/s, a 99.7 percentile (or once-in-
one-year) ow, are shown here.

The bottom shear stress coefcient of friction was calculated from Equation
5.30 with a z
0
of 100 µm and is shown in Figure 5.3. The minimum value is 0.002,
whereas the maximum value is 0.0032 and is located in the shallow regions. Fig-
ure 5.4 shows the velocity vectors and shear stress contours. As the ow enters
upstream at the DePere Dam, it makes a 90° turn into a wide, shallow region.
The region just below the dam has velocities of 35 cm/s and shear stresses up to
0.5 N/m
2
that rapidly drop off as the ow turns and slows. The currents and shear
stresses are higher in the channel than in the nearshore. Midway downstream,
the river makes a 45° turn and narrows. Due to the narrow cross-section, the ow
accelerates to 50 cm/s in some regions and produces bottom shear stresses up to
0.5 N/m
2
. At the East River, more water is added to the ow and the center chan-
nel narrows; this results in a maximum velocity of 65 cm/s and a corresponding
shear stress of 0.8 N/m
2
. The ow slows as the river widens into Green Bay. These
and similar calculations are used as the basis for the modeling of sediment trans-
port in the Fox River, as discussed in Section 6.4.
At a later time, between May 2003 and July 2004, the USGS conducted four
eld surveys that included measurements of vertical velocity proles at 30 loca-
tions in the Fox River from DePere Dam to Green Bay. This period included four
ow periods: low ow (100 m
3
/s), which was dominated by seiche motion from
DePere Dam

1
1
2
2
4
4
8
8
Green Bay
East River
4
FIGURE 5.2 Bathymetry of Lower Fox River from DePere Dam to Green Bay. Depth
in meters. For clarity, distances in vertical direction are scaled by a factor of 3 compared
to the horizontal.
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 193
Green Bay; two moderate ows (200 to 300 m
3
/s); and high ow (400 to 450 m
3
/s).
Calculated shear stresses ranged from 0.01 to 1.38 N/m
2
.
Essentially the same model as described above (but with more accurate
descriptions of the bottom shear stress and water depths) was then applied to
these four ow periods (Sea Engineering, 2004). Good agreement (correlation
coefcients of 0.80 or greater) between the measured and calculated verti-
cally averaged velocities was obtained (Figure 5.5). In this gure, the furthest
DePere Dam

0.003
0.0025
0.0021
0.003
East River
Green Bay
FIGURE 5.3 Coefcient of friction, c
f
, with z
0
= 100 µm. (Source: From Jones and Lick,
2000. With permission.)
60
50
40
30
20
10
0
0 102030
Measured Velocities (cm/s)
Modeled Velocities (cm/s)
40 50 60
FIGURE 5.5 Measured and calculated vertically averaged velocities for four events
between May 2003 and July 2004. (Source: From Sea Engineering, 2004.)
© 2009 by Taylor & Francis Group, LLC
Reference velocity vector
50 cm/s
0.05
0.1

0.4
0.4
0.2
Maximum shear stress = 0.8 N/m
2
figurE 5.4 Velocity vectors and shear stress contours for a constant ow rate of 280
m
3
/s. (Source: From Jones and Lick, 2000. With permission.)
194 Sediment and Contaminant Transport in Surface Waters
outliers generally occurred during periods of rapid ow variations or upstream
ow at the mouth.
5.2.3 WIND-DRIVEN CURRENTS IN LAKE ERIE
Lake Erie (Figure 2.2) is relatively shallow compared to the other Great Lakes.
Topographically, it is convenient to divide it into a Western Basin (average depth
of 7 m, or 24 ft); a Central Basin (average depth of 19 m, or 60 ft); and an Eastern
Basin (average depth of 26 m, or 80 ft). Because of the differences in the depths
of the basins, the hydrodynamics, wave action, and sediment and contaminant
transport are distinctly different for each basin. For this lake, a series of numeri-
cal calculations of the wind-driven currents, wave action, and the resulting sedi-
ment transport has been made (Lick et al., 1994a). The purpose was to illustrate
the basic characteristics of these processes. Results for the wind-driven currents
are shown here, for wave action in Section 5.5, and for sediment transport in
Section 6.5.
Calculations of wind-driven currents were made for a variety of wind direc-
tions and magnitudes, and also for a major storm. For the calculation presented
here, the period of the event was 13 days. It was assumed that the wind direction
was constant from the southwest (the dominant wind direction) throughout the
period. However, the wind speed was assumed to be variable and was constant
at 5 mph for 2 days, constant at 22.5 mph (moderately high winds) for 1 day, and

