Articles | Volume 18, issue 4
https://doi.org/10.5194/tc-18-1579-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/tc-18-1579-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
05 Apr 2024
Research article |
| 05 Apr 2024
| 05 Apr 2024
A microstructure-based parameterization of the effective anisotropic elasticity tensor of snow, firn, and bubbly ice
Kavitha Sundu
WSL Institute for Snow and Avalanche Research SLF, Davos, Switzerland
Johannes Freitag
Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar- und Meeresforschung, Bremerhaven, Germany
Kévin Fourteau
WSL Institute for Snow and Avalanche Research SLF, Davos, Switzerland
Univ. Grenoble Alpes, CNRS, IRD, Grenoble INP, IGE, 38000 Grenoble, France
WSL Institute for Snow and Avalanche Research SLF, Davos, Switzerland
Related authors
No articles found.
Audrey Goutard, Marion Réveillet, Fanny Brun, Delphine Six, Kevin Fourteau, Charles Amory, Xavier Fettweis, Mathieu Fructus, Arbindra Khadka, and Matthieu Lafaysse
EGUsphere, https://doi.org/10.5194/egusphere-2025-2947, https://doi.org/10.5194/egusphere-2025-2947, 2025
This preprint is open for discussion and under review for The Cryosphere (TC).
Short summary
Short summary
A new scheme has been developed in the SURFEX/ISBA-Crocus model, to consider the impact of liquid water dynamics on bare ice, including albedo feedback and refreezing. When applied to the Mera Glacier in Nepal, the model reveals strong seasonal effects on the energy and mass balance, with increased melting in dry seasons and significant refreezing during the monsoon. This development improves mass balance modeling under increasing rainfall and bare ice exposure due to climate warming.
Kévin Fourteau, Julien Brondex, Clément Cancès, and Marie Dumont
EGUsphere, https://doi.org/10.5194/egusphere-2025-444, https://doi.org/10.5194/egusphere-2025-444, 2025
Short summary
Short summary
The percolation of liquid water down snowpacks is a complex phenomenon, and its representation can sometimes be complicated for snowpack models. The goal of this article is to transpose some state-of-the-art strategies used for modeling liquid percolation in other media (such as rocks or soil) into snowpack models. With this, snowpack models can be made more efficient, requiring less time and power to perform their computation.
Léon Roussel, Marie Dumont, Marion Réveillet, Delphine Six, Marin Kneib, Pierre Nabat, Kevin Fourteau, Diego Monteiro, Simon Gascoin, Emmanuel Thibert, Antoine Rabatel, Jean-Emmanuel Sicart, Mylène Bonnefoy, Luc Piard, Olivier Laarman, Bruno Jourdain, Mathieu Fructus, Matthieu Vernay, and Matthieu Lafaysse
EGUsphere, https://doi.org/10.5194/egusphere-2025-1741, https://doi.org/10.5194/egusphere-2025-1741, 2025
Short summary
Short summary
Saharan dust deposits frequently color alpine glaciers orange. Mineral dust reduces snow albedo and increases snow and glaciers melt rate. Using physical modeling, we quantified the impact of dust on the Argentière Glacier over the period 2019–2022. We found that that the contribution of mineral dust to the melt represents between 6 and 12 % of Argentière Glacier summer melt. At specific locations, the impact of dust over one year can rise to an equivalent of 1 meter of melted ice.
Florian Painer, Sepp Kipfstuhl, Martyn Drury, Tsutomu Uchida, Johannes Freitag, and Ilka Weikusat
EGUsphere, https://doi.org/10.5194/egusphere-2025-633, https://doi.org/10.5194/egusphere-2025-633, 2025
Short summary
Short summary
Air clathrate hydrates trap ancient air in the deeper part of ice sheets. We use digital microscopy and automated image analysis to investigate the evolution of number, size and shape of air clathrate hydrates from 1250 m depth to the bottom of the ice sheet. We confirm the previously found relation of changes in number and size with past climate and find a connection of their shape to changes in ice deformation. The results will help to better understand air clathrate hydrates in deep ice.
Sindhu Vudayagiri, Bo Vinther, Johannes Freitag, Peter L. Langen, and Thomas Blunier
Clim. Past, 21, 517–528, https://doi.org/10.5194/cp-21-517-2025, https://doi.org/10.5194/cp-21-517-2025, 2025
Short summary
Short summary
Air trapped in polar ice during snowfall reflects atmospheric pressure at the time of occlusion, serving as a proxy for elevation. However, melting, firn structure changes, and air pressure variability complicate this relationship. We measured total air content (TAC) in the RECAP ice core from Renland ice cap, eastern Greenland, spanning 121 000 years. Melt layers and short-term TAC variations, whose origins remain unclear, present challenges in interpreting elevation changes.
