Models of subglacial drainage and of cavity formation generally assume that the glacier bed is pervasively hydraulically connected. A growing body of field observations indicates that this assumption is frequently violated in practice. In this paper, I use an extension of existing models of steady-state cavitation to study the formation of hydraulically isolated, uncavitated, low-pressure regions of the bed, which would become flooded if they had access to the subglacial drainage system. I also study their natural counterpart, hydraulically isolated cavities that would drain if they had access to the subglacial drainage system. I show that connections to the drainage system are made at two different sets of critical effective pressure, a lower one at which uncavitated low-pressure regions connect to the drainage system and a higher one at which isolated cavities do the same. I also show that the extent of cavitation, determined by the history of connections made at the bed, has a dominant effect on basal drag while remaining outside the realm of previously employed basal friction laws: changes in basal effective pressure alone may have a minor effect on basal drag until a connection between a cavity and an uncavitated low-pressure region of the bed is made, at which point a drastic and irreversible drop in drag occurs. These results point to the need to expand basal friction and drainage models to include a description of basal connectivity.

Subglacial drainage is often assumed to occur in part through a “distributed” drainage system: connected conduits that are not arborescent in their geometry

Large-scale models for subglacial drainage systems typically assume that the bed as a whole always remains hydraulically connected. Existing process-scale models for the evolution of subglacial cavities generally make the same assumption. In large-scale drainage models, cavities are represented by a water sheet thickness: a cavity depth averaged over a representative small area of the bed (that is, an area of the bed that is much larger than an individual cavity but much smaller than the glacier as a whole). The assumption of a connected bed here simply means that water can flow as soon as the sheet thickness exceeds zero (e.g.

In process-scale models, hydraulic connectedness typically occurs through the bed itself: the bed is highly permeable. Water sourced from an ambient drainage system at some given water pressure can force its way between the ice and bed as soon as compressive normal stress at the base of the ice drops to the water pressure in the ambient drainage system, causing a cavity to form

These assumptions are at odds with a growing set of observations (

Recent work in large-scale drainage modelling has attempted to address this issue

If only part of the glacier bed has access to the ambient drainage system, then isolated, uncavitated, low-pressure regions can form elsewhere, at normal stresses that would lead to ice–bed separation if water from the ambient drainage system had access. Conversely, these distant parts of the glacier bed can become flooded with water when connected cavities grow at low effective pressure. If the effective pressure in the drainage system increases again after that flooding, the intervening connections can become closed, leaving isolated cavities of fixed volume. These isolated cavities will generally be at different effective pressures than the connected drainage system.

In the present work, I have used a modified mathematical model for cavity formation to explore the physics involved. The basic physics of ice flow over an undulating bed, allowing for the possibility of ice–bed separation as water forces its way between the two, are the same as in existing models for subglacial cavity formation. However, only a pre-defined, highly permeable part of the bed, denoted by

The model comes in two flavours: first, a two-dimensional, purely viscous flow model for the ice assumes that the cavity roof is in steady state and that water pressure in each separate cavity is spatially uniform. Where a cavity is in contact with the permeable part

The two versions of the model are susceptible to solution by different methods, making the simpler, purely viscous, steady-state version a useful test case for the more complicated dynamic version. To make the presentation more manageable, I have split these two model versions across two separate papers, focusing here on the purely viscous steady-state model. The dynamic model is presented in a companion paper

Consider the possibility of isolated cavities in the two-dimensional, purely viscous, steady-state model of subglacial cavitation in

Definitions used in the model. The upstream and downstream cavity end points of the

To be definite, I also assume the domain to be periodic in

The previous work in

Here I abandon the assumption of a fully permeable bed. If parts of the bed are instead impermeable, there is no universally defined water pressure, and water will not force its way between the ice and bed simply because the normal stress drops locally to the water pressure in a distant drainage system. Water pressure is still assumed to be constant in each cavity while potentially differing between cavities, so the

To be more specific, I assume that only a part

Outside of the intervals

Also note that

The specification of a permeable bed portion

In any case, the modified steady cavity problem can be solved by a slight modification to the complex variable method in

In the next subsection, I consider a system of cavities that is in quasi-equilibrium, forced by a very slowly changing effective pressure

Figure

Note that, when expressed as functions of a scaled position

Cavity roof shape

With the bed geometry given by Eq. (

The cavities expand continuously as

Panel

The dependence of cavity size on

Cavity roof shape

The behaviour is somewhat different if I restrict the permeable portion

As before, the normal stress around the cavity is continuous at the detachment point at the upstream end of the cavity and singular at the reattachment point at the downstream end (Fig.

As

The newly expanded cavity roof now has a finite size gap above the smaller bed protrusion. It expands further, but now continuously, if effective pressure is lowered again. The expanded cavity is in fact identical in shape to the single merged cavity that forms for a fully permeable bed at the same effective pressure. More significantly, if

In the present two-dimensional model, re-contact with a limited permeable bed portion immediately leads to the formation of a second isolated cavity downstream of the right-hand bed protrusion, which I treat as retaining a constant volume

Conversely, if

Panel

The dependence of cavity end points on

In red is the effective pressure

In addition, I have plotted the effective pressure

Cavity roof shape

The ability of a cavity to expand across bed protrusions and subsequently create isolated cavities as described above depends on the position of the permeable portion of bed relative to prominent bed protrusions. Consider the same bed given by Eq. (

Note that the cavity is not able to expand upstream to the lee side of the bigger bed protrusion and only expands downstream past that bigger protrusion at the negative effective pressure

We can also ask how the formation of isolated cavities, and confinement of cavities, affects basal drag defined through

Friction law for the bed given by Eq. (

The standard assumption of a fully permeable bed

With a small

For the alternative case of

We, however, can also view

An oddity of the solution with

As a further caveat, note that for a fixed

The results above were computed either for completely permeable beds or for beds that had permeable sections located at normal stress minima prior to cavity formation. As pointed out, I view these permeable bed portions

In order to investigate the effect of choosing different permeable bed portions

Once formed, the cavity immediately has a finite size that extends beyond

Conversely, if

The only difference between the solutions in Fig.

