Modelling Ice-Dammed Lake Drainage

The drainage of ice-dammed lakes produces floods that can pose hazards, waste water resources and modulate ice flow. In this thesis I investigate several aspects of ice-dammed lake drainage through the development and analysis of mathematical models. After an introduction in the first chapter and a description of the mathematical background to the thesis in the second, the third chapter investigates the mechanisms behind observed variability in the size and timing of subglacial floods from ice-dammed lakes. In particular, I examine how environmental controls like the weather and the shape of glaciers affect floods. In the next chapter, I quantify how well simple models can predict the dates of floods from an ice-marginal lake in Kyrgyzstan. I find that incorporating environmental controls into models improves their prediction ability. Next I investigate the coupling between subglacial drainage and glacier motion during ice-dammed lake drainage by developing and analysing a model which couples a marginal lake, glacier sliding, subglacial drainage through a channel and subglacial drainage through a distributed system of cavities. I show how changes in lake level cause the rate at which a glacier slides to increase during the first half of floods and decrease during the second half. The next two chapters are concerned with two lake-drainage scenarios that involve water flowing as an open stream: firstly, the subglacial open-channel flow that occurs after a marginal lake drains completely during a flood, and secondly, the drainage of supraglacial lakes across the surface of ice sheets. I end the thesis with a summary of my findings and some suggestions of theoretical and field-based investigations that are worthwhile pursing in the future

Modelling the coupling of flood discharge with glacier flow during jökulhlaups

We explore a mathematical model that couples together a thermomechanically evolving subglacial channel, distributed cavity drainage, and basal sliding along a subglacial flood path fed by a jökulhlaup lake. It allows water transfer between channel and cavities and a migrating subglacial water divide or 'seal' to form between floods. Notably, it accounts for full coupling between the lake and subglacial drainage in terms of both discharge and pressure, unlike models that neglect the pressure coupling by imposing a known history of lake discharge at the channel inlet. This means that flood hydrographic evolution and its impact on glacier motion are consistently determined by our model. Numerical simulations for a model alpine lake yield stable limit cycles simulating repeating jökulhlaups, with the channel drawing water from the cavities at a varying rate that modulates basal sliding during each flood. A wave of fast sliding propagates down-glacier at flood initiation, followed by deceleration as the growing channel sucks water from the cavities. These behaviours cannot be correctly simulated without the full coupling. We show that the flood's peak discharge, its initiation threshold and the magnitude of the 'fast sliding' wave decrease with the background water supply to the cavities

Ice‐Shelf Basal Melt Channels Stabilized by Secondary Flow

Ice-shelf basal channels form due to concentrated submarine melting. They are present in many Antarctic ice shelves and can reduce ice-shelf structural integrity, potentially destabilizing ice shelves by full-depth incision. Here, we describe the viscous ice response to a basal channel - secondary flow - which acts perpendicular to the channel axis and is induced by gradients in ice thickness. We use a full-Stokes ice-flow model to systematically assess the transient evolution of a basal channel in the presence of melting. Secondary flow increases with channel size and reduces the rate of channel incision, such that linear extrapolation or the Shallow-Shelf Approximation cannot project future channel evolution. For thick ice shelves (> 600 m) secondary flow potentially stabilizes the channel, but is insufficient to significantly delay breakthrough for thinner ice (< 400 m). Using synthetic data, we assess the impact of secondary flow when inferring basal-channel melt rates from satellite observations

Modeling oscillations in connected glacial lakes

Mountain glaciers and ice sheets often host marginal and subglacial lakes that are hydraulically connected through subglacial drainage systems. These lakes exhibit complex dynamics that have been the subject of models for decades. Here we introduce and analyze a model for the evolution of glacial lakes connected by subglacial channels. Subglacial channel equations are supplied with effective pressure boundary conditions that are determined by a simple lake model. While the model can describe an arbitrary number of lakes, we solve it numerically with a finite element method for the case of two connected lakes. We examine the effect of relative lake size and spacing on the oscillations. Complex oscillations in the downstream lake are driven by discharge out of the upstream lake. These include multi-peaked and anti-phase filling–draining events. Similar filling–draining cycles have been observed on the Kennicott Glacier in Alaska and at the confluence of the Whillans and Mercer ice streams in West Antarctica. We further construct a simplified ordinary differential equation model that displays the same qualitative behavior as the full, spatially-dependent model. We analyze this model using dynamical systems theory to explain the appearance of filling–draining cycles as the meltwater supply varies

Long-period variability in ice-dammed glacier outburst floods due to evolving catchment geometry

