Numerical models have become an indispensable tool for
understanding and predicting the flow of ice sheets and glaciers. Here we
present the full-Stokes software package Underworld to the glaciological
community. The code is already well established in simulating complex
geodynamic systems. Advantages for glaciology are that it provides a
full-Stokes solution for elastic–viscous–plastic materials and includes
mechanical anisotropy. Underworld uses a material point method to track the
full history information of Lagrangian material points, of stratigraphic
layers and of free surfaces. We show that Underworld successfully reproduces
the results of other full-Stokes models for the benchmark experiments of the Ice Sheet Model Intercomparison Project for Higher-Order Models
(ISMIP-HOM). Furthermore, we test finite-element meshes with different
geometries and highlight the need to be able to adapt the finite-element
grid to discontinuous interfaces between materials with strongly different
properties, such as the ice–bedrock boundary.
Introduction
Numerical modeling has become a standard tool in the prediction of ice flow
in ice sheets and glaciers and has gained increasing importance due to the
quest to predict sea-level rise (Goelzer et al., 2017). Ice sheets and
glaciers on Earth consist of ice 1h, the crystallographic variant of water
ice that is stable under the conditions at the Earth's surface. Ice 1h is a
mineral with a hexagonal crystal symmetry that shows ductile or
crystal–plastic behavior (McConnell and Kidd, 1888; Nye, 1953; Glen, 1955;
Budd and Jacka, 1989) at differential stresses in the order of 0.01–0.1 MPa
that are typical for ice sheets.
The flow law of ice is generally assumed to be a power law (Glen, 1955; Budd
and Jacka, 1989), often termed “Glen's (flow) law” (Haefeli, 1961), in which
the strain rate is proportional to the differential stress to the power n,
the stress exponent. Usually modelers assume n=3 (see, e.g., Pattyn et al.,
2008), although several studies – including the original study of Glen
(1955) – assume that n≈4 probably best describes the rheology of
ice. This is confirmed by more recent studies (Goldsby and Kohlstedt, 2001;
Goldsby, 2006; Bons et al., 2018; Ranganathan et al., 2021).
Most rock-forming minerals also flow with a power-law rheology (Ranalli,
1987; Evans and Kohlstedt, 1995). Modelers of tectonic processes thus face
the same challenges related to nonlinear flow as those in the glaciological
community. Recent versions of the software package “Underworld” (Moresi et
al., 2007; Mansour et al., 2020; code available at
10.5281/zenodo.1436039, Beucher et al., 2022a) provide a Python API originally
developed to simulate geodynamics processes. Similar to Elmer/Ice
(Gagliardini et al., 2013), it solves the full-Stokes equations for
viscous–elastic–plastic deformation and is coupled to heat flow (Moresi et
al., 2003; Mansour et al., 2020). The latter is relevant considering the
potential impact geothermal heat flow may have on ice flow and ice streams
(Smith-Johnsen et al., 2020; Bons et al., 2021).
As with most minerals, the rheology of ice 1h strongly depends on a range of
factors, such as temperature and microstructure. In addition, ice 1h has a
strongly anisotropic rheology (Duval et al., 1983; Azuma, 1994), and it is
increasingly recognized that this plays a crucial role in the behavior of
flowing ice, especially at ice streams (Rathmann and Lilien, 2022).
In particular, airborne radar (Schroeder et al., 2020) has shown a rich
diversity in fold structures inside the ice sheets (NEEM community members,
2013; Wolovick et al., 2014; MacGregor et al., 2015; Bons et al., 2016; Cavitte et
al., 2016; Leysinger-Vieli et al., 2018). Radar data also allow for direct
measurements of the crystallographic and, hence, mechanical anisotropy in
ice (e.g., Young et al., 2021; Ershadi et al., 2022). As the mechanical
anisotropy, together with processes such as basal melting, is thought to
actively influence flow of ice and folding, there is an urgent need to
include it in ice flow models on various scales (Rathmann and Lilien, 2022).
Underworld includes mechanical anisotropy (Moresi and Mühlhaus, 2006; Sharples et
al., 2016). It employs the material point method (MPM) (Sulsky et al., 1994;
Moresi et al., 2003), where Lagrangian material points are combined with a
finite-element (FE) mesh. First and foremost, these material points allow
for tracking of the strain history and rheological or physical changes on
distinct Lagrangian points. Further, tracking of the material points allows
us to understand the deformation of individual volumes or layers within the
ice sheet and the evolution of the surface. Particles can also be used to
record the crystallographic preferred orientation (CPO) and thus the local
mechanical anisotropy of the material. This way, the mechanical anisotropy
can evolve as a result of the local deformation. The combination of both
anisotropic rheology and particle tracking has potential for the modeling of
large-scale folds of stratigraphic layers observed in ice sheets (Wolovick et al.,
2014; NEEM community members, 2013; Bons et al., 2016; Cavitte et al., 2016;
Leysinger-Vieli et al., 2018), in particular when the folding is a result of
the anisotropic rheology of ice (Bons et al., 2016).
Finally, Underworld can be coupled with other models to investigate surface
effects, such as sedimentation and erosion, and processes that affect the
base of the model, such as mantle deformation and heat flux (Salles et al.,
2018; Bahadori et al., 2022). These have their equivalents at the surface of
ice sheets in the form of snow precipitation, ablation, and both surface and
basal melting (e.g., Jacobson and Raymond, 1998; Smith-Johnsen et al., 2020).
