126
include the root inputs in many ecosystems if they
occur in a depth increment that is small compared
with the value of D. For Equation 8.17, with the
given boundary conditions, we obtain the solution
(from the mathematical software program
MATLAB®):
(8.19)
For 13C, the solution is:
C*(z) = fs~ ezFakiD
Jo.kD
(8.20)
where ~ is the 13C/ l2 C of plant inputs and a is a
fractionation factor, which if < 1 explains the 13C
enrichment of SaM with depth. Dividing Equation
8.20 by Equation 8.19 gives the steady state equation for the isotopic ratio of SaM as a function of
depth:
R(z) = Rp e[jk!DHzj;;:-I]
Fa
(8.21)
In Figure 8.3A total SOC is plotted versus depth
(Equation 8.19) and in Figure 8.3B the 013C value
of the SaM versus depth is illustrated (Equation
8.21). The equations have been fitted to the soil data
of O'Brien and Stout (1978), which are shown as
open circles. Figure 8.3A shows that the relatively
simple model solution (Equation 8.19) provides a
good representation of the SOC concentration in
the upper portion of the profile. Below 60 cm in
depth, the model fit deviates from the profile data.
This may be due to the need to consider a second
pool of more inert organic matter with a residence
time of centuries or millennia. O'Brien and Stout's
(1978) analysis of their radiocarbon data supports
this interpretation, suggesting that "old" SOC accounts for a small proportion of the total above 50
cm, but half of the total SOC at the lowest measured
depth.
The linear trend in 013C values with depth suggested by Equation 8.21 and plotted in Figure 8.3B
also appears to describe the data of O'Brien and
Stout (1978). The values of the plant material
( - 29%0) and fractionation associated with decomposition (10 3 ln a = - 2.5) that fit the data are well
within reasonable values observed in nature or in
experiments. However, it is not clear that this is the
only appropriate means of explaining the data.
Ronald Amundson and W. Troy Baisden
First, the effect of the decrease in the atmospheric
013C value during the twentieth century (approximately 1.3%0) is not considered. Second, the data
may also be explained by a shift in vegetation from
forest to pasture at the site, with a concomitant decrease in the 013C value of plant inputs of approximately 4%0. Moreover, it may be appropriate to
consider the isotope effects associated with the
transfer of SOC to a more inert pool that accounts
for the bulk of the SOC at lower soil depths. Finally, we note that we have neglected root inputs
without proving that this assumption is justified.
Although this model is relatively simple, it appears to capture the dynamics of the natural system.
It is likely that simple models of this type, balanced
with numerical solutions to more general models
such as Equation 8.14 that include below ground inputs, may prove useful in improving our understanding of SOC dynamics. In particular, these
models provide an important counterpart to ecosystem biogeochemistry simulations which do not
recognize soil depth as a continuous variable.
Multi-Box SOM Isotope Models
All the modeling approaches taken thus far assume
that plant inputs and SaM each act as homogeneous pools. It has long been recognized, from 14C
measurements, that the only way to reconcile observed isotopic changes of SaM with time is to
partition SaM into pools with different residence
times (Trumbore 1993). In using this approach, the
isotopic composition of inputs into all pools is considered to be the same, while the isotopic composition of losses varies with the SaM pool being
considered. For l2C:
n
dCldt
~ (~ - ~Ci)
(8.22a)
i=1
and for 13C:
n
dC*ldt
~ (R,Ii - kio.iC*i) (8.22b)
i=1
where Ii = inputs of soil organic matter to each
pool, and decay constants (k) and fractionation factors (a) refer to pool-specific values. More complicated models, involving transfers between pools,
require either matrix models (Baisden and Amund-
include the root inputs in many ecosystems if they
occur in a depth increment that is small compared
with the value of D. For Equation 8.17, with the
given boundary conditions, we obtain the solution
(from the mathematical software program
MATLAB®):
(8.19)
For 13C, the solution is:
C*(z) = fs~ ezFakiD
Jo.kD
(8.20)
where ~ is the 13C/ l2 C of plant inputs and a is a
fractionation factor, which if < 1 explains the 13C
enrichment of SaM with depth. Dividing Equation
8.20 by Equation 8.19 gives the steady state equation for the isotopic ratio of SaM as a function of
depth:
R(z) = Rp e[jk!DHzj;;:-I]
Fa
(8.21)
In Figure 8.3A total SOC is plotted versus depth
(Equation 8.19) and in Figure 8.3B the 013C value
of the SaM versus depth is illustrated (Equation
8.21). The equations have been fitted to the soil data
of O'Brien and Stout (1978), which are shown as
open circles. Figure 8.3A shows that the relatively
simple model solution (Equation 8.19) provides a
good representation of the SOC concentration in
the upper portion of the profile. Below 60 cm in
depth, the model fit deviates from the profile data.
This may be due to the need to consider a second
pool of more inert organic matter with a residence
time of centuries or millennia. O'Brien and Stout's
(1978) analysis of their radiocarbon data supports
this interpretation, suggesting that "old" SOC accounts for a small proportion of the total above 50
cm, but half of the total SOC at the lowest measured
depth.
The linear trend in 013C values with depth suggested by Equation 8.21 and plotted in Figure 8.3B
also appears to describe the data of O'Brien and
Stout (1978). The values of the plant material
( - 29%0) and fractionation associated with decomposition (10 3 ln a = - 2.5) that fit the data are well
within reasonable values observed in nature or in
experiments. However, it is not clear that this is the
only appropriate means of explaining the data.
Ronald Amundson and W. Troy Baisden
First, the effect of the decrease in the atmospheric
013C value during the twentieth century (approximately 1.3%0) is not considered. Second, the data
may also be explained by a shift in vegetation from
forest to pasture at the site, with a concomitant decrease in the 013C value of plant inputs of approximately 4%0. Moreover, it may be appropriate to
consider the isotope effects associated with the
transfer of SOC to a more inert pool that accounts
for the bulk of the SOC at lower soil depths. Finally, we note that we have neglected root inputs
without proving that this assumption is justified.
Although this model is relatively simple, it appears to capture the dynamics of the natural system.
It is likely that simple models of this type, balanced
with numerical solutions to more general models
such as Equation 8.14 that include below ground inputs, may prove useful in improving our understanding of SOC dynamics. In particular, these
models provide an important counterpart to ecosystem biogeochemistry simulations which do not
recognize soil depth as a continuous variable.
Multi-Box SOM Isotope Models
All the modeling approaches taken thus far assume
that plant inputs and SaM each act as homogeneous pools. It has long been recognized, from 14C
measurements, that the only way to reconcile observed isotopic changes of SaM with time is to
partition SaM into pools with different residence
times (Trumbore 1993). In using this approach, the
isotopic composition of inputs into all pools is considered to be the same, while the isotopic composition of losses varies with the SaM pool being
considered. For l2C:
n
dCldt
~ (~ - ~Ci)
(8.22a)
i=1
and for 13C:
n
dC*ldt
~ (R,Ii - kio.iC*i) (8.22b)
i=1
where Ii = inputs of soil organic matter to each
pool, and decay constants (k) and fractionation factors (a) refer to pool-specific values. More complicated models, involving transfers between pools,
require either matrix models (Baisden and Amund-