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Supersaturation. Saturation vapor pressures over water and ice use the Magnus/WMO forms ew=6.112·exp(17.62T/(243.12+T)) and ei=6.112·exp(22.46T/(272.62+T)) hPa. Humidity is entered relative to liquid water (what a hygrometer in a cloud reports), so the supersaturation over ice is σ = RH·ew/ei − 1. Water saturation at −15 °C already means σ ≈ 16 %, which is why dendrites thrive in mixed-phase clouds.
Crystal temperature and sunlight. Absorbed sunlight (≈3 % of the solar flux for millimetre ice, mostly near-infrared) is balanced by conduction to the air, ΔT = a·S·πr / (8kairfv). A warmer crystal sees a lower effective supersaturation σeff = (1+σ)·ei(T)/ei(T+ΔT) − 1, and if that goes negative it sublimates, tips first.
Pressure and diffusion. Vapor diffusivity follows D = 0.211 cm²/s · (T/273.15)1.94 · (1013.25/P). Thin air delivers vapor faster and more evenly, so the lattice runs extra diffusion relaxation passes in proportion to D, which suppresses branching — the low-pressure faceting Nakaya and Libbrecht both observed.
Fall speed, wind and ventilation. Terminal velocity uses a Locatelli–Hobbs power law corrected for air density and riming. Wind contributes turbulent relative flow (≈25 % of the mean wind). The Reynolds and Schmidt numbers give the Hall–Pruppacher ventilation factor fv, which raises the vapor supply; turbulence also adds fluctuations to the vapor field, the seed of imperfect arms.
Habit. Temperature selects the relative attachment kinetics of the prism (a-axis) and basal (c-axis) faces. Prism kinetics peak near −15 °C and −2 °C (plates, dendrites); basal kinetics peak near −6 °C and below −22 °C (needles, columns). This reproduces the Nakaya–Kobayashi morphology diagram drawn on the right.
Basal-plane growth. The top view is the Gravner–Griffeath local mesoscopic model on a hexagonal lattice: each cell carries diffusive vapor d, a boundary (quasi-liquid) layer b and ice c. Every step does diffusion with reflection at the ice, freezing b+=(1−κ)d, c+=κd, attachment (1–2 ice neighbours need b≥β; 3 need b≥1, or b≥α where nearby vapor is below θ; 4+ always attach), and melting back to vapor at rates μ, γ. Supersaturation sets the far-field vapor ρ held on a ring around the crystal; prism kinetics set β, α, θ and μ; temperature near 0 °C raises κ through the quasi-liquid layer.
Thickness. Each ice cell stores a c-axis thickness that grows at vb ∝ αbasal(T)·σeff·fv. New cells inherit neighbour thickness in proportion to how basal growth competes with prism growth, so columns stay prismatic while plates spread thin. When thin plate growth starts on a thick column, the new ice forms caps at both ends — capped columns and double plates fall out naturally.
Riming and melting. Supercooled droplets (taken as 10 µm radius) are collected at a rate of cross-section × fall speed × droplet number density × efficiency, with efficiency rising once the crystal passes ≈0.25 mm. They freeze as bumps; drops that land in a gap beside ice bridge it, which is how heavy riming fills a dendrite into graupel. Above 0 °C the crystal erodes from its exposed edges.
Hollow growth and split plates. When basal growth is diffusion-limited, ice near exposed edges thickens faster than ice deep inside the crystal, giving hollow plates and hollow column ends. Plates only cap ice that grew under column conditions; if the column is shorter than ≈30 µm the two caps compete, and in each 60° sector one plate shadows the other, producing split plates and split stars.
Twins and polycrystals. Twelve-branched stars, arrowhead twins, crossed plates, bullet rosettes, radiating assemblages and spatial dendrites are several lattices growing through the same air, each with its own orientation and anchor (stacked, end-on, edge-on, or seeded on a host arm). They do not compete for vapor with each other, a simplification. The micrograph and edge-on profile show the main crystal only.
What this is not. It is a 2.5-D model: real snowflakes are 3-D, and the mapping from (T, σ, P) to lattice parameters is calibrated to reproduce the observed habits rather than derived from first principles. Time, size and thickness are calibrated so that a stellar dendrite at −15 °C and water saturation reaches ≈3 mm in about 20 minutes, in line with laboratory and field growth rates.