Ice sheet flow with thermally activated sliding. Part 1: the role of advection

Abstract

Flow organization into systems of fast-moving ice streams is a well-known feature of ice sheets. Fast motion is frequently the result of sliding at the base of the ice sheet. Here, we consider how this basal sliding is first initiated as the result of changes in bed temperature. We show that an abrupt sliding onset at the melting point, with no sliding possible below that temperature, leads to rapid drawdown of cold ice and refreezing as the result of the increased temperature gradient within the ice, and demonstrate that this result holds regardless of the mechanical model used to describe the flow of ice. Using this as a motivation, we then consider the possibility of a region of ‘subtemperate sliding’ in which sliding at reduced velocities occurs in a narrow range of temperatures just below the melting point. We confirm that this prevents the rapid drawdown of ice and refreezing of the bed, and construct a simple numerical method for computing steady-state ice sheet profiles that include a subtemperate region. The stability of such an ice sheet is analysed in a companion paper.

1. Introduction

The patterning of flow into fast-moving ice streams separated by regions of much slower ice velocities is one of the most striking features of the Antarctic ice sheet [1–4], and to a lesser degree of other ice sheets and ice caps [5]. Several physical mechanisms that can drive fast flow in continent-scale ice sheet models have been identified [6], but the ability of ice sheet models to capture these mechanisms in a self-consistent way remains in question.

At its simplest, streaming flow can occur where troughs are incised into the bedrock that the ice rests on. While this explains some examples of fast flow, other ice streams are not controlled in a simple fashion by bed topography, principally those in the Siple Coast of West Antarctica [7–9]. Thermomechanical feedbacks are widely thought to be the cause of streaming flow in those cases.

First considered in detail by Clarke et al. [10] and Yuen & Schubert [11], these feedbacks rely on increased dissipation of energy in regions where flow velocity is higher. That additional heat can then, in turn, contribute to faster flow in two ways: ice becomes less viscous at higher temperatures [12] and sliding between ice and bed is facilitated when basal temperatures reach the melting point [13–15] and liquid water is supplied to the bed [16–18].

Attempts to incorporate these processes into ice sheet models have led to seemingly contradictory results, with no clear picture emerging of how thermomechanical feedbacks cause streaming flow, or even of what physics a minimal well-posed model needs to incorporate. The majority of studies of thermomechanically coupled ice sheet models have focused on the feedback between temperature and viscosity, as for instance in experiment F of the EISMINT II intercomparison in [19].

Models incorporating temperature-dependent viscosity frequently predict the formation of a ‘spoked’ pattern of alternating warm and cold regions at the ice sheet bed [6,20–22], and these have been interpreted as manifestations of ice streams [23]. The main, unresolved difficulty has been to determine whether typical thin-film viscous gravity current models for ice sheets (the ‘shallow ice approximation’) are ill-posed when coupled with the corresponding ‘shallow’ heat transport problem through viscous dissipation and a temperature-dependent viscosity. This is suggested by the dependence of thermal patterns on grid spacing in Payne & Baldwin [20] and Saito et al. [21], though Bueler et al. [22] provide a notable counterexample.

A linear stability analysis [24] also suggests that classical lubrication-type ‘shallow ice’ models lack a mechanism by which the instability can be suppressed at short wavelengths transverse to flow. Such a cut-off can be provided by the inclusion of lateral shear stresses [25–27], suggesting that so-called higher-order models for the mechanics of ice flow [28,29], or use of the underlying Stokes equations [30], are a panacea for the difficulties encountered in thermomechanical ice flow modelling.

Considerably less attention has been paid to the thermally regulated activation of sliding as a mechanism by which streaming flow can occur, and we will show in this paper that the onset of sliding causes fundamental difficulties for thermomechanical ice flow models regardless of the mechanical model used. In the EISMINT II intercomparison [19], only one experiment (H) addresses the case of temperature-dependent sliding. This receives only cursory attention in the accompanying paper by Payne & Baldwin [20], who view the temperature-viscosity feedback as the primary cause of pattern formation.

Since the flow of many ice streams in Antarctica is known to be associated with rapid sliding [31,32], with faster shearing due to lower ice viscosity a minor contributor to ice discharge, a legitimate question is whether the dissipation of heat can cause patterning even if we treat ice viscosity as temperature-independent. The thermomechanical feedback in that case is purely between sliding and dissipation. Investigating that feedback is the ultimate goal of this paper and its companions. It however turns out that more fundamental issues about the transition from no slip to sliding need to be answered first, as we shall see shortly.

The assumption in the EISMINT II experiment H, and in many ice sheet models, is that sliding satisfies a ‘hard switch’, in which sliding only commences when the bed reaches the melting point. In that case, the thermomechanical feedback is not smooth, since additional dissipation only occurs when sliding has commenced. Implicit in this assumption is the existence of a free boundary, separating regions of sliding at the melting point from regions with no slip below the melting point.

The physics of that putative transition has previously been considered by Fowler & Larson [33], Fowler [34] and Bueler & Brown [35]. Fowler & Larson [33] construct an argument that shows that a hard switch in the sliding law is incompatible with a shallow ice model if advection of heat is negligible in the ice. This argument is corrected in Fowler [34], and amounts to the following: as ice flows across a boundary between no-slip and sliding in one horizontal dimension, ice flux is locally conserved. As ice starts to slide, the amount of shearing must therefore be reduced, so that the vertically averaged velocity in the ice remains the same. That reduction in shearing requires a corresponding reduction in the pressure gradient driving the flow, and therefore in surface slope. A lower surface slope, however, implies that gravitational potential energy is lost at a slower rate, and therefore, potentially counter-intuitively, less heat is dissipated. The result is that, if the ice sheet starts to slide because its bed has just been warmed to the melting point, then the onset of sliding will immediately cause a reduction in the energy balance, and refreezing.

The restriction of that argument to the case of no advection (see electronic supplementary material, S5 for mathematical details) is of key importance: in fact, in that case, the ice carries no ‘memory’ of having been warmed across the transition from no slip to sliding, which is at odds with any realistic estimate of Péclet numbers in ice sheets [36]. Bueler & Brown [35] take a different perspective, identifying advection as the main culprit for difficulties in modelling the onset of sliding. Working in a ‘shallow’ limit of long horizontal length scales, they note that the abrupt onset of sliding in that limit corresponds to a discontinuity in horizontal velocity, and by implication, to a delta-function-like vertical advection term at the transition. To avoid the issue, Bueler & Brown [35] propose two remedies. First, they compute sliding velocities using an ice flow model that incorporates extensional stresses (and is therefore a variant of the ‘higher-order’ models referenced above). Second, they combine a shearing-dominated velocity field with their sliding velocity solution in an ad hoc fashion (their equation (21)) in order to create a smooth advection velocity.

We revisit the problem posed by Bueler & Brown [35], considering the putative ‘hard switch’ transition from no slip to sliding at the melting point as a boundary layer inside the ice sheet. We do so for the realistic case of an ice sheet that has O(1) effective Brinkman and Péclet numbers, defined at horizontal scales comparable with the full extent of the ice sheet. In other words, we consider the distinguished limit in which dissipation in the ice is sufficient to lead to O(1) changes in temperature (as it must if ice is ever to reach the melting point at the bed), and in which the effect of horizontal advection is comparable with that of diffusion in the vertical.

The paper is organized as follows: having laid out a continuum model in §2a, we consider its reduction to a shallow ice model at the ice sheet scale in §2b, identifying the need for a boundary layer at the transition from no-slip to slip in §2c. That boundary layer is analysed in §3, where we show that we are able to avoid the delta-function-like vertical advection term of Bueler & Brown [35] without any ad hoc devices. However, there is rapid downward advection of cold ice associated with horizontal speed-up at the onset of sliding, and this downward advection should cause immediate refreezing of the bed a short distance downstream of the onset of sliding.

We use this as motivation to reconsider the alternative remedy of Fowler & Larson [33] and Fowler [34], namely that a ‘hard switch’ should be physically impossible and must be replaced by a sliding law that permits significant sliding in a narrow range of temperatures just below the melting point. Their argument is that a significant part of the ice sheet bed should be within that temperature range, so as to ensure an extended transition from no slip to slip. As we go on to show numerically in §4 for our case of relatively significant advection, that extended transition spreads the downward advection of cold ice over a large area, ensuring that refreezing does not occur. The companion paper is then concerned with the stability of such a region of ‘subtemperate sliding’.

It is worth contrasting our work with that of the handful of other authors who have considered changes in sliding from a theoretical perspective. Hutter & Olunloyo [37], Barcilon & MacAyeal [38] and Moore et al. [39] worked on no slip-sliding transitions purely from a mechanical perspective. The common thread among these studies is that they describe sliding onset as a transition from no slip to free slip. This approach can be generalized to the case of finite friction on the temperate side (that is, a sliding law is applied instead of no shear stress at the ice-bed contact), which we demonstrate to collapse on free slip in close proximity to sliding onset (see electronic supplementary material, S2.3). Hindmarsh [26] considers a very gradual onset of sliding with temperature (in the process conflating the two thermomechanical feedbacks, arguing that shear in ice sheets is sufficiently concentrated near the bed that it can be treated mathematically as a form of ‘sliding’). His formulation, in which there is significant sliding at all temperatures, however ensures that there is no shearing anywhere in the ice sheet, even where temperatures are far below the melting point. This sidesteps the possibility of significant downward advection at a rapid transition from no slip to slip. Brinkerhoff & Johnson [27] also avoid a hard switch in sliding regimes by imposing a step change in a sliding parameter when temperature drops below the melting point, while still permitting sliding to occur. This implies that sliding is not affected by a further cooling of the bed, no matter how low the temperatures. In addition, the step change in the sliding law in Brinkerhoff & Johnson [27] still raises the possibility of a rapid speed-up of ice near the bed when the melting point is reached, although starting from a non-zero value. That speed-up should still lead to a drawdown of cold ice, but with the relatively coarse horizontal resolution of their model, it is difficult to identify whether any refreezing should occur locally in their model.

The only other papers that consider velocity patterning due to changes in sliding from a theoretical perspective are, to our knowledge, those of Fowler and Johnson [40] and Kirke-Smith et al. [41]; these focus on a thermomechanical feedback between sliding and heat dissipation mediated by water pressure at the bed, effectively requiring that larger water fluxes correspond to higher water pressures, but sidestepping the possibility of parts of the bed becoming frozen entirely (although the analysis by Calvo et al. [42] predicts that this can occur). While clearly a plausible avenue for patterning in ice flow, such hydraulically mediated patterning will in many cases still require a thermal transition to have occurred somewhere upstream in order for sliding to have commenced in the first place, and that is the concern of the present paper.

