The thermal state of an ice sheet is an important control on its past and future evolution. Some parts of the ice sheet may be polythermal, leading to discontinuous properties at the cold–temperate transition surface (CTS). These discontinuities require a careful treatment in ice sheet models (ISMs). Additionally, the highly anisotropic geometry of the 3D elements in ice sheet modelling poses a problem for stabilization approaches in advection-dominated problems. Here, we present extended enthalpy formulations within the finite-element Ice-Sheet and Sea-Level System model (ISSM) that show a better performance than earlier implementations. In a first polythermal-slab experiment, we found that the treatment of the discontinuous conductivities at the CTS with a geometric mean produces more accurate results compared to the arithmetic or harmonic mean. This improvement is particularly efficient when applied to coarse vertical resolutions. In a second ice dome experiment, we find that the numerical solution is sensitive to the choice of stabilization parameters in the well-established streamline upwind Petrov–Galerkin (SUPG) method. As standard literature values for the SUPG stabilization parameter do not account for the highly anisotropic geometry of the 3D elements in ice sheet modelling, we propose a novel anisotropic SUPG (ASUPG) formulation. This formulation circumvents the problem of high aspect ratio by treating the horizontal and vertical directions separately in the stabilization coefficients. The ASUPG method provides accurate results for the thermodynamic equation on geometries with very small aspect ratios like ice sheets.
Ice sheets and glaciers are important components of the climate system. Their evolution is one of the primary sources of sea-level change
Ice sheets and glaciers can exhibit a polythermal state that includes both cold (below the pressure melting point) and temperate (at the pressure melting point) domains, separated by the cold–temperate transition surface (CTS)
Modern state-of-the-art ice sheet models (ISMs) simulate the thermal state according to the enthalpy method originally formulated in
In ISMs, the governing thermodynamic equations are discretized, e.g. using the finite-element method (FEM). Special care has to be taken with regard to the parabolic thermodynamic equation as numerical instabilities inherent in the advection component of this equation tend to occur without stabilization. When employing the FEM, the standard Galerkin finite-element method is often stabilized with the popular streamline upwind Petrov–Galerkin (SUPG) method
ISMs deal with a low aspect ratio, since the ice vertical extent (up to
Beside the need for efficient stabilization in FEM, the phase change in the enthalpy formation leads to discontinuous thermal properties. This feature needs to be handled with care when seeking a numerical solution. Of particular concern are discontinuities of the thermal conductivity
We describe and analyse here recent developments designed to obtain an enthalpy formulation within the finite-element model ISSM that performs well over a wide range of grid aspect ratios in advection-dominated problems. The focus of this work is twofold: on the one hand, we focus on treatments of discontinuous conductivities at the CTS. Here, we test three formulations for the discontinuous conductivity proposed in
Let
In most cases, the liquid water fraction increases but temperature decreases towards the base because of the Clausius–Clapeyron relation. Therefore, the transport of latent heat down the liquid water fraction gradient (Eq.
Dirichlet boundary conditions are imposed at the upper surface in all setups. The type of basal boundary condition (Neumann or Dirichlet) is time dependent and follows the decision chart for local basal conditions given in
In ISSM
The stabilization parameter,
The standard stabilization techniques were initially developed for isotropic meshes, which essentially require that the elements have a similar size in all spatial directions. Once the elements become anisotropic, the local mesh parameter plays an important role in the calculation of stabilizing coefficients. Various definitions have been utilized based on, e.g., the maximum edge length, minimum edge length, circumradius of an element, and the element length aligned with the upwind direction
In order to develop a new SUPG stabilized method for anisotropic meshes, which accounts for geometrical information from the mesh, we consider a Cartesian three-dimensional mesh with prismatic elements. In doing so, we split the traditional SUPG formulation into a horizontal and vertical direction with the stabilization parameters
Since the conductivity is discontinuous at the CTS, special attention must be paid to the treatment of the effective conductivity The effective thermal conductivity is the weighted arithmetic mean: The effective thermal conductivity is the weighted harmonic mean: The effective thermal conductivity is given by the weighted geometric mean:
The weighting term
The applicability of the three models is controversial in the literature and depends strongly on the problem
Constants and model parameters used.
