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.

Introduction

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 . This paper introduces a moving-point method for the numerical simulation of ice sheets, especially the migration of their boundaries. In this paper we focus on the migration of continental ice sheet margins.

At the scale of an ice sheet or a glacier, ice is modelled as a flow which follows the Stokes equations of fluid flows , even though the flow is non-Newtonian. Solving this problem at that scale is costly. A 3-D finite element model called Elmer/Ice has been developed for this purpose numerically (see , for a detailed description of Elmer/Ice). Other models take advantage of the very small aspect ratio of ice sheets and use a thin layer approximation differing only in the order of the approximation. The oldest and numerically least expensive model used for ice flow is the Shallow Ice Approximation, or SIA . It gives an analytical formulation for horizontal velocities of ice in the sheet and for their vertically averaged counterpart. Although simple and fast, the SIA captures well the nonlinearity of the system due to shearing at large scales. However, the SIA is not designed to include basal sliding and is a poor approximation for small scales especially at the ice divide and the ice sheet margin. The SIA is, nevertheless, an excellent resource for testing numerical approaches, since moving-margin exact solutions exist in the literature .

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): ). Then the focus shifted to the simulation of grounding-line migration. The MISMIP (Marine Ice Sheet Model Intercomparison Project) and MISMIP3d projects have shown that fixed-grid methods perform poorly without high resolution around the grounding line or by enforcing the flow at the grounding line using asymptotic results from a boundary layer theory . This has led ice sheet modellers to develop adaptive and moving techniques to overcome this issue.

One approach to gain high resolution is to use automated adaptive remeshing , forcing the resolution to stay high around the ice sheet margin. A related approach is to use adaptive mesh refinement (AMR) techniques, which allow improved resolution to be achieved in key spatially and temporally evolving subregions . However, even with AMR, the ice sheet margin still falls between grid points, although by adapting the grid to increase the resolution near the margin the accuracy is kept high. Adapting the grid is, nevertheless, an expensive procedure, as areas where refinement is needed have to be regularly re-identified.

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 and to an ice sheet along a flowline. However, it is not easily translated into two dimensions.

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 location-based methods the positions of the nodes are redefined directly at each time step by a mapping from a reference grid . This is generally done by choosing a monitor function. This approach has been used by , the main difficulty being the definition of the monitor function. In velocity-based methods, on the other hand, the movement of the nodes is defined in terms of a time-dependent velocity, which allows the nodes to be influenced by their previous position . Currently, this approach has not been applied to the dynamics of ice sheets.

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 , with the difference that, instead of seeking a velocity intermediate between Lagrangian and Eulerian, the method uses both Eulerian and Lagrangian conservation to deduce the velocity and solution, respectively (see , and references therein). We derive a finite-difference moving-point scheme in a simplified context and verify the approach with steady states obtained from an analytic solution and with exact moving-margin transient solutions in the case of radially symmetrical ice sheets. We show in particular that the scheme is able to track the position of the ice sheet margin accurately. The paper is organised as follows: in Sect.  we recall the SIA and detail the simplified context of our study, in Sect.  we describe our velocity-based moving-point approach, and in Sect.  we verify our approach by comparison with exact solutions before concluding in Sect. .

Ice sheet modelling Ice sheet geometry and Shallow Ice Approximation

We consider a single solid-phase ice sheet whose thickness at position (x,y) and time t is denoted by h(t,x,y). We assume that the ice sheet lies on a fixed bedrock and denote by b(x,y) the bed elevation. The surface elevation, s(t,x,y), is then obtained as s=b+h. The evolution of ice sheet thickness is governed by the balance between the ice gained or lost on the surface, snow precipitation and surface melting, and ice flow draining ice accumulated in the interior towards the edges of the ice sheet. This is summarised in the mass balance equation ht=m-hUinΩ(t), where m(t,x,y) is the surface mass balance (positive for accumulation, negative for ablation), U(t,x,y) is the vector containing the vertically averaged horizontal components of the velocity of the ice, and Ω(t) is the area where the ice sheet is located.

