1. Introduction

The ice streams of the Siple Coast of the West Antarctic ice sheet are bands of rapidly moving ice that play a crucial role in the discharge of ice to the ocean and the mass balance of the ice sheet (Reference Alley and WhillansAlley and Whillans, 1991). The rapid streaming motion is enabled by lubrication of the base by pressurized water and weak subglacial till (Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990), which allows the rapid motion in spite of very low surface slopes and associated gravitational driving stress.

Examination of the force balance (Reference Van der Veen and WhillansVan der Veen and Whillans, 1989) has revealed that Siple Coast ice streams are supported in part from their lateral margins (Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994; Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others, 1994; Reference MacAyeal, Bindschadler and ScambosMacAyeal and others, 1995; Reference Jackson and KambJackson and Kamb, 1997; Reference Whillans and van der VeenWhillans and Van der Veen, 1997; Reference Harrison, Echelmeyer and LarsenHarrison and others, 1998). A related implication is that some of the energy from downslope motion will be dissipated by shearing in the ice near the margins rather than at the ice base in the central part of the ice stream.

The partitioning of energy dissipation between the margins and the bed is important in view of the role played by basal meltwater in lubrication of the bed. It becomes especially significant given that the balance of heat conducted to the base of ice streams could be negative (Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990). Net conduction of heat away from the bed would cause freezing and could operate to eliminate lubrication unless there were a compensating production of heat by the motion.

Reference RaymondRaymond (1995) showed that maximum basal heat production and generation of meltwater occurs for an intermediate level of lubrication, and defined regimes in which the acceleration of an ice stream by accumulation of basal water is limited by increasing capacity of the drainage system or alternatively by declining water production.

This paper uses a theoretical model of margin drag to develop the basis for these concepts more fully. I explore the partitioning between bed and margins of the motion-generated heat and examine the corresponding implications for the basal water balance that maintains lubrication in the central portion of an ice stream. Six cross-sections across Ice Streams B, D, E and Rutford Ice Stream are examined to identify characteristics for force balance and consequences for mechanical heating and associated melting on their beds.

2. Description of Processes

Figure 1 shows a schematic cross-section of an ice stream and adjacent ridges. The two important processes to quantify are motion over the base and deformation in the ice associated with lateral shearing across the margins.

Fig. 1. Descriptive cross-section of an ice stream and definition of coordinate system.

Motion over the base is described by a relationship between basal velocity u _b and shear stress τ _b given by

The parameter m allows a power relationship between u _b and τ _b. The parameter τ ₀ represents resistance to slip (an inverse lubrication) determined by the physical state of the bed as affected by roughness, till thickness, overburden stress, water pressure, amount of water storage and/or possibly other variables.

Equation (1) can represent basal motion derived from a variety of physical assumptions including sliding of ice over a rough hard bed (Reference WeertmanWeertman, 1957) or deformation of soft, water-saturated basal till (Reference AlleyAlley, 1989). Experimental investigations of till suggest a nearly cohesion-less, frictional-plastic behavior (Reference KambKamb, 1991; Reference Iverson, Hooyer and BakerIverson and others, 1998), which would be represented by m → ∞ The base of an ice stream may be heterogeneous, possibly contacting rock or locked till as well as deforming, dilated till. The basal motion may therefore involve a combination of sliding, till deformation, and ice deformation around sticky spots (Reference AlleyAlley, 1993; Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994; Reference MacAyeal, Bindschadler and ScambosMacAyeal and others, 1995; Reference Engelhardt and KambEngelhardt and Kamb, 1997). Thus, the basis for choosing specific values for τ ₀ and m in Equation (1) remains unclear. Equation (1) is adopted here because it allows the possibility that u _b increases monotonically with τ _b for a fixed level of lubrication because of stress-dependent deformation in the ice around sticky spots or in the till.

The inverse lubrication τ ₀ is used as an adjustable input variable to determine the locations and speeds of non-streaming and streaming flow. Outside of ice-stream boundaries, τ ₀ is taken to be effectively infinite, which suppresses slip. Under the ice streams, τ ₀ is taken to be small enough to predict significant u _b with low τ _b. The main purpose is to discover how the distribution of speed, stress and related energy dissipation is affected by the value of τ ₀ under an ice stream. Later it will be assumed that τ ₀ decreases monotonically with increasing water volume stored at the bed as described by an equivalent thickness δ (i.e. more water gives lower sliding resistance τ ₀ and higher lubrication 1/τ ₀).

The ice is assumed to deform in simple shear at strain rate in response to shear stress τ according to power-law creep (Reference PatersonPaterson, 1994, p.97)

∊ and τ are the shear strain-rate and stress magnitudes. A and n are the standard flow-law parameters. A depends on temperature. E is an enhancement factor for simple shear to account for ice fabric.

3. Force and Energy Balance

As an ice stream becomes more lubricated at its base, one expects several related consequences for the dynamics: the force balance is shifted more strongly toward support from the margins; the shear stress at the base of the ice stream drops; the speed increases. These effects on basal stress and speed have implications for generation of heat at the base of the ice stream in its central parts. Rate of mechanical generation of heat per unit area P is given by the product of basal velocity u _b and shear stress τ _b. Thus, it will be low for low lubrication, which causes u _b to be low. It will also be low for high lubrication, which causes τ _b to be low. Clearly, there will be a maximum at an intermediate level of lubrication (Reference RaymondRaymond, 1995). I investigate the dependencies of u _b, τ _b and P = u _b τ _b on the amount of lubrication (related to τ ₀ in Equation (1)) using methods in Reference RaymondRaymond (1996), henceforth referred to as R96.