For all these reasons, Underworld, which is already well established for the
simulation of complex tectonic processes (for instance, Sandiford et al.,
2020; Carluccio et al., 2019; Capitanio et al., 2019; Korchinski et al.,
2018), surface processes (Bahadori et al., 2022) and long-term ground water
motion (Mather et al., 2022), also seems well suited to simulate ice-sheet
and glacier flow.
Any numerical model needs to be validated or benchmarked. The Ice Sheet
Model Intercomparison Project for Higher-Order Models (ISMIP-HOM; Pattyn et
al., 2008, and Supplement or
https://frank.pattyn.web.ulb.be/ismip/welcome.html, last access: 30 May 2022) provides tests for the
comparison of computational ice-sheet flow models for different purposes.
“Higher-order” here refers to models that go beyond the shallow-ice
approximation (SIA) up to full-Stokes solutions (as Underworld does).
ISMIP-HOM includes both 2D and 3D experiments. The flow law is Glen's law
with a stress exponent n=3 and, in one experiment, Newtonian flow. In this
paper we publish the results for the full suite of experiments of the
benchmark. We focus on three issues: (i) the viability of the results as
compared to solutions provided by other models, (ii) the computation time
and (iii) the influence of the geometry of the underlying finite-element
grid. The tests are performed using the 2.10 release of the software package
Underworld. Finally, we provide one example of how mechanical anisotropy and
tracking of the stratigraphy can be incorporated in Underworld to illustrate
the potential of Underworld to simulate mechanically complex systems and the
resulting structures within a glacier or ice sheet.
MethodGoverning equations
The solution in Underworld is based on the Stokes equation of slow flow of a
Newtonian incompressible fluid:
1∂τij∂xj-∂P∂xi+ρgi=0,2∂vi∂xi=0.
Here τij is the deviatoric stress tensor, P the pressure, g the
gravitational acceleration and v the velocity (see Tables 1 and 2 for
symbols used). Simulations are based on Glen's flow law for viscous flow
(Glen, 1955), according to which the strain rate (ε˙ij) is proportional to the deviatoric stress (τij) to the
power n, the stress exponent. This flow law can be written as
ε˙i,j=AτIIn-1τij,
where A is the temperature-dependent rate factor and τII the second
invariant of the deviatoric stress tensor τij (Nye, 1953).
Based on Newtonian flow, where τij=2ηε˙i,j, we define an effective viscosity ηice after Eq. (3) as
ηice=12A-1/nε˙II1-n/n.
Parameters and their values, as prescribed by Pattyn et al. (2008)
for the intercomparison project.
SymbolParameterValueUnitAn=3ice-flow parameter for stress exponent n=310-16Pa-3 a-1An=1ice-flow parameter for stress exponent n=1 (Newtonian)2.140373×107Pa-1 a-1ρiceice density910kg m-3ρbedbed rock density2700kg m-3nexponent of Glen's flow law for ice3 or 1ηbedconstant bedrock viscosity1022Pa sηiceeffective viscosity of icePa sggravitational constant9.81m s-2Lmodel width5–160km
Symbols used in this paper and not listed in Table 1.
SymbolVariableTypical unitβ2basal friction coefficientε˙ijstrain rate tensora-1ε˙IIsecond invariant of the strain rate tensora-1madaption parameter for mesh deformationNnumber of nodes in a meshPpressurePapcoordinates of a vertex pointτxybhorizontal shear stress at the ice basis in x directionPaτijdeviatoric stress tensorPaτIIsecond invariant of the deviatoric stress tensorPavxb, vybvelocity at ice basis, x and y componentm a-1vxs, vysvelocity at ice surface, x and y componentm a-1x, yaxes parallel and vertical to the tilted surface, referred to as “horizontal”/“vertical”mCharacteristics of Underworld with regard to specific challenges in the
modeling of ice flow
Underworld is designed to solve some of the special problems relevant to
modeling geodynamic processes. Identical problems arise in the modeling of
ice. Some of these challenges are as follows:
the modeling of discontinuities in the material properties at layer
boundaries, for instance, at the ice–rock and ice–air interfaces;
gradients within the ice, for instance, due to strain softening or thermal
effects;
the tracking of the strain history;
the often extreme spatial extent of the modeled systems;
the very strong deformation of the material.
Underworld addresses these issues with the so-called material point method
(MPM) (Sulsky et al., 1994; Moresi et al., 2003), which is closely related
to the venerable particle-in-cell (PIC) method. MPM uses a Eulerian finite-element mesh in order to calculate the incremental development of the
velocity field and other field variables, such as, temperature
and pressure. In the MPM method, Lagrangian material points (“particles”)
carry the density, viscosity, thermal conductivity and other relevant
material parameters. They thereby record the history at their current
location at every time step and some historical properties like the stress
at previous time step for simulations of viscoelastic deformations
(Farrington et al., 2014). The underlying mesh provides solutions for the
incremental movement of the material points. The method is advantageous in
the modeling of the emergence of structures (e.g., folding; see Mühlhaus
et al., 2002) or where very strong deformation is involved, as in the
deformation across shear zones or near the base of an ice sheet. In MPM, the
mesh does not carry any history information other than deformation of the
boundary and therefore can be re-meshed at any time as required and without
loss of accuracy.
There is an unavoidable smoothing which comes from the coarseness of the
computational mesh relative to the material point density (Moresi et al.,
2003). While material boundaries are represented by a continuous interpolant
on the grid, they are necessarily discrete in the case of particles. This can
lead to fluctuations in the solution close to sharp rheological or
mechanical boundaries (Yang et al., 2021), for instance, at the interface
between ice and underlying rock. In the ice itself, a change in mechanical
parameters is usually more gradual and is controlled, for example, by the
temperature gradient.
