Phase-field computations of anisotropic ice crystal growth on a spherical surface

In this paper, we present a numerical method for the phase-field model of anisotropic ice crystal growth on a spherical surface. The mathematical model includes terms related to the anisotropic interfacial energy, which is defined by the interface angle with respect to a reference angle. One of the natural numerical methods on curved surfaces is a computational technique based on a triangular mesh for the surface in a three-dimensional space. However, it is difficult to compute terms with the interface angle on a triangular mesh. To resolve this problem, we solve the governing equation in Cartesian coordinates after rotating each vertex and the 1-ring neighborhood of the vertex on the triangular mesh. After rotation and interpolation, we numerically solve the phase-field model using a standard finite difference method. We present the results of several tests to demonstrate that the proposed algorithm can recover anisotropic ice crystal growth on a spherical surface.