The amount of ice discharged by an ice stream depends on its width, and the widths of unconfined ice streams such as the Siple Coast ice streams in West Antarctica have been observed to evolve on decadal to centennial timescales. Thermally driven widening of ice streams provides a mechanism for this observed variability through melting of the frozen beds of adjacent ice ridges. This widening is driven by the heat dissipation in the ice stream margin, where strain rates are high, and at the bed of the ice ridge, where subtemperate sliding is possible. The inflow of cold ice from the neighboring ice ridges impedes ice stream widening. Determining the migration rate of the margin requires resolving conductive and advective heat transfer processes on very small scales in the ice stream margin, and these processes cannot be resolved by large-scale ice sheet models. Here, we exploit the thermal boundary layer structure in the ice stream margin to investigate how the migration rate depends on these different processes. We derive a parameterization of the migration rate in terms of parameters that can be estimated from observations or large-scale model outputs, including the lateral shear stress in the ice stream margin, the ice thickness of the stream, the influx of ice from the ridge, and the bed temperature of the ice ridge. This parameterization will allow the incorporation of ice stream margin migration into large-scale ice sheet models.

The Siple Coast ice streams are fast-moving regions within the
West Antarctic ice sheet. They exhibit temporal changes on decadal to
centennial timescales in their spatial configuration, for example slowdown
and reactivation cycles and changes in ice stream width
(

Ice streams are bordered by slowly moving ice, called ice ridges, and the
close proximity of fast to slowly moving ice is reflected in a sharp gradient
in basal resistance between ridge and stream

However, if freezing in the bed is possible, a thermal barrier can form in
the bed which suppresses widening through subglacial drainage

In this scenario the outwards migration of ice stream margins requires
melting of the frozen sediment under the ice ridge. By contrast with a
narrowing ice stream, however, it is not necessary for the entire thickness of
the sediment column to melt out: only part of it needs to be unfrozen to
permit sliding, and we will later idealize this by assuming that sliding is
possible as soon as the melting point is reached at the bed. This, however,
also underlines the asymmetry between widening and narrowing of an ice
stream, which motivates us to focus on the harder problem of widening, which
requires heat to be transferred into the bed. Several studies show that a
strong gradient in basal resistance created by a thermal transition leads to
significant englacial heat production in the ice stream margins
(

Existing studies that derive a migration rate from this competition between
dissipation, conduction, and advection

In the presence of subtemperate slip, we expect significant changes to the velocity field, which is responsible for advection of heat, and to the spatial distribution of heat dissipation. In particular, heat is then dissipated at the frozen ice–bed interface. This is the very location where warming has to occur in order for the ice stream margin to migrate outwards. We therefore expect subtemperate slip to have a significant influence on the rate at which ice stream margins can migrate.

To determine the rate of margin migration, we have to consider the thermal
and mechanical transitions from ice ridge to ice stream flow, which take
place over a distance of just a few ice thicknesses. This is narrow in
comparison to the width of the ice ridge and the ice stream, and it can be
captured by a boundary layer model

This paper is laid out as follows: we state the model in
Sect.

The model for ice stream margins we use here is derived in

In contrast to typical “shallow” ice stream and ice sheet models
(

The asymptotic analysis in

The ice–bed interface is assumed to be flat and located at

Panel

We define the margin location

The boundary layer is effectively two-dimensional: the ice stream is much
longer than a single ice thickness, and therefore along-flow variations in
mechanical and thermal conditions in the

We assume that the thermal state of the ice–bed interface controls the basal
boundary conditions for the ice. This requires us to model the thermal
response not only of the ice but also of the bed. We therefore specifically
include the bed in the domain and apply a geothermal heat flux at

