Firstly, we generalize the geometry considered in by introducing aligned surface and basal cracks at the mid-way point x=W/2 as in Figure . Note that the intention here is to mimic the infinitely wide domain of and , and we use a large domain width W= 10, applying the same boundary conditions of vanishing elastic traction as .

Figure shows an example of a (db,dt) phase plane for this geometry. There are several non-standard qualitative features in this phase plane that turn out to be generic for the dynamical system (), in each case related to the maximum over KI-κ and zero taken on the right-hand side of Eq. (). Firstly, by construction, crack lengths can never shrink, so all orbits are either horizontal, vertical, or angled upwards to the right: if a stress intensity factor is less than the fracture toughness, the corresponding crack tip simply does not move.

Secondly, the dynamical system is non-smooth: the maximum function on the right-hand sides of Eq. () not only ensures that cracks cannot shrink, it generally renders those right-hand sides non-differentiable where KIb*=κ or KIb*=κ, even if KIb and KIt are smooth functions of (db,dt) (where the latter seems likely unless a new contact area is formed or a section of open crack fully disappears at that point; see Fig. 4a of ). For simplicity, we will refer to the sets of points for which KIb*=κ or KIb*=κ as marginal db and dt nullclines, respectively: these are the curves along which one of the elastostatic stress intensity factors is equal to the fracture toughness. This awkward terminology is necessary here because a nullcline is simply a set of points for which d˙b=0 or d˙t=0; for smooth dynamical systems, these sets are generally one-dimensional curves, which is not the case here, as we describe below. In a phase plane like Fig. , parts of the marginal nullclines are then the boundary between regions in which the orbits are purely horizontal or vertical (so KIt*<κ or KIb*<κ, respectively), and “curved” orbits, along with both crack lengths, change simultaneously. The change from curved to straight vertical or horizontal orbits is a graphical manifestation of the non-differentiability of the dynamical system.

Thirdly, equilibria of the dynamical system are generally not isolated but occupy regions of finite size, rendered with black dots in Fig. . These regions are again bounded by parts of the marginal nullclines defined above (where stress intensity factors are equal to the fracture toughness). Here, the marginal nullclines separate regions in which orbits are either vertical or horizontal from regions of steady states in which all orbits are fixed points. Inside the regions of steady states, both stress intensity factors are less than the fracture toughness.

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 H, the crack tips meet when 8db+dt=1, which we take to correspond to calving as marked in Fig. .

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 : if perturbed, the state variable (db,dt) stays nearby because it does not evolve. For equilibria on the boundary of a region of steady states (that is, equilibria on one of the marginal nullclines), we can distinguish between unstable and stable. The boundary is unstable if there are orbits that point away from it, which is the case for boundaries at the top or to the right of a region of steady states, and stable (again in the sense of Lyapunov) otherwise.

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. , although only one is clearly visible in the enlargement in the inset.

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 dt(0) and db(0), they will evolve towards this equilibrium solution provided the ratio dt(0)/db(0) is large enough and that dt(0) and db(0) are not too small. The last caveat arises because, for the small values of κ relevant to typical ice shelves, there is a small region around the origin in the phase plane (not visible in Fig.  due to its small size) in which neither crack will grow. When we state that orbits started near the origin will evolve towards the node, we have to add that they need to start outside that small region. The existence of short steady-state crack lengths has been discussed previously and is associated with low-stress intensity factors, scaling as d1/2τ for the surface crack and d1/2(τ-1+r-1) for the basal crack (see Appendix C1 of ): for small enough d, these are guaranteed to be less than fracture toughness κ. An analogous region of very short steady-state cracks can be seen, for instance, in Figs. 4 and 7 in .

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 dt(0)/db(0) that terminate at the boundary of a region of steady states as shown in the inset.

The third marginal fixed point is marked with a cyan triangle in Fig.  and again plays the role of a saddle in a standard phase plane. Here, one orbit (marked in cyan) emerges from the saddle point towards the calving boundary, while a second orbit (also marked in cyan) connects a fourth (cyan square) marginal fixed point that is almost on the dt axis to the cyan triangle saddle point; this orbit is analogous to a separatrix in a standard phase plane, and it divides initial conditions that lead to immediate calving from initial conditions that lead to stable, steady cracks of finite length that leave the ice slab intact.

