Lake Vostok is the largest subglacial lake in Antarctica. We show that as a result of its size tidal and atmospheric pressure forcings are large enough to induce vertical displacements of the ice surface above the lake. These effects have been modelled assuming equilibrium tides and an inverse barometer response based on mass conservation. Differences in the tidal potential on the area of the lake result in height changes with amplitudes of a few millimetres for the largest diurnal and semidiurnal tides with maximum values at the southern end of the lake. Air pressure differences above the lake cause a differential inverse barometer effect (IBE) with resulting height changes of up to 40 mm on timescales from days to weeks. Both effects could be verified by differential Global Positioning System (GPS) observations in the southern part of the lake during the 2001/2002 and 2002/2003 summer seasons. The measured amplitudes and phases of the main constituents correspond to the respective values of the differential equilibrium tides. ERS-1& 2 tandem interferograms of 1996 were used to analyse the spatial pattern of the surface displacement. For the southern part of the lake, these measurements reveal a bulge with a wide flexure zone of varying amplitude but a similar geometry. A quasi-simultaneous pair of interferograms from the northern and the southern part of the lake respectively, demonstrates surface height changes of opposite sign in both areas. Therefore, it can be concluded that the associated water redistribution as a result of tides and changing atmospheric pressure forms an additional component of the overall water circulation in the lake.
Subglacial lakes belong to the most interesting glaciological phenomena in Antarctica. Discovered just some decades ago, they could have existed since the inception of the Antarctic ice sheet. Thus, these unique environments could habour life that has been isolated from the direct exchange with the atmosphere for millions of years.
Lake Vostok is the largest and best explored of at least 100 known lakes (Priscu et al. 2003). It extends in length for approximately 280 km with a width of approximately 50 km. The lake has a depth of up to 1200 m and is covered by the up to 4300-m-thick ice sheet (Masolov et al. 2001). By a fortunate coincidence, the Russian Antarctic research station Vostok is situated above the lake (see Fig. 1) thus forming a convenient site for undertaking a variety of investigations of the geographic setting of the lake. The overlying ice sheet topography, ice thickness and water depth were determined by radar and laser altimetry, radio echo sounding, seismic and gravimetric surveys (Kapitsa et al. 1996; Siegert & Ridley 1998; Tabacco et al. 2002; Studinger et al. 2003, 2004). In contrast to these geometric properties, dynamic processes within the lake itself cannot be directly identified. The ice core of Vostok, originally designated to serve as a climate archive, retrieved 80 m of refrozen lake water, which is evidence of the interchange between the lake and the overlying ice sheet (Jouzel et al. 1999). Apart from this, numerical models have been the only means to study the circulation in the lake and the interaction of the water with the overlying ice (Wüest & Carmack 2000; Siegert et al. 2001; Williams 2001; Mayer et al. 2003). As a result of differences in ice thickness above the lake, the pressure-dependent melting point of the ice varies along the lake ceiling causing melting at some locations and freezing at others. The fresh water entering the lake by melting has a different density eventually resulting in a circulation. Depending on the assumed salinity and the resulting water density in the models, the flow regime changes considerably. Nevertheless, all circulation models result in a horizontal water exchange with timescales of years that do not involve a height change of the surface.
However, there are processes that manifest themselves through the approximately 4000-m-thick ice sheet. The response of the lake to external forces, such as tides and air pressure changes, can be traced at the surface because the floating ice takes part in any vertical motion of the water body similar to a floating ice shelf.
The ice shelves along the Antarctic coast are known to be affected by ocean tides. Height changes of the ice shelves as a result of tides have been observed by gravimetry (Doake 1992), SAR interferometry (e.g. Dietrich et al. 1998; Metzig et al. 2000; Rignot et al. 2000), Global Positioning System (GPS; King et al. 2000), as well as satellite altimetry (Fricker & Padman 2002). There are also ocean tide models dedicated to the ice shelves (e.g. Smithson et al. 1996; Robertson et al. 1998; Padman et al. 2002). An additional effect causing surface height changes on the ocean as well as on ice shelves is the inverse barometer effect (IBE), which has also been observed at Antarctic ice shelves (e.g. Dietrich et al. 1995; Padman et al. 2003).
