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6Arctic Centre, University of Lapland, Rovaniemi, Finland Open Access Open Access 7Uppsala University, Uppsala, Sweden The Cryosphere The Cryosphere Correspondence to: O. Gagliardini ([email protected]) Discussions

Received: 11 February 2013 – Published in Geosci. Model Dev. Discuss.: 4 March 2013 Revised: 8 June 2013 – Accepted: 12 July 2013 – Published: 22 August 2013

Abstract. The Fourth IPCC Assessment Report concluded preconditioned solver for the Stokes system. In this paper, all that ice sheet flow models, in their current state, were un- these components are presented in detail, as well as the nu- able to provide accurate forecast for the increase of polar merical performance of the Stokes solver and developments ice sheet discharge and the associated contribution to sea planned for the future. level rise. Since then, the glaciological community has un- dertaken a huge effort to develop and improve a new genera- tion of ice flow models, and as a result a significant number 1 Introduction of new ice sheet models have emerged. Among them is the parallel finite-element model Elmer/Ice, based on the open- Since the 2007 IPCC report (Solomon et al., 2007), theoret- source multi-physics code Elmer. It was one of the first full- ical glaciology has taken a big leap towards improved ice Stokes models used to make projections for the evolution sheet flow models, in order to provide reliable future es- of the whole Greenland ice sheet for the coming two cen- timates of the dynamical contribution of ice sheets to sea turies. Originally developed to solve local ice flow problems level rise. These models were originally designed to recon- of high mechanical and physical complexity, Elmer/Ice has struct the evolution of ice sheets over past glaciological cy- today reached the maturity to solve larger-scale problems, cles, neglecting short-term responses and local features. The earning the status of an ice sheet model. Here, we summarise new challenge of running ice sheet models to provide esti- almost 10 yr of development performed by different groups. mates of future sea level rise has created the need for a new Elmer/Ice solves the full-Stokes equations, for isotropic but generation of ice sheet models (Vaughan and Arthern, 2007; also anisotropic ice rheology, resolves the grounding line Gillet-Chaulet and Durand, 2010; Blatter et al., 2011; Kirch- dynamics as a contact problem, and contains various basal ner et al., 2011; Alley and Joughin, 2012). This new genera- friction laws. Derived fields, like the age of the ice, the tion of ice sheet models includes a set of requisites that are strain rate or stress, can also be computed. Elmer/Ice in- essential to provide a sufficiently accurate description of the cludes two recently proposed inverse methods to infer badly ice flow dynamics. known parameters. Elmer is a highly parallelised code thanks to recent developments and the implementation of a block

Published by Copernicus Publications on behalf of the European Geosciences Union. 1300 O. Gagliardini et al.: Elmer/Ice model

As a first requisite, these models must be able to describe more complex cases, the link between changes in the ocean the ice flow heterogeneity, and particularly the major con- and/or atmosphere and changes in the ice flow is indirect. In- tribution of individual ice streams to the total ice discharge. termediate processes (often not observable) are involved, as This requires the use of an unstructured mesh in the horizon- in the case for example of the link between surface runoff and tal plane (e.g. Gillet-Chaulet et al., 2012; Larour et al., 2012; basal sliding or ocean temperature and calving rate. Thus, Seddik et al., 2012) or of adaptive multi-grid methods (Corn- a dedicated model is required to describe the processes re- ford et al., 2013b). These mesh techniques are essential to sponsible for the transfer of these changes to the ice mass. produce hundred-metre-scale grid sizes in areas of interest, Driving this dedicated transfer model might require coupling especially near the coast, while for the interior regions where the ice sheet model with an atmosphere or an ocean model. variations in velocity gradients are small, classic grid sizes The last important requisite for a forecast model is to be can be kept to save computing resources. Grid refinement is able to simulate present-day observations with as much fi- even more essential when considering the dynamics of the delity as possible (Aschwanden et al., 2013). This point must grounding line, i.e. the boundary between the grounded ice be addressed clearly using data assimilation techniques and sheet and the floating ice shelf, because a grid size that is specific inverse methods to estimate the less well known pa- too large gives inconsistent grounding line dynamics (Du- rameters of the model (e.g. Heimbach and Bugnion, 2009; rand et al., 2009; Pattyn et al., 2013). Arthern and Gudmundsson, 2010; Morlighem et al., 2010). The second important requisite is to have an accurate de- Recent ice sheet model developments have started to fulfil scription of the complex state of stress prevailing in ice some of these priority requisites, leading the way toward the streams to solve the full-Stokes system, or at least to adopt new generation of ice sheet models (Bueler and Brown, 2009; a higher-order asymptotic formulation. As shown by the Pollard and DeConto, 2009; Rutt et al., 2009; Larour et al., ISMIP-HOM inter-comparison exercise (Pattyn et al., 2008), 2012; Leng et al., 2012; Winkelmann et al., 2011; Favier higher-order models are needed to describe the ice flow in ar- et al., 2012; Gillet-Chaulet et al., 2012). Among them, the eas where the basal topography and slipperiness vary greatly, Elmer/Ice model already includes many of these requisites. which are generally the most dynamic regions within ice Elmer/Ice is the glaciological extension of Elmer, the open- sheets. Higher-order models are also necessary to properly source finite element (FE) software developed by CSC in describe the dynamics of the grounding line. The MISMIP Finland (http://www.csc.fi/elmer/). Elmer is a multi-physics inter-comparison (Pattyn et al., 2012) indicated the need to code base from which it was possible to develop new spe- solve the full-Stokes equations near the grounding line to ob- cialised modules for computational glaciology while main- tain fully accurate results. taining the compatibility with the main Elmer distribution. The consequences of these first two requisites, i.e. high Thus, Elmer/Ice still benefits from the developments of the numerical resolution at places of interest and higher-order standard Elmer distribution. In this paper, for simplicity we formulations, are a high computing cost and the necessity to refer to Elmer/Ice even if some of the features described be- develop parallel codes, able to run on hundreds of CPUs. Re- long to the main Elmer distribution. Elmer/Ice was not orig- cent studies (Larour et al., 2012; Gillet-Chaulet et al., 2012; inally designed as an ice sheet model since the first appli- Seddik et al., 2012; Cornford et al., 2013b) have fulfilled cations were restricted to local areas of interest or glaciers these requirements and have shown that by deploying high- (Le Meur et al., 2004; Zwinger et al., 2007; Zwinger and performance computing (HPC) techniques this challenge can Moore, 2009). Elmer/Ice was primarily developed to solve be successfully taken on. In this context, Elmer/Ice takes ad- the flow of anisotropic polar ice and the evolution of its vantage of being backed by a large open-source community strain-induced fabric (Gillet-Chaulet et al., 2006; Durand that also develops new numerical and HPC techniques for the et al., 2007; Seddik et al., 2008, 2011; Ma et al., 2010; code (e.g. Malinen, 2007). Martín and Gudmundsson, 2012). It has since then been The third requisite, and from the physical point of view the used to model the flow of a cold firn-covered glacier using most challenging, is to implement physically founded bound- a dedicated snow/firn rheological law (Zwinger et al., 2007). ary conditions. These improvements are far more complex Elmer/Ice has been the only full-Stokes model to perform the and it will take more time to fully address them in the ice whole set of the ISMIP-HOM experiments (Gagliardini and sheet flow models. The recently observed changes in coastal Zwinger, 2008; Pattyn et al., 2008) and is still the only full- glacier dynamics (e.g. Moon et al., 2012) are certainly driven Stokes model to participate in the grounding line experiments by changes in ice sheet and ice shelf boundary conditions, MISMIP (Pattyn et al., 2012). Elmer/Ice was further used and consequently linked to changes in the ocean and atmo- as a reference for the later MISMIP3d experiments (Pattyn sphere components of the climatic system. In the simplest et al., 2013). Recently, data assimilation was implemented cases, changes in the climatic components directly drive the within Elmer/Ice (Jay-Allemand et al., 2011; Schäfer et al., changes at the boundaries of the ice mass. This is the case for 2012; Gillet-Chaulet et al., 2012) to infer poorly known pa- surface air temperature or ocean temperature which directly rameters such as basal drag. Today, Elmer/Ice is the only drive the temperature boundary condition of the upper sur- three-dimensional full-Stokes model that solves the ground- face or the bottom ice/ocean interface, respectively. In other ing line dynamics (Favier et al., 2012), and it will be the