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 dt axis (“almost” because there is in fact an imperceptible region near the dt axis in which db does not grow, as discussed above). For the dynamics of a single top crack, the red triangle and cyan square points were previously identified elsewhere (see e.g. Figs. 2a and 6a, b of ) as stable and unstable equilibria. That is, if we set db=0, then the evolution of short cracks towards the smaller (red triangle) of the two is physically explained by the effect of the imposed tensile stress τ in opening the short crack eventually being overcome by cryostatic pressure as the crack lengthens. In fact, in all of the phase planes shown in the paper, the dynamics along each of the coordinate axes reduce to the dynamics of a single crack, as previously discussed in and . As Fig.  shows, the simple rationale regarding the stability of single cracks developed previously in these papers, however, falters when coupling the surface crack with a basal crack: for instance, both the red triangle and cyan square equilibria are actually unstable to the growth of a basal crack.

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 db+dt=1. In Fig. , that is not the case: instead, cracks will evolve to the configuration represented by the red dot or one very close to it.

As in , we ask how the marginal fixed point at the red dot evolves under changes in forcing parameters and how such changes can themselves lead to calving. The same caveat applies here as in our earlier work (see Sect. 4.5 of ): in asking how the dynamical system changes under changes in parameters and how an existing crack configuration is affected, we are ignoring the fact that the viscous pre-stress will generally not remain of the same form as assumed in our model even over relatively short timescales (comparable with the Maxwell time of ice), and changes in domain geometry will occur as the result of viscous flow over long timescales so that results based on the rectangular geometry assumed here will eventually become misleading. We address these issues in the companion paper .

In Fig. , we plot phase planes analogous to Fig.  for a range of values of η and τ as indicated for each column and row of the grid. Figure  is reproduced in panel (c2). If we track the position of the (red) stable node-type marginal fixed point in Fig. , we find that it moves upwards and to the right (that is, in the direction of flow of the dynamical system) under increases in extensional stress τ, while it remains unchanged under an increase in η (that is, an increase in the depth of the water table). In other words, if τ is increased, (db,dt) will track the position of the node but conversely remain stranded in the region of steady states indicated by black dots under a subsequent reduction in τ. This is analogous to becoming stranded in the grey region of steady states in Fig. 6 of . The insensitivity to η by contrast is easy to understand in terms of the shallow depth of the top crack dt at the node: the tip of the surface crack is above the water level in columns 2–4 of Fig. .

Calving under increases in τ eventually occurs for each of columns 2–4 through the node meeting the diagonal calving boundary in row a, with the surface crack length dt remaining very shallow: calving occurs almost entirely by the propagation of the basal crack. The critical value of τ in each case lies somewhere between τ=0.03 and τ=0.04, which is consistent with the critical value of τ=0.039 for calving by basal crevasse propagation determined numerically in .

The saddle-type marginal fixed point marked by a cyan triangle moves downwards and to the right under increases in τ, and it moves downwards under decreases in η. An alternative calving mechanism, starting with a stable steady-state configuration like the node in panel (c2), is to raise the surface water level (that is, to decrease η) until the two marginal fixed points annihilate each other: this is analogous to the saddle-node bifurcations of Fig. 6a and b of (for instance, by going from panel (c2) to panel (c1) in Fig.  here). When that annihilation occurs due to a reduction in η, the length of the basal crack remains almost unchanged during the subsequent evolution of (db,dt): the orbit emerging from the saddle towards the calving boundary (shown in cyan in Fig. ) is nearly vertical; hence the motion in the phase plane from the equivalent of a saddle-node bifurcation is also nearly vertical. Calving driven by changes in water level occurs almost entirely by the propagation of the surface crack.