While the main part of the ice shelf moves freely up and down following ocean tides and IBE, a transition between the grounded and the floating ice occurs in the flexure zone, typically up to 10 km wide (Vaughan 1995). The exact width of this deformation zone depends on geometric parameters such as ice thickness and material properties namely the elastic modulus.
A similar situation can be expected for Lake Vostok and was first suggested by Dietrich et al. (2001). The lake water responds to external forces while the overlying ice sheet attenuates the effect along the margins.
The change of the equipotential surface as a response to the tidal force under idealized conditions is known as equilibrium tide. For ocean tides in the diurnal and semidiurnal tidal frequency band, the equilibrium theory is not appropriate because of the irregular shape of the oceans (coastline, bathymetry) and because the resonance frequencies of the global ocean basins are close to these frequencies (e.g. Pugh 1987). However, as a result of the smaller extent and therefore the shorter natural periods, lake tides can in most cases be assumed to show approximately an equilibrium response to the tidal forcing (Melchior 1983).
Because of the elastic properties of the earth, the tidal deformation of the solid earth as well as the resulting deformation potential have to be considered. The change Δh of an equipotential surface with respect to the deformed earth can be expressed by (e.g. Lambeck 1988)
For Lake Vostok, the resulting pattern of amplitudes and phases is similar for all constituents as it can be assumed from the equilibrium theory. Amplitudes and phases for the largest diurnal tide K1 are shown in Fig. 3. According to the model computation, the tides in the northern part of the lake are in phase with the tidal forcing, whereas at the southern tip the phase is opposite. A north–south mass exchange in the lake is therefore the main effect of tidal forcing. In the centre of the lake, each tidal constituent exhibits an amphidrome. Additional to the dominating north–south effect, the equilibrium tides cause a weak counter-clockwise forcing of the water circulation in the lake.
Looking at a specific location of the lake, a time-series of equilibrium tides can be generated, which allows a tidal analysis. The results for Vostok station (Table 1) reveal a maximum range of the tidal signal of approximately 20 mm with a dominant diurnal amplitude K1 of 4 mm. The tidal spectrum is shown in Fig. 4. The total rms of the tidal height variation is 4.1 mm with 3.8 mm in the diurnal and 1.3 mm in the semidiurnal band, respectively. This signal is the relative height change of an equilibrium lake surface with respect to the shore, in this case the bedrock and accordingly the grounded ice above the bedrock.
Atmospheric pressure/inverse barometer effect
Over the oceans, changes in air pressure cause an adjustment of the water level in such a way that increasing pressure results in a decreasing water level. Local pressure variations ΔP around the mean pressure over the ocean induce changes of the water level Δh relative to the mean level according to (e.g. Pugh 1987)King & Turner 1997). Assuming a linear pressure ramp justified by the restricted size of the lake and its shape with a dominating north–south orientation, the mean air pressure over the lake can be approximated by the difference of the pressure values in the north and the south of the lake, respectively. For a location at the southern end of the lake, this leads to Wüest & Carmack 2000). A positive pressure difference between north and south thus results in a positive height change in the southern part and a negative one in the northern part of the lake.
As the density of meteorological stations in the region is not sufficient to compute the air pressure gradient directly from observations, the National Centers for Environmental Prediction (NCEP) Reanalysis data (Kalnay et al. 1996) were used to derive the air pressure differences above the lake. According to eq. (3), the air pressure difference can be converted into the height change of the lake water surface under the assumption of an ideal IBE. The pressure variations lead to height changes in a maximum range of ±20 mm. The amplitude spectrum of this signal (Fig. 5) illustrates the random nature of a red noise structure superimposed with a weak diurnal periodicity. The total rms amounts to 7.2 mm. Most of the energy of the process is concentrated at low frequencies with 60 per cent of the variance occurring at periods longer than one week. However, for periods shorter than a week but larger than a day the height change shows an rms of 4.4 mm, whereas at higher frequencies only 2 per cent of the total variance result in an rms of 1.0 mm. Therefore, tidal and air pressure forcing are quite well confined to different frequency bands allowing us to distinguish between both effects by separating subdaily from day-to-day variations.
