2.1 Dynamic condition at the calving front or detachment location

Whether the calving front of the ice shelf lies exterior to its embayment (exterior calving) or interior to it (interior calving), the lateral stresses will cease to be exerted beyond a position denoted $x_{C}$ . These two situations are illustrated respectively in figure 2(a,b). Exterior calving characterises the Amery Ice Shelf, for example. Interior calving characterises the Ross and Filchner–Ronne ice shelves. For exterior calving, I identify $x_{C}$ as the position of detachment of the ice shelf from the lateral boundaries (a position coincident with the mouth of an outlet channel, as illustrated by experiments in Pegler Reference Pegler2016). For interior calving, I identify $x_{C}$ as the calving front itself.

At $x_{C}$ , I impose the stress condition

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}=\frac{\unicode[STIX]{x1D70C}g^{\prime }}{8}H\quad \text{at }x=x_{C}.\end{eqnarray}$$

For interior calving, $x_{C}$ depends on the calving processes that control the calving position. By treating $x_{C}$ as a parameter (cf. Gudmundsson et al. Reference Gudmundsson, Krug, Durand, Favier and Gagliardini2012), my analysis will determine the regimes that arise as a consequence of any specified calving position $x_{C}$ . An alternative calving model, such as a condition based on a prescribed calving thickness (Schoof et al. Reference Schoof, Davis and Popa2017) can be specified by replacing the imposition of $x_{C}$ by the implicit condition $H(x_{C})=H_{C}$ , where $x_{C}$ is then treated as an unknown and $H_{C}$ is a prescribed parameter. In my illustrative examples, I will initialise the system with no ice shelf and allow it to develop to an imposed calving position $x_{C}$ . During the development of the ice shelf, it will be assumed that calving does not occur. The front of the ice shelf $x_{N}(t)$ is then propagated according to ${\dot{x}}_{N}=u(x_{N},t)$ for $x_{N}(t)<x_{C}$ .

At the ice divide, I impose the symmetry conditions

(2.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}_{xx} & = & \displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}=0\quad \text{at }x=x_{D},\end{eqnarray}$$

(2.17) $$\begin{eqnarray}\displaystyle Hu & = & \displaystyle 0\hspace{44.39996pt}\text{at }x=x_{D},\end{eqnarray}$$

representing a no-stress and no-flux condition, respectively.

Figure 3.

Schematic of a marine ice sheet at large scales, illustrating the ice divide $x_{D}$ and the grounding line $x_{G}$ . The schematic illustrates the truncation of the domain at a position close to the grounding line $x_{0}$ , which underlies the steady-state simplification represented by (2.21).

2.2 Flow evolution and input specifications

The evolution of the thickness is described by the mass-conservation equation

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}t}+\frac{1}{w}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(wHu)=f(x,t),\end{eqnarray}$$

where $f(x,t)$ is the net ice accumulation per unit width (snowfall accumulation minus melting). For the main illustrative examples of this paper, I will specify a localised input concentrated at the ice divide,

(2.19) $$\begin{eqnarray}f(x)=2Q\unicode[STIX]{x1D6FF}(x-x_{D}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$ is the Dirac delta function (cf. Nowicki & Wingham Reference Nowicki and Wingham2008; Katz & Worster Reference Katz and Worster2010; Robison et al. Reference Robison, Huppert and Worster2010; Pegler & Worster Reference Pegler and Worster2012, Reference Pegler and Worster2013; Pegler et al. Reference Pegler, Kowal, Hasenclever and Worster2013; Kowal et al. Reference Kowal, Pegler and Worster2016; Pegler Reference Pegler2016). Ice sheets are typically fed in a distributed manner across their full extent, $f(x)\neq 0$ . However, the case (2.19) is useful within the context of illustrative examples because it confers the advantage of maintaining a specified control condition of the input flux. The examples will thereby be distilled to illustrate the effects of lateral stresses. The assumptions of a uniform basal-drag coefficient $C_{-}$ and flow width $w$ will be made likewise for distilling the illustrative examples to the effects of interest. An example of a distributed accumulation field $f(x)\neq 0$ will be provided in the supplementary examples of Pegler (Reference Pegler2018). With (2.19) imposed, equation (2.17) can be replaced by $Hu=Q$ at $x=x_{D+}$ , where the plus subscript denotes approach from the positive $x$ direction.

2.3 Steady-state reductions

By integrating the steady-state form of (2.18) and applying (2.17), we can determine the integral expression for the volumetric flux of ice per unit width in steady state,

(2.20) $$\begin{eqnarray}q(x)=Hu=\frac{1}{w(x_{G})}\int _{x_{D}}^{x_{G}}w(x^{\prime })f(x^{\prime })\,\text{d}x^{\prime }.\end{eqnarray}$$

The approximation of a nearly constant channel width, $w^{\prime }(x)=0,$ will be applied henceforth, but can be reinstated within the general balance equation developed for $w^{\prime }(x)\ll 1$ . The theoretical development of § 4 will be conducted in two stages: the first will involve a determination of the depth-integrated extensional resistive stresses associated with the steady states of the grounded region. The second will address the steady-state forces exerted by the floating region. For the latter, the full distribution field $f(x)$ can be incorporated directly into an expression for the buttressing force determined in § 4.2. For steady-state analysis of the flow in the grounded region, it is possible to focus on a demarked region of the ice sheet nearer the grounding line through which the variation in the flux (2.20) can be neglected to leading order. Let $x_{0}=x_{G}-{\mathcal{L}}$ denote the left-hand edge of a truncated numerical domain. The integral defining the flux (2.20) across the interval $[x_{0},x_{G}]$ can be split according to

(2.21) $$\begin{eqnarray}\displaystyle q(x)=\int _{x_{D}}^{x_{G}}f(x^{\prime })\,\text{d}x^{\prime }-\int _{x_{G}-x}^{x_{G}}f(x^{\prime })\,\text{d}x^{\prime }=q(x_{G})+O(f{\mathcal{L}}). & & \displaystyle\end{eqnarray}$$

Thus, if ${\mathcal{L}}$ is much less than the length scale of the ice sheet as a whole, ${\mathcal{L}}\ll (x_{G}-x_{D})$ , the flux across the demarked region $[x_{0},x_{G}]$ can be approximated as equal to $q(x_{G})$ . For the analysis of steady-state regimes and balances, one can therefore specialise to a domain closer to the grounding line, $[x_{0},x_{G}]$ , and apply a condition of uniform flux, $q_{0}=q(x_{0})$ , to leading order, where $x_{0}$ is chosen to be an order of ${\mathcal{L}}$ upstream of the grounding line. It should be noted that this option to specialise to a demarked region of the ice sheet nearer the grounding line is only generally applicable for steady states. A resolution of the transient evolution of the ice sheet depends on a consideration of the flow evolution across the full domain spanning ice divide to terminus. Thus, for the illustrative examples shown in this paper and the companion, where time-dependent solutions will be used to verify the steady-state results and their stability properties, a demarcation will not be used and the numerical domain will be specified as the full ice sheet, $[x_{D},x_{C}]$ . However, the possibility to demarcate in steady states to excellent approximation will be implicit in applying the steady-state analytical results of § 4.1 for situations involving a sheet-wide distributed accumulation. A similar approximation of neglecting accumulation was used in describing a quasi-steady inner region of the grounded ice sheet near the grounding line (Schoof Reference Schoof2007b ), and can be interpreted as a boundary-layer property. It should be noted that this property of neglecting small increases in flux near the grounding line is conceptually distinct from a boundary-layer property of there existing a region through which the terms in the governing equation (2.9) change their order of magnitude. The existence of a boundary layer of this second kind will be scrutinised in § 4.1.2.

2.4 Dimensionless model system

By forming scaling relationships between the extensional stress, basal stress and gravitational forces in the equation governing the grounded region (2.14a ), and the flux scale $hu\sim Q$ specified by (2.19), I determine the intrinsic scales of thickness and horizontal length given by

(2.22a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{H}}\equiv \left[\unicode[STIX]{x1D707}_{0}C_{-}^{m}\left(\frac{Q^{m}}{\unicode[STIX]{x1D70C}g}\right)^{m+1}\right]^{1/k_{m}},\quad {\mathcal{L}}\equiv \left[\frac{\unicode[STIX]{x1D707}_{0}^{m+2}Q^{m}}{\unicode[STIX]{x1D70C}g\,C_{-}^{m+1}}\right]^{1/k_{m}}, & & \displaystyle\end{eqnarray}$$

respectively, where $k_{m}\equiv (m+1)(m+2)-1$ . I also define the associated velocity and time scales, ${\mathcal{U}}\equiv Q/{\mathcal{H}}$ and ${\mathcal{T}}\equiv {\mathcal{L}}/{\mathcal{U}}$ . These scales are intrinsic to regions of the flow in which the stresses due to extension, basal drag and gravity are mutually comparable.

The mathematical specification also depends on the bed profile $b(x)$ appearing in (2.14a ). The dynamics will primarily be illustrated using a linear bed specified by $b(x)=b_{0}+ax$ , where $|b_{0}|$ is the depth of the ocean at the calving position and $a$ is the basal slope.

