We describe and test a two-horizontal-dimension subglacial hydrology model
which combines till with a distributed system of water-filled, linked
cavities which open through sliding and close through ice creep. The addition
of this sub-model to the Parallel Ice Sheet Model (PISM) accomplishes three specific
goals: (a) conservation of the mass of water, (b) simulation of spatially
and temporally variable basal shear stress from physical mechanisms based on
a minimal number of free parameters, and (c) convergence under grid
refinement. The model is a common generalization of four others: (i) the
undrained plastic bed model of

Any continuum-physics-based dynamical model of the liquid water underneath
and within a glacier or ice sheet has at least these two elements: the mass
of the water is conserved and the water flows from high to low values of the
modeled hydraulic potential. Beyond that there are many variations considered
in the literature. Modeled aquifer geometry might be a system of linked
cavities

Models have combined subsets of these different morphologies and processes

This paper describes a carefully selected model for a distributed system of
linked subglacial cavities, with additional storage of water in the pore
spaces of subglacial till. Water in excess of the capacity of the till passes
into the distributed transport system. In this sense the model could be
called a “drained-and-conserved” extension of the “undrained” plastic bed
model of

The cavities in our modeled distributed system open by sliding of the ice
over bedrock roughness and close by ice creep. These two physical processes
combine to determine the relationship between water amount and pressure.
Pressure is thereby determined non-locally over each connected component of
the hydrological system. No functional relation between subglacial water
amount and pressure is assumed

In cases where boreholes have actually been drilled to the ice base, till is
often observed

The major goals here are to implement, verify, and demonstrate this
two-dimensional subglacial hydrology model. The model is applicable at a wide
variety of spatial and temporal scales but it has relatively few parameters.
It is parallelized and it exhibits convergence of solutions under grid
refinement. It is a sub-model of a comprehensive three-dimensional ice sheet
model, the open-source Parallel Ice Sheet Model (PISM;

Channelized subglacial flow is widely assumed to occur in Greenland, based on
borehole and moulin evidence

Wall melt in the linked-cavity system, which is believed to be small

The structure of the paper is as follows: Sect.

We assume that liquid water is of constant density

Physical constants and model parameters. All values are configurable
in PISM; see Table

Functions used in the subglacial hydrology model (Eq.

The water source

The hydraulic potential

We have added the term “

Ice is a viscous fluid which has a stress field of its own. The basal value
of the downward normal stress, called the overburden pressure, is denoted by

Overpressure

Subglacial water flows from high to low hydraulic potential. The simplest
expression of this is a Darcy flux model for a water sheet:

Combining Eqs. (

We will construct our numerical scheme based on decomposition
Eq. (

From Eqs. (

The rate of change of the area-averaged thickness of the cavities in a
distributed linked-cavity system is the difference of opening and closing
rates

Till with pressurized liquid water in its pore spaces is expected to support much of the ice overburden. When present, such saturated till is central to the complicated relationship between the amount of subglacial water and the speed of sliding. Our model includes storage of subglacial water in till both because of its role in conserving the mass of liquid water and its role in determining basal shear stress.

We will assume throughout that liquid water or ice fills the pore spaces in the
till, and that there are no air- or vapor-filled pore spaces. When

The water in till pore spaces is much less mobile than that in the
linked-cavity system because of the very low hydraulic conductivity of till

As in

Deformation of saturated till is well modeled by a plastic (Coulomb friction)
or nearly plastic rheology

Let

While Eq. (

The void ratio

From Eqs. (

It follows from Eqs. (

Experiments on till suggest small values for cohesion

Measured till friction angles

The ratio

The till capacity parameter

Observe that the ice sliding velocity

Power-law Eq. (

The evolution equations listed so far, namely, Eqs. (

We first consider two simple closures which appear in the literature but
which do not use cavity evolution Eq. (

Setting the pressure equal to the overburden pressure is the simplest closure

Because the approximation

At an almost opposite extreme, our second simplified closure makes the water
pressure a function of the amount of water. Specifically,

One concern with form Eq. (

Simply requiring the subglacial layer to be full of water is also a closure

Equation (

Englacial systems of cracks, crevasses, and moulins have been observed in
glaciers

Using englacial porosity as a regularization, as in
Eq. (

Stiffness in these pressure equations ultimately follows from the
incompressibility of water and the relative non-distensibility
(i.e., hardness) of the ice and bedrock.

The major evolution equations for the model are mass conservation
(Eq.

The model includes these bounds on major variables:

A coupled weak formulation of Eq. (

In this model the pressure

As in Table

Four reductions (limiting cases) of model Eq. (

The zero till storage (

The bounds

The

The non-distributed “lumped” form of Eqs. (

The undrained plastic bed (UPB) model of

The above list does not imply that all possible subglacial hydrology models
are reductions of ours. For example, the subglacial hydrology model of

Two-dimensional models which include conduits

The steady form of model Eqs. (

We make three observations about solutions to
Eqs. (

From Eq. (

By Eqs. (

Radial nearly exact solutions can be constructed.

For the purpose of verifying numerical schemes we have built a
two-dimensional, nearly exact solution for

We solve the flat bed (

A nearly exact radial, steady solution for water thickness

To compute the nearly exact solution, we use adaptive numerical ODE solvers,
both a Runge–Kutta method and a variable-order stiff solver, with relative
tolerance

Verification results using the nearly exact solution appear in Sect.

Equations (

To set notation, suppose the rectangular computational domain has

A nearly exact radial, steady solution for pressure

We compute velocity components and flux components at the staggered
(cell-face-centered) points, shown in Fig.

