We present the result of the third Marine Ice Sheet Model Intercomparison Project, MISMIP+.
MISMIP+ is intended to be a benchmark for ice-flow models which include
fast sliding marine ice streams and floating ice shelves and in particular
a treatment of viscous stress that is sufficient to model buttressing,
where upstream ice flow is restrained by a downstream ice shelf. A set of idealized
experiments first tests that models are able to maintain
a steady state with the grounding line located on a retrograde slope due to buttressing and
then explore scenarios where a reduction in that buttressing
causes ice stream acceleration, thinning, and grounding line retreat.
The majority of participating models passed the first test and then produced similar responses to the loss of buttressing. We find that the most important distinction between models in this particular type of simulation is in the treatment of sliding at the bed,
with other distinctions – notably the difference between the simpler
and more complete treatments of englacial stress but also the differences between numerical methods – taking a secondary role.
Introduction
A number of ice-flow models have been developed in the last decade that simulate
fast-flowing ice streams and ice shelves as well as larger,
slower-moving ice masses. The key difference between this generation of models
and the previous generation is their choice of viscous stress balance equations.
All ice sheet models are based upon Stokes flow or, more commonly, one of several approximations to Stokes flow .
Models designed to simulate the creeping flow of continental ice sheets over glacial cycles
are typically based on the shallow ice approximation (SIA), which considers only
vertical shear stresses, while more complete approximations are needed for ice shelves
and ice streams. The simplest model that can be applied is the shallow-shelf/shelfy-stream
approximation , which includes horizontal normal and shear stresses
and requires the solution of vertically integrated, two-dimensional stress balance equations.
More complete models include the L1Lx class of vertically integrated models , which
resemble the SSA in many respects; the higher-order (HO) models , which require the solution of simplified three-dimensional
stress equations; and the complete Stokes models that include
all viscous stresses .
There have been several community exercises comparing ice sheet models where ice stream and shelf dynamics are important. These
can be divided into two types: exercises involving real-world ice flows, perhaps
forced with climate inputs from sophisticated atmosphere and ocean models ,
and exercises involving idealized settings with simple forcings.
This paper describes the results of an idealized exercise, which can be regarded
as sequential to three previous exercises.
The Ice Sheet Model Intercomparison Project for Higher-Order Model (ISMIP-HOM, ) quantified the differences
between SIA, SSA, higher-order, and full-Stokes models in time independent
settings with periodic bedrock topography and slipperiness.
The first Marine Ice Sheet Model Intercomparison Project (MISMIP, ) considered
a time-dependent but laterally unvarying problem and
highlighted the technical challenges faced by numerical
models of an ice stream with both grounded and floating portions, that is, with
a grounding line. Many models
failed to reproduce theoretically well understood properties of such systems.
Notably, that a laterally unvarying ice stream on a bedrock that slopes
monotonically down in the direction of flow has a single equilibrium
state where ice flux across the grounding line matches the total accumulation
upstream. The second Marine Ice Sheet Model Intercomparison Project (MISMIP3D, )
extended a subset of the MISMIP experiments to include perturbations with
some lateral variation and once again demonstrated that many models of
the time did not exhibit the expected unique equilibrium state. It did not, however,
feature strong buttressing of the kind that is important in Antarctic
ice shelf and ice stream systems. Recent real-world cases include applications to
Pine Island Glacier and Thwaites Glacier in western Antarctica and in Antarctica as a whole .
MISMIP+ explores the ability of ice sheet models to
simulate coupled ice sheet and ice shelf systems where the ice shelf strongly buttresses the flow upstream and to respond to a loss of buttressing caused by ice shelf ablation.
All of the experiments are based around an idealized ice stream, adapted from
, with ice sliding into an ice shelf along a
bedrock trough with steep walls (Sect. ). Part of the bedrock trough is retrograde – it slopes upward in the direction of flow –
and key model parameters are chosen so that,
in the absence of sub-ice-shelf melting, models should form a stable equilibrium state
with the grounding line crossing the center of the channel on this retrograde slope. The equilibrium state is only stable because of lateral variation in the flow field: it is well known that without such stresses
stable steady states form only when the grounding line lies on a prograde slope
. The experimental design was stated in ;
in this paper we recap the design for convenience and report the results from 15 distinct participants, several of whom carried out the experiments with multiple model configurations.
Relationship between the MISMIP+ Ice0, Ice1, and Ice2 experiments.
An upward sloping curve indicates an advancing grounding line, a downward sloping curve a retreating grounding line. All three experiments start from a common steady state. The Ice0 experiment
provides a control for the Ice1 and Ice2 experiments, which induce retreat with ice shelf melting or calving respectively for 100 a. Later stages of the experiments remove or continue the melting or calving.
MISMIP+ domain showing bedrock elevation zb(x,y) and boundary types. The spot
height (-644 m) indicates the beginning of a retrograde slope in the center of the channel.
Experimental design
Three groups of experiments (Fig. ) were carried out. The Ice0 experiments (Sect. )
were designed to show that models were close to steady state at the start of the experiments (time t=0).
The Ice1 experiments (Sect. ) saw the ice shelves subjected to ablation at the base of the ice, with a simple formula intended to resemble the gross features of melt rates computed by ocean
circulation models, with maximum melt rates close to (but not at) the grounding line.
Ice shelf ablation also drives the Ice2 experiments (Sect. ),
but in this case the imposed ablation is concentrated at the calving front and does not evolve over time.
Geometry
The MISMIP+ ice stream is set in a rectangular domain, spanning 640 km in the x direction
and 80 km in the y direction. Ice flows in a direction roughly parallel to the x axis from x=0 toward x=640 km, so we will refer to x as the along-flow direction and y as the lateral- or across-flow direction.
A no-slip boundary applies at x=0 and free-slip boundaries
apply at both lateral boundaries, while calving front boundary conditions apply at x=640 km.
This choice of boundary conditions, together with the bedrock geometry and mass sources and sinks,
results in solutions that have mirror symmetry about the lateral center of the ice stream. For that
reason, some participants chose to run their experiments in only one half of the domain,
a perfectly acceptable practice for the MISMIP+ and MISMP3d experiments, though not the related ISOMIP+ and
MISOMIP experiments , where ocean circulation results in nonaxisymmetric melt rates.
For convenience in this paper, we define y such that -40≤y≤40 km, in contrast
with , so that the axis of symmetry is the x axis.
The bedrock elevation zb, measured in meters, is given by
zb=max[Bx(x)+By(y),-720],
where
Bx(x)=B0+B2xx‾2+B4xx‾4+B6xx‾6,
and
By(y)=dc1+e-2(y-wc)/fc-1+1+e2(y+wc)/fc-1,
with the parameters value given in Table . Figure illustrates the main features
of Eq. (): a steep-walled channel around 48 km wide running parallel to the x axis, and a ridge
around x=505 km. Ice flows from the divide at x=0 toward the calving front at x=640 km, so
that portion of the ridge between x=390 km and the summit at x=505 km is retrograde: it slopes upward in the direction of ice flow.
Initial state
Participants were asked to compute a steady state given the bedrock geometry and boundary conditions
above, a constant rate of accumulation a=0.3 m a-1, and no melting
at the ice shelf base. No particular method was prescribed. Participants simply needed to show that their
initial state was sufficiently close to equilibrium that model drift in experiment Ice0 was small compared to
the response to the perturbations of the Ice1 and Ice2 experiments. An obvious if time-consuming method
is to carry out a spin-up, evolving the ice sheet from some simple initial state over tens of thousands of model years. Alternatives include taking
a steady state from some simpler model and relaxing it over a shorter period
of time – a method that might be useful, for example, when working with full-Stokes models.
Parameter values for the MISMIP+ bedrock geometry
ParameterValuex‾300 kmB0-150 mB2-728.8 mB4343.91 mB6-50.75 mwc24 kmfc4 kmdc500 m
One point of departure from the previous MISMIP designs is the specification of the initial-steady-state grounding line position, rather than a full set of model parameters.
Participants were asked to produce a steady state where the grounding line crossed
the center of the channel (y=0) at a point xgl(y=0)=450±10 km, that is, on the retrograde slope.
In order to meet this requirement, participants were free to set any value at all for the rate factor A, which affects stresses within the ice (Sect. ), and for the basal friction
coefficient β2, which affects stresses at the ice base (Sect. ).
This is a key part of the design for two reasons.
First, we wanted all models to begin with their grounding line in steady state on the partly
retrograde bed to test that all models could achieve this basic result. Second, we wanted to
more closely emulate real-world applications of ice sheet models, where the present day geometry
of the ice sheet might be well known, but parameters such as A and β2 are unknown and are found by some kind of calibration.
Englacial stresses
Participants were permitted to choose any approximation at all
for the englacial stresses. That said, the shallow-shelf/shelfy-stream approximation
(SSA, ) is the simplest approximation that includes the horizontal
normal and shear stresses that describe the coupling between floating and grounded ice, so that in particular we did not expect submissions based solely on the shallow ice approximation (SIA). See, for example, for a discussion of stress approximations. Whichever approximation was chosen, it was
expected to involve Glen's flow law. In the general case, strain rate components
Dij and deviatoric stress components τij
satisfy
τij=A-1/nDe1/n-1Dij,
where n=3 and 2De2=DijDji. Participants were
free to chose any constant A in order to realize a
steady state within the specified tolerance, though a suggested value was given: A=2.0×10-17 Pa-3 a-1 (6.34×10-25 Pa-3 s-1).
Basal friction
Participants submitted results from simulations carried out with one
or more of three basal friction laws. All three ensure that
the basal friction τb=0 for floating ice, but differ
in their approach to ice close to flotation, where the effective
pressure at the base, N, is low. The simplest of the three is based on the rule for sliding over hard beds, with
τb=β2|u|1/3|u|-1uN>00N≤0,
where u is the horizontal ice velocity at the bed.
Although well known, the sliding law is certainly not the
final word on glacier sliding .
Taking a pragmatic (and model-centered) view, it may not be applicable close to the grounding line, where τb will be discontinuous.
The two alternative rules considered here ensure that τb is continuous (but not differentiable)
A third rule, the Weertman–Budd rule considered in, e.g., , also ensures that τb is continuous across the grounding line but is not considered here.
.
and, later, considered the case of sliding with cavitation, over hard beds, leading to a rule,
τb=α2β2N|u|1/3β6|u|+α2N31/3|u|-1u,
which, among other features, ensures that |τb| does not exceed
some fraction α2 of the effective pressure but can be approximated by
the rule far from the grounding line where N
is large. considered a case where friction is due to either sliding over hard beds at high N and deforming beds when N is low. The resulting rule is
τb=min(α2N,β2|u|1/3)|u|-1u.
We refer to both modified rules as Coulomb-limited,
because the and rules both
imply |τb|≤α2N: in other words, that the magnitude of the basal
friction cannot exceed that given by a Coulomb law,
τb=α2N|u|-1u,
where α2<1 is a coefficient of friction.
uses the term regularized Coulomb, reflecting another benefit of this class of rules: that they permit Coulomb sliding but do not insist upon it over the whole domain, which would be problematic for two reasons. The first problem is physical: across much of the ice sheet N≳1 MPa, leading to a magnitude of
friction larger than observed. The second problem is mathematical, or at least pragmatic:
if |τb| does not increase with |u| anywhere in the domain, then
a Dirichlet condition for u must be imposed on at least some part of the horizontal boundary.
The modified rules avoid both of these problems by reverting to the familiar
rule of Eq. () where N is large (or ice flows slowly in the case of ).
In all three cases participants were free to modify the
parameter β2 in order to achieve the desired steady state: the suggested
value was β2=104 Pa m-1/3 a1/3 (3.16×106 Pa m-1/3 s1/3).
The Ice0 experiment
The Ice0 experiment is simply a test of the initial steady state.
Simulations ran from t=0 a to t=100 a, with zero sub-ice-shelf melt
and an upper surface accumulation rate a=0.3 m a-1. Since these are
the melt and accumulation rates defining the the initial steady state, models
should exhibit little or no variability during this process, or, if they
do exhibit some variation, it should result in little long-term drift. That is,
fluctuations in ice thickness were acceptable provided that the grounding line
did not advance or retreat and the ice volume did not grow
or shrink to any great extent compared to the other two experiments.
Participants were free to produce the initial state by any method; the
majority of participants chose to evolve their models with the stated parameters
for ∼10 ka in order to approach steady state.
