Geometry, kinemnatic and flow-law a assumptions

My purpose is to calculate the distribution of ice velocity and stress as a function of position in a cross-section perpendicular to the flow direction, supposing that the basal boundary condition varies across the flow direction but not parallel to this direction. To do the analysis I will use both dimensional variables (denoted by lower-case letters with a hat

and corresponding non-dimensional ones (denoted by un-hatted lower-case letters that are related through scales as defined in the text and listed in Table 1. Figure 1 shows the assumed geometry of the ice. With reference to the definitions of coordinates in Figure 1 and notation in Table 1, I assume that the flow is non-accelerating and rectilinear, such that ice moves only in the downslope direction at a speed û that depends on ŷ and ẑ, but not on or time. Since the assumed motion is parallel to the upper (ẑ = 0) and lower (ẑ = H) surfces, the geometry is independent of time.

Fig. 1. Schematic of geometry and coordinate system.

To deseribe the ice, I assume constant density

and a quasi-viscous flow law relating defarmation rate and deviatoric stress by

where

is an effective viscosity. To account for non-linear flow law, may depend on stress or deformation rate. A power relation between deformation rate and stress may be expressed by

where n and B are constants and

is an effective shear-strain rate related to the second invariant or . When n = 1, Equation (1b) gives , which corresponds to a Newtonian fluid. It is common to assume n = 3 for ice (Reference PatersonPaterson, 1994), By assuming B is constant, I neglect influences on the ice deformation that could come from variations in temperature or fabric.

These assumptions have some immediate implications that simplify the mathematical problem (e.g. Reference NyeNye, 1965). Mass is conserved automatically, since the rectilinear flow pattern is divergenceless and density is constant. The only non-zero deformation-rate components are

Thus, by Equations (1) the only non-zero shear-stress components are and these are independent of position along the flow. Conservation of momentum in the cross-flow directions ŷ and ẑ is then sa tisfied trivially by a hydrostatic distribution of normal-stress components The mathematical problem is then reduced to finding subject to the flow law and conservation or momentum in the flow direction Since the effect of temperature on the flow law is neglected, it is not necessary to consider conservation of energy.

If the flow law were to have a more complex non-linearity than allowed by Equations 1, then the simplifications of planar geometry and rectilinear flow could break down. For example, a flow law with “normal stress effects” (such tha t corresponding components of deviatoric stress and deformation rate are not proportional) could cause a more complex balance or momentum in the y and z directions, force cross-flow secondary motions and/or displace the upper surface to a non-planar configuration. Normal stress eflects are not commonly assumed for ice (Reference PatersonPaterson, 1994), and I wi11 not introduce that complexity, here.

Field equations and boundary conditions

With the simplifications discussed above, the flow law and conservation of momentum reduce to

Equations (2) and (3) are the field equations to determine

The upper surface (ẑ = 0) is shear-stress free, so th at

At the base (ẑ = H), the shear stress acting on the bed

and the basal speed are assumed to be related by

where

describes a resistance to slip.

For my purpose it is not necessary to distinguish between basal slip that arises from shearing in soft subglacial material, sliding associated with a discrete velocity discontinuity at the base of the ice or some combination. Either kind of process can be represented by Equation (4b), By allowing

to depend on Equation (4b) can be used to represent non-linear relationships between as in varios theoretical slip laws that have been proposed. Although realistic relationships between t _b and u _b are likely to be non-linear, the details of such non-linearity are unknown. Therefore, in most of the development, I consider only linear relationships subject to the requirement that t _b increases as u _b increascs The view I pursue here is that depends on ŷ because control variables, such as bed roughness, till thickness, till composition, temperature and/or water pressure, vary with ŷ and alter the way are related lalerally across the bed, At the end of the paper I consider the implications of non-linear slip laws.

Dimensionless formulation

I describe the slip resistance

as

where Ξ is a constant that sets the scale of

, and is a dimensionless description or the spatial pattern of cross-flow variation.

For the condition that

in Equations (5) (that is, is constant at value Ξ); Equations (2)–(4) lead straightforwardly to the one-dimensional solution

where

and

This solution describes a planar slab slipping over its base at a uniform rate. It is well known in the theory of fluid flow. For a non-slipping Newtonian fluid (n = 1, B = η, Ξ = ∞), it predicts a parabolic velocity profile (Reference BatchelorBatchclor, 1967). It is widely used (normally with n = 3 to approximate flow of glaciers (Reference PatersonPaterson, 1994). Equations (6) show T to be the basal shear stress

U _D to be the total deformational speed arising from shearing in the ice over the thickness H, and T/Ξ to be a uniform rate of slip at the base.

