Creep of ice

The creep-closure rate of a tunnel in ice is given by

where R is the tunnel radius, and A _i and n are the constants in Glen’s flow for ice

where and T are the second invariants of the strain-rate tensor and the stress deviator tensor T_ij). This result applies directly to the closure of a cylindrical borehole (Reference NyeNye, 1953) and is reasonably presumed to apply approximately in other geometric configurations.

Creep of till

Reference Boulton and HindmarshBoulton and Hindmarsh (1987) proposed a flow law for till of the form

where

is the shear rate, T is the shear stress, N is the effective pressure P _i – p _w and A _s,a,b are positive constants. This law, based on a fit to seven data points, has been criticized by other authors (e.g. Reference KambKamb, 1991; Reference BlakeBlake, 1992) but, in the present context, we consider it the simplest law which captures the idea that Equation 10 should increase with N, although the singularity at N = 0 is unphysical.

Reference Boulton and HindmarshBoulton and Hindmarsh (1987) conjectured, following Reference NyeNye’s (1953) solution for closure of a borehole in ice, that the consequent closure rate of a canal due to creep of till would be

where implicitly N_c = N, i.e. P _c = p _w With a = 1.33<b = 1.8 (as measured), sediment creep is then most effective at low values of N _c, which has an important bearing on the stability of the canal system, as we shall discuss later.

More recendy, Reference Fowler and WalderFowler and Walder (1993) have solved the Nye borehole-closure problem for the case of saturated, deformable till where in general p _c ≠ P _w. They showed that the closure rate depends on the parameter ^ a dimensionless measure of permeability. Estimates of ^ are in the range 10^-;6 < A < 1, for a permeability range 10^-;19–10^-;13m² (Reference Freeze and CherryFreeze and Cherry, 1979, p. 29) and an apparent till viscosity of 10⁹Pas (cf. Reference Blake and ClarkeBlake and Clarke, 1989; Reference BlakeBlake, 1992; Reference Humphrey, Kamb, Fahnestock and EngelhardtHumphrey and others, 1993). For most tills, then, the asymptotic case

applies and the correct generalization of Equation (3.2). is as long as

as long as

If the condition Δp_w is not satisfied, the analysis predicts the existence adjacent to the hole of a “failed” zone in which piping probably occurs and permeability is locally enhanced. An analogous situation arises subaerially where streams are recharged by ground water: as stage falls following a flood, the hydraulic gradient driving seepage through the stream banks may become great enough that piping and bank failure occur (e.g. Reference Bradford, Piest, Coates and VitekBradford and Piest, 1980). Reference Fowler and WalderFowler and Walder (1993) considered how the solution to the borehole-closure problem must be modified to account for this effect; they showed the “corrected” closure rate is not greatly different from Equation (3.4)., so for simplicity we assume Equation (3.4). is generally valid in the rest of our analysis.

Pore-water pressure

In general, we expect p _w > p _c: meltwater generated at the ice/till interface makes its way towards conduits (although the opposite situation could occur transiently if, for example, flooding occurs via an englacial waterway). If the till is very permeable, or the melt supply is low, then the water will be evacuated towards conduits through the till. If these are spaced a distance H apart and the till is of thickness H_t, then the excess pore-water pressure necessary to drain water supplied at a rate

per unit area of ice-till contact is where k is permeability and μ_w is viscosity (Reference ShoemakerShoemaker, 1986). Taking H _t = 10 m, , μ_w = 2 x 10^-;3Pas, k = 10^-;16m² (a median value), then μp_w = 1 bar; for H = 100 m, μp_w = 100 bar. In general, μp_w/N may be significant; for example, under Ice Stream B, N is only about 1 bar (Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990).

Various criteria may be offered for the initiation of conduits over a sediment bed (cf. Reference ShoemakerShoemaker, 1986). A physically reasonable criterion is that conduit spacing adjusts itself so that sediments at the edge of conduits are marginally stable against piping; from Equation (3.5)., this means N_c = aN. If conduits were closer, piping would occur, increasing the permeability of sediment near the conduit and thereby drawing down pore pressure p_w at the drainage divides until the condition N_c = aN was satisfied. The sediment creep law would then become

In subsequent analysis, we will adopt the form of Equation (3.6)..

