2.1. Base case: 1D flowline model – no buttressing

First consider a simple one-dimensional approximation to an ice shelf with the domain aligned along a flowline. This simple case has been considered by many previous authors (e.g. Schoof, Reference Schoof2007; Robison and others, Reference Robison, Huppert and Worster2010) and we include it here as the first-order representation of an ice shelf. Here ice-shelf thickness and velocity vary in one dimension only (Fig. 2a). In this case, the horizontal force-balance Eqn (1) reduces to

(2)$$4 {\partial \over \partial x} \left(\mu H {\partial u\over \partial x} \right)= \rho g' H {\partial H\over \partial x}\comma\; $$

with x aligned in the flow direction. This equation can be integrated along the length of the shelf, from the calving front, x _C back to any point within the ice shelf, x _i, (most commonly the grounding line, x _G) to determine the depth-integrated horizontal stress at that point,

(3)$$\mu H \left. {\partial u\over \partial x} \right \vert _{x_{i}} = \mu H \left. {\partial u\over \partial x} \right \vert _{x_{C}} + \left. {\rho g'\over 8} H^{2} \right \vert _{x_{i}} - \left. {\rho g'\over 8} H^{2} \right \vert _{x_{C}}\comma\; $$

where in each term variables are evaluated at x _C or x _i as denoted by the vertical bar to the right of each term. The depth-integrated horizontal stress at the calving front is equal to the hydrostatic pressure difference between the ice and the ocean

(4)$$\mu H \left. {\partial u\over \partial x} \right \vert _{x_{C}} = \left . {\rho g'\over 8} H^{2} \right \vert _{x_{C}}.$$

Therefore, the two terms cancel in Eqn (3), giving

(5)$$F=\mu H \left. {\partial u\over \partial x} \right \vert _{x_{i}} = \left . {\rho g'\over 8} H^{2} \right \vert _{x_{i}} = F_0.$$

The extensional stress at point x _i in the presence of this 1D ice shelf is equivalent to the extensional stress felt in the absence of the shelf. We consider this to be the base state in which no buttressing is generated by the ice shelf, and hence denote the depth-integrated stress F ₀. The following two sections describe fundamental ice-shelf geometries that generate buttressing.

Fig. 2. Fundamental ice-shelf geometries: (a) 1-D flowline. (b) Laterally confined ice shelf in a parallel embayment. (c) Axisymmetrical radially-spreading shelf. (d) Sector of an axisymmetric radially-spreading shelf, which in plan view forms an annulus.

2.2. Laterally confined ice shelf – shear-stress buttressing

Consider an ice shelf laterally confined within a parallel embayment. Ice flow is parallel to the side walls, aligned in the x-direction only, with y aligned perpendicular to flow. The ice-shelf thickness, H, is uniform across the width of the shelf (i.e. H(x, y) ≡ H(x)) (Pegler and others, Reference Pegler, Kowal, Hasenclever and Worster2013; Kowal and others, Reference Kowal, Pegler and Worster2016; Pegler, Reference Pegler2016; Wearing, Reference Wearing2016; Haseloff and Sergienko, Reference Haseloff and Sergienko2018), as shown in Figure 2b. The horizontal force-balance Eqn (1) reduces to

(6)$$4 {\partial \over \partial x} \left(\mu H {\partial u\over \partial x} \right)+ {\partial\over \partial y} \left(\mu H {\partial u\over \partial y} \right)= \rho g' H {\partial H\over \partial x}.$$

The additional term on the LHS is due to shear within the shelf, induced by friction or no-slip at the side walls of the embayment. Integrating along the length of the shelf

(7)$$\left [ \mu H \displaystyle{{\partial u}\over{\partial x}} \right ]^{x_{C}}_{x_{i}} = \int^{x_C}_{x_i} \left [ \displaystyle{{\rho g^{\prime}}\over{4}} H \displaystyle{{\partial H}\over{\partial x}} - \displaystyle{{1}\over{4}} \displaystyle{{\partial}\over{\partial y}} \left ( \mu H \displaystyle{{\partial u}\over{\partial y}} \right ) \right ] \, \rm d x,$$

