The governing equation is the : [ \frac\partial c\partial t = D \nabla^2 c ] where ( c ) = water vapor concentration. On the crystal surface, the Stefan condition applies: [ v_n = D , (\nabla c \cdot \mathbfn) ] with ( v_n ) = normal growth velocity. 4.1 Mullins–Sekerka Instability A flat interface is unstable: small protrusions grow faster because they encounter higher vapor gradients. This yields the characteristic dendritic arms.
| Model | Approach | Produces | |-------|----------|-----------| | | Random walkers stick to cluster | Fractal, noise-driven shapes | | Phase Field | Solves coupled PDEs for phase & concentration | Realistic anisotropy | | Cellular Automaton | Rule-based growth on hexagonal grid | Stellar plates, dendrites | Example: Anisotropic Laplacian Growth Introduce anisotropy in surface energy: [ \gamma(\theta) = \gamma_0 (1 + \epsilon \cos(6\theta)) ] The growth velocity becomes: [ v(\theta) \propto \gamma(\theta) + \gamma''(\theta) ] This explains why real snowflakes grow primary arms exactly at 60° intervals. 6. Classification & Mathematical Morphology Nakaya’s diagram (temperature vs. supersaturation) defines snowflake types mathematically: snowflake maths
Wavelength selection: [ \lambda \propto \frac1\sqrt\textsupersaturation ] Several mathematical models simulate snowflake growth: The governing equation is the : [ \frac\partial
The governing equation is the : [ \frac\partial c\partial t = D \nabla^2 c ] where ( c ) = water vapor concentration. On the crystal surface, the Stefan condition applies: [ v_n = D , (\nabla c \cdot \mathbfn) ] with ( v_n ) = normal growth velocity. 4.1 Mullins–Sekerka Instability A flat interface is unstable: small protrusions grow faster because they encounter higher vapor gradients. This yields the characteristic dendritic arms.
| Model | Approach | Produces | |-------|----------|-----------| | | Random walkers stick to cluster | Fractal, noise-driven shapes | | Phase Field | Solves coupled PDEs for phase & concentration | Realistic anisotropy | | Cellular Automaton | Rule-based growth on hexagonal grid | Stellar plates, dendrites | Example: Anisotropic Laplacian Growth Introduce anisotropy in surface energy: [ \gamma(\theta) = \gamma_0 (1 + \epsilon \cos(6\theta)) ] The growth velocity becomes: [ v(\theta) \propto \gamma(\theta) + \gamma''(\theta) ] This explains why real snowflakes grow primary arms exactly at 60° intervals. 6. Classification & Mathematical Morphology Nakaya’s diagram (temperature vs. supersaturation) defines snowflake types mathematically:
Wavelength selection: [ \lambda \propto \frac1\sqrt\textsupersaturation ] Several mathematical models simulate snowflake growth: