GMDGeoscientific Model DevelopmentGMDGeosci. Model Dev.1991-9603Copernicus PublicationsGöttingen, Germany10.5194/gmd-11-4563-2018Dynamically coupling full Stokes and shallow shelf approximation for marine ice sheet flow using Elmer/Ice (v8.3)Coupling ice flow models using Elmer/Ice (v8.3)van DongenEef C. H.vandongen@vaw.baug.ethz.chhttps://orcid.org/0000-0002-4290-3887KirchnerNinahttps://orcid.org/0000-0002-6371-5527van GijzenMartin B.van de WalRoderik S. W.ZwingerThomashttps://orcid.org/0000-0003-3360-4401ChengGonghttps://orcid.org/0000-0001-9171-6714LötstedtPerhttps://orcid.org/0000-0003-2143-3078von SydowLinaLaboratory of Hydraulics, Hydrology and Glaciology, ETHZ, Zurich, SwitzerlandDepartment of Physical Geography, Stockholm University, Stockholm, SwedenDepartment of Applied Mathematical Analysis, Delft University of Technology, Delft, the NetherlandsInstitute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, the NetherlandsBolin Centre for Climate Research, Stockholm University, Stockholm, SwedenCSC-IT Center for Science, Espoo, FinlandDivision of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, SwedenEef C. H. van Dongen (vandongen@vaw.baug.ethz.ch)16November201811114563457613December20177February201824October201825October2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://gmd.copernicus.org/articles/11/4563/2018/gmd-11-4563-2018.htmlThe full text article is available as a PDF file from https://gmd.copernicus.org/articles/11/4563/2018/gmd-11-4563-2018.pdf
Ice flow forced by gravity is governed by the full Stokes (FS) equations,
which are computationally expensive to solve due to the nonlinearity
introduced by the rheology. Therefore, approximations to the FS equations are
commonly used, especially when modeling a marine ice sheet (ice sheet, ice
shelf, and/or ice stream) for 103 years or longer. The shallow ice
approximation (SIA) and shallow shelf approximation (SSA) are commonly used
but are accurate only for certain parts of an ice sheet. Here, we report a
novel way of iteratively coupling FS and SSA that has been implemented in
Elmer/Ice and applied to conceptual marine ice sheets. The FS–SSA coupling
appears to be very accurate; the relative error in velocity compared to FS is
below 0.5 % for diagnostic runs and below 5 % for prognostic runs.
Results for grounding line dynamics obtained with the FS–SSA coupling are
similar to those obtained from an FS model in an experiment with a periodical
temperature forcing over 3000 years that induces grounding line advance and
retreat. The rapid convergence of the FS–SSA coupling shows a large
potential for reducing computation time, such that modeling a marine ice
sheet for thousands of years should become feasible in the near future.
Despite inefficient matrix assembly in the current implementation,
computation time is reduced by 32 %, when the coupling is applied to a
3-D ice shelf.
Introduction
Dynamical changes in both the Greenland and Antarctic ice sheets are, with
medium confidence, projected to contribute 0.03 to 0.20 m of sea level rise
by 2081–2100 . The main reason for the uncertainty in
these estimates is a limited understanding of ice dynamics. Thus, there is a
great need for improvement of ice dynamical models .
The gravity-driven flow of ice is described by the full Stokes (FS)
equations, amended by a nonlinear rheology described by Glen's flow law.
Model validation is required over centennial to millennial timescales to
capture the long response time of an ice sheet to external forcing
. However, the
computation time and memory required for an FS model to be applied to ice
sheets restricts simulations to sub-millennial timescales
. Therefore, approximations of the FS
equations are employed for simulations over long timescales, such as the
shallow ice approximation SIA;, the shallow shelf approximation SSA;,
Blatter–Pattyn , and hybrid models
.
Any ice sheet model accounting for ice shelves needs to resolve grounding
line dynamics (GLD). Despite many recent efforts, modeling GLD still poses a
challenge in numerical models, as illustrated by the wide range of results
obtained in the Marine Ice Sheet Model Intercomparison Project
MISMIP;. In MISMIP3d, GLD differ between FS models and
SSA models, with discrepancies attributed to so-called higher-order terms
which are neglected in SSA models but included in FS models
. Based on these model intercomparisons, it is
advised to use models that include vertical shearing to compute reliable
projections of ice sheet contribution to sea level rise
. On the other hand, it is not entirely clear how
much of the difference in GLD is due to the different numerical treatment of
the grounding line problem in shallow models. An updated version of the
hybrid SIA/SSA Parallel Ice Sheet Model (PISM) that uses a modified driving
stress calculation and subgrid grounding line interpolation showed GLD
comparable to an FS model . It should be noted
that the experiments in MISMIP3d consisted of laterally extruded idealized
2-D geometries with quite
small sideward disturbances, and MISMIP+ may give
more insight on realistic situations. Additionally, there is a recent
publication that sheds new light on a possible problem with the setup of
MISMIP experiments .
Solving the FS equations over large spatiotemporal domains is still
infeasible. However, solvers combining approximations (e.g., SIA or SSA) with
the FS equations allow the simulation of ice dynamics over long time spans
without introducing artifacts caused by the application of approximations in
parts of the domain where they are not valid. For instance,
coupled FS and SSA, in the framework of the Ice
Sheet System Model ISSM;. They apply the
tiling method which includes a blending zone of FS and SSA. Their result
looks promising with respect to both accuracy and efficiency but is limited
to diagnostic experiments. The Ice Sheet Coupled Approximation Levels (ISCAL)
method couples SIA and FS by a
nonoverlapping domain decomposition that dynamically changes with time. ISCAL is
implemented in Elmer/Ice , an open-source finite
element software for ice sheet modeling. Here, we present a novel coupling
between FS and SSA, also by the implementation of a nonoverlapping domain
decomposition in Elmer/Ice. The domain decomposition changes dynamically with
grounding line advance and retreat. GLD are modeled with FS and coupled to
SSA on the ice shelf via boundary conditions. The equations discretized by
the finite element method (FEM) are solved iteratively, alternating between
the FS and the SSA domain, until convergence is reached.