Sediment erosion and transport

The inward creep of till into a conduit is balanced by erosion of the sediment and its subsequent transport along the channel. We base our discussion on that of alluvial channels in non-cohesive sediment and give only the basic details: in particular, the formulae we use are very rough approximations.

Erosion occurs only if the shear stress T exerted by the water flow is larger than some critical stress T_c, which for sorted sediment depends on grain-size D_s. Beyond this critical stress, particles can be dislodged and they are transported downstream either as bed load if they are large enough (D_s > 0.1 mm) or as suspended sediment.

Various empirical expressions are available for bed-load transport; we take, for simplicity, the common Reference Meyer-Peter and MüllerMeyer-Peter and Müller (1948) equation, which we write as

where (a;)_(x)+ = max(x, 0), Q_b is the bed-load-transport rate (mass per unit time), K is a constant approximately equal to 10, l_s is the width of the stream bed, g is acceleration due to gravity, ρ_w is the density of water and μρ = ρ_s - ρ_w.

Theoretical justification for expressions such as Equation (3.7). has been developed by Reference Wiberg and SmithWiberg and Smith (1989). For stream beds with a mixture of sand-and gravel-sized clasts — probably a reasonable description for many subglacial streams (e.g. Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990; Reference Humphrey, Kamb, Fahnestock and EngelhardtHumphrey and others, 1993) — a condition of “equal mobility” holds, wherein all grain-sizes begin to move at about the same stress, namely,

where

is an empirically found, dimensionless constant, and D ₅₀ is the median grain-size (Reference Parker, Klingeman and McLeanParker and others, 1982).

Expressions for the rate of erosion of suspendable sediment are difficult to come by. Reference ParkerParker (1978a) emphasized that this is pardy because the usual theory of suspended-sediment concentration (e.g. Reference RichardsRichards, 1982, p. 101) is a local, steady-state theory in which erosion rate Ė and deposition rate Ḋ balance exactly. In principle, however, Ė may be prescribed as boundary conditions on the concentration profile. Reference ParkerParker (1978a) proposed that the erosion rate is given by

where K ₁ ≈ 0.1 and v _s is the particle-settling velocity, and that the deposition rate is

where c is the depth-averaged suspended-sediment concentration and K ₂ ≈ 6. The net-erosion rate of suspendable sediment is then

An important feature of subaerial stream beds is that they evolve their own shape through the processes discussed above. We can expect this to be true also for subglacial conduits cut into sediment. For non-cohesive sand/silt stream beds, Reference ParkerParker (1978a) argued that a stable bed shape could be maintained by a secondary lateral circulation of sediment, as follows. In the central parts of the stream, the shear stress acting on the bed is greatest and consequently greater erosion there leads to a larger suspended load than at the banks. This causes a lateral eddy diffusion of sediment towards the banks, which balances the bed-load transport that tends to bring sediment back down-slope toward the centre. Thus, even though the banks are being actively eroded, the river can maintain a constant cross-section via this dynamic equilibrium. In alluvial rivers, this leads to channel cross-sections with high aspect ratio (i.e. width l much greater than depth h). In gravel-bed rivers, the adjustment mechanism is more complicated. Since sediment is not being carried in suspension toward the banks, a stable channel cannot exist unless no erosion (and thus no bed-load transport) occurs on the banks. Accordingly, the shear stress must be great enough near the channel centre to transport the imposed sediment load but below critical on the banks. Reference ParkerParker (1978b) resolved this by showing that lateral turbulent flux of downstream momentum can produce the required shear-stress distribution. However, it is generally the case that T ≈ T_c for alluvial gravel-bed rivers. Again, large aspect ratios are typical.

