GlaDS is a widely used 2D continuum model of subglacial hydrology . It operates on an anisotropic mesh and includes a description of (i) distributed flow through a linked-cavity network system represented as a continuous sheet with variable thickness, and (ii) channelised flow through Röthlisberger channels that are located along element edges. Sheet elements exchange water with channels, and the cross-sectional area of these channels evolves through time due to the dissipation of potential energy, sensible heat exchange, and cavity closure rates due to viscous ice creep. Discharge in both the distributed (qs, m2s-1) and channelised system (Qc, m3s-1) is driven by gradients in the hydraulic potential, ∇ϕ (Pam-1), with steeper hydraulic gradients driving enhanced water discharge. In GlaDS, Qc is always non-zero along every internal edge, though few reach a discernible discharge. For ease of visualisation we follow in declaring edges where Qc≥1m3s-1 to be `meaningful channels' (e.g., Fig. c–e). Water in the distributed and channelised drainage components is assumed to be at the pressure melting point. An advantage of GlaDS for simulating jökulhlaup pathways is that the location of these meaningful channels is automatically determined and may evolve throughout a simulation without prior prescription: a behaviour not represented in previous 1D jökulhlaup models. Boundary conditions are applied in GlaDS either as a prescribed flux (typically for upstream inflow boundaries) or ϕ (for downstream outflow boundaries). We refer interested readers to Appendix  for an overview of key GlaDS equations, and to for a full description.

ISSM is a finite element higher-order ice-flow model which operates on anisotropic meshes and can be coupled to additional modelling components such as GlaDS . In addition to the ice-marginal lake modifications, which we describe below (Sect. ), the primary difference between the original GlaDS and the ISSM version used here is that rather than assuming water flow within the continuous sheet to be turbulent cf., we allow it to switch freely between laminar and turbulent flow according to the local Reynolds number, by implementing the transition parameterisation. We also prevent cavity expansion by ice creep when effective pressure, N<0 Pa (N=pi-pw, where pi is ice pressure and pw is water pressure) because GlaDS does not include mechanisms to represent uplift of basal ice (separation at the ice-bed interface) once water pressure exceeds ice overburden e.g.,. Without a physical representation of uplift, allowing cavity expansion by ice creep when N<0 Pa is undesirable and has been shown to inhibit channel formation in overpressurised regions of the bed .

To capture the influence of ice-marginal lake drainage on basal sliding, we include a two-way coupling between ice flow and subglacial hydrology. The GlaDS-derived effective pressure N modifies basal velocity Ub (ma-1) by governing frictional resistance at the ice-bed interface. In turn, the magnitude of Ub modifies the rate at which cavities open and thereby controls the thickness of the sheet (Eqs. –). These interactions featured in the model, but the use of ISSM–GlaDS here enables the resulting ice-dynamical variations to be fully captured in 2D. As in the model of , we assume water is at the pressure melting point.

Coupling between ISSM and GlaDS is a relatively new development, and the influence of basal friction parameters remains the subject of ongoing research e.g.,, though model response is known to be strongly influenced by the choice of friction law . We view this work as necessarily explorative, and do not undertake a full analysis of the modelling interactions between subglacial hydrology and ice dynamics. Rather, we seek to capture the broad influence that ice-marginal lake drainage cycles have upon ice dynamics. Accordingly, we present results using two widely applied friction laws: (i) a “Budd-style” friction law and (ii) “Schoof-style” friction law . The Budd-style friction law is implemented in ISSM in terms of basal stress with the form: 1τb=Cb2NrUbs, where τb is the basal stress magnitude (Pa), Cb is the Budd friction coefficient, N is effective pressure, Ub is basal velocity magnitude (the modulus of Ub), and both r and s are friction coefficients. The Schoof-style friction law is implemented in ISSM with the form: 2τb=Cs2Ubm1+Cs2Cmax⁡N1/mUbm, where Cs is the Schoof friction coefficient (note that Cs2, rather than Cs itself, is used to ensure a positive number when inverting for basal friction), m is a friction law exponent m=1/n, and Cmax is Iken's bound. In our experiments on a synthetic domain we use the Budd-style friction law (Eq. , Sect. ) to facilitate comparison to the model. In the Greenland experiment (Sect. ), with an observational constraint on Ub, we use the Schoof-style friction law (Eq. ). A Schoof-style friction law has been shown to more faithfully represent observed ice-dynamics with a GlaDS derived N than the Budd-style friction law . Following , and , we limit N to ≥ 3 % of ice overburden pressure with both friction laws to prevent instabilities as N→0 Pa.

