TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-791-2016Metamorphism during temperature gradient with undersaturated advective
airflow in a snow sampleEbnerPirmin PhilippSchneebeliMartinschneebeli@slf.chhttps://orcid.org/0000-0003-2872-4409SteinfeldAldoDepartment of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, SwitzerlandWSL Institute for Snow and Avalanche Research SLF, 7260 Davos Dorf, SwitzerlandMartin Schneebeli (schneebeli@slf.ch)7April201610279179717August201511September20156March201621March2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/791/2016/tc-10-791-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/791/2016/tc-10-791-2016.pdf
Snow at or close to the surface commonly undergoes temperature gradient
metamorphism under advective flow, which alters its microstructure and
physical properties. Time-lapse X-ray microtomography is applied to
investigate the structural dynamics of temperature gradient snow metamorphism
exposed to an advective airflow in controlled laboratory conditions. Cold
saturated air at the inlet was blown into the snow samples and warmed up
while flowing across the sample with a temperature gradient of around
50 K m-1. Changes of the porous ice structure were observed at
mid-height of the snow sample. Sublimation occurred due to the slight
undersaturation of the incoming air into the warmer ice matrix. Diffusion of
water vapor opposite to the direction of the temperature gradient
counteracted the mass transport of advection. Therefore, the total net ice
change was negligible leading to a constant porosity profile. However, the
strong recrystallization of water molecules in snow may impact its isotopic
or chemical content.
Introduction
Snow has a complex porous microstructure and consists of a continuous ice
structure made of grains connected by bonds and interconnecting pores
(Löwe et al., 2011). It has a high permeability (Calonne et al., 2012;
Zermatten et al., 2014) and under appropriate conditions, airflow through the
snow structure can occur (Sturm and Johnson, 1991) due to variation of
surface pressure (Colbeck, 1989; Albert and Hardy, 1995), simultaneous
warming and cooling, and induced temperature gradients (Sturm and Johnson,
1991). Both diffusive and advective airflows affect heat and mass transport
in the snowpack and influence chemical concentrations (Gjessing, 1977;
Waddington et al., 1996). Various airflow conditions in a snow sample occur,
namely, isothermal airflow, air cooling by a negative temperature gradient
along the airflow leading to a local supersaturation of the air, and air
warming by a positive temperature gradient along the airflow leading to a
local undersaturation of the air (Fig. 1). In general, in a natural snowpack
close to the surface (< 1 cm) two additional conditions can occur:
(1) warm air enters a snowpack with a positive temperature gradient, leading
to a supersaturation of the air at the entrance, and (2) cold air enters a
snowpack with a negative temperature gradient, leading to an undersaturation
of the air at the entrance. However, because snow has a high heat capacity
compared to the air, a high specific surface area, and therefore a high
convective heat transfer to the air, a quasi-thermal equilibrium (the term
“quasi” is used because normally the snow structure continuously changes
and therefore the equilibrium conditions change as well) is usually assumed
inside the snowpack (> 1 cm). In this paper, only conditions
deeper than 1 cm inside a snowpack are considered. Under isothermal
conditions, the continuous sublimation and deposition of ice is due to higher
vapor pressure over convex surfaces and lower vapor pressure over concave
surfaces, respectively (Kelvin effect) (Neumann et al., 2008; Ebner et al.,
2014). However, applying a fully isothermal saturated airflow across a snow
sample has been shown to have no influence on the coarsening rate that is
typical for isothermal snow metamorphism independently of the transport
regime in the pores at a physically possible Péclet number (Ebner et al.,
2015a). When applying a temperature gradient, the effect of sublimation and
deposition in the snow results from interaction between snow temperature and
the local relative humidity in the pores. If vapor is advected from a warmer
zone into a colder zone, the air becomes supersaturated, and some water vapor
deposits onto the surrounding ice grains. This leads to a change in the
microstructure, creating whisker-like crystals (Ebner et al., 2015b).
