The two-scale expansion for (vapor) mass and energy leads to the following set of equations for the vapor density ρv and the temperature T: 1(1-ϕi)∂∂tρv-∇⋅Deff∇ρv=-ρisvn‾,2(ρC)eff∂∂tT-∇⋅keff∇T=Lsvn‾. Here ϕi is the ice volume fraction, Deff the effective diffusion coefficient, ρi the constant density of ice, s the surface area density per unit volume, vn‾ the volume-averaged normal velocity of the ice–air interface indicating deposition or sublimation, (ρC)eff the effective volumetric heat capacity (times snow density), keff the effective thermal conductivity and L the latent heat of sublimation.

The surface area density s, reflecting the current state of the microstructure and principally evolving in time, is assumed to be constant here in accordance with .

The source term -ρisvn‾ on the righthand side (r.h.s.) of the vapor Eq. () quantifies phase changes, i.e., the net condensation rate. This implies the form of the energy source term Lsvn‾ through latent heat. In the simplest setting , the volume-averaged interface velocity is given by 3vn‾=1βρveq(T)(ρv-ρveq(T)), where β is the inverse growth velocity and ρveq(T) the saturation vapor density. Note that vn‾ strongly depends on the temperature, indicating the temperature feedback on the vapor mass balance. In order to close the system of Eqs. ()–(), parametrizations for the effective PDE coefficients must be provided (see below in Sect. ). The feedback of a spatiotemporally evolving ice phase ϕi(z,t) has not been considered in .

A similar coupling scheme of the vapor transport to the energy equation has been put forward in . Using the same notation as above, these equations can be written in the form 4(1-ϕi)∂ρveq∂T∂T∂t-∇⋅Deff∂ρveq∂T∇T=-c,5(ρC)eff∂∂tT-∇⋅keff∇T=cLρi. Here the r.h.s. of both equations is expressed in terms of the condensation rate c.

The comparison of both models reveals that the vapor mass balance of (Eq. ) can be obtained from the corresponding vapor mass balance of (Eq. ) using the following reasoning: the deviation of the vapor concentration δρv from its equilibrium value ρveq(T) will be mostly small. This suggests the use of an ansatz ρv=ρveq(T)+δρv in Eq. (). Assuming further that δρv diffuses faster than it relaxes locally, the deviation from equilibrium is kept nonzero only in the reaction term, which then implies Eq. (). support this derivation with a (non-rigorous) scaling argument to avoid two conflicting equations for the temperature T.

Though both presented models are strikingly similar in structure, it should be emphasized that the system of Eqs. ()–() is closed and can be solved for ρv and T using an arbitrary closure for the condensation rate. This is different for the system of Eqs. ()–(): due to the simplifying assumption outlined above, Hansen's model lacks an evolution equation for the (perturbed) vapor mass balance. Since both vapor mass balance (Eq. ) and energy balance (Eq. ) are actually formulated in terms of the temperature T, proceeds by consolidating both equations into a single equation which eliminates the condensation rate. 6((1-ϕi)∂ρveq∂T+ρiL(ρC)eff)∂T∂t-∇⋅Deff∂ρveq∂T+ρiLkeff∇T=0 The latter now poses a nonlinear yet closed PDE that can be solved for the evolving temperature. Back-substituting the resulting temperature into either Eq. () or Eq. () allows us to evaluate the condensation rate c, without the freedom remaining to impose yet another closure relation. An additional, important implication of the near-equilibrium assumption in is the impossibility to impose arbitrary boundary conditions for the vapor phase. By construction of the model, the boundary conditions are Dirichlet values for the equilibrium vapor concentration consistent with the boundary temperatures. Similar to the feedback of a spatiotemporally evolving ice-phase ϕi(z,t) is not considered in .

Despite similarities in the forms of the PDEs, both models have used different parametrizations for the transport coefficients Deff, keff and (ρC)eff and ρveq. While the two-scale homogenization presented in contains derived expressions for the effective properties in terms of the microstructure as a byproduct, mixture theory in relies on independently postulated parametrizations. The implications of these differences are presently still under debate .

In general, there is a broad agreement that all effective parameters are primarily influenced by the density or ice volume fraction ϕi. The goal of the present paper is not an exhaustive intercomparison of different formulations of the parametrized coefficients in the analyzed snow pack models. We rather want to focus on the intrinsic features and physical implications of the underlying process models. We therefore aim to keep differences due to specific flavors of the coefficients at the minimal level. To this end we use the same parametrization for the equilibrium vapor pressure and the effective heat capacity in both models but use different parametrizations for the effective thermal conductivity and diffusion constant. In the following we refer to these PDE parametrizations as the Calonne parametrization and the Hansen parametrization. All coefficients are stated explicitly in Appendix .

