The prescription of a simple and robust parameterization for calving is one of the most significant open problems in ice sheet modelling. One common approach to the modelling of crevasse propagation in calving in ice shelves is to view crevasse growth as an example of linear elastic fracture mechanics. Prior work has, however, focused on highly idealized crack geometries, with a single fracture incised into a parallel-sided slab of ice. In this paper, we study how fractures growing from opposite sides of such an ice slab interact with each other, focusing on different simple crack arrangements: we consider either perfectly aligned cracks or periodic arrays of laterally offset cracks. We visualize the dynamics of crack growth using simple tools from dynamical systems theory and find that aligned cracks tend to impede each other's growth due to the torques generated by normal stresses on the crack faces, while periodically offset cracks facilitate simultaneous growth of bottom and top cracks. For periodic cracks, the presence of multiple cracks on one side of the ice slab, however, also generates torques that slow crack growth, with widely spaced cracks favouring calving at lower extensional stresses than closely spaced cracks.

Iceberg calving is a key process in the dynamics of marine ice sheets, since it regulates the length of floating ice shelves and consequently the rate at which ice is discharged across the grounding line

Calving generally happens as the result of cracks growing to occupy the full thickness and width of an ice shelf. One common approach to modelling such cracks is to regard ice as an elastic medium on the short timescales associated with fracture propagation and to either employ a classical linear elastic fracture mechanics approach

Discrete element models

Here we attempt to bridge the gap between idealized classical fracture mechanics models and more complicated (and computationally much more expensive) discrete element and phase-field models by extending prior work on linear elastic fracture mechanics models to take account of multiple interacting cracks. We use the boundary element method described in

To keep the scope of our work tractable, we restrict ourselves to understanding simple interactions between basal and surface crevasses. In particular, we seek to identify how the spacing and alignment of crevasses on opposite sides of an ice shelf affect calving. Note that the study of interacting cracks has a long history, often involving complicated geometries in which the direction of crack propagation must be determined as part of the solution of the linear elastic fracture mechanics problem (e.g.

The paper is structured as follows: in Sect.

The basic model is described in

As in

We apply the same contact-type boundary conditions on crack faces as described in

At the lateral domain boundaries at

We have previously considered only a single surface or basal crack in

The elastostatic problem of

For a given position of the cracks along the domain, the domain geometry is fully specified by ice thickness

In that vein, we will treat Eq. (

To simplify the set of geometrical and forcing parameters, we non-dimensionalize the model using the same set of scales as in

Using the Green's function formulation in

Note that, for simplicity, we immediately omit the asterisk decorations, in the understanding that all variables and parameters used below are dimensionless. Any changes in forcing parameters are assumed to occur much more slowly than cracks propagate, so the dimensionless forcing and geometry parameters

Plotting orbits on a phase plane provides a simple graphical way of identifying the behaviour of the system for a given set of parameters and for all possible initial conditions. In that way, a phase plane is analogous to, for instance, Fig. 10 of

The ability to visualize evolution from arbitrary initial conditions using a phase plane also allows us to address how the dynamical system evolves under slow changes in forcing parameters (see also

Firstly, we generalize the geometry considered in

Geometry of the problem: the finite domain shown in panel

Figure

Phase plane diagram for two aligned cracks for

Secondly, the dynamical system is non-smooth: the maximum function on the right-hand sides of Eq. (

Thirdly, equilibria of the dynamical system are generally not isolated but occupy regions of finite size, rendered with black dots in Fig.

In addition, the phase plane here is bounded: crack lengths must be positive, and, for aligned cracks, their sum must be less than the ice thickness. When using dimensionless crack lengths scaled with

The usual notions of phase plane analysis, like identifying isolated fixed points and their stability, do not apply without modification due to the non-differentiability of the dynamical system and due to the fact that equilibria occupy extended regions of the phase plane. Equilibria inside these extended regions are stable in the sense of Lyapunov but not asymptotically stable

There are several equilibria that occupy a special role, namely those where two marginal nullclines intersect. We will refer to these equilibria as marginal fixed points below. There are five such equilibria in Fig.

One, marked with a red dot, is analogous to a stable node in standard phase plane analysis, and we will therefore refer to it as a “stable node” in a slight abuse of terminology. If we start the system with small surface and basal cracks of length

Even though the stable node is not an attractor in the strict sense (there are other equilibria arbitrarily close to the stable node), it does have a finite basin of attraction demarcated by the red and yellow orbits into the stable node. Note that the size of that basin of attraction is easy to overestimate visually due to the finite resolution used in computing the phase portrait. Close to the stable node is a marginal fixed point that is analogous to a saddle in standard phase plane analysis, marked with a yellow triangle in the inset. For this marginal fixed point, a single orbit ends at the saddle, while a second orbit connects saddle and node (both shown in yellow in the inset). Below the orbit leading up to the saddle, there are additional orbits starting with lower values of

The third marginal fixed point is marked with a cyan triangle in Fig.

The unstable node point marked by a cyan square is paired with a third saddle point marked as a red triangle that is also located almost on the

Suppose the system is started with only small seed cracks to initiate crevasse growth (where these seed cracks need to be large enough in order to start outside the region of steady states around the origin discussed above). Calving will then occur if there is an orbit connecting the near-origin initial conditions in the phase plane to the calving boundary at

As in

Gridded phase plane diagrams for two aligned cracks.

In Fig.

Calving under increases in

The saddle-type marginal fixed point marked by a cyan triangle moves downwards and to the right under increases in

In fact, if we suppose that a step from one panel in Fig.

In other words, in order to predict parameter combinations that lead to calving with aligned cracks, we can actually look at the dynamics of surface and basal cracks in isolation: our admittedly coarse sampling of parameter space strongly suggests that calving occurs from near the origin (a nearly unfractured initial domain) as it does from the node configuration (with a short surface and larger basal crack). One crack is always dominant, and the propagation of one crack does not significantly reinforce the propagation of the other.