Whisker-like crystals are very small (∼ 10–30 µm) elongated
monocrystals. A flow rate dependence of the deposition rate of water vapor
deposition at the ice interface was observed, asymptotically approaching an
average estimated maximum volumetric deposition rate on the whole sample of
1.05×10-4 kg m-3 s-1 (Ebner et al., 2015b).
Contrarily, if the temperature gradient acts in the same direction of the
airflow, the airflow through the snow brings cold and relatively dry air into
a warmer area, causing the pore space air to become undersaturated, and
surrounding ice sublimates. Here, we investigate specifically this last
effect.
Schematic of the ice–air interface transport processes.
(a) Under isothermal conditions, the Kelvin effect leads to a saturation
of the pore space in the snow but does not affect the structural change (Ebner
et al., 2015a); (b) air cooling by a negative temperature gradient
along the flow direction leads to a change in the microstructure due to
deposition (Ebner et al., 2015b); (c) air warming by a positive
temperature gradient along the flow has a negligible total mass change of the
ice but a strong reposition effect of water molecules on the ice grains,
shown in this paper.
Sublimation of snow is a fundamental process that affects its crystal
structure (Sturm and Benson, 1997), and thus is important for ice core
interpretation (Stichler et al., 2001; Ekaykin et al., 2009), as well as
calculation of surface energy balance (Box and Steffen, 2001) and mass
balance (Déry and Yau, 2002). Kaempfer and Plapp (2009) suggest that
condensation of water vapor will have a noticeable effect on the
microstructure of snow using a 3-D phase-field model, which is also confirmed
by a 2-D finite-element model using airflow velocities, vapor
transport, and sublimation rates of Albert (2002). Neumann et al. (2009)
determined that there is no energy barrier to be overcome during
sublimation, and suggested that snow sublimation is limited by vapor diffusion
into pore space, rather than by sublimation at the crystal surface.
In the present work, we studied the surface dynamics of snow metamorphism
under an induced temperature gradient and saturated airflow in controlled
laboratory experiments. Cold saturated air at around -14 ∘C
was blown into the snow samples and warmed up to around -12.5 ∘C while flowing across the sample. Sublimation of ice was
analyzed by an in situ time-lapse experiment with microcomputer tomography
(micro-CT) (Pinzer and Schneebeli, 2009; Chen and Baker, 2010; Pinzer et
al., 2012; Wang and Baker, 2014; Ebner et al., 2014) to obtain the
discrete-scale geometry of snow. By using discrete-scale geometry, all
structures are resolved with a finite resolution corresponding to the voxel
size of 18 µm.
Morphological and flow characteristics of the experiments: volume
flow (V˙), initial superficial velocity in snow (uD,0), initial
snow density (ρ0), initial porosity (ε0), specific
surface area (SSA0), initial mean pore size (dmean), average
inlet (Tin,ave) and outlet temperature (Tout,ave),
and the average temperature gradient (∇Tave), corresponding
Reynolds number (Re) and Péclet number (Pe).
Temperature gradient experiments with fully saturated airflow across snow
samples (Ebner et al., 2014) were performed in a cooled micro-CT (Scanco
Medical µ-CT80) at a cold laboratory temperature of Tlab=-14∘C. Cold saturated air was blown into the snow samples and
warmed up while flowing across the sample. Aluminum foam including a heating
wire was used to warm the side of the snow opposite to the entering airflow.
We analyzed the following flow rates: a volume flow of 0 (no advection),
0.3, 1.0, and 3.0 L min-1. Higher flow rates were experimentally not
possible as shear stresses by airflow destroyed the snow structure (Ebner et
al., 2015a). Nature-identical snow produced in a cold laboratory (Schleef et
al., 2014) was used for the snow sample preparation (water temperature: 30 ∘C; air temperature: -20 ∘C). The snow
was sieved with a mesh size of 1.4 mm into a box, and was sintered for 27 days at -5 ∘C to increase its strength. The sample holder
(diameter: 53 mm; height: 30 mm) was filled by cutting out a cylinder from
the sintered snow and pushing into the sample holder without mechanical
disturbance of the core. The snow samples were measured with a voxel size of
18 µm over 108 h with time-lapse micro-CT measurements
taken every 3 h, producing a sequence of 37 images. The innermost 36.9 mm of
the total 53 mm diameter were scanned, and subsamples with a dimension of
7.2 mm × 7.2 mm × 7.2 mm were extracted for further
processing. The imaged volume was in the center of the sample (Fig. 1c). A
linear encoder with a resolution of less than 1 voxel (< 2 µm)
was used to verify that the scans were taken at the same position.
