TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-11-179-2017A comparison of two Stokes ice sheet models applied to the Marine Ice Sheet Model
Intercomparison Project for plan view models (MISMIP3d)ZhangTongPriceStephensprice@lanl.govhttps://orcid.org/0000-0001-6878-2553JuLiliLengWeiBrondexJulienDurandGaëlGagliardiniOlivierhttps://orcid.org/0000-0001-9162-3518State Key Laboratory of Severe Weather (LASW), Chinese Academy of Meteorological Sciences, Beijing, ChinaFluid Dynamics and Solid Mechanics Group, Los Alamos National Laboratory, Los Alamos, NM, USADepartment of Mathematics and Interdisciplinary Mathematics Institute, University of South Carolina, Columbia, SC, USAState Key Laboratory of Cryospheric Sciences, Chinese Academy of Sciences, Lanzhou, ChinaState Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing, ChinaUniversité Grenoble Alpes, CNRS, IRD, IGE, 38000 Grenoble, FranceStephen Price (sprice@lanl.gov)25January201711117919019February201616March20167December201611December2016This 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/11/179/2017/tc-11-179-2017.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/11/179/2017/tc-11-179-2017.pdf
We present a comparison of the numerics and simulation results for two
“full” Stokes ice sheet models, FELIX-S and Elmer/Ice
. The models are applied to the Marine Ice Sheet Model
Intercomparison Project for plan view models (MISMIP3d). For the diagnostic
experiment (P75D) the two models give similar results (< 2 %
difference with respect to along-flow velocities) when using identical
geometries and computational meshes, which we interpret as an indication of
inherent consistencies and similarities between the two models. For the standard (Stnd),
P75S, and P75R prognostic experiments, we find that FELIX-S (Elmer/Ice)
grounding lines are relatively more retreated (advanced), results that are
consistent with minor differences observed in the diagnostic experiment
results and that we show to be due to different choices in the implementation
of basal boundary conditions in the two models. While we are not able to
argue for the relative favorability of either implementation, we do show that
these differences decrease with increasing horizontal (i.e., both along- and
across-flow) grid resolution and that grounding-line positions for FELIX-S
and Elmer/Ice converge to within the estimated truncation error for
Elmer/Ice. Stokes model solutions are often treated as an accuracy metric in
model intercomparison experiments, but computational cost may not always
allow for the use of model resolution within the regime of asymptotic
convergence. In this case, we propose that an alternative estimate for the
uncertainty in the grounding-line position is the span of grounding-line
positions predicted by multiple Stokes models.
Introduction
As Earth's largest reservoirs of fresh
water, ice sheets are important components of the global climate system.
Humans feel their impacts most acutely through changes in global sea level,
as ice sheets grow or decay in response to climate forcing and internally
controlled dynamics. While the rate of present-day sea-level rise is
dominated by ocean steric changes and eustatic changes due to shrinking
mountain glaciers, the eustatic contribution from the large ice sheets
(Greenland and Antarctica) has increased in recent decades and is expected to
continue increasing in coming decades and centuries . While
currently smaller than the sea-level contribution from mountain glaciers or
Greenland, the future sea-level rise contribution from Antarctica is of
particular concern; because of inherent dynamic instabilities associated with
marine-based ice sheets see, e.g.,, the
Intergovernmental Panel on Climate Change (IPCC) recently highlighted future
Antarctic ice sheet evolution as the largest uncertainty with respect to
projecting future rates of sea-level rise .
Largely to address these concerns, the international community has focused
intense efforts over the last decade on improving the predictive skill of
large-scale, whole-ice-sheet models. These improvements include increased
fidelity and accuracy with respect to the governing nonlinear Stokes-flow
equations, increased numerical and computational robustness and efficiency,
increased complexity and realism with respect to representation of relevant
physical processes, and increased efforts towards partial and full coupling
with earth system models (e.g., see models described in
).
Alongside and critical to advancing these efforts have been the development
of model intercomparison exercises, which have provided community-based
“benchmark” solutions for gauging the correctness of model output
e.g.,. While designed to be
simple, distilling a test for a particular model feature of interest down to
its essence, these exercises are still generally too complicated for the
application of formal model verification through the use of analytical or
manufactured solutions. Thus, these same model intercomparisons have become
increasingly dependent on the output from so-called full Stokes ice sheet
models – the highest fidelity representation of the equations governing the
momentum balance for ice flow – to provide a metric for the most accurate
model solutions available (one example is the Elmer/Ice model,
, which has taken part in all of the intercomparison
projects referenced above). One clear problem with this practice is that the
limited number of Stokes models (often only one) participating in the
intercomparison exercises has prohibited any systematic study of differences
in solutions from Stokes models.
Here, we apply a second Stokes model, the FELIX-S model of see also, to the Marine Ice Sheet
Model Intercomparison for plan view models(MISMIP3d) experiments
. We conduct a careful comparison of the numerical methods
used and the solutions produced by FELIX-S and Elmer/Ice. In a recent
contribution, show that both diagnostic and
prognostic grounding-line (GL) positions from Elmer/Ice exhibit substantial
sensitivity as a function of not only the across-flow mesh
resolution (along-flow mesh resolution has been previously explored and
discussed in detail, e.g., ) but also as a
function of seemingly arbitrary choices about how basal boundary conditions
are implemented in the model. Here, by “arbitrary” we mean in the sense
that it is not obvious if and why one choice should be superior to another.
Below, we show that a similar level of sensitivity is apparent when comparing
FELIX-S and Elmer/Ice output to marine ice sheet benchmark experiments, even
when using the same computational mesh with very high along-flow resolution.
While these differences clearly argue for a degree of caution when
interpreting Stokes model output as the metric for model solution accuracy,
we also show that the differences between Elmer/Ice and FELIX-S solutions
decrease as the mesh resolution increases. The consistency between these two
models at high resolution (e.g., 50 m along flow) lends support for
their use as a benchmark for lower fidelity models, provided these benchmark
solutions are generated using adequate grid resolution.
The paper proceeds as follows. First, we give a brief overview of the
governing Stokes-flow equations for ice flow, which are discretized and
solved by Elmer/Ice and FELIX-S. We then discuss in some detail the
implementation of boundary conditions – some specific to the problem of
simulating marine ice sheets – and how they are implemented in the two
models. A brief introduction to the MISMIP3d model setup is then given,
followed by a presentation of experimental results for the two models. We
then give an in-depth discussion of the similarities and differences between
results from the two models, our interpretation of where these differences
come from, and an assessment of their significance. We close with a summary and
concluding remarks.