Another complication in the numerical modeling of ice flow is the highly
anisotropic behavior of ice, created by the near-orthotropic properties of
the ice crystal. The possibility to model anisotropic flow is built into
Underworld (Mühlhaus et al., 2004). Like any other local material
property, the orientation of the anisotropy can be stored on the particle
level.
Underworld offers a variety of possible solvers, including the well-known
MUMPS, LU and multi-grid methods. We have carried out brief comparative
precision tests with these solvers and could not find any difference in
precision to the standard solver which is based on the multi-grid method.
Throughout this study we will generally use MUMPS for 2D models and
multi-grid for 3D models. The exception is tests of the computation time,
for which we contrast a variety of solvers.
Description of experiments
The ISMIP-HOM benchmark experiments have been described in detail in Pattyn
and Payne (2006) and Pattyn et al. (2008). We perform all the experiments as
described in these publications. For simplicity we also apply the same
alphabetical numbering scheme and refer to the experiments as Experiment A
to Experiment F. As we strive to focus on the essentials in the following
descriptions, we refer to the original publications for further technical
details if needed.
Experiments A, B, C and F are three-dimensional. Experiments D and
E only consider two spatial dimensions, and Experiment B has an
additional version in 2D. Experiments A through D are performed for a
variety of horizontal system dimensions L, with L=5, 10, 20, 40, 80 or 160 km. Experiments A through E use a flow law based on n=3, and Experiment F
applies Newtonian flow where n=1 (Table 1).
We tested the influence of the mesh geometry on the results and the CPU time
consumption as a function of the total degree of freedom using Experiment D and the 2D version of Experiment B.
Experiments A and B
A and B consider a slab of ice with a mean ice thickness H=1000 m, lying
on a sloping bed with a mean slope α=0.5∘. The
bedrock topography consists of a series of sinusoidal bumps (Experiment A)
or ripples (Experiment B) with an amplitude of 500 m (Fig. 1). The minimum
thickness of the rock layer is 500 m, and the total height of the model is
2000 m. The flow of ice is governed by Eq. (3). The bedrock viscosity is
constant, and ice is frozen to the bedrock. Relevant material parameters are
compiled in Table 1.
The surface elevation is described by the formula
zsx,y=-x⋅tanα.
Bedrock topography for Experiment A is described using zs by
zbx,y=zsx,y-1000+500sinωx⋅sinωy
and for Experiment B by
zbx,y=zsx,y-1000+500sinωx.
(a) 2D geometry of Experiment B. This is identical to a section
parallel X located at y^=0.25 in Experiment A (right). Sloping angle α is given in
degrees. Also depicted is the velocity field of the flowing ice, resulting
for a model width L of 5000 m from the simulations described below. Color and
arrow length visualize the amount of velocity. (b) Bedrock topography for
Experiment A and general naming scheme for the axes of 3D experiments.
Experiments C and D
Experiments C and D are similar to Experiments A and B, although the
topography of the bedrock is flat. Instead, the coefficient of basal
friction β2 varies in a sinusoidal manner. The ice thickness H is
constant at 1000 m. The slope angle α of the ice surface and of the
underlying bedrock is 0.1∘. Ice flow is governed by Eq. (3), and the
material parameters are summarized in Table 1. According to the benchmark
specifications, Experiment C is run exclusively in 3D and Experiment D
exclusively in 2D.
The basal friction coefficient relates the basal drag τb to the
basal velocity vb by
τb=vbβ2.
With ω=2π/L the coefficient of basal friction β2 for Experiment C is defined as
β2=1000+1000⋅sinωx⋅sinωy
and for Experiment D by
β2=1000+1000⋅sinωx.
Figure 2 shows the basal drag and the basal velocity calculated from Eq. (10) (Experiment D) and Eq. (8). Here, the velocity is calculated for a
constant basal shear stress τb=ρicegHsinα, according to the shallow-ice approximation (SIA) (Hutter, 1983).
Notice the singularity in the velocity field (Fig. 2b), which develops
because β2=0 at x=3/4L.
(a) Basal drag β2 and (b) basal velocity vxb
plotted according to Eqs. (9) and (10) and applying the SIA, plotted for L=5000 m. Notice the singularity in the velocity field, which develops because
β2=0 at x=3/4L, which is within the modeled domain.
Experiment E
Experiment E is a two-dimensional diagnostic experiment along the central
flowline of a glacier in the European Alps (Haut Glacier d'Arolla). The
basic experiment and geometry are described in Blatter et al. (1998) and by
Pattyn et al. (2008). The general geometry of the glacier profile as used in
the experiment is shown in Fig. 3. The experiment is run with two different
basal conditions: (1) the ice is frozen to the ground everywhere (β2=∞), or (2) a zone of zero traction (β2=0)
between x=2200 m and x=2500 m exists. Compare Eq. (8) for the meaning of
the basal friction coefficient β2.
Longitudinal profile of Haut Glacier d'Arolla (Pattyn et al., 2008). Blue
line: contact ice–rock. Orange line: contact ice–air. Red zone: area of
varying basal conditions, with either β2=∞ or β2=0.
Experiment F
Experiment F is a prognostic experiment in which a free ice surface relaxes
until a steady state is reached for a zero surface mass balance. The slab of
ice is resting on a bed with a mean slope α=3∘. The
bedrock plane parallels the surface but is perturbed by a Gaussian bump.
The initial bedrock (B0) and surface (S0) elevation are described
by
11S0x,y=012B0x,y=-H0+a0⋅exp-x2+y2σ2,
where σ=10H0.
