Numerical Modelling of the Cordilleran Ice Sheet

Home , Cassiar Mountains, Cordilleran Ice Sheet, Last Glacial Maximum, Pacific Coast Ranges, Skeena Mountains

Numerical modelling of the Cordilleran ice sheet

Julien Seguinot

Department of Physical Geography and Quaternary Geology Stockholm University Stockholm 2014 c Julien Seguinot

ISSN: 1653-7211 ISBN: 978-91-7447-973-7 Cover: Numerical reconstruction of the Cordilleran ice sheet at the Last Glacial Maximum. Bedrock topography, ice surface topography, ice surface velocity, and frozen bed areas. Layout: Julien Seguinot based on LATEX templates by Rickard Petterson (cover summary) and Copernicus Publications (Paper IV). Published papers typeset by Copernicus Publications (Paper I) and the International Glaciology Society (Papers II and III). Graphics created using matplotlib, cartopy, LATEX and Inkscape. Printed by Universitetsservice US-AB, Stockholm, Sweden, 2014. Doctoral dissertation 2014 Julien Seguinot Department of Physical Geography and Quaternary Geology Stockholm University

Abstract. This doctoral dissertation presents a study of the glacial history of the North American Cordillera using numerical ice sheet modelling calibrated against field evidence. This area, characterized by the steep topography of several mountain ranges separated by large inter-montane depressions, was once covered by a large-scale ice mass: the former Cordilleran ice sheet. Because of the irregular topography on which the ice sheet formed, geological studies have often had only local or regional relevance, thus leaving the Cordilleran ice sheet least understood among Pleistocene ice sheets in terms of its extent, volume, and dynamics. Here, I present numerical simulations that allow quantitative reconstructions of the former ice sheet evolution based on approximated physics of glacier flow. These simulations show that the geometry of the Last Glacial Maximum Cordilleran ice sheet was largely controlled by sharp contrasts in regional temperature, precipitation, and daily temperature variability associated with the presence of mountain ranges. However, this maximum stage appears short-lived and out of balance with contemporaneous climate. During most of the simulated last glacial cycle, the North American Cordillera is characterized by an in- termediate state of glaciation including isolated glaciers and ice caps covering major mountain ranges, the largest of which is located over the Skeena Mountains. The numerically modelled Cordilleran ice sheet appears in constant imbalance with evolving climate conditions, while the complexity of this transient response transcends that encapsulated in two-dimensional, conceptual models of ice sheet growth and decay. This thesis demonstrates the potential of numerical ice sheet modelling to inform on ice sheet history and former climate conditions over a glacial cycle, given that ice sheet models can be calibrated against field constraints.

Sammanfattning. I min avhandling presenterar jag den glaciala historien för de Nordamerikanska Kordillerabergen baserad på numerisk ismodellering kalibrerad mot fältdata. Området, som karaktäriseras av branta bergskedjor och mellanliggande lägre platåer, täcktes tidigare av en inlandsis, Kordilleraisen. Den oregelbundna to- pograﬁn över vilken inlandsisen bildades har bidragit till att tidigare fältstudier främst har haft ett lokalt eller regionalt fokus och bland de Pleistocena istäckena hör därför Kordilleraisen till de sämst utredda när det gäller inlandsisens utbredning, volym och dynamik. Jag presenterar numeriska simuleringar som är baserade på approximationer av isﬂödesfysik och som möjliggör kvantitativa rekonstruktioner av istäckets utveckling. Simuleringarna visar att Ko- rdilleraisen under dess senaste glaciala maxima till stor del styrdes av markanta regionala kontraster i temperatur- och nederbördsförhållanden samt dygnsmässiga temperaturvariationer. Inlandsisens största utbredning under den sista glaciala cykeln var kortlivad. Under största delen av den simulerade glaciala cykeln karaktäriserades nedisningsområdet av isfält och platåisar samt isolerade glaciärer i bergsked- jorna, och störst isvolym fanns över Skeenafjället. Den modellerade Kordilleraisen verkar ha varit i kon- stant obalans med det ständigt förändrande klimatet. Inlandsisens respons på dessa klimatförändringar var därför vidare mer komplext än vad som har presenterats i tvådimensionella konceptuella modeller av hur inlandsisen växte till och smälte bort. I min avhandling demonstrerar jag den potential som numerisk ismodellering har för att kunna dra meningsfulla slutsatser om nedisningshistoriken och forna klimatförhållanden över glaciala cykler, under förutsättning att modelleringen kan kalibreras mot fältdata. Numerical modelling of the Cordilleran ice sheet

Julien Seguinot

List of papers This doctoral dissertation consists of a cover summary and the following journal articles, which are referred to by their Roman numerals in the text.

I Seguinot, J., Khroulev, C., Rogozhina, I., Stroeven, A. P., and Zhang, Q.: The effect of climate forcing on numerical simulations of the Cordilleran ice sheet at the Last Glacial Maximum, The Cryosphere, 8, 1087–1103, doi:10.5194/tc-8- 1087-2014, 2014. II Seguinot, J.: Spatial and seasonal effects of temperature variability in a positive degree-day glacier surface mass-balance model, J. Glaciol., 59, 1202–1204, doi:10.3189/2013JoG13J081, 2013.

III Seguinot, J. and Rogozhina, I.: Daily temperature variability predetermined by thermal conditions over ice-sheet surfaces, J. Glaciol., 60, 603–605, doi:10.3189/2014jog14j036, 2014. IV Seguinot, J., Rogozhina, I., Stroeven, A. P., Margold, M., and Kleman, J.: Numer- ical simulations of the Cordilleran ice sheet through the last glacial cycle, manuscript.

Author contributions

I A. P. Stroeven initiated the Cordilleran ice sheet modelling project; J. Seguinot set-up the ice sheet model (PISM) and supercomputing workflow, ran simulations and wrote the first draft; C. Khroulev implemented necessary code changes in the model; I. Rogozhina contributed to experiment design; A. P. Stroeven helped with interpretation of the results; and Q. Zhang prepared some of the forcing data. All co-authors contributed to improve the text. II J. Seguinot analysed the weather data, wrote the positive degree day model (PyPDD), ran computations and wrote the manuscript. A. P. Stroeven contributed with editorial comments at the pre-submission stage and I. Rogozhina helped to assess the reviews. III I. Rogozhina provided the original idea; J. Seguinot analysed the weather data and developed a new parameterization of daily temperature standard deviation. I. Rogozhina and J. Seguinot wrote the manuscript. A. P. Stroeven contributed with editorial comments at the pre-submission stage. IV J. Seguinot integrated the new parameterization of standard deviation into PISM, ran the simulations and wrote the first draft; I. Rogozhina guided experiment design; A. P. Stroeven, M. Margold and J. Kleman took part in the interpretation and comparison of model results against geological evidence. All co-authors contributed to improve the text. Cover summary

Contents

1. Introduction 2

2. North American Cordillera 4 2.1. Topographic setting ...... 4 2.2. Climatic setting ...... 6 2.3. Glacial history ...... 7

3. Numerical ice sheet model 8 3.1. Overview ...... 8 3.2. Field equations ...... 8 3.3. Shallow approximations ...... 10 3.4. Basal sliding ...... 11 3.5. Bedrock response ...... 13 3.6. Surface mass balance ...... 13 3.7. Atmospheric forcing ...... 14 3.8. Numerical implementation ...... 15

4. Results summary 15 4.1. Simulations of the Last Glacial Maximum ...... 15 4.2. Role of daily temperature variability on melt ...... 17 4.3. Simulations of the last glacial cycle ...... 18

5. Sources of error 19 5.1. Interaction with the Laurentide ice sheet ...... 19 5.2. Past changes in precipitation ...... 19 5.3. Geothermal heat ﬂux patterns ...... 20

6. Conclusions 20

1 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

1. Introduction (Fig. 2). In several parts of the world, montane people and early explorers learned since long to read these traces, In our daily life, we most commonly experience ice and to understand that glaciers had once been more ex- as small, solid cubes that crackle on contact with re- tensive than today (e.g., Windham and Martel, 1744, freshing liquids. Yet, on geological time scales, and un- p. 21). With the emergence of the glacial theory came der high pressure, ice behaves quite differently. In the the idea that, under colder temperatures than today, ex- cold and mountainous regions of the globe, ice accumu- pansive ice sheets had once covered much of Europe and lates through cumulative snowfall until it reaches a crit- North America (Agassiz, 1840). ical thickness. Beyond this threshold, usually of several However, the glacial theory did not gain general tens of metres, but dependent on temperature, ice de- acceptance until the discovery and exploration of the forms under its own weight to flow downhill (Fig. 1). two present-day ice sheets on Earth, the Greenland and Glaciers, these bodies of moving ice, flow by a combi- Antarctic ice sheets, provided a modern analogue for the nation of meltwater-induced sliding at the base (de Saus- proposed European and North American ice sheets. The sure, 1786, §532), and viscous deformation within the ice first traverse of Greenland was performed by Nansen in body (Forbes, 1846). 1888, while Amundsen reached the south pole in 1911. As glaciers flow and slide across their bed, they trans- Meanwhile, in the European Alps, it was then discov- port rock debris and erode the landscape, thereby leav- ered from observations of the glacial landscape, that there ing geomorphological traces of their former presence had not been only one, but at least four, glacial periods (Penck and Brückner, 1909).

(a) (b) (c)

Figure 1. Illustration of various glacier ice behaviours: (a) surface crevasses formed by ice ﬂow over a hidden bedrock knob, Glacier Blanc, French Alps, 2004; (b) medial moraines illustrating downhill ice ﬂow on Susitna Glacier, Alaska Range, photograph by Austin Post, 1970 (NSIDC, 2009); and (c) mountain glacier calving into a small lake, revealing a folded structure formed by deformed debris layers in the ice, Glacier d’Arsine, French Alps, 2004.

(a) (b) (c)

Figure 2. Different indicators of past glaciation: (a) glacial valley, exhibiting a broad, U-shaped proﬁle characteristic of erosion by glaciers, Cirque du Fer-à-Cheval, French Alps, 2013; (b) erratic boulders, deposited in an improbable conﬁguration as the glacier melted underneath, Finse area, Norwegian Scandes, 2011; and (c) lateral meltwater channels, formed on a gentle slope as the lateral ice margin retreated towards the valley bottom, Tuya Lake area, Canadian Cordillera, 2012.

2 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

(a) 3 (a) ) ( 4 O 8 1

4 (b) (b)

0 ) K (

T 4

12 800 700 600 500 400 300 200 100 0 age (ka)

Figure 3. Palaeo-climate records of the last 800 ka. (a) Benthic oxygen isotope (δ18O) record from a compilation of globally- distributed deep sea sediment cores (Lisiecki and Raymo, 2005). Higher δ18O values correspond to periods of lower temperatures (Emiliani, 1955) or lower global sea level, and thus higher ice volume stored in glaciers and ice sheets on land Figure 4. Spatial extent of ice sheet and glacier cover in the (Shackleton, 1967). (b) Temperature anomalies (∆T ) recon- northern hemisphere (a) at present (Patterson and Kelso, 2014), structed from a central Antarctic ice core (EPICA Dome C, and (b) during the last glacial maximum (Ehlers and Gibbard, Jouzel et al., 2007). The grey bands indicate the approximate 2007). time span of the last glacial period of 110 to 10 ka. sive Greenland and Antarctic ice sheets, the Fennoscan- More recently, palaeoclimate records extracted from dian and Barents-Kara ice sheets in the Eurasian Arctic, deep sea sediments and ice cores have provided a much and the Innuitian, Laurentide, and Cordilleran ice sheets more detailed picture of the Earth environmental his- over North America (Fig. 4). tory (e.g., Emiliani, 1955; Shackleton and Opdyke, 1973; Nevertheless, the landform record is sparse in time Dansgaard et al., 1993; Augustin et al., 2004), indicat- and, most often, spatially incomplete. Palaeo-ice sheets ing neither one, nor four, but rather tens of glacial cycles. did not leave a continuous imprint on the lanscape, and During the last 800000 years (800 ka), glacial and inter- much of this evidence has been overprinted by subse- glacial periods have succeeded each other with a 100 ka quent glacier re-advances and other geomorphological periodicity (Fig. 3; Hays et al., 1976; Augustin et al., processes (Kleman, 1994; Kleman et al., 2006, 2010). 2004). However, much of the traces left by glaciers on Therefore, geomorphological reconstructions of former the landscape date from the most recent of these cycles, glacier extent require a substantial amount of interpreta- the last glacial cycle. tion. This is where numerical ice sheet models, incorpo- Together, the Greenland and Antarctic ice sheets cur- rating the processes of glacier sliding (e.g., Weertman, rently host almost all glacial ice on the planet, corre- 1957) and deformation (e.g., Nye, 1953) observed on sponding to 65.6 m of potential sea level rise, were these modern glaciers, can help. Although these models alone ice sheets to melt away entirely (Bamber et al., 2013; contain too many uncertainties to reproduce accurate ice Fretwell et al., 2013). At the apogee of the last glacial sheet reconstructions during former glacial cycles (e.g., period, global sea level was 120 to 135 m lower than Hebeler et al., 2008), they can be calibrated against ge- today (Clark and Mix, 2002), implying an additional, ological data where they exist (Napieralski et al., 2007), double amount of ice stored in ice sheets and glaciers while allowing quantitative reconstructions of ice sheet on land. Systematic mapping and compilation of glacial sectors where they do not, based on approximated physics landforms from all continents (e.g., Prest et al., 1968; of glacier ﬂow (e.g., Marshall et al., 2002; Tarasov and Boulton and Clark, 1990; Kleman et al., 1997; Hättes- Peltier, 2004). trand, 1998) permitted reconstructions of these former ice In addition, numerical ice sheet models provide quan- covers (Ehlers and Gibbard, 2007), indicating that the ex- titative insights on the four-dimensional distribution of cess ice volume was repartitioned between more exten- stresses and deformations in palaeo-ice sheets and allow

