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): 1σ=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.

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: 2∂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 3τij=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, 4τij=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 : 5μe=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, 6ϵ˙e=12ϵ˙ijϵ˙ij12. The strain rate tensor is defined by gradients in the components of the ice velocity vector ui: 7ϵ˙ij=12∂ui∂xj+∂uj∂xi,i,j=1,2,3. Finally, the rate factor A follows an Arrhenius relationship, 8AT*=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.

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.

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: 19∂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: 20∂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 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 .

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.

MALI includes a few simple methods for removing ice from calving fronts during each model time step.

All floating ice is removed.

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 59Rβ(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).

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 .

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

https://github.com/MPAS-Dev/geometric_features (last access: 10 Septemebr 2018).

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.

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.

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.

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 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.

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.