Ice sheet flow with thermally activated sliding. Part 2: the stability of subtemperate regions

Abstract

The onset of sliding in ice sheets may not take the form of a sharp boundary between regions at the melting point, in which sliding is permitted, and regions below that temperature, in which there is no slip. Such a hard switch leads to the paradox of the bed naturally wanting to refreeze as soon as sliding has commenced. A potential alternative structure is a region of subtemperate sliding. Here temperatures are marginally below the melting point and sliding velocities slower than they would if the bed was fully temperate. Rather than being controlled by a standard sliding law, sliding velocities are then constrained by the need to maintain energy balance. This thermal structure arises in temperature-dependent sliding laws in the limit of strong sensitivity to temperature. Here, we analyse the stability of such subtemperate regions, showing that they are subject to a set of instabilities that occur at all length scales between ice thickness and ice sheet length. The fate of these instabilities is to cause the formation of patches of frozen bed, raising the possibility of highly complicated cold-to-temperate transitions with spatial structures at short length scales that cannot be resolved in large-scale ice sheet simulation codes.

1. Introduction

Ice sheets can slide over their beds when the basal temperature is at the melting point. The extent to which sliding is also possible at lower temperatures remains somewhat unclear, but sliding velocities are likely to be significantly reduced for a given basal shear stress as temperatures are lowered [1–3]. The onset of sliding is a significant component in the dynamics of ice sheets and has been linked with the formation of finger-like patterns of fast flow known as ice streams [4–6], as well as with oscillations in ice thickness known as surges [7–9].

The interior of typical ice sheets is likely to be cold-bedded due to significant downward advection of cold ice near the central ice divide, with insignificant sliding velocities. Cumulative shear heating along flow lines can then progressively warm the ice eventually bringing it to the melting point, where sliding is known to occur. Here, we consider the physics of the transition in space between a no-slip boundary condition applied at cold temperatures and sliding on temperate portions of the bed.

Many ice-sheet models have assumed that sliding commences when temperature reaches the melting point, with a no-slip boundary condition at any lower temperature (e.g. [10]). In a companion paper [11], we have shown that such a ‘hard switch’ transition leads to a paradox, predicting that the rapid downdraw of cold ice associated with the horizontal speed-up that occurs as sliding is initiated should lead to refreezing of the bed (see also [12]), at least in the physically realistic parameter regime of a moderate Brinkman number (quantifying the strength of shear heating with respect to the background conductive cooling) and a large local Péclet number. This result confirms an analogous earlier one due to Fowler & Larson [13] and Fowler [14], who show that a hard switch is also impossible in the case of a zero Péclet number: in that case, a simple energy balance calculation confirms that not enough heat is dissipated on the temperate side of the ‘hard switch’ in order to maintain the bed at the melting point.

In the companion paper, we went on to show that the notion of a region of ‘subtemperate sliding’ as first suggested by Fowler & Larson [13] and Fowler [14] alleviates the problem of rapid downdraw of cold ice and permits a smooth transition from a non-sliding, cold to a sliding, temperate bed. Here, sliding occurs at temperatures below the melting point, with the rate of sliding dependent on the temperature depression. A clearly delineated ‘subtemperate region’, distinct from a cold bed, arises if we assume that significant sliding is possible only when temperatures are depressed very slightly below the melting point, so that the bed appears to be at the melting point, but sliding occurs at a slower rate than that predicted by a fully temperate sliding law. What sets the sliding velocity in that case at leading order is the need to maintain a vanishing energy balance at the bed, while the net energy balance at a fully temperate bed can be positive, corresponding to a finite rate of basal melting.

What remains unclear is whether this construction is stable, in the sense of whether small perturbations in basal heat flux and sliding velocity decay or grow over time: for purely cold or purely temperate ice sheet beds, the corresponding ice sheet flow is known to be stable, satisfying a diffusion problem for ice thickness [14]. The purpose of the present paper is to analyse the stability of ice sheets that contain a region of subtemperate bed, which the companion paper has identified as a plausible structure for the cold-to-temperate transition.

Although our aim ultimately is to use the tools developed here to address the question of patterning in ice flow, and hence of ice-stream formation, we consider stability purely in one horizontal dimension. The paper is structured as follows: first, we review the large-scale model for an ice sheet including a subtemperate region as constructed in the companion paper (§2a). Using the numerical tools developed in the companion paper, we consider the stability properties of that large-scale model, identifying two different types of instability associated with different sliding regimes (§2b). Both instabilities appear pathological, with discretized modes that exhibit gridpoint-to-gridpoint oscillations.

Next, we use a simplified slab geometry in order to identify the physics behind these numerically detected instabilities. We again use a linearization of the large-scale model, this time in its continuum guise, which can be solved in closed form for the slab geometry. We confirm the presence of a pathological instability that appears as backward diffusion in the model (§2c). This suggests that a stabilizing mechanism must be sought at shorter length scales, which leads us to reconsider the asymptotic limits used to construct the large-scale models as singular limits.

Rescaling to resolve that singular behaviour, however, turns out to open something of a Pandora's box, with a proliferation of additional instabilities, some of which are hidden at the large scale: not only do instabilities occur at all wavelengths from the ice sheet length scale to the ice thickness scale, they also occur at time scales that are much faster than the large-scale, advective time scale that one might naively assume to dominate the dynamics of the ice sheet as a whole (§§3 and 4).

We conclude our analysis of these instabilities with an attempt to track their evolution to finite amplitude numerically in §5. We find that the growth of the instability does not saturate before parts of the bed become ‘cold’ in the sense of suppressing sliding velocities almost entirely. The subsequent evolution of the resulting patches of alternating cold and subtemperate bed is left open here; among our more challenging findings is that there is poor separation between the length scales at which instabilities occur in ice flow with a subtemperate bed, and that future work may need to employ models that simultaneously resolve both the ice thickness and ice sheet length scale, without employing notions of ‘shallowness’ to simplify either model formulation, or to justify coarse resolution in Stokes flow models for ice flow.

2. Temperature-dependent sliding: a leading order approximation

(a). A continent-scale model

We start our analysis by re-stating the ice sheet scale model with subtemperate sliding derived in the companion paper (see §4 for a detailed derivation), which we give here in its non-dimensional form. Let us consider a laterally uniform ice sheet with surface elevation z = s(x, t) and bed elevation z = b(x), where z = 0 denotes sea level, as represented in figure 1. Here, x is the horizontal spatial coordinate, scaled with the length of the ice sheet, z is the vertical coordinate and t is time scaled with the natural advective time scale of the ice sheet. The ice has thickness h(x, t) = s − b and extends from a divide located at x = 0 to a margin located at x = x_g. The velocity field scaled with the shear velocity is u = (u, w), while T is temperature, scaled with the difference between the melting point and a representative surface temperature. At this large horizontal length scale, it is justified to take the limit of small ice sheet aspect ratio, ε → 0 (i.e. the so-called shallow ice approximation [15,16]), leading to a horizontal velocity u and basal shear stress τ_b

$$u=\frac{1}{2}[h^2-{(s-z)}^2]\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|+u_b\mathrm{and}{\tau }_b=h\left|\frac{\mathrm{\partial }s}{\mathrm{\partial }x}\right|,$$ 2.1a

where u_b = u(z = b) is the sliding velocity, which for now we regard simply as a function of basal shear stress and basal temperature. From the expression above, we compute the mass flux as

$$q={\int }_b^{s}u\hspace{0.17em}\mathrm{d}z=\frac{h^2}{3}{\tau }_b+u_{b}h;$$ 2.1b

then, along-flow changes in q drive the evolution of the ice surface through the diffusion equation

$$\frac{\mathrm{\partial }s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }q}{\mathrm{\partial }x}=\dot{b},$$ 2.1c

where $\dot{b}$ describes mass gain by surface accumulation (if positive) or loss by melt or sublimation (if negative). Boundary conditions for equation (2.1c) at the divide and grounding line are

$$q=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}x=0,h=-{\rho }_w{\rho }^{-1}b\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}x=x_{\mathrm{g}},q=Q_{\mathrm{g}}(h)\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}x=x_{\mathrm{g}},$$ 2.1d

whereby we have assumed that the ice sheet is symmetric at x = 0, and that it has a marine margin whose location is defined by incipient flotation, equation (2.1d₂), and by a flux condition, equation (2.1d₃), where ρ, ρ_w denote ice and water density, respectively, and the function Q_g is given by Schoof [17].

Figure 1.

Model geometry for a laterally uniform ice sheet with temperature-dependent sliding. The ice sheet extends from an ice divide at x = 0 to a grounding line at x = x_g, and undergoes a transition from frozen to temperate bed through a region of subtemperate sliding. The subtemperate region (x_s < x < x_t) is interposed between cold (x < x_c) and temperate (x > x_t) regions of the bed.

The vertical velocity w decouples from the leading order mechanical problem and is the solution to the local mass conservation

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}=0\text{on}\hspace{0.17em}b<z<s,$$ 2.2a

which can be integrated between the bed and the ice surface straightforwardly to compute w using (2.1a₁) for the horizontal velocity u, along with the basal boundary condition

$$u_b\frac{\mathrm{\partial }b}{\mathrm{\partial }x}-w=0\text{on}\hspace{0.17em}z=b.$$ 2.2b

Lastly, omitting terms of O(ε²), the leading order heat transport problem reads

$$\begin{array}{rl}& \mathit{P}\mathit{e}\hspace{0.17em}(\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }\mathit{t}}\mathit{+}\mathit{u}\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }\mathit{x}}\mathit{+}\mathit{w}\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }\mathit{z}})\mathit{-}\frac{{\mathrm{\partial }}^{\mathit{2}}\mathit{T}}{\mathrm{\partial }{\mathit{z}}^{\mathit{2}}}\mathit{=}\alpha \mathit{a}\mathit{(}\mathit{x}\mathit{,}\mathit{z}\mathit{,}\mathit{t}\mathit{)}\mathit{\text{for }}\mathit{b}<\mathit{z}<\mathit{s}\mathit{,}\end{array}$$ 2.3a

$$\begin{array}{rl}& \mathit{P}\mathit{e}\hspace{0.17em}\frac{\mathrm{\partial }\mathit{T}}{\mathrm{\partial }\mathit{t}}\mathit{-}\frac{{\mathrm{\partial }}^{\mathit{2}}\mathit{T}}{\mathrm{\partial }{\mathit{z}}^{\mathit{2}}}\mathit{=}\mathit{0}\mathit{\text{for }}\mathit{z}<\mathit{b}\mathit{,}\end{array}$$ 2.3b

with strain heating a = (∂u/∂z)². In the latter equations, Pe is the ice sheet scale Péclet number, which can be seen as the ratio of the divergences of advective and diffusive heat fluxes, where the latter is dominated by vertical diffusion at the ice sheet scale. The parameter α is the Brinkman number, which denotes the strength of strain heating compared to the background conductive heat flux in the ice. We regard both parameters as strictly of O(1).

Boundary conditions for the heat transport problem at the surface and in the far field of the bed are

$$\begin{array}{rl}& T=-T_s\text{on}\hspace{0.17em}z=s,\end{array}$$ 2.3c

$$\begin{array}{r}\\ & -\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\to \nu \text{as}\hspace{0.17em}z\to -\mathrm{\infty }.\end{array}$$ 2.3d

In order to specify boundary conditions at the bed, z = b, it is useful to first examine the physics of basal sliding, which we assume to be thermally controlled. In general, temperature-dependent sliding is modelled through an explicitly temperature-dependent sliding law of the form [1,18]

$$u_b={\gamma }^{-1}F\left(\frac{T}{\delta }\right){\tau }_b,$$ 2.3e

where γ∼O(1) is a non-dimensional friction coefficient; δ≪1 is a positive parameter that compares the range of temperatures below the melting point over which subtemperate sliding is O(1) to the typical temperature scale of the ice sheet. The dependence of sliding on temperature is described by the function F, which is monotonically increasing in T subject to the constraint that T ≤ 0 (where T = 0 is the melting point), and such that F(0) = 1. The fully temperate sliding law is recovered when the bed attains the melting point, while significant sliding occurs within a temperature range of δ below the melting point, so F(T/δ) ≈ 0 when |T|≪δ.