Additionally, a cross-correlation function was applied to suppress all
erroneous data from the data set. The reconstructed micro-CT images were
filtered by using a 3 × 3 × 3 median filter followed by a
Gaussian filter (σ=1.4, support = 3). The clustering-based
Otsu method (Otsu, 1979) was used to automatically segment the grey-level
images into ice and void phase. Morphological properties of the two-phase
system were determined based on the geometry obtained by the micro-CT. The
segmented data were used to calculate a triangulated ice matrix surface and
tetrahedrons inscribed into the ice structure. Morphological parameters such
as porosity (ε) and specific surface area (SSA) were then
calculated in subsamples of the size of 6.3 mm × 6.3 mm × 6.3 mm. An opening-based morphological operation was applied to
extract the mean pore size of each micro-CT scan (dmean) (Haussener et
al., 2012). As additional physical and structural parameter, the effective
thermal conductivity kcond was estimated by direct pore-level
simulation (DPLS) to determine the influence of changing microstructure.
DPLS determined the effective thermal conductivity by solving the governing
steady-state heat conduction equations within the solid phase and the
stagnant fluid phase (Kaempfer et al., 2005; Petrasch et al., 2008; Calonne
et al., 2011; Löwe et al., 2013).
Results
Time-lapse tomographic scans were performed with temperature gradients
between 43 and 53 K m-1 (Table 1). Small fluctuations of the measured
inlet and outlet temperature were due to temperature regulation both inside
the cold chamber and inside the micro-CT (Ebner et al., 2014). A shift of
Δt < 10 min between inlet and outlet temperature indicated
that a fast equilibrium between the temperature of the snow and the airflow
was reached (Albert and Hardy, 1995; Ebner et al., 2015b). The morphological
evolution was similar between all four experiments and only a slight
rounding and coarsening was visually observed during the 108 h experiment
(Fig. 2). The initial ice grains did not change with time, and the locations
of sublimation and deposition for ota3 and ota4 are shown in Fig. 3.
Sublimation of 7.7 and 7.6 % of the ice matrix and deposition of 6.0 and 9.6 % on the ice matrix were observed. The data were extracted
by superposition of vertical cross sections at 0 and 108 h with an
uncertainty of 6 %. The mass sublimated preferentially at locations of the
ice matrix with low radii and was relocated leading to a smoothing of the
ice surface and to an increase in the size of pores (Fig. 4a). The pore
size (uncertainty ∼ 6 %) increased by 3.4, 3.6,
5.4 and 6.5 % for ota1, ota2, ota3, and ota4, respectively.
Evolution of the 3-D structure of the ice matrix with applied
temperature gradient and advective conditions. Experimental conditions (from
left to right) at different measurement times from beginning to the end (top
to bottom) of the experiment. The cubes shown are
110 × 40 × 110 voxels
(2 × 0.7 × 2 mm3) large with 18 µm voxel
size (a high-resolution figure can be found in the Supplement).
Loss of ice due to sublimation could not be detected by the micro-CT scans
due to limited accuracy, and no flow rate dependence was observed during any
of the four experiments. The temporal evolution of the porosity, shown in
Fig. 4b, did not change with time and the influence of sublimation of
water vapor was not observed. Only ota2 showed a slight drop in the
temporal evolution of the porosity until 18 h into the experiment but kept
constant afterwards. This slight drop (≈ 0.5 %) was probably
caused by settling of the snow. Coarsening was observed for each experiment
but the influence of changing airflow was not visible, confirmed by the
temporal SSA evolution, shown in Fig. 4c.