In our model, a lake is defined as a specific Dirichlet boundary condition (see below) at n vertices along the boundary of the ice-lake contact. As in previous studies e.g.,, the lake depth, lh (m, Fig. ), evolves according to the continuity equation: 3dlhdt=1AL[Qin-Qr], where AL is the lake area (m2), Qin (m3s-1) represents extraglacial lake input (e.g., rainfall, surface runoff) and Qr (m3s-1) represents outflow from the lake (when Qr is negative there is inflow into the lake). Here, we also include the water exchange between the lake and the sheet unlikewho isolated cavities from the lake, so that Qr is given by: 4Qr(t)=∑i=1nqs,i(t)+Qc,i(t), or the sum of the normal component of the continuous sheet discharge qs, integrated along the ith boundary edge, qs,i (m3s-1), and channel discharge, Qc,i at the ice-lake contact vertices. As a boundary condition at and between vertices along the ice-lake interface, hydrostatic lake pressure fixes the hydraulic potential, ϕcontact to be: 5ϕcontact(t)=ρwgzb+ρwglh(t), at all times, where g is gravitational acceleration (ms-2) and zb is bed elevation (ma.s.l.). Therefore, lake filling/draining alters ϕcontact and thus ∇ϕ and sheet/channel discharge along the ice-lake contact boundary (see Eqs.  and ). Between neighbouring vertices which do and do not describe an ice-lake-contact boundary, the value of the Dirichlet boundary condition is interpolated. In the case of an ice–lake contact defined on a single vertex (as in the synthetic experiments described below), the sheet discharge in Eq. () is integrated over the combined half-edge length either side of the ice–lake contact. This maintains consistent dimensions (flux per unit length) and avoids double-counting in assembly. Finally, because GlaDS assumes that subglacial water temperatures are equal to the pressure melting point, we assume that the lake water also exists at the pressure melting point. The consequences of this decision are explored in Sect. .

At each new timestep, we solve for lh using a Picard fixed-point iteration scheme comprising an outer and inner loop. In the outer loop, we integrate Eq. () forward in time to estimate lh at the current (i.e., new) timestep given the current estimate of Qr (initialised from the previous timestep), using this lh to set ϕcontact. In the inner loop, GlaDS solves for the domain-wide ϕ until it satisfies a set convergence criteria, updating N as it does so . Following the inner loop converging, and given the estimated field of ϕ (X,Y), the model updates hs, Qc, and qs, before summing Qc and qs at lake boundary nodes to give a new estimate of Qr (Eq. ). The outer loop iterates until the maximum pointwise change in lh between successive iterations falls below a prescribed absolute tolerance.

In the reference simulation, the lake undergoes recurrent, stable, and asymmetric fill-drain cycles (Fig. a). Drainage is associated with a recurring flood with stable maximum amplitude and duration (Fig. b). The lake only partially drains in each flood cycle, and never reaches a height necessary to sustain flotation (N=0 Pa, Fig. f). In the initial 5 years of transient evolution, when the model adjusts to initial conditions, peak flood discharge increases with each successive flood, and the lake reaches successively lower post-flood lowstands (Fig. a, b). Thereafter, the flood cycles reach a stable periodicity and clockwise hysteresis (Fig. f) with flood cycles occurring approximately every 1.5 years. As will be discussed in Sect. , these simulated cycles are comparable to – yet with notable differences from – those of .

The evolution of the subglacial channel system through a single flood cycle is shown in Fig. c–e (an animated version is shown in Movie A1), and the width-averaged basal velocity response over the same flood cycle and the following one is shown in Fig. g. At the start of a flood cycle (Fig. c), as the lake approaches its highstand, channelised drainage is limited and ∇ϕ is controlled by the ice surface slope. Close to the ice–lake contact (X>9,km), contours of hydraulic potential form ridges pointing downglacier from the ice–lake contact. In GlaDS, sheet discharge, qs, is directed perpendicular to these contours of ∇ϕ and such ridges would induce sheet flow out of the lake. Four short (< 100 m), low-discharge (≥ 1 m3s-1) channels remain open following formation during the previous flood cycle, three at the terminus and one at the ice–lake contact.

During the lake filling phase, increasing lh drives ϕcontact up (Eq. ), resulting in a steeper ∇ϕ across the domain and promoting channelised drainage close to the ice–lake contact (Movie A1). During the lake drainage phase, initiated once Qr begins to rapidly exceed the lake refill rate, Qin (when lh≈75 m and t≈7.9 years), an arborescent channel structure originating from the lake extends across the full y-axis extent of the domain, perpendicular to the contours of ϕ, and deviating from the direction of ice-surface slope. Simultaneously, channels extend upglacier from the terminus, and contours of ϕ become increasingly convex downglacier. At the flood peak, channel tributaries coalesce into the three primary terminus channels which evacuate water from the system. As the lake level drops sharply during a flood, ϕcontact drops, reducing ∇ϕ, and in turn reducing Qc below the level necessary to sustain channels across the domain, including at the ice–lake contact. Once Qr<Qin the flood terminates and the cycle begins again. Each cycle takes ≈ 1.5 years with the rising and falling limb of the flood lasting ≈ 6 months.

Basal velocity, Ub changes significantly during a flood cycle (Fig. g), tracking the development of a significant channelised drainage system. The glacier accelerates during the year-long flood initiation phase before channelised drainage is widely established, then decelerates abruptly (in less than two months) as channels become well-established, approximately two months before the flood peak (Fig. g). Ice flow acceleration propagates from the ice–lake contact towards the terminus, with maximum velocity sustained within 2 km of the ice–lake contact.

