We present BrAHMs (BAsal Hydrology Model): a physically based basal hydrology model which represents water flow using Darcian flow in the distributed drainage regime and a fast down-gradient solver in the channelized regime. Switching from distributed to channelized drainage occurs when appropriate flow conditions are met. The model is designed for long-term integrations of continental ice sheets. The Darcian flow is simulated with a robust combination of the Heun and leapfrog–trapezoidal predictor–corrector schemes. These numerical schemes are applied to a set of flux-conserving equations cast over a staggered grid with water thickness at the centres and fluxes defined at the interface. Basal conditions (e.g., till thickness, hydraulic conductivity) are parameterized so the model is adaptable to a variety of ice sheets. Given the intended scales, basal water pressure is limited to ice overburden pressure, and dynamic time stepping is used to ensure that the Courant–Friedrichs–Lewy (CFL) condition is met for numerical stability.

The model is validated with a synthetic ice sheet geometry and different bed topographies to test basic water flow properties and mass conservation. Synthetic ice sheet tests show that the model behaves as expected with water flowing down gradient, forming lakes in a potential well or reaching a terminus and exiting the ice sheet. Channel formation occurs periodically over different sections of the ice sheet and, when extensive, displays the arborescent configuration expected of Röthlisberger channels. The model is also shown to be stable under high-frequency oscillatory meltwater inputs.

Subglacial basal hydrology is a potentially critical control on
basal drag and ice streaming. Furthermore, it is a clear control for
subglacial sediment production/transport/deposition processes

Many models relating to basal hydrology are either meant for short timescales (e.g., on the order of weeks to centuries), or are missing a key component of basal water flow (channelized flow). We present a computationally fast physics-based subglacial hydrology model for continental-scale ice sheet systems modelling over glacial cycles, which is meant to capture the relevant features of basal water flow for the above three contexts (including both distributed and channelized flow components).

This large spatio-temporal scale context places a high requirement on
computational speed and justifies certain simplifications compared to
glacier-scale models

Only a few subglacial hydrology models have been described in the literature
for continental-scale ice sheets. Of these models, some of the more advanced
include the models developed by

The original work of

The work of

The basal hydrology model described here combines features from the above
models to create a relatively fast subglacial hydrology model for
continental-scale contexts. Following the work of

A distinguishing feature of this model is the numerical time stepping scheme.
The model uses a combination of Heun's method and the leapfrog–trapezoidal
schemes, which are iterative predictor–corrector schemes. The latter scheme
has been used in the more demanding case of ocean modelling

The hydrological model has been incorporated into the glacial systems model

Subglacial drainage systems can be characterized as belonging to one of two categories: distributed drainage systems or channelized drainage systems.

There are several ways that water can be distributed underneath the ice:
it can be stored via a thin film

The channelized drainage system is, to a certain degree, the obverse of the
distributed drainage system. This system has a lot of water concentrated in a
small area of the glacial bed and transports water quickly. Since channelized
systems are efficient at draining water, they tend to decrease the water
pressure, which increases basal friction between the ice and the bed.
Thus, channelized systems are associated with slow flowing ice regimes and
are often seasonal.

For the analyses presented herein, the subglacial hydrology model is
passively coupled to the glacial systems model (GSM). Full two-way coupling
was turned off to isolate the dynamical response of the basal hydrology
model. The GSM is composed of a thermo-mechanically coupled ice sheet model
(using the shallow ice approximation), locally 1-D diffusive permafrost
resolving bed thermal model

The evolving temperature field (

Basal sliding uses Weertman type sliding relations (i.e., function of driving
stress) with power law 3 for hard bed and power law 1 for soft bed with
sliding onset linearly ramped up starting from 0.2

For brevity and clarity, this section discusses the physical and numerical
concepts of the hydrology model developed in

The dynamical evolution of distributed drainage is extracted from the mass
continuity equation. Written in conservative form, the equation is

The water flux,

The distributed flow of water beneath the ice sheet can come in many forms
(such as cavities, Nye-channels, thin film, and flow through porous
sediment). However, the extent to which the details of these flow mechanisms
matter under large spatial scale separation and mechanistic heterogeneity is
unclear. Therefore, we use the large difference in scale between glacial
cycle ice sheet model grid cells (

We use an empirical relation for water pressure from Flowers (

The channelized system is likened to a system of R-channels (tunnels incised
upward into the ice). Numerically, the model first calculates the water flux
from the Darcian flow (Eq.

To simulate the change between different drainage systems, at regular
user-defined intervals, the channel flow subroutine is called. Grid cells for
which the water flux exceeds the distributed flow stability criterion
(Eq.