Channels eroded into deforming subglacial till will differ from alluvial channels in one important aspect: whereas for an alluvial channel a steady-state cross-section exists only if there is no net erosion, for the subglacial channel a steady state will exist only if the net erosion balances the creep of till toward the channel. Thus, for a sand/silt-floored subglacial channel, there will not be a macroscopic balance between Ė and Ḋ, but rather

where the brackets

denote cross-sectional averages. For a gravel-floored subglacial channel, t will not drop below T_c on the banks, but rather T will exceed T_c by just enough to erode the banks and transport the inwardly creeping till at the appropriate rate. Accordingly, (Reference ParkerParker’s 1978a, Reference Parkerb) analyses of the shapes of equilibrium stream channels cannot strictly apply to the subglacial case; in fact, his analyses should give lower bounds on the magnitude of bed shear stress. Qualitatively, however, the processes described by Parker for subaerial streams should apply to subglacial ones,hence we fully expect subglacial streams over non-cohesive sediment to tend towards high-aspect ratio cross-sections.

Governing equations

Having described the constituent ingredients of the process, we now write down governing equations for the flow. We consider a conduit at the ice–till interface (Fig.1). In general, we expect the conduit to extend both upwards into the basal ice and downwards into the till. We can thus define cross-sectional areas S _i and S _B of ice-or till-bounded conduits, respectively, and the total cross-sectional area S by

The differential equations governing the physics are straightforward and are based on those elucidated by Reference RöthlisbergerRöthlisberger (1972), Reference NyeNye (1976), Reference Spring and HutterSpring and Hutter (1981) and Reference LliboutryLliboutry (1983) but, here augmented by relations describing sediment creep, erosion and trans port. They must properly be framed in the context of a two-phase flow and we assume the simplest such model, that of homogeneous flow; in other words, we assume the sediment moves with the same velocity as the fluid. Let s be the downstream longitudinal coordinate and t be time. Equations of conservation of mass are, for the water,

and for the sediment

where

is the ice-erosion (i.e. melting) rate per unit downstream distance; and are, respectively, the water- and sediment-supply rates due to flow from tributaries, moulins or veins; is the volume fraction of suspended sediment and Q_a is the suspended-sediment flux, defined by

where u is the mean flow rate; other symbols are as defined previously. We include the bed-load transport rate Q _b in Equation (3.15)., because we envisage the bed load as existing as a thin skin at the bed, occupying negligible volume.

Using Equations (3.2) and (3.6), the kinematics of conduit closure are described by

and

where l _i and l _s are defined in Figure 1 and K _i and K _s and O(1) shape factors, partly representing the geometric difference between the conduit in Figure 1 and the idealized cylindrical shape assumed in the closure-rate calculations of Reference NyeNye (1953) and Reference Fowler and WalderFowler and Walder (1993). Equations (3.17) and (3.18) are appropriate generalizations for non-cylindrical conduits. If the conduit is actually cylindrical, then

and the creep-closure rate as expected. For a wide, flat channel in ice (cf. Reference Hooke, Laumann and KohierHooke and others, 1990) with height h and width (thus l≈l _i), we expect thus , because the average closure rate depends only on the width of the “hole” into which ice is flowing. A similar argument applies to a wide, shallow conduit incised into till.

Neglecting inertial effects, the momentum balance for the flow can be replaced by a force balance between pressure forces and drag on the channel walls,

where sin θ is the bed slope and l is the channel perimeter, defined by

Assuming turbulent flow, which is valid except for very small tunnels (cf. Reference WeertmanWeertman, 1972), the average wall shear stress is given by

where the dimensionless friction factor fR is weakly dependent on the Reynolds number and is typically ~0.1 for subaerial streams. In general T_w (the average shear stress over the sediment bed only) and T may differ but we assume for simplicity that they are the same. This seems reasonable for pressurized flow but may be less realistic for open-channel flow.

We write the energy equation in the form

where Cp is specific heat, T is temperature, ΔT is the temperature drop across a thermal-boundary layer adjoining the ice wall (but not the stream bed, the temperature of which is unconstrained) and h _T is a heat-transfer coefficient. We neglect work done in eroding and transporting sediment (Reference BagnoldBagnold, 1961). The temperature T will be related to the temperature of the ice, which we expect to be at the pressure-melting point, by

where c _t is the rate of change of melting temperature with pressure and ΔT is the temperature step across the thermal boundary layer at the ice/water interface. The rate of heat transfer across the thermal boundary layer is given by

where L is the latent heat and we neglect any small temperature gradient in the ice (cf. Reference WeertmanWeertman, 1972).

