Data assimilation and prognostic whole ice sheet modelling ... · D. J. Brinkerhoff and J. V....

The Cryosphere, 7, 1161–1184, 2013www.the-cryosphere.net/7/1161/2013/doi:10.5194/tc-7-1161-2013© Author(s) 2013. CC Attribution 3.0 License.

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Data assimilation and prognostic whole ice sheet modelling with thevariationally derived, higher order, open source, and fully parallelice sheet model VarGlaS

D. J. Brinkerhoff and J. V. Johnson

University of Montana, Missoula, MT, USA

Correspondence to:D. J. Brinkerhoff ([email protected])

Received: 12 February 2013 – Published in The Cryosphere Discuss.: 8 March 2013Revised: 10 June 2013 – Accepted: 12 June 2013 – Published: 25 July 2013

Abstract. We introduce a novel, higher order, finite elementice sheet model called VarGlaS (Variational Glacier Simula-tor), which is built on the finite element framework FEniCS.Contrary to standard procedure in ice sheet modelling, Var-GlaS formulates ice sheet motion as the minimization of anenergy functional, conferring advantages such as a consis-tent platform for making numerical approximations, a coher-ent relationship between motion and heat generation, and im-plicit boundary treatment. VarGlaS also solves the equationsof enthalpy rather than temperature, avoiding the solution ofa contact problem. Rather than include a lengthy model spin-up procedure, VarGlaS possesses an automated frameworkfor model inversion. These capabilities are brought to bearon several benchmark problems in ice sheet modelling, aswell as a 500 yr simulation of the Greenland ice sheet at highresolution. VarGlaS performs well in benchmarking experi-ments and, given a constant climate and a 100 yr relaxationperiod, predicts a mass evolution of the Greenland ice sheetthat matches present-day observations of mass loss. VarGlaSpredicts a thinning in the interior and thickening of the mar-gins of the ice sheet.

1 Introduction

Models have become an important tool in the study of glacierand ice sheet physics, with applications to the predictionof cryosphere/climate interactions, sea level rise, and funda-mental questions of ice dynamics. In recent years, while con-ceptual and theoretical advances in the development of icesheet models have been made, computational constraints lim-ited practical ice sheet models to low-order asymptotic ap-

proximations of ice physics. Early models (and many modernones) relied upon the shallow ice approximation, sacrificingaccuracy in regions of complex flow for efficiency at largescales (Hutter, 1983; Schafer et al., 2008). Models includinghigher order physics were also used, but the increased com-putational demand made operating at high resolutions infea-sible. Additionally, many models were based on finite dif-ference schemes, which made variable resolutions over dif-ferent regions in the computational domain difficult. Recentincreases in computing power and the availability of paral-lel libraries, in tandem with the finite element method andunstructured meshes, have made possible the use of higherorder physical approximations coupled with the high resolu-tion necessary to resolve fine-scale features of glacier flow.The ice sheet modelling community has been quick to takeadvantage of these advances, for example, inLarour et al.(2012), Seddik et al.(2012), Gillet-Chaulet et al.(2012),Leng et al.(2012), and Bueler and Brown(2009), amongothers. In this work, we present a new thermomechanicallycoupled, prognostic ice sheet model called VarGlaS (Varia-tional Glacier Simulator). As stated above, there are alreadya few examples of so-called “next-generation” ice sheet mod-els in existence, but this model differs in implementationstrategy in several critical regards. Namely, these are the usea of a variational principle in the model formulation, the so-lution of an enthalpy equation, the use of a kinematic bound-ary condition, and the use of automatic differentiation fordata assimilation. The implementation of these features isstreamlined by building the model upon the framework of-fered by the open-source FEniCS project (Logg et al., 2012).FEniCS offers access to a variety of finite element solversand tools, as well as a very deep integration of automatic

Published by Copernicus Publications on behalf of the European Geosciences Union.

1162 D. J. Brinkerhoff and J. V. Johnson: Variational ice sheet model

differentiation, which makes many of the more unique capa-bilities of VarGlaS possible.

VarGlaS treats the solution to the momentum balance(Stokes’ equations) as the minimization of an energy func-tional. The existence of a variational principle for nonlinearStokes’ flow was shown byBird (1960). When applied toice sheets, the method consists in minimizing the dissipationof gravitational potential energy by viscous and frictionalheat generation. This approach has been suggested bySchoof(2006), Dukowicz (2012), andBassis(2010). This type oftreatment stands in contrast to the heretofore standard treat-ment of the velocity field, which is to explicitly account forthe balance of viscous stresses and forcing by gravity, as isdone in most other ice sheet models (e.g.Larour et al., 2012;Seddik et al., 2012; Leng et al., 2012; Bueler and Brown,2009; Rutt et al., 2009). Viewing the problem as a variationalminimization problem confers a number of advantages. Themost important advantage is that the momentum balance isuniformly derived from a single scalar conservation state-ment. When approximations to the physics are made, they aremade to the scalar quantity, and these changes are automati-cally propagated through the rest of the model. This is partic-ularly useful when automatic differentiation is available. Asnoted above, the FEniCS software offers strong automaticdifferentiation capabilities, with particularly strong supportfor the calculation of Gateaux derivatives, which is the nec-essary operation for deriving equations of motion from avariational principle (Dukowicz, 2012). The procedure ofgenerating the code for a new approximation to the Stokes’equations is as simple as making a change to the variationalprinciple. This makes an extension of the model to differentasymptotic approximations straightforward. Other potentialadvantages are that the variational principle is coordinate-independent, streamlining the transition to a curvilinear orgeographic coordinate system. Boundary conditions are alsoimplicitly defined within the variational principle, meaningthat the often complex process of imposing boundary con-ditions is simplified. The use of a variational principle alsoconfers several computational advantages. For example, theoperators derived from the variational principle are guaran-teed to be at least symmetric and semi-definite. Also, theseoperators are already in the appropriate form for use with thefinite element method, requiring no further manipulation.

Treatment of the energy balance by VarGlaS also differsfrom standard methods. Typically, temperature is the variableof interest in the energy balance (Larour et al., 2012; Seddiket al., 2012; Greve and Hutter, 1995; Rutt et al., 2009; Pattyn,2003). Computing the temperature field is a contact problemwhere the temperature must be constrained to remain belowthe phase boundary. Different methods have been employedto enforce this constraint, such as treating temperate and coldice as two separate fluids (Greve and Hutter, 1995), manip-ulating heat sources and sinks such that heat sources are ap-plied to the temperature equation when below the meltingpoint and to calculate a melt rate when the ice is at the pres-

sure melting point (Rutt et al., 2009), or solving a contactproblem (Zwinger et al., 2007). To avoid inconsistency andheuristics, we eschew the temperature formulation in favourof an enthalpy treatment that tracks total internal energy den-sity rather than sensible heat. This eliminates the need forspecial numerical treatment of the cold-temperate transitionsurface, at the expense of introducing a nonlinearity in en-ergy diffusion (Aschwanden et al., 2012). While enthalpy ismore straightforward computationally, the temperature fieldis still necessary for the computation of ice rheology and forinterpretation of model results. Temperature can be recov-ered from enthalpy in a straightforward way through a bijec-tion between enthalpy, temperature, and water content.

VarGlaS is equipped with a kinematic boundary conditionthat allows for the evolution of ice geometry. Many modelscalculate change in surface elevation as the flux divergence ofthe vertically averaged horizontal velocity field (Larour et al.,2012; Rutt et al., 2009; Bueler and Brown, 2009). We treatit as the advection of the ice surface by the surface velocityfield, as is done inSeddik et al.(2012) andLeng et al.(2012).The two forms are equivalent, and both are numerically un-stable. VarGlaS, similar to other modern finite element mod-els, uses streamline upwind Petrov–Galerkin finite elementsto stabilize the free surface problem (Larour et al., 2012; Sed-dik et al., 2012). Additionally, transitions between glaciatedand ice-free regions of the model domain produce numericalinstabilities which need to be addressed with methods be-yond upwinding in order to maintain higher order accuracy.To this end, VarGlaS also introduces a discontinuity captur-ing scheme to maintain stability in the presence of large gra-dients. VarGlaS uses a unique and fully explicit total vari-ation diminishing Runge–Kutta scheme for time discretiza-tion. This method guarantees that no new spurious extremaare generated by the time-stepping scheme, and provides sec-ond order in time accuracy without necessitating a complexand expensive coupling between the momentum balance andtime evolution.

