The incorporation of vapor transport has become a key demand for snowpack modeling in which accompanied phase changes give rise to a new, nonlinear coupling in the heat and mass equations. This coupling has an impact on choosing efficient numerical schemes for 1D snowpack models which are naturally not designed to cope with mathematical particularities of arbitrary, nonlinear partial differential equations (PDEs). To explore this coupling we have implemented a stand-alone finite element solution of the coupled heat and mass equations in snow using the computing platform FEniCS. We focus on the nonlinear feedback of the ice phase exchanging mass with a diffusing vapor phase with concurrent heat transport in the absence of settling. We demonstrate that existing continuum-mechanical models derived through homogenization or mixture theory yield similar results for homogeneous snowpacks of constant density. When snow density varies significantly with depth, we show that phase changes in the presence of temperature gradients give rise to nonlinear advection of the ice phase amplifying existing density variations. Eventually, this advection triggers a wave instability in the continuity equations. This is traced back to the density dependence of the effective transport coefficients as revealed by a linear stability analysis of the nonlinear PDE system. The instability is an inherent feature of existing continuum models and predicts, as a side product, the formation of a low-density (mechanical) weak layer on the sublimating side of an ice crust. The wave instability constitutes a key challenge for a faithful treatment of solid–vapor mass conservation between layers, which is discussed in view of the underlying homogenization schemes and their numerical solutions.

Neglecting vapor transport in the pore space of a snowpack for the overall mass balance
is considered as a serious uncertainty in snow modeling. Persistent temperature gradients throughout the season may contribute to the depletion of snow density at the bottom of the snowpack due to upward vapor fluxes, as has been hypothesized for shallow tundra snowpacks by

The governing equations of macroscopic vapor transport in snow have been used for a long time. The homogenized equations of heat and purely diffusive vapor transport including phase changes have been derived from mixture theory in early work

Recently

Due to the lack of analytical solutions for these nonlinear problems, confidence in orders of magnitudes of computed numbers can only be achieved via careful numerical experiments to address solver accuracy or mesh effects. This is naturally cumbersome within a full snowpack model. In addition, an explicit solution of the ice mass conservation equation is commonly avoided by using the Lagrangian frame of reference of the settling equation

It is the aim of the present paper to advance the understanding of coupled heat and mass transport in snow by a careful numerical analysis of existing homogenization schemes. To overcome the limited flexibility in existing snowpack models we have implemented a stand-alone solver for the PDEs using the finite element (FE) framework FEniCS

The paper is organized as follows. In Sect.

As a theoretical starting point we focus on two recently published homogenized formulations for an evolving vapor phase, namely

The two-scale expansion for (vapor) mass and energy

The surface area density

The source term

A similar coupling scheme of the vapor transport to the energy equation has been put forward in

The comparison of both models reveals that the vapor mass balance of

Though both presented models are strikingly similar in structure, it should be emphasized that the system of Eqs. (

Despite similarities in the forms of the PDEs, both models have used different parametrizations for the transport coefficients

In general, there is a broad agreement that all effective parameters are primarily influenced by the density or ice volume fraction

The consistent treatment of phase changes requires a dynamic ice phase that evolves through recrystallization alongside the vapor phase in a mass-conserving way. This was considered in neither

We summarize all symbols and parameter values of the models in Table

Symbols, defining equations and constants used in this study.

To minimize the coding overhead and focus on the physical problem while keeping access to advanced numerical adjustments, the coupled
PDE model was implemented using the python-based finite element framework FEniCS

Below we outline the spatial and temporal discretization for the weak formulation of both systems (Eqs.

The nonlinear PDE systems of interest can be rewritten in the form

Time derivatives are approximated using the theta method, also known as Rothe's method

Due to the different timescales of the involved equations, a monolithically coupled solution for the vector

We apply different theta values for each differential operator. For the diffusion operators in the vapor and energy equations we use

To evaluate the models, we tested them on three different

The first scenario is proposed by

The solutions of all three cases we obtain for this combination of IC and BC are shown in Fig.

Comparison of Calonne's and Hansen's model for the response to a transient temperature decrease at the boundary without ice-phase evolution: slight differences are observed if the models are used with their own formulation of coefficients (red and green line), while both yield virtually indistinguishable results if the same coefficients are used (red and turquoise line).

Next we investigate the test case envisaged by

Comparison of Calonne's and Hansen's model for the response to a piecewise-linear, layered profile under a temperature gradient without ice-phase evolution: the results for Hansen's model are indistinguishable even if different PDE coefficients are used (green and turquoise lines) but differ from the Calonne model (red line).