In fact, if we suppose that a step from one panel in Fig.  to a neighbouring panel corresponds to calving (for instance, starting from the node in (c2) and changing parameter values to those in (c1) or starting from the node in panel (b2) and changing parameter values to those in (b1)), then we can identify which crack will propagate to cause calving purely by looking at orbits that emerge near the origin in the panel in which calving occurs: in panel (c1), orbits emerging near the origin evolve predominantly upward. More significantly still, if we start an orbit near the origin with db=0, that orbit will evolve to the calving boundary at dt=1 while maintaining zero basal crack length. Similarly, orbits emerging near the origin in panel (b1) evolve predominantly to the right, and an orbit started near the origin with dt=0 will evolve to calving at db=1 with zero surface crack length.

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 , Appendix C2): the torque generated by extensional stress on and fluid pressure in the bottom crack induces a rotation on both sides of the crack (see also Fig. 10 of ) that tends to compress the ice surface and vice versa for the torque generated on a surface crack.

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 x=W/4 and x=3W/4 and switch to laterally periodic boundary conditions as in Eq. (): as a result, the model describes a periodic array of surface and bottom cracks in which the surface and bottom cracks are spaced a half-wavelength W/2 apart (Fig. ). Note that the assumption of vertical crack propagation is then consistent with a maximum hoop stress criterion (, Sect. 4.4.1), but we do not address the question of crack path stability , namely that a perturbed crack could evolve progressively away from a vertical orientation.

Figure shows a grid of phase planes analogous to Fig. . One obvious difference is that the “calving boundary” now consists of the lines db=1 and dt=1: offset cracks will no longer meet part-way through the ice. We find similar behaviour in Fig.  as in Fig.  for cases where crack evolution away from the origin does not lead to immediate calving, with a node-type marginal fixed point evident in panels (d2)–(e2), (c3)–(e3), and (b4)–(e4). The corresponding saddle-type fixed point does not always exist, being absent in panels (b4) and (c4) (which is also the case in Fig. , panels (b4) and (c4)).

A subtle but significant qualitative difference with Fig.  is the size of the surface crack at the node point: while basal crack length db is significantly larger than surface crack length dt, the latter is much bigger than in Fig. . As a consequence, the tip of the surface crack corresponding to the node point equilibrium is below the water level, and the stress field is actually affected by surface water pressure in several of the examples in Fig. . As a result, the node point moves upwards as η is decreased (the crack lengthens as water level rises) in several examples (for instance, in going from panel (c4) to panel (c3) or from panel (d3) to panel (d2) in Fig. ).

This has a knock-on effect on the style of calving under increases in extensional stress: in the aligned crack case of Fig. , an increase in extensional stress invariably leads to calving by basal crack propagation (see the discussion in Sect.  and columns 2–4 in Fig. ). In the offset crack case of Fig. , increases in τ by contrast lead to calving by top crack propagation through the annihilation of saddle and node points (panels (d2)–(c2) and (c3)–(b3)) more frequently than by the node point reaching the right-hand calving boundary (panels (b4)–(a4)), while the annihilation of saddle and node points in a saddle-node bifurcation only occurs as the result of a decrease in η in 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.  as compared with Fig.  is, however, that the values of extensional stress τ are significantly larger in the former than in the latter, even though we obtain qualitatively similar phase planes (with a node- and saddle-type marginal fixed point in some of them and therefore with parameter combinations that lead to finite crack growth but not to calving if starting near the origin). The fact that larger extensional stresses are required to produce a qualitatively similar result is not the result of having laterally offset versus aligned cracks but of using a periodic domain with a relatively short (W=1) length: in addition to surface and basal cracks interacting, neighbouring cracks on each of the surfaces also interact.

Here we analyze the interaction between neighbouring cracks, restricting ourselves to a periodic array of basal cracks for simplicity. The second equation in Eq. () then becomes redundant, and the dynamical system can be visualized simply by plotting KIb against db, identifying where KIb>κ. We show results in Fig. . Panel (a) shows how KIb depends on crack length db for fixed a value of W=2 as τ is varied, analogously to Fig. 7b in . For a given set of parameters, the crack will grow if KIb>κ (the latter being the broken straight line) and remain static otherwise. Calving occurs if, as in the curves for τ=0.12 in Fig. a, KIb>κ up to the limiting value db=1, at which the basal crack fully penetrates through the ice thickness. As shown in Fig. b, this is the case for larger τ, qualitatively mimicking the corresponding behaviour for a non-periodic domain studied in .