3 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

ARCTIC OCEAN GIS IIS

Alaska

CANADA

Hudson Bay CIS PACIFIC LIS Newfoundland OCEAN Prairies

Great

Lakes ATLANTIC

OCEAN USA

Figure 5. Relief map of northern North America showing a reconstruction of the areas covered by the Cordilleran (CIS), Laurentide (LIS), Innuitian (IIS) and Greenland (GIS) ice sheets during the last glacial maximum (21.4 to 16.8 cal 14C kyr BP, Dyke, 2004). The rectangular box delimits the model domain used in all simulations and most figures of this thesis. The background map consists of ETOPO1 (Amante and Eakins, 2009) and Natural Earth Data (Patterson and Kelso, 2014). for reconstructions of their past thickness and volume the Hudson Bay, and fully covered the Canadian Prairies, evolution, which are difficult to infer from geological ev- Eastern Canada, and the Great Lakes Basin. The Innuitian idence. ice sheet, much smaller, covered the northernmost Cana- In this thesis, I focus on numerical modelling of the dian archipelago. Finally, the Cordilleran ice sheet, inter- last Cordilleran ice sheet, which once covered much of mediate in size, capped the mountain ranges of the West- the mountainous terrain of western North America (Daw- ern Cordillera. son, 1888), a region where topographic complexity has presented a particular challenge to reconstructing glacial 2.1. Topographic setting history based on the geomorphological record (e.g., Kle- The North American Cordillera forms an elongated but man et al., 2010) or numerical ice sheet modelling (e.g., broad mountain system with an average east-west width Robert, 1991). of about 1000 km (Fig. 6). Its topography is a complex jigsaw of mountain ranges and interior basins. It 2. North American Cordillera was formed during multiple accretion phases associated with the closure of several back-arc oceanic trenches, During the last glacial cycle, much of North America has over a 200 million year (200 Ma)-long geological history been covered by ice (Fig. 5). During the Last Glacial (Sigloch and Mihalynuk, 2013). Maximum (LGM), peak of glaciation, three major ice The dominant reliefs occur along two major ribbons sheets had collided to form a continuous blanket of ice that run parallel to each other along the general orienta- that spanned from southern Alaska to Newfoundland, tion of the North American Cordillera. The western rib- connecting with the Greenland ice sheet in the Cana- bon is formed by the Pacific Coast orogenic belt and in- dian Arctic, and extending as far south as Chicago. The cludes, from north to south, the Alaska Range (Fig. 7), the Laurentide ice sheet, largest of the three, was centred on Wrangell and Saint-Elias Mountains, the Coast Moun- tains and the North Cascades. The eastern ribbon con-

4 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

ARCTIC OCEAN

Brooks Range

McKenzie Mts Alaska Range CANADIAN Selwyn Mts Wrangell Mts Cassiar Mountains PRAIRIES St Elias Mts Liard Lowland

Skeena Mts

Rocky Mountains Coast Mountains Fraser Q. Charlotte I. Plateau Columbia Mts

PACIFIC Vancouver Island

OCEAN N. Cascades

Puget Lowland

Figure 6. Topographic map of the northern American Cordillera, oriented north. Cumulative last glacial maximum ice cover (Dyke, 2004) is marked in red, and the model domain in black. The background map consists of ETOPO1 (Amante and Eakins, 2009) and Natural Earth Data (Patterson and Kelso, 2014).

Figure 7. The Alaska Range hosts several of the highest peaks in the North American Cordillera. Despite this fact and its northerly location, its north slope hosts relatively small glaciers only, which had not been much more extensive during the Last Glacial Maximum (Kaufman and Manley, 2004), indicating dry climatic conditions. The Alaska Range is tectonically active and rapidly eroding, as shown by landslide-covered valley sides on this photograph.

5 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

(a) (b)

Figure 8. Contrasted climates of the North American Cordillera: (a) Pacific Coast at Vancouver Island, western British Columbia, photograph by Martin Margold, 2010. (b) Fraser Valley, south-central British Columbia, 2010. sists of the Nevadan and Laramide belts, which are sepa- are clearly expressed in the landscape (Fig. 8). Due to the rated by the deep and strikingly linear Rocky Mountain / large latitudinal extent of the region, mean annual tem- Tintina Trench depression, and includes the Selwyn and peratures vary significantly from north to south (Fig. 9a). MacKenzie Mountains in the north, and the Rocky moun- About half of the model domain has mean annual temper- tains in the south. The Brooks Range of northern Alaska atures below freezing. However, temperature seasonality, has been covered by ice, but its glaciers remained disjunct here computed as the difference between the warmest and from the main body of the Cordilleran ice sheet (Kaufman the coldest months, varies between around 10 ◦C along and Manley, 2004). most of the Pacific Coast to above 40 ◦C for the continen- The North American Cordillera hosts several inter- tal Arctic sectors (Fig. 9b). There exists a partial correla- montane basins, the presence of which influenced former tion between cold regions and those that have a large sea- patterns of Cordilleran ice sheet flow. The two major de- sonality, so that many regions where mean annual tem- pressions are the Interior Plateau in south-central British peratures are well below freezing point experience warm Columbia, and the Liard Lowland in northern British summers. Columbia (Fig. 6). These two basins are separated by the Annual precipitation rates are highest on the west- Skeena Mountains, a mildly elevated, but extensive and ern slopes of the Pacific Coast ranges and decline quickly deeply incised mountain group. This is the most dom- on the eastern side of the mountains (Figs. 8 and 9c). In inant upland area in-between the dominant topographic terms of ice sheet mass balance, this contrast is even re- ribbons that form the North American Cordillera. inforced through the differences in timing of maximum The ubiquitously mountainous topography of the re- precipitation through the year (Fig. 9d). Coastal regions gion presents a serious obstacle to reconstructing the receive most precipitation during fall and winter, which palaeoglaciology of the Cordilleran ice sheet. Because is therefore largely delivered as snow. In contrast, many of the irregular arrangement of valleys and ridges that inland regions experience dry winters, and most of the an- formed the ice sheet’s subglacial landscape, ice moved nual precipitation consequently falls there as rain during along tortuous and intertwined pathways, sometimes fol- summer, further enhancing ice melt. lowing the major topographical troughs, and sometimes These topographically-induced, spatial variations in flowing across them (Davis and Mathews, 1944; Kleman temperature and precipitations tightly constrain the lo- et al., 2010). In terms of numerical ice sheet modelling, cation of potential ice sheet nucleation centres that lead the presence of steep and irregular subglacial topogra- to the formation of a Cordilleran ice sheet. The numer- phy yields complex flow patterns and stress distributions ical ice sheet model is therefore highly sensitive to the within the ice sheet and requires high-resolution and thus, choice of input topography and climate used (Sect. 3.7.). large computational grids. This forms the motivation for the study presented in Pa- per I, where the climatic setting of the North American 2.2. Climatic setting Cordillera is also described in further detail. The topographic complexity of the North American Cordillera is associated with distinctive patterns of temperature and precipitation, whose effects on vegetation

6 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

(a) (b) (c) (d)

12 6 0 6 12 0 10 20 30 40 0.1 1.0 10.0 DJF MAM JJA SON Mean annual temperature (°C) Temperature seasonality (°C) Annual precipitation (m) Precipitation peak season

Figure 9. Some characteristics of the Cordilleran climate according to the North American Regional Reanalysis (NARR, Mesinger et al., 2006).

2.3. Glacial history of the two, the Gladstone Glaciation, reached its maximum extent between 50.4 3.0 and 54.3 2.0 10Be ka Evidence from glaciofluvial deposits dated using cos- ± ± +0.20 (Ward et al., 2007), during Marine Oxygen Isotope Stage mogenic nuclides at 2.64−0.18 Ma (uncertainty range from 2.46 to 2.84 Ma) indicates the first presence of an (MIS) 4, a period of relatively large global ice volume. ice sheet on the Cordilleran mountains during the late However, evidence from the Gladstone Glaciation has Pliocene (Hidy et al., 2013). This indicates that the North not been documented along the southern ice sheet mar- American Cordillera has been glaciated many times since gin, making it difficult to seize its spacial extent. The the late Pliocene, with the first of these glaciations oc- younger one of the two last glacial cycle glaciations, the curring even before the mid-Pleistocene onset of 100 ka McConnell Glaciation, is associated with end moraines of oldest minimum apparent exposure ages of 15.7 1.5 glacial cycles (Fig. 3). However, only fragments of the ± and 17.7 1.6 10Be ka (Stroeven et al., 2014), which ap- last few glaciations can be reconstructed from the geo- ± logic record, implying that much of the older evidence pears to correspond to the last global ice volume max- has been erased by subsequent glacial re-advances and imum (MIS 2). In contrast to the Gladstone Glaciation, perhaps even by other, non-glacial geomorphological the McConnell Glaciation appears contemporaneous with processes. glacial advance of the southern ice sheet margin during In the northern sector of the Cordilleran ice sheet do- the Fraser glaciation, with a maximum extent attained 17.4 16.4 14C cal ka main, at least four glaciations have been identified (Duk- between and (Porter and Swanson, Rodkin, 1999; Ward et al., 2007, 2008; Briner and Kauf- 1998). Within errors inherent to the cosmogenic nuclide man, 2008; Demuro et al., 2012; Stroeven et al., 2010, exposure dating methods (Heyman et al., 2011), the Mc- 2014). The two older ones, often collectively referred to Connell and Fraser glaciations correspond to the same as the Reid and pre-Reid glaciations, pre-date the last glacial episode, which resulted in the LGM extent of the glacial cycle, and are therefore beyond the scope of this Cordilleran ice sheet. study. The two younger glaciations, however, occurred Apart from geological evidence of the final demise of both during the last glacial cycle. First, the older one the Cordilleran ice sheet, most field evidence relates to its LGM extent. It indicates that ice covered most of the

7 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Canadian Cordillera, and some of the adjacent regions The retreat of the ice sheet during the last deglacia- in north-western contiguous USA and Alaska (Fig. 6). tion was likely rapid. Most of the evidence for the style The LGM Cordilleran ice sheet originated from the co- and pace of deglaciation comes from the Interior Plateau alescence of several mountain-centred ice caps (Davis of British Columbia. Studies there have indicated that and Mathews, 1944). It is generally thought that the ice some of the highest uplands emerged from the ice sheet sheet configuration was short-lived (Clague et al., 1980; prior to the surrounding lowlands, possibly resulting in Clague, 1985; Cosma et al., 2008), a hypothesis which thin, stagnant ice remnants covering parts of the plateau can be studied through numerical modelling. (Fulton, 1967, 1991; Margold et al., 2011, 2013b). Less The southern margin of the ice sheet has been stud- is known about the deglaciation of the northern sector ied extensively, and is in several respects better under- of the Cordilleran ice sheet, but abundant glacial land- stood than its northern counterpart (Booth et al., 2003). forms from the Liard Lowland indicate a different mode It consists of coalescent piedmont lobes, each of which of deglaciation, with active retreat of the ice margin to- formed at the outlet of the main valleys. The Puget Lobe wards the surrounding mountain ranges (Margold et al., (Thorson, 1980; Porter and Swanson, 1998) was the west- 2013a). Patterns of deglaciation, as well as the possibil- ernmost and largest one. It covered the Puget Lowland, a ity for a late-glacial re-advance, are discussed in further region that comprises the cities of Vancouver and Seat- detail and compared with modelling results in Paper IV. tle. Further east, beyond the Cascade Range, the Okana- gan and Purcell lobes formed the southern Cordilleran ice sheet margin. The latter dammed the Clark Fork 3. Numerical ice sheet model River, inhibiting runoff and forming Glacial Lake Mis- soula (Pardee, 1910). As the ice sheet margin retreated, 3.1. Overview the lake repeatedly and catastrophically drained to the Pa- The simulations presented in this thesis employ the Par- cific Ocean (Bretz, 1923; Waitt, 1980). These floods con- allel Ice Sheet Model (PISM), an open-source, finite- tributed to shape a distinctive landscape, known as the difference, shallow ice sheet model (the PISM authors, Channeled Scabland. 2014). The model requires bedrock topography, sea level, The western margin of the Cordilleran ice sheet cov- geothermal heat flux and climate forcing as inputs, and ered part of the Pacific continental shelf. An indepen- computes the thermal and dynamic states of the ice sheet, dent ice cap covering Vancouver Island diverted flow and the associated thermal and deformational responses from the main body of ice to drain into the Strait of of an idealised lithosphere. However, the atmosphere and Georgia and then between Vancouver Island and the ocean are not part of the model. Instead, PISM includes Olympic Peninsula, where it formed the Juan de Fuca boundary models that provide simple parametrisations of marine ice lobe, calving into the Pacific Ocean (Herzer their interaction with the ice sheet across interfaces that and Bornhold, 1982). Further to the North, the ice mar- are assumed to be well defined (Fig. 10). gin encroached the Queen Charlotte Islands (Blaise et al., This section does not attempt to provide an extensive 1990). However, the archipelago may not have been fully description of PISM, which can be found in an on-line ice-covered, even during the maximum of glaciation. documentation (http://www.pism-docs.org, the PISM au- Submarine troughs traversing several portions of the con- thors, 2014). Instead, it aims to highlight the modelling tinental shelf (Fig. 6) indicate the erosional action of choices that I have made in this thesis, and to present an additional ice streams, yet it is not clear whether they overview of the physics involved in these choices. have formed during the last glacial cycle or during ear- Vectors are identified by bold arrow symbols (v~) and lier glaciations. second-order tensors by bold symbols (σ), while scalars Finally, the conditions at the eastern margin of the use a normal font (s). The divergence of a vector field is Cordilleran ice sheet are the least understood. Although written div v~, and grad~ s denotes the gradient of a scalar it is known that the Cordilleran ice sheet collided with the field. The divergence of a tensor field is characterised by Laurentide ice sheet at times during the last glacial cycle div~ σ and the gradient of a vector field by grad v~. The (e.g. Margold et al., 2013a,b), the nature of the interac- race of a tensor σ is written tr(σ). Parameter values are tion between the two ice sheets and the precise location of summarised in Table 1, page 12. this junction remains unclear (Gowan, 2013), while lim- ited information exists on its timing (e.g., Jackson et al., 3.2. Field equations 1997; Bednarski and Smith, 2007). Because the present The thermodynamic core of the ice sheet model relies on thesis focuses specifically on the Cordilleran ice sheet, I, four elementary field equations: the equation of incom- therefore, neglect buttressing effects of the Laurentide ice pressibility, the balance of stresses, the constitutive law sheet. for ice, and the conservation of energy. Firstly, ice flow