During the tunnel flow, no model time is stepped as tunnel drainage is computed diagnostically.

moves water down the path of steepest potential gradient (channelizing grid cells along that path) until there is no grid cell with a lower hydraulic potential (so forms subglacial lake) or the water exits the ice sheet. The solver considers all adjacent grid cells (including corner adjacency) when searching for the lowest hydraulic potential. Once the tunnel water transport is complete, the tunnels are assumed closed and the distributed flow algorithm continues.BrAHMs is highly modular and designed for asynchronous coupling at user specified time steps. Aside from basic grid information, for each call, the hydrology model requires the following input fields: ice thickness, basal elevation, sea level, basal ice temperature, basal melt rate, and basal sliding velocity of the ice. For two-way coupling, the relevant outputs from BrAHMS are basal water pressure and thickness.

Given that there is no lower limit to coupling time steps, synchronous coupling can also be implemented. For two-way coupling, sensitivity tests are recommended to determine the appropriate coupling time step for the relevant context.

The basal hydrology model was subject to several validation tests with synthetic axisymmetric ice sheets. The three set-ups are: symmetrical ice sheet on a flat bed, symmetrical ice sheet on a dilating (sinusoidally wavy) bed, and a symmetrical ice sheet on an inclined plane.

The continental-scale ice sheet model used in these tests has a profile that
follows a normal distribution from the centre of the ice sheet to the
terminus and is symmetric around the centre (i.e., bell-shaped ice sheet),
according to the following equation:

Model parameters used in validation studies.

The bed for the inclined plane (as a function of latitude) is given by

The dilating bed is given by

Simple synthetic ice sheet testing scenarios. Transects are plotted
as the absolute, normalized distance from the centre.

It should be noted that the model is based on spherical polar coordinates (as
it is designed for modelling large sections of the Earth's surface), and so
the figures presented here are akin to the Mercator projection

Set
between 17 and 40

Tables

In the model runs, the ice sheets starts from the ground (at

To facilitate the growth of the subglacial hydraulic system, a constant
melting at the base of the ice is applied in a “ring” of uniform thickness
near the terminus, with 0.6 m yr

Maximum mass balance discrepancy over time.

The first set-up tested was for the ice sheet on a flat bed. For the
bell-shaped ice sheet on a flat surface (see Fig.

Chosen values for the baseline model run for synthetic and North American test runs.

The transects in Fig.

Plots of the bell-shaped ice sheet on an inclined plane. This series also tests the convergence of the model. “Normal” refers to the model using the same resolution as the previous tests and “double” refers to the tests that has twice the resolution (finer) grid than the other tests.

The next test was designed to show the formation of lakes by the model. The
ice sheet for this study, as seen in Fig.

Figure

The transect plots show that there is a slight asymmetry that arises in the results (there appears to be two red curves). Under perfect symmetries, the tunnel solver will break symmetry in its down-slope search algorithm. While the results are not shown here for brevity, when the tunnel solver is turned off, the results do not show any discernible asymmetry. The asymmetry due to the inclusion of the tunnel solver is unlikely to be an issue in more realistic cases where the ice sheet would lack such symmetry.

Figure

Figure

Lastly, the model was tested for stability by shocking the system with sudden
changes in the meltwater production. For this test, the base case of the ice
dome lying on a flat bed was used (the same scenario as Fig.

Figure

Time series analysis of basal water level response to time-varying meltwater production, showing that the dynamic time stepping of the model is capable of providing stable, realistic results under sudden changes.

The North American ice sheet model used herein is from a large ensemble Bayesian calibration as detailed in Tarasov et al. (2012). Model runs start from 122 ka under ice free conditions.

Due to the complex nature of basal hydrology and the spatial and
temporal scales for the current context, there are many processes that
are approximated through parameterizations. As such, there are a number
of poorly constrained parameters in the hydraulic model (these are listed in
Table

The first parameter,

The simplified aquifer drainage of

Due to the small time steps (relative to glacial modelling) involved
in the basal hydrology model, it would become computationally
expensive to check for tunnels at each time step. As such,

For clarity of this initial analysis, results presented herein are
with a uniform basal sediment cover over the whole bed for the
duration of the run. The sediment cover was specified by

The water flux between grid cells is directly proportional to the hydraulic
conductivity of the sediment. For each run, the conductivity was
allowed to vary between a minimum and maximum value defined in the
range of

In Eq. (

As the base of the ice sheet becomes colder, the ice should begin to freeze
to the bed, preventing water from flowing there. Due to the 40 km resolution
of the grid, it is unlikely that the entire bed in a grid cell would be
frozen completely when the grid cell basal temperature crosses the pressure
melting point. Therefore, water could potentially flow through a frozen grid
cell (in the unfrozen places), but the water should have a harder time as it
has fewer pathways to flow across. In the hydrology model, this is
represented by parameter

Tunnel formation has a direct impact on basal water pressure. To further test
this impact, an enhancement factor,

Our choice of baseline model for the sensitivity analysis was based solely on mid-range values for parameter uncertainty ranges and not on any sort of tuning. As such, results presented here have an exploratory instead of predictive focus.