then constant at 5 mph for 10 days. Results of calculations at the end of 3 days
(end of high wind period) are shown in Figure 5.6 (currents) and Figure 5.7 (bot-
tom shear stresses).
In Figure 5.6, it can be seen that the nearshore ows are in the direction of the
wind, whereas the offshore ows are opposed to the wind. This is typical of wind-
driven currents in lakes. Vertically averaged velocities are approximately 5 to
10 cm/s. In Figure 5.7, constant bottom shear stresses (due to waves and currents)
of 0.1, 1.1, 2.1, and 3.1 N/m
2
are plotted. A shear stress of 0.1 N/m
2
is approxi-
mately the stress at which sediments just begin to be resuspended. The stresses in
most of the Central and Eastern basins but only a small part of the Western Basin
5 CM/S
FIGURE 5.6 Lake Erie. Vertically averaged, wind-driven currents at end of storm.
Winds are from the southwest at 22.5 mph for a period of 1 day. (Source: From Lick et al.,
1994a. With permission.)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 195
are below 0.1 N/m
2
; little or no resuspension therefore occurs in these regions for
this moderate wind. The highest shear stresses and therefore highest resuspen-
sions occur in narrow zones along the northeastern shore of the Central Basin and
the eastern shore of the Eastern Basin where the wave action is strongest. For all
wind speeds, the shear stress is primarily due to wave action, whereas the effects
of currents on the shear stress are generally only a small correction.
The fact that the highest resuspension generally occurs in the nearshore areas
of Lake Erie as well as other lakes is due to the high wave action and resulting

high shear stresses in these regions. This is in contrast to the Fox and other rivers
where the highest shear stresses and highest resuspension generally occur in the
central, deeper parts of the river; this difference is due to the higher currents that
are present there and that are the main cause of the bottom shear stress in rivers.
5.3 TWO-DIMENSIONAL, HORIZONTALLY
INTEGRATED, TIME-DEPENDENT MODELS
Two-dimensional, horizontally integrated (or horizontally averaged), time-
dependent models often are used to predict longitudinal and vertical variations
in velocities, temperatures, and/or salinities in stratied reservoirs and estuaries,
that is, in systems where vertical variations are signicant but where horizontal
velocities transverse to the longitudinal direction can be neglected compared with
longitudinal and vertical velocities. The conservation equations for this type of
model are obtained by integrating the three-dimensional equations in the trans-
verse direction. Numerous models of this type exist, for example, the Laterally
Averaged Reservoir Model (Buchak and Edinger, 1984a, b); the CE-QUAL-W2
model (Cole and Buchak, 1995); and a model by Blumberg (1975, 1977). The lat-
ter two models have been used most extensively.
In small lakes and reservoirs, the Coriolis force and the associated transverse
velocities can normally be neglected. However, in Lake Erie, because of its size,
this is no longer correct. In this case, a modied version of a two-dimensional,
time-dependent model that included the Coriolis force was used to investigate
the temperatures and currents in Lake Erie (Heinrich et al., 1981). In this investi-
gation, the emphasis was on the general characteristics of thermocline formation,
0.1
0.1
0.1
0.1
0.1
0.1
0.1

1.1
2.1
0.1
0.1
1.1
2.1
3.1
1.1
1.1
FIGURE 5.7 Lake Erie. Bottom shear stress (N/m
2
) at end of storm. Winds are from the south-
west at 22.5 mph for a period of 1 day. (Source: From Lick et al., 1994a. With permission.)
© 2009 by Taylor & Francis Group, LLC
196 Sediment and Contaminant Transport in Surface Waters
maintenance, and decay, and therefore the time scales of interest were weeks
and months rather than hours or even days. The purpose was to approximately
reproduce and understand the main features of the observed spatial and temporal
distributions of temperatures and currents in Lake Erie as an example of a large
stratied lake. In particular, calculations were made to predict conditions in a
vertical cross-section of the lake from Port Stanley in the north to Ashtabula in
the south (a north–south line just west of the Ohio–Pennsylvania border, Fig-
ures 2.2 and 5.8). Some of the results of this investigation are described here.
5.3.1 BASIC EQUATIONS AND APPROXIMATIONS
In the analysis, it was assumed that properties of the ow varied only in the ver-
tical direction, z, and one horizontal direction in the plane of the cross-section,
x, but not in the other horizontal direction perpendicular to the cross-section, y.
Because of this, derivatives of u, v, w, and T with respect to y could be neglected.
Coriolis forces were included. These forces induce velocities in the y-direction
that are approximately independent of y but are dependent on x and z. A net ux