To determine u _b and τ _b in the center of an ice stream, I use the basal-motion (Equation (1)) and ice-flow (Equation (2)) laws coupled through force balance in the direction of ice-stream motion (x) as given by

where τ_ij are components of deviatoric stress, p is mean compressive stress, p is ice density, g is gravitational acceleration, and α is the surface slope in the direction of ice-stream motion.

Of primary interest is stress and velocity on the bed in the ice stream, so I integrate Equations (3), (1) and (2) from the margin (y = 0) inward into the ice stream to the center (y = W) along the base (z = 0). The coupled equations are simplified by accounting for primary characteristics of the flow expressed by the following assumptions:

1. (1) Thickness H, slope α and lubrication (1/τ ₀) are all independent of horizontal position in the ice stream, and width does not vary along the length. Longitudinal gradients are not central to the problem addressed here. In reality they may be large locally because of topography or variations of τ ₀ on the bed, but on average they are generally small compared to the transverse gradients (Reference Bindschadler and ScambosBindschadler and Scambos, 1991; Reference Whillans and van der VeenWhillans and Van der Veen, 1993). In some cases the across-flow transition from sheet to streaming flow may be gradual (e.g. Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others, 1994). However, the primary characteristic is a sharp jump in lubrication beneath the margin as evidenced by the strong concentration of shear in the margins (Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994), and this can be captured with τ ₀ constant under the ice stream.

2. (2) Along-flow longitudinal (x) gradients in both deviatoric stress τ_xx and pressure p are negligible, so that

   (4)

   

   This follows from assumption (1).

3. (3) Lateral (y) gradients dominate vertical (z) gradients (∂u/∂y ≫ ∂u/∂z), so that

   (5)

   

   This assumption becomes valid over most of the width of an ice stream at high levels of lubrication appropriate for fast-moving ice streams (R96).

4. (4) Vertical shear stress gradient at the bed is equal to the average over the thickness, as described by

   (6)

   

   This assumption is consistent with the commonly assumed linear variation of shear stress with depth (Reference PatersonPaterson, 1994, p. 251).

Introduction of these assumptions into Equations (3), (1) and (2) gives the ordinary differential equation involving only the y variation of u _b along the bed z = 0,

where I use du _b/dy > 0 for y directed inward toward the ice-stream center. The function u _b (y) can be found by solving Equation (7) over y = [0, W] with boundary conditions u _b (0) = 0 and du _b (W)/dy = 0, where y = 0 is the margin and y = W is the center (Fig. 1). The function τ _b(y) can then be calculated from Equation (1). Since the primary concern here is with conditions near the center line y = W, I focus on u _b(W) and τ _b(W), which will henceforth be denoted simply as u _b and τ _b.

R96 (figs 7 and 8) showed that results from this approximate one-dimensional model of Equations (1–3) along the base are consistent with two-dimensional finite-element calculation of flow in the cross-section that fully includes both the lateral and vertical gradients in the force balance (Equation (3)) as long as the basal motion in the center is much larger than the speed produced by shearing through the thickness. This condition is appropriate for active ice streams moving over well-lubricated beds. While assumptions 3 and 4 may break down locally near the margin or center, those failures do not strongly affect the calculated shape of u _b(y) or more importantly u _b(W) derived from the integration from the margin to the center, which is of interest here.

It is useful to consider solution to Equation (7) for the limiting cases of no drag from the margins (low lubrication) and no friction on the bed (perfect lubrication). Straightforward solutions exist for both cases.

Low lubrication corresponds to no lateral gradients (d/dy = 0), and solution of Equation (7) for u _b gives

where

is the driving stress. T is the value that τ _b would have at the base of a sloped, planar ice slab with thickness H, slope α and no gradients parallel to the surface (Reference PatersonPaterson, 1994, p.239, 240). u _b is the basal speed that τ _b = T would produce with the basal motion law (Equation (1)). It is an alternative measure of how well the bed is lubricated.

Solution of Equation (7) for perfect lubrication (completely frictionless bed τ ₀ = 0) with constant A and E (Reference NyeNye, 1952) gives speed in the center y = W as

where

is the thickness-scaled half-width and

I will refer to U ^* as the frictionless speed. Here U _d represents specifically the frictionless speed in the center of a hypothetical ice stream of half-width W = H (w = 1 in Equation (10)). It is an alternative measure of the deformability of the ice for lateral shearing. When the ice is anisotropic and/or temperature varies spatially, then EA must be spatially averaged as indicated by . The appropriate averaging of EA for the transverse shearing of primary concern here is considered below. (If the ice were isotropic and isothermal, then U _d could also be interpreted as the velocity difference that would result from surface-parallel shearing through the vertical thickness of a slab of thickness H acted on by a base stress T (Reference PatersonPaterson, 1994, p. 251).)

The quantities T, U ^*, U _d and U _b will be useful as scales for stress and velocity for representing other solutions to Equation (7).