Oceanic forcing and the Ice1 experiments
The Ice1 experiments were intended to examine the response of models to intense ablation at the base of the ice shelf, with a spatial distribution reflecting the results of typical cavity circulation models .
The melt rate, measured in m a-1, was
m1=0.2tanhzd-zb75max(-100-zd,0),
where zd is the ice shelf draft and zd-zb is the cavity thickness, both measured in meters.
The resulting melt rate will generally increase with draft, reaching values of
around 100 m a-1 when zd≈700 m,
but will vanish at the grounding line where the cavity thickness is zero.
In the Ice1r experiment, the melt rate was applied to floating ice over the course of 100 a,
starting from the steady state at t=0.
This results in the loss of much of the original ice shelf thickness over the course of the experiment
and was expected to result in grounding line retreat, assuming that the ice shelf had
a role in buttressing the ice upstream. The two follow-on experiments, Ice1rr and Ice1ra, were
designed to start from the end of the Ice1 experiment at t=100 a and terminate at t=200 a.
Ice1rr is simply a continuation of Ice1r, with the same melt rate applied, while Ice1ra imposes zero melt rate, allowing the ice shelf to thicken, so that the grounding line should readvance.
Calving and the Ice2 experiments
The Ice2 experiments follow the same basic structure as the Ice1 experiments but impose a different
melt rate,
m2=100ma-1ifx>480km0otherwise,
applied to ice shelf regions only.
This choice of melt rate in Ice2 results in something like a large calving event, where a downstream
portion of the ice shelf, amounting to around half the total area, is removed over a short period of time. The majority of
the ice removed lies in a zone that provides little buttressing.
By allowing a thick ice shelf to
form in the wake of the retreating grounding line, the Ice2 experiments test a model's ability to form
a new stable steady state with the grounding line on a retrograde slope. They also test the
numerical implementation of the model, because there is often an abrupt
increase in melt rate immediately downstream from some portions of the grounding line, in contrast to the smooth
increase in melt rates seen in the Ice1 experiments.
Participating models
The participating models cover the same variety of englacial stress approximations
as the earlier MISMIP and MISMIP3d exercises,
with each model including some approximation of the horizontal normal and shear stress (membrane stress).
The most complete models make use of the full-Stokes equations, but the computational
expense entailed by solving the full 3D stress balance equation limits both the number of participants able to run such a model and the number of submissions from those participants, so that there are only two full-Stokes submissions. The most common class of model is based upon a 2D, vertically integrated hydrostatic stress balance equation, either through the shallow-shelf/shelfy-stream approximation (SSA), which neglects shear strains Dxz and Dyz above the bed, or an approximation that assumes a simple form for Dxz and Dyz, for example, the models of and . We will follow in labeling this second class of vertically integrated model “L1Lx”. Intermediate in complexity are the higher-order (HO) or first-order models, which include the hydrostatic Blatter–Pattyn models. HO models exploit the low aspect ratio of ice sheets to reduce the four 3D Stokes equations to a pair of 3D equations in the horizontal velocity components.
Finally, the HySSA models solve the 2D SSA stress balance equation but adapt that analytic expression derived for flow-line models with no buttressing to the general case by means of a heuristic buttressing factor: such models were able to produce results in earlier exercises that, in contrast to all other types, did not depend strongly on mesh resolution.
All of the participating models construct and solve their stress balance and mass transport equations
though a limited choice of methods. There are several finite-volume methods and finite-difference methods based on rectangular meshes, extruded
vertically, and several finite-element methods based on unstructured triangular meshes, also extruded vertically.
One model (MALI) takes a mixed approach, combining a finite-element discretization of the stress balance equation
with a finite-volume discretization of the mass transport equation. The majority of finite-volume and finite-difference methods
employ spatially uniform meshes, with two exceptions: WAVI employs a wavelet-based adaptive grid to reduce the computational expense
in solving the stress balance equation, while BISICLES makes use of a time-evolving adaptive block structured mesh in both the stress
balance and mass transport equations. The finite-element methods tend to employ spatially nonuniform meshes that do not change over the course of the simulation, with the exception of two of the several Ice Sheet System Model (ISSM) submissions, which update their meshes over time.
Models differ in their discretization of the stress balance equations in the region close to the grounding line.
One common class of techniques is the modification of the discretized basal friction term around the grounding
line. These techniques have been given at least two different names in the literature – grounding line parameterization
and sub-element parameterization – but they all represent a similar approach. We will use the terms
preferred by the individual model authors when referring to their submissions. The simplest
type constructs a piecewise linear approximation to the thickness above flotation (h-hf) and uses that to evaluate
a weight, w∈[0,1], associated with each grid cell or element that reduces the discrete approximation to the
friction term accordingly. None of these schemes introduce additional degrees of freedom, and they also have the same
order of accuracy with respect to mesh resolution, O(Δx), as the unmodified scheme.
Nonetheless, they have been seen to improve accuracy for a given mesh resolution in several cases .
A related – but more controversial – modification to standard methods applies a similar weighting to the basal melt rate term in the mass balance equation. Several participants employed such a modification in these exercises, with clear consequences in the Ice2 experiments (Sect. ).
The participating models are described briefly below,
ordered alphabetically by model name in Sect. to Sect. and listed in Table . The supplement
also contains a data sheet for each model.
Six submissions are based on BISICLES , a finite-volume model
that employs time-evolving adaptive mesh refinement to maintain fine resolution close
to the grounding line. Three vertically integrated stress approximations are included: the shallow-shelf approximation (SCO_BISICLES_SSA_Schoof_250m, SCO_BISICLES_SSA_Tsai_250m), the L1L2 approximation (SCO_BISICLES_L1L2a_Tsai_500m), and a modified L1L2 approximation that includes vertical shear in the effective viscosity but neglects it in the mass flux (the remainder). All three
basal friction rules are represented. Mesh spacing at the grounding line
is set to 250 m in most cases, but two coarser-resolution cases are included with
Δx≥500 m and Δx≥1 km. All of the submissions apply a
one-sided difference when evaluating the gravitational driving stress
at the grounding line, with no other parameterization, although
a subgrid friction scheme has been useful in other cases .
CISM
Two submissions are based on the CISM model. Both
employ the shallow-shelf approximation to describe englacial stresses, discretized according to the finite-element method, on a uniform mesh with 1 km horizontal resolution. Thickness transport is effected with an incremental remappping scheme : ice thickness and velocity data are stored at staggered locations. The difference between submissions is the choice of basal friction rule: one submission uses the scheme and the other uses the scheme. Both make use of a grounding line parameterization .
