Implications of flow law uncertainty for flow-driven ice-loss inGreenland under idealized warming pathwaysMaria Zeitz1,2,*, Ricarda Winkelmann1,2,*, and Anders Levermann1,2,3

1Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, 14412Potsdam, Germany2University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany3LDEO, Columbia University, New York, USA

Correspondence: Ricarda Winkelmann (ricarda.winkelmann@pik-potsdam.de)

Abstract. The mass balance of ice sheets is determined by the interplay of surface processes, basal melting and the ice flow

which in numerical modelling and glaciological theory is described by a flow law. Here, we take a deeper look at Glen’s flow

law, used in most continental ice-sheet models with mainly one set of flow parameters. On the basis a review of more than 50

studies based on laboratory and field experiments dating back from 1952 to present, we show that the literature is consistent

with a wide range of values for the activation energy and flow law exponent. So far these uncertainties are not reflected in5

model simulations performed to estimate future sea-level rise from the Greenland and Antarctic Ice Sheet.We illustrate the

importance of including the uncertainty of the flow parameters in exemplary simulations of the Greenland Ice Sheet with the

Parallel Ice Sheet Model. In idealized warming simulations of the Greenland Ice Sheet, the uncertainty in flow-law parameters

alone can lead to a threefold increase in flow-driven ice loss within this century. An improved understanding of the flow law is

therefore important for simulations of the ice-sheet evolution and estimates of future sea-level changes.10

1 Introduction

Understanding large-scale ice-dynamics is crucial for providing accurate sea-level rise projections which are necessary to pre-

pare for a future impacted by climate change (Goelzer et al., 2020; Seroussi et al., 2020). Due to various community efforts,

numerical ice-sheet models have improved greatly over the last years, allowing us to estimate future sea-level rise and the

associated uncertainties (Pattyn and Morlighem, 2020). Assessing the uncertainties associated with sea-level projections is15

central for adaption efforts (Pörtner et al., 2019). These are driven on the one hand by uncertainties in the boundary conditions,

e.g. climatological forcing (Barthel et al., 2020), and on the other hand by uncertainties in the models, through model param-

eters, which are difficult to constrain or through missing processes. The uncertainties posed by the stress-strain relation of the

ice-sheets and shelves, although central to numerical models, is usually not considered in future sea-level projections.

Recent isotope measurements suggest that Greenland was nearly ice free for extended periods within the past 1 280 00020

years (Schaefer et al., 2016). This implies that the Greenland Ice Sheet must have lost mass at a rate which is much faster than

we would expect with our current physical understanding of ice sheet dynamics: even considering self-amplifying feedbacks

(Levermann and Winkelmann, 2016), increased surface melting (Box, 2013; Shepherd et al., 2012) alone cannot explain these

1

faster rates of ice loss. Since the largest part of the ice volume is not in contact with ocean water, a near complete elimination

of the ice sheet can also not be explained by mere sliding of ice into the ocean, which is currently considered to be the fastest25

process for ice loss both in Greenland (Robinson et al., 2012; Pfeffer et al., 2008) and Antarctica (Pritchard et al., 2012). As

a consequence, it must be the flow of ice that is much more sensitive to temperature changes than so far assumed in ice-sheet

models and thus sea-level projections.

Most ice-sheet models rely on Glen’s flow law (Glen, 1958) to describe the flow of ice (Budd and Jacka, 1989; the PISM au-

thors, 2018; Winkelmann et al., 2011; Pattyn, 2017; Larour et al., 2012). It relates the deformation rate to stress30

ε=Aτn, (1)

where ε is the deformation rate, τ the driving stress, A the softness of the ice, and n the flow exponent. The softness is usually

described as a function of the pressure adjusted temperature T′

A(T ′) =A0 exp

(− Q

RT ′

), (2)

with the pre-exponential factor A0, the activation energy Q, and the gas constant R. Due to pre-melt processes the temperature35

sensitivity of the ice softness increases closer to the melting point. This is typically reflected by a larger activation energy

Qw for temperature above a threshold, and a lower activation energy Qc for temperatures below that threshold. The threshold

temperature is often considered to be at −10C (Cuffey and Paterson, 2010; Greve and Blatter, 2009). Ice-sheet simulations

show a strong response to changes in the critical temperature, at which the activation energy changes from warm to cold

conditions (Bueler et al., 2007). However, as of yet there is no rigorous consensus for the exact value of the temperature40

threshold.