Figure  plots the time series of lake height lh and lake discharge Qr from a total of 14 simulations across the maximum (red lines) and minimum (blue lines) parameter values, with the reference “mid-point” simulation in grey. The majority of simulations yield stable and recurring asymmetric lake filling and drainage cycles (Fig. a(i)–f(i)) with corresponding variations in Qr at the ice–lake contact (Fig. a(ii)–f(ii)). The parameter variations most strongly influence the amplitude of lake height and its rate of change, both of which influence the timing of flood initiation and peak flood discharge, with higher amplitude lake depth cycles leading to higher peak flood discharges .

Increasing channel conductivity, kc, in effect making channels more sensitive to ∇ϕ (see Eq. ), leads to lower lake highstands as the lake drains sooner. However, peak discharge is higher as the lake also drains more quickly and drains entirely. At maximum kc, there is a longer transient before the model reaches a fixed flood recurrence interval, draining at a higher frequency than the default value of kc until ∼11 years into the simulation, at which point flood cycles occur at approximately the same interval as in the default case (Fig. a(ii)). Conversely, at minimum kc, the capacity of channels to drain the lake is inhibited and flood onset is delayed. Because channels grow slowly, peak Qr is also dampened, leading to relatively long flood recurrence intervals (∼4 years, Fig. a(ii)).

Increasing the sheet conductivity, ks increases the capacity of the sheet to accommodate water and delays transmission to channels. This results in reduced lake highstands and delayed flood initiation as the onset of classical channel growth is delayed. Reducing ks increases the relative efficiency of channels and leads to more higher amplitude lake-fill drain cycles (Fig. b(i)), with shorter and higher discharge floods as a result (Fig. b(ii)). Changing hb has a similar effect to reducing ks, although at the maximum bump height the sheet becomes so efficient that it can accommodate all lake discharge and within 1 year the lake reaches a stable equilibrium where Qr=Qin (Fig. c(i)).

Changing the contributing width of the sheet to the channel (lc) or the representative horizontal spacing of cavities (lr) alters the relative efficiency of channelised and sheet drainage. Reducing lr and increasing lc reduces the lake highstand and lowers peak discharge, and vice versa when increasing lr and reducing lc. Finally, increasing the input of water into either the glacier via basal melt rate, Mr (Fig. f) or lake via Qin (Fig. g) enhances the amplitude of lh oscillations and in turn, peak flood discharge (Fig. f).

In all simulations, the maximum lake height remains below the maximum ice thickness (150 m) and does not surpass the effective pressure necessary to initiate ice flotation (N=0Pa). In most simulations the lake does not reach a lowstand below ∼ 10 m, only draining completely when kc=0.2m3/2kg-1/2, and when hb=0.05m.

The character of the fill–drain cycles has a strong influence on the spatiotemporal pattern of basal sliding across all experiments. In each case, modelled ice motion follows the reference case with velocity peaking before peak discharge is reached, and dropping rapidly thereafter during the falling limb of a flood (Fig. ). In general, shorter more intense jökulhlaup cycles lead to a more rapid but lower magnitude velocity response (e.g., Fig. a, e, f), whereas broad and less intense jökulhlaup cycles yield more gradual but higher magnitude velocity responses for a longer duration (e.g., Fig. b, d). The highest magnitude basal velocity (>300ma-1) is associated with the low kc case (Fig. d), characterised by muted lake filling–drainage cycles but a relatively high lake level (Fig. a(i)).

We discretise our Isunnguata Sermia domain derived from using an anisotropic mesh with an edge length of 150 m close to terminus, increasing to 2 km close to the upper margin of our domain at the approximate ELA (at H≈1600 m). The mesh is further manually refined to a typical edge length of 50 m close to the ice-marginal lake (Fig. c) in order to faithfully represent glacier geometry close to the ice margin. Bedmachine v5 is used for zb and H Fig. b, c. Our domain is clipped to exclude areas of the glacier where H<100 m to ensure numerical stability. Basal velocity, Ub, is initialised from the 2021 MEaSUREs annual surface velocity mosaic . Geothermal heat flux is taken from , and basal melt rates are taken from . The flow-law rate factor A within the ice-column is derived using surface ice temperatures from and assuming a temperature-dependent Arrhenius relation following .

Our model includes multiple boundary conditions in order to represent Isunnguata Sermia. For the hydrological solution, we impose two distinct Dirichlet boundary conditions at three locations: (i) at the two ice–lake contact boundaries ϕcontact is allowed to evolve with lh (see Eq. ) and (ii) a fixed atmospheric boundary is imposed at the glacier terminus. At all other vertices, we impose Neumann (zero-inflow) boundary conditions. The ice–lake contacts are manually delineated: seven vertices were identified at the western lake boundary and at the eastern margin we identified the lowest elevation vertex within a group of vertices close to the observed stream. At all ice-lake contact vertices we fix bed elevation equal to the minimum lake elevation (257.9 m a.s.l.). Because our model assumes a fixed lake area (AL=3km2) and cannot dynamically represent intermittent lake-glacier contact along the eastern margin, prescribing the same boundary conditions at the eastern ice–lake contact as along the western ice–lake contact provides a simple and consistent boundary treatment and ensures water can exchange between the subglacial system and the lake at both places. At the glacier terminus we set the Dirichlet boundary condition ϕ equal to atmospheric pressure at the lowest elevation boundary vertex (cyan triangle in Fig. c). Within the ice-dynamic solution, the boundary condition velocity is given by the initialisation Ub (Dirichlet), except at the western ice–lake contact where velocity is allowed to evolve freely (Neumann).