Experiment F applies a Newtonian (n=1) flow law, given in Table 1, so
that the effective viscosity ηeff=(2An=1)-1. The
experiment is run with two different values for the slip ratio c, so that c=1 and c=0 is applied. c is used to describe the basal friction
coefficient β2 (see Eq. 8) by
β2=cAn=1H0-1.
The FE mesh
The model domain is discretized by quadrilateral Q1/P0 elements, where
velocity is continuously linear, and pressure is discontinuously constant.
The pressure grid is offset from the velocity grid. A direct comparison of
the pressure at fixed locations to the results of other models is therefore
prone to some interpolation error. Periodicity of the in- and outflow
boundaries is applied in all experiments except Experiment E.
The ice–rock interface is defined by either of the two following methods,
depending on the type of experiment: (i) by particles or (ii) directly by the
mesh geometry.
FE grids with a rectangular hull
In most experiments we assume that the bedrock is identical to the basal
grid boundary. In experiments with a flat bedrock topography (C, D and
F), this means that the resulting shape of the mesh is rectangular.
An exception is the 2D version of Experiment B: in order to evaluate the
impact of the mesh geometry and of a particle representation of the bedrock
material, we define the sinusoidal bedrock topography using particles on a
FE grid with a rectangular hull. Different materials are represented by
particles with different rheological properties. Particles are assigned to
either ice or bedrock depending on whether their depth exceeds the local ice
thickness H. As pointed out, the material point method can lead to spurious
fluctuations in gradients at material boundaries, if particles of different
materials are both located in one element.
In order to test and improve this behavior, we define three different
internal grid geometries and compare the smoothness of the resulting basal
shear stress τb. These geometries are (i) a classic rectangular
grid, (ii) a grid where the grid resolution increases in the vicinity of
the ice–bedrock interface and (iii) a grid where the mesh perfectly fits
the rock surface (Fig. 4).
(a) Rectangular mesh. (b) Structurally conforming mesh, with
increased resolution at the ice–rock interface. (c) Mesh perfectly fits the
rock surface. Blue: ice. Red: bedrock. For visualization purposes, both mesh
resolution and distortion are reduced compared to the actual experiments.
The structure parameter m=0.5 for both adapted grids (see Eqs. 14, 15). Mesh resolution: x=128, y=64. In the case of the rectangular mesh, it is
additionally 256×128.
Rectangular grid geometry
The benchmark assumes a constant height of the model for each experiment,
while the width is varied in a series of runs during such an experiment. It
is typically expected that the accuracy of the solution and the computation
time are most optimal if the aspect ratio of cells is close to 1.
Structurally conforming grid
We adapt the rectangular mesh to the underlying topography defined by the
ice–rock boundary by vertically shifting its vertices, so that the model
resolution is significantly increased close to the bedrock surface. Using
the vertices of the regular rectangular grid as input, the new vertical
y coordinate p2′ of a vertex point p=[p1,p2]
from the regular mesh becomes
p2=sp1-Δy⋅Δysp1-Y0m if sp1>p2sp1-Δy⋅ΔyY1-sp1m if sp1≤p2.
We assume here that the grid in y direction originates at Y0=0 and
ends in Y1=1. The rock surface is defined by a function s(x).
Δy=s(p1)-p2 is the vertical distance between the rock surface
and p2. Both resolution and the geometry of individual cells
are adapted to the interface line as shown in Fig. 3b. m is a structural
adaption parameter, controlling the intensity of the mesh deformation. It is
worth pointing out that the resolution in x is not affected by this geometry.
Grid fitted to ice–rock interface
The mesh defined by Eq. (14) does not guarantee that the ice–rock interface
is aligned perfectly with finite-element edges. This may still introduce
stress perturbations in elements containing different materials with strong
viscosity contrasts (Yang et al., 2021). Therefore it makes sense to apply
another mesh structure whose mesh edges fit the ice–rock interface exactly.
We define the new vertical y coordinate p2′ of a vertex point
p=[p1,p2] from the regular mesh by
p2′=sp1-sp1-Y0⋅n0-nn0n0-nn0m if sp1>p2sp1-Y1-sp1⋅n0-nnt-n0n0-nnt-n0m if sp1≤p2.
The variable n denotes the nth node in the vertical y direction; n0 is a
predefined node, which is relocated exactly to the rock surface; and nt
is the total number of vertical vertexes. The difference between the
equations above is that we fix n0 in Eq. (14), while n0 varies along
the x direction for the case discussed in Eq. (15). As before, m is an adaption
parameter which controls the intensity of the mesh deformation. The adaption
of the grid geometry does not affect the position of nodes in the x direction.
FE grids with a non-rectangular hull
In all other experiments with an uneven bedrock topography (Experiments A
and E and the 3D version of Experiment B) we apply Eq. (15) with m=0
to the lower system boundary. In the case of Experiment D, the ice–air interface is not flat; therefore particles represent ice and overlying air
(Fig. 5).
Grid geometry and particle distribution used for Experiment E,
representing Haut Glacier d'Arolla. Particles carry the
rheological properties of ice and air. Blue: ice. Red: air.
Basal conditions
Underworld pre-implements no-slip and free-slip boundary conditions for
flat system boundaries. However, Experiments C through E require the
usage of a friction law. We realize the basal drag requirement by a basal
layer with Newtonian viscosity, whose viscosity is dependent on the basal
friction coefficient β2.
In the following, the relation between the Newtonian viscosity of this basal layer
(ηb) and β2 is derived. For the sake of simplicity, we
assume a flat horizontal surface, so that the usual notation can be used. In the case of an uneven base, x and y correspond to directions parallel or
perpendicular to the lower system boundary.
