Hooke R.L. Principles of glacier mechanics

Подождите немного. Документ загружается.

Horizontal velocity at depth in an ice sheet 81

Because glacier-surface slopes are normally small, sin α ≈tan α ≈α.

Thus, Equations (5.2a) and (5.2b) are nearly interchangeable, and for

small slopes we commonly write:

=−ρghα (5.2c)

Note that in a coordinate system in which the z-axis is directed upward,

would be positive.

Local stresses may be augmented or reduced by gradients in longitu-

dinal stress, σ

. Thus, Equations (5.2) provide only an estimate of σ

and they cannot be used to calculate changes in σ

over short distances.

As a rule of thumb, the values of h and α that are used in Equations (5.2)

should be averages over horizontal distances that are several times the

ice thickness.

Horizontal velocity at depth in an ice sheet

Demorest (1941, 1942)argued that the horizontal velocity in a glacier

should increase with depth. He thought that the pressure of the overlying

ice would soften the deeper ice, making it ﬂow faster. Nye (1952a),

however, pointed out that this concept was physically unsound because

the faster-moving deeper ice would exert a shear stress on the overlying

ice, and there would be no corresponding resisting forces to oppose this

shear stress. Therefore, the overlying ice must move at least as fast as that

below. We now know from numerous borehole deformation experiments

that Nye’s analysis was basically correct.

To pursue Nye’s reasoning quantitatively, we start with the ﬂow law,

Equation (4.5), and assume that strain rates other than ε˙

and ε˙

and

stresses other than σ

and σ

are negligible. Then, using Equations

(2.10) and (2.11), and making use of the symmetry of the tensor so that

ε˙

= ε˙

and σ

= σ

we obtain:

˙ε





(5.3)

By using an equation analogous to (2.6a) and assuming that all of the

shear takes place in the plane normal to z so ∂w/∂x = 0, the mode

of deformation that we identiﬁed as simple shear in Chapter 4 (see

Figure 4.14c), this becomes:

= 2





(5.4)

To integrate this, σ

must be expressed as a function of z.Wewill

use the coordinate system of Figure 5.3a and Equation (5.2a), and will

82 The velocity ﬁeld in a glacier

integrate from the surface down to depth z (Figure 5.4), thus:

u(z)



du =−2



ρg sin α





dz (5.5)

Carrying out the integration and rearranging terms yields:

u(z) = u

−

n + 1



ρg sin α



n+1

(5.6)

This is the desired solution for the velocity proﬁle. It was ﬁrst obtained

by Nye (1952b). Knowing u

, B, and α,wecan calculate the velocity as

a function of depth, u(z). If the total thickness, H,isknown, we can solve

Equation (5.6) for the velocity at the bed, u

, thus:

= u

−

n + 1



ρg sin α



n+1

(5.7)

Because n

∼

3, the velocity at the bed is quite sensitive to the values of

α, B, and H.

Figure 5.4. Parameters

involved in integrating

Equation (5.4).

This derivation is rigorously correct only for a glacier that is in the

form of a slab of inﬁnite extent on a uniform slope. If the glacier is

bounded laterally, drag on the sides must be considered in calculating

.Wewill take this up in the next section.Ifthe thickness is not

uniform in the longitudinal direction, there are likely to be gradients

in the longitudinal stresses that either augment or diminish σ

relative

to the values calculated from any of Equations (5.2). Normally, these

gradients are sufﬁciently small that this source of error is not of major

concern in comparison with some others. This is discussed further below

and in Chapter 10.

Avelocity proﬁle for an ice sheet 300 m thick with a surface slope

of 2.2

◦

, n = 3, and B = 0.2 MPa a

1/n

, calculated from Equation (5.6),

is shown in Figure 5.5. The proﬁle has a distinctive form; the velocity

is nearly independent of depth in the upper part of the glacier, and then

decreases rapidly near the bed. For comparison, the dashed line shows

the proﬁle for a linearly viscous (n = 1) material, with the value of B

adjusted to give the same velocity at the bed. The distinctive form of the

n = 3 proﬁle is a consequence of the “high” value of n.

Note also in Figure 5.5 that the velocity at the bed, u

,iscomposed of

two components. If the glacier is at the pressure melting point at the bed, it

can slide over its substrate (with speed u

), whether that substrate be hard

bedrock or unconsolidated material. If the substrate is unconsolidated

material such as glacial till, this substrate may also deform. This adds

a speed u

to the total. These contributions to the speed of a glacier are

discussed in detail in Chapter 7.

