Introduction
The determination of the Antarctic ice-sheet mass budget has been one of the major aims of the International Antarctic Glaciology Project (IAGP) since its inception in 1969. However, collection of sufficient field data has been a slow task and the interpretation made difficult by results which often exhibit wide scatter and errors.
Many previous mass-flux calculations have begun by proposing a zero net mass budget as a first approximation. Two questions follow: how well does this assumed balance condition correlate with real measurements over a large section of East Antarctica? Secondly, is it possible to better estimate flow-law parameters appropriate for the ice sheet?
Reference Budd, Budd, Jenssen and RadokBudd and others (1971) compiled a comprehensive work aimed at calculating previously unknown physical characteristics of the Antarctic ice sheet, although it was recognized at the time that there were large gaps in the data coverage.
Reference Budd and SmithBudd and Smith (unpublished) and Reference Morgan and JackaMorgan and Jacka (1979) also referred to the coarseness of mass-balance predictions when data are lacking. Many earlier studies (e.g. Reference LoeweLoewe, 1960; Reference Bardin and SuyetovaBardin and Suyetova, 1967) suggested that the net budget was difficult to estimate due to the character of errors but that accumulation probably exceeded ablation by an amount up to 100% (i.e. positive imbalance).
Recent Antarctic mass-balance studies have generally considered outlet glaciers or individual drainage basins. The variance reported illustrates that trends in particular coastal or inland regions may not be the same as each other, or as that of the Antarctic ice sheet as a whole. Reference Morgan and JackaMorgan and Jacka (1979) and Reference MeierMeier (1980) summarized the studies to that time, reporting positive imbalances in all cases. These summaries included the studies of: Reference AllisonAllison (1979), Lambert Glacier basin, imbalance +100%; Reference Shimuzu, Shimizu, Watanabe, Kobayashi, Yamada, Naruse and AgetaShimuzu and others (1978), Shirase Glacier and Soya Glacier, imbalance +40% and +100%, respectively; Reference Bogorodskiy, Bogorodskiy, Trepov and Sheremet’yevBogorodskiy and others (1977), Hays Glacier, Campbell Glacier, and Carnebreen Glacier, imbalance +13%; Reference YoungYoung (1979), Pionerskaya to Dome C, imbalance +21%; Reference Morgan and JackaMorgan and Jacka (1981), Kemp Land, imbalance +100%; Reference LoriusLorius (1962), Terre Adélie, imbalance +73%.
Reference Brooks, Brooks, Campbell, Ramseier, Stanley and ZwallyBrooks and others (1978) suggested that conventional surveys are an inadequate way of mapping the Antarctic ice sheet for dynamic studies and outlined a proposal for the use of repetitive satellite radar altimetry as a way of measuring directly the balance of ice sheets and monitoring possible surges.
Various authors have described computer models which seem to provide realistic assessments. However, computer models are usually limited by an inadequate coverage, smoothing of the input field data, and uncertainty about flow-law parameters derived from laboratory studies.
This paper draws on field measurements of ice velocity and other data from traverses within the IAGP study area bounded by long. 90°E. and 135°E., and north of lat. 80°S., to provide the most extensive direct check of the balance of the inland ice sheet so far available.
Measured surface velocity and ice thickness are multiplied to determine flux rates which are then compared to independently calculated balance fluxes. Balance fluxes were calculated using digitized surface elevations and accumulation rates for the entire Antarctic continent.
The sector of East Antarctica examined in this paper incorporates a number of drainage basins and was originally reported by Reference YoungYoung (1979) to show a positive imbalance in the net mass budget of +21%. However, because of an estimated error of around 20%, this result was described as being “close-to-zero”. Reference Kotlyakov, Kotlyakov, Dyurgerov and KorolevKotlyakov and others (1983) reevaluated the sector inland of the Mirny to Dome C line, with more recent accumulation data and calculated the imbalance to be +28%.
The relationship between the mean strain-rate term and the expression for down-slope shear stress is then examined to determine empirical values of power-law flow parameters for comparison with the previous work of Reference Budd and SmithBudd and Smith (1981) and Reference Cooper, Cooper, McIntyre and RobinCooper and others (1982).
