the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# A local model of snow–firn dynamics and application to the Colle Gnifetti site

### Carlo De Michele

The regulating role of glaciers in catchment run-off is of
fundamental importance in sustaining people living in low-lying
areas. The reduction in glacierized areas under the effect of
climate change disrupts the distribution and amount of run-off,
threatening water supply, agriculture and hydropower. The prediction
of these changes requires models that integrate hydrological,
nivological and glaciological processes. In this work we propose a
local model that combines the nivological and glaciological
scales. The model describes the formation and evolution of the
snowpack and the firn below it, under the influence of temperature,
wind speed and precipitation. The model has been implemented in two
versions: (1) a multi-layer one that considers separately each firn
layer and (2) a single-layer one that models firn and underlying
glacier ice as a single layer. The model was applied at the site of
Colle Gnifetti (Monte Rosa massif, 4400–4550 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$). We
obtained an average reduction in annual snow accumulation due to
wind erosion of 2×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ to be
compared with a mean annual precipitation of about
2.7×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$. The conserved
accumulation is made up mainly of snow deposited between April and
September, when temperatures above the melting point are also observed.
End-of-year snow density, instead, increased an average of
65 kg m^{−3} when the contribution of wind to snow compaction
was added. Observations show a high spatial and interannual
variability in the characteristics of snow and firn at the site and
a correlation of net balance with radiation and the number of melt
layers. The computation of snowmelt in the model as a sole
function of air temperature may therefore be one of the reasons for
the observed mismatch between model and observations.

Glacier ice covers almost 16×10^{6} km^{2} of the Earth's
surface, of which it is estimated that only 3 % is retained by
the mountains outside the polar regions (Benn and Evans, 2010). Despite
this small percentage, the amount of water stored in mountain glaciers
plays a key role in sustaining people living in low-lying areas
(Adhikary, 1993), influencing run-off on a wide range of temporal and
spatial scales (Jansson et al., 2003; Huss et al., 2010). Storing water
coming from precipitation in winter and delaying the time in which it
reaches the river network, mountain glaciers sustain streamflow in hotter and drier
periods when precipitation is lacking and when it is most needed for
agriculture and as drinking water (Fountain and Tangborn, 1985; Hagg et al., 2007).

Jost et al. (2012) studied a Canadian river basin, covered for only 5 % by glaciers, and they found that ice melt contributes up to 25 % to streamflow in August, up to 35 % to streamflow in September, and between 3 % and 9 % to total streamflow.

In high mountain river basins of the northern Tian Shan (central Asia), with areas of glaciation higher than 30 %–40 %, glacier melt contribution is 18 %–28 % of annual run-off but it can increase to 40 %–70 % during summer (Aizen et al., 1996).

The reduction in glacier volume observed over the past 150 years (Vaughan et al., 2013; Hock et al., 2019) will result in a change in the present distribution and amount of water storage and release, with implications for all aspects of watershed management (Hock et al., 2005) and with consequently high economic impacts (Huss et al., 2010). The prediction of these changes is therefore fundamental in order to assess and reduce their impacts, optimizing consequently the management of water resources. To accomplish this task, models that integrate hydrological, nivological and glaciological components and that consider a variable glacier extension and the transient response of glaciers to climate change are required (Luo et al., 2013).

Despite their importance, fully integrated glacio-hydrological catchment models are not common in the literature (Wortmann et al., 2019). Some examples of glacio-hydrological models are provided by the works of Huss et al. (2010), Naz et al. (2014), Seibert et al. (2018) and Wortmann et al. (2019).

Wortmann et al. (2019) grouped the main problems of glacio-hydrological models into two categories: integration and scale. With integration problems they refer to the simplified or absent description of the remaining catchment hydrology in models that describe glacier processes in detail. The decrease in the fraction of ice-covered areas requires a proper description of both components, even in basins that are currently highly glacierized. Another aspect is the integration of nivological and glaciological components: a joint simulation of glacier mass balance and snow accumulation and melt is required in order to avoid inconsistencies (Jost et al., 2012; Naz et al., 2014). The problems of scale arise from the different resolutions required by glacial, nivological and hydrological processes. Physically based models that consider all glacier processes (mass balance, subglacial drainage and ice flow dynamics) are often too computationally expensive to be used in a combined glacio-hydrological model that considers the entire catchment. In addition, they are characterized by a complexity higher than the one of many semi-distributed hydrological models. It is therefore necessary to develop glacier models with a degree of complexity similar to the one of hydrological models but that are still able to reproduce important processes (Seibert et al., 2018).

In the present work, we give our contribution proposing a local model that follows the transformation of snow into firn and glacier ice under the influence of meteorological variables (temperature, precipitation and wind speed). Existing firn densification models are, in general, forced by snow characteristics. In this sense, the presented model allows us to move the boundary of the firn densification models from surface accumulation and density to hourly meteorological series. When we do not assume stationarity in the climate, in fact, this is required to properly capture the effects of climate changes.

The core of the model was derived from mass balance, momentum balance and rheological equations, governing the evolution of snowpack and firn (depth and density of snow and firn, depth of water and refrozen meltwater and rain inside the snowpack). The calculations of the terms in the resulting equations were then approached looking at methods already used in the literature; for example, snowmelt mass flux was computed with a temperature-index approach and the run-off from the snowpack with a flow matrix approach. We present two versions of the model: (1) a version (multi-layer) that considers separately each firn layer and (2) a version (single-layer) that models firn and underlying glacier ice as a single layer. The latter consists of only six equations, and it is therefore more suitable for a possible application to a hydrological model. The former consists of four equations for the snowpack plus two equations for each firn layer. Providing a profile of density with depth, it captures better the influence of meteorological variables on snow and firn characteristics. Besides, it allows a better validation of the snow–firn model. The equations that describe the snowpack are derived from the work of De Michele et al. (2013) and later Avanzi et al. (2015), modified in order to take into account the contribution of wind erosion and the transformation of snow into firn. To model the firn component, both the densification model of Arnaud et al. (2000) and the one of Herron and Langway (1980) were implemented. In order to test the model, a high-altitude site, Colle Gnifetti, belonging to the Monte Rosa massif, was chosen.

The paper is organized as follows: we present the model in Sect. 2, illustrate the case study in Sect. 3, give the results in Sect. 4 and discuss them in Sect. 5. The conclusions are given in Sect. 6.

In this section, firstly the snowpack model, proposed by De Michele et al. (2013) and later modified by Avanzi et al. (2015) with the addition of the contribution of wind to snow transport, is illustrated and secondly the model with the integration of snow and firn processes is presented.

## 2.1 Snow model

The snowpack is modelled, according to De Michele et al. (2013) and
Avanzi et al. (2015), as a mixture of dry and wet constituents. The
solid deformable skeleton that consists of both snow grains and
pores has a total volume *V*_{S}, unit area, height *h*_{S}, mass
*M*_{S} and density *ρ*_{S}. The liquid water inside the pores has
a volume *V*_{W}, unit area, height *h*_{W}, mass *M*_{W} and constant
density ${\mathit{\rho}}_{\mathrm{W}}=\mathrm{1000}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$. The refrozen meltwater and
rain inside the pores has a volume *V*_{MF} with unit area, height
*h*_{MF}, mass *M*_{MF} and constant density
${\mathit{\rho}}_{i}=\mathrm{917}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$. It is also possible to define the
bulk snow density *ρ*, snow water equivalent SWE and
volumetric liquid water content *θ*_{W} as
$\mathit{\rho}=({\mathit{\rho}}_{\mathrm{S}}{h}_{\mathrm{S}}+{\mathit{\rho}}_{\mathrm{W}}{h}_{\mathrm{W}}+{\mathit{\rho}}_{\mathrm{i}}{h}_{\mathrm{MF}})/h$,
$\mathrm{SWE}=\left(\mathit{\rho}h\right)/{\mathit{\rho}}_{\mathrm{W}}$ and ${\mathit{\theta}}_{\mathrm{W}}={h}_{\mathrm{W}}/h$, where *h* is the
height of the snowpack equal to
$h={h}_{\mathrm{S}}+\langle {h}_{\mathrm{MF}}+{h}_{\mathrm{W}}-\mathit{\varphi}{h}_{\mathrm{S}}\rangle $
(Avanzi et al., 2015) with *ϕ* being the porosity and
〈〉 denoting the Macaulay brackets that provide zero when
the argument is negative and its value when it is positive. The height
*h* and *h*_{S} always coincide except at the end of the snowpack
existence when the liquid part and the solid part
due to refreezing become predominant (i.e. ${h}_{\mathrm{MF}}+{h}_{\mathrm{W}}>\mathit{\varphi}{h}_{\mathrm{S}}$). In this
case, *h*>*h*_{S} because a layer of water and/or ice forms on top of
the deformable skeleton.