8 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

law for ice (Nye, 1953) based on laboratory experiments (Glen, 1952) is used to relate these two quantities. It atmosphere reads: n−1 ˙ = A τe τ . (3) ice sheet 2 The equivalent stress, τe, is deﬁned by τe = IIτ = 1 2 − 2 tr τ , where IIτ is the second invariant of the stress ocean PISM tensor. The ice softness coefﬁcient, A, depends on ice temperature, T , pressure, p, and water content, ω, through bedrock an Arrhenius-type law (Paterson and Budd, 1982; As- chwanden et al., 2012, Eqs. 63–65),

Figure 10. PISM is a numerical model for ice and bedrock −Qc ( RT physics. Processes occurring at the interface with the atmo- Ac(1 + fω) e pa if T < Tc , A = −Qw (4) RT sphere and the ocean, that are progressive transitions in nature, Aw(1 + fω) e pa if T Tc , are parametrised along well-deﬁned boundaries in the model. ≥ After the PISM authors (2014). where Tpa is the pressure-adjusted temperature calcu- lated using the Clapeyron relation, T = T βp. R is pa − z the ideal gas constant, and Ac, Aw, Qc and Qw, are constant parameters corresponding to values measured below and above a critical temperature threshold T = 10 ◦C c − (Table 1; Paterson and Budd, 1982). The water fraction, σzz ω, is capped at a maximum value of 0.01, above which no τxz s τxz measurements are available (Lliboutry and Duval, 1985; σxx Greve, 1997, Eq. 5.7). h For a convenient derivation of the shallow shelf ap- ~v ~g proximation, developed in the next section (Sect. 3.3.), the constitutive law can be rewritten using an inverted formulation. Assuming ˙2 = 1 tr˙2 and B = A−1/n, b e 2 x Eq. (3) becomes

(1/n)−1 τ = B ˙e ˙ . (5) Figure 11. Sketch of notations used for the main variables of the ice sheet model. An infinitesimal ice parcel is moving at By analogy to Newtonian flow, an apparent viscosity, velocity v~, under the balance of stresses σ and gravity g~. ν, can then be defined by τ = 2ν˙, thus yielding B ν = ˙(1/n)−1 . (6) is assumed incompressible. Hence, ice density, ρ, is con- 2 e stant, and conservation of mass therefore implies a conservation of volume, which can be expressed in terms of Finally, the evolution of temperature within the ice conservation of energy the ice velocity vector, v~ (Fig. 11), by is governed by the . In PISM, the conservation of energy uses an enthalpy formulation in div v~ = 0 . (1) order to account for thermodynamic effects associated with internal phase changes in the glacier (Aschwanden Secondly, the balance of stresses is expressed by the et al., 2012, Eqs. 20–21). This enthalpy (H) formulation Stokes equation, thus assuming creeping flow. In other reads words, ice is considered as a slow-moving fluid whose de- ∂H ν˙2 formation is entirely controlled by internal friction, here ρ + v~ grad~ H = div q~ + e , (7) ∂t · − 4 expressed by the Cauchy stress tensor, σ, and the gravity vector, ~g: where q~ represents the heat flow and ν˙ 2/4 = tr(τ ˙) div~ σ + ρ ~g = ~0 . (2) e is a source term corresponding to strain heat release. In Thirdly, after defining the strain-rate tensor the case where ice temperature, T , is below the pressure- 1 T ˙ = 2 (grad v~ + grad v~ ), and the deviatoric stress melting point, the heat flow is modelled by Fourier’s law, tensor, τ = σ + p δ, where p = 1 tr(σ) is the hydro- q~ = k grad~ T , where k is the thermal conductivity of − 3 static pressure and δ is the identity tensor, a constitutive ice, while enthalpy can be expressed as H = c T , where

9 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET c denotes the ice speciﬁc heat capacity (Table 1). Conse- Neglecting atmospheric pressure, Eq. (11c) yields the quently, the enthalpy equation (7) can be rewritten using pressure, a perhaps more familiar, cold-ice, temperature formula- p = ρg(s z) , (12) − tion, ∂T k ν˙2 varying from zero at the ice surface, s, to ρgh at the base, + v~ grad~ T = ∆T + e . (8) b = s h (Fig. 11). This expression can be re-introduced ∂t · ρc 4ρc − into Eqs. (11a) and (11b), thus yielding the expression Note that, here, the term “enthalpy” is actually an abuse of the horizontal shear stresses, sometimes referred to as of language, commonly used in glaciology as a synonym driving stresses, of “interval energy” (Aschwanden et al., 2012). Eqs. (1), (2), (3) and (7) constitute the thermody- ∂s τ = ρg(s z) , (13a) namic basis of the model. However, their resolution in xz − − ∂x full form implies a computational demand too high for ∂s τyz = ρg(s z) . (13b) multi-millennial, continental-scale applications, such as − − ∂y the numerical modelling of the Cordilleran ice sheet Using the constitutive law (3), vertical derivatives of through the last 100 ka glacial cycle, thus requiring shal- horizontal velocities can be related to the horizontal shear low approximations. stresses:

3.3. Shallow approximations ∂vx 2 2 n−1 = 2A(τxz + τyz) τxz , (14a) PISM employs two well-documented approximations of ∂z ∂vy 2 2 n−1 the ice ﬂow equations: the shallow ice approximation = 2A(τxz + τyz) τyz . (14b) (SIA) and the the shallow shelf approximation (SSA). Al- ∂z though both approximations can be derived from a rigor- Thus, within the framework of the SIA, horizontal ve- ous scaling approach (Morland and Johnson, 1980; Hut- locities v~SIA can be directly expressed as a function of ter, 1983; Morland, 1987; Weis et al., 1999), I won’t ex- the slope gradient, pand these lengthy derivations here, assuming instead the − simpliﬁcations and focusing on their effect. ∂v ∂s ∂s n 1 ∂s x = 2A(ρg)n(s z)n + , Distinguishing the hydrostatic and deviatoric compo- ∂z − − ∂x ∂y ∂x nents of the stress tensor, the balance of stresses (2) can (15a) be expanded into n−1 ∂v ∂s ∂s ∂s y = 2A(ρg)n(s z)n + , div~ τ grad~ p + ρ ~g = ~0 . (9) ∂z − − ∂x ∂y ∂y − (15b)

Projecting along the three Cartesian axes, this reads which yields, in vectorial notation,

∂τxx ∂τxy ∂τxz ∂p ∂v~ + + = , (10a) SIA = 2A (ρg)n (s z)n grad~ s n−1 grad~ s . ∂x ∂y ∂z ∂x ∂z − − | | ∂τ ∂τ ∂τ ∂p (16) yx + yy + yz = , (10b) ∂x ∂y ∂z ∂y From Eq. (16) it can be seen that shallow ice velocities, v~SIA, can be directly computed from the geometry of ∂τzx ∂τzy ∂τzz ∂p + + = ρg . (10c) the ice sheet surface, s, after a single vertical integration ∂x ∂y ∂z ∂z − over the ice thickness, in order to account for variations The shallow ice approximation (SIA) assumes that of ice rheology, A, with depth. basal friction is high enough for the vertical shear stresses The shallow shelf approximation (SSA), in contrast τ , τ to predominate over all other components of to SIA, assumes that basal friction is low enough for { xz yz} the deviatoric stress tensor. In this framework, the pro- ice to deform uniformly within the entire ice column. In jected stress-balance can be simpliﬁed to this context, the stress-balance can be simpliﬁed to (Weis et al., 1999, Eqs. 4.10–4.12) ∂τxz ∂p = , (11a) ∂τ ∂τ ∂τ ∂p ∂z ∂x xx + xy + xz = , (17a) ∂τyz ∂p ∂x ∂y ∂z ∂x = , (11b) ∂τ ∂τ ∂τ ∂p ∂z ∂y yx + yy + yz = , (17b) ∂p ∂x ∂y ∂z ∂y 0 = ρg . (11c) ∂τ ∂p ∂z − zz = ρg , (17c) ∂z ∂z −

10 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET and horizontal velocities can be assumed independent of ice sheet ice stream ice shelf depth (Weis et al., 1999, Eq. 4.22), so that ∂v x = 0 , (18) ∂z ∂v ~vSIA y = 0 . (19) ∂z ~vSSA ~vSIA Once again neglecting atmospheric pressure, and re- membering that tr(τ ) = 0, Eq. (17c) yields the pressure ~vSSA p = ρg(s z) + τ , (20) − zz = ρg(s z) τ τ , (21) − − xx − yy ~v ~v ~v ~v ~v ~v which can be re-introduced into Eqs. (17a) and (17b): SIA SSA SIA ∼ SSA SIA SSA ∂τ ∂τ ∂τ ∂τ ∂s 2 xx + yy + xy + xz = ρg , (22a) Figure 12. Combination of Shallow Ice Approximation (SIA) ∂x ∂x ∂y ∂z ∂x and Shallow Shelf Approximation (SSA) velocities in PISM, ∂τ ∂τ ∂τ ∂τ ∂s yx + 2 yy + xx + yz = ρg . (22b) assuring a smooth transition between shear-dominant and lon- ∂x ∂y ∂y ∂z ∂y gitudinal stress regimes. After Winkelmann et al. (2011, Fig. 1). Using the inverse formulation of the constitutive law Not to scale. (5), components of the deviatoric stress tensor can be re- placed by corresponding velocity components, yielding In contrary to the the SIA velocities, v~SIA, the SSA velocities, v~SSA, can not be computed in a direct manner. ∂ ∂v ∂v ∂ ∂v ∂v Instead, Eq. (25) must be resolved numerically using an 2¯ν 2 x + y + ν¯ x + y ∂x ∂x ∂y ∂y ∂y ∂x iteration method, in order to obtain approximated values ∂τ ∂s for v~SSA. + xz = ρg , ∂z ∂x In PISM, the SIA and SSA velocities are eventually (23a) added (Fig. 12; Winkelmann et al., 2011, Eq. 15): ∂ ∂vx ∂vy ∂ ∂vx ∂vy ν¯ + + 2¯ν + 2 v~ = v~ + v~ . (26) ∂x ∂y ∂x ∂y ∂x ∂y SIA SSA ∂τyz ∂s + = ρg , Because v~SIA is small where v~SSA is large, and recipro- ∂z ∂y cally, we can expect the model to perform well in regions (23b) where conditions for applicability of either the SIA or the where ν¯ is the depth-integrated apparent viscosity defined SSA are met (Fig. 12). However, Eq. (26) is a heuristic, by and its validity in the zone of transition between shear 1 Z s flow (SIA) and longitudinal flow (SSA) has not been ν¯ = ν . (24) H b mathematically proven. A final integration over the entire ice column yields (Weis et al., 1999, Eqs. 7.7–7.8) 3.4. Basal sliding Although the SIA is valid only under non-sliding condi- ∂ ∂vx ∂vy ∂ ∂vx ∂vy 2¯νh 2 + + νh¯ + tions, the SSA requires a sliding law as basal boundary ∂x ∂x ∂y ∂y ∂y ∂x condition, which in fact constitutes the main control on ∂s + τbx = ρgh , SSA velocities and, in turn, the locations of ice streams ∂x in the model. (25a) The simulations presented here use a pseudo-plastic ∂ ∂v ∂v ∂ ∂v ∂v νH¯ x + y + 2¯νh x + 2 y sliding law, ∂x ∂y ∂x ∂y ∂x ∂y v~ = b τ~b τc q 1−q , (27) ∂s − vth v~b + τby = ρgh , | | ∂y where τ~b is the basal traction force, v~b is the basal veloc- (25b) ity, and vth is an arbitrary velocity threshold (Table 1). Note that, in the case of q = 0, Eq. (27) corresponds to a where τbx and τby correspond to x and y-components of the basal traction, and will be determined by a sliding law.