Model descriptionThe Stokes ice flow model
Consider the flow of a viscous, incompressible fluid (ice) in a low-Reynolds
number flow. Conservation of linear momentum is expressed by the balance
between the stress-tensor divergence and the gravitational body force,
∇⋅σ=ρig,
with σ representing the Cauchy (full) stress tensor,
ρi the density of ice, and g the acceleration due to
gravity.
The incompressibility of glacier ice is expressed as
∇⋅u=0,
where u=(u,v,w) denotes the ice velocity vector. For glacier ice,
the constitutive relation can be expressed as for a Newtonian fluid:
τ=σ+pI=2ηϵ˙,
where τ is the deviatoric stress tensor, p is the
isotropic ice pressure, I is the identity tensor,
ϵ˙ is the strain rate tensor, and η is the
“effective” viscosity, defined by Nye's generalization of Glen's flow law
as
η=12A-1/nϵ˙e(1-n)/n.
The flow-law exponent n is assigned a value of 3, with n>1 leading to a
“shear thinning”, non-linear rheology. A is the temperature-dependent
rate factor and ϵ˙e is the effective strain
rate (the square root of the second invariant of the strain-rate tensor).
Boundary conditions
At the upper surface, a stress-free boundary condition applies:
σ⋅n=0,
where n is the surface normal vector in a Cartesian reference frame.
The lower-ice surface consists of two different boundary conditions
. For the “grounded” part of the flow where ice is in
contact with bedrock (i.e., the ice–bedrock interface), the normal stress
exerted by the ice body is larger than ocean water pressure. Here we apply
the nonlinear friction sliding law prescribed for the MISMIP3d experiments
:
σnti+Cum-1u⋅ti=0(i=1,2),
where C is a friction coefficient that is non-zero for grounded ice only, ti (i=1,2)
are the two bedrock tangent vectors, and m is a friction-law exponent. For
the floating part of the flow, where ice is detached from the bedrock (i.e.,
the ice–ocean interface, or the ice–bedrock interface at minimal
floatation), the normal stress exerted by the ice body is smaller than
or equivalent to the ocean water pressure; therefore, we apply a
stress balance condition; normal stress, σnn=(n⋅σ)⋅n, is balanced by the pressure due to
buoyancy, Pw:
-σnn=Pw=ρwg(zw-z),
where zw is the sea level. For the case of z(x,y)=b(x,y), we
also need to consider a “contact problem” to decide the
actual location of the GL. We discuss the contact problem and its
implementation in Elmer/Ice and FELIX in more detail below.
The evolution of both the upper and lower free surfaces is determined by a
kinematic boundary condition:
∂zi∂t+u∂zi∂x+v∂zi∂y=w+a˙i
for i=s, b denoting the upper and lower surfaces, respectively, and
a˙i representing the surface or basal mass balance.
Consistent with the MISMIP3d experimental setup, the horizontal velocity is
set to u=0ma-1 at the ice divide (x=0km) and
free-slip conditions are applied at the two lateral boundaries
(y=0km and y=100km). Along the fixed ice shelf front at
the downstream end of the model domain, Eqs. ()
and () apply for ice above and below the water line,
respectively.
Values for all model constants and parameters, including those that
specifically apply to the MISMIP3d experimental protocols, are noted in
Table .
Lower-order approximations
Lower-order approximations to the full Stokes equations expressed above, such
as the “shallow-ice” approximation SIA; and
“shallow-shelf” approximation SSA;, come about via
geometric scaling arguments. These arguments can be used to show that, for
many locations on glaciers and ice sheets, specific gradient terms in the
stress and strain-rate tensor expressions above contribute negligibly to the
momentum balance. While omitting these terms leads to a significant reduction
in the numerical complexity and computational cost involved in solving the
momentum balance equations see, e.g.,,
the resulting errors may lead to non-negligible differences in dynamically
complex regions of the ice sheet, such as near GLs .
For this reason, full Stokes models are assumed to provide a better measure
of the most complete and accurate solution near GLs, against which solutions
from lower-order approximations may be compared in order to assess their
accuracy.
Parameters used in Elmer/Ice and FELIX-S.
SymbolConstantValue and unitρiIce density900 kgm-3ρwWater density1000 kgm-3gGravitational acceleration9.8 ms-2nFlow law exponent3AFlow law parameter10-25s-1Pa-3CBed friction parameter107Pam-1/3s1/3mBed friction exponent1/3a˙sAccumulation rate0.5 ma-1Comparison of model numerics
Both FELIX-S and Elmer/Ice discretize the Stokes-flow momentum balance
equations using the finite element method (FEM). Both models have undergone
extensive formal verification see, have
been subject to formal convergence studies
see, and have been shown to
compare very favorably to one another when applied to the Ice Sheet Model
Intercomparison for Higher-Order Models (ISMIP-HOM)
experiments (see, Figs. 6, 7, 10, 11, 13, and 14 in ).
Additional details (and references) for Elmer/Ice are given in
and , and for FELIX-S
additional details are given in . Here, we
provide a summary of several important similarities and differences between
the numerical implementations used by Elmer/Ice and FELIX-S, noting that we
view the differences as arbitrary. That is, there are not clear
arguments for why one choice is superior to the other and, in that sense, we
view both methods as equally valid.
The first significant difference between Elmer/Ice and FELIX-S is in the
choice of finite elements; Elmer/Ice uses hexahedral elements with P1–P1
basis functions (linear in velocity and pressure) and “bubble” function
stabilization, whereas FELIX-S uses tetrahedral, Taylor–Hood elements with
P2–P1 basis functions (quadratic in velocity, linear in pressure). The
second important difference is that Elmer/Ice and FELIX-S use different
“masking” schemes for identifying grounded vs. floating regions of the
lower surface; Elmer/Ice marks the nodes bounding each element whereas
FELIX-S marks the element faces. The third important difference, which is a
generic FEM implementation issue and not specific to the Stokes-flow problem,
is in how the value of the basal friction coefficient, C, is applied at the
Gaussian quadrature points. FELIX-S calculates the values of C at
quadrature points directly (with the accuracy of integration increasing with
the number of integration points), whereas Elmer/Ice interpolates the values
of C at Gaussian quadrature points from nodal values
(with the number of integration points needed for a
given degree of accuracy determined by the order of the basis function).
