Based on geophysical observations from extant ice-sheet beds, as well as the landform record of grounding-zone wedges, we know that deforming subglacial tills are transported across the grounding zone (Alley et al., 1986, 1989; Anandakrishnan et al., 2007). When they are deposited at a stable grounding-line position, advected sediments can form constructional landforms marking that position (e.g. Alley et al., 1989, 2007; Dowdeswell and Fugelli, 2012). A simple mechanism for producing tidally modulated landforms might arise from this concept of constant till deposition over the grounding zone, as the grounding line moves back and forth across it. The volume of till delivered to any given location will inversely depend on the speed of grounding-line migration across the area, which changes over the tidal cycle.

Suppose that the bed b(x) is initially smooth and that there is a constant sediment flux qs0 (in m3 s-1) delivered subglacially to the grounding line GL(t), as defined in Eq. (4) (Fig. 2a, b). Then, the final depth of sediment deposited at the grounding line (in metres) is given by 5∫-∞∞qs0×δx-GLtdt, where δ is the Dirac delta function. For shallow effective slopes, the range over which the grounding line migrates is large, so the speed of migration is slowest (and deposition is greatest) when migration changes direction. This occurs twice per tidal cycle, at high and low tides, thus generating two ridges per cycle (Fig. 2a, b). In the model for this simple mechanism, the ice sheet does not interact with the ridge after it is formed (e.g. Fig. 2b), leading to a complex pattern of superimposed ridges, unless the retreat rate is sufficiently large that the next low-tide position is inland of the high-tide position. We consider this highly unlikely, given ice-shelf tidal cavity widths have length scales of hundreds of metres to kilometres (Rignot et al., 2011; Mohajerani et al., 2021; Chen et al., 2023), far greater than the observed corrugation ridge spacing. More realistically, a grounding line that migrates seaward after high tide would be expected to interact with recently deposited sediments, eroding or reworking them. The high-tide ridge may be flattened or pushed forwards to the low-tide grounding-line position, as the ice re-settles on the sea floor and migrates downstream resulting in only one ridge per diurnal tide. These processes form the basis of the model in Sect. 2.1.2 and are described by the model code in Hogan et al. (2023).

Here, we consider the movement of till across the grounding zone into a ridge as ice at the grounding line re-settles on the sea floor (after high tide) and then migrates seaward (and pushes sediment) to the low-tide position (Fig. 2c, d). As a simple idealised model of this process, we assume that the ice has a fixed geometry di at its base and that this shape translates across the bed as the grounding line migrates: dix-GL(t). Considering the till as a yield-strength fluid (e.g. Boulton, 1987; Boulton and Hindmarsh, 1987) to allow the till to deform, and invoking mass conservation, we assume that any volume of till that the ice displaces as the grounding line migrates seaward is transported to the point where the ice lifts off from the bed. When the grounding line retreats inland, we assume that any available space below the ice becomes filled with water, and the displaced till remains fixed at the low-tide position.

We explore two possible sources of till to form each ridge with this mechanism: a constant till flux as in the previous mechanism (qs0; Fig. 2c) or mobilisation of existing bed sediments from close to the grounding line due to elastic deformation of the ice, leading to enhanced compression of the sediment by a depth dcomp (Fig. 2d; Sayag and Worster, 2011). Corrugation ridges will be produced as long as one of these processes occurs.

Whichever the source of till flux to the grounding line, as the tide is coming in, a new layer of till is left over the bed (as in the previous mechanism). This layer can then be pushed forwards into a ridge when the tide goes out (Fig. 2c). Thus, the total flux of till towards the grounding line is no longer constant but is given by 6qs=qs0+ddtds-di. For simplicity, since the spacing and volume of sediment in the ribs will not be sensitive to the exact shape taken for the ice lid, we use the steady shape of an ice sheet in hydrostatic balance with the ocean, resting on the till up to the grounding line, at which point the cavity opens at an angle α and (if dcomp>0) includes a highly simplified form of compression near the grounding line, so that 7di=-α⋅min⁡dcompα,xordi=0ifx>2dcompα.

Note that the cavity angle α will be the same as θeff if the ice-sheet–ice-shelf system is in hydrostatic equilibrium, which we shall assume in the results shown. As a result, the exact shape of the corrugation ridges this model produces closely reflects the choice of basal ice topography, but as we have not included any viscous or erosional mechanism that would smooth the shape of the ridges after their formation, we do not expect to be able to compare the detailed ridge morphology (modelled) with observations to evaluate this mechanism.