in the y-direction therefore results if a zero pressure gradient in the y-direction
is assumed. Over sufciently long time scales such that averaged seiche effects
are negligible (and these are the time scales that are of interest here), this net
ux in the y-direction must be zero. In the analysis, a pressure gradient in the
y-direction independent of x and z was therefore assumed such that this net ux in
the y-direction was zero. Because of this assumption, velocities in the transverse
direction are present and are functions of x, z, and t. The model is a simplica-
tion to two dimensions of a three-dimensional, time-dependent model previously
developed and used (Paul and Lick, 1974).
The basic equations used in the analysis are the usual hydrodynamic equa-
tions for conservation of mass, momentum, and energy, plus an equation of state.
With the above assumptions, these are as follows:
t
t

t
t
¤
¦
¥
³
µ
´

u
x
w
z
0
(5.59)

t
t

t
t

t
t

t
t

t
t
t
t
¤
¦
¥
³
µ
u
t
u
x
uw
z
fv
p
xx

A
u
x
r
H
2
1
R
´´

t
t
t
t
¤
¦
¥
³
µ
´
z
A
u
z
V
(5.60)
t
t

t

t

t
t

t
t

t
t
t
t
¤
¦
¥
³
µ
v
t
uv
x
vw
z
fu
p
yx
A
v
x
r

H
1
R
´´

t
t
t
t
¤
¦
¥
³
µ
´
z
A
v
z
V
(5.61)
t
t

p
z
gR
(5.62)
t
t


t
t

t
t

t
t
t
t
¤
¦
¥
³
µ
´

t
t
t
t
T
t
uT
z
wT
zx
K
T

xz
K
T
z
Hv
(
¤¤
¦
¥
³
µ
´
(5.63)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 197
S = S(T) (5.64)
R
vdxdz
¯¯
 0 (5.65)
where R denotes the area of the two-dimensional cross-section in the x-z plane.
In the analysis, the horizontal eddy viscosity and eddy diffusion coefcients
were assumed constant. From previous one-dimensional analyses and in analogy
to Equation 5.21, the vertical eddy diffusivity was approximated as
K
Kc ce
gDe T
z
v
S

zD
zD
w



t
t
¤
¦
¥
³
µ
´


()
/
/
12
2
22
1 S
A
T
332/
(5.66)
KK h
SS
w

 (,)T (5.67)
where the terms dependent on z were meant to approximate the generation of
shear by surface waves and near-surface currents. Similarly, the vertical eddy
viscosity was approximated as
A
Ac ce
gDe T
z
v
S
zD
zD
w



t
t
¤
¦
¥
³
µ
´


()
/
/
12

1
22
1 S
A
T
112/
(5.68)
where B is the thermal expansion coefcient for water; c
1
, c
2
, and D are empirical
constants and D is the same order of magnitude as the wave length of the surface
waves; A
S
is assumed equal to K
S
; both are functions of the wind stress, U
w
, and
water depth, h; and T
1
and T
2
are constants. When the temperatures were calcu-
lated, it sometimes was found that the colder, denser water was above warmer,
less dense water. In this statically unstable situation, the unstable region was uni-
formly mixed.
At the water-air interface, z = 0, a heat ux and a stress due to the wind
was specied


t
t
RK
T
z
q
v
(5.69)
RTRTA
u
z
A
v
z
vx
w
vy
w
t
t

t
t
, (5.70)
At the sediment-water interface, the conditions were those of no uid motion
and either zero heat ux or continuity of heat ux and temperature across the
sediment-water interface.
© 2009 by Taylor & Francis Group, LLC
198 Sediment and Contaminant Transport in Surface Waters

