Our approach is grounded in the combination of a Glacier Drainage System (GlaDS) subglacial hydrology model , implemented in the Ice-sheet and Sea-level System Model (ISSM) applied as forcing to the GraphSSeT sediment system model.

GlaDS resolves hydrological flow in two coupled components; an inefficient continuous sheet representing flow in a linked-cavity network , and an efficient channelised flow, represented by semi-cylindrical channels, called Röthlisberger-channels, or R-channels . In the finite-element-method (FEM) model the sheet-flow is resolved over the elements while the channelised flow is resolved along element edges. Coupling between these components is achieved by equalising the pressure between the channels and the adjacent sheet .

In GlaDS, the R-channels evolve in time through a thermodynamic balance between expansion through melting versus closure by viscous ice-creep . The opening rate is driven by heat-transfer from turbulent water flow and changes in the pressure-melting point, and varies with changes in the hydraulic potential gradient, the rate of water flow, and the water pressure see Eqs. 15 and 16 in. The closing rate is driven by ice deformation proportional to Nn where N is effective pressure, and n is the exponent in Glen's flow law, typically n= 3. In this study, the ice geometry is fixed, therefore channel evolution reflects the changing water flow and basal water pressure. With increased meltwater supply, mass conservation demands faster flow, in response to which channels will expand; in contrast decreasing water supply will cause slower flow, and channels will contract. Consequently, the conditions of pressurised high-velocity water flow needed to transport sediment may be quite variable through both space and time.

Taking a GlaDS model result as input, GraphSSeT translates the FEM mesh into a directed acyclic graph incorporating element edges and nodes, through which the channelised hydrology network is represented . Edges carry the information for channelised flow including water flow rate, channel cross-sectional area and hydraulic potential gradient. Nodes carry the information of the ice geometry including bed elevation, ice thickness, effective pressure and basal ice velocity. In GraphSSeT elements are not included and the role of hydrologic sheet flow in sediment transport is not considered explicitly.

The GraphSSeT model involves several processes computed in order. First, the sediment transport capacity is derived based on the basal shear stress τ which is proportional to the square of the mean water velocity uw in the channel. The transport capacity, in this case using the formulation of , is proportional to τ5/2 and the net effect is that transport capacity varies proportional to uw5. The transport capacity is grain size dependent, and is linearly inverse-proportional to d50 , the median grain size of sediment to be transported. Other factors are small influences on transport capacity . In the second step, sediment mobilisation from and demobilisation to the basal sediment layer is calculated. Here we consider the balance of sediment transport capacity versus sediment supply from upstream and local bed erosion, and the sediment available in the basal sediment layer . If sediment supply from upstream is less than the transport capacity, this will lead to mobilisation from the basal sediment layer, so long as sediment is available. On the other hand, sediment supply exceeding transport capacity will lead to demobilisation to the basal sediment layer (Fig. ). The third step considers transport at network-scale, which is managed to maintain the following conditions: sediment volume is always conserved; transport capacity is never exceeded; reasonable sediment velocity limits are never exceeded. In the last stage statistical distributions of grain size and detrital provenance are tracked on the network through volumetric-mixing equations. Detritus tracking is entirely passive, however, the evolving grain size can have significant impacts on transport capacity and so model evolution, where transport-capacity limited .

Paleoproxy-based reconstructions indicate that climate and climate seasonality impact was not uniform through the North Atlantic and Europe during the Younger Dryas, with different timing and character . This demonstrates its complexity, including ocean-atmospheric coupling, which is difficult to master from spatio-temporal perspective. Although these variations may have had very significant impacts on the hydrology and sediment systems, for this case study we seek to understand variable sediment dynamics using a single hydrology forcing. Therefore, we restrict analysis to the subglacial hydrology derived using the climate forcing applied in . In their work, the FLIDL extent is representative of the end of the Younger Dryas stadial (∼ 12 cal. ka), when complex seasonality is thought to have given way to warmer climate with similar seasonality to present-day conditions .