We combine a glacier outburst flood model with a glacier flow model to investigate decadal to centennial variations in outburst floods originating from ice-dammed marginal basins. Marginal basins can form due to the retreat and detachment of tributary glaciers, a process that often results in remnant ice being left behind. The remnant ice, which can act like an ice shelf or break apart into a pack of icebergs, limits a basin’s water storage capacity but also exerts pressure on the underlying water and promotes drainage. We find that during glacier retreat there is a strong, nearly linear relationship between flood water volume and peak discharge for individual basins, despite large changes in glacier and remnant ice volumes that are expected to impact flood hydrographs. Consequently, peak discharge increases over time as long as there is remnant ice remaining in a basin, and peak discharge begins to decrease once a basin becomes ice-free. Thus, similar size outburst floods can occur at very different stages of glacier retreat. We also find that the temporal variability in outburst flood magnitude depends on how the floods initiate. Basins that connect to the subglacial hydrological system only after reaching flotation depth yield greater long-term variability in outburst floods than basins that are continuously connected to the subglacial hydrological system (and therefore release floods that initiate before reaching flotation depth). Our results highlight the importance of improving our understanding of both changes in basin geometry and outburst flood initiation mechanisms in order to better assess outburst flood hazards and their impacts on landscape and ecosystem evolution.This project was supported by funding from the Alaska Climate Adaptation Science Center and the US National Science Foundation (OIA-1757348 and OPP-1743310). We thank Christian Kienholz for fruitful discussions that led to this study.Ye

Long-period variability in ice-dammed glacier outburst floods due to evolving catchment geometry

We combine a glacier outburst flood model with a glacier flow model to investigate decadal to centennial variations in outburst floods originating from ice-dammed marginal basins. Marginal basins can form due to the retreat and detachment of tributary glaciers, a process that often results in remnant ice being left behind. The remnant ice, which can act like an ice shelf or break apart into a pack of icebergs, limits a basin’s water storage capacity but also exerts pressure on the underlying water and promotes drainage. We find that during glacier retreat there is a strong, nearly linear relationship between flood water volume and peak discharge for individual basins, despite large changes in glacier and remnant ice volumes that are expected to impact flood hydrographs. Consequently, peak discharge increases over time as long as there is remnant ice remaining in a basin, and peak discharge begins to decrease once a basin becomes ice-free. Thus, similar size outburst floods can occur at very different stages of glacier retreat. We also find that the temporal variability in outburst flood magnitude depends on how the floods initiate. Basins that connect to the subglacial hydrological system only after reaching flotation depth yield greater long-term variability in outburst floods than basins that are continuously connected to the subglacial hydrological system (and therefore release floods that initiate before reaching flotation depth). Our results highlight the importance of improving our understanding of both changes in basin geometry and outburst flood initiation mechanisms in order to better assess outburst flood hazards and their impacts on landscape and ecosystem evolution.This project was supported by funding from the Alaska Climate Adaptation Science Center and the US National Science Foundation (OIA-1757348 and OPP-1743310). We thank Christian Kienholz for fruitful discussions that led to this studyYe

Glacial melt under a porous debris layer

In this paper we undertake a quantitative analysis of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris-layer airflow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Østrem curve (a plot of the melt rate as a function of debris depth). Specifically, we show that the turning point in the Østrem curve can be caused by two distinct mechanisms: the increase in the proportion of ice that is debris-covered and/or a reduction in the evaporative heat flux as the debris layer thickens. This second effect causes an increased melt rate because the reduction in (latent) energy used for evaporation increases the amount of energy available for melting. Our model provides an explicit prediction for the melt rate and the temperature distribution within the debris layer, and provides insight into the relative importance of the two effects responsible for the maximum in the Østrem curve. We use the data of Nicholson and Benn (2006) to show that our model is consistent with existing empirical measurements

Full-depth englacial vertical ice-sheet velocities measured using phase-sensitive radar

We describe a geophysical technique to measure englacial vertical velocities through to the beds of ice sheets without the need for borehole drilling. Using a ground-based phase-sensitive radio-echo sounder (pRES) during seven Antarctic field seasons, we measure the temporal changes in the position of englacial reflectors within ice divides up to 900 m thick on Berkner Island, Roosevelt Island, Fletcher Promontory and Adelaide Island. Recorded changes in reflector positions yield 'full-depth' profiles of vertical ice velocity that we use to examine spatial variations in ice flow near the divides. We interpret these variations by comparing them to the results of a full-Stokes simulation of ice-divide flow, qualitatively validating the model and demonstrating that we are directly detecting an ice-dynamical phenomenon called the Raymond Effect. Using pRES, englacial vertical ice velocities can be measured in higher spatial resolution than is possible using instruments installed within the ice. We discuss how these measurements could be used with inverse methods to measure ice rheology, and to improve ice-core dating by incorporating pRES-measured vertical velocities into age modelling