Formally derived by , the SIA is one of the most common approximations for large-scale ice sheet dynamics. Combined with Glen's flow law , the SIA provides (in the isothermal case) an analytical form for U as follows: U=-2n+1Aρignhn+1sn-1s. Parameters involved in this formulation are summarised in Table . Regarding the exponent n>1, its fixed value is classically set to 3 (see , for more details).

As a first step, we confine the study to limited-area ice sheets with radial symmetry, in other words, Ω(t)=[0,rl(t)]×[0,2π]. The ice sheet is centred on (0,0), and rl(t) denotes the position of the ice sheet margin (edge of the ice sheet) at time t (see Fig. , which shows a section through the ice sheet). The radial symmetry implies that the geometry of the sheet depends only on r, so h(t,x,y)=h(t,r), s(t,x,y)=s(t,r), and b(x,y)=b(r). The vector U can then be written in the radial coordinate system as U=Ur^,U=-2n+2A(ρig)nhn+1srn-1sr, where r^ is the unit radial vector, and the mass balance Eq. () simplifies to ht=m-1r(rhU)r. A symmetry condition is added at the ice divide (r=0): U=0andsr=0, and the ice sheet margin rl(t) is characterised by the Dirichlet boundary condition: h(t,rl(t))=0. We also assume that the flux of ice through the ice sheet margin is zero (no calving).

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

Parameter Meaning Value n Creep exponent in Glen's flow law 3 A Creep parameter in Glen's flow law 10-16Pa-3yr-1 ρi Density of ice 910kgm-3 g Gravitational acceleration 9.81ms-2

Under hypotheses regarding the regularity of the ice thickness near the margin (see ), we can differentiate Eq. () with respect to time. Using the mass balance equation Eq. () and h=0 at the margin, dhdt(t,rl(t))=ht+hrdrldt=m-1r(rhU)r+hrdrldt=0 and (rhU)r=hUr+rUrr+rUrhr=rUrhr at the margin. Realistically, since h=0 at the margin, h/r will not be zero there, and hence drldt=U(t,rl(t))-m(t,rl(t))hr-1. This velocity (Eq. ) will be used in the moving-point approach described in the next section.

Section of a grounded radially symmetrical ice sheet.

A moving-point approach

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. . This method is essentially a velocity-based (or Lagrangian) method relying on the construction of velocities for grid points at each time step. This allows the grid to move with the flow of ice. Moving points cover the domain only where the ice sheet exists, so that no grid point is wasted. Adjacent points move to preserve local mass fractions, and the movement is thus based on the physics . This conservation method has been applied to a variety of problems and is perfectly suitable for multi-dimensional problems (different examples are summarised in and references therein; see also for the special case of ice sheet dynamics). The key points of the method are given in the next paragraphs, and the numerical verification of the method is carried out in Sect. .

Conservation of mass fraction

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 θ(t) as θ(t)=2πρi0rl(t)rh(t,r)dr, where ρi is the constant density of ice. Since only mass fractions are considered in this paper and ρi is assumed constant, we can omit ρi without loss of generality.

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 m(t,r), and hence the rate of change of the total mass, θ˙, is given by θ˙(t)=2π0rl(t)rm(t,r)dr. We now introduce the principle of the conservation of mass fractions. Let r^(t) be a moving point and define μ(r^), the relative mass in the moving subinterval (0,r^(t)), as μ(r^)=2πθ(t)0r^(t)rh(t,r)dr. The rate of change of r^(t) is determined by keeping μ(r^) independent of time for all moving subdomains of [0,rl(t)]. Note that μ(r^)[0,1] is a cumulative function, with μ(0)=0 and μ(rl)=1.

