We introduce MPAS-Albany Land Ice (MALI) v6.0, a new variable-resolution land ice model that uses unstructured Voronoi grids on a plane or
sphere. MALI is built using the Model for Prediction Across Scales (MPAS)
framework for developing variable-resolution Earth system model components
and the Albany multi-physics code base for the solution of coupled systems of
partial differential equations, which itself makes use of Trilinos solver
libraries. MALI includes a three-dimensional first-order momentum balance
solver (Blatter–Pattyn) by linking to the Albany-LI ice sheet velocity
solver and an explicit shallow ice velocity solver. The evolution of ice
geometry and tracers is handled through an explicit first-order horizontal
advection scheme with vertical remapping. The evolution of ice temperature is
treated using operator splitting of vertical diffusion and horizontal
advection and can be configured to use either a temperature or enthalpy
formulation. MALI includes a mass-conserving subglacial hydrology model that
supports distributed and/or channelized drainage and can optionally be
coupled to ice dynamics. Options for calving include “eigencalving”, which
assumes that the calving rate is proportional to extensional strain rates. MALI is
evaluated against commonly used exact solutions and community benchmark
experiments and shows the expected accuracy. Results for the MISMIP3d
benchmark experiments with MALI's Blatter–Pattyn solver fall between
published results from Stokes and L1L2 models as expected. We use the model
to simulate a semi-realistic Antarctic ice sheet problem following the
initMIP protocol and using 2 km resolution in marine ice sheet regions. MALI
is the glacier component of the Energy Exascale Earth System Model (E3SM)
version 1, and we describe current and planned coupling to other E3SM
components.
Introduction
During the past decade, numerical ice sheet models (ISMs) have undergone a
renaissance relative to their predecessors. This period of intense model
development was initiated following the Fourth Assessment Report of the
Intergovernmental Panel on Climate Change ,
which pointed to deficiencies in ISMs of the time as being the single largest
shortcoming with respect to the scientific community's ability to project
future sea level rise stemming from ice sheets. Model maturation during this
period, which continued through the IPCC's Fifth Assessment Report
and to the present day, has focused on
improvements to ISM “dynamical cores” (including the fidelity,
discretization, and solution methods for the governing conservation
equations; e.g.,
),
ISM model “physics” (for example, the addition of improved models of basal
sliding coupled to explicit subglacial hydrology, e.g.,
; and
ice damage, fracture, and calving, e.g.,
), and the
coupling between ISMs and Earth system models (ESMs)
e.g.,.
These “next-generation” ISMs have been applied to community-wide
experiments focused on assessing (i) the sensitivity of ISMs to idealized and
realistic boundary conditions and environmental forcing and (ii) the
potential future contributions of ice sheets to sea level rise see,
e.g.,.
While these efforts represent significant steps forward, next-generation ISMs
continue to confront new challenges. These come about as a result of
(1) applying ISMs to larger (whole-ice-sheet), higher-resolution (regionally
O(1 km) or less), and more realistic problems, (2) adding new or
improved sub-models of critical physical processes to ISMs, and (3) applying
ISMs as partially or fully coupled components of ESMs. The first two
challenges relate to maintaining adequate performance and robustness, as
increased resolution and/or complexity have the potential to increase forward
model cost and/or degrade solver reliability. The latter challenge relates to
the added complexity and cost associated with optimization workflows, which
are necessary for obtaining model initial conditions that are realistic and
compatible with forcing from ESMs. These challenges argue for ISM development
that specifically targets the following model features and capabilities:
parallel, scalable, and robust linear and nonlinear solvers;
variable and/or adaptive mesh resolution;
computational kernels based on flexible programming models to allow for
implementation on a range of high-performance computing (HPC) architectures;
and
For example, traditional CPU-only architectures and MPI programming
models versus CPU + GPU, hybrid architectures using MPI for nodal
communication, and OpenMP or CUDA for on-node parallelism.
automatic differentiation capability for the computation of adjoint
sensitivities to be used in high-dimensional parameter field optimization and
uncertainty quantification.
Based on these considerations, we have developed a new land ice model,
MALI, which is composed of three major components: (1) model framework,
(2) dynamical cores for solving equations of conservation of momentum, mass,
and energy, and (3) modules for additional model physics. The model leverages
existing and mature frameworks and libraries, namely the Model for Prediction
Across Scales (MPAS) framework and the Albany and Trilinos solver libraries.
These have allowed us to take into consideration and address, from the start,
many of the challenges discussed above. We discuss each of these components
in more detail in the following sections.
MPAS Framework
The MPAS Framework provides the foundation for a generalized geophysical
fluid dynamics model on unstructured spherical and planar meshes. On top of
the framework, implementations specific to the modeling of a particular
physical system (e.g., land ice, ocean) are created as MPAS cores.
To date, MPAS cores for atmosphere , ocean
, shallow water
, sea ice , and land ice have been
implemented. At the moment the land ice model is limited to planar meshes due
to the planar formulation of the flow models; however, we have an
experimental implementation of the flow model for spherical coordinates that
enables runs on spherical meshes. The MPAS design philosophy is to leverage
the efforts of developers from the various MPAS cores to provide common
framework functionality with minimal effort, allowing MPAS core developers to
focus on the development of the physics and features relevant to their
application.
The framework code includes shared modules for fundamental model operation.
Significant capabilities include the following.
Description of model data types. MPAS uses a handful of
fundamental Fortran-derived types for basic model functionality. Model
variables specific to an MPAS core are handled through custom groupings of
model fields called pools, for which custom accessor routines exist.
Core-specific variables are easily defined in XML syntax in a
registry, and the framework parses the registry, defines variables,
and allocates memory as needed.
Description of the mesh specification. MPAS requires 36 fields
to fully describe the mesh used in a simulation. These include the position,
area, orientation, and connectivity of all cells, edges, and vertices in the
mesh. The mesh specification can flexibly describe both spherical and planar
meshes. More details are provided in the next section.
Distributed memory parallelization and domain decomposition.
The MPAS Framework provides needed routines for exchanging information
between processors in a parallel environment using a Message-Passing Interface
(MPI). This includes halo updates, global reductions, and global broadcasts.
MPAS also supports decomposing multiple domain blocks on each processor to,
for example, optimize model performance by minimizing the transfer of data from
disk to memory. Shared memory parallelization through OpenMP is also
supported, but the implementation is left up to each MPAS core.
Parallel input and output capabilities. MPAS performs parallel
input and output of data from and to disk through the commonly used libraries
of NetCDF, Parallel NetCDF (pnetcdf), and Parallel Input/Output (PIO)
. The registry definitions control which fields can
be input and/or output, and a framework streams functionality
provides easy run-time configuration of what fields are to be written to what
file name and at what frequency through an XML streams file. The MPAS Framework includes additional functionality specific to providing a flexible
model restart capability.
Advanced timekeeping. MPAS uses a customized version of the timekeeping
functionality of the Earth System Modeling Framework (ESMF), which includes a
robust set of time and calendar tools used by many Earth system models
(ESMs). This allows for the explicit definition of model epochs in terms of years,
months, days, hours, minutes, seconds, and fractional seconds and can be set
to three different calendar types: Gregorian, Gregorian no leap, and 360 day.
This flexibility helps enable multi-scale physics and simplifies coupling to
ESMs. To manage the complex date–time types that ensue, the MPAS Framework
provides routines for the arithmetic of time intervals and the definition of
alarm objects for handling events (e.g., when to write output, when the
simulation should end).
Run-time-configurable control of model options.
Model options are configured through namelist files that use
standard Fortran namelist file format, and input/output is configured
through streams files that use XML format. Both are completely
adjustable at run time.
Online run-time analysis framework. See Sect. for examples.
Additionally, a number of shared operators exist to perform common operations
on model data. These include geometric operations (e.g., length, area, and
angle operations on the sphere or the plane), interpolation (linear,
barycentric, Wachspress, radial basis functions, spline), vector and tensor
operations (e.g., cross products, divergence), and vector reconstruction
(e.g., interpolating from cell edges to cell centers). Most operators work on
both spherical and planar meshes.
Model meshes
The MPAS mesh specification is general enough to describe unstructured meshes
on most two-dimensional manifold spaces; however, most applications use
centroidal Voronoi tessellations on a sphere or plane. This
paper focuses on applications with planar centroidal Voronoi meshes,
with some additional consideration of spherical centroidal Voronoi meshes.
Voronoi meshes are constructed by specifying a set of generating points (cell
centers) and then partitioning the domain into cells that contain all points
closer to each generating point than any other. Edges of Voronoi cells are
equidistant between neighboring cell centers and perpendicular to the line
connecting those cell centers. A planar Voronoi tessellation is the dual graph
of a Delaunay triangulation, which is a triangulation of points in which the
circumcircle of every triangle contains no points in the point set. Voronoi
meshes that are centroidal (the Voronoi generator is also the center of mass
of the cell) have favorable properties for some geophysical fluid dynamic
applications and maintain high-quality cells because
cells tend towards equi-dimensional aspect ratios, and mesh resolution (where
nonuniform) changes smoothly. On both planes and spheres, Voronoi
tessellations tend toward perfect hexagons as resolution is increased. Note
that while the MPAS mesh specification supports quadrilateral grids, such as
traditional rectangular grids, they are described as unstructured, which
introduces significant overhead in memory and calculation over regular
rectangular grid approaches.
Because MPAS meshes are two-dimensional manifold spaces, they are convenient
for describing geophysical locations, either on planar projections or
directly on a sphere. Because they are unstructured, meshes can contain
varying mesh resolution and can be culled to only retain regions of interest.
Planar meshes can easily be made periodic by taking advantage of the
unstructured mesh specification and, for most operations, periodic cell
relationships are handled the same as for neighboring cell relationships.
MPAS meshes are static in time. The vertical coordinate, if needed by an MPAS
core, is extruded from the base horizontal mesh. Each MPAS core chooses its
own vertical coordinate system. A comprehensive suite of tools for the
generation of centroidal Voronoi tessellations on a plane or sphere has been
developed, as have tools for modifying existing meshes (e.g., removing
unneeded cells, coordinate transformations, etc.) and converting some common
unstructured mesh formats (e.g., Triangle; ) to
the MPAS specification.
The basic unit of the MPAS mesh specification is the cell. A cell
has area and is formed by three or more sides, which are referred to as
edges. The end points of edges are defined by vertices.
Figure illustrates the relationships between these mesh
primitives. The MPAS mesh specification utilizes 36 fields that describe the
position, orientation, area, and connectivity of the various primitives. Only
four of these fields (x, y, z cell positions and connectivity between
cells) are necessary to describe any mesh, but the larger set of fields in
the mesh specification provides information that is commonly used for routine
operations. This avoids the need for the model to calculate these fields
internally, speeding up the process of model initialization and integration.
MALI typically uses centroidal Voronoi meshes on a plane. Spherical Voronoi
meshes can also be used, but little work has been done with such meshes to
date. MALI employs a C-grid discretization for advection,
meaning state variables (ice thickness and tracer values) are located at
Voronoi cell centers, and flow variables (transport velocity,
un) are located at cell edge midpoints
(Fig. ). MALI uses a sigma vertical coordinate
(specified number of layers, each with a spatially uniform layer thickness
fraction; see , for more information):
σ=s-zH,
where s is surface elevation, H is ice thickness, and z is the vertical
coordinate.
A set of tools supporting the MPAS Framework includes tools for generating
uniform and variable-resolution centroidal Voronoi meshes. Additionally, the
JIGSAW(GEO) mesh-generation tool can be
used to efficiently generate high-quality, variable-resolution meshes with
data-based density functions. Density functions that are a function of
observed ice velocity or its spatial derivatives and/or distance to the
existing or potential future grounding line position have been used.