In general, the total rib volume is given by Vrib=qs0×ttotal+dcomp×distance  between  high-tide  positions, where ttotal is the time between high tides. We present results for the two limiting cases, either when dcomp=0 (all till is sourced from constant flux) or when qs0=0 (all till is sourced from compression at the grounding line).

A third mechanism for ridge formation involves the erosion and deposition of grains as water enters and exits the ice-shelf cavity with the tides. As described in Warburton et al. (2020), when the grounding line migrates downstream, water that was brought in by the incoming tide must drain out again. This flux of water could erode sediments from the water-saturated till that is exposed to the ocean at high tide and deposit it at the low-tide grounding-line position once the outgoing tidal flow ceases (Fig. 2e, f). This mechanism is somewhat based on the concepts of tidal pumping in the ice-shelf cavity and grounding-zone estuaries, both of which have been put forward as processes capable of eroding the grounding-zone sedimentary bed (Domack, 1990; Horgan et al., 2013).

As before, we start from a smooth bed b(x) and deposit a sediment flux qs at the low-tide grounding line. However instead of a constant flux, we now take qs to be made of the sediment eroded by the draining water, which is removed at a rate Q from the region between the high-tide grounding-line position and the current position and is given by 8qst=Q×maxT<tGLT-GLt. Following the calculation in Warburton et al. (2020), the shear stress τ exerted on the bed in the draining region is given by 9τ=2.34μU1/4B1/8ρwg5/8θeff, where U=dhtide/dt is the speed at which the ocean height lowers, and B is the bending stiffness of the ice sheet as an elastic beam, with flow in the draining region assumed to remain laminar throughout. For physical values of these parameters, the shear stress can reach up to a maximum of approximately 1 N m-2. For simplicity, we take the erosion rate Q in the draining region to be constant, although since U varies over the tidal cycle, a more detailed model could include the effect of variable Q.

The till flux mechanism produces mostly double-peaked or highly composite ridge forms, with no clear fortnightly pattern to either the spacing or ridge height (Fig. 3a, b). This strongly contrasts our Thwaites observational dataset (Tables 2, S1) with its one ridge per tidal cycle, as well as numerous other tidally modulated ridge landforms (Jakobsson et al., 2011; Graham et al., 2013; Dowdeswell et al., 2020; Supplement Text S1, Fig. S1). So, while this is the simplest conceptual model, we clearly must discount it.

The till extrusion models, with either a constant till flux or grounding-line compression, produce ridges that are arguably most similar to the very regular features observed at Thwaites (Fig. 3c, e, i), although some differences do exist. The observed 14-ridge periodicity (Sect. 2.2) is reproduced, as is the normal correlation between ridge height and spacing (Tables 2, S1). As stated in Sect. 2.1.2, we cannot assess the symmetry of ridges produced by this mechanism because we assume that each ridge forms with the same geometry: a landward-facing side reflects the ice-shelf base cavity angle (equal to θeff), and the seaward-facing side is vertical. This is shown clearly in the model results in Fig. 3c and e in comparison with the symmetric forms in Fig. 3i. More complex modelling of the extrusion process, viscous slumping of the ridge, and ice-till coupling during the push process for a subglacial till type of material would be required to allow a detailed comparison of ridge shape. However, we do note that when effective slopes most similar to the observed bed at Thwaites (θeff=0.02) are used, pronounced composite ridges are formed, resulting in a long wavelength (λ∼90 m) cyclicity in bed topography (Fig. 3c, e). By way of comparison with the Thwaites observations, although some corrugation ridges do appear to climb and descend longer wavelength topography, we do not see any clear evidence for either composite ridges or regular bed undulations in the sea-floor data.

