## Calculating basal shear stresses using a force balance

To a first approximation, the basal drag can be estimated from rb = pgha (or tb = Sfpgha in a valley glacier). However, if longitudinal forces are unbalanced, rb may be either greater or less than pgha. For example, in Figure 12.7, the body force, pgh, has a downslope component, pgha. In addition, there are longitudinal forces F and Fd. If Fu > Fd, as suggested by the lengths of the arrows in the figure, rb will clearly have to be greater than pgha in order to balance forces parallel to the bed, and conversely. We now explore this effect in greater detail. The first part of the development is a three-dimensional generalization of an approach suggested by B. Hanson (Hooke and Hanson, 1986, p. 268).

x surface

Figure 12.8. Coordinate system used in force balance analysis.

Because our goal is to calculate the drag exerted on the glacier by the bed, the momentum balance equations (Equations (9.32b)) are the obvious starting point for the analysis. The coordinate system to be used is shown in Figure 12.8. The x-axis is horizontal and in the direction of flow, and the z-axis is vertical. Writing out the momentum balance equations in the x- and z-directions, remembering that o^' = oj - SjP, leads to:

do' doyx dozx d P

dx dy dz dz

The procedure now will be to solve Equation (12.24b) for P, substitute the result into Equation (12.24a), and integrate over the depth to obtain oZx(z=h) (= Tb).

To solve Equation (12.24b) for P, separate variables and integrate over depth z:

/dP=— I ——dz—1 ——dz— I do! — I pgdz J d x J dy J zz J ™ 0 0 0 0 0

or, noting that P = 0 at the surface:

-dz — I -dz — a„ + a„ z-o — pgz d x J dy zz zz oo

Now take the horizontal derivative, assuming that Bazz'/dx\z=0 is negligible and noting that dz/dx = a, and substitute the result into Equation (12.24a), thus:

z z da' davr dazx f d2axz f d2avz da'

Equation (12.25) is interms of stresses atany givenlevel, z, inthe glacier, whereas we are interested in summing the stresses over depth to obtain rb. Thus, as just noted, we integrate over depth:

dx2 dxdy

Clearly, azx (z=h) (= rb) will be obtained from integrating the third term, and the last term is the familiar pgha. Some simplification is obviously desirable, however.

The double integral term in Equation (12.26), also sometimes referred to as the T-term, is difficult to interpret physically. In two dimensions, Budd (1969, p. 116) has shown that it can be approximated by:

Van der Veen and Whillans (1989) argue that this term is related to "bridging" effects, in which the pressure on the bed varies spatially owing to the influence of bed irregularities and, particularly, cavity formation. Because ice is "soft", they suggest that these bridging effects should be small compared with the average normal pressure. Thus, they neglect the T-term in force-balance calculations, and we shall follow their lead in this respect. They acknowledge, however, that the "physical implications" of doing so are "not conceptually straightforward".

Turning to the first term in Equation (12.26), a'zz can be eliminated by noting that, owing to the proportionality between deviatoric stress and strain rate, the incompressibility condition, sxx + % + ezz = 0, leads to a+ a'yy + a'zz = 0. In addition, because azx would be zero on a free horizontal surface, and is only slightly different from zero on the gently sloping glacier surface, the third term in Equation (12.26) is the desired basal drag, rb, as just noted. With these modifications, Equation (12.26)

becomes:

To get an intuitive sense for this relation, consider a situation in which a 'xx and ayx are independent of depth and a 'yy is negligible. Then Equation (12.28) reduces to:

In two dimensions (oyx = 0), this equation says that if the driving stress, pgha, exceeds the drag provided by the bed, Tb, the stretching rate (exx a aXx) will increase downglacier (da'xx/dx > 0), and conversely The second term on the right takes shear stresses on the valley sides into consideration. (Adjusting for the fact the we have neglected bridging effects and have taken the z-direction to be positive downward, Equation (12.28) is identical to Van der Veen and Whillans' (1989) Equation (12.25). The rest of our development follows theirs.)

Our objective now is to express Equation (12.28) in a form that will allow evaluation of tb from strain rate measurements at the glacier surface. To this end, we note that because an-1 = (e1 B)n-1

the flow law can be written:

Inserting this in Equation (12.28), reversing the order of differentiation and integration, and rearranging terms yields:

tb = Pgha - — (2exx + Syy)dz - — eyx dz (12.29)

Van der Veen and Whillans (1989) developed a numerical procedure to carry out the integration over depth, z. However, for simple applications we assume that strain rates are independent of depth, and express Equation (12.29) in finite-difference form, thus:

Here, the symbols | dwn, | up, | rgt, and | lft refer, respectively, to the downglacier and upglacier ends, and to the left and right sides (looking downglacier), of a "block" of the glacier of length Ax and width Ay.

