Introduction
Ice layers form a marked microstructural transition inside the snowpack
. Their formation is generally considered to be tightly
coupled to the presence of preferential flow in snow .
Despite their often small vertical extent, (thin)
ice layers may have a profound impact on large-scale processes in a snowpack,
such as liquid water, heat and vapour flow .
Many fields of study have addressed the issue of ice layers in snowpacks.
Water may flow laterally over ice layers or crusts, which reduces travel
times in catchments and has a significant impact on catchment-scale
hydrology; alternatively, preferential flow in snow may promote vertical
percolation instead . Recent studies have demonstrated
that the increased melt on the Greenland Ice Sheet during the last few
decades led to changes in the firn structure, particularly through the
formation of ice layers by percolating water in sub-freezing snow
. These ice layers can reach considerable vertical extents on
the order of 1 m and may reduce the storage capacity of
meltwater in the firn by making access to deeper firn layers more difficult.
Subsequent melt events may thus be accompanied by much more efficient runoff
due to lateral flow over these ice layers . Ice layers can
also have a profound impact on microwave emission from the snowpack, which is
used in remote sensing retrieval algorithms . For
rock stability of permafrost-affected regions, the presence of ice layers
near the base of the snowpack as well as inside the snowpack was found to
prevent liquid water from reaching joints in the rocks, thereby improving
rock stability . Ice layers in snowpacks also impact the
access to food resources for wild life in snow-covered areas (e.g.
). Climate change projections of future increases
in rain-on-snow events in high latitudes , increased snow melt
on ice sheets as well as more frequent melt events in alpine
snowpacks show urgency to be able to determine how these
changes affect the snowpack microstructure in the future.
For 1-D snow cover models, whether they are physics based or simple, it is
notoriously difficult to simulate the formation of ice layers. This can be
understood as most models do not consider preferential flow, which is a
crucial transport mechanism to allow downward propagating water flow in
sub-freezing snow. Liquid water can thereby reach areas in the snowpack where
the cold content is large enough to refreeze the percolating meltwater and
form ice layers . In early attempts by
and to describe preferential flow in
snowpack models, the water flow in snow is considered as a flow in multiple
flow paths. In , flow paths are defined that differ in
size and snowpack properties, which results in different percolation speeds
in the individual snow paths when applying Darcy's law. In ,
the snowpack is divided in flow paths of equal size and snowpack properties.
In their approach, it is determined how much of the total flux is transported in each
of the individual flow paths based on data from compartmented lysimeter experiments.
Both
approaches never found widespread adoption, probably because they require
a priori specification of the flow path variability . In
, a description of preferential flow for snowpack
models is proposed where water in excess of a threshold in saturation (for
example in ponding conditions inside the snowpack) is directly routed to the
soil below the snowpack. This approach improved the prediction of snowpack
runoff, but it is not able to simulate the formation of ice layers due to
percolating meltwater in preferential flow channels, as the water in
preferential flow is considered to have left the snow domain of the model.
The fact that many snow models neglect preferential flow, even when they are
used for hydrological studies where snowpack runoff is a primary process, may
be justified for describing seasonal runoff characteristics
. However, preferential flow may be crucial for understanding
the response of a snow cover on short, sub-daily timescales, for example
during rain-on-snow events .
Also, for wet snow avalanche formation the exact location at
which liquid water starts ponding can influence snow stability
and considering preferential flow may be important for the
exact timing of when weak layers are reached by water. Snowpack models
developed for avalanche warning purposes may
therefore also benefit from a description of preferential flow processes in
snow.
Multidimensional snow cover models have been developed to simulate
preferential flow , but those models
simplified and neglected several snowpack processes (i.e. snow settling,
snow microstructure evolution), meaning that they are not yet applicable to
natural snowpacks. Furthermore, multidimensional snow cover models generally
require more computational power, making them unsuitable for large-scale
deployment. However, those multidimensional model developments provide
crucial insights that allowed for a parametrisation of a dual domain approach
for preferential flow for the 1-D, physics-based, detailed SNOWPACK model
, which we present in this
study.
Schematic overview of the dual domain
implementation for the SNOWPACK model, in which the pore space that can be
occupied by liquid water is separated into a part for matrix flow
(θs,matrix) and a part representing preferential flow
(θs,pref). The numbers refer to processes described in the
text.
Dual domain implementation
To simulate preferential flow, we apply a dual domain approach as
schematically shown in Fig. . The pore space is
subdivided into a part that is involved in preferential flow and a part that
is representing matrix flow (labelled 1 in Fig. ). For
the construction of the domains and the exchange processes between both
domains, we exploit recent results from laboratory and model experiments as
well as applying concepts from hydrological modelling. The water flow in the
model is described for both the matrix and preferential flow domain by
solving the Richards equation () for both domains sequentially at the commonly used
SNOWPACK time step of 15 min. After solving the Richards equation for the matrix
domain, the exchange of water between the matrix and preferential flow domain
is determined and vice versa. If the pressure head exceeds the water entry
pressure head of the layer below (labelled 2 in Fig. ),
water moves from matrix to preferential flow (labelled 3 in
Fig. ). If the saturation in the preferential flow path
exceeds a threshold (labelled 4 in Fig. ), water moves
back to the matrix domain (labelled 5 in Fig. ). Only
the matrix part is allowed to undergo phase changes and ice layers form when
water moves back from preferential flow to matrix flow and refreezes.