Horizontal velocity in a valley glacier 83

Till

100

200

300

Depth, m

050

Velocity, m a

−1

n = 1

= 3

2.2

Figure 5.5. Velocity proﬁle

for an ice sheet with a surface

slope of 2.2

◦

and B = 0.2

MPa a

1/n

.Aproﬁle for a

linearly viscous material is

shown for comparison. The

thickness of the till layer at the

base is greatly exaggerated.

(a)

Figure 5.6. (a) Cross section

of a glacier in a semicircular

valley. (b) Cross section of a

realistic valley glacier.

(b)

Horizontal velocity in a valley glacier

In a valley glacier, some of the resistance to ﬂow, or drag, is provided by

valley sides. To see how this alters the situation, consider ﬁrst a glacier

in a semicircular valley of radius R (Figure 5.6a) and slope α. Balancing

forces on a cylindrical surface of radius r and of unit length parallel to

the ﬂow gives:

r =−ρg

r

sin α (5.8)

84 The velocity ﬁeld in a glacier

Here, r is the area of the surface and ρr

/2isthe mass of ice inside

the surface. The latter, multiplied by g sin α,isthe total force parallel

to the surface that must be resisted by a shear stress, σ

,onthe surface.

Thus, now:

=−

ρgr sin α (5.9)

Inserting this in Equation (5.4) with r in place of z as the depth dimension

and integrating as before yields:

u(r) = u

−

n + 1



ρg sin α



n+1

(5.10)

where u

is the velocity at the surface on the centerline. But for the

change to a cylindrical coordinate system, this result differs from that

of Equation (5.6) only in the factor of 2 in the denominator of the term

in brackets. However, as n ≈ 3, the difference in velocity between the

surface and a given depth is a factor of 8 less in the valley glacier. This

represents the effect of drag on the valley sides.

Semicircular cross sections are not common in nature, so let us

consider a more realistic shape (Figure 5.6b). By analogy with Equation

(5.8)wewrite:

P =−ρgAsin α (5.11)

Here, τ

is the drag exerted on the glacier by the bed, averaged over the

length of the ice–bed interface, P, and A is the cross-sectional area of the

glacier. Although τ

is a force per unit area and is often called the basal

shear stress, it is conﬁned to a plane and is thus a vector, not a tensor

quantity. Therefore, we will use the term drag and the symbol τ for it.

Dividing by P and multiplying the top and bottom of the right-hand side

by the thickness of the glacier at the centerline, H, yields:

=−ρg

H sin α =−S

ρgH sin α (5.12)

Here, we have deﬁned A/PH = S

. S

is known as the shape factor. The

reader will readily see that S

is 1 for an inﬁnitely wide glacier and

for

a semicircular glacier.

We now make the assumptions:

= τ

(5.13)

and

(5.14)

In these equations, the subscript “

L” stands for centerline. Assumption

(5.13)says that the basal drag at the centerline is equal to the average

over the cross section, and assumption (5.14)says that the shear stress

at the centerline varies linearly with depth and approaches τ

at the bed

Horizontal velocity in a valley glacier 85

Tr imline

Debris-covered

glacier tongue

Figure 5.7. Trimlines above an Alaskan glacier. Photo by B. G. Hooke.

without any discontinuity. With these assumptions, Equation (5.12) can

be rewritten as:

=−S

ρgz sin α (5.15)

Inserting this in Equation (5.4) and integrating as before yields:

u(z) = u

−

n + 1



ρg sin α



n+1

(5.16)

If one knows u

and can make reasonable estimates of H, B and S

Equations (5.15) and (5.16) can be used to calculate the basal drag and

speed, respectively. With such calculations, Nye (1952b) demonstrated

that a large fraction of the movement of temperate valley glaciers was due

to sliding (or till deformation) at the bed, and that despite a large variation

in thickness and surface slope, basal drags fell within a relatively narrow

range: 0.05 <τ

< 0.15 MPa. In practice, however, values of u

thus

calculated are not very reliable because small errors in τ

are ampliﬁed

when it is raised to the nth power (Equation (5.16)).

The narrow range in τ

is a consequence of the nonlinearity of the

ﬂow law. Small increases in H result in comparatively large increases in

, and hence in the rate at which mass is transferred from the accumu-

lation area to the ablation area. Thus, positive net balances may lead to

86 The velocity ﬁeld in a glacier

(a)

(b)

4A 2A 1A 3A 5A

Boreholes

Contours in m a

−1

100 m

Figure 5.8. (a) Contours of

longitudinal velocity in a

transverse cross section of

Athabasca Glacier. Contours

are based on measurements

in boreholes shown. All

boreholes reached the bed

except 4A, which was only

100 m deep. Dashed

contours are extrapolated.