Xavier Faïn, Sophie Szopa, Vaishali Naïk, Patricia Martinerie, David M. Etheridge, Rachael H. Rhodes, Cathy M. Trudinger, Vasilii V. Petrenko, Kévin Fourteau, and Philip Place
Atmos. Chem. Phys., 25, 1105–1119, https://doi.org/10.5194/acp-25-1105-2025, https://doi.org/10.5194/acp-25-1105-2025, 2025
Short summary
Short summary
Carbon monoxide (CO) plays a crucial role in the atmosphere's oxidizing capacity. In this study, we analyse how historical (1850–2014) [CO] outputs from state-of-the-art global chemistry–climate models over Greenland and Antarctica are able to capture both absolute values and trends recorded in multi-site ice archives. A disparity in [CO] growth rates emerges in the Northern Hemisphere between models and observations from 1920–1975 CE, possibly linked to uncertainties in CO emission factors.
Emma Pearce, Dimitri Zigone, Coen Hofstede, Andreas Fichtner, Joachim Rimpot, Sune Olander Rasmussen, Johannes Freitag, and Olaf Eisen
The Cryosphere, 18, 4917–4932, https://doi.org/10.5194/tc-18-4917-2024, https://doi.org/10.5194/tc-18-4917-2024, 2024
Short summary
Short summary
Our study near EastGRIP camp in Greenland shows varying firn properties by direction (crucial for studying ice stream stability, structure, surface mass balance, and past climate conditions). We used dispersion curve analysis of Love and Rayleigh waves to show firn is nonuniform along and across the flow of an ice stream due to wind patterns, seasonal variability, and the proximity to the edge of the ice stream. This method better informs firn structure, advancing ice stream understanding.
Julien Westhoff, Johannes Freitag, Anaïs Orsi, Patricia Martinerie, Ilka Weikusat, Michael Dyonisius, Xavier Faïn, Kevin Fourteau, and Thomas Blunier
The Cryosphere, 18, 4379–4397, https://doi.org/10.5194/tc-18-4379-2024, https://doi.org/10.5194/tc-18-4379-2024, 2024
Short summary
Short summary
We study the EastGRIP area, Greenland, in detail with traditional and novel techniques. Due to the compaction of the ice, at a certain depth, atmospheric gases can no longer exchange, and the atmosphere is trapped in air bubbles in the ice. We find this depth by pumping air from a borehole, modeling, and using a new technique based on the optical appearance of the ice. Our results suggest that the close-off depth lies at around 58–61 m depth and more precisely at 58.3 m depth.
Benjamin Walter, Hagen Weigel, Sonja Wahl, and Henning Löwe
The Cryosphere, 18, 3633–3652, https://doi.org/10.5194/tc-18-3633-2024, https://doi.org/10.5194/tc-18-3633-2024, 2024
Short summary
Short summary
The topmost layer of a snowpack forms the interface to the atmosphere and is critical for the reflectance of solar radiation and avalanche formation. The effect of wind on the surface snow microstructure during precipitation events is poorly understood and quantified. We performed controlled lab experiments in a ring wind tunnel to systematically quantify the snow microstructure for different wind speeds, temperatures and precipitation intensities and to identify the relevant processes.
Kévin Fourteau, Johannes Freitag, Mika Malinen, and Henning Löwe
The Cryosphere, 18, 2831–2846, https://doi.org/10.5194/tc-18-2831-2024, https://doi.org/10.5194/tc-18-2831-2024, 2024
Short summary
Short summary
Understanding the settling of snow under its own weight has applications from avalanche forecasts to ice core interpretations. We study how this settling can be modeled using 3D images of the internal structure of snow and ice deformation mechanics. We found that classical ice mechanics, as used, for instance, in glacier flow, explain the compaction of dense polar snow but not that of lighter seasonal snow. How, exactly, the ice deforms during light snow compaction thus remains an open question.
Julien Meloche, Melody Sandells, Henning Löwe, Nick Rutter, Richard Essery, Ghislain Picard, Randall K. Scharien, Alexandre Langlois, Matthias Jaggi, Josh King, Peter Toose, Jérôme Bouffard, Alessandro Di Bella, and Michele Scagliola
EGUsphere, https://doi.org/10.5194/egusphere-2024-1583, https://doi.org/10.5194/egusphere-2024-1583, 2024
Preprint archived
Short summary
Short summary
Sea ice thickness is essential for climate studies. Radar altimetry has provided sea ice thickness measurement, but uncertainty arises from interaction of the signal with the snow cover. Therefore, modelling the signal interaction with the snow is necessary to improve retrieval. A radar model was used to simulate the radar signal from the snow-covered sea ice. This work paved the way to improved physical algorithm to retrieve snow depth and sea ice thickness for radar altimeter missions.
