### General considerations

A sufficiently large basal shear stress will cause pervasive deformation in a subglacial sediment. Such deformation has been observed beneath Breidamerkurjökull (Boulton and Hindmarsh, 1987) and Columbia Glacier (Meier, 1989), and inferred elsewhere. The velocity-tlepth profile and thickness of a pervasively deforming till layer are important in determining ice and till fluxes, and may be used in some cases to constrain the flow law for till; however, these topics have received little study to date.

Here I calculate velocity-depth profiles for the likely range of till properties and for both lithostatic and hydrostatic water-pressure gradients. The solutions obtained also apply if water-pressure gradients less than hydrostatic are specified; however, I do not consider the estimation of such gradients, which has been modeled by Boulton and Hindmarsh (1987) and Clarke (1987), among others. I exclude from consideration extrusion flow of till, which might occur if till viscosities were reduced to values typical of surficial debris flows by water pressures essentially equal to overburden pressures. Such flows could not be steady unless restricted to channels narrow compared to the ice thickness, owing to the mechanical instability of extrusion flow (Nye, 1952). Extrusion flow has been modeled by Mellors and Whillans (1986) and Mellors (unpublished), and its geomorphic implications have been discussed by Boulton and Hindmarsh (1987). There is no direct evidence for the existence of such flow, however, and measured and inferred till viscosities at Breidamerkurjökull, Blue Glacier, UpB (Alley and others, 1987b), and Columbia Glacier (Fahnestock and Humphrey, 1988) are too high to allow significant extrusion flow (Alley and others, 1987b; Mellors, unpublished).

For this discussion, I adopt the coordinate axes shown in Figure 1, with the origin at the ice-till interface, *z* positive downward, and *x* positive in the direction of ice flow. The system is assumed to be two-dimensional: Boulton (1979) discussed three-dimensional effects.

Fig. 1. Coordinate axes for till-deformation calculations, with the origin at the ice-till inter face (or the top of the till if there is a significant water layer), z positive downward, and x positive in the direction of flow. The ice velocity is u_{i}, the sliding velocity is u_{y} the till velocity is u, the till velocity at z = 0 is u_{0}, and u_{i} = u_{s} + U_{0} if there is no ice deformation. The pervasively deforming till thickness is Z_{1}.

If a fluid occupies communicating pore spaces in a material and is not flowing vertically, then the effective pressure is

where *N*_{0}
is the effective pressure at *z =* 0 and *Δρ* is the difference between the bulk density, ρ_{b>} and the pore-fluid density.

In a till that is not deforming, the pore-fluid density is the density of water, *ρ*_{w}, and the effective pressure assumes hydrostatic values given by

Effective pressure increases more rapidly with depth than this if downward porous flow occurs to a deeper, efficient aquifer.

Studies of debris flows suggest that, if the debris framework is deformed rapidly enough, water and fine clasts form an effective fluid in which larger clasts float (Rodine and Johnson, 1976; Hampton, 1979). The fluid density then approaches the bulk density, and Δ ρ is reduced toward zero. In the limit of very fast deformation, Δ ρ = 0 and the fluid can be said to exhibit a lithostatic pressure gradient, giving

It is unknown whether equation (17), (18), or some intermediate case applies at the strain-rates that occur beneath glaciers lacking efficient deeper aquifers.

### Velocity-depth profiles

Most proposed flow laws for shear deformation of water-saturated sediments have the form

where *u* is the velocity in the *x* direction, K_{b} is a constant (strictly speaking, *K*_{b}
here is twice the value of *K*_{b}
in equation (1), part I), and τ*, the yield strength, is given by

In the thin-till, lithostatic-water case where τ_{b} and *N* are independent of *z,* the velocity varies linearly with depth. If τ_{b} is constant but *N* varies with depth, equation (21) generates an interesting family of curves.

