For over 2-years we have collected GPS data ∼40 km upstream of the grounding line on Rutford Ice Stream, West Antarctica. Here we examine deviations from mean downstream flow. Although the record is incomplete during winter, there is clear modulation of flow at semi-diurnal, diurnal, two-weekly, semi-annual, and annual ocean tidal frequencies. This is the first observation of ice stream flow variations over such a long time-period, and such a wide range of frequencies. The ice stream flows fastest at equinoxes when there are two semi-diurnal tides of equal magnitude, and slowest at solstices when one of the semi-diurnal tides has lower amplitude. The sensitivity of the downstream flow is greatest to the long-period forcing, which suggests that ice stream velocity may be affected by future changes in sea level. If so, this effect would provide a feedback whereby rising sea levels could increase ice stream velocity and hence discharge.
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 The volume of the West Antarctic Ice Sheet is primarily controlled by the dynamics of fast-flowing ice streams and outlet glaciers. A collapse of the ice sheet could result if they accelerated substantially. Some Antarctic ice streams have accelerated [Rignot et al., 2002], but others have decelerated [Joughin et al., 2005]. Remote sensing techniques are attractive for measuring ice stream flow, but their interpretation requires assumptions about temporal variations in flow rate. It is clear we need good long-term measurements of ice stream flow at high sampling rates and an understanding of the processes that modulate their flow if we are to predict future ice sheet mass balance.
 Recent developments in GPS technology and processing techniques [e.g., King, 2004] mean that flow rates can be measured at much higher temporal resolution than previously possible. Such measurements have shown that ice stream velocity is modulated in response to tidal forcing of the floating ice shelves [e.g., Bindschadler et al., 2003; Anandakrishnan et al., 2003; Gudmundsson, 2006]. Most observations have been made on the ice streams that flow into the Ross Ice Shelf, where the tides are dominated by diurnal variation. Stick-slip flow correlated to ocean tides on the ice shelf has been reported for the ice plain of Whillans Ice Stream (formerly ice stream B), with almost all motion occurring during short slip events [Bindschadler et al., 2003]. Bindschadler Ice Stream (formerly ice stream D) exhibits a 50% modulation of its flow speed at the grounding line which is 90° out-of-phase with the tidal motion (slower flow occurs during a rising tide and faster flow during a falling tide), persists up to 80 km upstream from the grounding line, and is progressively lagged upstream [Anandakrishnan et al., 2003]. Diurnal variation in basal seismicity has been recorded by passive seismic measurements on Kamb Ice Stream (formerly ice stream C), as well as diurnal variations in basal water pressures beneath Whillans Ice Stream at 300 km upstream from the grounding line [Harrison et al., 1993; Engelhardt and Kamb, 1997].
 Rutford Ice Stream, West Antarctica (Figure 1) flows into Ronne Ice Shelf which, in contrast to Ross Ice Shelf, has larger amplitude tides dominated by semi-diurnal frequencies. This results in a strong two-weekly beating: at the grounding line spring tides have peak-peak range up to ∼6.6 m and neap tides ∼1.8 m. The ice stream has two-weekly cycles in downstream flow [Gudmundsson, 2006] and basal seismicity (G. Aðalgeirsdóttir et al., Tidal influence on Rutford Ice Stream, West Antarctica: Observations of surface flow and basal processes from closely spaced GPS and passive seismic stations, submitted to Journal of Glaciology, 2007, hereinafter referred to as Aðalgeirsdóttir et al., submitted manuscript, 2007) which correlate with this spring-neap cycle. The ice stream flows by both sliding and sediment deformation, and conditions at the bed are both spatially variable and rapidly changing [Smith, 1997; Smith et al., 2007]. In this paper, we present a two-year record of motion from Rutford Ice Stream collected ∼40 km upstream of the grounding line. This record is the first published for any ice stream to span more than a few weeks in duration, and it allows us to show Rutford Ice Stream's flow rate is tidally modulated at a wide range of frequencies from sub-diurnal to annual. For the first time we are able to identify semi-annual and annual modulation in the rate and nature of the ice stream's downstream flow, and unequivocally identify the primary two-weekly modulation as the MSf tidal component (Table 1), which has a frequency of 14.77 days.
Table 1. The 12 Most Important Tidal Constituents in Terms of Their Amplitude in the Velocity Response, Plus Saa
Bold species are statistically significant in the T-TIDE model for the velocity response; italic constituents are significant for the vertical amplitude of the forcing on the ice shelf (based on a GPS record too short to resolve the longer period constituents). Details of the source of tidal constituents in the forcing are given in the text. Phase is relative to Greenwich. All forcing close to 0.5 days is assumed to be S2.
Phase difference is Response–Forcing.