2. Abrupt sliding onset: a continent-scale model

(a). Model formulation

We consider a two-dimensional ice sheet as shown in figure 1. We assume a Cartesian coordinate system (x₁, x₂) = (x, z) such that the z-axis is vertical and oriented upward, with z = 0 at sea level. The upper surface of the ice is at z = s(x, t) and the bed is at z = b(x), so that the ice has thickness is h(x, t) = s − b. Ice flow is governed by the Stokes equations

$$\frac{\mathrm{\partial }{\tau }_{ij}}{\mathrm{\partial }{x}_j}-\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}+\rho g_i=0,$$ 2.1a

where τ_ij is the deviatoric part of the stress tensor σ_ij = τ_ij − pδ_ij, p is pressure, ρ is the density of the ice and g_i is the component of gravity in the i direction. We use the summation convention throughout. Deviatoric stresses are related to the strain rate tensor D_ij through

$${\tau }_{ij}=2\eta D_{ij},\text{with}{D}_{ij}=\frac{1}{2}(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}),$$ 2.1b

with the velocity field u = (u₁, u₂) = (u, w) satisfying incompressibility

$$\mathrm{\nabla }\cdot \mathbf{u}=0,$$ 2.1c

and ∇ = (∂/∂x, ∂/∂z). In equation (2.1b), η is the viscosity, which we regard as constant. Disregarding the dependence of viscosity on temperature and strain rate implies that we rule out thermoviscous feedbacks as those described, among others, by Clarke et al. [10], consistently with our objective to isolate the effects of basal thermal transitions.

Figure 1.

Model geometry for a laterally uniform ice sheet with a basal thermal transition. Ice flows from a region of frozen bed, where basal no slip occurs (x < x_onset), to a region of temperate bed where ice slides on its base (x > x_onset) and a Weertman-type sliding law is applied.

The surface is stress-free and also a free boundary, that is

$${\sigma }_{ij}{n}_j=0,\frac{\mathrm{\partial }s}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }s}{\mathrm{\partial }x}-w=\dot{b}\text{on}\hspace{0.17em}z=s,$$ 2.1d

with

$$\mathbf{n}=(-\frac{\mathrm{\partial }s}{\mathrm{\partial }x},1)/\sqrt{1+{\left(\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right)}^2}.$$ 2.1e

The specific surface mass balance $\dot{b}$ is positive for net snowfall. The bed is a material surface where there may or may not be slip: here, we are concerned with the case of thermally controlled sliding, so we associate a frozen bed with no slip and a temperate bed with basal sliding. In the former case, we have

$$\mathbf{u}=0\text{on}\hspace{0.17em}z=b\text{if}T<T_m,$$ 2.1f

where T(x, z, t) is temperature and T_m is the melting point temperature, which we consider independent of pressure. If slip occurs, we assume that there is an applied shear stress τ_b at the base of the ice sheet that is a function of the basal velocity u_b through a linear slip law [43,44]

$${\tau }_b=C{u}_b,\mathbf{u}\cdot \hat{\mathit{n}}=0\text{on}\hspace{0.17em}z=b\text{if}T=T_m,$$ 2.1g

where C > 0 denotes the friction coefficient. Defining the normal vector to the bed as

$$\hat{\mathit{n}}=(-\frac{\mathrm{\partial }b}{\mathrm{\partial }x},1)/\sqrt{1+{\left(\frac{\mathrm{\partial }b}{\mathrm{\partial }x}\right)}^2},$$ 2.1h

the sliding velocity and basal shear stress, both parallel to the bed, are

$$u_b=u{[1+{\left(\frac{\mathrm{\partial }b}{\mathrm{\partial }x}\right)}^2]}^{1/2}{\tau }_b=\frac{[1-{\left(\mathrm{\partial }b/\mathrm{\partial }x\right)}^2]{\tau }_{xz}-2(\mathrm{\partial }b/\mathrm{\partial }x){\tau }_{xx}}{1+{\left(\mathrm{\partial }b/\mathrm{\partial }x\right)}^2}.$$ 2.1i

Even with a viscosity independent of temperature, we require a thermal model to know whether the bed is cold or temperate. Energy conservation for the ice and the bed reads

$$\rho c(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }T)-\mathrm{\nabla }\cdot \left(\kappa \mathrm{\nabla }T\right)={\tau }_{ij}{D}_{ij}\text{for}\hspace{0.17em}b<z<s$$ 2.2a

$${\rho }_{\mathrm{bed}}{c}_{\mathrm{bed}}\frac{\mathrm{\partial }T}{\mathrm{\partial }t}-\mathrm{\nabla }\cdot \left({\kappa }_{\mathrm{bed}}\mathrm{\nabla }T\right)=0\text{for}\hspace{0.17em}z<b,$$ 2.2b

where c is heat capacity, and κ and κ_bed are the thermal conductivity of the ice and of the bed, respectively. We prescribe a temperature T_surf at the surface, and a geothermal heat flux q_geo at large distance below the ice-bed contact,

$$T=T_s\text{on}\hspace{0.17em}z=s$$ 2.2c

$$-{\kappa }_{\mathrm{bed}}\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\to q_{\mathrm{geo}}\text{as}\hspace{0.17em}z\to -\mathrm{\infty },$$ 2.2d

while at the bed z = b we require

$${\kappa \frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z\to b^{+}}-{{\kappa }_{\mathrm{bed}}\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z\to b^{-}}={\left[T\right]}_{-}^{+}=0\text{if}\hspace{0.17em}T<T_m,$$ 2.2e

or

$$T=T_m\text{if}\hspace{0.17em}{u}_b>0,$$ 2.2f

where ${[f(z)]}_{-}^{+}=f(z\to z_0^{+})-f(z\to z_0^{-})$ denotes the difference between positive and negative limiting values of f across a prescribed surface z = z₀.

(b). Non-dimensionalization and simplification

We consider scales for the ice sheet length [x] = L, surface accumulation $[\dot{b}]$, temperature [T] = [T_s] − T_m as known quantities, and introduce the usual, shallow ice scale relationships (e.g. [45])

$$[w]=[\dot{b}],\hspace{0.17em}[u]=\frac{[w][x]}{[z]},\hspace{0.17em}[t]=\frac{[x]}{[u]},\hspace{0.17em}[p]=\rho g[z],\hspace{0.17em}[\tau ]=\frac{\rho g{[z]}^2}{[x]},[z]={\left(\frac{\eta [\dot{b}]{[x]}^2}{\rho g}\right)}^{1/4},$$ 2.3a

in addition to the non-dimensional parameters

$$\mathit{P}\mathit{e}=\frac{{[z]}^2}{\mathrm{\lambda }[t]},\hspace{0.17em}\alpha =\frac{[\tau ][u]}{\kappa [T]/[z]},\hspace{0.17em}\gamma =C\frac{[u]}{[\tau ]},\hspace{0.17em}\nu =\frac{q_{geo}[z]}{\kappa [T]},\epsilon =\frac{[z]}{[x]},$$ 2.3b

where Pe is the Péclet number, α is the Brinkman number, which compares the strength of strain heating to the background conductive heat flux, γ is a non-dimensional friction coefficient, ν is a non-dimensional geothermal heat flux, ε is the aspect ratio of the ice sheet, and λ = κ(ρc)⁻¹ is the thermal diffusivity of the ice. We consider a parameter regime reasonable for the Antarctic ice sheet; with a length of the ice sheet L = 3000 km, a scale for the accumulation rate $[\dot{b}]=0.1\hspace{0.17em}{\mathrm{m}\hspace{0.17em}\mathrm{yr}}^{-1}$, a scale for temperature [T] = 30 K, and a viscosity η = 10¹³ Pa s [46], along with physical parameters ρ = 920 kg m⁻³, g = 9.81 m s⁻², κ = 2.3 W m⁻¹ K⁻¹, c = 2 kJ kg⁻¹ K⁻¹, q_geo = 0.05 W m², C = 0.1 kPa yr m⁻¹, we obtain parameter estimates

$$\mathit{P}\mathit{e}\approx 6,\alpha \approx 2.3,\gamma \approx 0.8\nu \approx 1.7\mathrm{and}\epsilon \approx 8\times {10}^{-4}.$$ 2.3c

Hence in the following we treat the aspect ration ε as a small parameter, while all other parameters as O(1).

Equipped with these scales, we rescale model variables as

$$\begin{array}{rl}& x=[x]x^{\ast },z=[z]z^{\ast },t=[t]t^{\ast },s(x,t)=[z]s^{\ast }(x^{\ast },t^{\ast }),b(x)=[z]b^{\ast }(x^{\ast }),\end{array}$$ 2.3d

$$\begin{array}{rl}& u=[u]u^{\ast },w=[w]v^{\ast },\dot{b}=[\dot{b}]{\dot{b}}^{\ast },p=\rho g[z](s^{\ast }-z^{\ast })+\epsilon [\tau ]p^{\ast }\end{array}$$ 2.3e

$$\begin{array}{rl}& {\tau }_{xz}=[\tau ]{\tau }_{xz}^{\ast },{\tau }_{xx}=-{\tau }_{zz}=\epsilon [\tau ]{\tau }_{xx}^{\ast }=-\epsilon [\tau ]{\tau }_{zz}^{\ast },T=[T]T^{\ast }+T_m,\end{array}$$ 2.3f

and also assume that ice and bed have the same physical properties. Dropping stars for simplicity, and omitting terms of O(ε), we obtain a standard shallow ice model for u and s [45,47] (see electronic supplementary material, S1 for details)

$$\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}=\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\text{on}\hspace{0.17em}b<z<s$$ 2.4a

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}=0\text{on}\hspace{0.17em}b<z<s,$$ 2.4b

with boundary conditions at the ice surface

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }z}=0,\frac{\mathrm{\partial }s}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }s}{\mathrm{\partial }x}-w=\dot{b}\hspace{0.17em}\text{on }z=s,$$ 2.4c

and at the bed, z = b,

$$\begin{array}{rlr}& \text{either}u=w=0,& \text{if}\hspace{0.17em}T<0,\end{array}$$ 2.4d

$$\begin{array}{rlr}& \text{or}{u}_b={\gamma }^{-1}{\tau }_b,\hspace{0.17em}{u}_b\frac{\mathrm{\partial }b}{\mathrm{\partial }x}-w=0,& \text{if}\hspace{0.17em}T=0,\end{array}$$ 2.4e

with

$$u_b=u,{\tau }_b={\tau }_{xz}=\frac{\mathrm{\partial }u}{\mathrm{\partial }z}\text{on}\hspace{0.17em}z=b.$$ 2.4f

Note that, having scaled the horizontal velocity with the shearing velocity in the temperate region, we have implicitly assumed that sliding is at most asymptotically comparable with shearing. While this approach is consistent with our objective to study the onset of basal sliding, an alternative scaling based on the sliding velocity (e.g. [29]) would be required to capture a sliding-dominated flow.

