Swiss Federal Institute of Snow and Avalanche Research Swiss Federal Institute of Technology, Lausanne Laboratoire de Metallurgie Physique, EPFL Information about Switzerland Solothurn, my home town Davos, Graubuenden, Switzerland, where the SLF is Lausanne, where the EPFL is Picture gallery My homepage
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Thomas Kaempfer's Snow Page



Note: This is my snow research page. For recreational fun in the snow, check out my snow adventures in the picture gallery.


Observation of Snow Metamorphism by
X-ray Computed Micro-Tomography

Project Summary

Post-doctoral work within the snow physics group of the Swiss Federal Institute for Snow and Avalanche Research, SLF, Davos, Switzerland, under the direction of Martin Schneebeli.


Goal : Undesturbed observation of the metamorphosing micro-structure of natural snow samples subjected to different temperatures or temperature gradients. Simultaneous measurement of heat flow (heat conductivity) and structural analysis. The experiments provide input for micro-structural numerical models of snow metamorphism and snow physics and will, in combination with the numerical simulations, improve our understanding of snow metamorphism and the relation of the snow micro-structure to snow properties.
Method : A desktop X-ray micro-tomography (SCANCO mu-CT 80) with a resolution down to 18 mu-m is set up inside a cold room. It is instrumented to control the temperature, a possible temperature gradient, and simultaneously measure heat flux through a snow sample. Time-laps series of metamorphosing snow can be recorded.
Collaborators : Martin Schneebeli, Bernd Pinzer, SLF Davos
Sergey Sokratov, Moscow State University
Sponsor : Swiss National Science Foundation and SLF Davos, Switzerland
Duration : 2003-2006

Introduction

Snow is a highly porous medium consisting of an ice matrix and porous space containing water vapor. With time, snow undergoes metamorphism and its microstructure evolves. There is a strong interaction between the snow microstructure and snow pack properties. Understanding snow metamorphism is thus crucial for predicting mechanical properties used in avalanche forecasting, chemical composition associated with the interpretation of ice cores data, or thermophysical properties important for modeling the energy balance of snow-covered landscapes.

It is common to distinguish between isothermal and temperature gradient metamporphism. The former can be regarded as a sintering process of individual ice crystals and the dominating driving force is the sintering stress. In the later, a temperature gradient across the snowpack induces a water vapor gradient in the pore-space due to the dependency of saturation wapor pressure in air on temperature. The dominant mechanisms are then sublimation and condensation and associated water vapor diffusion.

Methods

A snow sample holder with fine temperature control that fits into a Scanco micro CT80 computer-tomograph was built at SLF (Figure). This setup allows for non-disturbed, time-laps imaging of metamorphosing snow and is the method of choice to observe temperature gradient metamorphism. Experiments can be performed for several days without removing the sample from the tomograph.

For isothermal metamorphism experiments, due to the considerably longer time-scale, the snow samples are stored in controlled isothermal environments for several weeks and only moved to the micro-CT for short measurement periods.

Scanco Micro-CT 80            Snow sample holder

Figure: X-ray computer micro-tomograph (Scanco micro-CT 80) installed in SLF Davos' Cold-lab (left) and the snow sample holder allowing for temperature gradient control (right).

Results

Metamorphosing snow under temperature gradient

If snow is subjected to a temperature gradient, the heat flow induces mass flow and thus an evolution of the ice-pore network (Figure); on the other hand, the microstructure influences heat flow as heat transport is governed by conduction in the ice and pores as well as phase change processes and water vapor transport in the pore space.

Controlled experiments in the snow-breeder inside the micro-CT at SLF Davos allow for the continuous observation of the dynamic sublimation and growth processes within the ice-matrix. Under strong temperature gradients, kinetic growth forms as facets develop (Figure).

Micro-CT of snow, rounded grains            Micro-CT of snow, depth hoar

Figure: Micro-CT image of a 4x4x4 mm cube of a snow pack collected in the field and consisting of rounded grains sintered together (left) and the same snow after 10 days under a temperature gradient of 50 K/m at T=-3.4 C (right).

Isothermaly metamorphosing snow

Isothermal metamorphism of snow is a sintering process between the ice grains forming the snow. Under the absence of temperature gradients, the driving force for snow metamorphism is the sintering stress, which tends to reduce the surface free energy of the complex ice-air interfaces and of the grain boundaries between the ice crystals. The energy reduction of the porous structure is achieved by mass redistribution. Mass transport processes include surface diffusion, volume diffusion, grain boundary diffusion, viscous flow, plastic flow, and evaporation-condensation with vapor transport.