11 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Table 1. Constant parameters of the ice sheet model. Not. Name Value Unit Source Ice material properties ρ Ice density 910 kg m−3 Aschwanden et al. (2012) g Standard gravity 9.81 m s−2 Aschwanden et al. (2012) n Glen exponent 3 - - −13 −3 −1 Ac Ice hardness coefficient cold 3.61×10 Pa s Paterson and Budd (1982) 3 −3 −1 Aw Ice hardness coefficient warm 1.73×10 Pa s Paterson and Budd (1982) 4 −1 Qc Flow law activation energy cold 6.0×10 J mol Paterson and Budd (1982) 4 −1 Qw Flow law activation energy warm 13.9×10 J mol Paterson and Budd (1982) R Ideal gas constant 8.31441 J mol−1 K−1 - Tc Flow law critical temperature 263.15 K Paterson and Budd (1982) f Flow law water fraction coeff. 181.25 - Lliboutry and Duval (1985) β Clapeyron constant 7.9×10−8 K Pa−1 Lüthi et al. (2002) −1 −1 ci Ice specific heat capacity 2009 J kg K Aschwanden et al. (2012) −1 −1 cw Water specific heat capacity 4170 J kg K Aschwanden et al. (2012) k Ice thermal conductivity 2.10 J m−1 K−1 s−1 Aschwanden et al. (2012) L Water latent heat of fusion 3.34×105 J kg−1 K−1 Aschwanden et al. (2012) Basal sliding q Pseudo-plastic sliding exponent 0.25 - Aschwanden et al. (2013) −1 vth Pseudo-plastic threshold velocity 100.0 m yr Aschwanden et al. (2013) c0 Till cohesion 0.0 Pa Aschwanden et al. (2013) α Effective pressure coefficient (Paper I) 0.95 - Aschwanden et al. (2013) δ Effective pressure coefficient (Paper IV) 0.02 - Bueler and van Pelt (2014) e0 Till reference void ratio 0.69 - Tulaczyk et al. (2000) Cc Till compressibility coefficient 0.12 - Tulaczyk et al. (2000) Wmax Maximal till water thickness 2.0 m Aschwanden et al. (2013) b0 Altitude of max. friction angle 0 m - b1 Altitude of min. friction angle 200 m - ◦ φ0 Minimum friction angle 10/15 - ◦ φ1 Maximum friction angle 30/45 - Bedrock and lithosphere −3 ρb Bedrock density 3300 kg m - −1 −1 cb Bedrock specific heat capacity 1000 J kg K - −1 −1 kb Bedrock thermal conductivity 3.0 W m K - −2 qG Geothermal heat flux 70.0 mW m - 21 νm Mantle viscosity 1×10 Pa s Lingle and Clark (1985) −3 ρl Lithosphere density 3300 kg m Lingle and Clark (1985) D Lithosphere flexural rigidity 5.0×1024 N Lingle and Clark (1985) Surface and atmosphere ◦ T0 Temperature of snow precipitation 0.0 C - ◦ T1 Temperature of rain precipitation 2.0 C - −3 −1 −1 Fs Degree-day factor for snow 3.04×10 m K day Shea et al. (2009) −3 −1 −1 Fi Degree-day factor for ice 4.59×10 m K day Shea et al. (2009) γ Air temperature lapse rate 6×10−3 K m−1 - purely plastic law (Winkelmann et al., 2011, Eq. 11), to water content by a simple parametrisation (Winkel- mann et al., 2011, Eq. 13), v~ τ~ = τ b . (28) b − c v~ W | b| N = ρgh(1 α ) , (30) − Wmax The yield stress τc is related to till properties by the Mohr-Coulomb criterion, where W is the amount of water at the bed, and varies from zero to Wmax = 2 m, a threshold above which further melt simply exits the system, thus assuming in- τc = c0 + N tan φ . (29) stantaneous drainage. Although mass-conserving water drainage is now implemented in PISM (Bueler and van The till cohesion c0 is assumed to be zero. The effective pressure on the till, N, is determined by the mod- Pelt, 2014), it is not included in the simulations presented elled amount of water at the bed. However, two different here for the sake of simplicity. parametrisations are used within this thesis. In Paper I, In Paper IV, using PISM 0.6, the effective pressure using PISM 0.5, the effective pressure is linearly related is determined by an empirical parametrisation based on laboratory experiments on Antarctic till (Tulaczyk et al.,

12 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

2000; Bueler and van Pelt, 2014), corresponds to the ice load (Bueler et al., 2007). Here, ∆2 = (div grad~ )2 designs the biharmonic (Laplacian (e0/Cc)(1−(W/Wmax)) N = δρgh 10 , (31) square) operator of linear elastic theory, while grad~ is | | a pseudo-differential operator defined through its Fourier where e is a measured reference void ratio, C a 0 c transform by Bueler et al. (2007, Eq. 6). On the left-hand measured compressibility coefficient, and δ is a non- side of Eq. (34), the first term accounts for mantle relax- dimensional parameter chosen as 0.02 based on simple ation, the second for local isostasy, and the third for elas- sensitivity tests on the Greenland ice sheet (Table 1). tic flexure. Due to high mantle viscosity, ν , there exists Finally, the till friction angle, φ (Eq. 29), is taken as m a time lag between the ice sheet growth and the isostatic a function of modern bed elevation, with lowest values bedrock response. occurring at low elevations, thus accounting for a weak- ening of till associated with the presence of marine sedi- 3.6. Surface mass balance ments (cf. Martin et al., 2011, Eq. 10),  At the interface between the ice sheet and the atmosphere φ0 if b b0 , (Fig. 10), surface mass fluxes are computed from monthly  ≤ b−b0 φ(x, y) = φ0 + (φ1 φ0) if b0 < b < b1 , mean surface air temperature, Tm, and monthly precip- − b1−b0  itation, Pm, using a temperature-index model (Fig. 13; φ1 if b1 b , ≤ (32) cf. Hock, 2003). Snow accumulation, As, equals precipitation when temperature is below T = 0 ◦C, and de- where b0 = 0 m is modern sea-level and b1 = 200 m is a 0 rough average of highest observed palaeo-shoreline ele- creases to zero linearly with temperature between T0 and ◦ vations in the model domain (Table 1; Clague and James, T1 = 2 C, 2002).  P if T T ,  m m ≤ 0 = T1−Tm 3.5. Bedrock response As Pm T −T if T0 < Tm < T1 , (35)  1 0 0 if T T , The modelled ice sheet interacts with the underlying 1 ≤ m bedrock in two ways. Firstly, bedrock temperatures are M computed to a given depth by appending the ice enthalpy Surface melt, , is assumed proportional to the num- model (7) with an underlying bedrock thermal model. ber of positive degree days (PDD), defined over an arbi- [t , t ] The only process accounted for is vertical heat conduc- trary time interval, 1 2 , as the integral of positive Cel- cius temperature θ = T T , tion, hence this model consists of a simple diffusion equa- − 0 tion, Z A ∂T k = b ∆T, (33) PDD = max(θ(t), 0) dt . (36) ∂t ρbcb 0 where k , ρ and c are the thermal conductivity, den- b b b For multimillenial simulations of palaeo-ice sheet sity and specific heat capacity of the bedrock (Table 1). evolution such as presented in this thesis, daily or hourly Bedrock temperature is conditioned at depth by a fixed temperature data is not available, forcing the PDD com- upwards geothermal heat flux q (Table 1). The bedrock G putation to rely on an idealised representation of the an- thermal model is necessary for modelling the Cordilleran nual temperature cycle, θ . Sub-annual temperature vari- ice sheet on multi-millennial time-scales, because tem- ac ability around the freezing point, however, significantly perature fluctuations caused by climate change and vary- affects surface melt on a multi-year scale (Arnold and ing degrees of isolation from atmospheric temperatures MacKay, 1964). It is then typically included in the mod- by ice sheet coverage penetrate several kilometres into els under an assumption of normal temperature distribu- the rock, and eventually affect basal ice temperatures. tion, using a standard deviation parameter σ in the Secondly, bedrock elevation evolves in response to PDD PDD computation (Braithwaite, 1984). PDDs can then the ice weight. The bedrock deformation model used in be computed using a double-integral formulation (Reeh, this thesis describes the flexure of an elastic lithosphere 1991), on top of an infinite half-space viscous mantle (Lingle and Clark, 1985). It can be described by a single pseudo- 1 differential equation of the bed elevation, b, PDD = σPDD√2π ∂b Z t2 Z ∞ 2 ~ 2 (θ θac) 2νm grad + ρlgb + D ∆ b = σzzb , (34) dt dθ θ exp − , (37) | | ∂t 2 2 × t1 0 − σPDD where νm is mantle viscosity, ρl is lithosphere density, D is the lithosphere’s flexural rigidity (Table 1), and σzzb

13 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET or more efﬁciently using an error function formulation (Calov and Greve, 2005), Compute snow accumulation, As Z t2 σ θ2 PDD = dt PDD exp ac 2 2 t1 √2π − σPDD θ θ Compute PDDs + ac erfc ac . (38) 2 −√2σPDD

In the simulations presented here, Eq. (38) is numer- Update snow depth ically approximated using week-long sub-intervals. For S S + As each sub-interval, PDDs, which serve as a proxy for the ← available energy for melt, define the amount of snow Compute potential snow melt melt through a proportionality factor, Fs. If all snow is Ms,p := PDD Fs melted, remaining PDDs are then used to melt ice, using × another proportionality factor, Fi (Fig. 13). The degree- day factors for snow, Fs, and ice, Fi (Table 1), are derived from mass-balance measurements on contemporary glaciers in the Coast and Rocky mountains of British No Yes M > H ? Columbia (Shea et al., 2009). In Paper I, the tempera- s,p s ture standard deviation, σPDD, is a constant model parameter. In Paper IV, it is computed from daily temperature values from the North American Regional Reanal- ysis (NARR, Mesinger et al., 2006), using a method ini- Melt some snow Melt all snow tially developed in Paper II, and further improved upon in Ms := Ms,p Ms := Hs Paper III. Although the PDD model is a crude parametrisation of processes happening at the surface of a glacier, Melt no ice Melt some ice its applicability has been demonstrated (e.g. Hock, 2003), Ms Mi := 0 Mi := (PDD F ) Fi while other, physically-based methods (e.g. Hock, 2005), − s × require a precise knowledge of the physical state of the atmosphere, which typically is highly uncertain for past Compute surf. mass balance SMB := As Ms Mi periods such as the last glacial cycle. − − 3.7. Atmospheric forcing Update snow depth Atmospheric forcing of the model consists of a present- S S Ms ← − day monthly climatology, T ,P , modified by { m0 m0} temperature-offset corrections, ∆TTS, and lapse-rate corrections, ∆TLR, Figure 13. Positive degree-day algorithm used to compute accumulation and melt at the ice sheet surface. Adapted from the Tm(t, x, y) = Tm0(x, y) + ∆TTS(t) + ∆TLR(t, x, y) , PISM authors (2014). (39a) Pm(t, x, y) = Pm0(x, y) . (39b) thus accounting for the evolution of ice thickness, h = s b, on the one hand, and for differences between − The present-day monthly climatology is computed the the ice flow model basal topography, b, and the cli- from near-surface air temperature and precipitation rate mate input topography, bref , on the other hand. All sim- fields extracted from observational data and atmospheric ulations use an annual temperature lapse rate of γ = ∆T reanalyses (Table 2). The lapse-rate correction, LR, is 6 K km−1 (Table 1). s computed from the ice sheet surface elevation, , in re- In Paper I, we test the sensitivity of the ice sheet lation to the climate input reference bedrock topography, model to five different input present-day climatologies, b ref , T ,P (Table 2), while adopting a range of constant { m0 m0} temperature-offset corrections, ∆TTS, between -10 and ∆TLR(t, x, y) = γ [s(t, x, y) bref (x, y)] (40) − − 0 K. In Paper IV, on the contrary, we test ice sheet model = γ [h(t, x, y) + b(t, x, y) − sensitivity to time dependent temperature-offset correc- b (x, y)] , (41) − ref

14 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Table 2. Sources of forcing data used to inform the ice sheet model. Name Type Original reference Employed in ETOPO1 Bedrock topography Amante and Eakins (2009) Paper I, Paper IV WorldClim Gridded observations Hijmans et al. (2005) Paper I NCEP/NCAR Atmospheric reanalysis Kalnay et al. (1996) Paper I ERA-40 ” Uppala et al. (2005) Paper III ERA-Interim ” Dee et al. (2011) Paper I, Paper II CFSR ” Saha et al. (2010) Paper I NARR ” Mesinger et al. (2006) Paper I, Paper IV GRIP Ice-core δ18O record Dansgaard et al. (1993) Paper IV NGRIP ” Andersen et al. (2004) Paper IV EPICA ” Jouzel et al. (2007) Paper IV Vostok ” Petit et al. (1999) Paper IV K0 ODP 1012 Sea-ﬂoor U37 record Herbert et al. (2001) Paper IV ODP 1020 ” Herbert et al. (2001) Paper IV