An extrusion-type model is the mechanism favoured by marine geoscientists (Dowdeswell et al., 2020; Graham et al., 2022) based on form analogy with other landforms produced at glacier and ice-sheet grounding lines. Recessional or De Geer moraines that form by push as a grounding line readvances over a bed of unconsolidated sediments (Boulton, 1986; Lindén and Möller, 2005) can also have very regularly spaced, repeating morphologies when produced annually by winter readvances of local glacier fronts (e.g. Ottesen and Dowdeswell, 2006; Todd et al., 2007; Burton et al., 2016). However, these moraines tend to be at least an order of magnitude larger in their dimensions (height, width, slope angle) and are often asymmetric in cross section due to the forward motion of grounding-line readvance steepening the ice-distal ridge face (Boulton, 1986). Here, the model only simulates the resettling of ice on to deformable (recently subglacial) sediments at the low-tide position squeezing sediment up into a small ridge, leaving an imprint of the grounding line on the sea floor. As such, we do not expect to recreate ridge asymmetry with this model but note that only the landward side of the corrugation ridge should reflect the shape of the ice base as it touches down onto the bed (Tables 2, S1). We also do not expect any significant modification of ridge shape by post-formation downslope processes (e.g. slumping), owing to the incredibly shallow slopes of their flanks: Graham et al. (2022) report median values of 0.05–0.2∘ for both flanks.

Supporting observational evidence for a till extrusion process of corrugation ridge formation is found in the remarkable consistency from one ridge to the next, i.e. as retreat progressed, at both Thwaites and in the Larsen Inlet (Dowdeswell et al., 2020; Graham et al., 2022). This ridge-to-ridge consistency is the case even when along-ridge form is quite variable, indicating that the small-scale (metres to decimetres) shape of the ice base did not vary as the grounding line retreated (Batchelor et al., 2020), which perhaps supports widespread dynamic thinning as a retreat mechanism over high rates of melting that might alter the shape of the ice-base variably along the grounding line. A process analogy to the squeezing out of sediment that we describe here may come from crevasse-squeeze ridge landforms, which are metres high, sharp-crested, and formed of till and are mostly observed in association with surge-type glaciers. For surging glaciers, the process is squeezing of subglacial till into basal crevasses that opened when high basal water pressures facilitate bottom-up crevassing (Rea and Evans, 2011); however, subglacial till extrusion into existing ice-margin crevasses have also been described for non-surging but radially spreading glaciers (Evans et al., 2016).

In terms of inferred sedimentological properties of the ridges, it is well known from both extant (sampled) glacier beds (e.g. Englehardt et al., 1990; Tulacyk et al., 1998; Christ et al., 2021) and formerly glaciated terrains (e.g. Boulton and Jones Paul, 1976; Evans et al., 2006; Demet et al., 2019) that subglacial sediments (or tills) typically have diamictic grain-size distributions. That is, most or all grain-size fractions are represented in a poorly sorted or homogeneous mixture, often lacking specific structures or textures (e.g. Eyles et al., 1983; Clarke, 1987). With the till extrusion mechanism, either with constant till flux or grounding-line compression, we expect that subglacial till was originally supplied to the grounding zone via a deforming “conveyor belt” at the ice-bed interface (Alley et al., 1987; Kamb, 2001). If the till used in ridge formation is sourced from a constant till flux to the grounding line during its retreat, then this till layer is “fresh”; conversely, if the till is sourced by grounding line compression of material over a former ice-sheet bed, then the till may pre-date the grounding-line retreat phase. In either case, we do not expect any textural or grain-size differences between the ridge and inter-ridge areas beyond the deformational sediment textures and structures typically associated with subglacial traction tills (e.g. van der Meer et al., 2003; Evans et al., 2006; Reinardy et al., 2011; Table S1). Similarly, we would not expect any shearing deformation or stacking of till layers associated with forward motion, or push, as might be found within recessional push moraines or thrusts (e.g. Menzies, 2000; Evans et al., 2007; Table S1).

Based on our model results we cannot properly distinguish between the constant till flux or grounding-line compression source for the ridge material. Both mechanisms produce a series of regular corrugation ridges, although the correlation of ridge height to spacing is stronger, and in line with our observations, for the constant till flux mechanism. We also tend to favour this mechanism for till supply because, when visualising a grounding line that retreats only 6 m d-1, we recognise that almost the same area of sea floor will be compressed every day, as the ice lifts and resettles with the tides. We posit that repeated compression of till across the grounding zone would make this material more difficult to mobilise over time (by compressing, dewatering, and stiffening). Basal properties derived from seismic datasets and inferred from numerical models provide some support for this notion. At Whillans Ice Stream, arguably the best-studied extant Antarctic grounding zone, seismic properties (density ρ, P wave velocity VP, S wave velocity VS) indicate significant stiffening of a 5 m till layer across a sharp transition at the grounding zone (Horgan et al., 2021). This stiff, low-permeability substrate at the grounding zone is consistent with models of ice-shelf flexure (for fixed and tidally migrating grounding lines) that predict compression and potentially dewatering of subglacial till immediately upstream of the grounding zone, as ice bends down into the substrate at high tide (Walker et al., 2013; Sayag and Worster, 2013). Whether this compression and dewatering is enough to prevent material squeezing out from the grounding line after the ice resettles on the sea floor is unknown.