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 nor . To investigate the feedback of an evolving ice phase on the two models from above we supply Eqs. ()–() and ()–() with a dynamic ice mass conservation equation . In the absence of settling but in the presence of phase changes, the continuity equation reduces to an ordinary differential equation for each location in space, 7∂∂tϕi=svn‾(Calonne)cρi(Hansen), to balance the source terms in the vapor equations above. In Calonne's model vn‾ can be evaluated from the closure relation (Eq. ), whereas Hansen's c has to be reconstructed from either Eq. () or Eq. () as outlined before.

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

The nonlinear PDE systems of interest can be rewritten in the form F(u)=0, where F(u) is a nonlinear differential operator, and u is the (vector valued) solution that comprises the unknown fields. Following a standard Galerkin approach, the PDE's weak formulation is obtained by multiplying F(u)=0 with test functions v and integrating over the domain. The Galerkin method approximates the solution as the sum over a discrete set of basis functions {vj}, i.e., u≈∑jujvj. This procedure allows us to apply the differential operator to the test functions. The test functions have a small support in which the values are nonzero, which simplifies the integral evaluations significantly. By choosing the test functions v from the basis {vj}, the PDE system is reduced to finding the roots of a (nonlinear) algebraic system. This standard Galerkin procedure is, for example, detailed in . The implementation in the open-source finite element software FEniCS is straightforward.

Time derivatives are approximated using the theta method, also known as Rothe's method . For a PDE of the form 8∂∂tu=F(u), the discretization reads 9un+1-unΔt=θF(un+1)+(1-θ)F(un)+O((1/2-θ)Δt,Δt2), where un denotes the state vector's solution at time tn. Note that for θ=0 and θ=1 this reduces to standard first-order explicit and implicit Euler methods, respectively, whereas for θ∈(0,1) it constitutes a weighted average of both.

Due to the different timescales of the involved equations, a monolithically coupled solution for the vector u=(T,ρv,ϕi) would be most consistent, yet it turns out to be inefficient. The vapor equation has by far the fastest dynamics, followed by the second fastest, the energy equation. The ice mass balance equations have much slower dynamics. The water vapor and energy equations are coupled diffusion–reaction equations and directly determine the r.h.s. of all equations. Therefore we solve them together in a first step and subsequently integrate the ice equation with its expected slower dynamics in a separate step. This operator splitting allows an independent fine-tuning of the numerical methods. Practically, we included the r.h.s. of the equations as an algebraic constraint in the state vector, i.e., u=(T,ρv,vn‾) for Calonne and u=(T,c) for Hansen, which are then solved in a fully coupled way. The ϕi-dependent coefficients of the vapor and energy equations use ϕi from the previous time step. Subsequently ϕi is updated via vn‾ (or c). Time step size is mostly determined through the vapor and energy dynamics. Changes in ϕi, and hence in the PDE coefficients, are therefore expected to be very small, which justifies the splitting-based coupling procedure.

We apply different theta values for each differential operator. For the diffusion operators in the vapor and energy equations we use θ=0.5 (Crank–Nicolson) which is known to be stable and convergent with second-order accuracy for linear operators. As the phase change (source) terms are expected to be stiff, a fully implicit, unconditionally stable backwards Euler scheme with θ=1.0 is applied.

The first scenario is proposed by and investigates the response of a homogeneous snow layer on transient heating. The initial conditions are 10T(z,0)=T0,11ρv(z,0)=ρveq(T(z,0)),12ϕi(z,0)=ϕi,0, with T0=273 K and ϕi,0=0.3. We employ a fixed Dirichlet BC at z=0 and a transient temperature drop at z=H=1m, viz. 13T(0,t)=T0,T(H,t)=T0-(TH-T0)tτΘ(t)-t-ττΘ(t-τ),ρv(0,t)=ρveq(T(0,t)),ρv(H,t)=ρveq(T(H,t)), where the transient drop is characterized by the Heaviside step function Θ with parameters τ=5h and TH=263 K.

The solutions of all three cases we obtain for this combination of IC and BC are shown in Fig.  for t=100 min. Notably, Calonne's and Hansen's model along with their own (different) parameterizations for the PDE coefficients deviate in all variables since Hansen's effective diffusion constant is higher. This difference does not depend on the underlying process models but is rather due to the formulations used of the PDE coefficients as shown by the mixed case (Han-Eq/Cal-Par) which coincides with Han-Eq/Han-Par. Only a small difference can be observed close to the right boundary, where the changes in T and ρv are the fastest, and therefore the assumption ρv≈ρveq(T) underlying is violated.