Taking the limit δ → 0 allowed us to simplify basal boundary conditions (on z = b) for the ice sheet scale model to (see §4c of the companion paper for an asymptotic justification)

$$\begin{array}{rl}& {\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}={\left[T\right]}_{-}^{+}=u_b=0\text{if}\hspace{0.17em}T<0,\end{array}$$ 2.3f

$$\begin{array}{r}\\ & {\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha u_b{\tau }_b=T=0\text{if}\hspace{0.17em}{u}_b<{\gamma }^{-1}{\tau }_b,\end{array}$$ 2.3g

$$\begin{array}{r}\\ & T=u_b-{\gamma }^{-1}{\tau }_b=0\text{if}\hspace{0.17em}m>0,\end{array}$$ 2.3h

with the basal melt rate m defined as

$$m={\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}+\alpha {\tau }_b{u}_b\text{on}\hspace{0.17em}z=b$$ 2.3i

and ${[g(z_0)]}_{-}^{+}=g(z\to z_0^{+})-g(z\to z_0^{-})$. Note that boundary conditions (2.3f)–(2.3h), previously proposed by Fowler [14], implicitly identify three distinct regions of the ice sheet: a cold-based region, where bed temperature is well below the melting point and no sliding occurs; a subtemperate region, where bed temperature is approximately at the melting point and O(1) sliding occurs, subject to the constraint that the ice-bed contact remains in thermal balance; a temperate-based region where the basal melt rate remains positive and fully temperate sliding occurs. We denote the cold–subtemperate and subtemperate–temperate boundaries as x = x_s and x = x_t, respectively, and we demand that at these boundaries, the boundary conditions (2.3f)–(2.3g) and (2.3g)–(2.3h) hold simultaneously, leading to

$$T(z=b)=u_b=0\text{at}\hspace{0.17em}x=x_{\mathrm{s}},{\gamma }^{-1}{\tau }_b=-{\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}{(\alpha {\tau }_b)}^{-1},\hspace{0.17em}m=0\hspace{0.17em}\hspace{0.17em}\text{at}\hspace{0.17em}x=x_t,$$ 2.3j

which fix x_s and x_t. Lastly, continuity of mass and temperature is ensured demanding

$${\left[q\right]}_{-}^{+}={\left[h\right]}_{-}^{+}={\left[\frac{\mathrm{\partial }T}{\mathrm{\partial }z}\right]}_{-}^{+}=0\text{on}\hspace{0.17em}x=x_{\mathrm{s}},x_t.$$ 2.3k

In §4d of the companion paper, we solved the steady-state version of the model (2.1)–(2.3) numerically. The method of solution of the steady state, which relies on the computation of the Jacobian of the discrete version of equations (2.1)–(2.3), allows us to assess the stability features of the model numerically, which we discuss in the next subsection. We anticipate that this analysis will show that the system is always unstable, thus raising the issue as to whether the numerical scheme used is unstable (and therefore inappropriate) but the underlying continuum model is well posed, or whether the model itself is fundamentally deficient. To address this issue, in §2c we derive and analyse suitably reduced versions of the continuum model that are useful to highlight specific features of the numerical stability analysis. The reader not interested in the technical details is referred to §2d, where we discuss our findings and their implications.

(b). Numerical stability analysis of the continent-scale model

Let us consider the continuum problem (2.1)–(2.3) and introduce a mapping to a stretched coordinate system (x, z, t) → (ξ, η, τ) such that each subdomain is mapped to a unit square and τ = t. Keeping the name of dependent variables unchanged for simplicity, we have now h(σ, τ), T(σ, η, τ), x_s(τ), x_t(τ) and x_g(τ); we then semi-discretize h and T on a regular grid while keeping a continuous dependence on time. A general form for the unsteady, semi-discretized problem will then be

$$\frac{\mathrm{\partial }}{\mathrm{\partial }\tau }[\mathcal{M}(\mathit{y})]+\mathcal{A}(\mathit{y})=0,$$ 2.4a

where y(τ) is a solution vector of the form

$$\mathit{y}=\{h_1,h_2,\dots h_{3N_h},T_1,T_2,\dots T_{3N_T},x_{\mathrm{s}},x_t,x_{\mathrm{g}}\},$$ 2.4b

while $\mathcal{M}$, $\mathcal{A}$ are discrete operators of size (3N_h + 3N_T + 3) × 1 derived from the spatial discretization of the continuum model (their form is discussed in the electronic supplementary material, S4, of the companion paper [11]), and N_h, N_T denote the number of nodes in the h- and T-grids for each subdomain.

We are now interested in the behaviour of small-amplitude perturbations about the steady state, which we computed numerically. Denoting steady-state variables with $\overline{\cdot }$, we expand y in the sum of a steady state and a small-amplitude, time-dependent perturbation,

$$\mathit{y}=\overline{\mathit{y}}+\beta {\mathit{y}}^{\prime }(\tau ),\beta \ll 1.$$ 2.4c

Then, linearizing about the steady state and also posing the ansatz ${\mathit{y}}^{\prime }(\tau )=\tilde{\mathit{y}}\exp (\sigma \tau )$, to O(β) we find the generalized eigenvalue problem

$$\sigma \hspace{0.17em}{\mathsf{J}}_{\mathsf{M}}\hspace{0.17em}\tilde{\mathit{y}}={\mathsf{J}}_{\mathsf{A}}\hspace{0.17em}\tilde{\mathit{y}},$$ 2.4d

where ${\mathsf{J}}_{\mathsf{M}}=\mathrm{\partial }\mathcal{M}/\mathrm{\partial }{y}_j{|}_{\mathit{y}=\overline{\mathit{y}}}$, ${\mathsf{J}}_{\mathsf{A}}=\mathrm{\partial }\mathcal{A}/\mathrm{\partial }{y}_j{|}_{\mathit{y}=\overline{\mathit{y}}}$ are the Jacobians of the discrete operators $\mathcal{M}$, $\mathcal{A}$, evaluated at steady state, with dimensions (3N_h + 3N_T + 3) × (3N_h + 3N_T + 3), and σ is the eigenvalue, generally complex. In order to solve the steady-state problem we had to compute already the matrix J_A, while the matrix J_M is obtained straightforwardly from the semi-discretized problem assuming an exponential dependence of the solution on τ, as detailed above.

The eigenvalue problem (2.4d) is (i) sparse, as a result of the finite volume discretization; (ii) large, because the number of grid nodes must be in the range 10⁴–10⁶ to ensure convergence of the steady-state solver; (iii) non-symmetric as a result of the coupling between thermal and mechanical problem; and (iv) singular, because the location of subdomain boundaries does not depend explicitly on time. We deal with the singularity of the mass matrix J_M by means of row-reduction, and we then solve the row-reduced eigenvalue problem with an implicitly restarted Arnoldi method as implemented in ARPACK for Matlab [19]. This allows us to compute a subset of the spectrum, which we select to be the 10 eigenvalues with largest real part, as those correspond to the most unstable modes.

Ice thickness perturbations $\tilde{h}$ for the most unstable mode are plotted in figure 2 for two different values of the friction coefficient γ, with the shaded area denoting the subtemperate region. In figure 2a, the contribution of shearing to the ice flux exceeds the contribution due to basal sliding everywhere in the subtemperate region, while the ice flux is sliding-dominated in some finite portion of the subtemperate region for figure 2b. We will refer to these two regimes as ‘slow sliding’ and ‘fast sliding’ throughout. The corresponding spectra are illustrated in figure 3 for varying grid sizes, while a reference steady-state solution is illustrated in fig. 4 of the companion paper. The full solution of the linearized problem, including perturbations of the sliding velocity and of the basal heat flux, is displayed in electronic supplementary material, appendix A, figure S7. Figure 2 illustrates that the fastest-growing eigenmodes in the discretized model come in two flavours: they are characterized either by gridpoint-to-gridpoint oscillations in the subtemperate region itself (fast sliding), or by localized structure around the transition from subtemperate to temperate sliding (slow sliding). In both cases, the growth rate of the fastest-growing modes increases as we refine the numerical grid (figure 3). The two cases correspond to different parameter regimes: the gridpoint-to-gridpoint oscillations occur for fast sliding, while the localized instability at the subtemperate–temperate boundary occurs for the opposite regime of slow sliding.

Figure 2.

Ice thickness perturbation associated with the fastest-growing eigenvalue for γ = 1.7 (a) and γ = 0.7 (b). The subtemperate region is shaded, and insets show a zoom near the subtemperate–temperate boundary. Parameters are α = 0.4, ν = 0.5, Pe  = 7.5, T_S = − 1, q₀ = 5. The number of computational grid points is N_h = 320, N_T = 320 × 320 for each subdomain. Note that the divide (not shown) is located at x = 0.

Figure 3.

Spectra (ten most unstable eigenvalues) as a function of grid spacing (Δξ, Δη = N⁻¹_h) for two different values of friction coefficient: (a) γ = 1.7; (b) γ = 0.7. The remaining parameters are the same as in figure 2. (Online version in colour.)

In the remainder of this section, we will demonstrate that this behaviour is not a consequence of the specific numerical scheme chosen, but rather that both modes have equivalents in the underlying continuum model. Specifically, we will derive a simplified model of the subtemperate region that is amenable to analytical treatment, and demonstrate the existence of unstable modes in the reduced continuum model that are associated with an eigenvalue that increases systematically as a length scale parameter is decreased, so that smaller features grow faster. The latter is one of the hallmarks of ill-posedness: we will, therefore, consider what ingredients may be missing in the ‘shallow’ continuum model that we are studying here that could correct this problem (§3).

(c). The stability of a continuum subtemperate ice slab

Motivated by the numerical results, we consider the subtemperate region in isolation and investigate its stability to perturbations with wavelength short compared with the ice sheet length. By ‘short’ we mean a horizontal length scale that is much larger than the ice thickness, so that the shallow ice approximation holds, but still small compared with the ice sheet length. In this limit, the mechanical model reduces to a parallel-sided slab of ice, thus enabling an analytical treatment of the stability problem. By analysing this problem, we seek to isolate the core ingredients responsible for the instabilities observed in the discrete model.

Let us consider the ice sheet scale model of §2a, and restrict ourselves to the subtemperate region. We then take the limit of a short horizontal length scale within the shallow ice approximation, which yields the slab model

$$\frac{\mathrm{\partial }s}{\mathrm{\partial }t}+\frac{\mathrm{\partial }q}{\mathrm{\partial }x}=0,$$ 2.5a

with mass flux (2.1b), and sliding velocity u_b defined by the basal energy budget

$$-Q+\nu +\alpha {\tau }_b{u}_b=0,$$ 2.5b

where Q =  − ∂T/∂z|_{z → b⁺}, and τ_b is given by equation (2.1a₂). Note that, as basal temperature is approximately at the melting point in the subtemperate region, the heat equation in the bed (2.3b) can be integrated analytically, leading to −∂T/∂z = ν for z < b; this justifies the form of the basal heat flux in the bed in equation (2.5b). Regarding the heat equation in the ice, it is important to note that at this shorter horizontal length scale the local Péclet number Pe_slab is large. As basal temperature is at the melting point in the subtemperate region, then the thermal model for the ice reduces to the Q-equation (see §3b of the companion paper [11])

$$\frac{\mathrm{\partial }Q}{\mathrm{\partial }t}+u_b\frac{\mathrm{\partial }Q}{\mathrm{\partial }x}-Q\frac{\mathrm{\partial }{u}_b}{\mathrm{\partial }x}=0,$$ 2.5c

which completes our slab model.