Anna Braun, Kévin Fourteau, and Henning Löwe
The Cryosphere, 18, 1653–1668, https://doi.org/10.5194/tc-18-1653-2024, https://doi.org/10.5194/tc-18-1653-2024, 2024
Short summary
Short summary
The specific surface of snow dictates key physical properties and continuously evolves in natural snowpacks. This is referred to as metamorphism. This work develops a rigorous physical model for this evolution, which is able to reproduce X-ray tomography measurements without using unphysical tuning parameters. Our results emphasize that snow crystal growth at the micrometer scale ultimately controls the pace of metamorphism.
Kévin Fourteau, Julien Brondex, Fanny Brun, and Marie Dumont
Geosci. Model Dev., 17, 1903–1929, https://doi.org/10.5194/gmd-17-1903-2024, https://doi.org/10.5194/gmd-17-1903-2024, 2024
Short summary
Short summary
In this paper, we provide a novel numerical implementation for solving the energy exchanges at the surface of snow and ice. By combining the strong points of previous models, our solution leads to more accurate and robust simulations of the energy exchanges, surface temperature, and melt while preserving a reasonable computation time.
Amy R. Macfarlane, Henning Löwe, Lucille Gimenes, David N. Wagner, Ruzica Dadic, Rafael Ottersberg, Stefan Hämmerle, and Martin Schneebeli
The Cryosphere, 17, 5417–5434, https://doi.org/10.5194/tc-17-5417-2023, https://doi.org/10.5194/tc-17-5417-2023, 2023
Short summary
Short summary
Snow acts as an insulating blanket on Arctic sea ice, keeping the underlying ice "warm", relative to the atmosphere. Knowing the snow's thermal conductivity is essential for understanding winter ice growth. During the MOSAiC expedition, we measured the thermal conductivity of snow. We found spatial and vertical variability to overpower any temporal variability or dependency on underlying ice type and the thermal resistance to be directly influenced by snow height.
Julien Brondex, Kévin Fourteau, Marie Dumont, Pascal Hagenmuller, Neige Calonne, François Tuzet, and Henning Löwe
Geosci. Model Dev., 16, 7075–7106, https://doi.org/10.5194/gmd-16-7075-2023, https://doi.org/10.5194/gmd-16-7075-2023, 2023
Short summary
Short summary
Vapor diffusion is one of the main processes governing snowpack evolution, and it must be accounted for in models. Recent attempts to represent vapor diffusion in numerical models have faced several difficulties regarding computational cost and mass and energy conservation. Here, we develop our own finite-element software to explore numerical approaches and enable us to overcome these difficulties. We illustrate the capability of these approaches on established numerical benchmarks.
Xavier Faïn, David M. Etheridge, Kévin Fourteau, Patricia Martinerie, Cathy M. Trudinger, Rachael H. Rhodes, Nathan J. Chellman, Ray L. Langenfelds, Joseph R. McConnell, Mark A. J. Curran, Edward J. Brook, Thomas Blunier, Grégory Teste, Roberto Grilli, Anthony Lemoine, William T. Sturges, Boris Vannière, Johannes Freitag, and Jérôme Chappellaz
Clim. Past, 19, 2287–2311, https://doi.org/10.5194/cp-19-2287-2023, https://doi.org/10.5194/cp-19-2287-2023, 2023
Short summary
Short summary
We report on a 3000-year record of carbon monoxide (CO) levels in the Southern Hemisphere's high latitudes by combining ice core and firn air measurements with modern direct atmospheric samples. Antarctica [CO] remained stable (–835 to 1500 CE), decreased during the Little Ice Age, and peaked around 1985 CE. Such evolution reflects stable biomass burning CO emissions before industrialization, followed by growth from CO anthropogenic sources, which decline after 1985 due to improved combustion.
Zhuo Wang, Ailsa Chung, Daniel Steinhage, Frédéric Parrenin, Johannes Freitag, and Olaf Eisen
The Cryosphere, 17, 4297–4314, https://doi.org/10.5194/tc-17-4297-2023, https://doi.org/10.5194/tc-17-4297-2023, 2023
Short summary
Short summary
We combine radar-based observed internal layer stratigraphy of the ice sheet with a 1-D ice flow model in the Dome Fuji region. This results in maps of age and age density of the basal ice, the basal thermal conditions, and reconstructed accumulation rates. Based on modeled age we then identify four potential candidates for ice which is potentially 1.5 Myr old. Our map of basal thermal conditions indicates that melting prevails over the presence of stagnant ice in the study area.
