As a first equation governing the evolution of a snowpack, we consider the energy conservation of snow, understood here as the combination of the ice matrix and the air within (and excluding potential liquid water). The temporal evolution of the snow energy is given by a classical conservation equation, i.e. 1∂ths+∇⋅Fcond=Qabs+Qfreeze, where hs is the energy content of snow (expressed in Jm-3), Fcond the heat conduction flux (in Wm-2), Qabs a volumetric energy source (in Wm-3) due to shortwave absorption within the snowpack , and Qfreeze the energy absorbed/released during freezing/melting (in Wm-3). Moreover, we classically assume that the heat conduction flux Fcond follows Fourier's law 2Fcond=-λ∇T where λ is the thermal conductivity of snow (in WK-1m-1) and T its temperature (K). These equations, and the value of the snow thermal conductivity in terms of the microstructure, can for instance be derived from homogenization methods e.g.. Moreover, the temperature and the energy content are related through the thermal capacity of snow 3hs=∫T0TcpdT=cp(T-T0) where T0 is a reference temperature taken as the fusion temperature of ice at standard pressure for simplicity, and cp is the volumetric thermal capacity of snow in Jm-3K-1;. Note that we have assumed for simplicity that cp does not depend on temperature, as regularly done in snowpack models e.g. in the Crocus model;. Using homogenization methods, the volumetric thermal capacity of snow can be shown e.g. to be 4cp=ϕici+(1-ϕi)ca≃ϕici where ϕi is the ice volumetric fraction, and ci and ca the volumetric thermal capacity of ice and air, respectively.

Since the heat budget and temperature of snow are directly related to the melting and freezing of water, it is necessary to also treat the liquid water budget. This can be done with the use of Richards' equation . Note that in this paper we only consider “matric” water flow, and do not consider fast preferential flow . This limitation is discussed in Sect. . Under these conditions, the mass conservation of liquid water is given by 5ρw∂tθw+ρw∇⋅Fw=-Mfreeze where θw is the volumetric liquid water content (LWC; expressed in m3 of water per m3 of snow), ρw the density of liquid water (assumed constant; in kgm-3), Fw is the liquid water flux (in m3 of water per m2 of snow per s), and Mfreeze is the rate of freezing water (counted positive in the case of freezing and expressed in kg of water per m3 of snow per s). The rate of freezing water Mfreeze is directly related to the rate of energy release/absorption Qfreeze of Eq. () through Qfreeze=LfusMfreeze, with Lfus the specific enthalpy of fusion of water (in Jkg-1).

The liquid water flux is assumed to follow the Darcy-Buckingham law 6Fw=-Ksatkr(∇ψ+cos⁡γ) where Ksat is the saturated hydraulic conductivity (in ms-1), which does not depend on the LWC but only the properties of the ice matrix, kr the relative hydraulic conductivity, that depends on the LWC, ψ the liquid matric potential (in m and that can be negative due to capillary forces), and γ the slope angle. Note that in the multidimensional case, the gravity term cos⁡γ should be replaced by the unit vector orientated with gravity. In the same way that the snow energy and temperature are related through Eq. (), the LWC and the matric potential are related through the use of a water retention curve (WRC). This WRC can be expressed with the use of, for instance, the van Genuchten or Brook models . In this article, and consistently with previous snow-related work , we use a van Genuchten model for the WRC in snow 7θw=θr+(θs-θr)(1+|αψ|n)1-nnif ψ<0θsotherwise where n and α (in m-1) are parameters characterizing the shape of the WRC and which depends on the snow microstructure , and θr and θs are referred to as the residual and saturation LWC, respectively. Physically, θs corresponds to the maximum amount of liquid water a snow sample can hold (thus typically assumed to correspond to the total porosity), and θr to the minimum amount of water that can be reached with draining through capillary and gravity forces.