ran a FLDIL-GlaDS simulation for 10 000 d (27.3 years) to describe subglacial drainage within the FLDIL. The model geometry comprised an initially parabolic ice surface that was allowed to relax over 10 000 years, and a representative basal topography created by subtracting sediment thicknesses from a modern terrain. The FLDIL-GlaDS model was forced using a modern precipitation record for the region, a modern temperature record depressed by 15 °C, and an elevation dependant lapse rate of 7.5 °C km⁻¹. A quadratic function was used to estimate monthly temperature variability , and a positive degree day scheme used to calculate total monthly melt rates . Melt was integrated within randomly distributed catchment areas and routed to the bed via 2500 moulins. A spatially uniform basal velocity of 150 m yr⁻¹ was imposed, and the FLDIL-GlaDS model ran with an adaptive timestep between 90 s and 1 h.

For input to the GraphSSeT model, we use the last 10.5 years of the model run. To reduce compute load, the 10.5 year model evolution is divided into beginning, middle, and end segments each of 3.5 years duration (Fig. ). GlaDS model outputs were extracted at ∼ 4 d intervals and for each, the channel water flux and channel area were spatially filtered to reduce abrupt changes in conditions between edges and ensure network connectivity (Fig. ).

The GraphSSeT geology model involves sediment in active transport, a basal sediment layer accessible to hydrology-forced mobilisation, and an underlying bedrock released into the transport system by glacial erosion (Fig. ). In the standard implementation, the basal sediment layer is variable in thickness between zero and an imposed upper limit (1 m here). In this work, we introduce a mixed-bed mode where thick sediment accumulations are available for mobilisation, but we retain the maximum limit for demobilisation, necessary to avoid potential for runaway sediment accumulations. In the standard implementation, the sediment cover of the area (Fig.) is considered as bedrock, requiring glacial erosion, while in the mixed bed model it is considered as basal sediment, able to be mobilised by hydrology. Glacial erosion potential is defined with a quadratic formula see; using the basal velocity of 150 m yr⁻¹ from the GlaDS model, giving erosion potential of ∼ 6 mm yr⁻¹.

For each ∼ 4 d GlaDS model output interval, the GraphSSeT implementation runs with constant forcing. GraphSSeT time steps were between 1 h and 1 d duration, adapting to the basal sediment thickness (H), targeting max(dH/dt)=H/2. At the end of the GlaDS interval, the graph is reformed, including any changes in the directions of edges and/or locations of outlet nodes and updating the forcing parameters. State variables including sediment thickness, active sediment flux density, grain size and detrital properties are transferred to the new graph. Summary model outputs are derived at the end of each GlaDS model output interval, and full model outputs are exported for selected intervals identified at inflections in the total channelised water flux (Fig. ).

The ensemble result for the low-H scenario (Fig. ) shows some key features of the sediment system. The sediment discharge rate is low, peaking at ∼ 0.23 m³ s⁻¹. During the melt season a systematic pattern is seen: The early melt season sees mobilisation of basal sediment and an increase in discharge. This early stage shows, in some runs, a distinct peak in grain size, while in others this peak is absent (Fig. a). This peak can be attributed to the high transport capacity of high-pressure flow in developing channels combined with the availability of previously-deposited sediment.

The melt season continues with, at peak discharge, reduced grain size, indicating selective transport of fine grained material from upstream sources. In this scenario the peak sediment discharge occurs well before the peak channelised flow conditions, and the decline is driven by the failure of the supply system. Post peak-discharge, sediment derived from the basal sediment layer reduces, and the grain size recovers to the mean value. By autumn a dominance of bedrock-derived sediment is seen at a low volume, paced by erosion. For this scenario the high-σ mode does not have increased discharge rate, although the sediments are finer. These characteristics indicate that the low-H scenario is strongly supply-limited. The detrital models all have a significantly higher volume of bedrock-derived sediment, which indicates ∼ 2/3 of sediment has experienced at least one cycle through the basal sediment layer.