We are interested in the behaviour of short-wavelength perturbations about the steady state of equations (2.5), which we study through a linear stability analysis. To this aim, we rescale

$$x=k^{-1}\hat{x}\mathrm{and}t=k^{-2}\hat{t},$$ 2.6a

where k⁻¹≪1 is a short horizontal length scale; in addition, we expand model variables about their steady state (denoted with $\overline{\cdot }$) as

$$h=\overline{h}+\beta k^{-2}{h}^{\prime },u_b={\overline{u}}_b+\beta k^{-1}{u}_b^{\prime },{\tau }_b={\overline{\tau }}_b+\beta k^{-1}{\tau }_b^{\prime },Q=\overline{Q}+\beta k^{-2}{Q}^{\prime },\beta \ll 1,$$ 2.6b

where $\overline{h}$, ${\overline{u}}_b$, ${\overline{\tau }}_b$, $\overline{Q}$ are uniform in space. Linearizing about the steady state, and dropping terms of O(βk⁻¹) and higher, we find the reduced model valid for k → ∞, β → 0

$$\begin{array}{rl}& \frac{\mathrm{\partial }{h}^{\prime }}{\mathrm{\partial }\hat{t}}+\frac{\mathrm{\partial }}{\mathrm{\partial }\hat{x}}[\frac{{\overline{h}}^2}{3}{\tau }_b^{\prime }+\overline{h}{u}_b^{\prime }]=0,\end{array}$$ 2.7a

$$\begin{array}{rl}& \frac{\mathrm{\partial }{Q}^{\prime }}{\mathrm{\partial }\hat{t}}-\overline{Q}\frac{\mathrm{\partial }{u}_b^{\prime }}{\mathrm{\partial }\hat{x}}=0,\end{array}$$ 2.7b

with basal energy budget

$$\overline{{\tau }_b}{u}_b^{\prime }+\overline{u}{\tau }_b^{\prime }=0$$ 2.7c

and perturbed basal shear stress

$${\tau }_b^{\prime }=-\overline{h}\frac{\mathrm{\partial }{h}^{\prime }}{\mathrm{\partial }\hat{x}}.$$ 2.7d

Fourier-transforming in $\hat{x}$ and also assuming an exponential dependence on time so that $f^{\prime }(\hat{x},\hat{t})=\hat{f}\exp (\mathrm{i}\hat{x}+\sigma \hat{t})$ we obtain the solution

$$\sigma =\frac{\overline{h}}{{\overline{\tau }}_b}(-\frac{{\overline{\tau }}_b{\overline{h}}^2}{3}+{\overline{u}}_b\overline{h}),{\hat{u}}_b=\mathrm{i}\frac{\overline{u}\overline{h}}{{\overline{\tau }}_b}\hat{h},\hat{Q}=\frac{\mathrm{i}\overline{Q}}{\sigma }{\hat{u}}_b.$$ 2.7e

This confirms that the two regimes identified on the basis of the numerical results behave differently also in the continuum model. In fact, with the eigenvalue of the continuum model given by (2.7e₁), and recalling the definition of the mass flux (2.1b), we recognize that the system is unstable (σ > 0) when sliding contributes more than shearing to the mass flux, while it is stable (σ < 0) in the opposite case. Given that our linear model is effectively a diffusion-only problem (this can be seen substituting u_b′ from (2.7c) into (2.7a)), and that such a mathematical model is either stable (for a positive diffusion coefficient, where the diffusion coefficient D is given by D = − σ), or unstable (for D < 0) and ill-posed (this is known as ‘backwards heat equation’; e.g. Renardy & Rogers [20]), we conclude that the fast sliding regime is ill-posed due to backward diffusion, while the slow sliding regime appears to be well behaved.

Physically, the instability we detect in the fast sliding regime relies on a feedback between perturbations of the basal shear stress, perturbation of the mass flux and ice thickness response. The key ingredient is the constraint of a basal energy budget in balance, which effectively serves as a sliding law for the subtemperate region. In the short-wavelength limit, the basal energy budget reduces to a balance between perturbations of frictional heating driven by changes in basal shear stress, and perturbations of frictional heating driven by changes in sliding velocity, equation (2.7c). Keeping the bed in thermal balance then requires that the sliding velocity is minimum where the basal shear stress is maximum, u_b′∼ − τ_b′. This is effectively a velocity-weakening friction law.

Even though velocity-weakening friction suggests that the model should be always ill-posed, having sliding and shearing contributing comparably to the mass flux implies that this is not the case, at least in a shallow ice model. For fast sliding, the perturbed mass flux (term in square brackets in equation (2.7a)) is dominated by sliding, q′∼u_b′∼ − τ_b′; with ${\tau }_b^{\prime }\sim \mathrm{\partial }{h}^{\prime }/\mathrm{\partial }\hat{x}$ this yields an increasing divergence of the mass flux where h′ decreases (h′ is in counterphase with respect to the divergence of the mass flux). Intuitively, this means that a perturbation of the ice surface will drive flow towards regions of thicker ice, thus leading to the growth of the perturbations in a runaway feedback. For slow sliding, the opposite is true: q′∼τ_b′, and a perturbation of the ice surface will drive flow towards regions of thinner ice, thus leading to damping of initial perturbations.

A stable subtemperate region for the slow sliding regime is in apparent contradiction with the numerical stability analysis (figure 3b), which finds an instability with wavelength-dependent growth rate. This raises the question as to whether a slower, but still wavelength-dependent instability exists in the subtemperate region that we overlooked in the analysis above, or the subtemperate–temperate boundary is also unstable.

As for the subtemperate region, it turns out that the scaling (2.6a) is only one out of two possible scalings for the short-wavelength limit, an alternative being $x=k^{-1}\hat{x}$, $t=k^{-1}\hat{t}$. In order to rule out any wavelength-dependent oscillations emerging from the subtemperate region in the slow sliding regime, we thus perform a linear stability analysis on the slab model (2.5) with the latter scaling (see electronic supplementary material, S1.2.4), and confirm that a wavelength-dependent growth rate is not to be expected for slow sliding. We are therefore left with the possibility of an instability emerging at the subtemperate–temperate boundary, as suggested by the localized structure in the numerical solution (figure 2a). In electronic supplementary material, S1.3, we demonstrate that the subtemperate–temperate boundary is indeed violently unstable to localized perturbations, with the growth rate increasing linearly as the length scale of those perturbations is decreased.

(d). Discussion

The analysis of the shallow ice model with subtemperate sliding presented in the last section raises a number of critical issues regarding the model itself, namely (i) the ill-posedness of the fast sliding regime and (ii) the existence of an instability associated with the subtemperate–temperate boundary such that the growth rate increases as the wavelength of perturbations is decreased. Identifying what ingredients are missing in our model that could correct these issues is precisely the purpose of the rest of the paper.

In this regard, our finding of backward diffusion in the shallow ice model (2.1)–(2.3) suggests that a rescaling at shorter length scales might fix the pathological behaviour associated with the subtemperate region. A natural choice for this shorter length scale is the ice thickness scale: here we are looking at a Stokes model for ice mechanics, as opposed to the lubrication approximation that we have adopted so far, while the thermal problem remains unchanged compared to the slab model (2.5) (a derivation of such ice thickness scale model with basal boundary conditions (2.3g) is presented in electronic supplementary material, S2.2).

Even though one might expect that relaxing the assumption of shallowness is sufficient to correct the pathology of the ice sheet scale model, it turns out that this is not the case. The motivation for this is that the constraint that the bed remains in thermal balance in the subtemperate region leads to velocity-weakening friction, and this remains ill-posed regardless of the mechanical model it is coupled to. To confirm this, we perform a stability analysis of the Stokes problem with basal boundary conditions (2.3g), and find that the model is pathological for both slow and fast sliding, with an infinite growth rate for perturbations with an O(1) wavelength (see electronic supplementary material, S3).

Having ascertained that relaxing shallowness alone does not amend the pathological behaviour, it is important to recall that in the derivation of the boundary conditions (2.3f)–(2.3h) we have made use of two asymptotic limits: that of a small ice sheet aspect ratio (ε → 0) and that of a small temperature range over which subtemperate sliding can make a leading order contribution to the ice flux (δ → 0). It turns out that both are singular limits, and we have to account for both in order to arrive at a well-posed model of subtemperate sliding.

To this aim, in §3a, we derive an ice thickness scale model with temperature-dependent sliding that accounts for the effects of finite deviations of basal temperature from the melting point. In doing so, we have to look at an asymptotic limit in both the aspect ratio ε and the sensitivity on temperature of basal sliding, δ. The obvious distinguished limit to consider turns out to be δ∼ε^1/2, and we restrict ourselves to that case. Analysing this model (§3b), we will find that an ice thickness scale slab with subtemperate sliding is indeed linearly unstable, but it shows no obvious signs of ill-posedness provided finite deviations of basal temperature from the melting point are resolved.

While accounting for finite δ resolves the pathology of the shallow ice model, this additional model component complicates the mathematical structure of the problem substantially. In fact, we will show in §3 that (i) the instability of the ice thickness scale slab with temperature-dependent sliding persists at long wavelengths, but does not share the same physics driving the instability in the short-wavelength limit of the shallow ice model discussed in §2c and (ii) the instability occurs at a fast time scale, ε times the natural time scale of the ice sheet. This on the one hand highlights the need for an intermediate length scale between the ice thickness and the ice sheet length, where a distinct dominant balance holds in the thermal problem near the bed. On the other hand, we will have to revisit the ice sheet scale problem of §2c in order to account for the fast time scale, which we do in §4. We anticipate here that a key conclusion of §4 will be that subtemperate sliding is unconditionally unstable over length scales ranging from the ice thickness to the ice sheet scale, where the latter ice sheet scale instability is overlooked by the simple model in §2 due to the singular nature of the limit $\delta \to 0.$

3. A higher order, ice thickness scale model for the subtemperate region

(a). Model

We now consider ice flow in the subtemperate region with small but finite δ, as opposed to the δ = 0 case described in §2, and seek to derive a leading order model for a subtemperate ice slab that resolves the ice thickness scale in the along-flow direction. A detailed derivation of our model is provided in electronic supplementary material, S2.3, while here we state only the relevant rescalings and the leading order model.

Let us consider a coordinate system (x, z) aligned with the bed, which we assume to be locally flat over the along-flow scales of interest and tilted of an angle μ to the horizontal; so in this new coordinate system the x-axis is parallel to the bed and pointing down-slope, while the z-axis is perpendicular to the bed and pointing upwards. Maintaining the same notation as before, the ice-bed interface is located at z = b, and the ice surface at z = s(x, t); h(x, t) = s − b is the ice thickness, u = (u, w) is the velocity field, p is pressure, and T is temperature. In the tilted coordinate system, the ice sheet scale problem for the subtemperate region (see electronic supplementary material, S2.1) takes the same form as the problem presented in §2a, except for (i) the explicitly temperature-dependent sliding law that now features at leading order as a result of dispensing with δ = 0 and (ii) a constant driving term proportional to tanμ in the along-flow momentum conservation that replaces the variable bed topography gradient.