To further detail the response to high-gradient regions in the ice phase, our third scenario investigates the response of the models to a smooth, small-scale density variation under a temperature gradient in a shallow snow sample of height

In the following we solely focus on differences between Cal-Eq/Cal-Par and Han-Eq/Han-Par and investigate this difference at different physical times. We refer to this scenario as a Gaussian crust. As IC we employ

Figure

Comparison of Calonne's and Hansen's model each with their own formulation of coefficients. Both models are coupled to the evolving ice phase for an initial Gaussian profile under a temperature gradient.

First, the Gaussian crust shows a quasi-advection in the ice phase towards the warm boundary despite the absence of an explicit advection term in the ice equation. During this quasi-advection, density gradients steepen on the cold side and flatten on the warm side. Far away from the crust a linear density gradient emerges in the domain as a consequence of the boundary conditions. The difference between the models lies in the apparent advection velocity. This is consistent with the observed differences in the phase changes since recrystallization rates differ approximately by a factor of 2.

Second, in both models oscillations emerge at the lower boundary. They are modest in the Calonne model and arise at a considerably later time with smaller magnitude. For the Hansen model these are apparent immediately.

These two observations are further detailed below.

The quasi-advection of density heterogeneities in the ice phase despite the absence of an explicit advection term in Eq. (

To understand the origin of this quasi-advection, we approach the coupled system (Eqs.

We stress the significance of this result. First, by using the
approximations for the heat and vapor transfer, the three coupled heat and mass conservation equations can be equivalently cast into a single continuity equation for the ice phase. Second, the form of this continuity equation is a nonlinear and non-local advection equation, which explains the nature of this quasi-advection as a variant of shock formation reminiscent of the nonlinear Burgers equation

To test the derived approximation we have compared the numerical solution of Eq. (

Comparison between Calonne's model coupled to an evolving ice phase and the derived advection Eq. (

Despite remaining differences, the approximation (Eq.

Similar to Eq. (

All numerical solutions so far are subject to oscillations

The following example shows simulations with varying mesh size (number of elements

Sensitivity of the numerical solution to mesh resolution and time step size:

As an additional check we have analyzed the residuals of the numerical solution (see Appendix

We use perturbation theory to further comprehend the oscillatory nature of the solution.
Pattern formation in nonlinear PDE systems can be generally understood by investigating the dynamics of perturbations around a known stationary state via linear stability analysis of the dynamical equations. To do so, we start from the full, nonlinear, coupled, transient situation (Calonne model) in the form

The system (

To carry out the stability analysis we use a vector notation and
combine the fields in the vector

Next we investigate the linear stability of the fixed point (

The matrix

Space–time plot of the perturbation field

We can compare the prediction of the stability analysis with the simulation of the full model for the Gaussian crust by analyzing the growth of amplitude of Fourier modes for wave number

Finally we conducted a few sensitivity studies to facilitate a discussion about the relevance of our findings in future snow modeling.

To confirm that observed wave patterns in the Gaussian crust are robust against variations of absolute values in temperature gradients and initial crust density, we simulated a denser Gaussian crust with different values for the temperature gradient. Figure

Simulation of the fully coupled system:

Second we subjected a smoothly varying density profile to a temperature gradient of 50 K m

Finally, we mimic the situation of a snowpack over a dry soil where no vapor can enter the system from below by imposing a zero-flux Neumann
BC on the vapor equation. The results are shown in
Fig.

We have revisited published models of coupled heat and (diffusive) vapor transport in snow. To investigate their numerical requirements we have implemented a stand-alone solver in the open-source software FEniCS for the sake of flexibility in numerical experiments involving spatial and temporal resolution, solution strategies, and accuracy.

From a mere physical point of view, our comparison of

For the present work, however, the precise numbers were of minor importance. The primary goal was an assessment of the nonlinearity that is contained in published vapor homogenization schemes and their numerical requirements in view of previously reported numerical issues (

We have discovered that both models

In view of numerical problems, we have shown that the dynamic coupling of the ice phase to heat and mass transport is equivalent to an advection of the ice phase (Sect.

After increasing mesh and time resolution to suppress numerical problems, the true solution converges to a smooth, traveling wave pattern (Figs.

We have shown that spatial heterogeneities in the density and ice volume fraction can amplify under the coupled thermodynamic description in snow. This has been pointed out before

In the present case, the observed instability may have far-reaching consequences, which comes as a (surprising) side-product of our numerical study: as the instability causes the depletion of density on the sublimating side of a crust (Fig.

The present work poses questions on the limits of validity of the homogenization schemes used that are relevant for a user of the equations.