In contrast to the exactly known tidal forcing function, this air pressure forcing is based on a model with only sparse data in the area of investigation. A comparison between the automatic weather station (AWS) at Vostok and the NCEP air pressure for the same location reveals a correlation of 0.97 and an rms of 2.1 hPa, thus indicating the high reliability of the NCEP model in case the weather station Vostok is operational.
However, the IBE is computed using the difference in air pressure above the lake area. Although the size of the lake is close to the resolution of this global model, the dominant long wavelength character of the pressure distribution over the continent should be captured. Furthermore, only temporal changes of air pressure differences have to be considered in order to calculate the differential IBE. This differentiation of the NCEP pressure field in space and time should lead to an amplification of the noise. Therefore, the application of this global pressure field to our regional investigation is certainly challenging.
The atmospheric loading, which also causes vertical deformation of the solid earth, acts on synoptic scales (1000–2000 km wavelength) in the interior of a continent (van Dam et al. 2002). Therefore, it is not relevant for this investigation as it influences the ice around and on top of the lake similarly and the differential effect can be neglected.
Influence of the Ice Sheet
Similar to ice shelves, the ice cover above Lake Vostok rests on top of the water and responds to height changes of the underlying water. Along the margin of the lake, this height change will be damped to form a continuous transition to the grounded ice, which does not experience any vertical forcing.
In the case of floating glacier tongues, the vertical deformation along a profile across the flexure zone has been described for instance using an elastic beam model (e.g. Holdsworth 1969; Vaughan 1995; Rignot 1998). To simplify the situation of Lake Vostok to a 1-D model does not seem to be appropriate as a result of the more complex geometry. Rabus & Lang (2002) considered several special cases, derived from the bending of an elastic plate. The special case of 2-D elastic bending for a grounding line of constant curvature and constant ice thickness is
To investigate the response of Lake Vostok to tidal and air pressure forcing, two different types of observations were selected. The precise positioning of observation sites in the area using GPS should yield time-series of differential height changes for these specific locations. Synthetic Aperture Radar Interferometry (InSAR) should reveal the areal extent of any vertical motion especially allowing us to investigate the transition between the floating ice above the lake and the grounded ice. Because GPS is discrete in space and continuous in time whereas InSAR is continuous in space and discrete in time, these two techniques complement each other perfectly.
GPS observations in the vicinity of Vostok station could be performed in the Antarctic summer seasons 2001/2002 and 2002/2003. The locations of the GPS sites (Fig. 2) were selected to enable the observation of the maximum possible displacement: site CNTR was placed according to the synthetic aperture radar (SAR) interferograms (see Section 4.2) to be least influenced by the flexure effects mentioned above; sites EAST and WEST are situated on grounded ice outside the lake area. The observation schedule (Table 2) ensured simultaneous data recording on the lake surface and on grounded ice. This concept allows us to directly monitor the height changes of the ice surface above the lake with respect to the grounded surrounding.
The GPS data were processed using the Bernese GPS software version 5.0 (Hugentobler et al. 2001). The International Terrestrial Reference Frame 2000 (ITRF2000), fixed to the Antarctic Plate at the respective mean observation epoch, served as reference frame for this analysis. In a first step, Antarctic stations of the International GPS Service (IGS) all located on bedrock near the coast were used to determine exact positions for the sites within the working area. These bedrock stations are McMurdo, Casey, Davis, Mawson and Sanae IV. The processing algorithm includes solid earth tide and ocean tidal loading corrections (McCarthy & Petit 2003). This approach provided a priori coordinates in the ITRF2000 for the sites around Vostok with 1-cm accuracy. In the second step, time-series of the baseline components for EAST–CNTR (32 km) and for EAST–WEST (20 km) were computed to investigate the surface height changes. Therefore, the influence of any imperfect modelling as e.g. tropospheric refraction, ocean and atmospheric loading is reduced to the differential effect for these baseline lengths.
For the interferometric analysis, SAR data of the European Remote Sensing Satellite (ERS)-1& 2 tandem mission recorded at Syowa and McMurdo in 1996 were utilized. Here, we show interferograms of the southern part of Lake Vostok (Fig. 2) and the complementary one for the northern part (see Table 3).
