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Updated: Dec 11, 2025
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Use of Principal Components for Scaling Up Topographic Models to Map Soil Redistribution and Soil Organic Carbon
Published on: October 16, 2018
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Variational analysis of landscape elevation and drainage networks
Milad Hooshyar1, Shashank Anand2, Amilcare Porporato3
1Princeton Environmental Institute and Princeton Institute for International and Regional Studies, Princeton University, Princeton, NJ, USA.
Proceedings. Mathematical, Physical, and Engineering Sciences
|August 25, 2020
Summary
Landscapes naturally form complex channel networks. This study reveals that without soil diffusion, landscape surfaces optimize average elevation, forming critical minima or maxima based on erosion parameters.
Area of Science:
- Earth Science
- Geomorphology
- Mathematical Modeling
Background:
- Landscapes evolve complex channel and ridge networks due to climate and tectonic forces.
- Minimalist models using partial differential equations capture essential landscape dynamics.
Purpose of the Study:
- To analyze variational principles governing minimalist landscape evolution models.
- To connect landscape evolution with optimal channel networks and diffusion's role.
Main Methods:
- Analysis of variational principles.
- Development of partial differential equation models for sediment and water balances.
- Investigation of steady-state surface properties based on erosion exponents.
Main Results:
- In the absence of diffusive soil transport, steady-state surfaces extremize average domain elevation.
- Critical surfaces are minima for 0 < m < 1, maxima for m > 1, and saddle points at m = 1, where m is the exponent in the erosion term.
- A link is established between landscape evolution models and optimal channel networks.
Conclusions:
- Variational principles provide a framework for understanding landscape evolution.
- The exponent in the erosion term critically determines the nature of steady-state landscape surfaces.
- Diffusion plays a significant role in the governing variational principles of landscape evolution.
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