1.1. Nivlisen

The ice shelves around Antarctica are important elements in the climate coupling of the Antarctic ice masses. The grounding line is the output gate for mass loss of the grounded ice sheet. The ice shelves themselves have intensive interactions with the atmosphere (through ice accumulation and ablation, radiation and albedo) and with the ocean (through bottom melting and freezing and iceberg calving). Hence, ice shelves are particularly sensitive to climatic influences and changes.

Here we investigate Nivlisen, one of a number of small ice shelves bordering Antarctica’s Atlantic sector. Figure 1 shows its geographic situation. The ice shelf extends over about 80 km in the south–north and 130 km in the west–east direction. Ice thicknesses vary from about 700 m in the southeast to 150m at the front. At some locations the ice rests on the bedrock, forming ice rises and ice rumples. The ice shelf is nourished by glaciers draining the inland ice of Wegenerisen through the mountains of Wohlthatmassivet. Typical flow velocities are in the order of 100 ma^–1.

Fig. 1. The region of Schirmacheroasen with Nivlisen. Grounding lines and the ice front (red lines), ice surface heights (green), the bare ice area (blue) and open rocks (brown) are indicated. The background image is the RADARSAT amplitude image (Reference JezekJezek and others, 2002). Dashed lines indicate two ERS-1/-2 SAR interferograms used in section 2.3.

The surface regimes, and, accordingly, firn and ice structures in the upper layers, vary significantly over Nivlisen. A bare ice region extends over about 100 km to the southeast. This is predominantly a large ablation zone preserved by dry katabatic winds and summer melting due to the reduced albedo. To the north, there is a transition to an extended accumulation zone. On the border between the inland ice and the ice shelf, there is an ice-free area of bedrock (a so-called oasis) maintained due to the ablation regime of the surrounding ice. This area is called Schirmacheroasen.

The ice shelf can be considered in hydrostatic balance at positions a few kilometres away from grounded ice. This property is supported both by mechanical modelling (Reference Rabus and LangRabus and Lang, 2002) and by observational evidence. Synthetic aperture radar (SAR) interferometry from the European Remote-sensing Satellite 1 and 2 (ERS-1/-2) tandem mission shows distinct zones of vertical deformation along the grounding lines, indicating that the ice shelf adapts to tidal and other sea-level displacements along these zones and keeps its bulk in hydrostatic balance (Reference Metzig, Dietrich, Korth, Perlt, Hartmann and WinzerMetzig and others, 2000; see also section 2.3). Reference Korth, Perlt, Dietrich and DietrichKorth and others (2000) used several kinematic global positioning system (GPS) tracks across the grounding zone to show in detail how the reaction to tidal displacements evolves from zero on the grounded side to the full tidal displacement over a distance of 2–4 km.

Not least due to the logistic basis provided by the Soviet/Russian, Indian and former East German (GDR) research stations in Schirmacheroasen, the region has been the subject of extensive geoscientific investigations for decades. Reference Bormann and FritzscheBormann and Fritzsche (1995) give a comprehensive review of research up to the beginning of the 1990s. Since that time, geodetic–glaciological field observations have been widely extended. New types of satellite-aided observations have become available, such as GPS measurements on the ground, and satellite altimetry and SAR interferometry from space.

This paper analyzes some of these newer observations which allow a more extensive, more detailed and more accurate description of Nivlisen’s glaciological regime than presented thus far. After a description of our basic observations and their processing (section 2), we present in section 3 the additional inferences from combining the individual types of data under two aspects: hydrostatic equilibrium and mass balance. (These fundamental links between major ice-shelf parameters are briefly reviewed in section 1.2 and 1.3.) In section 4 we discuss our findings, addressing the spatially varying firn conditions visible from the hydrostatic balance relation, basal processes and possible temporal changes, the interplay between snow accumulation and compaction and dynamical ice-flow conditions.

1.2. Hydrostatic balance of ice shelves

Free floating of an ice shelf implies the hydrostatic balance relation

where Z is the total ice thickness, h the surface height above sea level (freeboard height), ρ _w the mean density of the displaced sea-water column, and the mean density of the ice-shelf column.

