Erosion and weathering

[jasonwebb]

Erosion and weathering models simulate how natural forces—such as rainfall, flowing rivers, wind, and temperature fluctuations—mechanically break down, transport, and redeposit material across landscapes. Unlike simple heightmap noise, erosion algorithms introduce hydrological continuity, carving realistic river drainage basins, sharpening ridge lines, smoothing valley floors, and building talus slopes at the bases of cliffs. In generative design and 3D environment creation, these algorithms convert raw mathematical terrains into naturalistic, geomorphologically plausible landscapes.

Key Terms and Concepts

• Hydrological Flow Accumulation: Tracking water movement downhill along steepest gradients. As flow paths converge, water volume accumulates exponentially, transitioning broad sheet wash into distinct stream channels and deep river valleys.
• Sediment Transport Capacity ($C$): The maximum concentration of suspended soil or rock that a given volume of moving fluid (water or wind) can carry at a specific velocity.
  • Erosion: Occurs when current sediment load is less than capacity ($S < C$).
  • Deposition: Occurs when current sediment load exceeds capacity ($S > C$).
• Angle of Repose ($\theta_{\text{repose}}$): The maximum stable slope angle (typically $30^\circ$–$35^\circ$ for loose rock/dirt) an unconsolidated material can maintain without sliding downhill due to gravity.
• Advection-Diffusion: The dual process where suspended sediment travels along fluid velocity vectors (advection) while simultaneously dispersing into lower-concentration regions (diffusion).
• Thermal Weathering (Freezing/Thawing): Mechanical breakdown of rock caused by temperature shifts, turning sheer cliff faces into crumbling scree and forming smooth talus ramps below.

Key High-Level Algorithms

• Particle-Based Hydraulic Erosion (Droplet / Lagrangian Agent): Spawns discrete water agents on a 2D heightmap that move downhill along terrain gradients ($\nabla h$), eroding or depositing sediment based on local speed and slope before evaporating.
• Grid-Based Hydraulic Erosion (Continuous Eulerian Pipe / Shallow Water): Simulates fluid flow across an entire grid simultaneously using simplified shallow water equations to calculate hydrostatic pressure, velocity fields, and global sediment transport.
• Thermal Weathering & Talus Relaxation (Cellular Automata): A local grid-smoothing algorithm that compares height differences ($\Delta h$) against a material's critical angle of repose ($\tan(\theta_{\text{repose}})$) and redistributes excess material downslope.
• Stream Power Law (Macro-Scale Fluvial Incision): A geomorphological model that computes contributing drainage area ($A$) and erodes terrain directly using the fluvial incision equation ($\frac{\partial z}{\partial t} = K A^m |\nabla z|^n$) to sculpt large mountain chains and river networks.

Key Research & Literature References

• Musgrave, F. K., Kolb, C. E., & Mace, R. S. (1989): "The Synthesis and Rendering of Eroded Fractals" — The seminal SIGGRAPH paper introducing hydraulic and thermal erosion to computer graphics.
• Benes, B., & Forsbach, R. (2002): "Visual Simulation of Hydraulic Erosion" — Introduced grid-based shallow-water virtual pipe models for multi-layer terrain erosion.
• Stava, O. et al. (2008): "Interactive Terrain Modeling Using Hydraulic Erosion" — Optimized hydraulic erosion solvers to run efficiently on GPUs for real-time interactive painting.
• Cordonnier, G. et al. (2016): "Authoring Landscapes by Combining Feature Machines and Hydrological Modeling" — SIGGRAPH paper demonstrating stream-power law hydrology to generate artist-guided mountain ranges.

Notable Tools, Libraries, and Software Addons

• Gaea (QuadSpinner) & World Machine: Production-standard node-based landscape software built around advanced hydraulic, thermal, and directed erosion filters.
• SideFX Houdini (Heightfield Erosion SOP): Native GPU-accelerated terrain suite utilizing multi-layered hydraulic, thermal, and precipitation solvers.
• Sebastian Lague's Hydraulic Erosion: Widely referenced open-source C# / Unity implementation demonstrating particle-based droplet erosion.
• Fast Hydraulic Erosion (WebGL): Fragment-shader implementations on ShaderToy and GitHub demonstrating grid-based hydraulic erosion running in real time inside web browsers.

Fracture Equations & Physical Models (Modular References)

• Griffith Energy Criterion: $G = -\frac{\partial U}{\partial a} \ge G_c$
• Stress Intensity Factor ($K$) & LEFM: $\sigma_{ij} = \frac{K_I}{\sqrt{2\pi r}} f_{ij}(\theta)$
• Phase-Field Damage Regularization: Continuous scalar field ($\phi \in [0,1]$) variational energy model.
• Stream Power Incision Equation: $\frac{\partial z}{\partial t} = K A^m |\nabla z|^n$