SUPPLEMENTARY INFORMATION DOI: 10.1038/NGEO1887 Diverse calving patterns linked to glacier geometry J. N. Bassis and S. Jacobs 1. Supplementary Figures (a) (b) (c) Supplementary Figure S1 Schematic of different calving patterns observed in nature. a, Outlet glaciers with grounded calving fronts detach icebergs that are usually less than one ice thickness in length and often capsize after detaching. b, Some outlet glaciers form permanent or seasonal floating tongues. These glaciers often calve infrequent tabular bergs that do not capsize. When an ice tongue or ice shelf becomes sufficiently fractured, many small bergs detach and capsize, triggering an explosive disintegration of the ice shelf (c). The process in c mimics disintegration events observed in sections of the Larsen ice shelf. NATURE GEOSCIENCE www.nature.com/naturegeoscience 1 1
Supplementary Figure S2 Simulated detachment of an iceberg from a glacier with Helheim-like geometry (ca 2005-05). The scale bar represents 1 km, and red lines denote fractures. The iceberg that initially detaches and capsizes is narrow, with a characteristic length smaller than the ice thickness. In the process of capsizing, the berg collides with the glacier calving front, triggering a cascade of calving events that lead to retreat of the calving front by nearly 1 km. See Supplementary Online Material Movie S1 for an animation. Supplementary Figure S3 Effect of ice thickness on a simulated glacier with Helheim-like geometry (ca 2005-08). The scale bar represents 1 km, and red lines denote fractures. a, The original glacier exhibits iceberg detachment. b, A geometrically similar glacier with a uniform increase in ice thickness of 25 m experiences more intense crevassing but no iceberg detachment. 2
2. Supplementary Methods Particle positions and velocities in our discrete element model (DEM) are determined using Newton s law for linear and angular momentum: (S1) dv i dt = 1! F i, m i (S2)! i = M! i, I i where v i and m i are the velocity and mass of the ith particle, is the rate of rotation of the ith particle, I i is the moment of inertia of the ith particle and the summation is over all forces F i and torques M i acting on the ith particle. Displacements and velocities in our two-dimensional model are confined to the x-z plane, which we assume lies along a flowline. This treatment ignores lateral drag from the margins and is therefore most appropriate for glaciers with intense crevassing near the margins that limit the magnitude of lateral shear stresses. There are two types of forces that act on each particle: Body forces describe gravity and buoyancy-driven ice-water mechanical interactions and contact forces describe particle-particle interactions. We discuss each of these separately below.! i Body force: A particle submerged in water experiences an upward force equal to the acceleration due to gravity multiplied by the volume of water displaced by the submerged portion of the particle such that: "! (S3) b = 1! f w % i # $ & ' g,! ice 3
where f i is the fraction of the sphere that is submerged, and g, the acceleration due to gravity, is a vector directed downward in the negative z-direction. Granular forces: When the ith and jth particles collide, there is an elastic repulsion that acts perpendicular to the contact plane between particles given by: (S4) F e n = K ijn! ij n, where K n ij = 2k n i A i k n j A j A i k n i + A j k is the effective spring constant between the two particles, k n n i j is Young s modulus of the ith particle, A k =!r k 2, is the contact area of the kth particle and! ij n is the overlap between particles i and j. Elastic collisions are dissipated by a linear damping term defined as: (S5) F d n =!! A ij!!ij n, where γ is a damping coefficient,! ij n is the differential normal velocity between particles and A ij is the average contact area of the two particles. This damping force acts perpendicular to the contact plane. Forces tangent to the contact plane resist motion. These frictional forces are calculated using a common parameterization for friction of the form 24 : (S6) F s f =! min µf n e," #! t { ij }, where! t ij is the component of the differential velocity between the ith and jth particles tangent to the contact plane, µ is the coefficient of friction and β is the friction 4