Trajectories of moving points

We obtain the velocity of a moving point by differentiating Eq. () with respect to time, giving ddt2π0r^(t)rh(t,r)dr=μ(r^)θ˙(t). Carrying out the time differentiation using Leibniz's integral rule and substituting for h/t from the mass balance Eq. () gives ddt0r^(t)rh(t,r)dr=0r^(t)rm(t,r)dr+r^(t)h(t,r^(t))dr^dt-U(t,r^(t)) with boundary conditions (Eq. ) at r=0. From Eqs. (), (), and (), we can determine the velocity of every interior point as dr^dt=U(t,r^(t))+1r^(t)h(t,r^(t))μ(r^)0rl(t)rm(t,r)dr-0r^(t)rm(t,r)dr.

The point at r=0 is located at the ice divide, which does not move during the simulation. The point at rl(t) is dedicated to the ice sheet margin, which moves with the velocity obtained in Eq. (). We verify in Appendix  that the interior velocity calculated by Eq. () coincides with the boundary velocities calculated directly from the boundary conditions (see Eq. ).

Determination of the ice thickness profile

Once the velocities dr^/dt of the moving points r^(t) have been found from Eq. (), the points are moved in a Lagrangian manner. In addition, the total mass θ(t) is updated from Eq. (). The ice thickness profile is then deduced from Eq. () as follows. Differentiating Eq. () with respect to r^2, we obtain h(t,r^(t))=θ(t)πdμ(r^)d(r^2), which allows the ice thickness profile at time t to be constructed since dμ(r^)/d(r^2) is constant in time and therefore known from the initial data. Note that the positivity of the ice thickness is preserved since μ is by definition a strictly increasing function (see Eq. ).

Asymptotic behaviour at the ice sheet margin

As pointed out by and , singularities can appear with the SIA at the margin of grounded ice sheets. The singularity arises because of the vanishing of h at the margin and the steepening of the slope h/r. Nevertheless the ice velocity U defined by Eq. () can remain finite even if the slope is infinite. We give more details on this subject in this subsection. We also detail the influence of the singularity on the movement of the ice sheet margin.

At a fixed time and for points r sufficiently close to rl, we can write the ice thickness profile h(r) as the first term in a Frobenius expansion, h(r)=(rl-r)γϕl, to leading order, where ϕl=O(1). If γ=1, then h(r) is locally linear with slope ϕl, but if γ<1 the slope h/r is unbounded. Hence in the asymptotic region near the margin, in the case where the bedrock topography b(r) is constant, from Eq. () U=2n+2A(ρig)nγn(rl-r)(2n+1)γ-nϕl2n+1, which vanishes as r tends to rl if γ>n/(2n+1) and remains finite if γ=n/(2n+1).

Suppose that, in the evolution of the solution over time, γ(t)>n/(2n+1) initially so that rl(t) is constant and the boundary is stationary (waiting). If r^(t) follows a CMF trajectory, then, in the absence of accumulation/ablation, the velocity of the moving coordinate r^(t) is given by U=2n+2A(ρig)nγ(t)n(rl(t)-r^(t))(2n+1)γ(t)-nϕl(t)2n+1. Asymptotically, except at the boundary itself, this velocity is finite and positive, since U>0 and its spatial derivative U/r<0 sufficiently close to the boundary, showing that the distance rl(t)-r^(t) decreases with time.

In the absence of accumulation/ablation, therefore, the conservation of mass fractions from Eq. () implies that (rl2(t)-r^2(t))h(t,r^(t)) is constant in time. Thus, from Eq. (), for points r^(t) sufficiently close to the boundary (rl(t)+r^(t))(rl(t)-r^(t))γ(t)+1ϕl(t) is constant in time. Hence, since (rl(t)-r^(t)) is decreasing, γ(t) is also decreasing. When γ(t) reaches n/(2n+1), the boundary moves.

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 b/r at the margin (see ). The key point to notice is that the asymptotic behaviour depends on an infinite slope of h at the margin whereas b(r) always has a finite slope.

