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8 Transport in the Oceans and Coastal Zone
Figure 8.8a illustrates the distribution of concentration of four cross-sections
of the plume at distances 30 m, 75 m, 150 m, and 250 m from the release point,
respectively. The rate of contaminant release is Q = 10 kg/s and current speed
is ii = 0.2 m/s. Values of coefficients of turbulent diffusion Ky and Kz are 0.075
m 2 /s and 0.0042 m 2 /s, which correspond to values measured during Ozmidov's
experiments in the Black Sea (Ozmidov, 1986). The Gaussian type distribution of concentration gradually becomes flatter as distance from the release
point increases. The attenuation rate of the maximum concentration along the
central axis of the plume is proportional to x-I (see Eq. 8.67). Experimental
data from Black and Baltic Seas showed that attenuation of concentration is
slower, with the relationship x- n , where n = 0.66-0.82. The difference between
theoretical and observed attenuations is caused by the fact that coefficients Ky
and K z are not necessary constants but depend on the scale of the phenomenon
(Ozmidov, 1986).
In Fig. 8.8b, the corresponding vertical profiles of contaminant concentration
at four distances from the release point, and for plume axis of symmetry (y = 0)
are shown. Further from the origin, concentration is lower but the contaminant
penetrates deeper into the water column.
Finally, in Fig. 8.8c, a plan view of the contaminant plume is shown with
areas of particular concentrations varying from higher than 10 kg/m 3 to 1
kg/m 3 . As should be expected, the plume is spread mostly along the direction
of the surface current, but also spreads in the y direction due to the action of
diffusion.
Instantaneous Release of Substance. Random Walk Approach. Instead of solving a diffusion equation analytically, as above, it is possible to
simulate the process by numerically tracking a finite number of particles as
they are carried by the fluid. Then, a local concentration is calculated as the
limit b.m / b. V where b.m is the substance mass and b. V is the volume of space
under consideration. The final movement of particles is due to both advection and diffusion. Advection is accomplished by the local fluid velocity. The
diffusive part of the particles movement can be simulated by a random walk
procedure. Additionally, for the case of a non-conservative substance, decomposition and withdrawal of some number of particles from the fluid domain can
be simulated.
As was mentioned above, the concentration at a certain location is computed
from the number of particles that are contained in a known volume. By comparing the number of particles in a known volume, with the number of particles
in the same volume for a known concentration, we can calculate the concentration. For example, if we know that 10 particles in a small area (b.xb.y)
corresponds to 5 ppm, then 30 particles in another small area corresponds to
a concentration of 15 ppm.
A determination of the individual particle positions in the Lagrangian description is usually based on the Monte Carlo simulation technique. Let us
assume that there are n elements of substance at the sea surface where the
8 Transport in the Oceans and Coastal Zone
Figure 8.8a illustrates the distribution of concentration of four cross-sections
of the plume at distances 30 m, 75 m, 150 m, and 250 m from the release point,
respectively. The rate of contaminant release is Q = 10 kg/s and current speed
is ii = 0.2 m/s. Values of coefficients of turbulent diffusion Ky and Kz are 0.075
m 2 /s and 0.0042 m 2 /s, which correspond to values measured during Ozmidov's
experiments in the Black Sea (Ozmidov, 1986). The Gaussian type distribution of concentration gradually becomes flatter as distance from the release
point increases. The attenuation rate of the maximum concentration along the
central axis of the plume is proportional to x-I (see Eq. 8.67). Experimental
data from Black and Baltic Seas showed that attenuation of concentration is
slower, with the relationship x- n , where n = 0.66-0.82. The difference between
theoretical and observed attenuations is caused by the fact that coefficients Ky
and K z are not necessary constants but depend on the scale of the phenomenon
(Ozmidov, 1986).
In Fig. 8.8b, the corresponding vertical profiles of contaminant concentration
at four distances from the release point, and for plume axis of symmetry (y = 0)
are shown. Further from the origin, concentration is lower but the contaminant
penetrates deeper into the water column.
Finally, in Fig. 8.8c, a plan view of the contaminant plume is shown with
areas of particular concentrations varying from higher than 10 kg/m 3 to 1
kg/m 3 . As should be expected, the plume is spread mostly along the direction
of the surface current, but also spreads in the y direction due to the action of
diffusion.
Instantaneous Release of Substance. Random Walk Approach. Instead of solving a diffusion equation analytically, as above, it is possible to
simulate the process by numerically tracking a finite number of particles as
they are carried by the fluid. Then, a local concentration is calculated as the
limit b.m / b. V where b.m is the substance mass and b. V is the volume of space
under consideration. The final movement of particles is due to both advection and diffusion. Advection is accomplished by the local fluid velocity. The
diffusive part of the particles movement can be simulated by a random walk
procedure. Additionally, for the case of a non-conservative substance, decomposition and withdrawal of some number of particles from the fluid domain can
be simulated.
As was mentioned above, the concentration at a certain location is computed
from the number of particles that are contained in a known volume. By comparing the number of particles in a known volume, with the number of particles
in the same volume for a known concentration, we can calculate the concentration. For example, if we know that 10 particles in a small area (b.xb.y)
corresponds to 5 ppm, then 30 particles in another small area corresponds to
a concentration of 15 ppm.
A determination of the individual particle positions in the Lagrangian description is usually based on the Monte Carlo simulation technique. Let us
assume that there are n elements of substance at the sea surface where the