Preferential flow always remains in the liquid phase. Refreezing of
preferential flow water is mimicked by moving water from preferential flow to
the matrix flow domain (labelled 6 in Fig. ). Below, the
water exchange processes are described in more detail.
Defining the dual domains
For the dual domain approach, the pore space is subdivided in a matrix and
preferential flow domain (denoted with 1 in Fig. ). For
soils, the relative area involved in preferential flow is often found to be a
function of the ratio of system influx rate over saturated hydraulic
conductivity for a given soil texture. In the
experimental data on snow presented by , a more
pronounced dependence of the preferential flow area with grain size is found,
rather than with the system influx rate. We illustrate their experimental
results graphically in Fig. . Whereas the grain size shows
a distinct pattern of smaller preferential flow area for larger grain sizes,
the system influx rate showed a rather ambiguous pattern, where increased
influx did not always lead to a larger preferential flow area. The grain
sizes used in their experiments span over typical ranges found in natural
snowpacks and this dependence is important to take into account. It also has
to be noted that the infiltration rates in their experiments exceed typical
values in natural conditions. We therefore decided to determine the
dependence of preferential flow area on grain size using the lowest
experimental infiltration rate only. A fit to this selection of the data (see
Fig. ) provides the following expression for the
preferential flow area (F):
Relationship of the area involved in
preferential flow as a function of grain radius. Data points represent
laboratory experiments by , presented quantitatively by
. Data points are coloured based on the water influx
rate used in the experiments. The large black dots denote the data points
used for determining the fit (solid line), corresponding to the lowest influx
rate per grain size class.
F=0.0584rg-1.109,
where F is the preferential flow area fraction (–), and rg
is the grain radius (mm). The matrix flow domain is accordingly defined as
1-F. For fine grained snow (rg≈0.12 mm), did not observe preferential flow, in
contrast with the snow samples with rg≈0.21 mm and
larger. However, typically most parts of the snowpack consist of larger
grains than the smallest grain size used in the experiments. Thus, to provide
a continuum description of the matrix and preferential flow regime, Eq. () is used for all grain sizes. For numerical stability,
F is limited between 0.01 and 0.90. Generally grain size increases over
time in snow, and this may occasionally lead to a situation where the
preferential flow area in the next SNOWPACK time step is reduced below the
required one to accommodate for the liquid water present in the preferential
flow domain. We therefore additionally ensure that F is large enough to
contain all present preferential flow water.
For both domains, the relationship between pressure head and liquid water
content (LWC) is described by the van Genuchten parametrisation for snow as
experimentally determined by . For the matrix flow
domain, the saturated LWC is scaled by (1-F) and for the preferential flow
domain by F. Furthermore, we determine the residual water content for the
matrix flow domain using the approach described in , while
setting it to 0 for the preferential flow domain. Saturated hydraulic
conductivity is parametrised using the parametrisation for permeability
proposed by .
Water exchange between matrix and preferential flow domain
All liquid water input (snow melt, rainfall, condensation) is added to the
matrix flow domain. A prerequisite for the formation of an unstable wetting
front (i.e. flow fingering) is that the system influx rate is below the
saturated hydraulic conductivity of the medium , which is
generally fulfilled in snow . In order to initiate
preferential flow, we use the concept that preferential flow paths form when
the pressure head in the matrix flow domain exceeds the water entry pressure
of the layer below. This was found to be the case in laboratory experiments
and was successfully exploited in
numerical modelling to initiate preferential flow . The water entry pressure hwe (m) can be
expressed as a function of grain radius according to :
hwe=0.04372rg+0.01074.
One important condition to reach the water entry pressure is water ponding on
a microstructural transition inside the snowpack
. This is denoted with 2 in
Fig. . To achieve the high LWC value observed in
experiments , we use the geometric average to calculate the
hydraulic conductivity between snow layers . In our
implementation, the amount of water in the matrix part in excess of the
threshold corresponding to the water entry pressure of the layer below is
moved to the preferential flow domain in the layer below (denoted with 3 in
Fig. ). A capillary overshoot condition was found in
snow , which means that the capillary pressure in the
ponding layer decreases again after preferential flow forms. This increases
the liquid water content in the preferential flow paths, and to mimic this
effect we allow more water to flow from matrix flow to preferential flow
once the threshold is exceeded than only the amount of water above the
threshold. If after the water in excess of the threshold is moved and the
saturation (i.e. ratio of water volume to pore volume) in the layer in the
matrix domain is still higher than the saturation in the layer below in the
preferential flow domain, the saturation is equalised by an equivalent water
flow with the following approach. Equal saturation in a specific layer with
index i in the matrix domain and a layer with index j in the preferential
flow domain can be expressed as
θmi-θr,miθs,mi-θr,mi=θpj-θr,pjθs,pj-θr,pj,
where the subscripts m and p denote the matrix and preferential flow domain,
respectively, θ is the LWC (m3 m-3), θr
is the residual LWC (m3 m-3) and θs is the
saturated LWC (m3 m-3). In the model, layers are counted from
below, such that equalising saturation between the matrix domain in the layer
above and the preferential flow domain in the layer below corresponds to
j=i-1.