In terms of similarities, Elmer/Ice and FELIX-S use the same scheme for
evolving the free surfaces based on an FEM discretization of the kinematic
boundary Eq. () . The two models also use
nearly identical implementations of the contact problem. For FELIX-S, the
ocean water buoyancy pressure is compared to the normal stress of
the ice on the bed, and for Elmer/Ice, the ocean water buoyancy pressure is
first integrated and then compared to the normal force of the ice on
the bed . While both models solve the contact problem at
nodes, the information is used differently; Elmer/Ice uses it to decide if
nodes in contact with the bed are floating or grounded, whereas
FELIX-S uses it to decide if nodes in contact with the bed constitute an
element face that is floating or grounded. We return to the
discussion of these different schemes and their impact on model output in
Sect. .
Lastly, of the three potentially different ways for defining how the basal
friction coefficient C varies over the area of a grounded-to-floating
element – “last grounded” (LG), “discontinuous” (DI), and “first
floating” (FF) discussed in more detail in –
FELIX-S uses what amounts to the DI implementation (the C values are
discontinuous across the GL) (Fig. ). However, because the values
of C are intimately tied to the location of the GL, and because of the
different masking schemes used to decide on grounded vs. floating nodes (in
Elmer/Ice) or element faces (FELIX-S), a direct comparison based on the
implementation of the friction coefficient is really only meaningful for the
P75D (diagnostic) simulation. We also return to this discussion in more
detail in Sect. .
A schematic of the
different basal boundary masking schemes used by FELIX-S and Elmer/Ice and
their impact on the definition of the MISMIP3d basal friction coefficient
(C(x,y) is assumed uniform beneath grounded ice for illustrative purposes.
For floating ice, C(x,y)=0.). Circles denote the nodes at the ice–bed
interface, defining the basal finite element faces (triangular and
quadrilateral for FELIX-S and Elmer/Ice, respectively); open circles denote
floating nodes for which z(x,y,t)>b(x,y); and solid circles denote
grounded nodes for which z(x,y,t)=b(x,y) and -σnn>Pw. Numbers 1–8 identify triangular element faces of FELIX-S
and letters A–C identify specific nodes common to both models. As discussed
in Sect. , the different masking schemes lead to the
different grounding-line positions and also to the different nodal values of
C along profiles 1–3. The C profiles based on DI for Elmer/Ice (heavy
red line) and FELIX-S profiles (heavy blue line) are shown, as are the
corresponding Elmer/Ice FF (black dashed) and LG (black dotted) profiles.
Comparisons of across-flow-averaged velocities for lower (ub, vb, wb)
and upper (us, vs and ws) surfaces along
the x direction for FELIX-S (black solid line), Elmer/Ice FF (red dashed
line), DI (black dotted line), and LG (blue dotted line) cases for the
diagnostic experiment P75D. Where the black dotted line is not clearly
visible, Elmer/Ice and FELIX-S solutions are overlying.
Experimental setup
We provide a brief review of the MISMIP3d experimental setup, referring the
reader to for additional details. Three experiments are
conducted and reported on; the “standard” prognostic experiment (Stnd), the
prognostic, basal sliding perturbation experiments (P75S and P75R), and the
diagnostic experiment (P75D). The Stnd experiment is similar to that
conducted in the original, two-dimensional MISMIP experiment for flowline
models , where steady-state ice sheet GL positions are
examined for a uniform, downward sloping (non-retrograde) bed in the
along-flow (x) direction, with a uniform basal friction coefficient
and uniform bed properties in the across-flow (y) direction. The goal is to
compare three-dimensional model results to those from the two-dimensional
test case, for which analytic solutions are available . The
prognostic P75S experiment starts from the steady-state geometry of the Stnd
experiment and introduces a two-dimensional, Gaussian perturbation (a
slippery patch) to the basal sliding coefficient field, C(x,y), which
introduces changes to the model state (velocity and geometry fields). The ice
sheet geometry and GL are then allowed to advance for 100 years. The P75R
experiment, which starts from the final state of the ice sheet at the end of
the P75S experiment, returns the C(x,y) field to its original, uniform
distribution, inducing GL retreat. The model is then integrated forward in
time for another 100 years. The P75D experiment compares the diagnostic model
state when using the P75S geometry calculated by the Elmer/Ice model. Below,
we first report on the comparison between Elmer/Ice and FELIX-S for the P75D
experiment. We then follow with a comparison for the Stnd, P75S, and P75R
experiments.
For all experiments (unless otherwise noted) the vertical dimension in both
models is discretized with 10 layers. For the P75D and Stnd experiments, the
nodal coordinates used by Elmer/Ice and FELIX-S are identical, with
along-flow resolution of 50 m in the vicinity of the GL and
across-flow resolution of 2500 m. For the P75S and P75D experiments,
along-flow resolution is 50 m and across-flow resolution is varied
from 2500 to 625 m. In this case the Elmer/Ice and FELIX-S nodal
coordinates are not identical, as discussed further below in
Sect. (note that we distinguish identical nodal coordinates from identical meshes because the mesh can
also be considered a function of element type, which is different for
Elmer/Ice and FELIX-S).
ResultsThe diagnostic experiment P75D
We first compare the two models for the diagnostic experiment P75D
(Fig. ). Both models use the same parameters (e.g., A, C, and
m; see also Table ) and, despite the different element
types discussed above, have identical nodal coordinates over the entire model
domain. From Fig. , it is clear that the three velocity components
(u, v, and w) for Elmer/Ice and FELIX-S are in close agreement for both
the upper and lower surfaces, an indication of inherent consistencies between
the two models. For this experiment, the most direct comparison between
Elmer/Ice and FELIX-S is afforded by the DI results since, prior to determining
C, we directly interpolate the nodal basal boundary condition mask
from the Elmer/Ice diagnostic solution onto the element-face mask
used by FELIX-S. In general, for the x-component of the horizontal velocity
(u), the differences are relatively small (< 2 %) over the entire
model domain and relatively less near the ice divide, and they increase continuously
from the GL to the ice-shelf portion of the domain (Fig. ). For the
v and w velocity components, we observe relatively larger discrepancies
in the region of the GL (around km 535–555), but still very small
differences (< 5 %) over the majority of the domain (Fig. ).