In order to resolve the ice thickness scale in the along-flow direction, we rescale this ice sheet scale model as

$$X={\epsilon }^{-1}x,Z=z-b,\tau ={\epsilon }^{-1}t(U,W)=(u,\epsilon w),H=h,P=p,\mathit{\Theta }=T.$$ 3.1a

In addition, we expand dependent variables as

$$\begin{array}{rl}& (U,W)=(U^{(0)},W^{(0)})+O(\epsilon ),H=H^{(0)}+O(\epsilon ),P=H^{(0)}-Z+\epsilon P^{(1)}+O({\epsilon }^2),\\ & \mathit{\Theta }={\mathit{\Theta }}^{(0)}+O(\epsilon ),{\mathit{\Theta }}_b={{\mathit{\Theta }}_b}^{(0)}+\delta {{\mathit{\Theta }}_b}^{(1)}+O({\delta }^2),\end{array}\}$$ 3.1b

where the leading order ice thickness H⁽⁰⁾ is independent of X and Θ_b denotes basal temperature, and Θ_b⁽⁰⁾ = 0 meaning that the leading order bed temperature is still at the melting point, as a result of δ≪1. To an error of O(ε), we find the Stokes problem

$$\begin{array}{rl}\frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }X}+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }Z}& =0,\end{array}$$ 3.2a

$$\begin{array}{rl}\frac{{\mathrm{\partial }}^2{U}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{U}^{(0)}}{\mathrm{\partial }{Z}^2}-\frac{\mathrm{\partial }{P}^{(1)}}{\mathrm{\partial }X}+\tan (\mu )& =0,\end{array}$$ 3.2b

$$\begin{array}{rl}\frac{{\mathrm{\partial }}^2{W}^{(0)}}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2{W}^{(0)}}{\mathrm{\partial }{W}^2}-\frac{\mathrm{\partial }{P}^{(1)}}{\mathrm{\partial }Z}& =0,\end{array}$$ 3.2c

defined on the domain 0 < Z < H⁽⁰⁾, with boundary conditions

$$\begin{array}{rl}& \frac{\mathrm{\partial }{U}^{(0)}}{\mathrm{\partial }Z}+\frac{\mathrm{\partial }{W}^{(0)}}{\mathrm{\partial }X}=0,W^{(0)}=0,\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=H^{(0)},\end{array}$$ 3.2d

$$\begin{array}{r}\\ & W^{(0)}=0,{U_b}^{(0)}={\gamma }^{-1}F\left({{\mathit{\Theta }}_b}^{(1)}\right){T_b}^{(0)},\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=0,\end{array}$$ 3.2e

where U_b⁽⁰⁾ = U⁽⁰⁾|_Z=0, T_b⁽⁰⁾ = ∂U⁽⁰⁾/∂Z|_Z=0 are the sliding velocity and the basal shear stress, respectively.

With large scale Péclet number Pe∼O(1), the local Péclet number at the ice thickness scale, Pe _slab = ε⁻¹Pe , is large. Similarly to the case of the shallow subtemperate slab, and based on the fact that the horizontal velocity remains O(1) also near the bed, the large Pe _slab allows us to derive an advection-only leading order thermal problem for the ice, which we state below (see equations (3.3)). However, such an advection problem does not allow us to resolve finite deviations of bed temperature from the melting point, which are instead required by the sliding law (3.2e₂). In fact, in a strongly advective temperature field the bed must be an isotherm; with δ≪1 this isotherm will be the melting point one, up to an error of O(δ). This highlights the need for a separate thermal boundary layer near the bed, where temperature changes of O(δ) in response to O(1) changes in the heat flux imposed by the strongly advective temperature field above. It is then the advection–diffusion problem in this boundary layer (see equations ((3.5)) below) that determines the O(δ) deviation of bed temperature from the melting point.

We are now going to illustrate the leading order thermal problem. Taking the limit of Pe _slab → ∞ yields to leading order, and with an error of O(ε),

$$\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }\tau }+U^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }X}+W^{(0)}\frac{\mathrm{\partial }{\mathit{\Theta }}^{(0)}}{\mathrm{\partial }Z}=0\text{on}\hspace{0.17em}0<Z<H^{(0)},$$ 3.3a

where the leading order temperature profile must satisfy, with an error of O(δ),

$${\mathit{\Theta }}^{(0)}=0\text{on}\hspace{0.17em}Z=0,$$ 3.3b

while the heat equation in the bed simply demands that Θ(Z < b) is independent of τ up to O(ε). Given that to leading order the bed is a characteristic of the advection-only outer problem, equation (3.3a–3.3b) can be reduced to the standard Q-equation (for the derivation see §3b of the companion paper [11])

$$\frac{\mathrm{\partial }{\mathcal{Q}}^{(0)}}{\mathrm{\partial }\tau }+{U_b}^{(0)}\frac{\mathrm{\partial }{\mathcal{Q}}^{(0)}}{\mathrm{\partial }X}-{\mathcal{Q}}^{(0)}\frac{\mathrm{\partial }{U_b}^{(0)}}{\mathrm{\partial }X}=0,$$ 3.3c

which relates directly the basal heat flux ${\mathcal{Q}}^{(0)}=-\mathrm{\partial }{\mathit{\Theta }}^{(0)}/\mathrm{\partial }Z{|}_{Z=0}$ to the sliding velocity U_b⁽⁰⁾. Equation (3.3c) is the outer component of our thermal boundary layer model.

As discussed, the outer problem (3.3c) must be coupled to a boundary layer near the bed so as to resolve finite deviations of basal temperature from the melting point, Θ_b⁽¹⁾. Assuming the distinguished limit δ∼ε^1/2, and rescaling the ice thickness scale thermal problem as

$$\tilde{X}=X,\tilde{Z}={\delta }^{-1}z,\tilde{\mathit{\Theta }}={\delta }^{-1}\mathit{\Theta },(\tilde{U},\tilde{W})=(U,{\delta }^{-1}W),$$ 3.4a

and also expanding dependent variables as

$$(\tilde{U},\tilde{W})=({U_b}^{(0)}+O(\delta ),-\frac{\mathrm{\partial }{U_b}^{(0)}}{\mathrm{\partial }X}\tilde{Z}+O({\delta }^2)),\tilde{\mathit{\Theta }}={\tilde{\mathit{\Theta }}}^{(0)}+O(\delta ),$$ 3.4b

yields the following leading order problem correct up to an error of O(δ²):

$$\begin{array}{rl}\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tau }+{U_b}^{(0)}\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tilde{X}}-\frac{\mathrm{\partial }{U_b}^{(0)}}{\mathrm{\partial }X}\tilde{Z}\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tilde{Z}}-\frac{{\mathrm{\partial }}^2{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }{\tilde{Z}}^2}& =0\text{for}\hspace{0.17em}\tilde{Z}>0,\end{array}$$ 3.5a

$$\begin{array}{rl}\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }{\tilde{Z}}^2}& =0\text{for}\hspace{0.17em}\tilde{Z}<0,\end{array}$$ 3.5b

with boundary conditions at the bed

$${\left[\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tilde{Z}}\right]}_{-}^{+}+\alpha \hspace{0.17em}{U_b}^{(0)}\hspace{0.17em}{T_b}^{(0)}=0,{\left[{\tilde{\mathit{\Theta }}}^{(0)}\right]}_{-}^{+}=0,\hspace{0.17em}\text{on}\hspace{0.17em}\tilde{Z}=0,$$ 3.5c

and matching conditions

$$-\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tilde{Z}}\to {\mathcal{Q}}^{(0)}\hspace{0.17em}\text{as}\hspace{0.17em}\tilde{Z}\to +\mathrm{\infty },-\frac{\mathrm{\partial }{\tilde{\mathit{\Theta }}}^{(0)}}{\mathrm{\partial }\tilde{Z}}\to \nu \hspace{0.17em}\text{as}\hspace{0.17em}\tilde{Z}\to -\mathrm{\infty }.$$ 3.5d

The role of the boundary layer model (3.3c), ((3.5)) is to restore vertical diffusion to leading order near the bed. Physically, this decouples the outer problem (3.3c) from the basal energy budget of the bed (3.5c₁) by enabling deviations from a vertically uniform temperature gradient (as specified by the advection problem above the bed) in the near-bed region. Mathematically, restoring vertical diffusion provides an additional degree of freedom that permits to satisfy both the temperature-dependent sliding law, equation (3.2e₂), and the basal energy budget, equation (3.5c₁), at leading order. We will see in the next subsection that this is key to relieving the ill-posedness issue that we identified for δ = 0.

(b). Linear stability analysis

In order to understand the role of small but finite δ, we perform a linear stability analysis on the model (3b)–((3.5)). First of all, we assume without loss of generality a unitary ice thickness; then, to facilitate the solution of the mechanical model, we reformulate the Stokes problem (3b) in terms of a streamfunction ψ defined as

$$U^{(0)}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }Z},W^{(0)}=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }X},$$ 3.6

so that sliding velocity and basal shear stress are given by U_b⁽⁰⁾ = ∂ψ/∂Z|_Z=0, T_b⁽⁰⁾ = ∂²ψ/∂Z²|_Z=0, respectively. This leads to the biharmonic problem

$${\mathrm{\nabla }}^4\psi =0\text{on}\hspace{0.17em}0<Z<1,$$ 3.7a

where ∇ = (∂/∂X, ∂/∂Z), with boundary conditions

$$\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{Z}^2}-\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{X}^2}=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=1,\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{X}^2}=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=1,\hspace{0.17em}Z=0.$$ 3.7b

Then, recalling that the ice thickness is independent of τ and X, we expand dependent variables about their steady state (denoted by a bar) as

$$\psi =\overline{\psi }+\beta {\psi }^{\prime },T_b={\overline{T}}_b+\beta T_b^{\prime },{\mathcal{Q}}^{(0)}=\overline{\mathcal{Q}}+\beta {\mathcal{Q}}^{\prime },{\tilde{\mathit{\Theta }}}^{(0)}=\overline{\tilde{\mathit{\Theta }}}+\beta \widehat{\tilde{\mathit{\Theta }}},\beta \ll 1,$$ 3.8

where barred quantities do not depend on X and τ. To O(1), and assuming a spatially uniform steady state, we have

$${\overline{U}}_b=\frac{\overline{F}}{\gamma }{\overline{T}}_b,\overline{\tilde{\mathit{\Theta }}}=\{\begin{array}{ll}{\mathit{\Theta }}_{\mathrm{bed}}-\overline{\mathcal{Q}}\tilde{Z}& \text{for}\hspace{0.17em}\tilde{Z}>0\\ {\mathit{\Theta }}_{\mathrm{bed}}-\nu \hat{Z}& \text{for}\hspace{0.17em}\tilde{Z}<0\end{array},\overline{\mathcal{Q}}=\nu +\alpha {\overline{T}}_b{\overline{U}}_b,{\overline{T}}_b=\tan (\mu ),$$ 3.9

where $\overline{F}=F({\mathit{\Theta }}_{\mathrm{bed}})$, with Θ_bed∼O(1) a free parameter effectively fixing the location of the ice thickness scale slab in the large-scale subtemperate region.

