Fracture Field for Large-Scale Ice Dynamics

Journal of Glaciology, Vol. 00, No. 000, 0000 1

Fracture ﬁeld for large-scale ice dynamics Torsten ALBRECHT,1,2∗Anders LEVERMANN,1,2

1Earth System Analysis, Potsdam Institute for Climate Impact Research, Potsdam, Germany 2Institute of Physics, University of Potsdam, Potsdam, Germany

ABSTRACT. Recent observations and modeling studies emphasize the crucial role of fracture mechanics for the stability of ice shelves and thereby the evolution of ice sheets. Here we introduce a macroscopic fracture density field into a prognostic continuum ice flow model and compute its evolution incorporating the initiation and growth of fractures as well as their advection with the two-dimensional ice flow. To first approximation fracture growth is assumed to depend on the spreading rate only, while fracture initiation is defined in terms of principal stresses. The inferred fracture density fields compare well with observed elongate surface structures. Since crevasses and other deep reaching fracture structures have been shown to influence the overall ice shelf dynamics, we propose the fracture density field introduced here to be used as a measure for ice softening and decoupling of the ice flow in fracture-weakened zones. This may yield more accurate and realistic velocity patterns in prognostic simulations. Additionally, the memory of past fracture events links the calving front to the upstream dynamics. Thus the fracture density field proposed here, may be em- ployed in a fracture-based calving parameterizations. The aim of this study is to introduce the field and investigate which of the observed surface structures can be reproduced by the simplest physically motivated fracture source terms.

1. INTRODUCTION close to the surface where the compressive overburden stresses Ice shelves are the floating extensions of the large polar ice are small. They form in the presence of strong divergence or sheets and are predominantly found around Antarctica. The convergence of the ice flow (Hughes, 1977, 1983) or due to recent discussion on the rate of global sea-level rise empha- the buckling and shearing close to ice rises and side margins sizes the crucial role of ice shelves at the conjunction between (e.g., Glasser and others, 2009; Cuffey and Paterson, 2010). atmosphere, ocean and the large ice sheets. A shrinkage or Also the abrupt acceleration and cyclic tidal flexure (e.g., even a loss of buttressing ice shelves has only a relatively small Holdsworth, 1977; Lingle and others, 1981) in the transition direct effect on this sea level rise (Jenkins and Holland, 2007; zones between grounded or stagnant parts and fast flowing Shepherd and others, 2010), but it can cause an accelera- units of floating ice shelves support fracture formation. As a tion and thinning of the upstream tributaries, which increases basic approach the initiation and growth of large-scale frac- solid ice discharge from the ice sheet (Rott and others, 2002, tures close to the surface, so-called surface crevasses, can 2007; Rott and others, 2011; De Angelis and Skvarca, 2003; be considered in terms of horizontal strain rates or surface Rignot and others, 2004, 2008; Scambos and others, 2004; stresses. This is a simplification, since in general, brittle frac- Joughin and others, 2004; Cook and others, 2005; Rignot, ture formation is known to depend on the total stress ten- 2006; Gudmundsson, 2007; Shepherd and Wingham, 2007; sor across a crack (Lawn, 1993; Paterson and Wong, 2005). Bamber and others, 2007; Glasser and Scambos, 2008; Cook The presence of water strongly influences fracture growth. and Vaughan, 2010; Joughin and Alley, 2011). Numerical At the bottom of the ice shelf, crevasses can extend far up- studies are in principle able to reproduce this behavior (Schmeltz ward into the ice body due to seawater penetration (Weert- and others, 2002; Payne and others, 2004; Thomas and others, man, 1980; Jezek and Bentley, 1983; Van der Veen, 1998a; 2004). Rist and others, 2002; Luckman and others, 2011). At the surface meltwater-enhanced fracturing has been proposed as Gravitational driving stresses in ice shelves must be balanced one of the triggers of the catastrophic disintegration of sev- by drag at the lateral shelf margins due to the lack of basal eral ice shelves fringing the Antarctic Peninsula in the last traction. Fractures affect this stress balance by disturbing decades (Weertman, 1973; Doake and Vaughan, 1991; Rott the ice body’s mechanical integrity (e.g., Larour and others, and others, 1996; Scambos and others, 2000; MacAyeal and 2004; Larour and others, 2004; Vieli and others, 2006, 2007; others, 2003; Rack and Rott, 2004; Vieli and others, 2007; Khazendar and others, 2009; Humbert and Steinhage, 2011) Van der Veen, 2007; Khazendar and others, 2007; Glasser and thereby reducing the buttressing effect (Dupont and Al- and Scambos, 2008; Braun and others, 2009; Scambos and ley, 2005; Gagliardini and others, 2010). Hence, prognostic others, 2009). ice-sheet models need to incorporate at least the large-scale Considerable effort was made to incorporate fracture me- effects of the complex physics of fracture processes. chanics into regional diagnostic ice-shelf models: Rist and others (1999), Jansen and others (2010) and Hulbe and others Fractures develop from small-scale flaws that have sufficient (2010) use a mechanistic two-dimensional fracture criterion in initial size (Rist and others, 1996, 1999), most likely initiated diagnostic models to evaluate the stability of Filchner-Ronne and Larsen C Ice Shelves, respectively. Humbert and others (2009) identified shear zones and rifts in the Brunt Ice ∗Correspondence: [email protected] 2 Albrecht and Levermann: Fracture field for large-scale ice dynamics