MALI grids. (a) Horizontal grid with cell center (blue
circles), edge midpoint (red triangles), and vertices (orange squares)
identified for the center cell. Scalar fields (H, T) are located at cell
centers. Advective velocities (un) and fluxes are located at
cell edges. (b) Vertical grid with layer midpoints (blue circles)
and layer interfaces (red triangles) identified. Scalar fields (H, T) are
located at layer midpoints. Fluxes are located at layer interfaces.
The Albany software library
Albany is an open source, C++ multi-physics code base for the solution and
analysis of coupled systems of partial differential equations (PDEs)
. It is a finite-element code that can (in three spatial
dimensions) employ unstructured meshed comprised of hexahedral, tetrahedral,
or prismatic elements. Albany is designed to take advantage of the
computational mathematics tools available within the Trilinos suite of
software libraries and it uses template-based generic
programming methods to provide extensibility and flexibility
. Together, Albany and Trilinos provide parallel data
structures and I/O, discretization and integration algorithms, linear solvers
and preconditioners, nonlinear solvers, continuation algorithms, and tools
for automatic differentiation (AD) and optimization. By formulating a system
of equations in the residual form, Albany employs AD to automatically compute
the Jacobian of the discrete PDE residual, as well as forward and adjoint
sensitivities. Albany can solve large-scale PDE-constrained optimization
problems using the Trilinos optimization package ROL, and it provides
uncertainty quantification capabilities through the Dakota framework
. It is a massively parallel code by design and recently it
has been adopting the Kokkos programming model to provide
many-core performance portability on major HPC platforms.
Albany provides several applications including LCM (Laboratory for
Computational Mechanics) for solid mechanics problems, QCAD (Quantum Computer
Aided Design) for quantum device modeling, and LI (Land Ice) for modeling ice
sheet flow. We refer to the code that discretizes these diagnostic momentum
balance equations as Albany-LI. Albany-LI was formerly known as Albany/FELIX
(Finite Elements for Land Ice eXperiments) and is described by
and under that name. Here,
these tools are brought to bear on the most complex, expensive, and fragile
portion of the ice sheet model, the solution of the momentum balance
equations (discussed further below).
Conservation equations
The “dynamical core” of the MALI ice sheet model solves the governing
equations expressing the conservation of momentum, mass, and energy.
Conservation of momentum
Treating glacier ice as an
incompressible fluid in a low-Reynolds-number flow, the conservation of
momentum in a Cartesian reference frame is expressed by the Stokes flow
equations, for which the gravitational driving stress is balanced by
gradients in the viscous stress tensor, σij:
∂σij∂xj+ρgi=0,i,j=1,2,3,
where xi is the coordinate vector, ρ is the density of ice, and
g is acceleration due to gravity
In Eq. () and
elsewhere we use indicial notation, with summation over repeat indices.
.
Deformation results from the deviatoric stress, τij, which relates
to the full stress tensor as
τij=σij-13σkkδij,
for which -13σkk is the mean compressive
stress and δij is the Kronecker delta (or the identity tensor).
Stress and strain rate are related through the constitutive relation,
τij=2μeϵ˙ij,
where ϵij˙ is the strain rate tensor and μe is
the “effective” non-Newtonian ice viscosity given by Nye's generalization
of Glen's flow law :
μe=γA-1nϵ˙e1-nn.
In Eq. (), A is a temperature-dependent rate factor, n
is an exponent commonly taken as 3 for polycrystalline glacier ice, and
γ is an ice “stiffness” factor (inverse enhancement factor related
to the commonly used enhancement factor Ef by γ=Ef-1n) used to account for other impacts on ice rheology,
such as impurities or crystal anisotropy (see also Sect.
). The effective strain rate ϵ˙e is
given by the second invariant of the strain rate tensor,
ϵ˙e=12ϵ˙ijϵ˙ij12.
The strain rate tensor is defined by gradients in the components of the ice
velocity vector ui:
ϵ˙ij=12∂ui∂xj+∂uj∂xi,i,j=1,2,3.
Finally, the rate factor A follows an Arrhenius relationship,
AT*=Aoe-Qa/RT*,
in which Ao is a constant, T* is the temperature (relative to the
pressure melting point), Qa is the activation energy for crystal
creep, and R is the gas constant.
Boundary conditions required for the solution of Eq. () depend
on the form of reduced-order approximation applied and are discussed further
below.
Reduced-order equations
Ice sheet models solve Eqs. ()–() with varying
degrees of complexity in terms of the tensor components in
Eqs. ()–() that are accounted for or
omitted based on geometric scaling arguments. Because ice sheets are
inherently “thin” – their widths are several orders of magnitude larger
than their thickness – reduced-order approximations of the full momentum
balance are often appropriate see, e.g.,
and, importantly, can often result in considerable computational cost
savings. Here, we employ two such approximations, a first-order-accurate
“Blatter–Pattyn” approximation and a zero-order “shallow ice
approximation” as described in more detail in the following sections.
First-order velocity solver and coupling
Ice sheets typically have a small aspect ratio and small surface and bed
slopes. These characteristics imply that reduced-order approximations of the
Stokes momentum balance may apply over large areas of the ice sheets,
potentially allowing for significant computational savings. Formal
derivations involve nondimensionalizing the Stokes momentum balance and
introducing a geometric scaling factor, δ=H/L, where H and L
represent characteristic vertical and horizontal length scales (often taken
as the ice thickness and the ice sheet span), respectively. Upon conducting
an asymptotic expansion, reduced-order models with a chosen degree of
accuracy (relative to the original Stokes flow equations) can be derived by
retaining terms of the appropriate order in δ. For example, the
first-order-accurate Stokes approximation is arrived at by retaining terms of
O(δ1) and lower (the reader is referred to
, and , for additional
discussion
In practice, additional scaling parameters describing the
ratio of deformation to sliding velocity may also be introduced.
).
Using the notation of and
Vectors and tensors are given in bold rather than using indices.
Note that, in a slight abuse of notation, we have switched from using x1,
x2, x3 to denote the three coordinate directions to x, y, z.
,
the first-order-accurate Stokes approximation also referred to as the
Blatter–Pattyn approximation; see is expressed
through the following system of PDEs:
-∇⋅(2μeϵ˙1)+ρg∂s∂x=0,-∇⋅(2μeϵ˙2)+ρg∂s∂y=0,
where ∇⋅ is the divergence operator, s≡s(x,y) represents
the ice sheet upper surface, and the vectors ϵ˙1 and
ϵ˙2 are given by
ϵ˙1=2ϵ˙xx+ϵ˙yy,ϵ˙xy,ϵ˙xzT
and
ϵ˙2=ϵ˙xy,ϵ˙xx+2ϵ˙yy,ϵ˙yzT.
Akin to Eqs. () and (), μe in
Eq. () represents the effective viscosity but for the case
of the first-order stress balance with an effective strain rate given by
ϵ˙e≡ϵ˙xx2+ϵ˙yy2+ϵ˙xxϵ˙yy+ϵ˙xy2+ϵ˙xz2+ϵ˙yz212,
rather than by Eq. (), and with individual strain rate
terms given by
13ϵ˙xx=∂u∂x,ϵ˙yy=∂v∂y,ϵ˙xy=12∂u∂y+∂v∂x,ϵ˙xz=12∂u∂z,ϵ˙yz=12∂v∂z.
At the upper surface, a stress-free boundary condition is applied,
ϵ˙1⋅n=ϵ˙2⋅n=0,
with n the outward normal vector at the ice sheet surface,
z=s(x,y). At the bed, z=b(x,y), we apply no slip or continuity of basal
tractions (“sliding”):
u=v=0,no slip2μeϵ˙1⋅n+βum=0,sliding,2μϵ˙2⋅n+βvm=0,
where β is a linear friction parameter and m≥1. In most
applications we set m=1 (see also Sect. ).
On lateral boundaries, a stress boundary condition is applied,
2μeϵ˙1⋅n,ϵ˙2⋅n,0T-ρg(s-z)n=ρogmax(z,0)n,
where ρo is the density of ocean water and n the outward
normal vector to the lateral boundary (i.e., parallel to the (x,y) plane)
so that lateral boundaries above sea level are effectively stress free and
lateral boundaries submerged within the ocean experience hydrostatic pressure
due to the overlying column of ocean water.
We solve these equations using the Albany-LI momentum balance solver, which
is built using the Albany and Trilinos software libraries discussed above.
The mathematical formulation, discretization, solution methods, verification,
and scaling of Albany-LI are discussed in detail in .
Albany-LI implements a classic finite-element discretization of the
first-order approximation. At the grounding line, the basal friction
coefficient β can abruptly drop to zero within an element of the mesh.
This discontinuity is resolved by using a higher-order Gauss quadrature rule
on elements containing the grounding line, which corresponds to the
sub-element parameterization SEP3 proposed in .
Additional exploration of solver scalability and demonstrations of solver
robustness on large-scale, high-resolution, realistic problems are discussed
in . The efficiency and robustness of the nonlinear
solvers are achieved using a combination of the Newton method (damped with a
line search strategy when needed) and a parameter continuation algorithm
for the numerical regularization of the viscosity. The scalability of the
linear solvers is obtained using a multilevel preconditioner (see
) specifically designed to target shallow problems
characterized by meshes extruded in the vertical dimension, like those found
in ice sheet modeling. The preconditioner has been demonstrated to be
particularly effective and robust even in the presence of ice shelves that
typically lead to highly ill-conditioned linear systems.
Correspondence between the MPAS Voronoi tessellation and its dual
Delaunay triangulation used by Albany. Key MALI variables that are
natively found at each location are listed. Note that variables are
interpolated from one location to another as required for various
calculations.
The Albany-LI first-order velocity solver written in C++ is coupled to MPAS
written in Fortran using an interface layer. Albany uses a three-dimensional
mesh extruded from a basal triangulation and composed of prisms or tetrahedra
(see ). When coupled to MPAS, the basal triangulation is
part of the Delaunay triangulation, dual to an MPAS Voronoi mesh, that
contains active ice and is generated by the interface. The bed topography, ice
lower surface, ice thickness, basal friction coefficient (β), and
three-dimensional ice temperature, all at cell centers
(Table ), are passed from MPAS to Albany. Optionally,
Dirichlet velocity boundary conditions can also be passed. After the velocity
solve is complete, Albany returns the x and y components of velocity at
each cell center and layer interface, the normal component of velocity at
each cell edge and layer interface, and viscous dissipation at each cell
vertex and layer midpoint.
The interface code defines the lateral boundary conditions on the finite-element mesh that Albany will use. Lateral boundaries in Albany are applied
at cell centers (triangle nodes) that do not contain dynamic ice on the MPAS
mesh and that are adjacent to the last cell of the MPAS mesh that does
contain dynamic ice. This one element extension is required to support
the calculation of normal velocity on edges (un) required for
the advection of ice out of the final cell containing dynamic ice
(Fig. ). The interface identifies three types of lateral
boundaries for the first-order velocity solve: terrestrial, floating marine,
and grounded marine. Terrestrial margins are defined by bed topography above
sea level. At these boundary nodes, ice thickness is set to a small ice
minimum thickness value (ϵ=1 m). Floating marine margin triangle
nodes are defined as neighboring one or more triangle edges that satisfy the
hydrostatic floatation criterion. At these boundary nodes, we need to ensure
the existence of a realistic calving front geometry, so we set ice thickness
to the minimum of thickness at neighboring cells with ice. Grounded marine
margins are defined as locations where the bed topography is below sea level,
but no adjacent triangle edges satisfy the floatation criterion. At these
boundary nodes, we apply a small floating extension with thickness
ϵ. For all three boundary types, ice temperature is averaged from
the neighboring locations containing ice.