The extent of present-day ice shelves is limited to approximately 10 % of
the area of Antarctica . Therefore, one may question the
reduction in computational work by applying SSA to model ice shelves in
continental-scale simulations of marine ice sheets. However, the coupling is
targeted at conducting paleo-simulations, for which much larger ice shelves
have been present . In that case,
a large part of the interior of a marine ice sheet is modeled with SIA, SSA
is applied to the ice shelves, and the FS domain is restricted to ice streams
and areas around the grounding line.
An overview of the FS and SSA equations governing ice sheet and shelf
dynamics in three dimensions (3-D) is presented in Sect. ,
together with the boundary conditions. Memory and performance estimates of an FS–SSA coupling, independent of the specific coupling implemented, are
provided in Sect. . Section describes the
coupled FS–SSA model, hereafter “coupled model”. The coupling is applied to a
conceptual ice shelf ramp and marine ice sheet in Sect. . The simulation of a 3000-year long cycle of grounding
line advance and retreat (described in Sect. ) shows
the robustness of the coupling.
Governing equations of ice flow
Ice is considered to be an incompressible fluid, such that mass conservation
implies that the velocity is divergence-free:
∇⋅u=0,
where u=(u,v,w)T describes the velocity field of the ice with
respect to a Cartesian coordinate system (x,y,z)T, where z is the
vertical direction. For ice flow, the acceleration term can be neglected in
the Navier–Stokes equations . Therefore, the
conservation of linear momentum under the action of gravity g can be
described by
-∇p+∇⋅η(∇u+(∇u)T)+ρg=0,
where ∇ is the gradient operator, p pressure, η viscosity,
ρ ice density, and g gravity. Letting
σ denote the stress tensor, pressure p is the mean
normal stress (p=-1/3Σiσii) and D(u) is
the strain rate tensor, related by
σ=2ηD(u)-pI=η(∇u+(∇u)T)-pI,
where I is the identity tensor. Together, Eqs. ()
and () are called the full Stokes equations.
Observations by suggest that the viscosity depends
on temperature T and the effective strain rate D(u):
η(u,T)=12A(T)-1nD(u)1-nn,D(u)=12∂u∂x2+∂v∂y2+∂w∂z2+14∂u∂y+∂v∂x2+∂u∂z+∂w∂x2+∂v∂z+∂w∂y2‾,
where Glen's exponent n=3. The
fluidity parameter A increases exponentially with temperature as
described by the Arrhenius relation . This
represents a thermodynamically coupled system of equations. However, in the
current study, we focus on the mechanical effects and a uniform temperature
is assumed. Due to the velocity dependence of the viscosity in
Eq. (), the FS equations form a nonlinear system with four
coupled unknowns, which is time consuming to solve. Therefore, many
approximations to the FS equations have been derived in order to model ice
sheet dynamics on long timescales; see Sect. .
Shallow shelf approximation
Floating ice does not experience basal drag, hence all resistance comes from
longitudinal stresses or lateral drag at the margins. For ice shelves, the
SSA has been derived by dimensional analysis
based on a small aspect ratio and surface slope
. This dimensional analysis
shows that vertical variation in u and v is negligible, such that w and
p can be eliminated by integrating the remaining stresses over the vertical
and applying the boundary conditions at the glacier surface and base
(described in Sect. ). Then, the conservation of linear momentum,
Eq. (), simplifies to
∇h⋅2η‾Dh(u)+tr(Dh(u))I=ρgH∇hzs,
where the subscript h represents the components in the x–y plane,
η‾ the vertically integrated viscosity, H the thickness of
the ice shelf, and zs the upper ice surface; see
Fig. . The effective strain rate in
Eq. () simplifies to
Dh(u)=∂u∂x2+∂v∂y2+∂u∂x∂v∂y+14∂u∂y+∂v∂x2,
where w is eliminated using incompressibility; Eq. (). The
SSA equations are still nonlinear through η‾, but since w and
p are eliminated and vertical variation in u and v is neglected, the 3-D
problem with four unknowns is reduced to a 2-D problem with two unknowns.
Therefore, the SSA model is less computationally demanding than FS. The
horizontal velocities are often of main interest, for example when results
are validated by comparison to observed horizontal surface velocity. If
desirable, the vertical velocity can be computed from the incompressibility
condition.
Overview of the notations and domain decomposition for a conceptual
marine ice sheet. The vertical scale is exaggerated. The sea level at z=0
is dashed blue and the interface between the FS and SSA domains is solid red.
The bed elevation is denoted by b, the coupling interface by
xc, and the grounding line by xGL. The distance
between xc and xGL, defined in
Eq. (), is denoted by
dGL.
Boundary conditions and time evolution
The coupling is applied to a marine ice sheet, with bedrock lying (partly)
below sea level (see Fig. ), and involves boundaries
in contact with the bedrock, ocean and atmosphere. The only time dependency
is in the evolution of the free surfaces.