Model simulation necessitates knowledge of the spatial field of basal friction coefficient, C, which we assume to be time-invariant. Because we have an observational constraint on ice-surface velocity, we use a Schoof-style friction law (see Eq. , Sect. ), inverting for the basal friction coefficient Cs using an L-curve scheme following . We run an initial inversion for Cs using an empirically-derived N (assuming N=pi), before running a simple GlaDS-only model run over 10 years, which does not include an ice-marginal lake, surface melt or two-way coupling to ice dynamics. This allows us to derive a more realistic and spatially variable estimated field of N from which a second inversion for basal friction can be obtained. Such an approach has previously been demonstrated to yield more faithful basal inversions in a coupled hydrology model .

As climate forcing, we use the 10 km resolution daily Modèle Atmosphérique Régional (MAR, v3.14) runoff and accumulation data between 2007–2025 inclusive. Runoff is routed directly to the bed at all vertices. Whereas we keep Qin fixed in the synthetic experiments (Sect. ), here the extraglacial water input to the lake is likely to vary with time as a function of air temperature . To account for this, we use a simple temperature index as in, where Qin is given by: 6Qin=max(kT,1),T>01,T≤0, where the temperature-index coefficient k is equal to 2m3s-1K-1. We set Qin≥1m3s-1 to capture positive changes in lake elevation observed between melt seasons in Fig. g.

Since our objective is to assess the model's ability to reproduce the observed record of fill-drain cycles of an ice-marginal lake in Fig. g, we focus on achieving a simulation that yields the closest quantitative match. Guided by insights from the synthetic experiments (Sect. ), we choose the final parameters (Table ) as follows: a decrease in channel conductivity, kc, to 0.02 m32kg-12, an increase in sheet conductivity, ks to 0.05 Pas-1, a decrease in the basal bump height, hb to 0.05 m, and changes to both the channel sheet width, lc, and cavity spacing, lr to 10 m. Compared to the synthetic case, we also increase the englacial void ratio ev to 0.01. This has little influence on our synthetic case when tested to the same limits (Fig. ) because the model does not reach flotation pressures (Sect. ). However, in our Greenland experiments, increasing ev dampens the model sensitivity to sudden changes in meltwater input to the system and prevents unreasonably high water pressures (>300% of overburden pressure) in isolated pockets close to the domain boundary. Across sensitivity tests, parameter variations produce changes in the frequency, timing, and amplitude of modelled outputs without altering the qualitative drainage system behaviour.

We simulate the Greenland model forward in time from 1987 to 2025 (for 38 years at a 5 min timestep). This simulation consists of three phases, the first of which involves running the model (for twenty years of model time) to reach a winter steady state where the 95th percentile difference in sheet thickness between successive timesteps is below 1×10-6m. During the steady state phase, we allow the ice-marginal lake to freely evolve from an initial height of lh=70m, the mean lake height between 1987–2024 . The steady state phase allows the model to reach a condition approximating the anticipated wintertime hydrological conditions in which there is limited channel activity and water pressures across the domain are close to the flotation fraction (pw≈0.7–0.9pi). We then run a “spin-up” phase from 1997–2007 where we introduce a climate forcing by gradually increasing the 2007 runoff by 10 % each year. Following this spin-up, we run an unconstrained phase from 2007–2025 where we apply the full climate forcing and compare results with the 2007–2025 lake-level record.

The combined spin-up and unconstrained modelling phase is shown in Fig. d–g. The spin-up phase (dashed line in Fig. d–f) is characterised by three cycles of lake filling and partial drainage, each reaching successively higher lake highstands and having a shorter flood recurrence period than the previous one; these cycles track the development of a catchment-wide subglacial drainage system induced by successively more intense melt summers (Fig. d).

From 2007 onwards, once the model is subject to the full extent of annual melt-forcing, the lake repeatedly fills and drains with a recurrence interval between floods of ∼ 2–4 years. At the peak of a flood, using model year 2022 as an example, the simulated drainage system of Isunnguata Sermia is characterised by a channel that occupies the central trough and extends > 45 km upglacier (Fig. c). The majority of surface melt is routed through this “primary drainage axis” towards the glacier terminus, and an anabranching tributary system is visible ∼ 20 km upstream. Near the terminus, a second drainage axis, comprising a high discharge central channel and two lower discharge subsidiary channels, connects the western ice–lake contact to the primary drainage axis, converging ∼ 2 km from the terminus.