To solve Richards' equation, it is also necessary to provide an expression for the dependence of the relative hydraulic conductivity kr on the liquid water content. Following , in complement of the van Genuchten WRC, the hydraulic conductivity is usually taken as 8kr=S∫0SdSψ∫01dSψ2=S1-1-Snn-1n-1n2 where S=θw-θrθs-θr is referred to as the saturation degree. It can be noted that the relative hydraulic conductivity is an increasing function of θw that vanishes at the residual LWC θr and reaches unity value at water saturation.

However, Richards' equation presents specific problems in the case of water-saturated and fully-dry materials. In the case of a water-saturated material, the liquid water content reaches a maximum value, while the matric potential can keep increasing. In other words, in the saturated case, the equation becomes degenerate and can no longer be expressed in terms of liquid water content. This is a classical difficulty associated with Richards' equation, usually circumvented by using ψ as the primary variable to describe the material . It can also be treated using a variable switch technique, effectively changing from θw to ψ as a primary variable depending on the degree of saturation of the material . This latter technique is explored below in the article. The problem of a fully-dry material is more specific to snow, and requires a dedicated treatment.

Besides the energy and liquid water budgets, the process of water melting and freezing needs to be accounted for in the ice budget to properly close the mass budget. There are two potential ways to account for a melting or refreezing event in the ice budget while respecting mass balance: it can be either viewed as a decrease or increase in density at constant volume, or as a loss or gain of volume at constant density. Depending on the snowpack model, these two possibilities have been employed to represent melt. For instance, melt is treated as a decrease of thickness at constant density in the Crocus model and a decrease of density at constant thickness in SNOWPACK . The justification for the former choice follows the observation that melting snow is usually of a high density, and thus that the melting process should not act as a de-densification mechanism. Yet, as phase changes occur directly within the snow microstructure, at the surface of the porous ice matrix, we rather believe that both melting and refreezing impact the snow density, without direct impact on the thickness, as done in the SNOWPACK model. With this idea, the relatively high-density nature of melting snow could be attributed to the low viscosity of wet snow that easily compacts under mechanical stress. Therefore, for the rest of the paper we assume that both type of phase changes affect the snow density, and not directly the snow layer thicknesses. The ice mass budget within the snowpack is thus given by 10ρi∂tϕi=Mfreeze where ϕi is the ice volume fraction and ρi the ice density (in kgm-3).

In order to close the system of equations, it is necessary to provide an expression for the freezing rate Mfreeze. This is done by defining both the thermodynamical equilibrium between ice and liquid water and the dynamics with which this equilibrium is restored. However, to the best of our knowledge, there is no broadly accepted theoretical or experimental work providing the dynamics of melting or freezing of water in snow . As most snowpack models e.g., we assume (i) that liquid water and the snow are in thermodynamical equilibrium (which means that the melting/freezing dynamics can be assumed as infinitely fast) and (ii) that this equilibrium occurs at the single temperature T0. However, we note that these assumptions are not systematic in snowpack models. First, the recent article of proposes to relax the assumption of local thermal equilibrium and to introduce a finite rate of phase change, derived from the upscaling of the Frenkel-Wilson equation. This implies that the ice and liquid water temperatures in a snow sample are in general different and can be above or below the fusion point T0. This modeling framework, composed of four partial differential equations, is briefly presented in Appendix . However, as observed in the Appendix, the timescale of relaxation towards local thermodynamical equilibrium appears to be much smaller than the timescale of matric water movement and heat diffusion considered in this manuscript, which supports the standard assumption of local equilibrium in snowpack models. Also, due to capillary effects, the thermal equilibrium between the ice and liquid water phases technically occurs on a temperature range rather than at a single temperature. This effect is commonly taken into account in soil models through a so-called soil Freezing Characteristic Curve soil FCC;. Some snowpack models have proposed to introduce a similar FCC for snow . While the FCC of snow could in principle be computed from the WRC of Sect.  as done for instance in, this would represent a significant increase in the complexity of the snow representation. Indeed, the simple equilibrium condition that ice and liquid water can only coexist at T0 would have to be replaced by an implicit equation relating the temperature to the matric potential. However, we note that the computation of a FCC from a diverging WRC implies that thermodynamical equilibrium cannot be reached with a LWC below the divergence point , and thus that regularizing the WRC is a necessary step to model a dry material.