Bedrock
Where the ice is grounded (in contact with the bedrock), the interaction of
ice with the bedrock is commonly represented by a sliding law f(u,N),
that relates the basal velocity ub and effective pressure N to the
basal shear stress as
(ti⋅σ⋅n)b=f(u,N)u⋅ti,i=1,2,(u⋅n)b+ab=0,
where ti are the vectors spanning the tangential plane, n is
the normal to the bed, and ab describes basal refreezing or melt. A
sliding law suggested by is assumed, which depends
on ub and the height above buoyancy z* such that
f(u,N)=-β|ub|1n-1z*(N).
Here, the sliding parameter β is constant in time and space. In line
with , instead of modeling N, a hydrostatic
balance is assumed to approximate z*, implying a subglacial hydrology
system entirely in contact with the ocean:
z*(H)=Hif zb≥0,H+zbρwρif zb<0,
where zb is the lower ice surface and ρw the water density and the sea
level is at z=0. Equation () implies that z* equals zero
when the flotation criterion (Archimedes' principle) is satisfied, i.e., where
zs=1-ρρwH,zb=-ρρwH.
Ice–ocean interface
As soon as the seawater pressure pw at the ice base
zb is larger than the normal stress exerted by the ice at the
bed, the ice is assumed to float. For a detailed description of the
implementation of the contact problem at the grounding line in Elmer/Ice, see
. At the ice–ocean interface, the tangential friction
is neglected (f(u,N)≡0 in Eq. ) and
σ⋅n=-pwn where pw(z)=-ρwgz if z≤0,
and σ⋅n=0 above sea level (z>0). Calving at
the seaward front of the ice shelf is not explicitly modeled, but the length
of the modeling domain is fixed and ice flow from the shelf out of the
domain is interpreted as a calving rate.
Surface evolution
Ice surface (assumed stress-free, σ⋅n=0) and
ice base at zs and zb behave as free surfaces according to
∂zs/b∂t+us/b∂zs/b∂x+vs/b∂zs/b∂y=ws/b+as/b,
where as/b is the accumulation (as/b>0) or ablation
(as/b<0) in meter ice equivalent per year, at the surface or
base, respectively. By vertical integration of the incompressibility
condition, Eq. (), w can be eliminated using Leibniz
integration rule and substituting the free surface equations (Eq. ), which yields the
thickness advection equation:
∂H∂t+∂Hu‾∂x+∂Hv‾∂y=as-ab,
where u‾ and v‾ are the vertically integrated horizontal velocities.
Memory and performance estimates of an FS–SSA coupling
The reduction in the memory required for an FS–SSA coupling by domain
decomposition, compared to an FS model, can be estimated. This estimate is
independent of the specific implementation of the coupling between the
domains and concerns only the most ideal implementation in which no redundant
information is stored. The main advantage of the SSA model is that
uSSA is independent of z, such that the SSA equations can be
solved on a part of the domain with a mesh of 1 dimension fewer. Besides
that, there are fewer unknowns since p and w are eliminated. An
additional advantage of eliminating p is that the resulting system is
mathematically easier to discretize and solve. In particular, difficulties
related to a stable choice for the basis functions for the pressures and
velocities are avoided see, e.g., and there
is no need for specialized iterative solution techniques to solve the
so-called saddle-point problem that the FS equations pose
see.
Suppose that the computational domain Ω is discretized with Nz
nodes regularly placed in the z direction and Nh nodes in a horizontal
footprint mesh and is decomposed into two parts (ΩSSA and ΩFS; see Fig. ). The fraction of nodes in ΩSSA is
denoted as θ with 0<θ<1. The number of nodes in ΩFS is
then approximately (1-θ)NhNz, and in ΩSSA, it is θNh, neglecting shared nodes on the boundary. For a 3-D physical domain, SSA
has two unknowns (u and v) and FS has four unknowns (u, v, w, and p).
Hence, the memory needed to store the solution with a coupled model is
proportional to 2Nh(θ+2(1-θ)Nz). For a 2-D simulation in the
x-z plane, where FS has three unknowns and SSA only one, the memory is
proportional to Nh(θ+3(1-θ)Nz). The memory requirement for a
physical domain in d dimensions reduces to
qvar=coupled model memoryFS model memory=1-θ+θ(5-d)Nz,d=2,3,
when part of the domain is modeled by the SSA equations. The memory
requirements for mesh-related quantities reduce to
qmesh=1-θ+θ/Nz in both 2-D and 3-D. The quotients
qvar and qmesh are close to 1-θ if Nz≳10.
The computational work is more difficult to estimate a priori since it
depends on the implementation of the coupling. The dominant costs are for the
assembly of the finite element matrices, the solution of the nonlinear
equations, and an overhead for administration in the solver. The work to
assemble the matrices grows linearly with the number of unknown variables.
Suppose that this work for FS in 3-D is 4CFSNhNz in the whole domain,
for FS 4CFS(1-θ)NhNz in ΩFS, and for SSA
2CSSAθNh in ΩSSA. The coefficients CFS and
CSSA depend on the basis functions for FS and SSA and the complexity of
the equations. The reduction in assembly time for the matrix is
qass=1-θ+CSSAθ/2CFSNz. If CFS≈CSSA, then
the reduction is approximately as in Eq. (). The same
conclusion holds in 2-D. Therefore, the reduction in that part is estimated to
be similar to the reduction in Eq. ().
Method for coupling FS and SSA
All equations are solved in Elmer/Ice using the
FEM. First the velocity u (using FS or SSA) is solved for a fixed
geometry at time t. The mesh always has the same dimension as the physical
modeling domain, but uSSA is only solved on the basal mesh
layer, after which the solution is re-projected over the vertical axis. Then,
the geometry is adjusted by solving the free surface and thickness advection
equations using backward Euler time integration. The nonlinear FS and SSA
equations are solved using a Picard iteration. The discretized FS equations
are stabilized by the residual free bubbles method
, the recommended stabilization method in
. First, the coupling for a given geometry is
presented, followed by the coupled surface evolution, both summarized in
Algorithm 1.