The model solves the mass balance for the dry and liquid mass of the
snowpack and the momentum balance and rheological equation for the
solid deformable skeleton, resulting in four ordinary differential
equations (ODEs) with the variables *h*_{S}, *h*_{W}, *h*_{MF} and
*ρ*_{S}. The mass fluxes considered are (1) solid precipitation
events, snowmelt and wind erosion for the dry-snow mass; (2) rain
events, snowmelt, melt–freeze inside the snowpack and run-off for the
liquid mass; and (3) melt–freeze for the mass of ice. The dry-snow
density is obtained considering (1) the compaction of snow due to
compaction not driven by wind, (2) the increase in the densification rate
due to drifting snow settlement and (3) densification due to the
addition of new mass. The following system is thus obtained (see
Appendix Sects. A1–A2 for the derivation of the
system and a detailed description of the terms in the equations):

In Eq. (1a), *ρ*_{NS} is the density of fresh snow
(kg m^{−3}), *s* is the solid precipitation rate
(m h^{−1}), *a* is a calibration parameter
($\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{}^{\circ}{\mathrm{C}}^{-\mathrm{1}}$), *T*_{A} and *T*_{τ} are the air
temperature and the threshold temperature for melting
(^{∘}C), *I* is equal to $\frac{{h}_{\mathrm{S}}}{{h}_{\mathrm{S}}+k}$ with
*k* equal to 0.01 m if *T*_{A}≥*T*_{τ} and zero otherwise
(Avanzi et al., 2015), and *Q* is the mass of snow eroded by wind
($\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$). In Eq. (1b), *r* is the liquid
precipitation rate (m h^{−1}), *e* is a calibration
parameter, *I*^{*} is equal to $\frac{{h}_{\mathrm{W}}}{{h}_{\mathrm{W}}+k}$ if
*T*_{A}<*T*_{τ} and to $\frac{{h}_{\mathrm{MF}}}{{h}_{\mathrm{MF}}+k}$ if
*T*_{A}>*T*_{τ} (Avanzi et al., 2015),
$\mathit{\alpha}=\mathrm{1.9692}\times {\mathrm{10}}^{\mathrm{9}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.25em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$
(DeWalle and Rango, 2008), and *K*_{W} is the intrinsic
permeability of water in snow (m^{2}). In Eq. (1d),
$c=\mathrm{0.10}\cdot \mathrm{3600}\phantom{\rule{0.125em}{0ex}}\mathrm{s}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$, *A*_{1}=0.0013 m^{−1},
${A}_{\mathrm{2}}=\mathrm{0.021}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$, $B=\mathrm{0.08}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$
(Liston et al., 2007), *U* is the wind speed contribution
(m s^{−1}), and *T*_{S} is the average snow temperature
(^{∘}C) obtained assuming thermal equilibrium between the
constituents and a bilinear profile of temperature through depth (see
De Michele et al., 2013, for further details).

With respect to the model by De Michele et al. (2013) and Avanzi et al. (2015), the version presented in this work includes the contribution of wind erosion to mass balance and effect of wind on densification. This is important when the model is applied to high-altitude sites: Haeberli and Alean (1985), in fact, suggested that a major part of the decrease in accumulation with altitude in the Alps that occurs above about 3500 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$ may be due to wind effects.

In analogy with solid transport, snow is mobilized only when wind
velocity at the surface exceeds a given threshold that depends on
physical properties of the surface snowpack (Li and Pomeroy, 1997). Once
transport begins, snow can travel in two main modes: saltation and
suspension (Déry and Taylor, 1996; Pomeroy et al., 1997). The total snow transport
*Q* is computed by the model with the following assumptions: (1) only snow erosion occurs, and no deposition of snow eroded in other
positions is present; (2) measured wind speed is always referred to a 10 m height – i.e. the height of the snow on the ground is neglected; and (3) wind cannot erode snow that has experienced a temperature greater than
0 ^{∘}C for the presence of ice crusts or wet layers,
following Vionnet et al. (2018). These last two assumptions allow
us to compute the series of total snow transport *Q* decoupled from the
snow model since knowledge of snow height is not required. In order to
implement the routine, we followed, with some modifications,
Lehning et al. (2000), where a model of snowdrift was added to the
one-dimensional snow model SNOWPACK (further details about the
implementation of the routine are reported in Appendix Sect. A1).

## 2.2 Model of snow–firn dynamics

We propose here two versions of the snow–firn model. The first version (single-layer) models firn and underlying glacier ice as a single layer (Fig. 1, left panel). The resulting output is an average density and the total column height. The second version (multi-layer) considers separately each firn layer (Fig. 1, right panel), and it allows us to distinguish between layers of firn and glacier ice. The discretized profile of density with depth can be obtained from this second implementation. The snow layer is, instead, treated as a single layer in both versions.

In the model we neglected the amount of water percolation inside firn. The presence of water inside firn varies greatly depending on the type of glacier. At high altitudes, where maximum temperatures are rarely positive, the effects of percolation due to melting are limited (Smiraglia et al., 2000); at the cold site of Colle Gnifetti, where the model was applied, percolation occurs only in the few centimetres below the surface and does not involve previous-year layers (Alean et al., 1983). If needed, the structure of the model allows us to easily implement additional processes.

In order to separate snow from firn, we refer to firn's original definition, according to which firn is snow that has survived one melt season (Cuffey and Paterson, 2010).

### 2.2.1 One-layer modelling of firn

The model is composed of two layers: the snowpack (see
Sect. 2.1) and the column of firn below it. The firn is
modelled as a single impermeable layer of volume *V*_{F}, unit area,
height *h*_{F}, mass *M*_{F} and density *ρ*_{F}
(Fig. 1, left panel).

The model consists of six ODEs: the four equations of the snow model and in addition the mass balance and momentum balance of firn. The mass variation in firn is obtained considering firn melt, the effects of precipitation on firn and the transformation of snow into firn at the end of each hydrological year. The firn densification rate is obtained considering densification due to overburden stress and densification due to addition of new mass. The resulting system is thus as follows (see Appendix A for the derivation of the system and a detailed description of the terms in the equations):

The last terms in Eqs. (2a)–(2c) move, at the
end of each melt season, the remaining snowpack (if present) in the
firn layer; *t*_{i} is the time instant at the end of hydrological
year *i* and *δ*(.) is the Dirac delta function. In Eq. (2d), *I*_{F} is equal to $\frac{{h}_{\mathrm{F}}}{{h}_{\mathrm{F}}+k}$, with
*k* specified above if *T*_{A}≥*T*_{τ} and equal to zero otherwise. In Eq. (2f), $\frac{\mathrm{d}{\mathit{\rho}}_{\mathrm{F}}}{\mathrm{d}t}{|}_{\mathrm{comp}}$ is the
densification of firn due to compaction (see Sect. 2.4). Equations (2a)–(2f) are impulsive differential equations
(see e.g. Bainov and Simeonov, 1993, for mathematical details). This type of
differential equation involving the impulse effect is used to describe
the evolution of many physical phenomena that have a sudden change
in their states such as mechanical systems with impact; biological
systems such as heartbeats and blood flows; and population dynamics.

## 2.3 Multi-layer modelling of firn

Firn is modelled as a multi-layer column where each layer *j* has
volume *V*_{Fj}, unit area, height *h*_{Fj}, mass *M*_{Fj} and
density *ρ*_{Fj}.

The equations of the model change as follows:

where *j* goes from 2 to the total number of firn layers. Firn
layers that reach the ice density or whose height goes to zero are
removed from the model.

## 2.4 Firn densification

The densification of firn due to compaction is usually subdivided into
three stages: (1) a first stage dominated by the settling of grains
that allows us to reach densities of up to about 550 kg m^{−3}, (2) a
second stage dominated by sintering that extends up to the close-off
density (i.e. the density at which pores become isolated) of about
830 kg m^{−3} and (3) a last stage that ends when ice density is
reached in which further densification is driven by the compression of
the bubbles of air (Cuffey and Paterson, 2010). This last stage is in turn
subdivided into two phases, depending on if the pores are cylindrical or
spherical. Different models of firn densification are available in the
literature (see Lundin et al., 2017, for a review). Here we
implemented the model of Arnaud et al. (2000) with some of the
modifications proposed by Bréant et al. (2017) (we will refer
to it with the abbreviation AR) and the model of Herron and Langway (1980) (we will
refer to it with the abbreviation HL). Other models could also be
implemented. Both HL and AR were developed for polar sites. The HL model
was derived using ice cores with a mean annual firn temperature
between −57 and −15 ^{∘}C and a mean annual accumulation
between 0.022×10^{3} and
0.5×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$, while the AR model was derived
from cores with a mean annual firn temperature between −57
and −19 ^{∘}C and a mean annual accumulation between
0.022×10^{3} and
1.1×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$. In the model of AR,
densification equations are based on grain sliding and creep
deformations, even though they maintain empirical parameters. The model
of HL consists of empirical equations tuned with ice cores, based on
the assumption that a proportionality is present between the variation
in density and the variation in stress due to new accumulation.
Besides, the model of AR represents stresses explicitly, while in HL
the load is parametrized through annual surface accumulation.

The model of HL was already applied to non-polar ice cores by Huss (2013), where the model was recalibrated in order to match depth–density profiles of temperate and polythermal firn. In the presented application, the parameters were not recalibrated, despite the fact that the study site is an alpine site. This was motivated by the low mean annual firn temperature (MAFT) and low surface accumulation observed at Colle Gnifetti that resemble conditions of some polar sites.

