Glacier lake outburst floods are common glacial hazards around the world. How big such floods can become (either in terms of peak discharge or in terms of total volume released) depends on how they are initiated: what causes the runaway enlargement of a subglacial or other conduit to start the flood, and how big can the lake get before that point is reached? Here we investigate how the spontaneous channelization of a linked-cavity drainage system can control the onset of floods. In agreement with previous work, we show that floods only occur in a band of water throughput rates in which steady reservoir drainage is unstable, and we identify stabilizing mechanisms that allow steady drainage of an ice-dammed reservoir. We also show how stable limit cycle solutions emerge from the instability and identify parameter regimes in which the resulting floods cause flotation of the ice dam. These floods are likely to be initiated by flotation rather than the unstable enlargement of a distributed drainage system.

Glacier lake outburst floods or “jökulhlaups” are a glacial hazard in many parts of the world

While the mechanics of the main discharge phase of an outburst flood are relatively well understood

One possibility for flood initiation is that the lake simply fills to the level at which the ice dam starts to float, and a sheet flow emerges between ice and bed that subsequently channelizes

Motivated by the subglacial lake at Grímsvötn in Iceland,

Grímsvötn typically starts its flood when water levels are below flotation

While this mechanism successfully explains how limit cycles (stable, periodic oscillations in lake level) can emerge in the model, it also predicts that no water can leave the lake between floods. Tracer experiments conducted at Salmon Glacier in Canada

In this paper, we consider an alternative mechanism by which floods can initiate in a recurring fashion without the need for a sealed lake, but with continuous leakage throughout the flooding-and-refilling cycle. We show that a conduit that is able to switch spontaneously between the behaviour typical of an R channel and the behaviour of a linked-cavity system

The behaviour studied in the present paper was first documented in a model by

The original motivation for the work in

We use a hierarchy of model versions, employing both a spatially extended, one-dimensional drainage system model and a “lumped”, box-type model that provides additional physical insight, as well as being the appropriate limit of the spatially extended model in the case of a long flow path. The paper is laid out as follows: in Sect.

We use the one-dimensional continuum model for drainage through a single conduit stated in

Denoting conduit cross section by

Physically, the model represents a conduit whose size evolves due to a combination of dissipation-driven wall melting at a rate

Geometry of the problem: the conduit runs along the ice–bed interface.

Vanishing effective pressure at the end of the flow path

The upstream boundary condition at

For simplicity, we have neglected the possibility of a partially air filled conduit and the effects of overpressurization, where effective pressure becomes negative

There is an important simplification we have made, irrespective of the particular choice of

The assumption of an ice cliff, in addition, allows us to assume that

The two situations (an ice cliff and a seal) can be reconciled in the sense that a seal very close to the edge of the reservoir corresponds to a short, steep ice surface slope rather than an actual cliff. This results in a large, negative

Nevertheless, our simplifying assumption is relevant as the mechanism for initiating periodically recurring floods is fundamentally different from that in

By contrast, our model assumes that the reservoir always experiences some amount of leakage and requires no englacially supplied conduit: the water supply to the drainage system in our model is purely through the inflow term

The model (1) can be simplified if we assume that

Under these assumptions, we can relate the flux

In what follows, we proceed first with an analysis of the simpler, lumped model (2), to which we can bring to bear the theory of finite-dimensional dynamical systems

Nye's (

The basis of this instability is relatively easy to capture mathematically. To set the scene, we present a simplified version first before tackling an analysis of the complete model (2). If we assume a classical R channel in the sense of

In more abstract terms, if

Key to the instability mechanism above was the notion that effective pressure

Note that

The primary dependent variables in the model (2) are

We have deliberately written the left-hand side of Eq. (

The second term in the relation (

We have established that Nye's instability will occur if the conduit is channel-like and water storage in the system is sufficiently large, as well as if the system length

We can go further and determine explicitly the regions of parameter space in which this stability occurs. While we present our results here in dimensional form, note that it is possible to reduce the spatially extended and lumped models (1) and (2) to a four-dimensional parameter space by non-dimensionalizing them (Sect. S2.2 and S2.3 of the Supplement). These parameters are dimensionless versions of the inflow rate

Parameter values common to all calculations except those in Fig.