These results serve as an illustration of how the placement of drainage access can serve to complicate the computation of cavity extent. Note, however, that in many cases the solutions shown in Fig.

The effect of permeable bed patch location, using the same plotting scheme as Fig.

The results I have found above for the double-humped bed given by Eq. (

Figure

Panel

The critical effective pressure at which a cavity extends abruptly across a smaller protrusion in its lee is marked by dotted black lines in Fig.

Friction law – the equivalent of Fig.

The friction law for the triple-humped bed (Fig.

Effective pressure in the isolated cavities shown as the blue solution in Fig.

One behaviour that differs subtly between the two bed geometries considered here is the dependence of effective pressure in isolated cavities on the effective pressure in connected cavities. For the triple-humped bed (Fig.

The steady-state solutions in Sect.

Second, when such connections occur, they invariably extend the existing cavity in the downstream direction and never upstream. This has major implications for the evolution of connectedness of the bed and for the effective pressures that can be sustained. For cavities that are caused by drainage system access

Third, once a connection has been made and the lee of a smaller bed protrusion has become submerged, the cavity space on that lee side can subsequently become isolated due to an increase in effective pressure (or decrease in sliding velocity), which causes the cavity roof to be lowered. The critical value for the disconnection between the upstream cavity and newly isolated, cavity however, occurs at a higher critical value

The reader may wonder at this point why one would bother with considering isolated, low-pressure contact areas at the bed at all: since their flooding is irreversible, are they irrelevant, since they will connect sooner or later and henceforth remain flooded, even if they become hydraulically isolated again? The point here is that treating the bed as fully impermeable outside of the region

A second point that needs to be addressed here is the limitation imposed by using a two-dimensional domain. True hydraulic connections over distances longer than a single bed wavelength

For a fully permeable bed, the ratio

The first qualitative difference between a permeable and impermeable bed is that Iken's bound

If drainage system access

The most dramatic changes in basal friction occur when

Once connection has occurred, the friction law mimics the friction law for a fully permeable bed. This remains the case even if

Computation of steady-state friction

In fact, a prototype parameterization for the friction laws shown in Figs.

A friction law of this form in turn implies that subglacial drainage models may need to incorporate a description of the evolution of cavitation ratio. As I will show in Part 2, cavitation ratio and mean cavity depth (the variable commonly used to define cavity geometry in large-scale drainage models) are not simple proxies for each other, implying that the introduction of cavity ratio into friction laws and drainage parameterizations would indeed imply an increase in model complexity.

There is a second complication in the definition of a friction law that deserves to be stressed for an impermeable bed: the quantity that is commonly understood as “effective pressure”, overburden minus water pressure at the bed, is not uniquely defined but potentially varies from cavity to cavity. That is, effective pressure varies over length scales that are treated as microscopic in typical subglacial drainage models because water pressure differs between cavities. In the idealized model I use here, I define a unique “ambient drainage system effective pressure”

The effective pressure in the connected portion of the drainage system is likely to be the only useful effective pressure that can be defined, as it will in general vary smoothly in space and can therefore be modelled at the large scale, at least in principle. That observation does underline, however, the need to include additional degrees of freedom that capture the degree of cavitation in friction laws, since effective pressure is then meaningless in a part of the bed that is fully hydraulically isolated, with no drainage system access at all: there may still be isolated cavities in that case, and their presence will affect basal friction as discussed above. To compound matters, this situation also significantly complicates any attempts to constrain such a friction law observationally: while effective pressure in a connected drainage system can in principle be measured by borehole access to the bed, the presence and extent of isolated cavities at the bed are much harder to determine.

Using a simple extension of an existing purely viscous model for steady-state basal cavities in two dimensions, I have shown that uncavitated regions of the bed can persist indefinitely at low normal stress provided there is no drainage pathway along which water can reach them. Such drainage pathways are created under slow changes in forcing effective pressure

The main limitations to the work presented here derive from its assumption of quasi-steady conditions and its restriction to two dimensions. Dynamic cavity connections have significantly richer behaviour than the quasi-steady solution in the present paper suggests and are investigated in detail in a companion paper. Three-dimensional bed topography by contrast remains an open problem and holds the key to a more complete understanding of hydraulic connectivity. Connections at the bed are presumably more likely to occur when bed topography is three-dimensional: in a two-dimensional setting, connectivity along the entire model domain is only possible when ice–bed contact is lost completely, whereas this is not the case in three dimensions. Similarly, contact of the ice roof between two cavities in three dimensions does not necessarily make them disconnected, whereas it does in two dimensions.

The construction in

Let

The solution method followed here is that of

As in

In fact, tangential cavity roof detachment and re-contact are required not only by Eq. (

The point here is really to account for the independent number of constraints on the solution that arise from the tangential re-contact. In integrating Eq. (

Armed with this result, I can again follow the same solution procedure as in

In order for

Suppose that the

In practice, I introduce the smallest new cavity possible when the inequality (

Neighbouring cavities

In order to capture the effect of cavity isolation, I compute solutions by arc length continuation under increases in

Here I show that continuous stress at cavity detachment points and a stress singularity at reattachment points are natural consequences of the inequality constraints (

Consider the original Hilbert problem (Eq.

The same approach can be used near a re-contact point, but with different conclusions. Replacing

The code used is available from the author on request.

No data sets were used in this article.

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. I would like to acknowledge the helpful comments of two anonymous referees, which improved the paper significantly.

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.