398
13 Transport and Mixing in Coastal Ecosystems
and the centre of contaminant mass, xo, becomes:
11
00
Xo = -
c(x)dx.
Ct -00
(13.12)
When the diffusion coefficient, K x, is not constant, it can be estimated from
Eq. (8.34) as:
1 d0"2
K = __ x
x
2 dt .
(13.13)
More general, instead of Eq. (13.9) we can assume the variance 0"; in the form:
(13.14)
and respectively:
1
m-l
Kx = 2mat
,
(13.15)
in which a and m are constants that should be determined. A constant value
of Kx (with m = 1) corresponds to Fick's diffusion process.
Dye patches usually elongate with time and two orthogonal axes can be defined in the along-patch and across-patch directions with corresponding turbulent diffusion coefficients Kx and Ky. As was discussed in Sect. 8.3.3, the
elongation of a contaminant patch is the result of the interaction of a Fickian
diffusion with the advection due to the tidal- or wind-driven currents. Okubo
(1967) found that for steady shear (au/ay = Const) in the transverse direction,
the along- and cross-patch concentration variances 0"; and O"~ are as follows:
(13.16)
and
(13.17)
By combining variances O"x and O"y we can form the so called radial variance O"~:
(13.18)
and the corresponding turbulent diffusion coefficient, K h, can be determined
from Eq. (13.13).