Stability boundaries for reservoir systems with the parameter values in Table

Figure

In each case shown in Fig.

For large enough

The upper critical value of

The analysis above has been purely linear, identifying parameter regimes in which steady drainage is unstable. That linearization will eventually fail where unstable growth is predicted, and a non-linear model is needed to predict what the instability grows into. A key aspect of many outburst floods is that they are a recurring phenomenon. For the reduced model (2) with two dynamical degrees of freedom and steady forcing, such a recurrence must correspond to a stable periodic oscillation in the absence of time-dependent forcing

It is straightforward to demonstrate bounded growth in our model computationally. Figure

In fact,

An alternative visualization of the physics involved is a phase plane, plotting

Periodic oscillations in the system (2) with parameter values as in Table

The periodic solution of Fig.

Figures

As in Sect.

Bifurcation diagrams for

The period of stable periodic solutions where they exist for the same parameter values as in Fig.

In all cases, an unstable steady state corresponds to a limit cycle. As

As the upper critical value of

The nature of the Hopf bifurcations at the stability boundaries in Fig.

We can also compute the period of the drainage oscillations, that is, the recurrence interval of floods. Figure

There is a significant caveat here: in many cases, the limit cycle solutions we have computed predict that

The stability boundary of Fig.

It is important to know where in parameter space the outburst flood mechanism would in reality change away from conduit growth due to cavities becoming channel-like at the end of the refilling phase and instead involve water separating ice from bed at vanishing effective pressure. The boundary between these two regimes should correspond to the location in parameter space where the minimum effective pressure during the flood cycle is zero. Figure

We can also address limit cycle solutions through asymptotic methods in some parametric limits in our model. The most relevant limit for real glacier-dammed lakes is likely to be that of a relatively large reservoir that is filled relatively slowly, but where water supply is not so small as to allow the conduit to be cavity-like in steady state (in which case the reservoir would be drained steadily, without a flood cycle): this is the case of large

In brief, the asymptotic solution confirms that there is a periodic flood cycle that the system very quickly settles into and that the flood cycle consists of three distinct stages. During the main flood stage, the evolution of the conduit is rapid and dominated by dissipation-driven melt

Qualitatively, this solution is illustrated by the limit cycle in Figs.

One of the predictions of the asymptotic solution is that the amplitude of effective pressure oscillations should be insensitive to refilling rate

The same asymptotic solution also provides an estimate for the inflow rate

We are also able to construct a second asymptotic solution for the opposite case of a large water supply rate

The analysis above provides a comprehensive picture of the lumped model (2). Here, we consider how well that lumped model represents the behaviour of its more complete, spatially extended counterpart (1). We begin by recreating the stability boundary diagrams of Fig.

The stability boundaries of Fig.

Results are shown in Fig.

While stabilization at large water input rates

To understand this discrepancy better. we have solved for the non-linear evolution of the drainage system as described by the spatially extended model (1) for the parameter values indicated by magenta circles in Fig.

Each column represents one of the magenta markers in Fig.

The three smallest values of

The lower row of panels in Fig.

Finite-amplitude evolution of the instability for the magenta dot marked “D” in Fig.

The extended model remains unstable beyond the stability boundary of the lumped model, but our numerical solutions no longer support the conclusion that the system necessarily settles into a limit cycle. Figure

Finite-amplitude evolution of the instability for the magenta dot marked “E” in Fig.

For even larger

Once again, a key observation is that the spatially extended model predicts negative effective pressures long before any kind of limit cycle is reached for all but one of the solutions shown and that the discrepancies between lumped and spatially extended models occur only where this is the case. Moreover, they occur for water supply rates to the reservoir that far exceed values that would be plausible for typical glacier-dammed lake systems: the lumped model appears to be robust for the latter. This includes the prediction by the lumped system of where overpressurization of the drainage system (

This does not render some of the more exotic instabilities shown in Figs.

We have shown that a drainage system that switches between cavity-like and channel-like behaviour spontaneously is capable of supporting periodic outburst floods from a glacier-dammed reservoir. At issue here is the initiation of the flood: as discussed by

At the heart of the oscillatory behaviour of glacier-dammed lakes is that runaway growth, or instability, of a channel-like conduit, which prevents steady discharge of the water supplied to the reservoir

For typical glacier-dammed lakes, moderate inflows

As in

We have also investigated a mechanism previously identified in

Stability boundary plotted in the same way as in Fig.