To investigate this family of curves further, it is convenient to make some definitions. Above, I defined *z*_{2}
as the depth at which the yield strength would equal the shear stress if the till were homogeneous and at least as thick as *z*_{r}
Now I define *Z*_{1}
as the thickness of the deforming layer; that is,

where “bedrock]” is any material under a till with a significantly higher yield strength. Combining Equations (20),(23) gives the general flow law for till as

In addition, I define z_{0} to be the depth at which the effective pressure is twice the value at the ice-till interface,

or an appropriate modification if downward water flow occurs. The variable *z*_{0}
measures how rapidly effective pressure increases with depth, Z_{1} is the deforming thickness, and z_{2} is the depth at which deformation would cease because of yield strength of the deforming layer.

Returning to equation (24) with z < Z_{1}, Equations (20) and (25) allow the deformation to be described by

which integrates to

where *u*(0) *= **u*
_{0}. At z = *z*_{1}, u drops to zero. Letting Z = *Z*_{i}
in equation (27), solving for the constant terms and substituting back into equation (27) yields

A second integration gives the depth-averaged velocity,
*u*
, as

Equation (28) shows that the relative velocity *u/u*_{0}
occurs in a six-dimensional space defined by the powers *a* and *b,* and the depths z, *z*_{0}, Z_{1}, and z_{2}. I will restrict further attention to the case *a* = 1, appropriate to the Bingham and linear-viscous cases. Boulton and Hindmarsh (1987) measured *N, u*_{0}, and *z*_{1}, and estimated τ_{b} for several sites beneath Breidamerkurjökull, and fitted the data using equation (24) with *∂u/∂z = u*_{0}Z_{1}
and with τ* = 0 or τ* determined from their measurements. Their results were *a =* 1.33, *b =* 1.80 for τ* = 0, and *a =* 0.625, *b =* 1.25 for τ* > 0. They did not report errors on these determinations; based on my analysis of their data, I estimate standard errors of ±0.2–0.3 on a and *b.* If so, then *a =* 1 is consistent with their data. Also, *a* = 1 allows a good fit to the longitudinal profile of Ice Stream B (see part III). I wish to emphasize that I choose *a* = I as the mathematically simplest expression that is consistent with the available data, but I do not argue that *a =* 1 necessarily is exactly correct.

Taking *a =* 1, the integrations in Equations (28) and (29) then can be done for arbitrary values of the parameters *b, z*_{0}, Z_{1}, and z_{2}, and the variable z. Results are listed in Tables I and II in terms of dimensionless variables defined there. The general case is valid for *b ≠* 1,2,3; logarithmic terms arise in *b* for *b =* 1,2 and in
*u*
for *b =* 1,2,3, as shown. In the limit of Z_{1} = Z_{2} (ω = 1), deformation occurs to bedrock. In the limit of z_{2} → ∞ (ω →0), *u* and
*u*
are well behaved and give the behavior for a till with no yield strength.

Table I. Equations for velocity-depth profiles in till with *a =* 1

Table 2. Equations for average velocity in till with *a =* 1

The behavior for selected values of *b* and the dimensionless variables of Table I is plotted in Figures 2 and 3. Rapid velocity decrease with increasing depth near the top of the till is favored by large values of *b* and of *χ* = *z*
_{0}/*z*
_{1} which lead to relatively low values of ö7 and thus of till flux. The variable ω = *Z*_{1}/z_{2}
determines how rapidly the velocity decreases to zero near the base of the deforming layer; large ω requires that the deeper velocities be almost asymptotic to zero, requiring rapid velocity decrease with depth near the top of the deforming layer and thus small
*u*
However, the velocity-depth profile is less sensitive to ω than to *b* or x.