Combining and integrating the relations τ=2ηbε˙ (Newtonian flow) and ε˙=∂vx/∂y (definition of the strain
rate) leads to ηbx=h⋅β2x, with h the layer height. This relation is then used to define the
local viscosity of the Newtonian layer. At the top of this layer, the
velocity condition defined in Eq. (8) is satisfied.
General performance tests
Below, we will first examine the CPU time consumption as a function of the
grid resolution and the effect of the grid geometry on the smoothness of
results.
CPU time consumption
The mesh resolution is one of the most important parameters that controls
the precision of the solution. Since computation time increases with
resolution, it is desirable to establish a relationship between these two
quantities. Below we test and display the CPU time of the initial solution
of the 2D version of Experiment B. The computation time is not directly
linked to the mesh resolution but instead to the degrees of freedom (DOF)
of the system. For 2D-Stokes problems with quadrilateral elements, the
degrees of freedom are 3N, with N the total number of nodes (Gagliardini and
Zwinger, 2008). Figure 6 shows the relationship between the DOF and the
computation time in a log–log plot for a series of 13 simulations with
different resolutions and for different solvers (MUMPS, LU and multi-grid).
The ratio of width to height of the grid cells remains constant in all
experiments. On the hardware side, the simulations ran on a system with an
Intel Xeon E5 processor with eight cores and 32 GB RAM. Of these eight cores, only one
core was assigned to a trial run, and only one experiment was calculated at
a time.
The interpolation between N and the computation time (s) expressed by a power
regression is 0.00014N1.21 (multi-grid), 0.00037N1.06 (MUMPS)
and 0.00013N1.15 (LU). In the case of the multi-grid method, outliers from
the generally linear relation exist, which are related to the recursive
refinement of the grid. Outliers do not exist for the direct solvers MUMPS
and LU.
It is important to note, that – theoretically – the direct solvers should not scale better than N2. Therefore, the power regression cannot be
used in order to extrapolate these results to an arbitrary DOF. It can be
seen in Fig. 6 that a slight upward curvature of the data with regard to the
interpolation exists.
Computation time for Experiment B plotted vs. the degrees of
freedom and using different solvers. Blue: Mumps (0.00037N1.06). Red:
LU (0.00013N1.15). Green: multi-grid (0.00014N1.212).
Impact of the grid geometry
We tested the impact of the grid geometry based on using the flow law
parameters n=3 and A=10-16 Pa-3 a-1 (Pattyn et al., 2008).
In order to increase effects related to the geometry, the mesh resolution is
intentionally relatively small, at 128×64. Only for the rectangular grid is a larger resolution applied. The non-rectangular meshes use an adaption
parameter m=0.25.
The results for the surface velocity are generally smooth and virtually
identical in the three grid geometries. However, fluctuations of the shear
stress can become considerable, especially close to the ice–bedrock
interface (Fig. 7), depending on the type of FE grid. These fluctuations are
largest if a low-resolution rectangular mesh is used. Increasing the
resolution of the rectangular mesh does not fully eliminate the
fluctuations but is a way to reduce the problem (as highlighted in Fig. 7c). A more conforming grid with an increased resolution around the rock
surface (Eq. 1) reduces the fluctuations significantly but does not
completely eliminate them (Fig. 7b). The unambiguously smoothest results are
achieved using a grid fully fitted to the rock surface after Eq. (15) (see
Fig. 7a). This confirms that fluctuations become smaller when there is less
mixing of materials with different properties within an element.
The absence of fluctuations in the stress field is a measure for the ability
of the model to deal with discontinuous material boundaries. In the
following simulations for Experiment B, we will therefore exclusively apply
the fitted mesh. This is in line with the original benchmark paper by Pattyn
et al. (2008), who suggested testing the models with optimized settings.
The boundary between ice and the bedrock is usually the only nonplanar
material boundary in glaciers and ice sheets. The other material boundary,
between ice and air, can often be considered almost planar in the modeling
of large ice sheets, while rheological changes within the ice itself can
usually be treated as gradual, controlled, for instance, by the temperature
field or the crystallographic fabric. It is an important conclusion from
these results that a FE mesh fitted (even approximately) to the underlying
bedrock topography can significantly improve the accuracy of ice flow
simulations based on the material point method.
The basal shear stress τxyb (in kPa), calculated
on (a) a grid fitted to the rock surface, and (b) a structurally more conforming
grid, which increases mesh resolution at the rock surface and (c) a
rectangular grid. The experiments are based on a grid resolution of 128×64.
Impact of the grid resolution
Choosing a well-suited mesh resolution is always a balance between precision
requirements and computational limitations. The computation limits can be
related either to the hardware or to general time constraints, which do not
allow for very long computation times. Also mesh geometry has an impact on the
accuracy and may or may not require a larger resolution.
Given the existing time constraints and the number of simulations, this
allowed us to test resolutions of 64×32 and 128×64 up to 256×128. Even at
the relatively small resolution of 64×32, the solution for stress and
velocity at the surface of the model is smooth and virtually identical to
the results obtained with higher resolutions, independent of the mesh
geometry (Sect. 5.2). This was different in the case of the interface between
the ice and the underlying bedrock. While performance was good enough or
very good with a resolution of 128×64 for grids adapted to the interface,
a resolution of 256×128 was necessary to obtain usable results in the case of
a rectangular mesh. Two grid points above the interface, the solution was
smooth for all mesh geometries at 128×64 and did not differ from higher
resolutions.
