175
D
L
t 2
2
=
(5.8)
These are very counter-intuitive relations which
have some surprising consequences. From Eq. 5.7
one can calculate the ‘diffusion velocity’, which is
distance divided by time:
‘Diffusion velocity’
t
D
t
L
/
2
=
=
(5.9)
This leads to the curious conclusion that the
shorter the period over which we measure diffusion, the larger is its velocity and vice versa.
This is critical to keep in mind when working with
diagenetic models, because the different chemical
species have time and length scales of diffusion
and reaction which vary over many orders of
magnitude and which are therefore correspondingly difficult to compare.
Calculations from Eq. 5.8 of the diffusion times
of oxygen molecules at 10°C show that it takes
one hour for a mean diffusion distance of 3-4 mm,
whereas it takes a day to diffuse 2 cm and
1000 years for 10 meters (Table 5.1). For a small
organic molecule such as glucose, these
diffusion times are about three times longer. Over
the scale of a bacterium, however, diffusion takes
only 1/1000 second. Thus, for bacteria of 1 µm
size, one could hardly envision a transport mechanism which would outrun diffusion within a
millisecond. The transition between predominantly
diffusive to predominantly advective or turbulent
transport of solutes lies in the range of 0.1 mm for
actively swimming organisms and somewhat
higher for passively sinking marine aggregates
(Fig. 5.3).
5.2.3
Diffusive Boundary Layers
The transition from a turbulent flow regime with
advective and eddy transport to a small scale
dominated by viscosity and diffusional transport
is apparent when an impermeable solid-water
interface such as the sediment surface is approached (Fig. 5.4). According to the classical eddy
diffusion theory, the vertical component of the
eddy diffusivity, E, decreases as a solid interface
is approached according to: E = A ν Z
3-4
, where A is
a constant, ν is the kinematic viscosity, and Z is
the height above the bottom. An exponent of 3-4
shows that the eddy diffusivity drops very
steeply as the sediment surface is approached. In
the viscous sublayer, which is typically about
1 cm thick in the deep sea, the eddy diffusivity
falls below the kinematic viscosity of ca 10
-2
cm
2
s
-1
.
Even closer to the sediment surface, the vertical
eddy diffusion coefficient for mass falls below the
molecular diffusion coefficient, D, which is
constant for a given solute and temperature, and
which for small dissolved molecules is in the order
of 10
-5
cm
2
s
-1
. The level where E becomes smaller
than D defines the diffusive boundary layer, δ e ,
which is typically about 0.5 mm thick. In this layer,
molecular diffusion is the predominant transport
mechanism, provided that the sediment is
impermeable and stable.
The diffusive boundary layer plays an
important role for the exchange of solutes across
the sediment-water interface (Jørgensen 2001). For
chemical species which have a very steep gradient
in the diffusive boundary layer it may limit the flux
and thereby the rate of chemical reaction. This
may be the case for the precipitation of manganese on iron-manganese nodules (Boudreau 1988)
or for the dissolution of carbonate shells and
other minerals such as alabaster in the deep sea
(Santschi et al. 1991). For chemical species with a
5.2
Life and Environments at Small Scale
Table 5.1 Mean diffusion times for O 2 and glucose over
distances ranging from 1 µm to 10 m.
Diffusion
Time (10°C)
distance
Oxygen
Glucose
1 µm
0.34 ms
1.1 ms
3 µm
3.1 ms
10 ms
10 µm
34 ms
110 ms
30 µm
0.31 s
1.0 s
100 µm
3.4 s
10 s
300 µm
31 s
100 s
600 µm
2.1 min
6.9 min
1 mm
5.7 min
19 min
3 mm
0.8 h
2.8 h
1 cm
9.5 h
1.3 d
3 cm
3.6 d
12 d
10 cm
40 d
130 d
30 cm
1.0 yr
3.3 yr
1 m
10.8 yr
35 yr
3 m
98 yr
320 yr
10 m
1090 yr
3600 yr
D
L
t 2
2
=
(5.8)
These are very counter-intuitive relations which
have some surprising consequences. From Eq. 5.7
one can calculate the ‘diffusion velocity’, which is
distance divided by time:
‘Diffusion velocity’
t
D
t
L
/
2
=
=
(5.9)
This leads to the curious conclusion that the
shorter the period over which we measure diffusion, the larger is its velocity and vice versa.