Each modelled jökulhlaup lasts 3–4 months, far exceeding the typical flood duration of 5–8 d observed at Isunnguata Sermia . Correspondingly, our modelled peak Qr≈ 100–200 m³ s⁻¹ are markedly lower than that of the 2022 jökulhlaup described by , which had an estimated mean discharge of at least 480 m³ s⁻¹ and a likely considerably higher instantaneous peak.

The lake fill–drain cycles are characterised by slow-rising limbs (between jökulhlaups) and abrupt drawdowns (jökulhlaups), though their hydrograph shape is irregular. Lake drainages typically occur at the end of a melt season. Lake levels then remain low during the subsequent winter, before filling in an irregular sawtooth fashion as in, with the lake filling more rapidly during summer melt seasons and remaining stable, or lowering slightly, during the intervening winter period. Lake drainage is triggered following the second or third melt season, and the cycle begins anew.

The timing and water height of modelled lake highstands in our simulation phase align well with those in the observed record, capturing variability in both flood timing and the proportion of lake drainage. Eight lake highstands are evident in the observed record (Fig. g) between 2007–2024, seven of which correspond with a simulated lake highstand to within ∼ 2 months and ±20 m (Fig. f). Further, there is good overall agreement in the lake lowstand heights, with the model matching observations of partial (e.g., 2010–2011) and complete lake drainage events (e.g., 2007–2008). This match is not perfect (e.g., 2013) and may be influenced by gaps in the observational record .

The evolution of channels during a typical jökulhlaup cycle (2019–2023) is shown in Fig. . On reaching lake highstand in 2019, the development of a significant channel connection between the lake and the primary drainage axis initiates a jökulhlaup and the drop in lake level (Fig. b). As lh and ∇ϕ lowers at the western ice–lake contact, the lake draining channel is starved of supply and shuts down by creep closure. At the lake lowstand, once ϕcontact is close to atmospheric pressure, drainage into the lake from the eastern ice–lake contact begins in summer 2020 (Fig. c) and the lake quickly fills to ∼50m. In the 2021 summer (Fig. d) and 2022 summer, no significant channels develop upglacier of the eastern ice–lake contact, and Qin accounts for the rise in lh. At the end of the 2022 melt season (Figs. , e), a relatively high discharge jökulhlaup drains the lake via a series of channels at the wester ice–lake contact and connected to the primary drainage axis. Though seven vertices along the western ice–lake contact are connected to the lake, discharge at the lake boundary is concentrated within a single high-discharge channel. Midway along the flood path, ≈ 2.5 km from the lake, this channel diverges and branches into multiple lower discharge channels running sub-parallel to the main lake-draining channel, a behaviour also evident in the synthetic runs (see Sect.  and Fig. c–e).

Turning to the simulated basal velocity field, Ub=ub,vb, Fig.  shows the mean components of ub and vb between 2014–2025 and the anomalies, generated by subtracting the overall mean velocity from the mean velocity during a jökulhlaup, associated with the modelled 2022 flood (Fig. a–d) and the observed 2023 flood (Fig. e–h) from . Each velocity field is interpolated onto the same model mesh. The x- (ub) and y- (vb) components of modelled basal velocity show a mean value (Fig. a, b) that agrees closely with the mean observed velocity (Fig. e, f), although our model predicts a higher mean vb>50ma-1 close to the lake margin (Fig. b) than the vb≈25ma-1 observed Fig. f. The modelled and observed lake-drainage anomalies (Fig. c, d, g, h) agree well in terms of the overall pattern of slow-down and acceleration, but differ in detail and magnitude. In our model, changes in ub and vb during a jökulhlaup are lower in magnitude, and are more spatially restricted than observed (Fig. ). During a modelled jökulhlaup, ub decreases (corresponding to a westward acceleration) by 20ma-1 upglacier of the lake, and vb increases (corresponding to a northward acceleration) by ≈15ma-1, with the acceleration restricted to close to the lake boundary (Fig. d). In contrast, the observations indicate a more widespread westward acceleration across the full extent of the glacier tongue during a jökulhlaup, with ub decreasing by >100ma-1, and a northward component where vb increases by >100ma-1 towards the lake (Fig. ). The observed vb is also more spatially variable than the equivalent model output, with ice accelerating towards the ice-marginal lake across a more extensive portion of the lake-adjacent glacier. Observations additionally identify a southward acceleration in vb close to the glacier terminus that is not reproduced in our model.

The overall agreement between the synthetic results in this work and that of earlier work is unsurprising; the lakes are governed by closely related equations and the underlying hydrology models assume broadly similar physics. However, our inclusion of a spatially variable (2D) hydrology solution, and our decision to permit hydraulic connectivity between the lake and the adjacent linked cavities, do present behaviours not evident or possible to explore with the 1D model. In their model, a single channel is connected to an ice-marginal lake. The channel itself is connected to a linked-cavity system with longitudinal (but not transverse) hydrological variations, and the two drainage components maintain independent effective pressure. Water exchange between the drainage elements occurs at a rate proportional to the pressure differential as in. Crucially, in their model, the lake is hydraulically isolated from the linked cavity system.