Table 12.1. Force balance calculations

Time

Sxx (dwn), a 1 Term 2*, kPa Term3*,kPa tb, kPa Atb, %

Block A Winter July 1983 July 1983 May 1984 June 1984 June 1985

Block B Winter July 1983 July 1983 May 1984 June 1984 June 1985

0.006 Data incomplete Data incomplete 0.000 -0.002 0.002

0.008

The values of pgha were 115 kPa beneath block A and 145 kPa beneath block B.

"Terms 2 and 3 are the second and third terms on the right-hand side of Equation (12.30), the longitudinal and transverse terms, respectively.

The second term on the right represents the contribution to Tb of an imbalance in forces on the ends of the block, while the third represents the contribution of forces on the sides.

An example of an application of this procedure is provided by an experiment conducted on Storglaciaren, Sweden (Hooke et al., 1989). Some stakes on the glacier surface (Figure 12.9) were surveyed frequently between 1982 and 1985 to determine velocities (Figure 12.10). The pattern of stakes was such that longitudinal and transverse strain rates could be calculated at the upglacier and downglacier ends of the "blocks" labeled A and B in Figure 12.9, and shear strain rates could be calculated along the sides. Results of the calculations for six time periods are shown in Table 12.1. One time period represents mean winter conditions; Tb was then -82 kPa beneath block A and -92 kPa beneath block B. The other five time periods were those during which high velocity events occurred (Figure 12.10). During these events, Tb was reduced an average of nearly 30% beneath block A. Beneath block B the change in tb was more variable, but significant increases occurred during two events.

Study of the patterns of changes suggests that acceleration of block A was, in every case, accompanied by an increase in magnitude of

, Tarfala

* station

, Tarfala

* station

Figure 12.9. Map of Storglaciaren showing generalized surface and bed topography (solid and dashed contours, respectively), locations of stakes used for velocity measurements, velocities, and blocks used in force balance calculations. (Data from Hooke et al., 1989, Figure 1a. Base map courtesy of Peter Jansson.)

60 r

70 r

30 60

SD 05

Standard error for surveys 7 d apart

SD 05

Standard error for surveys 7 d apart

30 60

10 e

1982

1983

1984

1985

Figure 12.10. Time series of mean horizontal velocities of the three strain diamonds (SD) shown in Figure 12.9. Velocities are averages of those of the four (or five) stakes in each diamond. Mean daily temperature, smoothed using a five-day running mean, is shown in the bottom panel. (Modified from Hooke et al., 1989, Figure 3a. Reproduced with permission of the International Glaciological Society.)

10 e

1982

1983

1984

1985

Figure 12.10. Time series of mean horizontal velocities of the three strain diamonds (SD) shown in Figure 12.9. Velocities are averages of those of the four (or five) stakes in each diamond. Mean daily temperature, smoothed using a five-day running mean, is shown in the bottom panel. (Modified from Hooke et al., 1989, Figure 3a. Reproduced with permission of the International Glaciological Society.)

Term 2 (the second term on the right in Equation (12.30)). From the strain rate data, it can be seen that at the upglacier end of the block éxx became less compressive, and in two cases, even extending, while at the downglacier end it became more compressive in all but one case. Thus, the accelerations were not due to either push from upglacier or pull from downglacier. The clear implication is that they were a result of a reduction in resistive drag at the bed, presumably induced by increases in water pressure.

In the case of block B, the strain rate data indicate that the marked change in Term 2 reflects push from upglacier and, in the case of the June 1984 event, pull from downglacier. This combination of push and pull resulted in higher strain rates in the basal ice, and hence, owing to the proportionality between stress and strain rate, in higher basal drag.

Because we assumed that strain rates are uniform over the sides and ends of the blocks, and also owing to other uncertainties in the calculations, the values of tb obtained are only estimates. However, as the errors are probably of comparable magnitude and sign in all calculations, the direction and approximate magnitude of the changes in tb are probably reliable. These calculations thus help us understand the mechanisms

Figure 12.11. Coordinate system used in discussion of floating ice shelves.

by which the accelerations took place in these instances. Through such analyses, we can gain insight into spatial and temporal variations in factors controlling the velocity of a glacier.

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