The FS domain ΩFS contains the grounded ice and a part of the shelf
around the grounding line; see Fig. . The SSA domain
ΩSSA is restricted to a part of the ice shelf and starts at the
coupling interface xc at the first basal mesh nodes located at least
a distance dGL from the grounding line xGL, such that
||x-xGL||:=(x-xGL)2+(y-yGL)2+(z-zGL)2≥dGL for all x in ΩSSA.
Boundary conditions at the coupling interface
Horizontal gradients of the velocity are not neglected in the SSA equations
unlike in the SIA;. Thus, not only FS and
SSA velocities but also their gradients have to match, in order to allow a
coupling of the two. Therefore, one cannot solve one system of equations
independently, for use as an input to the other system, as done for a one-way
coupling e.g.,. Instead, the coupling of FS and
SSA is solved iteratively, updating the interaction between FS and SSA
velocities in each iteration to obtain mutually consistent results. SSA-governed ice shelf flow is greatly influenced by the inflow velocity from the
FS domain. Therefore, we start the first iteration of the coupled model by
solving the FS equations. A boundary condition is necessary at
xc; we assume that the cryostatic pressure acts on ΩFS at
xc:
σFS⋅n(xc,z)=ρg(zs-z)n,
where n is normal to the coupling interface xc. The FS
velocity at xc provides a Dirichlet inflow boundary condition to the SSA
equations. Then, the Neumann boundary condition in Eq. () has
to be adjusted based on the ice flow as calculated for ΩSSA. This
is done using the contact force denoted by fSSA, as explained
below.
The SSA equations are linearized and, by means of FEM, discretized. This leads
to a matrix representation Au=b, where u is
the vector of unknown variables (here, horizontal SSA velocities). In FEM
terminology, the vector b that describes the forces driving or
resisting ice flow is usually called the body force and A the
system matrix . In Elmer/Ice, Dirichlet conditions
for a node i are prescribed by setting the ith row of A to
zero, except for the diagonal entry which is set to be unity, and bi
is set to have the desired value . For an exact
solution of Au=b, the residual
f=Au-b is zero. If we instead use the system
matrix ASSA obtained without the Dirichlet conditions being
set, the resulting residual is equal to the contact force that would have
been necessary to produce the velocity described by the Dirichlet boundary
condition. Since the SSA equations are vertically integrated,
fSSA=ASSAuSSA-bSSA is the vertically
integrated contact force and needs to be scaled by the ice thickness H. In
Elmer/Ice, fSSA is mesh dependent and needs to be scaled by the
horizontal mesh resolution ω as well. For 2-D configurations,
ω=1. Using fSSA instead of explicitly calculating the
stress is advantageous since it is extremely cheap to find the contact force
if ASSA is stored.
To summarize the boundary conditions at xc, for FS, an external
pressure is applied:
σFS⋅n(xc,z)=ρg(z-zs)n+fSSA(xc)ωH,
where fSSA:=0 in the first iteration
(for its derivation, see Appendix ).
For SSA, a Dirichlet inflow boundary condition,
uSSA(xc)=uFS(xc,zb),
provides the coupling to the FS solution. Here we take the uFS at
zb, but any z can be chosen since xc should be located such
that uFS(xc,z) hardly varies with z. Every iteration,
fSSA, and uFS(xc,zb) are updated until
convergence up to a tolerance εc.
Surface evolution
The surface evolution is calculated differently in the two domains
ΩFS and ΩSSA. Equation () is applied
to ΩFS for the evolution of zs and zb, avoiding assuming
hydrostatic equilibrium beyond the grounding line, since the flotation
criterion is not necessarily fulfilled close to the grounding line
. The thickness advection equation, Eq. (), is used for ΩSSA, which is advantageous since
the ice flux q=HuSSA is directly available (because
uSSA does not vary with z) and no vertical velocity is needed.
Moreover, only one time-dependent equation is solved instead of one for the
lower and one for the upper free surface. The evolution of the surfaces zs
and zb for ΩSSA is then calculated from the flotation criterion,
Eq. (). At xc, HSSA=HFS is applied as a
boundary condition to the thickness equation. First the surface evolution is
solved for ΩFS; then ΩSSA follows.
The algorithm
The iterative coupling for one time step is given by Algorithm 1.
First, the shortest distance d to the grounding line is computed for all
nodes in the horizontal footprint mesh at the ice shelf base. Then, a mask is
defined that describes whether a node is in ΩFS, ΩSSA, or at
the coupling interface xc, based on the user-defined dGL.
Technically, the domain decomposition is based on the use of passive elements
implemented in the overarching Elmer code , which
allow for deactivating and reactivating elements. An element in
ΩFS is passive for the SSA solver, which means that it is not included
in the global matrix assembly of ASSA and vice versa.
Two kinds of iterations are involved, since computing either uFS,k
or uSSA,k for the kth coupled iteration also requires Picard
iteration by the nonlinearity in the viscosity. As the experiments will
show, calculating uFS,k dominates the computation time in the
coupled model. The coupled model is therefore more efficient if the total
number of FS Picard iterations (the sum of FS Picard iterations over all
coupled iterations) decreases. This is accomplished by limiting the number of
FS Picard iterations before continuing to compute uSSA,k, instead
of continuing until the convergence tolerance εP is reached,
since it is inefficient to solve very accurately for uFS,k if the
boundary condition at xc is not yet accurate. Despite interrupting
the Picard iteration, the final solution includes a converged FS solution
since the coupled tolerance εc is reached. Picard iteration
for uSSA,k is always continued until convergence since the
computation time is negligible compared to FS.