Nora Hirsch, Alexandra Zuhr, Thomas Münch, Maria Hörhold, Johannes Freitag, Remi Dallmayr, and Thomas Laepple
The Cryosphere, 17, 4207–4221, https://doi.org/10.5194/tc-17-4207-2023, https://doi.org/10.5194/tc-17-4207-2023, 2023
Short summary
Short summary
Stable water isotopes from firn cores provide valuable information on past climates, yet their utility is hampered by stratigraphic noise, i.e. the irregular deposition and wind-driven redistribution of snow. We found stratigraphic noise on the Antarctic Plateau to be related to the local accumulation rate, snow surface roughness and slope inclination, which can guide future decisions on sampling locations and thus increase the resolution of climate reconstructions from low-accumulation areas.
Fanny Brun, Owen King, Marion Réveillet, Charles Amory, Anton Planchot, Etienne Berthier, Amaury Dehecq, Tobias Bolch, Kévin Fourteau, Julien Brondex, Marie Dumont, Christoph Mayer, Silvan Leinss, Romain Hugonnet, and Patrick Wagnon
The Cryosphere, 17, 3251–3268, https://doi.org/10.5194/tc-17-3251-2023, https://doi.org/10.5194/tc-17-3251-2023, 2023
Short summary
Short summary
The South Col Glacier is a small body of ice and snow located on the southern ridge of Mt. Everest. A recent study proposed that South Col Glacier is rapidly losing mass. In this study, we examined the glacier thickness change for the period 1984–2017 and found no thickness change. To reconcile these results, we investigate wind erosion and surface energy and mass balance and find that melt is unlikely a dominant process, contrary to previous findings.
Ghislain Picard, Henning Löwe, and Christian Mätzler
The Cryosphere, 16, 3861–3866, https://doi.org/10.5194/tc-16-3861-2022, https://doi.org/10.5194/tc-16-3861-2022, 2022
Short summary
Short summary
Microwave satellite observations used to monitor the cryosphere require radiative transfer models for their interpretation. These models represent how microwaves are scattered by snow and ice. However no existing theory is suitable for all types of snow and ice found on Earth. We adapted a recently published generic scattering theory to snow and show how it may improve the representation of snows with intermediate densities (~500 kg/m3) and/or with coarse grains at high microwave frequencies.
Julien Westhoff, Giulia Sinnl, Anders Svensson, Johannes Freitag, Helle Astrid Kjær, Paul Vallelonga, Bo Vinther, Sepp Kipfstuhl, Dorthe Dahl-Jensen, and Ilka Weikusat
Clim. Past, 18, 1011–1034, https://doi.org/10.5194/cp-18-1011-2022, https://doi.org/10.5194/cp-18-1011-2022, 2022
Short summary
Short summary
We present a melt event record from an ice core from central Greenland, which covers the past 10 000 years. Our record displays warm summer events, which can be used to enhance our understanding of the past climate. We compare our data to anomalies in tree ring width, which also represents summer temperatures, and find a good correlation. Furthermore, we investigate an outstandingly warm event in the year 986 AD or 991 AD, which has not been analyzed before.
Xavier Faïn, Rachael H. Rhodes, Philip Place, Vasilii V. Petrenko, Kévin Fourteau, Nathan Chellman, Edward Crosier, Joseph R. McConnell, Edward J. Brook, Thomas Blunier, Michel Legrand, and Jérôme Chappellaz
Clim. Past, 18, 631–647, https://doi.org/10.5194/cp-18-631-2022, https://doi.org/10.5194/cp-18-631-2022, 2022
Short summary
Short summary
Carbon monoxide (CO) is a regulated pollutant and one of the key components determining the oxidizing capacity of the atmosphere. In this study, we analyzed five ice cores from Greenland at high resolution for CO concentrations by coupling laser spectrometry with continuous melting. By combining these new datasets, we produced an upper-bound estimate of past atmospheric CO abundance since preindustrial times for the Northern Hemisphere high latitudes, covering the period from 1700 to 1957 CE.
Konstantin Schürholt, Julia Kowalski, and Henning Löwe
The Cryosphere, 16, 903–923, https://doi.org/10.5194/tc-16-903-2022, https://doi.org/10.5194/tc-16-903-2022, 2022
Short summary
Short summary
This companion paper deals with numerical particularities of partial differential equations underlying 1D snow models. In this first part we neglect mechanical settling and demonstrate that the nonlinear coupling between diffusive transport (heat and vapor), phase changes and ice mass conservation contains a wave instability that may be relevant for weak layer formation. Numerical requirements are discussed in view of the underlying homogenization scheme.