The Equations (3.14), (3.15), (3.17)–(3.19) and (3.22) are six differential equations for the variables

and T; in addition, the seven algebraic relations in Equations (3.7), (3.10), (3.13), (3.16), (3.21), (3.23) and (3.24) determine the subsidiary variables . Conduit perimeter I is determined by Equation (3.20)., so provided that l _i and l _s are prescribed functions of S _i and S _s respectively, the model is complete.

Critical effective pressure

From (3.17) and (3.18), we have at steady state

whence the relative rate of ice and till creep is given by

where we define the critical effective pressure to be

Taking K _i ≈ K _s, l _i ≈ l _s, and with ρ_i = 9 x 10² kg m^-;3, ρ_s = 2.65 x 10³ kg m^-;3, A _i = 7.36 × 10_-;24 Pa^-;3 s^-;1 (cf. Reference LliboutryLliboutry, 1983), n = 3 and A _s = 3 × 10^-;5 Pa^b-as^-;1 (after Reference Boulton and HindmarshBoulton and Hindmarsh, 1987), a = 1.33, b = 1.8 we find

Thus, for

, till creep is larger, while for , ice creep is. This leads us to consider two end cases, when either one or the other creep process dominates conduit closure.

Conduit geometry

In order to proceed, we need to describe the “constitution” of the conduit. Since the conduit in Figure 1 has a boundary partly of ice and partly of till, the respective closure rates will generally be different, i.e.

. Since both till and ice are viscous, a boundary condition applies at their interface. In what follows, we assume a no-slip condition. In this case, the ice and sediment speeds towards the conduit at the interface, V _i and V _s, must be equal. In a slightly crude way, we identify

where k = i or s and θ(θ_s) is the angle of the ice roof (till flow) to the horizontal. It then follows that

and thus the two extreme cases

correspond to values , i.e. level-floored ice channels and flat-roofed till canals, respectively. That is to say, if , the conduit is essentially a Röthlisberger channel in ice, floored by a relatively stiff till, while if it is essentially a canal cut into soft till and roofed by relatively stiff ice.

Note that the determination of

depends critically on the assumed till rheology. However, the identification of two asymptotically distinct conduit shapes is robust.

Determination of water pressure–flux relationship

The way to solve the equations is now as follows. First, we recognize that, based on observations c

so , and the volume flux Q = uS is determined from Equation(3.14).. From Equation (4.1)., it is easy to calculate that melting due to frictional dissipation is likely to be eligible in most circumstances, and therefore the water flux Q(s) in a tunnel is essentially determined by the input from surface-water and basal tributaries. In effect, we can consider Q(s) to be a prescribed function. Equation(3.15). determines the suspended-sediment load but this is not needed for the solution of the rest of the problem.

Using Equations (3.23) and (3.24), the energy Equation (3.22). can be rewritten in the form

where

(Reference RöthlisbergerRöthlisberger, 1972), K _w = k _w/p_wC_p is the thermal conductivity. The Nusselt number Nu, a dimensionless measure of heat transfer at the ice–water interface, is defined by . Equation (4.8). determines in terms of T, u and p _c, and in addition we have Equations (4.1), (3.19), (3.21) and

to determine the five unknowns S, u, T, p_c and m, (given l _i and l_s ). It is straight-forward to combine these to form a single, non-linear, second-order differential equation for Pc but it is probably more illuminating, and certainly easier, to simplify the system using two judicious approximations. For the first of these, note that, since N_c = p _i – p _c and P _i = ρigd, where d is the ice depth, then

where α is the ice-surface slope (Reference WeertmanWeertman, 1972). For bedrock, that varies sufficiently slowly, the term

will be negligible in this expression. For example, if the bedrock slope changes significantly over a length l _f along the flowline, then

where ΔN _C is the variation of N _c. For an alpine glacier, we can expect ΔN _C <

is negligible if , typically about 10 ice thicknesses. On an ice sheet, we might expect , but for ice streams, at least, geophysical evidence from (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others 1986, Reference Blankenship, Bentley, Rooney and Alley1987) suggests strongly that . In either case, it is therefore useful as a rough estimate to neglect in Equation (4.10).. A similarly small error is made if we then approximate the hydraulic gradient using the surface slope, thus

This approximation breaks down near the glacier terminus, where variations in p _c are substantial (Reference FowlerFowler, 1987).

