Many large-scale subglacial drainage models implicitly or explicitly assume that the distributed part of the drainage system consists of subglacial cavities. Few of these models, however, consider the possibility of hydraulic disconnection, where cavities exist but are not numerous or large enough to be pervasively connected with one another so that water can flow. Here I use a process-scale model for subglacial cavities to explore their evolution, focusing on the dynamics of connections that are made between cavities. The model uses a viscoelastic representation of ice and computes the pressure gradients that are necessary to move water around basal cavities as they grow or shrink. The latter model component sets the work here apart from previous studies of subglacial cavities and permits the model to represent the behaviour of isolated cavities as well as of uncavitated parts of the bed at low normal stress. I show that connections between cavities are made dynamically when the cavitation ratio (the fraction of the bed occupied by cavities) reaches a critical value due to decreases in effective pressure. I also show that existing simple models for cavitation ratio and for water sheet thickness (defined as mean water depth) fail to even qualitatively capture the behaviour predicted by the present model.

Much of the interest in subglacial drainage is motivated by the effect of pressurized subglacial water on glacier sliding

The basal effective pressure that appears in the friction law is likewise defined as a local spatial average of the difference between local normal stress at the bed and basal water pressure. At the local process scale, actual normal stresses are heterogeneous but have a well-defined average that is generally close to local ice overburden. By contrast, basal water pressure is generally not assumed to be heterogeneous in the friction law. A spatially smoothly varying basal water pressure will result if there is a pervasive, connected subglacial drainage system that causes pressure differences to equilibrate rapidly.

A growing body of observational evidence, however, suggests that hydraulic isolation of significant parts of glacier beds is a common phenomenon, even during the summer melt season. Moreover, different parts of the glacier bed can switch back and forth between being connected and isolated

The justification for such a treatment of hydraulic connection is that small cavities can exist in the lee of bed bumps without being sufficiently connected to each other to allow water flow, as in a percolation problem

The present paper is part of an effort to dispense with that assumption of a perfectly permeable bed and instead study how cavities can expand dynamically along the ice–bed interface from an access point or set of access points where water is injected through the bed at prescribed pressure by an ambient drainage system. In a companion paper

Based on that assumption, Part 1 shows that connections between the ambient drainage system and previously uncavitated parts of the bed are made in a quasi-steady state at a set of critical effective pressures. The system of cavities also exhibits hysteresis. If cavity enlargement past a bed protrusion on its downstream side has occurred previously and cavity size has shrunk subsequently due to an increase in ambient effective pressure, then reconnection to the now isolated pre-existing cavities happens at a different set of higher effective pressure: reconnecting to an existing downstream cavity is easier than creating that downstream cavity by enlarging the upstream cavity past the bed protrusion separating the two.

In a time-dependent system, cavity connections are likely to be more complicated than the quasi-steady model suggests. To study dynamic cavity connections, I complement the work in Part 1 with a generalized dynamic model. Ice is treated as viscoelastic to account for the possibility that cavity expansion could be very rapid and occur on timescales that are short compared with the Maxwell time of ice. In addition, I explicitly account for the water pressure gradients that are necessary to move water around a cavity, in particular, into any ice–bed gaps that are newly created when rapid cavity enlargements occur. By using a mass conservation model for water with a water flux that vanishes when the ice–bed gap size goes to zero, the dynamic model not only allows the process of cavity expansion to be captured dynamically, but also allows for the dynamic evolution of isolated cavities.

The generalized model in the present paper is formulated in three dimensions. In principle, this avoids another of the limitations of the work in Part 1. Because the MATLAB code I have written is not suitable for full parallelization, I have not been able to run the model in three dimensions except for very coarse meshes, leaving an obvious avenue for future research.

The paper is structured as follows: in Sect.

