Regional topography

Figure 1 shows ice-sheet surface and bedrock profiles between Dome Circe and the Wilkes Land coast at 120 °E; the five spectra have been produced by applying a Fast Fourier Transform to overlapping, 200-km sections of the ice sheet’s upper surface. Regional ice thicknesses decrease in a generally uniform manner and ice velocities and temperatures may be assumed to increase fairly regularly with distance from the ice divide.

Fig. 1. Profiles of upper and lower surfaces of the ice sheet, from aircraft altimetry and radio echo-sounding, respectively. Data are from two, virtually coincident flights down a flowline between Dome Circe and the Wilkes Land coast at 120 °E. Spectra have been produced from 5 segments of bandpass-filtered altimetry of the ice-sheet surface, each 200 km long.

The spectra provide details of the increasing topographic roughness, with distance from the ice divide, a trend which can also be seen in the surface profile itself. Two particular points should be noted. First, there is a marked increase in the energy of each spectrum as one approaches the coast. The RMS deviation of the surface, the square root of the zeroth moment of each spectrum, increases from under 2 to 12.2 m. Secondly, there is the relative contribution of different wavelengths to deviations from the regional, ice-sheet surface changes. Near the divide, minor spectral peaks occur at several wavelengths, but, as velocities increase downslope, the contributions from wavelengths typically greater than 25 km become increasingly dominant. While long wavelengths occur in the coastal region, the dominant peak is at 12 km; considerable energy also exists at shorter periods.

These patterns of increasing roughness and dominance of shorter undulation wavelengths, as one approaches the thinner, faster-flowing ice, near the coast, are typical of much of East Antarctica, which conforms to a near-parabolic profile (Reference VialovVialov, 1958). They confirm the trends noted in Greenland and on the Wilkes ice cap by Reference Budd and CarterBudd and Carter (1971), but extend farther inland and encompass considerably larger undulations, wavelengths of 2 to 40 km being observed.

The transition in terrain across the ice sheet may also be considered in terms of bandpass-filtered gradients (Figure 2). Regional slopes (indicated by mean values) increase from zero, at the divides, to 2 or 3% at the margin, while deviations may rise to approximately 8% in extreme cases (Figure 2a to d).

Fig. 2. Frequency histograms of gradients of 1 km slope segments, calculated by fitting a linear regression to lowpass-filtered, altimetric profiles of the ice-sheet surface. Standard deviations (a) are given for each distribution. Gaussian curves have been added for comparative purposes.

The distribution of undulations in coastal parts of the Antarctic is far from uniform. The rough terrain, associated with converging ice flow through ice streams and outlet glaciers, is typically separated by a series of domed divides, on which surface features are smaller or absent. These appear to be glaciologically stagnant, with lower ice velocities and possibly lower basal temperatures than those of the more active outlets (Reference McIntyreMcIntyre, 1985). Mean gradients of the lower sections of outlet glaciers and ice streams (Figure 2e and f) are often less than those of adjacent areas of linear or divergent flow. Local slopes may, however, be substantial, with gradients of up to 10%. The contrasting effects of different flow regimes, thought to predominate in the thick, basally-lubricated Institute Ice Stream and the thinner, fast-flowing Thwaites Glacier, are clearly illustrated.

The preceding characterisation of surface topography is based on a near-parabolic ice sheet profile, such as that in Figure 1, which is representative of much of East Antarctica. Local effects, associated with variations in subglacial relief, ice velocity and viscosity, distort the regular alteration of surface topography across the ice sheet.

Undulation symmetry

The orthogonal intersection of altimetric flight lines enables investigation of the symmetry of surface undulations (Figure 3). The distribution of gradients, normal and parallel to flowlines, is similar in form and magnitude. The small departures from symmetry, even for undulations in the thinner ice of coastal regions (Figure 3b), suggest that ice flow over subglacial peaks produces similar stress gradients in longitudinal and tranverse directions. Thus, approximately half of the stress gradients involved in ice flow over subglacial features are absorbed by the lateral displacement of ice. This supports the suggestion (Reference Robin and MillarRobin and Millar, 1982) that three-dimensional motion can sometimes (although not always (Reference Whillans, Jezek, Drew and GundestrupWhillans and others, 1984)) be an important element of ice flow.

