We use several long (5–19 months) time series of ice shelf motion from GNSS deployments on RIS (Fig. 1). These records were collected during different time intervals between 2014 and 2019 (Table 1). GNSS data from all stations were processed with a precise point positioning (PPP) approach (Zumberge et al., 1997; Geng et al., 2012, 2019).

DRRIS 2015–2016. An array of 13 GNSS stations was deployed on RIS from November 2015 to December 2016 as part of the Dynamic Response of the Ross Ice Shelf to Wave-induced Vibrations (DRRIS) project (Bromirski et al., 2017; Klein et al., 2020). Three stations were deployed along the ice front and nine along a flowline from the central ice front station to about 400 km upstream. One station (RS03) was located 100 km to the west of the along-flowline array and another (RS08) was on grounded ice on the western margin of Roosevelt Island. Only one station (DR10) recorded position data for a full year; however, the intra-annual signals in positions and velocities at the other DRRIS stations on floating ice were highly correlated with DR10 observations (Klein et al., 2020, their Fig. 6).

WISSARD 2014–2016. An array of GNSS stations was deployed as part of the Whillans Ice Stream Subglacial Access Research Drilling (WISSARD; Siegfried et al., 2014a; Tulaczyk et al., 2014) project. We used the record from station GZ19 located about 3 km offshore of the Whillans Ice Stream grounding line, which acquired data between November 2014 and November 2016 (Begeman et al., 2020).

Antarctica PI continuous network 2017–2019. Two GNSS stations (BATG and LORG) acquired data in the northwestern RIS. We obtained the time series for these sites from the GNSS database processed by the Nevada Geodetic Laboratory (NGL; Blewitt et al., 2018). Station BATG was located about 100 km east of Minna Bluff and acquired data from February 2017 to August 2018. Station LORG was located about 100 km east of Ross Island and about 90 km from BATG; the station recorded from November 2018 to November 2019 with a few interruptions, for a total of 289 d. The vertical components of tidal variability at these stations were reported by Ray et al. (2021).

SSH can be estimated using satellite radar altimetry, and monthly SSH estimates are available for the period 2011–2016 for regions north of the Antarctic coastline and ice shelves using measurements from the European Space Agency's CryoSat-2 radar altimeter (Armitage et al., 2018). These SSH estimates cover fully open water (free of ice shelves) and leads in the ice pack but do not extend under the ice shelves. Measuring SSH variations in the ocean cavities under ice shelves is challenging because they are small compared with other contributors to height changes, such as uncertainties in seasonal cycles of basal mass balance (e.g. Stewart et al., 2019; Tinto et al., 2019), snow and firn density changes (e.g. Zwally and Jun, 2002; Arthern and Wingham, 1998), and penetration of radar signals into the surface snow and firn layers (Ridley and Partington, 1988; Davis and Moore, 1993). Therefore, it is not currently possible to accurately estimate SSH variability under ice shelves.

Instead, we investigated the representation of intra-annual variability in SSH from five existing ocean models with thermodynamically active ice shelves (Mathiot et al., 2017; Tinto et al., 2019; Naughten et al., 2018; Dinniman et al., 2020; Richter et al., 2022). We used their SSH output relative to the Armitage et al. (2018) open water data set to determine the most realistic model for analyses and to assess the likely variability in SSH under ice shelves. More information on these models, and assessment of their performance, is provided in the Supplement. From these analyses we determined that the Ross Sea regional model described by Tinto et al. (2019) provides a seasonal cycle that is most consistent with the Armitage et al. (2018) satellite-based results for the Ross Sea continental shelf north of RIS, suggesting that it is also the best model for SSH variability under RIS.