Field Measurements
The bulk of these measurements are from three major traverse lines within the IAGP study area (see Fig. 1). The three traverse lines are as follows. The eastern Wilkes Land route from lat. 68º 36’ S., long. 113º19’E. to lat. 68° 59’ S., long. 126º 56’ E., which crosses the general flow direction at an approximate elevation of 2000 m (Reference Jones and HendyJones and Hendy, 1985); the southern Wilkes Land route from lat. 68° 54’ S., long. 112º 02’ E. to lat. 74° 08’ S., long. 109° 50’ E., which is (Figure.1) approximately along a flow line (Reference Young, Young, Sheehy and HamleyYoung and others, 1982; Reference SheehySheehy, unpublished); finally, the Soviet traverse route from Mirny (lat. 66° 33’ S., long. 93° 00’ E.) to Pionerskaya (lat. 69° 44’ S., long. 95° 32’ E.), which is approximately along a flow line, and then from Pionerskaya to Dome C (lat. 74° 44’ S., long. 124° 22’ E,), which crosses the general flow direction at an approximate elevation of 3000 m (Reference HamleyHamley, 1985), Other measurements are taken from Reference Jones and DavisJones and Davis (1985), Reference MedhurstMedhurst (1985), and unpublished ANARE data.
Fig. 1. Location of ice-movement stations.
Altogether, 55 ice-movement stations have been considered.
At each station, geodetic coordinates were computed from the results of satellite doppler surveys with a mean error range (for latitude, longitude, and height) assigned on the basis of the number of passes collected. After a suitable length of time, each station position was re-measured, requiring a simple calculation to determine surface-ice velocity. The error range in velocity measurements is therefore dependent on the quality of position “fixes” and the length of time between those measurements.
In Figure 2, the measured surface–ice velocity along sections of major traverse lines is shown by upper and lower bounds equivalent to the mean solution plus and minus the estimated error.
Fig. 2. Velocity profiles along sections of major traverse lines. Location of individual measurements is indicated along with station numbers.
Ice thicknesses were obtained using a 100 MHz echo-sounding radar.
Measured flux rates were calculated from ice thicknesses (meaned over 20 km) and the measured surface velocities (see Fig. 3).
Fig. 3. Flux profiles along sections of major traverse lines. Location of individual measurements is indicated along with station numbers.
Balance Calculations
The calculation of balance fluxes for East Antarctica has been the result of a joint study between the Cooperative Institute for Research into Environmental Sciences (CIRES), University of Colorado at Boulder, Colorado, U.S.A., and the Meteorology Department, University of Melbourne, Australia. One of the primary aims was the determination of a variety of physical characteristics for the region in the manner of Reference Budd, Budd, Jenssen and RadokBudd and others (1971), updated by new techniques and recent data.
Surface-elevation data used in balance computations were digitized from the map series published by the Scott Polar Research Institute (Reference DrewryDrewry, 1983). Accumulation data were based on a preliminary SPRI compilation supplied by Drewry and Radok (personal communications). These data and the results of various calculations consist of 281 × 281 grid points covering the entire Antarctic continent. Each grid point represents the mean value for the surrounding 20 km × 20 km square.
Balance fluxes have been calculated for the sector of East Antarctica covered by ANARE IAGP traverses.
Previous determinations of balance fluxes have been based on a method in which a series of flow lines is delineated to define a drainage basin. In a flow-line technique, accumulation rates are summed over the enclosed areas and at selected points the integral is divided by the flow-line separation (Reference Budd, Budd, Jenssen and RadokBudd and others, 1971). Manual methods of this type have been adopted in the mass-balance studies of Reference AllisonAllison (1979) and Reference YoungYoung (1979).
A computer program has been developed which now enables balance terms to be calculated for an arbitrary grid on which surface elevations and accumulation rates have been defined. The main advantages of a grid-point program are that it removes the subjective element of defining flow lines and provides a uniformly dense representation in a relatively short time.
The program treats each grid point in turn from the highest to the lowest surface elevation and calculates the rate of mass outflow required to balance mass inflow plus surface accumulation. Outflow is partitioned according to the relative magnitudes of the down-hill slope components.