Shelf using observational data and applied additional flow focus of future studies. Here, we further do not take into ac- enhancement factors and a decoupling scheme for the stress count previous micro-scale stages of fracture formation such balance. Saheicha and others (2006) use the von-Mises crite- as subcritical crack growth or damage accumulation (Weiss, rion (Vaughan, 1993), which expresses a relationship between 2004; Pralong and Funk, 2005). Such processes depend on tensile strength and effective stress. The macroscopic effect the prevailing local stress and produce the flaws necessary to of fracture structures on the ice body of Filchner Ronne Ice initiate crevasse opening, mostly in firn-layers with distinct Shelf in the diagnostic model is parameterized by softening mechanical properties. Such flaws can hardly be detected at and decoupling in order to reproduce characteristic features the ice shelf surface and do not contribute directly to the of the observed velocity distribution. These models identify here defined fracture density. However, they do play an im- regions of potential crack opening for given boundary condi- portant indirect role for the fracture density field φ by serv- tions and evaluate its effect on the stress balance. Sandhäger ing as starter cracks, from which the crevasses actually are and others (2005) additionally introduced the advection of initiated. In order to focus on the macroscopic dynamics of a variable with the diagnostic velocity field accounting for a fracture growth, we assume a spatially homogeneous field of shear-zone depth ratio healed with a constant rate. micro-cracks. In order to represent the manifold physical role of frac- The fracture density, φ, is assumed to be isotropic and it tures on large-scale ice sheet dynamics we need a well defined is transported by horizontal advection with the vertically av- time-evolving field, which is advected with the ice flow. Here eraged velocity v calculated from the continuum mechanics we introduce this field as a measure for fracture density and stress balance in Shallow Shelf Approximation (SSA) imple- introduce first-order parameterizations for the relevant con- mented into the Potsdam Parallel Ice Sheet Model PISM- tributing processes, such as the initiation, growth or healing PIK (see Appendix A for a brief and Winkelmann and others of fractures. The respective importance of these processes and (2011) for a comprehensive description) as parameters is evaluated in a suite of numerical experiments. ∂φ Characteristics of fracture density along the flow line and in + v · ∇φ = fs + f = f(ε˙, σ, φ, ..) (3) ∂t h side shear regions are discussed for two idealized ice shelf geometries. The individual pattern of fracture density for re- with forcing term f as the sum of sources fs and healing terms alistic scenarios, such as for the large Filchner-Ronne Ice Shelf fh, which depend on a number of glaciological parameters. To (FRIS) or the smaller Larsen A+B Ice Shelf (LABIS) is com- leading order we assume here a sole dependence on the strain pared to a snapshot of observed locations of surface features rate, ε˙, stress, σ, and the fracture density, φ, itself. For the in these ice shelves. source term we write

fs = ψ · (1 − φ), (4) 2. FRACTURE DENSITY FIELD where ψ is the fracture growth rate. The second factor reduces We define a two-dimensional field, φ, in the entire ice shelf the source by (1 − φ) and represents the fracture interaction, region, which quantifies the mean area density of fractured bounding the resulting macroscopic fracture density below ice. The unit-less, scalar field variable, φ, is bounded by the threshold 1. The justification for this term is that the local stress field surrounding a single crevasse is relaxed up to 0 ≤ φ ≤ 1, (1) a distance of about 3–4 crevasse depths (Sassolas and others, 1996). For closely-spaced crevasse bands and hence for a rela- where the value φ = 0 corresponds to non-fractured ice. In tively high fracture density, additional fracture formation and observed ice-shelf fracture fields the fracture density, φ , is obs further fracture penetration is hindered accordingly (Smith, defined as the fraction of failed ice. That is ice that can not 1976; Weertman, 1977). be described appropriately by an ice-continuity condition, e.g. Stokes Equation. φobs thus increases with the number N and 2.1. Longitudinal spreading as source of the length l of growing crevasses. Crevasses are discontinu- fracture growth ities, which alter the local stress field within a zone of influence. For comparison with observational fields we assume Ice shelves spread under their own weight with a rate, which that the area of failed ice around crevasses is rectangular, is generally strong in the vicinity of the grounding line, where spanning a region of horizontal distance w around an indi- it goes afloat. In two horizontal dimensions we attempt here vidual crevasse. Groups of individual zones of influence can a formulation using merely the eigenvalues of the vertically averaged strain-rate tensor,ε ˙ andε ˙ , whereε ˙ ≥ ε˙ . These be quantified as a fraction of an ice shelf surface section as 1 2 1 2 eigenvalues represent the expansion (ε ˙1,2 > 0) or contraction N w l (ε ˙ < 0) along the principal directions. In isotropic ice the φobs = , (2) 1,2 ∆x ∆y initial opening of surface cracks is observed to be transversal where (N w) and l are limited to the extent of the discrete to the principal direction of strongest stretching (Holdsworth, mesh, ∆x and ∆y. For φobs = 1 the zone of influence covers 1969; Rist and others, 1999). For the purpose of this study, we the entire discrete grid cell surface (see example Fig. 1b). use a simple formulation in terms of the maximum spread- We consider here macroscopic structural glaciological frac- ing rate (not the total effective strain rate) to capture the ture features, growing from the surface both vertically and first-order effect and understand how much of the observed horizontally or closing in appropriate stress regimes. Although, fracture fields can be reproduced by this simple approach. the vertical dimension of penetrating crevasses is an essential Hence we formulate the fracture growth rate for the fracture issue in the discussion on ice shelf stability, it is not explic- density source Eq. (4) as itly included in our definition. Instead we focus here on the ψ = γ · ε˙ , (5) location of surface crevasses, often collocated in band struc- 1 tures revealing a depth-dependent density of crevasses. φ can, where γ is a unitless constant parameter, which in general however, be expanded to three dimension which will be the should be a function of flow law parameters, such as the Albrecht and Levermann: Fracture field for large-scale ice dynamics 3

a N E b Weddell Sea S W SIS F

BIR REIS R SFIS w HIR l KIR FP MIS EIS FIS IIS CIS RIS 5 km

Fig. 1. a) Observed fracture density φobs in the Filchner-Ronne Ice Shelf based on data by Hulbe and others (2010) and Eq. (2) with length-dependent width of the zone of inﬂuence, bounded by 1 km < w = l/4 < 10 km. Regions of vanishing fracture density are masked. Main inlets and ice rises are denoted by abbreviations. Thin black and grey-dotted lines show the position of the grounding line and observed stream lines. b) Closeup-view of observed fracture density, φobs, overlaid by observed fractures (black) with associate zone of inﬂuence (grey contour lines).

temperature-dependent ice softness. For this uniaxial case without healing (fh = 0) we can derive from Eq. (3)–(5) an exact analytical steady-state solution, which only depends on the velocity distribution u(x) along the ﬂow line,