RIS thickness decreases from ∼800 m close to the grounding line to ∼300–400 m at the ice front, over a distance of ∼800 km (see Tinto et al., 2019, their Fig. S2a). This results in mean thickness and surface gradients of about 5×10-4 and 5×10-5, respectively. Since we are interested primarily in along-flow variations in ice velocity, we calculate the along-flow Lie derivatives (Yano, 2020) of the ice shelf surface height (∇u^zs) and ΔSSH (∇u^SSH). Values of ∇u^zs range from 10-5 to 10-2 over most of the ice shelf (Figs. 6a and S4a). Gradients of ΔSSH in Tinto et al. (2019; SSH2002) can reach 10-6 to 10-5 in February (Figs. 6b and S4b). This means that local tilting of the ice shelf by ∇u^SSH can modify the local driving stress of the ice shelf (Eq. 2) typically by 0.1 %–1 % and sometimes up to several percent, with substantial spatial variability (Fig. 6c). ∇u^SSH also varies by month (not shown). For example, in February, about 30 % and 6 % of the ice shelf experiences a fractional change of driving stress exceeding 0.1 % and 1 %, respectively (Fig. 6d). The largest fractional change in driving stress occurs away from the grounding line where the ice surface height gradients are smaller than closer to the grounding line and where the SSH gradients are the larger.

The complex spatial variability of the along-flow derivatives of ΔSSH (Fig. 6b) arises from changes in orientation and magnitude of the sub-ice-shelf circulation relative to ice flow. This circulation is itself complex: see, for example, Supplementary Video 1 in Tinto et al. (2019).

For most months there is a strong along-flow gradient in SSH close to the ice front (Figs. 4b and 6b), which directly impacts driving stress (Eq. 2). These variations in driving stress lead to ice velocity changes, which we present as anomalies with respect to the annual average velocity field. In general, months with a regionally averaged (i.e. over the ice shelf) negative ΔSSH (e.g. January–March period in Figs. 4 and S2) that slows ice flow as the grounding line migrates seaward also experience a relative uplift of the surface close to the ice front, leading to an additional slowdown (Fig. 2a). Conversely, the months experiencing a regionally averaged positive ΔSSH generally show a relative surface drop close to the ice front and an upstream migration of the grounding line, both contributing to an acceleration of the ice shelf (Fig. 2b).

Our modelling indicates that the amplitude of the grounding line migration, ΔL, is the primary control on the amplitude of the seasonal velocity signal. In February, for example, the model ensemble using ΔLB2predicts the smallest amplitude of velocity deviation of the three cases, with ΔUB2∼-1 m a-1 over most of the ice shelf (Fig. 7a). Larger values of ΔL (parameterisations ΔLC and ΔLB2L) allow the grounding line to move farther downstream during summer, leading to deviations ΔU∼-3 m a-1 in the centre of the ice shelf (Fig. 7b and c). The largest differences between the effects of ΔLC and ΔLB2L are generally found close to the grounding line in the deep and narrow fjords such as the floating extension of Byrd Glacier where ΔLC leads to a slowdown ΔUC<5 m a-1 compared with ΔUB2L∼3 m a-1 (Fig. 7b and c). These are regions where true bed slopes are steeper than the average around the RIS perimeter and which are also more sensitive to the initial state as the ensembles show a larger standard deviation in these areas with respect to the rest of the domain (Fig. 8, bottom row).

We regard the ΔLB2 parameterisation, which yields small grounding line migration, as an approximation of ice shelf response to SSH gradients alone.

All ensembles forced with ΔSSH2002 exhibit a maximal seasonal negative flow speed anomaly during summer and maximal positive anomaly during winter (Fig. 8); however, ΔLB2 simulations tend to switch to positive anomalies later than simulations using ΔLC and ΔLB2L. Simulations using ΔLB2 produce maximal amplitudes of speed anomaly at the ice front that progressively decrease farther upstream, while ΔLC and ΔLB2L produce maximal speed anomaly amplitudes in the deep fjords along the base of the Transantarctic Mountains. The amplitudes of speed anomalies of ΔLB2 are about 2–4 times smaller than for ΔLC and ΔLB2L simulations, depending on location.

To validate the results of the three grounding zone parameterisations, we extracted the modelled ice velocity anomalies at the GNSS locations and compared these to velocity variations (Fig. 9) estimated from the time derivative of measured displacement anomalies (Fig. 3).