Superposition of vertical cross section parallel to the flow
direction at 0 and 108 h for ota3 (left panel) and ota4 (right
panel). Sublimation and deposition of water vapor on the ice grains were
visible with an uncertainty of 6 % (a high-resolution figure can be found
in the Supplement).
The repositioning of water molecules led to a smoothing of the ice grains,
but did not affect the thermal conductivity of snow. This quantity (standard
deviation ∼ 0.025 W m-1) slightly increased after
applying airflow to the temperature gradient, shown in Fig. 4d, but no
flow rate dependence was observed. Every third scan was used to extract the
thermal conductivity and a change of -2.6, 3.6, 2.2, and 2.7 % for ota1, ota2, ota3, and ota4 was detected.
Discussion
The rate of deposition onto the ice surface depends on the flow rate where
warm saturated air cooled down while flowing through the sample, as shown in
previous experiments (Ebner et al., 2015b). Its deposition rate
asymptotically reached a maximum of 1.05×10-4 kg m-3 s-1. In this study, changing
the temperature gradient leads to a warming up of a cold saturated flow, and results in a sublimation rate too
small for the analyzed period of the experiment to measure a flow rate
dependence by the micro-CT and an influence on the temporal density
gradient.
Temporal evolution of (a) the mean pore size,
dmean, of the snow samples obtained by an opening size
distribution, (b) the porosity, ε, obtained by the
triangulated structure surface method, (c) the specific surface
area, SSA, of the ice matrix obtained by triangulated structure surface
method, and (d) the effective thermal conductivity of the snow
sample, kcond, estimated by DPLS. The sizes of the
volumes used for the computation of each property are
350 × 350 × 350 voxels
(6.3 × 6.3 × 6.3 mm3).
A structural change of the ice grains and repositions of water molecules was
observed but the total net flux of the snow was not affected. The
superposition of a vertical cross section in Fig. 3 shows a big effect on
reposition of water molecules on the ice structure. However, the temporal
porosity (Fig. 4b) was not affected and the total water vapor net flux was
negligible for the analyzed volume. Continued sublimation and deposition of
water molecules due to the temperature gradient led to a saturation of the
pore space. The vapor pressure of the air in the pore was in equilibrium
with the water pressure of the ice, given by the local temperature. The
entering air warmed up, allowing vapor sublimating from the snow sample to
be incorporated into the airflow. As time passed, the snow grains in the
sample became more rounded as convexities sublimated. As a result of the
reduced curvature, the rate of sublimation decreased and less vapor was
deposited in concavities and therefore the surface asperities persisted
longer. Finally, the Kelvin effect had a longer impact on the structural
change of the ice grains and the reposition of water molecules. In addition,
the uptake of water molecules and their transport due to warming during
advection was counteracted by diffusion of water molecules due to the
temperature gradient. As thermally induced diffusion was opposite to the
airflow gradient, a backflow of water vapor occurred and the two opposite
fluxes counteracted each other. The Péclet numbers (Pe=uD×dmean/D, where D is the diffusion coefficient of water vapor in air),
describing the ratio of mass transfer between diffusion and advection,
measured during each experiment, showed that diffusion was still dominant
(Table 1). Therefore, water molecules were diffused along the opposite
direction to the temperature gradient and advected along the flow direction
leading to a back and forth transport of water molecules.
As a Péclet higher than 1 is not possible in natural snow (Ebner et al.,
2015a), advection of cold saturated air into a slightly warmer snowpack has
a significant influence, not on the total net mass change but on the
structural change of the ice grains due to redistribution of water vapor on
the ice matrix. Additionally, the increasing pore size has an influence on the flow
field, leading to a deceleration of the flow, and therefore the interaction of
an air parcel with the ice matrix in the pores increases due to higher
residence time. In addition, the diffusive transport rises, whereas the
advective transport decreases, changing the mass transport in the pores. Our
results support the hypothesis of Neumann et al. (2009) that sublimation is
limited by vapor diffusion into the pore space or kinetics effects (reaction
effect) at crystal faces. This is supported by the temporal evolution of the
porosity (Fig. 4b) and the SSA (Fig. 4c), as no velocity dependence was
observed and the structural changes were too small to be detected by the
micro-CT.