Given layer thicknesses Lmi and Lpj for the
layers in the matrix flow and preferential flow domain, respectively, the
total LWC in the matrix and preferential flow layer is defined as
θtot=θmiLmi+θpjLpj.
Under the requirement of an equal degree of saturation for a given total LWC,
we can solve Eq. () for θmi:
θmi=-θr,miθs,pj-θr,pjθs,miLmi+θr,pj-θs,pjθtotθs,mi-θr,miLmi+θs,pj-θr,pjLpj,
after which θpj can be found by applying
Eq. ().
Additionally, if the saturation in the matrix domain exceeds the saturation
of the preferential flow domain in a snowpack layer, saturation is equalised
using Eq. (), with i=j and consequently
Lmi=Lpj. This is motivated by the fact that
once snow is wet, no horizontal gradients in pressure head are expected to be
present in a snow layer; thus, following the van Genuchten water
retention curve, the saturation of the matrix domain is equal to the
saturation of the preferential flow domain.
Conceptually, water will leave the preferential flow domain and enter the
matrix domain if the pressure head inside the preferential flow domain
exceeds the water entry pressure of the dry snow around the preferential flow
path. This procedure was able to simulate water spreading on microstructural
transitions in the multidimensional snow model by .
However, in our study, this approach rarely succeeded in forming ice layers,
as the condition is rather seldom met. This fact can be interpreted in light
of the physics behind preferential flow. It was demonstrated that an
overshoot condition exists in flow fingers, which means that the tip of a
flow finger shows marked higher saturation than the tail. This flow behaviour
cannot be described by the Richards equation , although
the Richards equation continues to provide a correct description above and below
the wetting front. The reason why the condition worked in
may be due to the fact that the simulations involved
high water influx rates, much higher than experienced in natural snowpacks.
This would increase the amount of water accumulating on the capillary barrier
when liquid water flow over the transition is slower than the water flux
arriving from above. In the absence of a solution for this problem
, we simply apply a threshold in saturation
(Θth) of the preferential flow domain (denoted with 4 in
Fig. ). Once this threshold is exceeded, water will flow
back to the matrix domain (denoted with 5 in Fig. ). In
our approach, we first move as much water as freezing capacity is available
in the matrix domain. If after this approach the threshold is still exceeded,
we additionally equalise the saturation in the specific layer in the matrix
and preferential flow domain, using Eq. (). For the
lowest snow layer above the soil, the saturation is always equalised between
the matrix and preferential flow domain, regardless of whether the saturation
threshold is exceeded or not. This suppressed spiky snowpack runoff
behaviour. In soil layers, preferential flow is ignored by setting the
hydraulic conductivity for the preferential flow domain to 0 and the
preferential flow area to 2 %.
Refreezing preferential flow
In our approach, water in the preferential flow domain is not considered for
phase changes. However, in reality preferential flow is known to refreeze,
even forming ice structures in the shape of flow fingers inside the snowpack
. For simplicity,
we currently do not consider microstructural changes due to preferential
flow, although they may have a strong effect on the water flow in snow. Grain
growth and subsequent reduction of capillary forces as well as ice columns
may increase the efficiency of the preferential flow paths considerably.
Schematic representation of a
preferential flow path with radius r and surface F inside a 1 m2
snowpack (i.e. R=0.5 m), seen from above (not to scale), to approximate
∂x. The preferential flow path is assumed to be at melting
temperature T0, the rest of the snowpack at temperature
Te. R* is the radius such that surface areas A1 and
A2 are equal. When assuming a linear temperature gradient,
Te is found at distance R*.
For the thermal effects, we first describe the heat flux between the
preferential flow part and the matrix part by assuming a pipe with radius r
(m) at melting temperature T0 (K) in the middle of a 1 m2
snowpack at temperature Te (K) (see
Fig. ). If we assume that the horizontal
temperature gradient inside the snowpack is linear, then the temperature
Te is found at a radius R* (m) such that surface areas
A1 (m2) and A2 (m2) are equal. We can then approximate
Fourier's law for heat flow for the heat flux between the preferential flow
and matrix domain (QH,p→m,
J m-1 s-1) as
QH,p→m=κ∂T∂x≈κTe-T01+F2π-Fπ.
