Predicting the evolution of ice sheets requires numerical models able to accurately track the migration of ice sheet continental margins or grounding lines. We introduce a physically based moving-point approach for the flow of ice sheets based on the conservation of local masses. This allows the ice sheet margins to be tracked explicitly. Our approach is also well suited to capture waiting-time behaviour efficiently. A finite-difference moving-point scheme is derived and applied in a simplified context (continental radially symmetrical shallow ice approximation). The scheme, which is inexpensive, is verified by comparing the results with steady states obtained from an analytic solution and with exact moving-margin transient solutions. In both cases the scheme is able to track the position of the ice sheet margin with high accuracy.

Ice sheets are an influential component of the climate system whose dynamics lead to changes in terms of ice thickness, ice velocity, or migration of
ice sheet continental margins and grounding lines. Therefore numerical modelling of ice sheets needs accuracy not only of the physical variables but also in the
position of their boundaries. However, simulating the migration of an ice sheet margin or a grounding line remains a complex task

At the scale of an ice sheet or a glacier, ice is modelled as a flow
which follows the Stokes equations of fluid flows

Significant efforts have been invested in ice sheet modelling. These
have led ice sheet modellers to compare results obtained by various
models for the same idealistic test problems. They first started by comparing results obtained with fixed-grid models for grounded ice sheets using the
SIA (European Ice Sheet Modelling INiTiative (EISMINT):

One approach to gain high resolution is to use automated adaptive remeshing

Another possibility is to transform the moving domain. The number of
grid points is kept constant in time, but the accuracy is kept high by
the explicit tracking of the position of the ice sheet margin. This is
done by transforming the ice domain to a fixed coordinate system
via a geometric transformation. This approach has been successfully
applied by

We consider here intrinsically moving-grid methods. As in the case of
transformed grids, these methods allow explicit tracking of the ice
sheet margin. There exist a number of techniques for generating the
nodal movement in moving-grid methods. They can be classified into two
subcategories: location-based methods and velocity-based methods

In this paper, we apply a particular velocity-based moving-point
approach based on conservation of local mass fractions to continental
ice sheets. The method is in the tradition of ALE (arbitrary
Lagrangian–Eulerian) methods

We consider a single solid-phase ice sheet whose thickness at position

Formally derived by

As a first step, we confine the study to limited-area ice
sheets with radial symmetry, in other words,

Parameters involved in the computation of the vertically averaged horizontal components of the velocity of the ice.

Under hypotheses regarding the regularity of the ice thickness near the margin (see

Section of a grounded radially symmetrical ice sheet.

In the following paragraphs we describe the moving-point method that
we use to simulate the dynamics of ice sheets in the context of
Sect.

Moving-point velocities are derived from the conservation of mass
fractions (CMF). To apply this principle we first define the total
mass of the ice sheet

Since the flux of ice through the ice sheet margin is assumed to be
zero, any change in the total mass over the whole ice sheet is due
solely to the surface mass balance

We obtain the velocity of a moving point by differentiating
Eq. (

The point at

Once the velocities d

As pointed out by

At a fixed time and for points

Suppose that, in the evolution of the solution over time,

In the absence of accumulation/ablation, therefore, the conservation of
mass fractions from Eq. (

It is a technical exercise to show that this property extends to cases
with accumulation/ablation and with a general bedrock with a finite
slope

We now implement a numerical scheme using a finite-difference
method. The complete algorithm is detailed in Appendix

This section is dedicated to the verification of the numerical scheme
derived from the moving-point method detailed in Sect.

We consider
a surface mass balance

We check the ability of the CMF method to track either advancing or retreating ice sheet margins by performing three different model runs. In each case,
the numerical model has a grid with 21 points, uses the EISMINT surface mass balance, and is initialised using the following
profile:

a uniformly distributed initial grid with

an initial grid with

a uniformly distributed initial grid with

Evolution of the geometry (on the left) and overall motion of the grid points (on the right) for
three experiments with the EISMINT surface mass balance and initial profile
described by Eq. (

Figure

We also note that run (b) has no difficulty with a non-uniform initial grid and keeps the resolution high close to the margin. This stresses the flexibility of the CMF method to deal with various resolutions at the same time.

We then check the convergence of the three initial states to the same steady state. The calculated ice thickness at the ice divide and the position of the margin at
the final time are compared with reference values in Table

Comparison between reference steady state described in Sect.

We now perform the moving-margin experiment described in the EISMINT benchmark in order to both verify our numerical model in this case and compare our results
with those obtained by 2-D fixed-grid models used in

We first verify the result of our run with the steady state obtained in Sect.

We next compare these results with results from fixed-grid models involved in the EISMINT intercomparison project. We confine our comparison to 2-D fixed-grid models as we
only use radial symmetry (see

Comparison between intercomparison results for the EISMINT moving-margin experiment in steady state (see Table 5 in

We now study the rate of convergence of our method towards the reference solution in the EISMINT experiment. Rates of convergence are generally expressed in the
form

We calculate the absolute error for both the margin position and ice thickness at the ice divide from the results obtained in the EISMINT framework using an initial
uniformly spaced grid with

The steady state from the EISMINT moving-margin experiment compared
with our

Estimation of absolute errors from results obtained in the EISMINT framework using the moving-point method with an initial uniformly spaced
grid with

The steady-state approach of Sect.

We consider the following fixed bedrock elevation:

We also check the convergence of the estimated margin position at steady state towards its reference value by performing the same experiment with an initial
uniformly spaced grid and

Evolution of the geometry and overall motion of the grid points for
the non-flat bedrock (topography given in Eq.

In the previous paragraphs, steady states were used to verify our numerical CMF moving-point numerical method. However these experiments did not verify the transient behaviour of the ice sheet margin. To do so, we use exact time-dependent solutions.

Few exact solutions for isothermal shallow ice sheets have been
derived in the literature. Most are based on the similarity solutions
established by

Estimation of absolute error from results obtained with the non-flat bedrock described by Eq. (

We study in this section the accuracy of transient model runs in
comparison with the time-dependent exact solutions. The initialisation
of every experiment is done by using the exact time-dependent solution
(Eq.

The first experiment is conducted with the constant mass similarity
solution (

Rate of convergence of different errors between numerical
results obtained for time-dependent solutions at time

The reference ice sheet profile (

We begin by analysing the results obtained with a grid made up of

We then study the convergence of our scheme at a final time

The result obtained at final time

Evolution of the rms error and maximum absolute error in the ice
thickness, and absolute error in the position of the margin between the run obtained with 100 nodes equally distributed at initial
time

In this paper, we have introduced a moving-point approach for ice sheet
modelling using the SIA (including non-flat bedrock) based on the
conservation of local mass. From this principle we derived an
efficient finite-difference moving-point scheme. The scheme was verified by comparing
results with steady states from the EISMINT benchmark

As mentioned earlier, the conservation approach is suitable not only
for 1-D cases (flowline or radial) but also for 2-D scenarios. A first
application has been demonstrated in

We now verify that

The moving-point method is discretised on a radial line using finite
differences on the grid

Before giving the formula for every quantity calculated, we give the structure of the finite-difference algorithm in Algorithm 1.

At the initial time the user needs to provide the initial location of
each grid point

We confine the algorithm to

The velocity of interior nodes is obtained by discretising
Eq. (

The total mass

As in the case of the total mass, the position of the nodes is updated
by using an explicit Euler scheme:

The ice thickness for interior nodes

As in Sect.

This research was funded in part by the Natural Environmental Research Council National Centre for Earth Observation (NCEO) and the European Space Agency (ESA). Edited by: F. Pattyn