Sir,

Reference ShoemakerShoemaker (1992) proposed that gently sloping lobes of the Laurentide ice sheet (e.g. Reference MathewsMathews, 1974; Reference BegetBeget, 1986) were not steady-state features due to ice movement over weak, deformable sediment, but rather were transient features whose advance was triggered by giant subglacial floods of water, movingin a meters-thick sheet, released from subglacial storage. Although there is much geological evidence for episodic advance of gently sloping ice-sheet lobes (e.g. Reference Clayton, Teller and AttigClayton and others, 1985; Reference ClarkClark, 1993), Shoemaker’s proposed explanation is untenable, because thick water sheets are unconditionally unstable to formation of channels (Reference WalderWalder, 1982). In the remainder of this commentary, I elaborate this criticism and also remark on other problems in Shoemaker’s discussion of subglacial hydrology.

(Reference NyeNye 1976, p. 207) briefly considered whether an outburst flood might take the form of a water sheet. He concluded this was unlikely for two reasons. First, unless both the ice surface and the bed have no lateral slope whatsoever, water flowing in a sheet tends to be driven into channels. Secondly, lateral variations in water-sheet thickness tend to be accentuated by concomitant variations in frictional melting of the basal ice. Nye’s remarks on the latter issue motivated my analysis (Reference WalderWalder, 1982), demonstrating that sheet flow is unconditionally unstable to formation of channels. I went on to present a heuristic argument that the effect of bed roughness might nevertheless allow sheets up to a few millimeters in thickness to be quasi-stable. Reference Wcertman and BirchfieldWeertman and Birchfield (1983) subsequently argued that channelized flow itself is unstable if all meltwater is subglacially derived, due to the supposed inability of channels to collect water from large lateral distances. I am skeptical of this conclusion for several reasons; for present purposes, it is sufficient to note that their result applies only for rigid, impermeable beds. If the bed consisted of permeable sediment, as was certainly the case for marginal lobes of the Laurentide ice sheet, there would be no impediment to lateral water movement to channels. Thus the meters-thick water sheets proposed by Shoemaker are undoubtedly unstable to channel formation. Oddly, (Reference ShoemakerShoemaker 1992, p. 107) cited my 1982 analysis but failed to note its fundamental conclusion.

A second instability leading to channelization is likely to arise as well for parts of an ice sheet flowing over unconsolidated sediment, such as the marginal lobes of the Laurentide ice sheet. This instability would be formation of channels cut into the sediment. Subaerial sheet wash over a hillslope is unstable to rill formation (Reference Smith and BrethertonSmith and Bretherton, 1972; Reference LoewenherzLoewenherz, 1991), and I expect a similar phenomenon to occur subglacially (see also Reference Walder and FowlerWalder and Fowler, 1994).

(Reference ShoemakerShoemaker 1992, p. 107) asserted that channelized subglacial water flow is unstable relative to sheet flow if the ice-surface slope decreases in the downstream direction. He stated that “the reduction in the pressure gradient renders the…channel…incapable of carrying the discharge… a water sheet is created”. This statement seems to be based on the implicit, but erroneous, assumption that channel dimensions are fixed. The steady-state relation between discharge Q, cross-sectional area S and water pressure p _c may be written, following (Reference NyeNye 1976, p. 189), as

where ρ_w is the density of water, g is acceleration due to gravity, θ is the bed slope, θ is a coordinate along the channel and N depends on channel roughness and shape (but not size). The expression on the lefthand side of Equation (1) is the total hydraulic gradient. It is well known (e.g. Reference ShreveShreve, 1972; Reference WeertmanWeertman, 1972; Reference Walder and FowlerWalder and Fowler, 1994) that, in the case of gently sloping ice and bed, the hydraulic gradient is dominated by the ice-surface slope:

where ρ_i is the density of ice and α is the ice-surface slope. Equation (1) does not apply near the terminus. Substituting Equation (1) into Equation (1) gives

Thus, a downstream decrease in α can be accommodated by an increase in S. More generally, when the approximation (2) does not hold, the pressure gradient as well as the channel area may change in a reach where

. Reference ShoemakerShoemaker’s (1992) conclusion — that water must leak out from a channel if the ice-surface slope decreases downstream — is incorrect.

Finally, I wish to note Reference ShoemakerShoemaker’s (1992) improper application of Reference NyeNye’s (1976) approximate analysis of outburst-flood hydrographs. Nye elegantly showed that the rising limb of at least some outburst-flood hydro-graphs could be calculated by neglecting plastic closure of the outlet tunnel. To predict the entire hydrograph, it is necessary to consider plastic closure and other factors, including temperature of released water and the shape of the water reservoir (Reference ClarkeClarke, 1982). (Reference ShoemakerShoemaker 1992, p. 112), however, ignored plastic closure and other possible effects for the entire duration of the putative outburst. He used values of discharge so calculated (along with the erroneous conclusion that flood waters leak out of the tunnel to form a sheet) to determine the period of time during which flood water supposedly would form a meters-thick sheet at the glacier bed and thereby trigger rapid ice advance.

In conclusion, Reference ShoemakerShoemaker’s (1992) model is fundamentally flawed by two erroneous conclusions in regard to sheet flow versus channelized flow and by his misapplication of results on outburst-flood hydrographs. For this reason, I believe his results should be regarded with skepticism.

The accuracy of references in the text and in this list is the responsibility of the author, to whom queries should be addressed.