While subglacial hydrology is known to play a role in glacial dynamics on sub-annual to decadal scales, it remains unclear whether subglacial hydrology plays a critical role in ice sheet evolution on centennial or longer timescales. Furthermore, several drainage systems have been inferred, but it is unclear which is most applicable at the continental/glacial scale. More fundamentally, it is even unclear if the structural choice of subglacial hydrology truly matters for this context.

Here we compare the contribution to the surge behaviour of an idealized Hudson Strait-like ice stream from three subglacial hydrology systems. We use the newly updated BAsal Hydrology Model – BrAHMs2.0 – and provide model verification tests. BrAHMs2.0 incorporates two process-based representations of inefficient drainage dominant in the literature (linked cavity and poro-elastic) and a non-mass-conserving zero-dimensional form (herein termed leaky bucket) coupled to an ice sheet systems model (the Glacial Systems Model, GSM). The linked-cavity and poro-elastic configurations include an efficient drainage scheme while the leaky bucket does not. All three systems have a positive feedback on ice velocity, whereby faster basal velocities increase melt supply. The poro-elastic and leaky-bucket systems have diagnostic effective pressure relationships – only the linked-cavity system has an additional negative feedback, whereby faster basal ice velocities increase the dynamical effective pressure due to higher cavity opening rates. We examine the contribution of mass transport, efficient drainage, and the linked-cavity negative feedback to surging. We also assess the likely bounds on poorly constrained subglacial hydrology parameters and adopt an ensemble approach to study their impact and interactions within those bounds.

We find that subglacial hydrology is an important system inductance for realistic ice stream surging but that the three formulations all exhibit similar surge behaviour within parametric uncertainties. Even a detail as fundamental as mass-conserving transport of subglacial water is not necessary for simulating a full range of surge frequency and amplitude. However, one difference is apparent: the combined positive and negative feedbacks of the linked-cavity system yield longer duration surges and a broader range of effective pressures than its poro-elastic and leaky-bucket counterparts.

The role of subglacial hydrology at timescales longer than multiple decades and at ice sheet spatial scales is unclear. Previous studies have inferred that subglacial hydrology plays a strong role in internally (e.g.

Several subglacial hydrologic systems have been conceptualized

We ask a basic question: does subglacial hydrology matter on longer than decadal timescales? And if so, to what extent are the structural details of the hydrological system important for this context, especially given the rest of the system uncertainties? Taking a modelling approach, we focus these broad questions on the following: is subglacial hydrology needed to capture Hudson Strait-scale ice stream cyclicity? If so, should effective pressure be dynamically determined – based on fully mass-conserving lateral drainage? Or does a zero-dimensional meltwater volume balance with a diagnostic pressure closure suffice? Turning to the parametric uncertainties, which are most important?

Previous model-based tests of Hudson Strait ice stream surging (e.g.

Here we examine the contribution to ice sheet internal oscillations from the three most dominant forms of distributed subglacial hydrology – linked cavity

Simple configurations make system behaviours more interpretable (e.g.

Below, we first test the BAsal Hydrology Model (BrAHMs). This includes a demonstration of the mass conservation, convergence, and symmetry of BrAHMs2.0 and verification of its solutions against another prominent model, the Glacier Drainage System model (GlaDS,

In the context of continental-scale ice sheet modelling, resolving individual drainage elements and multiple topologies present within the domain is not computationally feasible. In this section we provide a brief overview of some structural choices made by others and present the options compared in this study, beginning with the current understanding of subglacial hydrology and progressing to increasingly approximate representations of it.

Water in the subglacial system flows either through inefficient drainage systems (pressure

In the inefficient drainage regime, flux and water pressure rise together. Several inefficient drainage systems have been theorized: thin film, poro-elastic media, and linked cavities. Of these, poro-elastic and linked-cavity systems (e.g.