Elmer/Ice
The two Elmer/Ice
submissions differ in their treatment of englacial stresses.
IME_ElmerIce_FS_Schoof_250m is a full-Stokes model, corresponding to the majority
of Elmer/Ice publications, for example . IME_ElmerIce_L1L2b_Schoof_250m is a vertically
integrated model. Both are finite-element models and apply a horizontal
resolution of 250 m (indicated by convergence studies to be adequate)
over the region swept out by the grounding line during the Ice1 and Ice2 experiments,
and a vertical discretization of 7 layers, with finer resolution toward the base.
Both models employ the modified basal friction law of Eq. ().
ISSM
Several contributions are based on the ISSM (Ice Sheet System Model), which can treat englacial stresses
with a variety of approximations, from the SSA to the full-Stokes equations, using the finite-element method .
The submissions included include one set of full-Stokes simulations (HYU_ISSM_FS_Weertman_500m),
treating basal friction with the Weertman power law model,
and two sets of Blatter–Pattyn approximation
simulations (HSE_ISSM_HO_Weertman_1km, JCH_ISSM_HO_Tsai_200m),
treating basal friction with the Weertman power law model and Coulomb-limited basal friction rules respectively.
The remaining submissions are all SSA configurations, which
use either the Weertman or the Coulomb-limited basal friction rules and differ in their
treatment around the grounding line. HSE_ISSM_SSA_Tsai_1km, HSE_ISSM_SSA_Tsai_500m, and HSE_ISSM_SSA_Weertman_1km
all make use of a fixed-in-time, nonuniform mesh of triangular elements that is refined to either 500 m or 1 km
across the region swept out by the grounding line and employ the SEP1 subelement parameterization.
CBO_ISSM_SSA_Tsai_500m also relies on a fixed, nonuniform mesh but chooses a different parameterization, SEP2 ,
when evaluating friction in partly grounded elements. Two further submissions, TDI_ISSM_SSA_Tsai_500m and TDI_ISSM_SSA_Weertman_500m,
differ from the others in their use of an evolving adaptive mesh , which is updated throughout the simulations to
maintain 500 m resolutions close to the grounding line and coarser resolution elsewhere.
MALI
The MALI (MPAS-Albany Land Ice) submission
treats englacial stresses with the Blatter–Pattyn approximation
and basal friction with the Weertman rule. The stress balance equation is discretized horizontally in space on an
unstructured mesh of triangular finite elements, while the mass conservation equation is discretized, using the finite-volume method,
on the corresponding hexagonal Voronoi tessellation. Time discretization is accomplished with the forward (explicit) Euler scheme.
Basal friction around the grounding line is evaluated by computing it and the flotation criterion
at the quadrature points of a fifth-order scheme, leading to a treatment comparable to SEP3 of . The experiments were carried out with 500 m spatial resolution, while the Ice1r experiment used resolutions from 4 km to 250 m.
PISM
PISM employs a uniform-mesh finite-volume method to discretize the mass conservation equation and a finite-difference method to discretize the stress balance equation. It uses either the shallow-shelf approximation (SSA), or the SSA + SIA approximation, which complements SSA sliding with SIA internal deformation.
All eight submissions employ a mesh spacing Δx=1 km and differ
in subgrid friction interpolation versus none , SSA versus SSA + SIA, Tsai versus Weertman friction rules, and the use or otherwise of an ice thickness transformation, η=H(2n+2)/n, to achieve better numerical results at ice sheet margins (, Sect. 5.2). All submissions apply a one-sided difference when evaluating the gravitational driving stress at the grounding line. Time integration is explicit, with the time step satisfying both an advection Courant–Friedrichs–Lewy (CFL) criterion determined from the SSA sliding speed and an additional constraint found by expressing the SIA as a diffusion equation (and so proportional to Δx2).
PSU3D
The two PSU3D submissions represent the only HySSA model that
took part in MISMIP+. This class of models is distinct from the more common types of vertically integrated models
in their direct imposition of a flux across the grounding line, derived from the analytic expression of , which provides consistent performance between ∼10
and ∼1 km. As a result, PSU3D has been able to simulate simulations of Antarctica
lasting millions of years . It is a finite-difference model, based on a staggered horizontal grid, and Runge–Kutta time integration. Note that the ice-cliff failure mechanisms included in
current version of this model do not arise in MISMIP+.
STREAMICE
STREAMICE is a physical package of the MITgcm climate model . It solves velocities via a finite-element method, using bilinear basis functions on quadrilateral elements on a regular grid; while its thickness is evolved via an explicit finite-volume method. Its vertically integrated stress model
is based on . Near the grounding line, basal drag is regularized using a sinusoid profile where thickness is within 5 m of flotation (either above or below) – as this treatment is found to yield reversible grounding line movement in the MISMIP3d experiments. However, melt is applied to cells only where cell-averaged thickness is below flotation, i.e., no melting is applied to any cells in which basal drag is nonzero.
TIMFD3
TIMFD3 solves the stress balance equations with the LTSML higher-order approximation together with
the modified basal friction rule, Eq. (), discretized
according to the finite-difference method. In comparison to the Blatter–Pattyn approximation, the LTSML approximation considers the vertical resistive stress Rzz in the momentum balance, and thus vertical longitudinal stresses are not hydrostatic.
The MISMIP+
experiments were carried out on a 1 km horizontal grid, with the vertical extent
of the ice sheet treated as nine terrain-following layers, more closely spaced at the base.
Coarser resolutions do not exhibit a stable grounding line. The initial state
is found by taking the output from an SSA model
(one of the BISICLES submissions with the same selection of A and β2) and
performing a 500-year relaxation from that point. The model does not employ any kind of subgrid interpolation.
Úa
Úa is an open-source finite-element ice-flow model based on a vertically integrated formulation of the momentum equations. The ice-flow equations are solved on an unstructured mesh consisting of linear, quadratic, or cubic triangular elements. Various meshing options are available, including automated mesh refinement and coarsening. When simulating the flow of marine ice sheets these meshing options allow, for example, the areas around grounding lines to be automatically highly resolved as the grounding lines migrate through the computational domain. Elements can be activated and deactivated. This enables the computational domain itself to change in the course of a run, for example when simulating the growth and decay of a large group of mountain glaciers. Ice thickness positivity is enforced using the active set method.