Given the above discussion, we prefer the idea of a constant till flux supplied by the “till conveyor” to form the ridges. With this in mind, we can compare previous estimates of the subglacial sediment flux (to the grounding line) with the volume of a ridge formed once per day at Thwaites Glacier to see if this mechanism of till supply is realistic. Estimated fluxes of 100 m3 yr-1 (per metre of ice stream width) (0.27 m3 d-1) for the extant grounding line of Whillans Ice Stream (Engelhardt and Kamb, 1997; Anandakrishnan et al., 2007), and 800–1000 m3 yr-1 (2.2–2.7 m3 d-1) for large, fast flowing Antarctic and Greenlandic palaeo-ice streams (Dowdeswell et al., 2004; Hogan et al., 2012, 2020a), suggest that daily fluxes of sediment to the grounding line are likely to be on the order of a few tenths of 1 m3 to a few cubic metres. If we consider the average dimensions of a triangular, symmetric corrugation ridge at Thwaites (0.2 m tall, 5 m wide at the base), then we can calculate that every 1 m section along the ridge will contain 0.5 m3 of sediment. Thus, it appears that the subglacial till conveyor is able to supply about the right amount of sediment to build a daily ridge of the same magnitude as the Thwaites corrugation ridges.

One potential contradiction for the till extrusion mechanism comes from the modelling results of Warburton et al. (2020), who found that low subglacial till permeability filters the grounding-line response to tidal forcing, essentially fixing the grounding line at the high-tide position. As pointed out by Graham et al. (2022), the strong tidal periodicity of the corrugation ridge dimensions should, therefore, indicate a high permeability substrate in the region of ridge formation, despite observational evidence that most subglacial tills are cohesive and likely to have low permeabilities. Graham et al. addressed this problem by suggesting that water may drain out of the grounding zone via a series of shallow canals, although another alternative is that the pattern of grounding-line migration at Thwaites is not controlled by fluid connectivity through the till, as has been suggested for the Whillans grounding zone (Horgan et al., 2021).

Like the till extrusion mechanism (with either till source), with real-world corrugation ridges, the resuspension of sediment in the tidal cavities mechanism (shortened to “resuspension mechanism” hereafter) produces a series of corrugation ridges with a 13–15 cycle periodicity in their amplitudes and spacings (Fig. 3g, h), although these are only weakly correlated (Tables 2, S1). The shape of the modelled ridges, which are sharp-crested, is also different from the observation of the symmetric ridge cross-sectional shape at Thwaites (Fig. 3j). Again, this is largely due to the simplicity of our model that preserves a vertical seaward slope.

Perhaps of note is that the resuspension mechanism is the only mechanism that would result in a variation in acoustic backscatter over the ridges due to a grain-size change (Tables 2, S1). Smaller particles would be preferentially eroded from the grounding-zone bed and eventually deposited to form the ridges, leading to finer grain sizes in the ridges versus inter-ridge areas. However, the backscatter response on the side-scan sonar images clearly shows higher reflectivity (i.e. brighter returns) on the ridges and lower reflectivity (darker returns) in the inter-ridge areas (Fig. 1b). This is the opposite of what we would expect if the ridges were composed of fine-grained material and the inter-ridge areas consisted of the remaining coarse particles (e.g. Lurton and Lamarche, 2015). In contrast, we can envision that squeezing out of till into a ridge at the grounding line (as in the till extrusion mechanism) may produce a rougher surface texture, producing relatively higher backscatter returns than the inter-ridge areas that have been compressed, or that the stronger returns on the ridges are purely a product of the topography of the ridge fronts. A detailed sampling campaign would be required to ground truth the sedimentology of ridge and inter-ridge areas and so to determine the cause of the backscatter variations.