 GPS measurements were made at 0.1 Hz sampling from 17 December 2004 for 750 days. This sampling strategy was designed to capture both longer term changes in flow and any stick-slip events. The instrument formed part of a strain grid until February 2005 (those 5 weeks of data are also presented by Aðalgeirsdóttir et al., submitted manuscript, 2007) when it was moved and configured for over-wintering. It was re-located to its initial position in January 2006 and 2007. Relocation ensures that each year's data are comparable, minimizing the effect of the instrument advecting downstream into new strain regimes. Each relocation resulted in missing data; further data are missing each winter due to power loss, however, the instrument switched on whenever sufficient power was available and patchy data were collected during both winters. Each spring, the instrument restarted successfully.
 The antenna position was calculated every 5 minutes using a Precise Point Positioning (PPP) technique [e.g., Zumberge et al., 1997; King and Aoki, 2003]. Solid Earth and ocean tide loading displacements were modelled. Expected position errors (95% confidence interval) based on similar positioning at a stationary site located off the ice stream are ∼0.02–0.03 m (horizontal) and ∼0.05 m (vertical). It is known that the PPP processing method can introduce small (<∼0.01 m) signals at solar-related frequencies higher than 2 day−1 due to unmodelled solar effects on the GPS satellites. Systematic errors with similarly small magnitude can also exist in the diurnal and semi-diurnal bands at solar-related frequencies, notably at S1 (period 1.0 day), K1, S2, and K2 (Table 1 lists the periods of the major tidal components).
 The position data were linearly detrended to remove the mean displacement rate, and ice stream velocity was calculated from the 5-minute data using a 5-point differential operator. Detrending was undertaken separately for each year of data because a small but significant change in flow rate between years was apparent. A standard tidal analysis of the velocity was undertaken using T-TIDE [Pawlowicz et al., 2002]. Because of the data gaps when the instrument was moved and during winter months, we generated a power spectrum using the Lomb-Scargle method of Press et al.  for irregularly-spaced data. Finally, to allow visual comparison of the velocity with tidal forcing, a filtered velocity record was calculated based on 2-hourly median positions. A tidal series at the location of the grounding line was constructed using periods shorter than 2-weekly from a GPS sensor on the ice shelf (G. H. Gudmundsson, personal communication, 2005), at 2-weekly period from the CATS02.01 model [Padman et al., 2002], and at longer periods from the model of Takanezawa et al. . The longer-period terms in the models are not well calibrated, but the magnitude should still be representative of the actual forcing. Due to the brevity of the ice shelf GPS record, not all tidal constituents are separable. To overcome this, we inferred [Pugh, 1987] P1 from K1, and K2 from S2 (periods listed in Table 1) using phase and amplitude relationships as defined in CATS02.01.
 The ice stream downstream velocity averaged 377.3 m/year. The detrended position (Figure 2) clearly shows downstream motion is modulated at a range of frequencies that include two-weekly variation (this primarily has a period of 14.77 days, i.e., is the MSf tidal component), and longer term (annual, Sa, and semi-annual, SSa) modulation. Higher frequency variation can also be seen, but is much more clearly revealed in the downstream velocity power spectrum (Figure 3). This power spectrum shows strongest amplitudes at two-weekly MSf and the semi-diurnal K2, M2 and S2 frequencies. The diurnal components P1, K1 and O1 are also clear. Table 1 gives the magnitudes of the strongest components of the downstream velocity response compared to the vertical forcing of the ice shelf. Table 1 also shows that the phase difference between the semi-diurnal and diurnal forcings and responses are broadly consistent; phases of other constituents of the forcing and response are not known with sufficient accuracy to assess this. The ice stream lateral and vertical velocities contain mainly variations at solar-related and high frequencies which are potentially dominated by GPS systematic error: thus we do not attempt to analyse them further.
 Comparison of ice stream velocity (Figure 4) with the predicted tides at the grounding line reveals:
 (1) Tidal modulation of ice stream downstream velocity at 40 km upstream of the grounding line of up to ∼20%.
 (2) This modulation is larger than previously reported because there are seasonal differences such that the velocity is lower and often less variable around the solstices (mid-summer (December 21), when previous measurements have been made, and mid-winter (June 21)), and higher and more variable around the equinoxes (March 20 and September 22). (Compare Figures 4a and 4d with Figures 4b and 4c).
 (3) A seasonal response to individual semi-diurnal tidal cycles with multiple peaks often occurring within a cycle at the solstices and single peaks at the equinoxes.
 (4) Lower and typically less variable neap tide velocities than those at spring tides.
 (5) Highest velocity in a spring-neap cycle occurring when both of the major semi-diurnal constituents (M2 and S2) are in phase, which results in nearly equal magnitude semi-diurnal tides (i.e., small tidal inequality).
 In the tidal record, the solstices are characterised by the largest overall tidal range at spring tide, but also large differences in the magnitude of the two semi-diurnal high tides (i.e., large tidal inequality). The equinoxes are characterised by smaller spring tides, with more nearly equal magnitude semi-diurnal tides.