Forward time integration can be done in a fully coupled manner, and the resulting nonlinear system is solved using the Newton–Raphson method.
WAVI
WAVI is a finite-volume model
that makes use of a wavelet-based adaptive grid to accelerate the solution
of the stress balance equation. Its vertically integrated stress model
is derived from and treats
both membrane and simplified vertical shear stresses. Subgrid interpolation
of the basal drag, gravitational driving stream, and sub-ice-shelf melt rates
are deployed in the finite volumes immediately adjacent to the grounding line.
Two complete submissions are included in this paper, differing only in their
uniform mesh resolution. A further two partial submission are included, covering only the Ice2r experiments. These restrict nonzero
sub-ice-shelf melt rates to finite volumes whose
cell center is floating.
Results
The majority of participants completed each experiment, leading to a large volume of results, many of which are in
close agreement. The results of each experiment (Ice0, Ice1, and Ice2) are summarized below, concentrating
on the spread of results rather than any individual model. In most cases, a substantial portion of
variability in the results can be explained by straightforward groupings of the participating models:
for example models that make use of the friction rule see slower grounding line migration
than models employing either the or rules.
At the same time, distinctions that have been seen to be important in other experiments, such as the
use of grounding line parameterization in MISMIP3d ,
appear unimportant here.
Grounding line contours for all models with Δx≤1 km at the start of the
Ice0 experiments (a) and variation over 100 years (b).
In panel (a), yellow contours correspond to SSA, L1Lx, and HO models
with the Coulomb-limited basal friction laws;
orange contours to SSA, L1Lx, and HO models with the Weertman basal friction law; red (ISSM) and cyan (Elmer/Ice) contours
to full-Stokes models; and magenta contours to HySSA models. The color map and black contours depict bedrock elevation.
In panel (b), the bold line shows the grounding line position in the center of the channel, xgl(y=0), mean averaged
over all models, and the shaded region depicts the maximum and minimum values. The dashed lines show the
range specified in the experimental design, xgl(y=0)=450±10 km
Ice0 experiments
All participating models produced a similar initial grounding line, crossing the channel in
a region where the bedrock slopes upward. Figure plots the grounding line for
all models with Δx≤1 km at the start (t=0 a) and the
grounding line position xgl(y=0,t) in the center of the channel, mean averaged over all models.
At the start of the experiments, each grounding line intersects with the channel centerline
close to xc=450 km, and the domain edge close to xe=500 km, covering most of the
distance between xc and xe around the channel walls where the bedrock slopes sharply in the y direction.
Most of the models also exhibit a narrow prominence at the top of this lateral bedrock slope, extending
downstream from xe.
There is little apparent change over time in any model, as required. Not every submission included this
test, although the majority did.
Variation between the models is fairly minor. SSA, L1Lx, and HO models that employ the Coulomb-limited basal friction
laws are grouped most closely, largely because there was no need to modify the default A=2.0×10-17 Pa-3 a-1 given
in . SSA, L1Lx, and HO models that include the Weertman basal friction law are more
widely spread, with some models increasing A by as much as 25 % to achieve the specified xgl=450±10 km and
others submitting results with the grounding line further upstream. Neither of the two full-Stokes models elected to modify A, but their grounding lines
are close to others. Finally, the one HySSA model does produce xc≈450 km, setting A=3.5×10-17 Pa-3 a-1 to do so, but
differs from the remaining models over the shallower bedrock with xe≈540 km.
Grounding line migration for all models with Δx≤1 km in the Ice1 experiments.
The Ice1r experiment starts from the Ice0 steady-state position at t=0 a (a). Grounding
lines migrate upstream while the melt rate of Eq. () is applied
until t=100 a (b). There are two branches for t>100 a: the Ice1rr branch
which continues with the same melt rate (c) and the Ice1ra branch with zero melt rate (d).
Yellow contours correspond to SSA, L1Lx, and HO models with the Coulomb-limited basal friction law;
orange contours to SSA, L1Lx, and HO models with the Weertman basal friction law; red (ISSM) and cyan (Elmer/Ice) contours
to full-Stokes models; and magenta contours to HySSA models. The color map and black contours depict bedrock elevation.
Ice1 experiments
Figure plots the grounding line for all models with Δx≤1 km over
the course of the Ice1 experiments. All models see retreat of their grounding line in the center of the channel while the melt rate of Eq. () is imposed (the Ice1r and Ice1rr experiments) and little change outside the channel walls beyond the erosion of the prominences.
The vast majority of the models also see their grounding lines readvance once the melt rate is
reduced to zero (the Ice1ra experiment), albeit at a much lower rate so that the initial grounding line is not regained by the end of the experiment at t=200 a.
Although all of the models are closely grouped at the start of the Ice1r experiments,
a considerable spread of grounding line contours is evident after 100 years.
The median retreat of midchannel grounding line position xgl(y=0) is in the region of 40 km,
but the range is rather larger – close to 100 km. Within groups of models, there is
a much smaller spread: around 20 km between the Coulomb-limited models, a rather larger
spread between the Weertman models, and an obvious outlier in the one HySSA model with Δx≤1 km , whose midchannel grounding line position
retreats by around 100 km.
Rates of change in midchannel grounding line position xgl(y=0) plotted against
rates of change in volume above flotation Vf(a) and grounded area Ag(b).
For all three of the Ice1r, Ice1ra, and Ice1rr experiments each point
Δxgl(y=0)Δt,ΔVfΔt lies close to a straight line, as do the points
Δxgl(y=0)Δt,ΔAgΔt, despite the considerable variation between models.
We will present the rest of the Ice1 results in terms of the
midchannel grounding line position xgl(y=0), since it represents the
other bulk quantities of interest well enough. The rate
of change in xgl(y=0) over time represents both the
rate of change in volume above flotation Vf and the rate of change in
grounded area Ag (Fig. ).
A single proportional relationship
links xgl(y=0) and Ag in the Ice1r, Ice1ra, and Ice1rr experiments.
Ice1r and Ice1rr also see proportional relationships between xgl(y=0) and Vf,
albeit with distinct constants. The exception is the Ice1ra experiment, where the readvance of the grounding line following
the retreat in Ice1r is associated with a continued, mild loss of volume above flotation.