In AR all three stages of firn densification are modelled. Equations are as follows:

In the first stage (${D}_{\mathrm{D}}\le {\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i}\le {D}_{\mathrm{0}}$), *P* is the
overburden pressure (Pa) and
$\mathit{\gamma}={\mathit{\gamma}}^{\prime}\mathrm{exp}\left(-\frac{{Q}_{\mathrm{1}}}{{R}_{\mathrm{G}}({T}_{\mathrm{F}}+\mathrm{273.15})}\right)$ in
which *R*_{G} is the gas constant, *Q*_{1} an activation energy equal
to $\mathrm{48}\times {\mathrm{10}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\mathrm{J}\phantom{\rule{0.125em}{0ex}}{\mathrm{mol}}^{-\mathrm{1}}$, *T*_{F} the average
temperature of firn (^{∘}C), and *γ*^{′} a parameter
whose value is set in order to have a continuous densification rate
between the first and second stage (estimated in Sect. 4.2). *D*_{D} is
the relative surface density, and *D*_{0} is the relative density at
the transition between the first stage and the second stage. In the
second stage (${D}_{\mathrm{0}}<{\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i}\le {D}_{\mathrm{c}}$),
$A={A}_{\mathrm{0}}\mathrm{exp}\left(-\frac{{Q}_{\mathrm{2}}}{{R}_{\mathrm{G}}({T}_{\mathrm{F}}+\mathrm{273.15})}\right)$ with
${A}_{\mathrm{0}}=\mathrm{2.84}\times {\mathrm{10}}^{-\mathrm{11}}$ ${\mathrm{Pa}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$, *a*_{c} is the
average contact area, *Z* is the number of particle contacts (see
Appendix Sect. A3 for the expression of *a*_{c} and *Z*) and
*Q*_{2} is an activation energy. The value of *Q*_{2} was set to
60×10^{3} J mol^{−1}, as in the model of
Arnaud et al. (2000), since it is the typical activation energy
associated with self-diffusion of ice. However, at higher temperature
(i.e. higher than −10 ^{∘}C) a higher activation energy
may be required to best fit density profiles with firn densification
models (Cuffey and Paterson, 2010; Arthern et al., 2010; Jacka and Jun, 1994). A
discussion of the thermal variation in the creep parameter and the
impact of the different sintering mechanisms on it can be found in
Bréant et al. (2017). Lastly, in the third stage
(${\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i}>{D}_{\mathrm{c}}$), *P*_{b} is the pressure inside the bubbles
equal to
${P}_{\mathrm{b}}={P}_{\mathrm{c}}\frac{({\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i})(\mathrm{1}-{D}_{\mathrm{c}})}{{D}_{\mathrm{c}}\cdot (\mathrm{1}-{\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i})}$
with *D*_{c} and *P*_{c} the relative density and pressure at the
transition between the second and third stage. In the single-layer
version, the overburden pressure *P* was computed as the overburden
of the snowpack layer plus half of the firn layer. In the multi-layer
version, we computed the overburden for each layer of firn as the
overburden of the snowpack plus the overburden of all the firn layers
above plus the overburden of half the firn layer considered.

In HL only the first and second densification stages are modelled. The equations are as follows:

where ${k}_{\mathrm{0}}=\mathrm{11}\mathrm{exp}\left(-\frac{\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{160}}{{R}_{\mathrm{G}}({T}_{\mathrm{F}}+\mathrm{273.15})}\right)$,
${k}_{\mathrm{1}}=\mathrm{575}\mathrm{exp}\left(-\frac{\mathrm{21}\phantom{\rule{0.125em}{0ex}}\mathrm{400}}{{R}_{\mathrm{G}}({T}_{\mathrm{F}}+\mathrm{273.15})}\right)$ and
*ω* is the annual snow accumulation ($\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$).
In HL the transition density between the first and second stage is fixed
and equal to 550 kg m^{−3}. In order to run the model of HL in
a dynamic way, for each year we computed the annual accumulation
averaging the ones modelled between the year of deposition of the firn
layer and the year before the one considered, following
Stevens et al. (2020).

Steady-state firn densification models are not applied to the
superficial snow where the metamorphism is more complex and
significantly influenced by air temperature. The original model of
Arnaud et al. (2000), for example, was used only for depths higher
than 2 m. In this case, we applied them only to densities
higher than a density *ρ*_{D}, which represents the average snow
density. For firn densities lower than *ρ*_{D}, the densification
equation of snow was adopted although neglecting wind contribution. In
this way, the transition between the two equations is driven by
density rather than associated with the end of a water
year. This is important, for example, when consistent fresh snow
falls over the snowpack at the end of the hydrological year.

### 2.4.1 Temperature profile

The energetic description of the volume was simplified assuming the
constituents were in thermal equilibrium and assuming a bilinear profile of
temperature through depth. Temperature was assumed to vary linearly
from the surface temperature *T*_{0} to the MAFT at the depth *z*_{M}
at which seasonal variation in temperature is negligible. Below
*z*_{M}, temperature was kept constant and equal to MAFT. In cold
glaciers the value of MAFT is close to the mean annual air temperature
(MAAT) when meltwater percolation is limited (Suter et al., 2001)
while in temperate glaciers it is equal to the melting temperature
(Cuffey and Paterson, 2010). Surface temperature was fixed equal to *T*_{A}
if *T*_{A}<0 ^{∘}C and zero otherwise. Huss (2013) has already assumed a bilinear profile of temperature in order to
study temperate firn densification, fixing *z*_{M} to 5 m since
it is the typical penetration of winter air temperature. The
temperature profile was then used to compute the average snow and firn
temperatures that influence snow and firn densification.

## 2.5 Numerical model

The model was solved using the forward Euler method with a constant
step size, Δ*t*, of 1 h. To also compute the last terms in
Eqs. (1d), (2f) and (3g) when
*h*_{S}, *h*_{F} and ${h}_{{\mathrm{F}}_{\mathrm{1}}}$ are zero, these terms were
calculated, following De Michele et al. (2013), as
$\frac{{\mathit{\rho}}_{\mathrm{NS}}\left(t\right)-{\mathit{\rho}}_{\mathrm{S}}}{{h}_{\mathrm{S}}\left(t\right)+s\left(t\right)\mathrm{\Delta}t}s\left(t\right)$,
$\frac{\mathit{\rho}\left(t\right)-{\mathit{\rho}}_{\mathrm{F}}}{{h}_{\mathrm{F}}\left(t\right)+h\left(t\right)}\frac{h\left(t\right)}{\mathrm{\Delta}t}$ and
$\frac{\mathit{\rho}\left(t\right)-{\mathit{\rho}}_{{\mathrm{F}}_{\mathrm{1}}}}{{h}_{{\mathrm{F}}_{\mathrm{1}}}\left(t\right)+h\left(t\right)}\frac{h\left(t\right)}{\mathrm{\Delta}t}$.
Regarding the vertical discretization, the firn component of the
multi-layer version of the model was discretized modelling one layer
for each hydrological year.

In the following section we will present the study area (Sect. 3.1), the data collection and handling (Sect. 3.2–3.3), and finally the calibration and site-specific parameters (Sect. 3.4–3.5).

## 3.1 Study area

The site of Colle Gnifetti (CG) is part of the summit ranges of the
Monte Rosa massif, Swiss–Italian Alps. It is the uppermost part of the
accumulation area of the Grenzgletscher (Border Glacier), and it forms a saddle that lies
between the Signalkuppe (4554 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$) and Zumsteinspitze (4563 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$) at an altitude of 4400–4550 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$
(Lüthi and Funk, 2000) (Fig. 2). The glacier at Colle
Gnifetti has a thickness of between 60 and 120 m and a MAFT of
−14 ^{∘}C (Wagenbach et al., 2012). The regime is that of
a high-altitude site, i.e. nearly persistent sub-zero air temperature,
a high precipitation total and high wind speed
(Suter et al., 2001). A mean annual precipitation of
2.7×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ with an interannual
variability of $\mathrm{0.8}\times {\mathrm{10}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$
(Mariani et al., 2014) was estimated for the period 1961–1993 from a
core extracted at the upper Grenzgletscher (Eichler et al., 2000).

Even though the site is characterized by high precipitation totals,
accumulation in the saddle is considerably lower and highly variable
over the glacier surface due to wind erosion, with values ranging from
about 0.15×10^{3} to
1.2×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ depending on the wind
exposure (Alean et al., 1983; Lüthi and Funk, 2000; Licciulli et al., 2020).
Alean et al. (1983) measured the accumulation at CG between 17 August 1980 and 23 July 1982 with a network of 30 stakes. For the
period between 14 August 1981 and 23 July 1982 the mass balance was
negative at all the stakes due to wind erosion, while the net
accumulation of the hydrological year 1980–1981 varied between
$+\mathrm{0.04}\times {\mathrm{10}}^{\mathrm{3}}$ and
$+\mathrm{1.18}\times {\mathrm{10}}^{\mathrm{3}}$ $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ with the highest values on
south-facing slopes. This occurs because the enhanced melting and
refreezing cause the formation of wet layers and ice crusts and
because higher temperatures are associated with faster densification,
and both these aspects reduce the possibility of wind eroding
snow. This also results in the fact that almost all the snow that
survives the melt season comes from summer events
(Bohleber et al., 2018; Schöner et al., 2002).