Next we investigate the test case envisaged by that comprises a piece-wise linear density profile as an approximation for a layered snowpack containing a crust with an additional ice layer at the bottom to constrain the vapor from below. The IC is given by 14T(z,0)=T0-TH-T0zH,15ρv(z,0)=ρveq(T(z,0)),16ϕi(z,0)=1-9.2425zz∈[0,0.08],0.2606z∈[0.08,0.64]0.2606+4.915(z-0.64)z∈[0.64,0.72]0.6538z∈[0.72,0.75]0.6538-4.915(z-0.75335)z∈[0.75,0.86]0.1295895z∈[0.86,1.0], with T0=273 K and TH=253 K. The BCs are given by 17T(0,t)=T0,T(H,t)=TH,ρv(0,t)=ρveq(T(0,t)),ρv(H,t)=ρveq(T(H,t)). The corresponding simulation results are shown in Fig.  at a simulation time of 38 h. Again, the fields ρv and T highly agree in homogeneous regions as long as the same parameterizations for the PDE coefficients are used. In contrast, the phase change term (r.h.s. of the equations) shows an agreement only if the same equations are used. This is revealed in the kink regions of the profile where Han-Eq/Han-Par coincides with Han-Eq/Cal-Par. This shows the different sensitivities of the T, ρv and phase change terms on the form of the homogenized equation when density gradients are present.

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 H=0.02m. This scenario investigates the response of the models to density changes occurring on small length scales, e.g., layer interfaces which can be monitored in tomography experiments . In contrast to the previous two scenarios, the ice phase is now evolving according to Eq. ().

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 18T(z,0)=T0-(TH-T0)zH, 19ρv(z,0)=ρveq(T(z,0)), 20ϕi(z,0)=ϕi,0+Δϕi,0exp⁡-(z-z0)22σ2, with T0=273 K, TH=253 K, ϕi,0=0.3, Δϕi,0=0.2, z0=0.01 m and σ2=5×10-7 m2. For the Gaussian crust we use the same boundary conditions as in Eq. ().

Figure  shows the simulation results for several points in time. Similar to before the solutions for ρv and T agree very well, and notable differences are observed for the phase changes in density transition regions. From this comparison for a smooth density variation, two additional observations can be made for both models. These are a consequence of the evolving ice phase.

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 following example shows simulations with varying mesh size (number of elements ne) and varying time steps (dt) for the numerical solution at a fixed physical time. The results are shown in Fig. . Figure a shows the entire domain, revealing oscillations at the left boundary and on the sublimating side of the crust. Figure b shows a close up of the left boundary, while Fig. c shows a close-up of the sublimating side of the crust. For very low mesh resolution (ne=100, corresponding to a mesh size of 0.2 mm) and large time step (1 min), the solution in the vicinity of the crust oscillates from node to node, while for sufficiently high resolution the numerical solution converges to a smooth wave pattern independent of temporal and spatial resolution. This suggests that these waves are an intrinsic feature of the Calonne model equations rather than an artifact of the numerical scheme.

As an additional check we have analyzed the residuals of the numerical solution (see Appendix ) which confirms that patterns persist even when enforcing the residuals to approach zero.

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 26(ρC)eff∂tT-∇zkeff∇zT=Lα(ρv-ρveq),(1-ϕi)∂tρv-∇zDeff∇zρv=-αρi(ρv-ρveq),∂tϕi=α(ρv-ρveq), which is only subject to the simplification that on the r.h.s. the division by ρeq(T) in Eq. () is subsumed in the prefactor α which is assumed to be constant.

The system () is, as before, equipped with Dirichlet BC for the two PDEs, 27T(0)=T0,T(H)=TH,ρv(0)=ρveq(T0),ρv(H)=ρveq(TH), at the bottom z=0 and top z=H of the domain. To proceed analytically, we need to make an additional assumption to obtain an exact stationary solution of the system. We assume that the equilibrium vapor concentration is a linear function in T. This is reasonable within thin layers where temperature differences are small and a linearization of the equilibrium vapor curve is always justified. This linearization is written in the form 28ρveq(T)=ρ0eq+ρ1eq(T-Tref) as an expansion around a reference temperature Tref (e.g., the mean temperature in the sample). With this assumption, a stationary solution of Eq. () can be obtained by inspection. It is easily verified that the system 29T(0)(z):=T0+(TH-T0)zH,ρv(0)(z):=ρveq(T(0)(z)),ϕi(0)(z):=ϕi,0, satisfies Eq. () with BC Eq. () for arbitrary, constant volume fraction ϕi,0.