This flow law assumes an isotropic medium although the ice present in glaciers and ice-sheets is non-isotropic. This is

often compensated by multiplying an enhancement factor E to the softness. In many large-scale ice-sheet models the flow

parameters, as E, n or Q are assumed to be constant in space and time. The flow enhancement factors can be adjusted locally

in order to capture ice anisotropies in a more explicit way, e.g. by using a ratio of 5-10 for ESIA/ESSA (Ma et al., 2010).45

While Glen’s flow law, as described in the equations above, is still the most widely used deformation law for ice-sheet

modeling, improvements e.g. other mathematical forms, have been discussed. The Goldsby-Kohlstedt law, for example, incor-

porates the two main processes of ice deformation, grain-boundary sliding and dislocation creep. It assumes a superposition of

two power laws, with a low flow exponent at low stresses and a high exponent at high stresses (Goldsby and Kohlstedt, 2001).

The processes at high temperatures, in particular in terms of dislocation dynamics, are captured in (Cole and Durell, 2001;50

Cole, 2020), a composite power law which explicitly includes the dislocation density as a function of stress and temperature.

Until now those approaches are rarely used in large-scale ice-sheet models (the PISM authors, 2018; Goelzer et al., 2020;

Seroussi et al., 2020) with only some exceptions (Pettit and Waddington, 2003; Pettit et al., 2011; Quiquet et al., 2018). A

composite flow-law has been used recently to describe the deformations observed at the NEEM ice core (Kuiper et al., 2020a,

b). However, it remains a challenge to bridge the gap between the processes on the micro scale, which influence the flow of55

ice, e.g. grain size and structure, and the large-scale modeling of continental ice-sheets over centuries to millennia (Montagnat

et al., 2014).

2

Within the scope of this study, we will focus on the uncertainties associated to the original form of Glen’s flow law as shown

in the equation (1), in particular on uncertainties of the flow exponent n, which governs the non-linear response to stress, and

the activation energies Qc and Qw, which determine the temperature sensitivity of the ice deformation.60

In this manuscript we first compile a synthesis of published flow parameters, then discuss the resulting uncertainty (in

section 2) and finally present the implications for modeling the Greenland Ice Sheet with the Parallel Ice Sheet Model PISM

(the PISM authors, 2018) (in section 4).

2 Review of measurements of flow parameters

Ice-sheets and glaciers slowly deform due to the long-term exposure to high levels of stress from their own ice load (Cuffey65

and Paterson, 2010). In order to measure the relation describing this ice creep, different types of laboratory experiments and

field measurements have been conducted. Generally, these can be divided into experiments measuring the strain (relative

deformation) under constant stress (Fig. A1a), or measuring stress under a constant strain rate (Fig. A1b). Ice is a complex

fluid, its deformation response to a given stress is also dependent on the deformation which has already occurred. This yields

the distinction into primary creep, during which the relation between stress and strain-rate is linear, secondary creep, during70

which the maximum stress for a given strain rate is reached, and tertiary creep, during which recrystallization reduces the stress

needed to maintain the constant strain rate. The higher the applied stresses and the higher the temperature, the faster the ice

deforms and secondary and tertiary creep are reached.

Since the 1950s, laboratory experiments have been conducted in order to establish a flow law and estimate the flow param-

eters for glaciers and ice-sheets. Although the laboratory provides a highly controlled environment for these experiments, it is75

difficult to reproduce realistic conditions, both concerning the composition of the ice probes, as well as the flow regime: the

results from lab experiments have mostly been taken from the secondary creep regime, while the ice in glaciers and ice sheets

typically experiences large strains and is therefore in the steady-state tertiary creep regime (Cuffey and Paterson, 2010; Greve

and Blatter, 2009). Measuring the flow parameters in field experiments comes with the advantage, that the ice itself is already

in the tertiary flow regime. However, it is difficult to control all variables, e.g. the temperature of the ice or the stress, which is80

due to the weight of the ice itself. Numerical Full-Stokes models and high quality data for the topography of the ice and of the

bedrock can constrain the stresses on the ice, but have not been historically available.