Under our assumptions, the total energy content of both the snow and liquid water h=hs+ρwLfusθw=cp(T-T0)+ρwLfusθw, the temperature T, and the LWC θw become directly related: in the case where h is below the fusion value, θw vanishes and T<T0; in the case where h is above the fusion value, θw is proportional to the excess of energy and T=T0. Specifically, the liquid water content and the temperature can be expressed as a function of the total energy 11θw=0if h<0hρwLfusotherwise and 12T=T0+hcpif h<0T0otherwise

Thus, the snowpack becomes governed solely by two PDEs: the total energy budget and the total mass budget. These equations can be obtained by combing Eqs. (), (), and (), which yields 13∂th-∇⋅(λ∇T)+ρwLfus∇⋅(Ksatkr(∇ψ+cos⁡γ))-Qabs=0ρw∂tθw+ρi∂tϕi-ρw∇⋅(Ksatkr(∇ψ+cos⁡γ))=0 Note that the rate of freezing/melting Mfreeze has been eliminated from the system of equations. It can still be derived as a diagnostic from the closure of Richards' equation. It physically corresponds to the amount of frozen and melted water required to maintain the local thermodynamic equilibrium.

A difficulty with the system presented in Eq. () (complemented with its material laws) is the presence of 4 distinct regimes, where parts of the equations behave differently and sometimes degenerate. The first regime corresponds to dry snow, when the temperature is below the freezing point. Here, the thermodynamical state of snow can be characterized by either its temperature or its energy content (the two being related by the thermal capacity). On the contrary, the LWC and the matric potential cannot be used to describe the state of snow, as they assume constant, degenerate, values. In the second regime, the snow is at its fusion point, but the LWC remains below its retention value. Here, the snow can be characterized by its energy content or its LWC, but not by its temperature nor its matric potential. The third regime corresponds to the classical unsaturated Richards' case, where the LWC is above the retention point. Here, the snow can be characterized by its energy content, its LWC, or its matric potential, but not by the temperature. The fourth regime corresponds to the water-saturated regime. Here, the snow can only be characterized by its matric potential. This regime corresponds to the saturated regime of Richards' equation. A summary of the different regimes and the variables that can be used to characterize them is given in the Table . As seen, there is no natural variable that can be used to describe the thermodynamical state of snow over the 4 different regimes. While the energy content could be used to characterize snow in regimes 1, 2, and 3, it degenerates in regime 4. Similarly, while the matric potential is a good candidate to describe regimes 3 and 4 as usually done when numerically solving Richards' equation;, it fails in regime 1 and 2.

To circumvent this issue, we rely on a variable switch technique by parametrization , through the use of a fictitious variable meant to behave as the energy content in the first three regimes and as the water matric potential in the last one . For this, we introduce a new variable χ (expressed in Jm-3) used to parameterize the {h,ψ} graph. Specifically, we choose χ such that 14h(χ,ϕi)=χif χ<θs(ϕi)ρwLfusθs(ϕi)ρwLfusotherwise15ψ(χ,ϕi)=ψ(θw(χ,ϕi))if χ<θs(ϕi)ρwLfusχ-θs(ϕi)ρwLfusβotherwise where β is an arbitrary constant, introduced to respect the unit homogeneity of Eq. (), and where the dependence of ϕi has also been made explicit. Thanks to this variable χ, we are now able to characterize the energy content and matric potential of snow in all regimes, and from it to derive the values of all other relevant quantities for snow (θw, T, kr, etc.).