In the poro-elastic formulation, water can drain through the pore space of some permeable surficial material (e.g. till). Increasing subglacial water pressures expand the pore space and modify the permeability of the porous medium to flowing water. The conceptual basis for this system is examined in greater detail by

In the linked-cavity system, cavities within the base of the ice open up as basal ice flows over and around bed protrusions – fast flow and larger objects beget larger cavities

The substrate type that controls which inefficient system dominates – i.e. till cover and roughness – is variable

The contrast between kilometre-order or larger model scales and the metre-order or smaller process scales permits inefficient flow to be described as a continuum at the macro scale. On the macro scale, flux is related to water thickness and hydraulic gradient as follows

Water sheet thickness is a continuum property used to describe the average amount of water in a grid cell. Changes in water thickness are given by the fluxes and the aggregate of sources and sinks,

Pressurized subglacial water flows through the pore space of a layer between ice and bedrock, conceptualized as the interstitial space between till grains. As water pressure increases, the permeability of the porous medium rises. Water pressure is related to subglacial water thickness by a non-linear function using pore-space saturation (Eq.

The Darcy flow law is Eq. (

As ice flows over protrusions in the bed, cavities open in the lee side. The faster ice flows and the higher the protrusion, the greater the opening rate. The weight of the overbearing ice acts to close the void through viscous creep. The tradeoff between these two rates determines the net cavity size change rate. These cavities are linked through smaller connections and form a drainage network whose throughput is controlled by orifice size and system tortuosity. As water flows more quickly in the drainage network, wall melting due to frictional heating at the ice–water interface further opens cavities and the interconnecting orifices, forming a more efficient system. The Darcy–Weisbach flow law for turbulent flux depends on the hydraulic gradient and subglacial water thickness. The pressure closure is based on cavity opening and closing velocities and mass balance. The Darcy–Weisbach flow law is Eq. (

wall melting term (

opening from sliding over bed protrusions (

closing due to overburden pressure (creep) (

In the efficient drainage regime, flux and water pressure are inversely related. Flux in the efficient system occurs in subglacial tunnels incised into overbearing ice

The conceptual basis for the efficient flow model herein is the R channel, which evolves out of the inefficient system based on high fluxes.

Table of parameter names, descriptions, their numerical ranges, and the subglacial hydrologic system they parameterize used in the ensembles for this study. LC corresponds to the linked-cavity system, PE to the poro-elastic system, and LB to the leaky-bucket system.

The model used here is a fully coupled system of hybrid SIA/SSA (shallow ice approximation, shallow shelf approximation) ice physics

The subglacial hydrology model – BrAHMs2.0 – is an extensive update to version 1.0

Table of subglacial hydrology configurations showing the drainage law used, whether the efficient drainage system is coupled in, and what effective pressure is used. LC corresponds to the linked-cavity system, PE to the poro-elastic system, and LB to the leaky-bucket system.

n/a: not applicable

While in the linked-cavity model the hydraulic conductivity is a single parameter, in the poro-elastic model

Numerically, hydraulic conductivity in both the poro-elastic and linked-cavity formulations is defined at the cell centres and is a function of cell temperature relative to the pressure melt point (

In order to assess the importance of transport vs. pressure determination in surging, we implement a non-mass-conserving zeroth-order leaky-bucket scheme: a constant drainage rate (

Fully modelling the process of efficient drainage of water through the channel system would require very short time steps due to Courant–Friedrichs–Lewy (CFL)

For details on the numerical solver used here, readers are invited to read Appendix

gives symmetric solutions given symmetric boundary conditions,

converges under increasing spatial and temporal resolution at a rate commensurate with the discretization schemes,

conserves mass,

gives similar solutions to another model using similar physics

The basal velocity is from either a hard- or soft-bed sliding rule. For the hard bed the basal sliding rule is a fourth-power Weertman sliding law

Using a simple setup without externally driven variability from topography, a complex land–sea mask, and an unsteady climate, system behaviour is due to the initial transient response and internal feedbacks. Our Laurentide Ice Sheet square (LISsq) setup includes broad features of the North American bed (Fig.

This map of the LISsq bed configuration shows the extent of the domain and the position of the Hudson Bay and Strait and southern soft beds. Gray hatched regions are hard bedded, beige dotted regions are soft bedded, and blue represents water where ice is ablated.

LISsq aims to probe the effect of large-scale hard- to soft-bed transitions characteristic of North America. This simplified setup allows separating out the internal feedbacks from the externally forced elements (e.g. variability from real topography and a land–sea mask and unsteady, spatially varying climate). The shorter run times of this setup also allow larger ensembles, giving a better probe of the parameter space. The simplicity helps with model verification as any variability in the model stems purely from the encoded physical processes.