Omitting terms of O(ε²), the leading order heat transport problem is

$$\begin{array}{r}\mathit{P}\mathit{e}(\frac{\mathrm{\partial }T}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+w\frac{\mathrm{\partial }T}{\mathrm{\partial }z})-\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}=\alpha a(x,z,t)\text{for}\hspace{0.17em}b<z<s,\end{array}$$ 2.5a

$$\begin{array}{r}\mathit{P}\mathit{e}\frac{\mathrm{\partial }T}{\mathrm{\partial }t}-\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{z}^2}=0\text{for}\hspace{0.17em}z<b,\end{array}$$ 2.5b

with strain heating a = (∂u/∂z)², and boundary conditions

$$\begin{array}{r}T=T_s\text{on}\hspace{0.17em}z=s,\end{array}$$ 2.5c

$$\begin{array}{r}-\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\to \nu \text{as}\hspace{0.17em}z\to -\mathrm{\infty }\end{array}$$ 2.5d

and

$$\begin{array}{rl}& \text{either}{\left[T\right]}_{-}^{+}={\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}=0\text{on}\hspace{0.17em}z=b\text{if}\hspace{0.17em}{u}_b=0,\hspace{0.17em}T<0,\end{array}$$ 2.5e

$$\begin{array}{rl}& \text{or}T=0\hspace{0.17em}\text{on}\hspace{0.17em}z=b\text{if}\hspace{0.17em}{u}_b>0.\end{array}$$ 2.5f

A further simplification arises from depth integration of the mechanical model, which can be reduced to a diffusion equation for s. Depth integration of the horizontal momentum balance equation (2.4a) with boundary conditions equation (2.4c)₁ and either (2.4d)₁ or (2.4e)₁ gives the horizontal velocity, h

$$u=\frac{1}{2}[h^2-{(s-z)}^2]\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|+u_b,{\tau }_b=h\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|.$$ 2.6a

Defining the mass flux as

$$q={\int }_b^{s}u\hspace{0.17em}\mathrm{d}z,$$ 2.6b

the ice surface evolves according to the diffusion equation

$$\frac{\mathrm{\partial }s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }q}{\mathrm{\partial }x}=\dot{b},$$ 2.6c

which we obtained from the kinematic boundary condition equation (2.4c)₂, along with mass conservation equation (2.4b) and the boundary conditions for w, equations (2.4d)₂ or (2.4e)₂. As usual, w decouples from the rest of the problem, and can be computed as a function of ice thickness and surface slope from equations (2.4b), (2.4d)₂ and (2.4e)₂.

(c). The boundary between cold- and temperate-bedded regions

Our model so far consists of two versions, one that applies where bed temperature is below the melting point and another that applies for temperate beds. We will now discuss how to constrain the location of the transition point between cold and temperate bed, as well as the coupling between the two versions of the model across the cold-temperate boundary.

The location of the cold-temperate boundary is governed by bed enthalpy e* = e/[e], with [e] = κ[t][T]/([z]ρL_f) and L_f the latent heat of fusion. In general, bed enthalpy can be thought of as the water content of the bed: where the bed is frozen no water can be stored at the bed (so e* = 0), while for a temperate bed e* > 0 and a meltwater flux q*_w = q_w/k₀ moves through the subglacial drainage system. Defining a non-dimensional effective pressure (defined as the difference between overburden and water pressure) N* = N/[p], and a scale for the meltwater flux k₀ = kρ²gε²L_f/(η_wκ[T]) (where k is the hydraulic conductivity of the drainage network and η_w is water viscosity), and dropping asterisks for simplicity, the non-dimensional bed enthalpy e satisfies

$$\frac{\mathrm{\partial }e}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{q}_w}{\mathrm{\partial }x}=m,$$ 2.7a

where

$$m={\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha {\tau }_b{u}_b,q_w=k(N)\frac{\mathrm{\partial }}{\mathrm{\partial }x}(N+{\mathit{\Psi }}_0),{\mathit{\Psi }}_0=-[s-b(\frac{{\rho }_w}{\rho }-1)],$$ 2.7b

with Ψ₀ a non-dimensional hydraulic potential. Recalling that N is bounded from above by overburden (which we denote as N_max), and that for high enough N drainage pathways tend to close due to overburden forming a hydraulic blockage, we demand that k(N) → 0 as N → N_max, that is q_w → 0 as N → N_max. To close the model, we allow e to depend on N: on physical grounds, we expect N = N_max on the frozen side (T < 0), and N < N_max on the temperate side (T = 0), with e = 0 for T < 0 and e > 0 for T = 0.

The latter considerations lead to a number of simplifications. First, they provide a constraint on the location of the frozen-temperate boundary, x = x_onset: assuming that T, N, e and u_b are continuous across the cold-temperate boundary, it follows that

$$T=u_b=m=0\text{on}\hspace{0.17em}z=b,x=x_{\mathrm{onset}}.$$ 2.7c

Second, with q_w → 0 for e → 0, and recalling that τ_b remains finite as u_b → 0, the enthalpy equation (2.7a) reduces to (2.5e)₂ for frozen conditions. Last, a simplified treatement of the temperate region that involves only a constraint on the melt rate m can be justified if we restrict ourselves to a region close to x_onset.

Let us consider the simplified case of a steady-state first. Then, the global form of the enthalpy equation (2.7a) integrated over an interval [x_onset, x_onset + Δx] (with Δx > 0) is

$$q_w(x_{\mathrm{onset}}+\mathrm{\Delta }x)={\int }_{x_{\mathrm{onset}}}^{x_{\mathrm{onset}}+\mathrm{\Delta }x}m\hspace{0.17em}\mathrm{d}x,$$ 2.7d

whereby it is clear that the sign of the integral of m over the interval [x_onset, x_onset + Δx] determines the sign of q_w(x_onset + Δx). We are now going to argue that q_w must be positive downstream of a frozen-temperate transition, or at most not very negative, and therefore m must also be globally positive.

Consider the definition of q_w (2.7b)_2,3, and examine first the role of ice sheet geometry, represented by Ψ₀. Given that ρ_w/ρ − 1≪1, the leading order control on q_w are gradients in surface elevation that drive water flow away from the transition downstream into the temperate region (−∂s/∂x > 0 across the ice sheet, with ∂s/∂x∼O(1)). As for N, boundedness along with the notion that freezing of the bed causes a hydraulic blockage suggest that ∂N/∂x < 0, or at most small and positive in such a way not to offset the geometric gradient. We thus expect q_w(x_onset + Δx) > 0. As a result, a reduced enthalpy equation that holds near the transition on the temperate side is

$$m>0\text{if}\hspace{0.17em}T=0\text{on}\hspace{0.17em}z=b,$$ 2.7e

which can be extended to the unsteady case by recognizing that a negative melt rate would lead to freezing in finite time.

Finally, boundary conditions for the conservation laws (2.6c), (2.5a), (2.5b) must be provided on x = x_onset. To ensure that mass and energy are conserved across the boundary we demand

$${\left[q\right]}_{-}^{+}={\left[h\right]}_{-}^{+}={\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}=0\text{on}\hspace{0.17em}x=x_{\mathrm{onset}}.$$ 2.7f

Equipped with this model, we can now turn the attention to the cold-temperate boundary. The continuity conditions (2.7f)_1,2 allow us to recognize an inconsistency in the shallow ice model (2.6): in fact, enforcing continuity of mass flux and thickness across a no slip-sliding transition leads to a break in surface slope, which is unphysical. The existence of this jump in surface slope is straightforward to demonstrate: let us compute the mass flux (see equation (2.6b)) on either side of the transition by depth-integration of the respective horizontal velocities u (given by (2.6a) and (2.4e)₁). Then, demanding that the so-computed mass fluxes balance at constant thickness, as demanded by the continuity statements ((2.7f)_1,2) leads to a discontinuous surface slope across the transition point, with a relation between upstream and downstream slope of the form

$${\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|}^{+}=\frac{1}{1+3{(\gamma h)}^{-1}}{\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|}^{-}\text{on}\hspace{0.17em}x=x_{\mathrm{onset}}.$$ 2.8

It then follows (from (2.6a)) that the horizontal velocity is similarly discontinuous there, even if its average over the ice thickness is not. Mass conservation (2.4b) then implies that w is ill-defined at x = x_onset, with w delta-function like.

A discontinuous vertical velocity at the cold-temperate boundary is likely to have significant consequences for heat transport: in fact, a delta function in w suggests that streamlines should be discontinuous at x = x_onset, and similarly isotherms may be discontinuous when viewed at the ice sheet scale, although precisely how is unclear. In reality, this appears to indicate the need for a boundary layer that resolves the ice thickness scale in the horizontal direction near the transition point. Such boundary layer would reintroduce extensional stresses that may act as a regularizer in the force balance, as previously suggested by Bueler & Brown [35].

3. Abrupt sliding onset: the near-onset region

(a). Model formulation

Consider ice flow across the cold-temperate transition located at x = x_onset and let us rescale variables as

$$X={\epsilon }^{-1}(x-x_{\mathrm{onset}}),\hspace{0.17em}Z=z-b(x_{\mathrm{onset}}),\hspace{0.17em}(U,W)=(u,\epsilon w),\hspace{0.17em}H=h,\hspace{0.17em}P=p,\hspace{0.17em}\mathit{\Theta }=T.$$ 3.1

We also assume that bed topography has structure only at the ice sheet scale, so in the boundary layer b = b(x_onset) + O(ε). In addition, we expand dependent variables as

$$(U,W)=(U^{(0)},W^{(0)})+O(\epsilon ),\hspace{0.17em}P=P^{(0)}+O(\epsilon ),\hspace{0.17em}H=H^{(0)}+O(\epsilon ),\hspace{0.17em}\mathit{\Theta }={\mathit{\Theta }}^{(0)}+O(\epsilon ),$$ 3.2

where surface elevation H⁽⁰⁾ must be independent of X at leading order to conserve the mass flux across the boundary layer. This simplifies the domain for the leading order problem to the strip −∞ < X < ∞, −∞ < Z < H⁽⁰⁾, while structure in surface elevation at the ice thickness scale appears only at O(ε).

