Upscaling ice crystal growth dynamics in snow: Rigorous modeling and comparison to 4D X-ray tomography data

Abstract

Presently a unified treatment of microstructure dynamics in terrestrial snow from principles of ice crystal growth is hindered by the lack of models for the evolution of the bicontinuous ice matrix. To this end we developed a rigorous microstructure upscaling scheme which is based on common pore-scale (vapor diffusion) principles of crystal growth to predict the averaged evolution of the interface morphology. We derived a coupled set of evolution equations for the (volume averaged) ice volume fraction, surface area per unit volume, Gaussian curvature and first and second moment of the mean curvature distribution and demonstrate their correctness by a comparison to interface tracking of idealized grains. In a second step we use the model as a benchmark tool without a-priori assumptions for a comparison to experiments of snow microstructure evolution via image analysis on 4D X-ray tomography data. The benchmarking allows quantifying uncertainties from local estimates of crystal growth velocities. Finally we demonstrate how the rigorous model facilitates a statistical assessment of common growth laws by combining 4D microstructure data with finite element numerics. The results show that the prediction of the surface area evolution from first principles demands further conceptual insight from ice crystal growth.

Elements of future snowpack modeling – part 1: A physical instability arising from the non-linear coupling of transport and phase changes

“…This was neither considered in (Calonne et al, 2014) nor in (Hansen and Foslien, 2015). To investigate the feedback of an evolving ice phase on the two models from above we supply ( 1)-( 3) and ( 4)-( 5) with a dynamic ice mass conservation equation (Bader and Weilenmann, 1992;Krol and Löwe, 2018). In the absence of settling, but presence of phase changes the continuity equation reduces to an ordinary differential equation for each location in space Hansen).…”

“…The diffusion terms are characterized by the effective diffusion constant and effective thermal conductivity in snow while the reaction (or source) terms describe the phase changes, i.e. the volume averaged, solid-vapor re-crystallization rates from metamorphism (Krol and Löwe, 2018). However, (Calonne et al, 2014) and (Hansen and Foslien, 2015) both neglect the feedback of phase changes through an evolving ice phase in their numerical experiments.…”

Microstructural characterization of snow, firn and ice

“…The technique was pioneered by Lundy and Adams in 1998 [29] at Montana State University for the examination of snow. Since then a micro-CT has been used in many studies of snow metamorphism [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. The principle of the micro-CT is relatively straightforward: X-ray absorption images are collected at various angles as the specimen is rotated from 0 to 180°typically 270 images are collected, which takes upwards of 15 min-and assembled in a computer to produce a 3D image.…”

Author final response to comments of Referee #2, tc-2019-63

“…AC 1: In principle yes. As shown in (Krol and Löwe, 2018)), evolution equations for microstructural parameters principally do contain material derivatives which are mainly governed by the settling velocity. By stating a Lagrangian form for the anisotropy evolution, as done here, and superimposing this formulation in SNOWPACK to individual layers, we actually assume implicitly that the Eulerian counterpart of the anisotropy equation is governed by such a Eulerian (PDE) form with a material derivative due to settling.…”