−γ φ(x) = 1 − (1 − φ0 )(u(x)/u0 ) , (6)

and a given inﬂow velocity u0 = u(x0 ). The boundary value

φ0 = φ(x0 ) is associated with preexisting cracks which have formed upstream in the tributaries or with crevasses initiated at the grounding line, e.g., due to tidal flexure (more in Sect. 2.3). In the unconfined flowline case the function φ(x) is strictly monotonic sinceε ˙1 is tensile everywhere, i.e., positive values meaning expansive flow. The parameter γ becomes an exponent here determining the curvature of the function. The larger γ, the faster φ approaches saturation (at φ = 1). wide narrow

Experiments with idealized ice-shelf geometry In this first-order formulation we calculate the two dimen- Fig. 2. Steady state fracture density φ for an ice shelf confined sional steady-state fracture density φ(x, y) with PISM-PIK in a rectangular bay of 300 km width and 150 km length (left) or for an ice shelf confined in an idealized rectangular bay with 150 km width and 250 km length (right), respectively with constant unidirectional inflow (Fig. 2). We chose a parameter value γ = 0.5. Ice flows into the bay at the top boundary with φ0 = 0.1, γ = 0.5 for the fracture growth rate and as boundary con- ice thickness 600 m and maximal 300 m a−1 speed. Along the sides 13 friction is parameterized by a viscosity of ηb = 5 × 10 Pa s. At dition φ0 = 0.1. The steady state profiles of ice thickness H, velocity u and hence the larger principal strain rateε ˙ the ice shelf front ice calves off for an ice thickness less than 175 m. 1 Melting and accumulation is neglected. Black-dashed lines show along the center line x of the wide ice shelf Fig. 3a are close lateral and cross sections along which profiles are plotted in Fig. 3 to the analytical flow-line solution of an unconfined ice shelf spreading in one direction (Van der Veen, 1999, dash-dotted) since in the wide geometry the buttressing effect of side drag 3 −18 −3 −1 is small in the inner part. Inserting this analytical velocity where C = [ρ g (ρw − ρ))/(4 ρw B0)] = 2.45 × 10 m s solution into Eq. (6) yields the fracture density comprises material constants, such as the ice density, ρ, sea water density, ρw, and the acceleration due to gravity, g, 3 −γ/4 4 CH0 as well as the constant ice stiffness rate factor, B0. Even φ(x) = 1 − (1 − φ0 ) x + 1 , (7) u0 though in the simulations a temperature-dependent ice stiff- 4 Albrecht and Levermann: Fracture field for large-scale ice dynamics ness, B = B(T ), is used the obtained fracture density along where a certain threshold is first reached. Based mostly on the center line is very close to the analytical solution of Eq. (7) laboratory experiments those critical values are often defined and increases up to a value of 0.4 at the mouth of the em- in terms of effective stress or similar rotational invariant flow bayment. At the side margins, where friction is prescribed by properties such as combinations of stress or strain rate eigen- a staggered boundary viscosity, strong shearing yields large values. Fracture initiation starts from the surface, where the magnitudes of both principal strain rate components, while overburden pressure vanishes. The mechanical properties of the minor component,ε ˙2 , is generally compressive here with the upper firn layer are distinct, but strain rates are mostly respect to the principal axes (Nye, 1952). The here presented constant with depth (Nath and Vaughan, 2003). A variety of parameterization leads to steady state values of the fracture phenomenological criteria like the von-Mises criterion (Vaughan, density up to 0.8 close to the side margins (right column 1993), the Coulomb criterion or even advanced mixed-mode Fig. 3a). crevassing with maximum circumferential stress criterion based Figure 3b shows the profiles of ice thickness and velocity of on Linear Elastic Fracture Mechanics (e.g., Kenneally and an ice shelf confined in a comparably narrow bay of 150 km Hughes, 2006) can identify regions, where creep failure and width and 250 km length. For the same lateral friction as for fracture initiation most likely occur. the wide-bay setup we find a strong buttressing effect also We choose the simple von-Mises criterion, which is based in the inner part of the ice shelf, indicated by much thicker on the so-called “maxiumum octahedral shear stress”, defined and slower ice compared to the less confined, i.e., wider case. in terms of surface-parallel principal stresses, σ+ and σ− , as This leads to smaller spreading rates in the upstream region q 2 2 and hence to a smaller fracture density there. When the ice σt = σ+ + σ− − σ+ σ−. (8) shelf leaves the narrow bay it can extend freely, which leads −11 −1 The range of literature values of the tensile strength, σ , that to maximal spreading rates up toε ˙1 = 7 × 10 s and t hence to a strong increase in fracture density close the value ice can support, is 90 k Pa to 320 k Pa (Vaughan, 1993). This of the wide ice shelf at this position. Accordingly, the profiles large range may be due to the uncertainty in stress/strain- across the mouth of the narrow bay are similar to the results rate conversion of observational data with the constitutive of the wide bay (compare right columns of Figs. 3a,b). flow law as well as due to the effects of temperature, density and firn structure. Tensile strength in Antarctic ice shelves Experiments with realistic ice-shelf geometry are found around the median of this literature range. On the In a realistic setup, such as for the Filchner-Ronne Ice Shelf basis of the von-Mises criterion we refine our fracture growth (FRIS), the resulting fracture density field can be phenomeno- rate (Eq. (5)) to logically compared with observations, i.e. the location of ar- f = γ · ε˙ · (1 − φ) · Θ(σ − σ ), (9) eas with many surface features for a certain snapshot can be s 1 t cr compared to features of the obtained fracture density pat- where the additional Θ(∆) is the Heaviside step function to tern, indicated by comparably high values. In contrast to the predict regions of potential fracture growth, defined as simulations with idealized geometry (Fig. 2) we prescribe the ( ice thickness in the simulations with realistic topography and 1, ∆ > 0 Θ(∆) = . (10) specify boundary conditions (see Sect. 2.2 and Appendix A 0, ∆ ≤ 0 for precise values). This is necessary in order to avoid significant errors from a dynamic calculation of the flow field. We In the following we discuss the effect of this critical value simulate the evolution of fracture density starting from an ini- on the steady state fracture density pattern in a model of tial zero state in the whole ice shelf domain. After reaching a the Filchner-Ronne Ice Shelf. In order to be close to observed steady state, fracture density is highest in the stagnant outer conditions ice thickness and all other boundary conditions are shear regions and downstream from regions that have been prescribed and the three-dimensional temperature evolution subject to shearing along the side margins or promontories has already reached an equilibrium state close to observa- (Fig. 4a). Separate flow units draining from different inlets tions. Some parameter must be determined to achieve the can be identified in the structure, but the steady state pat- most realistic flow pattern for the given model configuration. tern of the fracture field appears comparably smooth. While The lateral friction of the ice shelf, parameterized by the the location of some features compare well with those in ob- boundary viscosity ηB, strongly influences the strain rate and servations, pronounced bands of observed surface features, as stress fields and thereby the fracture density field. Higher evaluated by Hulbe and others (2010) from MODIS MOA friction leads to vanishing ice speed along the margins and image maps can hardly be identified in the fracture density to strong shearing but also to much slower speeds in the in- pattern. Fracture density is also high in areas where the flow terior and outer parts of the ice shelf, which can be compen- is comparably slow like upstream of ice rises. Since in reality sated by a larger enhancement factor. Comparison to recon- fractures are not expected to form in those low strain rate structed observational ice surface speed data (especially for areas, this suggests the introduction of a critical threshold lower speeds) suggests to use a comparably small side friction 15 for fracture initiation. Observations suggest physically rea- (Fig. 5a). It was set to ηB = 10 Pa s. This accounts for the sonable thresholds for crevasse initiation defined in terms of partly decoupling of large interior flow units from the sides stresses, i.e., at high effective stresses (e.g., Vaughan, 1993). along weak zones. As long as this feature is not captured in Below these thresholds fractures may close and heal due to the flow model (see Sect. 3), a suitable enhancement factor for the overburden pressure, accumulation or refreezing. the SSA calculation has to be chosen. Such an enhancement factor is generally used to account for anisotropy and grain 2.2. Crevasse initiation threshold size effects in the ice shelf (Ma and others, 2010). Hence, the Observed crevasse bands downstream of inlet corners, con- vertically averaged effective viscosity in the SSA-stress bal- −1/n verging streams and ice rises aligned in flow direction sug- ance scales with ESSA (see Eq. 10 in Winkelmann and others gest an initiation of fractures in the intense shearing regions (2011)). Best comparison with observations in the FRIS with Albrecht and Levermann: Fracture field for large-scale ice dynamics 5