The volumetric content that needs to be transferred from the preferential
domain to the matrix domain (denoted with 6 in Fig. ),
in order to satisfy the refreezing capacity provided by the heat flux QH,p→m over the outer area of the preferential flow path
can be subsequently expressed as
Δθw,p→m=NπFLeQH,p→mΔt,
where Le is the latent heat associated with freezing (3.34×105 J kg-1) and N is a factor describing the effect of
multiple flow paths forming area F. Often, numerous flow paths can be
identified per square metre of snowpack, as for example found in a field
study by . They report a flow path density between roughly
100 and 300 m-2. However, this number is not necessarily representative
for the number of flow paths actively and concurrently transporting water, as
often new preferential flow paths form in subsequent melt cycles
. found only three preferential flow
paths per square metre during the first wetting of a previously sub-freezing
snowpack. When more flow paths are present, the energy exchange will be more
efficient. Additionally, the gradients with the surrounding snow will be
larger. We use N as a tuning parameter in the model related to the number
of flow paths per square metre.
Data and methods
Data
We simulate 16 subsequent snow seasons (2000–2015) for the Weissfluhjoch
(WFJ) measurement site, located at 2536 m altitude in the Eastern Swiss
Alps. For this site, a dataset of biweekly snow profiles made in close
vicinity (< 25 m) of the meteorological station used to drive the
SNOWPACK model in this study is available .
The snow profiles contain information about grain size and type, judged by
the observer using a magnifying glass, as well as snow density in sections of
typically 20–50 cm height and snow temperature. Melt–freeze crusts (i.e.
parts of the snowpack that have been wet and froze again), as well as ice
layers, are explicitly marked as such in the profiles. Ice lenses (i.e. noncontinuous ice layers) are not marked as an ice layer but are reported in a
separate remark. As subsequent snow profiles need to be made in undisturbed
snow, they also sample spatial variability in addition to the temporal
evolution. Furthermore, judging whether a specific layer is a crust or an ice
layer is also partly subjective. This is also indicated in the data:
sometimes the same layer is not identified similarly in subsequent snow
profiles, although this may also indicate spatial variability. To account for
spatial variability at the measurement site, we select the highest modelled
dry snow density within a range of 20 cm above or below an observed ice
layer, when comparing simulated and observed ice layers.
For validating the snowpack runoff simulated by the model, we use the snow
lysimeter data from a 5 m2 lysimeter, as described in
. In that paper, it was discussed that a discrepancy between
measured and modelled runoff is particularly present at the beginning of the
melt season and involves the first ca. 5 % of seasonal snowpack
runoff. Here, we consider the measured snowpack runoff for the period 1 March
to 31 May only, and we particularly focus on the first 20 mm w.e. (water equivalent) runoff from
the snowpack. This period corresponds to the onset of snowpack runoff, while
preventing the statistics from being dominated by the main melt period. We
additionally exclude lysimeter data from snow season 2000 and 2005 from the
analysis due to suspected problems with the lysimeter in these seasons
.
Interpolated results of the sensitivity study for
the parameters N and Θth for the probability of
detection (POD) when modelled dry snow density exceeds
600 kg m-3 (a), 700 kg m-3 (b), or
800 kg m-3 (c) within 20 cm of the observed ice layer. For
runoff, the r2 for daily sums of runoff (d), the RMSE error for
daily sums of runoff (e) and the number of days difference between
modelled and measured passage of 20 mm w.e. since 1 March of each snow
season (f) are shown. The jump in colour scale from blue to red
in (d) and (e) marks the score achieved with matrix flow
only.
Dry snow density without considering preferential
flow (a) and with preferential flow using high-resolution
simulations (b), validation with field observations (c) and
liquid water content in the matrix and preferential flow domain for the
simulation with preferential flow (d), for snow season 2012.
In (c), modelled layers are shown when they are either a melt–freeze
crust or have a dry snow density exceeding 500 kg m-3. For
visibility, values of LWC in preferential flow below 0.1 % are ignored
in (d).
LWC in matrix domain (a), LWC in
preferential flow domain (b), snow temperature (c), snow
density (d), grain radius (e) and grain shape (f),
depicting a detail of Fig. . Only the upper part of the
snowpack is shown for the period 27 February to 15 March. In (c),
snow at melting temperature is coloured black to highlight wet parts;
in (e), grain radii smaller than 0.16 mm are coloured black,
denoting the smallest grain size class used in ; and
in (f), ice formations are defined as modelled dry snow density
exceeding 700 kg m-3.
