dorsal/arxiv
View SchemaWhy is topography fractal?
| Authors | Jon D. Pelletier |
|---|---|
| Categories | |
| ArXiv ID | physics/9705033 |
| URL | https://arxiv.org/abs/physics/9705033 |
Abstract
The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to -2: S(k) is proportional to k^-2. In addition, river networks are fractal trees that satisfy many power-law or fractal relationships between their morphologic components. A model equation for the evolution of the earth's topography by erosional processes which produces fractal topography and fractal river networks is presented and its solutions compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the local diffusivity proportional to the square of the discharge. The dependence of diffusivity on discharge follows from fundamental equations of sediment transport. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of relief and a Taylor expansion procedure is carried out to obtain a differential equation for the landform elevation which includes the spatially-variable diffusivity to first order in the elevation. The solution to this equation is a self-affine or fractal surface with linear transects that have power spectra S(k) is proportional to k^-1.8, independent of the age of the topography, consistent with observations. The hypsometry produced by the model is skewed such that lowlands make up a larger area than highlands as in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating... Abstract continued in paper.
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"abstract": "The power spectrum S of linear transects of the earth\u0027s topography is often\nobserved to be a power-law function of wave number k with exponent close to -2:\nS(k) is proportional to k^-2. In addition, river networks are fractal trees\nthat satisfy many power-law or fractal relationships between their morphologic\ncomponents. A model equation for the evolution of the earth\u0027s topography by\nerosional processes which produces fractal topography and fractal river\nnetworks is presented and its solutions compared in detail to real topography.\nThe model is the diffusion equation for sediment transport on hillslopes and\nchannels with the local diffusivity proportional to the square of the\ndischarge. The dependence of diffusivity on discharge follows from fundamental\nequations of sediment transport. We study the model in two ways. In the first\nanalysis the diffusivity is parameterized as a function of relief and a Taylor\nexpansion procedure is carried out to obtain a differential equation for the\nlandform elevation which includes the spatially-variable diffusivity to first\norder in the elevation. The solution to this equation is a self-affine or\nfractal surface with linear transects that have power spectra S(k) is\nproportional to k^-1.8, independent of the age of the topography, consistent\nwith observations. The hypsometry produced by the model is skewed such that\nlowlands make up a larger area than highlands as in real topography. In the\nsecond analysis we include river networks explicitly in a numerical simulation\nby calculating... Abstract continued in paper.",
"arxiv_id": "physics/9705033",
"authors": [
"Jon D. Pelletier"
],
"categories": [
"physics.geo-ph",
"cond-mat"
],
"title": "Why is topography fractal?",
"url": "https://arxiv.org/abs/physics/9705033"
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