(b) Theoretical distribution

of longitudinal velocity in a

parabolic channel, scaled to

cover approximately the

observed range of velocities in

(a). (After Raymond, 1971,

Figure 10. Reproduced with

permission of the author and

the International Glaciological

Society.)

signiﬁcant increases in speed, and hence to advances of glaciers, with

only modest increases in thickness, and conversely. Following retreat of

a glacier, evidence for this effect is commonly seen in vegetation bound-

aries, called trimlines, that reﬂect the former position of the ice surface

(see Figure 5.7). Near the terminus, such trimlines are typically high

above the present glacier surface, and they meet the valley bottom well

down-valley from the terminus. However, when traced up-valley, they

become quite close to the level of the present surface.

Comparison with measurements

It is revealing to compare the velocity distribution measured in a tem-

perate glacier with one calculated on the basis of arguments similar to

those leading to Equation (5.16). The measurements in Figure 5.8a were

made by Raymond (1971)onAthabasca Glacier in the Canadian Rocky

Mountains and are compared with calculations (Figure 5.8b) made by

Nye (1965a) for a glacier in a cylindrical parabolic channel with a similar

aspect ratio. Nye assumed that basal sliding was uniform over the cross

section.

There are some interesting discrepancies between the observed and

calculated distributions. First, the basal velocity is 80% to 90% of the

surface velocity over the central section of Athabasca Glacier, and then

decreases rapidly towards the valley sides. These large lateral gradients

in u

conﬂict with Nye’s assumption. The gradients are attributed to

lateral variations in water pressure at the ice–rock contact. The role of

Mean horizontal velocity and ice ﬂux 87

water pressure in sliding and till deformation will be discussed further

in Chapters 7 and 8.

Secondly, in the ﬁeld measurements, ∂u/∂y >∂u/∂z (where the

y-axis is transverse), whereas in the theoretical model the reverse is true.

Unless the ice is quite anisotropic, the contrast in strain rates indicates a

similar contrast in stress. The higher shear strain rates near the margin

of Athabasca Glacier indicate that the glacier is supported by drag on

the margin more than by drag on the bed.

Thirdly, although σ

increases approximately linearly with depth as

in the theory, τ

<τ

, contrary to our assumption (Equation (5.13)). In

other words, if S

is calculated from its deﬁnition, A/PH, and if this is

then used to calculate σ

at the centerline, the value of σ

will be too

high. Put differently, to calculate u(z) and τ

at the centerline of a valley

glacier, one should use a value of S

that is less than A/PH. The fact that

the velocity contours are nearly semicircular in shape, which is quite

different from the shape of the margin, suggests that S

would give a

better estimate of τ

. Furthermore, in some situations, the variation of σ

with depth may not be linear. Calculations with a numerical model have

shown that σ

depends on the basal water pressure, and that at high water

pressures it may actually decrease near the bed (Truffer et al., 2001).

Mean horizontal velocity and ice ﬂux

The ice ﬂux per unit width, q,isreadily obtained by integrating the

velocity proﬁle (Equations (5.6), (5.10), or (5.16)) over depth. We will

illustrate this with Equation (5.16), thus:



u(z)dz =





−

n + 1



ρg sin α



n+1



= u

H −

(n + 1)(n + 2)



ρg sin α



n+2

(5.17)

Possibly of greater use is the mean velocity over depth, u = q

/H :

u = u

−

(n + 1)(n + 2)



ρg sin α



n+1

(5.18)

Combining this with Equation (5.16) and simplifying the result leads to:

u = u

(n + 2)



ρg sin α



n+1

= u

n + 1

n + 2

(

− u

)

(5.19)

88 The velocity ﬁeld in a glacier

where the numerical values are calculated assuming n = 3. This relation

will be of use in Chapter 14.

Vertical velocity

Let us now consider the variation in vertical velocity with depth. Because

we are dealing with a two-dimensional situation, the incompressibility

condition (Equation (2.5)) becomes:

∂u

∂x

∂w

∂z

= 0 (5.20)

We will ignore the compressibility of ﬁrn near the surface, and also,

initially, assume that the longitudinal strain rate, ∂u/∂x,isindependent

of depth. Thus ∂u/∂x =−c,where c is a constant. Equation (5.20) then

reduces to ∂w/∂z =c.Finally, continuing to use the coordinate system of

Figure 5.4,weassume that w = 0onthe bed where z =H, thus ignoring

any contribution from the normally small rates of melting or refreezing.