Figure b similarly shows KIb against crack length for a fixed value of τ=0.04, just above the critical value of 0.039 for calving by basal crack propagation for a non-periodic domain in . The curve marked “NP” reproduces the corresponding result for that domain, given by the purple curve in Fig. 7b of . If we compare the case of a finite-width periodic domain with the case of the non-periodic domain, we find that KIb agrees between the two for short crack lengths db, for which the far field is relatively unimportant. The difference becomes noticeable as db grows, more so for short periodicities W. Only the non-periodic case experiences calving at τ=0.04 in Fig. b.

For each combination of parameters τ and κ and a given domain periodicity W, the points of intersection between the corresponding solid curve and the dashed curve in Fig.  correspond to marginal equilibria as defined in Sect. , with the smaller solution being unstable and the larger solution being stable if it exists. As extensional stress is increased, the larger, stable marginal equilibrium also increases until it disappears entirely. Figure a shows the corresponding steady states for W=10 as a continuous function of the parameter τ, by analogy with Fig. 8b in . The grey region consists of steady states for W=10 as an example to clarify that the curves show separate regions of steady states from regions of phase space in which db increases.

As Fig. a shows, calving for basal cracks occurs through the continuous growth of crevasse length with increasing τ until db=1 is reached at a critical value τcrit. For the case W=10 shown in Fig. a, τcrit=0.088, as compared with τcrit=0.039 for a non-periodic domain. In Fig. b, we plot the dependence of τcrit on W: the larger the W, the lower the critical stress. We conjecture (but have not attempted to prove) that the limit of τcrit for large W is given by the critical stress for a non-periodic domain with traction-free lateral boundaries, τcrit,∞=0.039. If so, the decrease in τcrit clearly continues to be significant beyond the range for which we have calculated τcrit in Fig. : for W=20, τcrit is a little under twice the critical value for a non-periodic domain. Note that, in the limit of a small crevasse spacing (much less than a single ice thickness), the effect of neighbouring crevasses observed here agrees with the previous results of and , who found a significant reduction in crack tip stress intensity factor for crevasses that are spaced closer than their depth of penetration relative to an isolated crevasse.

As in the non-periodic case considered in , calving occurs because KIb changes from vanishing as db→1 for τ<τcrit to becoming infinite for τ>τcrit (Fig. a). In Appendix C, we were able to associate that behaviour with the net torque generated by the crack on the remaining “neck” of ice immediately above the crack. In a non-periodic domain with vanishing lateral traction, the torque generated by forces on the crack face must be balanced by a torque generated by stresses in the neck of ice. The balance of torques dominates the stress field in the neck when the crack spans nearly the full ice thickness, and a positive stress intensity factor (diverging to +∞ as db→1) results if the net torque on the crack remains positive for basal crack lengths approaching unity.

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 W between cracks, the stronger that effect becomes, which we can visualize by looking at the vertical displacement u3 around the crack. Figure  shows the stress field σxx and the corresponding deformation of the domain for identical parameter values τ=0.1.

Clearly, stresses are larger around the crack tip for the non-periodic case in panel (a). Far-field displacements uz are also larger for the non-periodic case and linear in x; without any applied traction at the lateral boundaries, the far field undergoes a rigid body motion. By contrast, the periodic case has smaller far-field displacement with a finite, negative curvature: far from the crack, the stress and displacement fields are those for an elastic beam, with the curvature signalling a bending moment that resists the torque generated in the crack.

