In June 1992 the Department of Water Resources hired Gilbert
Bogle of Water Engineering and Modeling to develop a nonproprietary
particle tracking module that DWR could adapt to output and geometry
of its DSM1 model. The original module provided by Dr. Bogle
was a quasi two- dimensional model which simulated longitudinal
dispersion by utilizing a vertical velocity profile, vertical
mixing, and a dispersion coefficient (which was a function of
velocity, depth, and width of the channel). Because of the tidal
nature of the water system and because of the channel grid, some
complications occurred when implementing the module. After some
scrutiny, the model was further developed to be a quasi three-dimensional
model. Dr. Bogle continued to give further suggestions on ways
to improve the module. To contact Dr. Bogle, e-mail him at firstname.lastname@example.org.
The Particle Tracking model was originally written in Fortran.
Due to the nature of the model, the code was rewritten in C++
and Java to take advantage of using an object-oriented approach.
Among other developments, the code was modified to handle the
new output from DSM2-HYDRO. Additionally, the input system was
rewritten (in Fortran) to be consistent with the DSM2-HYDRO and
DSM2-QUAL input system.
The Particle Tracking Model (DSM2-PTM) simulates the transport
and fate of individual notional "particles" traveling
throughout the Sacramento-San Joaquin Delta. The model utilizes
velocity, flow, and stage output from a one-dimensional hydrodynamic
model (DSM2-HYDRO). Time intervals for these hydrodynamic values
can vary but are on the order of either 15 minutes or one hour.
Input into the hydrodynamic model include inflows at various
rivers, exports, agricultural return and diversions, and stage
The Delta's geometry is modeled as a network of channel segments
and open water areas connected together by junctions; the particles
move throughout the network under the influence of flows and
random mixing effects.
The location of a particle at any time step within a channel
is given by the channel segment number, the distance from the
end of the channel segment (x), the distance from the centerline
of the channel (y), and the distance from the channel bottom
(z) (see Figure 4.1).
Three-dimensional movement of neutrally buoyant particles
within channels is depicted in Figure
Transverse Velocity Profile
The average cross sectional velocity during a time step, which
is supplied by the hydrodynamics portion of DSM, is adjusted
by multiplying it by a factor which is dependent on the particle's
transverse location in the channel. This results in a transverse
velocity profile where the particles located closer to the shore
move slower than those located near the centerline in the channel.
The model uses a quartic function to represent the velocity profile.
Vertical Velocity Profile
The average cross sectional velocity is adjusted by multiplying
it by a factor which is dependent on the particle's vertical
location in the channel. This results in a vertical velocity
profile where the particles located closer to the bottom of the
channel move slower than particles located near the surface.
The model use the Von Karman logarithmic profile to represent
the velocity profile. The longitudinal distance traveled by a
particle is equal to a combination of the two velocity profiles
multiplied by the time step.
Particles move across the channel due to mixing. A gaussian
random factor, a transverse mixing coefficient, and the length
of the time step are used in the calculation of the distance
a particle will move during a time step. The mixing coefficient
is a function of the water depth and the velocity in the channel.
When there are high velocities and deeper water, mixing is greater.
Particles also move vertically in the channel due to mixing.
A gaussian random factor, a vertical mixing coefficient, and
the length of the time step are used in the calculation of the
distance a particle will move during a time step. The mixing
coefficient is a function of the water depth and the velocity
in the channel. As with transverse mixing, when there are high
velocities and deeper water, mixing is greater.
Particles can be inserted at any node location within the
History of each Particle's movement is available. In the model,
the path each particle takes through the Delta is recorded. Output
useful in determining the particle's movement includes:
- Animation. Particles are shown moving through the Delta Channels.
The effects of tides, inflows, barriers, and diversions are seen
at hourly time steps.
- Number of particles passing locations. The number of particles
that pass specified locations are counted at each time step.
- Each particle has a unique identity and characteristics that
can change over time. Since each particle is individually tracked,
characteristics can be assigned to the particle. Examples of
characteristics are additional velocities that represent behavior
(self-induced velocities), or the state of the particle, such
- Particles travel at different velocities at different locations
within the cross section. The Particle Tracking Model takes the
average one-dimensional channel velocity from the DSM hydrodynamics
model and creates velocity profiles from it where higher velocities
occur closer to the surface of the water and towards the middle
of the channel. Therefore, if particles are heavy and tend to
sink towards the bottom, they will move slower than if they were
neutrally buoyant. As a result, their travel time through the
channels is longer.
