The distribution of pressure on the vertical seaward front of an ice shelf has been shown to cause downward bending of the shelf if the ice is assumed to have vertically uniform viscosity. Satellite lidar observations show that many shelf edges bend upward and that the amplitude of upward deflections depends systematically on ice shelf thickness. A simple analysis is presented showing that upward bending of shelf edges can result from vertical variations in ice viscosity that are consistent with field observations and laboratory measurements. Resultant vertical variations in horizontal stress produce an internal bending moment that can counter the bending moment due to the shelf-front water pressure. Assuming a linear profile of ice temperature with depth and an Arrhenius relation between temperature and strain rate allows derivation of an analytic expression for internal bending moments as a function of shelf surface temperatures, shelf thickness and ice rheologic parameters. The effect of a power-law relation between stress difference and strain rate can also be included analytically. The key ice rheologic parameter affecting shelf edge bending is the ratio of the activation energy,

Ice shelf breakup is important since it could reduce the buttressing of ice sheets, leading to a speedup of ice sheet flow and therefore a rise of sea level (Scambos et al., 2004; Rignot et al., 2004; Schoof, 2007; Gudmundsson, 2013; Fürst et al., 2016). Long-term models of ice sheet flow predict accelerated sea-level rise caused by loss of ice shelves (e.g., DeConto et al., 2021). Ice shelf bending may lead to calving and breakup of shelves in two ways. Bending stresses may lead directly to crevasse formation and propagation (e.g., Wagner et al., 2016). Bending may also affect the routing and pooling of surface meltwater, which can facilitate crevasse growth and calving (e.g., Weertman, 1973; Lai et al., 2020; Buck, 2023).

Locations of topographic profiles and borehole temperature measurements.

The rheology of ice is critical to ice shelf calving and to flow of ice sheets and shelves (e.g., Cuffey and Paterson, 2010). Laboratory and theoretical analyses suggest that ice flow can be described as a non-Newtonian viscous fluid (Glen, 1955) with a strong temperature dependence (e.g., Weertman, 1983). However, there is great uncertainty in the parameters that describe ice flow (e.g., Cuffey and Paterson, 2010; Behn et al., 2021; Zeitz et al., 2020; Millstein, et al., 2022). Analysis of ice shelf bending may provide an additional constraint on ice rheology.

The downward bending of ice shelf edges is expected to result from the bending moment due to the pressure in water exerted on the shelf. Weertman (1957) derived an expression for this bending moment as a function of ice and water densities, assuming a uniform ice rheology with depth. Reeh (1968) numerically calculated the deflections due to this bending moment by treating the ice shelf as a uniform, thin, viscous plate and showed how the downward deflections of the edge would increase with time since the last calving event, as viscous stresses relax. Fully two-dimensional viscous (Mosbeux et al., 2020) and viscoelastic models (Christmann et al., 2019) of ice shelf bending, assuming uniform properties with depth, confirm the thin-plate predictions.

The down-bending of shelf edges predicted by the Weertman–Reeh theory has been seen in several locations such as the edge of the Ronne Ice Shelf and across several iceberg edges (Scambos et al., 2005, and see Fig.

The only published model to explain rampart–moat structures relates to erosion of part of the sub-aerial shelf by wave action (Scambos et al., 2005; Wagner et al., 2014; Mosbeux et al., 2020; Becker et al., 2021). The submarine remnant, termed a “foot” or “bench”, then acts as a load that pushes up the uneroded shelf edge (Fig.

Illustrations of the bench and internal moment models for deflection of a shelf edge. The arrow in

This study considers an alternative model to the submerged bench model that depends on vertical variations in the viscosity of an ice shelf. In some sense this work is a continuation of the analysis of Reeh (1968) in that it deals with bending moments causing an ice shelf to flex. In the pioneering paper on that topic, Reeh (1968) wrote the following:

a correct treatment of the problem would require consideration of the great variation (by a factor ten or more) of the viscosity. This, however, would involve enormous mathematical troubles.

Here I show that, as long as we can assume that the viscosity in an ice shelf varies exponentially with depth, the “mathematical troubles” are minimal. The exponential viscosity approximation is shown to be reasonable if the temperature dependence of viscosity can be described by an Arrhenius relation, and the temperatures increase linearly with depth in an ice shelf. That approximation allows derivation of scaling relations between ice rheologic parameters, ice surface temperatures and shelf edge deflections. For “great variations of viscosity” across an ice shelf, I show that the edge of the shelf should bend upward to make a rampart with a corresponding inboard moat. After that I consider the effect of non-linear temperature profiles on shelf bending. Before launching into a detailed analysis of this problem, I discuss some basic ideas about bending moments and layer bending.

