Sir,
In commenting upon my paper (Reference ShoemakerShoemaker, 1992a), Reference WalderWalder (1994) asserted that thick water sheets are unconditionally unstable to formation of channels and strongly implied that this negates the possibility of water-sheet outburst floods. However, Walder’s “fundamental conclusion” does not argue against the existence of water-sheet floods, as I will demonstrate.
Reference WalderWalder (1982) obtained the differential equation
by applying a standard perturbation analysis to the full equations for steady-state laminar flow of a water sheet. Here, A is the amplitude of growth of the perturbation A in the cross-section profile 
of a water sheet, where A is the average water-sheet thickness, у is measured transverse to the flow, z is vertical to the flow and 
. With the last term in Equation (1) negligible (Reference WalderWalder, 1982), and considering only the case 
, Equation (1) becomes
where
Here, α is the ice-surface slope (assuming a horizontal bed), ηw is the absolute viscosity of water, L is the specific latent heat of fusion, ηi is the effective viscosity of ice and ρ denotes density. The term 
.
If 1/s1 – 1/s2 > 0 grows exponentially in time and a broad water sheet will evidently collapse into many narrow water sheets which could eventually develop into channels operating at pressures less than the overburden pressure.
Table 1 values of 1/s1 are an extension of Reference WalderWalder’s (1982) table which applied only to the range α ≥ 0.01. Apparently, Walder was considering valley glaciers. However, for ice sheets, a more appropriate range,corresponding to sites between 500 and 1500 km from the terminus, is 
, for basal shear stresses between 8 kPa for a ponded soft bed (Reference ShoemakerShoemaker, 1991) and 50 kPa, the mean for contemporary ice sheets (Reference PatersonPaterson, 1981). I have truncated Table 1 at h = 70 mm because, depending upon α the transition to turbulent flow occurs for A between 30 and 70 mm. Assuming that the growth phase of a water-sheet outburst flood spans several weeks, it is clear from Table 1 that the water sheet can survive the laminar flood phase provided α is smaller than about 3 × 10−3.
Table 1. Time constant 1/s1 for laminar flow
Equation (2), with 1/s 2 = 0, is easily obtained by considering two independent water sheets of rectangular cross-section with water thicknesses h i and h 2, respectively. Use the formula for fully developed laminar flow
for flux Q (m3(m s)–1). Now, equate the viscous power dissipation which goes into ice melting to the rate of change of enthalpy giving 
. Apply this independently to the two sheets and let A = h
1 – h
2. Substituting for Q from Equation (4) and using 
with h fixed, as appropriate to the perturbation analysis, we arrive at Equation (2) with 1/2 = 0. The exponential growth A= exp(t/s
1) represents the growth of the difference in water thickness of the two sheets.
To extend this simple analysis to the turbulent flood phase, replace Equation (4) by the Manning’s equation as developed for a rectangular water sheet
where n(SI) is Manning’s roughness factor. We obtain
Note that 1/s2 in Equation (3) is proportional to h2 whereas l/s1 in Equation (6) is proportional to h 2/3 The friction factor decreases less rapidly with increasing discharge in turbulent flow than in laminar flow.
Table 2 extends the exponential time constant l/s2 into the turbulent range. From Tables 1 and 2,I conclude that a very thick water sheet, up to 100 m thick, can exist for weeks provided ɑ is suitably small.
Table 2. Time constant 1/s1 for turbulent flow
The term l/s2 in Equations (2) and (6) (not shown) can be important. The effective ice viscosity η i in Equation (3) decreases with increasing effective shear stress, τ (Reference PatersonPaterson, 1981). The dominant contribution to τ in Walder’s analysis was the basal shear stress, τb. But in the water-sheet problem the dominant stress after ice lift-off is the tensile stress, which could easily be an order of magnitude greater than τb (Reference ShoemakerShoemaker, 1992a, Reference Shoemakerb). If τ increases by a factor of 10, η i in Equation (3) decreases by a factor of 100 and 1/s 2 > 1/s 1 at all but very short wavelengths. Others may wish to investigate the effect this has upon water-sheet stability.
In conclusion, once the turbulent phase is included in the analysis, Walder’s main objection is refuted. Field evidence, particularly bed forms associated with turb-ulent flow, should be reviewed by those who reject water-sheet floods.
Walder’s second conclusion that a downstream decrease in α can be accommodated by an increase in channel cross-sectional area S is, of course, normally true. However, one can easily construct examples where this is not true. For these cases, a solution is obtained if a water sheet exists over one or more reaches. Walder’s conclusion is incorrect.
Regarding Walder’s third point that I ignored, plastic closure and other effects in predicting the hydrograph, this matter was better dealt with by Reference ShoemakerShoemaker (1992b) in which an argument was given for crudely estimating the duration of a flood. No hydrographic analysis was made because I have concluded that a meaningful analysis cannot be produced. Not enough is known, for example, about the mega subglacial lake that feeds a mega flood (Reference ShoemakerShoemaker, 1991).
The accuracy of references in the text and in this list is the responsibility of the author, to whom queries should be addressed.