The highest-resolution 3D simulations ran mostly on a high-performance computer (HPC)
cluster. Accuracy was systematically tested for Experiment B using a fitted
grid, by comparing the results of the 2D and 3D experiments. Under perfect
conditions, 2D and 3D simulations should yield identical results. We
found that the necessary resolution is highly dependent on the system size,
i.e., on the aspect ratio of the cells. For small system sizes of up to 20 km, we found that a minimum resolution of 256×128×32 was necessary to
produce similar results as for the 2D model. Larger systems already produced
satisfactory results with a resolution of 128×64×8.
Specific results
Below we will compare the results of the individual benchmark experiments
generated by Underworld to the results of alternative codes. Where
applicable, we run both the three-dimensional and the two-dimensional
version of the benchmark experiments. This applies to Experiments B and
C. For the latter, Experiment D is the associated 2D setup. All results
are compiled in the Supplement to this paper. Concerning the
output parameter Δp, which is the difference between the isotropic
and the hydrostatic pressure, the curve progression is very similar to that
of other full-Stokes models of the ISMIP but shows stronger
fluctuations between adjacent evaluation points. This is due to the internal
architecture of Underworld experiments, where the mesh used for the
calculation of pressure is a sub-mesh of the velocity mesh with a staggered
geometry. Fluctuations are a product of internal interpolation.
Experiments A and B
We fit the lower system boundary to the topography of the underlying bedrock
in the 3D experiments by applying Eq. (15) with m=0.2 to the lower system
boundary. The 2D version of Experiment B is based on a rectangular grid,
which is internally fitted to the rock surface as in Sect. 5.2 above. Other
relevant parameters are given in Table 1.
Results of Experiment A are shown in Figs. S1 and S2 in the Supplement and display the surface velocity and the basal shear stress.
Results of Experiment B are shown in Fig. S3 to S6 in the Supplement and show surface velocity and basal shear stress in a 2D version of
the experiment and at an arbitrary section paralleling the x axis of the 3D
experiment. Diagrams include the results of full-Stokes solutions published
in Pattyn et al. (2008) for comparison.
The surface velocity is controlled by the model width L and is in the range
from a few meters per year to more than 100 m a-1. Figure 8 compares our
results for Experiment B to results compiled from eight full-Stokes models
that were previously published (Pattyn et al., 2008). A full comparison of
the results is in the Supplement to this paper.
In Experiment B, the shape of the horizontal surface velocity for L=5 km differs significantly from that for the other cases. The surface velocity
is larger over the bump and thus anti-correlated with the ice thickness.
Gagliardini and Zwinger (2008) explain this as a mass conservation effect:
horizontal flux cannot be balanced by vertical flux because the vertical
velocity would be too large for the given system size. This phenomenon does
not occur in Experiment A, with the same system size, although the section
is identical at the chosen location. This can be explained because ice can
flow around the sides of the bump.
Figure 8 shows the maximum and minimum surface velocity vxys and the maximum basal shear stress τxyyb
calculated by the 2D version of Experiment B and compares it to other
full-Stokes solutions compiled by Pattyn et al. (2008). Maxima of both the
shear stress and the surface velocity show a tendency to be at the lower
end of the spectrum, while minima are virtually identical to the results
from the comparison models. However, a full comparison (compare Supplement)
shows a more diverse distribution of the results than the minima–maxima
comparison in Fig. 6 implies. The results of our Experiments show generally
a very good agreement with the majority of models, while some of the
comparison models display large deviations with regard to the entire curve.
Figures S5 and S6 in the Supplement compare the results of the
2D and the 3D setup, which should be ideally identical. The results for the
surface velocity are virtually identical, with exception of the L=5 km
case, where a deviation of ∼2 % for the maximum and the
minimum velocity exists if comparing 2D and 3D results. A similar effect
exists regarding the extrema of the basal shear stress, which are most
notable for L=5 km and L=10 km.
To shed some light on the difference between 2D and 3D results, we ran 3D
simulations with a resolution of 128×64×8 and of 256×128×32 for
these system sizes and compared them to the results of the 2D simulation
(256×128). Regarding the surface velocity, the higher-resolution 3D mesh indeed yields results closer to the 2D result than the lower-resolution
simulation (Fig. S6 in the Supplement). This is not the case
for the basal shear stress, where the deviation from the 2D result is
comparable for both resolutions.
Possible reasons are (a) different hardware, the 2D experiment, and
the low-resolution 3D experiment running on the same machine while the
high-resolution 3D experiment ran on a HPC cluster and (b) differences in
the mesh geometry. The low-resolution simulations use an exponent m=0.25
in Eq. (15). For technical reasons m is set to 0 for experiments on the cluster.
Maximum and minimum values of the horizontal surface velocity
(vxys) and of the maximum basal shear stress (τxyyb) in Experiment B and D, plotted for every model
size. The underlying FE mesh is fitted to the rock–ice interface. Results
are compared to the results of eight full-Stokes models (compiled and published
by Pattyn et al., 2008). Abbreviations of the comparison experiments are
compiled in Table 3.
Model abbreviations used in the diagrams, taken from Pattyn et al.,
2008. “Dimensions”: model dimensions. “Method”: numerical method: FE:
finite element, Sp: spectral method, MPM: material point method.