In order to assess the uncertainty with respect to flow parameters we focus on publications, which assume a power-law

(instead of a composite flow law) and either measure the parameters directly or test if observations of ice sheets and glaciers

are consistent with a given set of flow parameters. A total of 63 values for n and 49 values forQwere reported. A synthesis of all85

laboratory and field measurements shows a wide range of flow exponent n and activation energies Qw and Qc for temperatures

above or below -10C (Figure 1). Reported values of the flow exponent n range between 1 (often associated with small stress)

and 4.5. Values forQ ranges between 40-193 kJ/mol. Under cold conditions (T <−10C)Qc is between 42-84kJ/mol. Under

warm conditions Qw (warm conditions) ranges between 45 - 193kJ/mol. Field measurements suggest that n ranges between

1-4.5 and Q ranges from 40-83 kJ/mol.90

3

1 2 3 4Flow exponent n

60

120

180

Act

ivat

ion

ener

gyQ

(kJ/

mol

) Lab cold

Lab warm

Field

Tertiary creep

Simulations

Figure 1. Synthesis measured flow parameters n and Q, from laboratory and field experiments since 1952. The values were measured

in laboratory (blue circles for cold (T <−10C) and red squares for warm (T>-10C) conditions, purple lines for tertiary creep) and field

(green triangles) experiments. The color intensity indicates the publication date of each data point: the more intense the color, the more recent

the publication. For studies which measured only one of the flow parameters, either n or Q, the respective values are indicated by horizontal

and vertical lines. The flow parameters typically used in numerical ice-sheet simulations are n=3 and Qw =139 kJ/mol and Qc=60 kJ/mol

(black crosses). (lab cold: (Jellinek and Brill, 1956; Butkovich and Landauer, 1960; Mellor and Smith, 1967; Weertman, 1968; Mellor and

Testa, 1969; Muguruma, 1969; Barnes et al., 1971; Weertman, 1973; Steinemann, 1958; Bowden and Tabor, 1964; Mellor, 1959; Paterson,

1977; Fletcher, 1970; Goldsby and Kohlstedt, 1997; Song et al., 2006; Craw et al., 2018; Saruya et al., 2019) lab warm: (Glen, 1952, 1953,

1955; Jellinek and Brill, 1956; Butkovich and J.k, 1957; Raraty and Tabor, 1958; Glen, 1958; Butkovich and Landauer, 1960; Mellor and

Smith, 1967; Bromer and Kingery, 1968; Barnes et al., 1971; Steinemann, 1958; Mellor and Testa, 1969; Duval, 1974, 1976; Song et al.,

2006; Qi et al., 2017; Treverrow et al., 2012; Duval and Gac, 1982) field: (Nye, 1953, 1957; Gow, 1963; Paterson, 1962; Paterson and Savage,

1963; Holdsworth and Bull, 1970; Hansen and Landauer, 1958; Thomas, 1973; Raymond, 1973; Hobbs, 1974; Paterson, 1977; Martin and

Sanderson, 1980; Doake and Wolff, 1985; Blinov and Dmitriev, 1987; Naruse et al., 1988; Talalay and Hooke, 2007; Cuffey and Paterson,

2010; Bons et al., 2018) tertiary: (Craw et al., 2018; Qi et al., 2017; Cyprych et al., 2016; Treverrow et al., 2012) )

4

Despite these uncertainties, ice-flow models have mostly been using a single set of parameters, fixed at n=3,Qw=139 kJ/mol

and Qc=60 kJ/mol (marked by black crosses in Fig. 1).

3 Numerical model and experimental design

3.1 Ice flow in the Parallel Ice Sheet Model PISM

Simulations in this study were performed with the Parallel Ice Sheet Model (PISM) v1.1. PISM is an open-source, three-95

dimensional, thermo-mechanically coupled ice-sheet-shelf model. It solves the equations of the shallow-ice approximation

(SIA) and shallow-shelf approximation (SSA) in parallel. The SIA is typically dominant in regions with high bottom fric-

tion, where the vertical shear stresses dominate over horizontal shear stresses and longitudinal stresses. The shallow shelf

approximation is typically dominant for ice shelves, with zero traction at the base of the ice, and for fast-flowing ice streams

(Winkelmann et al., 2011). Basal sliding is parameterized as a pseudo-plastic power law:100

τb =−τcu

uqthres|u|1−q, (3)

where τb is the basal shear stress, u is the basal sliding velocity, τc is the yield stress and uthres a threshold velocity. Typically

uthres is chosen to be 100m/yr.