At DR10, the range of the observed velocity anomaly (ΔU) was about 10 m a-1 with a minimum in February–March and a maximum in July (Fig. 9a). The other DRRIS GNSS stations located in the centre of the ice shelf did not record during austral winter (see Fig. 5a), preventing us from properly identifying the timing of maximum velocity for these stations. The ΔLC and ΔLB2L ensembles both give similar ΔU estimates that are qualitatively similar to observations, with velocity variations of about 50 % to 70 % of the observed amplitude and minima and maxima in summer and winter, respectively. The ΔLB2 grounding zone parameterisation has a much lower amplitude than observations and gives a maximum velocity in October, about 2 months later than the other ensembles and 4 months later than the observations. However, the timing of the summer ΔU minimum is close to the observations and the other grounding zone parameterisations. Expanding our analysis to the entire GNSS array of DRRIS, similar seasonal phasing occurred at each GNSS station located approximately along the central flowline of the ice shelf. ΔU amplitude generally decreases with increasing distance from the ice front (Fig. 10), although with some variability that may result from the proximity of the DRRIS array to the Byrd Glacier flow and its impact on RIS flow.

At GZ19, close to the grounding line of Whillans Ice Stream, there is no seasonal cycle visible in the GNSS observations of displacement anomaly (Fig. 3b). The measured velocity anomaly (Fig. 9b) shows an overall slowdown, consistent with previous observations of slowdowns of Whillans and Mercer ice streams and the adjacent region of RIS over the last few decades (e.g. Joughin et al., 2005; Thomas et al., 2013), and shorter periods of deceleration and acceleration that could be due to the inherent variability in the two ice streams (e.g. Winberry et al., 2009). This trend was not captured by our ice flow models, which do not account for varying forcing other than the annual cycle of SSH. The modelled anomalies at GZ19 are weak, with ΔUB2∼±0.5 m a-1 and ΔUC∼ΔUB2L∼±1 m a-1 over the year.

At station BATG, about 100 km east of Minna Bluff, the velocity time series shows an approximately 6-month periodicity, with a ΔU range of about 2.5 to 3 m a-1 (Fig. 9c). ΔLB2 provides a poor fit to these observations, in both ΔU amplitude and phase, with the amplitude better reproduced by ΔLC and ΔLB2L. However, the pattern of observed velocity anomaly changes between the first and second year of the record. In the first year, the 6-month cycle shows a large velocity drop in July–August (reaching a minimum in September), corresponding to the second minimum of the year. In the second year, the observed velocity reached a maximum in May and remained relatively high until the end of August, fitting the modelled velocities. While the record terminated at the end of August, this marked a particularly long plateau of high velocities (from May to August), suggesting that the record includes a seasonal signal that is added to the 6-month cycle that we tentatively attribute to semiannual changes in tidal range (see Sect. 1).

The time series of ΔU at LORG (Fig. 9d) for the period November 2018 to November 2019 is highly correlated (p=0.95) with the time series at BATG over the second year (from November 2017 to October 2018). The predicted velocity anomalies for ΔLC and ΔLB2L at these two stations agree especially well with the observations over the entire LORG times series and the second year of the BATG time series. More specifically, the model is able to reproduce the month-to-month accelerations and decelerations and the overall longer span of positive anomalies visible in LORG observations.

