2.1. Model

Our model closely resembles that described by Reference FowlerFowler (1999). It consists of a subglacial channel hydraulically coupled (in terms of both water discharge and pressure) to an ice-dammed lake (Fig. 1). The most important difference between our model and Fowler’s is our addition of a seasonally varying lake input. Model variables and parameters are summarized in Tables 1 and 2.

Fig. 1.

Our model ice-dammed reservoir system. (a) A subglacial channel is hydraulically coupled to an ice-dammed lake that receives meltwater at a rate Q _i. The channel receives a constant and uniform supply of water along its length M. (b) The prescribed basic hydraulic gradient ψ (blue curve) is negative near the lake – a topographic seal. Also displayed is an example pair of ice surface and bed profiles (z _s and z _b; black and brown curves) that would produce this hydraulic gradient. In this example the bed slope φ _b = 0.01.

Table 1.

Summary of model variables

Table 2.

Model parameters and physical constants

The ice-dammed lake is vertically sided, with a surface area A = 5 km², and the ice dam has a constant thickness H =100 m. Its depth h _L evolves with time t due to input Q _i and water exchange with the subglacial channel (Fig. 1) according to

where Q is the discharge in the channel and the spatial coordinate s is zero at the lake and s ₀ (=10 km) at the glacier terminus. The lake’s depth provides the boundary condition on the channel’s effective pressure N (equal to the difference between the ice-overburden and water pressures) at the lake outlet: N(0,t) = ρ _i gH − ρ _w gh _L, where ρ _i is the ice density (900 kg m⁻³), ρ _w is the water density (1000 kg m⁻³) and g is the acceleration due to gravity (9.8 m s⁻²). One choice for the boundary condition on N at s = s ₀ is N(s ₀,t) = 0 (e.g. Reference NgNg, 1998; Reference KingslakeKingslake, 2013). However, to simplify the numerics, we assume that drainage is controlled by a boundary layer in the channel’s effective pressure near the lake. Hence this boundary condition is replaced by ∂N/∂s = 0 (Reference FowlerFowler, 1999). The qualitative behaviour of the model is not affected by this simplification (Reference EvattEvatt, 2006; Reference KingslakeKingslake, 2013).

The evolution of the channel’s cross-sectional area is controlled by the competition between melt enlargement and creep closure:

where n = 3 and K ₀ = 10⁻²⁴ Pa⁻³ s⁻¹ are temperate ice-rheology parameters (Reference FowlerFowler, 1999) and m is the rate of frictional channel-wall melting per unit distance ((ψ + ∂N/∂s)|Q|/L, where L is the latent heat of fusion of water (334 kJ kg⁻¹) and ψ is the basic hydraulic gradient). This assumes that the water is at the pressure-melting point and all the energy it gains as it flows through the total hydraulic gradient (ψ +∂N/∂s) is used locally for melting (Reference FowlerFowler, 1999). Ignoring a small contribution from the changing channel area, water mass continuity gives

where M is a constant, uniform input to the channel along its length (7 × 10⁻⁴ m² s⁻¹). We use Manning’s equation to parameterize momentum balance in the turbulent water flow:

where f is the hydraulic roughness (n′²(l ²/S)^2/3 ≈ 0.07 m^2/3 s² for a semicircular channel with wetted perimeter l and Manning’s roughness coefficient n′ = 0.1 m^1/3 s; Reference FowlerFowler, 1999; Reference Kingslake and NgKingslake and Ng, 2013a). Water flow is driven by the total hydraulic gradient ψ + ∂N/∂s. This contains a dynamic component ∂N/∂s and a component that depends on glacier shape – the basic hydraulic gradient ψ (= ρ _w g sin φ _b − ∂P _i/∂s, where φ _b is the channel slope and P _i is the ice overburden pressure). To encourage the simulation of stable flood cycles, the prescribed glacier shape is such that ψ is negative near the lake (Fig. 1b):

where ψ ₀ = 100 Pa m⁻¹.