Correspondence between MPAS and Albany meshes and the application of
boundary conditions for the first-order velocity solver. Solid black lines
are cells on the Voronoi mesh and dashed gray lines are triangles on the
Delaunay triangulation. Light blue Voronoi cells contain dynamic ice and gray
cells do not. Dark blue circles are Albany triangle nodes that use variable
values directly from the colocated MPAS cell centers. White circles are
extended node locations that receive variable values as described in the text
based on whether they are terrestrial, floating marine, or grounded marine
locations. Red triangles indicate Voronoi cell edges on which velocities
(un) are required for advection.
Shallow ice approximation velocity solver
A similar procedure to that described above for the first-order-accurate
Stokes approximation can be used to derive the so-called “shallow ice
approximation” (SIA) ,
in this case by retaining only terms of O(δ0). In the case
of the SIA, the local gravitational driving stress is everywhere balanced by
the local basal traction, and the horizontal velocity as a function of depth
is simply the superposition of the local basal sliding velocity and the
integral of the vertical shear from the ice base to that depth:
u=-2(ρg)n∫bzA(s-z)ndz|∇s|n-1∇s+ub,
where b is the bed elevation and ub is the sliding
velocity.
SIA ice sheet models typically combine the momentum and mass balance
equations to evolve the ice geometry directly in the form of a
depth-integrated, two-dimensional diffusion problem
. However, we implement the SIA as an
explicit velocity solver that can be enabled in place of the more accurate
first-order solver, while keeping the rest of the model identical. The
purpose of the SIA velocity solver is primarily for rapid testing, so the
less efficient explicit implementation of Eq. () is not a
concern.
We implement Eq. () in sigma coordinates on cell edges for which we
only require the normal component of velocity, un:
un=-2(ρg)nHn+1|∇s|n-1dsdxn∫1σAσndσ+ubn,
where xn is the normal direction to a given edge and
ubn is sliding velocity in the normal direction to the
edge. We average A and H from cell centers to cell edges.
dsdxn is calculated as the difference in
surface elevation between the two cells that neighbor a given edge divided by
the distance between the cell centers; on a Voronoi grid, cells edges are
midway between cell centers by definition. The surface slope component
tangent to an edge (required to complete the calculation of ∇s) is
calculated by first interpolating surface elevation from cell centers to
vertices.
Conservation of mass
Ice sheet mass transport and evolution is conducted using the principle of
conservation of mass. Assuming constant density to write the conservation of mass
in volume form, the equation relates ice thickness change to the divergence
of mass and sources and sinks:
∂H∂t+∇⋅Hu‾=a˙+b˙,
where H is ice thickness, t is time, u‾ is
depth-averaged velocity, a˙ is surface mass balance, and b˙ is
basal mass balance. Both a˙ and b˙ are positive for ablation
and negative for accumulation.
Equation () is used to update thickness in each grid cell on
each time step using a forward Euler, fully explicit time evolution scheme.
Eq. () is implemented using a finite-volume method such that
fluxes are calculated for each edge of each cell to calculate ∇⋅Hu‾. Specifically, we use a first-order upwind method that
applies the normal velocity on each edge (un) and an upwind
value of cell-centered ice thickness. Note that with the Blatter–Pattyn velocity
solver, normal velocity is interpolated from cell centers to edges using the
finite-element basis functions in Albany. In the shallow ice approximation
velocity solver, normal velocity is calculated natively at edges. The MPAS Framework includes a higher-order flux-corrected transport scheme
for which we have performed some initial testing, but is
not routinely used in MALI at this time.
Tracers are advected horizontally layer by layer with a similar equation:
∂Qtl∂t+∇⋅Qtlu‾=S˙,
where Qt is a tracer quantity (e.g., temperature; see below), l
is layer thickness, and S˙ represents any tracer sources or sinks.
While any number of tracers can be included in the model, the only one to be
considered here is temperature due to its important effect on ice rheology
through Eq. () and will be discussed further in the following
section.
Vertical advection of tracers is included through a vertical remapping
operation. On the upper and lower domain boundaries, the grid moves to follow
the material, and in the interior we maintain fixed layer fractions that need
to be updated on each time step after Eqs. ()
and () are applied. The model does not explicitly calculate
vertical velocity, but the appropriate vertical transport of tracers occurs
during this vertical remapping operation. We employ a first-order vertical
remapping method. Overlaps between the newly calculated layers and the target
sigma layers are calculated for each grid cell. Assuming uniform values
within each layer, mass, energy, and other tracers are transferred between
layers based on these overlaps to restore the prescribed sigma layers while
conserving mass and energy.
Conservation of energy
Conservation of energy within glaciers can be formulated in terms of
temperature or enthalpy (internal energy) .
The enthalpy formulation has the advantage of eliminating the need for
tracking the cold–temperate transition surface, as both cold (below the
pressure melting point) and temperate (at the pressure melting point) ice
regions are handled with the same equations. MALI includes both temperature
and enthalpy formulations. In both cases, an operator splitting technique is
used. At each time step, an implicit vertical solve accounting for the
diffusion and dissipation terms (described below) is performed, followed by
explicit advection of the resulting temperature or enthalpy field (described
above in Sect. ). We describe the temperature formulation
in detail, followed by a briefer description of the enthalpy formulation that
uses a similar procedure. Note that the thermal model described here shares a
common lineage with that of the Community Ice Sheet Model, and parts of the
description below are therefore similar to the documentation of the thermal
solver in the Community Ice Sheet Model .
Temperature formulation
Conservation of energy can be expressed in terms of temperature through the
three-dimensional, advective–diffusive heat equation:
∂T∂t=1ρc∂∂xik∂T∂xi-ui∂T∂xi+Φρc,
with thermal conductivity k and heat capacity c. In
Eq. (), the rate of temperature change (left-hand side)
is balanced by diffusive, advective, and internal (viscous dissipation; see
Eq. for Φ) source terms (first, second, and third
terms on the right-hand side, respectively). In MALI we solve an
approximation of Eq. (),
∂T∂t=kρc∂2T∂z2-ui∂T∂xi+Φρc,
in which horizontal diffusion is assumed negligible p. 280
and k is assumed constant and uniform. The viscous dissipation term Φ
is discussed further below.
Temperatures are staggered in the vertical relative to velocities and are
located at the centers of nz-1 vertical layers, which are bounded by
nz vertical levels (grid point locations). This convention allows for
conservative temperature advection, since the total internal energy in a
column (the sum of ρcTΔz over nz-1 layers) is conserved
under transport. The upper surface temperature Ts and the lower
surface temperature Tb, coincident with the surface and bed grid
points, give a total of nz+1 temperature values within each column.
As mentioned above, Eq. () is solved by first
performing an implicit vertical solve accounting for the diffusion and
dissipation terms (described below), followed by explicit advection of the
resulting temperature field. The method for evolving ice temperature and
default parameter value choices are adapted from the implementation in the
Community Ice Sheet Model , which is in turn
based on the Glimmer model . The choice of constant k with
a temperate ice value (Table ) will lead to
underestimation of conduction in cold ice. Relaxation of this assumption is
planned for future releases of MALI.
Vertical diffusion
Using a “sigma” vertical coordinate, the vertical diffusion portion of
Eq. () can be discretized as
∂2T∂z2=1H2∂2T∂σ2.
In σ coordinates, the central difference formulas for first partial
derivatives at the upper and lower interfaces of layer k are
∂T∂σσk=Tk-Tk-1σ̃k-σ̃k-1,∂T∂σσk+1=Tk+1-Tkσ̃k+1-σ̃k,
where σ̃k is the value of σ at the midpoint of layer
k, halfway between σk and σk+1. The second partial
derivative, defined at the midpoint of layer k, is then given by
∂2T∂σ2σ̃k=∂T∂σσk+1-∂T∂σσkσk+1-σk.
By inserting Eq. () into
Eq. (), we obtain the discrete form of the
vertical diffusion term in Eq. ():
26∂2T∂σ2σ̃k=Tk-1σ̃k-σ̃k-1σk+1-σk-Tk1σ̃k-σ̃k-1σk+1-σk+1σ̃k+1-σ̃kσk+1-σk+Tk+1σ̃k+1-σ̃kσk+1-σk.
To simplify some expressions below, we define the following coefficients
associated with the vertical temperature diffusion:
27ak=1σ̃k-σ̃k-1σk+1-σk,bk=1σ̃k+1-σ̃kσk+1-σk.
Viscous dissipation
The source term from viscous dissipation in Eq. () is
given by the product of the stress and strain rate tensors:
Φ=σijϵ˙ij=τijϵ˙ij.
The change to deviatoric stress on the right-hand side of
Eq. () follows from terms related to the mean
compressive stress (or pressure) dropping out due to incompressibility.
Analogous to the effective strain rate given in Eq. (), the
effective deviatoric stress is given by
τe=12τijτij12,
which can be combined with Eqs. ()
and () to derive an expression for the viscous dissipation
in terms of effective deviatoric stress and strain:
Φ=2τeϵ˙e.
Finally, an analog to Eq. () gives
τe=2μeϵ˙e,
which can be used to eliminate ϵ˙e in
Eq. () and arrive at an alternate expression for the
dissipation based on only two scalar quantities:
Φ=4μeϵ˙e2.
The viscous dissipation source term is computed within Albany-LI at MPAS cell
vertices and then reconstructed at cell centers in MPAS.
For the SIA model, dissipation can be calculated in sigma coordinates as
Φ(σ)=σgc∂u∂σ⋅∇s,
which can be combined with Eq. () to make
Φ(σ)=-2σgcρ(gσρ)n+1(H|∇s|)n+1A.
We calculate Φ on cell edges following the procedure described for
Eq. () and then interpolate Φ back to cell centers to
solve Eq. ().
Vertical temperature solution
The vertical diffusion portion of Eq. () is
discretized according to
35Tkn+1-TknΔt=kρcH2akTk-1n+1-(ak+bk)Tkn+1+bkTk+1n+1+Φkρc,
where ak and bk are defined in Eq. (), n is the current
time level, and n+1 is the new time level. Because the vertical diffusion
terms are evaluated at the new time level, the discretization is
backward Euler (fully implicit) in time.
The temperature T0 at the upper boundary is set to
min(Tair,0), where the mean annual surface air temperature
Tair is a two-dimensional field specified from observations or
climate model output.
At the lower boundary, for grounded ice there are three potential heat
sources and sinks: (1) the diffusive flux from the bottom surface to the ice
interior (positive up),
Fdbot=kHTnz-Tnz-11-σ̃nz-1;
(2) the geothermal flux Fg prescribed from a spatially variable
input file (based on observations); and (3) the frictional heat flux
associated with basal sliding,
Ff=τb⋅ub,
where τb and ub are 2-D bed-parallel vectors of basal
shear stress and basal velocity, respectively, and the friction law from
Eq. () becomes
Ff=β|ub|2.
If the basal temperature Tnz<Tpmp (where
Tpmp is the pressure melting point temperature), then the
fluxes at the lower boundary must balance,
Fg+Ff=Fdbot,
so that the energy supplied by geothermal heating and sliding friction is
equal to the energy removed by vertical diffusion. If, on the other hand,
Tnz=Tpmp, then the net flux is nonzero and is used to
melt or freeze ice at the boundary:
Mb=Fg+Ff-FdbotρL,
where Mb is the melt rate and L is the latent heat of melting.
Melting generates basal water, which may either be stored at the bed locally,
serve as a source for the basal hydrology model (see
Sect. ), or may simply be ignored. If basal water is
present locally, Tnz is held at Tpmp.
For floating ice the basal boundary condition is simpler: Tnz is
simply set to the freezing temperature Tf of seawater. Optionally,
a melt rate can be prescribed at the lower surface.