To carry out the stability analysis we use a vector notation and combine the fields in the vector u:=(T,ρv,ϕi). Then the PDE system () can be written in the matrix form 30C(u)∂tu-K(u)∇z2u-(∇zK(u))(∇zu)=Ru+f with state-dependent (3×3) matrices K(u),C(u), constant matrix R and constant vector f defined by 31C=(ρC)eff000(1-ϕi)0001,K=keff000Deff0000,R=-Lαρ1eqLα0ρiαρ1eq-ρiα0-αρ1eqα0,f=Lαρ1eqTref-Lαρ0eq-ρiαρ1eqTref+ρiαρ0eqαρ1eqTref-αρ0eq. Then, the (stable) stationary state of Eq. () is denoted by the vector 32u(0):=(T(0),ρv(0),ϕi(0)). The nonlinearities of Eq. () emerge from the dependence of the matrices K≡K(u) and C≡C(u) on u. Note that this dependence is solely through the third component of u (i.e., ϕi) which highlights the special role of the density (evolution).

Next we investigate the linear stability of the fixed point (). To this end we make an ansatz, 33u=u(0)+u(1), and investigate the dynamics of the deviation from the stationary state u(0) by deriving an equation for u(1) through a perturbation expansion of Eq. () in first order. This procedure (carried out in detail in Appendix ) yields an equation for the Fourier modes ũ(1)(k) of the perturbation for wave number k which can be written in the form 34∂tũ(1)(k,t)=Mkũ(1)(k,t), with a wave-number-dependent matrix, 35Mk:=-k2C(0)-1K(0)+ikC(0)-1V(0)+C(0)-1R, with constant matrix coefficients K(0),V(0) and C(0). The eigenvalues of Mk control the behavior of the system in the vicinity of the stationary state. The matrix Mk is non-Hermitian due to the “advection” matrix V(0) and the “phase change” matrix R. Thus these two processes can principally cause eigenvalues with positive real parts (causing a growth of perturbations) and nonzero imaginary parts (causing oscillatory behavior). In the case of V(0)=R=0, Eq. () predicts negative eigenvalues for all k>0 and no pattern formation as it should be for simple diffusion equations.

The matrix Mk can be diagonalized in closed form using the symbolic algebra software MAPLE and eigenvalues subsequently separated into real and imaginary parts. Evaluating the eigenvalues with the corresponding parameter values from Table  shows that one of the eigenvalues has positive real part and nonzero imaginary part indicating a wave instability. This is controlled by the matrix V(0) with the coefficients k1 and D1 (see Eq.  in Appendix ) which characterize the sensitivity of the effective coefficients on density. Thus the instability originates from the fact that the effective diffusivity or the effective conductivity of snow depend on ice volume fraction (or snow density), and it is triggered by transitions in snow density, i.e., layers.

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 k through the power spectral density of the perturbation. To this end we have computed the third component of the perturbation vector u3(1)=ϕi(z,t)-ϕi(z,0) as the deviation from the initial, oscillation-free state. The space–time plot of the perturbation in real space is shown in Fig. a which shows the emergence of the traveling wave. Discrete time steps are shown in Fig. b which displays the self-amplifying behavior of the density modulation in the layer-transition region, i.e., at the boundary of the crust. The power spectrum |ũ3(1)|2 of the perturbation is computed via fast Fourier transform and shown in Fig. c as a function of wave number for different times, together with the theoretically predicted range of instability (eigenvalues with positive real part and nonzero imaginary part, gray region) derived from the diagonalization of the matrix Mk. The comparison confirms that growing modes are consistently lying in the unstable range. This supports our theoretical explanation that wave-like patterns emerge in regions of high-density gradients as an inherent feature of the coupled heat and mass transport in snow with density-dependent material properties.

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 and has shown that both schemes yield similar results as long as the same parametrizations are used as closure for the PDE coefficients. The impact of purely diffusive vapor transport on macroscopic density changes is rather small (Fig. ), in agreement with . Accurate estimates will certainly rely on the choice for the parametrization of the transport coefficients which naturally cause differences in the results (Fig. ). In homogeneous parts of the snowpack these differences are small, and larger differences can be expected at layer transition regions (Fig. ) where phase changes sensitively react to the underlying homogenized process model equations.

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 (, , and ). While we have certainly challenged the numerical scheme by predominantly exploring high-temperature and high-density gradients, the observed time–gradient scaling (Fig. ) indicates that the underlying nonlinear mechanisms are generally robust against these details.