Four snow samples for isothermal metamorphism experiments at different temperatures were prepared in a similar way and from the same initial snow. During the whole experiment, except for the short times when the samples were measured, each sample resided inside a temperature controlled environment.

At the beginning of the experiment and roughly every month during one year, each sample was analyzed. The visual inspection is shown in the figure below. For the three warmer samples, rounding of the forms and coarsening is observed, proceeding at different rates depending on the temperature. At -54 degrees C, nearly no metamorphism is seen. These observations are confirmed by structural analysis and are presented in more detail in Kaempfer and Schneebeli (2007).

Isothermal
      snow metamorphism

Figure: Evolution of the 3D structure of the ice matrix during isothermal metamorphism. The samples at temperatures of -1.6. -8.3, -19.1, -54 degrees C (from left to right) at different measurement times (0, 34, 75, 118, 162, 211, 317 days) from the beginning to the end (top to bottom) of the experiment. The shown cubes are 200x200x200 voxels (2x2x2 mm) large. Framed images have the same specific surface area.

Heat conductivity

The evolution of the heat conductivity was analyzed for several snow samples undergoing metamorphism under different conditions (temperature, temperature gradients) and were published by Schneebeli and Sokratov (2004).

Publications

[2007] Kaempfer, Th. U. and M. Schneebeli, Observation of Isothermal Metamorphism of New Snow and Interpretation as a Sintering Process, J. Geophys. Res., 112 (2007), D24101, doi:10.1029/2007JD009047 Preprint as pdf .

[2004] Kaempfer, Th. U., S. A. Sokratov, M. Schneebeli, The effect of the structural evolution of snow on heat transfer, Proceedings of 3rd International Symposium on Two-Phase Flow Modelling and Experimentation, Pisa, Italy (2004), ed. G. P. Gelata, P. Di Marco, A. Mariani, R. K. Shah, Edizioni ETS, Pisa, ISBN 88-467-1075-4. Preprint as pdf .

References

[2004] Schneebeli, M. and Sokratov, S. A., Tomography of temperature gradient metamorphism of snow and associated changes in heat conductivity, Hydrol. Process.18(18) (2004), pp. 3655-3665, doi:10.1002/hyp.5800





Microstructural Heat Conductivity Simulations
through Snow and Firn

Project Summary

Post-doctoral work within the snow physics group of the Swiss Federal Institute for Snow and Avalanche Research, SLF, Davos, Switzerland, under the direction of Martin Schneebeli. Followup project with Zoe Courville and Mark Hopkins at CRREL, Hanover, USA, as research scholar through an appointment to the Research Participation Program admistered by the Oak Ridge Institute for Science and Education.


Goal : The relation between heat flow through snow and firn and microstructure is crucial for the comprehension and modeling of thermophysical, chemical, and mechanical properties of snow. The accuracy of heat conductivity measurements in the field using devices as the needle-probe are unclear. We investigate heat conduction through snow and firn using heat flux measurements combined with a microstructural numerical approach.
Method : At SLF Davos, a snow sample subject to a given temperature gradient can be measured by X-ray micro-tomography (CT) and the heat flow through it simultaneously measured. At CRREL we have data-sets of field heat conductivity measurements combined with CT-images of firn. Simple 3D heat conduction simulations allow for an assessment of the problems and an investigation of the relation between microstructure and heat conductivity.
Collaborators : Martin Schneebeli, SLF Davos
Zoe Courville, Mark Hopkins, CRREL Hanover
Sponsor : Swiss National Science Foundation, SLF Davos, Switzerland
United States Department of the Army AT-24 Research Program
Duration : 2003-2006 (SLF), 2007 (CRREL)

Introduction

Current heat transport models relate the effective heat conductivity of snow empirically to the snow density. However, measured heat conductivities differ up to five times between measurements made in snow that is similar both in density and in crystal type. It is thus necessary to find more relevant parameters related to the snow microstructure that govern the heat conductivity.

So far, the microstructural complexity always required simplifications, as for example uniformly packed ice spheres. Only recently did tomographic reconstructions lead to 3D representations of the real snow structure at the micro-scale.

We use the snow microstructure imaged by computed X-ray micro-tomography (micro-CT) to study heat transport through snow. We subject a snow sample to a constant temperature gradient, measure the passing heat flux, and determine the effective heat conductivity of the snow. Simultaneously, we image the snow microstructure by micro-CT, discretized the ice matrix by finite elements or finite differences, and solve the heat transport equation corresponding to the experimental setup.

