We formulate tracer particle transport and mixing in soils due to
disturbance-driven particle motions in terms of the Fokker–Planck equation.
The probabilistic basis of the formulation is suitable for rarefied particle
conditions, and for parsing the mixing behavior of extensive and intensive
properties belonging to the particles rather than to the bulk soil. The
significance of the formulation is illustrated with the examples of vertical
profiles of expected beryllium-10 (

Soils on Earth's surface are granular materials consisting of polymineralic clasts and individual mineral grains, organic matter and live biota. These materials experience patchy, intermittent mixing motions associated with disturbances due to bioturbation (Darwin, 1881; Shaler, 1891; Gabet, 2000; Reichman and Seabloom, 2002; Meysman et al., 2006; Wilkinson et al., 2009; Covey et al., 2010; Astete et al., 2015), the effects of frost and ice growth and thaw (Branson, 1992; Matsuoka and Moriwaki, 1992; Auzet and Ambroise, 1996; Branson et al., 1996; Harris et al., 1997; Matsuoka, 1998; Anderson, 2002), and the swelling and shrinking of certain minerals with wetting and drying (Eyles and Ho, 1970; Fleming and Johnson, 1975). In addition, these soil materials may undergo mixing motions in relation to the chronic creation and relaxation of disordered granular structures (Hsiau and Hunt, 1993; Utter and Behringer, 2004; Fan et al., 2015) associated with granular creep (Houssais et al., 2017; Ferdowsi et al., 2018).

Soil particle mixing is a key process in soil formation (Shaler, 1891; Birkeland, 1984; Wilkinson et al., 2009) and in its associated ecological role of “modifying geochemical gradients, redistributing food resources, viruses, bacteria, …and eggs” (Meysman et al., 2006), as well as being responsible for redistributing substances, including contaminants, attached to particles (Cousins et al., 1999; Covey et al., 2010; Astete et al., 2015). Moreover, the idea of disturbance-driven transport and mixing of soil particles is central to current treatments of soil creep (Culling, 1963; Roering et al., 1999, 2002; Gabet, 2000; Anderson, 2002; Gabet et al., 2003; Furbish, 2003; Roering, 2004; Furbish et al., 2009b, 2018a), the slow but steady bulk motion of soils on hillslopes, where the influence of gravity gives a downslope bias to particle motions. Because of the significance of soil particle mixing in numerous problems spanning ecological to geomorphic timescales, there is a continuing, compelling need to fully clarify the kinematics, and eventually the mechanical basis, of soil particle motions during transport and mixing (Furbish et al., 2009b, 2018a, b; BenDror and Goren, 2018; Ferdowsi et al., 2018).

Currently it is not possible to directly measure disturbance-driven particle
motions and associated mixing in the setting of a natural soil (although this
is entirely possible in experiments and numerical simulations of granular
creep, Utter and Behringer, 2004; Kamrin and Koval, 2012; Fan et al., 2015).
Moreover, we do not yet have a mechanical theory to describe these motions
given the complexity – notably the biotic complexity – of phenomena
involved in disturbances and associated particle displacements (Furbish et
al., 2009b, 2018a). Thus, as in studies of particle mixing associated with
marine bioturbation (Boudreau, 1986a, b; Boudreau and Imboden, 1987; Teal et
al., 2008; Lecroart et al., 2010), a key strategy to clarify the nature of
particle motions and mixing in soils involves using tracer particles
identified by specific physical or chemical properties. Two tracer properties
have emerged in the field of geomorphology as being of particular interest:
in situ cosmogenic radionuclide (CRN) concentrations and optically
stimulated luminescence (OSL) particle ages (Granger and Riebe, 2014;
Heimsath et al., 2002; Johnson et al., 2014). Cosmogenic nuclides continually
accumulate within minerals due to cosmic ray interactions with mineral atom
nuclei, for example, producing