Anna Simson, Henning Löwe, and Julia Kowalski
The Cryosphere, 15, 5423–5445, https://doi.org/10.5194/tc-15-5423-2021, https://doi.org/10.5194/tc-15-5423-2021, 2021
Short summary
Short summary
This companion paper deals with numerical particularities of partial differential equations underlying one-dimensional snow models. In this second part we include mechanical settling and develop a new hybrid (Eulerian–Lagrangian) method for solving the advection-dominated ice mass conservation on a moving mesh alongside Eulerian diffusion (heat and vapor) and phase changes. The scheme facilitates a modular and extendable solver strategy while retaining controls on numerical accuracy.
Helle Astrid Kjær, Lisa Lolk Hauge, Marius Simonsen, Zurine Yoldi, Iben Koldtoft, Maria Hörhold, Johannes Freitag, Sepp Kipfstuhl, Anders Svensson, and Paul Vallelonga
The Cryosphere, 15, 3719–3730, https://doi.org/10.5194/tc-15-3719-2021, https://doi.org/10.5194/tc-15-3719-2021, 2021
Short summary
Short summary
Ice core analyses are often done in home laboratories after costly transport of samples from the field. This limits the amount of sample that can be analysed.
Here, we present the first truly field-portable continuous flow analysis (CFA) system for the analysis of impurities in snow, firn and ice cores while still in the field: the lightweight in situ analysis (LISA) box.
LISA is demonstrated in Greenland to reconstruct accumulation, conductivity and peroxide in snow cores.
Sebastian Hellmann, Melchior Grab, Johanna Kerch, Henning Löwe, Andreas Bauder, Ilka Weikusat, and Hansruedi Maurer
The Cryosphere, 15, 3507–3521, https://doi.org/10.5194/tc-15-3507-2021, https://doi.org/10.5194/tc-15-3507-2021, 2021
Short summary
Short summary
In this study, we analyse whether ultrasonic measurements on ice core samples could be employed to derive information about the particular ice crystal orientation in these samples. We discuss if such ultrasonic scans of ice core samples could provide similarly detailed results as the established methods, which usually destroy the ice samples. Our geophysical approach is minimally invasive and could support the existing methods with additional and (semi-)continuous data points along the ice core.
Kévin Fourteau, Florent Domine, and Pascal Hagenmuller
The Cryosphere, 15, 2739–2755, https://doi.org/10.5194/tc-15-2739-2021, https://doi.org/10.5194/tc-15-2739-2021, 2021
Short summary
Short summary
The thermal conductivity of snow is an important physical property governing the thermal regime of a snowpack and its substrate. We show that it strongly depends on the kinetics of water vapor sublimation and that previous experimental data suggest a rather fast kinetics. In such a case, neglecting water vapor leads to an underestimation of thermal conductivity by up to 50 % for light snow. Moreover, the diffusivity of water vapor in snow is then directly related to the thermal conductivity.
Kévin Fourteau, Florent Domine, and Pascal Hagenmuller
The Cryosphere, 15, 389–406, https://doi.org/10.5194/tc-15-389-2021, https://doi.org/10.5194/tc-15-389-2021, 2021
Short summary
Short summary
There has been a long controversy to determine whether the effective diffusion coefficient of water vapor in snow is superior to that in free air. Using theory and numerical modeling, we show that while water vapor diffuses more than inert gases thanks to its interaction with the ice, the effective diffusion coefficient of water vapor in snow remains inferior to that in free air. This suggests that other transport mechanisms are responsible for the large vapor fluxes observed in some snowpacks.
Alexander H. Weinhart, Johannes Freitag, Maria Hörhold, Sepp Kipfstuhl, and Olaf Eisen
The Cryosphere, 14, 3663–3685, https://doi.org/10.5194/tc-14-3663-2020, https://doi.org/10.5194/tc-14-3663-2020, 2020
Short summary
Short summary
From 1 m snow profiles along a traverse on the East Antarctic Plateau, we calculated a representative surface snow density of 355 kg m−3 for this region with an error less than 1.5 %.
This density is 10 % higher and density fluctuations seem to happen on smaller scales than climate model outputs suggest. Our study can help improve the parameterization of surface snow density in climate models to reduce the error in future sea level predictions.