wide narrow

a b

Fig. 3. Obtained velocity, ice thickness, principal horizontal strain rates and the inferred fracture density field evolution for ice shelves confined in a rectangular bay of 300 km width and 150 km length (wide, left hand side) and of 150 km width and 250 km length (narrow, right hand side, compare to Fig. 2). Profiles are normalized to unity length and width and plotted in lateral view along the center line (left panel columns) and across the mouth of the bay (right panel columns). The black-dashed vertical line in the left panel columns shows the position of the mouth of the bay. The grey-dashed profile on the upper right panel shows the unidirectional inflow velocity applied at the upstream boundary. Steady state values in the wide case are close to the analytical solution of an unbuttressed flow line ice shelf plotted as dashed line, whereas in the narrow case profiles vary strongly. In the lower panels steady state fracture densities with φ0 = 0.1 are shown.

a root mean square deviation of the velocity vector compo- regions close to the front. In those regions the fracture interac- nents of only 108 m a−1 averaged over about 14000 points and tion factor, (1 − φ), has a significant effect, which is expected a cross correlation coefficient of r2 = 0.96 is achieved by a in the dynamics of real crevasse trains. relatively low value of ESSA = 0.4. For increasing fracture initiation threshold, σcr, we get The so inferred steady effective stresses are comparably more pronounced band structures downstream of ice rises and low and regions of likely creep failure identified with the von- prominent geographical features, which agrees with the loca- Mises criterion with a threshold below the literature range, tion of crevasse trains in observations (compare Figs. 4a,b). e.g., σcr = 70 kPa, cover only 4% of the total FRIS area, For even higher threshold values (within the range of the indicated by purple shaded areas in Fig. 5b. These shear re- literature values) some of the features are not reproduced gions are found at the major inlets, downstream of ice rises anymore as for instance fractures in the suture zone of the at flow unit junctions, but also at shearing margins, partic- merging flow units of Evans (EIS) and Rutford Ice Stream ularly in the narrow Filchner Ice Shelf. Thus, fractures grow (RIS) in Fig. 4c. Most relevant features of the observational according to the corresponding spreading rate, which is on data map are best represented for a threshold of σcr = 70kPa, −10 −1 average about ε˙1 = 3 × 10 s in those regions. In ob- shown in Fig. 4b. The redish bands (high fracture denisty) servations much higher strain rates are found in the confined coincide very well with observed fracture bands and streak suture zones and at shear margins. A softening of the ice in lines (except for those structures formed at ice rumples, that areas of high fracture density would account for this effect are not considered in our model, such as Kershaw ice rumples (as discussed in Sect. 3). between FP and KIR). The modeled fracture density value is associated with the If we choose a length-dependent width w = l/4 < 10 km frequency, length and width of the the zone of influence of for the zone of influence around an observed crevasse, we can fractures, but a precise quantitative validation can only be compute an observed fracture density snapshot, φobs on the made on the basis of widespread ice-penetrating radar data. 5 km grid following the definition in Eq. (2). Figure 1 shows In our basic formulation we have to choose a somewhat ar- the resulting bands of high crevasse density aligned in flow bitrary value for parameter γ, so that the resultant fracture direction and in shear regions, which can be compared to density produces a physically meaningful steady state pat- the obtained smooth model fracture density φ (e.g. Fig. 4). tern, reflecting relevant features of observed surface struc- The close-up view in Fig. 1 reveals the spatially varying dis- tures (Fig. 4 b–d). Maximum calculated fracture density val- tribution of φobs. A time average over several snapshots of ues peak around φmax = 0.8, particularly in the outer shear sufficient delay would give a much smoother distribution. 6 Albrecht and Levermann: Fracture field for large-scale ice dynamics