The remaining basal sediment at the end of the model runs shows a systematic depletion of sediment in channels, extending back ∼ 60 km from the ice margin (Fig. b). Variation in the sediment thickness among the ensemble members is low but is highest in the upper reaches of subglacial channels (Fig. c).

Results for the high-H scenario are fairly similar to the low-H scenario, but with a higher discharge rate and a lower proportion of bedrock-derived sediment (Fig. ). The peak sediment discharge still precedes the peak channelised flow conditions, indicating a supply limited system. In this case the high-σ mode has a slightly higher discharge rate at peak flow indicating a greater role for sediment transport capacity in controlling supply to the outlet from upstream sources. Correspondingly, the detrital models have approximately half the proportion from bedrock-derived sediment, indicating less recycling through the basal sediment layer than in the low-H scenario.

As before, the remaining basal sediment is systematically depleted in channels, extending back ∼ 60 km from the ice margin (Fig. b). Variation in the sediment thickness among the ensemble members remains low, and is highest in the upper reaches of subglacial channels (Fig. c).

For the mixed-bed scenario, the results are markedly different (Fig. ). Most notably, the sediment discharge rate is approximately ten times higher than for the low-H and high-H scenarios, and bedrock-derived sediment is a very small proportion. Median grain size still reduces during peak flow, but by much less, and has a less distinct relationship with discharge-rate. In comparison with the low-H and high-H scenarios, the peak sediment discharge is later, and the high-σ mode has markedly higher discharge rate at peak flow. These indicate a more sustained supply and a more significant role for transport capacity in the upstream region controlling supply to the outlet. Despite this, sediment discharge remains far below bulk sediment transport capacity, and supply remains the limiting factor. The detrital models indicate that barely any sediment has been cycled through the basal sediment layer.

The remaining basal sediment is systematically depleted in channels, reaching zero in many places, but with regions of higher thickness patchily preserved (Fig. b). Variation in the sediment thickness among the ensemble members, while still low, is overall higher and more varied than the other scenarios due to the variable initial sediment thickness (Fig. c).

The results for the low-H scenario, and to a slightly lesser extent the high-H scenario map out a strongly supply-limited regime where discharge during the melt season is limited by the erosion rate (summed over the year). In these conditions, the seasonal cycle shows a clear zonation. Here we describe the seasonal characteristics of the high-H scenario (Fig. ).

At the end of winter (Fig. a) the system is dominated by three zones: an inactive zone in the upstream catchment (lacking channels) where dH/dt≈0; a small erosional zone with 0 <dH/dt≤e˙ (e˙ is erosion potential); a transport zone with variable dH/dt and a frontal depositional zone with dH/dt>e˙. Spring (Fig. b) sees a different zonation characterised by the development of a mobilisation front. Upstream of the mobilisation front, dH/dt is positive, while downstream, channels have negative dH/dt as basal sediment is mobilized. Initially, the mobilisation front is close to the ice margin, but then recedes, and by summer is ∼ 20–50 km back from the ice margin (Fig. c). This is in line with the processes considered for the deposition of eskers and murtoos and murtoo-like landforms . A lower transport-zone with both positive and negative dH/dt is emergent in summer downstream from the mobilisation front. The back-propagation of the mobilisation front is consistent with the elevation-dependent positive-degree day forcing applied in FLDIL-GlaDS , which, together with ice-thickness exceeding 1 Km, dictates the upstream extent of channelisation. Continuing into autumn, the mobilisation front has reached the upper limit of the channel extents, and has narrowed as basal sediment is depleted downstream. A deposition zone is reformed near the ice margin with positive dH/dt (Fig. d). The low-H scenario model (Fig. ) has the following differences attributed to lower sediment availability: the inactive zone sees dH/dt≈e˙, mobilisation in the mobilisation front is more localised, and the deposition in front of the mobilisation front is reduced. The mixed-bed scenario (Fig. ) demonstrates the same behaviours, albeit with increased intensity and spatial variability.