regularization parameter. When particles have different coefficients of friction, the smallest coefficient of friction is used. Bond forces: When bonds exist between particles, the bond force consists of tensile components normal to the bond plane and shear components parallel to the bond plane with an increment in tensile and shear forces for a given deformation given by 25 : (S7a)!F b n = "k n A!U n, (S7b)!F b s = "k s A!U s. Here k n is Young s modulus for ice, and k s = 1 2 1+! k n is the shear modulus for ice, which depends on Poisson s ratio, ν=0.3, and A is the contact area of the glacier ice particles (bonded particles all have the same size and contact area A). The terms ΔU n and ΔU s represent the changes in bond length normal and tangential to the bond plane (in a given time step), respectively. Bonds fail in tension or shear when: (S8) F b n A! " n, or (S9) F b s A! " s. Particle pairs also experience a torque proportional to the shear force multiplied by the distance between particles. Because we allow bonds to bend, there is an additional torque proportional to the differential rotation between particles: (S10) M =!K n I b "# 5
where I b is the bending moment of inertia of the bond and Δθ=θ i -θ j is the differential rotation between the two bonded particles. To calculate the bond forces and torques, we use a procedure similar to one used for bonded rocks 25. Numerical method: We solve equations (S1)-(S10) using a velocity Verlet algorithm with a time step of 0.001 second. Using smaller time step sizes did not make a noticeable difference in our results. We chose the radii of the ice boulders to maintain a constant number density of particles per unit thickness so that r = H, where r is the particle 2N radius, H is the mean ice thickness and N 35 is the number of particles in a column of ice. For most simulations of glaciers with Helheim-like thicknesses, this required a particle radius of approximately 12.5 m. We chose other parameters (Table S1) to be as realistic as possible, with the exception of Young s modulus for ice, which we decreased to reduce the velocity of seismic waves. This allowed us to take larger time steps and run our simulations for longer periods of time than would have otherwise been possible. In simulations without fracture, the change in Young s modulus had no effect on the stress within the glacier. In simulations with fracturing, however, we had to decrease the time step by several orders of magnitude to maintain numerical accuracy. An important limitation of our model is that real glacier ice may be much more brittle and fragile than our simulations suggest. Sensitivity to particle size: To determine if our results were sensitive to particle size, we performed sensitivity studies with particle radii varying between 25 m and 6 m using Helheim Glacier-like geometry. In our simulation with the largest particle size (Fig. S4a), through penetrating fractures produce a narrow iceberg that detaches, but is too 6
wide to capsize. The medium particle size simulation (Fig. S4b) and smallest particle size simulation (Fig. S4c) both produce narrower iceberg that detach and capsize. The divergence between simulated fracture paths for different particle sizes is a consequence of the sensitive dependence of fracture on initial conditions, a feature that may also be present in real glaciers. Our simulations, however, show that the overall pattern of iceberg calving is robust to changes in particle size, despite variations in the individual fracture paths. Supplementary Figure S4 Effect of particle size on a simulated glacier with Helheim-like geometry (ca 2005-05). The scale bar represents 1 km, and red lines denote fractures. a, Snapshot from a simulation with particle radii 25 m. b, Snapshot from a simulation with particle radii 12.5 m. c, Snapshot from a simulation with particle radii 6.125 m. This suite of simulations shows that the broad pattern of ice fracture is insensitive to particle size. 7
Time scale of simulations: Individual iceberg calving events occur over a timescale of a few hours or less. These events, which often occur more frequently in the summer than in the winter, accumulate to produce a seasonal cycle of glacier advance and retreat. We do not seek to simulate the entirety of this cycle since that would require more frequent ice geometry data (e.g., ice thickness on a weekly time scale from a viscous ice dynamics model or observations). Our model is instead designed to resolve individual calving events and determine if a given geometry is stable or not. 8
3. Supplementary Tables Parameter Description Value g Acceleration due to gravity 9.8 m/s 2 ρ ice Density of ice 920 kg/m 3 ρ w Density of sea water 1020 kg/m 3 τ n Tensile strength 0.1 MPa τ s Shear strength 1 MPa K n Young s Modulus 0.1 GPa ν Poisson s ratio 0.3 µ Coefficient of friction 0.1 β Friction regularization 10 8 Nm γ Collisional damping 0.05 MPa/s λ Bond width parameter 1/8 dt Time step size 0.001 s m Ice boulder mass m = 4 3!r 3 " ice I b Bending moment of inertia I b = 1 4! ("r)2 I Moment of Inertia I = 2 5 mr 2 Table S1. Parameter description and numerical values used in simulations. 9