A sediment suspension mechanism is probably the easiest to interrogate using a combination of empirical datasets and established sediment dynamics theory. Our model simulates water rushing into (and out of) a 1 km wide grounding-zone cavity when set up in a Thwaites configuration and assumes that enough particles can be eroded from the bed of the grounding zone to build a ridge every day. Precedence for such a mechanism comes from the process of “tidal pumping”, whereby tidal flows in a sub-ice-shelf cavity winnow fine-grained particles from glacigenic debris as it melts out from the ice-shelf base near the grounding line and then transports these particles in suspension seaward, eventually depositing them as laminated clayey to fine sandy units. This tidal pumping is well established based on marine sediment core data from the Antarctic continental shelf (Domack and Harris, 1998; Domack et al., 1999) and provides some support for the notion that tidal water velocities can be at least high enough to transport grains up to fine sands in size (0.25 mm diameter). Thus, our mechanism here is similar to tidal pumping because we stipulate (in the model) that the outflows carrying fine-grained particles do not migrate vertically, exiting the grounding zone horizontally and immediately depositing their suspension load to form a ridge.

Accepting this and remembering that every 1 m section along a corrugation ridge will contain around 0.5 m3 of sediment, a basic calculation can be done to determine how much subglacial sediment would need to be eroded from the bed to produce a 0.5 m3 ridge after each tidal cycle (i.e. once per day at Thwaites). Subglacial tills recovered from the western Amundsen Sea in marine sediment cores have similar grain-size distributions, with around 90 % of grains (by weight) being smaller than 2 mm in diameter and around 65 % smaller than 0.25 mm (Smith et al., 2011; see Supplement Table S2). Using a wet bulk density of 1.575 g cm-3 for a standard marine sandy-silty clay (Hamilton and Bachman, 1982), we calculate that around 788 kg of material is required for each 1 m section of ridge. Assuming that grains are eroded across the entire grounding zone, and that only 65 % of them are small enough to be transported to form a ridge, then we calculate that grains would only need to be eroded from the upper ∼ 0.6 mm of the till (wet bulk density 2.1 g cm-3; see Supplement Text B) to provide enough material to form each ridge. This thickness is very small indeed and may indicate that there would have been a plentiful supply of fine-grained particles (from the bed) across the grounding zone with which to form the ridges. However, there are several potential pitfalls with the following simple calculation: (1) with a grounding-zone width of 1 km, tidal water velocities are extremely low, on the order of only a few cm s-1, and therefore below the speeds required for the erosion and transport of fine sands from the bed (e.g. McCave and Hall, 2006); (2) it is unlikely that particles would be resuspended from across the entire 1 km wide grounding zone because as the ice-shelf base becomes less confined (away from the grounding line) the influence of tidal flows would diminish; and (3) it is unclear how fine-grained particles would continue to be mobilised from sub-sea-floor depths once surface sediments had been winnowed, presumably to form either a hardground or lag deposit as is found when strong ocean bottom currents winnow sea-floor sediments (Anderson et al., 1980; Hillenbrand et al., 2003) or to create a granulated facies (with fines removed) similar to those found overlying subglacial tills elsewhere in Antarctica (Domack and Harris, 1998; Domack et al., 1999; Kirshner et al., 2012).

The first of these issues is probably the hardest to counter. Water velocities of only a few centimetres per second, which are in fact in line with the few existing hydrographic observations of extremely low-to-undetectable tidal current velocities in grounding-zone cavities (Begeman et al., 2020; Davis et al., 2023), would only allow for transport of the very finest particles (clay and fine silts < 10 µm). Velocities above ∼ 20 cm s-1 are required to keep sand-sized grains (diameters 63 µm–2 mm) mobile in suspension (McCave and Hall, 2006). Such velocities could be achieved if the grounding zone was much wider; for example, a 10 km wide grounding zone, which is at the upper bound of observed widths around Antarctica (see Brunt et al., 2011), could produce tidal water velocities of ∼ 23 cm s-1 and, therefore, transport sand-sized grains. The occurrence of granulated units elsewhere on the Antarctic continental shelf, including in the Amundsen Sea (Kirshner et al., 2012), might actually indicate that such velocities can be achieved. Turning to the second and third issues, if grains were only eroded from a 100 m wide area adjacent to the grounding line then all particles from the upper ∼ 6 mm of the bed would be required to build a ridge, and mobilising grains from sub-sea-floor layers becomes increasingly difficult to envision (recalling daily retreat rates of only 6 m, so the bed would be repeatedly eroded). If grains can only be mobilised from the newly exposed grounding-zone area (6 m wide) then the depth of erosion increases to an untenable 625 mm (62.5 cm). Even if we accepted this erosion process for some area of the grounding zone, it is also difficult to see how such a “dumping” mechanism would result in all grains being deposited instantaneously (the finest particles would surely be carried further seaward), or how this would produce such consistent ridge-to-ridge morphologies as is observed at both the Thwaites and Larsen Inlet sites (Table S1).