Once parameterized with χ our system of equation becomes 16∂th(χ,ϕi)-∇⋅(λ∇T(χ,ϕi))+ρwLfus∇⋅(Ksatkr(∇ψ(χ,ϕi)+cos⁡γ))-Qabs=0ρw∂tθw(χ,ϕi)+ρi∂tϕi-ρw∇⋅(Ksatkr(∇ψ(χ,ϕi)+cos⁡γ))=0 that can be solved searching for χ and ϕi. We have explicitly shown the dependency of h, T, ψ, and θw on both χ and ϕi, but one should keep in mind that the effective properties λ, Ksat, and kr also depend on the value of χ and ϕi. With this set of equations, we now have a consistent description of the energy and mass budgets in snowpacks that naturally applies to both dry and wet snow.

In order to be directly comparable, the models share portions of their numerical implementations. This includes compaction under mechanical stress (assuming a linear viscosity between stress and deformation and which is solved after the thermodynamics), the constitutive laws defining the material properties of snow, the spatial and temporal discretization, and the method for solving the resulting non-linear systems of equations.

For the first model, we use the regularized WRC and solve the energy and mass budgets in a tightly-coupled manner, that is to say that the system of Eq. () is solved concomitantly for both χ and ϕi. In this version, there is therefore no degree of operator-splitting when solving the thermodynamical state of the snowpack.

While the numerical implementation of model 1 appears to be the most consistent, it results in a large non-linear system that can be numerically expensive to solve. As a compromise, some degree of operator-splitting (i.e. sequentiallity) between processes can be introduced. For model 2, we thus used the regularized WRC with a decoupling of the computation of the energy and liquid water budgets from that of the ice budget. The motivation behind it is that the timescale for the evolution of the density is usually longer (of the order of a day, unless in case of abrupt melting) than that of the energy and LWC evolution (of the order of an hour or less). Concretely, in Eq. () we thus first solve the evolution of χ (assuming a fixed ϕi). Then, we find the evolution of ϕi by closing the mass budget.

Keeping with the idea of using operator-splitting to reduce the numerical cost and complexity of the model, the third implementation is based on a decoupling of the energy, liquid water, and ice budgets solutions. Specifically, we first evolve χ under the process of heat conduction and phase changes only. We then evolve χ again, this time under the process of matric liquid water flow. Finally, ϕi is updated by re-applying phase transition and closing the mass budget.

The last two models are meant to emulate the techniques used so far in snow models to handle the degeneracy of the retention curve in dry snow. Specifically, model 4 is based on the SNOWPACK implementation and model 5 on the Crocus implementation . The specificities of the implementation are given in Appendix , but we recall that both techniques are based on the idea of introducing a small amount of liquid water in cold snow, shifting the residual liquid water content value below it, and using ψ as the primary variable. Moreover, these models rely on a decoupled solution of the thermodynamical processes.

The first test case is meant to represent the situation of melting snowpack, releasing liquid water near the surface that then percolates downward. For the initialization of the snowpack, we rely on a simulation performed with the Crocus snowpack . This yields a realistic stratigraphy, with thinner cells near the surface in order to better capture the generally steep gradients of density, temperature, and liquid water content near the surface. For the initial state of the simulation, the initial temperature was decreased compared to the Crocus output in order to obtain a cold and dry snowpack near its peak snow water equivalent. This defines the ice volume fraction, temperature, and specific surface area at the start of our simulations. Note that the specific surface area is needed as it plays a role in the hydraulic properties of the snowpack . However, as we did not implement metamorphism laws in our toy model, the specific surface area is simply kept constant during the simulation.

For the top boundary conditions, we impose an incoming diurnal energy flux at the surface of the snowpack (oscillating between 310 Wm-2 at noon and 200 Wm-2 at night), encapsulating the effect of incoming longwave radiation and turbulent heat fluxes. We also impose diurnal shortwave radiation (peaking at 840 Wm-2 at noon and vanishing at night) that penetrates within the snowpack with an albedo of 0.7 and an e-folding depth of 5 cm;. No liquid, nor solid, precipitation is considered in this test case. As the bottom boundary condition, we impose a constant heat flux of 10 Wm-2, emulating the heat flux received from a warm ground below, and a free-drainage condition for the liquid water flux. For each model, simulations are run for 6 d, so that water percolates to the bottom of snowpack. Default timesteps vary between 112 and 7200 s (we recall that because of the adaptive timestep strategy, the actual timestep used for some process might be shorter than the prescribed default value).