The reason why this stabilization mechanism is relevant is that it may explain why much smaller water reservoirs that are typically not recognized as lakes but nonetheless provide storage capacity (such as large moulins) do not invariably generate outburst-flood type behaviour. The stability diagram of Fig.

The picture changes significantly when we look at shorter, steeper glaciers. We can reduce the size of the parameter space that we need to sample in order to understand different glacier geometries by scaling the problem (Sect. S2.2 of the Supplement). If we change the background hydraulic gradient

The most significant impact here is that of steepening

As our discussion above indicates, relatively small, isolated storage volumes could lead to instabilities either for steeper glaciers or for very long flow paths. The latter in particular are however an unlikely scenario: take the example of 10 m-by-10 m moulin at the head of a 250 km flow path. In reality, there would most likely be many moulins spread out along the flow path instead. With this as motivation, we discuss a second exotic instability next, in which we opt for the opposite endmember of possible storage geometries by spreading out storage capacity evenly along the flow path. Much of the necessary groundwork is already in place from the analysis in Sect.

In Sect.

Here we sketch how to build on Sect.

With a finite domain size and the proposed boundary conditions above, a steady-state solution to the model (

Linearizing as

We can identify two mechanisms for instability. The first is essentially the same as Nye's melt–drainage feedback: the more storage capacity there is, the smaller all the terms containing

A second instability mechanism can occur if the conduit is cavity-like with

To make these considerations more concrete, Fig.

For instance, for longest flow path of

Stability of steady-state solutions to the model (

As demonstrated previously using a much more restricted sweep of parameter space in

It is worth pointing out that part of our focus has been on the emergence of limit cycle solutions in order to identify how the flood initiation mechanism can prevent flood magnitude from increasing progressively from cycle to cycle as observed in

The fact that we illustrate this by showing evolution towards a limit cycle is not at odds with the chaotic behaviour observed by

The lower cut-off to the drainage instability that leads to outburst floods corresponds to the drainage system switching to a cavity-like state under steady flow conditions when water input to the reservoir can be drained steadily by those cavities. The mechanism for the cut-off at high water throughput rates is harder to identify. The lumped model predicts that the high sensitivity of water level in the reservoir to the evolution of the draining conduit will induce water pressure gradients that reduce flow as the conduit grows and therefore suppress its enlargement due to heat dissipation. The threshold at which this happens is however significantly underestimated by the lumped model relatively to its spatially extended counterpart, which develops more wave-like instabilities at higher water throughput rates.

In closing, we have also investigated how wave-like instabilities can occur when the water reservoir is not localized but spread out or “distributed” along the flow path (for instance, in the form of many small reservoirs like basal crevasses). This type of instability was first observed in

Future work is likely to focus on capturing the role of overpressurization of the drainage system in initiating and mediating the instability driving outburst floods, since flood initiation at water pressures below flotation is confined to a relatively small part of parameter space, and the model predicts that reaching zero effective pressure and initiation by partial flotation of the ice dam is likely to be common.

This appendix provides only a brief sketch of the derivation of an asymptotic solution for a limit cycle solution in the case where the reservoir volume is large and inflow is sufficient to ensure that the reservoir cannot be drained by a cavity-like conduit but also that the conduit size evolves much faster than the timescale over which the reservoir fills. Full details are given in Sect. S4 of the Supplement.

The solution is developed from a parametric limit of the model (2) with Eq. (

We assume that the exponents

The main flood phase is described by omitting terms of

The refilling phase is described by the rescaling

The refilling phase ends as

The MATLAB code used in the computations reported is included in the Supplement, except for the code used to solve the transient calculations displayed in Figs.

The supplement related to this article is available online at:

The author declares that there is no conflict of interest.

Discussions with Rob Vogt, Ian Hewitt and Garry Clarke are gratefully acknowledged, as are reviews by Mauro Werder and an anonymous referee. This work was supported by an NSERC Discovery Grant.

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 Daniel Farinotti and reviewed by Mauro Werder and one anonymous referee.