Comparisons of across-flow-averaged velocity differences for lower (Δub, Δvb, and Δwb) and upper
(Δus, Δvs, and Δws)
surfaces along the x direction for FELIX-S and Elmer/Ice for the diagnostic
experiment P75D. The blue dotted, black solid, and red dashed lines denote
the differences by subtracting Elmer/Ice LG, DI, and FF values from FELIX-S
values, respectively.
Despite efforts to make mesh, initial, and boundary conditions and parameter
settings identical between the two models, several non-negligible differences
discussed above are likely responsible for the small differences in
velocities shown in Fig. . The first likely cause for the small
differences is the different boundary masking schemes; as noted above,
FELIX-S marks the basal boundary faces in an element-wise manner vs. the
node-wise manner used by Elmer/Ice. To apply boundary settings that are as
similar as possible for the P75D test case, FELIX-S applies the nodal mask
from Elmer/Ice when generating its own element-based mask; element faces in
FELIX-S are marked as grounded only if all three nodes of a triangle are
marked as grounded according to the Elmer/Ice mask. Otherwise, the elements
are marked as floating (Fig. ) (we note that this is not the same
criteria that is used by FELIX-S in the remainder of the experiments to
determine the location of floating vs. grounded ice, as discussed further
below). This may lead to small differences when assembling the element
stiffness matrices and the right-hand-side vectors (for the Dirichlet
boundary conditions) as part of the FEM discretization of the Stokes system.
Another likely cause for the minor velocity differences in the P75D
experiment is the specification of the sliding coefficient C at Gauss
quadrature points, as discussed above in Sect. . Finally,
despite identical mesh coordinates, Elmer/Ice and FELIX-S use different
element types, basis functions, and interpolation schemes as discussed above.
Overall, for the P75D experiment FELIX-S results in larger horizontal
velocities (u) at the GL than Elmer/Ice does. As a result, FELIX-S exhibits
a slightly larger ice flux (1 %) through the GL than Elmer/Ice does. This
systematic difference between the two models is likely a combination of the
different numerical choices discussed above. Again, as these choices appear
arbitrary with respect to our current level of understanding, it is not clear
that the implementation and results from one model can be distinguished as
being superior to the other. In any case, we expect these differences to
disappear as the horizontal grid spacing approaches zero (as discussed below in
Sect. ).
Comparison between Elmer/Ice (LG, DI, and FF) and FELIX-S GL
positions for the Stnd and P75S experiments. The xG0 denotes
the steady-state GL position for the Stnd experiment. The rows for ΔxGLc and ΔxGLm denote the differences between
xG0 and the GL position at year 100 in the P75S experiment at
the centerline and margin, respectively. As it is invariant in the
across-flow direction, we do not explore the sensitivity of the Stnd
experiment to across-flow resolution. All GL positions and differences are
given in km.
The comparison of Elmer/Ice and FELIX-S diagnostic experiment results
demonstrate that model velocities are within several percent of one another
when using identical nodal-mesh coordinates but that different numerics
and/or implementations of boundary conditions result in non-zero differences
in the model solutions. In turn, the prognostic experiments demonstrate how
those biases accumulate and affect the time-integrated model solutions.
For the Stnd prognostic experiment, FELIX-S uses the same initial ice sheet
geometry (based on the boundary-layer theory solution of
) and the same along- and across-flow resolution in the
vicinity of the GL (50 and 2500 m, respectively) as Elmer/Ice. Moving
away from the ∼ 30 km wide region of high resolution near the GL,
along-flow mesh resolution linearly increases to several tens of kilometers based on
a geometric progression. From this initial condition, the forward model is
integrated for ∼ 1300 years, by which time the GL position is close to
equilibrium (according to the criteria that the relative rate of volume
change is <10-5, the same criteria used by Elmer/Ice;
). Both models demonstrate a continuous advance of the
GL, with FELIX-S reaching a steady-state GL position (xg) of
519.85 km (Table ) and Elmer/Ice reaching steady-state positions of xg=529.55, 526.80, and 522.35 km, for
LG, DI, and FF, respectively . Apparently, FELIX-S
produces a smaller equilibrium-sized ice sheet with a GL position that is
several to ∼ 10 km upstream from that of Elmer/Ice.
We attribute the different equilibrium GL locations to differences in the
numerical schemes already discussed above. While the overall retreated
grounding line of FELIX-S relative to Elmer/Ice is consistent with the minor
velocity differences observed – FELIX-S produces higher along-flow
velocities (and hence flux) upstream from, at, and downstream from the GL,
with the time-integrated result of thinner ice (and hence floatation)
occurring slightly farther inland relative to Elmer/Ice – we note that
Elmer/Ice velocities when using the FF scheme are significantly faster (up to
∼ 100 ma-1 downstream of the GL) than for FELIX-S
(Figs. and ), and yet the Elmer/Ice GL when using the FF
scheme is still advanced relative to that of FELIX-S
(Table ). Hence, other differences in the two numerical
schemes must be more important in contributing to the observed steady-state
GL location differences (we return to the discussion of these differences in
greater detail in Sect. ). Regardless of the reasons, we
note that the differences between the equilibrium positions for the FELIX-S
and Elmer/Ice GL locations, for both DI and FF, are very close to or within
the range of the estimated truncation error for Elmer/Ice at an along-flow
resolution of 50 m in the vicinity of the GL (see
, Fig. 6, and , Fig. 1c, and
related discussions therein). As in , we find that the
modeled equilibrium GL (ice thickness of 685 m) is
∼ 5 km upstream of that implied by the floatation condition
(ice thickness of 618 m) (Fig. ; compare to Fig. 2b in
).
We repeat the Stnd prognostic experiment with FELIX-S but starting from an
initially oversized configuration, allowing the ice sheet to shrink over
time and the GL to retreat to its equilibrium position (as opposed to
starting from an undersized initial configuration with an advancing GL). In
this case, an equilibrium GL position is reached after a forward model
integration time of ∼ 1500 years and we find xg=524.50km, approximately a 5 km difference relative to the case
with an advancing GL. This difference in equilibrium GL positions under
advanced vs. retreated initial configurations is consistent with that found
by and and is consistent with a
model truncation error of ∼ 5 km at an along-flow resolution
near the GL of 50 m (∼100Δx).
demonstrated that steady-state GL locations from an advanced or retreated
initial condition do converge with increasing grid resolution. Based on the
similar truncation error estimate at 50 m along-flow resolution and results
from the P75S and P75R experiments (discussed next), we speculate that the FELIX-S
would show similar behavior.
