Understanding the processes that govern ice shelf extent is important to improving estimates of future sea-level rise. In present-day Antarctica, ice shelf extent is most commonly determined by the propagation of through-cutting fractures called ice shelf rifts. Here, I present the first three-dimensional analysis of ice shelf rift propagation. I model rifts using the assumptions of linear elastic fracture mechanics (LEFM). The model predicts that rifts may be stabilized (i.e., stop propagating) when buoyant flexure results in the partial contact of rift walls. This stabilizing tendency may be overcome, however, by processes that act in the ice shelf margins. In particular, loss of marginal strength, modeled as a transition from zero tangential displacement to zero tangential shear stress, is shown to favor rift propagation. Rift propagation may also be triggered if a rift is carried with the ice flow (i.e., advected) out of an embayment and into a floating ice tongue. I show that rift stability is closely related to the transition from uniaxial to biaxial extension known as the compressive arch. Although the partial contact of rift walls is fundamentally a three-dimensional process, I demonstrate that it may be parameterized within more numerically efficient two-dimensional calculations. This study constitutes a step towards a first-principle description of iceberg calving due to ice shelf rift propagation.

Despite decades of progress, it remains unclear whether Antarctica will gain or lose mass by the year 2100. Although the rates of mass change over this timeframe are typically projected to be nearly linear

The largest modern ice shelves exist in coastal embayments. This basic observation has long prompted the notion that embayments stabilize ice shelves

Ice shelf margins are the part of the ice shelf grounding zone that is roughly parallel to flow (see Fig.

An investigation into the forces that drive rift propagation requires a careful accounting of the forces acting on the rift walls.

I model rifts using linear elastic fracture mechanics (LEFM). Although a number of previous studies have examined ice shelf rifts using LEFM, no previous study appears to have considered three-dimensional effects.

A final point of background concerns the relationship between the forces that drive fracture and the background ice flow. In real ice shelves, the state of stress is constantly evolving due to the change in geometry brought about by ice flow. Previous studies have examined the relationship between ice flow and fracture in several ways.

This paper is organized as follows. In Sect.

I consider the idealized ice shelf geometry shown in Fig.

Geometrically, I model a rift as a rectangular hole in the ice shelf. Fractures in three dimensions have a fracture tip defined by a two-dimensional curve rather than a point. Although I refer to a rift tip for brevity, this term actually refers to a rift-tip curve. In the treatment presented here, the rift-tip curve is taken to be a vertical straight line. The rift is uniformly 10 m wide over most of its length. Simulations show low sensitivity to the choice of this width (for example, stress intensity factors, described later, show changes less than or on the order of 1 % change in response to changing the rift width to 15 m).

I consider the equations of linear, homogeneous, isotropic, static, three-dimensional elasticity

The perturbation stress tensor is necessary for the following physical reason. Without subtracting the initial overburden pressure, the ice shelf experiences an initial volumetric contraction

With respect to the perturbation stress tensor, the equations of motion are

The ice front, rift walls, and top and bottom ice shelf surfaces are loaded by a depth-varying normal stress that is equal to the water pressure below the waterline and equal to zero above the waterline. These boundaries have zero shear stress. The water pressure condition may be written as

The geometries and boundary conditions considered in this study:

The boundary conditions applied in the three-dimensional model are found by combining Eqs. (

In all simulations, the surface of the ice shelf above the grounding line at

I solve Eqs. (

Fractures in elastic materials create displacement fields that are proportional to

The quantities

A basic tenet of fracture mechanics is that unstable crack growth occurs when the elastic strain energy available to drive fracture exceeds the energy required to create new fracture area

The fracture propagation criteria may then be stated as

The partial contact of rift walls is a nonlinear phenomenon because it involves solving for the shape of the contacting region and therefore changing the region over which different boundary conditions are applied

The stress intensity factors are calculated by solving Eqs. (

Typical three-dimensional stress intensity factors (SIFs) as a function of depth

Figure

I estimate the individual components of the stress intensity factors from the three-dimensional solutions (Fig.

I now examine a parameterized two-dimensional model with the goal of approximating Eqs. (

Bending terms

There is some discrepancy in the literature concerning the precise values of the function

Bending also creates a Mode III stress intensity factor. Assuming that this bending can also be described within Euler beam theory, the Mode III and Mode I stress intensity factors are related by a factor,

I find good agreement between two-dimensional calculations and the depth-averaged values from three-dimensional calculations. Table

Comparison between 2D and 3D calculations.

The analytical solution is not expected to perfectly match the finite element solution because the latter accounts for the full floatation condition (Eq.

I use the two-dimensional approach described in Sect.

Stress intensity factors for marginal rifts are plotted in Fig.

Stress intensity factors for central rifts are plotted in Fig.

Stress intensity factors for marginal rifts may reflect either stability or instability depending on the position of the rift. Three pieces of information, the

I have presented a three-dimensional LEFM analysis of ice shelf rift propagation. While this model has many potential applications, I have focused on the relationship between marginal strength and rift stability. In that regard, the main result of this analysis is that rifts originating in the margins of ice shelves become unstable if the ice shelf margin looses shear strength. This transition between a strong margin and a weak margin can be seen, for example, by comparing the red and yellow curves in Fig.

I have assumed that margins have either zero displacement or zero shear stress. In reality, margins likely experienced reduced but nonzero shear stress. I have also considered only two rift locations (marginal or central), only one ice shelf geometry (square), and only one rift geometry (a single rift, perpendicular to flow, and without curvature). I treat the entire ice column as having uniform material properties and therefore describe neither a firn layer nor its relation to partial contact of rift walls. Additional observed rift-wall processes such as brine infiltration, surface accumulation, and variable uplift could also be investigated

Same as Fig.

All boundary conditions considered here give rise to a compressive arch, defined as the region where an ice shelf transitions from uniaxial to biaxial extension (Fig.

The ice shelf compressive arch (thick black dashed line) is plotted for three boundary conditions:

I have modeled an ice shelf as a three-dimensional buoyantly floating elastic plate. I then show how these three-dimensional results may be captured in simplified two-dimensional calculations. Using the two-dimensional theory, I show that ice shelf rifts become unstable in the presence of marginal weakening or upon exiting an embayment. These results are a step towards modeling marine ice sheet evolution with a physics-based relationship between ice flow and an ice extent set by rift propagation.

The analysis code used in the text is available on the author's GitHub repository

The author declares that there is no conflict of interest.

This project began with a discussion with Colin Meyer at the Cambridge symposium of the International Glaciological Society in 2015. Jim Rice and Eric Dunham read earlier versions of this manuscript and gave helpful comments. Several discussions with Brent Minchew, Jan De Rydt, and Hilmar Gudmundsson also provided interesting feedback along the way. On a visit to C.U. Boulder organized by Jed Brown, David Marshall was critical of some early results; this feedback was helpful.

This paper was edited by Olivier Gagliardini and reviewed by three anonymous referees.