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ABSTRACT To accurately predict earth system response to global change, we must be able to predict the responses of important properties of that system, such as the depths over which plant roots are distributed. In 2008, H. J. Schenk proposed a model for the depth distribution of plant roots based on a simple hydrological scheme and the assumptions that plants will take up the shallowest water available first and will distribute their roots in proportion to long‐term mean uptake at each depth. Here, we derive an analytical solution to the Schenk model under an idealised climate (in which infiltration events are treated as a marked Poisson process), explore properties of the result and compare with data. The solution suggests that in very humid and arid climates, the soil wetting and drying cycles induced by root water uptake are generally confined to a characteristic depth below the surface. This depth depends on the typical magnitude of rainfall events (most strongly so in arid climates), the typical total transpiration demand between rainfall events (most strongly in humid climates) and the plant‐available water holding capacity of the soil. Root water uptake (and thus predicted root density) in very humid and arid landscapes decreases exponentially with depth at a rate determined by this characteristic depth. However, in a mesic climate, soils may be wet or dry to greater depths below the near‐surface, and the duration spent in each state increases with depth. Consequently, root water uptake and root density in mesic climates more closely resemble a power law distribution. When the aridity index is exactly 1, the characteristic depth diverges and the mean rooting depth approaches infinity. This suggests that the most skewed root depth distributions might occur in mesic environments. We compared this model to another analytical solution and a compiled database of root distributions (159 combined locations). For a larger comparison dataset, we also compared 99th percentile rooting depth to rooting depths modeled by two other frameworks and a database of observed rooting depths (1271 combined locations). Results demonstrate that the analytical formulation of the Schenk model performs well as a shallow bound on rooting depths and captures something of the nonexponential form of root distributions, and its error is similar to or less than that of other modeling frameworks. Errors may be partly explained by the deviation of real climate from the idealisations used to obtain an analytical solution (exponentially distributed infiltration events and no seasonality). 

Abstract. End-member mixing analysis (EMMA) is a method of interpreting stream water chemistry variations and is widely used for chemical hydrograph separation. It is based on the assumption that stream water is a conservative mixture of varying contributions from well-characterized source solutions (end-members). These end-members are typically identified by collecting samples of potential end-member source waters from within the watershed and comparing these to the observations. Here we introduce a complementary data-driven method (convex hull end-member mixing analysis – CHEMMA) to infer the end-member compositions and their associated uncertainties from the stream water observations alone. The method involves two steps. The first uses convex hull nonnegative matrix factorization (CH-NMF) to infer possible end-member compositions by searching for a simplex that optimally encloses the stream water observations. The second step uses constrained K-means clustering (COP-KMEANS) to classify the results from repeated applications of CH-NMF and analyzes the uncertainty associated with the algorithm. In an example application utilizing the 1986 to 1988 Panola Mountain Research Watershed dataset, CHEMMA is able to robustly reproduce the three field-measured end-members found in previous research using only the stream water chemical observations. CHEMMA also suggests that a fourth and a fifth end-member can be (less robustly) identified. We examine uncertainties in end-member identification arising from non-uniqueness, which is related to the data structure, of the CH-NMF solutions, and from the number of samples using both real and synthetic data. The results suggest that the mixing space can be identified robustly when the dataset includes samples that contain extremely small contributions of one end-member, i.e., samples containing extremely large contributions from one end-member are not necessary but do reduce uncertainty about the end-member composition. 

Process-based modelling offers interpretability and physical consistency in many domains of geosciences but struggles to leverage large datasets efficiently. Machine-learning methods, especially deep networks, have strong predictive skills yet are unable to answer specific scientific questions. In this Perspective, we explore differentiable modelling as a pathway to dissolve the perceived barrier between process-based modelling and machine learning in the geosciences and demonstrate its potential with examples from hydrological modelling. ‘Differentiable’ refers to accurately and efficiently calculating gradients with respect to model variables or parameters, enabling the discovery of high-dimensional unknown relationships. Differentiable modelling involves connecting (flexible amounts of) prior physical knowledge to neural networks, pushing the boundary of physics-informed machine learning. It offers better interpretability, generalizability, and extrapolation capabilities than purely data-driven machine learning, achieving a similar level of accuracy while requiring less training data. Additionally, the performance and efficiency of differentiable models scale well with increasing data volumes. Under data-scarce scenarios, differentiable models have outperformed machine-learning models in producing short-term dynamics and decadal-scale trends owing to the imposed physical constraints. Differentiable modelling approaches are primed to enable geoscientists to ask questions, test hypotheses, and discover unrecognized physical relationships. Future work should address computational challenges, reduce uncertainty, and verify the physical significance of outputs. 