Rarely, the solution for T may exceed Tpmp for a given
internal layer. In this case, T is set to Tpmp, excess energy
goes towards the melting of ice internally, and the resulting melt is assumed to
drain to the bed immediately.
If Eq. () applies, we compute Mb and adjust the
basal water depth. When the basal water goes to zero, Tnz is set to
the temperature of the lowest layer (less than Tpmp at the bed)
and flux boundary conditions apply during the next time step.
Temperature advection
Temperature advection in any individual layer k is treated using tracer
advection, as in Eq. () above, where the ice temperature
Tk is substituted for the generic tracer Q. After horizontal transport,
the surface and basal mass balance is applied to the top and bottom ice
surfaces, respectively. Because layer transport and the application of mass
balance terms results in an altered vertical layer spacing with respect to
σ coordinates, a vertical remapping scheme is applied to provide the
necessary vertical advection of temperature. This conservatively transfers
ice volume and internal energy between adjacent layers while restoring
σ layers to their initial distribution. Internal energy divided by
mass gives the new layer temperatures.
Enthalpy formulation
The specific enthalpy (internal energy), E, in ice sheets and glaciers can
be expressed as a combination of ice temperature (T) and liquid water
fraction (water content; ω) :
E=cT-Tref,E≤EpmpEpmp+ωL,E≥Epmp,
where Tref is the reference temperature, c is the heat capacity
of ice, L is the latent heat of fusion, and Epmp is the
specific enthalpy at the pressure melting point for different vertical
locations (Tpmp(z)) defined as
Epmp=cTpmp(z)-Tref.
The balance equation for enthalpy reads
∂E∂t=∂∂xiK∂E∂xi-ui∂E∂xi+Φ,
where K is the diffusivity of ice defined differently in cold and
temperate ice:
K=kρc,E<Epmpνρ,E≥Epmp,
where ν is the water diffusivity in temperate ice, which is generally
taken as an empirical small number due to a lack of knowledge
.
The implementation of the enthalpy model follows that of the temperature
model. Vertical diffusion is as described above but replacing T with E.
Viscous dissipation remains unchanged. Boundary conditions in the vertical
enthalpy solution follow those applied for the temperature formulation above,
but cast in terms of E. Advection is as described above for temperature but
replacing T with E. Verification of the enthalpy model is described below
in Sect. .
Additional model physics
Additional physical processes currently implemented in MALI are a
mass-conserving subglacial hydrology model and a small number of basic
schemes for iceberg calving. These are described in more detail below.
Subglacial hydrology
Sliding of glaciers and ice sheets over their bed can increase ice velocity
by orders of magnitude and is the primary control on ice flux to the oceans.
The state of the subglacial hydrologic system is the primary control on
sliding , and ice sheet modelers
have therefore emphasized subglacial hydrology and its effects on basal
sliding as a critical missing piece of current ice sheet models
.
MALI includes a mass-conserving model of subglacial hydrology that includes
representations of water storage in till, distributed drainage,
and channelized drainage and is coupled to ice dynamics. The model is based
on the model of but modified for MALI's unstructured horizontal
grid and with an additional component for channelized drainage. While the implementation follows closely that of , the
model and equations are summarized here along with a description of the
features unique to the application in MALI.
Till
The simple till component represents local storage of water in subglacial
till without horizontal transport within the till. The evolution of the effective
water depth in till, Wtill, is therefore a balance of the delivery of
meltwater, mb, to the till, the drainage of water out of the till at
rate Cd (mass leaving the subglacial hydrologic system, for
example, to deep groundwater storage), and overflow to the distributed
drainage system, γ:
∂Wtill∂t=mbρw-Cd-γt.
In the model, meltwater (from either the bed or drained from the surface) is
first delivered to the till component. Water in excess of the maximum
storage capacity of the till, Wtillmax, is instantaneously
transferred as a source term to the distributed drainage system through the
γt term.
Distributed drainage
The distributed drainage component is implemented as a “macroporous sheet”
that represents bulk flow through linked cavities that form in the lee of
bedrock bumps as the glacier slides over the bed
. Water flow in the system is
driven by the gradient of the hydropotential, ϕ, defined as
ϕ=ρwgzb+Pw,
where Pw is the water pressure in the distributed drainage system.
A related variable, the ice effective pressure, N, is the difference
between ice overburden pressure and water pressure in the distributed
drainage system, Pw:
N=ρgH-Pw.
The evolution of the area-averaged cavity space is a balance of the opening of
cavity space by the glacier sliding over bedrock bumps and closing through
creep of the ice above. The model uses the common assumption
e.g., that cavities
always remain water filled cf., so cavity space can be
represented by the effective water depth in the macroporous sheet, W:
∂W∂t=cs|ub|(Wr-W)-ccdAbN3W,
where cs is bed roughness parameter, Wr is the maximum bed bump
height, ccd is a creep scaling parameter representing geometric and
possibly other effects, and Ab is the ice flow parameter of the
basal ice.
Water flow in the distributed drainage system, q, is driven by the
hydropotential gradient and is described by a general power law:
q=-kqWα1|∇ϕ|α2-2∇ϕ,
where kq is a conductivity coefficient. The α1 and α2
exponents can be adjusted so that Eq. () reduces to commonly used
water flow relations, such as Darcy flow, the Darcy–Weisbach relation, and
the Manning equation.
Channelized drainage
The inclusion of channelized drainage in MALI is an extension to the model of
. The distributed drainage model ignores dissipative
heating within the water, which in the real world leads to the melting of the ice
roof and the formation of discrete, efficient channels melted into the ice
above when the distributed discharge reaches a critical threshold
. These channels can
rapidly evacuate water from the distributed drainage system and lower water
pressure, even under sustained meltwater input
.
The implementation of channels follows the channel network models of
and . The evolution of channel area,
S, is a balance of opening and closing processes as in the distributed
system, but in channels the opening mechanism is melting caused by
dissipative heating of the ice above:
dSdt=1ρL(Ξ-Π)-cccAbN3S,
where ccc is the creep scaling parameter for channels.
The channel opening rate, the first term in Eq. (), is
itself a balance of the dissipation of potential energy, Ξ, and sensible heat
change of water, Π, due to changes in the pressure-dependent melt
temperature. Dissipation of potential energy includes energy produced by flow
in both the channel itself and a small region of the distributed system along
the channel:
Ξ=dϕdsQ+dϕdsqclc,
where s is the spatial coordinate along a channel segment, Q is the
flow rate in the channel, and qc is the flow in the
distributed drainage system parallel to the channel within a distance
lc of the channel. The term adding the contribution of dissipative
melting within the distributed drainage system near the channel is included
to represent some of the energy that has been ignored from that process in
the description of the distributed drainage system and allows channels to
form even when channel area is initially zero if discharge in the distributed
drainage system is sufficient . The term representing
the sensible heat change of the water, Π, is necessitated by the assumption
that the water always remains at the pressure-dependent melt temperature of
the water. Changes in water pressure must therefore result in melting or
freezing:
Π=-ctcwρw|Q+lcqc|dPwds,
where ct is the Clapeyron slope and cw is the specific
heat capacity of water. The pressure-dependent melt term can be disabled in
the model.
Water flow in channels, Q, mirrors Eq. ():
Q=-kQSα1|∇ϕ|α2-2∇ϕ,
where kQ is a conductivity coefficient for channels.
Drainage component coupling
Equations ()–() are coupled together by
describing the drainage system with two equations, mass conservation and
pressure evolution. Mass conservation of the subglacial drainage system is
described by
54∂W∂t+∂Wtill∂t=-∇⋅(VdW)+∇⋅(Dd∇W)-∂S∂t+∂Q∂sδ(xc)+mbρw,
where Vd is water velocity in the distributed flow, Dd is
the diffusivity of the distributed flow, and δ(xc) is the
Dirac delta function applied along the locations of the linear channels.
Combining Eqs. () and () and making the
simplification that cavities remain full at all times yields an equation for
water pressure within the distributed drainage system, Pw:
55ϕ0ρwg∂Pw∂t=-∇⋅q+cs|ub|(Wr-W)-ccdAbN3W-∂S∂t+∂Q∂sδ(xc)+mbρw-∂Wtill∂t,
where ϕ0 is an englacial porosity used to regularize the pressure
equation. Following , the porosity is only included in the
pressure equation and is excluded from the mass conservation equation.
Any of the three drainage components (till, distributed drainage, channelized
drainage) can be deactivated at run time. The most common configuration
currently used is to run with distributed drainage only.
Numerical implementation
The drainage system model is implemented using finite-volume methods on the
unstructured grid used by MALI. State variables (W, Wtill, S,
Pw) are located at cell centers and velocities and fluxes
(q, Vd, Q) are calculated at edge midpoints.
Channel segments exist along the lines joining neighboring cell centers.
Equation () is evaluated by summing tendencies from discrete
fluxes into or out of each cell. First-order upwinding is used for advection.
At land-terminating ice sheet boundaries, Pw=0 is applied as the
boundary condition. At marine-terminating ice sheet boundaries, the boundary
condition is Pw=-ρwgzb, where ρw
is ocean water density. The drainage model uses explicit forward Euler
time stepping using Eqs. (), (),
(), and (). This requires obeying
advective and diffusive Courant–Friedrichs–Lewy (CFL) conditions for
distributed drainage as described by , as well as an
additional advective CFL condition for channelized drainage if it is active.
We acknowledge that the non-continuum implementation of channels can make the
solution grid dependent, and grid convergence may therefore not exist for
many problems . However, for realistic problems with
irregular bed topography, we have found that the dominant channel location is
controlled by topography, mitigating this issue.
Coupling to ice sheet model
The subglacial drainage model is coupled to the ice dynamics model through a
basal friction law. Currently, the only option is a modified Weertman-style
power law that adds a term for effective
pressure to Eq. ():
τbi=C0Nubim,i,j=1,2,
where C0 is a friction parameter. Implementations of a Coulomb friction law
and a plastic till law
are in development. When the drainage and
ice dynamics components are run together, coupling of the systems allows for the
negative feedback described by in which elevated water
pressure increases ice sliding and increased sliding opens additional cavity
space, lowering water pressure. The meltwater source term, m, is calculated
by the thermal solver in MALI. Either or both of the ice dynamics and thermal
solvers can be disabled, in which case the relevant coupling fields can be
prescribed to the drainage model.
Verification and real-world application
To verify the implementation of the distributed drainage model, we use the
nearly exact solution described by . The problem
configuration uses distributed drainage only on a two-dimensional,
radially symmetric ice sheet of radius 22.5 km with parabolic ice sheet
thickness and a nontrivial sliding profile. showed that
this configuration allows for nearly exact reference values of W and
Pw to be solved at steady state from an ordinary differential
equation initial value problem with very high accuracy. We follow the test
protocol of and initialize the model with the near-exact
solution and then run the model forward for 1 month, after which we
evaluate model error due to drift away from the expected solution. Performing
this test with the MALI drainage model, we find error comparable to that
found by and approximately first-order convergence
(Fig. ).
Error in subglacial hydrology model for radial test case with
the near-exact solution described by for different grid
resolutions. (a) Error in water thickness; x symbols indicate
maximum error, and squares indicate mean error. Average error in water
thickness decays as O(Δx0.97). (b) Error in
water pressure, with same symbols. Average error in water pressure decays as
O(Δx1.02).
To check the model implementation of channels, we use comparisons to other
more mature drainage models through the Subglacial Hydrology Model
Intercomparison Project
(SHMIP)
https://shmip.bitbucket.io/ (last access: 10 September 2018).