To examine the relative effect of the variations in driving stress and basal friction through grounding line migration, we consider a key region of RIS, the floating extension of Byrd Glacier near its grounding line. Byrd Glacier is the fastest and the deepest outlet glacier feeding RIS and is the region of RIS where the outputs from the three ensembles (ΔLB2, ΔLC, and ΔLB2L) deviate the most. Observations show that, over a time span of a few years, flow upstream of the grounding line can increase by 10 %, coinciding with the discharge of subglacial lakes lubricating the bed (Stearns et al., 2008; Yuan et al., 2023). At seasonal timescales, variations in Byrd Glacier remain poorly constrained due to the lack of year-round GNSS measurements; however, Greene et al. (2020) used feature tracking in satellite imagery to estimate ice velocities and characterise the magnitude and timing of seasonal ice dynamic variability. For a region close to the grounding line of Byrd Glacier, they estimated seasonal variability of ΔU with a range of roughly 45 m a-1. Our ensemble using the ΔLB2L representation (Fig. 9e) shows a phase that is consistent with Greene et al. (2020); however, our modelled range in ΔU is always less than 10 m a-1. This difference could be explained by the substantial uncertainty due to irregular, seasonally biased sampling of the satellite data (see Fig. 4 of Greene et al., 2020). Our modelling may also underestimate either the ΔSSH or the basal condition changes that the ΔSSH changes trigger. There may also be other processes at play that we do not account for in our modelling; for example, the change in seasonal melt (explored in Klein et al., 2020) which, while expected to be small, could slightly increase the ΔU signal.

The inverse model used to generate the initial steady-state solution is under-constrained. Because we infer two parameters with multiple constraints during the inversion, an initial state with a minimal velocity misfit will not necessarily lead to a minimal ice thickness rate of change. Different combinations of friction and viscosity parameters can lead to similar misfits. Using the ensemble Ω15, consisting of the 15 optimal initial states (see Sect. 3.3 and Appendix A1), to estimate the effect of the initialisation on the forward model helps quantify this effect. For the ensemble of simulations using the ΔLB2 parameterisation, the impact of the initial state on the velocity intra-annual cycle is minimal over the ice shelf, with an average relative standard deviation under ∼5 % over most of the ice shelf (Fig. 8 (bottom row) and Figs. 9 and 10). The ensemble responses for the ΔLC and ΔLB2L parameterisations, while providing more realistic estimates of intra-annual velocity changes, show more sensitivity to the initial state with year average relative standard deviations of ∼15 %–20 % (±0.1–0.15 m a-1) at DR10 and ∼25 % (±0.4 m a-1) at Byrd Glacier. We attribute the relatively high variance of the ensemble in these regions to the sensitivity of the model to the initial basal friction, while the relatively low variance of the ensemble over most of the ice shelf indicates low sensitivity of the model to the initial viscosity parameter.

The friction law used in the model will also influence ice flow response, even for the same value of ΔL. Friction laws of different complexity have been proposed in the literature (Weertman, 1957; Budd et al., 1979; Schoof, 2005; Tsai et al., 2015) and have been shown to have different impacts on grounding line dynamics (e.g. Brondex et al., 2019). In our study, we only used the most common friction law (Weertman, 1957).

The results described in Sect. 3.2 were obtained with a linear version (m=1) of Eq. (1); i.e. stress is proportional to velocity. We also tested the value m=3 (e.g. Brondex et al., 2019; Gudmundsson et al., 2019), which only changes modelled velocity anomalies by a few percent. More complex friction laws (e.g. Schoof, 2005; Tsai et al., 2015; Joughin et al., 2019) that include the impact of water pressure change at the ice base as the grounding line migrates could increase the amplitude of our intra-annual velocity variations. However, such friction laws introduce additional poorly constrained parameters (Gillet-Chaulet et al., 2016) and, therefore, are not considered in this study.

The parameterisation of ΔL directly controls the amplitude of the grounding line migration which, in turn, controls the change in the friction coefficient we apply at the grounding line (see Sect. 3.3 and Appendix B). ΔLB2 leads to small migration of the grounding line (typically a few metres) so that most of the impact of SSH variability on the ice flow comes from changes in ΔSSH gradients. While driving stress variations from these SSH gradients and small grounding line migration (ΔLB2) due to ΔSSH can slow down or accelerate the ice flow by about ±1 m a-1 (Figs. 6 and 7), these modelled variations are only ∼20 % of the observed ΔU at DR10. Incorporating a larger grounding line migration in the model (ΔLC and ΔLB2L) gives values of ΔU that are consistent with our GNSS observations. Such grounding line migrations with respect to the hydrostatic case (ΔLB2) are, arguably, too strong, but are in line with observations by Brunt et al. (2011) and the values used by Rosier and Gudmundsson (2020) on the Filchner–Ronne Ice Shelf. The surface and bed slope are key parameters of the grounding line migration parameterisation (Tsai and Gudmundsson, 2015; Appendix B). The bed slopes around the RIS perimeter, estimated by Brunt et al. (2011) by applying the hydrostatic assumption to observed migration of the inner margin of tidal ice flexure in repeat-track satellite altimetry, are likely to be biased low, based on the modelling of Tsai and Gudmundsson (2015). While the ΔL values given by ΔLC and ΔLB2L are in the upper range, they remain consistent with previous studies of tidal migration of the grounding line.