Methods
The simulation set-up of the SNOWPACK model for WFJ is equal to the
snow-height-driven simulations in , in which new snowfall
amounts are determined from increases in measured snow height. This ensures a
simulation that closely follows the measured snow height, which will enable a
correct comparison of simulated and observed snow profiles. Ice layers
observed in the field can range from a few millimetres to a few centimetres and up to 1 m in
firn on the Greenland Ice Sheet . To reduce
computational costs, the SNOWPACK model applies an algorithm to merge
elements when they exhibit similar properties. In default setting, this
procedure typically maintains the layer spacing around 1.5 to 3 cm, except
for certain special cases, like buried surface hoar or ice layers inside the
snowpack, which should be maintained irrespective of their thickness. This
means that in default set-up, with a typical layer spacing of 1.6 cm, the
formation of ice layers is coupled to relatively thick layers compared to ice
layers found in natural alpine snowpacks. Forming thinner ice layers requires
less water and energy to refreeze. We therefore performed high-resolution
simulations where we lower the threshold above which no merge is allowed from
1.5 to 0.25 cm. Further, for the high-resolution simulations, we
initialise new snow layers during snowfall in steps of 0.5 cm, instead of
the default value of 2 cm. This led to a typical layer spacing of 0.45 cm.
Results presented here are with the high-resolution simulations, although we
discuss the performance of the default resolution as well. Simulations with
matrix flow only took on average 2.3 min per simulated year to complete on
a typical desktop PC, using the default SNOWPACK settings. The dual domain
approach, which requires solving the Richards equation twice, increased the
computation time to 8.0 min yr-1. The high-resolution simulations,
which we show here, took 71 min yr-1 to complete.
Densities of ice layers in the field can vary over a wide range. For example,
report a range from 630 to 950 kg m-3, which makes
it ambiguous to determine above which threshold of modelled dry snow density
a layer should be considered an ice layer. In the default set-up, a layer with
a dry snow density exceeding 700 kg m-3 is considered an ice layer by
the SNOWPACK model. However, we apply different thresholds here to verify the
sensitivity of the choice of threshold on the results. Indeed, it may be that
simulated layers cannot reach the density of observed thin ice layers due
to their larger vertical extent in the model.
Results
Parameter estimation
In the preferential flow formulation we propose, two tuning parameters are
left: the threshold in saturation of the preferential flow domain
(Θth), above which water will flow back to the matrix
part, and a parameter related to the number of flow paths per square metre (N). To
determine an optimal set of parameters, a sensitivity study was carried out.
For Θth, values from 0.02 to 0.16 in steps of 0.02 were
used and for N, values 0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 3.0, 4.0, 5.0 were
used.
Figure shows the probability of detection (POD; i.e. the
ratio of observed ice layers that are reproduced by the model over the total
number of observed ice layers) for different thresholds that define a
modelled ice layer, as a function of both tuning parameters. When ice layers
are defined by higher densities, the POD decreases. Highest POD is achieved
for no or minor freezing (i.e. small values for N), and a saturation in
the preferential flow path around 0.1. The non-linear relationship in
Fig. arises from the delicate balance of refreezing water
that is not able to percolate deeper and the amount of ponding possible at
microstructural transitions, required for water to move back to the matrix
domain in order to freeze as an ice layer or lens. For snowpack runoff,
highest scores in terms of r2, RMSE or the arrival date of the first
20 mm w.e. are generally achieved with refreezing and low thresholds in
saturation of the preferential flow domain (see Fig. ). Both
slow down the progression of preferential flow water. It seems difficult to
find a set of parameters that will maximise both the reproduction of ice
layers, as well as snowpack runoff simulations. Nevertheless, even with
optimal settings for the formation of ice layers, the early stage of snowpack
runoff (i.e. the passage time of the first 20 mm w.e. of runoff) is better
reproduced than without considering preferential flow.
After executing all 80 SNOWPACK simulations for the sensitivity study, ranks
were determined for the POD of ice layers, using 700 kg m-3 as density
threshold for ice layers, and the r2 value for hourly snowpack runoff. The
combination of both parameters that provides the lowest sum of the ranks for
ice layer detection and snowpack runoff was considered the optimal
combination of coefficients. This procedure gave Θth=0.1
and N=0 and Θth=0.08 and N=0 for the normal- and high-resolution simulations, respectively, as optimal combination of tuning
parameters and this set of parameters will be used for the results.
Interestingly, it implies that for ice layer formation, refreezing of
preferential flow should be ignored (i.e. N=0).
Example snow season
Figure illustrates the difference in simulated snow density
between a simulation with only the Richards equation and the Richards equation
including preferential flow at high resolution for snow season 2012. Similar
figures for the other simulated snow seasons are shown in the online
Supplement. The overall density distribution is similar in both simulations,
but only with preferential flow are ice layers formed. The location is in
good agreement with observations of ice layers and crusts observed in the
snow profiles in the field. Figure shows detailed
simulation output for the period in the beginning of March 2012 and the upper
part of the snowpack only. The distribution of liquid water is showing that
the preferential flow (Fig. b) is percolating ahead of
the matrix flow (Fig. a). This partly is due to the
absence of phase changes for water in preferential flow, but also due to the
lower area, and thereby lower value for θs, such that
hydraulic conductivity increases faster with increasing LWC. In contrast to
matrix flow, preferential flow reaches areas where the snowpack is still
below freezing (Fig. c).