Until recently, PTM simulations have primarily been made using
neutrally buoyant particles. Some studies have been made where
settling velocities and mortality rates have been included. These
studies have concentrated on Striped Bass eggs and larvae. As
additional fish data become available, additional modifications
to the model will be made for future studies. These modifications
will be a function of the state of the particle and the particle's
environment. These modifications will require the particles react
to the following:
If it is known that food exists at the sides of channels,
then a transverse velocity component can be included so that
particles can move towards the shore.
- Example: Inland
Silversides may swim towards the shore for food.
When particles age, their behavior may change. If eggs, their
density may be different. They may sink, swim, or die.
- Example: Longfin larvae are found at the surface of
the water column. Juveniles are found towards the middle of the
Particles may react to a diurnal cycle. An option can be included
so that the particles will rise and fall depending on the time
of day. This will influence their longitudinal position.
Particles can react to the tidal velocity and direction of
- Example: Longfin and Striped Bass move up in the water
column to ride the
Particles can have an additional longitudinal velocity component.
- Example: Salmon smolts swim with the flow.
Particles' growth rate and mortality can be a function of
water quality. This can include temperature, dissolved oxygen
level, pesticides, and food abundance.
Particles can swim towards a certain water quality.
Example: Adult Salmon swim towards fresher water.
Movement Within a Channel
Advection within the model is represented by the one-dimensional
velocity determined by DSM2-Hydro. This velocity assumes that
in a cross section of the channel, the velocity is constant throughout
the cross section.
Longitudinal Dispersion is caused by shear at the bottom and
sides of the channel. This shear creates differences in velocities
and causes turbulence within the cross section. If a tracer is
injected throughout the cross section, at a distance further
downstream, its concentration can be approximated by a gaussian
distribution (see Figure 4.3).
This approximation is used to define the dispersion coefficient
K which is one half of the change in variance with respect to
time (see Figure 4.4).
Column 4 in Figure 4.5, from
Mixing in Inland and Coastal Waters shows observed dispersion
from various Rivers. (Columns 5 and 6 show theoretical and DSM1
ptm dispersion, which will be explained later.) Figure
In order to simulate dispersion, velocity profiles and mixing
are included in the model.
The vertical velocity profile is approximated using the Von
Karman Logarithmic Velocity Profile and the transverse velocity
profile is approximated using a quartic function. The quartic
function was chosen because it closely approximated velocity
profiles measured by USGS in the Delta.
Figure 4-7a shows the movement
in the x direction, the direction of the flow, that is caused
by the bottom and side shear of the channel. When FT and FV are
equal to 1, the particle is traveling at the average velocity
within the channel. Aq, Bq, and Cq are currently set to 1.62,
-2.22, and 0.6. Aq is used as the free parameter with Bq and
Cq being derived under the assumptions that velocity is zero
at channel sides and the average value of the function is 1.
(Figures 4.7a, 4.7b)
Mixing and movement in the vertical and transverse directions
are necessary in modeling dispersion. If only velocity profiles
resulting in movement in the x direction were modeled, K would
not be a constant as it is defined to be, but continually growing
The mixing coefficients are described similarly to the dispersion
coefficient in that they are defined as the rate of change of
the variance in position. Figure 4.8
shows how the z and y distance traveled is determined. Note in
the derivation, that the variance and standard deviation for
both position and velocity are shown. The subscripts v and w
indicate velocities and the subscripts Y and Z indicate position.
Using the gaussian random number R, a concentration distribution
is created with a standard deviation of (2Evdt)1/2. To expand,
assume that there are a large number of particles at a particular
point in the cross section at the beginning of a time step. At
the end of the time step, approximately 95 percent of the particles
will have moved a distance equal to or less than two standard
deviations or 2(2Evdt)1/2.
The derivation for the vertical mixing coefficient Ev is shown
in Figure 4.9. It is derived from
the Von Karman Logarithmic Velocity Profile and is a function
of depth and velocity.