For both diagnostic and prognostic modelling, it is impor-tant to constrain the model to match observed values of statevariables as closely as possible. To this end, VarGlaS pos-sesses tools for data assimilation, again based on the auto-matic differentiation capabilities of FEniCS. With this func-tionality in place, calculating the adjoint state of a model andthe gradient of a given objective function with respect to arbi-trary parameters under the constraint that a forward model besatisfied is simple and automated. We use this capability intwo complementary ways. First, we have been able to invertfor sliding velocity (or basal traction) to best match surfacevelocities, a classic problem in glacier modelling (MacAyeal,1993; Goldberg and Sergienko, 2011; Larour et al., 2005;Gudmundsson and Raymond, 2008; Morlighem et al., 2010;Brinkerhoff et al., 2011). In contrast to many implementa-tions of data assimilation in ice sheet modelling (MacAyeal,1993; Morlighem et al., 2010), VarGlaS uses an unsimplifiedadjoint (Goldberg and Sergienko, 2011; Brinkerhoff et al.,

The Cryosphere, 7, 1161–1184, 2013 www.the-cryosphere.net/7/1161/2013/

D. J. Brinkerhoff and J. V. Johnson: Variational ice sheet model 1163

2011). We can also minimize the total imbalance in masscontinuity with respect to basal topography (e.g.Morlighemet al., 2011), although this procedure is presented in a sepa-rate paper (Johnson et al., 2013). These methods are impor-tant for long-term simulations, as velocity assimilation repro-duces a plausible velocity that is of leading order importancein calculating surface rates of change, while mass conserva-tion assimilation helps to eliminate strong transients result-ing from estimates of basal topography that are incoherentrelative to the ice physics and observed data.

The paper is structured as follows. Section2.1 discussesthe continuum mechanical formulation of the model physics.Section2.2 deals with the numerical implementation of themodel physics, and the difficulties arising from their dis-cretization. Section3 involves the application of the model toa few numerical experiments including well-known bench-marks involving idealized geometry, as well as the entireGreenland ice sheet. Finally, in Sect.4 we discuss how thecombination of advances has allowed VarGlaS to success-fully simulate both numerical benchmarks and continental-scale ice dynamics, as well as the fundamental limitations ofthe model and what directions should be explored in order toovercome these.

2 Model

VarGlaS solves for the three-dimensional ice sheet veloc-ity, enthalpy, and geometry through time. All three of thesevariables are strongly coupled. We first present the contin-uum formulation of ice sheet physics, followed the numericaltreatment for each.

2.1 Physics

2.1.1 Momentum balance

Our development of a variational principle for the momen-tum balance largely followsDukowicz (2012). The varia-tional principle for a power law rheology with linear basalsliding under the constraints of incompressibility and bed im-penetrability is

A[u,P ] =

∫�

2n

n + 1η(ε2)ε2︸︷︷︸

Viscous Dissipation

+ ρg · u︸︷︷︸Potential

− P∇ ·u︸︷︷︸Incomp.

d�

+

∫0B

β2

m + 1hr(u · u)

m+12︸︷︷︸

Friction

+ Pu · n︸︷︷︸Impen.

d0 (1)

+

∫0E

−Peu · n︸︷︷︸Env. Pressure

d0,

whereu is the ice velocity andε the rate of strain tensor,P the pressure,η(ε2) the strain-rate-dependent ice viscosity,g the gravitational vector,β2 the basal sliding coefficient,h the ice thickness,r a factor determining the relationshipbetween basal traction and thickness,m a factor determin-ing the nonlinearity of the frictional term,n the outward nor-mal vector, andPe, the environmental pressure, either atmo-spheric or water (see Table1 for a complete listing of sym-bols used in this manuscript). The expression is minimizedover the ice domain� with boundaries0, where0B is thegrounded portion of the ice sheet.0E is the non-groundedportion of the ice sheet, and is defined by0E = 0 \ 0B .Each of the additive terms in Eq. (1) has a specific meaning.Terms integrated from left to right over� are viscous dis-sipation, gravitational potential energy, and the incompress-ibility constraint, respectively. Terms under the first bound-ary integral are frictional heat dissipation and the impenetra-bility constraint. Note that basal traction is scaled by thick-ness compared to the standard form of the sliding law. Forr = 1, this eliminates the covariance between basal tractionand pressure evident in previously computed basal tractionfields (e.g.Larour et al., 2012). For r = 0, it is obvious thatthis form reduces to the standard sliding law. The term underthe second boundary integral is the pressure at the ice–air orice–water interface. Note that, at this point, VarGlaS does notallow these boundary domains to change, implying a staticgrounding line. The constitutive relationship for ice given byGlen(1955) gives a viscosity of

η(ε2) = b(T ,ω)[ε2]

1−n2n , (2)

whereε2 is defined to be the square of the second invariantof the strain rate tensor, andb(T ,ω) is a temperature- andwater-content-dependent rate factor:

b(T ,ω) =

[Ea(T ,ω)e−

Q∗(T )RT ∗

]−1n

. (3)

Here,E is an enhancement factor, witha(T ,ω), Q∗(T ) andR parameters, andT ∗ is temperature-corrected for pressuremelting point dependence. The traditional momentum bal-ance form of the Stokes’ equations can be recovered (in weakform) by taking the variation of Eq. (1). This is the functionalthat VarGlaS minimizes in order to solve the Stokes’ prob-lem.

The Stokes’ functional is a relatively complete statementof ice physics (the only assumptions being negligible inertialterms), but it includes four degrees of freedom per computa-tional node (three velocity components and pressure) and isa saddle point problem due to the presence of the Lagrangemultiplier pressure terms. A considerable simplification canbe made to the Stokes’ functional by expressing vertical ve-locities in terms of horizontal ones through the incompress-ibility and bed impenetrability constraints, that is

www.the-cryosphere.net/7/1161/2013/ The Cryosphere, 7, 1161–1184, 2013


Table 1.Table of symbols.

Symbol Value Units Description

A J a−1 Stokes’ energy functionalA1 J a−1 First-order energy functionala(T ) Pa−n a−1 Temperature-dependent ice hardnessa m a−1 Surface accumulation/ablation rateα Regularization weighting factorB m Bed elevation

b Pa a13 Linear viscosity factor

β2 Pa am m−(m+r) Basal traction coefficientC CFL stability maximumCp 2009 J kg−1 K−1 Heat capacity of iceCw 4181 J kg−1 K−1 Heat capacity of liquid waterc Mesh cellc Compensatory accumulationD m Vector cell sizeD m Scalar cell sizeE Enhancement factore m a−1 Cell errorε a−1 Strain rate tensorε0 10−30 a−1 Strain rate regularizationF Nonlinear equation systemF Forcing function on model interiorG Forcing function on model boundaryg m s−2 Gravitational acceleration vectorγ 9.76× 10−8 K Pa−1 Melting point pressure dependence0 Model domain boundaryH0 J Atmospheric melting enthalpyHm J Pressure-corrected melting enthalpyH J EnthalpyHess m−1 a−1 Hessian of surface velocityh m Ice thicknessη Pa a Ice viscosityI ′ Unconstrained objective functionalI Constrained objective functionalJ Jacobian of nonlinear equation systemK m2 a−1 SUPG diffusivityKshock m2 a−1 Shock capturing diffusivityk 6.62× 107 J a−1 K−1 m−1 Thermal conductivity of cold iceκ m2 a−1 Diffusivity of temperate and cold iceL 3.35× 105 J Latent heat of fusion3 m−1 a−1 Eigenvalues of Hessλ m a−1 Adjoint velocityM m−1 a−1 Anisotropic error metricMb m a−1 Basal melt ratem Sliding law nonlinearity parametern Outward unit normal vectorn 3 Glen’s flow law parameterν 3.5× 103 kg m−1 a−1 Moisture diffusivity in temperate iceP Pa Isotropic pressureQ J m−3 a−1 Internal heat generationQ∗ J mol−1 Activation energyqgeo 42 mW m−2 Geothermal heat fluxqfriction mW m−2 Frictional heat flux



Symbol Value Units Description

R m a−1 Residual of free surface equationR 8.3 J mol−1 K−1 Universal gas constantR Newton relaxation parameterr Sliding law thickness exponentρ 910 kg m−3 Ice densityS m Surface elevationS m Analytic surface elevationT Tikhonov regularization functionalT K TemperatureT ∗ K Pressure-corrected temperatureTm K Melting temperatureT0 273.15 K Triple point temperatureτgls m2 Pa−1 a−1 Galerkin least squares stabilization parameterU i Solution vector at nonlinear iterationiu m a−1 Velocity vectoruobs m a−1 Observed velocity vectoru‖ m a−1 Horizontal velocity vectoru m a−1 Analytic velocity vectorV Eigenvectors of Hessw m a−1 Vertical component of velocity vector� Model domainω Water content

w(u‖) = −

z∫B

∇‖ · u‖ dz′ (4)

with boundary condition

wB = u‖B · ∇‖B. (5)