I use (2.22) to define dimensionless variables by

(2.23a-d ) $$\begin{eqnarray}(H,b)\equiv {\mathcal{H}}(\tilde{H},\tilde{b}),\quad u\equiv {\mathcal{U}}\tilde{u} ,\quad x\equiv {\mathcal{L}}\tilde{x},\quad t\equiv {\mathcal{T}}\,\tilde{t}.\end{eqnarray}$$

On dropping the tildes, the governing equation (2.14) becomes

(2.24a,b ) $$\begin{eqnarray}\displaystyle 4\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(\unicode[STIX]{x1D707}H\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}\right) & = & \displaystyle \left\{\begin{array}{@{}ll@{}}\displaystyle (1+SH)u^{m}+H\left(\frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}x}+\frac{\text{d}b}{\text{d}x}\right) & \text{if }H>d(x),\\ \displaystyle SHu^{m}+\unicode[STIX]{x1D6FF}\,H\frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}x} & \text{if }H<d(x),\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D707}=|\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x|^{m-1}$ , $\unicode[STIX]{x1D6FF}\equiv g^{\prime }/g\approx 0.1$ . The key dimensionless parameter is the dimensionless lateral shear drag coefficient

(2.25) $$\begin{eqnarray}\displaystyle S\equiv \frac{{\mathcal{H}}C_{+}}{wC_{-}}\equiv c\left(\frac{{\mathcal{L}}}{w}\right)^{(m+1)}, & & \displaystyle\end{eqnarray}$$

where $c\equiv 2(1+n/2)^{1/n}$ . The parameter $S$ sets the significance of lateral-drag stresses, with $S=0$ reducing the model to the one-dimensional, unbuttressed case. The parameter is affected by the channel half-width $w$ and bridges weak to strong lateral stresses. For the representative values of $C_{-}$ and $C_{+}$ given above and fluxes of $Q\sim 10^{-3}$ – $10^{-2}~\text{m}^{2}~\text{s}^{-1}$ , I estimate $S=0$ – $10^{-2}$ . The value of $S$ is small because the ratio of depth-integrated lateral drag to width-integrated basal drag is typically small. As noted above, lateral stresses in the ice shelf nonetheless have a major effect on the large-scale dynamics because of its specialised role in resisting flow across the grounding line: without it, the resistance to flow across the grounding line is exclusively reliant on the resistance to extensional stretching.

The input symmetry conditions at the ice divide, equations (2.16) and (2.17), and the stress condition of (2.15) become

(2.26) $$\begin{eqnarray}\displaystyle u & = & \displaystyle 0\hspace{21.60004pt}\text{at }x=x_{D},\end{eqnarray}$$

(2.27) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x} & = & \displaystyle 0\hspace{21.60004pt}\text{at }x=x_{D},\end{eqnarray}$$

(2.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x} & = & \displaystyle \frac{\unicode[STIX]{x1D6FF}}{8}H\quad \text{at }x=x_{C}.\end{eqnarray}$$

The evolution equation (2.18) becomes

(2.29) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}H}{\unicode[STIX]{x2202}t}=-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(Hu)+F(x,t), & & \displaystyle\end{eqnarray}$$

where $F(x,t)\equiv {\mathcal{L}}f(x,t)/Q$ is the dimensionless net accumulation.

Two further dimensionless parameters specify the linear bed profile,

(2.30) $$\begin{eqnarray}\displaystyle b(x)=-\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FC}x,\quad \text{where}~\unicode[STIX]{x1D6FC}\equiv ({\mathcal{L}}/{\mathcal{H}})a,\unicode[STIX]{x1D6FD}\equiv -b_{0}/{\mathcal{H}}, & & \displaystyle\end{eqnarray}$$

are a scaled bed slope and dimensionless terminal ocean depth, respectively. By considering sections of data for the depth of the Antarctic bedrock below sea level (Fretwell et al. Reference Fretwell and Pritchard2013), I estimate $|\unicode[STIX]{x1D6FC}|\approx 10^{-1}{-}10^{-3}$ and $\unicode[STIX]{x1D6FD}=O(10)$ .

For the choice of distribution (2.19), $F=0$ and (2.26) can be replaced by

(2.31) $$\begin{eqnarray}Hu=1\quad \text{at }x=x_{D+}.\end{eqnarray}$$

Figure 4.

Evolutions of a marine ice sheet as described by the numerical integration of (2.24)–(2.29) for Newtonian rheology $n=1$ and $x_{D}=-800$ . Case (a) represents a negative bed slope $\unicode[STIX]{x1D6FC}=-2\times 10^{-3}$ , a terminal ocean depth $\unicode[STIX]{x1D6FD}=2.8$ and a non-buttressing ice shelf, $S=0$ . Case (b) represents the same geometry but with a positive lateral-drag coefficient, $S=2\times 10^{-3}$ , producing a buttressing ice shelf. Case (c) shows a positive bed slope $\unicode[STIX]{x1D6FC}=2\times 10^{-3}$ and $\unicode[STIX]{x1D6FD}=1.4$ for an unconfined ice shelf, $S=0$ . Case (d) shows the corresponding buttressed case with $S=2\times 10^{-3}$ . In each panel, the dashed curve shows the initial state. The profiles are shown at times $t=10^{3},2\times 10^{3},4\times 10^{3},8\times 10^{3}$ and $16\times 10^{3}$ . The stable steady states in (a) and (b) and the front section of the unstable steady state in case (c), as predicted by (3.1a,b )–(3.2a,b ), are shown as dotted blue curves. No steady state exists in case (d) because of the invalidation of the only candidate steady state by secondary grounding, resulting in unconditional advance of the grounding line (a phenomenon to be discussed in § 6 and Pegler (Reference Pegler2018)). The right-hand plots show the grounding-line evolutions $x_{G}(t)$ , with the asymptotic steady states shown as horizontal dashed lines. In case (c), the evolution of $x_{G}(t)$ for a second solution initiated downstream of the unstable steady state is shown as a thinner curve. A blow-up showing the early-time retreat of the grounding line while the ice shelf forms is shown in case (d).

4.1 The extensional-resistance function $E$

For an unbuttressed marine ice sheet, the only resistance to flow across the grounding line is the stress due to longitudinal extension $E$ . In physical terms, the balancing of the depth-integrated hydrostatic pressure between the front of the grounded region and the hydrostatic pressure of the ocean is given by the viscous resistance to stretching the flow over a region very close to the front of the grounded region. For no ice-shelf buttressing, the flow across the grounding line is entirely resisted by the rate of stretching in this short section of the ice sheet. Similarly to a thread of honey falling from a spoon, the basal drag just upstream of this region acts to pin the ice to the bedrock. The control of the flow rate in front of the pinning point is then predominantly controlled by the resistance to extension. An unbuttressed steady-state grounding-line balance arises where the extensional resistance in this localised region balances the driving hydrostatic-pressure discontinuity between the ice sheet and the ocean, as represented by the right-hand side of (4.2). In considering how the extensional stress is generally controlled in steady-state balances, it is helpful to note first that the grounded region always assumes the same steady-state shape for any given value of $n$ (at least for the general situation representing the limits of $\unicode[STIX]{x1D6FC}\ll 1$ and $S\ll 1$ , which I will assume for the time being). The only degrees of freedom for variation in the grounded profile are the thickness at which it is truncated (i.e. the location of the grounding line), and the degree of horizontal translation, which are set by the stress condition (2.28) and the flotation condition, $H(x_{G},t)=d(x_{G})$ . A general relationship between the local extensional resistance $E$ and the ice-sheet thickness – applicable at any position in the grounded ice sheet including the grounding line – can be derived from a consideration of this ‘universal’ shape of the grounded region.

In order to determine this universal profile for $n=1$ for $S\ll 1$ and $b^{\prime }\ll 1$ , I integrate (3.1) forwards from an arbitrary thickness $H=H_{\infty }$ , chosen as $10$ . This forwards-marching integration attracts rapidly to the universal profile for any upstream condition on $H^{\prime }$ ( $H^{\prime }=0$ suffices, for example). The solution for $\unicode[STIX]{x1D6FC}=0$ , which is representative of shallow slopes, $\unicode[STIX]{x1D6FC}=|b^{\prime }|\ll 1$ , is shown in figure 5. The extensional resistance $E$ occurring at any given position in the steady state can now be ‘read off’ from this universal profile and related to the local thickness. To this end, I write $E$ as

(4.3) $$\begin{eqnarray}E[H(x)]=4\unicode[STIX]{x1D707}H\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}=-4\unicode[STIX]{x1D707}H^{-1}\frac{\text{d}H}{\text{d}x}=-4\unicode[STIX]{x1D707}H^{-1}\unicode[STIX]{x1D701}(H),\end{eqnarray}$$

where I have used the steady-state condition $u=1/H$ , and $\unicode[STIX]{x1D701}(H)=\text{d}H/\text{d}x(x)$ is the relationship between the thickness gradient and thickness determined from the universal profile. Using the relationship of $\unicode[STIX]{x1D701}(H)$ obtained above, I determine the extensional-resistance function $E=E[d(x_{G})]$ shown as a solid black curve in figure 6. This function can be interpreted as a database that returns the extensional stress at the grounding line for a steady state with any given grounding-line thickness $d(x_{G})$ . Its decrease with $d$ is consistent with the rate of extension of the ice sheet increasing as one move downstream. Each value of $n$ has a different extensional resistance function $E[d(x_{G})]$ . A more general expression for $E$ that includes a dependence on local basal slope and general $n$ is obtained in appendix A.