Other analytical solutions for Equation (7) are known for n = 1, m =1 (R96, equation (36)) and n = arbitrary, m = ∞ (e.g. R96, equation (39)). Solutions for other combinations of n and to can be found by numerical integration. The main numerical difficulty arises from the split boundary conditions and the singularity at y = W. These problems are most easily handled by integrating from y = W ₁ toward the edge with du _b(W ₁)/dy negligibly small and an assigned positive u _b(W ₁). The actual half-width W is found as W = W ₁ − W ₂ where u _b(W ₂) = 0 defines the location of the edge. Numerical exploration of the solutions for various W and τ ₀ is facilitated by using a dimensionless form with the scaling for u _b and y introduced in R96 (equations (4b), (5), (31), (34) and (35)).

I first compare results for a range of non-linearity in the basal motion law (Equation (1)) spanned by values for to from 1 to ∞. For this purpose τ ₀ (or 1/τ ₀) is not an appropriate measure of lubrication, since the dimensions of τ ₀ depend on m. Quantities related to u _b (e.g. R96) do not work for m = ∞ for which u _b is undefined by Equation (1). It is convenient to use τ _b itself as a lubrication measure in the form 1 − τ _b/T, which goes from 0 (low lubrication with the bed supporting the full driving stress) to 1 (fully lubricated, frictionless bed). Figure 2 shows solutions for u _b, τ _b and P vs this measure of lubrication for n = 3 with m = 1, m = 3 and m = ∞. An essentially equivalent representation for the heat generation for the case m =∞ has also been worked out by Reference Tulaczyk, Kamb and EngelhardtTulaczyk and others (2000).

Fig. 2. Velocity u _b scaled by frictionless velocity U^* (a), basal shear stress τ_b scaled by driving stress T (b) and heat production per unit area P scaled by TU^* (c) in the center of an ice stream as a function of fractional reduction in τ _b relative to T. Curves show results for n = 3 with m = 1 (dashed), m = 3 (solid) and m = ∞ (short dashed). (Note the scaling of the vertical axis in (c) by a factor of 10.)

In all of these cases, speed increases and basal shear stress decreases with increasing lubrication. A maximum in heat generation occurs at intermediate lubrication, as expected on a qualitative basis. The quantitative predictions are not strongly dependent on m in the basal motion law (Equation (1)).

Table 1 lists the maximum P and the associated u _b and basal τ _b that give the maximum as shown by comparison of Figure 2c with Figure 2a and b. I include results for n = 1 (ice assumed to be a linear fluid) for completeness, but below I consider only n = 3, which is more applicable to ice in the stress range of interest for marginal shearing. The maximum P occurs at a basal shear stress τ _b about 25–30% of the driving stress T, i.e. with a lubrication with somewhat more drag from the margins than the base. The speed at the maximum P is about 40% of the frictionless speed U ^* (Equation (10)). The maximum P is about 10% of the frictionless rate of potential energy loss U ^* T. These results show that dynamic feedback considerably diminishes dissipation of potential energy loss locally on the bed in the center of the stream.

To consider the implications of the general pattern displayed by Figure 2, I focus primarily on the case n = 3, m = 3. Figure 3 shows the solution for n = 3, m = 3 with bed lubrication represented by the dimensionless quantity U _b/U ^*.

Fig. 3. Velocity u _b scaled by frictionless velocity U^* (a), basal shear stress τ _b scaled by driving stress T (b) and heat production per unit area P scaled by TU^* (c) in the center of an ice stream as a function of width-scaled lubrication measured as U _b /U^* for n = 3, m = 3.

I choose to explore m = 3, because 3 is intermediate in the potential range of m [1, ∞]. Furthermore, if basal till is much weaker than τ _b, then the resistance to basal motion comes from bed roughness or strong “sticky” spots, and this resistance is probably accommodated by ice deformation that would tend to impose m = n.

For m = 3 (or more generally for m = ∞), I choose to represent bed lubrication by the dimensionless quantity U _b/U ^* for several reasons. Equations (8) and (10) show that

so it depends only on input variables about geometry (H, W, α), ice-flow law (A, E, n = 3) and slip law (τ ₀, m = 3). It has the advantage over τ _b used in Figure 2 that it does not depend on the solution itself. It has the advantage over to that it is dimensionless and accounts for the geometry of the ice stream. By reference to the limiting low- and perfect-lubrication cases described above, U _b/U ^* = 1 represents the transition from relatively low (U _b/U ^* ≪ 1) to relatively high (U _b/U ^* ≫ 1) lubrication.

Figure 3a–c define functions that give u _b/U ^*, τ _b/T and P/U ^* T in terms of the input variables that define U _b/U ^* (Equation (13)). Of particular interest below is the function relating P to τ ₀. With the above definitions and evaluation of T and U ^* using Equations (9) and (10), P can be described as

where is the function plotted in Figure 3c.

4. Drainage/Melt Interaction

The amount of water stored per unit area at the bed is here denoted by the thickness δ. It will be increased by melting m and decreased by flow divergence associated with the drainage system denoted by d. Water balance is given by

where both m and d are allowed to depend on δ. I assume that the same water influences m and d, and δ is the thickness of that water. This assumption is not necessarily correct. However, it is a reasonable starting-point for a representative location on the bed outside of through-going passageways.

The basal melting rate can be represented by

P is the mechanically generated heating rate per unit area (section 3, e.g. Equation (14)). Q is the heat-flux density conducted upward into the ice away from the ice base. G is the geothermal heat-flux density conducted upward from the rock to the base. Q − G is the net conductive heat loss. P − Q + G is then the total power per unit area that melts water thickness depending on the latent heat per unit mass L and the density of liquid water ρ _w.