For firn, we measured the heat conductivity in the field using a needle-probe and subsequently determined the micro-structure using micro-CT in the laboratory.

Results

Heat
      conductivity simulation
      in Firn

Figure: Simulation of the temperature distribution during a needle-probe heat conductivity measurement in firn. The structural input to the simulation is a micro-tomography image.

Publications

[2005] Kaempfer, Th. U., M. Schneebeli, S. A. Sokratov, A Microstructural Approach to Model Heat Transfer in Snow, Geophys. Res. Lett., 32, 2005, doi:10,1029/2005GL023873. Preprint as pdf .





A 3D Microstructure Based Photon Tracking Model
of Radiative Transfer in Snow

Project Summary

Work as research scholar through an appointment to the Research Participation Program at the Cold Regions Research and Engineering Laboratory, CRREL, Hanover, NH, USA, admistered by the Oak Ridge Institute for Science and Education.


Goal : Solar radiation is a key component of the energy budget of snow covered landscapes. An understanding of the interaction of solar radiation with snow is essential to the study of the snow thermodynamics, chemistry, hydrology, ecology, and remote sensing. We developed a microstructure based photon tracking algorithm to study radiative transfer in snow.
Method : The three-dimensional snow microstructure is provided either by a discrete element model defined by shape, size, and spatial arrangement of individual ice grains or by an X-ray micro-tomography image of a snowpack. The model uses refraction, Fresnel reflection, and absorption laws and the only optical input parameters are the complex index of refraction and absorption coefficient. The model follows individual photons through the microstructure, a porous network of ice and air, applying the fundamental optics laws at the ice-air interfaces and within the ice. By firing tens of thousands of photons, a detailed examination of the spectral radiance and irradiance above, below, and within the snowpack is possible.
Collaborators : Mark Hopkins, Don Perovich, CRREL Hanover
Sponsor : United States Department of the Army AT-59 Research Program
Duration : 2005-2006

Introduction

Solar radiation is a key component of the energy budget of snow covered landscapes. Even a thin snow-cover reflects most of the incident sunlight and transmits little. An understanding of the interaction of solar radiation with snow is essential to the study of the thermodynamics, chemistry, hydrology, ecology, and remote sensing of snow covered sea ice or landscapes.

To investigate this interaction we developed a microstructure based photon tracking algorithm. The three-dimensional snow microstructure is provided either by a discrete element snow model or by an X-ray micro-tomography (micro-CT) image of a snowpack. The model uses refraction, Fresnel reflection, and absorption laws and the optical input parameters are only the complex index of refraction and absorption coefficient. The model follows individual photons through the microstructure, applying the fundamental optics laws at the ice-air interfaces and within the ice. By firing tens of thousands of photons, a detailed examination of the spectral radiance and irradiance above, below, and within the snowpack is possible.

Methods

Snow representation

Discrete element modeling (DEM) is a technique for explicitly modeling the dynamics of assemblies of grains and allows for the use of complex particle contact physics. In particular, by defining appropriate collision rules, the DEM handles naturally the travel of a photon through an ice matrix formed of sintered snow grains. Our microstructural snow model is based on a DEM approach that uses axisymmetric particle shapes and is ideal for an efficient study of the impact of physical properties such as mean grain size or density on the optical properties.

The photon tracking model can be applied to non-regular particle shapes and to experimentally imaged snow by using a more universal 3D representation, a binary voxel (3D pixel) image. This is typically the image format produced by a micro-CT scan. However, a voxel representation cannot be directly used for optical computations due to the staggered ice-air interfaces. A smoothing algorithm applied to the ice-air interface overcomes this problem.

DEM
      snow representation            Micro-CT of snow with smooth 
      ice-air surface

Figure: Left: 3D representations of a model snow sample (from top, clock-wise): Cylindrical DEM, with triangulated surface, voxel representation, and voxel based triangulation after smoothing. DEM grain radius is 0.5 mm. Right: micro-CT image of snow with small rounded grains and density of 170 kg m-3, after smoothing the ice-air interfaces (resolution 40 micro-m, side length of cube 4 mm).

Modeling light transmission through snow

At visible and near infrared wavelengths, radiative transfer in snow is governed by two processes: absorption and scattering. Absorption is well understood and has a strong wavelength dependence. Scattering in snow results from differences in the real indices of refraction, n, between ice (n ~ 1.31) and air (n ~ 1.0), with little wavelength dependence but strong coupling to the intricate and highly variable microstructure. There has been considerable theoretical work on scattering in snow and it is typically represented by parameterizations.

