We consider the flow of marine-terminating outlet glaciers that are laterally confined in a channel of prescribed width. In that case, the drag exerted by the channel side walls on a floating ice shelf can reduce extensional stress at the grounding line. If ice flux through the grounding line increases with both ice thickness and extensional stress, then a longer shelf can reduce ice flux by decreasing extensional stress. Consequently, calving has an effect on flux through the grounding line by regulating the length of the shelf. In the absence of a shelf, it plays a similar role by controlling the above-flotation height of the calving cliff. Using two calving laws, one due to Nick et al. (2010) based on a model for crevasse propagation due to hydrofracture and the other simply asserting that calving occurs where the glacier ice becomes afloat, we pose and analyse a flowline model for a marine-terminating glacier by two methods: direct numerical solution and matched asymptotic expansions. The latter leads to a boundary layer formulation that predicts flux through the grounding line as a function of depth to bedrock, channel width, basal drag coefficient, and a calving parameter. By contrast with unbuttressed marine ice sheets, we find that flux can decrease with increasing depth to bedrock at the grounding line, reversing the usual stability criterion for steady grounding line location. Stable steady states can then have grounding lines located on retrograde slopes. We show how this anomalous behaviour relates to the strength of lateral versus basal drag on the grounded portion of the glacier and to the specifics of the calving law used.

In the theory of laterally unconfined marine ice sheet flow, a standard
result is that flux through the grounding line is an increasing function of
bedrock depth

There are a number of mechanisms that can alter the flux-to-bedrock-depth
relationship. These include the appearance of “hoop stresses” in an ice
shelf fringing the ice sheet (see

An open question is whether an evolving calving front can lead to a similar stabilization, as we are no longer guaranteed that a retreat in the grounding line position leads to the same increase in shelf length and, therefore, to the same increase in lateral drag. To answer that question conclusively, we would need a universal “calving law” that can robustly predict the location of the calving front. Such a calving law is currently not available.

We investigate how two particular calving laws that are relatively widely
used in models for tidewater glaciers affect buttressing in a simplified
flowline model. The ice flow model itself lacks the sophistication of models
that resolve the cross-channel dimension. Instead, it relies on a
parameterization of lateral drag in terms of the mean along-channel velocity

The rationale for the calving models used here is described in greater detail
in Sect.

We consider a flowline model for a rapidly sliding, channelized outlet
glacier with a parameterized representation of lateral drag. The model has
the same essential ingredients as those in

Note that we have included the melt rate

The parameters

The second term

It is worth noting however that Eq. (

Schematic of the model domain and variables used. Not shown is the
lateral dimension: the glacier occupies a channel of width

We denote the glacier terminus by

The process of calving remains relatively poorly understood, but several
calving laws have been developed on theoretical grounds. Our aim is to
illustrate how different calving laws can lead to qualitatively, rather than
quantitatively, different dynamics in the outlet glacier. We consider two
possible calving laws. The first is the CD model due to

Calving laws. In all three panels, a solid line refers to the CD
calving law, and a dashed line to the FL law. The grey shaded region refers to
parts of parameter space where the calving front is grounded for the CD law,
and white background to a floating calving front.

The CD model works based on the assumption that water in surface crevasses
affects the depth to which those crevasses can penetrate. When they penetrate
deeply enough to connect with basal crevasses, calving occurs. When they do
not, there is no calving and the ice front simply moves at the velocity of
the ice. Algebraic manipulation of the

As an alternative to the CD model, in which the function

Note that

To complete the notation for our model, we also define the grounding line
position

In our view, the CD model is a cartoon version of the linear elastic fracture
mechanics explored in by

The basic method in

Note that there is one inconsistency in the calving law at small

A second practical pitfall of the CD model is that it predicts no calving at
all if

More recently, others have extended the linear elastic fracture mechanics
approach of

The

Even when taking the

In the remainder of this paper, we will consider the problem Eq. (1) in
dimensionless form. The purpose of doing so is two-fold.
Non-dimensionalization (i) reduces the number of free parameters and
(ii) allows systematic approximations based on the relatively small size of
some dimensionless parameters. We assume that we know scales

The system Eq. (4) can be solved numerically as posed. In this paper, we
focus on solutions of the steady-state version of the problem by a shooting
method, which provides a straightforward alternative to a solution by more
established time-stepping methods. As our method has not been used previously
in this context, we sketch it here for completeness; results are presented at
the end of this section and in Figs.