Abstract Unsteady transit time distribution (TTD) theory is a promising new approach for merging hydrologic and water quality models at the catchment scale. A major obstacle to widespread adoption of the theory, however, has been the specification of the StorAge Selection (SAS) function, which describes how the selection of water for outflow is biased by age. In this paper we hypothesize that some unsteady hydrologic systems of practical interest can be described, to first‐order, by a “shifted‐uniform” SAS that falls along a continuum between plug flow sampling (for which only the oldest water in storage is sampled for outflow) and uniform sampling (for which water in storage is sampled randomly for outflow). For this choice of SAS function, explicit formulae are derived for the evolving: (a) age distribution of water in storage; (b) age distribution of water in outflow; and (c) breakthrough concentration of a conservative solute under either continuous or impulsive addition. Model predictions conform closely to chloride and deuterium breakthrough curves measured previously in a sloping lysimeter subject to periodic wetting, although refinements of the model are needed to account for the reconfiguration of flow paths at high storage levels (the so‐called inverse storage effect). The analytical results derived in this paper should lower the barrier to applying TTD theory in practice, ease the computational demands associated with simulating solute transport through complex hydrologic systems, and provide physical insights that might not be apparent from traditional numerical solutions of the governing equations. 

Abstract Spatially integrated transport models have been applied widely to model hydrologic transport. However, we lack simple and process‐based theoretical tools to predict the transport closures—transit time distributions (TTDs) and StorAge Selection (SAS) functions. This limits our ability to infer characteristics of hydrologic systems from tracer observations and to make first‐order estimates of SAS functions in catchments where no tracer data is available. Here we present a theoretical framework linking TTDs and SAS functions to hydraulic groundwater theory at the hillslope scale. For hillslopes where the saturated hydraulic conductivity declines exponentially with depth, analytical solutions for the closures are derived that can be used as hypotheses to test against data. In the simplest form, the hillslope SAS function resembles a uniform or exponential distribution (corresponding to flow pathways in the saturated zone) offset from zero by the storage in the unsaturated zone that does not contribute to discharge. The framework is validated against nine idealized virtual hillslopes constructed using a 2‐D Richards equation‐based model, and against data from tracer experiments in two artificial hillslopes. Modeled internal age, life expectancy, and transit time structures reproduce theoretical predictions. The experimental data also support the theory, though further work is needed to account for the effects of time‐variability. The shape and tailing of TTDs and their power spectra are discussed. The theoretical framework yields several dimensionless numbers that can be used to classify hillslope scale flow and transport dynamics and suggests distinct water age structures for high or low Hillslope number. 

Abstract This is the first of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory (also known as Dupuit or Boussinesq theory). Here, we examine the effect of lateral flow on the downward fluxes of water and solutes through perched groundwater at steady state. We derive analytical expressions describing the decline in the downward flux rate with depth. Using these, we obtain analytical expressions for water age in a number of cases. The results show that when the permeability field is homogeneous, the spatial structure of water age depends qualitatively on a single dimensionless number, Hi. This number captures the relative contributions to the lateral hydraulic potential gradient of the relief of the lower‐most impermeable boundary (which may be below the weathering front within permeable or incipiently weathered bedrock) and the water table. A “scaled lateral symmetry” exists when Hi is low: age varies primarily in the vertical dimension, and variations in the horizontal dimensionxalmost disappear when the vertical dimensionzis expressed as a fractionz/H(x) of the laterally flowing system thicknessH(x). Taking advantage of this symmetry, we show how the lateral dimension of the advection–diffusion‐reaction equation can be collapsed, yielding a 1‐D vertical equation in which the advective flux downward declines with depth. The equation holds even when the permeability field is not homogeneous, as long as the variations in permeability have the same scaled lateral symmetry structure. This new 1‐D approximation is used in the accompanying paper to extend chemical weathering models derived for 1‐D columns to hillslope domains. 

Abstract The advance of a chemical weathering front into the bedrock of a hillslope is often limited by the rate weathering products that can be carried away, maintaining chemical disequilibrium. If the weathering front is within the saturated zone, groundwater flow downslope may affect the rate of transport and weathering—however, weathering also modifies the rock permeability and the subsurface potential gradient that drives lateral groundwater flow. This feedback may help explain why there tends to be neither “runaway weathering” to great depth nor exposed bedrock covering much of the earth and may provide a mechanism for weathering front advance to keep pace with incision of adjacent streams into bedrock. This is the second of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory. Here, we show how a simplified kinetic model of 1‐D rock weathering can be extended to consider lateral flow in a 2‐D hillslope. Exact and approximate analytical solutions for the location and thickness of weathering within the hillslope are obtained for a number of cases. A location for the weathering front can be found such that lateral flow is able to export weathering products at the rate required to keep pace with stream incision at steady state. Three pathways of solute export are identified: “diffusing up,” where solutes diffuse up and away from the weathering front into the laterally flowing aquifer; “draining down,” where solutes are advected primarily downward into the unweathered bedrock; and “draining along,” where solutes travel laterally within the weathering zone. For each pathway, a different subsurface topography and overall relief of unweathered bedrock within the hillslope is needed to remove solutes at steady state. The relief each pathway requires depends on the rate of stream incision raised to a different power, such that at a given incision rate, one pathway requires minimal relief and, therefore, likely determines the steady‐state hillslope profile.