The accuracy of the method has been verified by tests with analytic solutions (personal communication from D. Jenssen); however, several points need bearing in mind when making comparison with actual measurements. For example, it is assumed that surface slopes determine the flow direction which may not always be the case if longitudinal stresses are significant or other dynamic effects cause streaming. Streaming may introduce a bias in calculated balance fluxes down a flow line which would otherwise cancel out if similar comparisons are carried out along a contour of sufficient length. Therefore, the southern line may not provide the most reliable comparison (for mass balance), since it approximates a true flow line.
Another factor affecting balance calculations is the quality of surface-elevation data which is not uniformly reliable. Airborne profiling by the SPRI provides good coverage east of around long. 118°E. and the map‒derived data show good agreement with the latest field results. However, there still remain relatively large areas where detailed surface-elevation data are lacking.
Accumulation data for the entire continent have been compared with those of Reference Budd and SmithBudd and Smith (1982) and are, on average, greater by about 10%. The accumulation map of Reference Budd and SmithBudd and Smith (1982) originates from Reference Kotlyakov, Kotlyakov, Barkov, Loseva and PetrovKotlyakov and others (1974) with updated modifications. Latest ANARE accumulation data along the eastern line, and Soviet accumulation data along the Mirny to Dome C line, correlate well on the large scale with values obtained from the preliminary SPRI compilations.
Figure 2 shows the maximum and minimum values of the measured and calculated balance velocities along the three traverse routes (i.e. the Mirny to Dome C, southern, and eastern lines, respectively). Note that the balance velocity has been obtained by dividing the calculated balance flux by the measured 20 km mean ice thickness (i.e. meaned over 20 km of the relevant traverse segment). Figure 3 shows the same comparison for balance fluxes.
Figure 4 shows the contour plot of balance fluxes. Major topographic features including Ridge B, Dome C, and Law Dome show up as areas of minimum flux. Major outflow areas correspond to Totten, Vanderford, and Denman Glaciers. Certain minor outflow areas are not associated with identifiable geographic features but reflect slight patterns of convergence evident in surface‒elevation contours.
Fig. 4. Balance fluxes for East Antarctica from grid-point computer calculations.
For each ice-movement station, a maximum and minimum calculated balance flux is plotted, corresponding to the range of values occurring within a radius of 20 km (one grid spacing) of the point. Note that measured fluxes are based on surface velocities and are strictly comparable only if there is no vertical velocity gradient (i.e. the ice sheet flows as a block).
In Figure 5 the plot of measured versus balance flux shows that a linear relationship is evident within the limits of scatter.
Fig. 5. Measured surface flux versus computed balance flux with line of best fit shown dotted. Station names GF10, GM05, etc. are labelled F10, M5, and so on.
Discussion
In general terms, Figures 3 and 4 illustrate that flux rates are typically less than 20 km3 a−1/100 km at elevations above 2500 m (i.e. the majority of inland Antarctica). In coastal zones where the surface slope increases rapidly, so too do the flux rates. Between the 2000 m to 1000 m contours, flux rates typically increase by up to an order of magnitude.
In some areas local imbalance is suggested by a significant difference between the measured and balance fluxes (Fig. 3). Local thinning appears to be occurring in the area south of Casey between GC12 and GC40 on the southern line and GC01 and GC02 on the eastern line. There seems to be evidence of thickening at the end of the southern line near GM13; however, these conclusions may be unreliable for reasons discussed previously.
Between GM13 and Dome C on the Soviet traverse route there is evidence of local thinning which is supported by the recent model results of Reference Alley and WhillansAlley and Whillans (1984). Elsewhere, both the Soviet and eastern lines indicate a state of balance apart from two points (GM05 and GF10), which are significantly at variance to the rest of the data.
Flow in the area of GM05 is known to be divergent, although this feature is not reflected by surface-elevation data. Likewise, GF10 lies at the head of Denman Glacier and, although convergence is apparent, the data may not define this effect in sufficient detail. The fact that these discrepancies are of opposite sign, yet occur in relative proximity, suggests that only the larger-scale features are represented in the data for this region of East Antarctica and that smaller-scale features remain to be resolved. The scatter of data along the eastern line in Figure 5 is significantly less than others which is probably a reflection of the better data coverage.
