Depth-dependent soil mixing persists across climate zones

Significance

Soil mixing occurs when sediment and organic matter are moved by biotic processes (burrowing mammals and invertebrates, root growth, and decay) and abiotic processes (movement of water and granular creep). Soil mixing plays a key role in carbon storage, nutrient and contaminant dispersion, and hillslope sediment transport, yet we lack quantitative information about soil mixing and its variation with depth in a soil profile. Here we present evidence showing that soil mixing systematically varies with depth in the soil. We also find that soils appear to be mixed similarly across climate zones, suggesting that soil mixing operates in a similar way over a range of different environments.

Abstract

Soil mixing over long (>10² y) timescales enhances nutrient fluxes that support soil ecology, contributes to dispersion of sediment and contaminated material, and modulates fluxes of carbon through Earth’s largest terrestrial carbon reservoir. Despite its foundational importance, we lack robust understanding of the rates and patterns of soil mixing, largely due to a lack of long-timescale data. Here we demonstrate that luminescence, a light-sensitive property of minerals used for geologic dating, can be used as a long-timescale sediment tracer in soils to reveal the structure of soil mixing. We develop a probabilistic model of transport and mixing of tracer particles and associated luminescence in soils and compare with a global compilation of luminescence versus depth in various locations. The model–data comparison reveals that soil mixing rate varies over the soil depth, with this depth dependency persisting across climate and ecological zones. The depth dependency is consistent with a model in which mixing intensity decreases linearly or exponentially with depth, although our data do not resolve between these cases. Our findings support the long-suspected idea that depth-dependent mixing is a spatially and temporally persistent feature of soils. Evidence for a climate control on the patterns and intensities of soil mixing with depth remains elusive and requires the further study of soil mixing processes.

Soils cover much of Earth’s terrestrial landscape and consist of mechanically mobile granular material such as sediment and organic matter. This mobile material can be mixed by biotic processes, such as burrowing by mammals and invertebrates or due to the growth of plant roots, and by abiotic processes such as movement of sediment by water and gravity. However, we cannot directly observe these subsurface soil mixing processes (1–3). This is problematic as soil mixing contributes to a variety of important processes. For example, soil mixing facilitates the fluxes and residence of carbon within the largest terrestrial carbon reservoir (4–6), it influences the rates of bedrock weathering and soil production (2, 7, 8), it can drive the migration and alteration of nutrients (1, 9), and it operates on hillslopes where soil disturbances and granular creep transport sediment and shape landscapes (10–12). Because soil mixing plays such a significant role in these and other important processes, understanding its mechanisms, rates, and patterns is critical to a variety of fields.

This paper addresses two particularly compelling questions concerning soil mixing. The first is this: How do the various depth-dependent soil mixing processes translate into the motions of soil material? The second is a corollary question: As soil mixing processes appear to be strongly influenced by climate, does the resulting soil mixing express differently across climate zones? Regarding the former, researchers have observed greater activity in biotic and abiotic processes closer to the surface than at depth (e.g., refs. 2, 3, and 13–15). However, translating these observations into quantitative rates of soil movement is challenging and persists as a knowledge gap due to a lack of methods. For example, measuring fallout radionuclides (e.g., ¹³⁷Cs) is a useful method to evaluate soil movement since the beginning of nuclear testing (16) but does not address the longer timescales involved with mixing processes such as tree falls, with recurrence on the order of centuries (2, 17, 18). Due to a lack of conclusive evidence, the extent to which biotic and abiotic soil mixing processes translate into the vertical motions of soil material has remained speculative since the days of Darwin (19).

The second corollary question concerns how soil mixing relates to climate. The idea that climate influences soil development appears in the earliest models in soil science (20, 21). Similarly, climate has long been considered to be one of the most important factors influencing soil mixing (3, 22, 23). Climate is thought to influence the composition and abundance of species that interact with soil and the availability of water, both liquid and frozen, that can move soil material (2, 14). As with the depth dependency question above, translating these climate related observations into quantitative predictions of soil mixing is complicated by a lack of methods, and quantitative comparisons of soil mixing rates across climate zones are rare.