Numerics

We now implement a numerical scheme using a finite-difference method. The complete algorithm is detailed in Appendix . In addition, we explain in Appendix  why our implementation respects the asymptotic behaviour of the ice sheet at its margin.

Numerical results

This section is dedicated to the verification of the numerical scheme derived from the moving-point method detailed in Sect.  and to the study of its behaviour. Every numerical experiment is performed with the parameter values given in Table .

Verification with steady states on flat bedrock Accurate estimation of steady ice thickness profile

We consider a surface mass balance m(r) independent of time. The steady state of an ice sheet occurs when the temporal change in ice thickness h/t is zero. In that case, from Eq. (), the following relationship is valid: rm=r(rhUr), with h(r) being the thickness of the steady ice sheet and U(r) its ice velocity. By integrating the previous equation and by including the boundary conditions (Eqs.  and ), the position of the margin rl can be obtained from 0rlrm(r)dr=0rlr(rhUr)dr=rhUr0=0. If the bedrock is flat, the profile of the steady ice sheet, from Eqs. () and (), is

h(r)=2(n+1)nρignn+22A12(n+1)rrl1r0rm(s)sds1ndrn2(n+1). This approach was already in use in the EISMINT intercomparison project with the following constant-in-time surface mass balance:

m(r)=min⁡0.5myr-1,10-2myr-1km-1450km-r. Eq. () has an analytical formulation with this surface mass balance. Therefore, rl is determined with machine precision by numerical root-finding algorithms (rl579.81km), and the profile of the steady state is accurately estimated from Eq. () by a single numerical integration (0rm(s)sds in Eq. () has an analytical form) using a composite trapezoidal rule (we take enough grid points to ensure that the error of the estimates is smaller than 0.01m).

Runs with different initial profiles

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: h(t0,r)=h01-rrl(t0)2p. For each of the three different runs, we take

a uniformly distributed initial grid with rl(0)=450km, h0=1000m, and p=3/7;

an initial grid with rl(0)=500km and with higher resolution near the margin, h0=1000m, and p=1;

a uniformly distributed initial grid with rl(0)=600km, h0=4000m, and p=1/4.

The model is run for 25 000yr with a constant time step Δt=0.1yr.

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. (). Top: initial uniform grid with rl(0)=450km, h0=1000m, and p=3/7; middle: initial grid with higher resolution near the margin with rl(0)=500km, h0=1000m, and p=1; bottom: initial uniform grid with rl(0)=600km, h0=4000m, and p=1/4.

Figure  shows the evolution of the geometry and the overall motion of the grid points for each run. In run (a) the margin is staying at its initial position until the ice sheet is large enough and the sheet front steep enough to make it advance. Run (b) shows a retreating margin at an early stage before advancing, and run (c) captures the opposite behaviour.

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 . In each case our numerical model has been able to approach the position of the margin with high accuracy (less than 400 m) at low resolution, as only 21 grid points have been employed.

Comparison between reference steady state described in Sect.  and results obtained after a 25 000 yr run using 21 moving points with the EISMINT surface mass balance and initial profile described by Eq. (). Exp 1(a): initial uniform grid with rl(0)=450km, h0=1000m and p=3/7; Exp 1(b): initial grid with higher resolution near the margin with rl(0)=500km, h0=1000m, and p=1; Exp 1(c): initial uniform grid with rl(0)=600km, h0=4000m, and p=1/4.

Ice thickness at Position of r=0 (in m) the margin (in km) Reference 2986.95±0.01 579.81±0.01 Exp 1(a) 3019.59 579.99 Exp 1(b) 3040.28 579.77 Exp 1(c) 3017.75 579.43
EISMINT moving-margin experiment

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 . Compared with the experiments performed in Sect. , the only differences are that we use an initial uniformly distributed grid with 28 nodes, an initial domain of length rl(0)=450km, and an initial ice thickness profile h(0,r)=Δt×m(r), where Δt=0.1yr is the constant model time step and m(r) is given by Eq. (). Then we run the model as in the EISMINT experiment for 25 000yr to reach the steady state.