Force balance can be separated into a downstream component, with

The ice stream imposes a lateral shear stress

We assume that basal melting has a negligible effect on ice velocities, so

To the extent that additional degrees of freedom (other than temperature) are involved in sliding, the main concern would presumably be water pressure at the bed or within the till, rather than the thickness of the unfrozen till layer. Our assumption of a free slip once the melting point is reached is best justified (see Haseloff et al., 2015) if we suppose that the unfrozen bed is hydraulically well connected, so that the water pressure in the parts of the bed that have just become unfrozen quickly equilibrates with water pressure elsewhere under the ice stream (and hence basal friction is comparable to the rest of the active ice stream). Shear stresses experienced by the margins of the ice stream are large compared with basal drag throughout the ice stream (Haseloff et al., 2015), and this implies that basal friction is small at leading order everywhere where the melting point is reached. There are undoubtedly other, more elaborate models for basal shear stress of the unfrozen bed; ours is the simplest possible case to analyze.

Where the bed is frozen, we consider two different possibilities. The first
assumes that no slip is possible, so that the basal boundary condition for

The upper surface is traction-free and flat at leading order. In practice,
this implies vanishing shear stress and normal velocity, with vanishing
normal stress accounted for by a first-order correction to the constant
leading-order surface elevation. If the actual upper surface is located at

Note that we have formulated the flow problem in such a way that it can be
solved without reference to the temperature field. Physically, however, we
require that the temperature

Advection from the ice ridge prescribes a far-field temperature profile

At the surface at

The equalities in Eq. (

The flux constraint in Eq. (

The inequality constraints serve the role of determining a unique migration
rate

Parameter values used in the sample calculations presented here. Ice
stream thickness

We solve the coupled mechanical and thermal system
Eqs. (

The solutions to the problem are uniquely determined by the lateral shear
stress

Influence of the subtemperate yield stress

We begin with solutions to the ice flow problem
(Eqs.

The downstream velocity

In the case of no slip on the cold side of the margin (

For decreasing

Figure

To solve the heat equation (Eq.

The effect of shear heating (represented by

In Fig.

Introducing subtemperate slip (decreasing

Note that all the temperature fields shown in Fig.

By solving the heat equation where

Importantly, the region of temperate ice in the bottom two rows of Fig.

We now turn to a systematic investigation of the dependence of the migration
velocity on the ice ridge and ice stream parameters. As we have pointed out,
the solution to the velocity and temperature problem is determined uniquely
once we know the applied lateral shear stress

It is clear that

The goal of this section is to express the model in the most succinct form
possible. To do so we introduce

Note that a large Péclet number is what we would expect in a spatially
confined region like an ice stream margin: conduction of heat is relatively
ineffective, and advection mostly dominates. Large

With the definitions above, the velocity in the downstream direction is
determined by the scaled version of Eq. (

The boundary conditions in the ice stream far field
(Eq.

For later convenience, we write the thermal problem in terms of a reduced
temperature

We have now arrived at a model in which a (unique) dimensionless margin
migration velocity

Dependence of non-dimensional migration velocity

In Fig.

Solutions of

We have seen in the discussion of the mechanical fields in Sect.

A noticable feature of Fig.

We initially restrict ourselves to the case of no subtemperate slip and
consider the case of large

Boundary layer structure for asymptotic calculations with large

Asymptotic behavior of

In what follows we give a brief description of how we can derive a model that
ties migration velocity to heat production and transport in the conductive
boundary layer. The reader not concerned with the technical details will find
the result of this analysis in Eq. (

The non-dimensional mechanical problem (Eqs.

For large

With these changes of variables and parameter definitions in place,
Eq. (

However, we are still missing conditions on

The limiting behavior (Eq.

A more general scenario is to consider

Finally, by plotting the converged values of

The migration rate in the limit of large

We can confirm computationally from solutions to
Eq. (

In the last section, we considered large dissipation rates and rapid
advection of ice but not subtemperate slip. The migration velocity

Panel

Consider a slip region that is similar in size to the conductive boundary
layer of Sect.

The resulting mechanical problem is detailed in Sect. S5. We do not go into detail here: the point is that the velocity

In other words, the conductive boundary layer problem now depends on an
additional parameter through

Owing to the high computational cost of solving Eq. (

As already observed in Sect.