The quantities T. U _D and Ξ can be defined more generally from Equations (5) and (7) independently of the particular spatial variation of

. The stress T is commonly termed the driving stress, from which the actual may depart when ξ depends on ŷ. I use H. T. U _D and Ξ as scales for length, shear stress, speed and slip resistance to define dimensionles variables listed in Table 1.

To describe the relative contributions of basal slip T/Ξ and internal deformation U _D to the motion, I define the slip ralio

With these definitions of scales and related dimension-less variables (Table 1 ), the field Equations (2) and (3) and boundary conditions (4) reduce to

Two parameters, the flow-law power n and the slip ratio r, appear in the non-dimensional equations. For the flow law, I will restrict atention to n = 1 (which allows simple analytical solulion) and n = 3 (which corresponds to realistic ice). The slip ratio r can in principle have an enormous range from 0 (no slip) to ∞ (frictionless base). A principal goal of this paper is to explore is to the size of r affects the character of the the solutions to Equations (9)–(11).

Equations for newtonian viscous deformation

With the simplification of Newtonian viscosity (n = 1) and the constraint that fluctuations in the slip resistance ξ(z) are small compared to mean conditions, it is possible to find an analytical solution to Equations (9)–(11) for an otherwise arbitrary to express the distributions of slip resistance, speed and stress as

When ξ^* = t_y ^* = t_z ^* = u ^* = 0, then Equations (13) are the solution to Equations (9) and (10), with ξ(y) = 1 (uniform slip), and are dimensionless representayions of Equations (6). The starred quantities t_y ^*, t_z ^* and u ^* represent departures of stress and speed that arise when ξ^*(y) ≠ 0.

Introduction of Equations (13) into Equalions (9) with n = 1 gives

and then into Equations (10) gives

Substitution of Equations (12) and (13) into Equations (11) gives the boundary conditions

where

The approximation in the analytical solution is to assume that ξ^* << 1 and u _b ^* << r, which holds when the variations in ξ and u _b are small in comparison to their means. With this assumption, the third (non-linear) term on the right of Equation (16b) can be neglected.

Approximate solution

The solution for _u ^*, t_y ^* and t_z ^* when ξ^* ≠ 0 can be sought using Fourier-transform methods in form

with inverse

In this formulation the transform variable ν is wave number described by inverse wavelength.

The momentum Equations (15) when transformed becomes

This has exponential solutions exp(−2πνz) and exp(+2πνz). The boundary condition on the upper surface (Equatio (16a)) requires that these exponentials be be combined in the form

where A(ν) is to be determined from the boundary condition at the base.

At this point several fearures of the solution can be identified that will be useful later. The speed on the upper surface is given by

The basal speed is given by

Equations (20a) show that the surface and basal velocities are related by

where

Basal shear stress

can be calculated from Equations (14a) and (19) as

Comparison of Equations (20b) and (22) shows that

where

To complete the solution it is necessary to satisfy the basal boundary condition as described by Equation (16b) in the approximation of neglecting the non-linear term. Substitution of Equation (23a) into Equation (16b) gives

where

With these results the solution in the wave number domain for u _b ^*, t _b ^* and u _s ^* resulting from a specified ξ^* can be summarized as follows:

where the the

are transfer functions given by Equations (21b), (23b) and (24b). The transfer functions are plotted in Figure 2. The solution in the wave-number domain for the full depth distribution of velocity can be found from Equations (19), using Equations (25c) and (20a) to determind A(ν).

Fig. 2. Wave-number and space representations of transfer functions for linear flow defined in Equations (21), (23) and (24). Numbers ib curves for F_sl give values of slip ratio r

The spatial forms of the transfer functions

and can be expressed analytically as

and

These expressions can be found from Equations (21b) and (23b) with wtandard manipulations using similarity, the derivative theorem and well-known Fourier-transform paris (Reference BracewellBracewell, 1978). I have not found an analytical expression for the spatial form of F _sl(y) from

but it can be found numerically using the discrete Fourier transform, Figure 2 shows the spatial forms of F_u (y), F_t (y) and F _sl(y). They can be viewed as space-domain filters that can be convolved with ξ^*(y) and u _b ^*(y) to define the solution in the space domain in a way alternative to the convolution in the wave-number domain defined by Equations (25).