An element may switch from ΩSSA to ΩFS, for example
during grounding line advance. Then, the coupled iteration either starts with
the initial condition for uFS if the element is in ΩFS
for the first time, or the latest uFS(t) computed in this element,
before switching to SSA.
Numerical experiments
To validate the coupled model, we first verify for a conceptual ice shelf
ramp that solutions obtained with the coupled model resemble the FS velocity
in 2-D and 3-D. Then the coupled model is applied to a 2-D conceptual marine ice
sheet (MIS). Whenever “accuracy of the coupled model” is mentioned, this
refers to the accuracy of the coupled model compared to the FS model.
Investigating the accuracy of the FS model itself is outside the scope of
this study. No convergence study of the FS model with respect to
discretization in either time or space is performed. Instead, equivalent
settings are used for the FS and coupled model, such that they can be
compared, and the FS model is regarded as a reference solution.
Ice shelf rampTwo-dimensional ice shelf ramp
A simplified test case is chosen for which the analytical solution to the SSA
equations exists in 2-D as described in . It consists
of a 200 km long ice shelf (see Fig. ), with a horizontal
inflow velocity u(0,z)=100 m yr-1 and a calving front at x=200 km
where the hydrostatic pressure as exerted by the sea water is applied. The
shelf thickness linearly decreases from 400 m at x=0 to 200 m at
x=200 km; zb and zs follow from the flotation criterion (Eq. ). By construction, the SSA model is expected to be a good
approximation of the FS model. The domain is discretized by a structured
mesh and equidistant nodes on the horizontal axis and extruded along the
vertical to quadrilaterals. All constants used and all mesh characteristics are
specified in Table .
Three models are applied to this setup, FS-only, SSA-only, and the coupled
model, for which the horizontal velocities are denoted as uFS, uSSA, and
uc, respectively. The relative node-wise velocity differences between
uSSA and uFS stay below 0.02 % in the entire domain. However,
computing time for the SSA solution only takes 3 % of that of the FS
solution, which is promising for the potential speedup of the coupled model.
The horizontal velocity uc (m yr-1) and node-wise
difference |uFS-uc|/uFS⋅100 (%) in
the coupled solution for the 2-D ice shelf ramp. The vertical scale is
exaggerated 100 times. The ice thickness ranges from 400 to
200 m.
The coupling location is fixed at xc=100 km, as no grounding line is
present to relate xc to. In the first coupled iteration, uc(xc,zb)=100 m yr-1, while in the final solution uFS(xc,zb)=4805 m yr-1. The cryostatic pressure applied to ΩFS at xc
buttresses the ice flow completely and the force imbalance of the hydrostatic
pressure at the calving front does not yet influence the velocity uc in
ΩFS. In the second iteration, when fSSA is applied, a
maximum difference of only 0.3 % between uFS and uc in the entire
domain remains. The coupling converges after three iterations; the velocity
uc and relative difference compared to FS are shown in Fig. . Convergence of the coupled model requires 31 FS Picard
iterations compared to 35 for FS-only. However, assembly time per FS
iteration almost doubles in the coupled model compared to the FS model, and
assembly time dominates the computational work in this simplified 2-D case.
Therefore, the coupled model needs almost twice as much computation time as
the FS model. This issue is due to the use of passive elements and is addressed
in the “Discussion” section (Sect. ).
Three-dimensional ice shelf ramp
The 2-D ice shelf ramp is extruded along the y axis (see Fig. ). On both lateral boundaries at y=0 and 20 km,
u⋅n=0. All other boundary conditions remain identical to
the 2-D case, and the coupling interface is located halfway
xc=(100,y) km. First the solutions of the FS and SSA model in
Elmer/Ice will be compared before applying the coupled model.
The limited width of the domain (20 km) in combination with the boundary
condition u⋅n=0 at both lateral sides yields a negligible
flow in the y direction (vFS<10-8 m yr-1). Despite
differences in the models, the relative difference in u is below 1.5 %.
Running the experiment with the SSA model takes only 0.8 % of the time needed
to run it with the FS model.
The maximum relative difference between uFS and uc is
1.4 %, which is of the same order of magnitude as the velocity difference
between FS and SSA. The mean assembly time per FS iteration is 6 % higher
than in the FS-only model, but the solution time decreases by 55 %.
Convergence of the coupled model requires 30 FS iterations compared to 27 for
FS-only. The total computation time decreases by 32 %.
Horizontal velocity uc (m yr-1) from coupled model
for the 3-D ice shelf ramp with
xc=(100,y) km.
Marine ice sheet
First, a diagnostic MIS experiment is performed in 2-D to compare velocities
for the initial geometry. After one time step, velocity differences between
the coupled and FS models yield geometric differences. In prognostic
experiments, velocity differences can therefore be due to the coupling and to
the different geometry for which the velocity is solved. Computation times
for the FS and coupled model are presented for the prognostic case only.
The coupled velocity uc (m yr-1) and relative
difference |uFS-uc|/uFS⋅100 (%), for
the diagnostic MIS experiment. The bedrock is shaded in grey,
xGL=730.8 km, and xc=763.2 km (the mesh resolution
yields ||xc-xGL||h=32.4 km). The
vertical scale is exaggerated 100 times with an ice thickness ranging from
1435 to 296 m.