In general, an interferogram includes phase contributions resulting from topography, displacement and atmosphere. To reduce the topographic component, a digital elevation model (DEM) and satellite orbit information are essential. We used the altimetric DEM of Rémy et al. (1999) and precise orbit products of Delft University (Scharroo et al. 1998) to account for topography.
The effect of atmospheric water vapour on signal traveltime in the interior of Antarctica is small because the troposphere is very cold, dry and homogeneous on scales covered by an interferogram. Therefore, the influence of the wet troposphere and in particular its variability in time and space on the InSAR observations can be neglected. In contrast, ionospheric refraction can show a high variability within a SAR scene and has to be examined in every case individually. After consideration of the effects mentioned above, the remaining signal can be interpreted as displacement.
As the sensor is only sensitive to displacements in the satellite line of sight (23° off vertical), horizontal and vertical displacement cannot be separated without additional information. In the case of ERS tandem pairs with a temporal baseline of only 1 d, the contribution of the horizontal displacement can be neglected. The ice flow velocity in the Vostok region was determined by GPS to be approximately 2 m a−1. The horizontal velocity difference between the sites EAST and CNTR amounts to 1 mm per day, causing only a minor and long-wavelength gradient in the interferogram. The normally necessary procedure of phase unwrapping is not necessary here because all phase changes are smaller than 2π, equivalent to one fringe. Finally, the remaining fringe pattern over the lake was converted into a height signal using the radar wavelength and the look angle with a phase change of 2π corresponding to a vertical displacement of approximately 30 mm.
Results and Interpretation
Time-series of height changes
As one result of the GPS analysis, high-resolution time-series of height differences for the baseline between the stations EAST and CNTR with a duration of 24 d in season 2001/2002 and of 42 d in season 2002/2003 were determined (Figs 6a and b). For this purpose, EAST was fixed to its nominal position according to its ice flow trajectory and the height changes of CNTR were estimated for time intervals of 2 hr. In order to enable a harmonic tidal analysis, the obtained time-series for the height of CNTR was high-pass filtered applying a filter conventionally used in earth tide analysis (Wenzel 1996). The duration of the time-series does not allow a separation of close frequencies as P1−K1 and S2−K2. Therefore, phases and amplitudes of the smaller waves were calculated relative to the larger ones. A common analysis of the two years for EAST–CNTR does not enhance the frequency selectivity as the observations were carried out during the same season adding no information for the separation of PSK1 and SK2. The analysis of these time-series reveals significant tidal signals (Table 4). Considering the obtained standard deviations, the results are not far from the equilibrium tides as computed in Section 2.1, except for SK2. After the subtraction of the derived tidal signals from the original time-series, their rms values reduce from 5.7 to 4.1 mm for the first year and from 5.7 to 4.5 mm for the second year.
In the second season 2002/2003, the baseline between EAST and WEST was additionally observed and treated in exactly the same way as it was done for baseline EAST–CNTR. Because both sites are located on grounded ice, there should be neither effects of tides nor IBE. The resulting time-series (Fig. 6c) shows indeed only minor signals with diurnal and semidiurnal frequencies. The amplitudes are 0.8 and 2.1 mm for K1 and S2, respectively. These periodicities have also been reported in other GPS-derived time-series as Earth rotation parameters and are, at least partly, explained as artefacts originating from the repeated GPS orbit configuration (Rothacher et al. 2001). An additional contribution to solar frequencies may come from meteorological effects, which would also manifest in deviations for K1 and S2. Conversely, the lack of any significant signals with lunar frequencies for the grounded baseline indicates a non-tidal origin of the values obtained for K1 and S2. The observed artefact in S2 could also explain the significant departure of the SK2 results for CNTR (Table 4) from the equilibrium tide. The original rms of 4.0 mm for the baseline EAST–WEST without any correction corresponds to the tidal-corrected values of the other baseline and provides thus a measure of the precision of the height difference.