ModelDimensionsMethodReferencetsa12MPMthis study, based on Mansour et al. (2020)Jla13MPMthis study, based on Mansour et al. (2020)aas23FEunpublishedcma13FEMartín et al. (2004)jvj13FEJohnson and Staiger (2007)mmr13FEunpublishedoga13FEGagliardini and Zwinger (2008)rhi13SpHindmarsh (2004)rhi33SpHindmarsh (2004)ssu12FESugiyama et al. (2003)Experiments C and D
Parameters for Experiments C and D are given in Table 1 for n=3. Results
are summarized in Figs. S6 and S7 (Experiment C) and Figs. S8 and S9
(Experiment D) in the Supplement.
In Experiment C, the surface velocity is strongly dependent on the system
size L. In the case of the smallest system size (L=5 km), the surface velocity is
close to constant at ∼16 m a-1. With L=160 km it ranges
from 8.8 up to 122.5 m a-1 (Fig. S7, Supplement). The shear
stress results of the Underworld simulations line up well with the
comparison simulations. They show a sinusoidal curve with a maximum at
x^=0.25 and a minimum at x^=0.75. With increasing model size, the maximum
gets progressively flattened. The peak downwards at the singularity value
x^=0.75 is far less impacted by model size, which means it gets more
dominant for bigger values of L.
In Experiment D, the maximum surface velocity increases with model width
L and is in the range from 16 m a-1 to more than 235 m a-1. Figure 9
shows the maximum and minimum surface velocity and the maximum shear stress
τxy. Results are generally in good agreement with the majority of
model results they are compared to. However, the general variation between
the results of the comparison models is generally larger than in the 2D
version of Experiment B, the only other 2D experiment of the benchmark.
The same statement applies to basal shear stress. Again, a full comparison
of the results is included in the Supplement to this paper.
Experiments E and F
Experiment E is calculated for two different setups, where either the
entire glacier is frozen to the underlying bedrock or where a traction-free
section close to the center exists. In both cases, the results of Underworld
lie within the range of the comparison models (see Figs. S10 and S11 in the
Supplement). Both the calculated surface velocity and the basal
shear stress are in the lower range of this range, which is true for the
majority of models. In particular, extreme values of the curve are a bit
smoother than in some of the other full-Stokes models.
The prognostic Experiment F calculates the surface velocity and the
surface topography. Results are compiled in Figs. S12 and S13 in the
Supplement. Since the majority of potential comparison models
are not capable of calculating a free surface, the results of Underworld can
be compared to only two other software packages. Due to the special
properties of the software, we only model the version of the experiment with
a basal no-slip condition. Other values for the basal traction would either
involve a very complex implementation or yield questionable results. We ran
the model until no further change in topography or velocity occurred and
interpreted this as the stable state.
When we compare our results with those of the two reference models with
respect to an analytical sample solution (Frank Pattyn, personal
communication, 2022), qualitatively similar results are obtained. One of the
reference models shows very good agreement with the theoretical surface
elevation but shows lower accuracy with respect to surface velocity.
Underworld predicts the surface velocity the best of the three models but
tends to develop a slightly less extreme topography than the analytical
solution.
Comparison experiments with anisotropic and isotropic ice
Ice is assumed to deform mostly by dislocation creep at natural strain rates
(Budd and Jacka, 1989), whereby slip along the basal planes is much easier
than slip along the other slip planes (Duval et al., 1983). This makes an
ice single crystal mechanically effectively transversely isotropic, with the
c axis that is oriented perpendicular to the basal plane as the symmetry
axis. The anisotropy of ice single crystals leads to a crystallographic
preferred orientation (CPO) in a deforming aggregate of ice crystals (Alley,
1988; Llorens et al., 2017). The mechanical anisotropy of an aggregate may
be of a lower symmetry than that of a single crystal but can be approximated
to a first order as orthotropic (Gillet-Chaulet et al., 2006).
One of the features of Underworld which makes it interesting for the
modeling of ice is its capability to model linear orthotropic viscosity,
which includes transverse isotropy as a special case. In order to
demonstrate its effect, we set up comparison experiments with anisotropic and
isotropic ice rheology, based on the setup of Experiment B: ice flows over
a sinusoidal surface, driven by a general 0.5∘ tilt of the model.
Isotropic flow is governed by Eq. (4), using the parameters given in Table 1.
The orientation of the anisotropy is stored as the c axis orientation on the
level of the particles in Underworld. The aggregate anisotropy of one mesh
element is calculated from the individual c axis orientations of the cloud
of particles in an element. Fabric evolution is simulated using the rotation
of the c axes in the flow field after, e.g., Gillet-Chaulet et al. (2006) or
Richards et al. (2021). However, for simplicity, in our example here we
assume that all basal planes are and remain horizontal. The anisotropy of
ice, in terms of the ratio of resistance to slip parallel to
crystallographic non-basal and basal planes, is about 60–80 in a single ice
crystal (Duval et al., 1983). However, as we simulate aggregates of
crystals here, we set the viscosity for shear non-parallel to the basal planes
only 10 times higher than the viscosity for shear parallel to the basal
planes, which we assume equal to that used for the isotropic flow law.
Figure 9 shows the shape of marker lines prior to deformation and after 750 years of flow for both iso- and anisotropic ice models. Marker lines inherit
their sinusoidal shape from the shape of the underlying topography and are
then folded according to the localization of the highest shear rates. In
isotropic ice (Fig. 9b), the axial plane of the shear fold is mostly
controlled by the underlying topography. In the case of anisotropic ice (Fig. 9c), the folding is more intense, and the axial plane is subhorizontal,
showing that the anisotropy has a distinct effect on resulting fold
structures.
Marker lines prior to (a) and after 750 years of flow of (b) isotropic
and (c) anisotropic ice. The axial plane of the resulting shear fold in
isotropic ice mimics the bedrock topography, while it is controlled by
shearing along a horizontal shear zone in the case of anisotropic ice. Green:
bedrock, flow to the right.