The second simplification results from Equation (4.8).. We note that Q/k _wNu is dimensionally a length, and is in fact the length over which

relaxes to its equilibrium value. Taking values Q = 1 m³ s^-;1, K _w = 10^-;6 m² s^-;1, Nu ≈ 0.1 Re⁰⁸ (Reference Spring and HutterSpring and Hutter, 1981), Re ≈ 10⁶ (based on u = l ms^-;1, l = 1 m), so that Nu ≈ 10⁴, we have Q/k _wNu ≈ 10² m, so the derivative of in Equation (4.8). is negligible almost everywhere. The other derivative term in Equation (4.8). is

It is consistent with our previous approximation that

and thus using p _c = p _i – N _c,

Finally, then Equation(4.8). can be approximated by

and we have only to solve algebraic equations to find N_c, since p _i and Q are prescribed functions of s. The square-bracketed term in Equation (4.16). represents that part of the viscously dissipated energy that does not go into melting the ice roof. Reference RöthlisbergerRöthlisberger (1972) showed that for the simplest case of a horizontal tunnel and constant discharge Q, this term would be a fraction γ of the total viscous dissipation. More generally, simple estimates of the square-bracketed term in Equation (4.16). suggest it is typically about 0.1–0.3 times the term Tul/L, so for algebraic convenience we neglect it.

Model summary

We now simply have five algebraic equations:

we take l _i ≈ l and

to find

where

Thus, in order finally to determine N_c in terms of Q, we must prescribe l in terms of S. It is at this point that the conduit shape becomes critical. Specifically, we now consider the two distinct cases

(channel) and (canal), and seek to prescribe l(5) appropriately in each case.

Channel drainage

In the case

, the tunnel protrudes into the ice, forming a Röthlisberger-like channel. We then expect a more or less semi-circular cross-section, and accordingly we choose

Equation (4.17). then yields

which is essentially Reference LliboutryLliboutry’s (1983) form of the standard Röthlisberger result. Note that as Q increases, N_c increases and thus p_c decreases, which is consistent with formation of an arborescent network of channels, as we discuss below. Because N_c depends weakly on Q, we can reasonably take Q ≈ constant to evaluate N_c.

Canal drainage

Suppose now that

. In this case, drainage is through a canal incised into the sediment. The mechanics of sediment erosion and transport now become very important in determining the cross-sectional shape. In particular, canals are typically wide and shallow, in which case Equation (4.19). no longer applies. If h is the canal depth, then

and the determination of h depends on the precise nature of the stream bed, gravel, sand/silt and clay beds having different prescriptions. We discuss these subsequently. For now, suppose that h is prescribed in terms of the bed composition. Then, using Equation (4.17)., we have

Note the crucial result that, in contrast to Röthlisberger channels, canals incised into sediment lead to a water pressure that increases with increasing water flux. This is consistent with a distributed rather than an arborescent drainage network as for flow in cavities (Reference WalderWalder, 1986; Reference FowlerFowler, 1987; Reference KambKamb, 1987). We emphasize that the physics leading to a non-arborescent subglacial canal system is distinct from what is involved in the instability causing braiding of subaerial streams. That instability involves details of the sediment-transport process, includ ing non-steady phenomena such as growth and destruction of sediment bars and passage of bed-load pulses (e.g. Reference ParkerParker, 1976; Reference AshmoreAshmore, 1991). The predicted non-arborescent nature of a subglacial canal system, on the other hand, is simply the result of the relation between Q and N_c.

Numerical estimates

The coefficients b ₁, b ₂ and b ₃ depend only on sin α, and taking n = 3 they are given by

Assuming

, Röthlisberger-like channels exist, and the relation in Equation (4.20). between N_c and Q may be expressed numerically as

where Q = [Q] m³ s^-;1 and N _c = [N _C] bar. Inverting Equation (4.24). gives

Therefore, for sin α = 0.1, [Q] = 1, we have [N_c] ≈ 38; for sin α = 10^-;3, [N_c] ≈ 4. Thus, we can expect that in fact

for valley glaciers, although open-channel flow may be common, as pointed out by Reference HookeHooke (1984). The calculated value of N_c is, however, probably less than for very small slopes, contrary to assumption, so R channels are unlikely to exist beneath ice streams and ice sheets, where these are underlain by deforming sedimentsor till.