Melting rate m depends on δ through P (Equation (16)). I assume that there is a monotonically decreasing relationship τ ₀(δ) (i.e. more water gives lower sliding resistance τ ₀ and higher lubrication 1/τ ₀). Introduction of τ ₀(δ) into Equation (14) gives P(δ). Since τ ₀(δ) is assumed to be monotonic, P(δ) will have the same qualitative character as the dependence of P on lubrication found in section 3 (i.e. maximum at an intermediate δ) as shown in Figures 2 and 3. The corresponding pattern m = m(δ) defined from Equation (16) is shown schematically in Figure 4 as solid curves. Although P is always positive, m can be negative, which occurs at low and high lubrication (now represented by δ) when there is net conductive heat loss (Q − G > 0).

Fig. 4. Schematic comparison of rates of meltwater production (solid curve) and drainage (dashed curve) depending on basal water storage δ in six alternative scenarios discussed in the text. It is assumed that heat conduction removes heat from the bed at rate (Q − G > 0) so that melting is turned off where P < Q − G (Equation (16)) as at low and high δ in (a–c, e, f). In (d) it is assumed that Q − G < 0, so that there is melting (m > 0) even when P ⇒ 0.

I will assume that the drainage d(δ) is a monotonically increasing function of water thickness δ, since larger δ would tend to facilitate water transfer. Schematic representation of d(δ) is shown in Figure 4 as dashed curves. Such a relationship is reasonable as long as the sinks for locally generated meltwater (e.g. relatively low-pressure canals (Reference Walder and FowlerWalder and Fowler, 1994) or other drainage passages) do not become more widely spaced, more highly pressurized or completely destroyed as δ increases (e.g. Reference KambKamb, 1987).

A quantitative development of the evolution of δ(t) requires specification of τ ₀(δ) and d(δ) (e.g. Reference KambKamb, 1991), but the knowledge to quantify these relationships is still limited. In addition, the evolution of fields of temperature (affecting Q and G), hydraulic potential on the bed (affecting d(δ)) and ice thickness (affecting all of the processes) must be tracked for a complete development. I will stick to the relatively secure qualitative features of the meltwater production and drainage as described above. I also assume that distributions of temperature, hydraulic potential and ice thickness are slowly evolving in comparison to rates of change in δ as driven by the dependencies of P and d on δ. With these limitations in mind, several possible qualitative joint relationships between m(δ) and d(δ) are illustrated schematically in Figure 4. The alternative scenarios illustrate potential ways in which δ(t) could evolve.

Where m(δ) > d(δ), δ increases (water builds up). That would allow acceleration of the ice stream. The opposite effect would follow from m(δ) < d(δ). A steady state requires m(δ ₀) = d(δ ₀), i.e. an intersection of the solid and dashed curves in Figure 4 at a particular water thickness δ ₀. A steady state defined by m(δ ₀) = d(δ ₀) may be stable or unstable, for example as found by Reference KambKamb (1991).

The stability of a steady state at water thickness δ ₀ can be examined based on the change in water-balance rate δ caused by a small shift of δ from δ ₀. To first order in δ − δ ₀, . An intersection such that m increases less rapidly than d as δ rises (m′(δ ₀) − d′(δ ₀) < 0) would be stable (e.g. as labeled in Fig. 4). In that case, an incremental increase in δ from the steady state (δ ₀) would result in a negative water-balance rate that would return δ toward δ ₀. An incremental decrease in δ would similarly return toward δ ₀. On the other hand, an intersection such that m increases more rapidly than d as δ rises (m′(δ ₀) − d′(δ ₀) > 0) would be unstable (e.g. as labeled in Fig. 4).

If m(δ) < d(δ) for all δ (Fig. 4a), there is no steady lubricated state, stable or unstable. Water drains away and lubrication cannot be sustained. Otherwise, there will be one stable steady state (all other panels) and possible associated unstable steady states (Fig. 4b, c, e and f).

The stable steady state is of primary interest. Two qualitative conceptual alternatives may be distinguished (Reference RaymondRaymond, 1995) : a “drainage-limited” regime, for which the stable steady state occurs at a lubrication less than produces maximum m (Fig. 4b); and a “production-limited” regime, for which it occurs at lubrication larger than produces the maximum m (Fig. 4c–f). In the drainage-limited regime, an increase in drainage d with increase in water thickness δ is required to prevent runaway of the ice-stream speed from the stable steady state. No such increase in d is required in the production-limited regime, where a drop in melt rate m can stabilize the speed by itself.

Except for Figure 4d, all panels in Figure 4 are drawn as if conduction removes heat from the bed (Q > G). A consequence is that melting cannot be sustained at very low or high δ where P is insufficient to balance Q − G and m < 0 (Equation (16)). If there were no drainage at all as suggested by Reference Tulaczyk, Kamb and EngelhardtTulaczyk and others (2000), then the unstable and stable steady states would be as in Figure 4e. To maintain a very high speed approaching frictionless speed U ^*, as in Figure 4f, would require import of water (d < 0, negative drainage) to balance freeze-on (m < 0). Although water import would certainly be a necessary condition, the question remains whether such import is possible or sufficient to maintain lubrication and fast motion (Reference Tulaczyk, Kamb and EngelhardtTulaczyk and others, 2000). A condition of positive conductive heat balance (Q < G) would require positive drainage independent of P (Fig. 4d); however, the same discussion concerning drainage- and production-limited regimes still holds unchanged.