This relation has been widely exploited in ice-shelf studies. See, for example, Reference RennerRenner (1969) for an early review, and Reference Coslett, Guyatt and ThomasCoslett and others (1975) and Reference Shabtaie and BentleyShabtaie and Bentley (1982) for the methodology adopted here. Through hydrostatic balance, surface heights were derived from ice thicknesses (Reference Bamber and BentleyBamber and Bentley, 1994), and ice thicknesses from surface heights (Reference VaughanVaughan and others, 1995), or geopotential models were locally evaluated by the hydrostatic balance requirement (Reference Shabtaie and BentleyShabtaie and Bentley, 1987; Reference Jenkins and DoakeJenkins and Doake, 1991). Apparent violations of hydrostatic balance gave indications for grounding (Reference Shabtaie and BentleyShabtaie and Bentley, 1987), for varying firn and ice structures (Reference Budd, Corry and JackaBudd and others, 1982; Reference Bamber and BentleyBamber and Bentley, 1994), for sediments transported by the ice (Reference Thyssen and GrosfeldThyssen and Grosfeld, 1988) or for marine ice frozen to the shelf ice bottom (Reference ThyssenThyssen, 1988; Reference Jenkins and DoakeJenkins and Doake, 1991).

Equation (1) may be reformulated in the form

where ρ _i is the density of pure ice and δ is defined by

That is, δ is the relative deviation of the ice-shelf density ρ from the pure ice density ρ _i, integrated along the ice column. Resolved for its three main constituents, Equation (2) reads

In the common case that an ice shelf consists of a mixture of pure fresh-water ice and air, the meaning of δ is plausible: a (fictive) rearrangement of the ice shelf into a body of solid ice and a layer of air would result in the solid ice thickness

and the air layer thickness δ. Since most of the air is contained in the upper firn layers, δ depends on the accumulation and densification regime but has little or no dependence on total ice thickness. The mean density , in contrast, depends on both. Values of air layer thicknesses δ reported in the literature cited in this section vary between 10 and 20 m

Unless the ice shelf consists only of fresh-water ice and air, δ is not the air layer thickness. Therefore, δ is called the ‘apparent air layer thickness’ in the following. If, for example, a marine ice layer of thickness Z _mar and density ρ _mar > ρ _i is frozen to the bottom of a meteoric ice part of thickness Z _met, Equation (3) reads

δ is now the sum of the air layer thickness δ _air and a negative correction δ _mar. However, since ρ _mar ≈ 925kgm^–3 exceeds ρ _i only slightly (see section 2.4), δ _mar ≈ —0.02 Z_mar tends to be small. Sediments enclosed in the lower ice-shelf layers are another possible phenomenon that adds a negative contribution to δ and may even cause δ to be negative in total. The apparent air layer thickness δ when computed from surface height and ice thickness using Equation (4) is a useful indication of density structures and of the validity of the free-float assumption. At the same time, it is sensitive to errors in the input data, in particular in Z and h. Gross errors in Z may arise if a marine ice layer is not recorded in the measurement, as may occur with radar techniques. Biases in the freeboard height h may arise from uncertainties in the geoid and sea surface topography (SST) data needed to reduce ellipsoidal heights to freeboard heights. Therefore, δ is frequently treated as a combination of the air layer thickness and an unknown height bias.

Once δ is known for a certain region of free floating, the relation between ice thickness and freeboard height is established and can be utilized.

1.3. Mass flux and mass balance on ice shelves

The mass conservation equation for a vertical column of an ice shelf (Reference PatersonPaterson, 1994) can be written as

Here, b _s and b _b (in units of mass per surface area per time, positive for mass gain) are the surface mass balance and the basal mass balance, respectively, Z _i is the solid ice thickness (Equation (7)), ρ _i δZ _i/δt is the local ice mass change with time, and u is the depth-independent horizontal flow velocity vector with components u and v in the x and y coordinate directions, respectively. According to Reference Bamber and BentleyBamber and Payne (2003), we call b _s + b _b the specific mass balance and ρ _i δZ _i/δt the local mass balance. Their integral effect over a certain region A can be computed through Gauss’ theorem by balancing the ice in-flux and out-flux across the region boundaries:

where K is the boundary curve oriented anticlockwise, and β is the (anticlockwise) angle between K and the velocity u . Conveniently, some sections of K follow flowlines where the integral vanishes, and other sections form the flux gates.

The basal balance b _b is commonly the most uncertain mass-balance term. The above equations may constrain it based on the geometric magnitudes of ice thickness, freeboard height and ice-flow velocity.