Grounding-line position vs. local floatation. The vertical
black dashed line marks the equilibrium GL position for the Stnd experiment
and the heavy blue and black lines denote the modeled ice sheet surface near
the GL (blue) vs. that determined by the floatation condition (black)
(compare with the inset in Fig. 2b from ).
Lastly, we conduct a “quasi-convergence” study for the Stnd experiment by comparing solution error against
mesh resolution. In order to control computational costs, the mesh is
modified slightly relative to that discussed above. First, the number of
vertical layers is reduced from 10 to 5. Second, the quasi-qualifier
indicates that we do not double the along-flow mesh resolution everywhere in
the domain at each step in the study, unlike in a true convergence study.
Rather, the number of across-flow elements is unchanged and resolution
doubles only over a particular region within the vicinity of the grounding
line (based on a geometric progression as in previous work, e.g.,
). Simulations are conducted with along-flow resolution
of 1600, 800, 400, 200, 100, and 50 m in this refined region. For the
highest along-flow resolution, which coincides with that of the Stnd
experiment discussed above, the equilibrium GL position is 519.55 km
(a difference of 0.30 km relative to when using 10 vertical layers).
Figure shows the Richardson estimate for the
solution error vs. the along-flow mesh resolution. Slight irregularities in
the GL position as a function of increasing resolution result from doubling
the mesh resolution in the along-flow direction and in the region of the GL,
rather than over the entire mesh. Regardless of these minor irregularities,
the GL position is seemingly convergent as a function of resolution, with a
convergence rate between linear and quadratic. At the finest along-flow
resolution of 50 m near the GL, the truncation error estimate is
∼ 300 m (∼6Δx).
Convergence of the Stnd experiment as a function of along-flow grid
resolution (circles), as discussed in Sect. . Error estimates
for grounding-line position are based on Richardson error estimation.
Black dashed and dash-dot lines show perfect linear and quadratic convergence
rates (respectively).
GL evolution on both the symmetry axis (upper curves) and free-slip
boundary (lower curves) for the P75S (solid curves) and P75R (dashed curves)
comparisons between FELIX-S (bold-black curves) and Elmer/Ice DI (a,
thin black curves), LG (b, thin blue curves), and FF (c,
thin red curves). The number of elements along the y direction is 20
(Δy=2500m). Note that GL positions are plotted relative to
their equilibrium positions in the Stnd experiment.
GL evolution on both the symmetry axis (upper curves) and free-slip
boundary (lower curves) for the P75S (solid curves) and P75R (dashed curves)
comparisons between FELIX-S (bold-black curves) and Elmer/Ice DI (a,
thin black curves), LG (b, thin blue curves), and FF (c,
thin red curves). The number of elements along the y direction is 40
(Δy=1250m). Note that GL positions are plotted relative to
their equilibrium positions in the Stnd experiment.
GL evolution on both the symmetry axis (upper curves) and free-slip
boundary (lower curves) for the P75S (solid curves) and P75R (dashed curves)
comparisons between FELIX-S (bold-black curves) and Elmer/Ice DI (a,
thin black curves), LG (b, thin blue curves), and FF (c,
thin red curves). The number of elements along the y direction is 80
(Δy=625m). Note that GL positions are plotted relative to
their equilibrium positions in the Stnd experiment.
Difference in FELIX-S and Elmer/Ice GL position changes
(ΔGLElmer-ΔGLFELIX) at the centerline
for the P75S experiment as a function of increasing across-flow (y)
resolution (resolution increases from 2500 to 625 m as Ny
increases from 20 to 80). Lines representing the differences relative to the
LG, DI, and FF implementations in Elmer/Ice are labeled. The black dashed line
shows the slope for a theoretical first-order convergence rate. The grey dashed
line shows the estimated Elmer/Ice truncation error of ∼ 2 km
from and .
P75S and P75R prognostic experiments
In the P75S and P75R prognostic experiments, we investigate advance and
retreat of the GL following a step-change perturbation in the basal friction
distribution for 100 years and a return to the initial basal friction
distribution for a further 100 years (the P75S and P75R experiments,
respectively), as discussed above in Sect. . The initial
condition for the P75S experiment is the steady-state GL position of the Stnd
prognostic experiment discussed above. To manage computational costs,
especially in experiments where sensitivity to mesh resolution is explored,
both models employ regional refinement near the GL. Initial mesh resolution
in this region is 50 m along flow near the GL and 2500 m
across flow for both models, but the area of refined mesh in FELIX-S and
Elmer/Ice is located in different regions because of the different equilibrium GL
positions for the Stnd experiment. Thus, the two meshes have the same
refined resolution around the GL but different nodal coordinates for this set
of experiments (i.e., the two model meshes are not identical as they are for
the P75D experiment). Over the course of the P75S and P75R experiments, the
centerline ice thickness at the GL varied by < 2 % of its equilibrium
value reported on in Sect. .
Similar to the Stnd experiment, FELIX-S predicts relatively less GL advance
(P75S) and/or relatively more GL retreat (P75R) than Elmer/Ice, as shown in
Figs. –. Similar to Elmer/Ice ,
FELIX-S shows a clear sensitivity to the across-flow resolution (Δy);
as the number of elements in the y direction increases from 20 to 80
(Δy decreases from 2500 to 625 m), the “reversibility” –
i.e., the return to the initial position – of the GL improves (note that we
expect ≫ 100 years to demonstrate full reversibility,
). More importantly, we also find that as the number
of elements in the y direction increases from 20 to 80, the agreement
between FELIX-S and Elmer/Ice increases for all of Elmer/Ice GL
implementations (i.e., LG, DI, and FF; Figs. –). For the
highest across-flow grid resolution, differences in the FELIX-S and Elmer/Ice
DI and FF grounding-line position changes are close to or below the published
truncation error for Elmer/Ice, and differences relative to Elmer/Ice LG
converge to that same value (Fig. ).
Discussion
As noted above, some fraction of the differences in the prognostic model
simulation results can likely be attributed to the small differences in the
model velocity fields, as seen in the P75D experiment. In turn, these
differences are likely related to the different type of finite elements and
basis functions used by the two models. However, we attribute the bulk of the
prognostic model simulation differences to differences in the treatment of
the contact problem, and more importantly, to the different masking schemes
used for the basal boundary conditions.