To simulate seasonal weather variations and drive a simple model of lake input Q _i we use a synthetic sinusoidal air temperature time series,

where t has units of years and integer t corresponds to the beginning of each calendar year. The 0.29 year offset ensures the maximum air temperature T _m occurs in midsummer. We model meltwater input to the lake Q _i by

where k = 2 m³ s⁻¹ K⁻¹, close to the value of a similar parameter derived empirically by Reference Kingslake and NgKingslake and Ng (2013b). Figure 2 plots our synthetic air temperature time series, as defined by Eqn (6), and the corresponding time series of meltwater input to the lake, as defined by Eqn (7).

Fig. 2.

Synthetic air temperature and lake input time series defined by Eqns (6) and (7).

2.2. Parameters

The model includes several physical parameters that are poorly constrained by observations (Table 2). The Manning’s roughness coefficient n′ = 0.1 m^1/3 s and the ice-rheology parameters n and K ₀ are chosen based on previous similar studies (Reference FowlerFowler, 1999; Reference Evatt, Fowler, Clark and HultonEvatt and others, 2006; Reference Kingslake and NgKingslake and Ng, 2013a). The constant supply of water to the channel along its length M = 7 ×10⁻⁴ m² s⁻¹ implies a reasonable background terminus discharge of s ₀ M = 7 m³ s⁻¹. We expect the results of our analysis not to depend qualitatively on these parameter values.

To parameterize the basic hydraulic gradient, Reference FowlerFowler (1999) proposed Eqn (5) based on observations of ice thickness from Vatnajökull, Iceland (Reference BjörnssonBjörnsson, 1992). An area of negative hydraulic gradient near the lake, or ‘topographic seal’, encourages stable flood cycles in the model via the formation of a subglacial hydraulic divide. Reference KingslakeKingslake (2013) and Reference Kingslake and NgKingslake and Ng (2013a) showed that a topographic seal is not a requirement for divide formation or for the simulation of stable flood cycles. We impose a topographic seal in this study to increase the range of parameter space over which stable flood cycles can be simulated. Qualitatively the model’s behaviour is independent of the details of the seal (Reference KingslakeKingslake, 2013).

The formation of the hydraulic divide does require a nonzero channel input M, because when M = 0, Q is uniform (Eqn (3)). Without the stabilizing effect of divide formation, simulated floods grow unstably (Reference NgNg, 1998; Reference FowlerFowler, 1999; Reference KingslakeKingslake, 2013; Reference Kingslake and NgKingslake and Ng, 2013a). This is unrealistic, therefore we use a positive channel input M to allow us to simulate stably repeating floods.

2.3. Numerics

To explore the behaviour of the model we solve its governing equations numerically. Space and time domains are discretized with a grid spacing of 100 m and time steps of 2.7 hours. Integrating Eqn (3) and combining the result with the terminus boundary condition on N and Eqn (4) yields

With Q calculated using this expression, N is found by integrating Eqn (4) from the lake, where N is determined by the lake depth (N(0,t) = ρ _i gH − ρ _w gh _L), to the terminus. With Q and N known everywhere, h _L and S are evolved forward in time using Eqns (1) and (2) and the forward Euler method, with Q _i given by Eqns (6) and (7). Reference FowlerFowler (1999), Reference EvattEvatt (2006) and Reference KingslakeKingslake (2013) provide more details.

2.4. Experimental design

We conduct a suite of numerical model simulations during which the parameter T _m is varied systematically (0.1 ≤ T _m ≤ 28°C) between simulations and kept constant during each simulation. During this exploration of T _m parameter space, all other parameters are kept constant. Unless otherwise stated, all the simulations start with h _L = 40 m and a uniform channel cross-sectional area ∼5m², and continue for 120 model years.

3.1. Flood cycles

Figure 3 plots flood cycles simulated using three different values of the midsummer air temperature T _m. Figure 3a, c and e display 20 year time series of discharge at the lake outlet, Q(0,t), extracted from the results of three simulations that use T _m = 10°C, T _m = 12.7°C and T _m = 15°C. Figure 3b, d and f show corresponding orbits in Q(0,t)−h _L(t) phase space.