2.1. Ice thickness

Figure 2 gives an overview of ice-thickness observations on Nivlisen. In this study, we use data from the three sources shown by tracks in Figure 2. A radar system operating from a helicopter was used by the German Federal Institute for Geosciences and Natural Resources (BGR) during the GeoMaud expedition in 1995/96 (Reference Damm and EisenburgerDamm and Eisenburger, in press). A similar instrumentation, but mounted on an airplane, was used by the Alfred Wegener Institute (AWI), Bremerhaven, in the same summer season (Reference Steinhage, Nixdorf, Meyer and MillerSteinhage and others, 1999; Reference Meyer, Steinhage, Nixdorf and MillerMeyer and others, 2005). Finally, radar measurements along a ground traverse crossing the east of the ice shelf were carried out by the AWI in 1992 (Reference FritzscheFritzsche, 2005).

Fig. 2. Ice-thickness and firn observations. The drill-site positions are after Reference Korotkevich, Savatyugin and MorevKorotkevich and others (1978). The positions of 15 m density profiles are estimates after the description by Reference KhokhlovKhokhlov (1984).

The relative precision of the ice-thickness measurements was estimated to be 1–2%. In a crossover analysis for all radar tracks (including grounded ice), crossover differences at 61 positions, divided by the absolute thicknesses, showed an rms value of 4.5%. This value is affected by errors in the measurement positions and in the interpolation to crossover points which both, in turn, depend on along-track variations of the measurements. The ice-thickness tracks on Nivlisen seem to have two fluctuation regimes, with a distinct border along approximately –70.35° latitude (see Fig. 8 for an example). In the south, fluctuations are high, with typically 10–20 m differences between track points 300 m apart. In the north, the thickness curves are smooth. Accordingly, the rms of nine relative crossover differences in the northern part is 1.9%, indicating a measurement precision of 1.9%/ i.e. typically 1.4% × 350 m = 4.9 m. In what follows, we will rely on thickness data with a reduced resolution of 3.7 km. For these data a relative precision of 1.4% seems realistic over the entire ice shelf.

Despite this precision, biases in the thickness data must be considered. The along-track fluctuations in the southern part may indicate a complex (e.g. crevassed) structure of the ice-shelf bottom, with associated ambiguities in the radar echo. The estimated inaccuracy resulting from these effects is up to 4.4% of ice thickness. Variance combination with the above 1.4% precision leads to a total uncertainty of in the southern part, i.e. typically 4.6% x 450m = 21m. Another bias in the initial ice-thickness data results from using the signal propagation velocity for ice (168 × 10⁶ms^–1) in the radar processing. In section 3.2 we apply the correction for the higher velocities in firn.

Ice-thickness data from other measurements (see Fig. 2) were analyzed, but not included in the further computations. During the 1995/96 GeoMaud expedition, ice thickness was measured at ten ground positions on Nivlisen by radar sounding and partly also by seismic sounding (Reference ReitmayrReitmayr, 1996). For part of the data the interpretation of the signal return from only a single point measurement is ambiguous. In 1975/76, Soviet Antarctic Expedition groups performed two ice-core drillings through the ice shelf, and a third through the nearby grounded ice (Reference Korotkevich, Savatyugin and MorevKorotkevich and others, 1978). In the same year, radio-echo soundings were performed from airplanes (Reference Kozlovskii and FedorovKozlovskii and Fedorov, 1983) and along ground traces (Reference Boyarskii, Nikitin and StepanovBoyarskii and others, 1983; Reference Eskin and BoyarskiiEskin and Boyarskii, 1985). Differences between these historical data and the models derived hereafter are further addressed in section 4.3.

2.2. Ice surface height

2.3. Ice-flow velocity and grounding zone

2.4. Surface mass-balance and density data

As mentioned in section 1.1, Nivlisen is divided into an ablation zone in the southeast and an accumulation zone in its remaining parts. Accumulation is due to both precipitation and drift snow. Part of the accumulated snow melts in summer and percolates into deeper horizons, forming ice layers and lenses (Reference Bormann and FritzscheBormann and Fritzsche, 1995).

