Morford, S.L., Houtlon, B.Z., Dahlgren, R.A. XXXX. Direct quantification of longterm rock nitrogen inputs to temperate forest ecosystems. Ecology.
Appendix S3: Derivation of conservation of mass equations used in the manuscript, including a
table describing model variables and a figure illustrating the concept of strain.
The conservation of mass equations used in this manuscript are adapted from a
system of mass balance equations developed in the geochemical, geomorphology, and soil
science literature. The equations used here have been formalized elsewhere (Brimhall and
Dietrich 1987, Chadwick et al. 1990, Brimhall et al. 1992, Egli and Fitze 2000), and are
similar to approaches developed by soil scientists (Muir and Logan 1982). For the sake of
completeness, and to facilitate better understanding of the approach applied in this
manuscript, we provide a full derivation of the conservation of mass equations presented
in the main text. Table C1 provides the units and brief explanation of equation variables.
Application of conservation of mass equations in soil systems requires the
measurement, and consideration, of both geochemical and geophysical variables.
Concentrations of elements in soil/regolith (Cj,w) and bedrock parent material (Cj,p) can be
readily measured, but the contribution of bedrock derived elements to the soil sample is
obfuscated due to volumetric and density changes that occur as bedrock weathers to soil.
Conservation of mass in a open soil system is defined as:
(eqn. C.1)
where Vp and Vw are the volume of rock parent material and soil/regolith respectively, and
ρp and ρw are the rock and soil/regolith density, respectively. Inputs and losses of element j
1
to the system are described by variable mj, with positive values representing inputs (such
as contributions of element j from dust or volatile atmospheric reservoirs) and negative
values representing losses (via leaching of soluble elements or gaseous loss of volatile
elements).
Solving eqn. C.1 for mj is not directly possible in soil because the deformation
relationship between Vp and Vw in the system is not defined, and thus the boundaries of the
system are unknown. In saprolite that has gone isovolumetric weathering Vp = Vw, mj can
be directly calculated. As bedrock becomes soil, it can either dilate (increase in volume)
from addition of organic matter and mineral hydration, or collapse (decease in volume) as
chemical weathering contributes to hydrological losses of soluble elements and the matrix
is compacted by overlying soil/regolith (Fig. C1). The deformation relationship between
bedrock parent material and soil/regolith is defined as strain (ε) and is given as:
e=
Vw -Vp Vw
= -1
Vp
Vp
(eqn. C.2)
Strain can be estimated geochemically if the system is closed for a particular element. High
field strength elements (HFSEs) are often immobile, or sparingly mobile, in soils, and their
mass balance can be treated as a closed system (mj =0). Selection of an appropriate
immobile element for strain calculations is dependent on the parent material mineralogy
and chemical weathering environment (Brantley and White 2009). Conservation of mass
for an immobile element is described as:
Vp r pCi, p = Vw rwCi,w
(eqn. C.3)
Equation 3 can be rearranged to give:
2
Vw r pCi, p
=
Vp rwCi,w
(eqn. C4)
and inserting eqn. C.4 in eqn. C.2 yields an alternative interpretation of ε that can be
calculated using density and immobile element concentration data in bedrock parent
material (Ci,p) and soil/regolith (Ci,w), respectively:
e=
r pCi, p
-1
rwCi,w
(eqn. C.5)
The fractional change of element abundance in the system relative to the initial mass in
bedrock parent material is defined as the mass transfer coefficient (τj):
tj =
mj
Vp r pC j, p
(eqn. C.6)
In cases where mj is positive, inputs have increased the mass of element j in the system, and
τj is > 0. Conversely, when element j is lost from the system (negative mj), τj is < 0.
Using the equations above, we can define τj solely in terms of changes in
concentration of immobile (Ci,p and Ci,w) and mobile (Cj,p and Cj,w) elements in bedrock
parent material and soil/regolith. Solving eqn. C.1 for mj and inserting into eqn. C.6 yields:
tj =
(V r C
w
w
j,w
-Vp r pC j, p )
Vp r pC j, p
(eqn. C7)
Simplifying eqn. C.7 yields:
tj =
Vw rwC j,w
-1
Vp r pC j, p
(eqn. C.8)
Inserting eqn. C.4 into eqn. C.8 and simplifying yields the commonly used geochemical
determination of τj :
3
tj =
Ci, p C j,w
-1
Ci,w C j, p
(eqn. C.9)
Next, we use the equations above to derive the equation for the element-specific
whole-profile mass flux (Mj) presented in the manuscript. Here we formulate the equations
in a manner that is amenable to standard soil sampling techniques where multiple samples
(i.e. incremental sampling by depth or horizon) are integrated to estimate the total mass
flux for the entirety of the soil profile. First, we define mass flux in terms of a single soil
sample:
m j = t jVp r pC j, p
(eqn. C.10)
For a soil that has undergone deformation during pedogenesis, Vp cannot be directly
measured. However, we can use ε in combination with Vw to calculate Vp in our system.