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1301 only full-Stokes model able to run forecast simulations for of polar ice can be strongly anisotropic, and that this re- the whole Greenland ice sheet for the coming AR5 IPPC re- sponse depends on the crystal orientation distribution, i.e. port, in the framework of both SeaRISE (Seddik et al., 2012) the ice fabric (e.g. Gagliardini et al., 2009). Elmer/Ice in- and ice2sea (Gillet-Chaulet et al., 2012; Shannon et al., 2013; cludes the classic isotropic Glen’s flow law as well as two Edwards et al., 2013) programmes. anisotropic flow laws. As shown in various applications, the In this paper, we summarise ten years of consistent devel- anisotropy of polar ice has a strong influence on the over- opments and present the current state of the new-generation all flow (Zwinger et al., 2013) and will in turn modify the ice sheet model Elmer/Ice (Elmer library version 7.0 SVN age–depth relationship (Gillet-Chaulet et al., 2006; Seddik revision 5955). We only focus on the past developments that et al., 2011). In central parts of ice sheets, ice anisotropy and are relevant for simulations of three-dimensional ice sheets. the development of fabric can explain the observed hetero- Specific developments regarding two-dimensional flow line geneity of the ice deformation along a drilling (Durand et al., or glacier applications are not presented here, but one can 2007). On the coastal area, due to the large contrast of the consult previous publications on these types of applications stress regimes for the grounded part and for the ice shelf, the (the complete list of Elmer/Ice publications can be found on ice anisotropy induces an apparent hardening of the ice up to http://elmerice.elmerfem.org/). Section 2 presents the gov- a factor 10 when ice moves from grounded to floating (Ma erning equations implemented in Elmer/Ice. The associated et al., 2010). boundary conditions are discussed in Sect. 3. Other useful When ice is assumed to behave as an isotropic material, equations, such as the equation to evaluate the age of the ice, its rheology is given by a Norton–Hoff power law, known as are presented in Sect. 4. Section 5 is dedicated to the inverse Glen’s law in glaciology, which links the deviatoric stress τ methods implemented in Elmer/Ice. Some technical aspects with the strain rate ε˙: related to the resolution of these equations in the framework of the FE method are discussed in Sect. 6. The efficiency τ = 2ηε˙, (3) of Elmer/Ice was verified by standard convergence and scal- ability tests described in Sect. 7. Finally, we provide some where the effective viscosity η is defined as insights into the future planned developments in Sect. 8. 1 − η = (EA)−1/nε˙(1 n)/n. (4) 2 e 2 Governing equations 2 2 In Eq. (4), ε˙e = tr(ε˙ )/2 is the square of the second in- 0 2.1 Ice flow equations variant of the strain rate and A = A(T ) is a rheological pa- rameter which depends on T 0, the ice temperature relative Ice is a fluid with an extremely high viscosity that flows to the pressure melting point, via an Arrhenius law. The en- very slowly so that inertia and acceleration terms entering hancement factor E in Eq. (4) is often used to account for the momentum equation can be neglected. Therefore, the anisotropy effects, by prescribing an ad hoc value depend- three-dimensional velocity field and the pressure field of an ing on the ice age and/or type of flow. Due to the state of ice mass flowing under gravity are obtained by solving the stress, E is expected to be greater than 1 for grounded ice Stokes equations over the ice volume . The Stokes equa- of polar ice sheets, whereas a value lower than 1 should be tions express the conservation of linear momentum used for floating ice shelves (Ma et al., 2010). A compress- ible form of Glen’s law (Gagliardini and Meyssonnier, 1997), divσ + ρg = divτ − grad p + ρg = 0, (1) well adapted to describe the flow of firn, is also implemented and the mass conservation in Elmer/Ice (Zwinger et al., 2007). Both implemented anisotropic flow laws depend on the ice u = ε˙ = . div tr 0 (2) polycrystalline fabric, which is described by its second- and In these equations, ρ is the ice density, g = (0,0,−g) the fourth-order orientation tensors a(2) and a(4), respectively, gravity vector, u = (u,v,w) the ice velocity vector, σ = τ − defined as pI the Cauchy stress tensor with p = −trσ/3 the isotropic (2) = h i (4) = h i pressure, τ the deviatoric stress tensor and I the identity ma- aij cicj and aijkl cicj ckcl , (5) trix. This system of equations of unknowns u and p is closed hi by adopting one of the rheological laws presented in the next where c is the crystal c axis unit vector and denotes the av- section. The conditions that are applied on the boundary 0 of erage over all the grains that compose the polycrystal. By the volume  are discussed in Sect. 3. definition odd-order orientations tensors are null, and the higher the order of the orientation tensor the better the de- 2.2 Rheological laws for polar ice scription of the fabric. However, it can be shown that with a linear flow law, knowing the second- and fourth-order ori- Even if most ice sheet models assume an isotropic rheolog- entation tensors is sufficient to uniquely define the macro- ical law for ice, it is well known that the viscous response scopic flow law (Gillet-Chaulet et al., 2005; Gagliardini www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1302 O. Gagliardini et al.: Elmer/Ice model et al., 2009). For random c axes distribution the non-zero en- can be used to tabulate the six viscosities ηr . The most re- (2) (2) = (2) = (2) = alistic polycrystalline response is obtained using the visco- tries of a are a11 a22 a33 1/3, for a single maxi- mum fabric with its maximum in the third direction, a(2) > plastic self-consistent model (VPSC, Castelnau et al., 1996, 33 1998), with the two crystal anisotropy parameters chosen so 1/3 and a(2) ≈ a(2) < 1/3, and for a girdle-type fabric in 11 22 that the experimentally observed polycrystal anisotropy is re- (2) (2) ≈ (2) the plane (x1,x2), a33 < 1/3 and a11 a22 > 1/3. In ad- produced (Gillet-Chaulet et al., 2006; Ma et al., 2010). When dition to three eigenvalues, three Euler angles are necessary the ice is isotropic, ηr = 0 and ηr+ = 1 (r = 1, 2, 3), then (2) 3 to uniquely define a with respect to a general reference the GOLF (6) reduces to Glen’s isotropic flow law (3) with frame. It can be shown analytically with a linear flow that E = 1. if the second- and fourth-order orientation tensors have the The second anisotropic flow law implemented in same eigenframe, the polycrystal behaviour will exhibit or- Elmer/Ice is the Continuum-mechanical Anisotropic Flow thotropic symmetries (Gillet-Chaulet et al., 2006). The equa- model based on an anisotropic Flow Enhancement factor tions for the fabric evolution are presented in Sect. 2.5. (CAFFE, Seddik et al., 2008; Placidi et al., 2010). The The first anisotropic flow law implemented in Elmer/Ice is CAFFE model assumes collinearity between the strain rate the non-linear General Orthotropic Flow Law (GOLF, Gillet- and deviatoric stress tensors, so that the general form of Chaulet et al., 2005; Ma et al., 2010). The GOLF provides Glen’s law (3) is not modified, but the enhancement factor a non-collinear and non-linear relation between strain rate E is a function of the polycrystalline deformability D such and stress, using the concept of structure tensors. In its initial that form, the ice was assumed to behave as a linearly viscous or- ( t thotropic material. In more recent works (Martín et al., 2009; (1 − Emin)D + Emin 1 ≥ D ≥ 0 , E(D) = 2 (7) Ma et al., 2010), the GOLF has been extended to a non- 4D (Emax−1)+25−4Emax 5/2 ≥ D > 1 , linear form by adding an invariant in the anisotropic linear 21 law. The simplest choice is either to add the second invariant with ˙ of the strain rate εe (Martín et al., 2009) or the second in-   2 = 2 8 Emax − 1 variant of the deviatoric stress τe (with τe tr(τ )/2, Pettit t = ,Emax ≈ 10 ,Emin ≈ 0.1 . (8) et al., 2007; Ma et al., 2010). No theoretical or experimental 21 1 − Emin results are available today to discard one of these two solu- The polycrystalline deformability D is a function of strain tions, and other solutions based on anisotropic invariants of rate and fabric. When D = 0, the minimal enhancement fac- the deviatoric stress and/or the strain rate are also possible. tor E is reached, which corresponds to an uni-axial com- In Elmer/Ice, both solutions are implemented. Using the sec- min pression on a single maximum fabric. For an isotropic fab- ond invariant of the deviatoric stress, for a given fabric and ric, D = 1 and the response is identical whatever the strain a given state of stress, the corresponding strain rate relative rate, whereas the maximal enhancement E is obtained for to the isotropic response is the same for the linear and non- max D = 5/2, which corresponds to a single maximum fabric un- linear cases. Using the strain-rate invariant in the same way dergoing simple shearing. The adopted form for the poly- as Martín et al. (2009) leads to an opposite definition of the crystalline deformability, which verifies the above criteria, anisotropy ratios: for a given strain rate, the corresponding reads stress relative to the isotropic response is the same for the linear and non-linear cases. When using the stress second in- ε˙ · a(2) − a(4) : ε˙
: ε˙ D = . variant, the GOLF reads 5 2 (9) ε˙e 3 n−1 Xh D Di 2.3 Evolution of the surface boundaries 2Aτe τ = ηr tr(Mr · ε˙)Mr + ηr+3(ε˙ · Mr + Mr · ε˙) . (6) r=1 For transient simulations, the upper and lower boundaries (2) of the domain are allowed to evolve, following an advec- The six dimensionless anisotropy viscosities ηr (a ) and (2) tion equation. Evolution of the upper surface z = zs(x,y,t) ηr+3(a ) (r = 1, 2, 3) are functions of eigenvalues of the second-order orientation tensor a(2), which represent a mea- is given by sure of the anisotropy strength. The three structure tensors ∂zs ∂zs ∂zs M are given by the dyadic products of the three eigenvec- + us + vs − ws = as, (10) r ∂t ∂x ∂y tors of a(2), which then represent the material symmetry axes. In the method proposed by Gillet-Chaulet et al. (2006), the where (us,vs,ws) are the surface velocities obtained from (2) six dimensionless viscosities ηr (a ) are tabulated as a func- the Stokes solution and as = as(x,y,t) is the accumu- (2) tion of the fabric strength (i.e. the ai ) using a micro-macro lation/ablation prescribed as a vertical component only. model. Various micro-macro models, from the assumption of Elmer/Ice provides a surface melting parameterisation based uniform stress within the ice polycrystal to the assumption on positive degree-day (PDD) method (Reeh, 1991), supple- of uniform strain rate, as well as different crystal anisotropy mented by the semi-analytical solution for the PDD integral