Ice shelves that are not heavily buttressed are under extension (Weertman, 1957). While ice shelves are typically assumed to have negligible vertical gradients in horizontal strain rates, significant vertical variation in viscosity generates vertical gradients in horizontal stress that cannot be neglected. The idea that stresses internal to a layer can cause it to bend is well known in engineering and can be illustrated with a bimetallic strip that bends as temperatures change. Such a strip consists of two metal layers with different thermal expansion coefficients that are welded together. At a certain temperature this layered strip can be flat, but it will “curl up” when the temperature is increased as long as the lower layer has a larger thermal expansion coefficient. This occurs because the lower layer expands more than the upper layer, producing vertical variations in horizontal stress.

Internal stresses have been considered in explaining some lithospheric bending observations. Parmentier and Haxby (1986) showed how vertical variations of horizontal stress within a strong layer might explain downward bending of the lithosphere at transform faults. In this case, the stresses arise due to greater rates of thermal contraction of a deeper lithosphere combined with yielding of a shallow lithosphere. Those authors note that gravity prevents lithospheric bending that is of a much longer wavelength than the effective flexural wavelength of the layer. At very long length scales gravity prevents the layer from bending. The vertical variation in stress can be thought of as an internal bending moment that is matched everywhere by adjacent bending moments except at the plate edge (transform fault) where stresses, and so the applied bending moment, are different.

The concept of internal bending moments was also applied to the formation axial relief at plate spreading centers. This was first suggested to explain “axial highs” that mark the plate boundary at most fast-spreading centers and typically rise 300–500

A major difference between an ice shelf and the lithosphere (or a bimetallic strip) is that the stresses in the ice are controlled by viscosity. The viscous, or more precisely the viscoelastic, response of an ice shelf to bending moments was recognized by Reeh (1968), who noted that the wavelength of the response after a calving event should decrease with time. This cannot be properly treated without a fully 2D numerical simulation, but, as discussed at the end of this paper, this effect should lead to slow growth of the amplitude of bending deflections. As noted by Reeh (1968) the top of an ice shelf is typically colder than the base (by as much as 30

An internal moment will not cause bending where the shelf is laterally uniform and continuous over distances much larger than the flexural wavelength. Imagine that a calving event just broke off a broad section of the shelf, making a new shelf edge. If the bending moment applied by air and water at the shelf edge is different from the internal bending moment, then the shelf edge should bend, much like the bimetallic strip described above. However, the bending will take time to develop as viscous flow causes the flexural wavelength to diminish with time.

In the absence of a bench, it is the internal stress distribution that determines whether the edge bends up or down. The key question for this paper is how the horizontal stresses internal to an ice shelf affect the bending of a shelf edge. To address this question, I follow the approach of Weertman (1957) and Reeh (1968) and calculate the contribution to the total applied bending moment due to the difference between water pressure and internal stress in the ice layer. The new twist is that I consider internal stress variations related to vertical viscosity variations in the ice shelf.

To determine how an ice layer will flex due to the water pressure distribution on the side of the layer (as shown in Fig.

Illustrations of stresses and stress differences affecting a floating ice layer. Panel

To consider the effect on layer edge bending for a range of possible distributions of the horizontal stress that satisfy Eq. (

The reference horizontal stress distribution is what one would obtain for a shelf with uniform rheologic properties so that the horizontal stress is lower than the vertical stress by a uniform amount,

For

The internal bending moment term

The simplest case is for an ice layer with vertically uniform properties and infinite yield strength (or infinite fracture toughness), so there is no opening of surface or basal crevasses, thus reducing the stresses in the layer. This implies a constant offset between the horizontal and vertical stresses in the layer such that

Reeh (1968) calculated that the applied bending moment increases by up to 30 % as the layer is deflected, because as the shelf edge moves down the water pressure on the end increases. However, this neglects the counter-effect of the change in pressure on the underside of the deflected layer inboard of the edge. Thus, the average horizontal stress in the ice should remain constant as long as the thickness of the shelf does not change or the top of the layer does not drop below the water surface. Here, I neglect any changes in the applied moment with layer deflection.

As noted above, viscosity in an ice shelf is expected to decrease with depth (e.g., Reeh, 1968). Two simplifying assumptions are used here to relate viscosity variations with depth to rheologic parameters and ice shelf surface temperatures. The first assumption is that temperatures linearly increase with depth. The base of ice shelf should be at the pressure melting point, while the surface must be colder (e.g., Cuffey and Paterson, 2010), and borehole measurements on some ice shelves indicate nearly linear temperature–depth profiles, as is discussed below.

The second assumption is the form of the ice flow law. There is debate about how the flow of ice varies with stress and temperature, and there is evidence that multiple processes require complex descriptions (e.g., Behn et al., 2021). However, a wide range of observations and laboratory data are well approximated with a power-law relation between stress and strain rate, such as Glen's flow law (Glen, 1955) and an Arrhenius relation between strain rate and temperature (e.g., Cuffey and Paterson, 2010). Then the strain rate

A constant strain rate is used because the horizontal strain rate should be constant with depth for a uniform thickness ice shelf far from the shelf edge (i.e., where

Illustration of stress differences for full flow law and exponential approximation.