Substituting the rescalings and expansions above into the ice-sheet-scale model of §2b,c, and omitting terms of O(ε) (a detailed derivation is provided in electronic supplementary material, S1.2–S2.1), we find the Stokes problem

$$\begin{array}{rl}& \frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }X}+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }Z}=0,\end{array}$$ 3.3a

$$\begin{array}{rl}& \frac{{\mathrm{\partial }}^2{U}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{U}^{(0)}}{\mathrm{\partial }{Z}^2}-\frac{\mathrm{\partial }{P}^{(0)}}{\mathrm{\partial }X}=0,\end{array}$$ 3.3b

$$\begin{array}{rl}& \frac{{\mathrm{\partial }}^2{W}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{W}^{(0)}}{\mathrm{\partial }{Z}^2}-\frac{\mathrm{\partial }{P}^{(0)}}{\mathrm{\partial }Z}=0,\end{array}$$ 3.3c

on 0 < Z < H⁽⁰⁾, with boundary conditions

$$\begin{array}{rl}\frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }Z}+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }X}=W^{(0)}& =0\text{on }Z=H^{(0)},\end{array}$$ 3.3d

$$\begin{array}{rl}{W}^{(0)}& =0\text{on }Z=0\end{array}$$ 3.3e

$$\begin{array}{rl}{U}_b^{(0)}& =0\text{on }Z=0,\text{if}\hspace{0.17em}{\mathit{\Theta }}^{(0)}<0,\end{array}$$ 3.3f

$$\begin{array}{rl}{U}_b^{(0)}& ={\gamma }^{-1}{T}_b^{(0)}\text{on }Z=0,\text{if}\hspace{0.17em}{\mathit{\Theta }}^{(0)}=0\end{array}$$ 3.3g

where U⁽⁰⁾_b = U⁽⁰⁾|_Z=0, and T⁽⁰⁾_b = ∂U⁽⁰⁾/∂Z|_Z=0 are the sliding velocity and basal shear stress, respectively. Matching conditions between the boundary layer above and the shallow ice problem (2.6) are

$$H^{(0)}=h(x_{\mathrm{onset}},t),(U^{(0)},W^{(0)})\to (u(x_{\mathrm{onset}},z,t),0)\text{as }X\to \pm \mathrm{\infty },$$ 3.3h

where u is given by equation (2.6a). We note that, even though the ice flux is conserved, u will differ on either sides of the boundary layer as a result of the discontinuous surface slope at the outer scale, as described by equation (2.8).

Energy conservation up to an error of O(ε) is

$$P{e}_{BL}(U^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }X}+W^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z})-(\frac{{\mathrm{\partial }}^2{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }{Z}^2})=\alpha a\text{for}\hspace{0.17em}0<Z<H^{(0)}$$ 3.4a

$$\frac{{\mathrm{\partial }}^2{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }{Z}^2}=0\text{for}\hspace{0.17em}-\mathrm{\infty }<Z<0,$$ 3.4b

with local Péclet number Pe_BL = ε⁻¹Pe≫1, and strain heating

$$a=2{\left(\frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }X}\right)}^2+2{\left(\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }Z}\right)}^2+{(\frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }Z}+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }X})}^2.$$ 3.4c

Boundary conditions are

$${\mathit{\Theta }}^{(0)}=-T_s(x_{\mathrm{onset}})\text{on }Z=H^{(0)},$$ 3.4d

$${\left[{\mathit{\Theta }}^{(0)}\right]}_{-}^{+}={\left[\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}\right]}_{-}^{+}=0\text{on }Z=0,\text{if }{\mathit{\Theta }}^{(0)}<0,$$ 3.4e

$${\mathit{\Theta }}^{(0)}=0\text{on }Z=0,\text{if }{m}^{(0)}>0,$$ 3.4f

$$-\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}=\nu \text{as }Z\to -\mathrm{\infty },$$ 3.4g

where the basal melt rate is

$$m^{(0)}={\left[\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}\right]}_{-}^{+}+\alpha T_b^{(0)}{U}_b^{(0)}\text{on}\hspace{0.17em}Z=0.$$ 3.4h

In addition, the outer thermal problem (2.5) prescribes the temperature profile at the inflow boundary,

$$\underset{X\to -\mathrm{\infty }}{lim}{\mathit{\Theta }}^{(0)}=T(x_{\mathrm{onset}},z,t).$$ 3.4i

The leading order model above is similar to the transverse flow problem derived by Haseloff et al. [48,49] for an ice stream shear margin that coincides with a frozen-temperate transition. In that case, the bed is frozen outside of the stream and temperate inside, and the margin itself can be described as a boundary layer connecting these two regions. The mathematical problem for the velocity components in the plane perpendicular to the margin is identical to the boundary layer problem stated above. However, despite the mathematical similarities, there are important differences: the margin case is three-dimensional, with the flow on the temperate (streaming) side of the boundary layer predominantly in the direction parallel to the margin itself. This implies that, for a shear margin, velocity gradients in the direction perpendicular to the boundary layer plane (so-called antiplane shearing) set the strength of strain heating (α), which is then decoupled from advection in the boundary layer plane. As a result, advective cooling due to the frozen-temperate transition in the direction normal to the margin can be balanced by strong heating (asymptotically, α≫1 and Pe_BL≫1), thus preventing refreezing. This is not possible in the sliding onset case, where the strength of heating is set by shearing in the boundary layer plane. We will see later on that this has implications with respect to the existence of solutions to the sliding onset boundary layer (§§3b,c)

In our problem, energy conservation at the ice thickness scale is dominated by advection, while strain heating remains small. In the following, we will discuss the implications of strong advection, and we will show that in general the boundary layer cannot be expected to have a solution as a result of such a strongly advective temperature field. The idea is that ice speeds up along the bed downstream of the transition, and mass conservation implies that cold ice from above is drawn towards the bed. In a strongly advective temperature field, the temperature gradient should steepen downstream of the transition as a result of the speed-up, hence causing the bed to refreeze. In the rest of this section, we first present a simple analytical argument that supports this (§3b), and then (§3c) confirm this result through direct numerical solution of the boundary layer problem (3c)–(3d).

(b). Analysis: an abrupt onset causes refreezing

Consider a layer near the bed in the temperate region (so X > 0), far enough downstream of sliding onset to keep advection large compared to diffusion also in the near-bed region. Under these circumstances, we can justify a simplified model for the basal heat flux that allows us to link changes in the basal heat flux to changes in the sliding velocity.

Let us consider the heat transport problem (3.4a) with basal boundary condition (3.4f), and take the limit Pe_BL → ∞ with U⁽⁰⁾, W⁽⁰⁾∼O(1). Neglecting terms of O(Pe⁻¹_BL), for heat conservation we have

$$U^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }X}+W^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}=0,$$ 3.5a

while (3.4f) remains unchanged. We now seek an approximation that holds in proximity of the bed, Z = 0. Differentiating equation (3.5a) with respect to Z and taking the limit of Z → 0 gives

$$U^{(0)}\frac{\mathrm{\partial }}{\mathrm{\partial }X}\left(\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}\right)+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }Z}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}=0\text{as}\hspace{0.17em}Z\to 0,$$ 3.5b

where we used the boundary conditions (3.4f) and (3.3e). Taylor-expanding U⁽⁰⁾ and W⁽⁰⁾ about Z = 0 and using equations (3.3a), (3.3e) and (3.3g) yields

$$U^{(0)}\sim U_b^{(0)}+O(Z),W^{(0)}\sim -\frac{\mathrm{\partial }{U}_b^{(0)}}{\mathrm{\partial }X}Z+O(Z^2)\text{for }Z\to 0,$$ 3.5c

which substituted into equation (3.5b) provides the desired relation between basal heat flux and sliding velocity,

$$U_b^{(0)}\frac{\mathrm{\partial }Q}{\mathrm{\partial }X}-\frac{\mathrm{\partial }{U}_b^{(0)}}{\mathrm{\partial }X}Q=0,\text{with}\hspace{0.17em}Q=-{\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}|}_{Z=0}.$$ 3.5d

We refer to (3.5d) as the Q-equation, which admits solution

$$Q=Q_0{U}_b^{(0)},$$ 3.5e

with Q₀ an integration constant of O(1) to be determined via matching with the frozen region upstream, X → 0. By replacing equation (3.4a) with the Q-equation we are effectively solving an advection-only problem. The fact that Θ⁽⁰⁾ = 0 at the bed allows us to fortuitously satisfy the basal boundary condition with an advection-only heat equation: technically, the bed is a characteristic of the advection-only problem, hence requiring a constant temperature along it.

In general, even though the ice thickness scale Péclet number Pe_BL is large, having U⁽⁰⁾_b → 0 as X → 0 implies the existence of a region near sliding onset where diffusion becomes important near the bed. This would be the region bounded by blue curves for X > 0 in figure 2. While the Q-equation holds along the bed above this region, one would expect instead that an advection-diffusion boundary layer (nested within the boundary layer around the cold-temperate transition that we have introduced through equations (3c)–(3d)) is necessary along the bed as X → 0. We are now going to state such a thermal boundary-layer model, and we will then discuss under what circumstances the Q-equation represents a valid approximation to it. In the last part of this section, we will provide an argument based on the Q-equation that demonstrates refreezing is to be expected immediately downstream of sliding onset as a result of the abrupt speed-up.

Figure 2.

The structure of the thermal problem near the ice sheet bed at a frozen-temperate transition. Ice flows across the transition from left (no slip, frozen bed) to right (sliding, temperate bed), with the grey line denoting a streamline. The thermal problem is advection-dominated above the blue lines, which mark the boundary of the advection-diffusion boundary layer near the base. The derivation of the boundary layer scalings is provided in the electronic supplementary material. (Online version in colour.)

