Hisslope evolution by diffusive processes: the problem of equilibrium and effects of climatic and tectonic changes

Abstract

The convex hilltops of soil-mantled landscapes have been attributed to the action of diffusive (slope-dependent) processes like creep, rainsplash, and biogenic activity. Although many models based on the diffusion equation have been proposed, little is known about the effects of climatic and tectonic oscillations on the convex form. Such oscillations are expected to impose changes on the transport rates and/ or the incision rates at the base of the hillslopes. I specifically focus on how equilibrium convex hillslope profiles respond, both in terms of form and sediment flux, to one-step and cyclic oscillations in the diffusion coefficient or incision rate. Numerical and analytical solutions of the one-dimensional diffusion-type equation are obtained for initially convex hillslopes evolving under diffusion coefficient and incision rate values derived from field measurements. One-step changes, either in the diffusion coefficient or in the downcutting rate, are then imposed and the time required for the new equilibrium condition to be attained (relaxation time) is estimated. By characterizing the time-scale of morphological adjustments of these convex hillslopes, and consequently their relaxation times, we can determine whether the hilltop convexities that we observe in the field today represent equilibrium or relict forms. ln addition, the effects of climatic and tectonic oscillations on diffusive hillslopes are modeled by imposing cyclic changes, in either the diffusion coefficient or in the incision rate, in the form of steps, sine waves, and 180-constrained oscillations. Two-fold step changes in the diffusion coefficient or in the downcutting rate result in relaxation times of approximately 70 thousand years and one million years, 2 for 25 m and 100 m long hillslopes, respectively. The time-scale of such morphological adjustments varies depending on whether the hillslope profile is tending to increase or decrease its curvature through time. Toe new equilibrium condition is first developed at the base of the hillslope and then propagates upslope. A dimensionless graph is presented which allows the estimation of the relaxation time of convex hillslopes from estimates of diffusion coefficient, incision rate, magnitude of change, and hillslope length. Such a graph could be used in field studies. Step and sine oscillations in the diffusion coefficient cause the sediment flux from the hillslope to eventually oscillate around the initial equilibrium value, which is set by the downcutting rate. When these oscillations happen in the downcutting rate, the sediment flux oscillates around a new equilibrium value located mid-way between the lower and upper equilibrium values, associated with the minimum and maximum imposed downcutting rates, respectively. The sediment flux is shown to be in phase with 180-based oscillations in the diffusion coefficient whil􀁕 cumulative effects are observed when these oscillations take place in the incision rate. Because the relaxation times estimated here are much longer than the frequency of the climatic oscillations observed in the last few million years, I argue that most of the convex hilltops of soil-mantled landscapes are likely to represent forms that are far from being truly time-independent morphologies. The results also suggest that these convex hilltops represent forms that, once formed, are difficult to be perturbed with modest, but reasonable, variations in the diffusion coefficient or in the incision rate. Consequently, these convex hilltops may represent, at least for the case of long hillslopes, forms that developed before the climatic and tectonic oscillations that took place during the Quatemary.