### Abstract

The power spectra S of linear transects of Earth's topography is often observed to be a power law function of wave number k with exponent close to -2: S(k) ∝ k^{-2}. In addition, river networks are fractal trees that satisfy several power law relationships between their morphologic components. A model equation for the evolution of Earth's topography by transport-limited erosional processes which produces fractal topography and fractal river networks is presented, and its solutions are compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the diffusivity constant on hillslopes and proportional to the three-halves power of discharge in channels. The dependence of diffusivity on discharge is consistent with sediment rating curves. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of elevation, 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) ∝ k^{-1.8}, independent of the age of the topography, consistent with observations of real topography. The hypsometry produced by the model equation is skewed such that lowlands make up a larger fraction of the total area than highlands as observed in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating the discharge at every point. We characterize the morphology of real river basins with five independent scaling relations between six morphometric variables. Scaling exponents are calculated for seven river networks from a variety of tectonic environments using high-quality digital elevation models. River networks formed in the model match the observed scaling laws and satisfy Tokunaga side-branching statistics.

Original language | English (US) |
---|---|

Article number | 1998JB900110 |

Pages (from-to) | 7359-7375 |

Number of pages | 17 |

Journal | Journal of Geophysical Research: Space Physics |

Volume | 104 |

Issue number | B4 |

State | Published - Apr 10 1999 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics
- Earth and Planetary Sciences (miscellaneous)
- Space and Planetary Science
- Atmospheric Science
- Astronomy and Astrophysics
- Oceanography

### Cite this

*Journal of Geophysical Research: Space Physics*,

*104*(B4), 7359-7375. [1998JB900110].

**Self-organization and scaling relationships of evolving river networks.** / Pelletier, Jon.

Research output: Contribution to journal › Article

*Journal of Geophysical Research: Space Physics*, vol. 104, no. B4, 1998JB900110, pp. 7359-7375.

}

TY - JOUR

T1 - Self-organization and scaling relationships of evolving river networks

AU - Pelletier, Jon

PY - 1999/4/10

Y1 - 1999/4/10

N2 - The power spectra S of linear transects of Earth's topography is often observed to be a power law function of wave number k with exponent close to -2: S(k) ∝ k-2. In addition, river networks are fractal trees that satisfy several power law relationships between their morphologic components. A model equation for the evolution of Earth's topography by transport-limited erosional processes which produces fractal topography and fractal river networks is presented, and its solutions are compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the diffusivity constant on hillslopes and proportional to the three-halves power of discharge in channels. The dependence of diffusivity on discharge is consistent with sediment rating curves. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of elevation, 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) ∝ k-1.8, independent of the age of the topography, consistent with observations of real topography. The hypsometry produced by the model equation is skewed such that lowlands make up a larger fraction of the total area than highlands as observed in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating the discharge at every point. We characterize the morphology of real river basins with five independent scaling relations between six morphometric variables. Scaling exponents are calculated for seven river networks from a variety of tectonic environments using high-quality digital elevation models. River networks formed in the model match the observed scaling laws and satisfy Tokunaga side-branching statistics.

AB - The power spectra S of linear transects of Earth's topography is often observed to be a power law function of wave number k with exponent close to -2: S(k) ∝ k-2. In addition, river networks are fractal trees that satisfy several power law relationships between their morphologic components. A model equation for the evolution of Earth's topography by transport-limited erosional processes which produces fractal topography and fractal river networks is presented, and its solutions are compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the diffusivity constant on hillslopes and proportional to the three-halves power of discharge in channels. The dependence of diffusivity on discharge is consistent with sediment rating curves. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of elevation, 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) ∝ k-1.8, independent of the age of the topography, consistent with observations of real topography. The hypsometry produced by the model equation is skewed such that lowlands make up a larger fraction of the total area than highlands as observed in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating the discharge at every point. We characterize the morphology of real river basins with five independent scaling relations between six morphometric variables. Scaling exponents are calculated for seven river networks from a variety of tectonic environments using high-quality digital elevation models. River networks formed in the model match the observed scaling laws and satisfy Tokunaga side-branching statistics.

UR - http://www.scopus.com/inward/record.url?scp=0033541423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033541423&partnerID=8YFLogxK

M3 - Article

VL - 104

SP - 7359

EP - 7375

JO - Journal of Geophysical Research: Space Physics

JF - Journal of Geophysical Research: Space Physics

SN - 2169-9380

IS - B4

M1 - 1998JB900110

ER -