Figure 5.

The universal solution describing the thickness profile of the grounded region of the ice sheet. The grounded region of a marine ice sheet will be described by this solution subject to horizontal translation and truncation at a certain thickness controlled by the buttressing force. The leading-order solution for $n=1$ and shallow bed slopes $|b^{\prime }|\ll 1$ is illustrated here. It is obtained by solving (3.1) numerically by marching forwards until $H=0$ , a location set here as the reference position $x=0$ . The minimum thickness of truncation in the unbuttressed case occurs at thickness $d_{0}\approx 2.345$ , which is illustrated by a vertical dashed line (the region to right of this line is therefore impossible to arise in steady states). The truncating position for a buttressing force $B=2$ is also illustrated, corresponding to the case of figure 6. The solution is used to determine the general relationship between the thickness and the thickness gradient, $\unicode[STIX]{x1D701}[H(x)]=\text{d}H/\text{d}x(x)$ , of a steady grounded region, and, in turn, the extensional-resistance function (4.3).

Figure 6.

The extensional-resistance function $E(d)$ as a function of grounding-line thickness $d$ for $n=1$ , as calculated in § 4.1 (the solid black curve). Its analytical approximation (4.9) is shown as a dotted black curve. Its intersection with $(\unicode[STIX]{x1D6FF}/2)d^{2}-B$ yields the unbuttressed steady-state grounding-line thickness $d_{G}=d_{0}\approx 2.345$ for $B=0$ and the larger balancing value $d_{G}\approx 3.45>d_{0}$ for  $B=2$ .

With $E(d)$ determined, one can explore the predictions of the grounding-line balance (4.2) for different levels of buttressing $B$ . For zero buttressing, $B=0$ , equation (4.2) simplifies to

(4.4) $$\begin{eqnarray}E(d)={\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d^{2}.\end{eqnarray}$$

In this special situation, all the steady-state grounding-line forces depend purely on the grounding-line thickness $d$ . Therefore, there is a special balancing value of $d$ given by the solution of the algebraic equation (4.4). The individual variations of the sides of the balance as functions of $d$ are shown in figure 6, where $(\unicode[STIX]{x1D6FF}/2)d^{2}$ is shown as a blue curve and $E(d)$ as a black curve. Their intersection yields the critical balancing value $d_{G}=d_{0}\approx 2.345$ for $n=1$ , which universally describes the steady thickness at an unbuttressed grounding line (for $S\ll 1$ and $b^{\prime }\ll 1$ ). The dimensional expression of this result, namely $d/{\mathcal{H}}=d_{0}$ , represents the scaling between grounding-line flux $Q$ and thickness $d_{G}$ given by Schoof (Reference Schoof2007b ). For the bed profile (2.30), the solution of $d(x_{G})=d_{0}$ is given by $x_{G}=[\unicode[STIX]{x1D6FD}-(1-\unicode[STIX]{x1D6FF})d_{0}]/\unicode[STIX]{x1D6FC}$ , which is within 1 % of the numerically determined grounding-line position for the examples of figure 4(a,c). The ‘universal profile’ derived above applies for $|b^{\prime }|\ll 1$ and $S\ll 1$ . The estimates given in § 2.4 indicate that these parameter limits may be broadly relevant to many geophysical settings. Nevertheless, the analysis above can be repeated for $\unicode[STIX]{x1D6FC},S\neq 0$ to yield a more general form of the extensional-resistance function $E[d(x),\unicode[STIX]{x1D6FC},S]$ that can differ from $E[d(x),0,0]$ . For example, the first-order correction to the unbuttressed balance $E[d(x),\unicode[STIX]{x1D6FC},0]=(\unicode[STIX]{x1D6FF}/2)d^{2}$ for $n=1$ yields $d_{G}(\unicode[STIX]{x1D6FC})\sim d_{0}+2.78\,\unicode[STIX]{x1D6FC}$ , for $\unicode[STIX]{x1D6FC}\rightarrow 0$ . This represents a correction to the prefactor of the grounding-line flux formula of Schoof (Reference Schoof2007b ) owing to the local basal slope. For $|\unicode[STIX]{x1D6FC}|=|b^{\prime }|\lesssim 10^{-2}$ , the case $E[d(x),0,0]$ shown in figure 6 provides an excellent approximation.

Figure 7.

Panel (a) shows the general relationship for $n=1$ between the balancing steady-state grounding-line thickness $d_{G}$ and the applied buttressing force $B$ , as described by the algebraic equation (4.5a ) in which the extensional-resistance function $E(d)$ is represented using its numerical solution obtained in § 4.1. The plot shows the increase of the grounding-line thickness $d_{G}$ from the special unbuttressed thickness $d_{0}$ given by the solution of (4.4). The simplified relationship implied by the strong-buttressing limit of (4.6) is shown as a dashed blue curve. Panel (b) shows the corresponding relationship between the relative buttressing $\unicode[STIX]{x1D6FA}$ defined by (4.5b ) and $d_{G}$ .

We can deduce some essential effects of ice-shelf buttressing by considering the function $E(d)$ in conjunction with the steady-state grounding-line balance equation (4.2) with (for now) a specified buttressing force $B$ ,

(4.5a,b ) $$\begin{eqnarray}\displaystyle E(d)+B={\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d^{2}\quad \text{or}\quad \frac{E(d)}{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d^{2}}+\underbrace{\frac{B}{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d^{2}}}_{\unicode[STIX]{x1D6FA}}=1, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}$ is referred to as the relative buttressing. Application of a numerical root finder to (4.5a ) yields the general relationship between the grounding-line thickness $d_{G}$ and the buttressing force $B$ for $n=1$ shown as a solid curve in figure 7(a). The plot illustrates the increase of the grounding-line thickness $d_{G}$ from the unbuttressed value $d_{0}$ as the buttressing force $B$ is increased.

For sufficiently strong buttressing, equation (4.5a ) predicts a dominant grounding-line balance between the buttressing force and buoyancy alone, namely,

(4.6) $$\begin{eqnarray}B\approx {\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d^{2}.\end{eqnarray}$$

This balance represents the strong-buttressing control of a grounding line underlying the reduced nonlinear diffusive flow-line model of Pegler et al. (Reference Pegler, Kowal, Hasenclever and Worster2013). This balance represents a fundamentally different mode of grounding-line control compared to the unbuttressed balance (4.4). A simplified interpretation of the nature of this control as stemming directly from the structure of the ice shelf will be reserved for § 4.3. It follows that there is a complete transition from the ice shelf playing no role whatsoever for a one-dimensional marine ice sheet to playing a potentially completely dominant role for two-dimensional dynamics.

A key question for marine ice sheet dynamics is the proportion of the driving buoyancy drop $(\unicode[STIX]{x1D6FF}/2)d^{2}$ that is balanced by buttressing, as measured by $\unicode[STIX]{x1D6FA}$ . The general relationship between $\unicode[STIX]{x1D6FA}$ and $d_{G}$ , as derived from the relationship between $B$ and $d_{G}$ determined numerically above, is plotted in figure 7(b). The parameter $\unicode[STIX]{x1D6FA}$ describes a spectrum of balances bridging two limits: weak buttressing, $\unicode[STIX]{x1D6FA}\approx 0$ to strong buttressing $\unicode[STIX]{x1D6FA}\approx 1$ . It is related to a quantity $\unicode[STIX]{x1D6E9}\equiv (1-\unicode[STIX]{x1D6FA})$ used in the literature to impose a buttressing force as a multiplicative reduction of the driving buoyancy force [ $E(d)=(\unicode[STIX]{x1D6E9}\unicode[STIX]{x1D6FF}/2)d^{2}$ ] (e.g. Dupont & Alley Reference Dupont and Alley2005). The relationship of figure 7(b) gives a prediction for $\unicode[STIX]{x1D6FA}$ in terms of the dimensionless grounding-line thickness $d_{G}$ . In this case, for ${>}$ 90 % of the resistance to flow across the grounding line to stem from extension, $d_{G}$ must lie in the narrow range $d_{0}=2.35<d_{G}<2.40$ . For ${>}$ 90 % to stem from buttressing, $d_{G}>3.8$ .