This second test case is meant to represent the slow infiltration of rain under a long but low-intensity event. The initialization is the same as in case 1, based on the output of the same Crocus simulation, but with a higher temperature in order to have a snowpack close to its melting point. For the top boundary condition, we impose a constant surface energy flux of 310 Wm-2, a shortwave radiation flux peaking at 420 Wm-2 at noon (lower than in the first test case to represent cloudiness), and a constant rain flux of 1 mmh-1 lasting the whole simulation. For the bottom boundary condition, we use the same conditions as in the test case 1 (constant heat flux and free-drainage). Simulations were performed with each model for 1 d, in order for shallow infiltration to occur. Default timesteps range from 112 to 7200 s.

The third test case represents the case of a short but high-intensity rain event. The initialization is also based on a Crocus simulation, with a temperature field intermediate between test case 1 and 2. The boundary conditions are taken as in test case 2, except for the incoming rain flux that is now 15 mmh-1 and lasts for 4 h. This results in an abrupt input of liquid water in the snowpack, rapidly percolating downward. Simulations were run for about 2 d, long enough for deep percolation to occur. Default timesteps range from 112 to 7200 s.

The initial conditions for the density, SSA, and temperature are displayed in Fig. . For all three test cases, reference simulations were performed with the fully coupled model (model 1) and a timestep of 1 s. The resulting LWC profiles over time are displayed in Fig. . As seen in the figure, the first test case shows diurnal surface melt with liquid water infiltrating deeper in the snowpack days after days. Note that a significant amount of liquid water is retained around the 20 cm horizon. This coincides with an abrupt drop in the snow SSA, which acts as a capillary barrier. In the second test case, surface melt is less pronounced due to the lower input radiative fluxes. The constant rain input produces a steady percolation of the liquid water that reaches about the middle of the snowpack after a day. In the last test case, the abrupt rain input results in a fast movement of the percolation front through almost the whole snowpack. As in the first test case, a significant amount of liquid water is retained around the 20 cm horizon. In these three cases, liquid water is produced directly at the bottom of the snowpack in response to the 10 Wm-2 heat flux from the warm ground. In the absence of such heat flux, the bottom of the snowpack would remain dry until liquid water percolates through the whole snowpack. Finally, we note that using model 1 as a benchmark is not neutral, as it places this model in a specific position compared to the others. This choice was made as we expect its physics to be the most cleanly defined (without diverging WRC and artificial displacement of the residual LWC to circumvent this divergence) and that the use of operator-splitting is known to introduce errors

The first difference in behavior between the different implementations is their respective numerical cost, i.e. the computation time required to perform a given simulation. In the presence of an adaptive timestep, required to accommodate the strong non-linearity of the liquid matric flow and of the WRC, we observed that the numerical cost of the models is, at first order, controlled by the ability of the models to stick to large adaptive timesteps. Figure  displays the adaptive timesteps required by the models in the three test cases presented above. While the models using a regularized retention curve with a variable switch (models 1, 2, and 3) are able to maintain a timestep of around 1 h for most of the simulations, models 4 and 5 require the adaptive timestep to regularly drop, sometimes below 10 s. In the end, models 4 and 5 require about one order of magnitude more steps to perform the same simulation, resulting in significantly increased numerical cost and computation time. For instance, in the case of the test case 3 with a default timestep of 3600 s, models 4 and 5 require a computation time of about 500 s, while models 1, 2, and 3 require about 130, 25, and 45 s, respectively. Note that these numbers should only be interpreted qualitatively to assess numerical complexity, as the precise computation time is also influenced by factors that we did not control for (such as the level of code optimization or memory-caching). Therefore, the use of a regularized WRC with a variable switch appears as a very favorable practice to increase the robustness of the numerical scheme and to decrease its numerical cost by accommodating larger timesteps. Our understanding of this behavior is that the combination of regularized WRC and not using the matric potential as the primary variable near the retention point hinders the presence of abrupt, and nonlinear, variations of the numerical solution, thus making the problem easier to solve. To the contrary, using the matric potential as the primary variable is known to be inefficient for low LWC materials and to require timesteps that are much smaller than the actual timescale of liquid water transport .