We can write the steady-state problem as a four-dimensional, first-order
autonomous system of differential equations if, in addition to

We assume there is an ice divide at

Steady-state profiles, with

Steady-state profiles, with

Figure

Figure

Our aim in what follows is to explain the results in
Figs.

If we take

The model Eq. (7) holds everywhere except near the grounding line and in the
floating ice shelf. Following

In order for the rescaling in

The result is a boundary layer model at leading order in

In order to make the balances in Eq. (9) work with

In that physical regime (termed a “distinguished limit”, in which all
physical processes are potentially active), we have to assume that the basal
drag coefficient in Eq. (

Asymptotic matching is the mathematical formalism by which the boundary layer
problem and the “outer” problem Eq. (7) for the dynamics of the bulk of the
glacier are connected

From the perspective of the model Eq. (

As in previous work

We give additional detail on how to compute that relationship between flux,
geometry and model parameters in Appendix

The practical use of this form is that it reduces the complexity of the flux
formula: for a given set of constants

Equation (

Solutions of the boundary layer problem.

As already suggested by the steady-state solutions to the full problem in
Fig.

By contrast, the flux always increases with flotation thickness in the
calving at floatation model, just as it does in laterally unconfined marine
ice sheet flow

Other features of our solutions are also shown in panels (a)–(c) of
Fig.

We can also confirm our boundary layer results by direct comparison with
numerical solutions of the full ice flow problem, computed by the method in
Sect.

There are two aspects of the CD model flux solution that we still need to
explain in more detail: (i) why flux decreases with increasing flotation
thickness

Comparison of the solution of the boundary layer problem with

Key to the flux–flotation-thickness relationship is that flux depends on the
extensional stress at the grounding line, and that extensional stress in turn
depends on the geometry of the calving front and floating ice shelf. For
relatively small extensional stresses

For the FL model, it is easy to extract an analytical formula for flux as a
function of channel width and depth to bedrock from
Eq. (

Figure

Limiting forms of ice flux near the critical ice thickness

Here, we are interested in the anomalous relationship between

Note that the anomalous decrease in flux with increasing flotation thickness
is most pronounced around the critical value

Maintaining constant ice thickness at the calving front while bedrock depth
changes has a significant effect on the extensional stress at the grounding
line. Consider the case of a grounded calving front when

Boundary layer solutions

The extensional stress perturbation occurs because the calving cliff ice
thickness has not changed at first order, but bedrock depth is shallower. The
calving cliff now protrudes further above the water line, and the
depth-averaged normal stress exerted on it by the water is smaller. As a
result, the extensional stress in the ice has to increase. This increase in
stress is what leads to the increase in flux caused by the decrease in
flotation thickness

This is consistent with the behaviour shown in Fig.

We can conversely take the case of

For small

The shelf length is dictated by

Again, we have given an ad hoc derivation for
Eq. (

However, as Fig.

It is relatively straightforward to estimate how large

Finally, consider the limiting case of very large basal friction coefficient
(i.e.

For larger

For a fixed value of

This situation was previously explored by

Consider the special case of no basal drag on the grounded part of the
glacier. We can show how Eq. (

Solutions for a long shelf, same plotting scheme and parameter
values as in Fig.

In this paper, we have applied the boundary layer analysis of

Such an anomalous relationship has significant consequences for stable
glacier margin positions. Consider the model Eq. (

Steady states can now be computed easily from Eq. (

Conversely, we may see grounding lines attain stable steady-state positions
on upward-sloping beds if

Our aim has not been to be authoritative in establishing the existence of an anomalous flux–depth relationship: our model contains at least three components that can be improved upon. First, the parameterized description of lateral drag should eventually be dispensed with, replacing our model with one that resolves the cross-channel dimension. The scaling that underlies our boundary layer model should still be applicable in that case, but the actual boundary layer model will consist of a set of coupled partial differential equations (as opposed to ordinary differential equations) and is likely to be much more onerous to solve for a large number of parameter combinations, as we have been able to do here.

