272
8 Transport in the Oceans and Coastal Zone
~ 1.6
C 1.4
'"
§
= 1.2
'j
.... 1.0
o
u
.....
o 0.8
§
·c
S 0.6
....
= 11)
§ 0.4
U
0.2
0.0
-2
o
t=O s
initial concentration
/
distribution
2
4
6
8
10
12
14
Distance from relative point (m)
Fig. 8.6: Spreading of contaminant in time and space in a steady uniform flow
(t = 2 s, 5 s, and 10 s) have been selected. As time increases, the contaminant
moves along the x axis and concentration becomes more widely spread. Peak
concentration (moving at speed u) decreases in the positive x direction. As
t-+ 00, C = 0 for all x. As there are no sources of sinks of mass, the amount
of contaminant at each time t is constant and equal to M.
If the mass of the tracer material is introduced initially at time t = 0, in a
very small area (0' -+ 0 in Eq. 8.53) around point x = 0, the concentration
approaches infinity in such a way that the area under the initial curve (which
represents the total mass of the substance) is constant and equal to M. This
property can be represented by a function known as the delta function 8 (x) (or
Dirac delta function). The delta function has two main properties:
{
00 when x = xo
8(x - xo) = 0 when x -I Xo,
and
i: 8(x - xo)dx = 1.
(8.56)
(8.57)
As in our case Xo = 0, the initial condition (8.53) becomes (with 8 -+ 0):
c(x,O) = M8(x),
(8.58)