The majority of the inferred Late Pleistocene Laurentide substrate has been hard bedded

The LISsq climate prescribes a linear background temperature trend with lapse rate feedback. The annually averaged surface temperature,

These temperatures are then used together with a positive degree day scheme (PDD) to simulate net seasonal contribution to accumulation and ablation for an annual average temperature. The positive degree day sum assumes a 100 d melt season length with temperatures 10

We use a subset of the full-featured GSM for this setup. Here we omit glacial isostatic adjustment, surface meltwater drainage, sediment transport and production, and ice shelves with grounding-line flux and calving model. This is in order to clearly show the effect of hydrology feedbacks on ice flow and ice thermomechanics.

In this section we justify chosen parameter ranges based on physical and heuristic arguments and current understanding in the literature.

The range of values appropriate for hydraulic conductivity varies according to whether the drainage system is assumed to be poro-elastic or linked cavity or whether the flux is assumed to be laminar (Darcian) or turbulent (Darcy–Weisbach). Hydraulic conductivity is not truly known at the continuum-level macro-scale. Here we use a range based on bounding subglacial hydrologic flow velocities, typical hydraulic gradients, and subglacial water thicknesses.

The velocity of water flow in the subglacial channel end-member imposes an upper bound on the linked-cavity end-member flow velocity in the bifurcated channel–linked-cavity system

Fast ice velocities (e.g.

Range of linked-cavity hydraulic conductivities based on basal water flow speed, hydraulic gradient, and basal water thickness ranges in the text – using

The height of bedrock protrusions relevant to subglacial cavity formation and its spatial variation lacks assessment in the literature and justified values are difficult to come by. The height of these protrusions, or terrain roughness, affects several basal processes in glaciated regions, including heat generation in basal ice, sliding, subglacial cavity opening, and bedrock quarrying. Length scales relevant to subglacial cavity formation have been estimated from chemical alteration of bedrock (the deposition of calcium carbonate precipitates)

In deglaciated areas with bed access, quantifying roughness at the ice sheet scale is a non-unique problem and measures abound. For example: standard deviation of elevation, power spectral density of elevation, and local bed slope. These are relative measures which do not identify the typical prominence of roughness features in a domain. What is needed for modelling linked cavities is the average height of bedrock protrusions relevant to the cavity scale (itself uncertain) at given wavelengths. How these heights vary spatially for previously glaciated regions has not been assessed. Identifying this as a gap in the current glaciological literature, we adopt similar scale values and probe a wide range in order to capture ice sheet sensitivity to the scale of cavity-forming bump height. As stated above,

This parameter is used to interpolate between a conducting (at 0

The flux threshold switch from inefficient to efficient drainage is given by the ratio of cavity opening due to sliding versus wall melting from viscous heating

This is the value used to normalize the effective pressure in the basal sliding velocity calculation and is set based on typical effective pressures. Effective pressures greater than this parameter values should slow sliding and less than that should hasten sliding. We set this range to

The soft and hard sliding factors used in Eqs. (

A range of

Precipitation and temperature values extracted from PMIP4

To understand the effect of hydrology, ensembles for different model configurations are compared (Table

Ice sheet geometries vary widely among runs for all model configurations. Maximum ice thickness ranges from 0 to

The importance of hydrology parameters for determining ice sheet geometry can be probed with sensitivity analysis. Local sensitivity analysis methods neglect interaction terms important for studying feedbacks in coupled models and so are not applicable here

Cumulative kernel density function difference sensitivity metric for the most sensitive parameter,

We develop a novel non-parametric method to measure sensitivity: we assess ice sheet geometry sensitivity to parameters by comparing the original uniform input parameter distribution with the parameter distribution corresponding to the sieved geometries (limiting the ensemble to those within geometric bounds). The non-parametric nature alleviates the need to make assumptions about the underlying parametric distribution class (e.g. variance is a normal distribution parameter). Using the impact of a sieve on parameter distribution to measure sensitivity means that assumptions about the sampling methodology are not required and that successive sieves can be applied to the ensembles to measure different aspects of model sensitivities. For example, in Sect.