Fig. 3. Depth-averaged velocity of pervasive deformation,
*u*
, normalized by u_{0}, plotted against x (= chi), for different values of x (ω= omega); all variables are defined in the caption to Figure 2. Each plot contains curves for b = 0, 1, 2, 3, and 5 (indicated to the lower right of the appropriate curve) assuming a = 1. The average velocity (and thus till flux) decreases with increasing b and ω and with decreasing x. The likely values of x, b, and ω discussed in this paper and in part III give
*u*
/u ≈ 0.1–0.5.

Fig. 2. Velocity-depth plots. Plots of u/u_{0} (horizontal till velocity normalized by velocity at top of till) are shown against relative depth, ψ (= psi = z/z_{1} where Z_{1} is the base of pervasive deformation and z is the depth). The exponent a = 1 in equation (24), and each column of plots is headed by the value of the exponent b used in that column; each row uses the value of ω shown at the left (ω = omega = Z_{1}Zz_{v} where z_{2} is the depth at which the shear stress would equal the strength of the pervasively deforming till). Each plot contains curves for x = 0.01, I, 10, 100, where x = Z_{0}/z_{1} and z_{0} is the depth at which the effective pressure, N, in the till is twice the value at the ice-till interface, N_{0}. Curves for b = 0 are independent of x as shown. For b ≥ 1, curves are identified by x values placed to their lower right, unless otherwise indicated. The limiting case of x = 0, b > 0 gives a curve plotting on the top and left coordinate axes (as does b →∞); the x = 0.01 curves for b = 5 are close to this behavior. In the limit of x → ∞, curves with b ≠ O become identical to the b = 0 curves; the x = 10 and x = 100 curves for small b are essentially at this limit and plot quite close together. All curves are monotonie and smooth; slight irregularities in the figure arise from the computerized plotting package used.

In principle, if till deformation obeys equation (24) with *a* = 1, then one can measure *z*_{0}, z_{1} Z_{2}
(possibly), and *u(z),* and determine *b* and *Z*_{2}
(if necessary) from comparison between the data and Figure 2. Although no-one has measured all of the necessary parameters in a deforming till and the applicability of equation (24) is open to question, analysis of the seminal data set of G.S. Boulton and co-workers from Breidamerkurjökull (e.g. Boulton and others, 1974; Boulton, 1979; Boulton and Hindmarsh, 1987) suggests that this method is workable but difficult.

In Figure 4, I show the velocity-depth data of Boulton and Hindmarsh (1987, fig. 2) re-plotted according to the variables used here. In each case, I have estimated the velocity at the top of the till, *u*_{0}, by interpolating linearly between the velocities of the lowermost marker in the ice and the uppermost marker in the till in Boulton and Hindmarsh (1987, fig. 2). I have also assumed that the thickness of the deforming layer (z_{1} = *z*_{2}) is the till thickness above the lowermost marker placed by Boulton and Hindmarsh (1987), although slow motion across discrete shear planes is reported to occur below the lowermost markers.

Comparison of Figures 2 and 4 shows immediately that the real world is more complicated than assumed in equation (24). Some of the “wiggles” in the velocity-depth profiles in Figure 4 are caused by proximity to large rigid clasts (e.g. arrays 3 and 4 near *Z/Z*_{1}
= 0.6, where the shear strain-rate decreases almost to zero; Boulton and Hindmarsh, 1987). Also, transverse flow may have minor effects on velocity-depth profiles (Boulton, 1979). However, the rapid downward decrease in velocity (large shear strain-rate) near the base of each curve requires a different explanation.

The inflection points (that is, the points where the shear strain-rates pass through minima, which I term inflection points for convenience (Fig. 4)) in the velocity-depth profiles occur at (array 4) or just above (arrays 1, 2, 3) the depths where Boulton and Hindmarsh (1987) observed a downward change in mode of deformation from pervasive creep to shear on discrete planes. The till porosity is relatively high and decreases slowly with increasing depth above the inflection point, but decreases rapidly with depth from near the inflection point to the bottom of the profile (Boulton and Hindmarsh, 1987, fig. 4a).