The transverse mixing coefficient was determined empirically.
Figure 4.11 shows the mixing
coefficients and how the vertical and transverse distance is
calculated in the model. The 0.06 and the 0.0067 can be changed
in the input file.
When the calculated distance of travel in the vertical or
horizontal is greater than the actual distance a particle can
move, the particle reflects off of the boundary the additional
distance that it would have moved if the boundary was not there.
For example, a channel is 10 ft .deep and the neutrally buoyant
particle is located at 9.5 ft. at the beginning of a time step.
The vertical mixing results in a movement of 0.7 ft. upward.
The particle moves up 0.5 ft. to the 10 ft. surface and then
"bounces" back 0.2 ft to the 9.8 ft. level.
To avoid excessive bouncing, smaller sub-time steps are used.
The sub-time steps are calculated based on the distance traveled
by particles during a time step. If the particles travel a distance
larger than ten percent of the width or the depth, then the time
step is reduced so that the distance traveled is equal to or
less than the limiting distance. For mixing, the distance traveled
is based on a gaussian distribution. Time step calculations are
made for particles that travel one standard deviation away from
the zero mean.
Adjustment of Position After Longitudinal
After the particle has moved in the longitudinal, x direction,
its position is adjusted to reflect the change in depth or width
of the channel.
- new z position = z(dnew/dold)
- new y position = y(wnew/wold)
Z and y are the calculated positions. The "old"
depth and width corresponds to the depth and width at the x position
at the beginning of the time step. The "new" depth
and width correspond to the depth and width at the x position
at the end of the time step.
Figure 4.12 shows the derived
dispersion coefficient. This calculation is not used in the model
but is used as a comparison to the dispersion the model generates.
To determine if the 3D formulation is adequately modeling dispersion,
tests are made of the formulation using one long rectangular
channel with a constant velocity. Dispersion is determined in
the model by calculating the variance of the concentration of
particles over time. K is checked to see if it remains relatively
constant (does not increase). Model K is also compared to the
derived K and the observed K shown in Figure
(The model Kshown in Figure
4.5 was determined using a different transverse
profile than what is currently being used in the model.)
Checking the validity of the model in the past has also included
comparing the results of the particle tracking model to results
of the mass tracking model on a Delta-wide scale. Presently DSM2-PTM
has been tested for bugs but has not been validated.
Movement at Junctions
When a particle reaches a junction, the decision has to be
made as to where the particle is to go. Flows out of nodes include
flows into channels, open water areas, agricultural diversions,
and exports. Within the model, these locations are referred to
as water bodies. The probability of a particle entering another
water body is proportional to amount of flow entering that water
Movement In and Out of Open Water Areas
Once a particle enters an open water area, it no longer retains
its x, y or z position. The open water area is considered fully
mixed. At the beginning of a time step the volume of the open
water area the volume of water leaving at each opening of the
open water area is determined. From that the probability of the
particle leaving the open water area is calculated.
Exports and Agricultural Diversions
Particles entering exports or agricultural diversions are
considered "lost" from the system. Their final destination
Once particles pass the Martinez boundary, they have no opportunity
to return to the Delta.
Bogle, Gilbert V. 1995. Simulation of Dispersion in Streams
by Particle Tracking, submitted for review to J. Hydraul,
Bogle, Gilbert V. 1997. Stream Velocity Profiles and Longitudinal
Dispersion, ASCE Jnl. of Hyd. Eng. Vol. 123 No. 9
California Dept. of Water Resources. 1994. Methodology
For Flow and Salinity Estimates in the Sacramento - San Joaquin
Delta and Suisun Marsh, Fifteenth Annual Progress Report to the
State Water Resources Control Board.
California Dept. of Water Resources. 1995. Methodology
For Flowand Salinity Estimates in the Sacramento - San Joaquin
Delta and Suisun Marsh, Sixteenth Annual Progress Report to the
State Water Resources Control Board.
Fischer, H. B., List, E. J., Koh, R.C. Y., Imberger, J., and
Brooks, N. H. 1979. Mixing in Inland and Coastal Waters,
Academic Press., San Diego, California.
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