For the wavelength considered the snow grain radii are much larger than the wavelength size parameter and we can model light transmission through snow using a photon tracking approach. We consider a ray of given initial irradiance and incident zenith angle as an assembly of photons. The interaction of the ray with a single snow grain (Figure) is governed by the following optical laws.
  • When a ray impinges on a snow grain, some of the incident energy is reflected and some is transmitted at angles defined by Snell's Law. When traveling from ice to air there is a critical angle above which all the light is internally reflected.
  • The radiances of the reflected and transmitted beams are determined from the incident radiance by the Fresnel equation and considering the two exceptions of incidence angle larger than the critical angle and incidence angle equal to zero.
  • As a beam travels through the ice, it will be attenuated according to the Bouguer-Lambert absorption law.
  • These laws are implemented in a probabilistic sense and to model a ray of light a large number of photons (typically 10000) is fired into the snowpack. Ultimately, each photon has three possible fates: exit at the top, exit at the bottom, or be absorbed inside the sample.

    The input parameters needed for the model are the complex index of refraction for ice and the angle, intensity, and position of the incident ray.

    Schematic
      of ray tracing                    Photon tracking through DEM snow

    Figure: Left: Schematic of ray tracing for an incident ray on a single grain (real numbers = radiance, integers = # photons/1000). Right: Photon tracking at 470 nm (yellow) and 1000 nm (red) through a cylindrical DEM snowpack.

    Results

    Results from the photon tracking model were compared to a 4 stream discrete ordinates model that has been validated against observations (Figure). We used spherical particles with 0.5 mm radius as well as cylindrical particles.

    Photon
        tracking compared to a 4-stream discrete ordinates  
        method continuum model

    Figure: Photon tracking (PT) compared to a 4-stream discrete ordinates method continuum model (sample density 270 kg m-3, snowpack thickness 7.8 cm, incidence angle 60 degrees).

    The photon tracking model, with its ability to examine in detail the angular distribution of reflected light, is also well suited for studies of the reflectance distribution function (Figure). This study can represent the direct solar beam on a sunny day or the beam from a laser altimeter. The typical forward scattering of snow is predicted.

    Angular
      dependence of reflected light

    Figure: The directional-conical reflectance factor (DCRF) at 470 and 900 nm for incidence zenith angles of zero and 65 degrees. The model was run on a spherical DEM model snowpack with grain radius of 0.1 mm, a density of 270 kg m-3, and a depth of 30 cm.

    Publications

    [2007] Kaempfer, Th. U., M. A. Hopkins, D. K. Perovich, A 3D Microstructure Based Photon Tracking Model of Radiative Transfer in Snow, J. Geophys. Res., 112, 2007, doi:10.1029/2006JD008239. Preprint as pdf .




    Modeling Heat and Mass Transfer in Snow
    at a Microstructural Level
    using a Phase-Field Approach

    Project Summary

    Work as research scholar through an appointment to the Research Participation Program at the Cold Regions Research and Engineering Laboratory, CRREL, Hanover, NH, USA, admistered by the Oak Ridge Institute for Science and Education.


    Goal : Snow is a highly porous medium consisting of an ice matrix and porous space containing water vapor. With time, snow undergoes metamorphism and its microstructure evolves. There is a strong interaction between the snow microstructure and snow pack properties. Recently, computed X-ray micro-tomography emerged as a tool to observe snow metamorphism at a microstructural scale. A numerical model operating at similar length scales and based on fundamental physics is highly desirable to study metamorphism in detail.
    Method : Snow metamorphism under an imposed temperature gradient is governed by heat and mass conservation laws, with possible phase change at ice-air interfaces. We use a phase field model that treats the multi-phase system with complicated interface conditions by tackling the problem continuously, inclusive of the interfacial region. This approach overcomes the problems associated with topology changes commonly found in interface tracking methods.
    Collaborators : Mathis Plapp, Ecole Polytechnique, Paris, France
    Mark Hopkins, CRREL Hanover
    Sponsor : United States Department of the Army Fundamental Research Program
    Duration : ongoing

    Introduction

    Snow is a highly porous medium consisting of an ice matrix and porous space containing water vapor. With time, snow undergoes metamorphism and its microstructure evolves. There is a strong interaction between the snow microstructure and snow pack properties. Understanding snow metamorphism is thus crucial for predicting mechanical properties used in avalanche forecasting, chemical composition associated with the interpretation of ice cores data, or thermophysical properties important for modeling the energy balance of snow-covered landscapes. The link between heat transport and metamorphism is particularly strong. On the one hand, heat flow through snow induces mass flow and thus an evolution of the ice-pore network (Figure); on the other hand, the microstructure influences heat flow as heat transport is governed by conduction in the ice and pores as well as phase change processes and water vapor transport in the pore space.