Building on the pioneering work of Lal (1991) concerning the relation between rock erosion rates and the in situ production of cosmogenic radionuclides, vertical profiles of CRN concentrations in soils and underlying saprolite are now used to calculate soil production rates (e.g., Heimsath et al., 1997, 2000, 2005, 2012; Small et al., 1999; Anderson, 2002; Wilkinson et al., 2005) as well as to infer the intensity of soil particle mixing in the presence of mechanical and chemical erosion (Small et al., 1999; Schaller et al., 2009; Granger and Riebe, 2014; Furbish et al., 2018b). Similarly, profiles of particle OSL ages are used to assess particle mixing (Heimsath et al., 2002; Wilkinson and Humphreys, 2005; Johnson et al., 2014; Furbish et al., 2018b). Because profiles of CRN concentrations and OSL ages inform descriptions of soil transport and interpretations of the delivery of CRNs to channels (Heimsath et al., 2002; Anderson, 2015; Furbish et al., 2018b), and associated interpretations of erosion rates at catchment scales (e.g., Brown et al., 1995; Bierman and Steig, 1996; Granger et al., 1996; Granger and Riebe, 2014; Granger and Schaller, 2014; Lukens et al., 2016), there is merit in further clarifying what these profiles reveal about particle mixing in soils.

It is now conventional to conceptualize certain soil particle mixing motions as a diffusion-like process (Furbish et al., 2009b, 2018a, b; Campforts et al., 2016), building on the pioneering work of Culling (1963), who first pointed to the idea that soil particles undergo Gaussian diffusion in response to small disturbances. Various studies have thus appealed to some form of a diffusion equation or an advection–diffusion equation (Cousins et al., 1999; Covey et al., 2010; Stang et al., 2012; Johnson et al., 2014; Furbish et al., 2009b, 2018a, b; Astete et al., 2015; Campforts et al., 2016; Gray, 2018) to describe transport and mixing for comparison with measured vertical profiles of tracer particles in soils, notably including in situ CRN concentrations and particle OSL ages. But herein arises a need for caution, and clarity.

As described in Sect. 2, natural tracer particles – quartz particles in particular – occur under rarefied conditions, where it is unclear that a description of particle mixing based on a diffusion or advection–diffusion equation formulated for continuum conditions is satisfactory. Moreover, we often are interested in the transport of quantities that are associated with the particles and are not in themselves subject to advection and diffusion as normally envisioned to occur in a continuum. This includes particle CRN concentrations and OSL ages. Rather, such quantities might experience advection and diffusion, but only indirectly via the motions of the particles with which the quantities are associated. Within this context, our objectives in this mostly theoretical contribution are fivefold.

First, we illustrate why quartz tracer particles in soils experience
transport and mixing under rarefied (non-continuum) conditions, and why it
therefore becomes important to treat transport and mixing probabilistically,
in a manner that formally appeals to the statistical mechanics idea of
ensemble-expected (average) quantities. Our focus on quartz particles is
purposeful, as these are ideal targets for in situ production of

Note that, in the formulations presented below, we use full functional notation throughout. This provides clarity in how random variables, parameters, and moments of random variables depend on position and time, as well as how random variables might covary.

Quartz particles targeted in sampling for

Definition diagram of soil-mantled hillslope with mechanically
active soil thickness

Consider a soil element with dimensions

For a soil developed from granitic bedrock, 20 % to 60 % of the volume of
particles are quartz particles, some larger than 1 mm in diameter and many
smaller. Per unit volume, the number of quartz particles targeted in sampling
for

We therefore must admit at the outset that the number concentration of target
quartz particles does not necessarily satisfy the continuum hypothesis.
Nonetheless, we wish to use continuum-like formulations of transport and
mixing of particle concentrations and associated quantities, that is, where
particle concentrations,

For an element of soil with dimensions

In the developments below, we also consider joint probability density
functions, for example, the joint density

Consider a set of tracer particles that are undergoing transport and mixing
within a soil. Here we initially restrict this set to chemically resistant
quartz particles. Nonetheless, this set could consist of particles defined by
other mineralogies, or it could be defined as the subset of quartz particles
of a given size that possess a specified