Cited articles
Baker, I.: Microstructural characterization of snow, firn and ice, Philos. T. Roy. Soc. A, 377, 20180162, https://doi.org/10.1098/rsta.2018.0162, 2019. a
Bobillier, G., Bergfeld, B., Capelli, A., Dual, J., Gaume, J., van Herwijnen, A., and Schweizer, J.: Micromechanical modeling of snow failure, The Cryosphere, 14, 39–49, https://doi.org/10.5194/tc-14-39-2020, 2020. a
Calonne, N., Geindreau, C., Flin, F., Morin, S., Lesaffre, B., Rolland du Roscoat, S., and Charrier, P.: 3-D image-based numerical computations of snow permeability: links to specific surface area, density, and microstructural anisotropy, The Cryosphere, 6, 939–951, https://doi.org/10.5194/tc-6-939-2012, 2012. a
Calonne, N., Flin, F., Lesaffre, B., Dufour, A., Roulle, J., Puglièse, P., Philip, A., Lahoucine, F., Geindreau, C., Panel, J.-M., du Roscoat, S. R., and Charrier, P.: CellDyM: A room temperature operating cryogenic cell for the dynamic monitoring of snow metamorphism by time-lapse X-ray microtomography, Geophys. Res. Lett., 42, 3911–3918, https://doi.org/10.1002/2015GL063541, 2015. a, b
Calonne, N., Milliancourt, L., Burr, A., Philip, A., Martin, C. L., Flin, F., and Geindreau, C.: Thermal Conductivity of Snow, Firn, and Porous Ice From 3-D Image-Based Computations, Geophys. Res. Lett., 46, 13079–13089, https://doi.org/10.1029/2019GL085228, 2019. a, b
Chaput, J., Aster, R., Karplus, M., and Nakata, N.: Ambient high-frequency seismic surface waves in the firn column of central west Antarctica, J. Glaciol., 68, 785–798, https://doi.org/10.1017/jog.2021.135, 2022. a, b
Cowin, S. C.: The relationship between the elasticity tensor and the fabric tensor, Mech. Mater., 4, 137–147, https://doi.org/10.1016/0167-6636(85)90012-2, 1985. a, b
Diez, A. and Eisen, O.: Seismic wave propagation in anisotropic ice – Part 1: Elasticity tensor and derived quantities from ice-core properties, The Cryosphere, 9, 367–384, https://doi.org/10.5194/tc-9-367-2015, 2015. a
Diez, A., Eisen, O., Hofstede, C., Lambrecht, A., Mayer, C., Miller, H., Steinhage, D., Binder, T., and Weikusat, I.: Seismic wave propagation in anisotropic ice – Part 2: Effects of crystal anisotropy in geophysical data, The Cryosphere, 9, 385–398, https://doi.org/10.5194/tc-9-385-2015, 2015. a, b
Eshelby, J. D. and Peierls, R. E.: The determination of the elastic field of an ellipsoidal inclusion, and related problems, P. Roy. Soc. Lond. A Mat., 241, 376–396, https://doi.org/10.1098/rspa.1957.0133, 1957. a
Fourteau, K., Martinerie, P., Faïn, X., Schaller, C. F., Tuckwell, R. J., Löwe, H., Arnaud, L., Magand, O., Thomas, E. R., Freitag, J., Mulvaney, R., Schneebeli, M., and Lipenkov, V. Ya.: Multi-tracer study of gas trapping in an East Antarctic ice core, The Cryosphere, 13, 3383–3403, https://doi.org/10.5194/tc-13-3383-2019, 2019. a, b, c, d
Freitag, J., Wilhelms, F., and Kipfstuhl, S.: Microstructure-dependent densification of polar firn derived from X-ray microtomography, J. Glaciol., 50, 243–250, https://doi.org/10.3189/172756504781830123, 2004. a
Frolov, A. D. and Fedyukin, I. V.: Elastic properties of snow-ice formations in their whole density range, Ann. Glaciol., 26, 55–58, https://doi.org/10.3189/1998AoG26-1-55-58, 1998. a
Fujita, S., Hirabayashi, M., Goto-Azuma, K., Dallmayr, R., Satow, K., Zheng, J., and Dahl-Jensen, D.: Densification of layered firn of the ice sheet at NEEM, Greenland, J. Glaciol., 60, 905–921, https://doi.org/10.3189/2014JoG14J006, 2014. a, b
Garboczi, E. J.: Finite element and finite difference programs for computing the linear electrical and elastic properties of digital images of random material, NISTIR 6269, US Department of Commerce, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=860168 (last access: 25 March 2024), 1998. a, b
Gaume, J., Chambon, G., Eckert, N., and Naaim, M.: Influence of weak-layer heterogeneity on snow slab avalanche release: application to the evaluation of avalanche release depths, J. Glaciol., 59, 423–437, https://doi.org/10.3189/2013JoG12J161, 2013. a
Hagenmuller, P., Matzl, M., Chambon, G., and Schneebeli, M.: Sensitivity of snow density and specific surface area measured by microtomography to different image processing algorithms, The Cryosphere, 10, 1039–1054, https://doi.org/10.5194/tc-10-1039-2016, 2016. a, b