The erosional, mobilisation and depositional zones show corresponding signatures in the grain size of active sediment (Fig. ). Grain size for freshly eroded sediments is sampled from the population distribution, and erosion-dominated areas tend towards the population mean. In contrast, mobilisation zones show more variation with finer-than-mean grain sizes predominant in Spring and Summer (Figs. b and c). In the context of increasing water flow rate, we infer that the drop in mean grain size at the outlets during spring (Figs. , and ) is due to the preferential mobilisation and transport of finer-grained components derived from the basal sediment layer, and that the post-peak return to mean grain size is due to the later predominance of freshly eroded material as upstream supply wanes. This mobilisation and transport bias is strongest in the low-H and high-H scenarios where basal sediment supply sources are located upstream, and less marked in the mixed-bed scenario, where greater volumes of basal sediment are available locally.

Sediment discharge, and sediment properties are tracked at outlet nodes allowing us to analyse a detailed time series of sediment output for specific channels across the model ensemble (Fig. ). In the low-H and high-H scenarios, we see significant variations in volume output, grain size and detrital character within each melt season. The melt season often begins with a sharp peak in volumetric discharge rate, during which relatively coarse grain sizes (medium-coarse sand) are seen in some cases. Early-season detritus comes from remobilisation of the basal sediment. Through the melt season the volumetric sediment discharge rate typically declines, the grain size reduces to fine or very fine sand, and the proportion of bedrock-derived sediment increases. This systematic behaviour has parallels with the diverse sedimentary settings observed in murtoos in the region, previously linked to variability in water velocity and sediment load . This behaviour is fairly consistent along the ice sheet margin, with no obvious spatial trend in the onset timing or duration of the sediment flux event (Fig. ). Both the low-H and high-H scenarios indicate a situation where sediments deposited during autumn and winter are remobilised in spring. Discharge is ultimately limited by supply, first as the local basal sediment is depleted, and then by waning supply from upstream sources.

The mixed-bed scenario is different, comprising broadly symmetric high-flux events during the melt season, with sediment discharge rate peaking more in line with peak hydrology flow. Grain size in this scenario is more erratic, and the detrital provenance during high flux events is almost entirely derived from the basal sediment layer (Fig. ). In this scenario the more consistent availability of sediment for some outlets leads to a dichotomy of behaviour in winter: Some outlets show ongoing dominance of basal sediment with fine (but highly variable) grain size (e.g. nodes 9598, 5066) albeit at low discharge rate. In contrast, more sediment-starved outlets are dominated by bedrock-derived sediments with grain size close to the mean (e.g. node 14353) (Fig. ).

Individual outlets can show significant interannual variability with alternating high- and low-flux seasons, while others show a more consistent series. High-flux seasons have a greater dominance of basal sediment detritus compared to low-flux seasons (Fig. ). With an erosion-rate limited system, such variations may occur where there is insufficient basal sediment accumulated over winter to sustain concurrent high-flux seasons. Moreover, the hydrology forcing is not consistent year-to-year and channels have variable extents, and sometimes are not formed for one or several years, and then re-established. This hydrologic variability impacts supply as “new” regions of the bed are exposed to sediment mobilising conditions. The more severe interannual variability seen in the low-H and high-H scenarios relative to the mixed-bed scenario points to variable access to the basal sediment layer as the main factor driving interannual variability.

In detrital provenance, (Fig. ) the discharge in all cases predominantly reflects the local bedrock in the vicinity of the outlet, with relatively little variation. On the basis of these major units, it would be difficult to distinguish between our scenarios, although major changes in the margin location or hydrological structure (for example, focusing of flow towards the north or south) could be observed. In some cases, minor units are seen only during the melt season, such as the volcanic units seen at nodes 6804, 2246 and 1731 (for locations, see Fig. ). These units, if observed in detritus, could act as markers of the extent and location of channelized flow.

This work set out to address three research questions, for which we may now review the progress made:

Can the sub- to pro-glacial sediment dynamics of the FLDIL be well represented in a hydrology-forced sediment system model?