Now suppose that there is evolution of ice-stream conditions along its length because of adjustment to climate or internally driven time-dependent processes. Such changes could affect drainage (e.g. pressure distribution or spacing of water sinks), the basal heat generation (e.g. stress and velocity driven by slope and thickness) and conductive heat balance at the bed (e.g. geothermal heat in response to temperature history of the ice base). The two-dimensional modeling in a cross-section considered here cannot predict such evolution. We can, however, consider the local response in a cross-section with the supposition that these processes shift the drainage and/or melt production curves (Fig. 4).

For an active ice stream the dynamic feedback from the margins allows a smooth response with the stable steady state migrating to higher or lower speed (e.g. Fig. 4b and c). There are some important differences in how drainage- and production-limited steady states could respond. A drainage-limited steady state will have higher sensitivity to equivalent shifts because of the narrow angle of intersection of the curves for d and m (Fig. 4). Clearly, a drainage-limited condition has much more potential for large increases in speed. In a production-limited steady state, increase in speed tends to be self-limiting because of reduction in meltwater production, and the ultimate upper limit of frictionless speed is not so distant.

A potential for large jumps in speed is also evident. If the drainage curve separates from the melt-production curve (Fig. 4a), there would be a jump to no lubrication and nearly stagnant condition with negligible heat generation similar to an inter-ice-stream location. An ice stream could be restarted from low speed only in situations with a positive conductive heat balance at the bed (Fig. 4d) or with water import from an upstream source (Fig. 4f). Either of these circumstances could lead to a jump to a high-speed steady state. These concepts illustrate the potential for jump-like transitions between stagnant and active phases with self-propagating and large changes in velocity once thresholds have been reached.

5. Example West Antarctic Ice Streams

A crucial question is, what limits the speed of active ice streams? I examine whether those active West Antarctic ice streams with sufficient observations appear to be drainage- or production-limited as defined above. To facilitate comparison of different ice-stream cross-sections in the context of force and energy balance for the center of each ice-stream cross-section, I construct a dynamic-feedback diagram (Fig. 5) showing contours of τ _b/T and P/U _d T = u _b τ _b/U _d T with w = W/H and u _b/U _d as coordinates.

Fig. 5. Contours of heat production rate P (heavy curves) and basal shear stress τ _b (light lines) in the center of an ice stream for varying width W and speed u _b . Labels for contours of P (boldface numbers) give log base 10 values of P in units of U_dT. Labels of contours for τ _b (italic numbers) give τ _b /T. Note also the scaling of axis variables W and u _b . The diagonal line τ _b /T = 0 gives the speed for a frictionless bed U^* = U_dw⁴. States above this line are not accessible. Similarly, states below the line τ _b /T = 1 are not accessible.

Figure 5 is constructed by using the functional relationships in Figure 3 (or Fig. 2 for n = 3, m = 3) and eliminating explicit reference to a measure of lubrication (τ _b, U _b, τ ₀ or δ). The limits of u _b lie between U _d (low lubrication, bed drag regime) and U ^* = U _d w ⁴ (perfect lubrication with frictionless velocity, side drag regime). The plot is restricted to u _b/U _d > 10, for which the simplified one-dimensional theory for lateral variations in section 3 is valid. For this range it is also reasonable to ignore the distinction between U _d as a scale for lateral shearing and shearing over the ice thickness that arises when the ice is non-isothermal and/or anisotropic (Equations (10) and (12)). For a fixed w in Figure 5, the contour patterns for τ _b and P represent the curves in Figure 3 with low τ _b at high u _b and low P at both low and high u _b with a maximum in between.

This strategy for examining the relationship amongst the four variables W, u _b, τ _b and P = u_b Tb is convenient for comparing measurements because W, H and u _b ≈ u _s are the most accessible variables for observation. U _d (Equation (12)) can be estimated from H and α with information about the distributions of temperature and enhancement E and the associated flow law of ice (Equation (2)). The dimensionless formulation of Figure 5 allows plotting of any cross-section based on the values of u _b/U _d and W/H. Then τ _b/T and P/U _d T can be read off from their respective contour patterns (keeping in mind possible error in U _d, which causes uncertainty in the vertical position of the plotted point). Similarly T can be estimated from H and α, so that dimensional values for τ _b and P can be determined (subject to errors in T and also U _d in the case of P).

Table 2 (rows 1–6) gives basic input data to analyze five sections across ice streams of the Siple Coast and a characteristic one for Rutford Ice Stream. Sources for the data are given in the table. The names in Ice Stream B follow common usage (UpB and DwnB). In Ice Streams D and E, the definitions in Reference Bindschadler, Vornberger, Blankenship, Scambos and JacobelBindschadler and others (1996) are used (D1, D2 and E2). (Note that Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others (1994) used D5 to denote D2.) The section in Rutford Ice Stream is based on representative data from sections adjacent to the upstream part of Fletcher Promontory (Reference Frolich and DoakeFrolich and Doake, 1988). The primary data are thickness H, half-width W, surface slope α and surface velocity u _s (rows 1–4). In these ice streams the surface u _s and bed u _b velocities are essentially the same. The surface temperature θ and accumulation rate a (rows 5 and 6) are also of interest for estimating the temperature distribution relevant to the flow law (Equation (2)).