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1303 by Calov and Greve (2005)(Seddik et al., 2012). The accu- where c⊥ is the calving rate. The latter is defined as the mulation/ablation distribution can also be inferred from a re- ice volume flux across the calving front, c⊥ = (u − wc) · nc, gional climate model either directly as in Gillet-Chaulet et al. where wc is the kinematic velocity of the calving front (2012) and Shannon et al. (2013) or using a surface elevation (Greve and Blatter, 2009). Implementation of calving laws to parameterisation as in Edwards et al. (2013). evaluate the calving rate c⊥ is part of the developments cur- The lower surface of an ice sheet is either in contact with rently ongoing in Elmer/Ice, as discussed more in details in the bedrock or the ocean. The evolution of the lower surface Sect. 8. Moving the mesh both vertically (upper and lower z = zb(x,y,t) is given as surface) and horizontally (calving front) induces additional terms in the convection part of equations and in turn techni- "  2  2#1/2 ∂zb ∂zb ∂zb ∂zb ∂zb + u + v − w = a ⊥ 1 + + , (11) cal issues that are discussed in Sect. 6.1. ∂t b ∂x b ∂y b b ∂x ∂y 2.4 Heat equation where (ub,vb,wb) are the basal velocities and ab⊥ = ab⊥(x,y,t) is the melting/accretion function, taken perpen- The temperature within the ice is obtained from the general dicular to the surface. balance equation of internal energy and reads Assuming a rigid, impenetrable bedrock z = b(x,y), the   following topological conditions must be fulfilled by zs and ∂T ρcv + u · grad T = div(κ grad T) + D : σ, (14) zb: ∂t zs(x,y,t) ≥ zb(x,y,t) ≥ b(x,y) ∀x,y,t. (12) where κ = κ(T ) and cv = cv(T ) are the heat conductivity and specific heat of ice, respectively. The last term in the heat The weak formulation of Eq. (10) or Eq. (11), in combina- equation represents the amount of energy produced by vis- tion with the constraints (12) forms a variational inequality. cous deformation. The ice temperature T is bounded by the Technically, it is solved using a method of imposed Dirichlet pressure melting point T , so that T ≤ T , or equivalently conditions that are released by a criterion based on the resid- m m T 0 ≤ 0, with T 0 = T − T being the homologous tempera- ual, as described in Sect. 6.5. In Gagliardini et al. (2010), m ture entering the Arrhenius law to estimate Glen’s parameter melting below the ice shelf was prescribed using a param- in Eqs. (4) and (6). This inequality, as well as temperature- eterised expression following Walker et al. (2008). As dis- dependent material properties, make the solution of the heat cussed in Sect. 8, a proper description of the basal melt- transfer equation a non-linear problem which is solved using ing below ice shelves will certainly require the coupling of an iterative method as presented in Sect. 6.5. Elmer/Ice with an ocean model or at least the implementa- tion of a plume-type model. 2.5 Fabric description and evolution The margin boundary of an ice sheet is either land- or marine-terminated, depending on whether the bedrock ele- Assuming that recrystallisation processes do not occur and vation at the ice front is located above or below sea level, that the ice fabric is induced solely by deformation, the evo- respectively. In both cases, the front position evolves with lution of the second-order orientation tensor a(2) defined by time and its evolution is governed by the imbalance be- Eq. (5) can be written as tween ice flux and ablation/basal melting/calving processes. Land-terminated fronts can be treated classically by adopting ∂a(2) + grad a(2) · u = W · a(2) − a(2) · W − ι(C · a(2) a minimal ice thickness hmin, so that the exact condition (12) ∂t (2) (4) is replaced by the less strict one zs(x,y,t) ≥ b(x,y) + hmin + a · C − 2a : C), (15) (and zb(x,y,t) = b(x,y)). Where the ice sheet is marine-terminated, this type of where W is the spin tensor defined as the antisymmetric part treatment cannot be applied because the sea water pressure of the velocity gradient. The tensor C is defined as and lateral buttressing forces would not be correctly taken = − ˙ + n−1 into account. The front boundary of a marine-terminated ice C (1 α)ε αksAτe τ. (16) sheet must therefore be allowed to move over time, as a func- The interaction parameter α controls the relative weight- tion of the calving rate and ice flux at the margin. ing of the strain rate ε˙ and the deviatoric stress τ in the fab- Assuming that the calving front is a vertical surface, ric evolution Eq. (15). When α = 0, the fabric evolution is = it can be described by the implicit function Fc(x,y,t) solely controlled by the state of strain rate, whereas in the = 0 (Greve and Blatter, 2009). Denoting by grad Fc case where α = 1 the fabric evolves under the influence of the = | | (∂Fc/∂x,∂Fc/∂y,0) its gradient, Nc grad Fc the norm deviatoric stress solely. In between, as for the VPSC, both the = and nc grad Fc/Nc the unit normal vector (assumed to strain rate and deviatoric stress contribute to the fabric evolu- point out of the ice), the calving front evolves as follows tion. Assuming ι = 1, an interaction parameter α = 0.06 is in ∂Fc ∂Fc ∂Fc accordance with the crystal anisotropy and the VPSC model + u + v = N c⊥, (13) ∂t ∂x ∂y c used to derive the polycrystalline behaviour (Gillet-Chaulet www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1304 O. Gagliardini et al.: Elmer/Ice model et al., 2006). Seddik et al. (2008, 2011) adopted instead α = 0 unit vector to the bedrock nb. Note that the boundary con- and a value of ι lower than 1. In Eq. (15), the fourth-order dition Eq. (18) for the Stokes problem is equivalent to the orientation tensor is evaluated assuming a closure approxi- free-surface Eq. (11). The effective pressure N is defined mation giving a(4) as a tensorial function of a(2) (Chung and as the difference between the ice normal stress and the wa- Kwon, 2002; Gillet-Chaulet et al., 2006). Theoretically, re- ter pressure, such as N = −σnn − pw with σnn = nb · σ nb. crystallisation processes, such as continuous and migration Equation (18) is the no-penetration condition accounting recrystallisation, can be included by adding terms in Eq. (15) for basal melting (ab⊥ < 0) or basal accretion (a⊥b > 0), to parameterise on the polycrystalline scale the phenomena whereas Eq. (19) stands for the general form of a friction occurring at the grain scale (Seddik et al., 2011). Because law. When ff = 0, the ice slides perfectly over the bedrock, experimental data are currently missing, these parameterisa- whereas when ff → +∞ basal sliding is null. The three dif- tions have not yet been validated and are not presented here. ferent friction laws implemented in Elmer/Ice are presented below. The first friction law linearly relates the basal shear stress 3 Boundary conditions to the basal velocity, such as For all the equations presented above, classic Dirichlet, Neu- + = = σnti βuti 0, i 1,2, (20) mann, Robin, symmetric and periodic boundary conditions can be applied on the boundary of the domain. In this sec- where β ≥ 0 is the basal friction parameter. As shown later, tion, we present the conditions to be applied on the different this simple law is used for data assimilation and in this case boundaries of an ice sheet for the main equations presented β is a control parameter. above, and we focus more specifically on the treatment of the The second law implemented in Elmer/Ice is a Weertman- basal boundary. type sliding law: + m−1 = = 3.1 Ice/atmosphere boundary σnti βmub uti 0, i 1,2, (21) where u is the norm of the sliding velocity u = u − (u · The upper free surface z = zs(x,y,t), also denoted 0s, is in b b contact with the atmosphere and is therefore a stress-free sur- nb)nb, βm is a sliding parameter and m an exponent. When = face, so that m 1, the Weertman-type friction law Eq. (21) reduces to the linear law Eq. (20). Theoretically, in the case of ice slid- σ ns = −patmns ≈ 0 for z = zs, (17) ing without cavitation over an undulating bed, m is equal to 1/n (Lliboutry, 1968), where n is Glen’s law exponent. where ns is the normal outward-pointing unit vector to the The third friction was proposed by Schoof (2005) from free surface. For the dating equation, fabric equations and all mathematical expansions and by Gagliardini et al. (2007) other transport equations, Dirichlet conditions are applied on from FE simulations. This law describes the flow of clean the upper surface only where the ice velocity enters the do- ice over a rigid bedrock when cavitation is likely to occur: main (mainly in the accumulation area). Where z = zs and − !1/n u · ns ≤ 0, the temperature is equal to the imposed surface σ χ u1 n nti + b u = 0 i = 1,2, (22) temperature, T (x,y,zs,t) = Ts(x,y,t), and the fabric is as- + q ti (2) CN 1 αq (χ ub) sumed to be isotropic, a (x,y,zs,t) = I/3. For the heat equation, a heat flux can be imposed at the upper surface to n n q−1 q where χ = 1/(C N As), αq = (q −1) /q , As is the slid- account for melt-water refreezing. ing parameter in the absence of cavitation and n Glen’s law exponent, resulting in a non-linear relation between the 3.2 Ice/bedrock boundary basal drag σnti and the basal sliding velocity uti . The max- imal value of σ is C and the exponent q ≥ 1 controls the The lower interface z = zb(x,y,t), also denoted 0b, may be nti in contact with either the sea or the bedrock, so two kinds of post-peak decrease. When the post-peak exponent q is equal boundary conditions coexist on a single surface. The condi- to 1, the basal drag tends asymptotically to its maximum tions to be applied where the ice is in contact with the sea are value C (no post-peak decrease). Note that in the limit case presented in the next section. Where the ice is in contact with where N  0, the sliding parameter As and the friction pa- rameter β are inversely proportional. As shown by Schoof the bedrock (i.e. zb = b), the following conditions apply: m (2005), the coefficient C should be chosen smaller than the u · nb + ab⊥ = 0, (18) maximum local positive slope of the bedrock topography at ≤ = = a decimetre to metre scale, so that the ratio σnti /N C ful- σnti ff (u,N)uti , i 1,2, (19) fills Iken’s bound (Iken, 1981). The friction law Eq. (22) is = · = · = where σnti ti σ nb and uti u ti (i 1,2) are the basal strongly related to the water pressure pw through the effec- shear stresses and basal velocities, respectively, defined in tive pressure N. The law Eq. (22) can then be used to cou- terms of tangent vectors ti and normal outward-pointing ple the hydrology and the ice dynamics. The hydrological