Ablation rates were observed over up to 5 years (1988–93) at stakes along two traverses through the grounded part of the ablation area (Reference Korth and DietrichKorth and Dietrich, 1996). The mean ablation at 47 stakes was 230 kgm^–2 a^–1 with a 100 kg m^–2 a^–1 standard deviation of the spatial variations. For the close vicinity of Schirmacheroasen, Reference SimonovSimonov (1971) reported 340 kg m^–2 a^–1 ablation. Accumulation rates along the traverse crossing the ice shelf (cf. Fig. 2) were measured at seven stakes in 1992/93. The mean value is 316 kgm^–2 a^–1. Reference SimonovSimonov (1971) reported a mean accumulation rate of 230–240 kg m^–2 a^–1 in the years 1959–65. Summing up these references, the surface mass balance is typically –250± 100 kgm^–2 a^–1 in the ablation zone and +300± 100 kg m^–2 a^–1 in the accumulation zone.

Density information is available down to 610m depth from a deep core drilling in the grounded accumulation area south of Schirmacheroasen (Reference Kislov, Manevskii and SavatyuginKislov and others, 1983). These data together with densities from a nearby shallow ice core (Reference FritzscheFritzsche, 1996) may be fitted by the exponential relation (Reference PatersonPaterson, 1994)

with the depth z, the surface snow density ρ _s = 420 kg m^–3 and the densification parameter C = 0.029 m^–1. The resulting air layer thickness from this fit is δ≈ (ρ _i–ρ _s)/(ρ _i C) = 18.7 m. On the ice shelf itself, densities down to 15 m depth were observed from three ice cores (see Fig. 2 for locations) by the 1982/83 Soviet Antarctic Expedition (Reference KhokhlovKhokhlov, 1984). To the south, in the ablation area, vertically homogeneous ice with a large amount of air bubbles and a mean density of 879 kg m^–3 was found. In the transition zone from the ablation to the accumulation regime, the ice below 12 m resembled the ice in the ablation zone, while the upper 12 m consisted of both ice and consolidated firn. The third core near the ice front contained frozen firn of 640 kg m^–3 density at 10–15 m depth. In the upper 10 m, ice layers dominated in the interchange with firn layers. These cores may illustrate the complexity and spatial variability of meteorological influences on the firn and ice structure.

For the densities of ocean water, ρ _w, and pure fresh-water ice, ρ _i, required for hydrostatic balance considerations, we use the values ρ _w = 1029 kg m^–3 and ρ _i = 917 kg m^–3. Ocean water densities applied, for example, by Reference Shabtaie and BentleyShabtaie and Bentley (1982), Reference Jenkins and DoakeJenkins and Doake (1991) and Reference VaughanVaughan and others (1995) at different ice shelves vary slightly between 1027.5 and 1030 kgm^–3, indicating an uncertainty of 1.5 kg m^–3 in our value. The pure ice density, ρ _i, applied in the works cited above, varies between 915 and 920 kg m^–3. We assign an uncertainty of 3 kg m^–3 to our pure ice density value. In hypothetical computations assuming a marine ice layer (sections 1.2 and 4.3) we use the marine ice density ρ _mar = 925 kgm^–3 according to Reference Jenkins and DoakeJenkins and Doake (1991).

3.1. Spatial variations of the hydrostatic balance parameter δ

To exploit the hydrostatic balance relation between ice thickness and freeboard height, we selected those thickness measurements lying in the domain of our freeboard height model and on the freely floating ice shelf according to the limits explained in section 2.3.3. For these positions we applied Equation (4) to compute the apparent air layer thickness δ.

Figure 8 illustrates the result for the data along one radar track. From the irregularly distributed values of δ a gridded model was derived by a continuous curvature spline algorithm (Reference Wessel and SmithWessel and Smith, 1991). This model, shown in Figure 9, is intended to represent δ in the sense of 3.7 km scale averages.

Fig. 8. (a) Measured ice thicknesses, (b) freeboard heights interpolated from the model in section 2.2.3, and (c) derived values of the ‘apparent air layer thickness’ 6, along the AWI airplane radar track going to the ice tongue at 11.2° E (cf. Fig. 2).

Fig. 9 Model of the ‘apparent air layer thickness’ δ (in metres) derived from ice-thickness and freeboard height data at the positions indicated in grey.

Clearly, δ shows a large spatial variation with a distinct pattern. In the southeast of the ice shelf, values generally vary between 1 and 4 m. To the northwest of this region, δ increases from 4 to 14 m within a distance of about 22 km. Northwest of this zone of increase, values are generally 14–20 m.