Focusing on models 1, 2, and 3 in Fig. , it appears that these three implementations require very similar timesteps, apart from a notable drop for model 3 in the first and second test case. Their differences in numerical cost should be controlled by the complexity of the assembly and solving of the matrix-form of the equations. According to our numerical test cases, models 2 and 3 have about the same numerical cost, while our implementation of model 1 requires a longer computational time. This is to be expected, as model 1 requires the largest matrix to be assembled and solved (2N×2N), while the other models handle smaller matrices (N×N). However, we also note that a large portion of the up-cost of model 1 comes from the computation of derivatives with respect to ϕi for the assembly of the Jacobian. In our implementation, computing these derivatives comes with a high cost, as we rely on generic automatic differentiation without any attempt at optimization. If these derivatives with respect to ϕi are ignored (thus transforming the scheme from a true Newton to a form of modified Picard method), the numerical solving of the coupled system of Eqs. () can be made more efficient. For instance, in the aforementioned test case 3 with a default timestep of 3600 s, the computational time can be lowered from 130 to about 50 s, i.e. in the same range as models 2 or 3.

Besides their numerical cost, the different implementations also yield different levels of error when compared to the reference simulations. While the error level generally tends to decrease with shorter timesteps, not all models perform the same for decreasing timesteps. Here, we thus investigate the degree error of each model, for default timesteps ranging from 7200 to 112 s. For that, we compute the Root Mean Square Difference (RMSD) between the models' outputs and the reference simulations. Figure  display the RMSDs in LWC and temperature for the three test cases as a function of the default timestep. Note that because of the presence of an adaptive timestep for solving liquid water percolation, the value of the default timestep should be interpreted with caution in Fig. . Indeed, as models 4 and 5 (and 3 to some degree) require small adaptive timesteps, their solving of liquid water percolation actually occurs with a timestep much lower than the default value. This gives them an advantage in terms of RMSD, as errors stemming from the temporal discretization of Richards’ equation are reduced in the process. Also, the RMSD is computed over the whole snowpack, while errors tend to be spatially located near the percolation front. Therefore, the RMSD value is smaller, typically by an order of magnitude, than the errors occurring at the percolation front. As expected, the overall trend is a decrease of the RMSDs with small timesteps. However, while the unregularized models 4 and 5 perform similarly as the other models with large default timesteps, they do not show a clear decrease of error at smaller timesteps. Our understanding is that the strategy of modifying the residual LWC based at the start of each timestep impacts the physics and changes the solution to which the models converge with small timesteps. This results in a plateau or even an increase of the associated RMSDs in Fig. . However, it further appears that model 3 does not show a clear convergence for vanishing timesteps as well, despite having in principle the same physics as model 2 and 3. This is visible in the tendency of the RMSD model 3 to plateau for timesteps below 225 s. After investigation, we found that model 3 actually diverges away from the benchmark solution for very small timesteps (of the order of the second). The same behavior for very small timesteps was seen for models 4 and 5 as well. This is puzzling as it suggests that either (i) models 3, 4, and 5 require timesteps well below 1 s to reach convergence, or that (ii) they do not have a converging solution with vanishing timesteps. Unfortunately, due to high numerical cost, we were not able to run the models well below the 1 s timestep to further explore this behavior. Concerning the regularized models (1, 2, and 3), our results suggest that model 1 consistently yields the lowest RMSDs for timesteps of 900 s and less. The comparison of the errors of models 2 and 3 for timesteps between 112 and 900 s depends on the specific test case and the variable of interest, with no clear advantage of one over the other. We note that under some circumstances (e.g. test case 2 and timestep of 900 s), one model can appear better in one metric (e.g. LWC RMSD) while it is outperformed in another metric (e.g. temperature RMSD).