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1305 model and its implementation in Elmer/Ice are presented in 4.1 Age equation de Fleurian et al. (2013). For the heat equation, the geothermal heat flux qgeo is im- The age A of the ice at each point of the ice sheet domain is posed where the basal temperature is lower than the pres- obtained by solving the following equation: sure melting point (T < T or T 0 < 0), and the following m ∂A Neumann-type boundary condition applies: + u · grad A = 1, (27) ∂t · | = + | | κ(T )( grad T nb) 0b qgeo σnti uti , (23) where z = zs and u · ns ≤ 0, the age of the ice is zero, i.e. | | A(x,y,zs,t) = 0 (Zwinger and Moore, 2009). By solving the where σnti uti is the heat energy induced by basal friction. age equation we can compute isochrones and determine dat- Where the temperature melting point is reached (T = Tm), the amount of melted water is estimated from the imbalance ing as a function of depth at an ice core (drilled or planned) of heat fluxes and surface production: location. Input parameters entering other equations might also be age-dependent, such as the enhancement factor for qgeo + |σnt ut | − κ grad T · nb a = i i , (24) example. b ρL 4.2 Depth and elevation where L is the latent heat of ice. It is often very useful to know the depth below the upper 3.3 Ice/sea boundary surface or the height above the bedrock at each point of the ice sheet domain. For example, it can be used to prescribe At the bottom surface z = z (x,y,t) where the ice is in con- b parameterisation of the temperature or the ice fabric fields tact with the ocean (i.e. z > b) and at the front of the ice b as a function of depth. With the FE method, using unstruc- sheet, the normal stress is equal to the sea pressure p (z,t), w tured meshes, the depth d(x,y,z,t) = z − z or the height which evolves vertically as follows: s h(x,y,z,t) = z − zb at any point M(x,y,z) cannot be esti- ( mated directly because nodes are not necessarily vertically ρwg(lw(t) − z), z < lw(t) pw(z,t) = (25) aligned. Therefore, we compute the depth d (or equivalently , z ≥ l (t), 0 w height h) field by solving the following equations: where ρw is the sea water density and lw the sea level. The ∂d ∂h Neumann condition applied on these ice/ocean interfaces is = −1 , or = 1, (28) ∂z ∂z thus with the boundary conditions d = 0 on z = zs or h = 0 on σ n = −p n . (26) c w c z = zb. h 3.4 Grounding line dynamics Effectively, we solve, here for the height , the following system: The position of the grounding line is part of the solution and − e · ∇ e · ∇h = , can evolve with time. Its position at each time step is deter- z ( z ) 0 (29) mined by solving a contact problem. The contact is tested ez · ∇h|∂ = 1, (30) by comparing at each node where zb = b the normal force with the boundary condition h|0 = 0 and the unity vec- Rn exerted by the ice on the bedrock and the equivalent wa- b tor ez in the vertical direction. The variational form is ob- ter force Fw. Rn is directly evaluated from the residual of tained after integrating Eq. (29) by parts and accounting for the Stokes system, whereas Fw is obtained by integrating the water pressure over the boundary elements using the bound- the boundary condition Eq. (30), leading to a degenerated Laplace equation of the form ary element shape functions. Then, if Rn > Fw and zb = b, the boundary conditions Eqs. (18) and (19) apply; whereas Z if Rn = Fw and zb = b, or zb > b, the boundary condition − ∇ (ez · ∇h) · ϕezd = Eq. (26) applies instead.  Z I (ez · ∇h)∇ϕ · ezd − ϕez · nd0. (31) 4 Auxiliary equations  ∂ The goal of an ice sheet simulation, usually, is to obtain infor- 4.3 Stress and strain rate mation on either the geometry, the age/depth relationship or simply the exerted stresses and forces on a particular surface Elmer/Ice includes solvers to compute the Cauchy stress, in contact with the ice. This section introduces the methods deviatoric stress or strain rate fields from the Stokes solu- needed to obtain such information. tion, and also includes eigenvalues of these tensor variables. www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1306 O. Gagliardini et al.: Elmer/Ice model

In addition, calculating of the stress from the velocity and The cost function that expresses the mismatch between the isotropic pressure fields is a matter of interest because dif- solutions of the two models is given by ferent methods can lead to noticeably different solutions. In Z N D N D Elmer/Ice, the components σij of the nodal Cauchy stress Jo = (u − u ) · (σ − σ ) · nd0, (35) field are obtained from an existing Stokes solution (u,p) zs by writing the variational version of the constitutive law in a componentwise manner as where superscripts N and D refer to the Neumann and Z Z Dirichlet problem solutions, respectively. σij 8d = ei · σ ej 8d The Gâteaux derivatives of the cost function Jo with re- η β η0 β0   spect to the parameters and for perturbations and , Z     respectively, are given by = · + T − ei η grad u grad u pI ej 8d. (32) Z 0  D 2 N 2  dηJo = 4η (ε˙e ) − (ε˙e ) d, (36) This results in solving six independent equations, one for  Z each of the six independent components of the stress tensor. 0  D 2 N 2 dβ Jo = β |u | − |u | d0, (37) In a similar manner, components ε˙ij of the nodal strain rate tensor are obtained from the following variational form: zb Z Z   where the symbol ε˙2 denotes the square of the second invari- ˙ = · + T e εij 8d ei grad u grad u ej 8d. (33) ant of the strain rate as defined for Eq. (4) and | · | defines the   norm of the velocity vector. Note that this derivative is exact only for a linear rheology and thus is only an approximation 5 Inverse methods within Elmer/Ice of the true derivative of the cost function when using Glen’s flow law Eq. (3) with n > 1 in Eq. (4). The ice effective viscosity η(x,y,z) in Eq. (3) and the basal friction coefficient β(x,y) in Eq. (20) are two particularly 5.2 Control inverse method important input fields when modelling the flow of real glacio- For a linear isotropic rheology (a scalar viscosity η indepen- logical systems. However, these two parameters are used dent of the velocity, i.e. n = 1 in Eq. 4), the Stokes system to represent complex processes, and their values in situ are of equations is self-adjoint. Denoting by λ and q the adjoint poorly constrained and can vary by several orders of magni- variables corresponding to u and p, respectively, they are so- tude with time and space. On the other hand, our knowledge lutions of the following equations: of some of the outputs of the model (surface velocity, surface elevations) has considerably increased recently with data ac- 2divηε˙ λ − grad q = 0, (38) quired by remote spatial observation. λ = Two variational inverse methods have been implemented trε˙ 0, (39) within Elmer/Ice to constrain η(x,y,z) and β(x,y) in diag- where ε˙ λ is the equivalent of the strain rate tensor constructed nostic simulations from topography and surface horizontal with λ. For a non-linear rheology, the operator used by the velocity data. Both methods are based on minimising a cost forward solver (Stokes operator) remains self-adjoint when function that measures the mismatch between the model and equipped with the Newton linearisation (Petra et al., 2012). the observations. The two methods are briefly described be- The cost function is chosen to measure the mismatch be- low and their implementation in Elmer/Ice is verified in tween the modelled and observed surface velocities Sect. 7. Z obs 5.1 Robin inverse method Jo = j (u − u )d0, (40)

0s This method, initially proposed by Arthern and Gudmunds- son (2010), consists in solving alternatively the natural Neu- where j is the mismatch measure function and uobs are the mann-type problem, defined by Eqs. (1) and (2) and the sur- observed surface velocities. The choice of j can be case- face boundary conditions (17), and the associated Dirich- dependent and will affect the boundary condition terms of let-type problem, defined by the same equations except that the adjoint system. For example, as the surface velocity di- the Neumann upper-surface condition Eq. (17) is replaced by rection is mainly governed by topography, we can discard the a Dirichlet condition where observed surface horizontal ve- error on the velocity direction and express j as locities are imposed, such that 2 obs 1  obs  obs obs j (u − u ) = |uH| − |uH | , (41) u = u and v = v for z = zs. (34) 2