## 3.2 Data collection

The stations whose data were used in this study are presented in
Fig. 3, and they are summarized in Table 1. Hourly data of air temperature and wind speed at
Capanna Regina Margherita (CM, Margherita Hut in English) were used as input for the model;
hourly data at Passo del Moro (PM, Monte Moro Pass in English) to reconstruct precipitation at CM;
and hourly and daily air temperature data at Macugnaga Pecetto (MP),
Passo del Moro, Bocchetta delle Pisse (BDP) and Ceppo Morelli (CPM) to
infill missing temperature data at Capanna Regina Margherita. Hourly
wind speed data at Valtournenche–Cime Bianche (CB) and Col du Grand St
Bernard (SB, Great St Bernard Pass in English) were used to infill missing wind speed data at CM.
Hourly data at the meteorological stations of Gressoney-Saint-Jean –
Weissmatten (GWm) and Gressoney-la-Trinité – Gabiet (GGm) and snow
water equivalent data (GWs and GGs) were used to calibrate and
validate the parameters *a* and *e* of the snow model. Snow water
equivalent is measured by the Aosta Valley Region during winter both at fixed
locations and at itinerant sites. For GGs, 4 years of measurements
was available with on average 24 data points for each winter. For GWs, 5 years was available with on average 8 data points for each winter.

The station of Capanna Regina Margherita, whose data were used to run the snow–firn model, was installed in 2002 by the Piedmont Region at the Regina Margherita Hut as part of a project that aimed to study the interaction between synoptic flow and orography. With its 4560 m of altitude, it can be considered the highest meteorological station in Europe, and its wind speed series can be considered representative of the synoptic conditions (Martorina et al., 2003). Due to its recent installation, the use of these data limits the length of the simulation and the number of cores with which our results can be compared. Nevertheless, we believe that, given the peculiar characteristics of the station, the use of these data may give added value to this study.

In Table 2 ice core data are reported (Fig. 2). Available data consist of some or all of the following information: depth in metres, depth in metres of water equivalent, density and dating. We recall that the first three variables are related, so one of them can be computed given the other two.

Gäggeler et al. (1983)Gäggeler et al. (1983)Sigl et al. (2018)Sigl et al. (2018)Ardenghi (2012)Licciulli et al. (2020)Licciulli et al. (2020)Licciulli et al. (2020)## 3.3 Data handling

The model requires as input a continuous series of air temperature, precipitation and wind speed.

Following the comparison presented by Henn et al. (2013), to fill missing hourly temperature data at Capanna Margherita, the MicroMet preprocessor (Liston and Elder, 2006) was adopted for gaps smaller than 24 h and a long-term lapse rate approach with five stations (CM, MP, CPM, PM, BDP) was adopted for longer gaps. MicroMet is a meteorological model that includes a data-fill procedure adopted here. The method distinguishes between three conditions: (1) for 1 h gaps the missing information is replaced with the average of the previous and next measurement; (2) for 2–24 h gaps each missing value is replaced with the average of the values recorded the next and previous day at the same hour; (3) for longer gaps an auto-regressive integrated moving average (ARIMA) model is used (Hyndman and Athanasopoulos, 2021). Here, for the third condition, the long-term lapse rate approach was used, as specified above. In the period 1 October 2002–13 August 2013, 0.37 % of hourly temperature data were missing. After the MicroMet procedure 0.23 % remained missing and were substituted with a long-term lapse rate approach.

Wind speed data measured at CM are characterized by repeated zero values that are not observed in nearby stations and that are probably due to the freezing of the anemometer. In the period 1 October 2002–13 August 2013 nearly 30 % of the wind speed data at CM were equal to zero, while 1.3 % were missing. By comparison, in the same period, there were 2 % zero values in SB series. These zero values were therefore considered missing. To fill missing wind speed data at CM, the MicroMet procedure was used for gaps smaller than 24 h. For gaps longer than 24 h, data were replaced using measurements at CB or, if wind speed data were also missing at CB, with data measured at SB. In both series, zero wind speed values recorded for more than 4 consecutive hours were set as missing. In order to take into account the different characteristics of the sites, we first computed for each of the three stations the mean and standard deviation for each hour of the year, and we removed this from the data. Missing data at CM were first replaced with the corresponding residual value measured at CB (or SB), and then the final value was obtained using the mean and standard deviation estimated at CM. Reconstructed negative wind speed values at CM were set to zero. Missing wind speed data at CM were set to zero if they were zero at CB (or SB).

The precipitation series at CM was reconstructed using hourly data
measured at PM. The station was chosen due to it being in the vicinity of CM and
its altitude of 2820 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}$. Using the formula proposed by
Alpert (1986) and considering a bell-shaped mountain, we
estimated for the Monte Rosa massif an altitude of maximum
precipitation of *z*_{m}=2547 m. The altitude of maximum
precipitation is away from the crest as is typical of large
mountains (Roe, 2005). We therefore expect to have similar
precipitation totals at CM and PM. The precipitation series measured
at PM needs to be integrated with snow depth data due to the under-catch of solid precipitation by the pluviometer or does not catch solid precipitation events
in winter. In order to reconstruct the total precipitation series, we
followed the routine presented by Avanzi et al. (2014). Solid
precipitation is obtained looking at the positive variations in snow
depth data, while rainfall is given by the difference, if positive,
between total precipitation and solid precipitation. Positive
variations in snow depth, however, may also be recorded when strong
temperature variations occur, thus introducing false events. Unlike
Avanzi et al. (2014), we approached this problem smoothing the snow
depth series with a moving average whose window size was calibrated
running the snow model at PM and looking for the best match between
simulations and observations. Even though PM has an altitude higher
than the estimated altitude of maximum precipitation, we obtained a
mean annual precipitation of about
2×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ for the period 2002–2019,
lower than the one estimated at CG by Mariani et al. (2014). We
suppose this may be due to wind erosion events; the procedure
implemented by Avanzi et al. (2014), in fact, may compensate for
snow depth variations due to wind erosion decreasing the estimated
solid precipitation. We therefore increased the resulting hourly solid
precipitation with a constant factor in order to match the observed
mean annual accumulation at CG of
2.7×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$. The total precipitation
series was then divided between solid and liquid precipitation using a
threshold of 1 ^{∘}C since this is the value generally
found in Europe (Jennings et al., 2018).

## 3.4 Calibration of model's parameters

The model requires the calibration of three parameters, namely *a*
and *e* in Eqs. (1a)–(1c) and *γ*^{′} in
Eq. (4).

The parameters *a* and *e* were calibrated running the snow model,
with the addition of the wind module, at Gressoney-Saint-Jean – Weissmatten, with an hourly time step
from 1 October 2015 to 30 September 2020. Input series were processed
as reported for PM in Sect. 3.3. The parameter *a*
governs the amount of snowmelt and consequently snow height and the
relative amount of snow and ice inside the snowpack, thus influencing
snow water equivalent and density. On the contrary, the parameter
*e* influences only the relative amount of snow and ice inside the
snowpack and does not contribute to snow height. The calibration
problem is therefore a multi-objective one, since we could optimize
the error in both snow height and density or SWE. We decided to
move from a multi-objective to a single-objective optimization
problem, aggregating the NSE (Nash–Sutcliffe efficiency) between
observed and simulated snow depth data and the NSE between observed
and simulated snow water equivalent data. We calculated the error
metrics considering together all the available years but computing the
measure only in the periods with snow depth higher than zero, and we
aggregated them, giving a weight of 0.7 to the first and 0.3 to the
second. In this way we took into account the higher uncertainty in SWE
data due to the shorter sample length and the non-coincidence between
the location of the meteorological station and the snow measurements.
The optimum parameters were then estimated for different moving-average windows, used to process solid precipitation input data, with
the use of a population-evolution-based algorithm, namely SCE-UA
(Shuffled Complex Evolution – University of Arizona)
(Duan et al., 1992, 1993). We thus obtained a pair of *a*
and *e* values for each window, and we selected the one maximizing the
objective function. The validation was performed applying the model
with the selected set of parameters at Gressoney-la-Trinité – Gabiet for the period 1 October 2017–30 September 2021.

The parameter *γ*^{′}, which governs the firn densification rate in AR,
was chosen in order to have a continuous densification rate between
the first and second stage of densification. For each of the available
ice cores, with the exception of CG11, we computed the parameter
*γ*^{′} running AR in a steady-state condition (Bader, 1954)
using the mean accumulation reported in Table 2. In
addition, the parameters of the firn densification model chosen may
need calibration if applied to sites significantly different from the
polar ones.

## 3.5 Site-specific parameters

In order to apply AR we use *D*_{0}=0.56 (Bréant et al., 2017),
${P}_{\mathrm{c}}=\mathrm{740}\times {\mathrm{10}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}\mathrm{Pa}$ (Lüthi and Funk, 2000) and *D*_{c}=0.9
since the precise value is not known at CG (Lüthi and Funk, 2000). Two
different firn temperatures, ${T}_{\mathrm{F}}=-\mathrm{14}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\mathrm{C}$ and
${T}_{\mathrm{F}}=-\mathrm{10}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\mathrm{C}$, that cover the observed ice temperatures
at CG (Lüthi and Funk, 2000) were tested together with different surface
densities, chosen looking at values already used in the literature at
CG. We selected three values: *ρ*_{D}=300 kg m^{−3},
*ρ*_{D}=360 kg m^{−3} and *ρ*_{D}=410 kg m^{−3},
the values already assumed by Licciulli et al. (2020) and
Lüthi and Funk (2000). The model of AR was run with a slight
modification. We used the first-stage densification equation up to a
relative density of 0.6, but we kept *D*_{0}=0.56 in the second-stage
densification equation. The latter, in fact, cannot be applied for
*D*=*D*_{0}, and it gives densification rates tending to infinity for
values tending to *D*_{0}. The other site-specific parameters of the
snow–firn model that require specification are *z*_{M}, set to
5 m (Haeberli and Funk, 1991), and the grain radius *R*, which
influences the threshold wind speed. It is defined as
$R=\mathrm{3}/\left({\mathit{\rho}}_{i}\mathrm{SSA}\right)$, where SSA is the specific surface area in
m^{2} kg^{−1}. SSA was computed adopting the parametrization
of Domine et al. (2007) for recent snow:
$\mathrm{SSA}=-\mathrm{16.051}\mathrm{ln}({\mathit{\rho}}_{\mathrm{S}}\times {\mathrm{10}}^{-\mathrm{3}})+\mathrm{7.01}$.