4. Supplementary Discussion Observations show that many calving glaciers retreat when water depth increases or when the glacier thins to near flotation, but this is inconsistent with the decreased fracture penetration depths that theory and our simulations predict (Fig. 1a). To examine this more closely, we performed additional experiments with Helheim-like glacier geometry (ca 2005-08). In our first experiment, we uniformly increased the ice thickness by ~25 m (Supplementary Fig S3). This experiment revealed that an increased ice thickness yields slightly more intense fracturing, but no iceberg detachment. In contrast to fracture, iceberg separation is most efficient for those glaciers that are close to flotation. For further study, we performed another sequence of simulations with Helheim-like glacier geometry in which we segmented the lower 5 km of the glacier into narrow rectangular blocks with random aspect ratios (ratios of berg length to ice thickness) uniformly drawn from the range 0.1-0.5. This choice of aspect ratio distribution was ad hoc but created a glacier consisting of icebergs that were naturally prone to capsizing. To isolate the role of water depth in promoting iceberg separation in these experiments, we did not allow additional fracturing. Movie S3 shows the evolution of a Helheim-like glacier (ca 2005-05) with water depth uniformly increased by ~10 m so that the entire lower portion of the glacier is near flotation. In this simulation, a narrow berg separates and rotates outward while capsizing. As the simulation progresses, another narrow berg capsizes and collides with stable bergs, causing a net drift of bergs away from the calving front. Some of the more stable bergs remain upright and scrape along the bottom in shallower portions of the fjord before eventually becoming grounded in shallow water. 10
This creates a mixture of iceberg debris similar to the mélange that clogs many fjords 9,25. Movie S4 shows the same initial configuration but with water depth decreased by 50 m so that the entire glacier is grounded more than 50 m above buoyancy. In contrast to the events of the previous simulation, bergs in this experiment do not separate or capsize. This leads to a quasi-stable glacier even though the glacier was constructed to be pervasively fractured. This suggests that the calving behavior previously hypothesized to depend only on height-above-buoyancy and water depth is also caused by the export of highly fractured terminus ice. When a glacier advects into a deep trough or thins to near flotation it is basal and lateral resistance that keeps icebergs in place and limits calving fluxes. The sequence of capsizing icebergs we observed when the glacier became buoyant in Movie S3 is similar to the mechanism proposed for the catastrophic disintegration of the Larsen B ice shelf. 5. Supplementary Equations We can analytically estimate the maximum ice thickness possible for a stable calving front as a function of water depth used in Fig 1. To do this, we note that the depthaveraged strength of ice is given by 19 : (S11)! c = C 0 + 1 2 µ" icegh, where C 0 is the shear strength of ice (1 MPa), µ is the coefficient of friction (0.1), ρ ice is the density of ice (920 kg/m 3 ), g is the magnitude of the acceleration due to gravity (9.8 kg/m 2 ) and H is the ice thickness. The maximum ice thickness is then given by: (S12) H = (1! r)" c # ice g + ( 1! r)" # ice g + # w D 2, # ice 11
where r is the fraction of the ice thickness penetrated by surface and basal crevasses, and H and D are the ice thickness and water depth near the terminus, respectively. We estimate the fraction of the ice thickness penetrated by crevasses, r, using the Nye zerostress model 19 : (S13) r = S xx! ice gh +! ice # S xx! w "! ice! ice gh "1+! w D $ %! ice H where ρ w is the density of sea water (1020 kg/m 3 ), and the deviatoric stress, S xx, is a function of ice thickness and water depth estimated based on a long-wavelength approximation that neglects lateral shear and basal friction: & ' (, (S14) S xx = 1 2! gh ) 1"! w # D & ice +! i $ % H ' ( * 2,.. - The above set of non-linear algebraic equations is solved using the Brent Bisection method. Agreement between the analytic approximation and our numerical model confirms the accuracy of our first-order stress regime and shows that our model is not contaminated by the long-wavelength approximation. This excellent quantitative agreement between theory and numerical simulations is partly a consequence of the collisional damping we used, which was poorly constrained by observations. The qualitative trend between theory, simulations and observations is, however, robust to changes in the material parameters, confirming the qualitative envelope of our results. Moreover, we anticipate that the uncertainty in our estimate of the stability of glaciers is dominated by the large uncertainty in fracture properties. 12