The sediment system was adequately represented for timescales of weeks to several years, yielding internally consistent results across ensemble members in each scenario and between the low-H and high-H scenarios. The seasonal behaviours of the sediment system were particularly well resolved and generate variations in sediment yield and grain size characteristics that compare well to sedimentary observations from both murtoos and eskers . The interannual record is more limited due to the length of the model run, however, the observed variations in sediment characteristics are consistent with “flow switching” under varying hydrological forcing as a driver of the geomorphological complexity seen in nature .

Both longer-term and shorter-term impacts on sediment transport may be important and are not included in this modelling study. In particular, diurnal melt cycles were not included in the GlaDS forcing. Prior modelling studies show that including diurnal melt cycles can increase net sediment transport, in line with peak daily flow conditions . Diurnal cyclicity could further impact the onset timing and duration of seasonal events and the areal extent of the bed exposed to active transport processes. In this case, net sediment transport over annual timescales would likely be similar due to the model scenarios being strongly supply limited. For this model study all scenarios reach a consistent discharge rate within 10 years. Supra-decadal influences may include longer-term drivers of sediment supply, including retreat and advance of the margin location and re-routing of subglacial hydrology due to changes in ice dynamics. Compared to the static ice margin here, a margin evolution including such variations would be expected to access a more extensive area of the bed over time and sustain sediment supply for longer.

A limitation of the GlaDS-GraphSSeT approach applied is that the sediment evolution does not feed back into the hydrology conditions, although in nature the evolving subglacial sediment distribution may be an important factor for controlling hydrology, for example the filling of channels during esker formation affecting area and shape , and interactions linked to transitions between distributed and channelised flow . In this study we have very low rates of sediment mobilisation and deposition relative to the size of the channels, and the effect of these small volumes on channel geometry are negligible over the model duration. Ice marginal landforms may start as subaqueous fans and, if sediment delivery is sufficient, may grow to form Hjulström-type or Gilbert-types of ice contact delta depending on water depth , these deltas potentially impacting the upstream hydrology. At the outlet, we assume for this study that the depositional system can accept all sediment without building up any constriction; again the volumes are very small over the model duration. Although the very small sediment volumes involved over 10 years would not have any significant impact on the hydrological system, their accumulation over long periods may lead to systematic long-term changes that this study does not resolve.

Future work may look to refine the approach and resolve some of the deficits identified in this work.

The choice of the initial conditions was very significant for model evolution, in particular, the thickness and distribution of pre-existing sediment cover was in this case the critical factor controlling sediment supply. For accurate volume calculations, this initial condition must be known well, which is challenging for both modern subglacial systems and past examples. A strong control on initial conditions requires robust knowledge of the ice sheet and its subglacial hydrology, constrained by the subglacial environment and its past history and the characteristics of glacial deposits. We show for the FLDIL, where these factors are reasonably well known, the forward-model approach allows us to reject unsuitable initial conditions. For less well constrained examples, data-constrained model ensembles may be sought using inverse-methods that couple geological and geophysical data with the model outputs.

The realistic, but relatively narrow grain size distribution used here focuses this study on sands, consistent with the majority (∼ 80 %) of sediment observed in the region, but does not allow to analyse the origins of the extensive gravel deposits. In particular, we do not test if they are subject to significant fluvial transport, and over what timescales this might occur.

Interannual variations were detected for many outlets, however, the 10-year model run, after the “burn in” period, did not allow to resolve this over multiple cycles. A longer model run could allow this to be more fully tested, and to investigate more fully the interaction between variable hydrology forcing and evolving sediment availability.

Eskers and sediment-free zones extending hundreds of kilometers beneath the ice are not a feature of our model. Such features are inconsistent with a model domain fixed at the Salpausselkä II, and would instead require a time-transgressive margin evolution see; over longer time periods than considered here. Additionally, longer-term transport of coarse-grained sediment by englacial processes are likely important in order to provide a consistent source of coarse-grained sediment to the ice margin .