In this study, the Patterson-Budd flow law with the same flow parameters Qc,Qw and n is used for both, SSA and SIA

flow. While the flow parameters are always global parameters in PISM, a local distinction could be achieved by using different105

parameters for SSA and SIA flow. The flow parameters Qc,Qw and n are understood as independent parameters while the

factors A0 are determined as detailed in (Zeitz et al., 2020).

3.2 Experimental design

3.2.1 Diagnostic simulations for present-day Greenland.

The initial state for diagnostic PISM simulations of present day Greenland Ice Sheet was created with a paleo-spin up with110

grid sequencing and flux correction during the last 1000 years (Aschwanden et al., 2016). For the flux correction the bed and

ice thickness topography was taken from BedMachine v3 remapped on a 1.8km grid (Morlighem et al., 2017).

All diagnostic simulations were started from the same initial state (with parameters n= 3, Qw = 139kJ/mol and Qc =

60kJ/mol). The parameter perturbation in n and Q is chosen in order to disentangle possible feedbacks between flow param-

eters the state of the ice sheet (e.g. temperature distribution, subglacial conditions). The stress regime combines SIA and SSA115

flow unless indicated otherwise. An ensemble of all possible combinations ofQw ∈ [120,130,200]kJ/mol,Qc ∈ [42,60,85]kJ/mol

and n ∈ [2,3,4] was computed and analysed with respect to the surface velocities for each of the ensemble members and the

area of ice streams, which is the area where the ice velocities are larger or equal to 100 m/yr. The velocities are only averaged

over areas, where the thickness of the ice is larger than 10 m.

5

3.2.2 Transient simulations with idealized warming120

Transient simulation with PISM start from the final state of the diagnostic simulations in section 3.2.1, remapped to 4.5 km

resolution. Here, only three combinations of activation energies are tested for illustrative purposes: The standard values (Qw =

139 kJ/mol, Qc = 60 kJ/mol), a "high" combination, where both activation energies are at the high end of the possible range,

(Qw = 200 kJ/mol,Qc = 85 kJ/mol) and a "low" combination with both activation energies at the low end of the possible range

(Qw = 139 kJ/mol, Qc = 60 kJ/mol).125

The spatial distribution of the surface mass balance and the air temperature to force PISM is provided by RACMO2.3p2.

(Noël et al., 2018) We perform a monthly average over the period 1958-1967 in order to keep the seasonal cycle. The climatic

mass balance and the air temperature fields are periodic in time with a period of one year. The temperature anomaly is derived

from one GCM in the CMIP5 project (Figure 2). IPSL-CM5A-LR (Dufresne et al., 2013) was chosen because the simulations

extends to 2300. Monthly near-surface temperatures were extracted and averaged over the Greenland domain (60°N to 85°N,130

1600km wide) and the local scalar temperature anomaly was obtained by subtracting the 1958-1967 average seasonal cycle.

After 2300 the temperature anomaly is kept constant. In the transient simulations, the temperature anomaly is applied to the

ice-surface temperature and the climatic mass balance is kept constant in order to disentangle flow related ice losses from melt-

related ice losses and feedback - no increased melting occurs. The mass loss occurs only due to changes in ice flow caused by

the warmer ice.

2000 2100 2200 2300

Year

0

5

10

15

20

Tem

per

atu

rean

omal

y(K

)

monthy anomaly

yearly mean

Figure 2. Temperature anomaly over the Greenland Ice Sheet as derived from CMIP5 run IPSL-CM5A- LR (Dufresne et al., 2013).

Grey: monthly anomalies with respect to the climatological mean. Those anomalies are applied in the simulations. Black: yearly anomalies

with respect to the climatological mean for illustration.