## 4.1 Parameters' estimation

We obtained a value of the parameters *a* and *e* of
2.94×10^{−4} m h^{−1} ^{∘}C^{−1} and 0.164,
respectively. The combined NSE in calibration is 0.82, with a NSE of
0.84 and 0.78 for snow depth and snow water equivalent data,
respectively. In validation, the combined NSE and the NSE values for snow
depth and snow water equivalent are 0.77, 0.84 and
0.61, respectively. We also computed the RMSE and mean bias error (MBE) in validation, which are equal
to 0.126×10^{3} and
0.0116×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ for snow water equivalent
and 0.26 and 0.0672 m for snow depth.
Avanzi et al. (2014) estimated the parameter *a* for a selection
of 40 sites with altitudes between 91 and 3389 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$
within the SNOTEL (Snow Telemetry) network, a network of automated
stations located in mountain basins of the western USA and operated by the
Natural Resources Conservation Service (NRCS). They obtained median
values of *a* of between 1×10^{−4} and
6×10^{−4} m h^{−1} ^{∘}C^{−1}. Regarding the
parameter *e*, values of 0.2 and 0.25 were estimated by
Avanzi et al. (2015) for two sites in Japan.

## 4.2 Steady-state firn densification

The depth–density profiles obtained using the model of AR and HL in a
steady-state condition are reported in Fig. 4 for a
surface density *ρ*_{D}=360 kg m^{−3}. Both HL and AR have
very good performances when applied to the KCC ice core. The worst
performances occur for CG03 with an underestimation of the densification
rate for all depths. For the remaining ice cores the models of AR and
HL have a good fit up to depths of about 20–30 m, but they
underestimate the densification rate below it. The profiles show in
general a better performance of HL. We recall that the model of HL was
derived also considering cores with MAFT and accumulation close to the
ones of CG, while AR was optimized for cores with lower MAFT.

## 4.3 Snow accumulation

The annual accumulation obtained from the snow–firn model is reported
in Fig. 5, along with the values retrieved from the three
available ice cores, the average value of the observations and its
95 % confidence interval. The RMSE between the model and the average
of the observations is equal to
0.22×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$, and the modelled and
observed average annual accumulation and standard deviation are
reported in Table 3.

In order to better understand the characteristics of the accumulation at CG, the monthly box plots of snow transport; solid precipitation; number of hours with ${T}_{\mathrm{A}}>\mathrm{0}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\mathrm{C}$, which in the model correspond to hours with melting; and monthly contribution to annual accumulation, computed for each month as 100 × (solid precipitation − eroded snow) $/$ solid precipitation, are provided in Fig. 6. Since snow is moved into firn at the end of September and wind is not allowed to erode firn, the fraction of conserved snow of September may be overestimated and the snow transport of October underestimated. We can see that annual accumulation is composed of snow deposited mainly between April and September, with June the month that on average contributes the most. The months in which solid precipitation is conserved are also the months in which temperature goes above the melting point; winter snow, instead, is completely removed.

## 4.4 Firn density

### 4.4.1 One-layer model version

The modelled firn density was compared with the density estimated
from KCC and CG15 ice cores. With the one-layer version, we obtain
one average value of firn density for each run of the model. We
therefore run the model multiple times, fixing the end year of the
simulation to the date of the core drilling and anticipating at each
run the starting date of 1 year. For each run, the corresponding
observed firn density was obtained averaging the density profile of
the ice core associated with the same range of years. The results
obtained implementing both AR and HL are reported in
Fig. 7. For KCC we fixed the MAFT to −14 ^{∘}C,
while for CG15 we fixed it to −10 ^{∘}C, looking for the best performance in
Fig. 4.

Regarding firn density, we have contrasting results depending on the core and the densification model adopted. Both model versions overestimate KCC density with a better performance when AR is implemented; on the contrary we obtained an underestimation of CG15 average density, with a better performance when HL is implemented. In all the combinations, however, we observed a reduction in the error when more firn layers are averaged.

Moving to firn depth, the model nearly always predicts higher depths, with more significant differences for the KCC ice core.

### 4.4.2 Multi-layer model version

The modelled density profile was compared with KCC and CG15 density
data, implementing in the model both AR and HL (Fig. 8)
and testing three different transition densities between snow and
firn. Profiles are reported as steps, where each step corresponds to
a firn layer. Focusing on the CG15 ice core, we modelled, with both
versions, lower densities in the first 4–5 m, with a layer
with a particularly low density not matched by the ice core data at
around 1–2 m from the surface. This marked decrease however is
reduced when a *ρ*_{D} of 410 kg m^{−3} is chosen. For
higher depths, the model with AR implemented underestimates CG15 density,
while with HL a better match of the profile is observed. The best
performance is obtained implementing HL and selecting
${\mathit{\rho}}_{\mathrm{D}}=\mathrm{410}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$, with a mismatch only in the first
metres.

Moving to the KCC ice core, the model with AR implemented results in an overestimation of density up to a depth of around 4 m and an underestimation below it. Implementing HL, the density is instead overestimated except for a layer at around 8 m of depth.

## 4.5 Comparison between multi- and single-layer model versions

In order to understand the approximation introduced modelling firn as
a single layer instead of a multi-layer column, we compare in Figs. 9–10 the average firn density
obtained with the single-layer version of the model or averaging the
density of each individual firn layer obtained with the multi-layer
version, weighted for their heights. The results for
${\mathit{\rho}}_{\mathrm{D}}=\mathrm{300}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$ are not reported since they were not
significantly different from the ones with
${\mathit{\rho}}_{\mathrm{D}}=\mathrm{360}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$. Setting
*ρ*_{D} to 360 kg m^{−3}, we have, implementing HL, a maximum
difference between the two average densities of 16.7 kg m^{−3},
obtained for the KCC ice core, and, implementing AR, a maximum difference of 7 kg m^{−3}, obtained
for the CG15 ice core. Higher differences are obtained moving to
${\mathit{\rho}}_{\mathrm{D}}=\mathrm{410}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$, with a maximum difference of
29 and 14 kg m^{−3} implementing HL and AR,
respectively, obtained for the KCC ice core. In all cases the multi-layer
version predicts higher average densities.

## 5.1 Steady-state firn densification

Figure 4 shows a variable performance of the firn
densification model depending on the ice core considered; with the
exception of KCC and CG15, which show a very good and a
very poor performance, respectively, for all the other cores we have a good fit up
to a density of about 600–700 kg m^{−3}. Bréant et al. (2017), who modified the original model of AR, also observed a variable agreement between the data and model, also for sites
with similar accumulation and temperature. They suggested that this
may be due to different flow regimes of the sites, since their 1D
model does not include this effect. Another consideration that emerges
from Fig. 4, also pointed out by
Bréant et al. (2017), is that the modelled profile results in
worse performances when the observed density profile does not show a
clear change in the densification rate near the critical density
*D*_{0}. The transition is, in fact, more evident for the KCC ice core, which is associated with the best fit. Finally,
Bréant et al. (2017) reported a tendency of the model to
overestimate the densification rate for lower densities and to
underestimate it for higher densities. This is coherent with the
results obtained, in which HL predicts lower densities before *D*_{0}
and higher densities after *D*_{0} if compared with AR. In order to
compare modelled and observed profiles in Fig. 4, it is
important to point out that the two models assume stationary
conditions; therefore they are not able to reproduce possible changes
in the glaciological characteristics. In some of the ice cores it is
possible to see a bend in the profile in correspondence to about
20–30 m. The reason could be a combination of ice flow and the
upstream effect, i.e. changes in snow accumulation upstream, and these
effects cannot be reproduced by a 1D model like the ones used.

## 5.2 Snow accumulation

Snow accumulation at CG is characterized by a high spatial variability
(Keck, 2001; Licciulli et al., 2020). The difference in net annual
accumulation of CG11 and CG15, which are about 50 m apart,
ranges from $+\mathrm{0.13}\times {\mathrm{10}}^{\mathrm{3}}$ to
$-\mathrm{0.266}\times {\mathrm{10}}^{\mathrm{3}}$ $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ in the period 2002–2012,
while the one of CG15 and KCC, which are about 120 m apart,
ranges between $+\mathrm{0.41}\times {\mathrm{10}}^{\mathrm{3}}$ and
$-\mathrm{0.15}\times {\mathrm{10}}^{\mathrm{3}}$ $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$. Given the high variability
in the accumulation rate, three ice cores may not be enough to fully
represent the site. In addition, ice core data are biased due to the
fact they are drilled preferentially in the north flank, where
accumulation is lower. While the modelled average annual accumulation
is in the range of the ones estimated for the north flank of CG
(Licciulli et al., 2020), the model is not able to reproduce the
observed spatial variability. Due to the lack of dependence on
topography in the presented model, we do not expect the model to
correctly follow one or the other core. The topography influences the
amount of solar radiation received, which in turn influences melting and
wind erosion. At the same time, the topography modifies wind speed, which in turn modifies topography itself. This results in a quasi-random spatial variation and a systematic temporal variation in
surface accumulation at a given location (Keck, 2001). Surface
snow temperature was set equal to air temperature, instead of solving
the full surface energy balance that would have required a higher
availability of data; surface temperatures, in fact, may also reach
0 ^{∘}C for air temperatures below 0 ^{∘}C,
mainly when calm conditions are present, or, on the contrary, melting
may not occur during positive air temperatures, particularly when wind
is present (Keck, 2001).