I suggest that these observations demonstrate the existence of a minimum strain-rate required to maintain dilation (Alley and others, 1986,1987a). Remember that till can be transformed from a compact, strong state to a dilated, weak state by strain in the presence of excess water (so that the material remains saturated) and this is reversible if the pore water can escape during compaction (Boulton and Dent, 1974). It is reasonable to expect that if the strain-rate in a dilated material becomes too low, then the collapse process will be dominant over dilation, causing the till to lose porosity and gain strength. The feed-back related to this (slow strain → collapse → stronger till → slower strain) should cause the transition zone between soft, dilated, pervasively deforming till and strong, collapsed till without pervasive deformation to be relatively thin. However, discrete shearing is possible under conditions that suppress pervasive deformation (see discussion on ploughing, above), and thus might occur in and just below this transition zone. Calculations by J.S. Walder (personal communication, 1988) suggest that such slow shearing at the base of a pervasively deforming till is possible over consolidated rock as well.

The minimum strain-rate hypothesis then predicts that pervasive deformation will cease when the strain-rate falls to some critical value (or the effective pressure and shear strength rise to some critical value compared to the shear stress), that the strain-rate just above this depth will exceed the expected strain-rate just above the depth where τ = τ* (z = *z*_{2}) in equation (24), that the porosity will decrease rapidly near this depth, and that discrete shearing may occur near and below this depth. The agreement between hypotheses and observations suggests strongly that a minimum strain-rate for dilation does exist and is reached near the inflection points shown in Figure 4.

This discussion suggests that tan ϕ (and possibly C and *N)* are inverse functions of *du/dz,* and that the true flow law is likely to be a very complicated entity, as discussed by Clarke (1987). It remains possible, however, that equation (24) (and its solutions in Table I) can be modified to provide a reasonable approximation of the actual behavior. Boulton and Hindmarsh (1987) showed that the porosity decreases downward only slowly in the upper *70% of the deforming layer before dropping rapidly into the zone of discrete shearing, and porosity should be a good proxy for till strength for a given till. Suppose we let *z ≡* Z_{3}, be the depth to the bottom of the relatively homogeneous upper part of the layer, so that the strain-rate falls to the minimum value to maintain dilation (
) at z = z_{3}. Assume further that tan ϕ, C, and z_{0} are independent of Z in this upper, pervasively deforming region. Then equation (23) can be rewritten as

and equation (24) still describes pervasive deformation.

Here Z_{1} is the thickness of the pervasively deforming layer, Z_{2} is the depth at which the yield strength of this pervasively deforming layer would equal the basal shear stress, and *z*_{3}
is the depth at which the pervasive deformation ceases (or would cease) because of strain-rate control. This means that *Z*_{1} ≤ Z_{3} ≤ Z_{2}.

With these assumptions, Figure 2 and Table I still describe the deformation, but the variable ω = *Z*_{1}/z_{2}
cannot be larger than *Z*_{b}/z_{2}
and thus is less than 1. Also, the velocity *u* must be understood as arising from pervasive deformation; discrete shear at the base of the pervasively deforming layer provides an additional velocity just as discrete shear at the top of the layer provides the additional sliding velocity.

The data of Boulton and Hindmarsh (1987) show that the sediment porosity (and by inference, the softness of the sediment) does decrease slowly downward in the pervasively deforming layer. Thus, calculations following equation (24) and using the appropriate values of constants will underestimate the curvature of measured velocitydepth profiles. Stated differently, if z_{0}, *z*_{1}
z_{2}, and z_{3} are known and *b* is chosen to provide the best match between calculated and observed velocity-depth profiles, then this value of *b* will place an upper limit on the true value.

