Agriculture Reference
In-Depth Information
such that the whole sediment volume is accounted for. The water in a burrow is
assumed to mix instantaneously with the overlying seawater, and solutes diffuse
radially between the burrow and the sediment surrounding it as well as verti-
cally between the sediment and overlying water. The corresponding continuity
equation for transport in the sediment influenced by a particular burrow is
D
∂C
∂z
rD
∂C
∂r
∂C
∂t
=
∂
∂z
1
r
∂
∂r
+
+
R
(
2
.
37
)
where
z
is the distance from the sediment surface,
r
is the radial distance from
the centre of the cylinder and
R
is the rate of production or consumption of
solute in the sediment.
In Equation (2.37), the first term on the right-hand side accounts for diffusion
in the vertical direction; the second term accounts for radial diffusion across the
cylinder. The following boundary conditions apply. At the sediment-water and
sediment-burrow interfaces, the concentrations are the same as in the overly-
ing water:
z
=
0
C
=
C
0
r
=
r
1
C
=
C
0
At the boundary between adjacent cylinders, there is effectively no transfer
of solute:
r
=
r
2
d
C/
d
r
=
0
where the radius of the cylinder,
r
2
,
=
1
/
√
πN
,where
N
is the density of worms
per unit sediment surface area. At the bottom of the cylinder, the flux of solute
is constant:
z
=
L
d
C/
d
z
=
B
The value of
B
is specified from empirical observations.
Aller (1980b) shows that if the mean distance between burrows is small com-
pared with their length, then a steady state
(∂C/∂t
=
0
)
will be attained rapidly,
and he provides an analytical solution of Equation (2.37) for the steady state
subject to the above boundary conditions. (The solution is complicated, involv-
ing Bessel functions, and is not reproduced here.) The mean concentration at a
particular depth is found by integrating the concentration across the cylinder of
sediment at that depth:
r
2
2
πrC.
d
r
r
2
r
1
C
z
=
(
2
.
38
)
2
πr.
d
r
r
1
Aller uses the model to explain seasonally fluctuating profiles of NH
4
+
concen-
tration in sediments in Long Island Sound. In this system NH
4
+
is produced
