A low‐dimensional model of bedrock weathering and lateral flow coevolution in hillslopes: 1. Hydraulic theory of reactive transport
2019
Harman, Ciaran J. | Kim, Minseok
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.
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