(8)$$\Rightarrow F = \mu H \left. \displaystyle{{\partial u}\over{\partial x}} \right \vert_{x_i} = F_0 + \int^{x_C}_{x_i} \displaystyle{{1}\over{4}} \displaystyle{{\partial}\over{\partial y}} \left ( \mu H \displaystyle{{\partial u}\over{\partial y}} \right ) \, \rm d x,$$

(9)$$= F_0 - F_{{\rm shear}},$$

(10)$$\Rightarrow F_{{\rm shear}} = - \int^{x_C}_{x_i} \displaystyle{{1}\over{4}} \displaystyle{{\partial}\over{\partial y}} \left ( \mu H \displaystyle{{\partial u}\over{\partial y}} \right ) \, \rm d x.$$

In this case, the depth-integrated horizontal stress, F, is reduced in comparison with the base case because of buttressing from shear stress, F _shear.

The level of ice-shelf buttressing can be quantified using the buttressing number (Gudmundsson, Reference Gudmundsson2013),

(11)$$B_N= {F_0 - F\over F_0}.$$

B _N is the difference between the depth-integrated horizontal stress in the absence of an ice shelf, F ₀, and the actual depth-integrated horizontal stress F, normalised by F ₀. While the value of F ₀ is the same in all directions, the orientation of F must be defined. At the grounding line, the depth-integrated horizontal stress of interest acts perpendicular to the grounding line. In this case, with F aligned perpendicular to the grounding line, B _N = 0 signifies that there is no buttressing generated by the shelf, while B _N = 1 implies that the downstream ice shelf provides complete buttressing such that extensional stress is zero.

The buttressing number can also be used to determine buttressing within the ice shelf. In this work we consider buttressing in the flow direction (flow buttressing Fürst and others (Reference Fürst2016)) and calculate the difference between F and F ₀, where F is aligned with flow. In contrast, Fürst and others (Reference Fürst2016) used the orientation and magnitude of the depth-integrated second principal stress (SPS) to define the maximum buttressing number within the ice shelf. The SPS is often not aligned in the flow direction (much of the time it is aligned perpendicular to the flow). If a section of the shelf is removed via calving, the flow and stress field within the shelf would be altered. Therefore, although B _N aligned with the SPS within the ice shelf may be a useful way to choose which sections of an ice shelf to remove in a numerical model, the corresponding value of B _N does not signify the magnitude of buttressing generated to resist ice flow upstream. As such, in Fürst and others (Reference Fürst2016) the maximum buttressing number varies between each section of passive ice. It is an open question whether a buttressing number is an effective way to measure buttressing within ice shelves, where flow and stress fields may be complex. We avoid this issue by considering simple geometries where the flow is perpendicular to the grounding line, such that buttressing in the flow direction acts across the grounding line. We use the flow buttressing number throughout the following analysis and hereafter refer to this as the buttressing number, B _N.

2.3. Unconfined radially-spreading ice shelf – hoop-stress buttressing

The third fundamental configuration we consider is an unconfined ice shelf, which is able to spread laterally in two directions. The simplest idealised example of this is an ice shelf spreading from a point source. In cylindrical coordinates, with the source at the origin, the flow is radially outwards and axisymmetric (symmetric with respect to rotation around the vertical axis) (Fig. 2c). This type of geometry has been considered previously by Morland and Zainuddin (Reference Morland, Zainuddin, Van Der Veen and Oerlemans1987), Pegler and Worster (Reference Pegler and Worster2012, Reference Pegler and Worster2013). With the radial coordinate, r, aligned in the flow direction and no variation in flow azimuthally (with respect to rotation about the vertical axis), the force-balance Eqn (1) becomes