We first verify the result of our run with the steady state obtained in Sect. . As shown in Fig. , absolute errors in the ice thickness profile mostly occur near the ice sheet margin, rising to 58.23m at the last grid point (compared to an rms error of 15.71m and an absolute error at the ice divide of 18.81m). Regarding the ice sheet margin, its position is again well estimated (with an absolute error of only 138.5m).

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 ). Regarding the ice thickness at the ice divide, our model result of 3005.76m is within the range of estimation given by the intercomparison: 2982.3±26.4m. These results are summarised in Table , showing that our moving-point method is able to achieve as good an equivalent estimation as classical fixed-grid methods with a small number of nodes while providing accurate tracking of the movement of the margin, in the context of a shallow grounded ice sheet.

Comparison between intercomparison results for the EISMINT moving-margin experiment in steady state (see Table 5 in ) and results obtained for the same experiment with the moving-point method with an identical number of grid points nr=28. The reference values are obtained from the accurate evaluation described in Sect. .

Ice thickness at Position of r=0 (in m) the margin (in km) Reference 2986.95±0.01 579.81±0.01 EISMINT/2d 2982.3±26.4 593.3±9.0 Moving point 3005.76 579.68
Rates of convergence with EISMINT moving-margin experiment

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 O((Δr)γ), with Δr being some mesh spacing. However, this approach is not appropriate in our case since the moving-point method has mesh spacings varying in time and space. Instead we present our estimated rate of convergence as a function of the number of grid points.

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 nr=20,30,40,60, and 80 grid points. From those results we estimate the rate of convergence for both errors. Results are summarised in Table . We observe that the error for the margin position decreases at an almost quadratic rate O(nr-1.95) and the error in the ice thickness at the ice divide at a linear rate O(nr-1.16). This confirms that our CMF method is well able to track the ice sheet margin without losing accuracy in the ice thickness profile.

The steady state from the EISMINT moving-margin experiment compared with our 25 000yr model run with 28 nodes, uniformly distributed at the initial time. The reference profile is obtained by a numerical integration of Eq. () using a composite trapezoidal rule. The error in the ice thickness occurs mostly near the ice sheet margin, as in other experiments (rms error is 15.71m and maximum error is 58.23m). The position of the margin is well determined as the absolute error is only 138.5m.

Estimation of absolute errors from results obtained in the EISMINT framework using the moving-point method with an initial uniformly spaced grid with nr=20,30,40,60, and 80 grid points. Rates of convergence are estimated directly from the calculated absolute errors.

Number of Absolute error in Absolute error in grid points nr the ice thickness at r=0 (in m) the position of the margin (in m) 20 29.46 233.58 30 17.29 139.02 40 12.51 62.17 60 7.97 28.49 80 5.86 16.33 Rate of convergence O(nr-1.16) O(nr-1.95)

The steady-state approach of Sect.  is still valid for an ice sheet lying on a non-flat bedrock. However, the experiments in such cases are quite limited as we only have the position of the steady margin from Eq. (). Nevertheless we carry out a few experiments in this context in order to demonstrate that the CMF moving-point approach is perfectly suitable for non-flat bedrock.

We consider the following fixed bedrock elevation: b(r)=2000m-2000mr300km2+1000mr300km4-150mr300km6. As in the previous section, experiments are performed with the EISMINT surface mass balance (Eq. ). At an initial time t=0 we prescribe a uniformly distributed grid with a margin located at rl(0)=450km and an initial ice thickness h(0,r)=Δt×m(r) for the constant time step Δt=0.1yr. The model is run for 25 000yr. The resulting evolution of the geometry and the overall motion of the grid points are shown for a grid of 20 points in Fig. . Regarding the position of the margin at steady state, our run has an absolute error of 127.7m. This is even better than the previous result obtained for a flat bedrock.