If, on the other hand, we assume

, then the expression in Equation (4.22). for water flux is, numerically,

where h = [h] m. For [Q] = 1, we find [N _c] = 85 [h] for sin α = 0.1, [N _C] = 4 [h] for sin α = 0.001. Thus, for steep valley glaciers, canals are only viable if h < 10 cm, while for ice sheets, canals can exist for stream depths up to about 2 m. In order to estimate what h is, we must consider further the mechanisms which determine it.

Determination of channel depth

Stability

In discussing stability, we distinguish between the stability of an individual solution branch (here used in a mathematical sense), i.e. the stability of a particular conduit to perturbations in that conduit, and what we will term collective stability, which is the stability of a particular drainage system against perturbation to another system, e.g. two (or more) channels, or two or more canals. First we discuss branch stability.

We suppose that each end-member solution (channel or canal) derived above is stable; we have no reason to suppose otherwise. Now consider what happens as Q increases from zero. For very small Q, Darcy flow through the till will suffice to drain all melt water. As Q increases, the till pore-water pressure will increase until N= 0, at which point flotation is initiated and a water film forms at the ice/till interface. As shown by Reference WalderWalder (1982), this sheet flow is always unstable, and we therefore expect that a bifurcating branch representing interfacial conduits exists with N_c > 0. This situation is portrayed in Figure 3.

Fig. 3.

Probable conditions for very low values of discharge. Darcy flow can drain all meltwater for sufficiently low discharge. For Q = Q _c, sheet-water flow commences at zero effective pressure (broken line) but this drainage configuration is always unstable relative to channelized drainage.

At larger Q, if canals and channels can exist, then we surmise that these solutions arise as bifurcating branches from an (unstable) solution in which

, with conduits more or less equally incised into ice and till. This branch will be the continuation of the bifurcating branch at low Q. The situation is portrayed in Figure 4. We have explored other possibilities, but this one seems the likeliest. There are certain immediate implications. In the case that channels and canals can coexist (as in Figure 4 ), then because the channel-solution curve intersects the line at a lower value of Q than the canal curve, channels will be the preferred solution as Q is increased. Moreover, if canals and channels were to coexist at larger Q (each system being in itself stable), then because the effective pressure is higher for channels (and hence the water pressure is lower), leakage through the till will drain water from canals towards channels. (This is an example of collective stability of channels.) The net effect of these factors is that if channels can exist, they will be the preferred stable drainage style, and this is likely to be the case for valley glaciers. If the surface slope is sufficiently small, as for ice sheets or ice streams, canals will form the stable drainage network.

Fig. 4.

Qualitative picture of the entire pressure vs discharge plot. The line

probably coincides (for large enough values of Q) with an unstable solution branch representing conduits inched equally into ice and till, and joins with the incipient-channel solution branch beginning at Q = Q_c. For small slopes (sin α~10^-;3), the channel solution does not exist, whereas for large slopes (sin α~10^-;1), the canal solution does not exist. At intermediate slopes (sin α~10^-;2), both solutions exist but canals are always unstable relative to channels.

Hydrological and geomorphological implications

Our analysis predicts that drainage at the base of ice sheets and ice streams should take place via canals of depth ~10 cm (plus, presumably, Darcy flow) at effective pressures of only a fraction of a bar, for fine-grained sediments. This seems consistent with basal water-pressure values at Ice Stream B, West Antarctica, inferred on the basis of geophysical measurements (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others, 1986, Reference Blankenship, Bentley, Rooney and Alley1987) and subsequently confirmed by borehole measurements (Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others, 1990), although we should emphasize that actual data on the geometry of the basal drainage system is practically nil. Our qualitative prediction is independent of the exact rheological properties of till; as long as there is some value of effective pressure below which till can creep more quickly into conduits than can ice, distributed canals as we have described will be the stable drainage type. Strictly speaking, this requires

. The results of Reference Fowler and WalderFowler and Walder (1993) give , but this particular value is by no means critical.