There are small differences in the way the contact problem is implemented in
FELIX-S vs. Elmer/Ice; while following the same physical basis for the
contact problem, FELIX-S compares the normal stress and the sea water
pressure acting at nodes, whereas Elmer/Ice compares the normal force and sea water force acting at
nodes . The result may be that, effectively, Elmer/Ice and
FELIX-S feel slightly different normal forces (or pressures) at basal nodes
of the ice–bed interface, resulting in slight differences when assessing
whether a node (Elmer/Ice) or element face (FELIX-S) is grounded or not.
Unfortunately, the different element types used by FELIX-S and Elmer/Ice do
not allow for a definitive confirmation of this hypothesis.
Of greater importance, however, are the different treatments of the basal
boundary condition masking schemes discussed in Sect. .
Figure provides a schematic summary of the differences in the
Elmer/Ice and FELIX-S basal boundary masking schemes and demonstrates how
those differences would impact the GL location in the two models for a
particular edge case. In the upper part of Fig. , the nodes
marked A and C are unambiguously floating (i.e., z(x,y,t)>b(x,y) so that
no contact problem needs to be considered for those nodes). Because FELIX-S
considers any element with one or more floating nodes to be floating,
elements 3, 4, and 8 are all marked as floating, with the resulting FELIX-S
GL position shown by the blue line in Fig. . For the same geometric
configuration, the node-based scheme used by Elmer/Ice defines a slightly
different position for the GL, shown by the red line in Fig. .
In addition to the slightly different grounding-line locations, the different
basal boundary masking schemes will lead to different profiles for C, as
shown schematically in the lower part of Fig. where we plot
approximate nodal C profiles for the two models. These differences come
about because, for FELIX-S, the nodal matrix coefficients contain the
contributions of C (and other variables) from the surrounding elements. As
an example, consider profiles 1 and 3 in Fig. . The C=0
contributions from elements 3 and 8, and assuming additional floating
elements to the north of element 3 and to the south of element 8, reduce the
matrix coefficients associated with the node along the centers of profiles 1
and 3 by a factor of approximately 2/6=1/3 (for these nodes, two of the six
surrounding elements are floating), relative to Elmer/Ice. This estimate is
only approximate because in reality the nodal coefficients contain
additional terms related to the ice velocity and the basis functions, which
are not uniform for all elements surrounding a node. Similarly, assuming that
additional elements downstream of the FELIX-S GL are also floating, the
coefficient at node B along profile 2 will be reduced by ∼5/6
relative to the Elmer/Ice value, since five of the six surrounding elements are
floating.
We attribute the majority of the differences observed in prognostic model
simulations to these slight differences in GL position, and more importantly
to these slight differences in the value of C. If we again consider
profile 2 in Fig. (and to a lesser extent profiles 1 and 3), the
relatively larger value of C for Elmer/Ice will lead to relatively less
basal sliding there and eventually relatively thicker ice. This in turn will
make it more likely that neighboring nodes may also eventually ground. The
overall, time-integrated result will be that, all other things being equal,
the Elmer/Ice masking scheme will favor grounding and/or grounding-line
advance relative to the FELIX-S scheme. This proposed difference in model
behavior is consistent with the differences observed when the two models are
applied to the prognostic experiments.
We further note that the differences between the nodal C profiles for
Elmer/Ice and FELIX-S shown in Fig. are broadly similar to the
differences between the DI and FF implementations in Elmer/Ice; despite the
DI-like implementation of C in FELIX-S, the different masking scheme
results in C values at nodes that effectively look more similar to the
FF implementation of Elmer/Ice (dashed C profile lines in Fig. ).
Simulations using Elmer/Ice with these two different implementations
demonstrate differences that are broadly similar to the FELIX-S and Elmer/Ice
differences observed here for prognostic simulations; the FELIX-S equilibrium
grounding line for the Stnd experiment is closest to that for Elmer/Ice when
using the FF implementation (Table ), the change in FELIX-S
GL in the P75S experiment is closest to that observed for Elmer/Ice when
using the FF implementation (Table ), and at all across-flow
resolutions the advance and retreat curves for FELIX-S in the P75S and P75R
experiments most closely resemble those for Elmer/Ice when using FF
(Figs. –).
Based on our understanding of these model-to-model differences and their
hypothesized impact on model simulations, we have a strong expectation that
the differences in model outputs will decrease as model resolution increases.
As the element size decreases, the differences in ice sheet geometry between
nodes – the primary cause for differences in the nodal- vs. element-based
masking schemes – will also decrease, and the two sets of model results
should converge. Indeed this is exactly what we see for the P75S and P75R
experiments. Similar to the observation of that the
LG, DI, and FF implementations in Elmer/Ice all converge to a similar
solution with increasing resolution, we demonstrate here that the FELIX-S
results also appear to converge to that same solution with increasing grid
resolution (Figs. –). When considering the two most
comparable implementations of the basal boundary condition masking schemes
(FELIX-S and Elmer/Ice FF), the two models agree for the P75S and P75R
experiments to within the estimated truncation error for Elmer/Ice at
all across-flow grid resolutions explored here (Fig. ).
For the Elmer/Ice DI and LG implementations, the differences with FELIX-S as
a function of grid resolution are also clearly converging (Fig. ).
The convergence study for the Stnd experiment suggests a GL position error of
∼6Δx at an along-flow grid resolution of 50 m
(∼ half the ice thickness). Conversely, the difference in the GL
position for the Stnd experiment when starting from a retreated vs. advanced
initial condition suggests a GL position error of ∼100Δx
(∼ 7 times the ice thickness). The discrepancy between these two
possible error estimates suggests that the more conservative of the two
truncation error estimates should be used.
Conclusions
We have conducted a first, detailed
comparison of two full Stokes ice sheet models, FELIX-S and Elmer/Ice,
applied to the MISMIP3d benchmark experiments. While previous informal
comparisons have suggested very close agreement between the two models
, here we explore the model similarities and differences much
more carefully, focusing on how differences in model numerics lead to
differences in model outputs when using identical mesh coordinates and
forcing, and in particular we focus on differences that are important for the
simulation of marine ice sheet dynamics.