In our model, with subglacial hydrology resolved in two dimensions, pressure continuity is maintained across all channel margins (i.e., at all vertices where channels contact their adjacent sheet); water exchange occurs at the rate necessary to maintain this condition and it is the hydraulic potential gradient ∇ϕ across neighbouring drainage elements that drives overall water flow. At the ice–lake contact, where the hydraulic potential at the ice–lake contact ϕcontact evolves through time (Eq. ), we maintain pressure continuity by allowing the linked cavities to exchange water with the lake, with sheet discharge integrated over an edge length centred on the single ice–lake contact (see Sect. ). This approach has been shown to match well to observations of jökulhlaups elsewhere, when flux is prescribed across the ice–lake contact e.g.,. However, as with the assumption of that the distributed system is entirely isolated from the lake, we recognise that our opposing assumption of continuous exchange between the lake and adjacent sheet is also an idealisation arising from our model formulation. In reality, it is likely that the ice–lake contact boundary is characterised by complex and highly variable conductivity in both space and time, complexity which is not possible to resolve within our current model.

The clearest distinction between our model and that of , is in the different response to changes in basal melt, Mr denoted by cavity water supply, Mc, in . In their model, an increase in basal melt reduces the peak flood Qr, whereas here, increasing the basal melt rate (substantially, from a base of 1 to 5 m a⁻¹) nearly doubles peak Qr from ≈50 to ≈90m3s-1 (Fig. f(ii)). In the model, basal melt is fed directly to the linked-cavity system. With independent effective pressures in the channel and the linked cavity system, higher basal melt lowers the effective pressure in the linked cavities only. Because sheet to channel exchange occurs at a rate proportional to the effective pressure difference, meltwater input accelerates cavity–channel exchanges, promotes earlier channel growth and triggers lake drainage at lower lake levels, with a lower peak Qr. In our model, with a lake connected to the continuous sheet, elevated basal melt lowers the system-wide effective pressure, reduces ∇ϕ at the ice–lake contact, inhibits lake leakage into the glacier and allows the lake to both fill more quickly and reach a higher lh before drainage. Once triggered, the greater water supply sustains enlarged channels with a higher peak Qr that drains the lake to a lower lh compared to the reference basal melt rate case. This effect in our model is asymmetric; reducing Mr below the baseline has negligible effect on flood recurrence interval or peak discharge (Fig. f(ii)), suggesting that below a threshold basal melt rate the glacier-wide effective pressure is governed primarily by the lake drainage cycle itself rather than by the ambient melt supply, and the effective pressure (and in turn basal sliding) becomes effectively decoupled from meltwater availability at the bed. Our results indicate that in glacier–lake systems characterised by relatively low basal meltwater availability, lake dynamics may become the dominant control on glacier sliding velocity over time.

Jökulhlaup characteristics and basal sliding response are strongly controlled by drainage efficiency. In our synthetic experiments, drainage requires an increased highstand and drains to a higher lowstand, with a longer, less intense flood if the efficiency of the distributed sheet is raised relative to the channelised system; for example, by raising the sheet conductivity (ks Fig. ), raising the basal bump height (hb Fig. ), or decreasing cavity spacing (lr, Fig. e). Longer duration, less intense jökulhlaups drive higher and more sustained increases in Ub. With a longer rising limb, greater subglacial discharge is accommodated within the sheet, sustaining the low effective pressure necessary to enable fast ice flow. Conversely, when channelised drainage is relatively efficient (e.g., Fig. a, d, f), jökulhlaups are shorter in duration and peak drainage is higher. In turn, the Ub response is shorter and lower in peak velocity because efficient drainage develops more easily, increasing effective pressure over wide-tracts of the glacier bed and inhibiting basal sliding. The parameters which control drainage efficiency in GlaDS and other comparable hydrology models are known to be highly uncertain . Additional observations of drainage efficiency, extent, and timing, are crucial for further reducing this uncertainty.

Subglacial flood channel locations are not predefined in our model; they emerge as part of the solution (Sect. ). In the synthetic case, the local control of the lake on ∇ϕ generates channels offset by 20–40° relative to the central axis of the glacier (Fig. c–e). This lateral expansion of the jökulhlaup through the subglacial system results in a distinctive pattern of ∇ϕ and an arborescent channel propagation which expands outwards from the central ice–lake contact position during a flood before collapsing into a few flow-parallel channels close to the glacier terminus. This behaviour contrasts sharply with the ice-flow parallel channels expected in the absence of significant lateral topographic variation whereby the basic hydraulic potential is governed by ice surface slope alone e.g., see Fig. 2 in. Lateral propagation of flood water within the subglacial system during a jökulhlaup has been invoked as the cause of short-term lateral deviations in GPS measurements towards the ice-margins during jökulhlaups at Gornergletscher, Switzerland .