Midchannel grounding line position plotted against time for
the Ice1 experiments. Panel (a) shows the mean (solid curves)
and the range (shaded regions) over all models. Panel (b) shows
the mean (solid curves) and range (shaded regions) for the main
subset, with individual curves (lines and symbols) for the remainder.
The majority of models behave similarly in the Ice1 experiments, so
we define a number of subsets to represent the spread of results.
The first of these is simply the set of all models, but since there
are some clear outliers we also define a main subset
comprising models which (1) completed all experiments, (2) place their
Ice0 steady-state grounding line at x=450±15 km
The requirement x=450±15 km was relaxed, since many participants neglected it.
, and
(3) see their grounding lines retreat along the center of the channel
to x=385±35 km. Figure
shows the spread in midchannel grounding line position against time
for all models, the main subset, and the remainder. Of the remainder, two are the HySSA models, and
the remaining two are L1Lx models. We also separate the
two full-Stokes models from the lower-order approximations. The main
subset then comprises HO, SSA, and L1Lx models.
Ice1 midchannel grounding line position plotted against time for
the main subset (a) and the Weertman and Coulomb-limited models (b).
Weertman (W) models retreat at two-thirds the rate of Coulomb-limited (S/T) models.
The larger spread in the Weertman models is attributed in large
part to the spread in initial states.
One major division within the main subset is the distinction between
models that employ the Weertman basal friction law (Eq. )
and models that utilize either of the Coulomb-limited rules (Eq. or Eq. )
(Fig. ). The mean rate of retreat seen in the
Weertman models is around 0.5 km a-1 versus 0.7 km a-1 in the Coulomb-limited models, and indeed
the ranges of these two subsets barely overlap. The Weertman subset also exhibits
a much larger range, but we attribute this to the larger range of initial states.
That larger range is due to only some participants altering the rate factor A
to place the initial grounding line within the specified range, rather than any inherent difficulty with implementing the Weertman friction law.
Ice1 midchannel grounding line position plotted against time for the PSU HySSA models compared to other models. The HySSA models (solid lines with symbols) both exhibit a retreat rate around twice as fast as the main subset mean (shaded regions, panel a), and 3 times as fast as the Weertman subset mean (shaded regions, panel b).
Although the two HySSA simulations achieve an initial state close to the other models, at least in terms of grounding line positions, their transient behavior differs. Figure shows both sets of HySSA results with the main subset and the Weertman subset. Their rate of retreat in Ice1r is around 1 km a-1, compared to 0.6 km a-1 for the main subset but it is more appropriate to compare
to the 0.5 km a-1 of the Weertman subset given that both HySSA calculations employ that rule. Something along the same lines was seen in the MISMIP3d experiments .
Note though that the HySSA simulations were computed with same model (PSU), albeit at different resolutions (1 and 10 km), so this may not be
a typical result. Note also that the 1 and 10 km HySSA simulations
are essentially the same, as has been the case for this class of models in other cases .
Ice1 midchannel grounding line position plotted for Stokes, higher-order, and other models. The two full-Stokes models (symbols and lines, left panel) see a comparable rate of retreat to
the main subset (shaded regions, panel a) in Ice1r but a much lower rate of advance in Ice1ra.
Within the main subset, higher-order models (HO, panel b) behave in essentially the same way as the
other models (non-HO, panel b).
Comparison of full-Stokes and vertically integrated models in the Ice1r (t≤100 a) and Ice1ra (t≥100 a) experiments. Panel (a) shows the change in grounded area ΔAg and panel (b) the change in volume above flotation ΔFf. “FS” indicates a full-Stokes model, “(S)” the sliding law, and “(W)” the sliding law.
The two full-Stokes models see their grounding lines retreat at the same average rate as
other models while the ice shelf is ablated, but they see a lower rate of
advance when the ice shelf regrows. Figure compares the
two full-Stokes models with the main subset, and the Ice1r retreat rate of one
(IME_ElmerIce_FS_Schoof_250m) lies at the mean of the main subset. The other
(HYU_ISSM_FS_Weertman_500m) retreats rather more rapidly to begin with but slows
to obtain the same position after 100 years – this may be related to this model's position
outside the main subset with regard to its initial state. Neither model sees rapid readvance
when ice shelf ablation ceases in the Ice1ra experiment.
Figure compares the Stokes models with their nearest vertically integrated counterpart: the Elmer/Ice full-Stokes and L1Lx models, which both use the sliding law, and the ISSM full-Stokes and SSA models, which both use the sliding law. The ISSM full-Stokes model exhibits a similar total decrease in grounded area and both a similar regrowth and similar rate of regrowth to its SSA counterpart in Ice1ra. In contrast, the Elmer/Ice full-Stokes model sees its grounded area abate in Ice1r in a similar fashion to its L1Lx counterpart but is the only model that continues to show retreat in the Ice1ra experiment. All four models lose volume above flotation in Ice1r and Ice1ra, but the Elmer/Ice full-Stokes model sees more rapid loss in Ice1ra than the other three.
There is little variation between the higher-order and vertically integrated hydrostatic models.
Figure shows that the mean rate of retreat in the Ice1r experiment is
close to 0.6 km a-1 over 100 years for both higher-order (HO) and vertically integrated (non-HO) models. Likewise, there is essentially no difference between
HO and non-HO models in either the mean rate of further retreat in Ice1rr or the mean rate of readvance in Ice1ra. There is rather more variability between the non-HO models than between the HO-models, but that can be attributed to the smaller number of HO models (or the larger number of non-HO models).
Ice1 midchannel grounding line position plotted for models
in the Weertman subgroup with varying numerical treatments. (a) Subgroups
with (SG) and without (non-SG) subgrid friction schemes show similar rates
of retreat and advance. (b) Likewise, the subgroup with Δx=1 km is little
different from the subgroup with Δx<1 km
At least for the Δx≤1 km models included in the main subset,
the choice between finer resolution or a subgrid friction scheme is unimportant.
Dividing the main subset into subgroups with and without a subgrid friction scheme
results in the same mean and range of grounding line migration rates, and
the same is true if the subset is divided into subgroups with Δx=1 km
and Δx<1 km (Fig. ). That is not to
say that neither a subgrid scheme nor a fine mesh is consequential in general. Apart
from the fact that all models in the main subset have Δx≤1 km,
of the 15 models that employ a subgrid scheme, only 5 have Δx<1 km,
while of the 11 models that do not employ a subgrid scheme, 6 have Δx<1 km.