Overall, we find close agreement between the two model outputs for cases
where the impact of rather arbitrary choices in the implementation of basal
boundary conditions can be minimized; for the P75D experiment, diagnostic
solutions (e.g., velocity fields) agree to within ∼ 2–5 %. While
it is difficult to attribute those small differences to particular numerical
choices made by the two models, it is likely that different element types and
basis functions and different implementations of the contact problem play a
role. More significant differences between the two sets of model results are
found for prognostic problems. Overall, we find that equilibrium grounding
lines for FELIX-S are relatively more retreated than those for Elmer/Ice (as
demonstrated by the Stnd experiment) and that FELIX-S is less inclined to
ground and hence less inclined to show grounding-line advance than Elmer/Ice
(as demonstrated by the Stnd, P75S, and P75R experiments).
A detailed look at the two models strongly argues that differences in the
basal boundary masking schemes and in the implementation of the basal
friction coefficient are the source of these differences. As we are currently
unable to judge whether or not one scheme is superior to the other, our
results urge caution when interpreting the results from full Stokes models as
a metric for accuracy in model intercomparisons, particularly if those
results are not obtained at grid resolutions demonstrated to be within the
regime of asymptotic convergence. In cases where an estimate for the model
truncation error is not available (e.g., due to model cost with increasing
resolution), we propose that an alternative estimate for the uncertainty in
the grounding-line position is the span of grounding-line positions predicted
by multiple Stokes models. Here, we are encouraged to find that (1) as the
grid resolution for both models increases, the differences between the two
models continues to decrease and (2) for their most comparable
implementations, the models agree to within the estimated truncation error
for one of the models. This finding suggests (but does not prove) that in the
limit of high grid resolution, multiple full Stokes models can be shown to
agree on a particular test case solution, despite small differences in their
numerics.
Future efforts could improve on the work presented here by confirming the
truncation error for the FELIX-S model, in order to understand if different
numerics might be a means for further reducing model truncation error. Also,
by running simulations at even finer grid resolutions, future efforts could
definitively confirm that the results from multiple Stokes models converge in
the limit of very fine grid resolution.
Data availability
The data used in this article are attached as a
Supplement.
The Supplement related to this article is available online at doi:10.5194/tc-11-179-2017-supplement.
Tong Zhang and Stephen Price initiated the study with
input from Olivier Gagliardini, Gaël Durand, and Julien Brondex.
Necessary modifications to the FELIX-S model were made by Tong Zhang with
input and guidance from original code authors Lili Ju and Wei Leng. Tong
Zhang conducted the FELIX-S simulations. Julien Brondex, Olivier Gagliardini,
and Gaël Durand conducted Elmer/Ice simulations, provided results for
comparison with FELIX-S, and provided insight when attributing simulation
differences to model differences. Tong Zhang and Stephen Price wrote the
paper with contributions from all co-authors.
Olivier Gagliardini is a member of the editorial board of
the journal. All other authors declare that they have no competing
interests.
Acknowledgements
The authors thank Steph Cornford, the three anonymous reviewers, and the
editor Hilmar Gudmundsson for suggestions that helped to clarify and improve
the paper. Support for Tong Zhang, Stephen Price, Lili Ju, and Wei Leng was
provided through the Scientific Discovery through Advanced Computing (SciDAC)
program funded by the US Department of Energy (DOE), Office of Science,
Advanced Scientific Computing Research and Biological and Environmental
Research. Tong Zhang was also supported by the National Basic Research
Program (973) of China under grant No. 2013CBA01804 and
CHINARE2016. Lili Ju was
partially supported by the US National Science Foundation under grant
No. DMS-1215659. Wei Leng was partially supported by the National 863 Project
of China under grant No. 2012AA01A309 and the National Center for Mathematics
and Interdisciplinary Sciences of the Chinese Academy of Sciences. Elmer/Ice
development and simulations presented here were partly funded by the Agence
Nationale pour la Recherche (ANR) through the SUMER, Blanc SIMI 6-2012.
FELIX-S simulations presented here used computing resources of the National
Energy Research Scientific Computing Center (NERSC; supported by the Office
of Science of the US Department of Energy under Contract
DE-AC02-05CH11231). Elmer/Ice simulations discussed in this paper used
computing resources of CINES (Centre Informatique National de l'Enseignement
Supérieur, France) under allocations 2015-016066 made by GENCI (Grand
Equipement National de Calcul Intensif). This study was inspired by
discussions with Frank Pattyn and Gaël Durand at the first MISOMIP
workshop, supported by the Center for Global Sea-Level Change at New York
University Abu Dhabi.
Edited by: G. H. Gudmundsson
Reviewed by: S. L. Cornford and two anonymous referees
References
Clark, P. U., Church, J. A., Gregory, J. M., and Payne, A. J.: Recent
Progress in Understanding and Projecting Regional and Global Mean Sea Level
Change, Current Climate Change Reports, 1, 224–246, 2015.Cornford, S. L., Martin, D. F., Graves, D. T., Ranken, D. F., Brocq, A.
M. L., Gladstone, R. M., Payne, A. J., Ng, E. G., and Lipscomb, W. H.:
Adaptive mesh, finite volume modeling of marine ice sheets, J. Computat.
Phys., 232, 529–549, 10.1016/j.jcp.2012.08.037, 2013.
Cuffey, K. and Paterson, W.: The Physics of Glaciers, 4th Edn., Elsevier,
Amsterdam, 2010.Dukowicz, J. K., Price, S. F., and Lipscomb, W. H.: Consistent approximations
and boundary conditions for ice-sheet dynamics from a principle of least
action, J. Glaciol., 56, 480–496, 10.3189/002214310792447851, 2010.Durand, G., Gagliardini, O., de Fleurian, B., Zwinger, T., and Le Meur, E.:
Marine ice sheet dynamics: Hysteresis and neutral equilibrium, J. Geophys.
Res., 114, F03009, 10.1029/2008JF001170, 2009a.