Box and whisker plot showing the
distribution of snow density from observations (obs), simulations with the
Richards equation (REQ) and simulations with the dual domain approach to
describe preferential flow (REQ+PF). On the left, simulated snow density
represents an aggregated snow density over multiple model layers to match the
measured layer thickness. On the right, snow density of an individual model
layer is shown. Boxes represent interquartile ranges (25th to 75th
percentiles), and thick horizontal bars in each box denote the median (50th
percentile), the value of which is shown directly below the bar. Whiskers (vertical lines
and thin horizontal bars) represent the highest and lowest value within 1.5
times the interquartile range above the upper or below the lower quartile,
respectively. Notches are drawn at ±1.58 times the interquartile range
divided by the square root of the number of data points. Outliers are shown
as individual dots.
Ponding at microstructural interfaces is occurring in both the matrix and the
preferential flow domain. In the example, jumps in snow density
(Fig. d) and grain radius
(Fig. e) around 165 and 210 cm in the snowpack mark
the layers where water accumulates, refreezes and forms ice layers
(Fig. d, f). Solving the Richards equation twice (for both
domains) appears to be able to identify those layers. Refreezing locally
increases the snow temperature to melting temperature
(Fig. c). Initially, the model identifies refreeze
inside the snow layer and marks the layer as a melt–freeze crust. Once dry
snow density exceeds 700 kg m-3, the layer is marked as an ice layer
(Fig. f). Note that Fig. e
shows that most of the snowpack consists of grain sizes for which
preferential flow was observed in the experiments by .
The smallest grain size class from those experiments, for which no
preferential flow was observed, is only found in the new snow layers during
snowfall (black coloured areas), after which metamorphism rapidly increases
grain size to regimes for which preferential flow was observed. This
justifies the application of Eq. () for the full range
of grain size in the model.
In addition to preferential flow, ice layers can also form by surface
processes. For example, rainfall in November 2003 in a sub-freezing snow
cover formed an ice layer at the surface and this ice layer was subsequently
observed during the rest of the 2004 snow season (see Fig. S5 in the
Supplement). This layer is not reproduced in the SNOWPACK model. Firstly, the
model did not recognize the precipitation as rainfall due to the low air
temperature during the event. Second, even when the model was forced to
interpret the precipitation as rainfall, the ice layer did not form at the
surface. The model solves for the heat and water flow sequentially in a
15 min. time step, whereas the formation of an ice layer during rainfall is
occurring on shorter timescales. Furthermore, we hypothesise that rain
droplets probably freeze directly upon contact with the snow surface,
creating an ice layer locally at the surface, whereas the SNOWPACK model
considers the rainfall as an incoming flux in the top layer. When the
available energy for freezing is not sufficient to freeze the full depth of
the top layer in the model, an ice layer is not formed. In reality, the
surface ice layer is possibly even hindering water entry to deeper layers,
which may thicken the surface ice layer. This particular ice layer in 2004
has been excluded in further analysis.
Density profiles
Figure shows the observed snow density distribution
in all snow profiles from snow seasons 2000–2015, typically representing
vertical sections of 20–50 cm and sometimes smaller sections. The
distribution of snow density for these sections is reproduced well by the
simulations, although the spread in simulated snow density is lower than the
observed spread. All simulations provide very similar snow density
distributions. The r2 value between observed and simulated density in the
measurement sections is highest (r2 = 0.74) for the simulations with
the
Richards equation only, and in it is shown that the
temporal evolution and vertical distribution of snow density is in good
agreement with measured snow density. With preferential flow, the r2
value reduces to 0.71. This reduction in model performance when using
preferential flow is also confirmed when using Willmott's index of agreement
, which was determined to be
0.84 and 0.83 for simulations with the Richards equation only and simulations
with preferential flow, respectively. Nevertheless, the simulations with
preferential flow maintain the overall snowpack density profile
generally well and the reduction in r2 value may be attributed to the fact
that calibration of snow settling functions was not performed by considering the
preferential flow model. Another reason may be that with preferential flow,
more water is moved downward and less water can refreeze in matrix flow in
the upper snow layers. It may be argued that an underestimation of snow
settling can be compensated for by an overestimation of refreezing water. In
any case, the simulations with preferential flow stand out when looking at
the highest snow density simulated in a layer within ±20 cm of an
observed ice layer. In this case, much higher snow densities are found in
individual layers under consideration of preferential flow.
As the manual snow density measurements in the field represent much larger
vertical sections (20–30 cm), these measurements cannot be used to verify
the much higher-resolution (1–2 cm or less) simulated densities on that
scale. Time series using other measurement techniques, for example snow
micro-penetrometry or measuring volume and mass of
excavated ice layers , may assist in a more in-depth model
verification in the future.
Contingency statistics as a function of
threshold in dry snow density that defines an ice layer in the simulations,
for both normal and high-resolution simulations including preferential flow
(REQ + PF) and normal-resolution simulations using the Richards equation only
(REQ).