We assume a periodic domain in X with period L∼O(1), and Fourier-transform perturbed variables as

$${\hat{g}}_n(Z,\tau )=\frac{1}{L}{\int }_0^L{g}^{\prime }(X,Z,\tau )\exp (-\mathrm{i}{k}_{n}X)\hspace{0.17em}\mathrm{d}X,k_n=\frac{2\pi n}{L}.$$ 3.10

Substituting into the biharmonic problem (3.7a), we obtain the following O(β) problem for the streamfunction:

$$k^4{\hat{\psi }}_n+\frac{{\mathrm{\partial }}^4{\hat{\psi }}_n}{\mathrm{\partial }{Z}^4}-2k_n^2\frac{{\mathrm{\partial }}^2{\hat{\psi }}_n}{\mathrm{\partial }{Z}^2}=0\text{on}0<Z<1,$$ 3.11a

with boundary conditions

$$\frac{{\mathrm{\partial }}^2{\hat{\psi }}_n}{\mathrm{\partial }{Z}^2}+k_n^2{\hat{\psi }}_n=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=1,{\hat{\psi }}_n=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}Z=1,\hspace{0.17em}Z=0,$$ 3.11b

and sliding law

$${\hat{U}}_{b,n}={\gamma }^{-1}(\overline{F}{\hat{T}}_{b,n}+{\overline{T}}_b{F}^{\prime }\widehat{\tilde{\mathit{\Theta }}}{|}_{\tilde{Z}=0}),$$ 3.11c

where $F^{\prime }=\mathrm{d}F/\mathrm{d}\mathit{\Theta }{|}_{\overline{\mathit{\Theta }}}$ is a strictly positive function. The O(β) outer thermal problem is

$$\frac{\mathrm{d}{\hat{\mathcal{Q}}}_n}{\mathrm{d}\tau }+\mathrm{i}{k}_n({\overline{U}}_b{\hat{\mathcal{Q}}}_n-\overline{\mathcal{Q}}{\hat{U}}_{b,n})=0,$$ 3.12a

while in the boundary layer we have

$$\begin{array}{rl}& \frac{\mathrm{\partial }{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }\tau }+\mathrm{i}{k}_n({\overline{U}}_b{\widehat{\tilde{\mathit{\Theta }}}}_n+\overline{\mathcal{Q}}{\hat{U}}_{b,n}\tilde{Z})-\frac{{\mathrm{\partial }}^2{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }{\tilde{Z}}^2}=0\text{for}\hspace{0.17em}\tilde{Z}>0,\end{array}$$ 3.12b

$$\begin{array}{r}\\ & \frac{\mathrm{\partial }{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }{\tilde{Z}}^2}=0\text{for}\hspace{0.17em}\tilde{Z}<0\end{array}$$ 3.12c

with boundary conditions

$${\left[\frac{\mathrm{\partial }{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }\tilde{Z}}\right]}_{-}^{+}+\alpha ({\overline{U}}_b{\hat{T}}_{b,n}+{\overline{T}}_b{\hat{U}}_{b,n})=0,{\left[{\widehat{\tilde{\mathit{\Theta }}}}_n\right]}_{-}^{+}=0\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}\tilde{Z}=0,$$ 3.12d

and matching conditions

$$-\frac{\mathrm{\partial }{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }\tilde{Z}}\to {\hat{\mathcal{Q}}}_n\hspace{0.17em}\hspace{0.17em}\text{as}\hspace{0.17em}\tilde{Z}\to +\mathrm{\infty },\frac{\mathrm{\partial }{\widehat{\tilde{\mathit{\Theta }}}}_n}{\mathrm{\partial }\tilde{Z}}\to 0\text{as}\hspace{0.17em}\tilde{Z}\to -\mathrm{\infty }.$$ 3.12e

The Fourier-transformed biharmonic problem (3.11a) has general solution

$${\hat{\psi }}_n=(A+BZ)\exp (-k_{n}Z)+(C+DZ)\exp (k_{n}Z),$$ 3.13a

with integration constants

$$B=-\frac{A[1-\exp (2k_n)]}{2},C=-A,D=\frac{A[1-\exp (-2k_n)]}{2},$$ 3.13b

where we set A = 1 in the light of the biharmonic problem being homogeneous. This yields a perturbed sliding velocity and basal shear stress of the form

$${\hat{U}}_{b,n}=-2k_n+\sinh (2k_n),{\hat{T}}_{b,n}=-4k_n\sinh {(k_n)}^2.$$ 3.13c

For the thermal problem (3.12), we seek a separable solution of the form

$$({\hat{\mathcal{Q}}}_n,{\widehat{\tilde{\mathit{\Theta }}}}_n)=({\hat{q}}_n,{\hat{\vartheta }}_n)\exp (\sigma \tau ),\sigma \in \mathbb{C}.$$ 3.14a

Direct integration of (3.12b)–(3.12c) with the separable ansatz above and boundary conditions (3.12d₂), (3.12e) yields the perturbed temperature field

$${\hat{\vartheta }}_n=\{\begin{array}{ll}{\vartheta }_0\exp {(-\tilde{Z}(\mathrm{i}{k}_n\overline{U_b}+\sigma )}^{1/2})-{\hat{q}}_n\tilde{Z}& \text{for}\hspace{0.17em}\tilde{Z}>0\\ {\vartheta }_0\exp ({\sigma }^{1/2}\tilde{Z})& \text{for}\hspace{0.17em}\tilde{Z}<0,\end{array}$$ 3.14b

where the perturbed outer flux and perturbed basal temperature are given by the Q-equation (3.12a) and the sliding law (3.11c), respectively, and read

$${\hat{q}}_n=\frac{\mathrm{i}{k}_n\overline{\mathcal{Q}}[-2k_n+\sinh (2k_n)]}{\mathrm{i}{k}_n{\overline{U}}_b+\sigma },{\vartheta }_0=\frac{-2k_n[\overline{F}(1-\cosh (2k_n))+\gamma ]+\gamma \sinh (2k_n)}{{\overline{T}}_b{F}^{\prime }},$$ 3.14c

and in order to ensure boundedness of the temperature solution (3.14b) we require

$$\mathrm{Re}[{\sigma }^{1/2}]>0,\mathrm{Re}[{(\mathrm{i}{k}_n\overline{U_b}+\sigma )}^{1/2}]>0.$$ 3.14d

Lastly, the basal energy budget (3.12d₁) yields a dispersion relation of the form

$$a_0-a_1(y^2+\mathrm{i}{k}_n{\overline{U}}_b)[y+{(y^2+\mathrm{i}{k}_n{\overline{U}}_b)}^{1/2}]+a_2(y^2+\mathrm{i}{k}_n{\overline{U}}_b)=0,\text{with}\hspace{0.17em}\hspace{0.17em}y={\sigma }^{1/2},$$ 3.14e

with the coefficients a₀(k_n), a₁(k_n), a₂(k_n) provided in electronic supplementary material, S4.1.2. After straightforward algebraic manipulation, equation (3.14e) can be seen as a fifth-degree polynomial in y subject to the constraints (3.14d), and a typical solution is displayed in figure 4. The absolute value of the growth rate, Re[σ], is displayed as a function of k_n for slow sliding in figure 4a and fast sliding in figure 4b, with colours marking different solutions. Note that axes are logarithmic, so a positive or negative Re[σ] is marked by a solid or dashed line, respectively.

Figure 4.

Growth rate from the linear model in §3b. Displayed is |Re[σ]|, with solid (dashed) line denoting Re[σ] > 0 (Re[σ] < 0). Variables and parameters are scaled with the steady-state ice flux q⁽⁰⁾ and thickness H⁽⁰⁾. (a) γ = 5H⁽⁰⁾⁻¹ (${\overline{U}}_b=0.273q^{(0)}{H^{(0)}}^{-1}$), (b) γ = H⁽⁰⁾⁻¹ (${\overline{U}}_b=0.652q^{(0)}{H^{(0)}}^{-1}$). Parameters are ν = H⁽⁰⁾⁻¹, α = (q⁽⁰⁾/H⁽⁰⁾)⁻², Θ_bed = − 0.5, with F = exp(Θ).

Our results in figure 4 show that at least one branch of the solution has positive growth rate (solid line) at all wavelengths, meaning that the system is unstable regardless of whether sliding is slow or fast. The fastest-growing wavelengths are for a wavelength comparable with the ice thickness (k_n∼O(1)), whereas the instability is damped for $k_n\to \mathrm{\infty }$. It is, however, worth noting that the linearized problem is not self-adjoint, and by separation of variables we find a finite number of eigenfunctions, so our solution is not complete. The short-wavelength modes we are able to compute do not hint at any ill-posedness (Re[σ]∼k^−1/2_n as k_n → ∞) but we cannot exclude that non-separable solutions to the linearized problem could lead to pathological behaviour at short wavelengths. We will return to the issue of growth of short-wavelength perturbations in the context of the nonlinear evolution of the subtemperate slab, which we discuss in §5.

(c). Processes

The stability analysis above illustrates that an ice thickness scale subtemperate ice slab is unstable. To isolate the physics driving this instability, we analyse the asymptotic behaviour of the linearized model in the short (k_n → ∞) and long (k_n → 0) wavelength limits. In the remainder of this section, we present the essential results of this analysis, while we refer readers to electronic supplementary material, S4.2 for the mathematical details.

For short-wavelength perturbations dynamics result from the interplay of two time scales: a fast time scale ∼k_n over which we see dynamics in the inner thermal problem for the ice, and a slow time scale ∼k^−1/2_n, over which we see dynamics in the inner thermal problem inside the bed and in the outer thermal problem. Since growth occurs at the slow time scale, to understand the physics leading to instability we need to disentangle the interaction between the two time scales, which we do using multiple scales expansions [21]. The analysis is described in electronic supplementary material, S4.3, while we discuss results below.

At the fast time scale, the solution is a travelling temperature wave that propagates within the ice in the near-bed boundary layer. This temperature wave affects only the temperature gradient within the ice, which however couples to the temperature field within the bed through the basal energy budget at the slow time scale. The diffusive propagation of the temperature wave within the bed then sets a small (O(k^−1/2_n)) perturbation of basal temperature that lags in phase the leading order temperature wave in the ice. This perturbation of basal temperature feeds back into the sliding law, leading to a similarly phase-shifted sliding velocity and basal shear stress. It is then such a phase shift in the sliding velocity that leads to growth at the slow time scale. In fact, the phase shift is such that the divergence of the perturbed sliding velocity is positive where the outer flux is maximum, which results in an amplified heat flux and growth according to the first-order Q-equation.

For long-wavelength perturbations (k_n → 0), stability depends on whether sliding is slow or fast: for slow sliding (figure 4a), the growth rate tends to zero, while for fast sliding (figure 4b), it remains constant and positive, thus suggesting that the instability continues at horizontal length scales much longer than the ice thickness. The physics of this instability are those of a thermal runaway feedback between basal temperature and basal heat flux. For long-wavelength perturbations advection near the bed is negligible, so for an O(1) perturbation of the basal temperature, boundedness of the temperature field implies that the net heat flux into the ice is reduced where the bed is warmest (basal temperature and net heat flux into the ice are in anti-phase). This will further warm the bed, leading to growth of perturbations.

Our argument so far would equally apply for slow and fast sliding; we are now going to illustrate that the requirement of a basal energy budget in balance renders this mode not viable for fast sliding. To this aim, it is important to recall that, with a bounded temperature field in the boundary layer, a basal energy budget in balance is possible only if the net heat flux into the ice is in anti-phase with respect to the perturbed frictional heating. As the perturbed basal shear stress and sliding velocity are in anti-phase for long-wavelength perturbations (with sliding velocity in phase with basal temperature), and given that their relative importance with respect to the phase of frictional heating depends on whether sliding is slow (perturbations of the sliding velocity dominate in the frictional heating term) or fast (perturbations of the basal shear stress dominate), it is apparent that only one regime will be viable. The sliding law then prescribes that the sliding velocity is in phase with the prescribed perturbation of basal temperature, while the perturbed basal shear stress is in anti-phase; given the role of perturbed basal shear stress and sliding velocity in determining the phase of frictional heating described above, this implies that the basal energy budget can only be satisfied for slow sliding. This argument explains why this unstable mode is a solution to the linear problem when sliding is slow, but it is not in the opposite case of fast sliding.