Substitution of these expressions intoA yields an uncon-strained and positive definite integro-differential functionalwhich is equivalent to Eq. (1). However, the integral termsthat result from the vertical integration of the mass conserva-tion equation are undesirable. Standard methods for the nu-merical solution of partial differential equations (PDEs) arenot equipped to handle integral terms of this type, so we seeka simplification that eliminates them. In order to derive thefunctional associated with the so-called “first-order” equa-tions of ice sheet motion (Blatter, 1995; Pattyn, 2003), twoassumptions must be made. First, bed slopes are small, whichis also equivalent to assuming cryostatic pressure. Second,horizontal gradients of vertical velocity are small comparedto other components of the strain rate tensor. This eliminatesvertical velocity terms in the interior of the ice, as well as atthe surface and grounded basal boundaries. After these as-sumptions and some manipulation, the first-order functionalis

A1[u‖] =

∫�

2n

n + 1η(ε2

1)ε21︸︷︷︸

Viscous Dissipation

+ ρgu‖ · ∇‖S︸︷︷︸Potential

d�

+

∫0B

β2

m + 1hr(u‖ · u‖)

m+12︸︷︷︸

Friction

d0 (6)

+

∫0E

Pw (u · n)︸︷︷︸Env. Press.

d�, (7)

whereu‖ is the velocity vector in the horizontal directions,S the elevation of the ice surface, andε2

1 the first-order strainrate tensor given byPattyn(2003) andDukowicz(2012). Thefirst-order assumption does not allow a decoupling of the ver-tical velocity where the ice is not grounded, which is a directconsequence of assuming that vertical resistive stresses arenegligible. Thus we assume, for the first-order model, thatany ice beyond the grounding line instantly calves and is nolonger considered. Thus we drop the last term in Eq. (6).Since the first-order equations are only associated with hor-izontal velocity components, this formulation yields signif-icant computational savings, as well as desirable numericalproperties such as guaranteed positive definiteness. Verticalvelocity for the grounded ice sheet is recovered from Eqs. (4)and (5).



2.1.2 Enthalpy

VarGlaS uses an enthalpy formulation of the energy balance(Aschwanden et al., 2012). Enthalpy methods track total in-ternal energy, rather than sensible heat, which correspondsbijectively to temperature for ice below the pressure meltingpoint, and to water content for ice at the pressure meltingpoint. The enthalpy equation is a typical advection–diffusionequation with a nonlinear diffusivity:

ρ (∂t + u · ∇)H = ρ∇ · κ(H)∇H + Q, (8)

whereH is enthalpy,ρ ice density, andQ strain heat gen-erated by viscous dissipation, given by the dissipative termin the momentum balance functional.κ is an enthalpy-dependent diffusivity given by

κ(H) =

{k

ρCpif cold

νρ

if temperate,(9)

wherek is the thermal conductivity of cold ice andCp is heatcapacity.ν is the diffusivity of enthalpy in temperate ice, andcan also be thought of as a parameterization of the subgrid-scale intraglacial flow of liquid water. It is not clear what thevalue ofν should be. BothHutter (1982) andAschwandenet al.(2012) have suggested that it be a function of both watercontent and gravity, but intra-glacial liquid modelling is be-yond the scope of this work, and we set this value toν �

kCp

.This implies that heat does not move diffusively within tem-perate ice, and that any heat generation immediately goes to-wards melting. The definitions for cold and temperate ice areas follows:{

cold (H − Hm(P )) < 0

temperate (H − Hm(P )) ≥ 0,(10)

where Hm is the pressure melting point expressed in en-thalpy,

Hm(P ) = −L + Cw(T0 − γP ), (11)

andCw is the heat capacity of liquid water,γ the dependenceof the melting point on pressure,T0 the triple point tempera-ture of water, andL the latent heat of fusion for water.

At the ice surface, we specify a Dirichlet boundary con-dition corresponding to surface temperature. At the basalboundary, we apply the Neumann boundary condition:

κ(H)∇H · n = qg + qf − MbρL, (12)

whereqg is geothermal heat flux, assumed known,qf fric-tional heat generated by basal sliding, andMb the basal meltrate. Frictional heat is given by the frictional term in the mo-mentum balance functional. Note that in temperate ice, whereκ(H) is nearly zero (no diffusion), this relation defines thebasal melt rate. In cold ice, a value must be specified for the

basal melt rate (which can be negative to account for basalfreeze-on). We usually take this to be zero.

Enthalpy is uniquely related to temperature and liquid wa-ter in the following way:

T (H,P ) =

{C−1

p (H − Hm(P )) + Tm(p) if cold

Tm if temperate

ω(H,P ) =

{0 if coldH−Hm(P )

Lif temperate,

(13)

whereω is fractional water content, whileHm andTm are thepressure melting points expressed in enthalpy and tempera-ture, respectively.

2.1.3 Dynamic boundaries

The ice sheet geometry evolves over time according to thekinematic boundary condition

(∂t + u‖ · ∇‖)S = w + a, (14)

wherea is the accumulation rate.

2.1.4 Marine outlet treatment

For the Stokes’ model, VarGlaS currently treats the ground-ing line in the simplest way possible, which is to keep itslocation fixed. At this point, ice can not become ungrounded.For transient runs, the geometry of calving fronts is fixed sothat mass loss due to calving is effectively proportional to thevelocity at the calving front. At the scale of outlet glaciers,this is a limitation and is a major priority in ongoing devel-opment, but the physics of the shelf is treated. As mentioned,the first-order approximation is not conducive to the treat-ment of a coupled sheet–shelf system, and so ice flowing pasta pre-specified grounding line is assumed to calve immedi-ately.

2.2 Numerical methods

In the following sections, we discuss how the continuum me-chanical equations discussed above are discretized in orderto be made computationally tractable.

2.2.1 Finite element discretization using FEniCS

VarGlaS is built upon the finite element package FEniCS(Logg et al., 2012). FEniCS is a powerful development en-vironment for performing finite element modelling, includ-ing strong support for symbolic automatic differentiation, na-tive parallel support and parallel interface with linear alge-bra solvers such as PETSc (Balay et al., 2012) and Trilli-nos (Heroux et al., 2005), and automatic code generation andcompilation for compiled performance from an interpretedlanguage interface. The Python scripting environment makesthe generation and linking of new code straightforward. We



find that this interface provides a level of extensibility thatmakes VarGlaS promising for distributed development andrapid prototyping of models for additional components of thecryosphere.

FEniCS has a large library of finite elements available. Weuse only one, the continuous, linear Lagrange finite element,defined over an unstructured triangular mesh. This choice ofelement is unstable for advection-dominated equations, suchas the kinematic boundary condition and the enthalpy equa-tion (in most cases), as well as for Stokes’ equations due tothe pressure term. We cover stabilization procedures in thefollowing sections.

The velocity field and enthalpy equations are both non-linear. These are each solved by using a relaxed Newton’smethod (e.g.Deuflhard, 2004) with a Jacobian calculated byautomatic differentiation:

J[U i]1U = −F [U i

], (15)

U i+1= U i

+ R1U, (16)

whereU i is the solution vector at thei-th iteration,1U asolution update,J the Jacobian matrix, andF the system ofnonlinear equations.R is a relaxation parameter that arbitrar-ily shortens the step size in order to improve numerical stabil-ity. The amount of damping required is specific to the prob-lem, but we find that a relaxation parameter between 0.7 and1.0 is typically sufficient to achieve convergence. We specifyboth a relative and absolute tolerance as convergence criteriafor Newton’s method. The solution is considered convergedif the L∞ norm of 1U is less than 10−6 m a−1 or theL∞

norm of 1UUn is less than 10−3.

In order to resolve the coupling between enthalpy and ve-locity, we use a fixed point iteration. Each of these nonlinearequations is solved independently, and the result is iterativelyused as input in calculating the other variable. Convergenceis assumed when both the velocity and temperature updatesare less than 10−6 m a−1 and 10−6 K, respectively.