We have discovered that both models exhibit oscillations in the numerical solution if they are dynamically coupled to an evolving ice phase (Eq. ). From our analysis of different combinations of initial and boundary conditions (Sect. ) we conclude that two types of oscillations can occur: node-to-node oscillations (indicating numerical problems) and smooth oscillations (features of the underlying equations)

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. ). The nonlinear nature of this advection causes a self-amplified increase in density changes and layer transitions, imposing special requirements on mesh resolution, time stepping and potentially shock-capturing schemes to avoid oscillations. The fact that node-to-node oscillations occur on the “outflow” boundary of the ice phase supports this. In most of our simulations we have imposed a Dirichlet boundary condition of the vapor phase which is strictly in equilibrium on the boundary. By virtue of Eq. () this implies that the snow density cannot change directly at the boundary, while in the interior of the domain the ice is piling up towards the boundary under the advection (Fig. ). This is reminiscent of outflow boundaries in fluid dynamics for high local Péclet numbers : the disagreement between the information transported through the domain from the cold boundary and the information prescribed on the warm boundary is numerically resolved by steep node-to-node oscillations (Fig. ). Similar issues can arise whenever abrupt changes in the advection velocity cannot sufficiently be resolved on a given mesh, as in Fig.  in the middle of the domain. This interpretation of node-to-node oscillations is supported by the fact that oscillations can be completely suppressed on the lower boundary by choosing a boundary condition, ∇ρv=∂/∂Tρveq∇T, for the vapor equation, which implies a vanishing gradient for the ice profile at the boundary. Our tests indicated that node-to-node oscillations can be eliminated by choosing a high resolution in space and time. A more efficient approach might be the utilization of stabilization methods for which our approximation (Eq. ) may serve to provide a local advection velocity of the ice phase which needs to be supplied, for example, to a streamline upwind Petrov–Galerkin stabilization scheme .

After increasing mesh and time resolution to suppress numerical problems, the true solution converges to a smooth, traveling wave pattern (Figs. , ). As confirmed by the theoretical analysis, these oscillations are true features of the nonlinear homogenization equations and are triggered by gradients in the density (Sect. ). These patterns can only form when the ice equation is dynamically coupled to the heat and vapor equation in a mass-conserving way. At the boundary, these physical oscillations are triggered as a consequence of the Dirichlet boundary condition as explained above: the competition of a fixed value of ϕi directly on the boundary for z=0 (implication of the imposed boundary conditions on the vapor phase for Eq. ) and the increase in ϕi in the interior of the domain gives rise to a transition layer at the boundary with a high-density gradient in the ice phase that triggers the instability according to the mechanism revealed in Sect. . This is again supported by the fact that the physical oscillations at the boundary can be suppressed by choosing the BC for the vapor equations such that the gradient in ice volume fraction vanishes. This behavior is reminiscent of the wave instability triggered by a Dirichlet boundary condition in a 1D nonlinear reaction–advection–diffusion system which travels upstream into the domain.

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 . We have investigated this phenomenon within the idealized scenario of a Gaussian crust where density gradients self-amplify under the nonlinear advective dynamics with a subsequent instability and the emergence of wave patterns. The eigenvalues of the linearized PDE system in our linear stability analysis have revealed the mechanism: a non-homogeneous density paired with the density dependence of the effective diffusion coefficients triggers the instability. Pattern formation following an instability is ubiquitous in nonlinear (diffusion–reaction) PDEs .

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. a), it explains why a low-density (mechanically weak) layer can form under these conditions. For high-density crusts, the parameters and model components used here predict a considerable reduction in the density in a sub-millimeter layer above the crust (Fig. a). This mechanism is solely a consequence of the continuum description of snow. We stress that this is different from a previously suggested reasoning , and it may occur as a superimposed effect on additional microstructural controls through near-crust metamorphism . In the future, it will be therefore important to assess how this instability is affected by adding mechanical settling, an evolving microstructure or enhanced mass transport from convection.

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 (keff,Deff) or boundary conditions, e.g., for more sophisticated exchange of vapor with the soil or the atmosphere. Alongside our study, we evaluated the FEniCS framework by comparing numerically different implementations. These experiences are shared for future reference.

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 ϕi in first order and ρv and T in second order caused large errors in the entire domain and also violation of the Dirichlet boundary condition. Adding auxiliary algebraic variables (as done here for the source terms) can improve the solvability of the system when convergence settings of the coupled system are adapted. Using so-called sub-functions in the formulation of the variational problem for coding modularity can introduce errors. We encountered deviations between a sub-function value and its hard-coded counterpart.

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 .