Diagnostic MIS experiment
The domain starts with an ice divide at x=0, where u=0, and terminates at
a calving front at x=L=1800 km. An equidistant grid with grid spacing
Δx=3.6 km is used. Other values of constants and mesh characteristics
are specified in Table .
showed that resolving grounding line dynamics with an FS model requires very
high mesh resolution around the grounding line. However,
showed that the friction law assumed in this study
(see Sect. ) reduces the mesh sensitivity of the FS model
compared to the Weertman friction law assumed in
, allowing the coarse mesh used here. The bedrock
(m) is negative below sea level and is given by
b(x)=200-900xL.
Basal melt is neglected, and the surface accumulation as (m yr-1) is a
function of the distance from the ice divide:
as(x)=ρwρxL.
This experimental setup is almost equivalent to ,
except that they applied a buttressing force to the FS equations. It is
possible to parameterize buttressing for the SSA equations as well through
applying a sliding coefficient . This was not
done here as it may introduce a difference between the FS and SSA models that
is unrelated to the coupling.
The diagnostic experiments are run on a steady-state geometry computed by the
FS model. First, the experiment “SPIN” in is
performed, starting from a uniform slab of ice (H=300 m), applying the
accumulation in Eq. () for 40 kyr, such that a steady state
is reached. The geometry yielded from these SPIN runs (which include
buttressing) is used in simulations without buttressing until a new steady
state (defined as a relative ice volume change below 10-5) is reached.
This removal of buttressing leads to grounding line retreat from 871.2 to
730.8 km (Fig. ).
Again, FS-only, SSA-only, and the coupled model are applied to this setup.
Where uFS≥5 m yr-1, the relative difference between uFS
and uSSA is below 1.8 %. The velocity uc is given in Fig. , with dGL=30 km such that 58 % of the nodes in the
horizontal footprint mesh are located inside ΩSSA (θ=0.58). The coupled model converges after 27 FS iterations on the restricted
domain ΩFS, compared to 24 Picard iterations in the FS model. The
relative difference between uFS and uc is below 0.5 % (Fig. ); this small difference shows that dGL=30 km is
sufficient. For this configuration, 4 % of the FS nodes are located between
xGL and xc with dGL=30 km; hence, decreasing dGL does not
affect the proportion of nodes in ΩFS significantly. Therefore,
dGL is kept equal to 30 km for the prognostic experiment.
Computation times for the MIS simulation of 3000 years with FS-only
and the coupled model. Model Ci denotes the coupled model, with i being the
maximum number of nonlinear FS iterations per coupled iteration; C5–C9 are
omitted for brevity. The assembly time for AFS is
denoted as tA. All relative computation times are given in
percentage of the total time ttot. The number of FS and coupled
iterations are averaged over the time steps.
Absolute difference |uFS-uc| (m yr-1)
after 3000 years. The vertical scale is exaggerated 100 times. The ice
thickness ranges from 1445 to 296 m.
Prognostic MIS experiment
The prognostic experiment aims to verify model reversibility as in
. Starting from the steady-state geometry, the ice
temperature T is lowered over a period of 500 years from -10 to
-30∘C and back according to
T(t)=-10(2-cos(2πt/500))∘Cfor 0≤t≤500years.
The resulting change in A (see Eq. ) induces a
grounding line advance and retreat and changes ΩSSA by Eq. (). Afterwards, T=-10∘C for 2500 years. Mass
balance forcing is kept constant throughout. The length of one time step is 1 year.
The maximum difference between uc and uFS after
3000 years is 10 m yr-1 (shown in Fig. ),
corresponding to a relative difference of 1.6 %. The time evolution of
xGL, ub(xGL), H(xGL), and the
grounded volume Vg is shown in Figs. and
. In general, ub is slightly higher in the coupled
model, with a maximum difference of 5.3 % in the entire experiment. The
grounding line advances to xGL=1036.8 km in the FS model and
xGL=1044 km in the coupled model. The FS model returns back to
the original xGL=730.8 km, but the coupled model yields
xGL=734.4 km, an offset of one grid point. The maximum
difference in thickness is 1 %. After 3000 years, Vg still
decreases but the relative difference is below 10-5 between two time
steps.
Time evolution of xGL (red) and ub(xGL) (blue) with solid lines for FS and dashed lines
for the coupled model.
To investigate the efficiency of the coupled model, the simulation is performed
with 10 different settings, where the maximum number of FS iterations per
coupled iteration is varied from 1 to 10. The assembly of the FS matrix takes
75 % of the computation time of the FS model (see tA in Table ), and assembly time per FS iteration is similar for the
coupled and FS model. Only 5 % of the computation time is used to solve the
linearized FS system (ts in Table ). For all coupled
simulations, assembling and solving the SSA matrix (tSSA) takes 4 %–6 %.
All the time that is left will be called overhead, to, which includes
launching solvers, i.e., allocating memory space for vectors and matrices, the
surface evolution, and solvers for post-processing. As expected, the total
number of FS iterations is the smallest when just performing one FS Picard
iteration per coupled iteration. However, the model then changes between
solvers more often, meaning that both overheads and the time to solve the SSA
model increase. It turns out that a limit of three FS Picard iterations per
coupled iteration balances minimizing to and tA, yielding a 10 %
decrease in computation time with respect to the FS model. This speedup comes
from a lower number of FS Picard iterations (Table ) and a
slight decrease in the time used to solve the linearized FS system (13 %
lower than the time that the FS model takes).
Time evolution of HGL=H(xGL) and grounded
volume Vg with solid lines for FS and dashed lines for the
coupled model.