Important general properties of the solution are displayed by the correlations amongst ξ^*(y), u _b ^*(y) and t _b ^*(y): u _b ^*(y) and ξ^*(y) are negatively correlated (Equations (24); Fig. 2); t _b ^*(y) and u _b ^*(y) are negatively correlated (Equations (23); Fig. 2). Thus, u _b ^*(y) is low and t _b ^*(y) is high where ξ^*(y) is high (i.e. bed is sticky).

Delocalization of slip-resistance response

The function F _sl. F_y and F_u describe how departures in slip resistance from a reference level affect the distributions of velocity and stress. The effect can be viewed in terms of spectral

or spatoal characteristics. I shall now focus on the spatial characteristics from the point of view of how a localized disturbance in slip resistance affects the velocity and stress on the base and in the ice over a broader width spread across the flow. The widths of the spatial forms of F give neasyres if delocalization.

The widths of F can be defined analytically in terms of the second moment

It can be evaluated in wave number domain from

evaluated at ν = 0 (Reference BracewellBracewell, 1978), which from Equations (21b), (23b) and (24b) gives

These spatial widths are apparent in Figure 2. The widths of F _u and F _t are scaled by thickness [H] and are otherwise independent of the flow dynamics. The width of F _sl depends on the flow dynamics through the slip ratio r. Implications of these general characteristics can be illustrated with some specific examples.

Figure 3a shows the solution for an impulse in ξ^*(y) = −δ(y) for the case r = 10. The solution for u _b ^* is given directly from the transfer function F _sl and is consistent with the √r dependence for the width of F _sl (Equation (28c)). The surface speed u _s ^* and base stress t _b ^* show only slight additional spreading associated with the unit widths of F _u and F _t.

Fig. 3. Solution for the departures in bed speed u _b ^*, basal shear stress y _b ^* and surface speed u _s ^* caused by a delta function impulse (a) and a slep function (b) in the slip resistance ξ.

Figure 3b shows results for a jump in ξ^*(y) = −0.5 sgn(y) fof the case r = 10. The transition in basal speed u _b produces a shear margin that is spread out over a boundary layer with a half-width of about 3, which corresponds to the width of F _sl for r = 10.

These solutions illustrate that the effects of a sticky of slippery zone can be spread out over a multi-thickness width. By reference to F _sl (Equations (24) and (28c); Fig. 2), the width of the spreading is larger, when the bed is more generally slippery in comparison to the deformability of the ice as described by r. This means, for example, that at a higher general level of u _b, the spatial variation of u _b reflects a more strongly smoothed response to spatiak patterns of ξ.

Stress concentration and relaxation at a shear margin

The results in Figure 3b illusurate some general features of a shear margin caused by a jump in slip resistance. I will refer to the location of the jump in ξ (y = 0) as the margin, There is a redistribution of basal shear stress from the slippery to the sticky side of the margin. Two zones of the boundary layer can be defined. On the slippery side (y > 0), sick drag provides partial support to the ice Column and t _b < 1. On the sticky side (y < 0). t _b > 1 because ice is being dragged along from the side. I will refer to these zones as the stress-relaxation and stress concentration zones.

Global force balance requires that the basal shear-force reduction in the relaxation zone, the shear force acting across the margin and the basal shear-force increase in the concentration zone are all equal, which expresses the transfer of driving stress from the relaxation zone through the margin to the concentration zone.

The discontinuous jump in ξ at y = 0 causes ∂u/∂y and t _y to be singular at the base and t _b to be discontinuos and singular. Thus, the stress relaxation and concentration can be very large locally near the margin.

The boundary layer has reflection symmetry. The margin (y = 0) is the location or maximum side-shear strain rate (∂u/∂y) and stress t_y , and the signs of ∂² u/∂² y and ∂u/∂y change across the margin, The marginal speed both at the surface u _s ≡ u(0, 0) and at the bed u _b(0) = u(0, 1) is the mean of the far field values on either side. The reflection symmetry arises because of the neglect of the non-linear cross-term in the basal-boundary condition (Equation (16b)). Although this symmetry is valid for small jumps in ξ, it is shown below that it breaks down for large jumps.

Assumptions for numerical solutions

The approximation or the foregoing solution for a step in ξ requires the step to be small compared to the mean i.e. ξ^* « 1 ). In reality, there can be large jumps, for example, at ice-stream margins, Solutions for large jumps were found numericaly using finite elements, The basa resistance ξ was chosen to be very large (effectively infinite) and constant outside of thc ice-stream boundary, which suppresses any basal motion. Following current terminology, I refer to this slow ice as ridge ice. The value of ξ was chosen relatively small and constant under the ice stream to allow significant slip rate.