Figure shows the POD for different dry snow density
thresholds that define an ice layer in the simulations. The POD decreases
with increasing threshold from 0.44 for 400 and 500 kg m-3 to 0.10
for a threshold of 800 kg m-3 for the high-resolution simulations.
When comparing with field observations, it is important to note that it is
not clear which density should be assigned to a layer that an observer would
classify as an ice layer. The probability of null detection, which in this case
is defined as the percentage of simulated profiles correctly simulating the
absence of ice layers in the full profile, is above 50 % for an ice
layer definition threshold of 600 kg m-3 in high-resolution
simulations. In normal-resolution simulations, the probability of null
detection is higher. The bias, which is the ratio of the number of simulated
ice layers over the number of observed ice layers, is generally below 1. This
indicates a slight underestimation of the frequency of ice layers in both
high- and normal-resolution simulations. It shows that our approach is neither
largely overestimating nor underestimating the presence of ice layers inside
the snowpack. The false alarm rate indicates that around half of the ice
layers that are simulated do not find a correspondence in the observed snow
pits. The results illustrate the general difficulty of observing ice layers
with often small vertical extent in the field and reproducing those ice
layers in the model due to a delicate interaction between water flow and the
ice matrix. However, the results also indicate that the model is able to
capture a significant proportion of ice layers that formed in natural
snowpacks, while maintaining the overall snowpack structure well. In
contrast, simulations with the Richards equation only generally do not reproduce
any layer with a dry snow density exceeding 600 kg m-3. The prediction
bias is correspondingly below 0.20 even for low thresholds of
400 kg m-3 for defining an ice layer. This indicates that the
failure of reproducing ice layers in those type of simulations cannot be
resolved by choosing low thresholds and that preferential flow seems to be a
crucial process to simulate the formation of ice layers.
r2 for both hourly and daily snowpack
runoff over the period 1 March to 31 May for each snow season.
Difference (in days) between modelled
and measured first 10 and 20 mm w.e. cumulative seasonal snowpack runoff (negative
values denote modelled runoff is earlier than measured runoff).
Snowpack runoff
In addition to ice layer formation, preferential flow is also assumed to impact snowpack runoff.
Interestingly,
Fig. shows that, for the melt period, there is no
consistent difference between r2 values for daily and hourly snowpack
runoff whether or not preferential flow is considered. On average both
simulations have equal r2 values of 0.81 and 0.90 for hourly and daily
snowpack runoff, respectively. However, as already noted in
, the effect of neglecting preferential flow on seasonal
timescales may be very limited. In that study, particularly the first
arrival of meltwater was noticeably underestimated when only considering
matrix flow with the Richards equation. As illustrated in
Fig. , the arrival time of the first 20 mm w.e.
in the melt season is much better reproduced by the dual domain approach. The
time difference between the arrival date of the first 20 mm w.e. changes
from 7.7 days too late for the Richards equation model to 2.9 days too early
in the dual domain approach. Generally, the average time difference between
modelled and measured first 10 mm w.e. cumulative snowpack runoff is even
more negative than the average time difference for 20 mm w.e. cumulative
runoff. This suggests that particularly earliest season snowpack runoff from
preferential flow is overestimated in the simulations. The standard deviation
of the time difference for 20 mm w.e. is slightly smaller for the
preferential flow formulation than for the matrix flow only. The fact that
the standard deviation is smaller indicates that yearly variability between
observed and simulated runoff is smaller and that the model is apparently
able to better explain yearly variability. In
additional analysis of the role of preferential flow in producing snowpack
runoff during rain-on-snow events shows that for these events, snowpack
runoff is better reproduced using the dual domain approach.
Although considering preferential flow improved the snowpack runoff
simulation, year-to-year variability in model performance is still large. The
difference between simulations with or without preferential flow are often
smaller than the year-to-year variability. For example, in melt season 2001
and 2010, r2 values are low in both simulations and the difference between
the simulations is smaller than the difference in r2 values with other
years. Explanatory factors for the year-to-year variability in model
performance for reproducing snowpack runoff were not found. Snowpack
characterising statistics, for example the observed number of ice layers or
observed number of jumps in grain size and hardness, did not correlate
significantly with r2 for snowpack runoff or the arrival date. This is
probably due to a combination of errors in meteorological forcing conditions,
observer bias in the biweekly snow profiles and the limited
representativeness of the snow lysimeter. Its surface area of 5 m2 may be
considered too small to capture a representative area for snowpack runoff,
such that randomness in the exact location where preferential flow paths form
may influence the measurements . Separating the
individual errors appears to be difficult.
Discussion
In the implementation of the dual domain approach, we attempted to stay close
to a physics-based process description. Laboratory experiments and
multidimensional snowpack models have provided crucial insights in the
preferential flow and water ponding processes. However, the number of
quantitative experimental studies is still limited and many aspects may be
refined in further studies. The model uses four criteria to specify the dual
domain approach: (1) the area involved in preferential flow, (2) a condition
to move water from matrix flow to preferential flow, (3) a condition to move
water from preferential flow to matrix flow and (4) a condition describing the
refreezing process of preferential flow. Two calibrating coefficients,
related to criterion (3) and (4), were used to optimise the simulations.