Table 3. Properties of the Canadian Atlas Lambert conformal 4. Results summary conic projection (EPSG code 3978) and the GRS 1980 ellip- soid, and corresponding PROJ.4 parameter names, as used in 4.1. Simulations of the Last Glacial Maximum all model runs, ice volume and area calculations, and most fig- Paper I — Seguinot, J., Khroulev, C., Rogozhina, I., ures. Parameter Name Value Stroeven, A. P., and Zhang, Q.: The effect of climate forcing on numerical simulations of the Cordilleran ice sheet Projection name +proj lcc at the Last Glacial Maximum, The Cryosphere, 8, 1087– Central meridian +lon_0 95◦W Latitude of origin +lat_0 49◦N 1103, doi:10.5194/tc-8-1087-2014, 2014. First secant parallel +lat_1 49◦N Second secant parallel +lat_2 77◦N As a first step towards a simulations of the Cordilleran +a Semimajor radius 6378137 m ice sheet through the last glacial cycle, I have opted to run Reciprocal flattening +rf 298.257222101 the ice sheet model under a range of constant temperature depressions, targeting the LGM as the period for which ice sheet limits are best known. These simulations, per- tions derived from six different palaeo-temperature proxy formed on a 10 km grid, use constant temperature depres- records (Table 2). sions from the present climate state, which is represented 3.8. Numerical implementation using five different input climatologies. All model runs start from ice free conditions and run for 10 ka, a period In PISM, the SIA and the SSA are resolved numerically chosen as representative of Cordilleran ice sheet build- using centred, explicit finite difference methods. Ice ve- up from assumed, negligible ice cover. Although this as- locities, v~SIA and v~SSA, are computed on a staggered sumption appears unjustified from results in Paper IV, the grid, i.e., a grid offset by half a grid cell in both di- simplistic set-up used in Paper I allowed an assessment rections of the x- and y-axes (Bueler and Brown, 2009; of model sensitivity to the choice of input climatologies, Winkelmann et al., 2011). The computation of SSA ve- without involving too many additional parameters. locities involves the numerical resolution of Eq. (25). Discrepancies between the five input data sets used After discretization, this can be thought of as the in- (Table 2) caused considerable variability in modelled ice version of a large but sparse linear system of the form sheet geometries (Paper I, Figs. 5–7). In particular, ice ~ Av~SSA = b. This sparse linear problem is resolved in growth in continental sectors of the modelling domain is parallel by numerical iteration methods implemented in strongly dependent on input precipitation patterns, which PETSc (Balay et al., 2014). The simulations presented are often poorly resolved by atmospheric reanalysis fields here use the generalized minimal residual method (GM- (Paper I, Figs. 10–14). The closest match between mod- RES, Saad, 2003). The vertically-averaged mass continu- elled ice sheet extent and the reconstructed LGM ice mar- ity equation is then treated as a diffusion-advection equa- gin (Dyke, 2004) was obtained using the North American tion, where SIA velocities are diffusive and SSA veloc- Regional Reanalysis (NARR, Mesinger et al., 2006) cli- ities advective (Bueler and Brown, 2009; Winkelmann matologies and a temperature depression of 7 K (Paper I, et al., 2011). The horizontal grid uses the Canadian At- Fig. 15). The relevant simulation is reproduced here, us- las Lambert conformal conic projection (Table 3). Time- ing a higher horizontal resolution of 3 km (Fig. 14). Fol- stepping is adaptive. lowing this sensitivity study, I have retained the NARR

15 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Figure 14. Constant-climate simulation of the LGM Cordilleran ice sheet using an horizontal resolution of 3 km. The map depicts modelled basal topography, cold-based areas (hatches), surface topography (100 m contours) and surface velocities, and a reconstruction of the approximate ice sheet margin at 16.8 cal ka from compiled geomorphological evidence (red contour; Dyke, 2004). The simulation is forced by atmospheric ﬁelds from the NARR (Mesinger et al., 2006) and a temperature depression of 7 ◦C, using the set-up described in Paper I. There are several issues with this run. First, all east-facing mar- gins are too extensive as compared to available reconstructions. Second, marine advance on the Paciﬁc Shelf is too restrictive, most notably in the south-western corner where the Puget and Juan de Fuca lobes are miss- ing. Third, en-glacial temperatures are anomalously low due to brutal application of cold climate conditions to ice-free bedrock at the inception of the run.

16 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET climatology as input to simulations of the Cordilleran ice sheet through the last glacial cycle (Sect. 4.3.; Paper IV).

4.2. Role of daily temperature variability on melt Paper II — Seguinot, J.: Spatial and seasonal effects of temperature variability in a positive degree-day glacier surface mass-balance model, J. Glaciol., 59, 1202–1204, doi:10.3189/2013JoG13J081, 2013.

Paper III — Seguinot, J. and Rogozhina, I.: Daily temperature variability predetermined by thermal conditions over ice-sheet surfaces, J. Glaciol., 60, 603–605, doi:10.3189/2014jog14j036, 2014.

Next to the atmospheric forcing, another critical source of uncertainty in palaeo-ice sheet modelling is the choice of parameters used by the surface mass balance model (e.g., Hebeler et al., 2008). In our case, these parameters are the degree-day factors for snow, Fs, and ice, Fi (Fig. 13 and Table 1), and the standard deviation of near-surface air temperature, σPDD (Eq. 38 and Table 1). Although the first two can only be deduced from surface mass-balance measurements on present-day glaciers (e.g., Shea et al., 2009) and assumptions that these values 0 1 2 3 4 5 5.3 are also valid for palaeo-simulations, σPDD can be esti- mated from temperature time series (e.g., Fausto et al., July PDD standard deviation (K) 2011). In Paper II, I used near-surface air temperatures from Figure 15. July, daily temperature variability in the Arctic, the ERA-Interim reanalysis (Dee et al., 2011) to com- as shown by monthly standard deviation of near-surface daily pute a global, monthly distribution of σ . The result- PDD mean air temperature computed from ERA-40 reanalysis (Up- ing grids show that, although often assumed constant in pala et al., 2005). Note the strong differences between continen- ice sheet modelling, σ exhibits significant spacial and PDD tal and maritime regions, notably within the modelling domain temporal variation, most particularly in the polar regions (black rectangle). Although daily variability is lowest in sum- (Fig. 15; Paper II, Fig. 1). Using a simplified surface mer, this is when it most importantly contributes to surface melt mass balance model, I then demonstrated the effect of this and, therefore, net mass balance. variation on the computation of positive degree-days (Pa- per II, Fig. 2), and surface mass-balance (Paper II, Fig. 3). Neglecting this variability in ice sheet modelling intro- σ = 0.15 T + 1.66 . (42) duces significant errors in modelled PDD and surface PDD − · m mass balance calculations. Therefore, the Cordilleran ice The effect of the PDD standard deviation, σPDD, on sheet simulations presented in Paper IV now include this simulations of the Cordilleran ice sheet is not shown in effect. any of the papers. However, following these two stud- The analysis is extended in Paper III, using re- ies, a natural step forward was to implement, in PISM, analysed air temperatures from ERA-40 (Uppala et al., spatially and seasonally variable σPDD input (PISM is- 2005). Although of coarser spatial resolution than the sue #179), and the parameterisation expressed by Eq. (42) ERA-Interim reanalysis used in Paper II, the ERA-40 over ice-filled grid cells (PISM issue #265). Their effect time series span over longer time, which is necessary to on modelled Cordilleran ice sheet surface elevations is compute standard deviation values that are representative demonstrated here using a constant climate set-up sim- of the long-term daily variability (Rogozhina and Rau, ilar to the one used in Paper I (Fig. 16). These simula- 2014). In Paper III, we demonstrated an inverse correla- tions show that the effect of daily variability is strongest tion between σPDD and monthly mean temperatures, Tm, in continental regions, where the σPDD values are high- over the Greenland ice sheet (Paper III, Fig. 1). This cor- est (cf. Fig. 15). When the results are compared to the relation could potentially be used as a parametrisation, mapped LGM ice limits (Fig. 6), which show a restricted and can be expressed as (Paper III, Eq. 3) ice extent in continental areas despite low temperatures

17 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

(a) (b) (c) (d)

1000 500 0 500 1000 100 50 0 50 100 elevation difference (m) 1000 500 0 500 1000

Figure 16. Effect of temperature variability on Cordilleran ice sheet simulations. Modelled basal topography and surface topography (250 m contours). The simulations use atmospheric forcing from the NARR (Mesinger et al., 2006), a temperature depression of 7 ◦C, and different treatments of temperature variability including (a) no daily variability, σPDD = 0; (b) a constant, mean summer (JJA), model domain-averaged standard deviation value, σPDD = 3.09 K; (c) monthly standard deviation grids derived from the NARR and (d) monthly grids for ice-free terrain and the Greenland ice sheet, ERA-40-based (Uppala et al., 2005) parametrisation in Paper III, σPDD = 0.15 Tm + 1.66. Differences in surface elevation are displayed relative to case (c). Note the different colour scales. − · at present (Fig. 9), it appears that including daily temper- cycle (Paper IV, Figs. 2–3). The resulting simulations, us- ature variability in the ice sheet model results in a closer ing horizontal resolutions of 6 and 10 km, all produced agreement with these limits (Fig. 16b and c). However, two major glaciations during the last glacial cycle, at applying the parametrisation derived for the Greenland MIS 4 and MIS 2. However, the differences in timing ice sheet (Eq. (42)) over ice-ﬁlled grid cells partly reverts and magnitude of these glaciations are considerable (Pa- this effect (Fig. 16d), indicating that this parametrization per IV, Figs. 3–4). may not be applicable to the former Cordilleran ice sheet. Geological evidence for the timing and extent of maximum glaciation during the LGM allowed me to select 4.3. Simulations of the last glacial cycle two simulations, driven by the GRIP (Dansgaard et al., Paper IV — Seguinot, J., Rogozhina, I., Stroeven, A. P., 1993) and EPICA (Jouzel et al., 2007) ice core records Margold, M., and Kleman, J.: Numerical simulations of (Paper IV, Figs. 5–6), as producing the most realistic re- the Cordilleran ice sheet through the last glacial cycle, constructions of the last glacial cycle. Using aggregated manuscript. model output from these two runs, I analysed the location of transient ice caps before and after the LGM, the The primary goal of my doctoral project has been longevity of major ice-dispersal centres, and the rates and to perform numerical reconstructions of Cordilleran ice patterns of deglaciation (Paper IV, Figs. 6–15), in rela- sheet history through the last glacial cycle, which forms tion to available geological evidence. Modelled ice vol- the topic of Paper IV. Here, monthly air temperature, ume evolution, expressed as metres of sea level equiv- precipitation, and PDD standard deviation climatologies alent, and snapshots of ice extent and ﬂow patterns from from the NARR are combined with palaeo-temperature the GRIP simulation (Fig. 17) show the growth and decay time-series from six proxy records to produce a time- of the Cordilleran ice sheet over its last expansion cycle. dependent climate forcing spanning over the last glacial An ice cap located over the Skeena Mountains acts as the

18 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

104 -42.95 ka -34.43 ka -29.65 ka -25.34 ka -19.14 ka -15.98 ka -13.0 ka -11.58 ka -10.0 ka surf. velocity (m/a)

101 (a)

9 (b) 8 7 6 5 4 3 2 1 ice volume (m s.l.e.) 0 120 100 80 60 40 20 0 model time (kyr)

Figure 17. (a) Modelled growth and decay of the last Cordilleran ice sheet. Basal topography, surface topography (1 km contours) and surface velocity (m/a) are model results. (b) Evolution of total ice volume during the last glacial cycle, expressed as metres of sea-level equivalent (m s.l.e.). The simulation is performed on a 6 km horizontal grid and adopts atmospheric fields from the NARR (Mesinger et al., 2006) and temperature offset time-series derived from the GRIP ice-core record (Dansgaard et al., 1993) as described in Paper IV. Note that, for this run, the Cordilleran ice sheet was almost as voluminous during MIS 4 as it was during the LGM. primary nucleation centre for the Cordilleran ice sheet. 5.1. Interaction with the Laurentide ice sheet After the LGM, the ice sheet shrinks rapidly such that its Certainly the most important assumption made in this last remnants reside once again in the Skeena Mountains thesis is that of a Cordilleran ice sheet evolution en- towards the end of the deglaciation in the early Holocene. tirely disconnected from its much larger neighbour, the Laurentide ice sheet. Geomorphological evidence from 5. Sources of error the suture zone indicates that, during the LGM, both Cordilleran and Laurentide ice flow were deflected (Kle- Through sensitivity studies presented in my thesis, I have man et al., 2010; Margold et al., 2013a,b), although it identified some of the predominant sources of uncertainty remains unclear for how long this suture condition has involved in numerical modelling of the Cordilleran ice lasted (Jackson et al., 1997; Bednarski and Smith, 2007; sheet, including the choice of present-day atmospheric Gowan, 2013). forcing fields, the effects of daily temperature variability, Nevertheless, during most of the last glacial cycle, the and the choice of palaeoclimate time series used to sim- two ice sheets have been disconnected (Kleman et al., ulate ice sheet growth and decay over glacial cycle time 2010). Thus, this assumption only affects numerical re- scales. The resulting simulations, combined with avail- constructions of the Cordilleran ice sheet during the LGM able geological evidence, yielded estimates of the LGM and subsequent deglacation. Numerical odelling results Cordilleran ice sheet extent and volume, identification of by Gregoire et al. (2012) indicate that the Laurentide ice the main ice-dispersal centres througout the last glacial sheet buttress resulted in a thicker Cordilleran ice sheet. cycle, and understanding of final deglaciation patterns. However, numerical modelling alone will neither eluci- However, there remains considerable room for model im- date the nature nor the duration of the interaction between provements beyond the ones achieved in this thesis. The of the Cordilleran and Laurentide ice sheets. For this, ad- foremost sources of error that remain unassessed in this ditional field constraints will be required. thesis include potential buttressing effects of the Lauren- tide ice sheet, changing precipitation patterns with evolv- 5.2. Past changes in precipitation ing topography, and regional variability of the geothermal Another important source of error is the treatment of heat flux. precipitation, to which the numerical ice sheet model