There is indirect sedimentological evidence for the existence of broad, shallow canals beneath ice sheets. Reference Brown, Hallet and BoothBrown and others (1987) studied deposits left by the Puget lobe of the Pleistocene Cordilleran ice sheet. Their detailed stratigraphie work in the Vashon Till of the Puget Sound region, Washington State, U.S.A., revealed the existence of lenses of water-laid, sorted sediment within the till. Brown and others interpreted these as having been deposited in subglacial conduits and estimate the width of such conduits as ranging from 0.1 to 8 m. They further inferred, from several lines of evidence, that water pressure beneath the Puget lobe was very near the ice-overburden pressure. Such low effective pressures in channelized flow would be consistent with canals but not with R channels in the ice. We further note that the discontinuous character of the water-laid subglacial deposits described by Reference Brown, Hallet and BoothBrown and others (1987) is consistent with a picture of broad, braided canals (similar to alluvial gravel-bedded streams), reflecting the constantly shifting geometry of the braids.

In northeastern England, an area of fine-grained till, Reference Eyles, Salden and GilroyEyles and others (1982) described “shoestring” deposits, which they interpreted as channel fills, and which may also have been deposited within a canal system. These deposits are elongated sub-parallel to the former direction of ice flow. They have nearly flat upper surfaces and irregular, concave-up lower surfaces, and are much wider than thick. Gravel lags at the deposit base are overlain mostly by sand but also include laminated clay. Reference Eyles, Salden and GilroyEyles and others (1982, p. 323) proposed that these deposits represent cut-and-fill structures formed by subglacial drainage, with grain-size variations within the deposit indicating variable discharge, including “ponding episodes possible through closure of the channel”. Diapiric deformation at the channel bases indicates low effective pressure in the till at the time of channel formation. Lenses of till that occur within some of these deposits may represent sediment masses that dropped into the channels from overlying ice.

Another line of geologic evidence supporting our picture of drainage beneath ice sheets comes from the distribution of eskers over the region of North America formerly covered by the Laurentide ice sheet. Eskers are very rare in areas where the ice flowed over thick till but ubiquitous where the ice-sheet bed comprised bedrock with only thin, discontinuous till cover. We would predict that, in the former instance, the stable drainage system would have comprised widespread, shallow canals incised downward into the till, whereas in the latter instance, with an essentially rigid bed, we would expect an arborescent R channel network, as in the classical view of subglacial drainage. Based on observed morphological and sedimentological aspects of eskers, we believe their formation would be very unlikely in the distributed-canal system, but definitely possible with the largest channels of an arborescent network, in accord with the observed distribution. This argument has been elaborated by Reference Clark and WalderClark and Walder (in press).

Reference ShoemakerShoemaker (1986) and (Reference AlleyAlley 1989, Reference Alley1992) have previously presented analyses of channelized drainage for sediment-floored glaciers but with different results. Reference ShoemakerShoemaker (1986) never considered creep of till into sediment-floored tunnels; furthermore, he did not consider the way in which sediment-transport mechanics affect conduit shape. Consistent with the neglect of these factors, Shoemaker further assumed that the relation between Q and N _c for sediment-floored conduits is simply Reference RichardsRöthlisberger’s (1972) result. We have demonstrated that this is not necessarily correct. (Reference AlleyAlley 1989, Reference Alley1992) dealt more carefully with the issues of till creep and sediment transport, but did not take into account the fact that sediment-transport mechanics may strongly affect conduit shape.

Finally, we remark that the present analysis does not point at any sort of glacier-surge mechanism associated with a switch between R channel and canal drainage. Unlike the case of R channels and cavities over a rigid bed (Reference FowlerFowler, 1987), we do not expect the two different types of drainage systems to coexist stably. Additionally, nothing in the present analysis indicates a dependence of the drainage-system characteristics on the ice-sliding speed (as for cavities). A surge mechanism that involves a switch between R channels and cavities is still conceivable for a glacier resting on deformable sediment. The glacier bed will not be flat and cavities can still exist to the lee of relatively large rock clasts that protrude into the ice. Thus, the potential for a linked-cavity system still exists for a deformable sediment bed, as pointed out by Reference KambKamb (1987).