First, solving eqn. C.2 for Vp yields:
Vp =
Vw
e +1
(eqn. C.11)
Substituting eqn. C.11 into eqn. C.10 allows us to define mj without directly measuring Vp:
m j = r pC j, p
t jVw
(e +1)
(eqn. C.12)
The mass flux per soil volume can then be calculated by rearranging eqn. C.12:
mj
t
= r pC j, p j
Vw
e +1
(eqn. C.13)
Multiplying both sides of the equation by soil sample depth/length (l) gives mass flux per
unit area (Mj):
Mj =
mj
t
l = r pC j, p j,w l
Vw
(e +1)
(eqn. C.14)
4
Equation 14 can be used directly in circumstances where a single bulk sample
representing the entire soil profile is used. However, when multiple depth-dependent
samples cannot be composited, the calculations must integrate changes in τj and ε over the
entire soil depth (from depth=0 to depth=lw):
M j = r pC j, p ò
lw
0
t j (l)
dl
(e (l) +1)
(eqn. C.15)
The numerical solution to eqn. C.15 yields the mass-flux equation presented in the
manuscript:
t j,k
(lk )
k=1 (e k +1)
n
M j = r pC j, p å
(eqn. C.16)
To calculate the initial mass of element j in the bedrock parent material (PMj) for a
particular soil, we derive a formula similar to eqn. C.16, with the exception that τj is
removed from eqn. C.10:
n
PM j = r pC j, p å
1
(lk )
k=1 (e k +1)
(eqn. C.17)
Finally, the depth integrated chemical depletion fraction (CDFj) over the whole soil profile
can be directly calculated from eqn. C.16 and eqn. C.17:
CDFj = -
Mj
PM j
(eqn. C.18)
We note that CDFj and -τj are often used interchangeably in the geochemical literature.
While the two calculations are similar, and typically approximate each other, there are
some notable exceptions. We refer the reader to (Brantley and Lebedeva 2011) for an
expanded discussion of these issues.
5
References
Brantley, S. L., and M. Lebedeva. 2011. Learning to read the chemistry of regolith to
understand the critical zone. Annual Review of Earth and Planetary Sciences
39:387-416.
Brantley, S. L., and A. F. White. 2009. Approaches to modeling weathered regolith. Reviews
in Mineralogy and Geochemistry 70:435-484.
Brimhall, G. H., O. A. Chadwick, C. J. Lewis, W. Compston, I. S. Williams, K. J. Danti, W. E.
Dietrich, M. E. Power, D. Hendricks, and J. Bratt. 1992. Deformational mass-transport
and invasive processes in soil evolution. Science 255:695-702.
Brimhall, G. H., and W. E. Dietrich. 1987. Constitutive mass balance relations between
chemical-composition, volume, density, porosity, and strain in metasomatic
hydrochemical systems - results on weathering and pedogenesis. Geochimica et
Cosmochimica Acta 51:567-587.
Chadwick, O. A., G. H. Brimhall, and D. M. Hendricks. 1990. From a black to a gray box — a
mass balance interpretation of pedogenesis. Geomorphology 3:369-390.
Egli, M., and P. Fitze. 2000. Formulation of pedologic mass balance based on immobile
elements: A revision. Soil Science 165:437-443.
Muir, J., and J. Logan. 1982. Eluvial/illuvial coefficients of major elements and the
corresponding losses and gains in three soil profiles. Journal of Soil Science 33:295308.
6
Table C1: Variables and units used in mass-balance calculations.
Variable Units
Description
Vp
length3
volume of bedrock parent material
Vw
length3
Volume of soil and/or regolith (weathering profile)
ρp
mass • length-3
Density of bedrock parent material
ρw
mass • length-3
Density of soil and/or regolith
Cj,p
unitless (massj • masssample-1) Concentration of element j in bedrock parent material
Cj,w
unitless (massj • masssample-1) Concentration of element j in soil and/or regolith
CI,p
unitless (massi • masssample-1) Concentration of immobile element i in bedrock
parent material
Ci,w
unitless (massi • masssample-1) Concentration of immobile element i in soil and/or
regolith
mj
mass
Mass flux of element j
ε
unitless (length3 • length-3)
Deformational strain
τj
unitless (mass • mass-1)
Mass transfer coefficient for element j
l
length
length (z direction) of soil and/or regolith sample.
Mj
mass • length-2
Mass flux of element j per unit area
PMj
mass • length-2
Initial mass of element j in soil and/or regolith profile
per unit area
CDFj
unitless (mass • mass-1)
Chemical depletion fraction: depletion (or
enrichment) of element j in soil/regolith relative to
bedrock parent material.
7
Volume of
soil/regolith
(Vw)
Volume bedrock
parent material
(Vp)
Dila on:
Vw > Vp
- Typically poorly developed
(young) soils & surface horizons.
- A ributable to incorpora on of
OM and eolian inputs,
biopedoturba on, and mineral
hydra on.
Collapse:
Vw < Vp
- Typically highly developed (old) soils
& subsurface horizons.
- A ributable to loss of soluble and
vola le elements with subsequent
compac on of original matrix.
Figure C1: Diagram illustrating volumetric changes (deformational strain) among bedrock
parent material and soils during chemical weathering and pedogenesis.
8