An important feature of this thermal runaway feedback is that the growth rate is independent of wavelength for k_n → 0, thus suggesting that the instability should persist at length scales much longer than the ice thickness. Interestingly, the physics of the thermal runaway feedback do not match the physics at the origin of backward diffusion for short wavelengths in the ice sheet scale slab problem (§2c). This apparent contradiction is resolved by introducing a horizontal length scale intermediate between ice sheet and ice thickness scale, which we do in §4.

As we shall see shortly, upon introducing this intermediate length scale, we will find that some aspects of the thermal runaway feedback detected in the long-wavelength limit of the ice thickness scale problem persist at this intermediate length scale and up to the ice sheet scale, provided the relevant fast time scale (ε times the natural advective time scale of the ice sheet) is taken into account. However, we will also find that the thermal runaway feedback is a solution for both slow and fast sliding at the ice sheet scale. Mathematically, perturbations of the basal shear stress become small at this longer scale, hence frictional heating is dominated by perturbations of the sliding velocity, meaning that the basal energy budget can always be satisfied. This unconditional instability, again with a constant growth rate, implies that we will also have to revisit the stability analysis for the ice sheet scale slab of §2c to account for the faster time scale, which we will do in §4b.

4. A higher order, large-scale model for the subtemperate region

(a). A length scale intermediate between ice thickness and ice sheet scale

The ice thickness scale stability analysis of the last section highlighted the necessity to resolve a length scale intermediate between the ice thickness and the ice sheet scale. Holding on to the distinguished limit δ∼ε^1/2, such length scale arises naturally from the heat transport problem near the bed in the form of a distinct dominant balance with x∼ε^1/2. At this longer horizontal length scale, the outer thermal problem is still advection-dominated, while the inner problem is purely diffusive both in the ice and in the bed, as opposed to the advection–diffusion problem at the ice thickness scale. In the following, we first state a leading order model that holds at this intermediate length scale, and then perform a linear stability analysis in order to clarify whether the thermal runaway detected at the ice thickness scale persists also at this longer scale.

(i). Model

To derive a leading order approximation that holds at this intermediate scale, once again we adopt a coordinate system aligned with the bed, with local bed slope μ, and rescale the relevant ice sheet scale model (described in electronic supplementary material, S2.1) as

$$\mathsf{X}={\epsilon }^{-1/2}x,\mathsf{z}=z-b,\tau ={\epsilon }^{-1}t(\mathsf{u}\mathsf{,}\mathsf{w})=(u,{\epsilon }^{1/2}w),\mathsf{h}=h,\mathsf{p}=p,\mathsf{T}=T.$$ 4.1a

We also expand dependent variables as

$$\begin{array}{rl}& \mathsf{u}=(\mathsf{u}\mathsf{,}\mathsf{w})={\mathsf{u}}^{(0)}+O({\epsilon }^{1/2}),\mathsf{h}={\mathsf{h}}^{(0)}+{\epsilon }^{1/2}{\mathsf{h}}^{(1)}+O(\epsilon ),\mathsf{p}={\mathsf{h}}^{(0)}-\mathsf{z}+{\epsilon }^{1/2}{\mathsf{h}}^{(1)}+O(\epsilon ),\\ & \mathsf{T}={\mathsf{T}}^{(0)}+O\left({\epsilon }^{1/2}\right),{\mathsf{T}}_b={{\mathsf{T}}_b}^{(0)}+\delta {{\mathsf{T}}_b}^{(1)}+O({\delta }^2),\end{array}\}$$ 4.1b

where the leading order ice thickness h⁽⁰⁾, which is independent of X and τ, is prescribed by the ice sheet scale problem, and T_b is basal temperature. For the sake of space the model derivation is discussed in electronic supplementary material, S2.3.2, while below we just state the leading order approximation.

Over this fast time scale, the leading order mechanical model is a shallow ice model that links the first-order correction of the ice thickness, h⁽¹⁾, to the leading order mass flux, q⁽⁰⁾. With an error of O(ε^1/2) we have

$$\frac{\mathrm{\partial }{\mathsf{h}}^{(1)}}{\mathrm{\partial }\tau }+\frac{\mathrm{\partial }{\mathsf{q}}^{(0)}}{\mathrm{\partial }\mathsf{x}}=0,$$ 4.2a

with leading order ice flux and horizontal velocity respectively defined as

$${\mathsf{q}}^{(0)}={\int }_{\mathsf{z}=0}^{\mathsf{z}={\mathsf{h}}^{(0)}}{\mathsf{u}}^{(0)}\hspace{0.17em}\mathrm{d}\mathsf{z}+{{\mathsf{u}}_b}^{(0)}{\mathsf{h}}^{(0)},{\mathsf{u}}^{(0)}=-\frac{1}{2}[{{\mathsf{h}}^{(0)}}^2-{({\mathsf{h}}^{(0)}-\mathsf{z})}^2][\frac{\mathrm{\partial }{\mathsf{h}}^{(1)}}{\mathrm{\partial }\mathsf{x}}-\tan (\mu )]+{{\mathsf{u}}_b}^{(0)},$$ 4.2b

where the sliding velocity is given by the temperature-dependent sliding law

$${{\mathsf{u}}_b}^{(0)}={\gamma }^{-1}F\left({{\mathsf{T}}_b}^{(1)}\right){{\mathsf{t}}_b}^{(0)},\text{with}\hspace{0.17em}\hspace{0.17em}{{\mathsf{t}}_b}^{(0)}=-{\mathsf{h}}^{(0)}\frac{\mathrm{\partial }{\mathsf{h}}^{(1)}}{\mathrm{\partial }\mathrm{\partial }\mathsf{x}}.$$ 4.2c

The leading order outer thermal problem simply states that T⁽⁰⁾ is independent of the fast time scale τ, both in the ice and in the bed, with T⁽⁰⁾ satisfying (with an error of O(ε^1/2))

$${\mathsf{T}}^{(0)}(\mathsf{z}={\mathsf{h}}^{(0)})=-T_s,{\mathsf{T}}^{(0)}(\mathsf{z}=0)={{\mathsf{T}}_b}^{(0)},-\frac{\mathrm{\partial }{\mathsf{T}}^{(0)}}{\mathrm{\partial }\mathsf{z}}\to \nu \text{as}\mathsf{z}\to -\mathrm{\infty },$$ 4.3

where T_b⁽⁰⁾ = 0 given that δ≪1. In order to accommodate O(δ) deviations of bed temperature from the melting point T_b⁽¹⁾, once again we resort to a thermal boundary layer near the bed. To this aim, we rescale the outer thermal model as

$$\tilde{\mathsf{x}}=\mathsf{x},\tilde{\mathsf{z}}={\delta }^{-1}\mathsf{z},\tilde{\mathsf{T}}={\delta }^{-1}\mathsf{T},\tilde{\mathsf{u}}=\mathsf{u},{\tilde{\mathsf{t}}}_b={\mathsf{t}}_b,$$ 4.4

where we put δ∼ε^1/2, and also expand dependent variables as

$$\tilde{\mathsf{u}}={{\mathsf{u}}_b}^{(0)}+O(\delta ),{\tilde{\mathsf{t}}}_b={{\mathsf{t}}_b}^{(0)}+O(\delta ),\tilde{\mathsf{T}}={\tilde{\mathsf{T}}}^{(0)}+O(\delta ).$$ 4.5

We will then have ${{\mathsf{T}}_b}^{(1)}={\tilde{\mathsf{T}}}^{(0)}(\tilde{z}=0)$, where the leading order problem for ${\tilde{\mathsf{T}}}^{(0)}$ correct to O(ε^1/2) is

$$\frac{\mathrm{\partial }{\tilde{\mathsf{T}}}^{(0)}}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{\tilde{\mathsf{T}}}^{(0)}}{\mathrm{\partial }{\tilde{\mathsf{z}}}^2}=0\text{for}\hspace{0.17em}-\mathrm{\infty }<\tilde{\mathsf{z}}<0\hspace{0.17em}\text{and}\hspace{0.17em}0<\tilde{\mathsf{z}}<+\mathrm{\infty },$$ 4.6a

with boundary conditions at the bed

$${\left[{\tilde{\mathsf{T}}}^{(0)}\right]}_{-}^{+}=0,{\left[\frac{\mathrm{\partial }{\tilde{\mathsf{T}}}^{(0)}}{\mathrm{\partial }\tilde{\mathsf{z}}}\right]}_{-}^{+}+\alpha \hspace{0.17em}{{\mathsf{u}}_b}^{(0)}{{\mathsf{t}}_b}^{(0)}=0,\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}\tilde{\mathsf{z}}=0,$$ 4.6b

and matching conditions

$$\frac{\mathrm{\partial }{\tilde{\mathsf{T}}}^{(0)}}{\mathrm{\partial }\tilde{\mathsf{z}}}\to 0\text{as}\hspace{0.17em}\tilde{\mathsf{z}}\to +\mathrm{\infty },-\frac{\mathrm{\partial }{\tilde{\mathsf{T}}}^{(0)}}{\mathrm{\partial }\tilde{\mathsf{z}}}\to \nu \text{as}\hspace{0.17em}\tilde{\mathsf{z}}\to -\mathrm{\infty }.$$ 4.6c

(ii). Linear stability analysis

We are interested in the response of the leading order model above to small-amplitude perturbations, which we study through a linear stability analysis. In the usual way, we expand dependent variables about their uniform-in-x steady state (denoted by a bar) as

$${\mathsf{h}}^{(0)}=\overline{\mathsf{h}},{\mathsf{h}}^{(1)}=\beta {\mathsf{h}}^{\prime },{{\mathsf{u}}_b}^{(0)}=\overline{{\mathsf{u}}_b}+\beta {\mathsf{u}}_b^{\prime },{{\mathsf{t}}_b}^{(0)}={\overline{\mathsf{t}}}_b+\beta {\mathsf{t}}_b^{\prime },{\tilde{\mathsf{T}}}^{(0)}=\overline{\tilde{\mathsf{T}}}+\beta {\tilde{\mathrm{T}}}^{\prime },\beta \ll 1,$$ 4.7

and we then Fourier-transform perturbed variables in x through (3.10). Assuming separation of variables, so that $\mathrm{\partial }{\hat{\mathsf{g}}}_n/\mathrm{\partial }\tau =\sigma {\hat{\mathsf{g}}}_n$, and linearizing about the steady state, to O(β) we find the following mechanical problem:

$$\sigma {\hat{\mathsf{h}}}_n+\mathrm{i}{k}_n\overline{\mathsf{h}}{\hat{\mathsf{U}}}_{b,n}+k_n^2\frac{{\overline{\mathsf{h}}}^3}{3}{\hat{\mathsf{h}}}_n=0,{\hat{\mathsf{U}}}_{b,n}={\gamma }^{-1}[\overline{F}{\hat{\mathsf{t}}}_{b,n}+{\overline{\mathsf{t}}}_b{F}^{\prime }{\widehat{\tilde{\mathsf{T}}}}_n(\tilde{\mathsf{z}}=0)],{\hat{\mathsf{t}}}_{b,n}=-\mathrm{i}{k}_n\overline{\mathsf{h}}{\hat{\mathsf{h}}}_n,$$ 4.8a

with $F^{\prime }=\mathrm{d}F/\mathrm{d}\mathsf{T}{|}_{\overline{\mathsf{T}}}$. The heat equation simplifies to

$$\sigma {\widehat{\tilde{\mathsf{T}}}}_n-\frac{{\mathrm{d}}^2{\widehat{\tilde{\mathsf{T}}}}_n}{\mathrm{d}{\tilde{\mathsf{z}}}^2}=0,\text{for}\hspace{0.17em}-\mathrm{\infty }<\tilde{\mathsf{z}}<0,0<\tilde{\mathsf{z}}<+\mathrm{\infty },$$ 4.8b

with boundary conditions

$$\frac{\mathrm{\partial }{\widehat{\tilde{\mathsf{T}}}}_n}{\mathrm{\partial }\tilde{\mathsf{z}}}\to 0\text{as}\tilde{\mathsf{z}}\to \pm \mathrm{\infty },{\left[\frac{\mathrm{\partial }{\widehat{\tilde{\mathsf{T}}}}_n}{\mathrm{\partial }\tilde{\mathsf{z}}}\right]}_{-}^{+}+\alpha ({\overline{\mathsf{t}}}_b{\hat{\mathsf{U}}}_{b,n}+{\overline{\mathsf{U}}}_b{\hat{\mathsf{t}}}_{b,n})={\left[{\widehat{\tilde{\mathsf{T}}}}_n\right]}_{-}^{+}=0\text{on}\hspace{0.17em}\tilde{\mathsf{z}}=0.$$ 4.8c