N E

a Weddell Sea S W b SIS F

BIR REIS R SFIS

HIR KIR MIS EIS FP FIS IIS CIS RIS

c d

Fig. 4. Steady state distribution of the fracture density in a realistic setup of the Filchner-Ronne Ice Shelf for varying fracture initiation thresholds σcr = 0 kPa (a), 70 kPa (b, d) and 90 kPa (c) in von-Mises fracture criterion with fracture growth parameter, γ = 0.3, and without contribution through the inlets, φ0 = 0. Results shown in (d) are obtained by choosing a fracture growth rate with a constant 15 ψ = γ ε˙1 . For a fixed ice thickness, a flow enhancement of ESSA = 0.4 and with parameterized side friction by ηB = 10 Pa s, the fracture density is transported with a steady SSA-velocity field. Overlaid, observed visible surface fractures, kindly provided by Hulbe and others (2010, Fig. 1).

The width w scales the observed fracture density as long Regarding the effects of fracture initiation, Θ(σt − σcr), as the overlap of individual zones of influence is small. In and fracture growth rate, ψ = γ ε˙1 , the resulting steady-state any case comparison of the model’s fracture field can only fracture-density pattern is apparently more sensitive to the be complete if the area of failed ice can be determined from fracture initiation threshold than to the growth rate. If we observations, since the here proposed geometrical derivation replace the spreading rate by its mean value found in the −10 −1 for surface crevasses is merely a rough approximation of the initiation regions, i.e., about ε˙1 = 3 × 10 s , we get a quantity that we proposed here as the fracture field φ. Large similar pattern (compare Figs. 4b,d). rifts observed close to the calving front are apparently not Rist and others (1999, Plate 1) provide a map of the FRIS captured by the model φ. Since no healing is applied so far, crevasse pattern using visible and radar imagery. Most of the the steady-state fracture density can also accumulate in very fracture features coincide with the map inferred by Hulbe slow moving sections as upstream of Berkner Ice Rise (BIR) and others (2010), except for fracture fields in the central supplied by the contributions of the Rutford (RIS) and Foun- Ronne Ice Shelf downstream of Korff (KIR) and Henry ice dation Ice Streams (FIS). rises (HIR), probably ice with densely distributed short and inactive fractures covered with snow and not seen by the vi- Albrecht and Levermann: Fracture field for large-scale ice dynamics 7

a b

Observed speed ice

PISM computed speed

Fig. 5. (a) Scatterplot shows a comparison of calculated versus observed surface speed values within the ice shelf domain for an optimal 15 parameter configuration of ESSA = 0.4 and ηB = 10 Pa s. (b) Distribution of calculated von-Mises effective stress σt in FRIS. Values larger than critical effective stress, σcr = 70 kPa, colored in purple, are found in confined shear regions at the inlets and some side margins −10 −1 and where flow units merge. A maximum spreading rate of aboutε ˙1 > 2.5 × 10 s is obtained in most parts of the indicated regions (thick brown contour lines). Thin brown lines delineate contours of lower spreading rate, i.e 1.5, 2.0 × 10−10 s−1.