Fig. 3. Frequency histograms of lowpass-filtered, ice-sheet surface gradients (derived as in Figure 2), both normal and parallel to flow lines. A: The head of Ice Stream D, in Marie Byrd Land. B: 40 km inland of the Wilkes Land coast at 120°E. Standard deviations (σ) and Gaussian curves are shown for each distribution.

Surface-undulation wavelength and ice thickness

Reference BuddBudd’s (1970) treatment of ice flow over bedrock perturbations predicted the dominance of surface wavelengths 3.3 times the mean ice thickness and has been confirmed by field evidence (Reference BeitzelBeitzel, 1970; Reference Budd and CarterBudd and Carter, 1971). This can be further tested with Antarctic data (previous section; Reference McIntyreMcIntyre, 1983), by plotting dominant surface wavelengths λ_d against mean ice thicknesses (Z) (Figure 4). There appears to be correlation, although not centred on λ_d= 3.3Z. Reference BuddBudd’s (1970) linear relationship does not describe adequately the full range of data presented here. Either a high-order curve, or the more vaguely-phrased conclusion, from Reference BuddBudd’s (1969) analysis, that ‘surface undulations of wavelengths 2 to 10 times the ice thickness may tend to predominate’, seems to be more appropriate.

Fig. 4. Relationship between mean ice thickness (from Reference DrewryDrewry, 1983) and dominant surface undulation wavelengths, from a variety of data sources. Spectral plots, traverse data, and Landsat data are from Reference McIntyreMcIntyre (1983). Data from Reference Budd and CarterBudd and Carter (1971) are from Greenland. The mean of data from Reference BeitzelBeitzel (1970) is given by one point. λ = 3.3Z (Reference BuddBudd, 1970) is shown by a dashed line and the limits 2Z <λd 10Z (Reference BuddBudd, 1969) by solid lines.

The data presented in Figure 4 come from a wide range of glaciological regimes and the lack of agreement with theory may result from violation of the model’s assumptions. In any circumstances, a certain amount of scatter of data points is to be expected, because of the averaging of ice thicknesses and the different errors associated with each source.

Surface response to bedrock undulations

A more thorough investigation of surface-bedrock relationships may be carried out with spectral analysis, (Reference BeitzelBeitzel, 1970; Reference Budd and CarterBudd and Carter, 1971). Cross-spectral analyses were carried out on bandpass-filtered profiles of surface, bedrock, and, where its continuity permitted, the lowest layer, the latter defining the upper limit of the basal echo-free zone (Reference MillarMillar, 1981). Frequency-response functions were calculated and are presented for three profiles (Figure 5), as non-dimensional gain factors. These indicate the proportionate magnitude of output (profiles of surface and lowest layer) for a unit change in input (profiles of lowest layer and bedrock), at any given frequency. They are, therefore, directly equivalent to the transfer functions of Reference Hutter, Legerer and SpringHutter and others (1981) and of Reference Whillans and JohnsenWhillans and Johnsen (1983). Relevant physical parameters are given in Table I.

Fig. 5. Frequency-response functions of bandpass-filtered profiles of bedrock, the lowest layer, and the ice-sheet surface, expressed as non-dimensional gain factors against undulation wavelength. Dashed lines show 95% confidence limits. Data are from flowlines downstream of Ridge B (78.5°S 102 E) and leading to the Lennox King and Reedy Glaciers (both in the southern Transantarctic Mountains). Vertical bars show the expected maximum response of the surface to bedrock undulations of 3.3 ice thicknesses (Reference BuddBudd, 1970), with a range of one standard deviation. Valid ranges are defined by Equation 1.