. Steady-state solutions of the drainage system
effective pressure, water fluxes, and channel development for an idealized
ice sheet with varying magnitudes of meltwater input (SHMIP experiment
suites A and B) compared between MALI and other models of similar complexity
(GlaDS, Elmer) are very similar.
To demonstrate a real-world application of the subglacial hydrology model, we
perform a stand-alone subglacial hydrology simulation of the entire Antarctic
ice sheet on a uniform 20 km resolution mesh (Fig. ).
We force this simulation with basal sliding and basal melt rate after
optimizing the first-order velocity solver to surface velocity
observations (Fig. a). We then run the subglacial
hydrology model to steady state with only distributed drainage active and
using standard parameter values and prescribed ice dynamic forcing. Though
this mesh is too coarse to provide scientifically valid results, the modeled
subglacial hydrologic state is reasonable. For example, the subglacial water
flux increases down-glacier and is greatest in fast-flowing outlet glaciers
and ice streams (Fig. b), as expected from theory and
seen in other subglacial hydrology models e.g.,.
Calibrating parameters for the subglacial hydrology model and a basal
friction law and performing coupled subglacial-hydrology–ice-dynamics
simulations are beyond the scope of this paper; we merely mean to demonstrate
plausible behavior from the subglacial hydrology model for a realistic
ice-sheet-scale problem.
Subglacial
hydrology model results for the 20 km resolution Antarctic ice sheet. Demonstration of subglacial hydrology model capability using a 20 km
resolution simulation of Antarctica (too coarse resolution for scientific
validity but sufficient for demonstrating model capabilities). (a) Grounded basal ice speed calculated by the first-order velocity
solver optimized to surface velocity observations. This field and the
calculated basal melt are the forcings applied to the stand-alone subglacial
hydrology model. (b) Water flux in the distributed system calculated
by the subglacial hydrology model at steady state. (Ice dynamics is
prescribed.)
Iceberg calving
MALI includes a few simple methods for removing ice from calving fronts
during each model time step.
All floating ice is removed.
All floating ice in cells with an ocean bathymetry deeper than a specified threshold is removed.
All floating ice thinner than a specified threshold is removed.
The calving front is maintained at its initial location by adding or removing ice after thickness evolution is complete.
When ice is completely lost in a grid cell through evolution, it is replaced
with a thin layer of ice (default value of 1 m). This does not
conserve mass or energy but provides a simple way to maintain a realistic ice
shelf extent (e.g., for model spin-up).
Eigencalving scheme . Calving front retreat rate,
Cv, is proportional to the product of the principal strain rates
(ϵ1˙,ϵ2˙) if they are both extensional:Cv=K2ϵ1˙ϵ2˙forϵ1˙>0andϵ2˙>0.The eigencalving scheme can optionally also remove floating ice at the
calving front with thickness below a specified thickness threshold
. In practice we find this is necessary to prevent
the formation of tortuous ice tongues and continuous, gradual extension of some
ice shelves along the coast.
Ice that is eligible for calving can be removed immediately or fractionally
each time step based on a calving timescale. To allow ice shelves to advance
as well as retreat, we implement a simple parameterization for sub-grid
motion of the calving front by forcing floating cells adjacent to open ocean
to remain dynamically inactive until ice thickness there reaches 95 % of
the minimum thickness of all floating neighbors. This is an ad hoc
alternative to methods tracking the calving front position at sub-grid scales
. In Sect. below,
we demonstrate the eigencalving scheme applied to a realistic Antarctic ice
sheet simulation. More sophisticated calving schemes are currently under
development.
To demonstrate a real-world application of the eigencalving parameterization,
we perform a 1000-year spin-up of Antarctica with evolving velocity, geometry,
and temperature and active eigencalving (Fig. ).
For the purposes of this demonstration, we use the same uniform, 20 km
resolution mesh used in Fig. , which is too coarse to
accurately resolve grounding line dynamics (see Sect. )
and therefore should not be interpreted as a scientifically realistic
simulation. We use an initial internal ice temperature field from
and the optimization capability described below
in Sect. (note that we optimize both β and γ
in this case), along with observed surface velocities from
,
to obtain a realistic model initial state
(Fig. a). For the 1000-year spin-up, we apply
steady forcing of present-day estimates for surface mass balance and
submarine melting (, and
, respectively). For temperature boundary
conditions, we apply the steady geothermal flux field from
and the surface (2 m) air temperature field from
. We apply eigencalving calving with the K2 parameter
tuned individually for large ice shelves and a minimum calving front
thickness threshold of 100 m. This spin-up, albeit much too short to come to
full equilibrium, allows ample time for migration of the calving front and
grounding line and removes a substantial portion of the largest model
transients. It demonstrates that a tuned eigencalving parameterization is
capable of maintaining stable and realistic calving front positions in MALI
during ice sheet evolution (Fig. a, b),
consistent with its implementation in other models
.
Demonstration of eigencalving capability using a 20 km
resolution simulation of Antarctica (too coarse resolution for scientific
validity but sufficient for demonstrating model capabilities).
(a) Modeled ice extent and surface speed after optimization. The white
contour line in each plot is the grounding line. Areas colored red exceed the
maximum speed shown in the color bar. Gray areas are ice-free regions of the
computational domain. (b) Modeled ice extent and surface speed after
1000 years with evolving velocity, geometry, and temperature and active
eigencalving, plotted as in (a).
Additional capabilitiesOptimization
MALI includes an optimization capability through its coupling to the
Albany-LI momentum balance solver described in Sect. . We
provide a brief overview of this capability here, while referring to
for a complete description of the governing equations,
solution methods, and example applications. In general, our approaches are
similar to those reported for other advanced ice sheet modeling frameworks
already described in the literature
e.g.,
and we focus here primarily on optimizing the model velocity field relative
to observed surface velocities. Briefly, we consider the optimization
functional
58J(β,γ)=∫Σ12σu2us-usobs2ds+cγ2∫Σ|γ-1|2ds+Rβ(β)+Rγ(γ),
where the first term on the right-hand side (RHS) is a cost function
associated with the misfit between modeled and observed surface velocities,
the second term on the RHS is a cost function associated with the ice
stiffness factor, γ (see Eq. ), and the third and
fourth terms on the RHS are Tikhonov regularization terms given by
Rβ(β)=αβ2∫Σ|∇β|2ds,Rγ(γ)=αγ2∫Σ|∇γ|2ds.σu is an estimate for the standard deviation of the uncertainty
in the observed ice surface velocities and the parameter cγ
controls how far the ice stiffness factor is allowed to stray from unity in
order to improve the match to observed surface velocities. The regularization
parameters αβ>0 and αγ>0 control the trade-off
between a smooth β field and one with higher-frequency oscillations
(that may capture more spatial detail at the risk of over-fitting the
observations). The optimal values of αβ and αγ can
be chosen through a standard L-curve analysis. The optimization problem is
solved using the limited-memory BFGS method, as implemented in the Trilinos
package ROL
https://trilinos.org/packages/rol/ (last access: 10 September 2018).
, on the reduced-space problem. The functional
gradient is computed using the adjoint method.
An example application of the optimization capability applied to a realistic,
whole-ice-sheet problem is given below in Sect. .
present another application to the assimilation of
surface velocity time series in western Greenland.
We note that our optimization framework has been designed to be significantly
more general than implied by Eq. (). While not
applied here, we are able to introduce additional observational-based
constraints (e.g., mass balance terms) and optimize additional model
variables
(e.g., the ice thickness). These are necessary, for example, when targeting
model initial conditions that are in quasi-equilibrium with some applied
climate forcing. These capabilities are discussed in more detail in
.
Simulation analysis
As with other climate model components built using the MPAS Framework, MALI
supports the development and application of “analysis members”, which allow
for a wide range of run-time-generated simulation diagnostics and statistics
output at user-specified time intervals. Support tools included with the code
release allow for the definition of any number or combination of predefined
“geographic features” – points, lines (“transects”), or areas
(“regions”) – of interest within an MPAS mesh. Features are defined using
the standard GeoJSON format and a large existing database of
globally defined features is currently
supported
. Python-based scripts are available for
editing GeoJSON feature files, combining or splitting them, and using them to
define their coverage within MPAS mesh files. Currently, MALI includes
support for standard ice sheet model diagnostics (see
Table ) defined over the global domain (by default)
and/or over specific ice sheet drainage basins and ice shelves (or their
combination). Support for generating model output at points and along
transects will be added in the future (e.g., vertical samples at ice core
locations or along ground-penetrating radar profile lines). In
Sect. below we demonstrate the analysis capability
applied to an idealized simulation of the Antarctica ice sheet.
Standard model diagnostics available for an arbitrary number of
predefined geographic regions.
DiagnosticUnitsNet ice area and volumem2, m3Net grounded ice area and volumem2, m3Net floating ice area and volumem2, m3Net volume above floatationm3Minimum, maximum, and mean ice thicknessmNet surface mass balancekg yr-1Net basal mass balancekg yr-1Net basal mass balance for floating icekg yr-1Net basal mass balance for grounded icekg yr-1Average surface mass balancem yr-1Average basal mass balance for grounded icem yr-1Average basal mass balance for floating icem yr-1Net flux due to iceberg calvingkg yr-1Net flux across grounding lineskg yr-1Maximum surface and basal velocitym yr-1Model verification and benchmarks
MALI has been verified by a series of configurations that test different
components of the code. In some cases analytic solutions are used, but other
tests rely on intercomparison with community benchmarks that have been run
previously by many different ice sheet models.
MALI currently includes 86 automated system regression tests that run the
model for various problems with analytic solutions or community benchmarks.
In addition to checking the accuracy of model answers, some of the tests
check that model restarts give bit-for-bit exact answers with longer runs
without restart. Some others check that the model gives bit-for-bit exact
answers on different numbers of processors. All but 20 of the longer-running
tests are run every time new features are added to the code, and these tests
each also include a check for answer changes. The verification and benchmark
descriptions below are the most important examples from the larger test
suite.
Halfar analytic solution
In , Halfar described an analytic solution for
the time-evolving geometry of a radially symmetric, isothermal dome of ice on
a flat bed with no accumulation flowing under the shallow ice approximation.
This provides an obvious test of the implementation of the shallow ice
velocity calculation and thickness evolution schemes in numerical ice sheet
models and a way to assess model order of convergence
. showed that the Halfar test is
the zero-accumulation member of a family of analytic solutions, but we apply
the original Halfar test here.
(a) Root mean square error in ice thickness as a function
of grid cell spacing for the Halfar dome after 200 years shown with black
dots. The order of convergence is 0.78. The red square shows the RMS thickness
error for the variable-resolution mesh shown in (b) with 1000 m
spacing around the margin. (b) Mesh with resolution that varies
linearly from 1000 m grid spacing beyond a radius of 20 km (thick white
line) to 5000 m at a radius of 3 km (thin white line). The ice thickness
initial condition for the Halfar problem is shown. This mesh requires
1265 cells for the 200-year duration Halfar test case, while a uniform
1000 m resolution mesh requires 2323 cells.
In our application we use a dome following the analytic profile prescribed by
with an initial radius of 21 213.2 m and an initial
height of 707.1 m. We run MALI with the shallow ice velocity solver and
isothermal ice for 200 years and then compare the modeled ice thickness to
the analytic solution at 200 years. We find that the root mean square error in
model thickness decreases as model grid spacing is decreased
(Fig. a). The order of convergence, 0.78, is somewhat lower than
expected from the first-order methods used for advection and time evolution.
We also use this test to assess the accuracy of simulations with variable
resolution. We perform an additional run of the Halfar test
using a variable-resolution mesh, which has 1000 m cell spacing beyond a radius of 20 km that
transitions to 5000 m cell spacing at a radius of 3 km
(Fig. b), generated with the JIGSAW(GEO) mesh-generation tool
. The root mean square error in thickness for
this simulation is similar to that for the uniform 1000 m resolution case
(Fig. a), providing confidence in the advection scheme applied to
variable-resolution meshes. The variable-resolution mesh has about half the
cells of the 1000 m uniform-resolution mesh.