Thickness-scaled half-width w is found from full width 2W and thickness H (Equation (11); Table 2, row 7). Although full width 2W is known fairly accurately, there is some ambiguity in specifying the half-width W. W is defined by boundary conditions in the analysis above to be the distance from the margin to the flow center line, where ∂u/∂y = 0. This location may not be halfway between the two margins because of asymmetry in cross-flow thickness and bed conditions. I take W to be half the full width. This choice should give an average picture for the two margins of the cross-section, although it may not represent either one accurately.

It is straightforward to calculate the driving stress T (Equation (9); Table 2, row 8) from thickness H and surface slope a. The primary uncertainty for T comes from α.

The vertical distribution of temperature was estimated following Reference RobinRobin (1955) using the recent surface temperature θ (Table 2, row 5), accumulation rate a (Table 2, row 6) and a basal heat-flux density into the ice Q adjusted to give melting temperature at the base (Table 2, row 9). This vertical temperature distribution was assumed to apply independent of lateral position.

The calculation of U _d (Equation (12); Table 2, row 10) is complicated by poorly known spatial variation of temperature and fabric affecting the flow law (Equation (2)). The spatial pattern of A(T) was estimated from a standard temperature-dependent representation of the flow law (Reference PatersonPaterson, 1994, p. 97) and the Robin temperature distribution. The following approach was used to estimate a limiting range for U _d arising from uncertainty in enhancement. First, an effective EA was determined by vertically averaging A(T) and E = 1. The vertical average of A produced by this method corresponds to A evaluated at a temperature of about −9°C, which is warmer than the vertically averaged temperature of about −16°C. Second, effective EA was estimated as above but including an enhancement E = 10 at the margin with a fall-off to E = 1 over about 30% of the half-width W. This spatial pattern of enhancement is similar to that estimated in the margin of Ice Stream B2 by Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others (1994). These two assumptions for enhancement (E = 1 everywhere and E = 10 in the margin zone) span from relatively small (Reference Jackson and KambJackson and Kamb, 1997; Reference Whillans and van der VeenWhillans and Van der Veen, 1997; Reference Harrison, Echelmeyer and LarsenHarrison and others, 1998) to substantial (Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994), arising from estimates of side drag at Upstream B (UpB) using a variety of measurements. The effect on U _d is a factor of about 3. Additional uncertainty in U _d arises from uncertainty in T, which extends the potential range of U _d typically by another factor of 2 or more.

With H, W, α, u _b, U _d and T in hand for a given cross-section as discussed above, it can be plotted in Figure 5, and derived quantities τ _b/T ⇒ τ _b (Table 2, rows 14 and 15) and P/TU _d ⇒ P (Table 2, rows 12 and 16) can be read from the figure.

This simple analysis for force balance (Fig. 5; Table 2, row 14) gives the following results. For cross-sections in Ice Streams B (UpB and DwnB), τ _b/T ≤ 0.5, so they are supported strongly from the sides. In all of the other cross-sections, τ _b/T ≥ 0.5, and they are not so strongly supported from the sides. However, side support is by no means negligible in any of the cross-sections. The results of this simple analysis are roughly consistent with other, more thorough, evaluations of force balance using more extensive data and analysis including attempts to account for enhancement (E > 1) and variable bed conditions (Reference Frolich and DoakeFrolich and Doake, 1988; Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994; Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others, 1994; Reference Jackson and KambJackson and Kamb, 1997; Reference Whillans and van der VeenWhillans and Van der Veen, 1997; Reference Harrison, Echelmeyer and LarsenHarrison and others, 1998), with the exception of section E2 (Reference MacAyeal, Bindschadler and ScambosMacAyeal and others, 1995) which is discussed further below.

The immediate consequence for heat generation on the bed is that P = τ _b u _b is less than TU _b as would be estimated without accounting for the force balance. Conditions in all of the analyzed cross-sections except D2 and E2 appear to be such that heat generation could be close to the maximum possible for their widths W. It would be very convenient if ice streams adjusted their speeds to achieve maximum P and corresponding melt rate m. Then the speed would be about 40% of the frictionless speed U ^* (Table 1; Table 2, row 11; Fig. 5), and could be easily calculated from the geometry (α, H, W) and ice flow law (A, E, n) (Equations (12) and (10)) without reference to the water drainage system. However, in the context of the discussion in section 4, there is no clear physical basis for maintaining a state of maximum heat production. That condition could lead to accumulation of basal water, increasing lubrication and acceleration of the ice stream (Fig. 4c or d) or the opposite (Fig. 4b) depending on the efficiency of the drainage system.

A production-limited regime is characterized by a speed that is larger than produces maximum heat generation. The upper range of scaled velocity estimated for Ice Stream B indicates the possibility that it is in a production-limited regime (Fig. 5). This would certainly be the case if side drag is very close to 100% as suggested by some of the examinations at Upstream B (Reference Jackson and KambJackson and Kamb, 1997; Reference Whillans and van der VeenWhillans and Van der Veen, 1997; Reference Harrison, Echelmeyer and LarsenHarrison and others, 1998). No drainage would be very convenient for modeling ice-stream motion (Reference Tulaczyk, Kamb and EngelhardtTulaczyk and others, 2000). A no-drainage state would be a special case of a production-limited regime. In view of the uncertainties, this is a possible condition for Ice Stream B.