5.3.2 TIME-DEPENDENT THERMAL STRATIFICATION IN LAKE ERIE
Calculations were rst made in an effort to understand the general effects of vari-
ous boundary conditions and parameters. A more realistic calculation then was
made to model general and essential characteristics of stratied ow during the
time of formation, maintenance, and decay of a thermocline. This latter calcula-
tion extended over a period of 180 days, from early spring to fall. The following
was assumed: (1) the heat ux was constant at 0.003 cal/cm
2
/s for 50 days and
then decreased linearly so that it became zero at 120 days and negative thereaf-
ter, an approximation to real conditions; and (2) the wind stress was constant at
0.075 N/m
2
and directed to the right (south) from 0 to 50 days, to the left (north)
from 50 to 70 days, to the right again from 70 to 130 days, to the left from 130
to 150 days, and then to the right from 150 to 180 days, an approximation to
show and include effects due to varying wind direction. Other parameters were
as follows; A
S
=K
S
=15 cm
2
/s, D = 900 cm, A
H
=10
6
cm
2
/s, K

H
=3×10
5
cm
2
/s,
T
1
= T
2
= 0.001875, c
1
= 0.5, c
2
= 1.5, and no heat transfer to the bottom. The ini-
tial temperature was 4°C throughout.
Some results of the computations are shown in Figures 5.8 to 5.11 for times
of 50, 60, 120, and 150 days. The general features of the temperature and current
distributions are similar to what is observed in Lake Erie during the spring, sum-
mer, and fall periods. Figure 5.8 shows the temperatures at 50 days. A thermo-
cline has formed at a depth of 16 to 18 m and is deeper in the downwind direction.
The hypolimnion temperatures are 10 to 11°C. The epilimnion temperature is
13°C in the middle of the lake and increases by 1 to 2°C in the nearshore areas.
In a narrow (Ekman) layer near the surface, there is a ow in the direction of the
wind. Below this, there is a broad return ow in the opposite direction.
The results for the temperature at 60 days (10 days after wind reversal) are
shown in Figure 5.9. The thermocline is quite at. The epilimnion temperatures












     












FIGURE 5.8 Temperature distribution after 50 days for a vertical cross-section of Lake
Erie. Wind is from left to right. (Source: From Heinrich et al., 1981. With permission.)
© 2009 by Taylor & Francis Group, LLC
Hydrodynamic Modeling 199
have increased by approximately 1.5°C, whereas the hypolimnion temperatures
have increased by less than 0.5°C. Wind reversal causes a time-dependent shift
in the orientation of the thermocline so that the depth of the thermocline tends to
increase in the downwind direction; that is, the hypolimnion waters are shifted in
the upwind direction. This motion of the hypolimnion is slow, and 10 to 20 days

are needed before a new quasi-steady state is reached. Although the location of
the hypolimnion is signicantly affected by wind reversal, the temperatures in the
hypolimnion change little.
The wind was again reversed at 70 days and remained constant until 120
days. At this time, the temperatures and velocities are shown in Figure 5.10. The
epilimnion temperatures (Figure 5.10(a)) have increased to 19°C, whereas the
hypolimnion temperatures have increased to only 12 to 14°C. Currents in the
x-z plane (Figure 5.10(b)) are now more restricted to the epilimnion, whereas the
coastal currents in the y-direction (Figure 5.10(c)) have increased in strength.
In the central basin, the thermocline is generally located quite close to the
bottom. The hypolimnion volume is therefore relatively small compared to the
epilimnion volume. Because of this, the temperatures in the hypolimnion are par-
ticularly sensitive to heat uxes from the epilimnion to the hypolimnion. When
stratication is strongest, the vertical ux of heat by diffusion is reduced, and the
convective ux of heat from the epilimnion to the hypolimnion in the nearshore
regions (where stratication is less) becomes signicant.
After 120 days, the heat ux is negative and cooling takes place. Cooling
occurs more rapidly near shore, eliminating the warm nearshore areas. At 150
days (Figure 5.11), a thermocline exists but is quite weak and getting weaker.
By 180 days, nearshore areas are somewhat cooler than offshore areas; however,
practically no vertical stratication exists and the ow is similar to that in a con-
stant-temperature basin.












     











FIGURE 5.9 Temperature distribution at 60 days. Heat ux is decaying linearly and
wind direction is from right to left from 50 to 60 days. (Source: From Heinrich et al., 1981.
With permission.)
© 2009 by Taylor & Francis Group, LLC