4.1.1 The force-balance structure

Let the four forces comprising the governing force-balance equation be denoted according to

(4.7) $$\begin{eqnarray}\underbrace{4(H\unicode[STIX]{x1D707}u_{x})_{x}}_{E_{x}}-H\unicode[STIX]{x1D70F}_{s}-\unicode[STIX]{x1D70F}_{b}=\underbrace{Hh_{x}}_{G_{x}}.\end{eqnarray}$$

This section will focus on discussing the force balance specifically in the grounded region and hence $\unicode[STIX]{x1D70F}_{s}$ will be set as zero for the time being (its effects on force-balance structure are reserved for § 5). The three remaining forces are plotted for the final steady state for the unbuttressed example ( $\unicode[STIX]{x1D70F}_{s}\equiv 0$ ) shown earlier in figure 4(a). Here, the divergence of the depth-integrated extensional stress $E_{x}$ is shown as a solid curve, the basal stress $\unicode[STIX]{x1D70F}_{b}$ is shown as a dashed curve and the gravitational term $G_{x}$ as a dotted curve. Upstream, the divergence of extensional stresses $E_{x}$ is small and hence the flow is dominated by a balance between the total drag and gravity, $\unicode[STIX]{x1D70F}_{b}\sim G_{x}$ , for which (2.24a ) reduces to the simplified flow equation

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{b}\approx -H\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

This drag-dominated balance is mathematically similar to the balance associated with the shallow-ice-approximation, which refers specifically to a leading-order balance between vertical shear stresses and gravity (Fowler & Larson Reference Fowler and Larson1978; Cuffey & Paterson Reference Cuffey and Paterson2010). Towards the grounding line, $E_{x}$ increases very slightly, but remains small. Therefore the approximation of (4.8) holds all the way up to and including the grounding line. Across the grounding line, basal stress vanishes abruptly, $\unicode[STIX]{x1D70F}_{b}\equiv 0$ and $E_{x}\equiv G_{x}$ for $x>x_{G}$ . The steady-state grounding-line position $x_{G}$ is set by where the extensional stress $E$ balances the buoyancy force in accordance with (4.4). It is therefore interesting that its divergence $E_{x}$ remains uniformly small throughout the grounded region. As a consequence, there is no boundary layer through which the basal-drag-dominated approximation of (4.8) breaks down. The lack of this boundary layer can be attributed to the smallness of the dimensionless density difference $\unicode[STIX]{x1D6FF}\approx 0.1$ , which appears as a prefactor to the gravitational term in (2.28) but not in (2.24a ). As a consequence, $E_{x}=O(\unicode[STIX]{x1D6FF}E)$ at the location where the grounding-line balance of (2.28) is satisfied, and is therefore small. By rewriting (2.28) as $H=8\unicode[STIX]{x1D6FF}^{-1}\unicode[STIX]{x1D707}\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x$ , it is explicitly seen that (2.28) is attained for a relatively larger thickness $H$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ . In other words, the universal profile of the grounded region shown in figure 5 is truncated further upstream for smaller $\unicode[STIX]{x1D6FF}$ . The geophysical value of $\unicode[STIX]{x1D6FF}=0.1$ is sufficiently small that the grounding line occurs at a thickness where $E_{x}$ is still small (the value of $E_{x}$ only becomes important in the ‘hypothetical region’ residing to the right of the vertical dashed line in figure 5). The limit of $\unicode[STIX]{x1D6FF}\rightarrow 0$ thus represents an asymptotic condition under which the basal-drag-dominated zone can be matched to the extensional ice shelf formally, and to excellent approximation, using a direct patching condition. In this regard, there is no boundary layer through which terms in the governing equation (2.14a ) change their order of magnitude. It should be emphasised that the notion of an extensional boundary layer discussed here is conceptually distinct from the notion of a boundary layer described in § 2.3, which is used only to neglect accumulation near the grounding line.

Figure 8.

The profiles of forces comprising the force-balance equation (4.7) along the length of an unbuttressed marine ice sheet, $S=0$ . The divergence of extensional stresses $E_{x}$ is shown as solid black curve, basal stress $\unicode[STIX]{x1D70F}_{b}$ is shown as a dotted blue curve and gravity $G_{x}$ is shown as a dashed black curve. Panel (a) illustrates the satisfaction of the basal-drag-dominated balance ( $\unicode[STIX]{x1D70F}_{b}\approx G_{x}$ ) given by (4.8) for $x<x_{G}$ . The enlargement of (b) shows the jump to an extension-dominated ice shelf ( $E_{x}\equiv G_{x}$ ).

It will be verified later in § 5.1 that the force balance throughout the grounded region is qualitatively similar with buttressing. Indeed, buttressing strengthens the drag-dominated approximation (4.8) even more because it causes the universal profile to be truncated at a yet thicker grounding line $d_{G}>d_{0}$ , as illustrated for $B=2$ in figure 5. However, lateral stresses fundamentally complicate the force-balance structure of the ice shelf (Pegler Reference Pegler2016), an aspect reserved for § 5.

4.1.2 Analytical approximation for $E$

By using (4.8) to evaluate the thickness-gradient function $\unicode[STIX]{x1D701}(H)$ in (4.3), one obtains the analytical approximation for the extensional-resistance function

(4.9) $$\begin{eqnarray}E(d)\approx 4d^{-3},\end{eqnarray}$$

given here for $n=1$ . This approximation is plotted alongside the numerical solution in figure 6 as a dotted curve. The generalisation of this result to other values of $n$ , and an effect of local slope $b^{\prime }$ , is obtained in appendix A. The approximation provides an excellent representation of the numerically determined curve. To illustrate its accuracy, note that a substitution of (4.9) into (4.4) and rearrangement for $d_{0}$ yields $d_{0}\approx (8/\unicode[STIX]{x1D6FF})^{1/5}$ , which is within 15 % of the exact value. The corresponding result for $n=3$ is within a mere 0.1 % of the exact result (see appendix A). Thus, the neglect of the divergence of extensional stresses, which is responsible for long-range communication of forces in the grounded region of an ice sheet, has practically no effect on the predictions of the SSA model (at least for steady or near-steady grounding-line balances).

4.2 The buttressing resistance function $B$

This section determines the steady-state buttressing force $B$ in terms of the parameters specifying the ice-shelf dynamics. In steady state, the integral form of the buttressing force defined in (4.1) is given by

(4.10) $$\begin{eqnarray}B[x_{G},x_{C},d_{G}]=\int _{x_{G}}^{x_{C}}H\unicode[STIX]{x1D70F}_{s}(u)\,\text{d}x=S\int _{x_{G}}^{x_{C}}Hu^{m}\,\text{d}x,\end{eqnarray}$$

where $u(x)$ is the solution to the steady-state system describing the ice shelf (3.1b ) integrated subject to a ‘prescribed’ grounding-line thickness $d_{G}$ and the frontal stress condition (3.2a ), namely,

(4.11a,b ) $$\begin{eqnarray}\displaystyle u(x_{G})=d_{G}^{-1},\quad u^{\prime }(x_{C})=[\unicode[STIX]{x1D6FF}/8u(x_{C})]^{1/m}. & & \displaystyle\end{eqnarray}$$

The system defined by (3.1) and (4.11a,b ) depends on $x_{G}$ , $x_{C}$ and $d_{G}$ , and hence the steady-state buttressing force (4.10) is a function of these variables alone. The function $B[x_{G},x_{C},d_{G}]$ can be interpreted as a ‘database’ giving the steady-state buttressing force for any given $x_{G}$ , $x_{C}$ and $d_{G}$ . A similar method of constructing database functions for steady-state buttressing forces was developed for a radially spreading ice shelf (Pegler & Worster Reference Pegler and Worster2012, Reference Pegler and Worster2013), for which a reduction to an algebraic system analogous to (4.2) was obtained.

The translational invariance of the system defined by (3.1) and (4.11a,b ) implies that the dependences of $B$ on $x_{G}$ and $x_{C}$ can be combined into a single dependence on shelf length, $L\equiv x_{C}-x_{G}$ . Integrating (3.1b ) subject to (4.11a,b ) numerically using a scheme similar to that described earlier in the text below (3.1) and evaluating the integral of (4.10) using quadrature over a range of $L$ , I determine the general relationship between $B$ and shelf length $L$ for two illustrative fixed values of $d_{G}=4$ and 16 in figure 9. The numerical similarity of the two curves indicates a very weak dependence of buttressing on grounding-line thickness $d_{G}$ . In other words, for each value of $n$ , there is a near-universal relationship between buttressing and shelf length spanning fully the limit of a short, weakly buttressing ice shelf to a long, strongly buttressing ice shelf.

Figure 9.

The relationship between the buttressing force (4.10) and shelf length $L\equiv x_{C}-x_{G}$ for grounding-line thicknesses of $d_{G}=4$ (solid) and 16 (dashed) and power-law exponent $n=3$ obtained by repeatedly solving (3.1) and (4.11) numerically and evaluating the integral in (4.10) using quadrature. The analytical approximation (4.13) is shown as a dotted blue curve, illustrating its excellent representation of the buttressing force.

For $n=1$ , equation (4.10) can be integrated directly to give (Pegler Reference Pegler2016)

(4.12) $$\begin{eqnarray}B=S(x_{C}-x_{G}),\end{eqnarray}$$

showing that, for linear rheology, the buttressing force is directly proportional to the length of the ice shelf. The possibility of this simple integration applies uniquely for $n=1$ because the integrand of (4.10) is uniformly proportional to the width-averaged flux $Hu=1$ . Expression (4.12) implies that there is no dependence of the steady-state buttressing force for $n=1$ on the grounding-line thickness $d_{G}$ .