Bearing this in mind, we now can attempt to fit the velocityiepth profiles for the pervasively deforming layer at Breidamerkurjökull, and thus to constrain the flow law for till. The inflection points in Figure 4, which I argue are good approximations of z_{3}, occur at ψ = z/z_{1} ≈ 0.8 (array 1), 0.6 (array 2), 0.75 (array 3), and 0.7 (array 4). Defining these points to be z_{s} and the deformational velocity, *u*, at these points to be zero yields the curves shown in Figure 5. From the data and calculations of Boulton and Hindmarsh (1987), the effective pressure at the ice-till interface is *N* ≈ 40 kPa, and the effective pressure would double at z_{0} ≈ 3 m. The pervasively deforming layer is Z_{1} ≈ 0.4 m thick, so x = *Z*_{0}/z_{l}
≈ 7.

Fig. 5. Curves from Figure 4 above the inflection points, recalculated so that u = 0, ψ = 1 at the inflection points. The curve labeled “model” is calculated from equation (28) with a = 1, b = 2, x=7, and ω = 0.9.

The reader can compare Figures 2 and 5 to estimate which values of ω and *b* in Figure 2 predict curves similar to those in Figure 5. I do not try a detailed, statistical curve-fitting exercise because of the large variability in the observed profiles and the uncertainties introduced by depth variation of till properties, the presence of large, rigid clasts, and the likely occurrence of transverse flow. However, by inspection, I find that *b* ≈ 2–3, ω ≈ 0.9 provides a good fit. Reducing ω requires an increase in *b* to maintain reasonable agreement between model and observed curves, but the quality of the fit deterioriates slowly from (ω, *b* = (0.9, 2) to (0.5, 5), and becomes very bad for smaller ω. It is notable that the values I find that give a good fit (0.9, 2–3) give *b* close to what we would expect based on the between-site fit of Boulton and Hindmarsh (1987), especially if we reduce *b* slightly from 2–3 to allow for the effect of slow variation of till properties with depth in the pervasively deforming layer. I therefore suggest that a curve-fitting exercise such as this is not a sensitive test for flow-law constants, but does aid in constraining these constants.

If we accept (ω, *b*) = (0.9, 2) as a working hypothesis, two further calculations suggest themselves. First, at *z =* Z_{1}, the slope of the model curve gives as an estimate for , the minimum strain-rate required to maintain pervasive till dilation and deformation.

Secondly, the estimate ω ≈ 0.9 can be used to constrain the yield strength of the dilated till. Because ω = *Z*_{1}Zz_{i}
and Z_{1} ≈ 0.4 m, Z_{2} ≈ 0.45 m. At this point, T_{b} = *N* tan *ϕ+ C* for the dilated till. Boulton and Hindmarsh (1987) (also see Boulton and Dent, 1974; Boulton and others, 1974; Boulton, 1979) measured C ≈ 4 kPa, tanϕ≈ 0.625, and presented data that allow us to estimate N(0.45 m) ≈ 46 kPa; however, the method used to measure tan ϕ probably over-estimates that number (Alley and others, 1987a). If T_{b} were known accurately, then we could use *N(Z*_{2}), C, and T_{b} to estimate tan ϕ. Unfortunately, the measurements of Boulton and Hindmarsh (1987) were made within a few ice thicknesses of the glacier terminus, where effects of longitudinal deviatoric stresses (Nye, 1967) or back-pressure from small push moraines can cause the basal shear stress to deviate significantly from the usual assumption τ_{b} = p_{1}gha, where ρ_{i}, *h,* and ρ are the ice density, thickness, and surface slope, respectively. (Boulton and Hindmarsh (1987) did not report the method they used to estimate τ_{b}; possible uncertainties from near-terminus effects probably broaden the likely errors in their determination of flow-law exponents beyond the limits I cited above.) Nevertheless, if we assume as an exercise that the usual basal shear-stress formula applies, then T_{b} ≈ 30 kPa and tan ϕ = 0.56. A similar calculation with ω = 0.5, z_{2} = 0.8 would give tan ϕ = 0.51. These calculations certainly involve large errors, but in principle calculations such as these can be used to constrain till strength better.