The unanswered questions above are important because various untested assumptions about soil mixing can be found in models of a variety of different environmental systems. For example, global carbon flux models have sometimes assumed that soil mixing is uniform across the soil depth, which greatly overestimates major carbon fluxes (8, 24, 25). Additionally, landscape evolution models using depth-independent soil transport rates may mischaracterize the flux of sediment, which limits their explanatory power (26). Furthermore, soil mixing plays a major role in contaminant and radioisotope dispersion (16, 27, 28), yet modeling efforts produce significantly different results depending on whether or how the vertical structure of soil mixing is handled (29, 30). Finally, as we do not know the relationship between soil mixing and climate, we lack an ability to predict how soil mixing may influence future and past climate change, due to uncertainty in estimates of soil carbon fluxes and resulting climate change (e.g., ref. 6).

To help resolve the uncertainty surrounding soil mixing, this paper uses mineral luminescence data together with soil-mixing theory to examine the depth patterns of soil mixing, with a focus on determining how the mixing intensity may vary over the soil depth and probing the hypothesis that climate influences the depth patterns and rates of soil mixing. To do this, we advance existing theory (31–33) to interpret compilations of soil luminescence data. Luminescence is a light-sensitive property of mineral matter that can be used to understand the long (>100-y) timescale movement of matter through soils (34, 35). Our analysis is restricted to the zone of mobile soil material above immobile in situ material (Fig. 1 and refs. 36–38). We examine previously published measurements of luminescence as a function of depth in soil material where mixing is either reported by the authors or is likely to have occurred. These datasets span a range of climate zones, which we simplify into four main Köppen–Geiger zones: temperate, tropical, arid, and cold (39). The processes that mix mobile soil material vary significantly across these datasets (SI Appendix, Table S1). A strong depth or climate control on soil mixing should be reflected in the luminescence of soil.

Fig. 1.

Definition diagram for this study. We conceptualize a zone of mobile material consisting of sediment and organic matter overlying a weather in situ parent material. Particles, such as sand grains, release from the parent material and undergo advective and diffusive motions as a function of soil mixing. As these periodically visit the surface, their luminescence changes through time. The resulting depth profile of particle luminescence is a function of soil mixing rates.

Luminescence as a Sediment Tracer

To examine soil mixing, we employ luminescence, a property that can cause sand grains to emit light when an observer applies heat, light, or pressure (40). In soil, the luminescence potential of a grain of sand increases over thousands of years while the grain is buried but will decrease rapidly when exposed to sunlight at the soil surface (5, 41). As sand grains migrate through a soil profile, their luminescence will change based on the history of surface exposure and burial (35). As various soil mixing processes move sediment, the depth profile of luminescence in soil evolves as a function of mixing rates (34, 42, 43). The ability to measure luminescence on geologically ubiquitous quartz and feldspar and the applicable timescales of thousands of years make luminescence a useful new tool to address soil mixing (37, 38, 44). A description of the physics of the trapped-charge phenomena behind luminescence is given in SI Appendix, section 1.

To interpret soil luminescence data in terms of mixing, one needs a theory that relates luminescence to the statistics of grain movement in soils. Furbish et al. (33) point out that the conditions of luminescence-suitable particles in soil, such as sand grains, are rarefied. This means that the volumetric number concentration of these particles is very low, and as such, the averaging length scale needed to describe the movement of these particles with a continuum-style formulation may approach the soil thickness. To avoid such averaging, we use a probabilistic formulation that considers a great number of nominally identical, but independent, soil/particle systems at a given moment in time. This ensemble-averaged probabilistic approach allows us to create a model that describes the time evolution of the probability distributions of particle positions and the resulting ensemble expected value of luminescence. We use this formulation to interpret compilations of luminescence data in terms of soil mixing (31–33), noting that it provides a description of statistically expected conditions for any individual realization.

Luminescence as a Random Variable in Soils.