sual inspection of MODIS images nor in the obtained fracture N density field. EBD E In a simulation of the northern Larsen Ice Shelf (A+B) we W use the same flow parameters as in the FRIS, which yields a A S velocity field, that differs from observed velocities with a cross RI correlation coefficient of r2 = 0.78. A varying combination Weddel Sea of flow enhancement and friction parameter does not much SNI improve this result. A fracture-density dependent rheology may correct for this underestimation of velocities as in many DG other smaller buttressing ice shelves. Nevertheless, the evolution of the fracture density with parameters σcr = 70kPa and γ = 0.5 produces a pattern (Fig. 6), which compares well with HG the advanced satellite image interpretations by Glasser and GG Scambos (2008, Fig.4). Elongated steady state bands initiated EG FP at the inlet corners of the main tributaries following the ice B CF flow down to the ice shelf front agree with crevasse patterns JG in the images (at inlet corners even rifts are formed). Similar CD JP longitudinal structures in observations separate transversally CG the stagnant regions around the Seal Nunataks (SNI), Cape Disappointment (CD), Jason Peninsula (JP) and Foyn Point Grahamland (FP) from the faster flowing interior units. LG 2.3. Contributing tributaries An important contribution to the observed elongated surface Fig. 6. Steady state distribution of the fracture density field for structures originates from the transition zone in the vicinity of γ = 0.5, φ0 = 0 and σcr = 70 kPa for a fixed geometry of the the grounding line (tensile modes) and at inlet corners (shear Larsen A+B Ice Shelf with steady SSA velocity field. modes) of the tributary glaciers. In the ice shelf region that is supplied by Evans Ice Stream (EIS), trains of deep crevasses are found all the way up to the grounding line (Hulbe and others, 2010). Such structures are created in localized high constant values here for the upstream contribution, since it is shear rate regions through interaction with bedrock features beyond the scope of this study to investigate fracture initia- (Glasser and Scambos, 2008). Since we discuss only the dy- tion and growth of grounded ice. Fig 7a shows how the bound- namics of floating ice shelves without the complex transition ary fracture density of φ0 = 0.4 affects the overall steady zone processes so far, pre-existing fractures along tributary state distribution. Basically the pronounced band structure inlet glaciers can be treated here only as a boundary condi- remains unchanged (comp. to Fig. 4b) while the fracture den- tion, φ0 , for the fracture density field. We choose arbitrary sity value is increased in a smooth way in areas between those 8 Albrecht and Levermann: Fracture field for large-scale ice dynamics bands. Note that in the wake of the ice rises HIR and KIR in in isotropic polychrystalline materials has been proposed in the interior of Ronne Ice Shelf the fracture density is very low. terms of linear-elastic fracture mechanics in combination with Low strain rates yield weak fracture initiation and growth in dislocation theory (e.g., Griffith, 1921; Weertman, 1996; Van this region. At the same time this area is shielded off from der Veen, 1998b; Kenneally and Hughes, 2002, 2004, 2006). the inflow of inlet fractures by the ice rises. This approach is not further considered here but could be implemented in terms of a fracture initiation function replacing 2.4. Healing and persistence of crevasses Θ(∆σ). Healing processes may close existing crevasses or generally transform weak and fractured ice to harder and compact ice. Studies concerning the dynamics of ice healing are rare. To 3. CONCLUSIONS AND PROSPECTS our knowledge, none provides a quantitative or observational We introduce a macroscopic fracture density field in ice shelves characterization of such processes. Pralong and Funk (2005) within the framework of an ordinary continuum flow model in assumed to first approximation that healing follows the dy- Shallow Approximation. In doing so, we take a minimalistic namics of crack growth,i.e., it can be expressed as a sink term approach in the sense that we aim to investigate which of in the fracture density evolution equation. We write the observed ice fractures can be reproduced by first-order fracture initiation, fracture growth and healing terms. The f = Θ(−ψ ) · ψ with ψ = γ (ε ˙ − ε˙ ) . (11) h h h h h 1 h fracture density in the applied simple formulation in terms of This expression is similar to the source in Eq. (4), but con- strain-rates and effective stresses yields a map of pronounced tributes for negative ψh only, i.e. below a certain threshold. zones of high values separating flow units of lower density Hence, ψh represents a healing rate, which is more effective, values, which qualitatively agrees with the pattern of surface the smaller the spreading rate is compared to a certain up- fractures in satellite visible imagery. Extensive deep-reaching per spreading rate threshold,ε ˙h . This formulation is a simple radar observations in fracturing ice shelf areas could pro- representation of the closure effect due to the ice overburden vide means for an actual quantitative relationship in terms pressure. For deep rifts or bottom crevasses also refreezing of depth, length and distance between fractures per area and and basal healing by marine ice along suture bands and in for a more general power law formulation of source or heal- the wake of promontories and island was suggested to play ing terms with additional parameters distinguishing between an important role as healing processes (Holland and others, fracture-related processes in different flow regimes in a more 2009; Glasser and others, 2009; Hulbe and others, 2010). Since process-oriented physical way. The aim here is to introduce a quantitative measurements which would allow a better pa- practical way to incorporate fracture into large-scale ice shelf rameterization of large-scale healing is not yet available we modeling that can be eventually expanded to ice streams and neglect here a dependence of healing on the local fracture den- ice sheets. Therefore only first-order terms are used. In future sity. Figure 7b shows the steady state fracture density with studies the fracture density is proposed to be implemented −10 −1 healing parameters ofε ˙h = 2.0 × 10 s and γh = 0.1. for decoupling regions of continuous ice flow through regions Contributions of the inlets to the fracture density vanish on in which the continuum assumption fails and to be used in the way downstream while along the shear zones the healing calving parameterizations at the ice margin. effect is negligible. As healing acts in slowly moving sections The implementation of active fracture-induced softening as over a prolonged time interval, e.g., upstream of Berkner Ice a dynamic feedback is proposed as an improvement towards Rise (BIR) supplied by the contributions of the Rutford (RIS) more realistic flow behavior. The inferred location and level and Foundation Ice Streams (FIS), fracture density vanishes of fracturing provides the relevant information to capture completely there. The numerical first-order explicit upwind its first-order macroscopic imprint on the main flow regime transport of the fracture density implemented into PISM- by modification of the ice rheology at these positions in or- PIK causes an artificial diffusion, which causes the fading of der to achieve more accurate flow simulations. Skrzypek and band structures downstream of fracture formation areas and Ganczarski (1999), Pralong and Funk (2005) and Pralong and represents numerical unintended healing. others (2006) suggest to use the constitutive equation for- Observed longitudinal surface features can form far up- malism of the continuous non-fractured material to apply a stream, where fast flowing glaciers begin and persist over long fracture-density dependent softness rate factor in weak frac- distances downstream (sometimes all the way down to the ture zones with regard to the equivalence principle. calving front). Considering the critical values for healing and Additionally, decoupling schemes as internal boundary con- fracture initiation, we can identify regions of intermediate ditions are needed to capture the partly mechanical separa- stresses, whereε ˙1 > ε˙h and σt < σcr is valid and fracture tion of the flow dynamics of adjacent ice shelf units. Time- density stays more or less constant. In reality, sufficient large dependent property changes in the evolution of zones of weak crevasses are observed to grow for strain rates below the ini- ice could be considered, such as meltwater-induced hydro tiation thresholds (Rist and others, 1999), which should be fracturing at the surface (Glasser and Scambos, 2008). Effects expressed by σcr = σcr(φ). Also temperature plays an impor- of reduced marine ice formation (Holland and others, 2009) tant role here, since brittle failure processes favor colder and and sea-water enhanced bottom cracking in suture zones are stiffer ice, particularly relevant for cold ice draining through suggested to play an additional important role in the precon- the inlets from the upstream high-altitude catchments. Weiss ditioning of rapid collapse events in the style of the Larsen (2004) introduced a concept of subcritical crack growth for B Ice Shelf in 2002. Our approach could be expanded to the stress values below the fracture formation threshold range simulation of fast flowing ice streams to capture the mani- given in Vaughan (1993). Depending on the changing stress fold observed mechanical decoupling from the side margins history while moving with the ice these flaws may grow at and hence a possibly stronger response of the plug-like flow comparably low rates and precondition accelerated unstable to perturbations. Hence, flow simulations capturing fracture crack growth further downstream in later stages of failure. An dynamics give a more reliable picture of the vulnerability of alternative approach to describe the formation of such cracks particularly the smaller ice shelves facing a warming climate. Albrecht and Levermann: Fracture field for large-scale ice dynamics 9

a b

Fig. 7. Steady state fracture density for Filchner-Ronne simulation with (a) fracture density boundary condition for the inlets φ0 = 0.4 −10 −1 or (b) healing rate γh = 0.1 and ψh = (ε ˙1 − 2 × 10 s ), but with φ0 = 0. Parameter for fracture initiation are chosen as σcr = 70 kPa and γ = 0.3 (compare to Fig. 4b).