2.2.2 Mesh refinement

The model domain is discretized using a tetrahedral meshwhich is unstructured in the horizontal dimensions, andstructured in the vertical. In order to equidistribute discretiza-tion error, we use the anisotropic error metric according toHabashi et al.(2000):

e(c) ∝ maxi∈ExTi Mxi, (17)

wheree(c) is a cell-wise error estimate,E a given mesh cell,xi an edge inE andM a metric tensor, in this case definedby

M = V T|3|V . (18)

V and3 are the respective eigenvectors and eigenvalues ofthe Hessian matrix of the field over which error is to beequidistributed (Habashi et al., 2000). For all the meshes pre-sented forthwith, we use the Hessian of an observed veloc-ity norm (either observed or modelled) in calculating errormetrics. A discrete approximation for each component of theHessian matrix is obtained iteratively for each level of meshrefinement by solving the variational problem∫�

Hessijφ d� = −

∫�

∂U

∂xi

∂φ

∂xj

d� +

∫0

∂U

∂xi

φnxid0, (19)

where Hessij denotes the components of the Hessian andU

is the surface speed. With error estimates in hand, we isotrop-ically refine all cells that are above a specified proportion ofthe average error. In order to account for the directional na-ture of the velocity field, we incorporate anisotropy by us-ing Gauss–Seidel iterations (Habashi et al., 2000) to solveapproximately an elasticity problem, with computed edge er-rors as “spring constants”. This mixed isotropic–anisotropictechnique yields high quality and efficient meshes with boththe structural simplicity of isotropic refinement, as well asthe better error to mesh size ratio of anisotropic techniques.An example of a mesh created with this method is shown inFig. 1. Note that this algorithm does not allow mesh coarsen-ing, so the refinement procedure is initialized with a coarsemesh.

2.2.3 Data assimilation and regularization

Many physical quantities of leading order relevance toglacier and ice sheet flow are either practically impossibleto collect, or are point measurements which cannot gener-ally be extrapolated to a broader spatial context. Examplesof the former include historic variables such as a detailedrecord of surface temperature or ice impurity content at de-position. Examples of the latter include basal water pressure,basal temperature, enhancement factors, and geothermal heatflux. A particularly important parameter which must usuallybe estimated is the coefficient of basal traction, which re-lates basal shear stress to sliding velocity. In many cases,sliding makes up nearly all of a glacier’s surface velocity(e.g.Weis et al., 1999). Thus, any model that wishes to re-produce plausible velocity and thermal structures must pa-rameterize traction. The availability of widespread surfacevelocity data, and the conceptually simple relationship be-tween surface and bed velocities have made the inversion ofsurface velocities for basal traction a popular choice for per-forming this parameterization (MacAyeal, 1993; Goldbergand Sergienko, 2011; Larour et al., 2005; Gudmundsson andRaymond, 2008; Morlighem et al., 2010; Brinkerhoff et al.,2011).

We have implemented basal traction inversion in VarGlaSusing a partial differential equation-constrained optimizationprocedure. In the following, we illustrate the method using



Fig. 1. Observed surface velocity projected onto an anisotropicallyrefined mesh of Greenland’s northeast ice stream.

surface velocity in the cost functional, and basal traction asthe control variable, but the procedure is analogous to anychoice of objective function of control variable. The funda-mental concept behind this method is to define a scalar ob-jective function, to calculate its gradient, and to use standardminimization techniques to find the minimum. We use a gen-eral form for the definition of the cost functionalI ′. Exam-ples include a linear cost functional:

I ′[u] =

∫0S

||u − uobs|| d0 (20)

or a logarithmic one

I ′[u] =

∫0S

[log

||u||

||uobs||

]2

d0. (21)

We require that the velocity field obtained by this functionalsatisfies the equations of motion by imposing the momentumequations via a Lagrange multiplier:

I[u,β2] = I ′

+ δA[u,β2;λ], (22)

whereδ implies the first variation operator, andA is one ofthe energy functionals defined in Sect.2.1.1. λ is a Lagrangemultiplier used to enforce the forward model as a constraint.Taking the variation ofI with respect tou, β2, andλ yields,respectively, a forward model, an adjoint model, and an ex-pression for the gradient of the objective function with re-spect toβ2, which is expressed in terms ofu andλ. Note that

no simplifying assumptions about the nature of the forwardmodel are made. In particular, when possible we use the fulladjoint, calculated via automatic differentiation, rather thanmaking the assumption that the viscosity does not dependon u, as is done in many inversion procedures (e.g.Gold-berg and Sergienko, 2011; Larour et al., 2012). In the casewhere strong mismatches between the modelled and surfacevelocity exist, stability of the inversion numerics necessitatesfixing the viscosity, and using an incomplete adjoint, as inGoldberg and Sergienko(2011), but only for the first few it-erations.

In order to impose a minimum bound on the smoothness ofthe solution, we add a Tikhonov regularization term, whichpenalizes wiggles in the control variable. This regularizationis of the form

T = α

∫0B

||∇β2· ∇β2

|| d0, (23)

where α is a positive weighting tensor. This value isdifferent for different objective functions and differentmodel domains, and the means to determine its value isnot obvious. In the glaciological literature, L-curve anal-ysis is a popular choice that involves solving the in-verse problem with increasing regularization until a notablechange in the rate of objective function increase is found(e.g. Gillet-Chaulet et al., 2012). It is assumed that this in-flexion point represents the optimal balance between regular-ization and minimization of the objective function. We preferto use heuristics, assuming (and imposing) a maximum rateof spatial variability in the basal traction field based on fac-tors such as ice thickness. Note that applying Tikhonov regu-larization on the gradient in this way is equivalent to applyingan anisotropic diffusion operator to the control variable.

With a means of efficiently computing the objective func-tion and its gradient with respect to the control variable inhand, we can use any number of optimization algorithms tominimizeI. We use the quasi-Newton algorithm LBFGSB(Nocedal and Wright, 2000). We used a parallel implementa-tion of the L BFGSB algorithm derived from that appearingin the dolfin-adjoint library (Farrell et al., 2012). Terminationcriterion for the optimization routine is essentially heuristic,with the optimization procedure terminating upon the objec-tive function reaching a valley. The definition of a reliableconvergence criterion is a subject of ongoing research, withthe methods ofHabermann et al.(2012) particularly promis-ing.

2.2.4 Time evolution

We use two different algorithms for the discretization of time.For the enthalpy equation, we use a semi-implicit

Crank–Nicholson time-stepping scheme, with an arbitraryLagrangian–Eulerian (ALE) treatment of the convective ve-locities in order to compensate for the moving mesh (Donea



et al., 2005) (ALE operates by subtracting the mesh veloc-ity from the fluid velocity). This semi-implicit method isacceptable because of the relatively minor nonlinear cou-pling between enthalpy and the velocity field; linearizationis achieved by calculating the nonlinear dependence with thevalues of the previous time step. Crank–Nicholson providessecond-order accuracy in time, provided that the Courant–Friedrich–Lewy criterion

1t max

(κ

1

D · D,u ·

1

D

)≤ C (24)

is satisfied, whereD is the vector of cell dimensions in eachdirection andC is a constant, usually taken to be12. Notethat this condition can be restrictive in some outlet glaciers,where the combination of 1 km-scale spatial resolution and> 1 km a−1-scale velocities requires time steps on the orderof a month.

For the free surface equation, the nonlinear coupling be-tween velocity and surface elevation is stronger, so lineariza-tion is not a desirable option, and solving the full nonlin-ear problem implicitly is inefficient. We instead choose touse a fully explicit total variation diminishing Runge–Kutta(TVD-RK)-type scheme of second or third order (Gottlieband Shu, 1998). The total variation diminishing property im-plies that no spurious oscillations should be created as a re-sult of the time discretization. This must be coupled with anon-oscillatory spatial discretization (such as streamline up-winding) in order to maintain a stable solution. The TVDproperty is important in suppressing spurious oscillationsnear sharp margins (such as those characterizing glacial ter-mini). Being a Runge–Kutta method, we must solve the mo-mentum balance once for each order of accuracy, but we findthat the added stability and accuracy of using a higher orderexplicit method makes this increase in computational over-head worthwhile.

2.2.5 Stabilization

Both the enthalpy and free surface equations are hyperbolic,and the standard centred Galerkin finite element methodgives rise to spurious oscillations. In order to provide sta-bilization, we apply streamline upwind Petrov–Galerkin(SUPG) methods (Brooks and Hughes, 1982). For the en-thalpy equation, this consists of adding an additional diffu-sion term of the form

ρ∇ ·K∇H, (25)

whereK is a tensor valued diffusivity defined by

Kij =D

2

uiuj

||u||, (26)

whereD is a cell size metric. Alternatively, we can view thisstabilization as using skewed finite element test functions

φ = φ +D

2

u

||u||· ∇φ (27)

to weight the advective portion of the governing equation.Since the time derivative is implicitly defined, there is noneed to apply upwind weighting to the time derivative orsource terms, and because linear elements are used, applyingthis weighting to the diffusive component would necessitatesecond derivatives of test functions, which are always zerofor linear elements.