Velocity field and strain rate field in isotropic ice. Large
strain rates and velocities occur in the vicinity of the bottleneck formed
by the crest of the hill. Green: bedrock. For velocity, red is 70 m a-1 and blue is
0 m a-1. For strain rate, red is 0.032 a-1, and blue is 0 a-1.
Velocity field and strain rate field in anisotropic ice. The
velocity field is vertically subdivided into a fast-flowing upper and an
almost stagnant lower part. The strain rate is thus at its maximum in the
horizontal shear zone that spans from crest to crest of the bedrock
undulations. Green: bedrock, velocity: red: 6.6 m a-1, blue: 0 m a-1,
strain rate: red: 0.031 a-1, blue: 0 a-1.
Figures 10 and 11 that show the velocity field and the strain rate field in
both materials allow for a better understanding of the deformation process. In the case of isotropic ice, the flow field is controlled by the underlying
topography. Hence the hill on the left acts as a bottleneck for the ice
flow, and the hill sides funnel ice in and out of the bottleneck region.
Consequently, the zone of high shear strain at the ice–rock interface
extends across the crest of the bedrock bump (Fig. 10a) and the maximum
velocity above it. Outside the bottleneck region, flow is comparatively
evenly distributed.
In the case of anisotropic ice, the flow regime and thus the shear folding
is quite different. Here, flow is strongly controlled by the inherent
anisotropy and far less by the bedrock topography. Looking at the velocity
field (Fig. 11a), it becomes clear that the flow field is internally
subdivided into a fast-flowing upper part and slow-flowing “dead ice” in the
lower part. Decoupling of the shallow and deep ice develops because the
anisotropy facilitates shear along the horizontal basal planes. The
resulting horizontal high-strain zone spans the entire system and produces
the distinct shear fold.
Outlook
One of the great advantages of the Underword2 software package from the
standpoint of the modeler is its great flexibility and traceability due to
its hybrid nature as both a particle and a finite-element model. Another
contribution to its flexibility is the easy extensibility of the core code
and the interactive development thanks to the existence of a Python API. The
compatibility of the API with the mathematical–scientific packages NumPy and
SciPy provides easy access to a wide range of numerical resources.
Anisotropic flow and heat flow are already part of the core package.
Given the current debates in the ice community about the role of the
anisotropy and hence of the fabric evolution of ice or the ideas for
improvements of the flow laws for ice (e.g., Kennedy et al., 2013; Llorens et
al., 2017; Richards et al., 2021; Martín et al., 2009), this type of
flexibility is an important precondition for a future proof numerical model
for glacier ice. One example was the implementation of the orthotropic
rheology and the fabric evolution in deformed ice following Gödert
(2003) and Gillet-Chaulet et al. (2006) by the authors of this text.
Possible improvements to the fabric evolution models – for instance, by the
SpecCAF model (Richards et al., 2021) – can be easily implemented and
tested.
A few desirable features are currently still missing but are on the road map
for future versions of Underworld. The soon-to-be-released Underworld3, for
instance, allows for greater flexibility of the mesh geometry, including the
triangulation of shapes and areas with arbitrary geometry.
Conclusions
The Underworld software is designed to solve deformation in complex
geodynamic systems with nonlinear elastic–viscous–plastic materials, for which
it provides a full-Stokes solution. It is therefore well suited for the
modeling of glacier and ice-sheet flow, as it includes heat flow and
anisotropic rheology. The combination of Lagrangian mass points (particles)
and a Eulerian finite-element solution allows for the tracking of individual
points as well as of inner and outer surfaces, such as deforming
stratigraphic layers, but also of the thermal–mechanical properties in
deforming materials. In the case of large rheological differences across
interfaces, the possibility to fit the grid to the interface greatly
improves the accuracy of stress field, compared to other grid types. In the case of ice flow experiments, it makes sense to fit the grid to the bedrock–ice
boundary.
We compared results of Underworld simulations with those of other modeling
approaches for the set of benchmark experiments provided by Pattyn et al. (2008). Our results match the full-Stokes solutions that are compiled in
that study. This means that Underworld is a viable alternative to other
full-Stokes models, in particular where the material point method is
advantageous, such as when accurate tracking of material volumes or
stratigraphic layers is desired. A further advantage is that, owing to the
built-in Python API, Underworld is very flexible and can be extended to be
applied to even the more complex processes which are involved in the flow of
ice sheets and glaciers.
Code and data availability
The program code that defines the experiments discussed in this article is included in the
Supplement to this article, along with graphical representations and text files of the results. They are
also available through Yang and Giordani (2022; 10.5281/zenodo.7384424). Underworld is
fully open-source. The version (v2.12.0b) used for this paper is available through Mansour et al. (2022; 10.5281/zenodo.5935717). The most recent version is available through
Beucher et al. (2022b; 10.5281/zenodo.6820562).
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-15-8749-2022-supplement.
Author contributions
TS and PB discussed and designed the first implementation of the
benchmark experiments. HY provided the code and the concept behind the
applied mesh geometries. TS and HY developed and tested the
code and performed some simulations. JL performed most of the 3D
experiments and compiled the related data files and data plots. TS
performed the 2D simulations and compiled the related data files and data
plots. TS prepared the manuscript with many contributions,
corrections and revisions from all authors. PB and LM provided
important guidance.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support
This open-access publication was funded by the University of Tübingen.
Review statement
This paper was edited by Mauro Cacace and reviewed by Frank Pattyn and one anonymous referee.
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