Then:



dw = c



w = c(z − H )

At the surface, z = 0, we have w = w

so c =−w

/H. Therefore,

w =

H − z

(5.21)

In other words, the vertical velocity decreases linearly with depth.

The key assumption in this derivation is that ∂u/∂x is independent

of depth. Because u is nearly independent of depth in the upper part

of a glacier (Figure 5.5), the upper parts of two deformation proﬁles

in locations some distance apart in the longitudinal direction will be

nearly parallel. Thus, the assumption that ∂u/∂x is independent of depth

seems like a reasonable ﬁrst approximation. This argument is stronger

in polar glaciers because the ice near the surface is colder, and hence

more viscous (higher B). The shear strain rate, ε˙

,isthus lower, so u is

nearly constant over a greater fraction of the ice thickness. In view of

these rationalizations and the simplicity of Equation (5.21), this approx-

imation is widely used in calculations, as we shall see in Chapter 6 and

elsewhere.

However, it is clear that if the ice is frozen to the bed, ∂u/∂x = 0

at the bed. Thus, if it is non-zero higher in the glacier, it must decrease

(in absolute value) with depth. We can incorporate this effect in the

Vertical velocity 89

following way. Setting u

= 0inEquation (5.7) and using the resulting

expression for u

in Equation (5.6) leads to:

u(z) = u



1 −





n+1



whence, ignoring small terms involving ∂H/∂x:

∂u

∂x

∂u

∂x



1 −





n+1



Substituting this into Equation (5.20) and integrating upward from the

bed where w = 0, z = H :



dw =−

∂u

∂x





1 −





n+1



leads to:

w =−

∂u

∂x



z −

n+2

(n + 2)H

n+1

− H +

n+2

(n + 2)H

n+1



(5.22)

At the surface, z = 0, this yields:

∂u

∂x



1 −

n + 2



(5.23)

Combining Equations (5.22) and (5.23)gives:

w = w



1 −



n + 2

n + 1

−

n + 1





n+1



(5.24)

a result ﬁrst obtained by Raymond (1983). Equations (5.21) and (5.24)

are plotted in Figure 5.9. The difference between them does not appear

large, but we will ﬁnd that it has important consequences.

Let us follow this line of inquiry somewhat further, giving special

attention to conditions at an idealized divide on an ice sheet. We will

assume there is no ﬂow parallel to the divide, so v = 0 and ε˙

, ε˙

ε˙

, and ε˙

are all 0. Ice on either side of the divide ﬂows symmetrically

away from it. Therefore, u is 0 at (and everywhere beneath) a divide, and

so, therefore, is ∂u/∂z.Finally, if the accumulation rate is symmetrical

across the divide, ∂w/∂x =0. Thus ε˙

and ε˙

are 0 (see Equation (2.6a)),

and ˙ε

= (1/

√

2)(˙ε

+ ˙ε

)

1/2

.Nowif|∂u/∂x| decreases with depth as

just suggested, |∂w/∂z| must also decrease (Equation (5.20)). Thus, ε˙

decreases. But from Equation (2.19), a decrease in ε˙

stiffens the ice,

effectively increasing the viscosity. This reduces |∂u/∂x| and |∂w/∂z|

still further, in a positive feedback.

If |∂w/∂z|is higher at the surface than it is at depth, a plot of w against

depth must be convex upward like the curve obtained from Equation

(5.24)inFigure 5.9.Wecannot derive the actual shape of the curve ana-

lytically because of the positive feedback effects and because the stresses

are not known well enough. Raymond (1983), however, has studied the

90 The velocity ﬁeld in a glacier

Figure 5.9. Variation in

vertical velocity, w, with

depth. The parabolic relation

is a good approximation

beneath a divide. Equation

(5.24)isappropriate at

distances greater than one ice

thickness from a divide. The

linear approximation is often

used for simplicity.

ﬂow ﬁeld by using a numerical computer model. Under isothermal con-

ditions, he ﬁnds that directly beneath the divide the vertical variation of

w is closely approximated by a parabolic curve:

w =



H − z



(5.25)

(Figure 5.9). Outward from the divide, the convexity decreases, so that

at a distance of one ice thickness from the divide, the variation of w

with depth given by the numerical model is very similar to that given by

Equation (5.24) (Raymond, 1983,Figure 3).

In reality, the stiffening resulting from the decrease in ε˙

is likely to

be partially offset by warming of the ice, with a consequent decrease

in B (Equation (2.19)). Raymond studied this effect, and found that the

convexity of the proﬁle, although diminished, was still present even when

the temperature difference between the surface and the bed was nearly

◦