I define the scale for slip resistance Ξ to be the value of

beneath the ice stream. The atio r defined by Equations (8) then corresponds to the slip speed in units of U _D that would occur at the base or the ice stream in the presence of the full driving stress of 1[T] undiminished by side drag.

The solution was carried out in a region of finite width extending 10[H] in to the slow-moving ridge ice and 30[H] into the fast-moving stream ice. The boundary codition ∂u/∂y = 0 was imposed at these lateral boundaries. The distance 10[H] into the ridge was sufficient that this edge or the solution region was decoupled from effects at the shear margin. The distance 30 [H] into the stream flow and the symmetry imposed by the boundary condition correspond to full ice-stream width of 60 [^H]. More generally I denote the half-width of the stream as w. which is 30 [H] in these specific numerical calculations.

Results from numerical calculations

Numerical calculations were done for a sequence of different values of slip ratio r to display the effect of r on the character of the solutions. Results were obtained for r = 1 × 10^k, 2 × 10 ^k and 5 × 10 ^k , where k = 1–5 for n = 1. and k = 1–7 for n = 3

Figure 4 shows an example of numerical results for n = 1 (Newtonian viscosity) and r = 10 (10 units of slip and 1 unit of deformational motion when side drag is absent). All solutions show a marginal boundary layer characterized by strong horizontal shearing that separates interior flows of the slow ridge and fast stream where horizontal gradients are absent or small.

Fig. 4. Finite-element solution for linear-flow and linear-slip laws with the shown variation of slip resistance ξ. The solution is for r = 10. Paths passing through triangular elements are contours of constant speed. Every other contour is highlighed in the slow (ridge) and fast (stream) zones. Contours in the shear zone are not highlighted because of the close spacing.

In Figure 4 the boundary layer lacks the reflection Symmetry or that for a small step in ξ shown in Figure 3b. The asymmetry can be understood fairly simply. The local effective value of r is small (large) in the ridge (stream) yielding a narrow (broad) stress-concentration (relaxation) zone. The increase in t _b in the stress-concentration zone is much larger than the drop in t _b in the stress-relaxation zone, which is expected from the different width scales and global force balance. Because of the essentially no-slip condition at the base under the ridge (y < 0), there is no expression of the stress concentration zone in u _b. However, the stress redistribution causes enhanced shearing in the overlying ice, so that the stress-concentration zone affects u _s with a width df about 1[H] associated with the transfer F _u (Equations (28)). The overall width of the boundary layer is dominated by the stress-relaxation zone with a scale determined primarily by the transfer F _sl (Equation (28c)).

Figure 5 shows the patterns of surface speed u _s(y) = u(y, 0), side-shear strain rate e(y, 0) == 0.5∂u(y, 0)/∂y, and basal speed u _b (y) = u(y, 1) for n = 1 (Fig. 5a–c) and n = 3 (Fig. 5d–f) with a range of choices for r Comparison of Figure 5a–c and Figure 5d–f shows that the overall width of the boundary layer is narrower for n = 3 than for n = 1. The boundary-layer asymmetry and width increase as r incrcases for both n = 1 and n = 3. The sie-shear strain-rate disturbance at the surface is spread further into the ridge (i.e. the expression of the stress concentration zone at the surface is broader) for n = 3 than for n = 1. This behavior is associated with the strain-rate softening arising from the flow-law non-linearity that causes stress guiding in a relatively soft, rapidly shearing ice near the base.

Fig. 5. Distributions of surface speed, surface shear strain rate and basal speed for linear n = 1 (a, b and c) and power n = 3 (d, e and f) flow law with selected values of slip ratio r. Numbers on curves indicate values of r in powers of 10. Surface shear strain rate in (b) and (e) is normalized by the maximum value e_m.

The maximum shear strain rate on the surface e _m appears to be just in board (y > 0) of the margin y = 0. For n = 3 and values of r less than 10⁵, the displacement is less than 0.5[H], which is the resolution of the numerical calculations. when r exceeds 10⁵ a small displacement appears to be resolved, but it remains less than 1[H]. The value of e _m depends systematically on τ (Fig. 6a) and call be represented approximately by

for 10¹ < r < 10² with n = 1 and for 10¹ < τ < 10³ with n = 3. The reason for this dependence becomes clear in the next section.

Fig. 6. Dependence of maximum shear strain rate (a) and surface speed at the margin (b) on slip ratio r.