We now turn to the case of

With an

Panel

Once more we confirm that the formula (Eq.

However, for small values of

One of the difficulties we still face in making our results directly
applicable to large-scale models is that we have a closed-form approximation
for the migration rate

There is one regime in which we can do better and reduce

Asymptotic behavior for large

Finding an approximation to

In this study, we have investigated how different physical processes
determine the widening of ice streams that are not topographically confined.
We have considered the case in which the transition from fast to slow or no
sliding that characterizes a typical ice stream margin is co-located with a
thermal transition at the bed. In this scenario, the often intense
dissipation of heat generated by the change in sliding behavior can cause the
corresponding transition from a temperate to a cold bed to move, and the main
objective of this study is to determine the corresponding rate of margin
migration into the cold region. This ice stream widening relies on a delicate
balance between heat dissipation, heat transport by advection, and conduction
to warm the initially cold bed outside the ice stream. We have specifically
excluded the case where heat loss dominates and the margin migrates into the
ice stream from consideration here, although similar physics would allow
inwards migration to be modeled

How the margin location is determined here differs from existing studies of
heat transfer processes in

To model the migration of ice stream margins, we solve a coupled model for ice flow and heat transport in the margin. In this model, the migration rate is determined by imposing constraints on the temperature and heat flux on the cold and warm side of the margin. The migration rate depends on material properties, ice geometry, lateral shear stress in the ice stream margin, and the velocity with which the ice enters the margin from the ridge. These dependencies can be expressed in terms of a small number of non-dimensional combinations of these parameters, although there is no closed-form solution, and the migration rate is expensive to evaluate on a case-by-case basis through the use of our model. In general, we have been able to establish that larger lateral shear stresses and less inflow of cold ice favor margin migration, as does a lower basal yield stress on the cold side of the margin.

To go further and provide quantitative parameterizations of the migration
rate, we have exploited the fact that heat dissipation is generally large.
This has allowed us to construct a number of approximate solutions for
migration velocities that we can give in closed form. Where the different
parameterizations we have derived apply depends on the amounts of
subtemperate slip, controlled in our model by basal yield stress

For an infinite yield stress, which is equivalent to a sharp transition from
no slip to free slip, the migration rate is
(Eq.

Equations (

Margin migration velocities

To illustrate how different geometric conditions alter the migration rate, we
assume that the lateral shear stress is determined by

Since the parameter ranges where Eqs. (

Naturally, migration rate increases with ice stream width, hinting at the
possibility of runaway widening of ice streams in a positive feedback. Note,
however, that plotting

In principle, incorporating ice stream widening with Eqs. (

Moreover, we have only considered the evolution of an ice stream that is
already fully evolved, but the physics governing the position of ice stream
tributaries and ice stream inception remain unclear. The position of ice
stream tributaries might be determined by geological factors

Our model as stated in Sect.

We also assume that the viscosity in the ice is independent of the ice
temperature (i.e.,

Finally, we have assumed that the dynamics of temperate ice can be
represented by a particular version of an enthalpy gradient model

All numerical calculations were done with the freely available finite-element solver Elmer/Ice (Gagliardini et al., 2013).

Here we summarize the behavior of the velocity and temperature fields close to a
transition from frictional to free slip, based on calculations given in full
detail in Sect. S2. We assume a constant viscosity

With the heat dissipation

If

The authors declare that they have no conflict of interest.

Marianne Haseloff was supported by a Four Year Fellowship at the University of British Columbia, NSERC grant 357193-13, and the Princeton AOS Postdoctoral and Visiting Scientist Program. Christian Schoof acknowledges NSERC grants 357193-13 and 446042-13. Numerical calculations performed on WestGrid facilities were supported by Compute Canada. We thank the editor Eric Larour and two anonymous referees for their thorough reviews, which have helped to improve the paper. Edited by: Eric Larour Reviewed by: two anonymous referees