Supplemental Digital Content 1 (SDC)
SDC1 – Deterministic ADR Model Derivation
The ADR model for 1 – dimension, z – direction downward into the soil column is Equation 3 in
the manuscript where all terms are defined. In geological time–periods the chemical weathering
breakdown, by mineral fluid reactions, occurs on rock strata, rock fragments, sand grains,
minerals, etc. so that R is non-zero. However, on the human agriculture time-scale in the
Anthropocene epoch R = 0 is a reasonable assumption. At steady-state with R = 0, Equation 3
becomes:
𝑂 = −𝐷𝑏∗
𝑑 2 𝜌𝑏
𝑑𝑧 2
+ 𝑉𝑏
𝑑𝜌𝑏
𝑑𝑧
(3a)
The negative sign appears because diffusion occurs downward into the soil. One integration
yields:
𝑁 = −𝐷𝑏∗
𝑑𝜌𝑏
𝑑𝑧
+ 𝑉𝑏 𝜌𝑏
(3b)
With N the particle flux (kg/m2∙s) being the arbitrary constant of integration. At position z=0 soil
particles are assumed not to exit the soil control volume (CV), but the surface plane position may
expands or contract with time but without erosion loss of particles from the surface. This requires
N=0 which further requires no loss of particle mass from the active CV. However, soil bulk
density ρb (z) is allowed to vary with position within the soil column, z (m). Equation 3b
becomes:
𝑂 = −𝐷𝑏∗
𝑑𝜌𝑏
𝑑𝑧
+ 𝑉𝑏 𝜌𝑏
(3c)
Re-arranging it yields the simple ordinary differential equation with soil bulk density, ρ b , being
the dependent variable and soil depth z, the independent variable:
𝑑𝜌𝑏
𝜌𝑏
𝑉
= 𝐷𝑏∗ 𝑑𝑧
(3d)
𝑏
1
The indefinite integral of ρb (z) is:
ℓ𝑛𝜌𝑏 =
𝑧∙𝑉𝑏
𝐷𝑏∗
+𝐶
(3e)
with C the arbitrary constant. This result suggest that soil bulk density increases exponentially
with depth z(m).
Oxygen, moisture, food and nutrients are readily available at, on or near the
soil/atmosphere interface thus for use by the macro fauna that reside in the surface layer. Passive
transport processes, such as molecular or Brownian in pore water are much too slow for
providing the necessities of life. The macro fauna must aggressively seek these necessities and
they move up and down to acquire them or they creates open pathways to increase soil porosity
(i.e. burrow tubes) for nutrients to move downward from the surface or for waste materials to be
expelled upward to the surface. Their movements and burrows result in soil particle
translocation. Here the translocation is quantified as a random particle mixing process. Random
mixing is a useful and correct term for describing the mechanism and a Fickian–like diffusion
mathematical model is commonly invoked for this ecological system (Okubo, k. and Levine,
2001). Normally diffusion requires a gradient of the quantity being transported. However, in this
case the particle density gradient is the result of organism activities within the soil, apparently.
Nevertheless particle flux does occur, and D*b (m2/s) is the kinetic parameter needed to quantify
the numerical magnitude of the biological organism activities perturbs the soil particle into being
transported, and hence, the origin of the word “bioturbation”.
Returning to Equation 3e, the other model parameter is Vb (m3/m2 s). In a confined space
such as the volume element of soil positioned below the interface, the law of conservation of
mass must be satisfied which means that the biodiffusion of particle mass upward (or downward)
2
must be countered ostensibly by a flowing soil velocity (i.e. flux) Vb (m/s) in the downward (or
upward) direction.
In combination and being intimately connected by the mass balance, the two soil
processes control the magnitude variation of soil bulk density, ρb (g/m3) with depth z (m) as
given by Equation 3e. Although there is some evidence to support Equation 3e (Peterson et al.,
1992; Cousins et al., 1999a) it is a testable hypothesis and requires further testing and evaluation
in the context of known active soil bioturbation. Nevertheless, with lower and upper limits of
integration a definite integral for ρb results, it is:
𝜌𝑏2 𝑑𝜌
∫𝜌
𝑏1
𝜌
= ℓ𝑛(𝜌𝑏2 /𝜌𝑏1 )
One also exist for the independent variable z as:
𝑧
2
∫𝑧 𝑑𝑧 = 𝑧2 − 𝑧1
1
When these are entered into Equation 3e the result is:
ℓ𝑛(𝜌𝑏2 /𝜌𝑏1 ) =
(𝑧2 − 𝑧1 )𝑉𝑏
⁄𝐷∗
𝑏
(3f)
If at the surface, z1 = 0, with ρb1 (0) and z2 = h, depth below the surface, ρb2 (h), solving Equation
4f for the biodiffusion coefficient yields:
𝐷𝑏∗ = 𝑉𝑏 ℎ/ℓ𝑛(𝜌𝑏2 /𝜌𝑏1 )
(3g)
This is also Equation 4 in the manuscript.
This ADR model result is an advancement in bioturbation theory. The final equation
provides a basis for hypothesis testing of the macro fauna driven processes in soils that underpin
the theoretical concepts. Having a mass balanced base algorithm for quantifying the biodiffusion
coefficient is a key factor. It is a single response parameter for the macro fauna that is a function
of three measurable soil properties. As the equation represents, measurements of the soil
3
volumetric flux or soil “turnover’ velocity across two soil layers with differing measured bulk
density and separated by a known distance yields the effective numerical values of the
biodiffusion coefficient. Such simultaneous measurements in bioturbated surface soils are absent
in the public literature.
DISCUSSION. Typically, bulk density is related to porosity by ρb=ρP (1-ε) where ε is
volume void fraction (i.e. porosity) and ρP is the solid particle density (kg/m3). It is arguable that
the macro fauna do alter the void volume in the vertical dimension while maintaining their
lifestyle and that ρb2>ρb1 is possible. It should be appreciated that Vb is a vector quantity, so it has
magnitude and direction. Theoretically, Vb may be positive or negative, and this aspect is
captured in Figure 1, note the directed large vectors. Being directed upward or downward implies
that the concentration gradient must correspondingly respond and be either positive or negative
for diffusion to occur. Mathematically, the equation adjusts and in either case D*b must be
positive. For all cast production data Vb is positive. Due to the absence of density measurements
in all data sets, Equation 5 was not be used in obtaining numerical estimates of D*b, except as an
example calculation.
4