The second possible application of the fracture density field the fracture density field by introduction of a proportionality is large-scale calving. The calving front position with respect constant kφ, which depends on the fracture density field and to the confining coast geometry affects the overall ice dy- thereby incorporates the material properties of the ice to first namics (Hughes, 1983; Dupont and Alley, 2005, 2006). Most order. calving processes are initiated by tensile crevasse formation c ≡ k ε˙ ε˙ andε ˙ > 0 , (12) when surface and bottom fractures penetrate a significant φ 1 2 1,2 portion of the ice thickness (Reeh, 1968; Hughes, 1988). If with kφ evaluated at the calving front in units [m s]. Such the ice shelf spreads beyond its anchoring confinements the a calving law can not predict individual intermittent calving flow regime changes structurally (Doake, 2001) as transversal events (Bassis, 2011). But it links average calving probability spreading occurs with a rate that may exceed the longitudinal at the terminus to ice dynamics in the entire ice region and motivates the idea of the fracture density field discussed in spreading (per definitionε ˙2 ≤ ε˙1 is still valid, see also Fig. 3) and tensile cracks may open and widen in two directions of this paper. Fracture penetration depth plays an important the principal axes in the vicinity of the central front (Nye, role in the precondition of calving and will be considered in a 1952; Hambrey and Müller,1978). Fractures are observed to three-dimensional extension of the fracture-density definition. grow and become deep crevasses parallel and perpendicular The implementation of such continuous calving rates in finite to the calving front, which even can penetrate the whole ice difference schemes requires a sub-grid representation of the thickness influenced by weak zones (Hulbe and others, 2010). calving front as introduced by Albrecht and others (2011). Along intersecting rifts tabular icebergs are finally released Thus the fracture density field introduced here is meant to (Rist and others, 2002; Larour and others, 2004; Joughin and serve as a practical tool to incorporate first-order effects of MacAyeal, 2005; Hammann and Sandhäger,2005; Kenneally ice fracturing into large-scale ice modeling. and Hughes, 2006). In this purely expansive region we de- fineε ˙ > 0 as the condition to identify ice shelf region 1,2 4. ACKNOWLEDGEMENTS promoting unstable rift growth and supporting calving. In We thank Ed Bueler and Constantine Khroulev (University of a first-order approach by Levermann and others (2011), as a Alaska, USA) for providing the sophisticated model base code two-dimensional model-applicable extension of the observa- and numerous advice, as well as Maria A. Martin and Ricarda tional relationship inferred by Alley and others (2008), the Winkelmann for a critical discussion of the study. We are very average calving rate at the front, c, is parameterized by the grateful to Douglas MacAyeal for helpful comments and sug- product of the two strain-rate eigenvalues, which represents gestions. The data of observed surface features on FRIS was the calving kinematics, multiplied with a constant k, which kindly provided by Christina L. Hulbe and Christine LeDoux. comprises all material properties. In this form the calving Torsten Albrecht is funded by the German National Academic law has already been used in simulations of the Antarctic Ice Foundation (Studienstiftung des deutschen Volkes). Sheet forced with present day boundary conditions (Martin and others, 2011) as well as in higher-resolution simulations of the Larsen and Ross Ice Shelves. This “eigencalving” param- REFERENCES eterization, however, only represents the kinematic first-order contribution to calving. It needs to be complemented by the Albrecht, T., M. A. Martin, R. Winkelmann, M. Haseloff and material properties of the ice. This can be achieve through A. Levermann, 2011. Parameterization for subgrid-scale motion of ice-shelf calving fronts, The Cryosphere, 5, 35–44. 10 Albrecht and Levermann: Fracture field for large-scale ice dynamics