Fig. 3.

Simulated flood cycles. Results from three simulations using seasonally varying air temperatures to drive lake input, with (a, b) T _m = 10°C, (c, d) T _m = 12.7°C and (e, f) T _m = 15°C. (a, c, e) Time series of discharge at the lake outlet Q(0,t) (blue curves) and lake input Q _i(t) (green curves) for 20 model years after transients have ended. (b, d, f) Solution orbits in Q(0,t)−h _L(t) phase space from the complete 120-year-long simulations.

When T _m = 10°C and 15°C, after transients, the system approaches limit cycles with repeat times of 2 years and 1 year respectively (Fig. 3a and e). The sequence of lake filling, channel enlargement through melt, lake highstand and drainage has been described elsewhere (Reference NyeNye, 1976; Reference FowlerFowler, 1999, Reference Fowler2009; Reference Ng and BjörnssonNg and Björnsson, 2003; Reference Kingslake and NgKingslake and Ng, 2013a). Note that the discharge at the lake outlet is negative between floods; a hydraulic divide forms in the channel near the lake. Reference FowlerFowler (1999) showed how the dynamics of this divide cause flood size to increase with Q _i when this input is kept constant (Reference KingslakeKingslake, 2013: ch. 3). Contrary to expectations based on these previous analyses, the warmer simulation (T _m = 15°C; Fig. 3e and f), with the higher mean lake input, produces flood cycles with a lower peak discharge (cf. Fig. 3a and e).

When T _m = 12.7°C, flood cycles are simulated but they do not approach limit cycles (Fig. 3c and d). Instead the cycles appear random, never reaching a repeating orbit in Q(0,t)−h _L(t) phase space (Fig. 3d). We will show that these cycles exhibit chaotic characteristics.

3.2. Mode locking

Figure 4 plots the results of many simulations during which we record the peak discharge of each flood. Each point in Figure 4 plots one flood’s peak discharge (vertical axis) against the value of T _m used in the simulation that produced that flood (horizontal axis). The colour of each point indicates the day of the year on which the flood peak occurred.

Fig. 4.

T _m parameter space. (a) Peak discharge of floods recorded during 120 year simulations plotted against the value of the midsummer air temperature T _m used in each simulation. The first ten flood peaks of each simulation are omitted to remove transients. The black box indicates the region shown in more detail in (b). Horizontal quantization visible in (a) (T _m > 16°C) is the result of a coarse search through this region. The vertical lines in (b) correspond to orbits plotted in Figure 9. Lines that are one point in vertical extent (e.g. 6 ≤ T _m ≤ 11°C and T _m > 19°C) correspond to limit cycles, where all the recorded floods have the same peak discharge; slightly thicker lines (e.g. 14 ≤ T _m ≤ 17°C) correspond to simulations during which transients lasted longer than ten flood cycles; and large blocks of points indicate dense, chaotic orbits. The colour of each point indicates the day of the year on which the flood peak occurs. In (a) the green arrows indicate two regions where floods occur progressively later in the year as T _m decreases.

In the regions of T _m parameter space around T _m = 10°C and T _m = 15°C, floods occur progressively later in the year as T _m decreases (green arrows; Fig. 4a). Moreover, instead of floods occurring less often as T _m decreases, the mean time between floods (hereafter referred to as the repeat time) remains constant and peak discharges decrease to compensate for the decrease in time-averaged lake input (Fig. 4a).

This behaviour is analogous to ‘mode locking’ of periodically forced nonlinear oscillators, when the frequency of an oscillator’s response stays locked to that of the frequency of the forcing. This can occur when the frequency of the forcing is comparable to an oscillator’s natural frequency. Similarly, the mode-locking behaviour of our model only manifests when the timescales of the lake-depth and subglacial-channel evolution, predominantly controlled by the lake surface area and glacier geometry respectively (Section 2), are comparable to the periodicity of the time-varying meltwater input (1 year).