Table I Physical Parameters Relevant to the Three Profiles to which Cross-Spectral Analysis has been Applied (Figure 5). Most Measured Values are from Reference DrewryDrewry (1983). Basal Temperatures have been Derived using the Model of Reference BuddBudd (1969).

The main limitation of the present data involves incomplete migration of the radio echo-sounding profiles. Layering within the ice mass, with amplitudes several hundred metres greater than those of the bedrock, suggests that the true basal profile has not been produced by migration and that echoes from valleys are still partly obscured by hyperbolae from adjacent peaks. This is supported by gain factors greater than 1.0 (Figure 5, B). Also, radio echo-sounding cannot detect depressions in the bedrock profile less than a critical wavelength, λ_c, such that the radius of curvature at the bottom of the hollow equals the range to that point (r). Thus

(1)

where b is the bedrock amplitude (Reference BuddBudd, 1969)

Gain factors for the three bedrock-surface cross-spectra (Figure 5, A, D, and G) show general trends of increasing response with undulation wavelength, as predicted by the models of Reference Hutter, Legerer and SpringHutter and others (1981) and Reference Whillans and JohnsenWhillans and Johnsen (1983). The latter two cross-spectra, in particular, show peaks in response at several wavelengths, but in none of the three is there a maximum at about three ice thicknesses, as suggested by Reference BuddBudd (1970).

Gain factors for the profile downstream of Ridge B show there to be strikingly different responses to basally-induced stresses in the lower (Figure 5, B) and upper (Figure 5, C) parts of the ice mass. The thin, basal echo-free zone has a very high response throughout, confirming that the basal ice appears to be a zone of low viscosity (Table I), which reacts sensitively to the bedrock profile. The very much colder, upper body of the ice mass, however, reacts more as a rigid slab to basally-generated stresses, in which all but the longest wavelengths (those over 20 km) are filtered out. Thus, where there is a significant discontinuity in the ice column, as appears to be the case downstream of Ridge B, the major stress variations are governed primarily by the higher, and more uniform, viscosity and velocity of the upper layers (Reference BuddBudd, 1969). These different responses indicate the effect of vertical variations in temperature and rheology on ice flow and the ice sheet’s resulting surface topography (Reference Hutter, Legerer and SpringHutter and others, 1981; Reference Whillans and JohnsenWhillans and Johnsen, 1983; Reference Reeh, Johnsen, Dahl-Jensen, Langway, Oeschger and DansgaardReeh and others, 1985; Reference Dahl-JensenDahl-Jensen, 1985).

The surface response of the profile leading to Reedy Glacier (Figure 5, G), with maximum gain factors of 0.04, is particularly low, compared with the other profiles. Its severe basal topography, steep surface slope, relatively thin ice, and higher balance velocity (Table I) would suggest, by all models discussed above, a higher surface gain than for the transects downstream of Ridge B and leading to the Lennox King Glacier, The surface response appears to be limited by the over-riding effects of vertical variations in temperature. While the other two profiles seem to be frozen to bedrock throughout, the profile leading to Reedy Glacier is at pressure melting point. The dominant effect of vertical temperature variations, in attenuating the transfer of bottom undulations to the surface, was recognised by Reference Hutter, Legerer and SpringHutter and others (1981), Reference Whillans and JohnsenWhillans and Johnsen (1983), Reference Reeh, Johnsen, Dahl-Jensen, Langway, Oeschger and DansgaardReeh and others (1985) and Reference Dahl-JensenDahl-Jensen (1985). In addition, the ratio of velocity to thickness (Table I), proportional to the mean shear strain rate through the ice column, indicates a considerably higher, vertical strain-rate gradient for the Reedy Glacier profile than for the other two. A substantial, basal echo-free zone, in this region of considerable basal relief, may indicate the presence of shear zones within the ice mass, which elsewhere have been shown to separate the upper body of the profile from less active ice in topographic lows (Reference Colbeck, St Lawrence and GowColbeck and others, 1978). This would result in a smoother, false bedrock and the low response of the surface, as observed.