As seen above, the use of a tightly-coupled solution in numerical models usually provides a slightly better precision and a better robustness , at the expense of a more complex numerical solution. Indeed, by ensuring physical consistency between the different variables, tightly-coupled solutions are known to prevent overshooting, which could develop into numerical oscillations or even divergence . One of the motivation behind the investigation of the impact of operator-splitting on the numerical solution of the energy and mass budgets with liquid water matric flow was thus to examine whether a tightly-coupled solution was warranted in this case. To our surprise, the problem of heat conduction, liquid matric flow, and phase changes, appears to be quite robust to operator splitting. Indeed, even in the case of a quite large timestep of 4 h, we did not observe any clear sign of numerical instabilities or very large overshoots in the models using operator-splitting. Our understanding is that this overall insensibility to operator-splitting comes from the nature of the involved processes, which do not play in the same regimes. Indeed, the process of heat conduction only occurs in dry snow, the process of matric flow in wet snow, and the process of phase change at the wet-dry interface. The possibility of interaction between processes thus remains limited and the large physical inconsistencies, that can be problematic with operator-splitting, did not emerge in most situations. However, as explained in Sect. , it appears that the models with phase changes decoupled from heat conduction and liquid percolation (i.e. models 3, 4, and 5) do not yield well-converged solution with timesteps of 1 s. While we were not able to precisely explain this behavior, the fact that it appears in models 3, 4 and, 5 might suggest that it is linked with the use of operator-splitting for phase changes. However, this point needs to be further investigated. Therefore, the standard practice of operator-splitting seems to be overall well-justified, at least with timesteps of the order of 900 s and when representing the interaction of matric flow with heat conduction and phase changes in snowpack models.

In this article, we considered the coupled mechanisms of heat conduction, matric liquid water percolation, and liquid-solid phase changes. While we focused on these mechanisms, as they can be found in all snowpack models (with more or less degrees of complexity), other thermodynamical mechanisms could be included as well. This is for instance the case of water vapor diffusion and vapor phase changes. To the best of our understanding, vapor-related mechanisms could be easily added to the systems of the thermodynamical equations governing snowpacks. For that, one would need to (i) introduce a new partial differential equation governing the mass balance of water vapor, (ii) introduce a water vapor component in the energy balance equation, and (iii) close the system by prescribing the phase change term for the gas phase (or assuming water vapor thermodynamical equilibrium with the other phases) and parametrizing the diffusivity of vapor. This could be done based on works of , , , and . Similarly, an important mechanism relating to the transport of liquid water in snowpacks is the process of preferential (i.e. non-matric) flow . While the exact mechanisms, and therefore description, of preferential flow in snowpacks remain unclear at this point , the presence of fast flowing, and out-of-equilibrium with the rest of the snow layer, liquid water appears as a prerequisite for the formation of internal horizontal ice layers. In this picture, the preferential flow transports liquid water through cold snow layers down to a capillary barrier, where the liquid water can then horizontally spread and refreeze as a horizontal crust . This is illustrated by the close relation between refrozen preferential paths (i.e. ice columns) and internal horizontal crusts, for instance illustrated in picture #58 of . As percolation chimneys are likely to play a key role for preferential flow, the use of a so-called dual-porosity/dual-permeability model e.g. has notably been proposed to take into account preferential flow in the SNOWPACK model . As preferential flow shares many similarities with matric flow (being essentially a faster version), the governing equations are subjected to the same degenerate behaviors in the case of a dry or a fully-saturated medium. Therefore, the techniques explored in this article, namely the regularization of the WRC to handle dry media with the use of a switch variable, could also benefit the numerical implementation of preferential flow. Moreover, as preferential flow is a fast and out-of-equilibrium process, it can introduce stiffness in the equations governing snowpacks , potentially resulting in overshoots and oscillations when using operator splitting with a large timestep . As discussed above, the derivation of a fully-consistent system of equations, that applies from dry to water-saturated snow, enables a tightly-coupled numerical solution, which could be key to handle stiff and fast systems of equations .