The heat equation (4.8b) can be integrated straightforwardly along with (4.8c_1,3) yielding

$${\widehat{\tilde{\mathsf{T}}}}_n=\{\begin{array}{ll}{\mathsf{T}}_0\exp (-\tilde{\mathsf{z}}{\sigma }^{1/2})& \text{for}\tilde{\mathsf{z}}>0\\ {\mathsf{T}}_0\exp (\tilde{\mathsf{z}}{\sigma }^{1/2})& \text{for}\tilde{\mathsf{z}}<0,\end{array}$$ 4.9a

where boundedness of temperature in the far field ($\tilde{\mathsf{z}}\to \pm \mathrm{\infty }$) is ensured provided Re[σ^1/2] > 0, while from equation (4.8a) we get

$${\mathsf{T}}_0=\mathrm{i}{\hat{\mathsf{h}}}_n\frac{[{\overline{\mathsf{h}}}^2{k}^2(3\overline{F}+\overline{\mathsf{h}}\gamma )+3\gamma \sigma ]}{\left(3k\overline{\mathsf{h}}{\overline{\mathsf{t}}}_b{F}^{\prime }\right)},{\hat{\mathsf{U}}}_{b,n}=\mathrm{i}\hat{\mathsf{h}}\frac{({\overline{\mathsf{h}}}^3{k}^2+3\sigma )}{\left(3\overline{\mathsf{h}}k\right)}.$$ 4.9b

Lastly, the eigenvalue can be computed from the basal energy budget (4.8c₂), which can be seen as a third degree polynomial in y = σ^1/2. Further details on the computation of σ are provided in electronic supplementary material, S5.1, while results are displayed in figure 5. The absolute value of Re[σ] is plotted against k_n for slow (figure 5a) and fast sliding (figure 5b), with colours marking different solution branches. Note that axes are logarithmic, so solid and dashed lines are meant to identify Re[σ] > 0 and Re[σ] < 0, respectively.

Figure 5.

Growth rate of the model (4.8). Displayed is |Re[σ]|, with solid (dashed) line denoting Re[σ] > 0 (Re[σ] < 0). Parameters are as in figure 4.

(iii). Processes

Figure 5 illustrates that the behaviour of the system depends on the relative importance of sliding with respect to shearing both in the short (k_n → ∞) and long (k_n → 0) wavelength limits. We discuss the relevant physics in each limit below, while we refer readers to the electronic supplementary material for the mathematical details.

In the short-wavelength limit, $k_n\to \mathrm{\infty }$, we expect that some of the physics correspond to the long-wavelength limit of the ice thickness scale problem described in §3b. An asymptotic analysis in this limit (electronic supplementary material, S5.1.3) confirms that this is the case, with the unstable branch that appears for slow sliding (red solid curve in figure 5a) being the continuation at the intermediate scale of the thermal runaway branch identified in the long-wavelength limit of the ice thickness scale problem (red curve in figure 4a).

In the long-wavelength limit, $k_n\to 0$, we identify two distinct modes: one has growth rate ∼k²_n (yellow curve in figure 5a, yellow and orange curves in figure 5b), and corresponds to the short-wavelength limit of the ice sheet scale slab discussed in §2c, with backward diffusion for fast sliding (yellow curve in figure 5b) and stability for slow sliding (figure 5a). The second mode (red curve in figure 5a,b) is unconditionally unstable for both slow and fast sliding, and arises from the same thermal runaway feedback between basal temperature and basal heat flux identified in the long-wavelength limit of the ice thickness scale problem. However, at this longer length scale an O(1) change in basal temperature yields only an O(1) perturbation of sliding velocity, while perturbations of the basal shear stress remain small. As a result, frictional heating and net heat flux into the ice are in anti-phase for both slow and fast sliding, so the requirement of a basal energy budget in balance can be always satisfied. This is indeed the reason why this unstable feedback yields a solution for both cases at this longer length scale.

Assuming that the long-wavelength limit of the intermediate length-scale problem corresponds to the short-wavelength limit of the ice sheet scale problem, our results suggest that even a leading order model at the ice sheet scale should remain unstable as a result of the thermal runaway feedback identified above. In this regard, it is important to recall that the thermal runaway develops over a fast time scale, ε⁻¹ times the natural advective time scale of the ice sheet. The leading order subtemperate sliding model in §2 uses the limit δ → 0 to justify a pseudo-steady thermal state close to the bed, while the fast time scale instability we have just encountered indicates that this is in fact a singular limit that needs to be revisited at the ice sheet scale too. We will do exactly that in the next subsection, showing that (i) the thermal runaway instability persists for subtemperate sliding at all wavelengths regardless of whether sliding is slow or fast and (ii) that our earlier conclusion that subtemperate sliding might be stable provided sliding is slow is fallacious.

(b). The ice sheet scale revisited

Let us consider an ice sheet scale subtemperate ice slab for which we want to resolve finite deviations of bed temperature from the melting point. Holding on to the distinguished limit δ∼ε^1/2, and using the fast time scale τ = t/ε, a leading order approximation of the thermo-mechanical problem can be obtained from the ice sheet scale model for the subtemperate slab described in electronic supplementary material, S2.1, following the procedure outlined for the intermediate scale in §4a. In the following, we first state the leading order approximation that holds at this scale (details are provided in electronic supplementary material, S2.3.3), and then perform a linear stability analysis in order to clarify whether the thermal runaway detected at the intermediate scale persists in an ice sheet scale model.

(i). Model

We proceed as in §4a, but replace the rescalings (4.1a) with a simple rescaling of time τ = ε⁻¹t; note that we do not rename the remaining dependent and independent variables (which we leave unscaled) in order not to complicate notation. Expanding dependent variables as

$$\begin{array}{rl}& \mathit{u}={\mathit{u}}^{(0)}+O({\epsilon }^2),h=h^{(0)}+O(\epsilon ),p=h^{(0)}-z+O(\epsilon ),\\ & T=T^{(0)}+o(1),T_b={T_b}^{(0)}+\delta {T_b}^{(1)}+O({\delta }^2),\end{array}\}$$ 4.10

it is straightforward to show that neither the leading order mechanical problem, nor the outer thermal problem evolve over this fast time scale, that is

$$\frac{\mathrm{\partial }{h}^{(0)}}{\mathrm{\partial }\tau }=\frac{\mathrm{\partial }{T}^{(0)}}{\mathrm{\partial }\tau }=0,$$ 4.11

while dynamics are restricted to the near-bed thermal boundary layer at these long horizontal length scales. Once again, we obtain the leading order problem in the thermal boundary layer by rescaling the outer thermal problem as

$$\tilde{x}=x,\hspace{0.17em}\hspace{0.17em}\tilde{z}={\delta }^{-1}z,\hspace{0.17em}\hspace{0.17em}\tilde{T}={\delta }^{-1}T,\hspace{0.17em}\hspace{0.17em}\tilde{u}=u,\hspace{0.17em}\hspace{0.17em}{\tilde{\tau }}_b={\tau }_b,$$ 4.12

and expand as

$$\tilde{u}={u_b}^{(0)}+O(\delta ),\hspace{0.17em}\hspace{0.17em}{\tilde{\tau }}_b={{\tau }_b}^{(0)}+O(\delta ),\hspace{0.17em}\hspace{0.17em}\tilde{T}={\tilde{T}}^{(0)}+O(\delta ),$$ 4.13

where the sliding velocity u_b⁽⁰⁾ is given by the sliding law

$${u_b}^{(0)}={\gamma }^{-1}F\left({T_b}^{(1)}\right){{\tau }_b}^{(0)},\text{with}\hspace{0.17em}{{\tau }_b}^{(0)}=-h^{(0)}\frac{\mathrm{\partial }{h}^{(0)}}{\mathrm{\partial }x},$$ 4.14

and h⁽⁰⁾ is the solution of the leading order ice sheet scale problem at the natural, advective time scale of the ice sheet. Then the leading order boundary layer problem correct up to O(ε) is

$$\frac{\mathrm{\partial }{\tilde{T}}^{(0)}}{\mathrm{\partial }\tau }-\frac{{\mathrm{\partial }}^2{\tilde{T}}^{(0)}}{\mathrm{\partial }{\tilde{z}}^2}=0\text{for}\hspace{0.17em}-\mathrm{\infty }<\tilde{z}<0\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}0<\tilde{z}<+\mathrm{\infty },$$ 4.15a

with boundary and matching conditions

$${\left[{\tilde{T}}^{(0)}\right]}_{-}^{+}=0,{\left[\frac{\mathrm{\partial }{\tilde{T}}^{(0)}}{\mathrm{\partial }\tilde{z}}\right]}_{-}^{+}+\alpha \hspace{0.17em}{u_b}^{(0)}{{\tau }_b}^{(0)}=0\text{on}z=0$$ 4.15b

and

$$\frac{\mathrm{\partial }{\tilde{T}}^{(0)}}{\mathrm{\partial }\tilde{z}}\to 0\hspace{0.17em}\text{as}\hspace{0.17em}\tilde{z}\to +\mathrm{\infty },-\frac{\mathrm{\partial }{\tilde{T}}^{(0)}}{\mathrm{\partial }\tilde{z}}\to \nu \text{as}\hspace{0.17em}\tilde{z}\to -\mathrm{\infty }.$$ 4.15c

(ii). Stability and processes

It is now clear that the thermal problem at the ice sheet scale is identical to the intermediate scale one, equations (4.6), and consists only of a time-dependent diffusion problem. The key difference is that at the ice sheet scale there are no dynamics in the mechanical problem at this fast time scale, so the stability of the system is entirely driven by the thermal boundary layer.

A linear stability analysis on the problem (4.15) is straightforward (details of the analysis are presented in electronic supplementary material, S5.2), and yields the positive eigenvalue

$$\sigma ={\left(\alpha F^{\prime }\frac{{\overline{\tau }}_b}{2}\gamma \right)}^2,$$ 4.16

which is independent of wavelength. It can be easily shown that the latter expression for σ corresponds to the leading order approximation of the thermal runaway branch for the intermediate length-scale problem in the limit k_n → 0 (see electronic supplementary material, S5.1), thus confirming that the thermal runaway feedback identified there survives in the ice sheet scale model. This excludes the possibility that a stable, steady subtemperate region exists, regardless of whether sliding is slow or fast.

5. Nonlinear dynamics at the ice thickness scale

Having determined that a subtemperate sliding region as envisaged by Fowler [14] is unconditionally unstable, though in different ways for the slow and fast sliding regimes, we still cannot say what the cold-to-temperate transition in a fully evolved ice sheet would look like, as this requires us to solve the coupled ice flow–heat transport problem beyond its linearized form.