(12)$$2{\partial\over \partial r} \left(\mu H \left(2 {\partial u\over \partial r} + {u\over r} \right)\right)+ 2\mu H {\partial\over \partial r} \left({u\over r} \right)=\rho g' H {\partial H\over \partial r}.$$

The full derivation of this equation is given in Appendix A. The first term on the LHS is the radial derivative of the vertically-integrated isotropic stress and along-flow (radial) deviatoric stress (identical to the LHS of Eqn (2), but in cylindrical coordinates). The second term on the LHS of (12) is the depth-integrated radial gradient in the azimuthal strain rate, multiplied by the effective viscosity. This is the additional component from the divergence of the strain-rate tensor, which emerges in cylindrical coordinates, and is required to ensure the flow is incompressible when the shelf flows radially (Appendix A). Equation (12) can be integrated radially back from the ice front (r _C) to a point in the ice shelf (r _i) to give

(13)$$F = \displaystyle{{\mu H}\over{2}} \left ( \left. 2 \displaystyle{{\partial u}\over{\partial r}} + \displaystyle{{u}\over{r}} \right) \right\vert_{r_{i}} = \left. \displaystyle{{\rho g^{\prime}}\over{8}} H^{2} \right \vert_{r_{i}} + \int^{r_C}_{r_i} \displaystyle{{\mu H}\over{2}} \displaystyle{{\partial}\over{\partial r}} \left (\displaystyle{{u}\over{r}} \right ) \, \rm d r,$$

(14)$$= F_0 - F_{{\rm hoop}},$$

(15)$$F_{{\rm hoop}} = - \int^{r_C}_{r_i} \displaystyle{{\mu H}\over{2}} \displaystyle{{\partial }\over{\partial r}} \left (\displaystyle{{u}\over{r}} \right ) \, \rm d r,$$

(16)$$= - \int^{r_C}_{r_i} \displaystyle{{\mu H}\over{2r}} \left [\displaystyle{{\partial u}\over{\partial r}} - \displaystyle{{u}\over{r}} \right ] \, \rm d r,$$

as in Pegler and Worster (Reference Pegler and Worster2012). Therefore, hoop stresses make a positive contribution to buttressing where the azimuthal extension u/r exceeds the radial extension ∂u/∂r. Conversely, negative hoop-stress contributions are made where azimuthal extension is less than radial extension. When the hoop-stress buttressing is negative the viscous resistance to azimuthal extension acts to increase radial extension and pulls ice downstream. From the integrand it is clear that the magnitude of hoop-stress contributions increases for increasing effective viscosity (μ) and thickness (H), but decreases with a radial distance.

Physically, hoop stresses act as the viscous resistance to spreading in the azimuthal direction as the ice shelf flows radially. For an axisymmetric shelf there is no azimuthal thickness gradient, so azimuthal spreading must be induced by the radial thickness gradient. As with the shear-stress case, the integrated effect of this viscous deformation back from the ice front produces buttressing and may reduce the extensional stress at the grounding line.

For an axisymmetric ice shelf extending to infinity, Pegler and Worster (Reference Pegler and Worster2012) established analytically an upper bound for the radius of curvature at the grounding line that would provide positive buttressing; $L \equiv \lpar \mu Q_0/2 \pi \rho g^{\prime} H_0^{2}\rpar ^{1/2}$, where H ₀ and Q ₀ are the ice thickness and flux at the grounding line. Pegler and Worster (Reference Pegler and Worster2012) estimated that for a typical ice shelf L ≈ 5 km. Furthermore, for ice shelves with radius of curvature at the grounding line greater than 6L, hoop stresses provided negative buttressing – generating additional extension at the grounding line. This analytical work was accompanied by laboratory-scale analogue experiments.