The time-series of daily GPS solutions shows the height variations between the grounded and the floating ice (EAST–CNTR), which are mainly caused by the IBE (Fig. 7). The time-series of both seasons exhibit a significant correlation between air pressure difference and height change, whereas the control baseline EAST–WEST on grounded ice shows no correlation.
However, while the regression coefficient m of 4.7 mm hPa−1 for the time-series in 2001/2002 corresponds to the expected value (eq. 3) indicating full adjustment of the surface to the forcing, the coefficient for the following season is considerably smaller. As the AWS at Vostok was operational in both seasons, there is no reason to assume a change of the accuracy of the air pressure differences from one year to the other. Effects from the GPS data processing can be excluded because both time-series were generated identically. It is interesting to note that the amplitudes of the diurnal tides are also smaller in the second season compared with the first one. Although the reason for this difference remains unknown and should be investigated by further observations, the existence of a response of the lake to tidal and air pressure forcings is clearly shown by the analysed GPS time-series.
Spatial pattern of the displacements
Especially the variation of the vertical deformation as a function of position in general and as a function of distance from the grounded ice in particular can be investigated using SAR interferometry. Four examples of interferograms are displayed in Fig. 8. Each data set shows a distinct signal, which is restricted to the lake area. Whereas scene 228 reveals an uplift between the successive days, the other three indicate a lowering of different amplitudes within the 24 hr between the respective acquisitions. Evidence for a direct exchange of water from north to south of the lake comes from Fig. 9. The two interferograms of track 272 and 273 shown there were acquired 100 min apart (see Table 3). Both forcings, tides as well as air pressure differences, can be assumed to have changed only marginally within this time span. While there is an uplift of the surface in the northern interferogram, the southern one shows a lowered surface between its acquisitions. A direct comparison of the involved water volume in the north and in the south cannot be performed because the scenes do not cover the whole surface of the lake.
In scene 258 (Fig. 8b), an ionospheric streak is superimposed on the deformation producing an apparent uplift signal of approximately 5 mm near the GPS site EAST, also affecting the signal on the lake area. After correction of this disturbance by assuming a constant magnitude along the streak, the maximum displacement changed to 15 mm. Additionally, the maximum of the deformation shifts to approximately the same position as in the other scenes. Near this location, the GPS site CNTR was established. Therefore, the displacement signals in the interferograms (the difference between the maximum displacement on the lake and the level of the surrounding ice) can be directly compared to the same forcings as the GPS baseline EAST–CNTR. Table 5 summarizes these effects for all analysed scenes. The remaining signal on the grounded ice can be used as a rough measure of accuracy for the vertical displacements, which leads to an estimate of approximately 1 to 2 mm.
The main contribution to the signal measured in the interferograms comes from the IBE as a result of changes in the pressure gradient above the lake in the time between the first and second data acquisition. Fig. 10(a) illustrates the evolution of the north–south pressure difference around the acquisitions of the interferograms in Figs 8 and 9. For the assessment of consistency of modelled and measured displacements, it has to be taken into account that Vostok station was abandoned in 1996 during the acquisition of the SAR scenes. Therefore, the reliability of the NCEP air pressure has to be considered to be lower than during the GPS campaigns. For most scenes the consistency is reasonable, but for scene 272 the model predicts a raised surface while a sinking one is observed in the interferogram. However, only a minor shift in the timing of the temporary low in air pressure difference (Fig. 10a) would change the sign in the prediction.
Because of the time separation of 24 hr, the influence of the tides is highly reduced in the interferograms. In fact, the S2 constituent is completely cancelled because it is sampled at exactly the same phase in both acquisitions combined to the respective interferograms. Moreover, the frequency of the largest diurnal constituent K1 (1.0027 cycles per day) is close to this time interval. Thus, also K1 exhibits little difference in elongation within an interferogram.
Fig. 10(b) shows the regression between the modelled displacement of IBE and tides and the observation in the interferograms. The linear orthogonal regression coefficient of 1.2 signifies that the observations tend to be larger than the model. However, a population of 12 samples does not admit a statistically valid result. Compared to the GPS results, the observed surface displacements are approximately the same order of magnitude.