From the box plots in Fig. 6 we observe that the conserved snow is made up mainly of summer precipitation and that the conserved fraction of solid precipitation reflects the number of hours with greater-than-zero temperature rather than the seasonality of precipitation. The accumulation is, in fact, mainly governed by the wind erosion (Wagenbach et al., 1988) and the presence of wet layers or ice crusts, as well as a faster compaction when temperatures are higher, which protects snow from wind erosion. This is well known in ice core studies at CG, since it results in isotope records that are biased towards precipitation of the warm season (Schöner et al., 2002; Bohleber et al., 2013, 2018). To our knowledge, no models have been applied at CG that try to model the influence of wind speed and temperature on snow accumulation. The only confirmation of this behaviour we have, other than from ice core analysis, comes from temporary measurements of the snow height in nearby sites or observations at CG. For example, at Seserjoch, 4300 $\mathrm{m}\phantom{\rule{0.125em}{0ex}}\mathrm{a}.\mathrm{s}.\mathrm{l}.$, the snow height was measured between 1998 and 2000 by Suter et al. (2001), and a main accumulation from about April to November, with practically no accumulation in high winter, was observed. Assessing the link between temperature, wind speed and snow accumulation may be important under scenarios of climate warming for glacial sites, like CG, whose behaviour is strongly regulated by wind activity. At CG, our results suggest that higher temperatures may lead to a counterintuitive response in the snow mass balance, as already conjectured by Alean et al. (1983). In fact, an increase in melting due to a warmer climate may be compensated for, or exceeded, by the consequent reduction in snow erosion, therefore resulting in a net snow accumulation increase.

## 5.3 Firn density

### 5.3.1 One-layer model version

The modelled densities have an opposite behaviour when compared with
CG15 or KCC. In the first case, the average density is underestimated
and in the second case overestimated. The multi-layer version of
the model allows us to better understand the reasons for this
mismatch. An error in the density of an individual layer will, in
fact, affect all the modelled average firn densities that contain that
individual layer. This is the case for CG15, where the underestimation
of the density of the most superficial layers (Fig. 8)
influences the average density of the whole firn column. This
influence however decreases when more annual layers are averaged. In
the case of CG15, this is probably related to the characteristics of
the CG15 location since underestimation was also observed when
applying the firn densification model alone in a steady state and not
inside the presented snow–firn model. On the contrary, the
steady-state application of HL and AR resulted in a good match with
the KCC ice core (Fig. 4). Hence, it is reasonable to
suppose that the mismatch is related to the modelled snow
characteristics. The discrepancy may be explained by the different estimated
surface accumulations; the model, in fact, results in an average snow
accumulation of 0.46×10^{3} against the
0.22×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ accumulation of KCC. Since
the firn densification is driven by overburden stress, this may lead
to a systematic bias in the results.

### 5.3.2 Multi-layer model version

Implementing HL inside the model, we obtained in general a better
agreement and a higher variability in the density. The latter is
probably related to a higher sensitivity of the equations to surface
density, which has instead a low influence on AR as also reported by
Bréant et al. (2017). Choosing
${\mathit{\rho}}_{\mathrm{D}}=\mathrm{300}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{3}}$ and looking at the KCC ice core, we
observe a decrease in the density at around 7–8 m in both
cores. This low density value corresponds to the layer deposited in
the 2006–2007 hydrological year. This year was characterized by a
solid precipitation event that occurred at the end of the hydrological
year. As a consequence, the modelled average snow density decreased
from 412 kg m^{−3} at the beginning of 25 September 2007 to
260.6 kg m^{−3} at the end of 27 September 2007. Due to the low
amount of snow that is not eroded, new solid precipitation events, in
fact, may have a high influence on average snow density. The choice of
adopting firn densification equations only after a certain density is
reached rather than at the end of the water year allows us to better
model this circumstance, in which at the end of the water year the
layer has characteristics that are more similar to recent snow
than to 1-year-old snow. In comparing the observations and model
results, it is also important to point out that, due to the
differences in firn height between model and ice core, the ages of the
modelled and observed steps do not correspond.

The present model does not resolve a full energy balance to compute
surface snow temperature, thus not taking into account the effect of
the different amount of solar radiation in the different ice core
locations, which is very likely responsible for the significantly
different behaviours of the two ice cores. The ice core CG15 is closer
to the axis saddle and presumably in a location with a higher
exposure to solar radiation than KCC since accumulation is
doubled. Consequently, melting events may be more frequent in the CG15 ice
core. The differences in the model parameters when run for the KCC or CG15
ice core are the values of MAFT, *ρ*_{D} and *γ*^{′} when AR is
implemented, and this limits the ability of the model to adapt to the
different ice cores.

In this study we have proposed a local model that combines snow and firn dynamics. It was derived from the mass balance, momentum balance and rheological equations of snow and firn, combined with semi-empirical and empirical approaches proposed in the literature to model the included mass fluxes. It requires for input hourly (or sub-hourly) series of precipitation, temperature and wind speed with which the series of snow, water and ice inside the snowpack and firn height along with dry-snow and firn density are computed. Two versions of the model were proposed: (1) a version (multi-layer) that considers separately each firn layer and (2) a version (single-layer) that models firn and underlying glacier ice as a single layer. The two implementations allow us to cover different purposes. A simpler model is, in fact, more suitable to reproducing firn inside a hydrological model, where the whole depth–density profile is not necessary and where a reduced number of equations may allow an easier integration. On the other hand, a model that resolves density with depth allows us to assess the influence of meteorological variables on snow and firn characteristics. In order to obtain this, we integrated two existing firn densification models (other firn densification models could possibly be implemented) into a wider model, therefore moving the boundary of the model from surface accumulation and density to hourly meteorological series. While this may not be needed to retrieve past climatological information from ice cores, it is required to assess the response of the system to present and future changes in the climate.

Both models' versions require the calibration of three
parameters: *a*, *e* and *γ*^{′}. In addition, the
parameters of the firn densification equation chosen may also need
calibration if applied to a temperate site. The modularity of the
model allows us to easily test different modelizations of the included
fluxes. Depending also on the availability of measured data at the
application site, less empirical approaches could be adopted for the
mass fluxes. Also, depending on the specific application, some
processes presented in the model may be neglected with a reduction in
the total number of parameters. A high number of parameters are, for
example, associated with wind effects that may be neglected in lower-altitude sites but that are likely to be important in high-altitude
sites (Haeberli and Alean, 1985). At Colle Gnifetti, for example,
we saw that observed surface accumulations and densities cannot be
explained if wind effects are not considered. The modelled reduction
in annual snow accumulation due to wind erosion is on average
2×10^{3} $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ while the increase in end-of-year snow density is on average 65 kg m^{−3}. The strong wind
erosion also results in a greater correlation between the amount of
snow preserved in each month and the number of days with above-zero
temperature rather than with the solid precipitation seasonality. This
behaviour may be important in understanding the response of the site
to a warming climate. All these elements result in a strong spatial
and temporal variability in snow density and accumulation at CG that
is not captured by the model. This would, in fact, require taking into
consideration the influence of topography on wind speed and the effect
of solar radiation on surface snow temperature.

The aim of this study was to illustrate the new snow–firn modelling and to present its potentiality through a case study. In order to integrate it into a hydrological model, further steps are required; in particular fluxes among the different columns should be properly considered regarding both run-off and snow transport. In the present modelling, for example, we considered only the snow erosion. At CG the deposition can be neglected due to its characteristics. The site is a saddle with west–east orientation where the predominant wind direction coincides with the saddle orientation. Besides, an ice cliff at the end of it works as a perfect sink for the eroded snow. A distributed modelling approach needs to take into account the presence of sinks and sources of snow transport. This will also improve the representation of wind effects on snow accumulation since, in order to include snow transport in a 1D framework, approximation of the process is unavoidable for the nature of the process itself.