Following Furbish et al. (32, 33), we model the dimensionless depth profiles of ensemble-averaged luminescence using a Fokker–Plank equation, a probabilistic form of the Master equation, adapted to describe the ensemble-averaged movement of luminescence of soil particles under rarefied conditions. First, our approach treats the number of trapped electrons as a conserved quantity (45), in a manner similar to the treatment of the concentration of ¹⁰Be atoms in soil (33). We then use luminescence as a proxy for the trapped electrons by defining luminescence as the sensitivity-corrected luminescence intensity normalized by the sensitivity-corrected luminescence intensity at saturation for a given soil depth. Sensitivity is the number of photons of luminescence produced per unit radiation, and it varies grain by grain unless corrected for during measurement (46). Saturation is defined as the state in which all available electron traps are filled by displaced electrons. This normalized luminescence varies from 0 to 1 and is treated as a proxy for the normalized concentration of trapped electrons, which also varies from 0 to 1. This treatment departs from previous uses of luminescence ages as a soils tracer (32–34, 37, 38, 41, 44) and has the advantage of allowing luminescence to be directly treated as a property that evolves with progressive soil mixing.

The governing equation for ensemble-averaged luminescence (SI Appendix, section A) is

$$\frac{\partial }{\partial t}\left[L\left(z,t\right)\right]=\frac{\partial }{\partial z}\left[w_p\left(z,t\right)L\left(z,t\right)-{\kappa }_z\left(z,t\right)\frac{\partial L\left(z,t\right)}{\partial z}\right]+\frac{D_R}{\overline{D_0}}\gamma \left(1-L\left(z,t\right)\right),$$ [1]

where $L$ is the ensemble-averaged luminescence per the definition above, $t$ is time, $z$ is height above the soil/saprolite contact, $w_p$ is the ensemble-averaged vertical particle velocity, ${\kappa }_z$ is the ensemble-averaged vertical particle diffusivity, $D_R$ is the averaged dose rate for the soil, $\overline{D_0}$ is the arithmetic average characteristic dose for soil particles, and $\gamma$ is a constant relating the arithmetic and harmonic averages of $D_0$.

We follow Anderson (47) and Furbish et al. (31) and examine three cases for the motions of particles in soil: 1) one-dimensional (1D) vertical motion with uniform mixing $\left({\kappa }_z\left(z\right)={\kappa }_{z_0}\right)$ where surface erosion removes particles from the soil when they travel vertically through the column $\left(w_p\left(z\right)=w_0\right)$, 2) 1D vertical motion with a linear depth dependency for soil mixing $\left({\kappa }_z\left(z\right)={\kappa }_{z_0} \frac{z}{h}\right)$, and 3) 2D motion with linear soil mixing and no surface erosion where the downslope movement of soil satisfies continuity with the vertical advection of soil $\left(w_p\left(z\right)=w_0\left(1-\frac{z^2}{h^2}\right)\right)$. Here ${\kappa }_{z_0}$is a maximum soil diffusivity at the surface, $w_0$ is the soil production rate at the soil/saprolite interface, and h is soil mobile layer thickness. Finally, to eliminate the need to choose values for various parameters and to create a comparable case with the data, we use the dimensionless quantities $\widehat{t}=\frac{t}{h/w_0}$ and $\widehat{z}=\frac{z}{h}$. We then obtain

$$\text{Uniform}\hspace{0.17em}1\text{D}:\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{t}}=\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{z}}+\frac{1}{Pe}\frac{{\partial }^{2}L\left(\widehat{z},\widehat{t}\right)}{\partial {\widehat{z}}^2}+D_n\left(1-L\left(\widehat{z},\widehat{t}\right)\right),$$ [2]

$$\text{Linear}\hspace{0.17em}1\text{D}:\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{t}}=\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{z}}+\frac{1}{Pe}\left[\frac{{\partial }^{2}L\left(\widehat{z},\widehat{t}\right)}{\partial {\widehat{z}}^2}+\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{z}}\right]+D_n\left(1-L\left(\widehat{z},\widehat{t}\right)\right),$$ [3]

$$\text{Linear}\hspace{0.17em}2\text{D}:\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{t}}=\left(1-{\widehat{z}}^2\right)\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{z}}+\frac{1}{Pe}\left[\frac{{\partial }^{2}L\left(\widehat{z},\widehat{t}\right)}{\partial {\widehat{z}}^2}+\frac{\partial L\left(\widehat{z},\widehat{t}\right)}{\partial \widehat{z}}\right]+D_n\left(1-L\left(\widehat{z},\widehat{t}\right)\right),$$ [4]

where the following dimensionless numbers appear:

$$Pe=\frac{h{w}_0}{{\kappa }_{z0}},$$ [5]

$$D_n=\frac{h\gamma }{w_0}\frac{D_R}{\overline{D_0}}.$$ [6]

Each of these dimensionless numbers represents a different control on the system. The first, $Pe$, is a Peclet number representing a ratio between the rate of advection, equal to the soil production rate, and the strength of soil mixing. The Peclet number serves as a ratio of the advective transport to diffusive (mixing) motions. The number $D_n$ represents the rate of advection relative to the rate of luminescence generation and is defined as the ratio between the time needed to advect soil across the soil thickness and the time needed to regenerate luminescence.