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 nr grid points, nr=20,30,40,60, and 80. Absolute errors are summarised in Table . As in Sect.  we observe that the absolute error for the margin position decreases at a nearly quadratic rate O(nr-1.83). This corroborates the ability of the moving-point method to track the ice sheet margin even for non-flat bedrocks.

Evolution of the geometry and overall motion of the grid points for the non-flat bedrock (topography given in Eq. ) with the EISMINT surface mass balance. At steady state, the observed error for the position of the margin is 127.7m.

Verification with time-dependent solutions

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.

Similarity solutions

Few exact solutions for isothermal shallow ice sheets have been derived in the literature. Most are based on the similarity solutions established by for a zero surface mass balance. extended this work to non-zero surface mass balance and established a new family of similarity solutions by adopting the following parameterised form for the surface mass balance: m(ε)(t,r)=εth(ε)(t,r), with ε being a real parameter in the interval -12n+1,+. Assuming that t>0, this leads to the following family of similarity solutions: h(ε)(t,r)=1tα(ε)h0,12n+1n-Λ(ε)rtβ(ε)n+1nn2n+1 for r0,tβ(ε)Θ(ε), α(ε)=2-(n+1)ε5n+3,β(ε)=1+(2n+1)ε5n+3 and Λ(ε)=2n+1n+1(n+2)β(ε)2A(ρig)n1n,Θ(ε)=h0,12n+1n+1Λ(ε)-nn+1. The total mass of such ice sheets, as defined in Eq. (), is θ(ε)(t)=β(ε)-2n+1tεW1, where W1 is a constant independent of ε: W1=2π0Θ(1)sh0,12n+1n-Λ(1)sn+1nn2n+1ds.

Estimation of absolute error from results obtained with the non-flat bedrock described by Eq. () using the moving-point method with an initial uniformly spaced grid with nr=20,30,40,60, and 80 grid points. Rates of convergence are estimated directly from the calculated absolute errors.

Number of Absolute error in grid points nr the position of the margin (in m) 20 127.74 30 98.44 40 43.96 60 18.23 80 12.39 Rate of convergence O(nr-1.83)
Results

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. ), and, at each time step, the surface mass balance is evaluated at each moving node by using the relationship m=εth from Eq. (). When ε is non-zero, some feedback between the surface mass balance and the ice thickness is expected . Each model run in this section uses a fixed time step of Δt=0.01yr.

The first experiment is conducted with the constant mass similarity solution (ε=0) between t=100yr and t=20 000yr for the reference period. Rapid changes occur in the state of the similarity solution between t=100yr and t=1000yr; then the dynamics dramatically slow (see Fig.  for the evolving ice thickness profile of the similarity solution). The ice thickness at the ice divide decreases at a rate t-1/9, and the position of the margin increases at a rate t1/18.

Rate of convergence of different errors between numerical results obtained for time-dependent solutions at time t=20 000yr. The different estimated rates of convergence are obtained by performing experiments with nr=10,20,40,60,80,100, and 200 grid points for different configurations of surface mass balance (Eq. ).

ε=0 ε=-1/8 ε=1/4 ε=3/4 rms error on h O(nr-1.07) O(nr-1.10) O(nr-1.10) O(nr-1.12) Max error on h O(nr-0.57) O(nr-0.60) O(nr-0.59) O(nr-0.60) Error in rl O(nr-1.32) O(nr-1.41) O(nr-1.38) O(nr-1.41) Error in total volume O(nr-1.24) O(nr-1.43) O(nr-1.43)

The reference ice sheet profile (ε=0) obtained from transient similarity solutions (see Sect. ) is displayed for t=100years, for t=1000years, and at 1000-year intervals thereafter. Rapid changes occur in the state of the sheet at the beginning of the simulation; then the dynamics dramatically slow. The ice thickness at the ice divide decreases at a rate t-1/9, and the position of the margin increases at a rate t1/18.

