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Universiteit van Amsterdam (2006)

Description of pattern formation in semi-arid landscapes by weakly non-linear stability analysis

Marieke Jesse

Titre : Description of pattern formation in semi-arid landscapes by weakly non-linear stability analysis

Auteur : Marieke Jesse

Université de soutenance : Universiteit van Amsterdam

Grade : Doctor of Philosophy (PhD) 2006

In semi-arid areas, which are defined as having a precipitation of 250-500 millimeter per year, vegetation cover can be patterned such that distinct bands of vegetation are visible on hillsides, whereas the vegetation is distributed as mosaics in flat areas. These patterns have been described from e.g. Africa, Australia and Mexico. Due to their scale it is difficult to see these patterns from the ground ; hence they have only been discovered via aerial photography. The hills on which these patterns occur generally have slopes of approximately 0.25%. The vegetation is concentrated in bands of 100 to 250 meter in width and are separated by gaps, in which vegetation is absent or sparse. These gaps can have widths ranging from 200 meters to 1 kilometer. There is discussion on the exact cause for these banded patterns. However there is a broad consensus that the key factor is competition for water, Sherrat [7]. The hypothesis about how these patterns arise on hillsides can be described as follows. The water is flowing downhill, it does not infiltrate the bare areas, to a region with vegetation where it is absorbed and supports plant growth. The water is exhausted at the downhillside of the vegetation stripe causing the next gap to occur and the band to be maintained uphill of the stripe. The soil just uphill of the vegetation band is moist and is able to support vegetation. As a result, bands are reported to gradually migrate uphill. The hypothesis has not been completely tested yet. Although Klausmeier [3] proposed a model from which pattern formation in semi-arid areas can be studied, this model has been studied with only linear stability analysis by Sherrat [7]. A non-linear stability analysis has not been applied yet, this will be done in this thesis. In the following of this chapter I will use the techniques of non-dimensionalizing and scaling to simplify Klausmeiers model to the model I am going to work with. Linear stability analysis will then be applied in the second chapter. The stability of the equilibria will be determined in the case that a) only advection but no diffusion is present and b) in the case that both advection and diffusion are present. Using linear stability analysis, parameter values can be be determined for which pattern formation is possible. However, we do not know what these patterns will look like. In Chapter 3 I apply a weakly non-linear stability analysis. With this analysis we will derive the Complex Ginzburg-Landau equation which describes the pattern by an amplitude function. In Chapter 4 the solution of the Ginzburg-Landau equation will be subjected to small perturbations in order to study the stability of the equilibri

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