The surface speed at the margin u _s(0) = u(0,0) is related to the pattern of side shearing on the surface. It is smaller than the average of the distant speed on either side of the boundary layer, which is 1 + r/2 (Fig. 6b), The results determine that approximately

for 10¹ < r < 10² with n = 1 and for 10¹ < r < 10³ with n = 3.

As r becomes large (> 10² for n = 1, > 10³ for n = 3 ), the boundary-layer width exceeds the half-width of 30[H] assumed in the numerical solution (Fig. 5), and the relationships described by Equations (29) and (30) break down (Fig. 6). This behavior arises because the boundary layers from each side bridge across the full stream width. Side drag then affects the full flow width, and speed in the center is less than expected in the absence of side drag (Fig. 5a and d).

Boundary-layer characteristics

The departure from reflection symmetry, boundary-layer width and bridging are all interrelated. This section quantifies how these characteristics depend on slip ratio τ To explore this dependence, I introduce

For n = 1, R ₁ = (r/2)^0.5. For n = 3, R ₃ = (r/4)^0.25. It turns out to be convenient to compare solutions for different r and n on the basis of R_n . The rationale for this particular, variable is developed in the next section.

There are various alternatives for describing the width l of the boundary layer. For simplicity I use the distribution of u _b, and define l to be the distance from the margin (y) = 0) to the point where u _b reaches 0.8 times the actual speed at the center u _b(w), that is

Reference to u _b does not differ significantly from u _s, because of the domination of the width by the stress-relaxation zone and the small difference ≈ 1 [U _D] between u _s and u _b for the values of slip ratio r under consideration. The factor 0.8 was choscn with the following considerations. A factor 1.0 is not practical, since the approach of u _b to the limiting value of r is asymptotic. A factor much less than 1 would be sensitive only to the details of shearing very close to the margin.

Figure 7a shows how l is related to r drscribed in terms of l/w and R _n/w as found from the numerical calculations with w = 30 (Fig. 5). For R_n /w < 0.1, the numerical results for n = 1 and n = 3 approximately coincide, and for this low range of R_n/w,

Equations (31) and (33) imply that l is proportional to r1/(n+1). For n = 1, l ≈ 0.9r ^1/2, which is consistent with the analytical result for small jumps in ξ (Equation (28c)). The corresponding result for n = 3 is l ≈ 0.9r ^1/4. This dependence of l on r implies that the mean side-shear strain rate should scale as r/l = r ^n1/n , which explains the power in the dependence or e_m on r (Equations (29)).

Fig. 7. Dependence of boundary layer width l (a) and center-line basal shear stress t_b(w) = u_b(w)/r(b) on slip ratio r expressed through R_u (Equations (31)) for linear (n = 1) and non-linear (n = 3) flow laws. Points show results from numerical calculations with w = 30. Sikud curves show results from simplified boundary-layer model (Equations (34), (35) and (11b)). In (a), dashed curve shows Equations (33).

The effect of bridging is illustrated in Figure 7b, which shows u b/r = t b in the center (y = w = 30). Keep in mind that in the absence of any side drag ∂t _y/∂y = 0, t _b = 1 and u _b = r. For small values of r such that R_n /w < 0.1, side drag does not affect u _b(w) and t _b(w). For large values of r such that R _n/w > 1, u _b(W) and t _b(w) are reduced well below values for no side drag, and the dynamics in the center are dominated by side drag. In the super-high r regime with R_n /w » 1, the stream flow is essentially like an ice shelf where t _b ≈ 0 and support is entirely by side drag. With n = 3, side and basal drag are equal for R ₃/w = 0.5, which defines a transition between side- and basal-dominated regimes.

Formulation

To generalize the numerical results to widths other than 30[H], I use a one-dimensional boundary-layer model to predict the variation of basal speed u _b(Y) under the stream flow. The model is based on the flowlaw (Equation (9)) and conservation of momentum (Equations (10)) with two assumptions about stress and strain rate at the base

The first means that is the same as its average over depth −t _b/1. The second is resonable over much of the width of the boundary layer, which is characterized by dominantly horizontal gradients. It is significant only in the case that n ≠ 1, when ∂u/∂x affects the calculation of effective strain rate and viscosity. To analyze the one-dimensional model I make the substitutions

With assumptions (i) and (ii) and substitutions (Equations (34)), Equations (9) and (10) reduce to

This equation is to be solved for µ as a function of ν on

with the boundary conditions of no motion at the margin (µ(0) = 0) and no side shear in the center When t _b depends on u/r = µ (e.g. Equation (11b)), Equations (34) and (35) show that u/r is a function of y/R_n . This is the rationale for the introduction of R_n (Equations (31)) and its use in the scaling of axes in Figure 7.