Hashin, Z.: Theory of mechanical behavior of heterogeneous media, Appl. Mech. Rev., 17, 1–9, 1964. a
Hashin, Z. and Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity, J. Mech. Phys. Solids, 10, 335–342, https://doi.org/10.1016/0022-5096(62)90004-2, 1962. a, b, c
Hellmann, S., Grab, M., Kerch, J., Löwe, H., Bauder, A., Weikusat, I., and Maurer, H.: Acoustic velocity measurements for detecting the crystal orientation fabrics of a temperate ice core, The Cryosphere, 15, 3507–3521, https://doi.org/10.5194/tc-15-3507-2021, 2021. a, b
Hill, R.: Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, 11, 357–372, https://doi.org/10.1016/0022-5096(63)90036-X, 1963. a
Ishimoto, H., Adachi, S., Yamaguchi, S., Tanikawa, T., Aoki, T., and Masuda, K.: Snow particles extracted from X-ray computed microtomography imagery and their single-scattering properties, J. Quant. Spectrosc. Ra., 209, 113–128, https://doi.org/10.1016/j.jqsrt.2018.01.021, 2018. a
Klatt, M. A., Schröder-Turk, G. E., and Mecke, K.: Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative, Med. Phys., 44, 3663–3675, https://doi.org/10.1002/mp.12280, 2017. a
Krol, Q. and Löwe, H.: Relating optical and microwave grain metrics of snow: the relevance of grain shape, The Cryosphere, 10, 2847–2863, https://doi.org/10.5194/tc-10-2847-2016, 2016. a, b
Leinss, S., Löwe, H., Proksch, M., and Kontu, A.: Modeling the evolution of the structural anisotropy of snow, The Cryosphere, 14, 51–75, https://doi.org/10.5194/tc-14-51-2020, 2020. a, b, c, d
Montagnat, M., Azuma, N., Dahl-Jensen, D., Eichler, J., Fujita, S., Gillet-Chaulet, F., Kipfstuhl, S., Samyn, D., Svensson, A., and Weikusat, I.: Fabric along the NEEM ice core, Greenland, and its comparison with GRIP and NGRIP ice cores, The Cryosphere, 8, 1129–1138, https://doi.org/10.5194/tc-8-1129-2014, 2014. a
Moreno, R., Borga, M., and Smedby, Ö.: Generalizing the mean intercept length tensor for gray-level images, Med. Phys., 39, 4599–4612, https://doi.org/10.1118/1.4730502, 2012. a
Moser, D. E., Hörhold, M., Kipfstuhl, S., and Freitag, J.: Microstructure of Snow and Its Link to Trace Elements and Isotopic Composition at Kohnen Station, Dronning Maud Land, Antarctica, Front. Earth Sci., 8, 487823, https://doi.org/10.3389/feart.2020.00023, 2020. a, b
Nemat-Nasser, S. and Hori, M.: Universal Bounds for Overall Properties of Linear and Nonlinear Heterogeneous Solids, J. Eng. Mater.-T. ASME, 117, 412–432, https://doi.org/10.1115/1.2804735, 1995. a
Odgaard, A.: Three-dimensional methods for quantification of cancellous bone architecture, Bone, 20, 315–328, https://doi.org/10.1016/S8756-3282(97)00007-0, 1997. a
Parnell, W. and Calvo-Jurado, C.: On the computation of the Hashin-Shtrikman bounds for transversely isotropic two-phase linear elastic fibre-reinforced composites, J. Eng. Math., 95, 295–323, https://doi.org/ 10.1007/s10665-014-9777-3, 2015. a
Petrenko, V. F. and Whitworth, R. W.: Physics of Ice, Oxford University Press, ISBN 9780198518945, https://doi.org/10.1093/acprof:oso/9780198518945.001.0001, 2002. a, b, c
Picard, G., Löwe, H., and Mätzler, C.: Brief communication: A continuous formulation of microwave scattering from fresh snow to bubbly ice from first principles, The Cryosphere, 16, 3861–3866, https://doi.org/10.5194/tc-16-3861-2022, 2022. a
Proksch, M., Löwe, H., and Schneebeli, M.: Density, specific surface area, and correlation length of snow measured by high-resolution penetrometry, J. Geophys. Res.-Earth, 120, 346–362, https://doi.org/10.1002/2014JF003266, 2015. a, b
Roberts, A. P. and Garboczi, E. J.: Computation of the linear elastic properties of random porous materials with a wide variety of microstructure, Proceedings of the Royal Society A: Mathematical, Phys. Eng. Sci., 458, 1033–1054, https://doi.org/10.1098/rspa.2001.0900, 2002. a, b
Salomon, M. L., Maus, S., and Petrich, C.: Microstructure evolution of young sea ice from a Svalbard fjord using micro-CT analysis, J. Glaciol., 68, 571–590, https://doi.org/10.1017/jog.2021.119, 2022. a