Table 3 summarizes an error assessment for δ obtained by variance propagation from the errors of the individual inputs of Equation (4). We consider the total error as a sum of the error of the spatial variations and the error of the absolute level. In particular, errors in ρ _w and ρ _i induce errors in δ which are proportional to Z–h, i.e. positive everywhere. Therefore, the error propagated from ρ _w and ρ _i is the sum of a constant part associated with the mean value of Z–h and a spatially varying part associated with the spatial variations of Z–h. Errors of interpolating and slightly extrapolating δ are roughly estimated from the local variability of δ. The resulting estimated errors in the spatial variations of δ are 2.8 m or less. The discovered variation of δ is, hence, much larger than its uncertainty and must reflect a geophysical signal.

3.2. Improved ice-thickness and height information

In section 4.1 we argue that the obtained model for δ can in fact be considered to represent the air layer thickness. In anticipation of this outcome, we use the information on firn density inherent in δ to determine the firn correction ΔZ for the ice-thickness measurements (see section 2.1). It can be given as

(Reference Jenkins and DoakeJenkins and Doake, 1991), with the pure ice refractive index n _i = 1.78. Strictly speaking, the values for δ computed with the uncorrected ice thicknesses must also be corrected. With δ _m denoting the values computed with the uncorrected ice thicknesses, we have

where the correction Δδ is given through Equation (4) by

From the last three equations we find

That is, in the northwestern part of Nivlisen, ΔZ is about 7 m. Corrected values for Z and δ are used in the further analyses as well as in Figures 8 and 9.

Now, by the parameter δ, we know the spatially varying relationship between ice thickness and ice surface height and can use this knowledge to transform heights into ice thicknesses and vice versa, thus improving our information on both.

Thus, an ice-thickness model (Fig. 10) was generated merging direct observations and height data at floating positions converted to ice thicknesses (Equation (6)) using the established model of δ. The model domain was extended to the grounded ice east of Nivlisen, where a satisfyingly dense coverage with original thickness measurements is available. The resolution is again about 3.7 km.

Fig. 10. Ice-thickness model (in metres) derived from direct ice-thickness observations and from height observations converted to ice thickness through the hydrostatic balance relation. Contour interval is 25 m.

For an error estimation of the obtained ice thicknesses, a simple error propagation from all inputs of Equation (6) would mean an overestimation because, by the genesis of the δ model, systematic errors of δ are correlated with errors in ρ _w, ρ _i and the absolute level of the freeboard correction (N + ζ _SST + ζ _Tides) such that they largely cancel out in Equation (6). Therefore, only the errors in the spatial variations of h and δ (see Tables 1 and 3) were propagated. The resulting estimated error in ice thicknesses derived from heights is 23 m in general and 27 m near the model margin and the ice rumple at about –70.4º S, 10.7° E. The latter value is also reasonable for the eastern ice-shelf margin where thicknesses were interpolated over larger distances. For a model of the solid ice thickness Z _i = Z–δ the accuracy is practically the same as for Z.

Analogously, an ellipsoidal height model (Fig. 11) was generated from both direct height measurements and ice-thickness data converted to heights using Equation (5). In this way the model described in section 2.2.3 was extended, in particular at the eastern ice-shelf margin. At gridcells without original thickness observations, we used the ice-thickness model.

Fig. 11. Ellipsoidal height model (in metres) derived from direct height observations and ice-thickness observations converted to heights. Contour interval is 2.5 m.

Concerning the errors of heights converted from thicknesses, an analogous argumentation applies as for thicknesses obtained from heights. Propagating only the errors of Zand the errors in the spatial variations of δ, we estimate the error of the obtained heights to be 3.8 m

3.3. Mass flux and mass balance

Figure 12 shows the horizontal ice mass flux on Nivlisen computed from the models of solid ice thickness and of horizontal flow velocity. Visual inspection suggests that the ice body loses mass while flowing northwards.

Fig. 12. Horizontal ice mass flux (absolute value in 10⁶ kgm^–1 a^–1) computed from the ice-thickness model and the ice-flow velocity field.

For a quantitative assessment of this loss, the net effect of specific mass balance and local mass balance, b _s + b _b – ρ _i ∂Z _i/∂t, was evaluated using Equation (9). Figure 13a shows the result. Its spatially varying accuracy is indicated by dashed brown contours.

