The following images are frames from a model simulating the formation of a parabolic high surface from a straight hillslope (slope=0.2) bounded by rapidly lowering glaciated valleys. Initially, there is zero soil cover, and the bedrock has a short wavelength (order 10 m) correlated random topography with a mean amplitude of 10 m. The physics of this simulation include: 1) mass conservation, 2) a rule for the production of soil from bedrock that is a function of local soil thickness, and 3) soil flux proportional to slope with a proportionality constant that is a function of local soil thickness.[click image to download movie]
This is a plot of the rule describing soil production as a function of soil thickness used in the simulation below. Importantly for both the formation of parabolic high surfaces and the long term stability of tors, the soil production function is a maximum at a thin but finite soil thickness.
This image and the associated movie show the entire simulation area, including the edges, which are lowering at a rate of 50 meters per million years by processes associated with glacial erosion. The interior cells are lowering more slowly, primarily by perigalcial processes. The cliff edge (first interior cell) is maintained free of soil, and thus is constrained to lower at the bare bedrock rate given by the soil production function, while the interior can lower at higher/lower soil covered rates. The bare bedrock edge and the peak in the soil production function at finite soil thickness decouple the high surface from the glacial valley (i.e. the high surface only feels the effect of the bare bedrock edge, not the great depth to the valley floor).
This image and the associated movie are from the same simulation as shown above; however, only the interior cells are shown such that characteristics of the high surface and its evolution can be examined. Additionally, in order to explore surface evolution on various time-scales, the frame rate of the movie changes from 1000 years between frames for the first 50,000 years to 10,000 years between frames for the next 450,000 years, then to 100,000 years between frames for the remainder.
On the 1000 year time-scale, erosion and filling in of the high frequency, random, correlated topography occurs. On the 10,000 year time-scale, the high surface adjusts itself to a parabolic shape leaving bedrock tors at the crest. If the surface is in steady state, then it must be lowering at the same rate everywhere; since the cliff edge and the crest tors are lowering at the bare bedrock rate, the steady state lowering rate is the bare bedrock rate everywhere. Where soil cover exists, the soil production rate and soil thickness must be given by the intersection of the red line and the blue curve in the first figure. The persistence of bedrock tors to steady state relies on the bare bedrock soil production being equal to or less than the surrounding soil mantled slope. This can only be the case if the peak in the soil production function occurs at a finite soil thickness. As seen in the first movie, on the 100,000 year time-scale, the high surface decouples from the glacial valleys, retaining its topography as it lowers.
Indian Peaks Glacier Simulations