Coupling ice-marginal lake evolution to a 2D description of subglacial hydrology is an essential advance over previous work, because the resulting model can be applied to specific glaciers to study jökulhlaup behaviour in a realistic manner. Model application to Isunnguata Sermia (Sect. ), shows that the evolution of lh through time in our model (Fig. f) closely matches the observed record reported by Fig. ,g. Between 2007 and mid 2024, eight significant floods from the ice-marginal lake are recorded from remote sensing records (Fig. g). Of these floods, seven are hindcasted by our model (Fig. f), closely agreeing with observations in terms of timing and the amplitude of change in lh. The eighth flood (in 2023), not hindcasted by our model, is characterised by a short return interval relative to previous floods (see Sect. ).

We caution that the successful hindcasting of lh here is not in and of itself validation of our modelling approach or parameter choice. The nonlinear evolution of lh and Qr through time (Fig. f) is driven in part by forcing our model with a realistic, time-varying climate, which drives both the lake input Qin and ∇ϕ through the glacier-wide melt input to the bed (see Sect. ). In contrast, flood cycles are cyclical in our synthetic model when the system is forced with constant basal melt and lake input (see Sect. ). Elsewhere, nonlinear variability in flood discharge has been shown to arise when a transient climate is imposed . Given a fixed lake area and realistic inflow, it is reasonable to expect that the lake will eventually drain once it reaches a highstand, irrespective of the state of the subglacial system and the nature of lake–glacier coupling. However, the timing, magnitude and variability of drainage depend on the evolving state of the subglacial system adjacent to the lake. In our model, drainage occurs when the hydraulic potential gradient across the ice–lake contact becomes sufficiently large to drive substantial water flux from the lake into the subglacial drainage system. The resulting reduction or reversal in hydraulic potential gradient across the ice–lake contact then promotes flood termination. Thus, while the imposed climate and lake geometry contribute to the successful hindcasting of lh, the timing and character of individual drainage events emerge from the coupled lake–subglacial hydrology.

As well as a close agreement in terms of lh, our results corroborate additional observations from Isunnguata Sermia. A series of elevation anomalies identified close to the glacier terminus were interpreted to have been caused by the filling and draining of subglacial lakes . Three subglacial lakes labelled SGL 1–3 in lie due south of the primary drainage axis Figs. c, and b–e and a fourth, more recently identified labelled Anomaly 1 in, lies to the southwest of the ice-marginal lake (Figs. c and b–e). Between June and September 2019, Anomaly 1 experienced rapid subsidence (up to 38 m) after a long period of slow uplift; a pattern consistent with the fill–drain cycles of active subglacial lakes elsewhere . The sudden subsidence of Anomaly 1 was coincident with the complete drainage of the ice-marginal lake during 2019 (see Fig. g), and the jökulhlaup is thought to have triggered the release of water along the subglacial flood-path . In our model, the predicted jökulhlaup flowpath passes directly through Anomaly 1 (as shown in Figs. c and b–e) supporting the hypothesis that ice-marginal lake drainage is a triggering mechanism for the sudden drainage of subglacial lakes.

Finally, our model results reproduce some of the observations relating to ice-dynamics at Isunnguata Sermia. Both our model and observations show acceleration along the primary flow direction during a flood and a component of acceleration towards the lake concentrated along the western ice–lake contact (Fig. ). attributed the observed acceleration to (i) ice inflow into the lake as a result of a sudden drop in water level and change in force balance at the ice front, and (ii) a transient reduction in basal friction as the sudden influx of flood water overpressurises remaining channels and the nascent distributed system becoming re-established at the end of the melt season. In our model, flood duration is longer than observed (see Sect. ), resulting in a longer duration change in effective pressure and thus a more muted velocity response than is observed (Fig. ). In addition, the boundary conditions in our ice-dynamic solution remain fixed through time and we do not simulate the change in frontal forces as the lake drops during a flood; in effect, our Neumann boundary condition corresponds to a permanently empty lake (see Sect. ). As a result we overestimate the background vb into the lake (Fig. b) and underestimate the vb anomaly associated with lake drainage (Fig. d). Nonetheless, our modelling does reproduce limited acceleration into the lake during a flood as a result of low effective pressures close to the ice–lake contact during a jökulhlaup.

The primary discrepancy between our model and observations is that, as in the earlier 1D work e.g.,, our experiments produce unrealistically long jökulhlaup durations extending multiple months. This stands in clear contrast to the days–weeks more typically associated with jökulhlaups e.g., and is much longer than the 5–8 day jökulhlaup duration observed at Isunnguata Sermia itself . Results in Sect.  are presented from one model run at Isunnguata Sermia, chosen as the best match to lake level through time (Fig. ); parameter testing was not exhaustive, and untested combinations may improve agreement with jökulhlaup duration. However, we expect the discrepancy to be attributable, at least in part, to limitations in our modelling setup (Sect. –) and with GlaDS more generally .