Equations (35) Illustrates the underlying cause for a finite-width boundary layer, which stems from the property of the slip law (Equation (11b) that t _b also rises. As u _b approaches r, t _b approaches 1, and side drag must drop to nil.

For n = 1 and t _b given by Equation (11b) with ξ(y) = 1 for y > 0, the solution to Equations (35) and (34) for u _b(y) is straightforwardly found to be

For n = 3, I have not succeeded in finding a useful analtic representation of the solution to Equations (35). I therefore resorted to numerical integration to explore characteristics of its solutions.

Test against full numerical model

The predictions of the one-dimensional model for the dependence of l and u(w) on r and w as expressed through Rn/w are shown in Figure 7, where they are compared to the numerical results from the full equations for w = 30. In spite of the assumptions. the one-dimensional model predicts the boundary-layer width and the onset of bridging very accurately. The difference between the simplified and full numerical model is most evident for small values of r, where both calculations begin to break down from lack of resolution when the boundary layer gets narrow in the case of the full numerical calculations or of the failure of assumptions (i) and (ii) in the simplified model. Except for this restriction, the numerical results in Figure 7 can then be generalized to w ≠ 30.

Figure 8 shows the shape of the cross-profile of speed predicted by the one-dimensional model for n = 3, when there is no significant bridging (calculated for w/R_n = 20, yielding t _b (w) = 0.99). The results from the numerical calculations for which there was no significant bridging (r = 10, 20, 50, 100) are also plotted in Figure 8 as points. The points represent u _b/r calculated at grid points with y scaled by R ₃. With this rescaling of y, the distinct curves of u _b(y) in Figure 5f collapse on to approximately a single curve that describes the boundary-layer structure.

Fig. 8. Basal speed versus distance for different basal slip laws (Equations (36) and (37)).. Numbers on curves give power m in Equations (36). Point give results from numerical calculations for n = 3 and r = 10, 20, 50, 100.

Comparison of the one-dimensional model and the full numerical model shows a small, systematic difference near the margin. It arises because the ice directly over the margin. (y = 0) in the full model is free to move (e.g. u _s in Figure 5), whereas u = 0 is implicitly imposed by the assumptions of the one-dimensional model. Although the agreement is not perfect, it is quite good.

In Figure 8, one can see that the width of the boundary layer l (Equations (32)) is about 1.3 in units of R ₃ in accordance with Equations (33). It also illustrates the extreme plug-like flow associated with a narrow boundary layer introduced by a velocity-dependent drag at the base.

Non-linear slip

The precision of the one-dimensional model suggests that it can be used to explore the implications of non-linearity in the slip condition. For this purpose, I assume a slip law of the form

When m = 1, Equations (37) is equivalent to the slip law used earlier (Equation (11b)). When m = 2, it gives the dependence u _b ∝ t _b ² commonly assumed for sliding over hard beds as originally proposed by Reference WeertmanWeertman (1957) on the basis of motion past scaleless bed roughness by a combination of creep and regelation. If there are only large roughness elements present, then m = 3 is indicated. Equations (37) preserves the condition that t _b = 1 when u _b = r.

I also consider a basal-yield condition such that

This slip law is an extreme non-linear version of Equations (37) and would approximate slip over a perfectly plastic subglacial material with a distinct yield stress t ₀. The glacial till beneath ice streams by itself could have a near-perfectly plastic behavior (Reference KambKamb, 1991). Under the stream where u ≠ 0. Equations (38) requires that 0 < t₀ < 1 and t _b = t ₀ everywhere. Correspondingly, the support of all ice columns by side drag is the same (1 − t ₀). The solution for this plastic slip law in the simplified model is easily derived by integration of Equations (35) with t _b = t ₀ and substitution back into Equations (34), which gives

For this solution, the boundary-layer width l defined by Equations (32) is l/w = 0.55 for n = 1 and l/w = 0.33 for n = 3. These values predict the limiting values l/w for very large r (Fig. 7a), for which t _b approaches zero everywhere under the stream flow (Fig. 7b).

Figure 8 compares the variation of u _b found for n = 3 and w/R_n = 20 using m = 1 and 3 in Equations (37). A profile for u _b predicted from Equations (38) is also shown for the case that w and t ₀ are adjusted to give the same u _b(w).