Appropriate rescalings for the near-bed, advection-diffusion thermal boundary layer in the temperate region (X > 0) are

$$X=r\stackrel{`}{X},Z=P{e}_{BL}^{-1/2}{r}^{1/4}\stackrel{`}{Z},U^{(0)}=r^{1/2}\stackrel{`}{U},W^{(0)}=P{e}_{BL}^{-1/2}{r}^{-1/4}\stackrel{`}{W},{\mathit{\Theta }}^{(0)}=r^{3/4}\stackrel{`}{\mathit{\Theta }},$$ 3.6a

where r ≤ O(1) marks distance from the origin. Substituting into (3d), we find a leading order heat equation in the ice of the form

$${\stackrel{`}{U}}^{(0)}\frac{\mathrm{\partial }{\stackrel{`}{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\stackrel{`}{X}}+{\stackrel{`}{W}}^{(0)}\frac{\mathrm{\partial }{\stackrel{`}{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\stackrel{`}{Z}}-\frac{{\mathrm{\partial }}^2{\stackrel{`}{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }{\stackrel{`}{Z}}^2}=0\text{for}0<\stackrel{`}{Z}<+\mathrm{\infty },$$ 3.6b

with

$${\stackrel{`}{\mathit{\Theta }}}^{(0)}=0\text{on}\hspace{0.17em}\stackrel{`}{Z}=0\mathrm{and}-\frac{\mathrm{\partial }{\stackrel{`}{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\stackrel{`}{Z}}\to Q\text{as}\hspace{0.17em}\stackrel{`}{Z}\to +\mathrm{\infty },$$ 3.6c

and the outer flux Q evolving according to equation (3.5d). Note that the full problem and a detailed derivation are provided in electronic supplementary material, S2.2–S2.7.

With a constant bed temperature, as prescribed by the Dirichlet condition (3.4f), this boundary layer is not necessary if the temperature gradient at its upstream end is constant across the boundary layer thickness. That gradient simply evolves according to the Q-equation along the boundary layer, ensuring that the diffusive term in (3.6b) vanishes everywhere. Under those conditions, the boundary layer model (3.6) and the Q-equation (3.5d) are effectively equivalent.

Let us momentarily assume a linear temperature profile at some small distance downstream of the transition, so that the Q-equation (3.5d) holds, and consider the implications of an O(1) increase in the sliding velocity. In a steady state, the solution of the Q-equation (equation (3.5e)) shows that Q is proportional to the sliding velocity, where the latter increases from zero to a finite amount downstream of the cold-temperate transition. If the Q-equation held all the way up to the origin with a finite heat flux there, this would imply an infinite heat flux at any finite distance downstream of the origin. As discussed, this is not the case: in fact, near the origin the velocity becomes very small while velocity gradients are large. This implies that the vertical conductive flux re-appears at leading order in the energy balance there. Nonetheless, we can take the Q-equation to hold up to a region close to the origin—the matching region with a separate thermal boundary layer centred on X = 0 (the circular region in figure 2). Here, a finite flux Q corresponds to a small sliding velocity U⁽⁰⁾_b, and consequently we expect the heat flux to become very large at an O(1) distance downstream of the transition.

In reality, the solution of the Q-equation, equation (3.5e), is modulated by the advection-diffusion boundary layer described in equations (3.6) as a result of the temperature profile at its upstream end being nonlinear. This however does not alter the qualitative result from the Q-equation, as shown by an asymptotic analysis of the bed boundary layer that accounts for diffusion near the origin. The structure of the thermal problem including this boundary layer is illustrated in figure 2; we refer readers interested to the full asymptotic analysis to electronic supplementary material, S2.2–S2.7. The fundamental insight from that analysis is that the heat flux on the temperate side of the transition is large and scales as ∼Pe^1/2_BL. The asymptotically large flux cannot be compensated for by any other terms in the energy balance of the bed, which must therefore refreeze.

This is a mathematical explanation of the physical effect that we alluded to above: rapid increase of basal velocity causes cold ice to be drawn down to the bed; the resulting steepening of the temperature gradient then causes the bed to refreeze. The problem we encounter here is not the result of a velocity discontinuity, and it is not alleviated by introducing extensional stresses in the ice flow problem (suggested as a suitable regularization of the velocity discontinuity at the outer scale by Bueler & Brown [35]): our boundary layer model manifestly makes no simplifications to the force balance in the ice, and preserves continuity of the velocity field across the transition point. Refreezing is caused by the significant change in basal velocity over a length scale that is short compared to the outer scale, regardless of how that basal velocity is modelled.

Despite the mathematical similarities previously discussed, our result differs substantially from the analysis in Haseloff et al. [48,49], where the no-slip-to-sliding transition inherent to the ice stream shear margin is found to have well-behaved solutions. This discrepancy is explained by the role of strong lateral flow in the shear margin setting: this lateral flow (where by ‘lateral’ we mean in the antiplane) causes large strain heating (so α≫1) that balances the cooling due to the speed-up in the along-flow direction, thus preventing refreezing. In the absence of lateral flow, strain heating remains of O(1), so the bed necessarily refreezes.

Last, we note that an argument supporting the impossibility of an abrupt transition is also provided in Fowler [34] (see their eqns (47)–(54)) and Fowler & Larson [33], who analyse the case of strong diffusion at the ice sheet scale, Pe → 0. For completeness, their argument is revisited in electronic supplementary material, S2.9. Here, we recall that it is unclear that this argument can be extended to the more realistic case of a finite Péclet number we are concerned with, since their argument would require well-posedness of the solution to an advection-diffusion equation with two boundary conditions specified on a space-like boundary.

(c). Direct numerical simulations confirm refreezing

To confirm the insight from the asymptotic analysis above, we solve the ice thickness scale problem (3c)–(3d) numerically. The problem (3c)–(3d) describes ice flow in a narrow boundary layer across sliding onset; this boundary layer therefore couples to the outer problems at the ice sheet scale, specifically through ice thickness, ice flux and temperature.

The outer thermal problem (2.5a) is an advection problem with diffusion only in the vertical. At large boundary-layer Péclet number Pe_BL, the thermal problem in the ice-thickness-scale boundary layer is advection-dominated. Therefore, the temperature field at the inflow of the boundary layer is dictated by the outer thermal problem upstream, while the temperature field at the inflow of the outer region downstream of the boundary layer is dictated by the boundary layer itself.

For the sake of the numerical implementation, it also important to recall that the transition between cold and temperate beds is a free boundary. A change in the location of the transition in the outer problem corresponds to a change in the location of the boundary layer, and therefore of the inflow temperature field as well as of ice flux and ice thickness: effectively, solving the boundary layer problem is necessary to determine a relationship between these three quantities. Without reference to the inequality constraints (3.4e)₂, (3.4f)₂, it would be possible to solve the boundary layer problem for an arbitrary inflow temperature profile, whereas we expect a unique solution if the constraints are enforced [50]. For a prescribed ice flux and thickness H⁽⁰⁾ and q⁽⁰⁾, we can therefore regard the inequalities (3.4e)₂, (3.4f)₂ as a constraint on the inflow temperature profile.

With Pe_BL≫1, the only important aspect of the inflow temperature profile is the temperature $T_{b,\mathrm{\infty }}=\underset{X\to -\mathrm{\infty }}{lim}{\mathit{\Theta }}^{(0)}{|}_{Z=0}$ at the bed in the far field; then, the energy balance at the bed dictates that the temperature gradient at the bed is geothermal, so once the bed temperature is known, so is the far-field temperature profile near the bed. As the inequality constraint on the temperate side refers precisely to basal temperature, we can effectively take the cold-side inequality as a constraint on T_b,∞. This is analogous to Haseloff et al. [48,49] and Schoof [50]; however, instead of computing a migration velocity of the transition from the inflow temperature profile, we are constraining the temperature profile based on the assumption of a zero migration velocity.

In what follows, we solve the boundary layer problem (3c)–(3d) with prescribed ice thickness H⁽⁰⁾ and flux q⁽⁰⁾, and treating the inflow temperature profile as a one-parameter degree of freedom. Above the bed, we arbitrarily prescribe a temperature profile exponentially decreasing towards the surface so as to satisfy equation (3.4d) (see details in electronic supplementary material, S2.8). Then we adjust T_b,∞ until both constraints are satisfied locally around the transition at X = 0, as they must (see the local analysis of the thermal problem near the origin in electronic supplementary material, S2.5 for a demonstration of this).

We solve the ice thickness scale problem numerically using Elmer/Ice [51]. The computational domain is −7.5 < X/H⁽⁰⁾ < 7.5, −4 < Z/H⁽⁰⁾ < 1; we use a triangular mesh locally refined near the bed and around the transition point, with a smallest grid size of 10⁻⁷. Figure 3 shows an example of the solution of the ice-thickness-scale problem. The velocity field is shown in figure 3a, with streamlines in light grey; note that the streamlines converge steeply towards the bed around X = 0 as a result of the basal speed-up. The temperature field is shown in figure 3c, and a close-up near the origin in figure 3d; similarly to streamlines, isotherms (in black) within the ice also converge towards the bed around X = 0, regardless of the singular shear heating there (figure 3b). In fact, a local analysis of the heat transport problem near the origin (see electronic supplementary material, S2.6) shows that, for X → 0, heating remains a small correction to the dominant balance in the heat transport problem, which is diffusion-dominated for |X|≪Pe^−2/3_BL.

Figure 3.

Solution of the boundary layer problem (3c)–(3d). (a) Magnitude of the velocity field, with contours denoting stream lines. (b) Strain heating near the origin. (c) Temperature field. The bold line denotes the melting point isotherm. (d) Close-up of (c) near the origin. Parameters are α = 0.5(q⁽⁰⁾/H⁽⁰⁾)², γ = 0.1H⁽⁰⁾, ν = 0.9, T_s =  − H⁽⁰⁾, Pe_BL = 100, T_b,∞ =  − 0.0512. Note that parameters and dependent variables are scaled with the leading order ice thickness and mass flux. (Online version in colour.)

Figure 4 illustrates basal temperature (top row) and basal energy budget (bottom row) for different values of bed temperature at the inflow, with a close-up of the solution near the origin in the right column. We find a unique T_b,∞ =  − 0.0512 such that the inequality constraints are satisfied locally near the transition point (see close-up in figure 4b,d), while inequalities are violated for different values of T_b,∞: for more negative T_b,∞ net freezing occurs everywhere for X > 0 (dark blue curve in figure 4c), thus violating the temperate side inequality, while for less negative T_b,∞ temperature is above the melting point everywhere upstream of X < 0 (light blue curve in figure 4a), thus violating the cold side inequality. This behaviour is consistent with Schoof [50], who shows that the local flow problem near the transition point is well-posed regardless of the singular heating rate.