For $n>1$ , one can develop the generalised analytical approximation

(4.13) $$\begin{eqnarray}B\approx \frac{\unicode[STIX]{x1D6FF}}{2}\left\{\left[\frac{NS}{\unicode[STIX]{x1D6FF}}(x_{C}-x_{G})+H_{C}^{N}\right]^{2/N}-H_{C}^{2}\right\},\end{eqnarray}$$

where $N\equiv (n+1)/n$ , $H_{C}\approx 8^{1/N^{2}}\unicode[STIX]{x1D702}$ is the calving thickness and $\unicode[STIX]{x1D702}\equiv \unicode[STIX]{x1D6FF}^{-1/N}S^{1/nN^{2}}$ . This excellent approximation follows from substitution of the analytical solution for the prevailing interior of the ice shelf derived in Pegler (Reference Pegler2016) into the buttressing integral (4.10), as detailed in appendix A. The result of (4.13) is shown as a dotted blue curve in figure 9 and closely approximates the two numerical curves. The lack of a dependence of (4.13) on $d_{G}$ corroborates the very weak dependence of the numerical solutions on $d_{G}$ . The weak dependence occurs because $d_{G}$ only affects flow in the transitional boundary layer in front of the grounding line, which provides a relatively small contribution to the buttressing integral (4.10). For $n>1$ , equation (4.13) shows that a shear-thinning rheology causes $B$ to depend nonlinearly on the shelf length $(x_{C}-x_{G})$ and introduces a dependence on the calving thickness $H_{C}$ (a known quantity that is determined as part of the analytical solution of the universal profile (A 4)).

Figure 10.

Panels (a–c) illustrate the profile of the marine ice sheet as the dimensionless lateral-drag coefficient $S$ is increased through 0, $2\times 10^{-3}$ and $10^{-2}$ , respectively, for topography defined by $\unicode[STIX]{x1D6FC}=-2\times 10^{-3}$ and $\unicode[STIX]{x1D6FD}=2.8$ . The plot illustrates the change in form of the ice sheet from zero buttressing to positive buttressing. Panel (d) shows the corresponding grounding-line position $x_{G}$ across this range. The strong-buttressing prediction of (4.6) is shown as a curve of blue circles. Panel (e) shows the corresponding variation of the relative buttressing parameter $\unicode[STIX]{x1D6FA}$ defined by (4.5). The threshold (4.15) for large $S$ is shown as a horizontal dashed line.

Net accumulation in the ice shelf has a direct dynamical effect via its control of ice-shelf buttressing. By integrating the steady-state form of (2.18), one obtains the non-uniform flux per unit width $q(x)=Hu=1+\int _{x_{G}}^{x}F(x^{\prime })\,\text{d}x$ . With this, the generalisation of (4.13), derived in appendix A, is

(4.14) $$\begin{eqnarray}B\approx \frac{\unicode[STIX]{x1D6FF}}{2}\left\{\left[\frac{N}{\unicode[STIX]{x1D6FF}}\int _{x_{G}}^{x_{C}}Sq(x^{\prime })^{(1/n)}\,\text{d}x^{\prime }+H_{C}^{N}\right]^{2/N}-H_{C}^{2}\right\},\end{eqnarray}$$

which relates the distribution of melting $F(x)$ to the buttressing force. This expression embodies the key link between the oceanographic control of melting (e.g. Jenkins Reference Jenkins1991) and glacial dynamics. The result shows that buttressing generally depends on the integral of the flux along the ice shelf $q(x)$ . An interesting implication is that melting near the grounding line, which can be the most significant along the ice shelf, has a larger effect on the buttressing than melting further downstream because of the longer region over which it reduces the flux of the ice shelf.

4.3 Dynamical variation across the spectrum of relative buttressing strength

When the function (4.12) (or one of its generalisations, (4.13) or (4.14)) is incorporated into (4.2) along with $E[d(x)]$ , the full equation provides a closed algebraic equation for the position of steady-state grounding lines $x_{G}$ . The equation bridges the complete spectrum connecting unbuttressed to strongly buttressed grounding lines. This equation incorporates ice-shelf buttressing predictively, contrasting with a direct prescription of $\unicode[STIX]{x1D6E9}$ (e.g. Dupont & Alley Reference Dupont and Alley2005).

To demonstrate the application of the balance equation (4.2), I solve it for $x_{G}$ for the illustrative example of the negative slope $\unicode[STIX]{x1D6FC}=-2\times 10^{-3}$ and reference ocean depth $\unicode[STIX]{x1D6FD}=2.8$ . The predicted grounding-line position $x_{G}$ is plotted as a function of the dimensionless lateral-drag coefficient $S$ in figure 10(d), showing the positioning of the grounding line in deeper water for larger $S$ . Panels (a–c) illustrate the flow profile of a marine ice sheet as $S$ is increased, showing the transition from the broadly concave ice-shelf profile of a non-buttressing ice shelf (van der Veen Reference van der Veen1983) to the more wedge-shaped profile of a buttressing ice shelf (Pegler Reference Pegler2016). The corresponding relative buttressing $\unicode[STIX]{x1D6FA}$ is shown in figure 10(b). The plot shows a sharp increase in the relative buttressing with $S$ towards the regime of strong buttressing. As $S$ is increased, $\unicode[STIX]{x1D6FA}$ asymptotes towards the value 0.7 shown as a horizontal dashed line. Perhaps surprisingly, this value is less than unity. This upper bound on the relative buttressing arises because the buttressing cannot balance the hydrostatic pressure drop between the calving front and the ocean immediately in front of it. This property will be discussed further in the text below (5.4). The asymptotic value of $\unicode[STIX]{x1D6FA}$ is obtained by substituting the flotation depth of the ocean at the calving front, $d_{C}\equiv \unicode[STIX]{x1D6FD}/(1-\unicode[STIX]{x1D6FF})$ , into (4.5b ) to determine that

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}\leqslant \unicode[STIX]{x1D6FA}_{C}=1-\frac{E(d_{C})}{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}d_{C}^{2}}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FD}=2.8$ , $\unicode[STIX]{x1D6FA}_{C}\approx 0.7$ , in agreement with the asymptotic behaviour of $\unicode[STIX]{x1D6FA}$ determined numerically. The large- $S$ solution can be approximated by the strong-buttressing limiting balance of (4.6), which is shown as a curve of blue circles in figure 10(d).

As $S$ is increased, there is a fundamental switch in the dominant forces controlling the grounding line. To illustrate this, I express the limiting end-member balances predicted by (4.2) in their dimensional forms. For $n=1$ , these are given by

(4.16a,b ) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}E(d)=\displaystyle \frac{4\unicode[STIX]{x1D707}_{0}Q^{2}C_{-}}{\unicode[STIX]{x1D70C}gd(x)^{3}}\sim \displaystyle \frac{\unicode[STIX]{x1D70C}g^{\prime }d(x)^{2}}{2}\quad (\unicode[STIX]{x1D6FA}=0),\\ B(x_{C}-x_{G})=\displaystyle \frac{3\unicode[STIX]{x1D707}_{0}Q}{w^{2}}(x_{C}-x_{G})\sim \displaystyle \frac{\unicode[STIX]{x1D70C}g^{\prime }d(x)^{2}}{2}\quad (\unicode[STIX]{x1D6FA}\approx 1).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The extension-dominated unbuttressed balance of (4.16a ) recovers the expression for $Q$ as a function of grounding-line thickness $d$ given by Schoof (Reference Schoof2007b ). This first result shows that an unbuttressed grounding-line position is influenced by the viscosity of the ice $\unicode[STIX]{x1D707}_{0}$ and the basal-drag coefficient $C_{-}$ . In this case, the balance can be interpreted as an expression for the grounding-line thickness, as represented by the special dimensionless thickness $d_{0}$ discussed above. The pure dependence of (4.16a ) on the single dependent variable $d(x)$ is consistent with the simplified situation of an unbuttressed grounding being controlled directly by the local grounding-line thickness. The strong-buttressing expression of (4.16b ) represents a completely different physical control of the grounding-line position dominated instead by the ice-shelf dynamics. This includes an inherent dependence on the calving position $x_{C}$ and width of the ice shelf $w$ , and a loss of dependence on the basal drag coefficient $C_{-}$ . Thus, there is a full switch in dynamical control from the calving position playing no role whatsoever for a one-dimensional or non-buttressing marine ice sheet to playing a dominant control under the effect of even relatively weak lateral stresses on the ice shelf.

It should be noted that the application of the augmented expression for grounding-line flux in a form that incorporates the multiplicative reduction in the hydrostatic-pressure drop owing to buttressing $\unicode[STIX]{x1D6E9}$ (Schoof Reference Schoof2007b ), and used in simulations (e.g. DeConto & Pollard Reference DeConto and Pollard2016), becomes degenerate for sufficiently strong buttressing. The general balance equation in the form (4.2) will seamlessly incorporate both unbuttressed and strongly buttressed situations.

Figure 11.