For a constant temperature gradient

Relations between model parameters.

For a thin floating layer, the horizontal strain rate far from the sides (i.e., many layer thicknesses) should be constant so that the difference between the horizontal stress and the vertical stress is well approximated as

The relationship between

We do not need to know the strain rate to find

Analytic model predictions for a range of surface temperatures. Values of the rheologic parameter

Figure

The internal bending moment given by Eq. (

Equation (

The error in the analytic approximation was analyzed by carrying out numerical integration of the stress differences for the full ice flow law (Eq.

Internal moment model topography. Deflections versus distance from a shelf edge are calculated for a thin plate subject to an applied moment at

To estimate the deflections expected for the bending moments derived here, we can use the thin-plate flexure approximations. Many studies of ice shelf bending treat an ice shelf using the plate approximations either with elastic, viscous or viscoelastic rheologies (e.g., Reeh, 1968; MacAyeal and Sergienko, 2013; Olive et al., 2016; Wagner et al., 2016; Banwell et al., 2019; MacAyeal et al., 2021). Vertical deflections of a thin, semi-infinite plate with a moment

Reeh (1968) and Olive et al. (2016) find that, for a viscous or viscoelastic plate with a uniform viscosity

Figure

The thin-plate approximations can also be used to illustrate the predictions of the bench model. The effect of the load

Several effects, including accretion or melting of the surface or base of an ice shelf, can contribute to non-linearity of temperatures with depth, and this should affect viscosity, stresses and internal moments. Observations of temperature profiles are limited since they require boreholes through ice shelves, and Fig.

Borehole temperature measurements for parts of three Antarctic ice shelves. Panels

To estimate the possible generation of such non-linear temperature profiles and their effect on ice shelf internal bending moments, I use a standard ice shelf thermal model. The approach, described by Robin (1955), assumes that pure shear thinning of the layer maintains a uniform shelf thickness,

Effect of surface or basal accretion on ice shelf temperature profiles and internal bending moments.

Using a temperature–depth distribution given by Eq. (

Figure

The simple analysis presented here shows how stresses internal to a floating ice shelf can affect the bending of the shelf edge. It shows that Reeh (1968) was right to be concerned about variations of viscosity through an ice sheet, since those stresses determine whether the edge of a shelf bends up or down. For small variations of viscosity across an ice shelf (less than about a factor of about 5), the edge bends down, while for larger viscosity variations the edge bends up. Assuming a fairly standard ice flow law and a linear temperature gradient through a shelf, these viscosity variations are controlled by two parameters: the surface temperature,

Significant departures of ice shelf temperature–depth profiles from linearity, as seen for parts of some ice shelves (see Fig.

The edge of the Ross Ice shelf is the only place where a systematic study of shelf bending has been done along an entire shelf front, and it offers a good first test of this model for several reasons: RIS shows rampart and moat structures along most of the front (Becker et al., 2021), it has a low surface temperature (e.g., MacAyeal and Thomas, 1979), and no benches have been reported there. The last point is significant since shelf edge benches have been seen for several other ice shelves (e.g., Scambos et al., 2005), including recent studies using ICESat-2 lidar (Philipp Arndt, personal communication, 2023).

Becker et al. (2021) show strong and systematic variations in the height of upward bending (or in the difference in elevation between the moat and rampart,

The horizontal offset between ramparts and moats along the RIS front also may be easier to explain with the internal moment model than with the bench model. This offset is slightly less than the layer thickness as estimated by Becker et al. (2021). As shown in Fig.

This analysis makes several significant approximations that can be considered in numerical simulations. Such simulations are now being done (e.g., Glazer and Buck, 2023) and can treat fully two-dimensional deformations, non-uniform vertical temperature gradients, variations of ice density with depth and the time dependence of viscoelastic deformation, among other factors. Of particular importance will be the calculation of the evolution of the effective flexural wavelength. However, the present analysis can guide those model studies since it suggests testable scaling relations, including the dependence of deflection amplitude on the rheologic parameter

Figure

Further numerical and observational tests of the internal moment model may allow new constraints to be placed on these rheologic parameters. Since models of ice sheet and ice shelf flow depend on these parameters, there should be interest in doing such tests. One important observation is to measure the distribution and size of ice benches since they certainly can affect shelf bending (e.g., Wagner et al., 2014, 2016). Key observational tests should also involve comprehensive studies of shelf edge bending for all the ice shelves of Antarctica and Greenland. Surface temperatures vary for different ice shelves. If the bending characteristics vary systematically with surface temperature, this offers hope of giving new constraints on the temperature dependence of ice rheology.

No data sets were used in this article.

The author has declared that there are no 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.

Thanks go to Emily Glazer, Ching-Yao Lai, Till Wagner, Nicolas Sartore, Phillip Arndt, Niall Coffey, Kirsty Tinto and Andrew Hoffman for helpful discussions and comments. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

This paper was edited by Benjamin Smith and reviewed by two anonymous referees.