EISMINT
The European Ice Sheet Modeling Initiative (EISMINT) model intercomparison
consisted of two phases designed to provide community benchmarks for
shallow ice models. Both phases included experiments that grow a radially
symmetric ice sheet on a flat bed to steady state with a prescribed surface
mass balance. The EISMINT intercomparisons test ice geometry evolution and
ice temperature evolution with a variety of forcings.
describe an alternative tool for testing thermomechanical shallow ice models
with artificially constructed exact solutions. While their approach has the
notable advantage of providing exact solutions, we have not implemented the
non-physical three-dimensional compensatory heat source necessary for its
implementation. While we hope to use the verification of
in the future, for now we use the EISMINT intercomparison suites to test our
implementation of thermal evolution and thermomechanical coupling.
The first phase (sometimes called EISMINT1) prescribes
evolving ice geometry and temperature, but the flow rate parameter A is set
to a prescribed value so there is no thermomechanical coupling. We have
conducted the Moving Margin experiment with steady surface mass balance and
surface temperature forcing. Following the specifications described by
, we run the ice sheet to steady state over 200 kyr. We
use the grid spacing prescribed by (50 km),
but due to the uniform Voronoi grid of hexagons we employ, we have a slightly
larger number of grid cells in our mesh (1080 vs. 961). At the end of the
simulation, the modeled ice thickness at the center of the dome by MALI is
2976.7 m compared with a mean of 2978.0±19.3 m for the 10
three-dimensional models reported by . MALI achieves
similar good agreement for basal homologous temperature at the center of the
dome with a value of -13.09∘C compared with -13.34±0.56∘C for the six models that reported temperature in
.
The second phase of EISMINT (sometimes called EISMINT2)
uses the basic configuration of the EISMINT1 Moving Margin experiment but
activates thermomechanical coupling through Eq. (). Two
experiments (A and F) grow an ice sheet to steady state over 200 kyr from an
initial condition of no ice, but with different air temperature boundary
conditions. Additional experiments use the steady-state solution from
experiment A (the warmer air temperature case) as the initial condition to
perturbations in the surface air temperature or surface mass balance forcings
(experiments B, C, and D). Because these experiments are thermomechanically
coupled, they test model ice dynamics and thickness and temperature
evolution, as well as their coupling. There is no analytic solution to these
experiments, but 10 different models contributed results, yielding a range
of behavior against which to compare additional models. Here we present MALI
results for the five such experiments that prescribe no basal sliding
(experiments A, B, C, D, F). Our tests use the same grid spacing as
prescribed by (25 km), again with a larger number of grid
cells in our mesh (4464 vs. 3721).
report results for five basic glaciological quantities
calculated by 10 different models, which we have summarized here with the
corresponding values calculated by MALI (Table ). All
MALI results fall within the range of previously reported values, except for
volume change and divide thickness change in experiment C and melt fraction
change in experiment D. However, these discrepancies are close to the range
of results reported by , and we consider temperature
evolution and thermomechanical coupling within MALI to be consistent with
community models, particularly given the difference in model grid and
thickness evolution scheme.
A long-studied feature of the EISMINT2 intercomparison is the cold “spokes”
that appear in the basal temperature field of all models in experiment F and,
for some models, experiment A
. MALI with shallow ice
velocity exhibits cold spokes for experiment F but not experiment A
(Fig. ). argue that these spokes are a
numerical instability that develops when the derivative of the strain heating
term is large. demonstrate that the model VarGlaS
avoids the formation of these cold spokes. However, that model differs from
previously analyzed models in several ways: it solves a three-dimensional,
advective–diffusive description of an enthalpy formulation for energy
conservation; it uses the finite-element method on unstructured meshes;
and conservation of momentum and energy are iterated on until they are consistent
(rather than lagging energy and momentum solutions as in most other models).
At present, it is unclear which combination of those features is responsible
for preventing the formation of the cold spokes.
EISMINT2 results for MALI shallow ice model. For each experiment,
the model name EISMINT2 refers to the mean and range of models reported in
, and we assume that the range reported by
is symmetric about the mean. For experiments B, C, and D reported values are
the change from experiment A results. MALI results that lie outside the range
of values in are italicized.
(a) Basal homologous temperature (K) for EISMINT2
experiment A. (b) Same for experiment F. Figures are plotted
following .
Enthalpy benchmarks
present a set of benchmark experiments to test numerical
models of ice sheet enthalpy evolution. The experiments use designs that
allow for comparison to analytic solutions. Here we only give very brief
descriptions of the enthalpy benchmark experiments. Details can be found in
. The benchmark includes two different experiments
for testing the capability of transient behavior and horizontal and vertical
advection of the enthalpy model. Both experiments use a parallel-sided ice
slab with constant thickness and inclination and prescribed ice dynamics
decoupled from thermodynamics, resulting in effectively one-dimensional
vertical experiments.
The result of (a) basal temperature (Tb),
(b) basal melt rate (ab), and (c) basal water
thickness (Hw) for enthalpy benchmark experiment A. The green line
from 150 to 170 ka in (b) indicates the analytical results
(overlapped with model results).
Experiment A
Both heat advection and frictional heating are neglected in this experiment.
Heat diffusion is the only controlling process in the redistribution of
enthalpy; i.e., the enthalpy balance equation simplifies to
∂E∂t=∂∂zK∂E∂z.
To test the transient ability of the enthalpy model, this experiment was
designed to run for a time period of 300 kyr. During the run, the geothermal
heat flux (G) is constant over time, but the surface temperature (upper
Dirichlet boundary condition) changes in three different time intervals.
Ts=-30∘C,0<t≤100ka-5∘C,100<t≤150ka-30∘C,150<t≤300ka
(Ts=-5∘C, from personal communication with Thomas Kleiner,
compared to Ts=-10∘C, incorrectly stated in
.)
During the time period of 100–150 ka, when the surface temperature rises to
-5∘C, the glacier base becomes temperate and the basal
boundary condition changes from Neumann type (∂T/∂z=G) to
Dirichlet type (T=Tpmp). Then the surface temperature switches back
to its initial value, -30∘C, for testing the reversibility
of the enthalpy model. In this experiment the basal water content produced by
basal melting is allowed to freely accumulate in order to test the basal melt
rate calculation.
From Fig. a, we can clearly see that the basal
temperature becomes steady (-10∘C) at around 50 ka, and
then starts to rise to the pressure melting point after the prescribed change
in surface temperature to -5∘C at 100 ka. The model then
keeps the glacier base temperate until 225 ka, with the prescribed change in the
surface temperature boundary condition back to -30∘C at 150 ka. The
subglacial water layer starts to accumulate from around 110 ka when the
basal ice starts to melt (Fig. b) and reaches a maximum
layer thickness of around 133.9 m at around 155 ka
(Fig. c). After the surface gets colder again, it
gradually decreases and disappears completely at 225 ka. From the comparison
with the analytical basal melt rate result during 150–170 ka (green line),
we can clearly see that the enthalpy model of MALI captures the features of
basal melting (water content production).
Experiment B
In experiment B, a 200 m thick, 4∘ downward-inclined slab is used as
the model domain. A particular objective of this experiment is to test the
model ability to find the correct position of the cold–temperate ice
transition interface. The horizontal velocity is given as an analytical
(shallow ice approximation type) expression, and the vertical velocity is set
to be constant (i.e., thermomechanically decoupled). In addition, the
geothermal heat flux is set to be zero during the model run so that the
englacial strain heating is the only energy source in the enthalpy balance
equation. Initialized from an isothermal field (-1.5∘C),
our model spins up for several thousand years with a constant surface
temperature of -3∘C until a steady-state temperate ice
layer thickness is achieved (Fig. ).
The vertical distribution of (a) enthalpy (E),
(b) temperature (T), and (c) water content (ω) for
enthalpy benchmark experiment B. The green lines indicate the analytical
results (overlapped with model results).
From Fig. 9 we can see that our enthalpy model can predict very close
enthalpy, temperature, and basal water content results (blue lines) compared
to analytical solutions (green lines; almost overlapped). Using a uniform
vertical resolution of 1 m, MALI simulates a temperate ice layer thickness
of 19 m, nearly identical to the analytical output. This experiment also
shows that MALI can accurately compute the englacial enthalpy distribution in
the presence of nonzero horizontal ice advection.
ISMIP-HOM
The Ice Sheet Model Intercomparison Project-Higher Order Models (ISMIP-HOM)
is a set of community benchmark experiments for testing higher-order
approximations of ice dynamics .
describe results from the Albany-LI velocity solver for ISMIP-HOM experiments
A (flow over a bumpy bed) and C (ice stream flow). For all configurations of
both tests, Albany-LI results were within 1 standard deviation of the mean
of first-order models presented in and showed
excellent agreement with the similar first-order model formulation of
. These tests only require a single diagnostic solve
of velocity, and thus results through MALI match those of the stand-alone
Albany-LI code it is using.
Grid resolution convergence for the MISMIP3d Stnd experiment with (gray
squares) and without (black circles) grounding line parameterization.
Results of the MISMIP3d P75R and P75S experiments from MALI at
increasing grid resolution: (a) 2000 m, (b) 1000 m,
(c) 500 m, (d) 250 m. Results for 250 m without
grounding line parameterization (e) are also shown for reference.
Plots follow conventions of Figs. 5 and 6 in . Upper plots
show steady-state grounding line positions for steady-state
spin-up (black), P75S (red), and P75R (blue) experiments. Lower plots
show grounding line position with time for P75R (red) and P75S
(blue) at y=0 km (top curves) and y=50 km (bottom curves). 500 and
250 m results are nearly identical. 250 m results without grounding line
parameterization are intermediate of those at 1000 and 2000 m resolution
with grounding line parameterization.
MISMIP3d
The Marine Ice Sheet Model Intercomparison Project-3d (MISMIP3d) is a
community benchmark experiment testing the grounding line migration of marine ice
sheets and includes nontrivial effects in all three dimensions
. The experiments use a rectangular domain that is 800 km
long in the longitudinal direction and 50 km wide in the transverse
direction, with the transverse direction making up half of a symmetric
glacier. The bedrock forms a sloping plane below sea level. The first phase
of the experiment (Stnd) is to build a steady-state ice sheet from a
spatially uniform positive surface mass balance, with a prescribed flow rate
factor A (no temperature calculation or coupling) and prescribed basal
friction for a nonlinear basal friction law. A marine ice sheet forms with an
unbuttressed floating ice shelf that terminates at a fixed ice front at the
edge of the domain. From this steady state, the P75R perturbation experiment
reduces basal friction by a maximum of 75 % across a Gaussian ellipse
centered where the Stnd grounding line position crosses the symmetry axis.
The perturbation is applied for 100 years, resulting in a curved grounding
line that is advanced along the symmetry axis. After the completion of P75S,
a reversibility experiment named P75R removes the basal friction perturbation
and allows the ice sheet to relax back towards the Stnd state.
report results from 33 models of varying complexity
applied at resolutions ranging from 0.1 to 20 km. Participating models used
depth-integrated shallow shelf or L1L1/L2L2 approximations, hybrid
shallow ice–shallow shelf approximation, or the complete Stokes equations;
there were no three-dimensional first-order approximation models included.