In most applications, luminescence is reported in terms of the equivalent amount of absorbed radiation dose. This equivalent dose, $D_E$ (Joules per kilogram), is the amount of radiation needed to produce the observed amount of natural luminescence and is the most commonly reported measurement of luminescence in our data compilation. The equivalent dose is obtained from modeled dimensionless luminescence using

$$\widehat{D_E}\left(\widehat{z},\widehat{t}\right)=-ln\left(1-L\left(\widehat{z},\widehat{t}\right)\right),$$ [7]

where $\widehat{D_E}$ is equivalent dose nondimensionalized by dividing by $\overline{D_0}$.

The structure of Eqs. 2–4 provides a useful means to broadly evaluate the general depth profiles of soil mixing intensity. When normalized over the thickness of the zone of mobile soil material and by the equivalent dose of the lowermost sample, the relationship of ensemble-averaged luminescence versus depth displays either a concave profile in the case of a depth-independent soil mixing profile, that is, uniform soil mixing, or a convex profile in the case of a depth-dependent depth profile (Fig. 2). This difference in concavity serves as our basis for distinguishing between depth-independent and depth-dependent soil mixing profiles in our compiled dataset.

Fig. 2.

Results of the modeling of Eqs. 2–4 for $Pe$ values over 5 orders of magnitude (line patterns in legend). Lower $P_e$ values indicate a stronger influence of mixing. (A) Modeled nondimensional (N.D.) luminescence $L$ versus N.D. depth $\widehat{z}$ for a uniformly mixing soil (Eq. 2) and (B) resulting N.D. equivalent dose $\widehat{D_E}$ (Eq. 7). (C) Plot of equivalent dose normalized by maximum depth and maximum equivalent dose for comparison with the compiled data in Fig. 3. Black line shows linear relationship with slope of 1 for visual aid. Visible convexity is present in the normalized equivalent doses (red lines). (D–F) Equivalent plots for modeling of linear mixing depth profile. Note concavity in F versus C. (G–I) Equivalent plots for modeling of exponential mixing profile. Note concavity in I versus C. Compare C, F, and I with Fig. 3.

We numerically modeled the dimensionless Eqs. 2–4 using an explicit finite-difference scheme programmed in Python 3.4. In all model runs, we use a Dirichlet condition for the soil surface of $L\left(1\right)=0$ (32). For the 1D uniform case, we use a Neumann condition at the soil/saprolite boundary $\widehat{q}\left(0\right)=L\left(0\right)-\frac{1}{Pe}\frac{dL\left(0\right)}{d\widehat{z}}=1$. For the 1D and 2D linear cases, the diffusivity is set to 0 at the soil/saprolite boundary, and as such, the lower boundary is fixed at 1: $\widehat{q}\left(0\right)=L\left(0\right)=1$. We ran the model for each mixing profile over five orders of magnitude for Pe and then converted the dimensionless luminescence $L\left(\widehat{z},\widehat{t}\right)$ into dimensionless equivalent dose $\widehat{D_E}\left(\widehat{z},\widehat{t}\right)$ using Eq. 7.

To compare with the field data compilation, we normalized the model results by either the maximum $\widehat{D_E}\left(\widehat{z},\widehat{t}\right)$ value or by 3 if the maximum $\widehat{D_E}\left(\widehat{z},\widehat{t}\right)$ is greater than 3. The reason for using 3 as a normalizing value is that $\widehat{D_E}\left(\widehat{z},\widehat{t}\right)$ is the dimensional equivalent dose divided by $\overline{D_0},$ and $3\overline{D_0}$ represents effectively 95% of the saturated state (i.e., $\overline{n_p}=\overline{n_s}$). From a measurement standpoint, it is unlikely that equivalent doses higher than $3\overline{D_0}$ are analytically resolvable, and therefore, equivalent doses higher than $3\overline{D_0}$ can be regarded as saturated.