Second, we have neglected the effect of basal melting on the shelf here. This is tractable in the framework we have developed here with a simple, prescribed basal melt rate, but doing so still introduces sufficient complications to lie beyond the scope of a single paper; a second manuscript that incorporates melting into our theory is in preparation.

Third, the calving law we have employed is relatively poorly constrained by observation and is based on a number of simple assumptions about how cracks form near a calving front. Furthermore, it relies entirely on water depth in surface crevasses as a control parameter that should itself be determined by additional physics governing the drainage of surface meltwater.

We have chosen to take the calving model at face value, simply prescribing
the crevasse water depth as a control parameter. This is worth emphasizing as
the dependence of calving cliff height on flotation thickness predicted by
the calving model turns out to be key to the anomalous flux–depth
relationship. It is likely that other, more sophisticated calving models (for
instance one based on the formulation in

For a floating ice shelf, calving cliff height in the CD model is simply
proportional to crevasse water depth and independent of depth to bedrock. In
other words, the CD model can then be thought of as a generic calving model
that imposes a fixed thickness at the floating glacier terminus. Moving the
grounding line to a location with greater flotation thickness (or,
equivalently, depth to bedrock) therefore leads to a longer ice shelf forming
before it can reach the prescribed calving cliff height. If the mechanical
effect of the ice shelf is primarily to provide lateral drag, then a longer
shelf leads to a greater reduction in extensional stress between calving
front and grounding line, and therefore to lower ice flux despite a greater
depth to bedrock at the grounding line. Whether this occurs or not is a
function of basal drag on the grounded part of the glacier: if basal drag
upstream of the grounding line is moderate compared with lateral drag, then
the surface slope and driving stress of the floating shelf will be small, so
the effect of the shelf is mostly to generate lateral drag. By contrast, if
basal drag is large upstream of the grounding line, then the floating shelf
will be relatively steeply sloped and lateral drag will play a lesser role in
force balance there, leading to the possibility that the floating shelf does
not cause a reduction in extensional stress and hence flux through the
grounding line. The changeover between the two regimes happens when, in the
notation of Sect.

As we have indicated, the thickness of floating calving fronts in the CD
model is uniquely controlled by the crevasse water depth parameter and does
not depend on depth to bedrock. The same generic relationship between ice
flux and depth to bedrock at the grounding line will therefore be obtained
for any other calving law that fixes the height of a floating calving front
independently of depth to bedrock. By contrast, the CD model results are
unlikely to be robust in the same way for grounded calving fronts.
Specifically, for a grounded calving front, the

This contrasts with an alternative “calving-at-flotation” (FL) calving law, in which calving front height is always proportional to depth to bedrock and no floating shelf forms. In that case, extensional stress at the grounding line increases with depth to bedrock, and so does ice flux.

We close by noting that our approach can potentially be used to study the effect of other calving laws relatively simply in future, by replacing the function that specifies ice thickness at the calving front. Since an anomalous flux-depth-to-bedrock relationship may be possible and would have significant consequences for stable outlet glacier configurations, it may be worth testing this before embarking on simulations of actual glaciers using different calving laws.

The MATLAB code used in the computations reported is included in the Supplement.

At issue is the uniqueness of the orbit that emerges from the fixed point of
the dynamical system Eq. (6), at which

The simplified forms of the flux law in Eq. (10) can be derived by a
transformation of the boundary layer problem Eq. (9), using

Deriving that functional relationship between

The authors declare that they have no conflict of interest.

Christian Schoof acknowledges the support of NSERC grants 357193-13 and 446042-13. Andrew D. Davis's work is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under award number DE-SC0010518. Tiberiu V. Popa was supported by the NSERC CREATE AAP programme. We are grateful to the three anonymous referees and the editor for their thorough scrutiny. Edited by: Olivier Gagliardini Reviewed by: three anonymous referees