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1307

where subscript H refers to the horizontal component node i = (1,...,Np) and Wi is the nodal weight associated of the velocity vectors (Gillet-Chaulet et al., 2012). The with node i and computed following the standard integration Gâteaux derivatives of Eq. (40) with respect to η and β are scheme. The sum of all weights is the volume (or area) of obtained as follows: the FE mesh. The discrete gradients of Jo at each mesh node i Z used for the minimisation are then given by Wi∇ Jo and ac- 0  λ  γ dηJo = −2η ε˙ : ε˙ d, (42) count for the volume or area surrounding each node.  The minimisation of the cost function Jo with respect to Z η or β is done using the limited memory quasi-Newton rou- = − 0 · i i dβ Jo β u λd0. (43) tine M1QN3 (Gilbert and Lemaréchal, 1989) implemented in 0b Elmer/Ice in reverse communication mode. This method uses an approximation of the second derivatives of the cost func- 5.3 Regularisation tion and is therefore more efficient than a fixed-step gradient When working with non-perfect (noisy) data, it is neces- descent. sary to add a regularisation term in the cost function to im- How we define the inner product used to compute the prove the conditioning of the inverse problem and ensure the Gâteaux derivatives affects the definition of the Fréchet existence of a unique minimum. The regularisation term is derivatives, and could affect the convergence of the min- based on a priori information on the solution either from imisation, but does not affect the minimum we are seek- measurements, from analytical solutions (Raymond Pralong ing to achieve. As for glaciological applications, velocities and Gudmundsson, 2011), or from assumptions on the spa- and strain rates can vary by several orders of magnitude in- tial variations of the variable. In Elmer/Ice, a smoothness side the domain, and we have observed that including the constraint on a variable α can be imposed in the form of nodal weights in the definition of the Fréchet derivatives a Tikhonov regularisation penalising the first spatial deriva- leads to good convergence properties when using an unstruc- tives of α as in Morlighem et al. (2010), Jay-Allemand et al. tured mesh where large elements correspond to low-velocity (2011), and Gillet-Chaulet et al. (2012): areas, and vice versa. Possible alternatives are, for the con- trol inverse method (Morlighem et al., 2010), to use a cost ! 1 Z ∂α 2 ∂α 2 ∂α 2 function that measures the logarithm of the misfit or, for the J = + + d0. (44) reg 2 ∂x ∂y ∂z Robin inverse method (Arthern and Gudmundsson, 2010), to 0b use a spatially varying step size rather than a fixed step in the gradient descent algorithm, as proposed in Schäfer et al. The Gâteaux derivative of Jreg with respect to α for a per- turbation α0 is obtained by (2012). Z ∂α ∂α0  ∂α ∂α0  ∂α ∂α0  d J = + + d0. (45) α reg ∂x ∂x ∂y ∂y ∂z ∂z 0b 6 Numerical implementation and specificities The total cost function to minimise then reads 6.1 Mesh and deforming geometry Jtot = Jo + λJreg, (46) Ice sheets and ice caps have a very small aspect ratio, hori- where λ is a positive ad hoc parameter. The cost function zontal dimensions being much larger than the vertical dimen- minimum is therefore no longer the best possible fit to obser- sions, and therefore meshing requires special care. The strat- vations, but a compromise (through the tuning of λ) between egy commonly adopted in Elmer/Ice for meshing glaciers, fitting with observations and smoothness in α. ice sheets and ice caps is to mesh first the horizontal 2-D footprint and then extrude it vertically. These meshes are then 5.4 Minimisation vertically structured with the same number of layers over the whole domain, whereas the horizontal dimension can be The Gâteaux derivatives of Jo are given by a continuous scalar product represented by the integral terms in Eqs. (37) meshed using an unstructured mesh. This is one of the main and (42). When discretized on the FE mesh, these equations advantage of a FE ice flow model in comparison to the clas- are transformed into a discrete Euclidean product as follows: sically used finite difference or volume methods for which the grid has the same size over all the domain, unless a mesh adaptive method is implemented (Cornford et al., 2013a). Z Np The unstructured mesh of the footprint can be created = ∇ 0 ≈ X ∇i 0 dγ Jo γ Joγ Wi γ Joγi , (47) using triangle-shaped elements of various sizes to account i=1 for the spatial heterogeneity of the variables gradient. The where γ represents η or β, ∇γ Jo is the continuous Fréchet horizontal size of the elements can be controlled using, derivative of Jo, the expression of which is given by com- for example, a metric constructed from the Hessian ma- i parison with Eqs. (37) and (42), ∇γ Jo is its value at mesh trix of observed surface speed (Gillet-Chaulet et al., 2012). www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1308 O. Gagliardini et al.: Elmer/Ice model

Technically, optimising the mesh sizes according to this met- For instance, if we solve Eq. (14), we should take any in- ric is done using the freely available anisotropic mesh adap- duced mesh velocity Eq. (50) into account. This is done by tation software YAMS (Frey and Alauzet, 2005). Because of the arbitrary Lagrangian–Eulerian (ALE) formulation, which the overall size of ice sheets, the mesh is then partitioned and is based on the Reynolds transport theorem (e.g. Greve and all partitions are solved in parallel using the Message Passing Blatter, 2009). Conservation simply demands that in the gen- Interface (MPI). In Elmer/Ice, the mesh can be generated ei- eral reference frame of a moving mesh, Eq. (51) changes into ther by extrusion as a preprocessing step or by a built-in mesh extrusion feature which operates on the parallel level. This d9 ∂9 internal procedure efficiently removes some of the possible = + (u − um) · grad 9. (52) dt ∂t bottlenecks in preprocessing as the maximum mesh size is no longer constrained by serial operations. Also, in the case A slight deviation from this is for the kinematic boundary of an extruded mesh, certain operations become trivial, as condition Eq. (10), as the convection term is only in the hor- for example modifying the geometry or computing the depth izontal plane, i.e. or elevation, which efficiently becomes a one-dimensional ∂zs ∂zs ∂zs problem. + (us − um) + (vs − vm) − ws = as. (53) For transient simulation, the geometry of the ice sheet is ∂t ∂x ∂y evolving with time and the mesh has to be deformed to follow The same is applied to Eq. (11) for the evolution of zb. these changes. The common approach to deform geometries A special case of Eq. (52) is when the surface is considered in Elmer/Ice, if dealing with unstructured meshes, is to rear- to move horizontally (it does vertically by definition) at the range the nodes by solving a pseudo linear elasticity problem. speed of the fluid particles, i.e. um = us. This, for instance, is Any mesh displacement, 1x, in Elmer/Ice is relative to the needed if dealing with advancing fronts in marine-terminated initial mesh position, x0, i.e. x(t) = x0 +1x(t). A deforma- glaciers. In this case, the new position of the surface is deter- tion of the surface, for instance, can be induced by a chang- mined by x(t + 1t) = x(t) + u(t)1t. In terms of the abso- ing free-surface elevation, h. Hence, the prescribed vertical lute mesh update 1x, this means that 1x(t +1t) = 1x(t)+ · = − = deformation here is 1x(t) ez h(t) h(t 0). Inside the u(t)1t, which simply reflects Eq. (50) under um = us and bulk mesh, the corresponding deformation is then obtained as = 0. by solving 6.2 Variational formulations  2Y κ Y  −∇ ·  + λ ∇ · 1xI = 0, (48) (1 − κ)(1 − 2κ) 2(1 + κ) Elmer/Ice relies on the FE method, and all the equations pre- sented above are solved using a discretised variational form, where Y and κ are respectively a pseudo Young’s modulus leading in turn to solving a linear system for which the un- and Poisson ratio, describing the resistance against the de- knowns are the nodal value of the variable. The aim of this formation and its directional ratio. Further,  describes the section is to present some technical details for the variational symmetric strain tensor forms of both the Stokes and transport equations. 1    = ∇1x + (∇1x)T . (49) 6.2.1 Stokes equations 2 The discrete variational form of the Stokes system Eqs. (1) In consequence, the induced mesh velocity from the re- and (2) is obtained by integration over the ice domain  computation of 1x by Eq. (48) from one discrete time-level using the vector-valued weight function 8 and the scalar + t to t 1t then is given by weight function 9, 1x(t + 1t) − 1x(t) Z um = . (50) 9divud = 0, (54) 1t  The continuum equations, as presented above, strictly, are Z Z I valid only in a fixed reference frame. In ice sheets or glaciers, τ : grad 8d − p div8d − n · σ 8d0 however, the geometry by nature is not fixed. In a fixed ref-   ∂ erence frame, if u is the fluid velocity, the total change of Z a scalar property, 9, = ρ g · 8d. (55)  d9 ∂9 = + u · grad 9, (51) In the relation given above, the left-hand-side term in the dt ∂t momentum equation given in Eq. (1) has been integrated by consists of the local change and of a convective part parts. One part is re-formulated by applying Green’s theo- u · grad 9. rem, transforming it from an integral over the domain  into