We begin by analysing the results obtained with a grid made up of 100 nodes, uniformly distributed at the initial time. In terms of thickness, errors mostly occur near the ice sheet margin as is the case with fixed-grid methods (see ). For example, at the final time t=20 000yr, a maximum error of 134m in the ice thickness is obtained at the computed margin while the interior of the sheet has errors less than 10m (see Fig. ). We also notice that errors in the ice thickness (both maximum and rms errors) decrease as the ice sheet slows down (see Fig. ). Regarding the margin, even if the absolute error in its position increases in time, it is kept under one kilometre (880m at the final time t=20 000yr). This confirms the combined ability of our method to model accurately the evolution of the ice thickness profile and to track precisely the movement of the ice sheet margin in transient behaviour.

We then study the convergence of our scheme at a final time t=20 000yr when the number of grid points is increased. We perform the same analysis for ε=-1/8,1/4, and 3/4. Rates of convergence for different errors (rms error and maximum error for ice thickness profile, absolute error for the position of the margin and the volume of the ice sheet) are summarised in Table . These demonstrate the ability of the scheme to achieve accurate results for the position of the margin and the ice thickness profile for transient behaviour even with a small number of nodes.

The result obtained at final time t=20 000yr with 100 nodes equally distributed at initial time t=100 yr, and a fixed time step Δt=0.01 yr is compared to the reference transient similarity solution with ε=0 (see Sect. ). A maximum error of 134m on the ice thickness is obtained at the computed margin, while the interior of the sheet has errors less than 10m. The position of the margin is obtained with an error of 880m.

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 t=100yr and a fixed time step Δt=0.01y and the reference transient similarity solution with ε=0 (see Sect. ). Errors in the ice thickness decrease as the ice sheet slows down. The errors in the position of the margin increase in time, but their evolution is slower when the dynamics are slower.

Conclusions

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 and time-dependent solutions from . Accurate results have been achieved with a small number of grid points in both cases. In particular our approach has been able to track the position of the ice sheet margin with high accuracy without compromising the estimation of the ice thickness profile. Hence the comparison shows that the approach has considerable potential for future investigations. Whilst this paper uses a vertically averaged horizontal ice velocity given by the shallow ice approximation, the moving-mesh scheme is independent of the form of the ice velocity used here and could be used as a solver for mass balance alongside more complex vertically integrated approximations (see e.g. ).

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 and will be the subject of a new paper. The conservation approach can also be applied to marine ice sheets. In these cases, different kinds of boundaries have to be considered: e.g. grounding line, shelf front, and continental margin. Preliminary results with the moving-point method have been obtained in . However, the problem of initialisating such a model for use in real applications remains open. The incorporation of various data assimilation procedures is currently being investigated in this context.

Consistency of the moving-point approach at boundaries

We now verify that dr^/dt tends to the velocity obtained from Eq. () at the ice margin when r^(t) tends to rl(t). Assuming the continuity of h/r and m in the vicinity of the ice sheet margin, by L'Hôpital's rule lim⁡r^(t)rl(t)dr^dt=U(t,rl)+lim⁡r^(t)rl(t)θ˙θr^h(t,r^)-r^m(t,r^)h(t,r^)+r^hr(t,r^). This gives lim⁡r^(t)rl(t)dr^dt=U(t,rl)-m(t,rl)hr(t,rl)-1. The limit is consistent with the velocity of the moving margin obtained in Eq. (). The same approach can be used to show that dr^/dt tends to 0 when r^(t) tends to the ice divide r=0.

A finite-difference algorithm

The moving-point method is discretised on a radial line using finite differences on the grid r^i, i=1,,nr, where 0=r^1(t)<r^2(t)<<r^nr-1(t)<r^nr(t)=rl(t). The approximation of h(t,r) at r^i(tk)=r^ik is written hik, and that of the ice velocity U(t,r) as Uik. The velocity of the points is represented by vik. The symbol θk designates the numerical approximation of the total mass, and the constant mass fractions are represented by μi for every μ(r^ik).

