A low‐dimensional model of bedrock weathering and lateral flow coevolution in hillslopes: 1. Hydraulic theory of reactive transport

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 dimension x almost disappear when the vertical dimension z is expressed as a fraction z∕H(x) of the laterally flowing system thickness H(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.

A low‐dimensional model of bedrock weathering and lateral flow coevolution in hillslopes: 2. Controls on weathering and permeability profiles, drainage hydraulics, and solute export pathways

“…In order to make use of the theory developed in the companion paper, Harman and Kim (), we will assume that the impermeable surface (whether it is at the base of the weathering zone or at some deeper level) has a relatively small relief. This allows us to assume that the flow system is characterized by a small hillslope number (Hi ), and so most of the hydraulic gradient driving flow is provided by the mounding of the saturated zone within the hillslope, and not imposed by the geometry of an underlying surface.…”

Lagged rejuvenation of groundwater indicates internal flow structures and hydrological connectivity

“…The discharge is proportional to the width of the transect profile. The simulation of transport is performed in 2D (Harman & Kim, 2019; Pollock, 1988; Strack, 1984). We applied the equations to a representative hillslope that has been generated from a DEM with a resolution of 5 m by taking all cells with the same distance to the stream and calculating a mean elevation, and the corresponding width (Figure 2).…”

Water within, moving through, and shaping the Earth's surface: Introducing a special issue on water in the critical zone

“…How hydrology shapes the CZ structure in the long term was directly and indirectly addressed by several papers in this collection. Direct coupling of hydrology and CZ development underlies three major contributions to the special issue: Anderson et al (), Harman and Kim (), and Harman and Cosans (). In a recent review and synthesis on process controls of deep CZ development, Riebe et al () proposed four hypotheses that states (H1—tectonic) tectonic stress interacts with topographic stress to determine the depth of rock damage (St. Clair et al, ), (H2—climatic) rock damage from frost action is deeper on the shady side of a ridgeline (Anderson et al, ), (H3—hydrologic) groundwater drainage through bedrock, modulated by stream incision, sets the depth for physical and biogeochemical weathering (Rempe & Dietrich, ), and (H4—geochemical) the weathering front is set by the kinetics of mineral dissolution and removal in the subsurface and its balance with the rate of physical erosion on the surface (Lebedeva & Brantley, ).…”

“…Harman and Kim () and Harman and Cosans () presented a two‐part series describing a model that coupled vertical (1D) chemical weathering and porosity development (H4—geochemical, Riebe et al, ) and lateral (2D) subsurface hydrology under a hillslope (H3—hydrologic, controlling the kinetics of chemical weathering) and insights gained from this coupling. Part 1 (Harmon and Kim, 2019) discusses lateral flow processes in hillslopes and shows how lateral dimensions of advection–diffusion–reaction equations can be collapsed, yielding a 1D vertical equation in which the downward advective flux declines with depth.…”

“…The second is the need to integrate conceptual and numerical models that are developed separately by CZ scientists trained in different disciplines. The contributions by Anderson et al (), Harman and Kim (), and Harman and Cosans () represent significant breakthroughs in demonstrating that it is indeed possible to construct simple, first order, but critically linked process models at geologic time scales that shape the evolution of the CZ. These models are not intended to simulate a particular site but serve as numerical laboratories to test hypotheses on the leading controls of CZ development, as exemplified by Anderson et al (), Harman and Kim (), and Harman and Cosans ().…”