## A1 Mass balance equations

The mass balance equations of snow (*M*_{S}), liquid water in snow
(*M*_{W}), refrozen meltwater and rain in snow (*M*_{MF}), and firn
(*M*_{F}) are as follows:

*P*_{S} and *P*_{R} are the mass of solid and liquid precipitation
events, and they are equal to ${P}_{\mathrm{S}}=s\cdot {\mathit{\rho}}_{\mathrm{NS}}$ and
${P}_{\mathrm{R}}=r\cdot {\mathit{\rho}}_{\mathrm{W}}$. The fresh-snow density was calculated as
proposed by Liston et al. (2007):
${\mathit{\rho}}_{\mathrm{NS}}={\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{0}}}+{\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{w}}}$. Following
Anderson (1976), ${\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{0}}}=\mathrm{50}$ kg m^{−3} if the
air temperature ${T}_{\mathrm{A}}<-\mathrm{15}\phantom{\rule{0.125em}{0ex}}{}^{\circ}\mathrm{C}$ and
${\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{0}}}=\mathrm{50}+\mathrm{1.7}\cdot ({T}_{\mathrm{A}}+\mathrm{15}{)}^{\mathrm{1.5}}$ kg m^{−3}
otherwise. The second term gives the increase in fresh-snow density
due to wind, and it is computed as
${\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{w}}}={D}_{\mathrm{1}}+{D}_{\mathrm{2}}(\mathrm{1.0}-\mathrm{exp}(-{D}_{\mathrm{3}}({u}_{\mathrm{2}}-\mathrm{5})\left)\right)$, where *u*_{2} is the
wind speed at a 2 m height, *D*_{1}=25 kg m^{−3},
*D*_{2}=250.0 kg m^{−3} and
*D*_{3}=0.2 s m^{−1}. When *u*_{2}≤5 m s^{−1},
${\mathit{\rho}}_{{\mathrm{NS}}_{\mathrm{w}}}=\mathrm{0}$ kg m^{−3}.

*M* is the snowmelt mass flux that was computed with a temperature-index
approach (Hock, 2003): $M=(I\cdot a)({T}_{\mathrm{A}}-{T}_{\mathit{\tau}}){\mathit{\rho}}_{\mathrm{S}}$.

*F* is the melt–freeze mass flux
that was modelled with a coupled melt–freeze temperature-index
approach: $F=({I}^{*}\cdot e\cdot a)({T}_{\mathrm{A}}-{T}_{\mathit{\tau}}){\mathit{\rho}}_{\mathrm{W}}$.

The run-off *O* was modelled with a matrix flow approach, and it is
equal to *O*=*ρ*_{W}*α**K*_{W}, where
$\mathit{\alpha}={\mathit{\alpha}}^{\prime}(\mathrm{5.47}\times {\mathrm{10}}^{\mathrm{5}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$)
(DeWalle and Rango, 2008) with ${\mathit{\alpha}}^{\prime}=\mathrm{3600}\phantom{\rule{0.125em}{0ex}}\mathrm{s}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$.
Following Colbeck (1972), *K*_{W} was computed as
${K}_{\mathrm{W}}=K{S}^{*\mathrm{3}}$, in which *K* is the intrinsic permeability of snow
in square metres and *S*^{*} is the effective saturation degree of the
mixture equal to ${S}^{*}=({S}_{\mathrm{r}}-{S}_{{\mathrm{r}}_{i}})/(\mathrm{1}-{S}_{{\mathrm{r}}_{i}})$ with ${S}_{{\mathrm{r}}_{i}}$
being the irreducible saturation degree equal to
${S}_{{\mathrm{r}}_{i}}=\mathrm{0.02}{\mathit{\rho}}_{\mathrm{S}}/\left({\mathit{\rho}}_{\mathrm{W}}\mathit{\varphi}\right)$ (Kelleners et al., 2009)
and *S*_{r} the average saturation degree of the porous matrix
equal to 1 when *h*_{W}≥*ϕ**h*_{S} and equal to ${h}_{\mathrm{W}}/\left(\mathit{\varphi}{h}_{\mathrm{S}}\right)$
otherwise (Avanzi et al., 2015). The intrinsic permeability is
obtained using the parametrization proposed by Calonne et al. (2012),
and it is equal to $K=\mathrm{3}{R}^{\mathrm{2}}\mathrm{exp}(-\mathrm{0.013}{\mathit{\rho}}_{\mathrm{S}})$, in which *R* is
the equivalent sphere radius. The radius *R* is defined as
$R=\mathrm{3}/\left(\mathrm{SSA}{\mathit{\rho}}_{i}\right)$, where SSA is the specific surface area in
m^{2} kg^{−1} that was computed by Avanzi et al. (2015)
adapting the formula proposed by Domine et al. (2007)
with $\mathrm{SSA}=-\mathrm{30.82}\mathrm{ln}({\mathit{\rho}}_{\mathrm{S}}\times {\mathrm{10}}^{-\mathrm{3}})-\mathrm{20.60}$. When
*S*_{r}>0.5, to avoid numerical instability, the run-off is
calculated with a kinematic wave approximation
(De Michele et al., 2013) as ${\mathit{\rho}}_{\mathrm{W}}{\mathit{\theta}}_{\mathrm{W}}{h}_{\mathrm{W}}^{\mathrm{1.25}}$.

The firn melting *O*_{F}, which may occur only when the snowpack is absent,
was modelled with a temperature-index approach, and it is equal to
${O}_{\mathrm{F}}=({I}_{\mathrm{F}}\cdot a)({T}_{\mathrm{A}}-{T}_{\mathit{\tau}}){\mathit{\rho}}_{\mathrm{F}}\mathit{\delta}\left({h}_{\mathrm{S}}\right)$.

*P*_{F} is the effect of rain on firn that, when the snowpack is
absent, causes an increase in *M*_{F} when *T*_{A}<0 ^{∘}C
because rainfall is chilled to the firn temperature and a decrease
when *T*_{A}>0 ^{∘}C because the energy supplied by rain
will be used to melt ice. In the first case ${P}_{\mathrm{F}}={\mathit{\rho}}_{\mathrm{W}}\cdot r$,
while in the second case *P*_{F} was set to zero due to its small
contribution to mass balance (Doyle et al., 2015).

The terms *E*_{S}, *E*_{W} and *E*_{MF} move the mass of the snowpack
still on the ground at the end of each melt season inside the firn, and
they are equal to ${E}_{j}={\sum}_{i}{\mathit{\rho}}_{j}\frac{{h}_{j}}{\mathrm{d}t}\mathit{\delta}(t-{t}_{i})$
with *j* denoting S, MF and W.

*Q* is the mass of snow eroded by wind obtained from snow transport.
The latter was computed adopting the parametrization proposed by
Pomeroy et al. (1993) as the sum of a transport in saltation and a
transport in suspension.

The saltation transport rate *Q*_{salt} ($\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$)
occurs only when wind exceeds a given threshold, and it is computed as
follows:

where *ρ*_{a} is the atmospheric density (kg m^{−3}) and
*u*^{*} and ${u}_{\mathrm{t}}^{*}$ are the atmospheric friction
velocity and the friction velocity applied to the snow surface at the
transport threshold, respectively (m s^{−1}). To move from the measured wind
speed *u* to *u*^{*}, knowledge of the aerodynamic roughness height
*z*_{0} is required. This passage is not straightforward since the
value of *z*_{0} during blowing snow events is different from the one
during non-transport conditions and it depends on friction velocity
(Pomeroy and Gray, 1990). In order to avoid an iterative procedure, we
adopted the approximation proposed by Pomeroy and Gray (1990), using
${u}^{*}\approx \mathrm{0.02264}{u}^{\mathrm{1.295}}$ and
${z}_{\mathrm{0}}=\frac{\mathrm{0.1203}{u}^{{*}^{\mathrm{2}}}}{\mathrm{2}g}$, where *u* is the 10 m wind
speed (m s^{−1}).

Suspension transport, which occurs only when particles are already in saltation, was computed as follows:

where *Q*_{susp} is in $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$, *κ* is the
von Kármán constant equal to 0.4, *h*^{*} is the lower boundary for
suspension equal to ${h}^{*}={c}_{\mathrm{H}}{u}^{{*}^{\mathrm{1.27}}}$
(Pomeroy and Male, 1992) with
${c}_{\mathrm{H}}=\mathrm{0.08436}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{0.27}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{\mathrm{1.27}}$, *z*_{b} is the top of the
surface boundary layer for suspended snow and *η*(*z*) is the mass
concentration of suspended snow (kg m^{−3}) at height *z*.
The mass concentration can be approximate as
$\mathit{\eta}\left(z\right)=\mathit{\eta}\left({z}_{\mathrm{r}}\right)\mathrm{exp}(-{A}_{Q}(({B}_{Q}{u}^{*}{)}^{-\mathrm{0.544}}-{z}^{-\mathrm{0.544}}))$
(Pomeroy and Male, 1992), where *η*(*z*_{r}) is the reference mass
concentration for suspension set to 0.8 kg m^{−3}
(Pomeroy and Male, 1992), *A*_{Q} is equal to 1.55 m^{0.544}
and *B*_{Q} is equal to 0.05628 s^{−0.544}. *z*_{b} was set to
5 m since its value is typically between 5 and
10 m (Déry and Taylor, 1996). The exact value is unimportant because
of small mass fluxes at this height (Pomeroy et al., 1993). The snow
erosion in the control volume of the model was set equal to the sum of
these two transports.