Discussion
The presented coupling is dynamic, since the coupling interface xc
changes with grounding line changes, but the distance dGL that defines
xc has to be chosen such that the FS velocity at the interface is
almost independent of z. In the experiments described in Sect. 4, this is
already the case at the grounding line. We propose that further studies let
ΩSSA be determined automatically, for example, by a tolerance for
the vertical variation in the horizontal velocities, which should be close to
zero in order to allow for a smooth coupling to SSA. Another option is to use
a posteriori error estimates based on the residual
.
The current implementation in Elmer/Ice does not give as much speedup as
expected from computation times of the FS- and SSA-only models for the ice
shelf ramp (tSSA=0.03tFS) and from the performance estimates in
Sect. 4.2. This is due to an inefficient matrix assembly. The assembly of the
system matrix AFS restricted to ΩFS currently takes
at least as much time as the assembly for the full domain Ω, even
though the domain ΩFS is much smaller than Ω; in Eq. (13),
θ=0.5 for the ice shelf ramp and θ=0.58 for the diagnostic
MIS experiment. Since the assembly time dominates the total solution time in
simple 2-D simulations, this is problematic. The inefficient assembly is
caused by the use of passive elements implemented in the overarching Elmer
code , which allow the de- and reactivation of elements.
A passive element is not included in the global matrix assembly, but every
element must be checked to determine if it is passive. The inefficient
assembly can be overcome by implementing the coupling on a lower level,
hard-coded inside the FS solver. This was done for the coupling of SIA and FS
in ISCAL , which showed significant speedup when restricting
the FS solver to a smaller domain. However, using passive elements is more
flexible, since the coupling is independent of the solver used to compute
velocities outside ΩSSA. One is free to choose between the two
different FS solvers in Elmer/Ice see or to
apply ISCAL. The latter is irrelevant in the experiments presented here since
both the grounded and floating ice experiences low basal drag, and SIA is not
capable of representing ice stream and shelf flow. Only a preliminary 3-D
experiment is performed here, since the current implementation is not
sufficiently efficient to allow extensive testing in 3-D. If the coupling is
implemented efficiently such that the time spent on solving the FS equations
on the restricted domain ΩFS scales with the size of ΩFS,
the computational work is expected to decrease significantly (see Sect. 4.2).
Conclusions
We have presented a novel FS–SSA coupling in Elmer/Ice, showing a large
potential for reducing the computation time without losing accuracy. At the
coupling interface, the FS velocity is applied as an inflow boundary
condition to SSA. Together with the cryostatic pressure, a depth-averaged
contact force resulting from the SSA velocity is applied as a boundary
condition for FS. The main finding of this study is that the two-way coupling
is stable and converges to a velocity that is very similar to the FS model in
the tests on conceptual marine ice sheets, and it yields a speedup in 3-D.
In diagnostic runs, the relative difference in velocity obtained from the
coupled model and the FS model is below 1.5 % when applying SSA at least 30 km seaward from the grounding line. During a transient simulation, where the
coupling interface changes dynamically with the migration of the grounding line,
the coupled model is very similar to the FS model, with a maximum difference
of 5.3 % in basal velocity at the grounding line. An offset of 3.6 km remains
in the reversibility experiment in Sect. 4.3, which is within the range of
the expected resolution dependence for FS models .
In experiments involving areas where SIA is applicable, this new FS–SSA model
can be combined with the ISCAL method in that
couples SIA and FS in Elmer/Ice. This mixed model is motivated by
paleo-simulations, but reducing computational work by the combination of
multiple approximation levels is also convenient for parameter studies,
ensemble simulations, and inverse problems.
The code of Elmer/Ice is available at
https://github.com/ElmerCSC/elmerfem/tree/elmerice (last access:
13 November 2018). An example of the coupling is provided at
10.5281/zenodo.1202407. The version of the
Elmer/Ice code that includes the coupling discussed in the paper can be
accessed by using the hash qualifier linked to the commit of the coupling
code at
https://github.com/ElmerCSC/elmerfem/archive/ba117583defafe98bb6fd1793c9c6f341c0c876.zip
(last access: 13 November 2018).
Derivation of the interface boundary condition
The boundary condition in Sect. between the FS and the
SSA domains is derived following a standard procedure in FEM using the weak
formulation of the equations. Let ΩFS∈Rd, d=2,3 denote the open FS domain in two or three dimensions with the boundary
ΓFS. After multiplying Eq. () with a test function
v and integrating over the domain ΩFS, the weak form of Eq. () is
-∫ΩFSv⋅∇⋅σ=∫ΩFSρv⋅g.
Use the definition of σ and the divergence theorem to
rewrite Eq. ():
∫ΩFSηD(u):D(v)-∫ΩFSp∇⋅v=∫ΩFSρv⋅g+∫ΓFSv⋅σ⋅n.
The operation A:B denotes the sum ∑i,jAijBij. The test function v vanishes on the inflow boundary
Γi, has a vanishing normal component on the bedrock boundary
Γb, and lives in the Sobolev space [W1,1/n+1(ΩFS)]d, i.e.,
v∈V0={v∈[W1,1/n+1(ΩFS)]d|v|Γi=0,v|Γb⋅n=0}.
The space V0 has this form because the boundary conditions on
Γi and Γb are of Dirichlet type. Furthermore, there is a
lateral boundary Γℓ for ΩFS∈R3, where
the normal component also vanishes (v|Γℓ⋅n=0), and we assume a vanishing Cauchy-stress vector for unset boundary conditions
to velocity components, such that the integral over Γℓ vanishes.