Saruya, T., Fujita, S., Iizuka, Y., Miyamoto, A., Ohno, H., Hori, A., Shigeyama, W., Hirabayashi, M., and Goto-Azuma, K.: Development of crystal orientation fabric in the Dome Fuji ice core in East Antarctica: implications for the deformation regime in ice sheets, The Cryosphere, 16, 2985–3003, https://doi.org/10.5194/tc-16-2985-2022, 2022. a
Scapozza, C.: Entwicklung eines dichte- und temperaturabhängigen Stoffgesetzes zur Beschreibung des visko-elastischen Verhaltens von Schnee, PhD thesis, ETH Zurich, Zürich, Technische Wissenschaften, Eidgenössische Technische Hochschule ETH Zürich, Nr. 15357, https://doi.org/10.3929/ethz-a-004680249, 2004. a
Schlegel, R., Diez, A., Löwe, H., Mayer, C., Lambrecht, A., Freitag, J., Miller, H., Hofstede, C., and Eisen, O.: Comparison of elastic moduli from seismic diving-wave and ice-core microstructure analysis in Antarctic polar firn, Ann. Glaciol., 60, 220–230, https://doi.org/10.1017/aog.2019.10, 2019. a, b, c, d
Sigrist, C.: Measurement of fracture mechanical properties of snow and application to dry snow slab avalanche release, PhD thesis, ETH Zurich, Zürich, https://doi.org/10.3929/ethz-a-005282374, 2006. a
Sundu, K., Freitag, J., Fourteau, K., and Löwe, H.: Effective, anisotropic elasticity tensor of snow, firn, and bubbly ice, EnviDat [code, data set], https://doi.org/10.16904/envidat.462, 2023. a
Thomsen, L.: Weak elastic anisotropy, Geophysics, 51, 1954–1966, https://doi.org/10.1190/1.1442051, 1986. a
Torquato, S.: Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties, Appl. Mech. Rev., 44, 37–76, https://doi.org/10.1115/1.3119494, 1991. a, b, c
Torquato, S.: Effective stiffness tensor of composite media–I. Exact series expansions, J. Mech. Phys. Solids, 45, 1421–1448, https://doi.org/10.1016/S0022-5096(97)00019-7, 1997. a, b, c
Torquato, S.: Effective stiffness tensor of composite media : II. Applications to isotropic dispersions, J. Mech. Phys. Solids, 46, 1411–1440, https://doi.org/10.1016/S0022-5096(97)00083-5, 1998. a
Torquato, S.: Statistical Description of Microstructures, Annu. Rev. Mater. Res., 32, 77–111, https://doi.org/10.1146/annurev.matsci.32.110101.155324, 2002b. a, b
Wautier, A., Geindreau, C., and Flin, F.: Linking snow microstructure to its macroscopic elastic stiffness tensor: A numerical homogenization method and its application to 3-D images from X-ray tomography, Geophys. Res. Lett., 42, 8031–8041, https://doi.org/10.1002/2015GL065227, 2015. a, b, c, d, e, f, g, h
Wautier, A., Geindreau, C., and Flin, F.: Numerical homogenization of the viscoplastic behavior of snow based on X-ray tomography images, The Cryosphere, 11, 1465–1485, https://doi.org/10.5194/tc-11-1465-2017, 2017. a
Weng, G.: Explicit evaluation of Willis' bounds with ellipsoidal inclusions, Int. J. Eng. Sci., 30, 83–92, https://doi.org/10.1016/0020-7225(92)90123-X, 1992. a
Willis, J.: Variational and Related Methods for the Overall Properties of Composites, vol. 21, in: Advances in Applied Mechanics, 1–78, Elsevier, https://doi.org/10.1016/S0065-2156(08)70330-2, 1981. a
Wu, F., Li, J., Geng, W., and Tang, W.: A VTI anisotropic media inversion method based on the exact reflection coefficient equation, Front. Phys., 10, 926636, https://doi.org/10.3389/fphy.2022.926636, 2022. a, b
Zysset, P. and Curnier, A.: An alternative model for anisotropic elasticity based on fabric tensors, Mech. Mater., 21, 243–250, https://doi.org/10.1016/0167-6636(95)00018-6, 1995. a
Zysset, P. K.: A review of morphology–elasticity relationships in human trabecular bone: theories and experiments, J. Biomech., 36, 1469–1485, https://doi.org/10.1016/S0021-9290(03)00128-3, 2003. a, b
Short summary
Ice crystals often show a rod-like, vertical orientation in snow and firn; they are said to be anisotropic. The stiffness in the vertical direction therefore differs from the horizontal, which, for example, impacts the propagation of seismic waves. To quantify this anisotropy, we conducted finite-element simulations of 391 snow, firn, and ice core microstructures obtained from X-ray tomography. We then derived a parameterization that may be employed for advanced seismic studies in polar regions.
Ice crystals often show a rod-like, vertical orientation in snow and firn; they are said to be...