A plausible physical explanation of this behaviour is provided by the torques generated on each crack face (see also

Motivated by our conjecture that torques generated by a crack on one side of the domain affect the propagation of a crack incised into the opposite side of the ice, we consider whether the dynamics of misaligned (or laterally offset) cracks differ qualitatively from those of aligned cracks: presumably, if a basal crack causes compression at the upper surface, that compression is strongest above the basal crack and decreases with horizontal distance.

To maintain the requisite symmetry for laterally offset cracks to act as mode I crack and propagate vertically, we assume that they are located at

A periodic domain with offset cracks at

Figure

Gridded phase plane diagram for two offset cracks with periodic boundary conditions and

A subtle but significant qualitative difference with Fig.

This has a knock-on effect on the style of calving under increases in extensional stress: in the aligned crack case of Fig.

The results in the previous section suggest that the presence of offset instead of aligned cracks may promote the simultaneous growth of both cracks: specifically, the motion of the node-type marginal fixed point under changes in parameter values involves significantly greater growth of the surface crack.

A second notable feature of Fig.

Here we analyze the interaction between neighbouring cracks, restricting ourselves to a periodic array of basal cracks for simplicity. The second equation in Eq. (

Figure

For each combination of parameters

As Fig.

As in the non-periodic case considered in

In the present case, the torque generated by one crack is balanced not only by the torque in the overlying neck of ice but also by torques due to all the other cracks in the periodic array, which reduces the net torque and therefore stresses generated in the neck of ice. The smaller the spacing

Clearly, stresses are larger around the crack tip for the non-periodic case in panel (a). Far-field displacements

The stress distribution for a basal crevasse with

As in the simpler case of a single crack studied by

A more complicated problem is the construction of a calving law that allows slow changes in the forcing parameters: by slow, we mean as compared with the natural timescale of crack evolution in the dynamical system (

In the confines of our model, the difference between calving by such slow changes in forcing parameters and instant calving as discussed above is that the cracks will have a significant initial length at the instant that a critical parameter combination permitting calving is reached. The analogous situation for a single crack was discussed in Sect. 5 of

Consider the case of offset cracks as shown in Fig.

The node-type marginal fixed point, if present, must be at

Any movement of the boundaries of the region of steady states at

Mathematically, we can capture the implied motion of the phase point

We have not attempted to complete the calculation suggested at the end of the last subsection, namely to compute in detail how the boundary of the region of steady states depends on forcing parameters. The reason is that Fig.

The absence of an optimal, finite crevasse spacing at which crack propagation occurs most easily (in the sense of calving occurring at the lowest possible extensional stress

There are several other limitations in addition to not accounting for the effect of buoyancy on elastic stresses. For a given elastic pre-stress, the linear elastic fracture mechanics problem solved here relies on the same weakly inertial propagation rate prescription due to

There are two other major complications that need to be addressed. The first is the computation of the viscous pre-stress itself. In the present work, we have insisted on a parallel-sided slab as the basic domain into which cracks are incised, with viscous pre-stress that corresponds to a completely unfractured ice slab. In practice, we expect these assumptions to break down over different timescales: firstly, once cracks have propagated to a steady-state configuration in which they span only part of the ice thickness, the elastic stress built up during fracture propagation will decay viscoelastically over a single Maxwell time, likely on the order of hours in an ice shelf

The second major issue is the prescription of surface hydrology through a surface water table elevation, as we do through the parameter

A simpler end member is, of course, the case of a dry ice shelf surface (

In this study, we modelled the simultaneous growth of both basal and surface crevasses, which has received less attention in the literature due to a lack of tabulated Green's functions. Here, a semi-smooth two-dimensional dynamical system is used to display crack growth rate,

For two aligned crevasses in the shelf, we observed that the top crack dominates when the water level is high and when there is a significant amount of water in it. When the shelf is intact, the basal crack grows rapidly and reaches a steady state where the hydrostatic pressure is equal to the cryostatic pressure. Meanwhile, the surface crack, despite containing a large amount of water, remains small and stable until a certain point and then becomes unstable and grows to span the entire thickness of the shelf. Our study has shown that no set of parameters can result in both cracks growing simultaneously from an undamaged shelf to calving while considering that only one crack is insufficient to cause calving. The model indicates that one crack is always dominant, and the scenario of two aligned cracks growing at the same time to a steady state is not predicted. This behaviour can be attributed to the difference in crack growth speed and stress intensity factor at the crack tips. The results support the conclusion that the calving criterion established by

Driven by our hypothesis that torques produced by a crack on one side of the domain can impact a crack on the opposite side of the ice, we investigated laterally offset cracks. To ensure that misaligned cracks maintain the necessary symmetry to function as a mode I crack and propagate vertically, we assume that they are located at regular intervals. We consider that periodic boundary conditions and the frequency of cracks on a shelf can improve stability and resistance to crack propagation under higher extensional stress. In this case, the domain length and crack spacing become effective, and, as the domain grows and the crack separation increases, the stress intensity factors also increase. The model proposed in this study, similar to

The code for calculations is available from the corresponding author upon request.

No data sets were used in this article.

MZ executed the research. MZ and CS designed the project and wrote the paper. MZ and AP developed the numerical method and the code. All authors edited the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Maryam Zarrinderakht has been supported by the ArcTrain NSERC CREATE graduate training program. Christian Schoof and Anthony Peirce have received support from NSERC Discovery Grants (grant nos. RGPIN-2018-04665 and RGPIN-2015-06039).

This paper was edited by Caroline Clason and reviewed by Jeremy Bassis and one anonymous referee.