For the free surface equation, a similar procedure is used,where modified Galerkin test functions

φ = φ +D

2

u‖

||u‖||· ∇‖φ (28)

are used to discretize the equations. Note that in this case, dueto the explicit time-stepping scheme, the augmented weight-ing function is applied consistently to the entire residual, in-cluding the time derivative. In addition to streamline upwind-ing, we apply a shock-capturing artificial viscosity in order tosmooth the sharp discontinuities that occur at the ice bound-aries, where the model domain switches from ice to ice-freeregimes (Donea and Huerta, 2003). This additional term isgiven by

∇ ·Kshock∇S, (29)

whereKshock is the nonlinear residual-dependent scalar

Kshock=D

2||u||[∇‖S · ∇‖S]

−1R2. (30)

Here,R is the residual of the original free surface equation.For the Stokes’ equations to remain stable, it is neces-

sary either to satisfy or to circumvent the Ladyzhenskaya–Babuska–Brezzi (LBB) condition. The typical way of doingthis is to use a mixed second order in velocity, and first orderin pressure finite element (the Taylor–Hood element). WhileVarGlaS has the capacity to use this formulation, we find thatthe additional degrees of freedom introduced by the higherorder elements lead to an unacceptable loss of computationalperformance. Instead, we circumvent this condition by usinga Galerkin least squares (GLS) formulation of the Stokes’functional

A′[u,P ] =A−

∫�

τgls(∇P − ρg) · (∇P − ρg) d�, (31)

whereτgls is a stabilization parameter (Baiocchi et al., 1993).For a linear Stokes’ problem, the usual value forτgls is

τgls =D2

12η. (32)

Sinceτgls is a function of the ice viscosity,τgls should rightlybe nonlinear. However, we have found through experimen-tation that ignoring the strain rate dependence of the viscousterm yields acceptable results and much better numerical sta-bility. Thus we use

τgls =D2

12η, (33)



where η is some linear estimate ofη. We have foundη =

103× b(T ) to yield an appropriate blend of fidelity to the

governing equations and stabilization. Note that this has theeffect of adding a diffusive term over pressure to the conser-vation of mass equation.

2.2.6 Parallelism

VarGlaS has been developed to take full advantage of the in-nate parallel capabilities of PETSc (Balay et al., 2012) andFEniCS (Logg et al., 2012), on which it is built. All compu-tationally intensive components of the model are compatiblewith parallel usage, such as the nonlinear solvers, time step-ping, and optimization. VarGlaS exhibits good scaling be-tween 1 and 16 cores, the largest cluster to which the authorshave access. Parallel efficiency for a nonlinear solution of thefirst-order equations over all of Greenland for over a milliondegrees of freedom is shown in Fig.2.

2.2.7 Verification with the method of manufacturedsolutions

We use the method of manufactured solutions (MMS,Salariand Knupp, 2000) to construct analytical solutions to the mo-mentum balance and surface evolution equations in order todetermine the extent to which our numerical solutions repro-duce the exact results. This procedure is known as verifica-tion, which ensures that the given equations are being solvedcorrectly and consistently (this is held in contrast to valida-tion, which is the procedure of ensuring that the appropriateequations for a given physical scenario are being solved). Theuse of MMS in the context of glaciological models has beenpreviously developed byBueler et al.(2005), with more re-cent results bySargent and Fastook(2010) andLeng et al.(2013). We adopt a similar approach to these works, but likeLeng et al.(2013) and unlikeBueler et al.(2005), we explic-itly forego verifying thermomechanical coupling.

The main idea behind MMS is to select an arbitrary solu-tion, and then construct a source function that produces thatsolution. To that end, we select the following functions forthe components of the ice velocity and free surface:

u(x, t) = ud

[1−

S(x, t) − z

S(x, t) − B(x)

]+ ub, (34)

v(x, t) = 0, (35)

w(x, t) = −

z∫B(x)

[∂u

∂x(x′, t) +

∂v

∂y(x′, t)

]dz′, (36)

S(x, t) = −x tanα + Amp(1− e−tτ )

× sin

(2π

Lx

)sin

(2π

Ly

). (37)

Also, we have thatB(x) = S(x, t = ∞)− 103, L = 104, andAmp= 500, which corresponds to the geometric set-up of

0 2 4 6 8 10 12 14 16N Processors

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Para

llelE

ffici

ency

4× 104 DOF9× 104 DOF9× 105 DOF

Fig. 2.Parallel efficiency computed for one complete Newton solveof the first-order equations for meshes with different degrees offreedom. All linear solves were performed using parallel GMRES(generalized minimal residual method) preconditioned with Hypre-AMG.

ISMIP-HOM A (Pattyn et al., 2008). We setud = 20 m a−1

andub = 10 m a−1

We begin by verifying the computation of the velocityfield, which is a pseudo-steady-state calculation; the veloc-ity field’s only dependence on time is through the problemgeometry. This implies that we can verify the velocity fieldindependent of time, and for this procedure sett = 0.

We seek vector-valued functionsF(x, t) andG(x, t) suchthat the functionsu(x, t) that we have selected minimize thefunctional

A[u] =

∫�

[2n

n + 1η(ε2)ε2

+ G · u

]d�

+

∫0

u diag(G) uT d0, (38)

where diag(G) is a diagonal matrix with the components ofgon the diagonal. Note that we have eliminated the Lagrangemultiplier P from the functional, sinceu is constructed tobe divergence-free. Taking the variation with respect tou

yields the Stokes’ equations with arbitrary forcings withinthe model domain and at the boundary. Sinceu is analytical,we can invert forF and G. The procedure is the same forfinding the forcing functions for the first-order equations, butwe use the first-order functional Eq. (6).

While the functions used in the MMS are arbitrary, wenote a few features of our selection. We have eliminatedP

from the formulation by a specific choice of velocity field.



One may view the pressure as itself being an arbitrary forc-ing function, and in this case it becomes part ofF and G.Also, we have chosenv = 0 because it greatly simplifies theanalytical source terms, and also because it allows us to carryout an important test of rotational symmetry by swapping thedefinitions ofu andv (and their spatial arguments).

With the analytic forcing functionsF andG in hand, aswell as their corresponding analytical solutions, we can solvethe discrete model over progressively finer meshes and eval-uate whether the error in the numerical solution tends to-wards zero as the discrete domain better approximates thecontinuum. In practice, computational intensity limits themaximum level of refinement, and it is sufficient to showthat a discretization converges towards the analytical solu-tion at close to the theoretical order of accuracy, which forlinear finite elements isO(1x2) (Salari and Knupp, 2000).We solved both of these problems over meshes ranging from1x = 5×103 m to1x = 40 m, two orders of magnitude. Theresulting error profiles are shown in Fig.3. The slope of therelativeL2 norm for both approximations is close tom = 2in logarithmic space, which is the theoretically correct resultfor linear elements. This confirms that our discretization pro-cedure shows theoretical order of accuracy convergence andthus correctly solves the governing equations for the velocityfield.

Verification for our free surface evolution scheme pro-ceeds similarly, except that refinements to the computationaldomain are carried out both in time and space. We seek acompensatory accumulation fieldc, such that the kinematicboundary condition

∂t S + u∂S

∂x+ v

∂S

∂y= w + c (39)

holds for the pre-defined, analytical functionsS and u.Again, it is a simple matter to determine a closed form ex-pression forc. We use the same analytical functions as de-fined above, but allowt to vary. To test the convergence ofour time-stepping scheme, we progressively refine1t andrun the model tot = τ using the compensatory accumulationc in place ofa in Eq. (14) and evaluate the error in surfaceelevation at that time. Note that because we are using an ex-plicit time-stepping algorithm, the Courant–Friedrich–Lewy(CFL) condition applies, limiting the minimum cell size thatcan be used for a given time step. Simultaneously, we requirea certain spatial accuracy such that errors in the temporal dis-cretization are not overwhelmed by errors resulting from thespatial discretization. Thus, as we refine the time step, wealso refine the spatial resolution such that the Courant num-berC = 0.1 for all simulations. This ensures that instabilitydue to a violation of the CFL condition does not occur, andthat an appropriately fine spatial resolution is used for a giventime step.

We refine between1t = 50 a and1t = 0.1 a, which corre-spond to spatial resolutions between1x = 104 m and1x =

101 102 103 104

∆x (m)

10-5

10-4

10-3

10-2

10-1

100

101

|u−

u| (

ma−

1)

L2 (FO Measured)

L2 (FO Best Fit, m=2.01)

L∞ (FO Measured)

L∞ (FO Best Fit, m=1.88)

L2 (Stokes Measured)

L2 (Stokes Best Fit, m=1.96)

L∞ (Stokes Measured)

L∞ (Stokes Best Fit, m=1.65)

10-1 100 101 102

∆t (a)

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1|S

(x,τ

)−S(x,τ

)| (m

)L2 (Measured)

L2 (Best Fit, m=2.79)

L∞ (Measured)

L∞ (Best Fit, m=2.80)

Fig. 3. Error between numerical and analytical solutions versusmesh cell size and time step. The slopes of these lines represent theconvergence rates of the spatial and temporal discretization methodsused by VarGlaS.