Reference Budd and MclnnesBudd and McInnes (1979) and Reference Budd, Budd, Jenssen and RadokBudd and others (1970) have stated that around major ice-shelf basins balance velocities tend to be higher (by about a factor of two) than measured values, thus indicating a slow build up in the interior. Similar evidence is apparent for several stations in the area of Vanderford Glacier (shown by solid black dots in Figure 1); however, the results indicate that this may only be a localized feature.
Analysis of the data in Figure 5 shows that the correlation between balance calculations and field measurements is on average very close after an allowance is made for the conversion from surface to column velocity. The value for the slope of the line of best fit (0.89) agrees favourably with suggested values of between 0.85 and 0.92 given by Reference Budd, Budd, Jenssen and RadokBudd and others (1971, p. 161) for a flow line in this region, and the value of 0.9 given by Reference YoungYoung (1979). We therefore propose that this section of East Antarctica is generally in balance with local areas possibly subject to imbalance as previously described.
Figure 6 presents the relationship between the mean strain-rate term (V s/Z) and the down-slope shear stress (ρgᾱZ) where V s = measured surface velocity, Z = 20 km mean ice thickness, ρ = density of ice, g = acceleration of gravity, ᾱ = surface slope averaged over the surrounding SPRI data grid points (approximately 20 km). A regression analysis of this relationship was used to determine empirical values for coefficients used in the general form of the power flow law of ice which may be expressed as follows:
where τb = ρgZsin α.
Fig. 6. Strain-rate versus down‒slope shear stress with regression line shown dotted. Station names GF10, GM05, etc. are labelled F10, M5, and so on.
Regression analysis of data shown in Figure 6 resulted in values for the flow-law parameters (Equation (1)), of n = 3.21 and k = 0.023 bar−n m−1 with a coefficient of determination r = 0.79. This compares well with the results of Reference Budd and SmithBudd and Smith (1981) who, using data from a variety of other sources, obtained values for n = 3.5 and k = 0.025 bar−n m−1 for polar ice. Reference Cooper, Cooper, McIntyre and RobinCooper and others (1982) compared balance velocities from Wilkes Land with calculated shear stresses and found n = 2.51 and k = 0.0085 bar−n m−1 with a correlation coefficient of 0.81.
For an isothermal ice mass with a power flow law, the theoretical value of the ratio of velocity at a depth z is given by
which gives
where V = average column velocity and V s = surface velocity.
Therefore, if n = 3.21, then V /V s = 0.81.
For cold ice sheets, however, the increase in temperature towards the base tends to increase the effective value of n (cf. Reference BuddBudd, 1969). The change in ice crystallography can also greatly influence the ratio of averge to surface velocity (Reference Russell-Head and BuddRussell‒Head and Budd, 1979). If the increasing temperature towards the base is taken into account, then the ratio can be higher as given by Reference Budd, Budd, Jenssen and RadokBudd and others (1970). Values between 0.85 and 0.92 might be expected.
Therefore, the value obtained from data shown in Figure 5 for the ratio of balance velocity to surface velocity, of 0.89, is as close as could be expected for a balanced state.
Conclusions
A comparison between field measurements of velocity and ice thickness with calculated balance fluxes in East Antarctica suggests that the IAGP study area bounded by long. 90°E. and 135°E. and north of lat. 80°S. is unlikely to be significantly out of balance (i.e. more than ±10% of balance).
Indications are that the balance state applies in general over the entire region, except in some areas where local imbalance might be suggested.
The ratio of average column to surface-ice velocity for this section of East Antarctica is estimated to be 0.89.
Analysis of the mean shear strain-rate versus down-slope shear-stress relationship suggests the following approximate values of the power flow-law parameters are appropriate for the bulk flow of the ice sheet in this region:
Acknowledgements
The authors gratefully acknowledge the efforts of the numerous expeditioners in both ANARE and SAE traverses through the years in which field measurements have been collected. Reduction of satellite doppler information into point positions was undertaken by the Australian Division of National Mapping, Department of Resources and Energy. We are grateful to Dr U. Radok and the United States Department of Energy for supporting the computing component of this work as part of contract number DE‒AC02‒84ER60197. Thanks are due to Professor W.F. Budd and I. Allison for helpful discussions and guidance throughout.