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1309 one over the closed boundary of the domain 0, for which Dirichlet conditions have to be applied on all the bound- Neumann or Newton boundary conditions can be set (e.g. aries of  where the flow velocity is directed inside the ice vanishing deviatoric surface stress components). The numer- domain. Because of the missing diffusion terms in Eq. (59), ical solution of Eq. (54) is obtained by either using the sta- the classic Galerkin method is unstable. This transport equa- bilised method from Franca and Frey (1992) or the residual tion is either solved using the discontinuous Galerkin method free bubbles method in Baiocchi et al. (1993). proposed by Brezzi et al. (2004) or a semi-Lagrangian For non-linear rheology, e.g. n = 3 in Glen’s law (Eq. 4) method (Staniforth and Côté, 1991; Martín et al., 2009). and Eq. (54) is non-linear and needs to be solved iteratively. For the (k + 1)-st iterations of the non-linear loop, the effec- 6.3 Preconditioned linear solvers tive viscosity ηk+1 is estimated from Eq. (4) using the pre- The discretisation and linearisation of the varying viscos- viously computed velocity field uk (fixed-point method or Picard method) or a Newton linearisation such as ity Stokes system lead to linear systems which cannot be solved efficiently by using standard linear solvers. A special ∂η ηk+1 = ηk + (uk+1 − uk) · . (56) preconditioned version of the generalised conjugate resid- ∂u ual (GCR) method has therefore been implemented recently Convergence is obtained much faster with the Newton iter- into Elmer/Ice to obtain effective parallel solutions of these ation than with the Picard method, but the former algorithm is systems. This new preconditioner utilises the natural block known to diverge when starting too far from the solution. In structure of the associated linear algebra problem and is de- practice, the solution is initially approached by performing rived from approximating the associated pressure Schur com- some Picard iterations, and then activating the Newton lin- plement matrix S as S ≈ (1/η)M, where η denotes the vis- earisation for the last iterations. Because the convergence is cosity corresponding to the current non-linear iterate and M problem-dependent and depends on the initial solution, there is the mass matrix corresponding to the pressure approxi- is no general rule stating when to start the Newton iteration mation (for similar solvers for varying viscosity flows see scheme. The efficiency of the Newton method is illustrated Grinevich and Olshanskii, 2009; Burstedde et al., 2009; Gee- on a test case in Sect. 7.2. nen et al., 2009; ur Rehman et al., 2011). Results of scal- ability tests done with this block preconditioned solver are 6.2.2 Transport equations presented in Sect. 7.3. Note that in conjunction with this Stokes solver version, we employ a bubble stabilisation strat- Assuming A = (A) or A = (a(2),a(2),a(2),a(2),a(2)), equa- 11 22 12 23 13 egy based on utilising bubble basis functions corresponding tions for the age of ice (Eq. 27) or the ice fabric evolution to the high-order version of the finite element method. (Eq. 15) can be expressed using the generic form ∂A 6.4 Normal consistency i + div(A u) + K A = F , (57) ∂t i i i i All boundary conditions involving vector (velocities) or ten- where i = 1, K = 0 and F = 1 for the dating equation and sor values (stress) are in need of a consistent description of where i = 1,...,5, and K and F are vector functions of C, the surface normals. Nodal normals, by nature of the discreti- W, a(2) and a(4) for the fabric evolution equations. For in- sation applied in the FE method, are not uniquely defined, es- compressible fluids (ice), div(A u) simplifies to u· grad (A ). i i pecially with linear elements. Thus, the representation of sur- Equation (57) is a first-order hyperbolic equation, and is non- faces has non-continuous derivatives at nodes. For ice sheet linear when solving for the fabric evolution. boundary conditions, this is a problem occurring typically The variational formulation of the transport equations is at the bedrock interface in the presence of sliding where a obtained by multiplying Eq. (57) by the test function 8 and Dirichlet-type non-penetration condition has to be applied, integrating over the ice volume . Because in the case of i.e. u·n = 0. Depending on the nodal definition of n, this con- a vector solution A the set of equations is solved iteratively dition can lead to artificial source and sinks for mass and mo- for each component independently, the variational formula- mentum in very uneven parts of the bedrock. Gillet-Chaulet tion is presented for a scalar A, and reads et al. (2012) have shown that using the average of the nor- Z ∂A Z Z Z 8d + div(Au)8d + KA8d = F 8d. (58) mal to the elements sharing the node to estimate the nodal ∂t normal can lead to an artificial mass loss of up to 10 % of     total ice discharge at the margin of Greenland. Recently, the The second term is then integrated by parts, so that mass-conserving way of deducing the nodal surface (Walkley Z Z Z ∂A ∂8 et al., 2004) has been implemented in Elmer. For a node xj , 8d − A ukd + KA8d ∂t ∂xk with an element-correlation number Nj , the surface normal    is derived by Z I = F 8d − Au · n8d0. (59)  ∂ www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1310 O. Gagliardini et al.: Elmer/Ice model

This algorithm is repeated as long as there is no change in N the active node set and the convergence criteria imposed for 1/N P j R n(x )ϕ dV j i=1 k j k the solver are met. nj = . (60) PNj R In a converged state, the physical meaning of the resid- k1/Nj n(xj )ϕkdV k i=1 k ual can be interpreted. For the heat equation Eq. (14), the The relation above constructs the nodal surface normal, residual represents the additional cooling needed to comply nj , as a sum of the normals evaluated at the adjacent ele- with the inequality T < Tm. For the free-surface equations ments, k, using the same weighting functions, ϕk, as for the Eqs. (10) and (11), the residual can be interpreted as the per- momentum equation. node additionally needed accumulation/ablation to meet the constraint Eq. (12). 6.5 Accounting for inequality

As presented in Sect. 2, the ice temperature T in Eq. (14) is 7 Elmer/Ice efficiency bounded by the pressure melting point Tm, and the free sur- 7.1 Convergence tests face zb and zs in Eqs. (10) and (11) must fulfil the inequali- ties z (x,y,t) ≥ b(x,y)+h and z (x,y,t) ≥ b(x,y). The s min b Convergence of the Stokes solver is tested by running the variational inequality is solved using a method of imposed same problem with an increased mesh resolution. The pur- Dirichlet condition that are released by a criterion based on pose of this exercise is to verify the model and compare the the residual. Let efficiency of the various elements and associated stabilisa- A · h = a (61) tion methods available within Elmer. For three-dimensional geometries, the Stokes equation can be solved using 8-node be the matrix equation of the unconstrained system. In FE (linear) or 20-node (quadratic) hexahedron elements, or 6- method terminology, A is the system matrix, h the solution node (linear) wedge elements. Stabilisation of the Stokes vector and a the body force. We then have to solve Eq. (61) equations is done either using the stabilised method (Franca under a constraint vector hmin. Here, we choose a minimum and Frey, 1992) or the residual free bubbles method (Baioc- value, hence a lower constraint, but the same method works chi et al., 1993). also with upper bounds, like it is applied in the case for The Stokes solver is verified using the manufactured ana- constraining the temperature with the local pressure melting lytical solution first proposed in Sargent and Fastook (2010) point. This lower constraint reads and subsequently modified and corrected in Leng et al. (2013). Here, we use exactly the same geometry and set of h > hmin . (62) parameters as in Leng et al. (2013). All meshes are structured and defined by the number of elements in x, y and z direc- In order to enforce Eq. (62) we apply the same method as tions. The mesh discretisation is made to vary from a very in Zwinger et al. (2007): coarse mesh (20 × 20 × 5, 10 584 degrees of freedom for the – A node i violating hi > hmini is set as “active”. linear element) up to a fine mesh (160 × 160 × 40, 4 251 044 degrees of freedom). The modelled domain is the bumpy bed – For each “active” node a Dirichlet condition hi = hmini of ISMIP-HOM experiment A (Pattyn et al., 2008) with the is introduced into Eq. (61). This is achieved by setting 80 km length. For this geometry, the finer mesh does corre- the ith row of the system matrix to Aij = δij , where δij spond to horizontal and mean vertical resolutions of 500 and is the Kronecker symbol, and the ith entry of the force 25 m, respectively. The convergence rate for the various el- vector to ai = hmini. Doing so for all active nodes re- ements and stabilisation methods is obtained as the slope of sults in an altered, constrained system the L2 relative error norm function of the grid size refine- 2 0 0 0 ment. The L relative error norm of two arbitrary vectors u A · h = a . (63) and v is defined as | − | – Instead of Eq. (61) we are now solving Eq. (63), obtain- 2 u v δu,v = . (65) ing a solution vector h0. |u + v| 0 For simplicity, it is plotted here as a function of the cu- – h in turn is inserted into the unconstrained system bic root of the inverse of the degrees of freedom, which is Eq. (61), defining the residual proportional to the grid size refinement. These curves for the R = A · h0 − a . (64) three components of the velocity and isotropic pressure are plotted in Fig. 1. They show that whatever the element type – If an earlier “active” node is found to comply with and stabilisation method, the rates of convergence (slopes of Eq. (62), it is taken of the list if, and only if, Ri < 0. the curves) are similar and close to 3 for velocity and pres- sure. A rate of convergence of 3 is greater than the theoretical

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1311

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2 -5 10 L -7 10 -3 -6 0 10 20 30 40 10 -3 -2 -110 -3 -2 -1 10 10 10 10 10 10 dof−1/3 dof−1/3 Iteration 2 2 Fig. 2. Evolution of the2L relative error norm between two con- Fig. 1. Results of the convergence tests: L relative error norm be- Fig. 2. Evolution of the L relative error norm between two consecutive solutions of the Stokes system, δup,up+1 , as secutive solutions of the Stokes system, δup,up+1 , as a function of tween Elmer/Ice and analytical solutions for the 3 components of a function of the non-linear iteration, for a switch criterion from the Picard to the Newton scheme of 10−6 (Picard the velocity (u,v,w) and the pressure, p, as a function of the grid the non-linear iteration, for a switch criterion from the Picard to the Newtononly, black scheme curve), of red 1010−6−(Picard2, green only,10−1, black blue 1 curve),and magenta red 102−(Newton2, green only), for the ISMIP HOM experiment A005 size (which is proportional to the inverse of the cubic root of the −1 degrees of freedom) for Franca and Frey (1992) stabilisation with 10with initial, blue conditions 1 and magenta (circle) 2 assuming (Newton zero only), velocity for the and ISMIP-HOM pressure and (triangle) estimated from the SIA solution. experiment A005 with initial conditions (circle) assuming zero ve- (black) 6-node wedge element, (red) 8-node hexahedron element The colour of the dot-dashed lines indicates the value of the switch criterion of the corresponding colour curve. and with (blue) 20-node hexahedron element; and for the residual locity and pressure and (triangle) estimated from the SIA solution. free bubbles stabilisation (Baiocchi et al., 1993) with (green) 8-node The colour of the dot-dashed lines indicates the value of the switch hexahedron element and with (magenta) 20-node hexahedron ele- criterion of the corresponding colour curve. ment. The black dashed line indicates a rate of convergence of 3.