First, as detailed in

Second, the two-scale homogenization

Any snowpack contains variations in its properties that may be described either by continuous profiles containing large gradients (as done here) or by a discontinuous stacking of layers. We want to point out here that if a continuous description of density variations were to be replaced by discontinuous layers, the investigated wave problem will not simply disappear. In a hypothetical discontinuous layer description, as commonly pursued in snowpack models, the mass continuity of the ice phase at the layer interface would require us to derive a dynamic (nonlinear) equation for the migration of the layer interface on which the continuity of temperature and heat and mass fluxes were to be imposed. From the continuous description used here, no firm conclusion can be drawn on the behavior of the interface evolution between discontinuous, homogeneous layers. We hypothesize though that the wave instability of the continuous PDE formulation may translate into an oscillatory instability for the position of the interface. In view of the mathematical overhead of tracking continuity conditions at moving interfaces, we tend to recommend a continuous description in future snow modeling with numerical schemes that cope with arbitrary gradients in the properties.

We have used FEniCS for the stand-alone implementation to minimize the finite element method implementation effort while retaining full control of the numerical solution. Overall FEniCS provides a convenient, modular setup for exchanging
PDE coefficients (

We found that integration by parts in the weak formulation is not only necessary to apply Neumann boundary conditions but also increases the precision, regardless of the order of interpolation polynomials. Operator splitting turned out to be of limited value, the decrease in precision was not outweighed by a decrease in numerical complexity. As expected, a non-dimensionalization of the equations and the corresponding re-scaling to values of order unity did not impact the solution, as long as solver convergence settings are adapted. Increasing the polynomial order of the test functions was equivalent to
increasing the mesh resolution with the corresponding number of nodes. However, we experienced large errors if the polynomial order of the variables of the coupled equations was not the same: solving

While FEniCS has provided an excellent numerical framework for the present study, a clear drawback of FEniCS would however emerge if mechanical settling was to be considered as a necessary, future extension. In the presence of settling, the ice-phase conservation equation is an advection-dominated problem which is notoriously difficult to solve on a Eulerian FE mesh without numerical smoothing. Re-meshing is currently not supported in FEniCS. To this end we have explored another, fully different numerical route to enable a flexible coupling of transport and phase changes to mechanical settling which is presented in Part 2 of this companion paper

We have shown that the widely accepted form of homogenized vapor transport equations in snow predicts

Combining numerical experiments with theoretical considerations we have shown that the wave instability originates from the dependence of the effective heat and vapor diffusion coefficients (

The instability is triggered by high-density gradients which either (pre-)exist in snowpacks in the form of layers or are generated from the self-amplification of density gradients under the coupled dynamics. This amplification is a consequence of the effective advection of the ice phase due to phase changes. This is explained within the derivation of an approximate, nonlinear and non-local advection equation. Given the observed (approximate) equivalence between time and imposed temperature gradient, the system always undergoes a self-propelled evolution into its own instability. The instability might be practically irrelevant as long as smooth density profiles are considered. But the instability will certainly become relevant if a snowpack model should be applicable to simulate a sublimating side of a crust as a potential origin of weak layer formation.

We have outlined open questions and limitations of the present study related to the homogenization scheme, the numerical scheme and the concept of discontinuous layers. While the present study required a stand-alone numerical implementation, it seems to be of key importance that future snow models will be flexible enough for conducting advanced numerical studies. Only then will re-implementations of stand-alone numerical experiments become obsolete and the rich, nonlinear behavior of snow be able to be predicted from a snow model alone.

For the equilibrium vapor pressure we used the parametrization from

For the effective heat capacity we used a volume-averaged formulation given in

In the Calonne case we used for the effective diffusion parameters Eq. (12) from

In the Hansen case we used Eqs. (87) and (88) from

Due to the lack of an analytical solution for the nonlinear problem, we used the nodal residuals as an indicator for the solution error. FEniCS does not provide access to the nodal residuals to verify convergence in the

Results upon lowering the residuals of all equations. While some pattern is still visible, the order of magnitude is small enough to assume small errors in the solution as well.

Comparison between node-to-node oscillation patterns for the regular setup and the low-residual setup.

Carrying out the perturbation expansion of the PDE system (Eq.

First, for any matrix

Second, the expansion around the fixed point of an arbitrary matrix

Now we can state the expansion of the PDE system (Eq.

Equating zero- and first-order terms in

Generally, it seems feasible (though tedious) to carry out an expansion of Eq. (

The python FEniCS code for running all simulations and output analysis to reproduce the key Fig. 5 is available via

The study was designed by HL and JK. Implementation, simulations and figures were done by KS with contributions and supervision from HL and JK. Derivation of the quasi-advection and the linear stability analysis was done by HL. The manuscript was written by HL with contributions from KS and JK.

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

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

We thank Andy Hansen for discussions and sharing his experience on his numerical implementation.

This paper was edited by Mark Flanner and reviewed by two anonymous referees.