As above, let

This formulation assumes Gaussian diffusion of particles. Interestingly, Culling (1963) first pointed to the idea that soil particles undergo Gaussian diffusion in association with particle concentration gradients, in response to small disturbances. Culling developed his ideas from kinetic theory and statistical mechanics, borrowing the description of Brownian motion due to Einstein (1905) and the formulation of a particle diffusion-like equation due to Chandrasekhar (1943), both of which start from the master equation (Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, b). Culling's formulation has for decades provided the inspiration for conceptualizing what now are referred to as “disturbance-driven” particle motions associated with bioturbation, freeze–thaw cycles, etc. (Darwin, 1881; Shaler, 1891; Eyles and Ho, 1970; Fleming and Johnson, 1975; Matsuoka and Moriwaki, 1992; Auzet and Ambroise, 1996; Harris et al., 1997; Matsuoka, 1998; Gabet, 2000; Anderson, 2002; Reichman and Seabloom, 2002; Meysman et al., 2006; Wilkinson et al., 2009; Covey et al., 2010; Astete et al., 2015), and numerous authors have applied some form of a diffusion equation to describe transport and mixing of soil particles (Cousins et al., 1999; Furbish et al., 2009b, 2018a, b; Covey et al., 2010; Johnson et al., 2014; Astete et al., 2015; Campforts et al., 2016; Gray, 2018).

We emphasize that Eq. (1) is basically an advection–diffusion equation. As
written, it is purely kinematic, as nothing is specified mechanically about
the velocity

We reemphasize a point made above: currently it is not possible to
directly measure particle displacements

Let

The double integral on the left side of Eq. (6) defines the expected value of
the product

We now multiply Eq. (4) by the product

In the absence of advection and diffusion, the joint probability density

In turn, integrating Eq. (16) with respect to

In principle, the experimentally determined OSL burial age of a particle is independent of its size. In addition, as previously mentioned, quartz particles targeted for single-grain OSL analysis have a relatively narrow range of sizes (0.35–0.425 mm). For these reasons we may neglect particle volume in the following formulation.

Let

In the absence of advection and diffusion, the joint probability density

Because of its significance for sampling of particles for OSL analysis, here
we consider the variance of particle OSL ages. Let

In turn, multiplying Eq. (20) by

The Fokker–Planck equation is basically an advection–diffusion equation. But
here we reemphasize that the

With respect to a soil column with dimensions

Similarly,

We write the product

Schematic diagram of

We now turn to a benchmark situation inspired by the pioneering work of
Lal (1991) and Lal and Chen (2005) concerning CRN profiles within rock, and
within well-mixed soils above rock, undergoing steady surface erosion. With
reference to Fig. 2, we imagine the idealized situation involving a
one-dimensional vertical mean motion of particles through a soil column,
where steady surface erosion plus any chemical mass losses matches the rate of
soil production at the base of the column (e.g., Mudd and Yoo, 2010; Dixon
and Riebe, 2014; Granger and Riebe, 2014). Although idealized, given that
surface erosion rates generally are not steady (e.g., Small et al., 1997;
Parker and Perg, 2005; Schaller et al., 2009), this benchmark nonetheless
represents a valuable starting point for assessing actual conditions in field
settings, including the possibility of a sudden change in surface erosion
(Granger and Riebe, 2014), and as a contrast for two-dimensional transport by
soil creep (Small et al., 1999; Anderson, 2015; Furbish et al., 2018b). With
respect to cosmogenic nuclides –

We note that quartz enrichment (Small et al., 1999; Granger and Riebe, 2014) due to chemical weathering and mass loss may occur during any transient approach to steady conditions, but under steady conditions this enrichment does not impact the mechanical transport and mixing of quartz particles. In addition, we are for simplicity neglecting the vertical variation in soil bulk density that can occur with bioturbation (e.g., Furbish et al., 2009b; see Fig. 4 therein).

In this steady problem, note that

Following Furbish et al. (2018b), we assume that particles experience a
constant radiation dose rate

With respect to the source

If the magnitude of the cosmic dose rate is similar to that of the external dose rate near the soil surface, then the total dose rate is approximately uniform (Fig. 2c). If, however, the cosmic rate does not fully offset the decrease in the external rate, we nonetheless suggest that the assumption of a uniform dose rate is a reasonable starting point for comparison with deviations in OSL age profiles that might be expected from a nonuniform dose rate, particularly under conditions of moderate to strong particle mixing, whose effects likely mask spatial variations in the total dose rate (e.g., Furbish et al., 2018b). That is, this is a parsimonious assumption – that the effects of mixing of ages outweigh any consequence of a nonuniform dose field. Previous studies using luminescence to examine soil mixing show relatively uniform total dose rates (e.g., Heimsath et al., 2002; Johnson et al., 2014).