We ran several experiments with the emphasis on testing our modifications in ISSM regarding accuracy and regarding stability. The discontinuous conductivity treatments are verified against an analytical solution within a polythermal-slab experiment. As this experiment results effectively in a one-dimensional vertical experiment, it is not suitable to test the SUPG parameter choices. Therefore, we set up a synthetic second ice dome experiment with variations in the topography. Constants and model parameters used in the experiments are summarized in Table
We repeat the well-established polythermal sided slab experiment proposed in
List of employed stabilization approaches.
An analytical solution for the steady-state enthalpy profile based on the solution of
In this experiment, a more realistic setup than the polythermal-slab experiment is considered with a three-dimensional ice dome based on the Vialov profile
In this experiment, a thermo-mechanical coupling is considered. The Glen–Steinemann power-law rheology
For the dynamical model, we employ the higher-order Blatter–Pattyn approximation
For the thermal model, we impose a Dirichlet condition at the surface:
To investigate the sensitivity of over- and under-stabilization, we perform experiments with three different stabilization formulations (Table
To study whether the stabilization is dependent on different mesh resolutions and the amount of advection, we vary the horizontal grid size and the amount of sliding. Here, we use a base mesh of 20 km in the interior, which is subsequently refined to
The final steady-state CTS elevations for all simulations are shown in Fig.
Difference of simulated steady-state CTS elevations to the analytical CTS elevation for different values of the temperate ice conductivity,
Simulated steady-state profiles of the enthalpy
Root-mean-square error (RMSE) for the polythermal-slab experiment. The RMSE is computed between the modelled enthalpy result and the analytical solution for different vertical grid resolutions
Simulated enthalpy (kJ kg
Simulated enthalpy (kJ kg
The steady-state results of the three conductivity models are verified with the analytical solution of the vertical enthalpy profile. Figure
The accuracy of the simulations with the lowest conductivity ratio is measured with the root-mean-square error (RMSE) to the analytical solution. The RMSE as a function of vertical resolution is shown in Fig.
The different behaviours highlight the dependency of the solution on the CTS implementation details. As already identified by
In this experiment, we explore the impact of the parameter choices in the SUPG formulation on the reliability and accuracy of the results. In Fig.
Simulated depth-averaged enthalpy (kJ kg
Simulated steady-state profiles of the enthalpy
Surprisingly, SUPG maxK and ASUPG are visually indistinguishable and result in qualitatively similar results. However, when re-running the polythermal-slab experiment with the three SUPG formulations, distinct differences in the simulated enthalpy are obtained (Fig.
Our results demonstrate that choosing the stabilization parameter in a heuristic or ad hoc manner, without knowledge of the possible effects, can impact the solution significantly. Choosing a sub-optimal value for the stabilization parameter can affect the accuracy of the solution and result in over- or under-stabilization. The viability of the SUPG formulation strongly depends on appropriate parameter choices, and in a worst-case scenario, the oscillations could cause unphysical values or the solver to diverge. However, we have not investigated how the solution differences propagate to other components of an ice sheet model, e.g. by coupling to the evolution of the ice thickness.
Since the above-presented solutions for the ASUPG method are excellent, the parameter choices for the local mesh parameters
We presented extended enthalpy formulations within the ice flow model ISSM compared to
Additionally, we tested various SUPG stabilization formulations regarding their ability to deal with the high aspect ratio of 3D elements in glaciological applications. We found that the traditional parameters in the SUPG stabilization coefficients are susceptible to stabilization parameter choices, here the local mesh parameter, which is easily adjustable. We propose a novel anisotropic SUPG (ASUPG) method that circumvents the high aspect ratio problem in ice sheet modelling by treating the horizontal and vertical direction separately in the stabilization coefficients. The ASUPG method provides accurate results for the thermodynamic equation on geometries with very small aspect ratios like ice sheets.
The ice flow model ISSM version 4.17
MR conducted the study supported by the other authors. MR set up the experiments conducted with ISSM and analysed the experiments. MM and HS provided technical ISSM support. MR wrote the paper together with the other authors.
The authors declare that they have no conflict of interest.
Martin Rückamp acknowledges support of the Helmholtz Climate Initiative REKLIM (Regional Climate Change). We thank Stephen Cornford and Alexander Robinson for the reviews that helped to improve the paper. For discussions and suggestions we thank Vadym Aizinger (University Bayreuth), Yonqi Wang (University Darmstadt), and Luca Wester (University of Erlangen).
The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
This paper was edited by Philippe Huybrechts and reviewed by Alexander Robinson and Stephen Cornford.