Then, the boundary integral in Eq. () consists of a sum
of the remaining boundary terms:
∫ΓFSv⋅σ⋅n=∑i=1d-1∫Γbfu⋅tiv⋅ti-∫Γwpwn⋅v+∫ΓFSintv⋅σ⋅n,
given by the boundary conditions on Γb in Eqs. () and
(), on the ocean boundary Γw in Eq. (), and the internal boundary ΓFSint between the FS
and the SSA domains. The force σ⋅n on
ΓFSint is determined by the SSA solution.
The open SSA domain ΩSSA∈R2, coupled to
ΩFS∈R3, has the boundary
ΓSSA=ΓSSAint∪ΓCF∪Γℓ where
ΓSSAint is adjacent to ΩFS and partly coinciding with
ΓFSint (but of one dimension less) and ΓCF is at the
calving front. Let B have the elements
B11=4η‾∂u∂x+2η‾∂v∂y,B12=B21=η‾∂u∂y+η‾∂v∂x,B22=2η‾∂u∂x+4η‾∂v∂y,
when d=3. If d=2, then B=4η‾∂u/∂x. Then the
SSA equations Eq. () can be written as follows:
∇h⋅B=fg,
where fg=ρgH∇hzs and ∇h is the horizontal
gradient operator. The boundary condition on ΓSSAint is the
Dirichlet condition (Eq. ), and the force due to the water
pressure at the calving front ΓCF is fCF, as in Eq. () but integrated over z. Define the two test spaces
W={v∈[W1,1/n+1(ΩSSA)]d-1|v|Γℓ⋅n=0},W0={v∈W|v|ΓSSAint=0}.
Multiply Eq. () by v∈W0 and integrate.
The weak form of Eq. () is
∫ΩSSAv⋅(∇h⋅B)=∫ΩSSAv⋅fg.
Apply the divergence theorem to Eq. () to obtain
-∫ΩSSA∇hv:B+∫ΓSSAv⋅B⋅n=-∫ΩSSA∇hv:B++∫ΓCFv⋅fCF+∫ΓSSAintv⋅fSSA=∫ΩSSAv⋅fg.
A mesh is constructed to cover ΩFS and ΩSSA with nodes at
xi. In the finite element solution of Eq. (), the
linear test function vi∈W0 is non-zero at xi
and zero in all other nodes. The integral over ΓSSAint vanishes
when v∈W0. The finite element solution uh of
Eqs. () and () satisfies
-∫ΩSSA∇hvi:B(uh)+∫ΓCFvi⋅fCF-∫ΩSSAvi⋅fg=0,xi∈ΩSSA∪ΓCF.
It follows from Eq. () that with a test function
vi∈W that is non-zero on ΓSSAint and the
solution uh from Eq. ()
∫ΓSSAintvi⋅fSSA==∫ΩSSA∇hvi:B(uh)-∫ΓCFvi⋅fCF+∫ΩSSAvi⋅fg,xi∈ΩSSA∪ΓCF∪ΓSSAint.
The first integral in Eq. () corresponds to
(ASSAuSSA)i in Sect. and
bSSAi to the second and third integrals. By Eq. (), the
right-hand side of Eq. () vanishes for all xi in
ΩSSA and on ΓCF, but for a node on the internal boundary,
xi∈ΓSSAint, the force fSSA from the ice due to
the state uh in ΩSSA is obtained. The internal pressure in
the ice in ΩSSA is assumed to be cryostatic as in Eq. (). The total force on ΓFSint consists of one
component due to the state uh at ΓSSAint and one due to
the cryostatic pressure there. Let ΩSSA∗ denote the mesh on
ΩSSA, which is extruded in the z direction. The common boundary
between ΩFS and ΩSSA∗ is ΓFSint, and let
fSSA∗ be the stress force there, independent of z. Since
∫zbzsfSSA∗=fSSA at ΓFSint, we
have fSSA∗=H-1fSSA. Let vi be a test
function on ΩFS⋃ΩSSA∗ which is non-zero on
ΓFSint and zero in all other nodes. Then the weak form of the force
balance at ΓFSint is
∫FSintvi⋅σ⋅n=∫FSintfSSA∗⋅vi-∫FSintρg(zs-z)n⋅vi=∫FSintH-1fSSA⋅vi-∫FSintρg(zs-z)n⋅vi,
and the corresponding strong form of the boundary condition at
ΓFSint is
σ⋅n=H-1fSSA-ρg(zs-z)n;
cf. Eq. (). Thus, by computing the residual as in Eq. (), the two finite element solutions in ΩFS and
ΩSSA are coupled together at the common boundary ΓFSint
and ΓSSAint.
Numerical values of the constants used in the ice shelf ramp
experiment. Since the shelf is afloat, there is no sliding at the base.
NK, ECHvD, RSWvdW, MBvG, PL, and LvS designed the study. ECHvD
implemented the coupling and carried out the numerical simulations, with
support from TZ and GC. ECHvD drafted the paper with support from NK, and
all authors contributed to the final version.
The authors declare that they have no conflict of interest.
Acknowledgements
This work has been supported by FORMAS grant 214-2013-1600 to Nina Kirchner.
Thomas Zwinger's contribution was supported by the Academy of Finland (grant
number 286587). The computations were performed on resources provided by the
Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High
Performance Computing at KTH. We are grateful to Mika Malinen, Peter Råback, and Juha Ruokolainen for advice in developing the coupling, to
Rupert Gladstone for providing the setup as in , and
to Felicity Holmes, Guillaume Jouvet, and Daniel Farinotti for their feedback
on a draft of the paper. We wish to acknowledge the constructive
comments of two anonymous reviewers, which contributed to improving the
paper.
Edited by: Didier Roche
Reviewed by: two anonymous referees
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