Here we consider hoop-stress buttressing generated by unconfined sections of ice shelves of finite length, found where ice shelves flow out past lateral pinning points, such as Amery and Fimbul Ice Shelves, Figure 1. The downstream sections of these ice shelves can be approximated as sectors of an annulus with radius of curvature at the upstream boundary (r _E) dependent on the lateral divergence of the flow, Figure 3. With no shear against lateral boundaries, these sections can be approximated as azimuthally symmetric. We first consider some idealised examples, determining the magnitude of hoop-stress buttressing produced in each case. Then, using geophysical, data we assess hoop-stress buttressing in unconfined sections of Antarctic ice shelves.

Fig. 3. Approximating an unconfined section of an ice shelf as an annulus diverging from an imaginary origin (r = 0). The ice shelf flows out from an area of confinement and diverges into open ocean. The radius of curvature at the exit of the embayment (r _E) and at the calving front (r _C) are labelled.

3.1. Example 1

Figure 4 shows the steady-state results for an ice shelf of length 50 km, with radius of curvature at the upstream boundary of 70 km. The input thickness is 400 m and the input velocity is 500 m a⁻¹. For an angle of divergence of 10°, the shelf diverges from a width of 20 to 40 km along its length. Throughout the shelf the azimuthal strain rate is greater than the radial strain rate, implying a positive hoop-stress contribution everywhere. This can been seen from the increase in the buttressing number back from the calving front to r _E where it reaches 0.2; i.e. 20% of the hydrostatic stress is balanced by hoop stresses.

Fig. 4. Example of an idealised ice shelf with rate factor A ₋₁₀. The radius of curvature at the upstream boundary is 70 km and shelf has a length of 50 km. (a) Ice speed. (b) Ice-shelf thickness profile (blue) and value of the local thickness-profile exponent, a (red), for function $H=\tilde {H}r^{-a}$, calculated at each point by fitting the function to an 11-km interval around each point. (c) Radial strain rate (blue), azimuthal strain rate (red), difference; radial minus azimuthal strain rate (orange). (d) Depth-integrated stresses; radial extension (blue), hydrostatic driving stress (red dashed) and hoop stress (dotted yellow). (e) Buttressing number along the length of the shelf.

3.2. Example 2

A similar shelf is shown in Figure 5, however the radius of curvature at the upstream boundary has been increased to 400 km and the shelf is now 100 km long. Due to the large radius of curvature the width of the shelf only increases from 140 to 170 km along its length, for an angle of divergence of 10°. Figure 5c shows that the azimuthal strain rate is greater than the radial strain rate in the downstream section of the shelf (>435 km) only. This leads to a transition from positive to negative hoop-stress buttressing contributions in the upstream direction. The peak in buttressing number is just downstream of this transition (Fig. 5e) because the ice is thinner and the driving stress (hydrostatic pressure difference between ice and ocean) decreases downstream. In this case the buttressing number at the upstream boundary (r _E = 400 km) is very small and the shelf provides insignificant buttressing via hoop stresses (<1.5%).

Fig. 5. Same as Figure 4, but with a 400 km radius of curvature at the upstream boundary and shelf of 100 km in length. Hoop-stress buttressing remains positive throughout the length of the shelf. However, negative hoop-stress contributions are made in the upstream section and hence the peak in hoop-stress buttressing is located at r ≈ 440 km.

3.3. Positive and negative hoop-stress contributions

For steady-state ice shelves the sign of the hoop-stress buttressing contribution is dependent on the local ice-shelf thickness profile. For a radially spreading shelf, with angle of divergence θ at the origin, the flux at radius r in the steady state is

(19)$$\theta r u\lpar r\rpar H\lpar r\rpar = q_0 + {1\over 2} b \theta r^{2}\comma\; $$

where q ₀ is the input flux at r = 0 and the second term on the RHS is the net mass gain/loss between the origin and r, due to surface or basal processes. The ice must thin in the downstream direction and we approximate the local ice thickness profile as