20 m. Surface elevation errors are shown in Fig.3. The theo-retical behaviour of our RK2 algorithm implies a single-steperror ofO(1t3) and a cumulative error over time ofO(1t2)

(Gottlieb and Shu, 1998). These errors correspond to a slopeof m = 3 andm = 2 respectively, and we expect the slope ofthe error in surface height at timeτ with respect to1t to bebetween these two values. Indeed, we find thatm = 2.8 for



both theL2 andL∞ norms. This result shows that our time-stepping scheme possesses theoretical convergence proper-ties, and can be considered verified.

We noted earlier that our choice of velocity fields couldbe used to test symmetry. We performed both of the aboveconvergence tests for a swappedu and v (with associatedswapped coordinatesx andy), and produced convergence re-sults that were identical to machine precision. Thus we con-clude that VarGlaS possesses the appropriate properties ofrotational symmetry.

3 Numerical experiments

In order to assess the correctness and efficiency of model,we apply it to a number of well-known ice sheet modellingbenchmark experiments, before turning it towards a large-scale data assimilation and prognostic time-stepping simula-tion for the whole Greenland ice sheet.

3.1 ISMIP-HOM

The Ice Sheet Model Intercomparison Project–Higher OrderModel (ISMIP-HOM) benchmarks are a widely used test ofhigher order model capabilities (Pattyn et al., 2008). In orderto verify our model performance, we run ISMIP-HOM testsA, B, C, D, and F using both the first-order and Stokes’ equa-tions for the momentum balance. All meshes for the ISMIP-HOM experiments used structured grids with 20 elements ineach horizontal dimension, and 10 elements in the verticaldimension, for 4851 degrees of freedom.

ISMIP-HOM A and ISMIP-HOM B simulate steady iceflow with no basal slip over a sinusoidally varying bed withperiodic boundary conditions. B varies from A in that thebasal topography does not vary in the transverse direction,whereas in A it does. Note that despite the fact that B is uni-form in one spatial dimension, we still treat it as a three-dimensional problem and use the same discretization andboundary conditions (periodic, namely) as for A. Figures4and5show the simulated surface velocity for all length scalesoutlined in the benchmarks.

ISMIP-HOM C and ISMIP-HOM D simulate steady iceflow with sinusoidally varying basal traction over a flat bedwith periodic boundary conditions. Again, D varies from Cin that only C is transversely varying. Figures6 and7 showthe simulated surface velocity for all length scales outlinedin the benchmark. This experiment specifies thatr = 0 in thesliding law.

Experiments A–D were run in serial with a 2.2 GHz laptopcomputer. Runs using the first-order approximation took ap-proximately 10 s, while runs using the Stokes’ approximationtook around 60 s.

After running the ISMIP-HOM C experiment forward, wehave in hand the velocity field predicted by the model fora given basal traction. We used this as an opportunity to

test the inverse capabilities of the model, and to invert for aknown basal traction. We performed the inversion using thefirst-order approximation. Starting from an initial guess of auniform basal traction of 103 Pa a m−1, we allow the inversemodel to predict the basal traction field that produces the ve-locity field of the forward model (which we know to be asinusoid). In assessing misfits, we used the linear cost func-tional Eq. (20), because the velocity scale varies relativelylittle over the domain. This problem was unregularized, soα = 0. Figure8 shows both the rate of convergence as wellas the “observed” and modelled basal tractions and surfacevelocities along theL

4 transect of the ISMIP-HOM modeldomain using the results for the first-order approximation attheL = 10 km length scale. Results for other length scales aresimilar.

ISMIP-HOM F simulates unsteady ice flow over a Gaus-sian bump with periodic boundary conditions under slip andnon-slip conditions, and evaluates the surface geometry andvelocity as they relax to steady state. Experiment F also man-dates a linear rheology, son = 1. Figure9 shows the simu-lated surface velocities and elevations for the slip and non-slip cases, using a time step of1t = 1 a. These experimentswere run on a 4-core machine with 8 GB of memory and2.2 GHz per processor. Using the first-order approximation,the 500 a run took approximately 50 min. Using the Stokes’approximation, the run took 5.5 h.

3.2 EISMINT II

The European Ice Sheet Model INTercomparison II (EIS-MINT II) benchmarks are an older set of benchmarks thanISMIP-HOM, and were designed for use with models em-ploying the shallow ice approximation (Payne et al., 2000).While ISMIP-HOM generally tests the accuracy of the mo-mentum balance scheme over varying length scales, theEISMINT experiments are more focused towards assessingthe time-dependent mass and energy balances. EISMINT IIstresses the dynamic evolution of the system and provides atest for the long-term stability of our time-stepping scheme.We know of no other higher order model operating on anunstructured grid that has demonstrated a capacity to run for-ward on such timescales.

In EISMINT II A, a radially symmetric surface mass bal-ance and temperature field are imposed on an initially ice-free, flat bed. The model geometry and temperature field areallowed to evolve for 200 ka. Our chosen mesh had a cellsize of 25 km in the horizontal dimension, and 10 verticallayers, for a total of approximately 32 000 degrees of free-dom. We used a time step of 10 a, and on a 16-core machinewith 400 GB of memory and 2.9 GHz per processor, this run(and EISMINT II F) took approximately 65 h. At the end ofthis period, the total energy and mass were changing by lessthan 104 % a−1, implying near-steady-state conditions. Fig-ure 10 shows the resulting thickness and basal temperaturefields. Thickness and basal temperature at the centre point of



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ISMIP-HOM Experiment A

Normalized X coordinate

IceSpeed(m/a)

FOA OutputStokes Output

Stokes MeanStokes Std. Dev.

First Order MeanFirst Order Std. Dev.

Fig. 4. ISMIP-HOM A performed using first-order and Stokes’ approximations.

the ice sheet were 3647 m and 255.4 K respectively. Theseare important metrics for intercomparison performance, andlie near the benchmark means of 3688 m and 255.605 K.

EISMINT II F is identical to experiment A, except that im-posed surface temperatures are 15◦ C colder throughout themodel domain. The results of this experiment are similar tothose of experiment A, albeit with a thicker ice sheet, anda slightly different basal temperature profile. The resultingthickness and basal temperature fields are given in Fig.10Aninteresting difference between the resulting temperature fieldhere and that documented in the EISMINT II paper is thatVarGlaS does not predict the breakdown in radial symmetrythat occurred in all of the finite difference models of the inter-comparison. We suspect that this is due to VarGlaS using anunstructured grid, which alleviates some of symptoms of griddependency seen in the original experiment. This hypothesisis supported by the work ofSaito et al.(2006), who providesevidence that suggests that the “spoking” phenomenon seenin Payne et al.(2000) is numerical in origin. It could be the

case that using an unstructured grid removes the grid-inducedpathways that allow these spokes to form.

3.3 Greenland

We applied VarGlaS to a large-scale problem in glaciology,namely the transient simulation of the Greenland ice sheet.The strategy in so doing was to initialize the model us-ing measured present-day geometry, apply data assimilationtools to obtain an initial estimate of the basal traction field,and then allow Greenland’s geometry, velocity, and temper-ature to evolve over 500 yr. We performed all simulations ofGreenland using the first-order approximation for the mo-mentum balance (see Eq.6).

3.3.1 Data

We relied on the SeaRISE model set-up for input data(SeaRISE, 2012). Bedrock and surface geometry were fromBamber et al.(2001) with updated basal topography in theJakobshavn region from CReSIS, surface temperatures from



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ISMIP-HOM Experiment B


IceSpeed(m/a)


Full Stokes MeanFull Stokes Std. Dev.


Fig. 5. ISMIP-HOM B performed using first-order and Stokes’ approximations.

Fausto et al.(2009), basal heat fluxes fromShapiro and Ritz-woller (2004), and surface mass balances fromEttema et al.(2009). We used InSAR-derived 2007–2008 average surfacevelocities fromJoughin et al.(2010) for a surface velocitytarget. The Joughin data set is incomplete. The gaps werefilled with balance velocities, with gradients between the tworeduced by systematically exploring the uncertainties in theaccumulation rate. Specifically, we used a steepest descentalgorithm to minimize the misfit between the edges of theInSAR velocities and the corresponding balance velocitiesby varying the surface mass balance subject to the constraintthat it remains within its reported point-wise error bounds.