This relatively simple experiment revealed a number of key features necessary
to accurately model even a simple marine ice sheet. Insufficient grid
resolution prevented the reversibility of the steady-state grounding line
position after experiments P75S and P75R. Reversibility required a grid
resolution well below 1 km without a sub-grid parameterization of grounding
line position and grids a couple of times coarser with a grounding line
parameterization . The steady-state
grounding line position in the Stnd experiment was dependent on the stress
approximation employed, with the Stokes model calculating grounding lines the
farthest upstream and models that simplify or eliminate vertical shearing
(e.g., shallow shelf) having grounding lines farther downstream by up to
100 km. With these features resolved, numerical error due to grounding line
motion is smaller than errors due to parameter uncertainty
.
We find that MALI using the Albany-LI Blatter–Pattyn velocity solver is able to
resolve the MISMIP3d experiments satisfactorily compared to the
benchmark results when using a grid resolution of 500 m
with grounding line parameterization. Results at 1 km resolution with
grounding line parameterization are close to fully resolved. We first assess
grid convergence by comparing the position of the steady-state grounding line
in the Stnd experiment for a range of resolutions against our highest-resolution configuration
of 250 m (Fig. ). With the
grounding line parameterization, the grounding line positions at 500 and
250 m resolution are very similar (differing by less than the grid
resolution), whereas without the grounding line parameterization the
grounding line positions in our two highest-resolution simulations still
differ by 6 km. The converged grounding line position for the Stnd
experiment with MALI is 533 km. The grounding line position from our
three-dimensional first-order stress approximation model falls between that
of the L1L2 and Stokes models reported by , consistent with
the intermediate level of approximation of our model. The dissertation work
by reported similar results for the Blatter–Pattyn
velocity solver in the Community Ice Sheet Model when
using a sub-grid grounding line parameterization.
Reversibility of the P75S and P75R experiments shows the same grid resolution
requirement of 500 m, while the 1 km simulation with grounding line
parameterization is close to reversible (Fig. ). Our highest-resolution 250 m
simulation without grounding line parameterization does
show reversibility at the end of P75R (not shown), but the results differ
somewhat from the runs with grounding line parameterization due to the
differing starting position determined from the Stnd experiment. Thus for
marine ice sheets with similar configuration to the MISMIP3d test, we
recommend using MALI with the grounding line parameterization and a
resolution of 1 km or less.
The transient results using the MALI three-dimensional first-order stress
balance look most similar to those of the “SCO6” L1L2 model presented by
in that it takes about 50 years for the grounding line to
reach its most advanced position during P75S. In contrast, the Stokes models
took notably longer and the models with reduced or missing representation of
membrane stresses reached their furthest advance within the first couple of
decades .
In addition to MISMIP3d, we have used MALI to perform the MISMIP+
experiments . These results are included in the
MISMIP+ results paper in preparation and not shown here.
To demonstrate a large-scale, semi-realistic ice sheet simulation that
exercises many of the model capabilities discussed above, we describe MALI
results from the initMIP-Antarctica experiments. The overall goal of initMIP
is to explore the impact of different ice sheet model initialization
approaches on simulated ice sheet evolution. Following model initialization
– via spin-up, optimization approaches, or both – three 100-year forward
model experiments are conducted in order to examine model response to (1) an
unforced control run, (2) an idealized surface mass balance perturbation, and
(3) an idealized sub-ice-shelf melt perturbation. Here we show results from
simulations 1 and 3 for brevity. Additional details of the experiments are
described in http://www.climate-cryosphere.org/wiki/index.php?title=InitMIP-Antarctica
(last access: 10 September 2018), and additional
details on the broader Ice
Sheet Model Intercomparison Project (ISMIP6, part of CMIP6) that initMIP is a
part of are described in and .
Additional realistic applications of earlier versions of MALI to Greenland
simulations are discussed in and .
The Antarctica model configuration we use here has 2 km resolution near
grounding lines and in regions of marine (below sea level) bedrock in West
Antarctica and regions of East Antarctica where present-day ice thickness is
less than 2500 m to ensure that the grounding line remains in the fine-resolution region even under full retreat of West Antarctica and large parts
of East Antarctica. The resolution then slowly coarsens to 20 km elsewhere,
but maintaining no greater than 6 km resolution on ice shelves
(Fig. a). The 2 km resolution in regions of
potential grounding line migration was chosen as a balance between acceptable
accuracy (Sect. ) and computational cost. The mesh has
1 642 490 horizontal grid cells and uses 10 vertical layers, which are
finest near the bed (4 % of total thickness) and coarsen towards the
surface (23 % of total thickness).
Basal friction (β) and ice stiffness factor (γ) fields are
optimized as described above in Sect. to best allow the
model to match observed surface velocities from . Calving
position is fixed as described in option 2 in Sect. . To
avoid energy conservation issues related to maintaining a fixed calving front
position, we use a steady-internal ice temperature field from
. Over the short timescales investigated here
(200 years), we expect the assumption of fixed ice temperature to be a minor
uncertainty . From this initial state, we perform a
99-year relaxation with evolving velocity and geometry
(Fig. ), applying steady forcing of present-day
estimates for surface mass balance and submarine melting
(, and , respectively). This
relaxation removes a substantial portion of the largest model transients
(Fig. c, d). The model state at this point serves as
our initial condition from which we run the control and sub-ice-shelf melt
perturbation experiments mentioned above. In this initial state, the volume
above floatation mass loss for the entire Antarctic ice sheet is
602 Gt yr-1. This is substantially larger than the current best
estimate for Antarctic ice sheet mass loss of 109 Gt yr-1 for the
1992–2017 period, as well as the larger value of 219 Gt yr-1 for the
2012–2017 period , and results largely from retreat and
thinning in the Thwaites Glacier basin during the 99 years of relaxation
(Fig. d).
(a) Mesh resolution used for initMIP simulations. The black
line is the grounding line at the end of the relaxation and the white line is the
bed topography contour at sea level. (b) Modeled ice surface speed
at the end of relaxation. The black line is the grounding line.
(c) Thickness rate of change at the start of relaxation.
(d) Thickness rate of change at the end of relaxation (99 years).
The control simulation lost mass above floatation equivalent to a 167 mm sea
level rise after 100 years. The sub-ice-shelf melt perturbation experiment
yielded the equivalent of a 250 mm sea level rise, an 83 mm sea level rise
addition beyond the control run. During the control run, the Ross and
Filchner–Ronne ice shelves experienced a modest slowdown with some
acceleration near the grounding line (Fig. a). The Amery
and Thwaites ice shelves exhibited significant speedup, with the acceleration
propagating inland from Thwaites. Thickness changes in the control run are
modest with pronounced thinning occurring only on Thwaites Glacier and inland
of Cook Ice Shelf (Fig. b). The only noticeable
grounding line changes in the control run are a slight retreat at Thwaites
Glacier and a slight advance at Princess Ragnhild Coast in Queen Maud Land.
In the sub-ice-shelf melt perturbation experiment there is much more
pronounced speedup at Thwaites and Pine Island glaciers, propagating far
inland (Fig. c), with corresponding ice thinning up to
1000 m (Fig. d). All other ice shelves show increased
thinning or reduced thickening relative to the control run, consistent with
the application of increased ice shelf basal melt rates. The regions of the Ross
and Filchner–Ronne ice shelves near the grounding line experience greater
ice speedup than in the control run and some associated grounding line
retreat, but little additional ice thinning. The Totten Glacier region
exhibits significant ice speedup and thinning in the sub-ice-shelf melt
perturbation experiment.
Speed and thickness change during 100-year initMIP simulations.
(a) Change in surface speed at 100 years relative to initial
condition for control simulation. (b) Change in ice thickness at
100 years relative to initial condition for control simulation.
(c) Change in surface speed at 100 years relative to initial
condition for sub-ice-shelf melt perturbation simulation. (d) Change
in ice thickness at 100 years relative to initial condition for sub-ice-shelf
melt perturbation simulation. In all panels, the black line indicates the
grounding line at the initial time, the gray line is the grounding line at
year 100 in the control simulation, and the green line is the grounding line
at year 100 in the sub-ice-shelf melt perturbation simulation.
To summarize the disparate regional behavior seen in the experiments,
simulation statistics for selected drainage basins for the control and
sub-ice-shelf melt perturbation experiments are shown in
Fig. using the analysis capability described in
Sect. . The applied basal melt forcing for selected basins
can be seen in Fig. a. The prescribed sub-ice-shelf melt
perturbation according to the initMIP protocol increases for 40 years and
then remains constant. The continued growth in total ice shelf basal melt
after year 40 in Fig. a reflects increasing ice shelf area
as the grounding line retreats, while we have forced the ice shelf calving
front to remain fixed. Ice shelf thinning from increased basal melt results in
increased flux across the grounding line, with the largest increase in the
Thwaites–Pine Island catchment where the ice shelf basal melt perturbation
was largest (Fig. b). In turn, increased flux across the
grounding line leads to ice sheet mass loss (Fig. c) and
grounding line retreat (Fig. d). The largest changes occur
in the Thwaites–Pine Island catchment. Mass loss and grounding line retreat
occur in all basins for the control run but are greater in the perturbation
experiment, as expected. Note that the initMIP experiments use schematic
forcing, and results should not be interpreted as realistic ice sheet or sea
level projections. Our aim here is to demonstrate that when applied to large-scale, whole-ice-sheet simulations on realistic geometries, MALI is robust
and evolves reasonably during multi-century-length, free-running simulations.
Model results for initMIP control (thin lines) and sub-ice-shelf
melt perturbation (thick lines) experiments for selected drainage basins.
Basins are composed of their respective ice shelves and the IMBIE basins
flowing into them from upstream. (a) Total ice
shelf basal melt; (b) total flux across grounding line;
(c) change in volume above floatation from initial time;
(d) change in grounded area from initial time.
Coupling to Energy Exascale Earth System Model
MALI is the current land ice model component of the US Department of
Energy's Energy Exascale Earth System Model (E3SM). E3SM is an Earth
system model with atmosphere, land, ocean, and sea ice components linked
through a coupler that passes the necessary fields (e.g., model state, mass,
momentum, and energy fluxes) between the components. E3SM, which branched
from the Community Earth System Model (CESM, version 1.2 beta10) in 2014,
targets high-resolution global simulations, and all components have a
variable-resolution mesh capability. The ocean
and sea ice
components are also built on the MPAS Framework. Because the coupling between
E3SM and MALI is currently still fairly rudimentary, we include only a few
additional details below and leave a more detailed description to future
work. Having all three of these E3SM components in the MPAS Framework has
simplified adding and maintaining them within E3SM because developments in
the component driver code and build and configuration scripts made by one
MPAS component can easily be leveraged by the others. Note that each
component of E3SM can be run on differing numbers of processors within the
coupled model, including the individual MPAS cores.
Physics at the ice sheet–atmosphere interface are handled by the snow model
within the E3SM land model (ELM; ). ELM's snow
model calculates ice sheet surface mass balance using a surface energy
balance model and, at each coupling interval, MALI passes the current ice
sheet extent and surface elevation through the coupler to ELM. The coupler
then returns the surface mass balance and surface temperature calculated by
ELM to MALI. These fields are used within MALI as boundary conditions to the
mass and thermal evolution equations (Sect.
and ). Currently, runoff from surface melting is
calculated within ELM and routed directly through E3SM's runoff model, rather
than being passed to and used by MALI. The subglacial discharge model
discussed above in Sect. is not currently coupled
to the rest of E3SM.
Ongoing and future work on MALI and E3SM coupling includes the following: passing
subglacial discharge at terrestrial ice margins to the land runoff model in
E3SM; passing surface runoff calculated in E3SM to the land ice model (for
use as a source term in the subglacial hydrology model); two-way coupling
between the ocean and a dynamic MALI model
Coupling to a static
Antarctic ice sheet with ocean circulation in sub-ice-shelf cavities is
supported in E3SM version 1.0.0.