The influence of diffusion of water vapor in the direction of the
temperature gradient and the influence of the residence time of an
air parcel in the pores were also confirmed by a low mass change at the
ice–air interface. Overlapping two consecutive 3-D images, the order of
magnitude of freshly sublimated ice was detected. The absolute mass change
at the ice–air interface (kg m-3 s-1) estimated by the
experimental results is defined as
Sm,exp=ρiΔ1-εΔt,
where Δ(1-ε) is the change in the porosity between two
images separated by the time step Δt, and ρi is the
density of ice. Albert and McGilvary (1992) and Neumann et al. (2009)
presented a model to calculate sublimation rates directly in an aggregate
snow sample:
Sm=hmSAV(ρsat-ρv),
where SAV is the specific surface area per volume of snow, and
hm is the mass transfer coefficient (m s-1) given by Neumann et
al. (2009):
hm=(0.566×Re+0.075)×10-3,
assuming that the sublimation occurs within the first few millimeters of the sample.
Re (Re=uD×dmean/ν, where ν
is the kinematic viscosity of the air) is the corresponding Reynolds number
of the flow. The absolute sublimation rate is driven by the difference
between the local vapor density (ρv) and the saturation vapor
density (ρsat) (Neumann et al., 2009; Thorpe and Mason,
1966). Table 2 shows the estimated absolute sublimation rate by the
experiment (Eq. 1) and the model (Eq. 2). The very small change in porosity
due to densification during the first 18 h for ota2 was not taken into
account. The estimated sublimation rates by the experiment were 2 orders of
magnitude lower than the modeled values and also 2 orders of magnitude
lower than during the experiment of negative temperature gradient along an airflow
(Ebner et al., 2015b). As the air in the pore space is always saturated
(Neumann et al., 2009), the back diffusion of water vapor in the opposite
direction of the temperature gradient led to a lower mass transfer rate of
sublimation. The flow rate dependence for the model described is shown by the
mass transfer coefficient (Eq. 3), increasing with higher airflow. However,
the values calculated from the experiment showed a different trend.
Increasing the flow rate led to a lower mass transfer rate due to a lower
residence time of the air in the pores. Transfer of heat toward and water
vapor away from the sublimating interface may also limit the sublimation
rate. In general, the results of the model by Neumann et al. (2009) have to
be interpreted with care, as his model was set up to saturate dry air under
isothermal conditions. Ice crystals sublimated as dry air enters the snow
sample; water vapor was advected throughout the pore space by airflow until
saturation vapor pressure was reached, preventing further sublimation. The
model by Neumann et al. (2009) does not consider the influence of a
temperature gradient and the additional vapor pressure gradient. However, our
results concluded that a positive temperature gradient along the airflow has
a significant impact on the sublimation rate, decreasing the rate by 2
orders of magnitude.
Estimated sublimation rate Sm using the mass transfer
coefficient hm determined by Neumann et al. (2009) and the corresponding
average surface area per volume SAV,ave. Sm can be compared with
the measured sublimation rate of the experiment Sm,exp (Eq. 1).
In the experiments by Neumann et al. (2009), sublimation of snow using dry
air under isothermal condition showed a temperature drop for approximately
the first 15 min after sublimation started and stayed constant because the
latent heat absorption of sublimation for a given flow rate and heat
exchange with the sample chamber equalized each other. Such a temperature
drop was not observed in our experiments. In the experiments by Neumann et
al. (2009) the amount of energy used for sublimation was between -10 and -40 J min-1 for saturation of dry air. Using the expected mass change at
the ice–air interface Sm,exp (Eq. 1) and the latent heat of
sublimation (Lsub≈2834.1×103 J kg-1),
the energy needed for sublimation ranged between -2 and -12 J min-1 for
our experiments. Our estimated values are a factor up to 5 lower than the
estimated numbers of Neumann et al. (2009), because the entering air was
already saturated (with reference to the cold temperature) at the inlet. The energy
needed for sublimation could be balanced between the sensible heat
carried into and out of the sample, and the exchange of the snow sample with
the air stream and the surroundings prevented a temperature drop.