4. Discussion and Conclusions
 The downstream flow of Rutford Ice Stream 40 km upstream from the grounding line is clearly affected by tidal displacement of the ice shelf at a wide range of forcing frequencies. Because of this, velocities measured at different times in the tidal cycle (i.e., between spring and neap tides or solstice and equinox) will differ, complicating interpretation of changes in ice stream velocity derived from remotely sensed data [e.g., Doake et al., 2002]. Deriving satellite interferometric velocities requires data usually separated by 1, 3 or 6 days [e.g., Frolich and Doake, 1998; Rignot, 1998]. Differential processing to separate velocity and topography effects uses a second interferogram either 3 or 35 days later, and assumes that the velocity is constant between the two image pairs. The velocity at this location between such interferograms will vary by ∼15%, and will be greatest when the first scene is collected at either spring or neap tide. Using our GPS data, in conjunction with existing more spatially extensive but shorter-term data sets [Gudmundsson, 2006], it may be possible to model the Rutford Ice Stream velocity with sufficient accuracy to reduce this error to negligible levels, although that would not be possible for other ice streams until similar datasets are available.
 The magnitude of the downstream velocity response at each frequency does not scale linearly with the amplitude of the vertical tidal forcing (Table 1). The maximum forcing is at semi-diurnal frequencies, but the velocity response is apparently 30–50 times more sensitive to the two-weekly and semi-annual forcing. An important observation is that, at all frequencies which the data set is long enough to resolve well, there appears to be very little energy in the ice stream velocity at other than tidal frequencies (Figure 3). Thus it would appear that other possible drivers of velocity fluctuations (for example, snowfall or atmospheric pressure changes at the ice shelf, which could change ice shelf elevation in addition to the tides [Padman et al., 2003]) do not contribute significantly. We believe the sensitive low frequency response most likely involves propagation within the basal water system and basal sediments. It shows the ice stream is slow to adapt fully to sea level changes, even at annual periods.
 A number of possible models have been presented for tidally controlled ice stream flow at frequencies around one or two per day. (1) On tidewater glaciers, fastest flow occurs at low tide correlating with reduced backstress at the grounding line [Thomas, 2007]. (2) Basal resistive stress is reduced at high tide by partial ungrounding of ice from pining points [Heinert and Riedel, 2007] or reduction of effective stress at sticky spots. These effects would lead to velocities being in-phase with tidal height at the grounding line. (3) Gudmundsson  suggests that due to non-linear till rheology, basal sliding of Rutford Ice Stream is non-linearly related to water depth, so that velocity is increased more by high tides more than it is decreased by low tides, meaning that velocity is non-linearly modulated by tidal range. However, Rutford Ice Stream appears to have different sensitivity to the different forcing frequencies; neither the velocity at our site nor at the grounding line (G. H. Gudmundsson, personal communication, 2007) appears to be simply related to tidal height or tidal range as these models require. (4) Doake et al.  show the horizontal motion of the Brunt Ice Shelf is tidally modulated by 50–100% despite there being no significant ice streams feeding it, probably due to ocean currents causing sub-ice shelf friction. It may be, therefore, that Rutford Ice Stream's tidally modulated flow results partially from being pulled by horizontal motions of the Ronne Ice Shelf at tidal frequencies.
 None of these models can reproduce the differences in the record between solstices and equinoxes, which change the nature of the diurnal velocity response. We suggest that the more complex multi-peak response at the solstices (Figures 4a and 4d) is likely to be the result of two processes acting in tandem. One possibility is that the larger high tide causes partial ungrounding which operates in opposition to other processes. Note that in Figure 4 the response is most likely to be complex (multiple peaks per cycle) during the spring tides returning to a simpler 1 peak per cycle around neaps. Rignot  reports that interferograms reveal “numerous areas of partial grounding on the ice shelf”. Changes in backstress at the grounding line thus seem a plausible explanation of the observed seasonal variation in the nature of the velocity response.
 With rising sea levels, the flow velocity sensitivity to long period forcings could lead to faster ice stream flow, greater discharge and mass loss to the oceans. Additionally, the observation that ice stream flow is sensitive to long period forcing lends support to the notion that changes in tidal amplitudes resulting from long-term changes in sea levels were instrumental in causing paleo ice stream flow instabilities that led to major discharge of ice during Heinrich events in the North Atlantic [Arbic et al., 2004].
 This work was funded by UK NERC and BAS. TMs participation in fieldwork was funded by a Leverhulme Fellowship. MAK was funded by a NERC Postdoctoral Research Fellowship. R. Hindmarsh, T. O'Donovan and H. Pritchard are thanked for maintaining equipment in the field. H. Sykes produced the SAR mosaic in Figure 1.