; and discharge of icebergs (solid ice flux from
MALI) to the coupler and from there to the ocean and sea ice models.
Model performance
Detailed analysis of the performance and scalability of the Albany-LI
velocity solver for idealized test cases and realistic high-resolution
applications to Greenland and Antarctica has been reported by
and . Because the
momentum balance solver is ≥95 % of the cost of a typical
forward model time step outside of I/O, the previously reported model
performance is generally representative of overall MALI performance. To
provide some additional context we summarize MALI performance for the high-resolution
Antarctica application described in the previous section. That
mesh contained 1 642 490 horizontal grid cells and 11 vertical interfaces
(10 vertical levels) at which the two horizontal components of velocity are
solved. The simulations described above were run on the Edison Cray
XC30 supercomputer
More information about Edison can be found at
http://www.nersc.gov/users/computational-systems/edison/configuration/ (last access: 10 September 2018).
at the National Energy Research Scientific
Computing Center (NERSC). Computational nodes on Edison each contain
two 12-core Intel “Ivy Bridge” processors operating at 2.4 GHz and 64 GB
DDR3 1866 MHz memory. Simulations were done using 6400 processors. The
control simulation averaged 5.26 simulated years per wall-clock hour (SYPH)
over 2031 time steps, and the sub-ice-shelf melt perturbation experiment
averaged 4.61 SYPH over 2181 time steps. The differing performance is
partially due to the higher number of time steps required by the perturbation
experiment due to faster maximum ice velocity forcing the adaptive time
stepper to take smaller time steps, but may also be a symptom of varying
machine performance; performance during different segments of the simulations
varied from 3.25–7.38 SYPH, presumably due to the usage of different node
layouts on the machine and varying I/O performance. On average the velocity
solve took 91.9 % of the computational time, writing output took
7.5 %, and all other operations the remaining 0.6 %. Total simulation
cost for the 100-year simulations was 122 000 core hours for the control
run and 139 000 core hours for the perturbation simulation. For
reference, the high-resolution E3SM configuration (25 km resolution in
atmosphere and land components, varying 18 to 6 km resolution in ocean and
sea ice components) runs at 0.12 SYPH using 52 000 processors on
Edison. While MALI could be run with substantially fewer processors to
match the slower throughput of E3SM, the current optimal processor layout for
high-resolution E3SM could run MALI at the processor count we have done here
without incurring any additional expense due to latent processors during
model time stepping. At the resolution described here, MALI's computational
cost of 1400 processor hours per simulated year would be insignificant
compared to the cost of high-resolution E3SM at 448 000 processor hours per
simulated year. Of course running MALI within E3SM would restrict simulation
lengths to those used for the coupled model (decades to centuries), which are
too short for the investigation of many ice sheet science questions. At E3SM
low resolution (100 km resolution in atmosphere and land components, varying
60 to 30 km resolution in ocean and sea ice components), the computational
cost of MALI within E3SM remains a minor cost.
Conclusions and future work
We have described MPAS-Albany Land Ice (MALI), a higher-order,
thermomechanically coupled ice sheet model using unstructured Voronoi meshes.
MALI takes advantage of the MPAS Framework for the development of unstructured
grid Earth system model components and the Albany and Trilinos frameworks for
a parallel, performance-portable, implicit solution of the challenging
higher-order ice sheet momentum balance through the Albany-LI velocity
solver. Together, these tools provide an accurate, efficient, scalable, and
portable ice sheet model targeted for high-resolution ice sheet simulations
within a larger Earth system modeling framework run on tens of thousands of
computing cores, and MALI makes up the ice sheet component of the Energy
Exascale Earth System Model version 1.
MALI includes three-dimensional Blatter–Pattyn and shallow ice velocity
solvers, a standard explicit mass evolution scheme, a thermal solver that can
use either a temperature or enthalpy formulation, and an adaptive time
stepper. Physical processes represented in the model include subglacial
hydrology and calving. The model includes a mass-conserving subglacial
hydrology model that can represent combinations of water drainage through
till, distributed systems, and channelized systems, and it can be coupled to ice
dynamics. A handful of basic calving schemes are currently implemented,
including the physically based eigencalving method.
We have demonstrated the accuracy of the various model components through
commonly used exact solutions and community benchmarks. Of note, we presented
the first results for the MISMIP3d benchmark experiments using a
Blatter–Pattyn model, and the results are intermediate to those of Stokes and
L1L2 models, as might be expected. We also showed simulation results for a
semi-realistic Antarctic ice sheet configuration at coarse resolution, and
this capability was facilitated by the optimization tools within Albany-LI.
A number of model improvements are planned over the next 5 years, focused
heavily on improved representation of ice sheet physical processes and Earth
system coupling. An implicit subglacial hydrology model based on such existing
models is under development using the
Albany framework. It will include optimization capabilities, a technique
that has yet to be applied to subglacial hydrology beyond a
spatially average, zero-dimension application . The
difficulty in subglacial drainage parameter estimation remains one of the
primary reasons drainage models have yet to be widely applied in ice sheet
models.
Improved calving schemes are also under development using a continuum damage
mechanics approach . Additionally,
solid Earth processes affecting ice sheets are planned for future development,
including gravitational, elastic, and viscous effects. Higher-order advection,
through the flux-corrected transport and/or incremental
remapping schemes that are
already implemented in MPAS, and semi-implicit time stepping are planned.
Finally, a high priority is completing the coupling between ice sheet, ocean, and
sea ice models in E3SM.
Code availability
MPAS releases are available at
https://mpas-dev.github.io/ (last access: 10 September 2018) and model code is maintained at
https://github.com/MPAS-Dev/MPAS-Release/releases (last access: 10 September 2018). MPAS-Albany Land Ice is included in MPAS
version 6.0. The digital object identifier for MPAS v6.0 is
10.5281/zenodo.1219886. MPAS is openly developed at
https://github.com/MPAS-Dev/MPAS-Release (last access: 10 September 2018). The Albany library is developed openly at
https://github.com/gahansen/Albany (last access: 10 September 2018), and the Trilinos library is developed openly at
https://github.com/trilinos/Trilinos (last access: 10 September 2018). Region definitions for analysis are openly maintained at
https://github.com/MPAS-Dev/geometric_features (last access: 10 September 2018).
Physical constants.
SymbolDescriptionStandard valueUnitsgGravitational acceleration9.81m s-2ρDensity of ice910kg m-3RGas constant8.3145kg m2 s-2 K-1 mol-1ρwDensity of freshwater1000kg m-3ρoDensity of ocean water1028kg m-3cHeat capacity of ice2009J kg-1 K-1kThermal conductivity of ice2.1W m-1 K-1LLatent heat of fusion of water3.35×105J kg-1ctPressure melt coefficient7.5×10-8K Pa-1cwHeat capacity of water4.22×103J kg-1 K-1
General variables and parameters.
SymbolDescriptionUnitsx, y (x1, x2)Horizontal coordinatesmz(x3)Vertical elevationmnzNumber of vertical layerstTimesσ“Sigma” coordinateunitlessHIce thicknessmsUpper surface elevationmbLower surface elevationma˙Surface mass balancem s-1b˙Basal mass balancem s-1QtTracer quantitylLayer thicknessS˙Tracer sources and sinks
Ice dynamics variables and parameters.
SymbolDescriptionUnitsσijStress tensorPaτijDeviatoric stress tensorPaτeEffective deviatoric stressPaδijKronecker deltaϵ˙ijStrain rate tensors-1ϵ˙eEffective strain rates-1μeEffective ice viscosityPa sγIce stiffening factornGlen's flow law exponentmBasal friction law exponentu, uiHorizontal ice velocity vectorm s-1unComponent of horizontal advective ice velocity normal to cell edgesm s-1ubBed-parallel basal slip velocity vectorm s-1u‾Depth-averaged velocitym s-1AIce flow rate factors-1 Pa-nA0Ice flow rate factor constants-1 Pa-nβBasal friction coefficientPa yr m-1
Ice thermodynamics variables and parameters.
SymbolDescriptionUnitsTIce temperatureKEEnthalpyJ kg-1T*Absolute ice temperatureKTpmpPressure melting temperatureKQaActivation energy for crystal creepkg m2 s-2 mol-1ΦViscous dissipationPa s-1FdDiffusive flux at ice baseW m-2FfGeothermal heat fluxW m-2FfFrictional heatingW m-2
Subglacial hydrology variables and parameters.
SymbolDescriptionUnitsWtillWater layer thickness in tillmWWater layer thickness at bedmmbBasal melt ratem s-1CdTill drainage ratem s-1γtOverflow rate from tillm s-1ϕBasal hydropotentialPaPwBasal water pressurePazbBed elevationmNIce effective pressurePacsBasal roughness parameterm-1WrMaximum bed bump heightmccdCreep scaling parameter for distributed drainageAbIce flow rate factor for basal ices-1 Pa-nqWater flow in distributed drainage systemm2 s-1kqConductivity coefficient for distributed flowm2α2-α1 s2α2-3 kg1-α2α1Exponent on water thickness for water flowα2Exponent on water pressure for water flowSSubglacial channel aream2ΞDissipation of potential energy in water flowJ s-1 m-2ΠSensible heat change of waterJ s-1 m-2cccCreep scaling parameter for channelized drainageQWater flow in channelized drainage systemm3 s-1kQConductivity coefficient for channelized flowm2α2-α1 s2α2-3 kg1-α2qcWater flow in distributed drainage system along a channelm2 s-1lcDistance perpendicular to a channel where channel isminfluenced by distributed flow dissipationVdWater velocity of distributed flowm s-1DdDiffusivity of distributed flowm2 s-1δDirac delta functionϕ0Notional englacial porosityC0Basal friction parameter(s m-1)m
Calving variables and parameters.
SymbolDescriptionUnitsϵ˙1,ϵ˙2Horizontal principal strain ratess-1CvCalving velocitym s-1K2Eigencalving parameterm s
Optimization variables and parameters.
SymbolDescriptionUnitsJOptimization functionalσuStandard deviation of uncertainty in observed velocitym s-1cγControl parameter for ice stiffness factorαβ, αγRegularization parametersCompeting interests
The authors declare that they have no conflict of
interest.
Disclaimer
This paper describes objective technical results and analysis.
Any subjective views or opinions that might be expressed in the paper do not
necessarily represent the views of the US Department of Energy or the
United States Government.
Sandia National Laboratories is a multi-mission laboratory managed and
operated by National Technology and Engineering Solutions of Sandia, LLC., a
wholly owned subsidiary of Honeywell International, Inc., for the US
Department of Energy's National Nuclear Security Administration under
contract DE-NA-0003525.
Acknowledgements
We thank additional contributors to the MALI code, Michael Duda,
Dominikus Heinzeller, Benjamin Hills, and Adrian Turner, as well as
the developers of the MPAS, Albany, and Trilinos libraries. Support for this work
was provided through the Scientific Discovery through Advanced Computing
(SciDAC) program and the Energy Exascale Earth System Model (E3SM) project
funded by the US Department of Energy (DOE), Office of Science, Biological
and Environmental Research, and Advanced Scientific Computing Research
programs. Development of the subglacial hydrology model was supported by a
grant to Matthew J. Hoffman from the Laboratory Directed Research and
Development Early Career Research Program at Los Alamos National Laboratory
(20160608ECR). This research used resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science user facility supported
by the Office of Science of the US Department of Energy under contract
no. DE-AC02-05CH11231, and resources provided by the Los Alamos National
Laboratory Institutional Computing Program, which is supported by the US
Department of Energy National Nuclear Security Administration under contract
no. DE-AC52-06NA25396. Edited by: Jeremy
Fyke Reviewed by: Stephen Cornford and Thomas Zwinger
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