In order to present our results below in a manner that highlights the effects
of differences in the intensity and depth dependence of particle mixing, it
is convenient to define the following dimensionless quantities denoted by
circumflexes:

The analytical results presented in the next two sections involving

Assuming that

Recall that the number concentration

For uniform mixing with

Also for reference below, the column-averaged particle OSL age

We now turn to numerical simulations of particles undergoing random-walk
motions within the soil column, during which they accumulate

First, the random-walk motions implied by the probabilistic formulations
above are in principle straightforward to implement numerically, and it is
important to demonstrate that such computational results match the analytical
results presented. In doing this, the simulations reveal important
information that is not readily apparent in the analytical results. This
includes an illustration of the variability in

Second, numerical simulations of particle motions within soils offer important opportunities to examine phenomena that cannot readily be treated analytically, for example, effects of particle residence times on mineral weathering or effects of a nonuniform radiation dose rate. So, spinning our first objective around, any numerical simulation of random-walk motions must be able to correctly reproduce benchmark (analytical) solutions before being applied to more complex situations, for example, two-dimensional motions and unsteady conditions. The simulations presented here highlight important aspects involved.

Following Furbish et al. (2018a, b), we adopt a straightforward
Eulerian–Lagrangian algorithm to simulate particle motions in a mass-conserving manner. Particles are numerically introduced to the base of the
soil column (

Each particle accumulates

The lower boundary (

All simulated

Plot of dimensionless

Plot of dimensionless

The simulated, expected

Example histograms representing the distribution

With both uniform and nonuniform mixing, the distribution

Plot of dimensionless OSL age

Plot of dimensionless OSL age

The simulated, expected OSL ages closely match the theoretical results for
different values of the Péclet number

Example histograms representing the distribution

With both uniform and nonuniform mixing, the distribution

Plot of dimensionless variance

The simulated second moment

Exceedance probability plots of dimensionless particle OSL age

Plot of dimensionless column-averaged OSL age

The simulations suggest that particle OSL ages within the entire soil column
are distributed approximately exponentially for both uniform and nonuniform
mixing (Fig. 10), where the column-averaged age

We emphasize that, in contrast to continuum formulations of advection and
diffusion of material (e.g., mass) measured as an intensive quantity (e.g.,
concentration) of the continuum, the extensive and intensive particle
properties

In the case of particle OSL ages, the formulation similarly describes the
behavior of the expected value (and the variance) of individual particle OSL
ages at a position

An essential lesson is this: when the quantity of interest can be expressed
as a total value within an interval

Throughout we have emphasized that

We cannot avoid this issue of legacy (or “inheritance”), namely, the
likelihood that what is being measured reflects only the recent history of
transport and mixing as opposed to conditions consistent with an imagined
behavior averaged over longer timescales, as represented by the expected
profiles in Figs. 3, 4, 6 and 7 above. In the case of measured profiles of

We now take the ensemble average of relaxation timescales

This points to the need to avoid over-interpreting the forms of profiles from
individual soil pits in terms of what these forms might reflect about the
vertical structure of mixing (e.g., uniform versus depth-dependent mixing).
Unfortunately, this issue is exacerbated by the reality that digging soil
pits and sampling for

Momentarily assuming that mixing conditions are reasonably reflected by the
expected particle OSL age profile

Let is assume that within a small interval of

The results of the numerical simulations as depicted in Figs. 3, 4, 6 and 7
provide an important perspective on the nature of production of

Combining Eqs. (4) and (14), neglecting particle volume and the decay of

We normally envision that local production of a quantity implies a local
increase in the quantity. But this is not necessarily so when viewed in the

The numerical simulations suggest that the overall particle OSL age
distribution is approximately exponential (Fig. 10), consistent with field
data (see data of Heimsath et al., 2002 as described by Furbish et al.,
2018b). This result awaits a theoretical explanation. Meanwhile, as described
by Furbish et al. (2018b), the distribution

The emergence of a maximum average OSL age

That the numerical simulations mimic analytical solutions for the benchmark situation of a one-dimensional mean motion involving both uniform and nonuniform mixing with varying mixing intensities lends confidence in applying the numerics to more complicated situations. Such situations might be motivated by questions concerning consequences of transient conditions of surface erosion and soil production, aeolian inputs to the soil, particle weathering in relation to particle aging, accumulation of luminescence signals with nonuniform dose rates, and the structuring of tracer particles under depositional conditions. Our experience suggests the need to implement the numerics of boundary conditions carefully, ensuring consistency with global particle conservation.