19 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET is highly sensitive (Paper I). In all sensitivity studies, the lineation patterns we observe within these zones of we base precipitation patterns on current fields derived consistent cold-based deglaciation could be older fea- from atmospheric reanalyses. Although the distribution tures, formed during an earlier phase of advance or re- of precipitation has undoubtedly been affected by past treat of the ice sheet. However, without a better control changes in global climate and the intermittent presence on the geothermal heat flux, such analyses are perhaps of ice sheets and ice caps in the Northern Hemisphere, too premature. such changes are difficult to assess. In the North Ameri- can Cordillera, much of the continental interior was pre- sumed dryer than today due to the presence of ice caps 6. Conclusions on the Pacific Coast ranges, reinforcing the current topographic barrier to eastward moisture transport. How- In this thesis, I present numerical reconstructions of the ever, some regions were probably wetter as a side effect former Cordilleran ice sheet to a level of detail that has, to of modified atmospheric circulation patterns. my knowledge, not been attained before. Although math- These effects are typically analysed using atmo- ematically uncertain, the coupled SIA-SSA approxima- spheric circulation models. However, I have shown that tion, combined with massively parallel computing, al- different atmospheric reanalyses — circulation models lowed for ice sheet modelling at large spatial scales, high using data assimilation techniques — fail to agree re- spatial resolution, and long time scales characteristic of garding the present-day pattern of precipitation over the ice sheet growth and decay. modelling domain. These discrepancies, which arise as From the outcome of sensitivity studies and the com- a consequence of the highly mountainous terrain, have parison of model output to field constraints, I conclude dramatic effects on the ice sheet model response. Con- the following: sidering the lack of observational data for modelling past 1. Although spatial patterns of glaciation in the North climates, the increased computational demand, and a de- American Cordillera are sometimes poorly under- creased familiarity to processes at play, we must expect stood, they can be largely explained by the present- that, for ice sheet simulations forced by different palaeo- day distributions of temperature and precipitation, climate records, the uncertainty associated to the choice provided an accurate treatment of seasonality, oro- of atmospheric forcing will be even higher. Alternative graphic precipitation, and daily temperature vari- methods of coupling between ice sheet evolution and pre- ability in the model, and a simple temperature-index cipitation changes need to be developed. accumulation and melt model (PDD). 5.3. Geothermal heat flux patterns 2. The LGM Cordilleran ice sheet was a transient fea- Geothermal heat flux forcing, a major contributor to the ture, out of balance with contemporaneous atmo- modelled distribution of englacial temperatures, and the spheric temperatures. Its maximum extent is the re- location of basal melt areas, was assumed constant over sult of a 50 ka-long evolution. Peak glacial tem- the modelling domain. However, the geology of the study peratures were lower than those needed to sustain area consists of an amalgamation of former subduction a constant ice sheet geometry because ice sheet zones, resulting in complex geothermal patterns (Black- growth was finally inhibited by the deglacial tran- well and Richards, 2004). I am now at the stage to start in- sition to warmer temperatures. Therefore, this max- cluding this variability as a driver to the ice sheet model, imum stage is not representative of average condi- which will offer new opportunities to study its output in tions, as shown in the modelling by persistent ice terms of ice flow patterns, consistent basal freezing and caps throughout most of the last glacial cycle. basal melting patterns, regions prone for glacial erosion, 3. Patterns of growth and decay of the Cordilleran and regions characterised by glacial deposition. ice sheet vary spatially and temporally. Two- I expect that inclusion of geothermal heat patterns dimensional, conceptual models of growth and de- will allow modelled basal flow directions and distances cay (e.g. Fulton, 1991; Margold et al., 2013b) do to be compared with glacial lineations, which is one of not encompass such diversity. In order to deci- the main sources of geomorphological evidence used to pher the complexity of geological evidence, the reconstruct former ice sheets (Boulton and Clark, 1990; Cordilleran ice sheet needs to be thought of as a Kleman et al., 1997, 2010). For instance, in our sim- three-dimensional body of moving ice in perpetual ulations, large portions of the Cordilleran ice sheet re- transformation towards evolving climate conditions. main cold-based throughout the deglaciation (Paper IV, Fig. 15). As cold-based glaciers typically do not impact their beds significantly, this could indicate that some of

20 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

References C., Ruth, U., Schwander, J., Siegenthaler, U., Souchez, R., Stauffer, B., Steffensen, J. P., Stenni, B., Stocker, Agassiz, L.: Etudes sur les glaciers, Jent et Gassmann, T. F., Tabacco, I. E., Udisti, R., van de Wal, R. S. W., Soleure, 1840. van den Broeke, M., Weiss, J., Wilhelms, F., Winther, J.-G., Wolff, E. W., and Zucchelli, M.: Eight glacial cy- Amante, C. and Eakins, B. W.: ETOPO1 1 arc-minute cles from an Antarctic ice core, Nature, 429, 623–628, global relief model: procedures, data sources and anal- doi:10.1038/nature02599, 2004. ysis, NOAA technical memorandum NESDIS NGDC- 24, Natl. Geophys. Data Center, NOAA, Boulder, CO, Balay, S., Abhyankar, S., Adams, M. F., Brown, J., doi:10.7289/V5C8276M, 2009. Brune, P., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Rupp, Andersen, K. K., Azuma, N., Barnola, J.-M., Bigler, K., Smith, B. F., and Zhang, H.: PETSc Web page, M., Biscaye, P., Caillon, N., Chappellaz, J., Clausen, http://www.mcs.anl.gov/petsc, 2014. H. B., Dahl-Jensen, D., Fischer, H., Flückiger, J., Fritzsche, D., Fujii, Y., Goto-Azuma, K., Grønvold, Bamber, J. L., Griggs, J. A., Hurkmans, R. T. W. L., K., Gundestrup, N. S., Hansson, M., Huber, C., Hvid- Dowdeswell, J. A., Gogineni, S. P., Howat, I., Moug- berg, C. S., Johnsen, S. J., Jonsell, U., Jouzel, J., inot, J., Paden, J., Palmer, S., Rignot, E., and Stein- Kipfstuhl, S., Landais, A., Leuenberger, M., Lorrain, hage, D.: A new bed elevation dataset for Greenland, R., Masson-Delmotte, V., Miller, H., Motoyama, H., The Cryosphere, 7, 499–510, doi:10.5194/tc-7-499- Narita, H., Popp, T., Rasmussen, S. O., Raynaud, D., 2013, 2013. Rothlisberger, R., Ruth, U., Samyn, D., Schwander, J., Bednarski, J. M. and Smith, I. R.: Laurentide and mon- Shoji, H., Siggard-Andersen, M.-L., Steffensen, J. P., tane glaciation along the Rocky Mountain Foothills of Stocker, T., Sveinbjörnsdóttir, A. E., Svensson, A., northeastern British Columbia, Can. J. Earth Sci., 44, Takata, M., Tison, J.-L., Thorsteinsson, T., Watan- 445–457, doi:10.1139/e06-095, 2007. abe, O., Wilhelms, F., and White, J. W. C.: High- resolution record of Northern Hemisphere climate ex- Blackwell, D. D. and Richards, M.: Geothermal Map of tending into the last interglacial period, Nature, 431, North America, Am. Assoc. Petr. Geol., Tulsa, OK, 147–151, doi:10.1038/nature02805, data archived at 2004. the World Data Center for Paleoclimatology, Boulder, Colorado, USA., 2004. Blaise, B., Clague, J. J., and Mathewes, R. W.: Time of maximum Late Wisconsin glaciation, West Arnold, K. C. and MacKay, D. K.: Different methods of Coast of Canada, Quaternary Res., 34, 282–295, calculating mean daily temperatures, their effects on doi:10.1016/0033-5894(90)90041-I, 1990. degree-day totals in the high Arctic and their signiﬁ- Booth, D. B., Troost, K. G., Clague, J. J., and Waitt, R. B.: cance to glaciology, Geogr. Bull., 21, 123–129, 1964. The Cordilleran Ice Sheet, in: The Quaternary Period Aschwanden, A., Bueler, E., Khroulev, C., and Blatter, in the United States, edited by Gillespie, A., Porter, H.: An enthalpy formulation for glaciers and ice sheets, S., and Atwater, B., vol. 1 of Dev. Quaternary Sci., J. Glaciol., 58, 441–457, doi:10.3189/2012JoG11J088, pp. 17–43, Elsevier, Amsterdam, doi:10.1016/s1571- 2012. 0866(03)01002-9, 2003. Aschwanden, A., Aðalgeirsdóttir, G., and Khroulev, C.: Boulton, G. S. and Clark, C. D.: A highly mobile Lauren- Hindcasting to measure ice sheet model sensitiv- tide ice sheet revealed by satellite images of glacial lin- ity to initial states, The Cryosphere, 7, 1083–1093, eations, Nature, 346, 813–817, doi:10.1038/346813a0, doi:10.5194/tc-7-1083-2013, 2013. 1990. Augustin, L., Barbante, C., Barnes, P. R. F., Barnola, Braithwaite, R. J.: Calculation of degree-days for glacier- J. M., Bigler, M., Castellano, E., Cattani, O., Chap- climate research, Z. Gletscherk. Glazialgeol., 20, 1–8, pellaz, J., Dahl-Jensen, D., Delmonte, B., Dreyfus, 1984. G., Durand, G., Falourd, S., Fischer, H., Flückiger, J., Bretz, J. H.: The Channeled Scablands of the Columbia Hansson, M. E., Huybrechts, P., Jugie, G., Johnsen, Plateau, J. Geol., 31, 617–649, doi:10.1086/623053, S. J., Jouzel, J., Kaufmann, P., Kipfstuhl, J., Lambert, 1923. F., Lipenkov, V. Y., Littot, G. C., Longinelli, A., Lor- rain, R., Maggi, V., Masson-Delmotte, V., Miller, H., Briner, J. P. and Kaufman, D. S.: Late Pleistocene moun- Mulvaney, R., Oerlemans, J., Oerter, H., Orombelli, G., tain glaciation in Alaska: key chronologies, J. Quater- Parrenin, F., Peel, D. A., Petit, J.-R., Raynaud, D., Ritz, nary Sci., 23, 659–670, doi:10.1002/jqs.1196, 2008.

21 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Bueler, E. and Brown, J.: Shallow shelf approxima- Dawson, G. M.: III.— Recent observations on the glaciation as a “sliding law” in a thermodynamically cou- tion of British Columbia and adjacent regions, Geol. pled ice sheet model, J. Geophys. Res., 114, F03 008, Mag., 5, 347–350, doi:10.1017/s0016756800182159, doi:10.1029/2008JF001179, 2009. 1888.

Bueler, E. and van Pelt, W.: Mass-conserving sub- de Saussure, H.-B.: Voyages dans les Alpes, précédés glacial hydrology in the Parallel Ice Sheet Model, d’un essai sur l’histoire naturelle des environs de Geosci. Model Dev. Discuss., 7, 4705–4775, Genève, vol. 1, Fauche-Borel, Neuchâtel, 1786. doi:10.5194/gmdd-7-4705-2014, 2014. Dee, D. P., Uppala, S. M., Simmons, A. J., Berrisford, Bueler, E., Lingle, C. S., and Brown, J.: Fast com- P., Poli, P., Kobayashi, S., Andrae, U., Balmaseda, putation of a viscoelastic deformable Earth model M. A., Balsamo, G., Bauer, P., Bechtold, P., Beljaars, for ice-sheet simulations, Ann. Glaciol., 46, 97–105, A. C. M., van de Berg, L., Bidlot, J., Bormann, N., Del- doi:10.3189/172756407782871567, 2007. sol, C., Dragani, R., Fuentes, M., Geer, A. J., Haim- Calov, R. and Greve, R.: A semi-analytical solution berger, L., Healy, S. B., Hersbach, H., Hólm, E. V., for the positive degree-day model with stochas- Isaksen, L., Kållberg, P., Köhler, M., Matricardi, M., tic temperature variations, J. Glaciol., 51, 173–175, McNally, A. P., Monge-Sanz, B. M., Morcrette, J.-J., doi:10.3189/172756505781829601, 2005. Park, B.-K., Peubey, C., de Rosnay, P., Tavolato, C., Thépaut, J.-N., and Vitart, F.: The ERA-Interim reanal- Clague, J., Armstrong, J., and Mathews, W.: Advance ysis: conﬁguration and performance of the data assim- of the late Wisconsin Cordilleran Ice Sheet in south- ilation system, Q. J. Roy. Meteor. Soc., 137, 553–597, ern British Columbia since 22,000 Yr B.P., Quaternary doi:10.1002/qj.828, 2011. Res., 13, 322–326, doi:10.1016/0033-5894(80)90060- 5, 1980. Demuro, M., Froese, D. G., Arnold, L. J., and Roberts, Clague, J. J.: Delaciation of the Prince Rupert – Kitimat R. G.: Single-grain OSL dating of glacioﬂu- area, British Columbia, Can. J. Earth Sci., 22, 256– vial quartz constrains Reid glaciation in NW 265, doi:10.1139/e85-022, 1985. Canada to MIS 6, Quaternary Res., 77, 305–316, doi:10.1016/j.yqres.2011.11.009, 2012. Clague, J. J. and James, T. S.: History and isostatic effects of the last ice sheet in southern British Columbia, Duk-Rodkin, A.: Glacial limits map of Yukon Territory, Quaternary Res., 21, 71–87, doi:10.1016/s0277- Geol. Surv. of Can., Ottawa, ON, Open File 3694, 3791(01)00070-1, 2002. doi:10.4095/210739, 1999.