Parameters ranked by relative sensitivity by sieving for geometry (

The sensitivity metrics for all parameters in Fig.

The ranked parameter sensitivity for each model in Fig.

The most influential hydrology parameter in the LC setup (Table

The two most obvious measures of internal oscillation are amplitude and period. This highly non-linear system does not exhibit sinusoidal behaviour, but we can pick surge metrics which approximate these measures. To this end, each surge type was evaluated in two ways – the number of surge events (an indication of periodicity, the number of red dots in Fig.

Evolution of the ice sheet and idealized Hudson Strait ice stream showing repeated surge events and how metrics are extracted from a sample run. HS basal speed is shown as dashed blue line – which is used to pick surge peaks and estimate prominences – along with the area fraction of warm-based ice within the HS (dash–dotted green line) and its Hudson Bay source region (solid orange line). The red dots show picked event peaks, the vertical purple lines give their “strength” (prominence), and horizontal purple lines show the event duration. HB represents Hudson Bay, and HS represents Hudson Strait.

The background sliding speed of the actual Hudson Strait Ice Stream (HSIS) in the non-surging state is unknown. While this study does not aim to replicate the actual Hudson Strait (HS), we are studying the behaviour of an ice stream and sheet with similar dimensions to the HS and Laurentide ice sheets. As such, labelling and measuring the strength of a surge event needs to be agnostic of quiescent-phase conditions between events. Ice stream acceleration at scales comparable to the HS has not been observed in the modern period. Though significantly smaller than the HSIS and its catchment, the Vavilov ice cap did accelerate from 12 to 75 m yr

A typical run with surge events which passes the

Sieves used to select runs from each ensemble for analysis.

The sieves used for sensitivity analysis are shown in Table

Surge event metric distribution across parameterizations by model configuration for runs in the main geometry sieve (

The duration of HS surge events highlights a difference between the three hydrologies: the linked-cavity system yields longer duration events and the trend in duration with increasing event frequency diverges between the linked cavity and the other two hydrology systems. As the duration of surge events necessarily depends on the frequency of those events (having more events in a time period decreases the maximum possible duration of those events), we examine surge duration as a function of the number of events (as shown by the horizontal purple lines in Fig.

This figure shows the trend in surge event duration at different frequencies (number of surge events in a run). The scatter plot shows trends in median duration with an increasing surge frequency. A 10-run minimum was required at each surge frequency level. The no-hydrology setup falls below this requirement after the three-event level, and the linked-cavity setup shows a divergence from the other two subglacial hydrology systems at this point.

As the frequency of surge events in each run increases, the median duration of surges in those runs stays largely flat, perhaps decreasing slightly for both the poro-elastic and leaky-bucket hydrologies. This is not so for the linked-cavity system: the duration of surges increases up to seven surge levels, where it roughly doubles that of the poro-elastic and leaky-bucket hydrologies. This relationship is stronger still when selecting thinner ice sheets with a mean maximum thickness between

Applying a sieve on surge frequency in addition to the geometry sieve (

Surge frequency sensitivity to model parameters. Parameters are ranked by relative sensitivity by sieving for surge frequencies (3 to 12 events) relative to the geometry sieve (

In Fig.

Two-dimensional logarithmic histogram of effective pressure and velocity solutions for all warm-based points across the parameter–space–time domain. Fields are output every 100 years. Marginalized distribution for effective pressure and velocity shown along side, sharing the respective axes.

The increased incidence of surge behaviour in the hydrology cases is not due to increased sliding – the no-hydrology ensemble exhibits higher basal velocities than the three hydrology ensembles in Fig.

As we show above through sensitivity analysis and ensemble comparison of surge frequency and amplitude, subglacial hydrology is an important process that contributes to the feedbacks which govern Hudson Strait-scale ice stream surging. While the process as a whole matters, the details matter less so – though it does depend on the aspects of ice stream surging under scrutiny. Across the three hydrology setups, the same range of HS basal velocity increase occurs: the magnitude of ice stream speedup is not dependent on the form of the subglacial hydrologic system, and the three models can attain the same velocities within parametric uncertainty. This means that for model experiments looking to realistically capture ice stream surges, a leaky-bucket hydrology (the computationally cheapest of the three) is sufficient. Additionally, the range of frequency of HS surge occurrences is quite similar across the three hydrologies. However, the no-hydrology case falls short of covering the range inferred for actual Heinrich events attributed to HS surging

Plausibly, one might expect that simply increasing the sliding coefficient in the no-hydrology case would generate more surges. We therefore compared the basal velocity distributions between the configurations (Sect.