As in the simpler case of a single crack studied by , , and , we can attempt to create a calving law for the case of interacting cracks incised into opposite sides of the ice. Identifying which parameter combinations (τ,η,κ) result in calving given an initial state with only minimal crack lengths (db(0),dt(0)) is relatively simple. Firstly, the initial crack lengths need to be sufficient for the stress intensity factor to overcome the typically small fracture toughness κ. As the crack lengths are short, the effect of interactions between cracks is minimal in that case: the results of , , and suffice, with both and furnishing the relevant criteria. Secondly, the ensuing crack propagation needs to result either in the cracks meeting inside the ice as in Sect.  or in one crack propagating fully through the ice as in Sect. . That outcome is easy to identify through the absence of the node-type marginal fixed point in the relevant phase planes, as shown in Figs.  and . The relevant region of parameter space can be delineated crudely from Figs.  and directly, which could be improved upon by identifying conditions under which the fixed point marginally exists (defining the boundary of the region of parameter space in which calving occurs): for instance, when it first appears at the “calving boundary” (the diagonal db+dt=1 in Fig.  or the right-hand edge db=1 in Fig. ) or when it first forms at a saddle-node bifurcation (for instance, between the parameter values used in panels (d2) and (c2) of Fig. ).

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 (), which is likely the case in any realistic setting, where extensional stress τ and water level η typically evolve on the advective timescale for ice transport across an ice shelf and on the seasonal timescale associated with surface melt, respectively. In that situation, cracks may propagate only part-way across the ice. Calving can occur because of subsequent changes in forcing parameters to cause them to propagate further and eventually meet in the ice or reach the opposite ice surface.

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 and led to a relatively simple prescription for calving (Eq. 37 of ), but the present case is more complex to capture.

Consider the case of offset cracks as shown in Fig. . The simplest approximation is to assume that the region of steady states indicated by black dots as in panels (d2)–(e2), (c3)–(e3), and (b4)–(e4) always takes the form of a rectangle with its sides aligned with the (db,dt) axes. Let the upper and lower boundaries of the region be at dtm+(τ,η,κ) and dtm-(τ,η,κ) and the left-hand boundary at dbm(τ,η,κ). If any one of these boundaries is not present (as is the case for dtm+ in panels (b4) and (c4)), then simply set the corresponding value of d⋅m to unity.

The node-type marginal fixed point, if present, must be at (dtm-,dbm), and any initial evolution is to that fixed point, at least to a close approximation. Subsequent movement of the boundaries of the rectangle of steady states can have one of four consequences. Firstly, the node can move upwards and to the right, in which case the crack configuration (db,dt) tracks the location of the node. Secondly, the node can move downwards and to the left, leaving the crack configuration (db,dt) static and inside the region of steady states. Thirdly, the node can move down and to the right, in which case db tracks the left-hand edge dbm of the region of steady states, while dt remains static. Fourthly, the node can move upwards and to the left, in which case dt tracks the bottom of the region of the steady states at dtm-, while db remains static.

Any movement of the boundaries of the region of steady states at dtm+, dtm-, and dbm that occurs later leads to a similar pattern of crack evolution: (db,dt) will either move horizontally to stay on the left-hand edge of the region of steady states, move vertically to stay on the bottom edge, move diagonally to track the corner that is the node-type marginal fixed point, or remain inside the region of steady states if none of the other alternatives are possible. That is until either (db,dt) reaches the right-hand edge of the unit square, and calving occurs by the propagation of the bottom crack all the way to the upper ice surface, or the upper boundary of the region of steady states reaches the location (db,dt). The latter outcome can occur either as a result of the upper boundary reaching a location (db,dt) that had previously ended up on the inside of the region of steady states or because the upper boundary dtm+ reaches the lower boundary at dtm- and the two marginal fixed points annihilate each other in a saddle-node bifurcation. In either event, calving occurs by the propagation of the surface crack to the base of the ice.

Mathematically, we can capture the implied motion of the phase point (db,dt) under changes in parameter values by the following statement, analogous to Eq. (37) in . Let 9Sb(τ,η,κ)={d:d>dbm(τ,η,κ)},St(τ,ηκ)={d:dtm-(τ,η,κ)≤d<dtm+}∪{1}. Then, if T is a slow time variable associated with the change in parameters (as opposed to the fast time variable with respect to which the dynamical system () evolves), we can write 10db(T)=inf⁡{d:d∈Sb(τ(T),η(T),κ(T)andd>db(T′)for allT′<T},11dt(T)=inf⁡{d:d∈St(τ(T),η(T),κ(T)andd>dt(T′)for allT′<T}. All that is then required is to know how the boundaries of the region of steady states depend on τ, η, and κ.