Grounding line migration for all models with Δx≤1 km in the Ice2 experiments.
The Ice2r experiment starts (as does the Ice1r experiment; see Fig. ) from the Ice0 steady-state position at t=0a(a). Grounding
lines migrate upstream while the melt rate of Eq. () is applied
until t=100 a (b). There are two branches for t>100 a: the Ice2rr branch
which continues with the same melt rate (c) and the Ice2ra branch with zero melt rate (d).
Yellow contours correspond to SSA, L1Lx, and HO models with the Coulomb-limited basal friction laws;
orange contours to SSA, L1Lx, and HO models with the Weertman basal friction laws; red contours
to the full-Stokes model (ISSM only); and magenta contours to HySSA models. The color map and black contours depict bedrock elevation.
Midchannel grounding line position plotted against time for
the Ice2 experiments. Panel (a) shows the mean (solid curves)
and the range (shaded regions) over all models. Panel (b) shows
the mean (solid curves) and range (shaded regions) for the main
subset.
Domain edge grounding line position plotted against time for
the Ice2 experiments. Panel (a) shows the mean (solid curves)
and the range (shaded regions) for the main subset. Panel (b) shows
the mean (solid curves) and range (shaded regions) for two subgroups, Ice2A and Ice2B.
Models in subgroup Ice2A see little or no grounding line retreat along
the domain boundary.
Ice2 experiments
The Ice2 experiments are characterized by lower and diminishing rates of retreat compared to the Ice1 experiments. Figure shows grounding line positions at the start of the Ice2r experiment, before the calving perturbations at x=480 km are applied, 100 years later, and 200 years later both with and without sustained calving at x=480 km.
Figure shows the grounding line positions in the center of the channel from t=0 to t=100 a. The average rate of retreat in the first 25 years of the Ice2r experiment is around 0.4 km a-1, but by the last 50 years it has dropped to 0.1 km a-1 as a thick ice shelf is formed downstream of the retreating grounding line. Compare this to the Ice1 experiment, where only a thin ice shelf is formed: the average rate of retreat is initially similar (0.6 km a-1) but does not decay substantially over time.
The major difference between models in this case is related to numerical methods. The main subset can be split into two subsets: those that exhibit
no grounding line retreat along the domain wall and those that
see some retreat. The first group, Ice2A, consists solely
of models that do not apply a subgrid interpolation scheme when computing
melt rates. The second group, Ice2B, includes models that do apply such a scheme
and some that do not but have some treatment special to calving
fronts (PISM variants with “eta” in their name).
Figure shows the variation in grounding line position
at the domain wall between these two groups. Some of the Ice2B submissions have
their grounding lines retreat to exactly x=480 km
– the limit of nonzero melt rate in these experiments –
along the wall. All of these do employ a subgrid interpolation scheme when computing
melt rates and, notably, see a retreat rate that grows with their nominal mesh spacing
Δx. For example, the complete WAVI submissions (with subgrid interpolation
of melt rates) exhibit this retreat, whereas the supplementary results
(without subgrid interpolation) do not.
Midchannel grounding line position (a) and grounded area (b) plotted against time
for the Ice2A and Ice2B models. The Ice2B models exhibit
greater change in grounding line area than Ice2a, but
that change is restricted to thin ice outside the channel.
The impact of this grounding line retreat along the domain wall, when it does occur, is limited in this case (but is not necessarily limited in others). Figure shows both
the grounding line position in the center of the channel and the grounded
area as they evolve over the course of the experiment. While the loss of grounded area is significantly greater in the Ice2B models, with the mean rate of loss over the Ice2B group comparable to the maximum rate in the Ice2A group, the vast majority of the additional loss in the grounded area is restricted to the thin ice outside of the channel, and there is little difference between the two subgroups in the center of the channel.
Discussion
One clear result of the MISMIP+ exercise is the degree of similarity between models.
The majority of models demonstrated a similar and apparently stable equilibrium with a grounding line crossing a partly retrograde bed, an outcome which should not occur in laterally unvarying ice streams but does occur in buttressed systems . Those that did not had simply omitted the Ice0 experiment. Every model then exhibited ∼1 km a-1 grounding line retreat in response to widespread ablation of the shelf, and nearly every model exhibited a less pronounced response to ablation localized at the calving front. Likewise, nearly every model exhibited grounding line readvance once ablation ceased and ice shelves thickened. One obvious reason for such agreement is the similarity of the models, with most models featuring similar physics and mesh spacing Δx∼1 km – which most
demonstrated to be adequate.
This stands in contrast to the previous MISMIP and MISMIP3d experiments, where models were often limited by their numerical methods.
Inasmuch as the MISMIP+ experiments are representative, the choice of
basal friction law has more impact on ice stream models than the
choice of englacial stress model. Higher-order models deliver
essentially the same results as vertically integrated treatments, and
there is also little variation between the shallow-shelf/shelfy-stream approximation (SSA)
models and vertically integrated (L1Lx) models that include simplified vertical shear streams.
The MISMIP+ experiments are, however, rather
biased towards this conclusion, with fast sliding evident across much of the domain.
This result appears to extend to the full-Stokes models, though we note that
only two models of this type were included.
On the other hand, models that employ the Coulomb-limited basal friction laws produce faster
grounding line retreat and advance compared to models based on the Weertman rule.
A similar conclusion in the context of Thwaites Glacier is reached by
The level of agreement between full-Stokes models and other types is less clear. Both participating full-Stokes models exhibited time-averaged grounding line retreat in line with other models in response to ice shelf
ablation over 100 a, but one model (ISSM in full-Stokes mode) saw more variation over that period than other models. The other
model (Elmer/Ice in full-Stokes mode) saw its grounding line continue to retreat, albeit slowly, for 100 a after ablation ceased, while ISSM saw readvance that was slower than the average across all models but within their range. In other words, where the full-Stokes models disagree with the approximate models, they also disagree with one another.