Clark, P. U. and Mix, A. C.: Ice sheets and sea level of Dyke, A. S.: An outline of North American deglacia- the Last Glacial Maximum, Quaternary Res., 21, 1–7, tion with emphasis on central and northern Canada, doi:10.1016/s0277-3791(01)00118-4, 2002. in: Quaternary Glaciations - Extent and Chronology — Cosma, T., Hendy, I., and Chang, A.: Chronological con- Part II, edited by Ehlers, J. and Gibbard, P. L., vol. 2 of Dev. Quaternary Sci. straints on Cordilleran Ice Sheet glaciomarine sedi- , pp. 373–424, Elsevier, Amster- mentation from core MD02-2496 off Vancouver Island dam, doi:10.1016/S1571-0866(04)80209-4, 2004. (western Canada), Quaternary Sci. Rev., 27, 941–955, Ehlers, J. and Gibbard, P. L.: The extent and chronology doi:10.1016/j.quascirev.2008.01.013, 2008. of Cenozoic Global Glaciation, Quaternary Int., 164- Dansgaard, W., Johnsen, S. J., Clausen, H. B., Dahl- 165, 6–20, doi:10.1016/j.quaint.2006.10.008, 2007. Jensen, D., Gundestrup, N. S., Hammer, C. U., Hvid- berg, C. S., Steffensen, J. P., Sveinbjörnsdottir, A. E., Emiliani, C.: Pleistocene Temperatures, J. Geol., 63, Jouzel, J., and Bond, G.: Evidence for general insta- 538–578, doi:10.1086/626295, 1955. bility of past climate from a 250-kyr ice-core record, Nature, 364, 218–220, doi:10.1038/364218a0, data Fausto, R. S., Ahlstrøm, A. P., van As, D., and Steffen, archived at the World Data Center for Paleoclimatol- K.: Present-day temperature standard deviation param- ogy, Boulder, Colorado, USA., 1993. eterization for Greenland, J. Glaciol., 57, 1181–1183, doi:10.3189/002214311798843377, 2011. Davis, N. F. G. and Mathews, W. H.: Four Phases of Glaciation with Illustrations from Southwest- Forbes, J. D.: Illustrations of the Viscous Theory of ern British Columbia, J. Geol., 52, 403–413, Glacier Motion. Part III, Philos. T. R. Soc., 136, 177– doi:10.1086/625236, 1944. 210, doi:10.1098/rstl.1846.0015, 1846.

22 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Fretwell, P., Pritchard, H. D., Vaughan, D. G., Bamber, Herbert, T. D., Schuffert, J. D., Andreasen, D., J. L., Barrand, N. E., Bell, R., Bianchi, C., Bingham, Heusser, L., Lyle, M., Mix, A., Ravelo, A. C., R. G., Blankenship, D. D., Casassa, G., Catania, G., Stott, L. D., and Herguera, J. C.: Collapse of the Callens, D., Conway, H., Cook, A. J., Corr, H. F. J., California Current During Glacial Maxima Linked Damaske, D., Damm, V., Ferraccioli, F., Forsberg, R., to Climate Change on Land, Sience, 293, 71–76, Fujita, S., Gim, Y., Gogineni, P., Griggs, J. A., Hind- doi:10.1126/science.1059209, data archived at the marsh, R. C. A., Holmlund, P., Holt, J. W., Jacobel, World Data Center for Paleoclimatology, Boulder, Col- R. W., Jenkins, A., Jokat, W., Jordan, T., King, E. C., orado, USA., 2001. Kohler, J., Krabill, W., Riger-Kusk, M., Langley, K. A., Leitchenkov, G., Leuschen, C., Luyendyk, B. P., Mat- Herzer, R. H. and Bornhold, B. D.: Glaciation and post- suoka, K., Mouginot, J., Nitsche, F. O., Nogi, Y., Nost, glacial history of the continental shelf off southwestern O. A., Popov, S. V., Rignot, E., Rippin, D. M., Rivera, Vancouver Island, British Columbia, Mar. Geol., 48, A., Roberts, J., Ross, N., Siegert, M. J., Smith, A. M., 285–319, doi:10.1016/0025-3227(82)90101-3, 1982. Steinhage, D., Studinger, M., Sun, B., Tinto, B. K., Welch, B. C., Wilson, D., Young, D. A., Xiangbin, C., Heyman, J., Stroeven, A. P., Harbor, J. M., and Caffee, and Zirizzotti, A.: Bedmap2: improved ice bed, surface M. W.: Too young or too old: Evaluating cosmogenic and thickness datasets for Antarctica, The Cryosphere, exposure dating based on an analysis of compiled boul- 7, 375–393, doi:10.5194/tc-7-375-2013, 2013. der exposure ages, Earth Planet. Sc. Lett., 302, 71–80, doi:10.1016/j.epsl.2010.11.040, 2011. Fulton, R. J.: Deglaciation studies in Kamloops region, an area of moderate relief, British Columbia, Hidy, A. J., Gosse, J. C., Froese, D. G., Bond, J. D., vol. 154 of Bull., Geol. Surv. of Can., Ottawa, ON, and Rood, D. H.: A latest Pliocene age for the ear- doi:10.4095/101467, 1967. liest and most extensive Cordilleran Ice Sheet in northwestern Canada, Quaternary Res., 61, 77–84, Fulton, R. J.: A Conceptual Model for Growth and Decay doi:10.1016/j.quascirev.2012.11.009, 2013. of the Cordilleran Ice Sheet, Géogr. phys. Quatern., 45, 281–286, doi:10.7202/032875ar, 1991. Hijmans, R. J., Cameron, S. E., Parra, J. L., Jones, P. G., and Jarvis, A.: Very high resolution interpolated cli- Glen, J.: Experiments on the deformation of ice, J. mate surfaces for global land areas, Int. J. Climatol., Glaciol., 2, 111–114, 1952. 25, 1965–1978, doi:10.1002/joc.1276, 2005. Gowan, E. J.: An assessment of the minimum timing of ice free conditions of the western Lau- Hock, R.: Temperature index melt modelling in mountain rentide Ice Sheet, Quaternary Res., 75, 100–113, areas, J. Hydrol., 282, 104–115, doi:10.1016/S0022- doi:10.1016/j.quascirev.2013.06.001, 2013. 1694(03)00257-9, 2003. Gregoire, L. J., Payne, A. J., and Valdes, P. J.: Hock, R.: Glacier melt: a review of processes and Deglacial rapid sea level rises caused by ice- their modelling, Prog. Phys. Geogr., 29, 362–391, sheet saddle collapses, Nature, 487, 219–222, doi:10.1191/0309133305pp453ra, 2005. doi:10.1038/nature11257, 2012. Hutter, K.: Theoretical Glaciology, Math. approaches to Greve, R.: A continuum-mechanical formulation for shal- geophys., Reidel, Dordrecht, 1983. low polythermal ice sheets, Philos. T. R. Soc. A, 355, 921–974, doi:10.1098/rsta.1997.0050, 1997. Jackson, L. E., Phillips, F. M., Shimamura, K., and Little, E. C.: Cosmogenic 36Cl dat- Hättestrand, C.: The glacial geomorphology of central ing of the Foothills erratics train, Alberta, Ca series and northern Sweden, vol. 85 of , Sveriges Canada, Geology, 25, 195, doi:10.1130/0091- Geologiska Undersökning, Uppsala, 1998. 7613(1997)025<0195:ccdotf>2.3.co;2, 1997. Hays, J. D., Imbrie, J., and Shackleton, N. J.: Variations in Jouzel, J., Masson-Delmotte, V., Cattani, O., Dreyfus, the Earth’s Orbit: Pacemaker of the Ice Ages, Sience, G., Falourd, S., Hoffmann, G., Minster, B., Nouet, J., 194, 1121–1132, doi:10.1126/science.194.4270.1121, Barnola, J. M., Chappellaz, J., Fischer, H., Gallet, J. C., 1976. Johnsen, S., Leuenberger, M., Loulergue, L., Luethi, Hebeler, F., Purves, R. S., and Jamieson, S. S.: The D., Oerter, H., Parrenin, F., Raisbeck, G., Raynaud, impact of parametric uncertainty and topographic er- D., Schilt, A., Schwander, J., Selmo, E., Souchez, R., ror in ice-sheet modelling, J. Glaciol., 54, 899–919, Spahni, R., Stauffer, B., Steffensen, J. P., Stenni, B., doi:10.3189/002214308787779852, 2008. Stocker, T. F., Tison, J. L., Werner, M., and Wolff,

23 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

E. W.: Orbital and Millennial Antarctic Climate Vari- Lliboutry, L. A. and Duval, P.: Various isotropic and ability over the Past 800,000 Years, Sience, 317, 793– anisotropic ices found in glaciers and polar ice caps 796, doi:10.1126/science.1141038, data archived at the and their corresponding rheologies, Ann. Geophys., 3, World Data Center for Paleoclimatology, Boulder, Col- 207–224, 1985. orado, USA., 2007. Lüthi, M., Funk, M., Iken, A., Gogineni, S., and Truf- Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., fer, M.: Mechanisms of fast flow in Jakobshavns Is- Deaven, D., Gandin, L., Iredell, M., Saha, S., bræ, Greenland; Part III: measurements of ice defor- White, G., Woollen, J., Zhu, Y., Leetmaa, A., mation, temperature and cross-borehole conductivity Reynolds, R., Chelliah, M., Ebisuzaki, W., Hig- in boreholes to the bedrock, J. Glaciol., 48, 369 – 385, gins, W., Janowiak, J., Mo, K. C., Ropelewski, doi:10.3189/172756502781831322, 2002. C., Wang, J., Jenne, R., and Joseph, D.: The NCEP/NCAR 40-year reanalysis project, B. Am. Margold, M., Jansson, K. N., Kleman, J., and Meteorol. Soc., 77, 437–471, doi:10.1175/1520- Stroeven, A. P.: Glacial meltwater landforms of 0477(1996)077<0437:TNYRP>2.0.CO;2, 1996. central British Columbia, J. Maps, 7, 486–506, doi:10.4113/jom.2011.1205, 2011. Kaufman, D. S. and Manley, W. F.: Pleistocene Max- imum and Late Wisconsinan glacier extents across Margold, M., Jansson, K. N., Kleman, J., and Stroeven, Alaska, USA, in: Quaternary Glaciations - Extent A. P.: Lateglacial ice dynamics of the Cordilleran and Chronology — Part II, edited by Ehlers, J. Ice Sheet in northern British Columbia and south- and Gibbard, P. L., vol. 2 of Dev. Quaternary Sci., ern Yukon Territory: retreat pattern of the Liard Lobe pp. 9–27, Elsevier, Amsterdam, doi:10.1016/S1571- reconstructed from the glacial landform record, J. 0866(04)80182-9, 2004. Quaternary Sci., 28, 180–188, doi:10.1002/jqs.2604, 2013a. Kleman, J.: Preservation of landforms under ice sheets and ice caps, Geomorphology, 9, 19–32, Margold, M., Jansson, K. N., Kleman, J., Stroeven, A. P., doi:10.1016/0169-555x(94)90028-0, 1994. and Clague, J. J.: Retreat pattern of the Cordilleran Ice Sheet in central British Columbia at the end of the last Kleman, J., Hättestrand, C., Borgström, I., and Stroeven, glaciation reconstructed from glacial meltwater land- A.: Fennoscandian paleoglaciology reconstructed us- forms, Boreas, 42, 830–847, doi:10.1111/bor.12007, ing a glacial geological inversion model, J. Glaciol., 2013b. 43, 283–299, 1997. Marshall, S. J., James, T. S., and Clarke, G. K.: Kleman, J., Hättestrand, C., Stroeven, A. P., Jansson, North American Ice Sheet reconstructions at the Last K. N., De Angelis, H., and Borgström, I.: Recon- Glacial Maximum, Quaternary Res., 21, 175–192, struction of Palaeo-Ice Sheets – Inversion of their doi:10.1016/s0277-3791(01)00089-0, 2002. Glacial Geomorphological Record, in: Glacier Sci- ence and Environmental Change, edited by Knight, Martin, M. A., Winkelmann, R., Haseloff, M., Albrecht, P. G., pp. 192–198, Blackwell, Malden, MA, T., Bueler, E., Khroulev, C., and Levermann, A.: The doi:10.1002/9780470750636.ch38, 2006. Potsdam Parallel Ice Sheet Model (PISM-PIK) – Part 2: Dynamic equilibrium simulation of the Antarctic ice Kleman, J., Jansson, K., De Angelis, H., Stroeven, A., sheet, The Cryosphere, 5, 727–740, doi:10.5194/tc-5- Hättestrand, C., Alm, G., and Glasser, N.: North Amer- 727-2011, 2011. ican ice sheet build-up during the last glacial cycle, 115–21 kyr, Quaternary Sci. Rev., 29, 2036–2051, Mesinger, F., DiMego, G., Kalnay, E., Mitchell, K., doi:10.1016/j.quascirev.2010.04.021, 2010. Shafran, P. C., Ebisuzaki, W., Jovic,´ D., Woollen, J., Rogers, E., Berbery, E. H., Ek, M. B., Fan, Y., Lingle, C. S. and Clark, J. A.: A numerical model of inter- Grumbine, R., Higgins, W., Li, H., Lin, Y., Manikin, actions between a marine ice sheet and the solid Earth: G., Parrish, D., and Shi, W.: North American re- application to a West Antarctic ice stream, J. Geophys. gional reanalysis, B. Am. Meteorol. Soc., 87, 343–360, Res., 90, 1100–1114, doi:10.1029/JC090iC01p01100, doi:10.1175/BAMS-87-3-343, 2006. 1985. Morland, L. W.: Unconfined ice-shelf flow, in: Dynam- Lisiecki, L. E. and Raymo, M. E.: A Pliocene- ics of the West Antarctic ice sheet, edited by van der Pleistocene stack of 57 globally distributed ben- Veen, C. J. and Oerlemans, J., vol. 4 of Glaciology and thic δ18O records, Paleoceanography, 20, PA1003, Quaternary Geology, pp. 99–116, Springer, Dordrecht, doi:10.1029/2004pa001071, 2005. doi:10.1007/978-94-009-3745-1_6, 1987.