The interferograms also provide an insight into the spatial structure of the deformation. Except for scene 258 (Fig. 8b), which is partly contaminated by the ionospheric signal, all scenes show the same pattern of a displacement damped towards the shore of the lake. It is hard to decide if the maximum displacement in the interferograms corresponds to free floating. There is no constant level of vertical deformation in the centre, which would provide clear evidence for an adjusted surface. Therefore, profiles across the lake in the interferograms of scenes 229 and 458 (Fig. 11) were analysed independently to describe the damping quantitatively. By adopting the method of Rabus & Lang (2002), the flexure of the ice above the lake can be modelled considering a constant curvature of the grounding line. The boundary value problem (eq. 4) was solved numerically for a range of values of amplitude A and damping factor β. From a multitude of solutions, the one with minimum rms difference to the data was selected. A similar approach has been applied earlier by Vaughan (1995). This solution is not very sensitive to changes in the parameters, so the errors in amplitude and damping factor are large and difficult to quantify. The adjusted amplitudes for both profiles correspond to the maximum displacement recorded in the interferograms within the expected errors suggesting that the displaced surface is in equilibrium with the forces. The elastic modulus can be estimated from eq. (5) to be E= 2.1 GPa using a mean damping factor of 0.122 km−1, a mean ice thickness of 3900 m, a water density of ρ= 1016 kg m−3 and ν= 0.3. A relative error of 40 per cent has to be attributed to this estimate as a result of large uncertainties in the various terms. Hence, this value for the elastic modulus lies within the range reported in the literature, e.g. 0.88 GPa by Vaughan (1995) or 3.0 GPa by Rignot (1996).
The ice surface above Lake Vostok experiences vertical motions that have been observed by differential GPS and SAR interferometry. These height variations can be explained as a response of the lake to tidal and atmospheric pressure forcing. The GPS time-series reveal amplitudes of up to 4 mm for the largest diurnal and semidiurnal tides. Considering the standard deviations, the measured amplitudes and phases agree in general with the ones expected from equilibrium theory. Only for the SK2 constituent larger deviations occur, which could be an artefact from the repeated GPS orbit configuration.
There is a distinct correlation of the daily height solutions with pressure differences above the lake demonstrating the differential IBE. However, the rate of vertical displacement per unit of pressure difference and some tidal amplitudes vary between the data sets of the two field seasons. Thus, the existing field observations give a first insight into the response of the lake to external forces, but more field data are essential to quantify this response in more detail.
The areal extent of the deformation was analysed using ERS-1& 2 tandem interferograms from 1996. They clearly indicate a coherent movement, which is damped towards the lake edge as expected in analogy with ice shelves at the Antarctic coast. The eastern and western flexure zones adjoin without admitting a constant deformation level. Nevertheless, the numerical analysis of deformation profiles suggests free floating in the centre.
More important than the fact of a vertical motion of the ice surface itself in the order of a centimetre is the associated water mass redistribution within the lake. Acting as an external pump, tidal and atmospheric pressure forcing induce a water volume exchange in the lake between the northern and the southern part in the order of magnitude of 107 m3 with diurnal and semidiurnal periods of the tides as well as in a broader range varying from days to weeks for the IBE. Because tidal currents and IBE are well known for the open ocean, it is also worth considering these forcing terms when developing more sophisticated models of water circulation for Lake Vostok.
The research was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) and the Heiwa-Nakajima Zaidan Fund, Japan. The authors would like to thank Yukio Haruyama from National Space Development Agency of Japan (NASDA), who made efforts to realize ERS tandem reception at Syowa station, and the Japanese Antarctic Research Expedition (JARE-37) Syowa satellite receiving team. We are grateful to ESA for the ERS data recorded in McMurdo and made available under the VECTRA project and AO3-231. Furthermore, we thank the members of the team at Vostok station for their cooperation and assistance during both field seasons.
F. Rémy kindly provided the DEM of the Lake Vostok area. NCEP Reanalysis data were provided by the NOAA-CIRES Climate Diagnostics Center, Boulder, Colorado, USA, from their web site (http://www.cdc.noaa.gov/). The figures were produced with the Generic Mapping Tools (GMT).
Finally, we thank R. Coleman and M. King for their constructive review, which improved the paper considerably.