A drainage-limited regime is characterized by a speed that is smaller than produces maximum heat generation. Even with the substantial range of uncertainty, all of D1, D2, E2 and Rutford are evidently in a drainage-limited regime (Fig. 5). Drainage must play a crucial role in regulating these ice streams.

The predicted rate of mechanical generation of heat for each case in Figure 5 can be computed in dimensional form P (Table 2, row 16) by introducing the unit U _d T (Table 2, row 12). There is a wide range. Under Ice Stream B, P appears to be less than the likely geothermal heat flux G (60 mW m⁻² or larger). Under D1 and D2, Rutford and E2, P is distinctly larger than expectable G.

Comparison of P with the upward flow of heat into the ice (denoted Q, Table 2, row 9), estimated by the thermal analysis described above, predicts the minimum geothermal heat G _min required to maintain melting at the ice base (Table 2, row 17). For this comparison it is important to keep in mind that the actual temperature distribution in the ice at any specific location will be influenced by along-flow advection of heat and non-steady conditions that are not taken into account by the Reference RobinRobin (1955) analysis. Therefore, there is considerable uncertainty in the estimated value of Q. Nevertheless, where it can be checked (at UpB) there is agreement between this prediction (Q = 81 mW m⁻²) and measurement (Q = 0.08 W m⁻² found by Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990).

For D1, D2, E2 and Rutford, P > Q, and no geothermal heat is needed. This holds even though Q is predicted to be relatively high in D1 and D2, which are relatively thin, and Rutford, which has high accumulation rate. The important factor is that P is high in these sections. Basal melting is expected in all of them. At DnB G needed to maintain melting is estimated to be large (80–100 mW m⁻²), because of a combination of low P (associated with large side drag and production-limited regime) and high Q (associated with low thickness). Normal G would be distinctly smaller than this value, but higher values cannot be entirely ruled out in view of the history of rifting and low crustal thickness (Reference ElliotElliot, 1985). If the geothermal flux is near normal (about 60 mW m⁻²), the fast motion at DnB would then appear to be maintained even in the presence of basal freezing. The speed at DnB is presently decelerating (Reference Bindschadler and VornbergerBindschadler and Vornberger, 1998), and perhaps basal freezing is a contributing factor.

Section E2 in Ice Stream E deserves special discussion. In this analysis the basal shear stress τ _b predicted at E2 is about 80–90% of the driving stress T (Table 2, row 14). The substantial τ _b compared to other cross-sections follows from relatively large T and W, but nevertheless similar u _b (Table 2, rows 2, 3, 8 and 4). This result is distinctly different from conclusions in the rigorous inverse-theory analysis of the spatial pattern of velocity on Ice Stream E of Reference MacAyeal, Bindschadler and ScambosMacAyeal and others (1995) in an area that encloses E2. The preferred interpretation of these authors is a mean Tb of 14 kPa that is only 29% of a mean T of 49 kPa. They suggest an upper limit for the fraction of support from the base of about 60%. While there is reasonable agreement about the size of τ _b, T and the fraction of support from the sides appear to be distinctly different in the two analyses. Reference MacAyeal, Bindschadler and ScambosMacAyeal and others (1995) include large-scale horizontal support both laterally from the sides and longitudinally along the length without distinguishing between them. My analysis accounts only for lateral gradients, not longitudinal ones. Considering the range of uncertainty given in each analysis and the neglect of longitudinal stress gradients in mine, the two analyses could be reconciled with as little as 20% longitudinal support, which is a very reasonable possibility. On the other hand, reconciliation of the two analyses with the preferred interpretation of Reference MacAyeal, Bindschadler and ScambosMacAyeal and others (1995) requires about 50–60% longitudinal support. It would be surprising if longitudinal gradients supported such a large proportion of T, since they are not larger than lateral gradients in the area of the cross-section E2 based on data in Reference MacAyeal, Bindschadler and ScambosMacAyeal and others (1995) and Reference Bindschadler, Vornberger, Blankenship, Scambos and JacobelBindschadler and others (1996). The disagreement on the size of T would presumably derive mainly from a difference in assumed surface slope α (here taken to be in the range 1.6–2.3 × 10⁻³ based on the recent digital elevation model of Reference Liu, Jezek and LiLiu and others (1999)). However, introducing a larger surface slope in my analysis increases the disparity in the fraction of support from the bed. While it appears that the results from the two analyses can be reconciled by considering the outer limits estimated in each case and reasonable longitudinal gradients, the differences are disquieting. Neither D. MacAyeal (personal communication, 1999) nor I have a firm resolution. It is an important question that does point to the difficulties for assessing side and bed drag.

This discussion promotes caution in applying a generic model of cross-section flow that does not account for details of the lateral variations of thickness and velocity or longitudinal gradients to interpret specific cross-sections with many actual complexities. However, I have attempted to account for these problems using a reasonable range of variables to show that the general pattern of comparison between the different ice-stream sections is robust. It is clear that the ice-stream dynamics with respect to drainage-regulated lubrication and feedback from side drag are diverse. At a minimum level we can divide the sections investigated here into two groups: [B] and [D, E, Rutford].

The cause of the differences is of considerable interest with regard to the potential for large changes in discharge from the West Antarctic ice sheet. Ice Stream B is largely supported from its sides, and the potential for increase in speed by altering conditions at its base is limited. If the lubrication of the beds of Ice Streams D and/or E could be increased, the consequent increase in speed could be very large with positive feedback of increased basal melting. Are Ice Streams D and E different from Ice Stream B because the bed geology and/or morphology are different in a way that prevents a high level of lubrication? This would suggest stability of their discharges. Or are Ice Streams D and E in a different stage of evolution such that bed conditions could become like those at Ice Stream B? This would suggest possible instability of their discharges.