Suites of steady-state profiles of a marine ice sheet for (a) zero lateral stress, $S=0$ , and (b) a positive lateral stress, $S=2\times 10^{-3}$ shown as the calving position $x_{C}$ is varied. The plot illustrates the complete independence of the grounding line from the calving position $x_{C}$ for case (a) and the considerable sensitivity to it in case (b). Overlaid as a red dotted curve in (b) is a solution with the same parameters as the case $x_{C}=400$ but with the basal stress set identically equal to zero to leave just the lateral drag, illustrating the effective independence of the grounding line from the nature of the drag force in the grounded region.

The key switch in parametric dependence from the ice shelf playing a non-existent role to a dominant role is illustrated explicitly by the variation of the steady-state solutions for a selection of calving positions $x_{C}$ shown in figure 11 for (a) $S=0$ and (b) $S=2\times 10^{-3}$ . For the unbuttressed case (a), there is no change in the grounding-line position $x_{G}$ . In the buttressing case (b), a considerable change in grounding-line position occurs. As the calving position is varied, the change in grounding-line position is in fact larger than the change in the ice-shelf length. In this same plot, I illustrate as a red dotted curve a solution in which the basal stress, given by the third term in (3.1a ) is set identically equal to zero for the case $x_{C}=400$ . The grounding-line position is essentially unaffected despite the complete removal of basal stress. The grounding line is therefore being controlled completely by the ice shelf.

5.1 Effect of lateral stresses on the structure of the ice sheet

To illustrate the force-balance structure with lateral stresses, I show the profiles of the four individual forces comprising the continuum balance (4.7) for the illustrative case of $S=2\times 10^{-3}$ in figure 12, where the lateral stress $\unicode[STIX]{x1D70F}_{s}$ is indicated by a red dashed curve, which is horizontal for $x>x_{G}$ . In common with the unbuttressed case shown earlier in figure 8, the grounded region is basal-drag-dominated, with the divergence of extensional stress $E_{x}$ being negligible along its full length. However, the ice shelf forms two distinct regions: the ‘input boundary layer’ in which extensional stresses are important and the prevailing ‘universal profile’ of the ice shelf further downstream (Pegler Reference Pegler2016) in which the transverse shear stresses provide the dominant resistance to flow.

Figure 12.

Panel (a) shows the numerical solution for $\unicode[STIX]{x1D6FD}=2.8$ , $\unicode[STIX]{x1D6FC}=2\times 10^{-3}$ and $S=2\times 10^{-3}$ , and illustrates the three-component structure of a buttressed marine ice sheet. The profiles of forces, as defined in (4.7), are shown in (b). Here, the divergence of extensional stresses $E_{x}$ is shown as solid black curve, basal stress $\unicode[STIX]{x1D70F}_{b}$ is shown as a dotted blue curve, the dimensionless depth-integrated lateral stress $H\unicode[STIX]{x1D70F}_{s}$ is shown as a dotted red curve and the gravitational force $G_{x}$ is shown as a dashed black curve. The plots illustrate the basal-drag-dominated grounded balance ( $\unicode[STIX]{x1D70F}_{b}\approx G_{x}$ ), the extensional inlet boundary layer in front of the grounding line, $x_{G}<x<x_{G}+\unicode[STIX]{x1D6E5}$ , and the lateral-drag-dominated region downstream ( $H\unicode[STIX]{x1D70F}_{s}\approx G_{x}$ ). The enlargement given in (c) shows the extensional balance ( $E_{x}\approx G_{x}$ ) in the inlet boundary layer.

To analyse this asymptotic structure, I first recall the general analytical solution describing the confined ice shelf for $n=1$ given by solving (3.1) subject to (4.11a,b ), namely,

(5.1) $$\begin{eqnarray}H(x)=\unicode[STIX]{x1D702}\left[\text{e}^{\tilde{x}^{2}/4}\left(\tilde{d}_{G}^{-2}\text{e}^{-\tilde{x}_{G}/4}+\frac{1}{2}\int _{-\tilde{x}/2}^{-\tilde{x}_{G}/2}\text{e}^{-\unicode[STIX]{x1D709}^{2}}\,\text{d}\unicode[STIX]{x1D709}\right)\right]^{-1/2},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}\equiv (S/\unicode[STIX]{x1D6FF}^{2})^{1/4}$ , $(\tilde{x},\tilde{x}_{G})\equiv S^{1/2}(x,x_{G})$ , $\tilde{d}_{G}\equiv \unicode[STIX]{x1D702}^{-1}d_{G}$ (Pegler Reference Pegler2016). The solution (5.1) is illustrated for two shelf lengths $L=100$ and 500 in figure 13(b,c). The solution for $L=100$ clearly illustrates the inlet boundary layer of rapid thinning. The prevailing downstream region is independent of the grounding-line thickness $d_{G}$ and dominantly controlled by the calving position $x_{C}$ . By considering the asymptotic structure of (5.1), one can determine the solution for the prevailing downstream region,

(5.2) $$\begin{eqnarray}H_{U}(x)=\unicode[STIX]{x1D702}\left[{\textstyle \frac{1}{4}}\sqrt{\unicode[STIX]{x03C0}}\,\text{erfcx}\,(-{\textstyle \frac{1}{2}}\tilde{x})\right]^{-1/2}\quad (\unicode[STIX]{x1D6E5}\lesssim x\leqslant x_{C}),\end{eqnarray}$$

where $\text{erfcx}\,\unicode[STIX]{x1D701}\equiv \text{e}^{\unicode[STIX]{x1D701}^{2}}(1-\text{erf}\,\unicode[STIX]{x1D701})$ and $\unicode[STIX]{x1D6E5}$ is the length of the inlet boundary layer, which varies depending on the solution (Pegler Reference Pegler2016). The downstream solution (5.2) is shown as a dashed blue curve in figure 13(b,c) and is confirmed to describe the prevailing shape. The corresponding analytical result for power-law fluid (Pegler Reference Pegler2016) is reviewed in appendix A. If the ‘matching thickness’ of the universal profile (5.2) extrapolated to the grounding line, namely $H_{+}\equiv H_{U}(x_{G})$ , is less than the grounding-line thickness, $H_{+}<d_{G}$ , then the flow through the inlet boundary layer is characterised by thinning and acceleration. This situation, referred to as ‘over-thick’, corresponds to the profile of an ice shelf typically observed in nature. On this basis, it was hypothesised in Pegler (Reference Pegler2016) that the predictions of a full marine ice sheet model for steady-state balances should only produce this situation. If instead, $H_{+}>d_{G}$ , then the inlet involves thickening and deceleration, and was referred to as ‘under-thick’. An under-thick input is a theoretical possibility only if one has full control over the specification of the inlet thickness and position, and may never arise in the glaciological context. However, for input configurations involving the direct input at an imposed position or along a back wall, an under-thick flow is possible; this situation was observed in the laboratory experiments of Pegler (Reference Pegler2016) and may occur in float-glass manufacture.

Figure 13.

Panel (a) shows the steady-state grounding-line thickness $d_{G}$ as a function of the shelf length $L=x_{C}-x_{G}$ for $S=2\times 10^{-3}$ , compared alongside the value of the universal profile (5.2) extrapolated to the input $H_{+}\equiv H_{U}(x_{G})$ . It is found that $d_{G}>H_{+}$ , implying that only over-thick regimes of input occur for marine ice sheets. The vertical dashed line indicates the characteristic length $2S^{-1/2}\approx 44$ above which the distinction between the universal profile and the inlet boundary layer is clearly developed. The inset shows the relative buttressing parameter $\unicode[STIX]{x1D6FA}$ defined by (4.5) plotted as a function of shelf length and the associated approximation based on (5.4). Panels (b,c) show the analytical solution (5.1) (solid black curve) and the universal profile (5.2) (dashed, blue curves) for $L=100$ and 500, respectively.

To illustrate the variation in the ice-shelf structure as the shelf length $L$ is increased from zero, I show the grounding-line thickness $d_{G}$ predicted by the grounding-line balance equation (4.2) alongside the matching thickness $H_{+}$ predicted by (5.2) against $L$ for $S=2\times 10^{-3}$ in figure 13(a). The results show that $d_{G}>H_{+}$ , confirming the hypothesis that only over-thick input boundary layers can arise in the glacial context (at least in steady state). It should be noted that the inlet boundary layer is clearly defined only for a sufficiently long ice shelf, $L\gtrsim S^{-1/2}\hat{L}\approx 44$ , a value indicated by the vertical dashed line in figure 13(a) (this corresponds to the length of the ice shelf being greater than approximately one third of its width). For sufficiently large $L$ , figure 13(a) shows that the extrapolated and grounding-line thicknesses become similar, $H_{+}\approx d_{G}$ . The grounding line then occurs to excellent approximation where (5.2) intersects the bedrock, as illustrated in figure 13(c), implying that the grounding line is being controlled independently by the lateral-drag-dominated section of the ice shelf (cf. Pegler et al. Reference Pegler, Kowal, Hasenclever and Worster2013).