The area involved in preferential flow (condition 1) is currently
parametrised with grain size only. Given observations from soil physics
(e.g. ), a dependence on the water influx rate is to be
expected. Currently, laboratory settings, or field experiments with rainfall
generators, have generally large water input rates of typically
20 mm h-1 or more . It turns out to be difficult to have controlled,
constant and spatially well-distributed water input rates typically observed
in nature (rainfall and melt rates of 1–5 mm h-1). The absence of
studies at low water input rates makes the general validity of condition 1 we
implemented uncertain. Furthermore, preferential flow was not observed for
the finest grain size in , which, as shown by the black
coloured areas in Fig. e, exists only for short
periods of time in new snow. Although regimes with stable flow have been
identified for soil , using larger snow samples to
investigate preferential flow in snow could exclude the possibility that
finger width exceeds the snow sample size for the finest grain size class.
We consider condition 2 to be a relatively well-founded approach, as the role
of water entry suction in forming preferential flow was clearly identified in
laboratory experiments and turned out to be crucial in
forming preferential flow in multidimensional models in agreement with
laboratory experiments . However, also here, the exact
parametrisation of water entry suction may be different for lower water
influx rates.
Condition 3 may be one of the most uncertain ones. Understanding the LWC
distribution in a preferential flow path cannot be achieved by the Richards
equation . The other issue is that infiltration in an
initially dry porous medium is not accurately described by the Richards equation.
We consider the assumption we made here that water will move from
preferential flow to matrix flow based on the exceedance of a threshold in
saturation one of the least supported by experimental results.
The refreezing of preferential flow (condition 4) is mainly limited by
knowledge about the number of preferential flow paths that are actively
transporting water, which in itself is dependent on snowpack conditions.
Laboratory experiments at low input rates and with initially sub-freezing
snow, using detailed temperature measurements and dye tracer to follow the
wetting, may help here to develop a better understanding of the heat exchange
processes between preferential flow paths and the surrounding snow matrix, as a
function of preferential flow area and the number density of active
preferential flow paths. Results from laboratory experiments and
multidimensional snowpack models may thereby allow to quantify the amount of
refreezing of percolating meltwater in flow fingers. This knowledge is of
crucial importance, as it determines the efficiency of preferential flow to provide heat to deeper layers of the snowpack. Furthermore, refreezing of preferential
flow probably slows down the downward propagation of the fingers.
The term preferential flow can be interpreted ambiguously. Two phenomena are
known to cause deviations from a matrix infiltration pattern: flow fingering
and macropore flow. Here, we consider flow fingering purely as the result of
instabilities of the wetting front, which can occur in porous media with a
uniform pore space distribution. Generally the prerequisite for this effect
(coarse grains and low infiltration rates) is fulfilled for snow. However,
once flow fingering is occurring in snow, microstructural changes of the snow
grains in preferential flow paths will change the pore space distribution to
a bimodal or multimodal one. This has its equivalent in soils in, for
example, worm holes, root channels and cracks. This effect is not considered
in the dual domain approach we propose, although it may have a profound
impact on the efficiency of preferential flow paths. Modifications to the
parameters of the preferential flow domain can be imagined to better
represent a multimodal pore space distribution. In contrast, snow
microstructure inside and around preferential flow paths may not always
consist of an ice matrix where the Richards equation would be a good description
of water flow. However, a dual domain approach does not require both domains
to be solved with the Richards equation, and another description of water flow in
the preferential flow domain may be more appropriate.
Our simulations have a relatively low reproduction success of observed ice
layers. The sensitivity study has revealed that one factor is the delicate
balance between refreezing and further percolation. This is expected to be
particularly delicate in alpine snowpacks, where the cold content is low and
the ice layers are often thin. For cold regions, for example the Greenland
Ice Sheet, the abundance of ice layers observed in ice cores may be easier to
reproduce in simulations, as microstructural transitions formed by summer
melt–freeze crusts below cold new snow from the accumulation period are more
easy to capture in simulations. In contrast to alpine snowpacks, where the
ground heat flux often maintains melting conditions at the snowpack base,
firn temperatures are generally well below freezing and create a large
refreezing capacity.
Another factor contributing to the low POD is the small
vertical and sometimes small horizontal scale on which ice layer formation
happens in alpine snowpacks, which is difficult to capture in simulations. A
correct simulation of the snow microstructure is thereby a prerequisite for
simulating ice layer formation, although it is difficult to achieve. As an
example, buried surface hoar may provide a marked microstructural transition
on which liquid water may pond and build ice layers. Whether or not the
simulation is able to simulate correctly the burial of surface hoar
contributes to the failure or success in reproducing ice layers. Such a
failure or success will remain throughout the rest of the snow season.