Growth kinetics and morphology of snowflakes in supersaturated atmosphere using a three-dimensional phase-field model

Simulating ice crystal growth is a major issue for meteorology and aircraft safety. Yet, very few models currently succeed in reproducing correctly the diversity of snow crystal forms, and link the model parameters to thermodynamic quantities. Here, we demonstrate that the new three-dimensional phase-field model developed in Demange et al. [npj Comput. Mater. 3, 1 (2017)] is capable of reproducing properly the morphology and growth kinetics of snowflakes in supersaturated atmosphere. Aside from that, we show that the growth dynamics of snow crystals satisfies the selection theory, consistently with previous experimental observations. Finally, we link the parameters of the phase-field model to atmospheric parameters.

Figure 1

Inverse plot of the anisotropy function $A\left(\mathit{n}\right)$, for ${\epsilon }_{xy}=0.2$ and ${\epsilon }_z=0.7$. Top left: $1/A(\theta ,\pi /4)$, limit angle ${\theta }^m(\pi /4)$, and tangent construction between ${\theta }^m$ and $-{\theta }^m$. Top right: $1/A(\pi /12,\psi )$, limit angle ${\psi }^m(\pi /12)$, and tangent construction between ${\psi }^m$ and $-{\psi }^m$.

Figure 2

2D Wulff construction in the horizontal plane for ${\epsilon }_{xy}=0.1$ and ${\epsilon }_z=0.3$ in $\psi =\pi /2$ (a) and $\psi =\pi /4$ (b), and in the vertical direction for ${\epsilon }_{xy}=0.02$ and ${\epsilon }_z=0.6$, in $\theta =0$. (d)–(f) Corresponding ECS after regularization. (g)–(i) Corresponding polar plot before (outside curve in blue), and after (inside curve in red) regularization. (j)–(l) Corresponding inverse polar plot before (inside curve in blue), and after (outside curve in red) regularization.

Figure 3

Growth process of simulated snowflakes. (a) Contour representation ($\varphi =0$) of a fern dendrite growth. (b) Early growth of a fern dendrite and supersaturation field $u$. (c) Contour representation ($\varphi =0$) of a capped column growth. Time in ${\tau }_0$ units. Visual rendering uses the software Blender.

Figure 4

Side branching process for highly anisotropic surface tension. The simulations correspond to the fern dendrite. Time is in ${\tau }_0$ unit.

Figure 5

Terrace growth and terrace-step-kink propagation in our simulations. The simulations correspond to the sectored plate. Time is in ${\tau }_0$ unit. The interface width is close to 3.

Figure 6

Principal growth stages of (a) a sectored plate and (b) a stellar dendrite. Blue line: real snowflake from [46]. Red circles: phase-field simulation [30]. 1,2,3: growth stages.

Figure 7

Simulation of the simultaneous growth of multiple snowflakes. (a) Three fern dendrite snowflakes grow simultaneously in the plane. (b) 12 arm star. Surface plot uses the software Blender for visual rendering.

Figure 8

Tip morphology of a fern dendrite. (a) Contour representation (blue line) of a fern dendrite in the horizontal direction, from experimental (1) snowflakes [22, 46] and our simulations (2). Inset: zoom on the dendrite front most tip. (b) Fern dendrites (1) and (2) in the vertical direction, and column snowflake from experimental (3) snowflakes [46] and our simulations (4). Red circles: Ivantsov parabola fit of dendrite tip. Green line: envelope for side branching.

Figure 9

Time evolution of the tip velocity $V$ and tip curvature radii $R$, for the column growth (blue circle) and the fern dendrite in the horizontal (red triangle) and vertical (black square) directions. Vertical growth: $u_0=0.4$. Horizontal growth: $u_0=0.65$.

Figure 10

Steady-state values $V^s$ and $R^s$ of the tip velocity and tip curvature radii, as a function of the supersaturation $u_0$, for the column growth (blue circle) and the fern dendrite in the horizontal (red triangle) and vertical (black square) directions. Green diamond in (b): geometric mean radius $\sqrt{R_1^s{R}_2^s}$. Continuous line: linear fit. Dashed line: regression based on Ivantsov solution and the stability theory (each curve differs from the other by a translation only).

Figure 11

Relation between $R^s{V}^s$ and $u_0$ based on 3D Ivantsov solution, for the needle (a), the horizontal curvature radius of the dendrite (b), and the vertical curvature radius of the dendrite (c), and the geometric mean (d) radius. Straight line in (a): linear regression. Dashed line in (b)–(d): fitting with Ivantsov solution.

Figure 12

Steady-state stability parameter $A/\left(V^s{\left[R^s\right]}^2\right)$, as a function of the supersaturation $u_0$, for the column growth (blue circle), and the fern dendrite using the horizontal (red triangle), the vertical (black square), and the geometric mean curvature radius (green diamond). Dashed line: power law regression.

Figure 13

$\mathrm{\Gamma }-u_0$ morphology diagram for simulated snowflakes.

Figure 14

Schematic representation of the main parameter impact on snowflake morphology.

Figure 15

$T-\rho$ morphology diagram for snowflakes. The shapes of experimentally observed snowflakes are indicated by different marks, and the domains for the different morphologies obtained in PF simulations are shown by different colors.