The critical threshold above which snow transport occurs was computed adopting the formula proposed by He and Ohara (2017). The critical shear stress for snow movement can be therefore computed as follows:

where *R* is the grain radius (m); *t*_{d} is the time since
deposition in seconds; *C*_{c}, *C*_{d}, *C*_{g} and *C*_{l} are
dimensionless coefficients set to 1, 4, $\mathrm{1.3}\mathit{\pi}/\mathrm{6}$ and 3.4
(He and Ohara, 2017); *ς* is the stress caused by cohesion of ice
computed as $\mathit{\varsigma}=\mathrm{1.51}\mathrm{exp}\left(\mathrm{0.44}\right({T}_{\mathrm{A}}+\mathrm{9}\left)\right)+\mathrm{6.8}$
(Hosler et al., 1957) for temperatures between −20
and 0 ^{∘}C with *ς* in newtons per square metre and
*T*_{A} in degrees Celsius; and
$S=\mathrm{arcsin}\left({\left(\frac{C}{{R}^{m}}{t}_{\mathrm{d}}\right)}^{\mathrm{1}/n}\right)$.
*C*, *m* and *n* are parameters that influence the rate of ice
sintering, modelled following Maeno and Arakawa (2004). In particular,
$C={C}_{\mathrm{0}}\mathrm{exp}\left(\frac{-{Q}_{s}}{{R}_{\mathrm{G}}({T}_{\mathrm{A}}+\mathrm{273.15})}\right)$, in which
*R*_{G} is the gas constant and *T*_{A} is computed as the average air
temperature since deposition,
${C}_{\mathrm{0}}=\mathrm{4.14}\times {\mathrm{10}}^{\mathrm{19}}$ m^{3} s^{−1}, and
${Q}_{s}=\mathrm{1.965}\times {\mathrm{10}}^{\mathrm{5}}$ J mol^{−1}. Finally, *m* and *n*
are empirical parameters set to 2.9 and 5, respectively, following the
results of He and Ohara (2017). Once the critical shear stress is obtained,
it is possible to move to the critical friction velocity as follows:
${u}_{t}^{*}=\sqrt{{\mathit{\tau}}_{t}/{\mathit{\rho}}_{\mathrm{a}}}$. If wind speed is lower than the
critical threshold, no erosion occurs and *Q* is set to zero.

To implement the snow erosion routine, we proceeded as explained in
the following. When the first solid precipitation event occurs in a
time step, the amount of new snow on the ground at the end of the time
step, *S*_{A} ($\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$), is saved along with the time
of deposition and *ρ*_{NS} of the event. During the following
step, four different situations are possible: (1) a new snow event
occurs in the time step. In this case *S*_{A} is moved into a vector
*S*_{R} with its time of deposition and *ρ*_{NS}. *S*_{A} is
recomputed as *ρ*_{NS}⋅*s*. (2) *T*_{A}>0 ^{∘}C. In this case *S*_{A} and *Q* are set to
0 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$, and all the old snow events memorized in
*S*_{R} are removed. (3) *T*_{A}<0 ^{∘}C and
*Q*<*S*_{A}. In this case *S*_{A} is set to ${S}_{\mathrm{A}}={S}_{\mathrm{A}}-Q$. (4) *T*_{A}<0 ^{∘}C and *Q*>*S*_{A}. In this case, if
*S*_{R} has no elements, *Q* is set equal to *S*_{A} and *S*_{A}
to 0 $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$; otherwise the difference between *Q*
and *S*_{A} is subtracted from the most recent event in *S*_{R},
given that wind speed is higher than the threshold recomputed with the
characteristics of that event, and this event is removed from
*S*_{R}. This is repeated until an event in *S*_{R} that cannot be
eroded by wind is encountered or the total amount of snow eroded in
that time step reaches *Q*. In the latter case the actual transport
is *Q*, while in the former *Q* is given by the total amount of
snow eroded before reaching the non-erodible layer. The new *S*_{A} is
the amount of snow associated with the last event considered.

Given that *M*_{j}=*ρ*_{j}*h*_{j} and *ρ*_{k}=const with
*j* denoting S, MF, and W and *k* denoting MF and W, after some algebra we can move from
Eqs. (A1a)–(A1d) to Eqs. (2a)–(2d).

## A2 Snow densification

The densification of dry snow due to compaction was modelled adopting the formula proposed by Liston et al. (2007):

where $c=\mathrm{0.10}\cdot \mathrm{3600}\phantom{\rule{0.125em}{0ex}}\mathrm{s}\phantom{\rule{0.125em}{0ex}}{\mathrm{h}}^{-\mathrm{1}}$,
*A*_{1}=0.0013 m^{−1}, ${A}_{\mathrm{2}}=\mathrm{0.021}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}$,
$B=\mathrm{0.08}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}$ and *U* is the wind speed contribution
(m s^{−1}). For wind speeds $\ge \mathrm{5}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$,
$U={E}_{\mathrm{1}}+{E}_{\mathrm{2}}(\mathrm{1.0}-\mathrm{exp}(-{E}_{\mathrm{3}}({u}_{\mathrm{2}}-\mathrm{5.0})\left)\right)$, with *E*_{1}, *E*_{2} and
*E*_{3} equal to 5.0 m s^{−1}, 15.0 and
0.2 s m^{−1}, respectively, and *u*_{2} is the wind speed at a
2 m height. For wind speed $<\mathrm{5}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$,
$U=\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathrm{m}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$. Adding the densification due to mass
variation (see De Michele et al., 2013), the total densification rate
can be computed as follows:

where we assumed that melting events and snow erosion occur at
*ρ*_{S}=const.

## A3 Arnaud et al. (2000) model

The model of AR separates the densification of firn into three
stages. The first stage is governed by settling, and it is modelled by
Bréant et al. (2017) adapting the equation proposed by
Alley (1987). The second stage, which starts when the relative
density $D={\mathit{\rho}}_{\mathrm{F}}/{\mathit{\rho}}_{i}$ equals *D*_{0}, is dominated by power law
creep, and it is modelled following Arzt (1982) and
Arzt et al. (1983). Grains are considered spheres, and each sphere
is allowed to increase in radius around fixed centres. Starting from
an initial radius *l*, the new radius *l*^{′} (in units of the
initial particle radius *l*) is ${l}^{\prime}\left(D\right)=(D/{D}_{\mathrm{0}}{)}^{\mathrm{1}/\mathrm{3}}$. The growth
of spheres increases the number of particle contacts *Z* from the
initial value *Z*_{0} to $Z\left(D\right)={Z}_{\mathrm{0}}+b({l}^{\prime}-\mathrm{1})$, in which *b*=15.5.
The overlap due to the growth of particles produces an excess volume
of material. This excess is distributed uniformly around the portion
of the surface of the spheres not in contact. From this excess volume,
it is possible to calculate the new radius *l*^{′′} as

The average contact area (in units of *l*^{2}) can be obtained
averaging over all existing contacts:

The value of *Z*_{0} for a given value of *D*_{0} is obtained, as
proposed by Arnaud et al. (2000), assuming that the effective stress
${P}^{*}=\left(\mathrm{4}\mathit{\pi}P\right)/\left({a}_{\mathrm{c}}ZD\right)$ approaches *P* as *D* tends to 1. The
third stage begins when pores start becoming isolated (*D*>*D*_{c}) and
densification is calculated considering the deformation of ice shells
surrounding cylindrical pores (Wilkinson and Ashby, 1975). As for
Eq. (2e), the total densification rate is obtained adding
the densification due to new mass addition (Eq. 2f).

Meteorological data measured by the network of stations of the Aosta Valley Region can be downloaded from https://presidi2.regione.vda.it/str_dataview (last access: 11 March 2022). Snow water equivalent and snow density data must be requested directly from the Aosta Valley Region. Hourly meteorological data of the network of stations in Piemonte must be requested from the region, and they are free of charge for research purposes. Wind data at Col du Grand St Bernard come from from the Integrated Surface Dataset (ISD), maintained by the National Oceanic and Atmospheric Administration (NOAA), and they can be accessed from https://www.ncei.noaa.gov/metadata/geoportal/rest/metadata/item/gov.noaa.ncdc:C00532/html (NOAA National Centers for Environmental Information, 2001). B76 and B77 ice core data are published in the following paper: https://doi.org/10.3189/S0022143000005220 (Gäggeler et al., 1983). CG03 and CG15 are available on the PANGAEA repository (https://doi.org/10.1594/PANGAEA.894787, Sigl et al., 2018). CG11 ice core data are published in the following thesis: http://hdl.handle.net/10579/1924 (Ardenghi, 2012). The other ice core data were requested from the authors.

CDM and FB conceived the model. FB took care of data and developed the case study. FB wrote a first draft of the manuscript. FB and CDM reviewed the manuscript.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank ARPA Piemonte for the meteorological data used in this study, with particular thanks to Manuela Bassi, and Aosta Valley for the snow water equivalent data. Gratitude is also due to Carlo Licciulli and Josef Lier for ice core data and to Pascal Bohleber for the information about Colle Gnifetti. Thank you to Scenari Digitali and Rifugi Monterosa for additional information about Capanna Regina Margherita. We would also like to thank the editors, Olaf Eisen and Michiel van den Broeke, and the three anonymous referees for their comments and feedback about the manuscript (including about a previous submission of the manuscript).

This paper was edited by Michiel van den Broeke and reviewed by two anonymous referees.

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- Abstract
- Introduction
- Methodology
- Study area and data
- Results
- Discussion
- Conclusions
- Appendix A: Complete description and derivation of the snow–firn model
- Appendix B: List of symbols and abbreviations used
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Review statement
- References

- Abstract
- Introduction
- Methodology
- Study area and data
- Results
- Discussion
- Conclusions
- Appendix A: Complete description and derivation of the snow–firn model
- Appendix B: List of symbols and abbreviations used
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Review statement
- References