The flow of large ice sheets and glaciers can be simulated by solving the full Stokes equations using a finite element method. The simulation is particularly sensitive to the discretization of the grounding line, which separates ice resting on bedrock and ice floating on water, and is moving with time. The boundary conditions at the ice base are enforced by Nitsche's method and a subgrid treatment of the grounding line element. Simulations with the method in two dimensions for an advancing and a retreating grounding line illustrate the performance of the method. The computed grounding line position is compared to previously published data with a fine mesh, showing that similar accuracy is obtained using subgrid modeling with more than 20-times-coarser meshes. This subgrid scheme is implemented in the two-dimensional version of the open-source code Elmer/ICE.

Numerical simulation of ice sheet flow is necessary to assess the future sea-level rise (SLR) due to melting of continental ice sheets and glaciers

The distance that the GL moves may be long over paleo-timescales.
In

When the ice rests on the ground and is affected by large frictional forces on the bed, the ice flow is dominated by vertical shear stresses.
On the other hand, when the ice is floating on water, it is the longitudinal stress gradient that controls the flow of the ice. The GL is in the transition zone between these two types of flow with a gradual change of the stress field

The most accurate ice model in theory is based on the full Stokes (FS) equations.
A simplification of the FS equations by integrating the depth of the ice is the shallow shelf (or shelfy stream) approximation (SSA)

The evolution of the GL in simulations is sensitive to the model equations and the basal friction law.
In the Marine Ice Sheet Model Intercomparison Project (MISMIP)

The friction laws at the ice base depend on the effective pressure, the basal velocity, and the distance to the GL in different combinations in

Parameters in the numerical methods used to simulate ice sheet flow influence the GL migration.

A subgrid scheme introduces an inner structure in the discretization element or mesh volume where the GL is located.
Such schemes have been developed for simplifications of the FS equations.
A subgrid model for the GL is tested in

From the above we conclude that it seems crucial that the ice model include equations with vertical shear stress in the neighborhood of the GL, and one way to avoid the fine meshes that are otherwise needed close to the GL is to introduce a subgrid scheme in the discretization element where the GL is located. In this study, we develop such a numerical method for the FS equations in two dimensions, introducing a subgrid scheme in the mesh element where the GL is located. Since the subgrid scheme is restricted to one element in a 2D vertical ice, this is computationally inexpensive. In an extension to 3D, the subgrid scheme would be applied along a line of elements in 3D. The results with numerical modeling will always depend on the mesh resolution but can be more or less sensitive to the mesh spacing and time steps. It depends on the equation, the mesh size, the mesh quality, and the finite element spaces in the approximation.

We solve the FS equations in a 2D vertical ice with the Galerkin method implemented in Elmer/ICE

The paper is organized as follows. Section

To simulate flow in a 2D vertical cross section of an ice sheet, we use the FS equations with coordinates

A two-dimensional schematic view of a marine ice sheet.

At the boundary

The ocean surface is at

The boundaries

The

The thickness of the ice is denoted by

The 2D vertical solution of the FS equations in Eq. (

In

Introducing the notation

In Sect.

In this section we state the weak form of Eq. (

We start by defining the mixed weak form of the FS equations.
Introduce

The last term in

We employ linear Lagrange elements with Galerkin least-square (GLS) stabilization

The mesh is constructed from a footprint mesh on the ice base and then extruded with the same number of layers equidistantly in the vertical direction according to the thickness of the ice sheet.
To simplify the implementation in 2D, the footprint mesh on the ice base consists of

In general, the GL is somewhere in the interior of an interval

The resulting system of nonlinear equations forms a nonlinear complementarity problem

The complementarity condition also holds for

Similar implementations for contact problems using Nitsche's method are found in

The nonlinear equation Eq. (

The advection equations for the moving ice boundary in Eqs. (

The advection equations Eqs. (

A numerical stability problem in

The relation between

Assuming that

Much longer stable time steps are possible at the surface and the base of the ice with a semi-implicit method (Eq.

The basic idea of the subgrid scheme for the FS equations in this paper follows the GL parameterization (SEP3) for SSA in

The numerical solutions

Schematic figure of the GL in case i, with the arrows indicating the direction of the net forces in the vertical direction.
The light blue line is the analytical solution of the ice sheet with the analytical GL position

Schematic figure of the GL in case ii, with the arrows indicating the direction of the net forces in the vertical direction.
The colors of the lines follow those in Fig.

Depending on how the mesh is created from the initial geometry and updated during the simulation, the first floating node at

These two cases are similar to the LG and FF cases in

The GL moves toward the ocean in the advance phase and away from the ocean in the retreat phase.
First, we consider case i in the

To determine the position

Another situation in the advance phase is case ii, shown in Fig.

The implicit treatment of the ice base has the consequence that only case ii occurs in the

Since we have

The domains

The penalty term in Nitsche's method restricts the motion of the element in the normal direction.
It is only imposed on an element which is fully geometrically on the ground in case i.
In contrast, in case ii the GL element

The slip coefficient

A summary of the numerical treatment of the GL is as follows:

advance phase

retreat phase

The algorithm for the GL element is as follows:

Equations (

The numerical experiments follow the MISMIP benchmark

The dependence on

The MISMIP 3a retreat experiment with

The GL position during the transient simulations in the advance and retreat phases is displayed in Fig.

The distance between the steady-state GL positions of the retreat and the advance phases is shown in Fig.

The MISMIP 3a experiments for the GL position when

The MISMIP 3a experiments at the final time

We observe oscillations at the ice surface near the GL in all the experiments as expected from

Details of the solutions for the advance experiment with

The ratio between the thickness below sea level

The result for

The surface and the base velocity solutions from the advance experiment are displayed in Fig.

The velocities

The convergence of the steady-state GL position toward the reference solutions in

Our method can be extended to a triangular mesh covering

An alternative to a subgrid scheme is to introduce static or dynamic adaptation of the mesh on

A subgrid scheme at the GL was developed and tested in the SSA model for 2D vertical ice flow in

The FS subgrid model is implemented based on Elmer/ICE

GC developed the model code and performed the simulations. GC and PL contributed to the theory of the paper. GC, PL, and LvS contributed to the development of the method and the writing of the paper.

The authors declare that they have no conflict of interest.

We are grateful to Thomas Zwinger for advice and help in the implementation of the subgrid scheme in Elmer/ICE. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High Performance Computing, KTH Royal Institute of Technology. We also thank the anonymous referees for their helpful comments.

This research has been supported by the Svenska Forskningsråadet Formas (grant no. 2017-00665) to Nina Kirchner and the Swedish strategic research program eSSENCE.

This paper was edited by Alexander Robel and reviewed by three anonymous referees.