Figure 4.

Inequality constraints as a function of bed temperature at the inflow, T_b,∞. (a,b) Basal temperature. (c,d) Basal energy budget. Parameters as in figure 3.

Even if the inequality constraints can be satisfied locally near X = 0 for T_b,∞ =  − 0.0512, our solution shows that they are not satisfied globally. In fact, the basal energy budget turns to negative at small distances downstream of the origin, while melting occurs immediately upstream (figure 4a,c). We therefore conclude that the boundary layer problem admits no solution, as predicted by our asymptotic analysis.

(d). Relaxing the assumption of a steady onset

In the last section, we have shown that a steady, sharp transition from a frozen to a temperate bed is not possible in our geometry. However, our analysis so far does not exclude the possibility of a migrating transition, similar to a travelling wave. If true, this would however be problematic, as it would indicate that the transition should continue to migrate until the entire ice sheet bed is either frozen or temperate. While not an impossible conclusion, we expect that cold and warm-bedded portions of the ice sheet can indeed coexist.

To show that such a migrating transition cannot exist, we extend our work of §3b,c to the case of a moving frozen-temperate boundary. Assuming a constant migration speed V , we introduce a mapping to a travelling coordinate system τ = t − V X. We also again restrict ourselves to the limit of Pe_BL≫1. Under these assumptions, we obtain the leading order heat transport problem in the onset boundary layer

$$(U^{(0)}-V)\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }X}+W^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}=O(P{e}_{BL}^{-1})\text{for}\hspace{0.17em}Z>0$$ 3.7a

with

$${\mathit{\Theta }}^{(0)}=0\text{on}Z=0,$$ 3.7b

while in the bed to leading order we have Θ⁽⁰⁾ =  − νZ. As before, the advection problem in the temperate region (X > 0) can be reduced to the Q-equation:

$$(U_b^{(0)}-V)\frac{\mathrm{\partial }Q}{\mathrm{\partial }X}-\frac{\mathrm{\partial }{U}_b^{(0)}}{\mathrm{\partial }X}Q=0,$$ 3.7c

with solution

$$Q=Q_0(U_b^{(0)}-V).$$ 3.7d

On the cold side (X < 0) we have instead Q = ν. Recalling that U⁽⁰⁾_b → 0 as ∼X^1/2 for X → 0⁺ (electronic supplementary material, S2.3), flux continuity at the transition point (valid up to O(Pe⁻¹_BL)) yields the integration constant Q₀ =  − ν/V . In the travelling coordinate frame, the bed enthalpy must satisfy

$$-V\frac{\mathrm{\partial }e}{\mathrm{\partial }X}=m^{(0)},$$ 3.7e

with m⁽⁰⁾ given by equation (3.4h), and where we have also assumed that the bed is well drained near the transition point.

For a prescribed direction of migration of the cold-temperate transition, the enthalpy equation (3.7e) cannot be satisfied, leading to the conclusion that a migrating transition is impossible. To see this, consider some fixed location near the transition in the temperate-based region; here, using (3.7d) the enthalpy equation can be rearranged as

$$-\frac{\mathrm{\partial }e}{\mathrm{\partial }X}=\frac{\alpha }{V}{T}_b^{(0)}{U}_b^{(0)}+\frac{\nu }{V^2}{U}_b^{(0)},$$ 3.7f

where $T_b^{(0)}{U}_b^{(0)}\sim {\left(U_b^{(0)}\right)}^2$.

Note that by prescribing the direction of migration (that is, the sign of ∂e/∂t at our prescribed location in the temperate region), we implicitly specify the sign of the melt rate, which must be positive if migration is into cold (V < 0, ∂e/∂t > 0) and negative if migration is into warm (V > 0, ∂e/∂t < 0).

Keeping this in mind, and moving to equation (3.7f), for V > 0 we can satisfy ∂e/∂t < 0 if −∂e/∂X < 0. However, the right-hand side of equation (3.7f) is strictly positive when V > 0, thus we conclude that V > 0 does not allow the enthalpy equations to be satisfied. For the case of migration into the cold region (V < 0), once again we can satisfy ∂e/∂t > 0 if −∂e/∂X < 0. With the right-hand side of equation (3.7f) behaving as ∼V⁻²U⁽⁰⁾_b for U⁽⁰⁾_b → 0, which is strictly positive, it is straightforward to see that we arrive once again at an inconsistency. We have thus demonstrated that a migrating onset is not compatible with the energy budget of the bed, regardless of the direction of migration.

4. Ice sheet flow with subtemperate sliding

(a). The necessity of a gradual transition

Above, we have assumed that a discontinuous change in boundary conditions occurs at the location where the melting point is reached. Mechanically, this corresponds to a rapid increase in sliding velocity over a distance comparable with the ice thickness (the mechanical component of the boundary layer solution dictates as much). We have concluded this is incompatible with a bed that remains unfrozen: the rapid onset of sliding necessarily leads to refreezing.

To fix this inconsistency, we first seek to understand the role of the discontinuity in basal boundary conditions. To this aim, let us consider the effects of altering the sliding coefficient γ so that it is no longer constant, in such a way as to make the onset of sliding more spread out. We are now going to show that, even with a continuous sliding coefficient, refreezing would necessarily happen if the transition from no slip to O(1) sliding occurred over a distance much shorter than the ice sheet scale.

In the case of an abrupt sliding onset, refreezing is the result of having an advection-dominated temperature field: the horizontal speed-up of ice flow near the bed draws down cold ice. This relies on our assumption that the Péclet number Pe for the ice sheet as a whole is O(1). Specifically, by Péclet number, we mean here the ratio of the divergences of advective flux and diffusive fluxes, the latter being dominated by diffusion in the vertical (see equation (2.3b)). Over any horizontal length scale shorter than the ice sheet scale (but long enough for horizontal diffusion to be irrelevant), the correspondingly defined Péclet number becomes large: in other words, the advective heat flux now changes over a shorter length scale, while the vertical scale remains unchanged. This is why Pe_BL = Pe/ε is large in the boundary layer model derived in §3a.

Considering now the case of a sliding coefficient that changes continuously over a distance much shorter than the ice sheet scale, we understand that the derivation of the Q-equation (3.5d) still holds in the temperate-bedded part of the ice sheet, because the energy balance is dominated by advection at these short spatial scales. Then, by the same argument as for the discontinuous friction coefficient case, the sliding velocity cannot increase by O(1) from a near-zero value over these shorter length scales without causing refreezing.

(b). Temperature-dependent sliding

A non-constant sliding coefficient and a gradual onset of sliding are instead possible if sliding initiates below the melting point and depends of the deviation of bed temperature from the melting point itself. This is commonly referred to as subtemperate sliding. Such subtemperate sliding is not just a mathematical convenience: observational evidence suggests that some amount of sliding occurs even when T < T_m (in dimensional terms) at the bed (e.g. [13–15]) as a result of regelation and premelting.

Overall, the physics underlying subtemperate sliding imply that the sliding velocity should increase as bed temperature approaches the melting point. In fact, the thickness of premelted water films increases with temperature [52]. Increased film thickness reduces the contact surface between ice and bed, hence facilitating sliding. Moreover, premelting also plays a role in facilitating regelation [53], which is essential to basal sliding (e.g. [43]). A general form for a (dimensional) sliding law like (2.1g) that accounts for subtemperate sliding is u = f(τ_b, T) [13,54]. f is a monotonically increasing function of T and τ_b, subject to the constraint that T ≤ T_m. The fully temperate sliding law is recovered when the bed attains the melting point; in our case, this implies that f(τ_b, T_m) = C⁻¹τ_b. Moreover, we expect significant subtemperate sliding to occur only within a temperature range T₀ of the melting point, which amounts to f(τ_b, T)∼0 when T_m − T≫T₀. One suitable choice for f is, for instance,

$$f({\tau }_b,T)=C^{-1}\exp [(T-T_m)/T_0]{\tau }_b\text{for}T\le T_m.$$ 4.1

With sliding occurring also below the melting point, basal boundary conditions for the velocity field in the cold region (2.1f) are replaced by the sliding law (4.1), along with bed impermeability (2.4e₂), while boundary conditions for the heat equation (2.2e) are replaced by

$${\left[T\right]}_{-}^{+}=e=0\text{if}\hspace{0.17em}T<T_m.$$ 4.2

This implies a jump in heat flux across the bed, whose magnitude is constrained by the requirement that energy is conserved at the bed itself.

We non-dimensionalize the new model components as described in §§2b. The scaled version of the sliding law is then u* = F(τ*_b, T*/δ), with δ = T₀/[T] a new non-dimensional parameter that compares the range of temperatures over which subtemperate sliding is significant to the temperature scale of the ice sheet. For (4.1) (omitting asterisks on dimensionless variables once more), we get

$$u_b={\gamma }^{-1}\exp (T/\delta ){\tau }_b\text{for}T\le 0$$ 4.3

where the sliding coefficient increases continuously from near-zero values to the temperate sliding coefficient γ⁻¹ as bed temperature increases from very negative values to the melting point. In this sense, subtemperate sliding is a physically based regularization of the hard switch.

The parameter δ controls the range of subfreezing temperatures over which sliding is significant, meaning that sliding velocities are O(1) when T is O(δ). Based on experimental evidence [55], we expect δ to be small, meaning that the range T₀ of temperatures over which subtemperate sliding is possible is small compared to the natural scale [T] for temperature variations in the ice sheet. This suggests that subtemperate sliding velocities of O(1) only occur when the bed temperature is close to the melting point.

It is worth noting that the temperature-dependence alone does not change the insight provided by the hard switch case: a rapid speed-up remains impossible. When $\delta \to 0$, the bed temperature has to remain approximately at the melting point in the region where significant subtemperate sliding takes place. As basal temperature remains constant throughout this region, the Q-equation holds with the same limitations discussed in §3b. Then, by the same argument as in §3b, we expect the basal energy balance to be violated if the distance over which the sliding velocity changes from near-zero to O(1) is short compared to the ice sheet scale. This is also the case if δ∼O(1), as illustrated in electronic supplementary material, S3.1. The only option left is therefore that the transition region extends over a distance comparable with the ice sheet length, in which case the local Péclet number would remain O(1).