3.3. Resonance

As T _m decreases from 15°C to 10°C, the peak discharges of floods increase and their timing in the year shifts abruptly (Fig. 4). When the lake’s total annual input V _A is slightly too large to allow it to persist for 2 years before draining (e.g. when T _m = 15°C), relatively small floods occur every year. When T _m is smaller (e.g. when T _m = 10°C) the lake can persist for 2 years before draining. Although the time-averaged lake input is now lower, the flood repeat time is twice as large (2 years instead of 1 year). Hence the total input to the lake between each flood is increased. This increases flood peak discharges.

Floods are largest when the climatic forcing allows the lake to fill completely in an integer number of years. A lake’s ‘resonant climatic forcing’ is any value of T _m that allows the lake basin to fill completely over n years, where n = 1, 2, 3, etc. Therefore, there exist multiple resonant values of T _m corresponding to V _A = V _F/n, where V _F is the lake’s volume when full. This is reflected in Figure 4 by a second resonance peak at T _m ≈ 5.5°C and a third at T _m ≈ 2°C (not clearly visible at the resolution of Fig. 4). Integrating Eqns (6) and (7) yields V _A =3.15 × 10⁷ kT _m/π. Assuming the lake fills to the flotation level (9/10H) before emptying completely during each flood (V _F = 9/10HA), the nth resonant climatic forcing is given by T _m ⁿ = 0.9πHA/(3.15 × 10⁷ kn) = {22.4, 11.2, 7.5, 5.6,…}°C. The second resonant T _m value (11.2°C) matches the numerically simulated resonance peak (Fig. 4). However, when T _m < 10°C, a significant volume of water is left in the lake after floods and this simple analysis fails.

3.4. Chaotic dynamics

Mode-locked regions of parameter space are separated by densely populated regions (Fig. 4) corresponding to dense periodic orbits that never repeat. These orbits are sensitive to their initial conditions. Figure 5a plots time series of lake depth from two simulations that used T _m = 12.7°C, with two slightly different initial lake depths, 40.00 m and 40.01 m. Figure 5b and c plot the trajectories of the two solutions in three-dimensional (3-D) discharge–lake-depth–channel-size (Q(0,t)−h _L(0)−S(0,t)) phase space, while Figure 5d shows the normalized separation of these orbits. The solutions track each other for ∼15 years before diverging. However, their separation does not grow unboundedly; they remain in the same region of phase space (Fig. 5b and c), but despite approaching each other at t ≈ 65 years (Fig. 5d) they never collapse onto a common orbit. This behaviour is indicative of chaotic dynamics and has been studied thoroughly in other fields (e.g. Reference DrazinDrazin, 1992; Reference OttOtt, 2002).

Fig. 5.

Sensitivity to initial conditions. (a) Time series of lake depth from two simulations which used T _m = 12.7°C. Simulations are identical except for a slight difference in their initial lake depth h _L(0). The blue curve corresponds to h _L(0) = 40.00 m, and the red curve corresponds to h _L(0) = 40.01 m. (b, c) Corresponding orbits in Q(0,t)−h _L(0)–S(0,t) phase space, with the initial and final conditions shown in green and red respectively. (d) Time series of the normalized separation between the orbits in (b, c). Normalized separation is calculated by differencing each component of the orbits’ positions normalized with the maximum value of the corresponding variable. The Pythagorean sum of the normalized separation is plotted. Note the different time axes in (a) and (d).

Another indicator of chaotic dynamics is topological mixing of phase space, where orbits fold and bifurcate, visiting every part of a specific region of phase space. Figure 6 plots the trajectory of one simulation (with T _m =12.7°C) in another 3-D phase space: discharge–lake-depth–time (Q(0,t)−h _L(t)−t) space. Figure 6b displays two-dimensional Poincaré sections taken along Q(0,t)−h _L(t) planes, perpendicular to the t-dimension, at nine instants in the annual cycle. The trajectory possesses a clear structure; together the points form a shape called an attractor. The ‘Nye attractor’ rotates, bifurcates and folds over on itself as the sections progress through the annual cycle. This depiction of topological mixing is reminiscent of equivalent plots from studies of nonlinear oscillators (e.g. the Duffing oscillator; Reference Novak and FrehlichNovak and Frehlich, 1982).

