TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-1-2016A moving-point approach to model shallow ice sheets: a study case with radially symmetrical ice sheetsBonanB.b.c.bonan@reading.ac.ukhttps://orcid.org/0000-0002-8808-2201BainesM. J.NicholsN. K.PartridgeD.School of Mathematical and Physical Sciences, University of Reading, Reading, UKB. Bonan (b.c.bonan@reading.ac.uk)15January20161011146July20157August201527November20158December2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/1/2016/tc-10-1-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/1/2016/tc-10-1-2016.pdf
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 modellingIce 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
∂h∂t=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+1∇sn-1∇s.
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).
Radially symmetrical ice sheets
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+1∂s∂rn-1∂s∂r,
where r^ is the unit radial vector, and the mass
balance Eq. () simplifies to
∂h∂t=m-1r∂(rhU)∂r.
A symmetry condition is added at the ice divide (r=0):
U=0and∂s∂r=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.
ParameterMeaningValuenCreep exponent in Glen's flow law3ACreep parameter in Glen's flow law10-16Pa-3yr-1ρiDensity of ice910kgm-3gGravitational acceleration9.81ms-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))=∂h∂t+∂h∂rdrldt=m-1r∂(rhU)∂r+∂h∂rdrldt=0
and
∂(rhU)∂r=hUr+r∂Ur∂r+rUr∂h∂r=rUr∂h∂r
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))∂h∂r-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)=2πρi∫0rl(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)=2π∫0rl(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
ddt2π∫0r^(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
ddt∫0r^(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 bedrockAccurate 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(rh∞Ur∞),
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
∫0rl∞rm(r)dr=∫0rl∞∂∂r(rh∞Ur∞)dr=rh∞Ur∞0∞=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)⋅∫rrl∞1r′∫0r′m(s)sds1ndr′n2(n+1).
This approach was already in use in the EISMINT intercomparison project with the following constant-in-time surface mass balance:
m(r)=min0.5myr-1,10-2myr-1km-1⋅450km-r.
Eq. () has an analytical formulation with this surface mass balance. Therefore, rl∞ is determined with machine precision by numerical
root-finding algorithms (rl∞≈579.81km), and the profile of the steady state is accurately estimated from Eq. () by a single
numerical integration (∫0r′m(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 atPosition ofr=0 (in m)the margin (in km)Reference2986.95±0.01579.81±0.01Exp 1(a)3019.59579.99Exp 1(b)3040.28579.77Exp 1(c)3017.75579.43EISMINT 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 atPosition ofr=0 (in m)the margin (in km)Reference2986.95±0.01579.81±0.01EISMINT/2d2982.3±26.4593.3±9.0Moving point3005.76579.68Rates 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 ofAbsolute error inAbsolute error ingrid points nrthe ice thickness at r=0 (in m)the position of the margin (in m)2029.46233.583017.29139.024012.5162.17607.9728.49805.8616.33Rate of convergenceO(nr-1.16)O(nr-1.95)Steady states with non-flat bedrock
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-2000m⋅r300km2+1000m⋅r300km4-150m⋅r300km6.
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 r∈0,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=2π∫0Θ(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 ofAbsolute error ingrid points nrthe position of the margin (in m)20127.743098.444043.966018.238012.39Rate of convergenceO(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=0ε=-1/8ε=1/4ε=3/4rms error on hO(nr-1.07)O(nr-1.10)O(nr-1.10)O(nr-1.12)Max error on hO(nr-0.57)O(nr-0.60)O(nr-0.59)O(nr-0.60)Error in rlO(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
limr^(t)→rl(t)dr^dt=U(t,rl)+limr^(t)→rl(t)θ˙θr^h(t,r^)-r^m(t,r^)h(t,r^)+r^∂h∂r(t,r^).
This gives
limr^(t)→rl(t)dr^dt=U(t,rl)-m(t,rl)∂h∂r(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=π2∑j=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=π2θ0∑j=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)3h4∂b∂r3+35∂(h5)∂r∂b∂r2+13∂(h3)∂r2∂b∂r+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^ikhikμi∫0r^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+Δtπ∫0r^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
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