(c). The limit of significant subtemperate sliding close to the melting point

In this section, we are concerned with deriving the simplest possible ice sheet scale model with temperature-dependent sliding. We start our analysis from the subtemperate region, and we will then demonstrate that within the same framework we recover the boundary conditions we previously had for the frozen and temperate regions in the appropriate limits.

Consider a portion of the ice sheet where subtemperate sliding occurs. We are concerned with the case of a smooth sliding onset, so the length of this region has to be asymptotically comparable with the ice sheet scale. Over these long distances, shallow ice mechanics are appropriate, so equations (2.6) hold, and so does also the large-scale thermal model (2.5) with basal boundary conditions (4.2) and the temperature-dependent sliding law (4.3).

Motivated by the scaling argument for δ discussed in §§4b, we consider the case of significant sliding close to the melting point, and seek to derive a set of simplified boundary conditions for the subtemperate region. To this aim, let us take the limit of $\delta \to 0$ and expand temperature as T = T⁽⁰⁾ + δT⁽¹⁾ + o(δ), while for basal shear stress, basal heat flux, and sliding velocity we put

$${\tau }_b\sim {\tau }_b^{(0)}+O({\epsilon }^2),u_b=u_b^{(0)}+O(\delta ),{\frac{\mathrm{\partial }T}{\mathrm{\partial }z}|}_{z=b}\sim {\frac{\mathrm{\partial }{T}^{(0)}}{\mathrm{\partial }z}|}_{z=b}+O(\delta ),$$ 4.4

where leading order basal shear stress and basal heat flux are strictly ∼O(1). Since in the subtemperate region sliding is O(1) when basal temperature is O(δ), we put

$$T^{(0)}=0\text{on}\hspace{0.17em}z=b,\text{if}{u}_b^{(0)}<{\gamma }^{-1}{\tau }_b^{(0)},$$ 4.5a

where the latter inequality results from the function f(τ_b, T) in (4.3) being monotone in T. Then, the basal energy budget (4.2₂) yields

$${\left[\frac{\mathrm{\partial }{T}^{(0)}}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha {\tau }_b^{(0)}{u}_b^{(0)}=0\text{on}\hspace{0.17em}z=b\text{if}{u}_b^{(0)}<{\gamma }^{-1}{\tau }_b^{(0)}.$$ 4.5b

The boundary conditions (4e) apply in a finite region of the bed that we expect to separate the regions in which the cold and fully temperate boundary conditions stated previously hold. This is what we refer to as the ‘subtemperate region’.

Assuming instead T⁽⁰⁾(z = b)∼O(1), it is straightforward to see that we recover the boundary conditions for the frozen region, (2.5e₂). In fact, from the sliding law equation (4.3) we get

$$u_b^{(0)}\sim \exp (-{\delta }^{-1})\approx 0\text{for}\delta \to 0,$$ 4.6a

while the basal energy budget reduces to the flux continuity statement

$${\left[\frac{\mathrm{\partial }{T}^{(0)}}{\mathrm{\partial }z}\right]}_{-}^{+}=0\text{on}\hspace{0.17em}z=b\text{if}{T}^{(0)}<0,$$ 4.6b

which serves as basal boundary condition for the thermal problem (2.5a)–(2.5b) along with temperature continuity (2.5e₁). Last, fully temperate sliding is recovered automatically from (4.3) when basal temperature reaches the melting point.

In summary, taking the limit of δ → 0 in the temperature-dependent sliding law allows us to partition the ice sheet into a frozen, subtemperate and temperate region (figure 5). At the bed (z = b), the following set of boundary conditions applies,

$${\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}=u_b=0\text{if}T<0,$$ 4.7a

$${\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha u_b{\tau }_b=T=0\text{if}0<u_b<{\gamma }^{-1}{\tau }_b,$$ 4.7b

$$T=u_b-{\gamma }^{-1}{\tau }_b=0\text{if}{\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha u_b{\tau }_b>0.$$ 4.7c

These show that a region of subtemperate sliding (equation (4.7b)), in which sliding velocities are slower than they would if the bed was fully temperate, is interposed between frozen (equation (4.7a)) and temperate (equation (4.7c)) region, where the velocities are constrained by the need to maintain the bed in thermal balance. Last, we refer to the location of the cold-subtemperate boundary, and of the subtemperate-temperate boundary as x = x_s and x = x_t, and assume that the continuity statements (2.7f) hold at each of these boundaries.

Figure 5.

Model geometry for a laterally uniform ice sheet with temperature-dependent sliding. A subtemperate region for x_s < x < x_t is now interposed between cold (x < x_s) and temperate x > x_t regions of the bed, differently from the hard switch case illustrated in figure 1.

Looking at the limiting behaviour for δ → 0 allows us to highlight some peculiar features of the subtemperate region. First of all, a consequence of δ≪1 is that the leading order bed temperature is approximately at the melting point uniformly in the subtemperate region. This serves as bed boundary condition on the heat equation, which can then be used to compute the basal heat flux. Second, the energy balance of the bed links the basal heat flux directly to the sliding velocity, subject to the requirement that the bed remains in thermal balance. Practically, the energy balance of the bed dictates the sliding velocity in the subtemperate region, while the sliding law decouples from the leading order model. In fact, the actual O(δ) deviation of basal temperature from the melting point can be computed a posteriori using the full sliding law (4.3) from the leading order sliding velocity and basal shear stress, but that computation is purely diagnostic and does not couple back into the ice or heat flow problem at leading order.

A final note is that our asymptotics in the δ → 0 limit lead to the same set of boundary conditions and inequality constraints that were previously proposed by Fowler [34], although our approach to derive these constraints is different.

(d). Numerical implementation of the reduced model

To our knowledge, the boundary conditions (4.7) have not been implemented in this form in a computational solution of an ice flow model. The rest of the paper is concerned with doing exactly that, in the context of the ice sheet scale model (2.5a)–(2.5b) and (2.6).

Since we intend to model the whole ice sheet, a few additional specifications are required. First of all, in order to avoid singular behaviour at the terminus, we consider a marine ice sheet that ends at a grounding line located at x = x_g, as illustrated in figure 5. Here, we impose ice thickness at flotation and a condition on the ice flux

$$h=-\rho {\rho }_w^{-1}b,q=Q_g(h),\text{on}\hspace{0.17em}x=x_g,$$ 4.8

where the function Q_g is given by Schoof [56]; to the current non-dimensionalization this reads:

$$Q_g=q_0{\gamma }^{-1/3}{h}^{5/2}\mathrm{and}{q}_0={\epsilon }^{-1}{[(1-{\rho }_w^{-1}\rho )/8]}^{1/2}\sim O(1).$$ 4.9

We also assume there is a symmetrical ice divide at x = 0, where horizontal velocity and hence surface slope vanishes, so that

$$q=0\text{on}\hspace{0.17em}x=0.$$ 4.10

Last, we restrict ourselves to the case of a prescribed, uniform, surface mass balance $\dot{b}$ and surface temperature T_s.

We solve the model numerically in a stretched coordinate system that maps the cold, subtemperate and temperate subdomains onto unit squares (see electronic supplementary material, S4.1). Model variables are thickness h and temperature in the ice T, as well as the end points x_s, x_t and x_g of the cold, subtemperate and temperate subdomains. Since we are concerned with steady states, the heat equation in the bed can be integrated analytically leading to T =  − νz for z < b in all subdomains. Our computational domain therefore restricts to the ice only.

At the boundaries between the subdomains, ice thickness, velocity and temperature must be continuous. In addition, the inequality constraints (4.7a)–(4.7b) for the cold-subtemperate boundary, as well as (4.7b)–(4.7c) for the subtemperate-temperate boundary must be satisfied simultaneously, implying that T(z = b) = u_b = 0 at the cold-subtemperate boundary, x = x_s, and γ⁻¹τ_b = (ατ_b)⁻¹(∂T/∂z + ν) and m = 0 at the subtemperate-temperate boundary, x = x_t. These equalities in practice serve to fix the free boundary locations. Finally, it is worth noting that as we are not solving for the enthalpy of the bed, we cannot enforce directly a positive basal energy budget in the temperate region as demanded by (4.7c). Rather, we verify a posteriori that solutions satisfy this constraint.

We implement a finite volume discretization, with a second-order centred scheme for the mass flux and vertical heat fluxes. As the x-direction is time-like for the heat equation, we implement a first-order upwind scheme for the horizontal heat flux. In our finite volume discretization, the vertical heat flux at the bed is computed at cell centre locations, while ice velocity is computed at cell boundaries. However, in the subtemperate ‘sliding law’, basal heat flux must be computed at these cell boundary locations; we do so by effectively ‘upwinding’ as well: the heat flux at the cell boundary is taken to be equal to flux at the cell centre location for the cell immediately upstream (see electronic supplementary material, S4.2).

We have developed a time-stepping routine based on a backward Euler step as well as a separate steady-state solver, both based on Newton's method. In this paper, we focus purely on the steady-state problem, and consider time-dependent behaviour and the stability of steady states in the companion paper. While we have computed the discrete Jacobian for the complete problem, we employ a dimensional reduction for the steady-state problem, which we describe below.

For a given set of locations for the free boundaries x_s, x_t and x_g as well as a given ice thickness h₀ at the centre of the ice sheet, ice thickness, temperature and ice velocity can be solved for sequentially, one column of grid cells at a time. In steady state, ice flux is known at every horizontal cell boundary from the integral of surface mass balance. With basal heat flux at the cell boundary set equal to its upstream cell centre value, the known ice flux corresponds to a unique velocity field and, through the dependence of the velocity field on surface slope, we can compute ice thickness at the next cell centre downstream. With the upwind scheme for advection, the computation of temperature in the next row of cells is then equivalent to a backward Euler step in the horizontal coordinate, with a prescribed velocity. For a given guess for (h₀, x_s, x_t, x_g), we can therefore compute the residual of the continuity conditions at subdomain boundaries as well as the residual in the flotation and flux conditions at the grounding line. Solving the steady-state problem amounts to finding the appropriate roots that reduce all four residuals to zero. The full procedure is described in electronic supplementary material, S4.3.