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.  (on which the procedure described was based) was computed for a specific value W of the spacing between crevasses, and we demonstrated in Sect.  that our results are highly sensitive to that spacing. More specifically, we found no preferred finite spacing between crevasses at which calving occurs most easily (Fig. b). It appears that basal crevasses propagate more easily the further apart they are spaced, as the torques exerted on each other by neighbouring cracks have the effect of acting to close the tracks (Fig. ), and this makes the exercise of trying to define a calving rule for finitely spaced crevasses an exercise of limited value.

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 τ) likely points to the same deficiency in the model as previously discussed in Sect. 6.3 of : based on the assumption of small-strain elasticity, the model neglects the effect of vertical elastic displacements on fluid pressure at the domain boundary. In other words, the effect of elastic deformation on buoyancy is excluded from the model, even though, at large lateral distances, we can expect sufficient vertical displacements to have a leading-order effect. We anticipate that incorporating the feedback between displacement and fluid pressure at the boundary will lead to additional torques generated by vertical displacements in the far field, suppressing crack growth for very large crack spacings (see also ). We leave a study of this effect to future work.

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 that was previously used in . Since the cracks under consideration are typically fluid-filled, it is likely that dynamic propagation is controlled by the retarding effect of fluid flow in the fractures , which requires a significantly more complicated hydrofracture model that is unlikely to permit a comprehensive study of parameter space or even of fracture evolution for different initial conditions as in Figs. , , and . In addition, the assumption of purely vertical crack propagation is contingent on the highly specific crack orientations considered here, which ensure that we have purely mode I crack propagation. In reality, there are likely to be many interacting and potentially curved cracks, which we will address with a future iteration of the model.

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 . After that viscoelastic relaxation, the form of the pre-stress assumed here and in , , and will no longer be appropriate; a more complicated recomputation of the viscous pre-stress on the now partially fractured domain becomes necessary . Secondly, over the much longer timescale associated with significant viscous deformation of the ice shelf (typically decades or longer), even the fractured ice domain will no longer take the form of a parallel-sided slab with narrow cracks incised into it: the cracks will be widened by viscous flow, significantly altering the domain in which the linear elastic fracture mechanics problem considered here needs to be solved. We consider both of these complications in the companion paper .

The second major issue is the prescription of surface hydrology through a surface water table elevation, as we do through the parameter η. In reality, near-surface aquifers or surface hydrological systems are unlikely to be spatially uniform, or they are unaffected by the evolution of the crevasses that they feed: drainage of water into a crevasse should introduce an additional dynamic timescale and set of processes that the prescription of a fixed water level completely ignores. This is unlikely to be a trivial issue, as ice shelves are unlikely to be isothermal even if the near-surface supports an aquifer: if water is supplied slowly and crevasse propagation is slowed, then the refreezing of water in a newly formed crack (a process we ignore here on the basis that we assume rapid crack propagation) could suppress crevasse growth.

A simpler end member is, of course, the case of a dry ice shelf surface (η=1 in our notation). This is not a case that we have dwelt on, as we have focused on the richer dynamics generated by a finite water table depth. The rightmost columns of Figs.  and , however, provide a guide to what happens as we keep lowering the water table: surface crack propagation is progressively inhibited. For aligned cracks as in Fig. , a small surface crack generally forms, but its tip remains above water level unless initial conditions to the contrary are specified, and calving happens due to bottom crevasse propagation at sufficiently large extensional stress τ. For offset cracks, the top crack in column 4 of Fig.  still penetrates below the water level, but the actual depth to which it penetrates appears to have no bearing on steady-state bottom crack lengths, and calving once more occurs due to bottom crevasse propagation at sufficiently large τ. In other words, for progressively drier top cracks (meaning larger values of η approaching unity), we only expect a model for bottom crevasse formation to predict calving adequately.