5.2 Relationship between the buttressing force and the structure of the ice shelf

Analogously to the simplifications arising from the basal-drag-dominated approximation in the grounded region (4.8), analytical results can be determined using the uniform smallness of $E_{x}$ throughout the component of the ice shelf downstream of the inlet boundary layer. Similarly to that case, the divergence of extensional stresses $E_{x}$ remains small throughout the flow downstream of the input boundary layer, as shown in figure 12. It was noted in § 4.1.2 that this could be justified on the basis of the smallness of $\unicode[STIX]{x1D6FF}$ . However, in the purely floating context there is no parameter limit analogous to $\unicode[STIX]{x1D6FF}\rightarrow 0$ to justify the uniform smallness of $E_{x}$ . Despite this, the numerical value of $E_{x}\approx 10^{-2}E$ at $x=x_{C}$ for $n=3$ , is verifiably small (in a sense, ‘ $\unicode[STIX]{x1D6FF}=1$ ’ is small enough). Thus, the analytical approximations for the universal profiles resulting from the neglect of $E_{x}$ , as derived in Pegler (Reference Pegler2016) and reviewed by (A 4) in appendix A, provide practically exact descriptions of the prevailing region of the ice shelf downstream of the inlet boundary layer (there is a maximum error of just 2 %).

The neglect of $E_{x}$ in (2.24b ) yields the lateral-drag-dominated flow approximation

(5.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{s}\approx -\unicode[STIX]{x1D6FF}H^{\prime },\end{eqnarray}$$

which is the floating analogue of (4.8). Since this approximation applies uniformly downstream of the inlet boundary layer (Pegler Reference Pegler2016), one can evaluate (4.13) as

(5.4) $$\begin{eqnarray}B\approx \int _{x_{G}+\unicode[STIX]{x1D6E5}}^{x_{C}}H\unicode[STIX]{x1D70F}_{s}\,\text{d}x\approx \left[-\frac{1}{2}\unicode[STIX]{x1D6FF}H^{2}\right]_{x_{G}+\unicode[STIX]{x1D6E5}}^{x_{C}}\approx \frac{1}{2}\unicode[STIX]{x1D6FF}[H_{U}(x_{G})]^{2}-\frac{1}{2}\unicode[STIX]{x1D6FF}H_{C}^{2}\end{eqnarray}$$

on neglecting the contribution over $[x_{G},x_{G}+\unicode[STIX]{x1D6E5}]$ . The result of (5.4) shows that the buttressing force can be related straightforwardly to the difference in depth-integrated hydrostatic pressure between the ends of the lateral-drag-dominated region of the ice shelf. The function $H_{U}(x)$ used to determine these thicknesses is given by the analytical solution for the drag-dominated region (A 4). The implied relative buttressing, $\unicode[STIX]{x1D6FA}=d_{G}^{-2}(H_{+}^{2}-H_{C}^{2})$ , where $H_{+}\equiv H_{U}(x_{G})$ is illustrated by the green dashed curve in the inset of figure 13(a) and provides an excellent approximation. The result shows that the relative significance of buttressing can be linked to the three thicknesses $d_{G}$ , $H_{+}$ and $H_{C}$ defining the ice-shelf structure. In addition to providing an insightful representation of the buttressing force, equation (5.4) also provides a helpful analytical shorthand for the evaluation of $B$ once $H_{U}(x)$ has been determined (this was employed in evaluating (5.4), as discussed in appendix A). The effects of rheology, calving position and flux variations due to melting all encapsulate within the solution for $H_{U}(x)$ given analytically by (A 4). A simple physical interpretation of the effect of the first term on the right-hand side of (5.4) is that the thickness at the start of the lateral-drag-dominated region, $H_{U}$ ‘blocks’ a certain proportion of the driving buoyancy force at the grounding line, $(\unicode[STIX]{x1D6FF}/2)d_{G}^{2}$ . In a sense, the residual between the driving hydrostatic stress and the ‘blocking thickness’ $H_{U}$ represents the remaining effective hydrostatic-pressure drop available to drive flow across the grounding line. The second term in (5.4) represents a further negative contribution to the buttressing stemming from the hydrostatic-pressure drop at the calving front. This term can be interpreted as a contribution to the overall depth-integrated pressure drop that remains unresisted by the lateral stresses in the ice shelf. Thus, in view of the fact that the blocking thickness cannot exceed the grounding-line thickness, $H_{U}<d_{G}$ , and a calving front $H_{C}$ typically has an appreciable non-zero thickness (in accordance with observations and the analytical result of (A 4)), it follows that a buttressing force can never fully balance the hydrostatic-pressure drop at the grounding line completely.

7.1 Steady states and the removal of hysteresis

The first step is to determine the solutions to the grounding-line equation (4.2) with the expressions for the extensional-resistance function (A 1) and the buttressing function (4.13), for $x_{G}$ . Solving this algebraic equation using a numerical root finder, I obtain the steady-state grounding-line position $x_{G}$ as a function of $S$ shown as a thick solid curve in figure 17(a). With the candidate steady states determined, the second step is to determine which are rendered inconsistent by secondary grounding. To determine the locus of critical grounding-line positions for which secondary grounding occurs, I apply the numerical method outlined below (6.1). The grounding-line positions that would be affected by secondary grounded are overlaid on figure 17(a) in blue, with types I and II shown by the lighter and darker shades, respectively. The invalidated steady states are shown as dotted curves extended into the blue region.

For the unconfined case, $S=0$ , equation (4.2) reduces to $d(x)=d_{0}=7.96$ . The solutions to this equation represent the intersections between $b(x)$ and the line $z=-d_{0}(1-\unicode[STIX]{x1D6FF})$ , shown in figure 17(b). There are a total of four such states. As $S$ is increased, the first effect of lateral stresses is to cause the unbuttressed steady states on positive bed slopes to move backwards and those on negative bed slopes to move forwards. Another effect is to introduce a fifth steady state far upstream (not shown). At certain critical values of $S$ , certain pairs of steady states meet and ‘annihilate’ one another. The annihilations represent saddle-node bifurcations, which are a feature of dynamical systems with multiple steady states. The solutions near the bifurcation points in this example are invalidated by type-II secondary grounding causing an intersection between the ice shelf and one of the topographic maxima. The invalidations occur sharply at critical values given by $S=4.6\times 10^{-5}$ and $S=1.4\times 10^{-4}$ . For values of $S>1.4\times 10^{-4}$ , there is just one steady state remaining.

The existence of multiple steady states (there can be a total of five in the example above) implies the possibility for hysteresis effects, where the system will be attracted to a different state depending on its initial condition. For example, if a change in climate were to induce a reduction in ice flux $Q$ , a grounding-line retreat can be stimulated. If the flux were to reinstate in future then, if the grounding line has migrated upstream of an unstable steady state, then it will not return to its former state. Instead, it will continue to retreat upstream, potentially leading to either irreversible runaway retreat or stabilisation towards a new position further upstream. The steady-state plot of figure 17 shows that, with just one stable state remaining for $S>1.4\times 10^{-4}$ , lateral stresses have the effect of removing the possibility for hysteresis. Above this value, the system converges unconditionally towards this stable steady state. Above this threshold, the possibility for hysteresis has been eliminated. A general analysis of the conditions necessary to maintain the global stability of a marine ice sheet under the effects of ice-shelf buttressing and secondary grounding forms the subject of the companion paper (Pegler Reference Pegler2018).

7.2 Steady secondary contact

An ice rise or rumple represents a localised region of contact between an ice shelf and the bedrock downstream of the primary grounding line. The position of the primary grounding line and the ice rumple for such a state are not predicted by (4.1) because of the underlying assumption that the ice shelf forms a single region of flotation. To inspect the possibility of states that sustain a region of secondary grounding in steady state, I conducted a comprehensive search of the solutions to the full steady-state equation (3.1). This was achieved using a method of shooting over the thickness upstream, and iterating it bisectively until the frontal stress condition (3.2) is adequately satisfied. Unlike the solutions based on (4.2) alone, this general method of solution describes several additional solutions involving two disjoint regions of flotation. The additional branches of steady states are shown as red curves in figure 17(a). The additional solutions initially appear to continue the branch of solutions of the steady states with a fully floating ice shelf for values of $S$ above the critical values at which secondary grounding switches on, before abruptly turning back at saddle-node bifurcations. New solutions involving a secondary contact can occur for values of $S$ both above the threshold at which secondary grounding is predicted and below it. The profile of such a state, occurring at the blue circle in figure 17(a), is shown in figure 17(c). The solution exhibits a brief ‘glancing’ contact with the bedrock, which can be interpreted as an ice rumple. Thus, multiple additional steady states can be produced by allowing for a secondary contact. As shown in figure 17(c), the contacts create a sharp jump in the thickness gradient of the ice shelf between the start and end of the secondary region of contact, implying a considerable impact of the contact on the morphology of the ice shelf. This includes a considerable thickening of the region of the ice shelf between the point of secondary contact and the grounding line. Assuming an alternating pattern between stable and unstable states, I anticipate that a subset of these additional states are stable and the others unstable. By contributing to a larger interior thickness, lateral stresses can thus induce the development of an ice rise or rumple. A dedicated analysis could be conducted to investigate the local stability of these additional states and is left for future consideration.