The nonlinear effective conductivity

Numerical schemes Eqs. (

Define

Our scheme for approximating mass conservation Eq. (

Assuming no error in the flux components

We test two flux-discretization schemes, namely, a first-order upwind scheme
and the Koren flux-limited third-order scheme

For a flux-limited scheme, the following formulas apply in the cases

The first-order upwind scheme simply sets

For either scheme, if the water input

Pressure evolution Eq. (

Let

A sufficient condition for stability of mass-conservation scheme
Eq. (

These time-step restrictions can be understood by considering an example. We
ran the model on a

However, the time-step restriction from the pressure-equation scheme is
typically shorter than either

Recalling Eq. (

If implicit time stepping were instead used for the pressure equation, which
would require overt variational inequality treatment to preserve physical
pressure bounds

Mathematical model Eqs. (

For convenience only we denote the ice geometry, bed geometry, and sliding
speed (i.e.,

One time step follows this algorithm:

Start with values

Get

Get

Get time step

Using Eq. (

Get flux divergence approximations

If

If

If

If

Update time

This algorithm goes with a reporting scheme for mass conservation. Note that in
steps (ii) and (ix) water is lost or gained at the margin where
either the ice thickness goes to 0 on land (margins), or at locations
where the ice becomes floating ice (grounding lines). Because such loss/gain may
be the modeling goal – users want hydrological discharge – these amounts are
reported. This reporting scheme also tracks the projections in step
(x), which represent a mass-conservation error which goes to 0 in
the continuum limit

Option

The correspondence between the notation in this paper and PISM's configurable
parameters is shown in Table

By using the coupled, steady-state, nearly exact solution
(Sect.

Correspondence between PISM parameter names and symbols in this
paper (Table

Average water thickness error

This convergence evidence suggests that we have implemented the numerical
schemes in Sect.

The rates of convergence for average errors are nearly identical for the higher-resolution flux-limited scheme and for the first-order upwinding scheme (not shown). Because our problem is an advection–diffusion problem in which both the advection velocity and the diffusivity are solution-dependent, it is difficult to separate the errors arising from numerical treatments of advection and diffusion. The first-order upwinding scheme for the advection has much larger numerical diffusivity but this diffusivity is masked by the physical diffusivity. Based on our verification evidence it is reasonable to choose the simpler first-order upwind method for applications, as it requires less interprocess communication.

We now apply our hydrology models to the entire Greenland ice sheet at 2 km grid resolution. This nontrivial example demonstrates the model at large computational scale using real ice sheet geometry, with one-way coupling from ice dynamics giving realistic distributions of overburden pressure, ice sliding speed, and basal melt rate.

The inputs to the hydrology model are the modeled basal melt rate

The PISM dynamics and thermodynamics model

The spin-up grid sequence was to run 50 ka on a 20 km grid, 20 ka on a 10 km grid, 2 ka on a 5 km grid, and finally 200 a on a 2 km grid, with bilinear interpolation at each refinement stage. The final 2 km stage, on a horizontal grid of 1.05 million grid points, used uniform 10 m vertical spacing so that the ice sheet flow was modeled on a structured 3-D grid of 460 million velocity–temperature points. This whole spin-up used 2800 total processor hours on 72 2.2 GHz AMD Opteron processors, a small computation for modern supercomputers.

The results of this spin-up were validated by comparing results to
present-day observations. In the last 100 a of this run the ice sheet volume
varied by less than 0.04 %, so the model is in nearly steady state,
though the actual Greenland ice sheet may not be as close to steady. The
spun-up ice sheet volume of

The spun-up initial state includes, in particular, modeled ice thickness

We used fields

Outputs from the

In the runs, variables

Adaptively determined time steps reached a steady level of about 4
model hours for the

The final

The continuum limit of the model would have concentrated pathways of a few meters to tens of meters width. These concentrated pathways could be regarded as minimal “conduit-like” features of the subglacial hydrology. As noted in the introduction, however, our model has no “R-channel” conduit mechanism, in which dissipation heating of the flowing water generates wall melt back.

The final values of

Detail of transportable water

Recall that

Outputs from the

Scatter plots of

We can examine the local relationship between water layer thickness

This paper documents additions made to the Parallel Ice Sheet Model (PISM) in its
0.6 version released February 2014. It describes and demonstrates a
subglacial hydrology model which is novel in having these
features:

a 2-D parallel implementation of a coupled till-and-linked-cavities model;

a pressure-equation regularization, using notional englacial porosity, which eases implementation and improves numerical performance;

a scheme for maintaining physical pressure bounds (

verification using a nearly exact solution of the coupled mass-conservation and pressure equations, in the steady radial case;

demonstration at high resolution and whole ice-sheet scale on a million-point hydrology grid.

Furthermore, the comprehensive exposition here clarifies the relationship
among several pressure-determining “closures” (Sect.

The current paper only demonstrates one-way coupling, in which the PISM ice flow and thermodynamics model feeds basal melt rate and sliding velocities to the hydrology model. Two-way coupling will appear in future work.

Relative to the time-dependent model Eqs. (

We define a scaled basal sliding speed which has units of pressure; it is a
scale for the pressure drop from cavitation:

We now consider how the steady-state water velocity

The steady-state function

The graph of

Now note that Eqs. (

The source code for all versions of PISM is available through host website

Comments by editor Dan Goldberg, reviewer Tim Bartholomaus, and two anonymous reviewers improved the focus and quality of the paper. Detailed comments by Andy Aschwanden and Martin Truffer were much appreciated. Constantine Khroulev helped with the PISM implementation. The first author was supported by NASA grant #NNX13AM16G. This work was supported by a grant of high-performance computing resources from the Arctic Region Supercomputing Center. Edited by: D. Goldberg