(Picard only), 10−2, 10−1, 1 and 2 (Newton only). The non- −6 linearity is assumed to be resolved when δu ,u < 10 . value expected (e.g. Ern and Guermond, 2004), especially for p p+1 The evolution of δup,up+1 as a function of the non-linear it- the linear element and pressure. Surprisingly, the same rate eration indices is presented in Fig 2. As expected, the Newton of convergence is obtained for linear and quadratic elements, scheme is quadratically convergent, while Picard converges and for a given discretisation the quality of the solution is only linearly (Paniconi and Putti, 1994). When the initial even better using linear elements, so that the use of quadratic condition is null velocity and pressure, it takes 40 Picard it- elements is not recommendable, at least in this particular ex- erations to converge, whereas with Newton’s method alone, ample. For this application and the quadratic 20-node hexa- it requires only 10 iterations. Surprisingly, even36 if it takes hedron element, the residual free bubbles method is found to less Picard iterations to converge for the SIA initial condi- be less accurate than the stabilisation method of Franca and tion, the convergence of the Newton solver is only obtained Frey (1992). −1 if Picard iterations are performed from δup,up+1 < 10 . This example shows that Newton’s method can diverge if the ini- 7.2 Picard versus Newton linearisation tial condition is too far from the converged solution. A switch −2 criterion δup,up+1 < 10 is found to work in most cases and Picard and Newton schemes for the non-linear solution of it reduces the non-linear iterations by a factor about 2. Be- the Stokes equations are compared by performing the ISMIP- cause the CPU consumption is almost proportional to the HOM experiment A005 (Pattyn et al., 2008; Gagliardini and number of non-linear iterations performed within one time Zwinger, 2008) for two different initial conditions. The first step, switching from Picard to Newton iterative schemes can one assumes null velocity and pressure, whereas the second reduce CPU time by the same factor. initial condition is equal to the SIA solution for this problem. The switch from the Picard to the Newton iterative scheme 7.3 Elmer/Ice scalability 2 is controlled by a criterion on δup,up+1 , the L relative error norm, Eq. (65), between the previous p and current p + 1 The scalability of the new block preconditioned solver is velocity fields of the non-linear iteration loop. The same di- tested and compared with the parallel sparse direct solver agnostic simulation is repeated for switch criteria of 10−6 MUMPS (Amestoy et al., 1998). For this purpose, the www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1312 O. Gagliardini et al.: Elmer/Ice model

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E ffi ciency (%) Fig. 4. Efficiency for a weak scalability experiment using the block 50 preconditionerFig. 4. forEfficiency an approximate for a weak number scalability of nodes experiment in all meshesusing the of block preconditioner for an approximate number of 4200 andnodes meshes in all from meshes 0.708 of 4200× 10and6 nodes meshes up from to 4.580.708× 10×610nodes.6 nodes up to 4.58×106 nodes. 0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720 Number of cores new block preconditioned method for the strong scalability, whereas for a number of partitions greater than 100, MUMPS Fig. 3. Acceleration (top) and efficiency (bottom) for strong scala- Fig. 3. Acceleration (Up) and efficiency (Bottom) for strong scalability experiments usingwas alwaysthe block found preconditioner to scale badly due to an increase of the re- bility experiments using the block preconditioner for meshes with for meshes with (red •) 2.400 × 106 nodes, (red ) 1.142 × 106 nodes and (red ) 0.708 × 106 nodes, and the (red •) 2.400 × 106 nodes, (red ) 1.142 ×106 nodes and (red ) Nquired memory. This new solution strategy using the block  6 N 6 0MUMPS.708 × 10 solver6 nodes, for meshes and the with MUMPS (blue ) 1 solver.142 × 10 for meshesnodes and with (blue (blueN) 0.708 ×preconditioned10 nodes. The dashed solver line clearly opens the door to applications indicates a theoretical6 efficiency of 100%. 6 one order of magnitude larger in mesh size than what we ) 1.142 × 10 nodes and (blue N) 0.708 × 10 nodes. The dashed line indicates a theoretical efficiency of 100 %. were able to achieve so far using a direct solver like MUMPS. diagnostic Stokes solution is computed using the present-day 7.4 Inverse methods validation Greenland geometry. The Greenland footprint is first meshed We test the two inverse methods previously presented in a 2- using regular triangle elements and then vertically extruded. D example resembling a calving glacier. As our objective Different meshes are constructed by varying the horizontal is to validate the numerical implementation, we use a linear element size from 5 km to 10 km, but all have 20 vertical lay- 38 rheology for which the two inverse methods implemented are ers. The size of the tested meshes varies from 708 000 up exact. to 4 580 000 nodes. Temperature and basal drag37 are imposed Our domain is 20 km long, the bed elevation is constant using the same fields as in Gillet-Chaulet et al. (2012). New- and equal to −900 m. The free-surface elevation decreases ton iterations are used after the convergence criterion reaches linearly from 200 m at x = 0 km to 100 m at x = 20 km. 5 × 10−2. The free surface is stress-free, we prescribe an homogeneous The results obtained for strong scalability, i.e. a constant Dirichlet condition of 50 ma−1 for the horizontal velocity at problem size with different partitionings, and for weak scala- x = 0 km, we apply a Neumann condition (hydrostatic sea bility, i.e. a constant load per CPU using different mesh sizes, pressure) at x = 20 km, and we apply a linear sliding law and are presented in Figs. 3 and 4, respectively. If the elapsed a non-penetration condition at the bedrock. time is t for a number of partitions n, then, for a strong scal- n We generate a reference solution with ing test, the scalability of a solver can be characterised ei- −3 −1 ther by its acceleration tref/tn or its efficiency (n/nref)tref/tn, β = 10 (1.0 + sin(2πx × 2/L)) MPa m a, (66) where ref stands for the reference simulation (often the one η = 10(1.5 + sin(2πx × 6/L)) MPa a. (67) with the smallest mesh size). For a weak scaling test, the ac- celeration depicted in Fig. 4 is defined as (n/nref)tref/tn . The surface velocities computed from this reference so- The weak scalability experiment uses a constant number lution are then used as perfect synthetic observations (twin of 4200 nodes per partition in combination with an increas- experiments). ing number of partitions from 168 up to 1092. Weak scala- The first step in the validation process is to assess the bility is found to be greater than 60 % even for the largest ability of each solver independently to reconstruct the syn- test case. Efficiency greater than 100 % is obtained with the thetic observations. For the sliding coefficient β, we start