Increasing the lapse rate to non-physical bounds can increase the incidence of HS surge events in the no-hydrology case. In the main experiments, the lapse rate is limited to the range

Surge initiation at peak velocity for Hudson Strait-scale ice streams as soon as the pressure melt point is reached is physically implausible. Basal velocity increases after ice becomes warm based and the effective pressure decreases. Inclusion of subglacial hydrology in the coupled system accomplishes this. The accommodation of increasing amounts of basal meltwater and pressurization (in the case that channelization does not occur) acts as a system inductance, and the ice stream continues to speed up after becoming warm based. This inductance does not require the lateral transport of meltwater, only the balance of meltwater and a pressure closure dependence on subglacial water thickness.

Though periodicity and strength of surges are similar between the three hydrology-bearing experiments, an interesting distinction occurs when examining the duration of events at varied frequencies. The stabilizing negative feedback of increasing effective pressure at higher basal velocities in the linked-cavity pressure closure gives surge durations longer (up to double the time, depending on frequency) than those of the diagnostic pressure closure of the poro-elastic and leaky-bucket hydrologies. This feedback also results in a bimodal effective pressure distribution (i.e. Fig.

It is not possible to simulate fully dynamic channelized drainage at the scale studied here; the CFL criterion

Dynamical changes in flow through the efficient system occur on diurnal to seasonal timescales, while the timescales of system features examined here are centennial to millennial. This separation in scale by several orders of magnitude makes it unlikely that dynamical changes in the efficient system (requiring a dynamic model) would be a significant control on the longer-scale variability. However, in a non-linear system, such a control across scales cannot be fully ruled out.

While the treatment of efficient drainage in the model makes it more difficult to closely examine its role in the overall surging system, it is possible to evaluate its role at the ensemble level. At this level it is apparent that efficient drainage does not play a significant role in surging at this scale. Three points bring this to light:

The impact on effective pressure from the down-gradient tunnel routing scheme is exaggerated as its modification of the basal water distribution is immediate instead of smooth. This modifies the effective pressure field in both the poro-elastic and linked-cavity systems through the

The tunnel-switching criterion is well established from a physical mechanistic standpoint

Though the efficient system is not included in the leaky-bucket configuration, there is little difference in the range of surge frequency and amplitude with respect to the other two systems. The distinction in surge duration stems from the dynamic pressure closure of the linked-cavity system and its direct two-way feedback with sliding velocity.

The model presented herein passes multiple verification tests and as such is dependable for comparing the effects of structural choices of subglacial hydrology. The sensitivity analysis and ensemble comparison shows that subglacial hydrology is an important control on both ice sheet geometry and on the surging of major ice streams similar in scale to the Hudson Strait Ice Stream. However, depending on the characteristics of interest, the process details do not matter within current parametric uncertainties. The details do not matter for surge periodicity nor strength, but when studying the surge duration the hydrologic details are essential.

Surge behaviours can be produced in the absence of modelling a subglacial hydrology system, but this requires unrealistic assumptions: pushing lapse rates to unrealistic ranges or implementing an un-physical sudden thaw in a large grid cell when the temperature reaches the pressure melt point. Subglacial hydrology provides a system inductance necessary for realistic ice speedup at the temperate transition. The critical components are the accommodation of meltwater and a meltwater pressure closure, not the mass-conserving meltwater transport itself.

Surge event metric distribution across parameterizations by model configuration for runs in the thinner-geometry sieve (

Surge event duration at different frequencies for thinner ice sheet sieve (

Surge event metric distribution across parameterizations by model configuration for runs in the thicker-geometry sieve (

Surge event duration at different frequencies for the thicker ice sheet geometry sieve (

BrAHMs2.0 solves the conservative transport equation for the distribution of subglacial water (Eq.