Fig. 6.

The Nye attractor and topological mixing of phase space. (a) A solution orbit in Q(0,t)−h _L(t)−t phase space of a simulation that used T _m = 12.7°C and h _L(t = 0) = 40 m. The blue curve’s distance from the vertical green line denotes the lake depth h _L(t), its position along the vertical axis denotes the discharge at the lake Q(0,t) and its rotation around the green line denotes the fractional part of time t multiplied by 2π (one rotation corresponds to 1 year). (b) Poincaré sections taken perpendicular to the t-dimension at nine different stages of the year, denoted by the day of the year d. The points locate the intersection of the orbit with each section. Each section’s location in Q(0,t)−h _L(t)−t space is indicated by the boxes in (a). The colour of the points indicates simulation time t.

3.5. Bifurcations

The transition from limit cycles to chaotic cycles around 13 < T _m < 14°C involves two types of bifurcation (Fig. 4). The first, at T _m ≈ 13.88°C, is an abrupt transition from period-1 orbits (i.e. limit cycles that complete one orbit before repeating) to period-2 orbits (i.e. limit cycles that complete two orbits before repeating). Figure 7 displays the orbits in Q(0,t)−h _L(t) phase space of two simulations which lie on each side of this transition in parameter space. Figure 8 plots corresponding time series of Q _i and Q(0,t).

Fig. 7.

Orbits in Q(0,t)−h _L(t) phase space of solutions with (a) T _m = 13.888000°C and (b) T _m = 13.888002°C, which lie on each side of an abrupt bifurcation in T _m parameter space (Fig. 4) for 20 ≤ t ≤ 90 years. Points are colored to indicate simulation time t and are separated by ∼4.5 model days.

Both simulations start on almost identical unstable period-2 orbits (blue points in Fig. 7), in which every second flood is double-peaked (Fig. 8a; note that the two orbits appear as one curve in Fig. 8a). The onset of melt in spring occurs just after the first peak and instigates the second (Fig. 8a). Larger floods occur every third winter. The simulations leave these orbits in two different ways.

Fig. 8.

Time series of lake input Q _i and discharge through the lake outlet Q(0,t) from two simulations that lie close to the abrupt bifurcation at T _m ≈ 13.88°C. (a) Initial Q _i(t) and Q(0,t) time series from two simulations that used T _m = 13.888000°C and T _m = 13.888002°C. (b, c) Long-time time series of the same variables from simulations that used T _m = 13.888000°C and T _m = 13.888002°C respectively.

During the warmer simulation (T _m = 13.888002°C), floods occur progressively earlier each year and the double-peaked flood evolves into two floods of equal size (Fig. 8c). Meanwhile, the originally larger flood shrinks. The result is limit cycles with a repeat time of 1 year consisting of equally sized floods (red points, Fig. 7b; Fig. 8c). During the cooler simulation (T _m = 13.888000°C), floods occur progressively later until the first peak of the double-peaked flood shrinks and disappears, resulting in limit cycles where every other flood is roughly twice as large as the others and the mean flood repeat time is 1.5 years (red points, Fig. 7a; Fig. 8b).

After this abrupt bifurcation, further decrease in T _m leads to a cascade of period-doubling bifurcations. Figure 9 plots four solution orbits in Q(0,t)−h _L(t) phase space corresponding to T _m values that bracket several period-doubling bifurcations (vertical solid lines in Fig. 4b). As T _m is decreased past each bifurcation, the orbit split into two branches and the period of the orbit doubles. Many such bifurcations lead to the densely populated chaotic region on the left of Figure 4b.

Fig. 9.

Period-doubling bifurcations. Long-time solution orbits (transients have been removed) in Q(0,t)−h _L(t) phase space of four simulations that used (a) T _m = 13.701°C, (b) T _m = 13.401°C, (c) T _m = 13.201°C and (d) T _m = 13.151°C. (a–d) correspond respectively to the positions in T _m-parameter space indicated in Figure 4b by the vertical lines A–D.