Thus far, we have only considered the transport of grains in suspension and not how easy it is to erode them from the bed. The matrix of subglacial tills, including those recovered from the Amundsen Sea area close to Thwaites Glacier (Smith et al., 2011, 2013; Table S2) and those observed directly at the few Antarctic grounding zones to have been accessed (Langovde Glacier, East Antarctica: Sugiyama et al., 2014; Mackay Glacier: Powell et al., 1996), are dominated by fine grain sizes (typically ∼ 55 %–70 % clays and silts). As a result, subglacial tills behave cohesively (with respect to erosion by water), and the fine grains within them are able to withstand larger values of shear stress without being eroded (Mier and Garcia, 2011). Unidirectional flume experiments designed specifically to assess erosion of glacial till or till-like sediments in rivers or coastal environments have returned critical shear stress values of 4–9 N m-2 to initiate erosion (McNeil et al., 1996; Mier and Garcia, 2011; Pike et al., 2018), whereas our model predicts a shear stress across the grounding-zone bed of only ∼ 1 N m-2. Although this may intuitively indicate that greater shear stresses (or water velocities) are required to erode glacial till across the grounding zone, the additional complexities of being in a grounding-zone setting (tidal currents switching direction and lift-off and re-settling of the ice sheet, compressing but also disturbing the bed every 12 h for diurnal tides as in the Thwaites area) mean that this assumption is not straightforward. In addition, we note that there is a growing body of observations showing that some Antarctic grounding zones are in fact kilometres wide (Mohajerani et al., 2021; Milillo et al., 2022; Chen et al., 2023), an order of magnitude greater than predicted for ice shelves in hydrostatic equilibrium (Mohajerani et al., 2021). Such a wide grounding zone would significantly increase tidal flow velocities, especially in narrow cavities, and potentially allow for much greater erosion and sediment transport. Furthermore, current erosion of the bed may not be the only source of sediment to the tidal cavity. Meltout of basal debris from the ice-sheet base as the cavity opens could be an additional source of particles (see the tidal pumping mechanism described above), although this may be limited to areas very close to the grounding line (Domack and Harris, 1998; Smith et al., 2019), and the disturbance or ploughing of the bed by a rough ice base swiftly followed by lift off as the tidal cavity opens is another process that may mobilise fine-grained sediments into suspension, therefore lowering the current velocities required to transport material out of the grounding zone.

The tidal resuspension mechanism is eminently testable with direct sampling. Sediment samples from an area of corrugation ridges should comprise normally sorted, fine-grained deposits in the ridges and winnowed subglacial sediments (with fines removed at least in their uppermost layers) in intervening bed areas (Supplement Table S1). Although it appears that there is sufficient supply of fine-grained sediments to the grounding zone for this mechanism to be plausible, the cohesive nature of glacial tills and low predicted tidal current velocities make it difficult to see how enough material could be mobilised from the bed every day to produce even the subtle (low amplitude) ridges described from Thwaites or the Larsen Inlet.

A final yet important consideration is how the models perform with realistic ice-bed geometries, which is represented in the models by θeff. Both the till extrusion and resuspension mechanisms produce tidally correlated ridges for the larger values of θeff but increasingly large composite ridges with a decreasing number of individual ridges, quite unlike the observations that consistently show individual ridge forms were produced at each tidal cycle on low-gradient bed slopes (i.e. smaller values of effective slope; see Sect. 2.2). This conflicts with the suggestion in Graham et al. (2022) that the corrugation ridges at Thwaites may have formed in ice-plain conditions (θeff close to 0) because the thinning or melting rate required to sustain rapid retreat is proportional to θeff and is thus easier to achieve for lower values of effective slope. However, the control on composite ridge formation is the balance between the daily retreat rate (which we take to be directly observable) and the changes in grounding-line migration range over the tidal cycle, which depends only on the tides and the effective slope. While it is possible that present tides do not represent the tidal amplitudes during the time of ridge formation, we assume that θeff is left as our only control parameter. We therefore suggest that it would be impossible to form the observed ridges at small θeff, implying that a large thinning rate was truly necessary to drive the retreat. As such, we reiterate that, in addition to the melting at the grounding line quantified by Graham et al. (2022), dynamic thinning following ice-sheet acceleration likely also contributed to the total thinning rate.