Fig. 13. Net effect of specific mass balance and local mass balance, b _s + b, — piд/дtZ·i. (a) Map computed from the local ice mass-flux divergence. Dashed brown contours show the spatially varying accuracy. (b) Areas A–C of the integrated mass-balance evaluation with areally mean values colour-coded. Areas I–III (dashed brown lines) are referred to in section 4.4.

For this accuracy estimation, we expressed the velocity components u and v of Equation (9) through the two original observables r (velocity in the radar line of sight) and α (flow direction; see section 2.3.3). Then we performed an error propagation from the errors in r, α and Z _i (23 m, 2.6 ma^–1 and 5°, respectively, according to sections 2.3.3 and 3.2) and from the errors in their spatial derivatives. Those errors were roughly estimated to be 2 m km^–1 for ∂Z _i/∂x, ∂Z _i/∂y, 0.2 m a^–1 km^–1 for ∂r/∂x, ∂r/∂y, and 0.7° km^–1 for ∂α/∂x, ∂α/∂y. The largest contributions to the error in b _s + b _b – ρ _i ∂Z/∂t originate from the spatial derivatives of flow direction and of velocity.

The mass-balance magnitudes in Figure 13a are negative in most places, but show a variable pattern and have large uncertainties. To obtain more significant information, we performed a mass-flux evaluation (Equation (10)) over areas A-C shown in Figure 13b. Table 4 lists the results. Accounting for the known surface mass balance (section 2.4), we find that, on average over the three areas, the combined effect of basal mass balance and temporal change is –654 ± 170 kg m^–2 a^–1.

A rigorous error propagation for the integration in Equation (10) would require spatial correlation information on the error of the integrand |ρ _i Z _i u|sin β. Our error assessment for each area assumes uncorrelated errors of the fluxes through its four boundary sections: upstream, downstream and the two lateral flowline boundaries where zero cross-flux is expected but subject to flow direction errors (‘lateral in- and out-flux’ in Table 4). Along each section, we compute the cross-flux error as the integral of the one-sigma error thus allowing for full error correlation within the section. equals for flowline sections and for the perpendicular flux gates.) Area C covers part of the flow across the grounding line. Surface velocities of the grounded ice are known with 5ma^–1 accuracy from the interferometric analysis of SAR observations from the ERS-1/-2 tandem mission (Reference Baessler, Dietrich, Shum, Jekeli and ShumBaessler and others, 2003). Here, the depth dependence introduces an additional uncertainty in the depth-averaged velocity. Since likely values are 80–100% of the surface velocity (Reference PatersonPaterson, 1994, p.252) we applied 90% ± 10%.

4.1. Firn structure

In section 1.2 we explained that the magnitude δ, derived from Equation (4) at free-floating positions, represents the amount of air contained in the ice shelf (the ‘air layer thickness’) if, first, the ice shelf consists of a mixture of pure ice and air only, and second, there are no gross errors in the observations used. There is no indication of a violation of these assumptions. On the contrary, the values of δ and their spatial variations as shown in Figure 9 are in remarkable agreement with external information.

For comparison, we estimate δ from density information via Equation (3). In the bare ice region, the upper 15 m contribute only 0.6 m to S, based on the 879 kg m^–3 mean density cited in section 2.4. For the density below 15 m we apply two different models: a density profile of the Ross Ice Shelf, Antarctica, (Reference GowGow, 1963) starting from 879 kg m^–3, and an exponential densification analogous to Equation (14): with Z ₀ = 15 m, ρ ₀ = 879 kg m^–3 and C = 0.029 as obtained in section 1.2. The resulting total air layer thicknesses are 3.6 and 2.0 m, respectively, and compare well with our model.

The increase of δ to the north corresponds to the reported northward-growing firn layer (section 1.1). Moreover, the isolines of δ appear parallel to the blue-ice area boundary as seen in satellite imagery (e.g. Figs 1 and 2). Our δ values north of the increasing zone are in the 10–20 m range reported by other authors (cf. section 1.2) and they compare with the 18.7 m value derived from grounded ice-core data from the region (section 2.4).

We conclude that δ can be interpreted as air layer thickness to reflect the variations of firn conditions over Nivlisen.