The results in Figure 8 show that the boundary layer is broader for the non-linear slip laws compared to the linear one. The non-linearity (m > 1) makes it difficult to raise t _b to 1[T]. In the case of a basal-yield condition (Equations (38)) with t ₀, < 1. it is not possible to raise t _b to 1 [T] for any u _b. An alternative viewpoint is that the non-linearity in the slip law results in a smeared-out variation of an effective slip resistance that broadens the boundary layer.

Comparison with observations

The spatial scale of spreading l should be apparent in observations of cross-variation of velocity in ice streams. The theoretical size of l can be predicted from Equations (8), (31) and (33), when deformational speed U _D can be calculated (Equations (7)) and basal speed can be backed out from velocity measurements (Equation (6c)). The direct expression of l in observations can be found from measurements of velocity through the marginal shear zones and Equation 32.

For the Siple Coast ice streams, the thickness, driving stress and central surface speed have typical values: H ≈ 10⁰ km, T ≈ 10¹ kPa or less and

or more. On the basis of a flow law (Reference PatersonPaterson, 1994, table 5.2) evaluated for the vertically averaged ice temperature of −15°C (Reference Engelhardt, Humphrey, Kamb and FahnestockEngelhardt and others. 1990), Equation (7b) predicts a deformational speed U _D of order 10⁻¹ m a⁻¹ or less. Comparison of U _D to indicates that u _s and u _b are essentially the same and are order 10⁴ or more. The slip ratio r, which is the bed speed in the absence of side drag, is then order 10⁴ or larger when allowance is made for the reduction in actual speed by side drag (Fig. 7b). Equations (31) and (33) then predict R ₃ ≈ 7 or more and l ≈ 10 or more. The corresponding dimensional scale of spreading introduced by thickness H is then or more. An implication of these evaluations is that lateral variations in u _s on these ice streams should occur at scales that are about 10 km or longer.

In some places on the margins of Siple Coast ice streams, the shear zones are wider than

Examples have been found in Ice Streams E (Reference Bindschadler, Stephenson, MacAyeal and ShabtaieBiudschadler and Scambos, 1991, Fig. 6), B2 (Reference Whillans, Jackson and TsengWhillans and others, 1993, upstream part of transect), and D (Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others, 1994, profile 5). A wide shear zone can be explained by laterally distributed variations in the slip resistance arising from spatial changes in variables that control ξ. Slip-law non-linearity could also contribute to a gradational variation of an effective slip resistance (Fig. 8). Indeed, the irregularity of the surface velocity variations found on Ice Stream E indicates that similarly complex variations occur in the basal-boundary condition (Reference Bindschdler and ScambosBindschadler and Scambos, 1991; Reference MacAyealMacAyeal. 1992). Extreme examples are locales within a broader marginal zone of increasing speed where the cross-profile of speed is concave toward die down-flow direction and die ice motion is actually being dragged forward rather than being resisted from the side. Thus, from the point of view of side drag alone, the interior ice is isolated from the drag from the margins and instead responds to more local influences from the side (and probably along the length). These kinds of observation show that the change from sticky under a ridge to slippery under an ice stream can be transitional and irregular.

In other places on the margins of Siple Coast ice streams, the marginal shear is concentrated in zones that are narrow compared to

Examples have been found in Ice Streams B (Reference Bindschadler, Stephenson, MacAyeal and ShabtaieBindschalder and others, 1987, DwnB; Reference Whillans, Jackson and TsengWhillans and others, 1993, downstream part of B2 transect; Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994. UpB), D (Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others, 1994, profile 1) and E (Reference Bindschdler and ScambosBindschadler and Scambos. 1991). The boundary-layer widths as defined from Equations (32) are 1.4 km at UpB, 4 km at D1 and 5 km at DwnB. The distortion of crevasses in some locations points to bands of shearing its narrow as 1 km (Reference Merry and WhillansMerry and Whillans. 1993. Fig. 3).

Within the context of this analysis, such narrow shear zones would require values of r that are much smaller than could be consistent with the measured u _s and U _D expected from the normal flow law for ice. The discrepancy is very large at UpB, which is seen straightforwardly from the observations of Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others (1994): H = 1.1 km.

and By reference to Equations (32), the last is estimated as the lateral distance between locations on the upper surface of maximum shear strain rate and 80% of center speed, which utilizes the expectations that the margin (i.e. the location of a jump in slip resistance at the bed) lies beneath the maximum in shear strain rate and that the surface and bed speeds in the ice stream are nearly the same. It follows that Figure 7a and Equations (31) then imply that R ₃ = 1 and r = 5, which is orders of magnitude too small given the measured and any reasonable U _D. The situation is not so extreme for D1 where the same reasoning leads to l/w ≈ 0.26, and the implied values for R ₃ ≈ 6 and r ≈ 5 × 10³ are closer to the range of possibility (r > 10⁴) but still distinctly too low.

Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others (1994) and Reference Scambos, Echelmeyer, Fahnestock and BindschadlerScambos and others (1994) show that these narrow shear zones can be explained by a localized softening (shear enhancement) of the ice that localizes the shearing. However, it is important to recognize that localization of shear at scales less than order 10 ice thicknesses does not, in general, prove that there is local softening of the ice. That is only the case when r is high (slip much larger than deformation).

Implications for inversions

The above discussion motivates consideration of the inverse problem of deducing the distribution of slip resistance ξ from measurements of surface speed u _s. Realistic inversions of measurements should account for both longitudinal and lateral variations in slip conditions and non-linearity of the ice-flow law (Reference MacAyealMacAyeal, 1992). It also seems important to include variations in ice viscosity arising from fabric and/or temperature (Reference Echelmeyer, Harrison, Larsen and MitchellEchelmeyer and others, 1994). Nevertheless, some specific problems associated with inversion arising from lateral variations can be identified based on the general analysis for a linear flow law with small variations in ξ. It is useful to consider the inversion as a two-step process: first, determine u _b from u _s (Equation (21a)); second, determine ξ from u _b (Equation (24a)), which in combination are expressed as Equation (25c).

The first step is illustrated by Equations (21) and Figure 2. High frequency (ν > 1[H ⁻¹]) variations in u _b are strongly attenuated F _u ≈ 0) at the free upper surface. Thus, within the physics of the model, any high-frequency fluctuations in u _s, however small, would have to come from very large fluctuations in u _b. The calculation of t _b from u _s is similarly unstable (Equations (23) and (21)). Noise in u _s from measurement errors or inadequacy of the physical model, such as shear enhancement of the type discussed above, would tend to obscure any actual signal associated with the bed, and make accurate inversion for u _b and t _b at spatial scales smaller than 1[H] hopeless (Fig. 2). The length 1[H] then separates two spatial scales: long » 1[H](F _u ≈ 1 in Equations (21) and short < 1 [H](F _u ≈ 0 in Equations (21)) for which inversion is practical or not.

The behavior for lateral variations in u has some similarity to features known for longitudinal variations (e.g. Reference Balise and RaymondBalise and Raymond, 1985). but it is fortunately less complex. For longitudinal variations, there is an intermediate scale for which surface and basal variations in u have opposite sign. There is no corresponding behavior for the lateral variations. The differences arise largely because longitudinal variations involve blocking along flow paths that cause interactions imposed by continuity. Lateral variations interact by shear alone and are free of such continuity constraints.

The second step is motivated by the realization that variations in u _b, arise from variations in ξ, and therefore may be expected to have certain properties. In the high r range (high mean basal slip), a localized disturbance in ξ is converted into a relatively broader disturbance in u _b and t _b (Equations (24) and (25); Figs 2 and 3) with a width l » 1[H]). Furthermore, even though the variations in ξ and u _b may be small compared to their respective means, the long-scale variations in u _b can be much larger than the deformational speed U _D.

Now consider both steps in combination for high r, Induced variations u _b will have long spatial-scale features. At this scale, they can be transferred to the surface with only minor attenuation to appear as a signal in u _s. Moreover, the transfer from the bed to the surface is not attenuated by a depth-varying speed (e.g. Figs 2 and 3, Equations (21) with F _u ≈ 1). These results then support inversion for slip resistance at long spatial scales with a depth-integrated model such as used by Reference MacAyealMacAyeal (1992). However, inversion for short-scale features in slip resistance is confounded by spatial spreading both over the bed and through the ice dial smooths their effects at the location of potential observation on the upper surface.

A corollary of this discussion is that any substantial lateral variation of u _s with a scale much shorter than l must arise from physics that is not included in the model, for example, short-scale, lateral variations in the properties of the ice.

Although the above discussion is cast in terms of the theory for a linear fluid, the numerical calculations show the same features occur for a power-law fluid like ice with the quantitative difference that the spreading is mediated by the scale R _n (Equations (31)) with n = 3 (Figs 7 and 8).