(20)$$H \approx \tilde{H} r^{-a}\comma\; $$

where the exponent a is dependent on the local curvature of the ice thickness and in turn the deformation of the shelf at this point. Therefore the radial velocity takes the form

(21)$$u\lpar r\rpar = {q_0 + \displaystyle{{1}\over{2}} b \theta r^{2}\over \theta \tilde{H} r^{1-a}}.$$

This can be used to determine the radial and azimuthal strain rates, such that hoop-stress buttressing contributions are positive when

(22)$$\displaystyle{{u}\over{r}} \gt \displaystyle{{\partial u}\over{\partial r}},$$

(23)$$\Rightarrow \quad(2-a) \left ( q_0 + b \theta \displaystyle{{r^{2}}\over{2}} \right ) \gt b \theta r^{2-a}.$$

In the case of b = 0, this reduces to a < 2. Therefore, in the steady-state with b = 0, at locations where the ice thickness profile can be approximated by $\tilde {H} r^{-a}$ with a < 2, positive hoop-stress buttressing is generated. The exponent for the local ice-thickness profile, a, is calculated at each point along the shelf using the MATLAB curve fitting function fit, which uses the ice thickness within an 11 km interval centred at each point to determine a. The spatially-evolving value of a is shown in Figure 5b, where there is a transition from a > 2 to a < 2 at r ≈ 435 km, where the magnitude of the radial and azimuthal strain rates are equal. For a > 2 hoop-stress buttressing contributions are negative. In contrast, in Figure 4b, hoop-stress contributions are positive throughout the length of the shelf and a < 2 everywhere.

3.4. Varying parameters

As identified by Pegler and Worster (Reference Pegler and Worster2012), the magnitude of buttressing from hoop stresses is dependent on the radius of curvature at r _E, the shelf viscosity, the input thickness and input velocity. In Figure 6 these parameters, along with the length of the ice shelf, are varied to assess the impact on buttressing.

Fig. 6. Varying parameters in the idealised model. Each panel shows the buttressing number (vertical axis): (a) along the length of the shelf when increasing the shelf length; (b) at the upstream boundary (r _E) of a 75 km shelf, when the radius of curvature at the upstream boundary (r _E; horizontal axis) and rate factor (A _X; coloured curves) are varied; (c) and (d) along the length of the shelf for varying input thicknesses (fixed velocity 500 m a⁻¹) (c) and varying input velocities (fixed thickness 400 m) (d).

As the length of a shelf is increased the buttressing number at r _E increases (Fig. 6a). However, the magnitude of this increase decreases with shelf length, and the peak in buttressing number is shifted downstream. This implies that hoop-stress buttressing increases at a reduced rate in comparison with the driving hydrostatic pressure as the ice thickens (it is the ratio of hoop-stress buttressing to driving stress that determines the buttressing number). It may also suggest that the upstream section of the shelf begins to produce negative hoop-stress contributions. However, further assessment shows that in these cases the azimuthal extension remains larger than radial extension and the thickness-profile exponent a remains less than 2. Therefore, hoop-stress contributions remain positive.

Figure 6b shows the effects of varying both the radius of curvature at the upstream boundary and the rate factor for an ice shelf of fixed length (75 km). Buttressing is reduced as the radius of curvature at the upstream boundary, r _E, increases and the lateral divergence of the shelf decreases. There is a transition to slightly negative buttressing once r _E increases past a critical point. This is dependent on the rate factor in the ice rheology. For decreasing rate factor, and therefore increasing effective viscosity, hoop stresses provide greater buttressing.

Figures 6c and d show the buttressing number for varying input fluxes at the upstream boundary. Increasing the input thickness and maintaining input velocity leads to reduced buttressing. This is because thicker ice produces a larger driving stress, which generates large radial strain rates. Conversely, for a fixed input thickness and increased input velocity, buttressing increases. This is because the hydrostatic driving stress remains the same, but the azimuthal strain rate increases like u/r.