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Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

Let $\xi =z-Vt$ increase into the liquid, with the planar solidification front at $\xi =0$. Denote the common thermal conductivity by $K=\rho c_p\kappa$. A bounded travelling temperature profile satisfies $\kappa T^{\prime \prime }+V{T}^{\prime }=0$. In the solid, boundedness as $\xi \to -\infty$ forces $T_s=T_m$; in the liquid,The Stefan condition then givesThus the travelling solidification front in a pure supercooled melt releases just enough latent heat to warm the initially supercooled liquid to $T_m$. There is no separate imposed heat flux at infinity in this bounded travelling-wave problem.

For the interface perturbation write $\xi =\zeta (y,t)=\hat{\zeta }{e}^{i\alpha y+\sigma t}$. Use the Gibbs--Thomson relation for curvature-induced melting-temperature depressionHere $\gamma$ is the solid-liquid surface energy, $T_m$ in this expression is absolute temperature in kelvin, and $\mathcal{K}$ is the sum of principal curvatures, positive for solid convex into liquid. For this graph, $\mathcal{K}=-{\zeta }_{yy}={\alpha }^2\zeta$. If mean curvature is defined as half this sum, $\mathcal{K}=2H$; this fixes the capillary factor unambiguously.

Under the small-wavelength thermal approximation for a solidification front, take harmonic perturbation temperatures that decay away from the interface:This approximates their temperature equation by Laplace's equation, neglecting perturbation time dependence and the moving-frame advection. Expanding the interfacial temperature at the displaced surface givesThe second expression includes displacement through the negative base liquid gradient. Linearizing the Stefan condition also requires evaluation of the base gradient at the displaced surface:Using $S=1$ gives the requested reduced dispersion relation:orThe destabilizing $V\alpha$ term describes a protrusion entering colder liquid. The cubic term expresses stabilization by the curvature-induced temperature decrease.

There is an important accuracy qualification. The harmonic approximation gives the leading short-wave thermal exponents, but the retained constant term is not a uniformly controlled next-order correction to the full moving-boundary problem. If perturbation diffusion is retained, its exponents areSubstitution into exactly the same linearized boundary balances gives the full thermal stability of a travelling solidification front:For example, with capillarity removed, $\sigma =V\alpha$ solves this relation exactly, whereas the harmonic reduction gives $V\alpha -V^2/\kappa$. The difference is smaller than the leading term when $\alpha \gg V/\kappa$, but demonstrates why the reduced formula should not be asserted as an exact dispersion law or a precise onset calculation. To neglect perturbation time dependence one also needs $|\sigma |\ll \kappa {\alpha }^2$; extremely large capillary damping is outside that approximation.

To analyse the displayed cubic, defineThen $\kappa \sigma /V^2=g(q)=-1+q-2q^3/R_c$. It is negative at zero and at sufficiently large $q$. Its positive stationary point is the unique maximum:Thus the capillary threshold of a reduced solidification dispersion iswith equality giving a neutral maximum and strict inequality giving a band of unstable positive wavenumbers. At equality, however, $q_{\ast }=3/2$, not $q_{\ast }\gg 1$. The number $27/2$ is therefore the threshold of the stated reduced cubic, not a controlled threshold of the full thermal problem under the stated short-wave ordering. Far above this threshold its fastest-growing scale does satisfy $q_{\ast }\gg 1$, making the short-wave mechanism meaningful.

The analogy for a snowflake is dendritic growth of a snowflake. A protrusion has better access to the driving supersaturation or undercooling and grows faster than a sheltered region; this amplifies tips and promotes branching into crystal dendrites. Interfacial capillarity suppresses arbitrarily fine tips. In an actual snowflake, the main growth is deposition from water vapour, so the diffusion field and its driving variable differ from this pure-melt thermal example. The hexagonal ice crystal anisotropy supplies six equivalent preferred arm directions; an isotropic planar stability calculation alone cannot predict six arms. Nonlinear tip selection, vapour transport and side branching are needed for the later detailed morphology. The analysis explains the competition between diffusion-driven protrusion growth and capillary smoothing, with this physically necessary qualification.