As a continuum model, representing smoothly distributed drainage properties, GlaDS is not ideally suited to represent rapid, localised changes in subglacial water pressure or hydraulic connectivity . In ISSM–GlaDS, there is no physical representation of elastic uplift of ice once water pressure exceeds ice overburden pressure. This, together with our decision to prevent cavity expansion when effective pressure is negative (effectively fixing maximum sheet thickness to the basal bump height), means that our model is unable to replicate the instantaneous changes in sheet transmissivity expected once water pressure exceeds ice overburden pressure e.g.,. The englacial void ratio in our Greenland experiment (Table ) partially compensates for this missing phenomena and buffers water pressure during flood propagation, producing weeks–months long flood durations (e.g., Fig. e, f). Work by on subglacial lake Grímsvötn, albeit with a prescribed rate of flux from the lake, found that a model able to represent sheet thicknesses on the order of ten metres can explain rapid escalations in jökulhlaup discharge observed over hours to days at the site . Recent repeat satellite observations identified ∼1m of uplift along a jökulhlaup pathway, suggesting that instantaneous ice uplift in a high water-pressure regime is crucial to enabling high jökulhlaup discharge . A simple model of elastic deformation has been included in similar hydrology models elsewhere e.g.,, and is worth exploring in future work. Here, with no physically-informed representation of rapid cavity expansion, our model is unable to reproduce this specific mode of fast flood onset and instead floods primarily evolve as a result of channel expansion.

However, uplift alone is unlikely to account for our excessive flood durations because our model also underpredicts the rate of channel expansion. In our synthetic experiments on a relatively small glacier resting on a sloping bed (Sect. ), the steep hydraulic gradient means that water is efficiently evacuated from the system and water pressure does not exceed flotation at the outlet (Fig. f), precluding uplift triggered rapid drainage. Even with a model of elastic uplift, floods in our synthetic experiment would therefore remain unreasonably long.

In our model, we assume water in the subglacial system is always at pressure melting point (Sect. ). In turn, we effectively assume that the temperature of water entering the subglacial drainage system from the lake is also at the pressure melting point. Observations indicate ice-marginal lakes can exceed this temperature e.g., and that high lake water temperature increases the heat energy available for channel expansion via wall melting, resulting in higher peak discharge and shorter duration floods see Fig. 6a in. Inclusion of this factor in our model may simultaneously resolve our underprediction of flood peak discharge and our overprediction of flood duration (discussed above). On an alpine glacier with an analogous supraglacial lake, and using the equations for channel enlargement, found that increasing the lake temperature from 1 to 4 °C increases peak discharge from 380 to 960 m3s-1.

Besides the limitations of GlaDS inherited by our model, we also make other assumptions which influence our Greenland model outcomes and likely dampen the peak flood discharge. Lake area, AL remains fixed in time, reducing the lake to a vertically-walled reservoir and ensuring that the scaling relationship between flood discharge and lake depth remains constant. This simplification likely biases our lake drawdown rates towards being too low; in reality as a lake drains its areal footprint decreases, which would amplify the rate of lake-level decrease during a jökulhlaup. We assume a perfect connection between the ice-marginal lake and the distributed sheet, which may dampen the flood hydrograph by allowing the lake to constantly leak into the subglacial system. In our ice dynamics model, we fix ice thickness through time. This was a deliberate modelling choice made in the absence of a detailed understanding of lake calving dynamics to avoid the unphysical build up of ice at the lake boundary where surface mass balance alone is unable to account for observed mass balance changes. As a result, we do not capture the decreasing trend in peak lh over time (Fig. g), which is attributed to glacier thinning close to the terminus of Isunnguata Sermia . Glacier thinning, and a reduction in lh necessary to initiate flooding, may contribute to the short return interval of the 2023 flood , which is notable as the only flood in the observed record without a corresponding match in the modelled record. Future work will seek to implement a calving law, to allow ice thickness evolution necessary to account for changes in H through time. Finally, as described in Sect. , our Neumann boundary condition in our ice-dynamic solution at the western lake margin remains constant in time, effectively corresponding to a permanently empty lake. Recent work with a prescribed lake depth record has shown changing lake level can promote complex feedbacks in lake-terminating glacial systems as a result of the loading force a water body imparts on a vertical ice front .

Notwithstanding these limitations, and recognising that our model successfully reproduces flood timing across a multi-decadal record without tuning to individual events, we view the discrepancies in duration and peak as a significant but tractable modelling challenge, rather than a fundamental limitation. The recently described finite-difference formulation of provides a promising avenue for future development, incorporating an implicit representation of ice uplift, pressurised flow, and free-surface flow. The framework presented here is readily extensible, and the lake model is parameterised by relatively few, geometrically-interpretable quantities (e.g., lake area, ice thickness at the ice–lake contact, and extraglacial inputs) meaning the model could be straightforwardly applied to other ice-marginal lake systems, including systems with multiple lakes, without modification of the numerical scheme. In addition, our results suggest that periodically draining ice-marginal lakes may represent promising alternative benchmarks for evaluating machine-learning-based emulators of current and future models of subglacial hydrology e.g.,. Ice-marginal lakes offer spatially integrated, remotely observable records that are less impacted by the substantial point-to-point variability that characterises borehole and moulin datasets which have been previously used to evaluate emulators of subglacial hydrology e.g.,.