Model Assumptions.

The model involves several assumptions. First, we focus on sand-sized (90- to 250-μm-diameter) grains due to their favorable luminescence behavior. Soil mixing is thought to vary by grain size, with finer grain sizes having higher mobility and a tendency to migrate upward in a soil profile; coarse sizes, such as gravel, may tend to migrate downward (14). As such, our soil mixing results will be applicable to the finer soil material. Next, due to the level of abstraction necessary to compare soil mixing across climate zones, we remain largely agnostic to the specific processes mixing soil. This assumption is expressed as treating the particle motions as advective and diffusive terms having finite first and second moments (definitions of $w_p$ and ${\kappa }_z$ in the derivation in SI Appendix, section A). How individual processes produce mixing and alter luminescence is beyond the scope of this study but offers avenues for future research.

Next, using a probabilistic approach toward luminescence sometimes requires ensemble averaging, which in turn can produce covariance terms that are difficult to constrain (48). To minimize any covariance terms, we limit our analysis to a single mineralogy, quartz, which demonstrates generally straightforward luminescence behavior, e.g., no anomalous fading, and can often be described with a single saturating exponential as described below. In addition, quartz has the advantage of resistance to chemical weathering within soils, which minimizes any changes in grain size, and thus grain mobility, over time. Furthermore, quartz has a negligible internal dose rate, meaning that the total dose rate experienced by the grain is largely controlled by external factors: concentrations of radiogenic elements, water content, and cosmic ray flux $\left(D_R\right)$. This external control suggests that there is unlikely to be significant covariance between the dose rate $D_R$ of the soil and properties internal to the grain such as the e-folding scale of luminescence versus dose, $D_0$. The $D_R$ is treated as being uniform over the soil thickness following Furbish et al. (33). We assume that grain sizes used are in a narrow range (e.g., 250- to 180-μm-diameter grains), such that grain size effects on $D_0$ (e.g., ref. 49) are small compared to the natural variability of luminescence in soils based on random walk simulations (32, 33).

It is important to note that quartz may not always be the ideal mineral for soil tracer applications (34). Quartz extracted from crystalline magmatic or metamorphic bedrock can show a range of unfavorable luminescence behaviors, including very low signal-to-noise ratios, measurement irreproducibility, and thermal instability, among others (50–52). Potassium feldspars appear to circumvent many of these problems with greater probability of favorable luminescence properties, limited sensitivity change, and high internal dose rates (34) and demonstrate value as a soils tracer (37, 38). Here we do not consider explicit treatment of feldspars in our ensemble-averaged approach because 1) all but two of our datasets are quartz; 2) there is significant uncertainty on how to treat feldspar-specific properties such as the variable growth curve kinetic order (53) and anomalous fading (54); and 3) as feldspar weathering is a primary mechanism of soil production (36), feldspar grains decrease in diameter, and thus may increase in mobility, with time in the soil column (55). How each of these terms should be treated in a probabilistic ensemble-averaged derivation is unclear as we lack the data needed to inform an approach. However, as there are only two datasets in our compilation that use feldspar (34, 37, 38) and given that they use postinfrared infrared stimulated luminescence methods that greatly reduce anomalous fading (56) and appear to follow quartz-like exponential growth curves (34), we conclude that any differences with quartz methods are unlikely to affect our conclusions.

Finally, we note that measurements of the luminescence of multigrain aliquots can be weighted by the luminescence sensitivity (luminescence per unit absorbed radiation dose) of individual grains (57–59). We compare both single-grain and multigrain aliquot datasets and find that any effects of the sensitivity-weighing neither are resolvable in our analysis nor affect the conclusions of this paper (Fig. 3). This is consistent with the idea that only a few grains contribute most of the luminescence in any given aliquot (58).

Fig. 3.

Luminescence versus soil depth sourced from the data compilation (SI Appendix, Table S1). Left shows single aliquot data. Right shows single-grain data. The term “single aliquot” refers to measurements made on multiple grains at once and contains an additional averaging step compared to the single-grain measurements, which are made on individual grains. Points and lines show the equivalent dose and uncertainty respectively as reported by authors. Depth values are normalized by the depth of the lowermost sample. Data from ref. 68.