24 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Morland, L. W. and Johnson, I. R.: Steady motion of ice Rogozhina, I. and Rau, D.: Vital role of daily tempera- sheets, J. Glaciol., 25, 229–246, 1980. ture variability in surface mass balance parameteriza- tions of the Greenland Ice Sheet, The Cryosphere, 8, Napieralski, J., Hubbard, A., Li, Y., Harbor, J., Stroeven, 575–585, doi:10.5194/tc-8-575-2014, 2014. A. P., Kleman, J., Alm, G., and Jansson, K. N.: To- wards a GIS assessment of numerical ice-sheet model Saad, Y.: Iterative methods for sparse linear systems, Soc. performance using geomorphological data, J. Glaciol., for Ind. and Appl. Math., Philadelphia, PA, 2 edn., 53, 71–83, doi:10.3189/172756507781833884, 2007. 2003. NSIDC: Glacier Photograph Collection, Natl. Snow and Ice Data Center, Boulder, CO, Saha, S., Moorthi, S., Pan, H.-L., Wu, X., Wang, doi:10.7265/N5/NSIDC-GPC-2009-12, 2009. J., Nadiga, S., Tripp, P., Kistler, R., Woollen, J., Behringer, D., Liu, H., Stokes, D., Grumbine, R., Nye, J. F.: The Flow Law of Ice from Measurements Gayno, G., Wang, J., Hou, Y.-T., Chuang, H.-Y., Juang, in Glacier Tunnels, Laboratory Experiments and the H.-M. H., Sela, J., Iredell, M., Treadon, R., Kleist, D., Jungfrauﬁrn Borehole Experiment, Proc. R. Soc. Lon- Delst, P. V., Keyser, D., Derber, J., Ek, M., Meng, J., don, Ser. A, 219, 477–489, 1953. Wei, H., Yang, R., Lord, S., van den Dool, H., Ku- mar, A., Wang, W., Long, C., Chelliah, M., Xue, Y., Pardee, J. T.: The Glacial Lake Missoula, J. Geol., 18, Huang, B., Schemm, J.-K., Ebisuzaki, W., Lin, R., Xie, 376–386, doi:10.1086/621747, 1910. P., Chen, M., Zhou, S., Higgins, W., Zou, C.-Z., Liu, Paterson, W. S. B. and Budd, W. F.: Flow parameters for Q., Chen, Y., Han, Y., Cucurull, L., Reynolds, R. W., ice sheet modeling, Cold Reg. Sci. Technol., 6, 175– Rutledge, G., and Goldberg, M.: The NCEP climate 177, 1982. forecast system reanalysis, B. Am. Meteorol. Soc., 91, 1015–1057, doi:10.1175/2010BAMS3001.1, 2010. Patterson, T. and Kelso, N. V.: Natural Earth. Free vector and raster map data, http://naturalearthdata.com, 2014. Seguinot, J.: Spatial and seasonal effects of temperature variability in a positive degree-day glacier sur- Penck, A. and Brückner, E.: Die alpen im Eiszeitalter, face mass-balance model, J. Glaciol., 59, 1202–1204, Tauchnitz, Leipzig, 1909. doi:10.3189/2013JoG13J081, 2013. Petit, J. R., Jouzel, J., Raynaud, D., Barkov, N. I., Barnola, J.-M., Basile, I., Bender, M., Chappellaz, J., Seguinot, J. and Rogozhina, I.: Daily temperature Davis, M., Delaygue, G., Delmotte, M., Kotlyakov, variability predetermined by thermal conditions V. M., Legrand, M., Lipenkov, V. Y., Lorius, C., Pépin, over ice-sheet surfaces, J. Glaciol., 60, 603–605, L., Ritz, C., Saltzman, E., and Stievenard, M.: Cli- doi:10.3189/2014jog14j036, 2014. mate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature, 399, 429– Seguinot, J., Khroulev, C., Rogozhina, I., Stroeven, A. P., 436, doi:10.1038/20859, data archived at the World and Zhang, Q.: The effect of climate forcing on numer- Data Center for Paleoclimatology, Boulder, Colorado, ical simulations of the Cordilleran ice sheet at the Last USA., 1999. Glacial Maximum, The Cryosphere, 8, 1087–1103, doi:10.5194/tc-8-1087-2014, 2014. Porter, S. C. and Swanson, T. W.: Radiocarbon age constraints on rates of advance and retreat of the Seguinot, J., Rogozhina, I., Stroeven, A. P., Margold, Puget Lobe of the Cordilleran Ice Sheet during M., and Kleman, J.: Numerical simulations of the the last glaciation, Quaternary Res., 50, 205–213, Cordilleran ice sheet through the last glacial cycle, doi:10.1006/qres.1998.2004, 1998. manuscript.

Prest, V. K., Grant, D. R., and Rampton, V. N.: Glacial Shackleton, N.: Oxygen Isotope Analyses and Pleis- map of Canada, Geol. Surv. of Can., Ottawa, ON, “A” tocene Temperatures Re-assessed, Nature, 215, 15–17, Series Map 1253A, doi:10.4095/108979, 1968. doi:10.1038/215015a0, 1967. Reeh, N.: Parameterization of melt rate and surface temperature on the Greenland ice sheet, Polarforschung, Shackleton, N. J. and Opdyke, N. D.: Oxygen isotope and 59, 113–128, 1991. palaeomagnetic stratigraphy of Equatorial Paciﬁc core V28-238: Oxygen isotope temperatures and ice vol- Robert, B. L.: Modeling the Cordilleran ice sheet, Géogr. umes on a 105 year and 106 year scale, Quaternary phys. Quatern., 45, 287–299, doi:10.7202/032876ar, Res., 3, 39–55, doi:10.1016/0033-5894(73)90052-5, 1991. 1973.

25 NUMERICAL MODELLING OF THE CORDILLERAN ICE SHEET

Shea, J. M., Moore, R. D., and Stahl, K.: Deriva- E., Arpe, K., Balmaseda, M. A., Beljaars, A. C. M., tion of melt factors from glacier mass-balance Berg, L. V. D., Bidlot, J., Bormann, N., Caires, S., records in western Canada, J. Glaciol., 55, 123–130, Chevallier, F., Dethof, A., Dragosavac, M., Fisher, M., doi:10.3189/002214309788608886, 2009. Fuentes, M., Hagemann, S., Hólm, E., Hoskins, B. J., Isaksen, L., Janssen, P. A. E. M., Jenne, R., Mcnally, Sigloch, K. and Mihalynuk, M. G.: Intra-oceanic subduc- A. P., Mahfouf, J.-F., Morcrette, J.-J., Rayner, N. A., tion shaped the assembly of Cordilleran North Amer- Saunders, R. W., Simon, P., Sterl, A., Trenberth, K. E., ica, Nature, 496, 50–56, doi:10.1038/nature12019, Untch, A., Vasiljevic, D., Viterbo, P., and Woollen, J.: 2013. The ERA-40 re-analysis, Q. J. Roy. Meteor. Soc., 131, Stroeven, A. P., Fabel, D., Codilean, A. T., Kleman, J., 2961–3012, doi:10.1256/qj.04.176, 2005. Clague, J. J., Miguens-Rodriguez, M., and Xu, S.: In- Waitt, R. B.: About Forty Last-Glacial Lake Missoula vestigating the glacial history of the northern sector of Jökulhlaups through Southern Washington, J. Geol., the Cordilleran ice sheet with cosmogenic 10Be con- 88, 653–679, doi:10.1086/628553, 1980. centrations in quartz, Quaternary Sci. Rev., 29, 3630– 3643, doi:10.1016/j.quascirev.2010.07.010, 2010. Ward, B. C., Bond, J. D., and Gosse, J. C.: Evi- dence for a 55–50 ka (early Wisconsin) glacia- Stroeven, A. P., Fabel, D., Margold, M., Clague, tion of the Cordilleran ice sheet, Yukon Ter- J. J., and Xu, S.: Investigating absolute chronolo- ritory, Canada, Quaternary Res., 68, 141–150, gies of glacial advances in the NW sector of the doi:10.1016/j.yqres.2007.04.002, 2007. Cordilleran ice sheet with terrestrial in situ cosmogenic nuclides, Quaternary Sci. Rev., 92, 429–443, Ward, B. C., Bond, J. D., Froese, D., and Jensen, B.: Old doi:10.1016/j.quascirev.2013.09.026, 2014. Crow tephra (140 10 ka) constrains penultimate Reid ± Tarasov, L. and Peltier, W. R.: A geophysically glaciation in central Yukon Territory, Quaternary Res., constrained large ensemble analysis of the 27, 1909–1915, doi:10.1016/j.quascirev.2008.07.012, deglacial history of the North American ice-sheet 2008. complex, Quaternary Sci. Rev., 23, 359–388, Weertman, J.: On the sliding of glaciers, J. Glaciol., 3, doi:10.1016/j.quascirev.2003.08.004, 2004. 33–38, 1957. the PISM authors: PISM, a Parallel Ice Sheet Model, Weis, M., Greve, R., and Hutter, K.: Theory of shal- http://www.pism-docs.org, 2014. low ice shelves, Continuum Mech. Therm., 11, 15–50, Thorson, R. M.: Ice-Sheet Glaciation of the Puget doi:10.1007/s001610050102, 1999. lowland, Washington, during the Vashon Stade Windham, W. and Martel, P.: An account of the glacières (late pleistocene), Quaternary Res., 13, 303–321, or ice alps in Savoy, in two letters, one from an En- doi:10.1016/0033-5894(80)90059-9, 1980. glish gentleman to his friend at Geneva; the other from Tulaczyk, S., Kamb, W. B., and Engelhardt, H. F.: Peter Martel, engineer, to the said English gentleman, Basal mechanics of Ice Stream B, west Antarc- as laid before the Royal Society, London, printed for tica: 1. Till mechanics, J. Geophys. Res., 105, 463, Peter Martel, 1744. doi:10.1029/1999jb900329, 2000. Winkelmann, R., Martin, M. A., Haseloff, M., Albrecht, Uppala, S. M., Kållberg, P. W., Simmons, A. J., Andrae, T., Bueler, E., Khroulev, C., and Levermann, A.: The U., Bechtold, V. D. C., Fiorino, M., Gibson, J. K., Potsdam Parallel Ice Sheet Model (PISM-PIK) – Part Haseler, J., Hernandez, A., Kelly, G. A., Li, X., Onogi, 1: Model description, The Cryosphere, 5, 715–726, K., Saarinen, S., Sokka, N., Allan, R. P., Andersson, doi:10.5194/tc-5-715-2011, 2011.

26 Acknowledgements

This work would not have been possible without the help and involvement of many friends and colleagues. I am grateful to Arjen Stroeven, my main supervisor, for his continuous support through good and bad times. I most particularly thank him for his scrutinized work and tough criticism on all my texts, which is hopefully reflected in the quality of this thesis. I also would like to thank Irina Rogozhina for her fearless jump into my supervising team last year and constant support since then. She has been a great mentor, a good friend and a fantastic colleague, and I hope that we will continue to work together. I thank Qiong Zhang for her positiveness and encouragement, technical help with meteorological data and useful advice in my early writing steps, and Johan Kleman for few but always inspiring discussions. I also wish to particularly thank Jakob Heyman and Martin Margold, who have been great tutors on glacial geomorphology and many other things during the first years of my PhD. Likewise, I am immensely grateful to Ed Bueler and Constantine Khroulev for dedicated support on PISM during the last three years and for precious advice during my short visit at the University of Alaska Fairbanks. To my office mates; Carl Lilja, Markus Nathansson, Ping Fu and Natacha Gribenski; thank you for cultivating a friendly working atmosphere. I also wish to thank colleagues from the glaciology group for many entertaining paper discussions, empowered by the incorruptible commitment and delicious cookies of Patrick Applegate. Thanks to the administrative staff, and most particularly Susanna Blåndman, Carina Henricksson and Linus Richert, for having been constantly available for help, as well as to Helle Skånes, for guiding me through the last steps towards my doctoral defense. To colleagues from the German Research Centre for Geosciences, and especially to Maik Thomas, Jorge Bernales and Tonio; thank you for supporting my visit and welcoming me in Potsdam. To Ola Fredin and Lena Rubensdotter; thank you for introducing me to glacial geology during my stay in Trondheim in 2008, for encouraging me to pursue this path and for unforgettable fieldwork memories. To my friends from Stockholm and Potsdam; Max, Nathalie, Juri, Lisa, Reto, Peng, Ying, Nikki, Thomas, Viola; thank you for making these years enjoyable. Finally, I thank my family, and most specially my parents Lydie and Christophe, for having given me the chance to pursue a higher education, and for all those summers spent mountain-hiking in the Alps. Thủy, your determination to push me to the finish line had no limits. Thank you for providing this amazing support, and for sharing my frustrations and my joys along the way.

Funding sources My salary was provided by Stockholm University, and research grants from the Swedish Research Council (VR) and the German Academic Exchange Service (DAAD). Additional funds for conference participations, course attendances, collaborative travels and ﬁeldwork were provided by the Carl Mannerfelt Fund, the Bolin Centre for Climate Research, the Ahlmanns and Lagrelius Funds for Geographic Research, the De Geer Foundation for Quaternary Research and the Wallenberg Foundation.