Definitions used in the paper. Beige is used throughout the paper to indicate the connection portions

The model in Part 1 is based on the approximation of small bed slopes

I assume ice occupies a domain defined by

At the lower boundary of the ice,

To close the problem, I require one additional boundary condition. I consider two alternatives. First, I consider the standard assumption in dynamic models of subglacial cavity formation, namely that the bed is rigid yet highly permeable, with a prescribed water pressure

The boundary conditions above do not permit the formation of hydraulically isolated cavities or of underpressurized contact areas that remain hydraulically isolated as in Part 1. As an alternative to the boundary conditions (10), I therefore consider a bed that is perfectly impermeable except in specific locations at which water from an ambient drainage system can enter or exit the ice–bed gap. As in Part 1, I assume that there is a (typically small) highly permeable portion

Specifically, I assume that there is a water column of evolving height

Note that Eq. (

To avoid the negative fluid pressure singularities common to hydrofracture models

The first possibility, condition (

As far-field boundary conditions, I consider prescribed normal and shear stress in the form

The basal boundary conditions for the classical cavitation problem with a permeable bed consist of Eqs. (

By contrast, the equivalent set of boundary conditions for an impermeable bed given above introduces local fluid pressure

The counting argument of the previous paragraph is of course simplistic: the determination of

From the gap width relations (

In practice, only very small pressure gradients should be required in order to move water fast enough to fill the ice–bed gap as the latter evolves due to ice flow. That situation corresponds to the limit of a large gap permeability

Significant simplifications can be obtained by considering flow over “shallow” bed roughness, meaning a bed

With these scales in hand, I can define dimensionless variables as

Going forward, I will assume that

I omit the asterisk decorations immediately for improved readability. To an error of

As in other models of basal sliding with small-slope bed roughness

At first sight, this may not seem all that different from the original model of Sect.

Computationally, the problem defined in the previous section is well-suited to solution by mixed finite elements in order to handle both the viscoelastic rheology and the inequality-constrained boundary conditions as described in

I use a coordinate frame moving at the sliding velocity

To handle the mass conservation problem (

Note that in similar elastic problems solved elsewhere, water can never be completely removed from a pre-existing gap, though it can become arbitrarily thin

The use of piecewise constant finite volumes for

All inequality constraints that are part of the boundary conditions for either the impermeable or permeable bed case can be written as complementarity problems in discretized form, of the generic form

The code is written in MATLAB and uses neither adaptive time stepping (beyond automatic step size reduction when the Newton solver fails to converge to a prescribed tolerance for a given backward Euler step) nor adaptive meshing (although the mesh used is non-uniform, with nodes concentrated near the bed). Both of these features would be desirable future improvements. Although the code is written so it can be used for both two- and three-dimensional domains, the lack of adaptive meshing still leads to a relatively coarse resolution along the bed and restricts any realistic use of the code to two dimensions.

In the numerical results reported here, I use the model (

I use a finite domain depth

For the purpose of visualizing results, I focus mostly on several easy-to-identify scalar attributes of the solution and their evolution in time, plotting only selected cavity profiles. I identify cavity end points

The dynamic model of Sect.

There is an important qualification to the meaning of “steady state” here: I simply compute a numerical solution of the model (

The solutions of the dynamic model (plotted against the original, as opposed to moving, coordinate

Figure

Comparison of steady cavity roof geometry (denoted by

The purpose of introducing a dynamic model is precisely to study the transient behaviour leading up to the eventual steady state. Figure

Dynamic cavity evolution under step changes in forcing effective pressure

As expected from Part 1, the steady-state mean cavity size

The overshoot becomes large once there is only a single contact area with a cavitation ratio close to unity (between

Conversely, if there is limited cavity extent with only the lee of the larger bed protrusion cavitated and

The most sustained oscillations occur when there is a single small contact area in each bed period at low effective pressure

These variations in vertical velocity are presumably the reason for the significant oscillations in

In its simplest form, this mechanism is what happens if one rigid corrugated surface is dragged over another (imagine two pieces of corrugated sheet roofing moving relative to each other); in the present case, the ability of the ice to deform is significant, and the lower surface of the ice does change shape to adapt to the rigid bed underneath, which accounts for the approach to a steady state. It is then perhaps not surprising that low effective pressure

The generic behaviour shown in Fig.

