We revisit three variants of the well-known Stommel diagrams that have been used to summarize knowledge of characteristic scales in time and space of some important hydrologic phenomena and modified these diagrams focusing on spatiotemporal scaling analyses of the underlying hydrologic processes. In the present paper we focus on soil formation, vegetation growth, and drainage network organization. We use existing scaling relationships for vegetation growth and soil formation, both of which refer to the same fundamental length and timescales defining flow rates at the pore scale but different powers of the power law relating time and space. The principle of a hierarchical organization of optimal subsurface flow paths could underlie both root lateral spread (RLS) of vegetation and drainage basin organization. To assess the applicability of scaling, and to extend the Stommel diagrams, data for soil depth, vegetation root lateral spread, and drainage basin length have been accessed. The new data considered here include timescales out to 150 Myr that correspond to depths of up to 240m and horizontal length scales up to 6400 km and probe the limits of drainage basin development in time, depth, and horizontal extent.

The fate of water and water-soluble toxic wastes in the subsurface is of high importance for many scientific and practical applications. Although solute transport is proportional to water flow rates, theoretical and experimental studies show that heavy-tailed (power-law) solute transport distribution can cause chemical transport retardation, prolonging clean-up time-scales greatly. However, no consensus exists as to the physical basis of such transport laws. In percolation theory, the scaling behavior of such transport rarely relates to specific medium characteristics, but strongly to the dimensionality of the connectivity of the flow paths (for example, two- or three-dimensional, as in fractured-porous media or heterogeneous sediments), as well as to the saturation characteristics (i.e., wetting, drying, and entrapped air). In accordance with the proposed relevance of percolation models of solute transport to environmental clean-up, these predictions also prove relevant to transport-limited chemical weathering and soil formation, where the heavy-tailed distributions slow chemical weathering over time. The predictions of percolation theory have been tested in laboratory and field experiments on reactive solute transport, chemical weathering, and soil formation and found accurate. Recently, this theoretical framework has also been applied to the water partitioning at the Earth's surface between evapotranspiration, ET, and run-off, Q, known as the water balance. A well-known phenomenological model by Budyko addressed the relationship between the ratio of the actual evapotranspiration (ET) and precipitation, ET/P, versus the aridity index, ET₀/P, with P being the precipitation and ET₀ being the potential evapotranspiration. Existing work was able to predict the global fractions of P represented by Q and ET through an optimization of plant productivity, in which downward water fluxes affect soil depth, and upward fluxes plant growth. In the present work, based likewise on the concepts of percolation theory, we extend Budyko's model, and address the partitioning of run-off Q into its surface and subsurface components, as well as the contribution of interception to ET. Using various published data sources on the magnitudes of interception and information regarding the partitioning of Q, we address the variability in ET resulting from these processes. The global success of this prediction demonstrated here provides additional support for the universal applicability of percolation theory for solute transport as well as guidance in predicting the component of subsurface run-off, important for predicting natural flow rates through contaminated aquifers.

Soil depth tends to vary from a few centimeters to several meters, depending on many natural and environmental factors. We hypothesize that the cumulative effect of these factors on soil depth, which is chiefly dependent on the process of biogeochemical weathering, is particularly affected by soil porewater (i.e., solute) transport and infiltration from the land surface. Taking into account evidence for a non-Gaussian distribution of rock weathering rates, we propose a simple mathematical model to describe the relationship between soil depth and infiltration flux. The model was tested using several areas in mostly semi-arid climate zones. The application of this model demonstrates the use of fundamental principles of physics to quantify the coupled effects of the five principal soil-forming factors of Dokuchaev.

In his paper on net primary productivity of terrestrial communities predicted from climatological data, Rosenzweig (1968) argued that variability in productivity is well accounted for by (evapo)-transpiration, and that water from transpiration is, on global scales, the most variable component in the photosynthesis reaction. The goal of this paper is to investigate whether variability in plant growth on local scales and within species is primarily related to transpiration under several scenarios including different terrain curvature, slope aspect, soil characteristics, and climate ranges. We test the hypothesis that this relationship exists because root growth into the surface soil layers (0–2 m) tends to follow paths with minima in resistance, which in turn maximizes water flow and nutrient delivery rates that regulate growth. The set of all connected paths with individual pore-to-pore flow resistances less than a critical, percolating, value forms a cluster with mass fractal dimensionality, df. We propose that roots follow paths through the 2D percolation cluster, defining the set of all optimal flow paths, making the 2D value of df from percolation relevant to root fractal dimensionality. The tortuosity of such optimal paths as defined in percolation theory should then relate root length to root radial extent, linking the parameters of root tortuosity and plant productivity. Our analysis of large data sets across species implies that root radial extent and tree height are both proportional to cumulative transpiration until trees approached maximum height, and their growth rates are proportional to the transpiration rate, not to the moisture content. Local variations in tree height as functions of the variables investigated appear generally consistent with deduced variations in transpiration. Here this correlation is investigated more closely in the context of studies addressing individual tree species.

The fate of water and water-soluble toxic wastes in the subsurface is of high importance for many scientific and practical applications. Although solute transport is proportional to water flow rates, theoretical and experimental studies show that heavy-tailed (power-law) solute transport distribution can cause chemical transport retardation, prolonging clean-up time-scales greatly. However, no consensus exists as to the physical basis of such transport laws. In percolation theory, the scaling behavior of such transport rarely relates to specific medium characteristics, but strongly to the dimensionality of the connectivity of the flow paths (for example, two- or three-dimensional, as in fractured-porous media or heterogeneous sediments), as well as to the saturation characteristics (i.e., wetting, drying, and entrapped air). In accordance with the proposed relevance of percolation models of solute transport to environmental clean-up, these predictions also prove relevant to transport-limited chemical weathering and soil formation, where the heavy-tailed distributions slow chemical weathering over time. The predictions of percolation theory have been tested in laboratory and field experiments on reactive solute transport, chemical weathering, and soil formation and found accurate. Recently, this theoretical framework has also been applied to the water partitioning at the Earths surface between evapotranspiration, ET, and run-off, Q, known as the water balance. A well-known phenomenological model by Budyko addressed the relationship between the ratio of the actual evapotranspiration (ET) and precipitation, ET/P, versus the aridity index, ET0/P, with P being the precipitation and ET0 being the potential evapotranspiration. Existing work was able to predict the global fractions of P represented by Q and ET through an optimization of plant productivity, in which downward water fluxes affect soil depth, and upward fluxes plant growth. In the present work, based likewise on the concepts of percolation theory, we extend Budykos model, and address the partitioning of run-off Q into its surface and subsurface components, as well as the contribution of interception to ET. Using various published data sources on the magnitudes of interception and information regarding the partitioning of Q, we address the variability in ET resulting from these processes. The global success of this prediction demonstrated here provides additional support for the universal applicability of percolation theory for solute transport as well as guidance in predicting the component of subsurface run-off, important for predicting natural flow rates through contaminated aquifers.