3.3.2 A mesh for Greenland

The boundary of Greenland was digitized using the 1 m con-tour of theBamber et al.(2001) thickness data. We createdan initial two-dimensional (map plane) mesh by imposing a2 km element size at the margins, grading to a variable butmuch coarser resolution at the centre of the ice sheet. This

ensured that the mesh captured the complexity of the bound-ary while maintaining appropriate coarseness in the interior.We refined the mesh using the techniques of Sect.2.2.2. Weextruded this footprint over 10 vertical layers. Greenland,when more highly resolved in the vertical dimension, demon-strates convergence problems during Newton’s method solu-tion process. Similar issues have been reported by other high-resolution, higher order models, such as Elmer/Ice (Seddiket al., 2012) and ISSM (Larour et al., 2012). Larour et al.(2012) in particular demonstrated that this convergence issueis a result of very low aspect ratio elements producing poormatrix conditioning. This is a significant limitation, and at-tempts to overcome it are ongoing. Preliminary experimen-tation suggests that using the shallow ice approximation asa physically motivated preconditioner may help amelioratethis problem, but we have thus far been unsuccessful in ap-plying this scheme at a continental scale. The generated meshwhich is used for the data assimilation experiment has around4.4× 105 nodes, corresponding to 8.8× 105 degrees of free-dom for the first-order model. The horizontal resolution for



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IceSpeed(m/a)




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this mesh is variable, but grades from 500 m at the edges ofthe domain to approximately 100 km over the interior of theice sheet. We used a coarser mesh with 4× 104 nodes for ourtransient run, corresponding to 8× 104 degrees of freedomfor the first-order model. This corresponds to a minimumhorizontal resolution of 5000 m up to approximately 100 kmin the interior.

3.3.3 Data assimilation

We calculated a basal traction field using the techniques ofSect.2.2.3. We begin by calculating steady-state velocity andenthalpy fields for an arbitrary basal traction (with an initialguess of 4 Pa a m−2), with r equal to unity (which impliesthat basal traction is linearly scaled by thickness; this effec-tively eliminates the dependence of sliding speed on normalforce, and eliminates the covariance betweenβ2 andh). Withthis initial state in hand, we ran the BFGS algorithm using afixed viscosity (and an incomplete adjoint) for the first 10iterations, before switching to a full adjoint. The selection

of the regularization parameterα is motivated by the resultsof Balise and Raymond(1985), which indicate that belowthe one ice thickness length scale, variations in basal tractiondo not propagate to the surface. As such, we selectα = h2,which corresponds to applying smoothing with a Gaussiankernel with standard deviationh to the basal traction field.The temperature field was also recomputed every 50 evalua-tions of the objective function in order to maintain thermalequilibrium. After the first 10 iterations, the velocity fieldwas visually indistinguishable from that of the data, and theconvergence between temperature and velocity fields becamea fixed point iteration on the enthalpy field. The BFGS algo-rithm was allowed to run for 200 evaluations of the objectivefunction, which took approximately 3.5 h on a 16-core ma-chine with 2.9 GHz per processor and 400 GB of memory.Convergence of the algorithm is shown in Fig.11. The ob-served and modelled velocities, along with basal traction andtemperature fields, are shown in Fig.12. To illustrate someof the fine-scale detail of both the mesh and the data assimi-lation result, Fig.13 shows a close-up of Kangerdlugssuaq



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glacier in eastern Greenland. The modelled velocity fieldmatches the observed field closely, and the RMS error forthe whole ice sheet is approximately 36 m a−1. For outletglaciers like Kangerdlugssuaq, we see that surface velocitycan be explained by spatially variable basal traction fieldcomposed of both low traction streaming features, and stickypinning points that slow flow. Basal temperature is also re-lated to basal traction, where fast sliding is associated with amelted bed.

3.3.4 Prognostic run

After performing the data assimilation procedure outlinedabove, we allowed the ice sheet to evolve through time for500 yr under the C1 (constant climate) SeaRISE experiment,with a time step of 0.1 a. On a 16-core machine with 400 GBof memory and 2.9 GHz per processor, this run took 27 h.The present-day measured surface elevation of Greenland isnot in a steady state with respect to the basal topographyand the inverted velocity field, which is unsurprising given

the errors associated with the basal topography, the surfacevelocity field, and the approximations made by the modelphysics. Because of this mismatch, large transient signalspropagate through the system at the beginning of the run.We monitored the size of these transients as theL∞ normof the ∂tS field, given by Fig.14. Note the exponential de-cay rate; this gives some estimate of how much relaxationa model requires in order to eliminate transients. Resultsfrom Pritchard et al.(2009) show that the average∂tS of theGreenland ice sheet (GrIS) is−0.84 ma. An ice sheet modelshould have relaxed to around this level before any conclu-sions should be drawn from additional forcing being appliedto it. For VarGlaS, with the specified data inputs, this pro-cess takes about 100 a. We also monitored the mass of theentire ice sheet throughout time. After the initial transient pe-riod of ice increase, we found an annual average total ice de-crease of 10−3 % a−1 (2.63× 103 Gt a−1), which is in order-of-magnitude agreement with the GRACE-derived ice lossof approximately 6× 10−3 % a−1 (17.7× 103 Gt a−1) (Bauret al., 2009). As seen in Fig.15, the qualitative distribution



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of the basal temperature field changes relatively little over500 yr. Although relative to the magnitude of their initial val-ues, thickness and velocity also change relatively little; theyshow an interesting qualitative pattern that explains the massloss of Fig.14. We see that over time the combination ofsurface mass balance and basal traction tends towards thin-ning the interior of the ice sheet, while thickening the edges.This change is most notable where ice stream development isapparent, such as Jakobshavn and Kangerdlugssuaq, as wellas the outlet glaciers of the southern GrIS. This increase inthickness generates a steeper ice surface at outlets, elevatingthe flux through the lateral boundaries. Since ice loss due tosurface mass balance is constant (because surface mass bal-ance itself is constant), the change in total mass for the GrISis driven wholly by this increased boundary flux. This patternof thickness and velocity evolution is also seen in compara-ble simulations of the GrIS, such as inSeddik et al.(2012)

andLarour et al.(2012). This pattern is problematic, becauseit suggests an opposite redistribution of mass from that mea-sured through remote sensing (Pritchard et al., 2009; Bauret al., 2009). More work is needed to determine if this massloss rate is believable.

4 Discussion

VarGlaS performs well in a number of standard benchmark-ing experiments and can also simulate the evolution of theGrIS using higher order physics and thermomechanical cou-pling, as well as an advanced treatment of time evolution.

For the diagnostic and isothermal ISMIP-HOM bench-mark experiments, both VarGlaS first-order and Stokes’solvers perform well with respect to existing benchmark re-sults, with our first-order solver generally predicting valuesclose the the Stokes’ mean, and our Stokes’ solver generallypredicting velocities slightly slower than those reported bythe benchmark. Our data assimilation procedure is able to re-produce effectively these simple imposed basal traction fieldsthrough model inversion. Both first-order and Stokes’ solversperformed well on the prognostic ISMIP-HOM F, yieldingvelocity and surface elevation fields that are in good agree-ment with other models. Results for the EISMINT II exper-iments are similar, with model results comparing favourablyto mean values from the original publication. Also, thicknessand temperature fields compare well with the results ofPat-tyn (2003) andSaito et al.(2003), who also applied higherorder models to these experiments. Note that this is the firsttime that the EISMINT II experiments have been performedwith a higher order finite element model using an unstruc-tured mesh.

VarGlaS’ behaviour over the entire Greenland ice sheetechoes what has been determined by various investigatorsin the past, which is that after the relaxation of a strongtransient signal derived from the incompatibility of flawedbasal topography, surface velocities, and surface mass bal-ance data, the ice sheet seems to be losing mass on the orderof 10−3 % a−1, which is in agreement with contemporary es-timates, although the qualitative distribution of this mass lossis not entirely believable. Performing simulations of this tem-poral length and at this spatial resolution has only been madepossible by a combination of advances in ice sheet modellingtechnology, namely variable spatial resolution, data assimila-tion, and parallelism.

We used an adjoint model derived from automatic differ-entiation to invert the model at continental scale, yieldingplausible velocity fields. Inversion was a critical step for afew reasons; insofar as the measured velocity and surface el-evation fields are correct and the geometry of the ice sheet isself-consistent, the inversion procedure minimizes transientsignals, and allows the model to reach a self-consistent statemore quickly than would be otherwise possible with a lesssophisticated starting procedure. This reasoning extends to



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the calculation of a starting enthalpy field, which would beof a much lower quality without incorporating the very sig-nificant heat source due to friction at the bed.