Figure

While the dynamic behaviour of the fully permeable bed case is similar to the impermeable bed, there are two notable differences. First, as in the case of reconnection of a previously isolated cavity for the impermeable bed case in Fig.

The cavitation ratio is very close to unity (typically around

A very small number of nodes in contact with the bed raises the question of numerical artefacts. A comprehensive study of mesh size effects is beyond the scope of the work presented here. Due to the limitations of working in a MATLAB coding environment, it is difficult to refine the mesh significantly beyond what is used in the computations reported above. For the case of a fully permeable bed (which typically permits larger time steps), I have been able to refine the mesh to double the number of nodes on the bed for a relatively short computation. A comparison for a shortened version of the computation in Fig.

Using the same plotting scheme as in Fig.

Using the same plotting scheme as in Fig.

Using the same plotting scheme as in Fig.

The rather long time interval over which the solution in Fig.

In Fig.

Cavity connection under different step changes in

Immediately after the drop in

Importantly, this initial “hydrofracture” (which is not a hydrofracture in the true sense, as it corresponds to a pre-existing fracture being re-opened) has a very limited extent. In fact, the same initial fracture occurs every time that

In interpreting the results for

The second phase of slower cavity growth is the result of viscous deformation. Only once the downstream cavity end point has advanced significantly downstream does the rapid expansion (or connection) of the cavity past the smaller bed protrusion occur, marked as “rapid connection” in Fig.

The subsequent rapid expansion of the cavity (following the second phase of slower cavity growth and corresponding to the “drowning” of the smaller bed protrusion) can be separated into two parts: an initial advance of the cavity end point from

Figure

The gap then thickens more slowly (panel e), leading to oscillatory behaviour (panels f–j; these later times are not shown in Fig.

I illustrate the oscillation mechanism further in Fig.

Cavity shapes for a step change from

Cavity height

In the computations above, I have focused on a permeable bed section

Two solutions are plotted, both of them identical up to

Using the same plotting scheme as in Fig.

In Part 1, I showed that steady-state effective pressure in isolated cavities is remarkably insensitive to changes in the effective pressure

Here, I give three examples in which the connected cavity in the lee of the larger bed protrusion (with

In column 1, a relatively small isolated cavity forms before periodic behaviour is established. That cavity then remains isolated throughout the pressure cycle. The effective pressure

When the forcing oscillations have a somewhat lower frequency (column 2), there are more significant changes in the ice–bed contact area on the smaller bed protrusion. An isolated cavity now forms during every other period of the forcing pressure oscillation (that is, the solution is periodic with a periodicity twice that of the forcing). In each case, the cavity roof makes contact with that protrusion after a maximum in

The forcing pressure oscillation in column 3 is even slower and of larger amplitude than those in columns 1 and 2. Here, the solution has the same periodicity as the forcing, with a contact area forming on the smaller bed protrusion upstream of

When the cavities become fully disconnected,

Using the same plotting scheme as in Fig.

In Part 1, I showed that for a steady-state model, connections between cavities are created and destroyed at critical values of

One might therefore be tempted to parameterize cavity connection in large-scale drainage models in terms of effective pressure

This result is at least consistent with the previous modelling approach of

A plausible alternative to having a simple critical value

These observations point to a need to extend drainage models to describe the evolution of not only

Any attempt to amend subglacial hydrology models along these lines, however, faces another conundrum: as currently formulated, existing subglacial drainage models use an evolution equation for

Figures

It is conceivable that a model of the form (

If, on the other had, the overshoot oscillations are an important part of the evolution of the drainage system, then the set of variables that an extended model needs to consider is most likely larger than simply

The ad hoc addition of dynamical variables is clearly a disturbing prospect in the absence of a clear roadmap for how closure should be achieved. Once a set of such dynamical variables is identified, then perhaps the obvious next step would be to try to arrive at a closed set of equations for the evolution of these dynamical variables not by means of qualitative physical insight and subsequent parameter fitting, but by treating their evolution as being governed by a dynamical system that can be represented by a neural network, which in turn can be trained on output from a detailed process-scale model such as that described here