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Shepherd, A. and D. Wingham, 2007. Recent Sea-Level Contribu- ditions, especially at the calving front, yields vertically inte- tions of the Antarctic and Greenland Ice Sheets, Science, 315, grated velocities. Calculations are running in parallel using 1529–. the Portable, Extensible Toolkit for Scientific computation Shepherd, A., D. Wingham, D. Wallis, K. Giles, S. Laxon and A. V. (PETSc by Balay and others (2008)). Transport of fracture Sunda, 2010. Recent loss of floating ice and the consequent sea density on the fixed rectangular grid is approximated with a level contribution, Geophys. Res. Lett, 37, L13503. first-order upwind scheme based on a finite volume method Skrzypek, J. and A. Ganczarski, 1999. Modelling of Material Dam- (Morton and Mayers, 2005). Time stepping is explicit and age and Failure of Structures, Springer. adaptive. The code applies a sub-grid representation of the Smith, R. A., 1976. The application of fracture mechanics to the calving front as introduced by Albrecht and others (2011). problem of crevasse penetration, Journal of Glaciology, 17, 223–228. Thomas, R., E. Rignot, P. Kanagaratnam, W. Krabill and APPENDIX B. COMPUTATIONAL G. Casassa, 2004. Force-perturbation analysis of Pine Island SETUP Glacier, Antarctica, suggests cause for recent acceleration, An- nals of Glaciology, 39(1), 133138. In our experiments we used four different computational se- Van der Veen, C. J., 1998a. Fracture mechanics approach to pen- tups. For parameter studies, simple geometries, such as an etration of bottom crevasses on glaciers, Cold Regions Science ice shelf confined in a rectangular bay with unidirectional in- and Technology, 27(3), 213 – 223. flow, are used. For a narrow embayment (I.) of 150 km width Van der Veen, C. J., 1998b. Fracture mechanics approach to pen- and 250 km length and a wide bay (II.) of 300 km width and etration of surface crevasses on glaciers, Cold Regions Science 150 km length (Fig. 2) we define Dirichlet boundary condi- and Technology, 27(1), 31 – 47. tions for the vertically averaged velocity in SSA as a distribu- Van der Veen, C. J., 1999. Fundamentals of Glacier Dynamics, −5 −1 tion u(x0 ) with umax(x0 ) = 10 m s in the center decaying A.A. Balkema Publishers. to zero at the sides with the 4th power (for flow law exponent Van der Veen, C. J., 2007. Fracture propagation as means of rapidly n = 3 as in Dupont and Alley (2006, Eq. 6), shown as a grey transferring surface meltwater to the base of glaciers, Geophys- line in upper right panels of Figs. 3a,b). The ice thickness at ical Research Letters, 34, 1501–+. this boundary position is constant H(x ) = 600 m and for Vaughan, D. G., 1993. Relating the occurrence of crevasses to sur- 0 fracture density we assume φ(x ) = φ = 0.1. Mean annual face strain rates, Journal of Glaciology, 39, 255–266. 0 0 surface temperature is T = 247 K and at the bottom at pres- Vieli, A., A. J. Payne, Z. Du and et al., 2006. Numerical modelling and data assimilation of the Larsen B ice shelf, Antarctic sure melting temperature. Melting and accumulation are not Peninsula, Royal Society of London Philosophical Transactions considered here. Along the bedrock side friction is prescribed 13 Series A, 364, 1815–1839. by a viscosity of ηb = 5 × 10 Pa s, which is more realis- Vieli, A., A. J. Payne, Z. Du and A. Shepherd, 2007. Causes of pre- tic than forcing boundary velocities to zero. At the ice shelf collapse changes of the Larsen B ice shelf: Numerical modelling front ice calves off at ice thickness 175 m. The quadratic setup and assimilation of satellite observations, Earth and Planetary measures 201×201 grid cells of 2500 m length each. Vertical Science Letters, 259, 297–306. levels are unequally spaced and on average 6 m thick. Weertman, J., 1973. Can water-filled crevasse reach the bottom Simulations setups with realistic topography are used for surface of a glacier?, Int. Assoc. Sci. Hydrol. Publ., 95, 139– comparison with observational data. For the Filchner-Ronne 145. Ice Shelf (III. FRIS), data from the SeaRISE dataset defined Weertman, J., 1977. Penetration depth of closely spaced water-free on a 5 km grid (Le Brocq and others, 2010) was applied on crevasses, Journal of Glaciology, 18, 37–46. a 201×201 mesh. The lateral boundaries of the ice shelf are Weertman, J., 1980. Bottom crevasses, Journal of Glaciology, 25, inferred according to bed topography and floatation criterion 185–188. applied to the ice thickness. Only floating ice was modeled. Weertman, J, 1996. Dislocation based fracture mechanics, Hacken- At the inlets Dirichlet boundary conditions for the SSA- sack, NJ: World Scientific Publishin. velocity calculation were defined on basis of observed veloci- Weiss, J., 2004. Subcritical crack propagation as a mechanism of ties, determined using InSAR and speckle-tracking (Joughin, crevasse formation and iceberg calving, 50(168), 109115. 2002; Joughin and Padman, 2003). Boundary position at the Winkelmann, R., M. A. Martin, M. Haseloff, T. Albrecht, E. Bueler, C. Khroulev and A. Levermann, 2011. The Potsdam inlets have been adjusted to locations where velocity data Parallel Ice Sheet Model (PISM-PIK) Part 1: Model descrip- were available. Hence its position varies partly compared to tion, The Cryosphere, 5(3), 715–726. the grounding line in the overlaid data by Hulbe and others (2010). Vertical levels are equally spaced with ∆z = 35 m. Mean annual surface temperature was adopted accordingly (Comiso, 2000) and ranges from -24 ◦C to -19 ◦C, basal tem- APPENDIX A. PISM-PIK perature is at pressure melting point. Basal melting or accu- The fracture density parameterization was implemented in mulation is not applied in these experiments, the ice thickness the Potsdam Parallel Ice Sheet Model, PISM-PIK, which is is prescribed. based on the thermo-mechanically coupled open-source Par- As a second example we model the significantly smaller allel Ice Sheet Model stable 0.2 (Bueler and Brown, 2009; northern Larsen Ice Shelf (IV.) fringing the Antarctic Penin- PISM-authors, 2011). Most modifications compared to PISM, sula between 64.5 ◦S and 66 ◦S. The small ice shelves fringing as described in Winkelmann and others (2011), concern the the large ice sheets are of essential concern, since they reveal ice shelf dynamics and have been tested in a simulations of a strong buttressing effect on the tributaries and seem to be the Antarctic Ice Sheet forced with present day boundary much more sensitive to changing climatic boundary condi- conditions (Martin and others, 2011). Most modifications are tions. We prepared a computational setup of Larsen A+B on now merged into PISM stable 0.4. Solving the discretized a regular 201×201 mesh with all data regridded to 1 km res- stress balance in Shallow Shelf Approximation (SSA, Mor- olution. Surface elevation and velocity data have been raised land, 1987; MacAyeal, 1989) with appropriate boundary con- in the Modified Antarctic Mapping Mission (MAMM) by the Albrecht and Levermann: Fracture field for large-scale ice dynamics 13

Byrd Center group (Jezek and others, 2003) based on observations of the years 1997–2000 (before the catastrophic disintegration event in Larsen B) provided on a 2 km grid in polar stereographic projection grid by Dave Covey from the Univer- sity of Alaska, Fairbanks (UAF). Gaps in the data were filled by averaging over neighboring grid cells. Ice shelf thickness is calculated from surface elevation, assuming a firn-corrected hydrostatic relation (Lythe and others, 2001, Eq. 2). Grounding line position is assumed to be located where the surface gradient exceeds a critical value (also for ice rises). Reconstructed velocities and ice thicknesses were set as a Dirichlet boundary condition at the grounding line in inlet regions simulating the ice stream inflow through the moun- tains. The annual mean surface temperature data was taken from present-day Antarctica SeaRISE Data Set available on a 5 km grid (Comiso, 2000; Le Brocq and others, 2010), regridded to 1 km resolution, ranging from -13 ◦C to -11 ◦C. Accu- mulation and Basal Melting is neglected since ice thickness is prescribed in our experiments to reduce tuning parameters.