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

Initialisation

At the initial time the user needs to provide the initial location of each grid point {r^i0} and the initial ice thickness {hi0} there. By definition, we assume that r^10= 0 and hnr0= 0. We estimate the total mass of the ice sheet at the initial time by using a composite trapezoidal rule approximating Eq. (). This gives θ0=π2j=1nr-1hj0+hj+10r^j+102-r^j02. We derive the numerical approximation for the mass fractions μi by discretising Eq. () following the same principle: μ1=0,μi=π2θ0j=1i-1hj0+hj+10r^j+102-r^j02.

Ice velocities

We confine the algorithm to n=3 for the creep exponent in the Glen flow law. Then Eq. (), giving the ice velocity, can be expanded by using the binomial theorem |U(t,r)|=25A(ρig)3h4br3+35(h5)rbr2+13(h3)r2br+27343(h7/3)r3. We choose to rewrite the radial form of Eq. () in this way in order to ensure that the ice velocity at the ice sheet margin computed with a finite-difference scheme can be non-zero as noted in Sect. . The bedrock elevation b and its derivative are known for every location of the domain. The sign of Uik (U1k=0) is obtained by calculating the sign of sik-si-1k (approximating the sign of the surface slope by an upwind scheme). We also approximate the derivatives of hp for any p>0 by an upwind scheme: (hp)rr=rik=hikp-hi-1kprik-ri-1k.

Approximate nodal velocities

The velocity of interior nodes is obtained by discretising Eq. () as v1k=0,vik=Uik+12r^ikhikμi0r^nrkm(tk,r)dr2-0r^ikm(tk,r)dr2, where the integrals in Eq. () are approximated by a composite trapezoidal rule. For the velocity of the ice sheet margin, Eq. () is discretised by using a first-order upwind scheme, namely, vnrk=Unrk-mtk,rnrkr^nrk-r^nr-1khnrk-hnr-1k.

Time stepping

The total mass θk+1 is updated by using an explicit Euler scheme: θk+1=θk+Δtθ˙k=θk+Δtπ0r^nrkm(t,r)dr2. Again the integral is approximated by a composite trapezoidal rule.

As in the case of the total mass, the position of the nodes is updated by using an explicit Euler scheme: r^ik+1=r^ik+Δtvik. Δt is taken small enough to preserve the node order in Eq. () and to avoid oscillations in the ice thickness profile. In practice we have never observed node overtaking since spurious oscillations always appear first. This behaviour is similar to that observed with explicit schemes for fixed staggered grid methods .

Approximate ice thickness

The ice thickness for interior nodes hik+1 is recovered algebraically at the new time using a second-order midpoint approximation of Eq. (), namely, hik+1=θk+1πμi+1-μi-1r^i+1k+12-r^i-1k+12. The ice thickness at the ice divide h1k+1 is obtained by using the first-order upwind scheme h1k+1=θk+1πμ2-μ1r^2k+12-r^1k+12.

Behaviour of the approximate ice velocity at the ice margin

As in Sect. , assuming that the topography of the bedrock is flat in the vicinity of the margin, the asymptotic form of the radial ice velocity is U=2n+2A(ρig)nγn(rl-r)(2n+1)γ-nϕl2n+1. Hence the leading term in the numerical approximation (Eq. ) to the ice velocity at the approximation hl to the ice margin is -25sgnsnr-snr-1A(ρig)3373hnr7/3-hnr-17/3r^nr-r^nr-13=-25sgnsnr-snr-1A(ρig)3373hnr-17/3r^nr-r^nr-13 since hnr=0. But from Eq. () the asymptotic analytic ice velocity (when n=3) is 25A(ρig)3373(rnr-r)7γ-3ϕl7=25A(ρig)327343h(r)7/3rnr-r3 by Eq. (). Hence the numerical approximation to the ice velocity has the same asymptotic behaviour as the asymptotic analytic ice velocity with n=3. The result also holds for general creep exponent n.

Acknowledgements

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