This is critical to keep in mind when working with
diagenetic models, because the different chemical
species have time and length scales of diffusion
and reaction which vary over many orders of
magnitude and which are therefore correspondingly difficult to compare.
Calculations from Eq. 5.8 of the diffusion times
of oxygen molecules at 10°C show that it takes
one hour for a mean diffusion distance of 3-4 mm,
whereas it takes a day to diffuse 2 cm and
1000 years for 10 meters (Table 5.1). For a small
organic molecule such as glucose, these
diffusion times are about three times longer. Over
the scale of a bacterium, however, diffusion takes
only 1/1000 second. Thus, for bacteria of 1 µm
size, one could hardly envision a transport mechanism which would outrun diffusion within a
millisecond. The transition between predominantly
diffusive to predominantly advective or turbulent
transport of solutes lies in the range of 0.1 mm for
actively swimming organisms and somewhat
higher for passively sinking marine aggregates
(Fig. 5.3).
5.2.3
Diffusive Boundary Layers
The transition from a turbulent flow regime with
advective and eddy transport to a small scale
dominated by viscosity and diffusional transport
is apparent when an impermeable solid-water
interface such as the sediment surface is approached (Fig. 5.4). According to the classical eddy
diffusion theory, the vertical component of the
eddy diffusivity, E, decreases as a solid interface
is approached according to: E = A ν Z
3-4
, where A is
a constant, ν is the kinematic viscosity, and Z is
the height above the bottom. An exponent of 3-4
shows that the eddy diffusivity drops very
steeply as the sediment surface is approached. In
the viscous sublayer, which is typically about
1 cm thick in the deep sea, the eddy diffusivity
falls below the kinematic viscosity of ca 10
-2
cm
2
s
-1
.
Even closer to the sediment surface, the vertical
eddy diffusion coefficient for mass falls below the
molecular diffusion coefficient, D, which is
constant for a given solute and temperature, and
which for small dissolved molecules is in the order
of 10
-5
cm
2
s
-1
. The level where E becomes smaller
than D defines the diffusive boundary layer, δ e ,
which is typically about 0.5 mm thick. In this layer,
molecular diffusion is the predominant transport
mechanism, provided that the sediment is
impermeable and stable.
The diffusive boundary layer plays an
important role for the exchange of solutes across
the sediment-water interface (Jørgensen 2001). For
chemical species which have a very steep gradient
in the diffusive boundary layer it may limit the flux
and thereby the rate of chemical reaction. This
may be the case for the precipitation of manganese on iron-manganese nodules (Boudreau 1988)
or for the dissolution of carbonate shells and
other minerals such as alabaster in the deep sea
(Santschi et al. 1991). For chemical species with a
5.2
Life and Environments at Small Scale
Table 5.1 Mean diffusion times for O 2 and glucose over
distances ranging from 1 µm to 10 m.
Diffusion
Time (10°C)
distance
Oxygen
Glucose
1 µm
0.34 ms
1.1 ms
3 µm
3.1 ms
10 ms
10 µm
34 ms
110 ms
30 µm
0.31 s
1.0 s
100 µm
3.4 s
10 s
300 µm
31 s
100 s
600 µm
2.1 min
6.9 min
1 mm
5.7 min
19 min
3 mm
0.8 h
2.8 h
1 cm
9.5 h
1.3 d
3 cm
3.6 d
12 d
10 cm
40 d
130 d
30 cm
1.0 yr
3.3 yr
1 m
10.8 yr
35 yr
3 m
98 yr
320 yr
10 m
1090 yr
3600 yr