The discussion above has focused on the implications of the local-scale model results in the present paper for large-scale subglacial models. The same results also have implications for the interpretation of field observations: a perhaps obvious consequence of hydraulic isolation of the bed is that the usual basal water pressure may no longer be smoothly varying in space and in fact has no physical meaning in areas of ice–bed contact. For a highly permeable bed, a pressure sensor in a borehole that terminates on an ice–bed contact still measures the water pressure in any surrounding cavities, since water from those cavities can readily access the borehole through the bed. This is no longer true for an impermeable bed. Measuring borehole water pressure where a borehole terminates on an ice–bed contact area then records the peculiarities of pressure evolution in the isolated borehole, which itself is of unknown shape and must preserve its volume (assuming the borehole has closed, as is typically the case; see e.g.

By contrast, when a sensor is connected to an isolated cavity, the pressure measurement records the water pressure in the cavity. The latter again needs to preserve its own volume as modelled in the present paper. The pressure response of isolated cavities to temporally varying forcing pressures is sensitive to the timescales involved. For high-frequency oscillations in forcing (faster than the deformation timescale

For slower forcing oscillations, temporary connections between cavities can be established, corresponding to “switching events” observed in borehole pressure records

There are likely to be many areas in which the model described here can be improved, ranging from a careful analysis of the numerical method used to practical implementation issues such as the use of a potentially more suitable finite-element basis, adaptive time stepping, and adaptive meshing as well effective parallelization. In addition, there are physical processes that the present work has been unable to consider.

The most obvious among the latter is the effective solution of the model in three dimensions to capture changes in hydraulic connectivity: in a two-dimensional model, it is impossible to establish connectivity from one end of the domain to the other unless ice–bed contact is lost everywhere, which is not a physically reasonable situation. It is only in three dimensions that full end-to-end connectivity (meaning water is free to flow from one side of the domain to the other) can coexist with continuing ice–bed contact. Similarly, I have focused purely on the hydrological aspects of dynamic cavity evolution and do not attempt to address the question of a friction law for dynamically evolving subglacial cavities, which would be a worthwhile addition in its own right

In this paper, I have formulated a viscoelastic model for ice sliding over a rigid and mostly impermeable bed, allowing for the formation of cavities in which water is dynamically redistributed by an active local drainage system. The model is capable of describing the dynamic extension of subglacial cavities as bed obstacles progressively become submerged by water sourced from a localized water supply connected to an ambient drainage system at prescribed effective pressure. In the same vein, the model is capable of capturing the formation and evolution of isolated subglacial cavities that trap a fixed water volume after becoming isolated. Its steady-state results agree well with the results of a simpler, two-dimensional, and purely viscous steady-state model that is solved by an entirely different numerical method.

The model lends some credence to existing approaches to modelling hydraulic isolation of the glacier bed in large-scale models using a threshold in mean cavity size to define connectivity, but it also suggests that significant modifications to those models may be required. For instance, it suggests that the cavitation ratio measuring the horizontal extent of ice–bed separation needs to be considered separately from the mean ice–bed gap thickness, especially when modelling the rapid expansion of cavities as previously uncavitated low-pressure regions of the bed are flooded by water: the cavitation ratio evolves faster and is a better predictor of subglacial connectivity than ice–bed gap thickness, and the two variables are not simple proxies for one another

The code used to compute the results in this paper is available on request from the author.

The supplement related to this article is available online at:

The author has declared that there are no competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by NSERC Discovery Grant RGPIN-2018-04665 and by computing resources made available by the Department of Earth, Ocean and Atmospheric Sciences as well as the Digital Research Alliance of Canada. I would also like to acknowledge the constructive comments of two anonymous referees.

This research has been supported by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2018-04665).

This paper was edited by Nanna Bjørnholt Karlsson and reviewed by two anonymous referees.