8.1 The variation in parametric control of the grounding line

In order to visualise the new parametric dependences introduced by lateral stresses in the general balance equation (4.2), it is insightful to express it in its dimensional form. With the power-law results of (A 2) and (4.14) incorporated, this equation reads

(8.1) $$\begin{eqnarray}\underbrace{4\left[\frac{(MQ)^{n+1}C_{-}^{n}}{\unicode[STIX]{x1D707}_{0}^{n}}H_{G}^{(n^{2}-3n-1)}\right]^{1/n^{2}}}_{\text{Extensional resistance}}+\underbrace{\frac{\unicode[STIX]{x1D6FF}}{2}\left(\frac{M^{\prime }Q}{\unicode[STIX]{x1D6EC}}\right)^{2/n+1}\hat{B}\left(\frac{x_{G}}{\unicode[STIX]{x1D6EC}},\frac{x_{C}}{\unicode[STIX]{x1D6EC}}\right)}_{\text{Ice\text{-}shelf buttressing resistance}}=\frac{\unicode[STIX]{x1D6FF}}{2}H_{G}^{2},\end{eqnarray}$$

where $H_{G}\equiv d(x_{G})$ is the grounding-line thickness, $M\equiv (\unicode[STIX]{x1D707}_{0}/\unicode[STIX]{x1D70C}g)^{n}$ , $M^{\prime }\equiv (\unicode[STIX]{x1D707}_{0}/\unicode[STIX]{x1D70C}g^{\prime })^{n}$ , $\hat{B}$ is the dimensionless form of the buttressing force given by (4.13) or (4.14), which depends purely on the geometrical specification of the ice shelf, and $\unicode[STIX]{x1D6EC}\equiv (\unicode[STIX]{x1D706}_{+}w^{n})^{1/(n+1)}$ is a length formed from the channel half-width $w$ . For negligible softening and $n=3$ , for example, $\unicode[STIX]{x1D6EC}=0.47\,w$ . Equation (8.1) shows that the extensional resistance and ice-shelf buttressing forces depend individually on the grounding-line flux $Q$ . Therefore, the relationship between $Q$ and thickness $H_{G}$ depends on the solution of the implicit algebraic equation (8.1). With $Q=q(x_{G})$ represented in terms of the grounding-line position using (2.20) and $H_{G}=d(x_{G})$ equated to the bedrock flotation profile, equation (8.1) forms an algebraic equation that can be solved directly for the steady-state grounding-line position $x_{G}$ .

For zero buttressing, $B=0$ , equation (8.1) can be rearranged to give

(8.2) $$\begin{eqnarray}Q\approx \frac{1}{M}\left[\left(\displaystyle \frac{\unicode[STIX]{x1D6FF}}{8}\right)^{n^{2}}\left(\frac{\unicode[STIX]{x1D707}_{0}}{C_{-}}\right)^{n}H_{G}^{(n^{2}+3n+1)}\right]^{1/(n+1)}\quad \text{(unbuttressed)},\end{eqnarray}$$

which recovers the unbuttressed relationship for grounding-line flux for a sliding ice stream (Schoof Reference Schoof2007b ). In the opposing limit of strong buttressing, equation (8.1) predicts the distinct expression

(8.3) $$\begin{eqnarray}Q\approx \frac{\unicode[STIX]{x1D6EC}^{n+1}}{M^{\prime }(Nl)^{n}}H_{G}^{n+1}\quad \text{(strongly buttressed)},\end{eqnarray}$$

where $N\equiv (n+1)/n$ and $l\equiv x_{C}-x_{G}$ is the shelf length. For the purpose of illustrating the new parameter dependences arising for strong buttressing mathematically, I have here neglected the contributions to $\hat{B}$ due to $H_{C}$ (though it should be noted that this is not a good approximation in general and can be straightforwardly retained). The result of (8.3) illustrates that the calving position $x_{C}$ , as contained in the shelf length $l$ , and the half-width of the ice shelf, $w\propto \unicode[STIX]{x1D6EC}$ become central controls. Further, the absence of $C_{-}$ in (8.3) shows that basal conditions have no important effect on the dynamics of a sufficiently buttressed grounding line, thus confirming the control illustrated earlier in figure 11.

The removal of a dependence of the grounding line on the basal stress clarifies that the control of a marine ice sheet is based on two distinct physical processes: one is the control of the grounding-line position; the other is the control of the degree of ‘pile-up’ upstream of that position. Importantly, the physics controlling these independent processes result in two necessary physical conditions for maintaining a large marine ice sheet: one is to support a sufficiently advanced grounding line; the other is to sustain a sufficient drag on flow in the grounded region. The failure of either one of these physical constraints can form a ‘weak link’ that is sufficient for collapse of a marine ice sheet. It might be anticipated that the maintenance of an advanced grounding line is the weakest of these two links for sustaining the WAIS. The split in necessary physical conditions for maintaining a marine ice sheet explains why lateral stresses in an ice shelf are so much more important than those in the grounded region. Since ice-shelf buttressing directly provides an independent control of the grounding line, with a magnitude of force that can readily exceed the extensional resistance to flow across a grounding line relied upon in the absence of buttressing, it has a direct impact on the sustainment of the first of these weak-link conditions (the sustainment of an advanced grounding line). Lateral stresses in the grounded region, by contrast, only significantly influence the second necessary condition representing the sustainment of a thick and steep ice sheet upstream of the grounding line. This separation of physical controls explains why the maintenance of a stable marine ice sheet can be dictated by the sustainment of ice-shelf buttressing.

Figure 18.

The relative buttressing parameter $\unicode[STIX]{x1D6FA}$ evaluated using (8.4), which measures the proportion of the resistance to grounding-line flow stemming from ice-shelf buttressing. The ranges encompass the variations in parameters for the ice-shelf systems listed in table 1.

Table 1.

Estimates of the thickness at the grounding line $d$ , distance between grounding line and calving front $l$ , ice-shelf width $w$ and volumetric flux per unit width $Q$ for five ice-shelf regions. The given ranges for $d_{G}$ and $w$ encompass variations in values for a given ice shelf. The relative buttressing parameter $\unicode[STIX]{x1D6FA}$ evaluated using (8.4) is listed in the final column.

8.2 Assessing the significance of ice-shelf buttressing

To indicate the locations of a selection of ice shelves on the spectrum of relative buttressing strength, I evaluate the parameter $\unicode[STIX]{x1D6FA}$ , defined by (4.5). Using the analytical result of (4.13) for power-law fluid and negligible net accumulation, $\unicode[STIX]{x1D6FA}$ can be evaluated as

(8.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}\equiv \frac{B}{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}g^{\prime }d_{G}^{2}}=\frac{1}{d_{G}^{2}}\left[\left(\frac{\unicode[STIX]{x1D707}_{0}}{\unicode[STIX]{x1D70C}g^{\prime }}\right)^{n}\frac{Q}{\unicode[STIX]{x1D6EC}}\right]^{2/(n+1)}\left[\left(H_{C}^{N}+\frac{Nl}{\unicode[STIX]{x1D6EC}}\right)^{2/N}-H_{C}^{2}\right].\end{eqnarray}$$

The relative buttressing $\unicode[STIX]{x1D6FA}$ is evaluated for a selection of ice-shelf systems listed in table 1. The main central component of the Ross ice shelf is considered separately to its smaller, channelised component east of Roosevelt Island (denoted ROS $^{\ast }$ ). The predicted $\unicode[STIX]{x1D6FA}$ are listed in the final column and plotted in figure 18. Significant variations in its value are apparent. The smallest $\unicode[STIX]{x1D6FA}$ applies to the Ronne ice shelf, with ${<}$ 15 % of its resistance stemming from buttressing. This may be attributable to its considerable width, which reduces the transverse shear stresses. The largest $\unicode[STIX]{x1D6FA}$ applies to the confined region of the Ross ice shelf for which 65 % of grounding-line resistance is predicted to stem from buttressing.

8.3 Applicability

On the basis of the numerical comparisons given in the supplementary document, it is likely that the model describes a large range of marine ice sheet configurations from those channelised in fjords in addition to larger, broader ice shelves. However, a number of caveats should be noted. Primary anticipated limitations include: the existence of islands or ice rises, which will change the buttressing forces generated by the ice shelf; the more complex two-dimensional flow that may be generated by strongly localised ice-stream inputs, which can be more typical for large ice shelves; and the effects of significant transverse variations in basal conditions. A limitation of the steady-state results, as represented by the general balance equation (8.1), is their accuracy if the ice shelf or grounding line are sufficiently unsteady.

The examples of this study have focused on certain specialised situations. These include the assumption of a localised input directly at the ice divide specified by (2.19), and the assumption of a uniform basal coefficient of drag and flow width. These assumptions have been made for the sake of illustrating the results and can all be relaxed within the steady-state framework developed, which is more versatile than the specifications considered in these examples. The framework can also accommodate more complex rheological specifications, basal conditions and calving laws. With these generalisations, the framework provides an efficient complement to direct numerical analysis, with the numerical saving affording scope for thorough exploration of scenarios and sensitivity analysis in case studies.