Another explanation for our need for a large migration of the grounding line is the relatively low value of the basal friction coefficients we inferred at the grounding line during the model initialisation. Our initialisation scheme relies on the optimisation of the friction coefficient C in Eq. (1). This friction law does not include a direct dependency on the effective pressure as a Coulomb law would (e.g. Brondex et al., 2019; Urruty et al., 2022). However, as C is determined by inversion, it includes a dependency on the effective pressure close to the grounding line and reduces the value of C at the grounding line to match observations of velocity and thickness rate of change (e.g. Urruty et al., 2022). The inferred value represents an average annual value of the friction coefficient. The distribution of the seasonal variation around this annual average cannot be exactly determined without proper knowledge of the subglacial hydraulic system, but one can assume that the variation could be larger than the variations we estimate through our hydrostatic parameterisation (i.e. a change in C directly proportional to ΔL). Seawater intrusion at the ice–bed interface and in sediments has been shown to have a high impact on the ice flow response (e.g. Robel et al., 2022). Subglacial models depending on subglacial water pressure decrease effective pressure significantly near the grounding line, leading to an increased sensitivity for a given power in the sliding law (e.g. Kazmierczak et al., 2022). Seawater intrusion could also be enhanced by a highly retrograde slope (e.g. Byrd Glacier; see Fig. S6). Retrograde bed slope will enhance both the migration of the grounding line and the intrusion of seawater in the subglacial hydrologic system. The consequences of ΔL on this effective pressure are difficult to determine, but incorporating such a mechanism in our modelling could lead to a larger impact of ΔSSH on the ice flow, even in the purely hydrostatic case.

A final explanation relates to the potential effect of ΔL on the subglacial water system. If the grounding line retreats elastically over a short period, before relaxing to a position closer to the hydrostatic equilibrium, then this short retreat could modify the subglacial water system over a long distance upstream the grounding line. The proper modelling of the water system is largely out of the scope of this study but could help validate our theories in the future.

The SSH anomalies (ΔSSH) computed in the different ocean models (see Sect. 2.2 and Supplement) result from temporal variability in ocean currents driven by wind stress and lateral density gradients. However, these models do not account for the steric changes due to thermal and haline expansion and contraction or the ocean's response to atmospheric pressure variations. Both the ROMS and NEMO modelling frameworks (see Supplement) use the Boussinesq approximation based on the Navier–Stokes equations: the models conserve volume rather than mass and therefore do not properly account for steric changes. At the same time, variations in atmospheric pressure also lead to isostatic adjustments of the ocean (e.g. Goring and Pyne, 2003), while ice shelves have been shown to respond similarly (Padman et al., 2003). This effect, known as the “inverse barometer effect” (IBE), is not considered in the simulations used in this study. Combining the effect of Boussinesq SSH variations (ΔSSHboussinesq), the steric effect (ΔSSHsteric), and the IBE (ΔSSHIBE), we obtain the total ΔSSH monthly deviation: 3ΔSSH=ΔSSHboussinesq+ΔSSHsteric+ΔSSHIBE. Some efforts were made in the 1990s to evaluate the effect of steric sea level due to thermal expansion, concluding that a globally uniform, time-dependent correction of sea level can correct a non-Boussinesq solution (e.g. Greatbatch, 1994). Mellor and Ezer (1995) showed that the seasonal variation in this term is about 1 cm in the Atlantic Ocean, which represents about 10 % of our modelled amplitude of SSH variation over the ice shelf. At the spatial scale of RIS, this correction is roughly spatially uniform and, therefore, would not modify the driving stresses over the ice shelf but could affect the grounding line migration.

