Snow Crystal Growth (3D)

What is modelled

A snow crystal grows from a single ice seed in air that is supersaturated with water vapour. The simulation uses the Gravner-Griffeath hexagonal cellular model: space is a lattice of hexagonal cells, vapour diffuses from cell to cell, freezes onto the crystal edge as a thin layer of quasi-liquid water, and the edge cell joins the crystal once enough water has gathered. Tips stick out into richer air, take up vapour faster and race ahead, which makes dendritic arms; low supersaturation lets the crystal stay a compact hexagon.

Rules (one step)

• Diffusion: every cell that is not ice becomes the average of itself and its six neighbours. Ice cells and the domain wall reflect vapour, so no vapour passes through the crystal.
• Freezing: a boundary cell (next to ice) moves its vapour d into boundary water b and crystal water c in the fractions 1 - κ and κ.
• Attachment: a boundary cell with one or two ice neighbours freezes when b ≥ β; with three or more it freezes if b ≥ 1, or if b ≥ α while the vapour around it is below θ.
• Melting and noise: a small share of boundary water returns to vapour (μ, γ), and the vapour is perturbed by a small random factor σ.

From the Nakaya diagram to the model

The temperature and supersaturation you choose give the far-field vapour ρ and the attachment threshold β through a smooth map (high supersaturation near -15 C gives low β and dendrites). The map was fitted by sweeping (ρ, β): the filled fraction falls from 100 % (plates) to about 45 % (dendrites). The habit names follow the Nakaya diagram: thin plates near 0 C, columns and needles near -5 C, plates and dendrites near -15 C, columns again below -22 C. In the 3D page the ratio of c-axis to a-axis growth, 0.12 + 3.3 e-((T+5.5)/2.6)^2 + 1.9 e-((T+29)/5.5)^2 + ..., sets how thick a plate becomes or how long a column grows. That curve is a schematic of the measured behaviour (Libbrecht 2005), not a molecular theory.

Checks run in this page

• Water is conserved by diffusion, freezing and attachment in a closed domain (difference of order 1e-11).
• On the hexagonal lattice a point pulse spreads with mean-square radius 6/7 lattice spacings squared per step, exactly as random-walk theory gives.
• Without noise the crystal is exactly six-fold symmetric: the ice set maps onto itself under a 60 degree rotation.
• The habit map reproduces the column, plate and column regions of the Nakaya diagram.

Honest simplifications

• The model is two-dimensional (the basal face); thickness is added only as a schematic in the 3D page. Real crystal growth is controlled by surface molecular kinetics and the quasi-liquid layer, which the knobs β, α, θ, κ, μ, γ only imitate.
• The map from the diagram to the knobs is a fit chosen to give the right appearance, not a derivation. Needles, hollow columns, capped columns and twelve-branched crystals are not produced.
• Vapour beyond the active region is held at the far-field value; the lattice spacing is arbitrary and time is counted in model steps.