4.2. Mass balance and basal processes

Although the spatial distribution of the mass balance calculated in section 3.3 (Fig. 13a) is uncertain so that its interpretation would be speculative, we have found that on average over a 2400 km² subregion of Nivlisen (34% of its total area) the net effect of basal mass balance and temporal ice mass change is –654 ± 170 kg m^–2 a^–1 or, equivalently, millimetres water equivalent per year. Under the steady-state assumption ∂Z _i/∂t = 0, this would mean average basal melting of 654 kg m^–2 a^–1. Basal melt rates at other Antarctic ice shelves reach several metres per year (Reference RignotRignot, 2002), so our value is within the expected range. An additional temporal change would change the basal melt rate. For example, a thinning by 100 kgm^–2 a^–1 would enhance the melt rate by another 100kgm^–2 a^–1, while a thickening would accordingly reduce it.

A dominance of bottom melting appears likely from our results. Melting is also reported from the two Soviet ice cores on Nivlisen (see Fig. 2). Texture analysis of the basal ice at drilling site 2 showed that no bottom freezing took place and that the ice was of meteoric origin (Reference Korotkevich, Savatyugin and MorevKorotkevich and others, 1978). For site 3 also, texture and isotope analyses indicated lasting thawing processes (Reference Gordienko and SavatyuginGordienko and Savatyugin, 1980; Borman and Fritzsche, 1995, ch.6). On the other hand, basal processes may vary considerably over small scales. It has been estimated (Borman and Fritzsche, 1995, ch. 4) that 13.4 × 10⁶m³a^–1 of summer meltwater runs off into the lakes in Schirmacheroasen which are linked to the ocean water beneath the ice shelf. A similar meltwater entry may occur at the tidal fractures along the grounding zone. This fresh water, being lighter than sea water, is likely to refreeze quickly on the ice-shelf bottom. Bottom melting in other zones may then overcompensate this local freezing and lead to the negative net effect.

Independent observations also indicate a possible ice thinning with time. The bare ice area on its grounded part next to the ice shelf showed a negative mass imbalance of –100 to –200 kg m^–2 a^–1 in the years 1991–98 (Reference Korth, Perlt, Dietrich and DietrichKorth and others, 2000). In Schirmacheroasen, a decrease in snow-patches on a decadal scale has been observed (Reference Bormann and FritzscheBormann and Fritzsche, 1995, ch.6). Extrapolation of these signals to the ice shelf, with its peculiar regimes of ice dynamics, meteorology and ice–ocean interaction, is of course not justified. On the other hand, ice shelves may respond even more rapidly to mass imbalances than grounded ice (Reference VaughanVan der Veen, 1986). If temporal changes in the ice-shelf thickness occurred, their implications for the overall evolution of the ice shelf would be an interesting question. For example, Reference Shepherd, Wingham, Payne and SkvarcaShepherd and others (2003) observed a progressive thinning of Larsen Ice Shelf, Antarctica, prior to its partial collapse. A comparison of the present ice-thickness data to historical observations is tempting but very questionable (see section 4.3).

In conclusion, no quantification can be given from the present data on how bottom melting and temporal ice-mass changes contribute to their negative net effect. A dominance of bottom freezing, however, seems unlikely, at least for central and eastern Nivlisen.

4.3. Discrepancies with historical ice-thickness observations

In 1975/76, Soviet Antarctic Expedition groups performed three ice-core drillings (see Fig. 2) to the bottom of the ice (Reference Korotkevich, Savatyugin and MorevKorotkevich and others, 1978). The ice thickness reported at core 1 (374 m) agrees well with our nearby radar measurements, but the reported thicknesses at cores 2 (357 m) and 3 (447 m) are 65 and 61 m larger, respectively, than in our ice-thickness model. This discrepancy is striking. On the one hand, drilling ought to be a direct and accurate thickness observation, but on the other, the differences far exceed the uncertainties estimated for our data.

Geophysical variations alone are an unlikely explanation. Temporal variations of >60m in 20 years would be very large and are not supported by comparison of our data to radar observations during the ice drilling (Reference Kozlovskii and FedorovKozlovskii and Fedorov, 1983). Spatial variations at smaller scales than our model resolution (as assessed from variations along our radar tracks) rarely exceed 20 m in the vicinity of core 3 and are much smaller near core 2.