The verification of this scheme and its implementation is presented in Appendix

The physics of the linked-cavity system are highly non-linear. As such, a set of simplifying assumptions is required to make numerical modelling of this framework feasible.

Wall melting is not a control on cavity size until tunnels are opened. Drainage systems switch from inefficient to efficient for a given value of flux.

At timescales of continental-scale ice sheets, tunnels drain water instantaneously. The timescale of drainage through subglacial tunnels is less than a single melt season, much shorter than the centennial to millennial-scale changes this model is applied to. This assumption alleviates CFL violations from fast tunnel flux, which would render modelling on the long timescales of glacial cycles infeasible.

Cavities are filled with water. Consider the timescale for the closure of a recently drained cavity given various combinations of ice sheet overburden (thickness, m) and sliding velocities. This timescale for closure (from Eq.

Cavity closure times at varied ice sheet thickness and sliding speeds.

Following others (e.g.

The results presented in this section were reached in an effort to expose errors in the models, the lowest-hanging fruit in gaining confidence in the model solutions. The verification strategy in this section is to satisfy the following:

model solutions are symmetric given symmetric input,

model solutions converge under increasing spatial and temporal resolution,

mass is conserved, and

models using similar physics should have similar solutions.

Using simplified setups, expected behaviours are straightforward and in some cases may be calculated by hand (though hand calculations are not shown here). By using a progression of most simple to increasingly complex model setups for testing, model behaviour can be verified against expected behaviour and shown capable of simulating increasingly realistic environments. Here we demonstrate that the model correctly solves the equations. A progression of forcings and couplings were used, of which the transient, two-way coupled solutions from the least stable parameters (while still physical) are shown.

Parabolic surface topographies haven been used to approximate non-streaming ice sheet topographies

Testing of the linked-cavity system with a Darcy–Weisbach flux model configuration (Eqs.

The basal sliding velocity is determined by the effective pressure from Eq. (

Spatial symmetry at each spatial resolution was calculated as the sum of the difference between the two ice sheet halves across the divide. This difference is 0 for all fields showing perfect symmetry.

Here we test the effect of changing the length of the time step in the basal hydrology on model solution using the SHMIP

As a first test of convergence under increasing temporal resolution (decreasing time step length), the hydrology model was run to steady state under SHMIP scenario A (constant

Convergence with decreasing time step. Each field is normalized with the normalization factor shown in the legend (maximum). The points are fitted with a degree-2 polynomial to show the approximately quadratic rate of convergence.

Here we show the effect of varying spatial resolution on the model solution. The model was run to steady state with prescribed melt and basal velocity (

Difference in mean flowline solutions for unsteady SHMIP square root ice sheet topography at increasing spatial resolution, at end of 10 kyr run. The points are fitted with a line to demonstrate the match with the order of the numerical scheme.

Figure

Mass conservation is demonstrated by comparing flux at the margin to source rates of water or sediment within the ice sheet: the integral of the melt rate over the ice sheet less the total flux through the margin will give the change in basal water volume over time. Integrating this change up to each time step will give the basal water volume at each time step, which can be compared to model-calculated basal water volume in order to assess mass conservation.

Ice sheet configuration used in SHMIP with basal temperature (black

To test mass conservation with unsteady input, we applied a sinusoidal meltwater forcing,

A net volume of basal water time series was calculated by time-integrating the net of input and output, net

Assessment of mass conservation for subglacial hydrology model given steady square root ice sheet topography, flat basal topography, and sinusoidal ice sheet basal meltwater generation (m yr

Comparison of our model solution with the SHMIP tuning set, which used output from the model of

The dynamic model outputs from this test are summarized in Fig.

Results for this model are compared with output of the Glacier Drainage System model (GlaDS,

Here we use the time-varying water pressure calculation of

Analysis and data are available at Zenodo (

MD performed subglacial hydrology model development, experimental design and execution, analysis, and writing. LT maintains the GSM, advised on experimental design, and edited the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The figures in this work were generated with Matplotlib

This work was supported by NSERC Discovery (grant no. RGPIN-2018-06658), the Canadian Foundation for Innovation, and the German Federal Ministry of Education and Research initiative for Research for Sustainability through the PalMod project.

This paper was edited by Alexander Robinson and reviewed by two anonymous referees.