In this regard, the most relevant feature of the stability analyses of §§3 and 4 is the poor wavelength selection coupled with a very poor separation of length scales: an unstable branch of the spectrum spans all wavelengths from the ice thickness to the ice sheet scale, with the fastest-growing wavelength at the ice sheet thickness scale. This constitutes a major challenge to solving the transition problem numerically: the poor scale separation implies that one potentially has to resolve simultaneously the ice thickness and the ice sheet scale to predict the thermal state that the ice sheet evolves towards. In particular, the poor scale separation precludes us from applying a multiple scales expansion [21] to parametrize the effect of instability at the local scale in a simpler, large-scale model.

By contrast, there is some hope in principle of such a method being applicable for the case of fast sliding. This is conceptually analogous to Reynolds averaging in turbulent flows, the idea being to be able to derive expressions for spatially and temporally averaged heat transport at the inner scale in terms of outer variables only [21]. While being possible in principle, the averaging requires us to solve the nonlinear dynamics at the inner scale, and determining what state the local instability evolves into. To this aim, we solve the ice thickness scale problem (3.2)–((3.5)) numerically.

We solve the time-dependent thermo-mechanical problem (3.2)–((3.5)) on a periodic domain, using a finite volume approach with a centred, second-order accurate spatial discretization for both the Stokes problem and the inner and outer thermal problems, and an implicit time stepping scheme based on a backward Euler's step (for details see electronic supplementary material, S6). We force the system with a small-amplitude, noisy perturbation of basal temperature added to a uniform-in-X steady state, and construct an initial condition for ψ, ${\mathcal{Q}}^{(0)}$ and ${\tilde{\mathit{\Theta }}}^{(0)}$ by linear superposition of the Fourier modes derived for the linearized model in §3b. The noisy forcing has a white noise spectrum, which we cut off at some wavenumber intermediate between the fastest-growing wavenumber and the Nyquist wavenumber. For all simulations, we assume an exponential dependence of the sliding velocity on temperature below the melting point, that is ${U_b}^{(0)}={\gamma }^{-1}\exp ({\tilde{\mathit{\Theta }}}^{(0)}){T_b}^{(0)}$, as suggested by Shreve [1] and Fowler [18].

Figure 6 illustrates a typical solution for the fast sliding case. Figure 6a shows basal temperature along the slab for different moments in time, plotted with a vertical stagger for ease of visualization. The simulation is interrupted at τ = 64.5 when the sliding velocity reaches zero, with the final state displayed in the rightmost column of figure 6 (basal temperature in figure 6a, sliding velocity in figure 6b, outer heat flux in figure 6d. Note that our model breaks down when the bed freezes because we are considering an asymptotic reduction of the thermal problem based on the limit Pe _slab → ∞ (as opposed to solving the full thermal problem and making Pe _slab≫1) which no longer holds when the sliding velocity becomes small. Lastly, for this simulation (as well as for its slow sliding counterpart in figure 7), we have chosen model parameters consistently with §3b, so the spectrum of the corresponding linearized model is the one plotted in figure 4b.

Figure 6.

Numerical solution for the time-dependent problem (3.2)–((3.5)) for slow sliding. (a) Basal temperature as a function of X at time intervals of 2.7 plotted with a vertical stagger for ease of viewing. The final state is at τ = 64.5. (b–d) Basal temperature, sliding velocity and outer heat flux at the final time step. Note that variables are scaled with leading order ice flux q⁽⁰⁾ and ice thickness H⁽⁰⁾ to reduce the number of degrees of freedom. Parameters are ν = 1/H⁽⁰⁾, α = (q⁽⁰⁾/H⁽⁰⁾)⁻², γ = 1/H⁽⁰⁾, Θ_bed = − 0.5; the numerical time step is dτ = 0.007, while grid spacing is uniform in X and Z and equal to 0.01.

Figure 7.

Numerical solution for the time-dependent problem (3.2)–((3.5)) for slow sliding. (a) Basal temperature as a function of X at time intervals of 1.8 plotted with a vertical stagger for ease of viewing. The final state is at τ = 37.8. (b–d) Basal temperature, sliding velocity and outer heat flux at the final time step. Note that variables are scaled with leading order ice flux q⁽⁰⁾ and ice thickness H⁽⁰⁾ to reduce the number of degrees of freedom. Parameters as in figure 6 except for γ = 5/H⁽⁰⁾.

The main insight from this solution is that initially small perturbations grow unboundedly until the bed eventually freezes in places. This is not unique to the fast sliding regime, but occurs also for slow sliding (see results from an analogous simulation illustrated in figure 7), thus suggesting that patchy refreezing is a robust feature of the underlying continuum model, with the size of the frozen patches comparable in size with the linearly most unstable wavelength. Note that these results differ from the patches computed by Wolovick et al. [22]: we are concerned here with sliding that remains entirely subtemperate in our calculations, whereas Wolovick et al. [22] allow for patches of unfrozen bed, and as a result track the water content of the bed.

Unbounded growth of the instability until freezing shows that a more sophisticated thermal model is needed to predict the evolution of the frozen patches, and no simple closure can be constructed to model the effect of the instability at the ice thickness scale on the flow at the large scale. Furthermore, when the velocity near the bed becomes asymtotically small, we expect a different dominant balance in the thermal boundary layer, with horizontal diffusion and strain heating featuring at leading order. This balance must be accounted for to capture the evolution of the frozen patches, which is not possible in our model. We conclude by noting that, despite these complications, our numerical solution for ice flow in the subtemperate region shows no sign of ill-posedness, thus confirming our former conclusion that accounting for a finite δ in the temperature-dependent sliding law fixes the pathological behaviour observed in Fowler's model [14] (further details on this are discussed in §2).

6. Discussion and conclusion

Having identified in a companion paper [11] that a laterally uniform steady-state configuration for an ice sheet with basal thermal transitions appears to require an extended subtemperate region as predicted by Fowler & Larson [13], the present paper studied the stability of such a steady state. In §2, we analysed the stability of a horizontally one-dimensional, shallow ice model with subtemperate sliding to small-amplitude perturbations. In particular, we followed Fowler & Larson [13] and Fowler [14] in assuming that the temperature range over which significant sliding can occur below the melting point is very small, and used this to compute sliding velocity directly from a basal thermal balance.

Our analysis in §2b,c shows how such a formulation for subtemperate sliding gives rise to a velocity-weakening friction law, rendering the shallow ice model as a pathological, backward diffusion problem for ice thickness. These results highlight the need for a more sophisticated model for the subtemperate region. Ill-posedness at the ice sheet scale could suggest that the instability really occurs at a shorter length scale not captured by the ice sheet-scale model.

We pursued this in §3b, where we considered simultaneously the effect of two limits invoked in the construction of the large-scale shallow ice model, namely that the velocity regularization parameter in the sliding law is very small (δ≪1), as is the aspect ratio (ε≪1). Both are singular limits that we must account for to arrive at a well-posed model for subtemperate sliding.

Rescaling the horizontal coordinate to the ice thickness scale, we find that finite (∼O(δ)) deviations of bed temperature from the melting point eventually cause growth of perturbations to be suppressed at very short wavelengths due to a thermal boundary layer near the bed (§3a). In practice, restoring vertical diffusion to leading order near the bed adds one degree of freedom compared with the singular limit $\delta \to 0$, thus allowing both the temperature-dependent sliding law and the requirement that the basal energy budget remains in balance to be satisfied at leading order.

At the opposite end of the range of possible length scales, recognition of the fact that δ≪1 is a singular perturbation immediately leads to the conclusion that the sliding velocity can never be controlled by the energy balance of the bed as envisaged by Fowler & Larson [13] and Fowler [14], but that the relevant instability is artificially suppressed in their formulation for subtemperate sliding. Retention of an explicit temperature dependence in the sliding law (corresponding to finite δ) shows instead that subtemperate sliding is also unstable at infinite wavelength, although with a growth rate that does not diverge. Long-wavelength instability is simply due to a thermal runaway: since sliding becomes faster as basal temperature approaches the melting point, an increase in basal temperature augments the sliding velocity, which feeds back into the basal energy budget of the bed through frictional heating. As a result, the bed becomes even warmer, thus leading to a runaway growth that is only curbed when the melting point itself is reached. We are therefore left with a set of instabilities that span all length scales, ranging from the ice thickness scale to the horizontal ice sheet extent.

As discussed in §4, instability occurs on a fast (∼ε⁻¹) time scale at three distinct spatial scales: the ice thickness scale, the ice sheet scale, and the geometric average of the two. Each of these length scales corresponds to a different leading order model, with different physical balances. However distinct, these physical balances do not support any true separation of length scales: in fact, the spectrum of the linearized problems considered here is nearly flat across all these length scales. By this, we mean that the growth rate does not go to zero at wavelengths intermediate between ice thickness scale and ice sheet length scale, at least for sliding velocities that are slower than the shear velocity (which must be the case somewhere for a continuous transition from essentially no slip to fully evolved, temperate sliding). This lack of scale separation rules out the use of multiple scales techniques to average over the local scale, which could otherwise have permitted a simplified large-scale problem formulation in which the local scale velocity and temperature problem does not have to be solved for at the same time as the large-scale problem of evolving ice thickness.

This conclusion contrasts with the results of Hindmarsh [23], who finds that a smooth, stable transition from slow to fast sliding is possible without short-wavelength oscillations. The likely reason of this discrepancy is that the pathology is linked to the singular limit of small δ, while Hindmarsh [23] employs a sliding law in which the transition of slow to fast sliding occurs over a large range of temperatures, effectively putting δ = O(1); his model employs this device in order to avoid having to deal with shearing in the ice.

Our analytical results have only demonstrated that simple steady-state solutions of the subtemperate sliding problem are not stable, but do not provide information about what that instability evolves into. We consider the nonlinear evolution problem in §5, where we show that growth of the instability does not saturate at a finite amplitude before parts of the bed refreeze, in the sense of temperatures becoming low enough to effectively suppress all sliding.

This leaves open the question of how the transition between a cold, non-sliding bed and a fully temperate, sliding bed should work. In one horizontal dimension, a full numerical solution of the subtemperate sliding problem considered here may be required, which may reproduce the moving temperate patches of Wolovick et al. [22], although the latter were forced by an injection of water at the upstream end of their domain, rather than being the result of an instability. The model in Wolovick et al. [22] further has the odd feature that the bed can be warmed by injection of water into previously frozen areas (see their electronic supplementary material), rather than a frozen bed presenting a barrier to water flow, which is likely to affect the propagation of patches in their results. That said, it turns out that introduction of a second, transverse horizontal dimension significantly alters the conclusions we have come to here. As we will describe in a follow-up paper (in preparation), several of the stumbling blocks to a smooth transition from a cold to fully temperate bed can be avoided if flow is organized into an intrinsically three-dimensional structure, though not without significant challenges to implementation in large-scale ice sheet models.

Data accessibility

Source code for the numerical simulations presented in the paper is available at https://github.com/elisamantelli/subtemperate_sliding_rspa_2019.

Authors' contributions

The authors contributed equally to the research and jointly drafted the structure of the paper. E.M. wrote the paper and electronic supplementary material. The authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

The authors declare no competing interests.

Funding

E.M. acknowledges financial support from Stanford's SiGMa and RadioGlaciology groups, and NSERC Discovery (grant no. RGPIN 357193-13). C.S. acknowledges NSERC Discovery Grant (grant nos. RGPIN 357193-13 and RGPIN-2018-04665). Compute Canada provided computing resources.