6. Global Energy Balance

Thus far, I have been concerned with force balance and consequences for generation of basal heat in the central parts of an ice stream. Although heat generation falls off in the center of an ice stream as speed increases in the high-speed, production-limited regime, the rate of total potential energy loss continues to increase toward the frictionless potential energy loss U ^* T, which can be very large (Table 2, row 13). This energy is deposited toward the margins rather than in the center of the ice stream.

The lateral variation of mechanical energy dissipation and associated heating per unit area can be estimated analytically using the solution for a plastic bed (m = ∞) (e.g. R96, equation (39)). Figure 6 shows the partitioning between the center and very edge of such a model ice stream derived in this simplified way. The estimate includes heating at the bed and through the ice column by side shearing. As the bed becomes more lubricated, an enormous amount of energy can be focused into the margins. This energy redistribution leads to shear heating as observed in the margin of Ice Stream B (Reference Harrison, Echelmeyer and LarsenHarrison and others, 1998). The vertical distribution of the heating in the ice and its effect on the temperature pattern may be complex (Reference Jacobson and RaymondJacobson and Raymond, 1998). Therefore, it is not straightforward to estimate what proportion of the heat is exhausted to the atmosphere and what causes melting. This question is not taken up here. Nevertheless, there is a theoretical potential for melting at the bed in and even beyond the margin (Reference Jacobson and RaymondJacobson and Raymond, 1998). This opens the possibility that the margins may play an important role in setting the paths of water drainage in ice streams.

Fig. 6. Comparison of heat generation rate per unit area of bed at margin (dashed line) and center (solid line) derived assuming a laterally constant basal shear stress (m = ∞ and τ₀ = constant in Equation (1)). The value of P is integrated overfull thickness and includes shearing in the ice as well as generation at the bed.

7. Conclusion

The cross-flow transmission of force from an ice stream to its margins causes the heat generation on the bed in the center of a well-lubricated ice stream to be substantially smaller than would be deduced from the driving stress and measured speed. The production of meltwater for maintaining the lubrication is correspondingly less. The rate of basal heat generation is a maximum for an intermediate level of lubrication for which velocity is about 40% of the speed associated with a frictionless bed and base stress is about 25% of the driving stress. The maximum rate of heat generation is only about 10% of the rate given by the driving stress times the velocity over a frictionless bed.

Steady-state lubrication is achieved with equality of meltwater production and drainage. Both stable and unstable balances between meltwater production and drainage on the bed are possible. A stable steady state with a speed less (more) than that giving maximum heat generation is termed drainage- (production-)limited, since an increase in speed would lead to increased (decreased) basal melting and must (need not) be balanced by increased drainage. It is shown that a gradual continuous augmentation of the basal water drainage system or decline in basal melting can cause a discontinuous jump from a fast-moving steady state to a slow, stagnant state associated with a threshold in the balance between drainage and melting.

The redistribution of heat is particularly important with regard to the thermal control of ice streams. These effects arising from horizontal stress gradients across the flow are large (approaching 100%) compared to effects on the conductive heat balance from varying thickness associated with surface and bed topography. Evidently, thermomechanical modeling of an ice sheet with ice streams must include the horizontal redistribution of stress and in particular the associated heating to capture thermal regulation of ice-stream expansion, contraction and shutdown.

Available information about the geometry and speed for six cross-sections from Ice Streams B, D, E and Rutford Ice Stream shows them to be diverse in the level of support and corresponding shift of mechanical heating sideward from their central parts. Ice Streams D and E and Rutford Ice Stream appear to be in a drainage-limited condition, where acceleration of speed would increase mechanical heat dissipation on the bed and thereby increase melting. Ice Stream B could possibly be in a production-limited condition, where increase in speed would actually reduce basal melting rate. This possible condition for the upper part of Ice Stream B arises because the reduction in basal heat generation from side drag is especially significant there.

The conduction of heat upward into the ice is also expected to vary amongst the different ice streams, with the highest upward conduction in Ice Stream D and the lower part of Ice Stream B, which are relatively thin.

In examined sections of Ice Streams D and E and Rutford Ice Stream, the mechanical generation of heat at the bed is larger than conducted upward into the ice, so that no geothermal heat is required to maintain their bases at melting and produce meltwater. This condition is predicted even though basal heating is significantly reduced by side drag and/or the heat loss into the basal ice is large. On the other hand, in the lower part of Ice Stream B, the combination of side-drag-induced reduction of mechanical heating at the bed and high upward conduction into the ice requires a very high geothermal heat (more than about 80–100 mW m⁻²) to maintain a thawed condition. The lower part of Ice Stream B is a likely location where the dynamic reduction of mechanical heating on the bed contributes to basal freezing, which could promote slowing of speed and ice discharge. Ice Streams D and especially E have relatively high driving stress and large width, and the simplified analysis here identifying relatively low level of side drag suggests a potential for large increases in ice speed and discharge by enhanced lubrication of the bed. An already large rate of basal melting suggests that it may be difficult to develop lubrication with meltwater, possibly because the morphology of the bed is different than under Ice Stream B.