We employed an anisotropically refined, variable resolu-tion mesh in order to minimize superfluous degrees of free-dom in slowly varying regions of the ice sheet while main-taining detailed solutions in regions of large velocity gradi-ents. Although variable resolution modelling is not impos-sible with finite differences (Colella et al., 2000; Cornfordet al., 2013), it is applicable to finite elements in a straight-forward way.

Parallelism was another critical component in modellingthe whole Greenland ice sheet. The number of degrees offreedom is simply too large for one processor to handle in areasonable amount of time. We found that we retained bet-ter than 50 % parallel efficiency for nearly 1 million degreesof freedom and 16 processors, using an iterative solver. Thiscorresponds to a speed-up of around a factor of 10. With theincreasing availability of large computers with many proces-sors, the benefit of incorporating parallelism into model de-

sign is clear. Diagnostic modelling of very large ice sheets atresolutions similar to those of contemporary data products ispossible with even modest computers. For example, in thispaper, we perform data assimilation over the entire Green-land ice sheet at the horizontal scale of a few ice thicknessesusing 16 cores in 3.5 h, which we believe is reasonable. Forprognostic simulations with this same resolution, particularlypast the century timescale, larger clusters are necessary. Forexample, for prognostic simulations with a 0.1 a time stepand the same spatial resolution as the data assimilation exer-cise, using the same machine, we can simulate one year ofmodel time in one day of real time. This is not practical. For-tunately, there now exist many clusters with many thousandsof processors, and this type of computing power is becomingmore available, such that, with sufficient parallel efficiency,solutions of problems at long timescales and at high spatialand temporal resolution are plausible.

VarGlaS currently does not possess a detailed treatmentof marine terminal processes, namely grounding line migra-tion dynamics and a prognostic calving law. Each of these



Fig. 10.Thickness and basal temperature fields for EISMINT II A and F at 200 ka.

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presents its own computational and theoretical challenges.It is well known that grounding line dynamics operates ata scale that is typically sub-grid relative to the whole icesheet field equations (Nowicki, 2007; Favier et al., 2012).In many cases, relatively small changes in the position of thegrounding line probably do not greatly affect the evolution ofan ice sheet at a continental scale, particularly in cases likeGreenland where outlets are spatially localized, and topog-raphy is the primary control on outlet configuration. Never-theless, scenarios such as the potential collapse of the WestAntarctic ice sheet due to a fundamentally unstable bed ge-ometry provide a strong motivation for getting the physicsright, and higher order physics is mandatory in so doing. Var-GlaS is in a good position to incorporate a detailed treatmentof grounding line dynamics. The detailed spatial resolutionnecessary for capturing the physics can be managed by theexisting mesh refinement code. Additionally, VarGlaS alsopossesses robust free surface stabilization that will certainlybe necessary for performing simulations of grounding linemigration on complex real world topography. A prognostic



Fig. 12. Modelled and measured surface velocities, basal traction,and basal temperature of the GIS after assimilation of surface ve-locity. RMS velocity mismatch is 36 m a−1.

calving would also be necessary to simulate accurately thegrounding line dynamics. VarGlaS currently does not havethe capacity to move the lateral bounds of its computationaldomain. Leaving the thickness at the lateral boundaries fixedis equivalent to making the assumption that any flux throughthe boundary in excess of the present rate calves immedi-ately, or is otherwise removed from the ice sheet, and thatthe height of that boundary is fixed. This may be valid forcontinental scale model runs where the geometry of the icesheet is not expected to adjust dramatically. This treatmentis certainly insufficient for regional-scale experiments suchas modelling the response of inland glaciers to ice shelf col-lapse.

Data assimilation is an essential part of correctly mod-elling ice surface velocities. Without using inverse methodsto estimate the value of the basal traction field, the observedpattern of surface velocities is not well reproduced, and the

Fig. 13. Modelled and measured surface velocities, basal traction,and basal temperature of Kangerdlugssuaq glacier in eastern Green-land after assimilation of surface velocity.

present-day ice configuration is not (and should not be) cap-tured. Additionally, without relying on inverse methods, longand computationally expensive spin-up procedures are re-quired, which are not feasible from a processing standpoint,even with the efficient structure of VarGlaS and other mod-ern ice sheet models. Simultaneously, we must recognize thelimitations of inverting for the basal traction field. The inver-sion is ill-posed and often lacks a physically motivated stop-ping point for optimization algorithms. Even with the inclu-sion of a regularization term, there generally exist multiplesolutions for the basal traction field that produce plausiblesurface velocity results (although many qualitative featuresmust exist in all solutions). This lack of uniqueness makesdrawing conclusions about basal conditions at specific pointsfrom inverted models tenuous. Also, the inversion proceduredoes not allow for time dependency of the basal tractionfield. Basal traction is believed to be fundamentally linked



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to subglacial water routing and pressure. Under changingclimate scenarios, delivery of water to the bed may changedramatically, and this could fundamentally alter basal trac-tion. Similarly, changes in ice sheet geometry are expectedto change the pattern of basal traction through changes in iceoverburden pressure as well as surface-elevation-forced wa-ter input. For a non-trivial portion of the ice sheet, basal trac-tion is a zero-order control on all model physics. Long-termprognostic simulations involving major changes in climate orice sheet geometry must include a mechanism for estimatingchanges in basal traction.

With the major increases in efficiency gained from par-allelism and anisotropic mesh refinement, it is tempting tomodel on increasingly detailed meshes, simply due the phi-losophy that this will give us more detailed and thus moremeaningful results. In an ideal scenario, the data from whichwe draw our surface elevations, bed elevations, surface ve-locities, and surface mass balances are effectively error-freeand at spatial resolutions much finer than the grids on whichwe model. This may have been the case in the past, whenthe errors derived from using the shallow ice approximationwere larger than those inherent in data, and low spatial res-olution grids could average from a number of data pointsfalling within a given pixel. It is certainly not the case now.It is simple to generate meshes with a horizontal resolutionof near an ice thickness, but it is not simple to determinehow to interpolate accurately from 5 km thickness data downto this resolution. Nevertheless, in some heavily studied re-

Fig. 15. Velocity, surface elevation, and basal temperature changethrough a 500-year simulation of the Greenland Ice Sheet.

gions of the GrIS, data at thickness-scale horizontal resolu-tion exist (e.g. Jakobshavn, Russell–Isunnguata Sermia), andthese should be incorporated when possible. For the future,we intend to incorporate error metrics over all of VarGlaS’sinput data into our mesh generation procedure, in order toavoid obtaining spurious results and using unnecessary com-putational resources as a result of over-resolving in regionswhere the data do not have a commensurate level of detail.

5 Conclusions

In this paper, we have introduced a novel next-generation icesheet model called VarGlaS. VarGlaS is built upon the finiteelement package FEniCS, and borrows heavily on the innatecapabilities of FEniCS, with such features as automatic dif-ferentiation of the adjoint state, an interface to a variety of



efficient solvers, and demonstrably scalable parallelism. Var-GlaS eschews the common stress balance formulation of iceflow physics in favour of formulation as the problem of min-imizing a scalar variational principle representing the con-version of gravitational potential energy into heat under theconstraint of incompressibility. We use an enthalpy formula-tion for the energy balance equations, exchanging the tem-perature equation’s contact problem for additional nonlinear-ity. VarGlaS treats conservation of mass using the kinematicboundary condition.

We applied VarGlaS to the ISMIP-HOM benchmarks forhigher order models, as well as several of the EISMINT IIexperiments. VarGlaS performed well in all of these bench-marks, proving that it correctly solves the field equations.Additionally, we applied VarGlaS’s data assimilation to oneof the benchmark experiments, showing that it can recover asimple basal traction field from an observed velocity field.

We then turned model to diagnostic and prognostic simu-lation of the Greenland ice sheet. We began by solving forbasal traction using interferometrically derived surface ve-locities. Using this basal traction field, and other data fromthe SeaRISE data set, we solved for the geometry, temper-ature, and velocity of Greenland 500 yr into the future. Wefound that, after a brief period of relaxing transient signals,the model predicts a 10−3 % a−1 decrease in total mass overthe 500 yr period.

Code repository

A developmental version of VarGlaS including many ofthe experiments discussed above is available athttp://code.launchpad.net/um-feism.

Acknowledgements.D. J. Brinkerhoff was supported by NASAgrant NASA EPSCoR NNX11AM12A. J. V. Johnson was sup-ported by NASA grant NASA EPSCoR NNX11AM12A andNational Science Foundation grant NSF 0504628. This manuscriptwas greatly improved due to the comments of Andy Aschwan-den, Steve Price, two anonymous reviewers, and the editor,Olivier Gagliardini.

Edited by: O. Gagliardini

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