Durand, G., Gagliardini, O., Zwinger, T., Le Meur, E., and Hindmarsh,
R. C. A.: Full Stokes modeling of marine ice sheets : influence of the grid
size, Ann. Glaciol., 50, 109–114, 2009b.Favier, L., Durand, G., Cornford, S. L., Gudmundsson, G. H., Gagliardini, O.,
Gillet-Chaulet, F., Zwinger, T., Payne, A. J., and Le Brocq, A. M.: Retreat
of Pine Island Glacier controlled by marine ice-sheet instability, Nature
Climate Change, 5, 117–121, 10.1038/nclimate2094, 2014.Feldmann, J. and Levermann, A.: Collapse of the West Antarctic Ice Sheet
after local destabilization of the Amundsen Basin, P. Natl. Acad. Sci. USA,
112, 14191–14196, 10.1073/pnas.1512482112, 2015.Gagliardini, O. and Zwinger, T.: The ISMIP-HOM benchmark experiments
performed using the Finite-Element code Elmer, The Cryosphere, 2, 67–76,
10.5194/tc-2-67-2008, 2008.Gagliardini, O., Zwinger, T., Gillet-Chaulet, F., Durand, G., Favier, L.,
de Fleurian, B., Greve, R., Malinen, M., Martín, C., Råback, P.,
Ruokolainen, J., Sacchettini, M., Schäfer, M., Seddik, H., and Thies, J.:
Capabilities and performance of Elmer/Ice, a new-generation ice sheet model,
Geosci. Model Dev., 6, 1299–1318, 10.5194/gmd-6-1299-2013, 2013.Gagliardini, O., Brondex, J., Gillet-Chaulet, F., Tavard, L., Peyaud, V., and
Durand, G.: Brief communication: Impact of mesh resolution for MISMIP and
MISMIP3d experiments using Elmer/Ice, The Cryosphere, 10, 307–312,
10.5194/tc-10-307-2016, 2016.
Hutter, K.: Theoretical Glaciology: Material Science of Ice and the Mechanics
of Glaciers and Ice Sheets, Reidel, Dordrecht, 1983.
IPCC: Climate Change 2013: The Physical Science Basis, Contribution of
Working Group I to the Fifth Assessment Report of the Intergovernmental Panel
on Climate Change, Cambridge University Press, Cambridge, UK and New York,
NY, USA, 2013.Leng, W., Ju, L., Gunzburger, M., Price, S., and Ringler, T.: A Parallel
High-Order Accurate Finite Element Nonlinear Stokes Ice Sheet Model and
Benchmark Experiments, J. Geophys. Res., 117, 2156–2202,
10.1029/2011JF001962, 2012.Leng, W., Ju, L., Gunzburger, M., and Price, S.: Manufactured solutions and
the verification of three-dimensional Stokes ice-sheet models, The
Cryosphere, 7, 19–29, 10.5194/tc-7-19-2013, 2013.Leng, W., Ju, L., Gunzburger, M., and Price, S.: A parallel computational
model for three-dimensional, thermo-mechanical Stokes flow simulations of
glaciers and ice sheets, Commun. Comput. Phys., 1056–1080,
10.4208/cicp.310813.010414a, 2014.
Morland, L.: Unconfined ice-shelf flow, in Dynamics of the West Antarctic Ice
Sheet, Reidel, Dordrecht, 1987.Pattyn, F. and Durand, G.: Why marine ice sheet model predictions may diverge
in estimating future sea level rise, Geophys. Res. Lett., 40, 4316–4320,
10.1002/grl.50824, 2013.Pattyn, F., Perichon, L., Aschwanden, A., Breuer, B., de Smedt, B.,
Gagliardini, O., Gudmundsson, G. H., Hindmarsh, R. C. A., Hubbard, A.,
Johnson, J. V., Kleiner, T., Konovalov, Y., Martin, C., Payne, A. J.,
Pollard, D., Price, S., Rückamp, M., Saito, F., Soucek, O., Sugiyama, S.,
and Zwinger, T.: Benchmark experiments for higher-order and full-Stokes ice
sheet models (ISMIP–HOM), The Cryosphere, 2, 95–108,
10.5194/tc-2-95-2008, 2008.Pattyn, F., Schoof, C., Perichon, L., Hindmarsh, R. C. A., Bueler, E.,
de Fleurian, B., Durand, G., Gagliardini, O., Gladstone, R., Goldberg, D.,
Gudmundsson, G. H., Huybrechts, P., Lee, V., Nick, F. M., Payne, A. J.,
Pollard, D., Rybak, O., Saito, F., and Vieli, A.: Results of the Marine Ice
Sheet Model Intercomparison Project, MISMIP, The Cryosphere, 6, 573–588,
10.5194/tc-6-573-2012, 2012.Pattyn, F., Perichon, L., Durand, G., Favier, L., Gagliardini, O., Hindmarsh,
R. C., Zwinger, T., Albrecht, T., Cornford, S., Docquier, D., Fürst,
J. J., Goldberg, D., Gudmundsson, G. H., Humbert, A., Hütten, M.,
Huybrechts, P., Jouvet, G., Kleiner, T., Larour, E., Martin, D., Morlighem,
M., Payne, A. J., Pollard, D., Rückamp, M., Rybak, O., Seroussi, H.,
Thoma, M., and Wilkens, N.: Grounding-line migration in plan-view marine
ice-sheet models: results of the ice2sea MISMIP3d intercomparison,
J. Glaciol., 59, 410–422, 10.3189/2013JoG12J129, 2013.Schoof, C.: Ice sheet grounding line dynamics: Steady states, stability, and
hysteresis, J. Geophys. Res., 112, F03S28, 10.1029/2006JF000664, 2007a.Schoof, C.: Marine ice-sheet dynamics. Part 1. The case of rapid sliding,
J. Fluid Mech., 573, 27–55, 10.1017/S0022112006003570, 2007b.
Schoof, C. and Hewitt, I.: Ice-sheet dynamics, Annu. Rev. Fluid Mech., 45,
217–239, 2013.
Schoof, C. and Hindmarsh, R. C. A.: Thin-Film Flows with Wall Slip: An
Asymptotic Analysis of Higher Order Glacier Flow Models, Q. J. Mech. Appl.
Math., 63, 73–114, 2010.Seroussi, H., Morlighem, M., Larour, E., Rignot, E., and Khazendar, A.:
Hydrostatic grounding line parameterization in ice sheet models, The
Cryosphere, 8, 2075–2087, 10.5194/tc-8-2075-2014, 2014.Tezaur, I. K., Perego, M., Salinger, A. G., Tuminaro, R. S., and Price,
S. F.: Albany/FELIX: a parallel, scalable and robust, finite
element, first-order Stokes approximation ice sheet solver built for advanced
analysis, Geosci. Model Dev., 8, 1197–1220, 10.5194/gmd-8-1197-2015,
2015.Zhang, T., Ju, L., Leng, W., Price, S., and Gunzburger, M.:
Thermomechanically coupled modelling for land-terminating glaciers:
a comparison of two-dimensional, first-order and three-dimensional,
full-Stokes approaches, J. Glaciology, 61, 702–712,
10.3189/2015JoG14J220, 2015.