Seasonal changes in surface air pressure take place over the Antarctic continent, resulting in a decrease in surface pressure (loss of atmospheric mass) from January to April and an increase in surface pressure (gain of atmospheric mass) from September to December (Parish and Bromwich, 1997). Since most of the ocean models we presented use ERA-Interim reanalysis (Dee et al., 2011) as an atmospheric forcing, we therefore use ERA-Interim surface pressure over RIS to estimate the IBE effect contribution to ΔSSH and its potential effect on the ice flow. ERA-Interim is an older product than the currently recommended ERA5-Land surface air pressure (Hersbach et al., 2020), but both give similar surface pressures over RIS for the period we study, which limits the uncertainty of the IBE effect.

We simulate the effect of IBE on ΔSSH following Eq. (2) and apply the full ΔSSH as a forcing to the ice flow model. Due to the smaller isostatic adjustment of ice shelves to ΔSSHIBE close to the grounding line, we do not include its effect in the grounding line migration. The relative effect of the IBE on the seasonal ice flow is maximal at DR10 and BATG due to their relative proximity to the ocean. In contrast, GZ19 and the region of Byrd Glacier are less affected, since the IBE does not impact grounding line migration (Fig. 10). Overall, accounting for the IBE modifies the peak-to-peak amplitude of ice flow variations by up to ∼1.5 m a-1 (Fig. 10) without significantly impacting the seasonal pattern and phase of the ice flow velocity change. We note that if the IBE was to have a significant impact on the grounding line migration on average, it would most likely increase the amplitude of the grounding line migration with a similar phase to the one we observe without IBE. On a 38-year record of IBE (Fig. S5) the negative inverse barometer signal observed from December to June would lead to downstream migration of the grounding line and a deceleration in the ice shelf flow. Conversely, the positive signal observed from July to November would lead to an upstream migration of the grounding line and an acceleration in the ice shelf flow.

Our ice flow model uses the shallow-shelf approximation (SSA), a viscous rheology, which is well suited for studying long-timescale mechanisms involving ice creep (more than a few days). At the same time, two of our parameterisations of the grounding line migration are based on an elastic rheology, which is more appropriate for short-timescale mechanisms such as tidal effects (less than a few days). In reality, both rheologies are at play but either can sometimes be disregarded with respect to the other, depending on the Maxwell time: 4tM=ηE, with E being Young's modulus and η the characteristic viscosity of ice. Using a value η∼40 MPa a-1 (obtained from the inferred viscosity parameter and strain rates averaged over the ensemble Ω15 for RIS) and E=103-104 MPa (Cuffey and Patterson, 2010) gives tM∼2 d to ∼2 weeks. Given the seasonal timescale of the variability under consideration in this paper, our viscous ice flow model should adequately represent the real viscoelastic rheology of ice. The elastic migration of the grounding line is, therefore, less representative of the actual viscoelastic rheology for the timescale changes we are observing (SSH anomalies remain relatively stable in periods shorter than a month). However, the elastic parameterisation has previously been successfully applied in a viscoelastic ice flow model studying ice flow response to fortnightly tidal forcing (Rosier and Gudmundsson, 2020). As mentioned in Sect. 4.2., the elastic parameterisation is also a proxy to simulate unrepresented mechanisms that might trigger SSH-induced basal stress change in the grounding zone. Moreover, although the use of an elastic rheology to study a viscous problem usually requires decreasing the effective Young's modulus of ice (which could decrease ΔL), Tsai and Gudmundsson (2015) suggest that their parameterisation of the grounding line migration may also apply to a purely viscous case. This could also explain why grounding line positions in Stokes models, which are not constrained to the hydrostatic approximation, are generally more sensitive than in SSA models such as the one used in this study (e.g. Pattyn et al., 2013).