Observations of Soil Luminescence across Climate Zones

We compiled datasets of OSL equivalent dose depth profiles in soils in environments ranging from temperate forests in eastern Australia to the tropics of southern Ghana (SI Appendix, Table S1, and refs. 5, 34, 37, 41–44, and 60–66). All these datasets stem from locations with well-developed soils where soil mixing is driven by mixing agents such as burrowing fauna (e.g., termites, ants, and mammals) and dense vegetation with roots penetrating into the soil (SI Appendix, Table S1). In one case, we extract depth profiles from an ancient buried soil where the deposit was sampled at high density (67). We use data points from samples that are taken either directly from soil material as identified by authors or from deposits where the authors indicate visible evidence of mixing such as modified or overprinted sedimentary structures. These datasets consist of measurements performed on single grains and/or on multigrain aliquots of quartz or feldspar sand-sized (90- to 250-μm) grains. The key difference between the single-grain and multigrain datasets is that the multigrain datasets include averaging during measurement (57). Depending on data availability, the data we plot in Fig. 3 show the mean equivalent dose or the central age model (68). To get the data into a form suitable for comparing with the model, we normalize each sample depth by the depth of the lowermost sample and normalize each equivalent dose by the lowermost equivalent dose. All data points are then scaled between 0 and 1 for both depth and equivalent dose.

The global data compilation in Fig. 3 shows a convex relationship between depth and equivalent dose for both the single-grain and single-aliquot datasets. When compared with the model results in Fig. 2, the convexity of the field data provides evidence for depth-dependent mixing profiles. This depth dependency appears irrespective of climate zone, with temperate, subtropical, and tropical datasets falling into the same approximate area in Fig. 3. The mixing processes at these climatically diverse sites are varied. Despite variation in local biota and mixing mechanisms, depth-dependent mixing appears consistent among the soils observed in our datasets. Furthermore, there does not appear to be any particular trend between apparent values of the Peclet number Pe and climate zone, such that no climate zone is uniquely associated with a specific Pe value.

Implications

With regard to the questions posed in the introduction, all of the compiled data are consistent with a depth-dependent mixing model, thus providing evidence that depth-dependent soil mixing processes do indeed result in an overall depth-dependent pattern of soil mixing intensity. Per the second corollary question, there appears to be no clear grouping of luminescence data by climate zone, with most of the datasets appearing to fall within a limited Peclet number range $\left(Pe=\left[0.1,1\right]\right)$. This is an interesting result as the effects of climate have long been hypothesized to exert a strong control on soil mixing processes (2, 3, 23), yet we do not see a clear differentiation of datasets by climate zone. We discuss below the consistency and implications of a depth-dependent soil mixing profile across our datasets and then address the lack of clear differentiation of the luminescence datasets by climate zone.

First, a depth-dependent soil mixing model appears to best explain our results. A key result from our model is that the form of normalized luminescence depth profiles between the uniform and depth-dependent soil mixing rates should display different concavity (Fig. 2). The compiled soil luminescence data points almost completely fall within the zone of the plot that is consistent with depth-dependent mixing, in this case, soil mixing intensity that decreases linearly with depth (Fig. 3). The data are also consistent with a model in which soil mixing decreases exponentially with depth (SI Appendix, Fig. S2) but not with a model with uniform mixing throughout the soil profile.

Our results are derived from fine sand-sized grains (90- to 250-μm diameter). We interpret our results as reflecting the broader mixing of the soil across most mobile material but note that some grain sizes may not be reflected in these results. For example, very coarse material such as gravel may tend to move downward in a soil profile during mixing, creating stone lines (14). Furthermore, very fine material in the silt and clay size ranges may be easier to move via mixing processes. The applicability of our luminescence method may depend on the grain size composition of the soil (in particular whether sand is present) and how specific mixing processes sort grain sizes. We assume that the fine sand size range is appropriate for evaluating mixing, but future work should test this assumption.

