8.3 Concentration of Matter for Molecular and Turbulent Diffusion
271
8.3.4 Some Workable Solutions of the Diffusion Equations
The diffusion equation (8.30) is widely used for the determination of concentration of particles of various types of contaminants in ocean waters. The general
form of Eq. (8.30) can only be solved using numerical techniques (Fisher, 1981;
Ozmidov, 1986; Noye, 1987; Koutitas, 1988; Partheniades, 1992). However, in
some simpler cases, this equation can be solved analytically. Although simplifications are introduced into the equation, analytical solutions remain very
useful for a quick estimation of the required concentration, as well as serving as test references for numerical schemes. In this chapter we will discuss a
few examples of such simplified solutions of the diffusion equation. Computer
programs, ready for use, are given on the CD (see Appendix D).
Instantaneous Release of Substances in Steady Uniform Flow. Let us
assume a uniform one-dimensional current with mean velocity, ii, flowing in the
x direction. At time t = 0, a mass M of ~iOme contaminant is instantaneously
released at the ocean surface, in the vicinity of point x = O. The distribution
of the initial concentration c( x, 0) takes the form of the Gaussian curve, i. e.:
M
(x2)
c(x,O) = rc;- exp - - 2 '
V 27r0'
20'
(8.53)
in which M (kg/m) is the mass of contaminant per unit width, and 0' (m) is the
parameter characterizing the spreading of the initial contaminant distribution.
The contaminant has the same specific gravity as the ambient water, and the
coefficient of turbulent diffusion for the contaminant, K x , is constant. Under
these assumptions, a governing equation for c(x, t), resulting from Eq. (8.30),
becomes:
(8.54)
The solution of this equation, with initial condition (8.53), takes the form
(Noye, 1987):
(8.55)
Solution (8.55) is illustrated in Fig. 8.6. The mass of contaminant (for example, such as solution of fluoresceine dye) released is 1 kg/m and coefficient 0',
controlling the initial distribution, is assumed to be equal to 0.125 m. The
coefficient of turbulent diffusion, K x, is set as Kx = 500 cm 2 /s = 0.05 m 2 /s.
This value is very close to that observed during experiments with fluoresceine
in the Black Sea (Ozmidov, 1986). It was also assumed that current velocity
ii is constant and equal to 1 m/s. In Fig. 8.6, three particular time instants
271
8.3.4 Some Workable Solutions of the Diffusion Equations
The diffusion equation (8.30) is widely used for the determination of concentration of particles of various types of contaminants in ocean waters. The general
form of Eq. (8.30) can only be solved using numerical techniques (Fisher, 1981;
Ozmidov, 1986; Noye, 1987; Koutitas, 1988; Partheniades, 1992). However, in
some simpler cases, this equation can be solved analytically. Although simplifications are introduced into the equation, analytical solutions remain very
useful for a quick estimation of the required concentration, as well as serving as test references for numerical schemes. In this chapter we will discuss a
few examples of such simplified solutions of the diffusion equation. Computer
programs, ready for use, are given on the CD (see Appendix D).
Instantaneous Release of Substances in Steady Uniform Flow. Let us
assume a uniform one-dimensional current with mean velocity, ii, flowing in the
x direction. At time t = 0, a mass M of ~iOme contaminant is instantaneously
released at the ocean surface, in the vicinity of point x = O. The distribution
of the initial concentration c( x, 0) takes the form of the Gaussian curve, i. e.:
M
(x2)
c(x,O) = rc;- exp - - 2 '
V 27r0'
20'
(8.53)
in which M (kg/m) is the mass of contaminant per unit width, and 0' (m) is the
parameter characterizing the spreading of the initial contaminant distribution.
The contaminant has the same specific gravity as the ambient water, and the
coefficient of turbulent diffusion for the contaminant, K x , is constant. Under
these assumptions, a governing equation for c(x, t), resulting from Eq. (8.30),
becomes:
(8.54)
The solution of this equation, with initial condition (8.53), takes the form
(Noye, 1987):
(8.55)
Solution (8.55) is illustrated in Fig. 8.6. The mass of contaminant (for example, such as solution of fluoresceine dye) released is 1 kg/m and coefficient 0',
controlling the initial distribution, is assumed to be equal to 0.125 m. The
coefficient of turbulent diffusion, K x, is set as Kx = 500 cm 2 /s = 0.05 m 2 /s.
This value is very close to that observed during experiments with fluoresceine
in the Black Sea (Ozmidov, 1986). It was also assumed that current velocity
ii is constant and equal to 1 m/s. In Fig. 8.6, three particular time instants