Here we return to our starting point. Our use of the Fokker–Planck equation
assumes Gaussian diffusion of tracer particles. As described above, this is a
parsimonious choice whose consequences, and veracity, must be judged by its
consistency with measurable outcomes of mixing, including profiles of CRN
concentrations and OSL ages as emphasized here, but possibly to include other
soil properties. We suggest that a Gaussian model of particle mixing is
robust inasmuch as this mixing behavior is insensitive to the form of the
probability distribution of particle displacements,

Plot of dimensionless expected concentration

Plot of dimensionless expected particle OSL age

Here we focus on results for the one-dimensional benchmark case (Sect. 4,
Fig. 2) – specifically the profiles of expected

As described above, these profiles systematically vary with the Péclet
number,

Not surprisingly, with weak mixing the

These profiles highlight that uniform particle mixing is not synonymous with
the idea of complete mixing, and why a uniform profile of

Here we step back and look at published data. We first note that, whereas our
benchmark case involves a steady one-dimensional mean motion, available
field-based measurements of

As an important backdrop to the benchmark case examined here,

Relatively uniform

An OSL age profile

In all cases summarized above, the profiles suggest moderate (

To our knowledge there are no available measurements of
profiles of

The code for simulating particle motions is written for Matlab and is available by request from any of the authors.

To further illustrate the significance of the probabilistic formulation of
conservation in relation to rarefied versus continuum conditions, here we
start with the familiar example of Brownian motion, the initial formal
description of which is separately attributable to Einstein (1905) and von
Smoluchowski (1906). With reference to Fig. A1, let

Plot of coordinate position

Histograms of particle positions

Let us now imagine an arbitrarily great number

Now select a system with a modest number

Let us now consider a great number

Histogram of particle positions

To complete the picture, suppose that the

To place these ideas within the context of soil tracer particles, including
the practical assessment of rarefied versus continuum conditions, let us now
imagine a soil column of thickness

Let us now measure the number of tracer particles within each of 100 cm

Plot of proportion

Now imagine that, due to randomness in the particle mixing process, the
number of tracer particles

Consider first the idea of describing this one realization as a continuum,
that is, where we might imagine that

Consider now the idea of expanding either (or both) the horizontal dimensions

Finally, consider the idea of taking an ensemble average, where we return to
the small 1 cm

With respect to developments in the text, the Fokker–Planck equation
describes the time evolution of the probability density

Assuming

We first rewrite Eq. (B2) in terms of the flux

In this problem, the concentration

With uniform mixing (

At this point we assume that the upper boundary flux is purely advective.
Physically this means we are imagining that the rate of surface erosion

With nonuniform mixing (

Like the results in Sect. A1 above, the concentration gradient at the soil
surface,

The number concentration

Under steady conditions the total number of particles with finite
(measurable, non-saturated) OSL age within the soil element remains fixed. A
particle entering the soil cannot attain a finite OSL age until it reaches
the surface and is bleached, and then it becomes buried and exposed to the dose
field. Thus, even though particles that eventually possess a finite OSL age
continuously enter the element through its lower boundary, this boundary must
be considered a zero flux boundary, as no particle with finite age can be
added to the soil. Particles at the soil surface with zero OSL age are
removed by erosion. The erosion rate matches

With uniform mixing (

Plot of dimensionless OSL particle concentration

Let

We now rewrite Eq. (D2) as

We now write Eq. (D4) as

With uniform mixing (

With nonuniform mixing (

For a set of particles possessing finite OSL ages within any interval

For steady conditions we start with Eq. (40) in the text, namely,

With

With

All authors contributed to conceptualizing the problem and its technical elements. DF and RS contributed to the analytical analysis and the numerical simulations. RS wrote the final code. DF wrote much of the paper with contributions by RS and AKZ.

The authors declare that they have no conflict of interest.

We acknowledge support by the US National Science Foundation (EAR-1420831 to David Jon Furbish and EAR-1734299 to Rina Schumer). We appreciate continuing discussions with Peter Haff, Joshua Roering and Mark Schmeeckle concerning rarefied particle behavior, and Dan Morgan concerning cosmogenic radionuclide systematics. Edited by: Eric Lajeunesse Reviewed by: two anonymous referees