While we have tried to thoroughly assess the uncertainties in our data, it is difficult to assess the reliability of the historical ice-drilling observations. Tilted boreholes may have led to core lengths exceeding the ice thickness (e.g. a 65 m excess length at core 2 could result from a 35° tilt). Some discrepancies within the reports on the 1975/76 ice drilling and radar activities remain unexplained. Drilling-site positions reported by Reference Korotkevich, Savatyugin and MorevKorotkevich and others (1978) and by Reference Eskin and BoyarskiiEskin and Boyarskii (1985) differ by about 6 and 4 km for cores 1 and 3, respectively. These two publications agree on the position of core 2, but Reference Kozlovskii and FedorovKozlovskii and Fedorov (1983), in their ice-thickness map derived from radar observations, mark it about 14 km away. Otherwise they would have revealed a difference in the order of –100m between their mapped thickness and the drilled thickness.

Accepting, nonetheless, the reported ice thicknesses at drilling sites 2 and 3 leads us to two questions which we cannot plausibly answer: how are those large ice thicknesses compatible with our observed freeboard heights in view of the hydrostatic balance? and what is the horizon observed by various radar observations? A hypothesis on a marine ice bottom layer does not answer either question. First, the meteoric origin of the bottom ice was reported from ice-core analyses (see section 4.2). Second, with a marine ice layer of, say, 65 m (corresponding to the difference between ice-core and radar observations), the resulting higher total ice thickness would be incompatible with the observed freeboard height because the marine ice reduces δ only slightly (cf. Equation (8)) and, accordingly, only slightly changes the hydrostatic thickness-to-height relation (Equation (5)). Denser material transported by the ice (e.g. sediments) could resolve the problem of the hydrostatic balance but would not explain the internal horizon seen by radar observations.

We conclude that if no gross error is responsible for the discrepancy described, it is most likely a peculiar coincidence of several effects such as borehole tilts, position errors, errors in our ice-thickness model, small-scale variations not included in this model, and temporal changes. In any case, we are sceptical about including the reported borehole thicknesses in a geophysical interpretation.

4.4. Snow accumulation and compaction

The evolution of the air layer thickness δ along the ice flow can be used to constrain snow accumulation and firn compaction parameters. Air contained in the ice shelf is added in accumulating snow. For example, with the accumulation rate b _s = 300 kgm^–2 a^–1 (section 2.4) and a typical fresh snow density ρ _s = 300 kg m^–3, the air layer thickness added by snow accumulation can be derived from Equation (3) to be b _s (ρ ^—1 – ρ _i ^–1) = 0.67 ma^–1. On the other hand, contained air escapes from the ice shelf when snow and firn are compacted, either by the continuous transformation of snow to ice or by melting with subsequent refreezing. Much of the compaction occurs shortly after accumulation. For example, Reference Korth and DietrichKorth and Dietrich (1996) use a 600 kgm^–3 mean density of the previous year’s accumulation layer. With b _s and ρ _s as above, this implies that at the end of a year the accumulated snow has already been compacted by 0.5 m and retains no more than 0.17 m of air.

In analogy to the ice mass-flux and -balance methodology (section 1.3), the net effect of contained air gain and loss is described by the divergence of the horizontal flux of contained air, div(δ u). We computed the integral effect of this divergence for three areas by balancing the in- and outflow of contained air through their boundaries. We chose areas I–III marked in Figure 13b since they represent different glacio-logical regimes (cf. section 4.1). The obtained average divergence of contained air was 0.01, 0.04 and 0.02 ma^–1, respectively. That is, the effect of accumulation slightly predominates over the effect of compaction, and this dominance is largest in the central area II. The compaction effect in area II is in the order of 0.04–0.67 = –0.63 ma^–1, or, excluding the first-year compaction, 0.04–0.17 = –0.13ma^–1.

We speculate that the obtained qualitative pattern, irrespective of its uncertainties, reveals a geophysical signal induced by spatial variations of accumulation and compaction. A higher accumulation rate in area II could result from drift snow transported by katabatic winds from further south across the ablation area. A more rapid compaction in areas I and III could result from enhanced melting and refreezing in the vicinity of the ablation area (area I) and the coast (area III). In addition, area III has a thicker firn body with a larger air content, so there is more firn to be compacted.

4.5. Ice-dynamical features

The maps of flow velocity, ice thickness and ice mass flux (Figs 7, 10 and 12) show clearly how the ice flow across Nivlisen is channelled between areas of local grounding or, conversely, how these grounding areas form obstacles to the ice flow, leading to lower ice thicknesses in their lee. This underlines the complexity of the ice-dynamical regime of Nivlisen which may be sensitive to changing grounding conditions induced by changes in either ice thickness or sea level.