(e). Results

Figure 6 shows the details of a single steady-state solution, with shaded areas denoting the subtemperate subdomain, while figure 7 illustrates how the subdomain lengths depend on the model parameters γ and Pe. All solutions shown satisfy the constraint (4.7c). In fact, despite enforcing a zero basal melt rate at the subtemperate-temperate transition rather than the inequality (4.7c), our solver never produced solutions violating the latter. We therefore conclude that a transition with an extended region of subtemperate sliding effectively prevents refreezing of the bed, as suggested by our former analysis.

Figure 6.

Steady-state solution of the ice sheet scale model with subtemperate sliding and δ = 0. Parameter values are α = 0.4, γ = 0.7, ν = 0.5, Pe = 4, $\dot{b}=0.5$, T_s =  − 1, q₀ = 5, shaded areas denote the subtemperate region. (a) Horizontal velocity; contours spaced by 0.5. (b) Temperature field; contours spaced by 0.2. The bold line denotes the isotherm T = 0. (c) Streamlines. (d) Basal energy budget. (e) Velocity u and heat flux −∂T/∂z along the bed, z = b. The inset shows a zoom near the subtemperate-temperate boundary. (Online version in colour.)

Figure 7.

Domain length scaled with grounding line position as a function of friction coefficient (a) and Péclet number (b). Parameter values are α = 0.4, ν = 0.5, a = 0.5, q₀ = 5, T_s =  − 1; Pe = 7.5 in (a), γ = 0.7 in (b). Note that grounding line position is a function of γ. (Online version in colour.)

We now examine the solution depicted in figure 6 in more detail. A notable feature of the subtemperate region is that the ice surface becomes concave at some point, with ∂²h/∂x² < 0: since the ice flux in steady state increases monotonically in x, this implies that there is a region in which the flux increases with decreasing surface slope, contrary to the usual diffusive behaviour of shallow ice models. We will consider the stability implications of this feature in the companion paper.

A second consideration is that, by construction, the horizontal component of ice velocity is continuous throughout the ice column at both cold-subtemperate and subtemperate-temperate boundaries (figure 6a, subdomain boundaries marked with dotted lines). As a result, streamlines (in figure 6c) and isotherms (white curves in figure 6b) are also continuous. This behaviour differs substantially from the abrupt sliding onset case discussed in §2: there the horizontal velocity was discontinuous at sliding onset in the ice sheet scale model, and so were streamlines and isotherms.

Nevertheless, the strain rate ∂u/∂x and gradient of the basal heat flux ∂[∂T/∂z|_z=b]/∂x remain discontinuous at subdomain boundaries (see inset of figure 6e). We demonstrate in the electronic supplementary material, S4.4 that this is indeed to be expected. Correspondingly, the vertical velocity component is also discontinuous, thus streamlines are continuous but not differentiable across these boundaries. This issue is related to the fact that we are solving for velocity at the ‘outer’ scale in a shallow model; in reality, a local vertical velocity field would presumably still result if one were to introduce passive boundary layers at subdomain boundaries.

Last, figure 7 illustrates how the length of cold, subtemperate and temperate regions depend on the friction coefficient γ (figure 7a) and on the large scale Péclet number (figure 7b). We observe that the lengths of cold and temperate region depend monotonically on the parameters, whereas the subtemperate regions has maximum length for some O(1) values of γ and Pe. To interpret these results, it is worth recalling that an increase in Pe at constant γ increases the strength of advection at the expenses of strain heating, while an increase in γ makes the bed less slippery. However, the dependence of the subdomain lengths on the parameters relies on complex feedbacks between ice geometry, grounding line position, and strength of advection, whose detailed analysis is beyond the scope of the present work.

5. Discussion and conclusion

Basal thermal transitions are a widespread feature of ice sheets [57,58], and are likely related to the onset and dynamics of fast ice flow through thermally activated sliding. In spite of this, the physics at play in these transitions have remained largely unexplored so far, and so have the mathematical properties of the models used to represent such transitions in ice sheet models.

In the first part of the paper (§§2 and 3), we analysed the case of an abrupt sliding onset, where by ‘abrupt’ we mean that fully temperate sliding occurs where the bed attains the melting point. In this case, which is equivalent to having a discontinuous friction coefficient, the acceleration from no slip to an O(1) sliding velocity is localized over a distance comparable with the ice thickness, hence the onset can be treated as a boundary layer in a shallow ice model.

A numerical and asymptotic analysis of the boundary layer model (§3c) allowed us to show that, along a flow line, refreezing on the temperate side of the transition is to be expected, and that an abrupt sliding onset cannot exist. Physically, speed-up of basal velocity causes cold ice to be drawn down to the bed; the resulting steepening of the basal temperature gradient then causes the bed to refreeze. We have also shown that this result holds true for any acceleration from zero to a finite sliding velocity that occurs over a distance short enough that heat transport in the ice is advection-dominated, regardless of how the sliding velocity is modelled.

The exception to this case occurs where dissipation rates near the transition are large enough to offset the effects of downward advection; in the two-dimensional configuration studied here, there is no mechanism by which such large dissipation rates can occur, which differs from ice stream shear margins [48–50]. There, antiplane shear can generate very large amounts of heat without affecting advection. In the near-parallel flow configuration commonly observed upstream of ice stream initiation this regime is not accessible, thus suggesting that different physical processes lead to the onset of basal sliding.

The problem we encounter with the abrupt onset is purely of a thermal nature, and is unrelated to the mechanics of the sliding transition. In fact, our boundary layer model makes no simplifications to the stress tensor near the transition point (which would be the inner problem), where we solve a full Stokes model. Our analysis therefore demonstrates that extensional stresses do not act as an effective regularization of an abrupt sliding onset, as previously suggested by Bueler & Brown [35]. This also holds for a discontinuous friction coefficient around sliding onset, as for instance used by Brinkerhoff & Johnson [27]: locally, speed-up of ice near the bed over a short length scale (comparable with ice thickness) will still cause drawdown of cold ice, potentially leading to refreezing of the bed. This presents significant problems for large-scale ice sheet models (or rather, their numerically implemented versions). While coarse horizontal resolution may obscure the drawdown of cold ice at sharp sliding transitions, with grid spacings larger than ice thickness typical of many such models, this does not mean that the unphysical ‘solutions’ that result are to be trusted.

Our analysis for the hard switch suggests that the change in basal slipperiness must instead be more spread out, in such a way as not to localize the speed-up over distances short enough that energy conservation becomes advection-dominated. We therefore considered the role of temperature as a regularization of an abrupt sliding onset in the second part of the paper (§4), and demonstrated that sliding has to initiate below the melting point and depend on basal temperature in order for basal thermal transitions to be possible. This is what is commonly referred to as subtemperate sliding, and relies on regelation and premelting to lubricate the ice-bed contact at subfreezing temperatures.

Mathematically, subtemperate sliding is usually described by explicitly temperature-dependent sliding laws of the form u_b = f(τ_b, T), with f monotonically increasing with T. In this sense, temperature acts as a regularizer of the abrupt sliding onset: the length of the transition is now determined by the structure of the temperature field, subject to the requirement that the bed remains in thermal balance until the melting point is attained and net melt can occur. Physically, such subtemperate sliding occurs within a range of temperatures T₀ below the melting point that is much smaller than the temperature scale [T] of the ice sheet. This is quantified by the non-dimensional parameter δ = T₀/[T], which we take to be small in order to construct a leading order model for the subtemperate region valid at the ice sheet scale (§4c).

Taking the δ → 0 limit leads to the same basal boundary conditions as in Fowler [34], with a subtemperate region with the properties above interposed between the cold and temperate part of the ice sheet. The extent of the cold, subtemperate, and temperate portions of the ice sheet is unknown a priori, and depends on the temperature field and geometry of the ice sheet. In the last part of the paper (§4d,e), we implement these basal boundary conditions in an ice sheet scale model, and numerically solve the resulting free boundary problem. Our results show that no refreezing occurs in this case, thus confirming that a gradual transition with subtemperate sliding is potentially an admissible configuration for the ice sheet.

We conclude by noting that, even though observational evidence of sliding at subfreezing temperature has long been available (e.g. [13–15]), its importance in continent-scale ice sheet modelling has been unclear so far. Our work suggests that inclusion of subtemperate sliding is necessary to describe a fully fledged ice sheet where frozen and temperate regions coexist.

There are several complications to the ice sheet scale model with subtemperate sliding that we have derived, which we will address in the companion paper. First of all, the solutions we presented in §4 are for a steady state, hence our conclusions about the existence of a well-behaved solution to the leading order model are restricted to this case. We will dedicate the first part of the companion paper to the study of the dynamics and stability of these steady states, both numerically and analytically. In this respect, it is worth noting that the numerical method we have implemented for the solution of the steady state (which includes a backward Euler step, and therefore requires the computation of the full Jacobian) lends itself to a numerical stability analysis, and will allow us to assess the dynamical properties of the ice sheet scale model with subtemperate sliding.

Last, it is important to recall that the ice sheet scale model with subtemperate sliding we derived is effectively a zeroth-order approximation valid for δ = 0. The main advantage of this approximation is that it allows us to take basal temperature uniformly at the melting point in the subtemperate region, thus decoupling the thermal problem from the mechanical problem. Even though a small δ is physically justified, we will show in the companion paper that δ = 0 is a singular perturbation, and small deviations of bed temperature from the melting point must be resolved instead. Nonetheless, the insight gained by setting δ = 0 remain largely valid, particularly that temperature-dependent sliding is an essential model component when seeking to describe basal thermal transitions, as well as our conclusion that the frozen-temperate transition must span distances comparable with the ice sheet length when lateral flow is negligible.

Ethics

The research described here did not have any experimental or data collection components.

Data accessibility

This article has no additional data. Source code for the numerical simulations presented in the paper is available at: https://github.com/elisamantelli/subtemperate_sliding_rspa_2019.

Authors' contributions

E.M. carried out the research and wrote the paper. M.H. carried out the calculations with Elmer/Ice (figures 3 and 4). C.S. designed the project. E.M. and C.S. jointly drafted the structure of paper and electronic supplementary material. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

E.M. acknowledges financial support from Stanford's SiGMa and RadioGlaciology research groups. E.M. and M.H. were supported by NSERC Discovery grant no. RGPIN 357193-13. C.S. acknowledges NSERC Discovery grant nos. RGPIN-2018-04665 and RGPIN 357193-13. Compute Canada provided computing resources.