Thermal conductivity changed insignificantly in these experiments,
especially for ota1. This indicates that air warming by a positive
temperature gradient along the airflow and an open system reduces or
suppresses the increase in thermal conductivity usually observed by
temperature gradient metamorphism (Löwe et al., 2013; Calonne et al.,
2014); the timescales are also quite different between experiments. Compared
to the closed temperature gradient experiment, the applied temperature gradient
induced an air movement and therefore reduced the impact on the snow
metamorphism and its thermal conductivity, at least in the short term. As previously
mentioned, the thermal conductivity has been numerically estimated from the
geometrical information of the sample only and no air movement was taken
into consideration.
Summary and conclusion
We performed four experiments of temperature gradient metamorphism of snow
under saturated advective airflow during 108 h. Cold saturated air was blown
into the snow samples and warmed up while flowing across the sample. The
temperature gradient varied between 43 and 53 K m-1 and the snow
microstructure was observed by X-ray microtomography every 3 h. The micro-CT
scans were segmented, and porosity, specific surface area, and the mean pore
size were calculated. Effective thermal conductivity was calculated by direct
pore-level simulation (DPLS).
Compared to deposition (shown in Ebner et al., 2015b), sublimation showed a
small effect on the structural change of the ice matrix. A change in the
pore size was most likely due to sublimation of ice crystals with small
radii but a significant loss of water molecules of the snow sample and mass
transfer away from the ice interface due to sublimation and advective
transport could not be detected by the micro-CT scans and no flow rate
dependence was observed. The interaction of mass transport of advection and
diffusion of water vapor in the opposite direction of the temperature
gradient and the influence of the residence time of an air parcel in the
pores led to a negligible total mass change of the ice. However, a strong
reposition of water molecules on the ice grains was observed.
The energy needed for sublimation was too low to see a significant
temperature drop because the energy needed was balanced between the sensible
heat carried into and out of the sample, and the exchange of the snow sample
with the air stream and the surroundings.
This is the third paper of a series analyzing an advective airflow in a
snowpack in depths of more than 1 cm. Previous work showed that (1) under
isothermal conditions, the Kelvin effect leads to a saturation of the pore
space in the snow but does not affect the structural change (Ebner et al.,
2015a); (2) applying a negative temperature gradient along the flow
direction leads to a change in the microstructure and the creation of
whisker-like structures due to deposition of water molecules on the ice
matrix (Ebner et al., 2015b); and (3) a positive temperature gradient along
the flow had a negligible total mass change of the ice but a strong
reposition effect of water molecules on the ice grains, shown in this paper.
Conditions (1) and (3) showed that they have a negligible effect on the
porosity evolution of the ice matrix except when sintering is concerned.
Porosity changes can be neglected to improve models for snow compaction and
evolution at the surface, however, mechanical processes like compaction
strongly impact porosity. In contrast, condition (2) showed a significant
impact on the structural evolution and seems to be essential for such
snowpack models and other numerical simulations. Nevertheless, the strong
reposition of water molecules on the ice grains observed for all conditions (1)–(3) can have a significant impact on atmospheric chemistry and
isotopic changes in snow.
The Supplement related to this article is available online at doi:10.5194/tc-10-791-2016-supplement.
Acknowledgements
The Swiss National Science Foundation granted financial support under
project no. 200020-146540. The authors thank the reviewers E. A. Podolskiy
and F. Flin for the constructive reviews and M. Jaggi, S. Grimm, and H. Löwe
for technical and modeling support.
Edited by: G. Chambon
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