There is one more notable exception to the rule that basal friction physics matter more
than englacial physics. The HySSA submissions showed the same qualitative behavior as but
greater rates of grounding line retreat and advance than all of the other model types. This was also observed in the earlier MISMIP3d
exercise , but the
reasons are not necessarily the same. The HySSA
models produced the same steady-state grounding line position as properly resolved SSA
models in MISMIP3d with the same value for A, because the boundary layer approximation they rely upon
was derived for that exact case, with no buttressing. In this exercise, the
HySSA submissions have A around twice as large, presumably because the
buttressing formulation is an informal extension to the boundary layer approximation. It may well be the case that the faster retreat is at least in part due to the slacker grounded ice (with the
HySSA models setting the rate factor A=3.5×10-17 Pa-3 a-1 as opposed to the value A=2.0×10-17 Pa-3 a-1 typical of other models), but we note that
the result might still carry over to realistic problems because it will be equally
necessary to tune A or its equivalent when matching observations. However, recent modifications to the HySSA model bring its MISMIP+ grounding line migration rates into the envelope of the results presented here .
Model initial state – in this case the initial ice thickness – is also
a key determinant of the response to ice shelf thinning.
Models using the Weertman law had a larger spread of initial states
than models based on the Coulomb-limited laws, due to model tuning, and they also showed
a larger range of retreat rates when subject to strong melt rates.
This is within a group of models that all began in similar states,
with similar grounding line shapes crossing a retrograde slope, separated
by less than 20 km: a greater difference still would be anticipated
if any of the models had started from a more obviously distinct state, for example
with the grounding line positioned on an entirely pro-grade slope.
Subgrid treatment of basal friction appears to be of minor importance in these
experiments but does offer considerable benefit in other circumstances. Around half of the participating models employ some sort of modification to the discretization of the basal friction term in the
stress balance equation, and around half do not, but there is essentially
no difference between the results of these two groups. This stands in contrast to the MSIMIP3d experiment, where numerical error associated with
the abrupt change in basal friction at the grounding line is a major
source of differences between models at mesh resolutions comparable to
and finer than the Δx∼1 km chosen by most MISMIP+ participants . Convergence studies submitted by participants to MISMIP+ (see the model data sheets in the supplement) typically indicate that the choice
of Δx∼1 km is adequate for MISMIP+, subgrid friction scheme notwithstanding, while similar studies based around MISMIP3d reach the opposite conclusion: that a subgrid friction scheme permits Δx∼1 km and even coarser resolutions while the lack of such a scheme
required far finer resolutions . The most obvious difference between the MISMIP3d and MISMIP+ experiments is in the shape of the grounding line and ice shelf, with MISMIP+ having a larger zone of stress transfer between floating and grounded ice, but another possible cause is the
colder, stiffer ice and greater basal friction of MISMIP3d. MISMIP3d imposed A≈3.0×10-18 Pa-3 a-1 versus A≈2.0×10-17 Pa-3 a-1 in MISMIP+ and β2≈3×104 Pa m-1/3 a1/3 as opposed to β2≈1×104 Pa m-1/3 a1/3, so that shorter length scales in strain rates are to be expected within the grounding zone.
Subgrid treatment of melt rates – as opposed to subgrid treatment of
basal friction – can result in major numerical errors, even at the moderately fine resolutions employed by the majority of participating models. The Ice2 experiments show that imposing melt on even a single cell or element that is partly grounded leads to an erroneous melt rate proportional to the mesh spacing Δx, which is not small compared to the errors caused by under-resolution
of the ice dynamics. In other words, if the mesh is fine enough to
obviate the error caused by subgrid melt schemes, it is also fine enough
to resolve the dynamics. On the basis of these experiments, then, such treatments should be avoided. They are not to be confused with attempts to treat, for example, tidal variation in the grounding
line, which may well result in strong melt rates upstream from the
mean grounding line but should not produce a retreat rate
that vanishes as Δx→0.
Summary
We have presented the results of the third Marine Ice Sheet Model Intercomparison Project, MISMIP+. It is distinct from its predecessors in that its initial state includes a floating ice shelf that buttresses upstream ice, so that when the ice shelf is thinned the grounded ice upstream
accelerates and the grounding line retreats, and the converse when the ice shelf is allowed to regrow.
More than thirty distinct submissions using 11 model programs produced similar results:
an apparently stable equilibrium with a grounding line located on a retrograde slope, rapid grounding line retreat
on the ablation of the ice shelf, and slower readvance on ice shelf regrowth. We found that the most important distinctions between models were the basal friction model
and the initial state. In contrast, the distinction between models employing several small aspect ratio approximations to the Stokes equations was minor, at least with the limited range of model resolutions tested.
The two full-Stokes models themselves do exhibit some distinct behavior, but where they agree with one another
they also agree with other models.
Another exception was the HySSA model, which makes use of an explicit expression
for the flux across the grounding line to permit coarse (∼ 10 km) resolution and thus simulation over longer time intervals. It exhibited substantially faster
grounding line migration than any other model, but recent modifications bring it within the range of other models .
Data availability
All data used in the paper are available at 10.5281/zenodo.3611936.
Author contributions
SLC conducted the analyses and wrote the manuscript, with support from HS, XSAD, and GHG. All authors contributed model results and contributed to the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Special issue statement
This article is part of the special issue “The Ice Sheet Model Intercomparison Project for CMIP6 (ISMIP6)”. It is not associated with a conference.
Acknowledgements
We thank the editor, Ayako Abe-Ouchi, and three referees, Ralf Greve, Frank Pattyn, and Fuyuki Saito, for their constructive criticism leading to an improved manuscript.
Financial support
The work of Thomas Kleiner has been conducted in the framework of the PalMod project (FKZ: 01LP1511B), supported by the German Federal Ministry of Education and Research (BMBF) as the Research for Sustainability initiative (FONA). Support for Matthew Hoffman and the MALI model was provided through the Scientific Discovery through Advanced Computing (SciDAC) program and the Energy Exascale Earth System Model (E3SM) project funded by the U.S. Department of Energy (DOE), Office of Science, Biological and Environmental Research, and Advanced Scientific Computing Research programs. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science user facility supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-05CH11231, and resources provided by the Los Alamos National Laboratory Institutional Computing Program, which is supported by the U.S. Department of Energy National Nuclear Security Administration under contract DE-AC52-06NA25396. The material provided for the CISM model is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under cooperative agreement no. 1852977.
Review statement
This paper was edited by Ayako Abe-Ouchi and reviewed by Frank Pattyn, Ralf Greve, and Fuyuki Saito.
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