Basal water profiles for

Basal hydrology fields for the baseline model near the last glacial maximum (LGM)
are shown in Fig.

As the water is removed from 22 to 18 ka, some of the areas experience a large increase in basal effective pressure.

Sensitivity plot at

To account for dependence on baseline amounts of basal water, our sensitivity
tests consider both the 22 and 18 ka time slices. Figure

The sediment thickness parameter (

The runs with the basal freezing value closer to the PMP have about a
12 (%) increase in basal water volume, as expected due to the increased
likelihood of ice frozen to the bed hindering the flow of water. In
comparison to other parameter results, varying the value of the basal
freezing parameter,

The tunnel criterion scaling factor,

The bedrock bump height,

The results of changing the range of hydraulic conductivity (

Sensitivity plot at

The plot of the average basal effective pressure, in Fig.

During the low water storage times, the other parameters become important to
the basal effective pressure. The impact of saturated sediment thickness on
basal effective pressure appears to be relatively insensitive to the amount
of water storage, as the two plots in Fig.

Figures

One important test result is the low sensitivity of the average basal
water thickness and effective pressure to the maximum allowable time
step (

As a caveat, these initial sensitivity tests likely hide spatially localized parametric sensitivities. More critically, feedbacks in a two-way coupled ice sheet and basal hydrology model configuration may strongly change relative sensitivities to basal hydrology parameters. These analyses will be better placed in a future study examining fully coupled dynamics.

This paper presents a physically based hydrology model for numerical
simulations over a glacial cycle at continental scales. The model
considers two types of drainage systems: a distributed system that
slowly drains basal water, and a fast draining channelized system.
The distributed hydrology system is modelled with Darcy's law

The model was tested over a set of synthetic ice profiles and topography. The results of these tests show that the model is mass conserving and that the water flows down the hydraulic potential gradient where it can exit the ice sheet or form subglacial lakes.

With the model validated using the synthetic ice sheets,
the model was then one-way coupled to the GSM for testing on the North American
ice sheet complex at LGM. The sensitivity results in Figs.

The hydrology model also identified areas of low effective pressure, indicating areas of potentially fast flowing ice. These results were self-consistent with the GSM's parameterized areas of the fast-flowing ice.

The hydrology model presented here has been shown to be stable and
robust for the range of parameters used in this study. The coupled model
generally takes 5–8 h to run for a North American glacial cycle
(0.5

As an initial implementation of a 2-D basal hydrology solver, there
were several simplifications made to facilitate the initial study of
the basic properties of the subglacial water dynamics. One
simplification was that the aquifer drainage parameter was used instead of
a real aquifer drainage system

Basal hydrology code with validation drivers is freely
available at

The model uses the mass continuity equation (Eq.

Equation (

Variations of subglacial hydraulic conductivity,

This then simplifies to

Using the approximation

The model presented in this paper uses two predictor–corrector methods with
different predictors but identical corrector. The first predictor method,
based on Heun's method

The first step in Heun's method is to take some initial conditions
(

When the previous time step value of

Regardless of which predictor equation is active, the
trapezoidal scheme is applied to give the corrected value,

The Darcian flux,

If we consider the case of the flux on the westward edge of the grid cell,

Following the rules of

To simplify the flux equation, the upwind scheme

Hydrology model flow chart highlighting the processes involved in simulating basal water flow.

JH, BN, and TB performed the majority of the source code reconfiguration. BN provided overall oversight. JH designed the experiments, and carried them out with assistance from YM and DG. JH and BN prepared the manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We thank Jesse Johnson and Julien Seguinot for their thoughtful and helpful reviews. This work also significantly benefited from a review of the associated Thesis by Jesse Johnson. Entcho Demirov offered some helpful numerical suggestions. This work was supported by a NSERC Discovery Grant (Lev Tarasov), the Canadian Foundation for Innovation (Lev Tarasov), and the Atlantic Computational Excellence Network (ACEnet). Edited by: Philippe Huybrechts Reviewed by: Jesse Johnson and Julien Seguinot