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d Fig. 6. Ratio 1(h) obtained with the Robin (black curves) and con- | Fig. 6. Ratio ∆(h) obtained with the Robin (black curves) and Control (red curves) inverse methods for perturbations -6 trol (red curves) inverse methods for perturbations of the enhance- 10 ofment thefactor enhancement to the viscosity factor to theEη viscosity(x,y). Eη(x,y). 0 100 200 300 400 Iterations for the initial conditions of the previous twin experiments.39 Fig.Fig. 5. 5.EvolutionEvolution as a functionas a function of the number of the ofnumber iterations of of iterations (top) the cost of (top) function relativeWe toalso the initial test costthe function implementation of the Tikhonov regular- andthe (bottom) cost function the norm relative of the to gradient the initial vector cost relative function to the and initial (bottom) gradient the norm forisation, the Robin Eq. (black (44 curves)), by and choosing the cost function such as norm of the gradient vector relative to the initial gradient norm for R 2 −9 Control (red curves) inverse methods for the inversion of β (solid curve), E (x,y) (dashedJ = curve)0.5(∂α and Ei/∂x)(x,y,zd0) where αi = 1.0 (1.0+sin(2.0π × the Robin (black curves) and control (red curves) inverse methodsη 0b η (dotted curve). 4x/L)). for the inversion of β (solid curve), Eη(x,y) (dashed curve) and 0 For each solver, we use 10 random perturbation fields αi Eη(x,y,z) (dotted curve). 0 where each nodal value of αi is a random number between 0 10 −50 % and 50 % of the mean value of αi. The gradient com- − puted from the two inverse methods is verified using the ratio from the initial guess β = 10 3. The viscosity η is expressed i 1(h) defined as -1 as 10η = Eηη0, and the optimisation is done on the viscos- = 0 ity) enhancement factor Eη with initial guesses Eη 1 and (J (α + hα ) − J (α ))/h − ∇J h i i i αi η0 = 15. It is possible a priori that several distributions of Eη, 1(h) = . (69) ∆ ( -2 ∇J especially10 in the vertical direction, can lead to the same sur- αi face velocities; therefore, we made it possible in the model An example of the evolution of 1(h) as a function of h is -3 to10 invert Eη only in the horizontal plane (x,y) when using shown in Fig. 6 for the perturbation of the enhancement fac- vertically extruded meshes. The gradient of J with respect to tor to the viscosity Eη(x,y). For both inverse methods and all Eη at-4 a given position (x,y) is then obtained as the vertical 10 experiments, i.e. perturbation of β (not shown), Eη(x,y,z) -6 -5 -4 -3 -2 -1 0 sum10 of the nodal10 gradients10 at10 position10 (x,y)10. 10 (not shown), Eη(x,y) (Fig. 6) and the Tikhonov regularisa- h To ensure that β and Eη remain positive during the opti- tion (not shown), the ratio 1(h) is found to decrease as h α misation, β is expressed as β = 10 1 and Eη is expressed as decreases and it reaches a value typically lower than 10 %. Fig. 6.=Ratio2 ∆(h) obtained with the Robin (black curves) and Control (red curves) inverse methods for perturbations Eη α2. The optimisation is then done with respect to the Such values are already satisfactory; nonetheless we could of the enhancement factor to the viscosity E (x,y). αi (i = 1,2). The evolution of theη cost function and norm of obtain even more accurate gradients by automatically deriv- the gradient obtained for each test is given in Fig. 5. Both the ing the code itself. cost function and the gradient norm decrease39 and tend toward Figure 7 illustrates the difference of results obtained when zero with the number of iterations. inverting for Eη only in the horizontal plane or in the whole The second step in the validation process is to verify the ice volume. As can be seen, the two inferred fields Eη(x,y) following approximation: and Eη(x,y,z) are significantly different even if surface ve- locity and observation are in comparable agreement. This J (α + hα0) − J (α ) i i i = ∇ + indicates that a non-unique solution can be obtained when Jαi o(1). (68) h the number of control parameters starts to be larger than the 0 number of observation points. For a given perturbation αi, the left-hand-side term is eval- + 0 uated by computing J (αi hαi) with the direct model for several values of h, and the right-hand-side term is com- puted directly from the nodal gradients. This test is done www.geosci-model-dev.net/6/1299/2013/ Geosci. Model Dev., 6, 1299–1318, 2013 1314 O. Gagliardini et al.: Elmer/Ice model

1.8 of the stress distribution at the front of glaciers and of the submarine melting distribution, as well as a reliable ice dam- 15 age model (e.g. Pralong, 2005; Jouvet et al., 2011), are the re- 1.2 quired ingredients to describe calving at the front of marine- terminated glaciers. In Elmer/Ice, the already implemented 0.9 ALE formulation for the free surface accounts for moving ice 0.6 sheet margin boundaries. Because Elmer/Ice solves the full- 0.3 Stokes system, all components of the stress field are known and can therefore be used to evaluate ice damage. Work is in Fig. 7. Comparison of the obtained inverted enhancement factor progress to implement an ice damage rheological law follow- Fig.field 7. forComparison the control of the method obtained assuming inverted enhancement only changes factor in fielda horizon- for the Controling methodPralong assuming(2005 only) with changes the aim of using damage iso-surface intal a planehorizontal (Eη plane(x,y) (E, left)η(x,y or), changesleft) or changes in all in3 directions all 3 directions (Eη ((x,y,z)Eη(x,y,z, ), right).to locate The inverted the calving field Eη(x,y surface) in and move accordingly the front theright). left is The virtually inverted identical field toE theη(x,y) one prescribedin the left to is obtain virtually the observed identicalvelocities to surface. (Eq. (66)). the one prescribed to obtain the observed velocities (Eq. 67). Melting from beneath the ice shelves is certainly one the most important triggers of the observed recent ice stream ac- celerations (e.g. Payne et al., 2004; Dupont and Alley, 2005). 8 Outlook Not only is the total amount of basal melting important, but also its spatial distribution (Gagliardini et al., 2010). For nu- A number of the requisites for an ice sheet model as dis- merical and technical reasons, coupling an ocean model with cussed in the Introduction have already been implemented in an ice sheet model is still a challenging issue. An interme- Elmer/Ice, and especially those necessary to accurately de- diate approach we would like to explore as a preliminary scribe the flow of polar ice. Nevertheless, as for other ice step towards a complete coupling with an ocean model is sheet models, the physical processes at the boundaries and the implementation within Elmer/Ice of a plume-type model their coupling with the other components of the climate sys- (Holland and Feltham, 2006). tem can still be improved in the near future. This is the pre- requisite for running any forecast simulations of ice sheets, and not only sensitivity experiments based on more-or-less 9 Conclusions crude parameterisations that link changes in the atmosphere and the ocean to changes at the boundaries of the ice sheet. We have presented in detail the Elmer/Ice ice sheet flow Our efforts in the near future will be dedicated to improv- model, from the equations implemented to the way they are ing the physical description of the ice/atmosphere,40 ice/ocean solved using the FE method. Elmer/Ice contains a high me- and ice/bedrock boundaries, as well as the models describing chanical description of ice flow: it solves the complete Stokes how the pertinent variables at these interfaces are distributed. equations without any approximation, includes two complex For the basal boundary condition, numerical modelling anisotropic flow laws, resolves the grounding line dynamics (Schoof, 2010) or direct measurements (Sole et al., 2011) as a contact problem and incorporates a basal friction law ac- seem to indicate a very complex relation, most certainly non- counting for cavitation. Temperature and fabric fields within linear, between changes in surface runoff and modulation the ice sheet domain can be determined in a coupled man- of basal sliding. Two ingredients would then be required to ner with the flow solution. Other equations allowing for the fully account for the complexity of basal processes in rela- derivation of secondary variables from the Stokes solution, tion with changes in surface runoff: (i) a proper basal friction such as the age of the ice, the stress or strain rate fields, law depending on the effective basal pressure (i.e. Eq. 22), are also implemented. Two recent inverse methods have been and (ii) an associated hydrological model to describe the implemented in Elmer/Ice that make it possible to infer the basal water pressure distribution. This hydrological model poorly known parameters to construct the initial state of the is currently under development and will be presented in de ice sheet. From a technical point of view, Elmer/Ice reaps Fleurian et al. (2013). the benefits of the FE method, and provides an easy mesh Changes in the front position of marine-terminated adaptation method to focus on areas of interest. Elmer/Ice is glaciers seem to have a great influence on the upstream ice a highly parallelised code and, as a recent important improve- flow by modulating the buttressing force (Vieli and Nick, ment, the new block preconditioned solver will in the near fu- 2011). Determining the rate at which icebergs are calved for ture lead to increase significantly the size of the solved prob- many different configurations remains an open question in lems. As listed in the previous section, there is still a need glaciology. Submarine melting acting at the calving front of for future improvements and new developments, particularly glaciers certainly increases calving rate by undercutting the by linking more tightly the pertinent variables controlling the ice (Rignot et al., 2010; O’Leary and Christoffersen, 2013). flow at the boundaries, like the basal water pressure, with the A general calving law, especially for 3-D configurations, still other components of the climatic system. This next step is needs to be formulated (Benn et al., 2007). Better knowledge the requisite for driving ice sheet forecast simulations and

Geosci. Model Dev., 6, 1299–1318, 2013 www.geosci-model-dev.net/6/1299/2013/ O. Gagliardini et al.: Elmer/Ice model 1315 furnishing reliable estimates of ice-sheet-induced sea level Bueler, E. and Brown, J.: Shallow shelf approximation as a ’sliding rise for the coming centuries. law’ in a thermomechanically coupled ice sheet model, J. Geo- phys. Res. Earth Surf., 114, F03008, doi:10.1029/2008JF001179, 2009. Burstedde, C., Ghattas, O., Stadler, G., Tu, T., and Wilcox, L.: Paral- Acknowledgements. We would like to thank both an anonymous lel scalable adjoint-based adaptive solution of variable-viscosity reviewer and E. Larour for their positive and constructive comments Stokes flow problems, Comput. Methods Appl. Mech. Eng., 198, regarding our work, which greatly improved the initial version of 1691–1700, 2009. the manuscript. The development of Elmer/Ice was supported by Calov, R. and Greve, R.: Correspondence. A semi-analytical solu- various fundings, in particular the ice2sea programme from the tion for the positive degree-day model with stochastic tempera- European Union 7th Framework Programme, grant number 226375 ture variations., J. Glaciol., 51, 173–175, 2005. (Ice2sea contribution number 153), and the Nordic Centre of Castelnau, O., Duval, P., Lebensohn, R. A., and Canova, G.: Vis- Excellence SVALI, “Stability and Variations of Arctic Land Ice”, coplastic modeling of texture development in polycrystalline ice funded by the Nordic Top-level Research Initiative (TRI, SVALI with a self-consistent approach : Comparison with bound esti- contribution number 23). R. Greve and H. Seddik were supported mates, J. Geophys. Res., 101, 13851–13868, 1996. by a Grant-in-Aid for Scientific Research A (no. 22244058) Castelnau, O., Shoji, H., Mangeney, A., Milsch, H., Duval, P., from the Japan Society for the Promotion of Science (JSPS). 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