The finding that a depth-dependent model best fits the luminescence data provides a key link between the observations and interpretations of various processes that decrease in intensity with depth in soil (2, 3, 13–15, 26) and the actual resulting movement of mobile soil material. Namely, the structure of mixing intensity is depth-dependent following mixing that occurs via various biotic and abiotic soil mixing processes. Note that we are observing the time-integrated effects of various soil mixing processes. It is beyond the scope of this study to examine how individual mixing processes translate into particle motions, although the methods presented here could potentially be used to examine process-specific soil mixing in future research. Our results provide evidence supporting assumptions regarding the structure of mixing that underpin several areas of study, including the transport and weathering of sediment down hillslopes (13, 26), carbon cycling and storage in soil (8, 69), ecosystem engineering in soil ecology (1, 2), and contaminant transport modeling (16, 28).

Our second corollary question asks whether the influence of climate factors on soil mixing processes manifests in the resulting soil mixing structure and $Pe$ values. For the former, these results provide evidence that depth-dependent soil mixing is a potentially widespread phenomenon that is present across climate zones. The presence of depth-dependent soil mixing across climate zones implies that the processes that lead to soil mixing are either insensitive to climate with respect to the depths at which they operate or, despite climate-driven variations in mixing process, all produce the same net result of a mixed soil. From a biotic perspective, a possible explanation is that there may be a consistent life strategy such that whatever species are present for a given climate, the biota mix soil in a similar depth-dependent manner during their search for resources and shelter. From an abiotic perspective, the athermal granular creep of soil material (70) can induce depth-dependent mixing via shear, which proportionally follows the exponential-like velocity profiles of granular flow (15). Likewise, water-related mixing processes, such as shrink–swell of clay-bearing soils or the dilation and collapse of pore space by freezing, both likely have a depth dependence due to water delivery via precipitation (3, 14, 47). Finally, the depth-dependent mixing of soils appears to contrast with the reported mixing of agricultural soils by humans, which appears to show uniform mixing with depth (71).

There appears to be a lack of a clear differentiation in the Peclet number $Pe$ by climate zone among the datasets. For example, many of the luminescence datasets fall in the range of $Pe=\left[0.1,1\right]$. This may suggest that the link between soil mixing and soil production is strong enough that $Pe$ is broadly consistent as an increase in mixing $\left({\kappa }_{z0}\right)$ is matched by an increase in advection $\left(w_0\right)$. There is evidence for connections between soil mixing and soil production due to the liberation of particles from saprolite by rooting and burrowing and exposure of fresh weatherable bedrock surfaces by processes such as tree throw (2, 72). It is reasonable to hypothesize that changes in soil mixing by climate zone also result in changes in soil production rate. However, the relationship between climate and soil production rate appears subtle in compilations of soil production rates where production rate increases from arid to temperate conditions, although more data may be needed to resolve the relationship through the scatter (18). Similarly, the diffusion-like coefficient of hillslope sediment transport, a likely function of both soil production rate and soil mixing rate, increases with increasing moisture and vegetation, but the relationship becomes much less pronounced at high moistures and vegetation densities, and significant scatter is present (73). While the observation of our data falling into a limited Peclet number range hints at a possible relationship between soil mixing, soil production, and climate, the datasets presented here may be too few to resolve such subtleties given the natural scatter observed in the studies above. One way to test such a hypothesis is to expand the number of datasets on soil mixing.

Finally, the similarity in Pe distributions across disparate climate zone may indicate that nonclimate factors play the largest role in mixing. From a biotic perspective, an example could be the role of ubiquitous microfauna such as ants, termites, and earthworms (SI Appendix, Table S1). Regarding the former, ants are prolific bioturbation agents (74), capable of moving thousands of kilograms of soil per year (75), and are present in every climate zone. Combined with termites and earthworms, it is not unreasonable that significant amounts of soil mixing could occur by this biotic, but not climate-sensitive, process. Although the focus on soil mixing is largely on bioturbation (2), a significant amount of mixing occurs by abiotic processes (3), and it is important to note that shear in a granular material always induces mixing (12, 15, 70). As our data cannot distinguish between specific processes, only the result of mixing, the relative efficacy of various soil mixing processes arises as an outstanding question. Further collection of soil mixing data across locations characterized by different types of biotic and abiotic processes is needed to address this research need.

Data Availability.

The data used in this paper are available in the cited source publications. No data were generated in this paper. The Python code used to generate Fig. 2 is available in the Community Surface Dynamics Modeling System code repository: https://csdms.colorado.edu/wiki/Model:LumSoilMixer.