    Difference
      images of metamorphosing snow

    Figure: Difference of two micro-tomography images of a snowpack evolving under a temperature gradient: the temperature gradient induces a water vapor gradient in the pore space that drives mass flow (blue represents sublimated, green freshly condensated ice during one day).

    Phase field models treat multi-phase systems with complicated interface conditions by tackling the problem continuously, inclusive of the interfacial region). This continuous variation across the interface is realized using an order parameter, the phase field function, which describes the phases thermodynamically.

    Sharp
      vs. diffuse interface approach

    Figure: Sharp interface (left) and diffuse interface (right) with the phase field function phi.

    In a classical formulation the basic equations have to be written for each medium and the interface boundary conditions must be explicitly tracked. In diffuse-interface theory the basic equations, with supplementary phase field terms, are deduced from a free energy functional for the whole system and interface conditions do not occur. In fact, they are replaced by a partial differential equation for the phase field.

    Methods

    Snow metamorphism under an imposed temperature gradient is governed by heat and mass conservation laws, with possible phase change at ice-air interfaces. In the diffuse interface approach, we use the phase field function to continuously express the physical parameters as the heat conductivity or water vapor diffusivity by interpolation. The conservation equations, including one for the phase field parameter, are deduced from a free energy functional of the two-phase system.


    Results

    We applied the phase-field model to a 2D section of an X-ray micro-tomography image of natural snow. We extracted a 3 voxel thick section from the tomography image (Figure) and combined it into one plane to obtain the computational domain.

    Construction of 2D computational domain

    Figure: Construction of 2D computational domain.by extracting and combining 3 slices of an X-ray micro-tomography image of snow.

    We subjected the snow sample to a vertical temperature gradient of and observed the heat and mass fluxes and the evolution of the ice-matrix. The temperature distribution is considerably disturbed from a linear one, due to the different heat conductivities of ice and air (Figure, left). Even though the ice matrix is not fully connected, we observe that the heat fluxes concentrate along the ice structure (Figure, center). Moreover, strong inhomogeneities in the heat fluxes in the pore space are observed, leading consequently to similar inhomogeneities in the induced water vapor fluxes. Such effects have also to be expected in 3D. Using the phase field function , we can easily follow the microstructural evolution including topological changes and identify regions where the ice matrix sublimates or regions of crystal growth (Figure, right).

    simulated temperature distribution in
      metamorphosing snow                    simulated heat flow in
      metamorphosing snow                  simulated structural change in
      metamorphosing snow

    Figure: Phase field computation on a 2D slice of tomographed snow: A vertical temperature gradient was imposed, the ice matrix is shown in overlayed white. Temperature distribution (left), heat fluxes (center), and ice matrix after one day compared to the initial condition (right).

    Publications

    [2007] Kaempfer, Th. U. and M. Plapp, Modeling Heat and Mass Transfer in Snow at a Microstructural Level using a Phase-Field Approach - First Results, Proceedings of the 64th Annual Meeting of the Eastern Snow Conference (2007), May 2007, St. Johns, Newfoundland, Canada. Preprint as pdf .

    [2009] Kaempfer, Th. U. and M. Plapp, Phase-field modeling of dry snow metamorphism, Phys. Rev. E, 79, 2009, doi:10.1103/PhysRevE.79.031502. Preprint as pdf .




    Modeling of Destructive Metamorphism in Thin Snow
    and Impact on Optical Snow Properties.

    Project Summary

    Work as research scholar through an appointment to the Research Participation Program at the Cold Regions Research and Engineering Laboratory, CRREL, Hanover, NH, USA, admistered by the Oak Ridge Institute for Science and Education.


    Goal : We extend our micro-structure based snow model to include sintering mechanisms as grain-boudary and surface diffusion on an grain-scale level. These mechanisms become dominant in a snow-pack under little or no temperature gradient and are also particularly important during the initial, destructive metamorphism from dendritic to rounded grain snow. The impact of destructive snow metamorphism on the physical properties of a snow pack with emphasis on optical properties will be studied.
    Method : Multi-phase field models are emerging for the simulation of sintering powder particles. We will apply and further develop such models for metamorphosing snow and we will couple them to the existing heat and mass transfer model and the discrete element and optics modeling framework.
    Collaborators : Mark Hopkins, CRREL Hanover
    Sponsor : United States Department of the Army Fundamental Research Program
    Duration : ongoing