135

6

4 Impact of uncertainties in flow parameters on the Greenland Ice Sheet

4.1 Impact on simulated present-day velocities

Fast flowing icewith Qc, Qw (kJ/mol)

42, 120

85, 200

100

101

102

103

Surfac

eve

loci

ty(m

/yr)

a b

42,120

60,120

85,120

42,139

60,139

85,139

42,200

60,200

85,200

Activation Energies Qc, Qw (kJ/mol)

0

25

50

75

100

125

Rel

ativ

ech

ange

(%)

Sta

ndar

d

Average velocity

Ice stream area

Figure 3. Illustration of the influence of activation energy on modeled surface velocities for present-day Greenland. a) Color shading

indicates fast-flowing ice (surface velocities larger than 100 m/yr), as simulated with PISM. These results were derived for present-day

climatic boundary conditions and topography. Blue shading: Qc = 42 kJ/mol and Qw = 120 kJ/mol. Red shading: Qc = 85 kJ/mol and Qw =

200 kJ/mol. For comparison, observed surface velocities are given by grey-scales (Rignot and Mouginot, 2012). b) Relative change in surface

velocities of the Greenland Ice Sheet due to different flow parameters, as simulated with PISM. Solid bars show the relative change of the

area with fast-flowing ice (here defined as surface velocities larger than 100 m/yr). The hatched bars show the corresponding change in

surface velocity averaged over the whole ice sheet.

In order to estimate the sensitivity of flow to the spread of flow law parameters and the magnitude of resulting uncertainties

of ice velocities exemplary simulations of the Greenland Ice Sheet with the Parallel Ice Sheet model (PISM) were used. A

topographic set up for Greenland was initialized with the standard set of parameters and a paleo-spin-up (see Section 3.2.1).140

Afterwards the response of the velocity field to a change in the flow parametersQc andQw and nwas diagnosed, while keeping

all other parameters fixed. The sensitivity of the ice dynamics to uncertainties in those parameters can be seen as a relative

change in the area with fast-flowing ice (here defined as areas with an ice surface velocity of more than 100 m/yr) (S/S0− 1)

and as a relative change in surface velocity averaged over the whole ice sheet (v/v0−1), both compared to standard conditions

(Figure 3).145

7

The area with fast-flowing ice S at the upper range of Qc and Qw is 2.7 times larger than at the lower range of Qc and

Qw (Figure 3a). Compared to the standard values S by 133% for the parameter choice with the highest Qc and Qw. For the

parameter choice with the highest Qc and Qw the surface velocity averaged over the whole ice sheet increases by more than

64% compared to the standard value. Low values of Qw reduce surface velocities while high values of Qw increase surface

velocities. The velocities are less sensitive to changes in Qc (Figure 3b).150

While changes in the flow exponent n do not change the effect of the activation energies qualitatively, a smaller flow exponent

of n= 2 reduces the surface velocities compared to the standard case and a higher flow exponent of n= 4 increases surface

velocities (Fig. 4). With n= 2 the relative difference in diagnosed velocities is up to -27% for the lowest activation energies and

up to +45% for the highest activation energies compared to the standard parameter set. Compared to diagnostic simulations

with the same combination of Qw and Qc and n= 3, the relative difference in diagnostic velocities is less pronounced for155

n= 2. With n= 4 the upper end of the relative difference in diagnosed velocities for the highest activation energies increases

to 92% (and only -1% for the lowest activation energies). At the lower end of activation energy parameters, n= 4 leads to

positive relative difference in surface velocities, in contrast to n= 2 and n= 3. Overall n= 2 decreases surface velocities

while n= 4 increases surface velocities.

At high bottom friction and dominant vertical shear stresses in the shallow ice approximation only regime, the effect ofQ on160

the ice dynamics is amplified. In this regime sliding is inhibited and the ice flows due to internal deformations only. The area

with fast flowing ice then increases more than 25 times due to variations in activation energies alone over the whole range of

allowed parameters (Fig. 5). The area with fast-flowing ice increases by +1130% for the parameter choice with the highest Qc

and Qw compared to the standard values. For the parameter choice with the highest Qc and Qw the surface velocity averaged

over the whole ice sheet increases by +265%.165

High activation energies consistently lead to high ice velocities, in particular the activation energy for warm ice Qw is

important. In a large ice sheet the warm ice is at the base where the internal deformation is strongest. This is where especially

the Qw plays a crucial role for ice flow. That is to say increasing Qw increases the softness A close to the pressure melting

point while Qc has much less impact on A.

4.2 Impact on projections of flow-driven ice-loss due to warming170

In order to disentangle the flow-driven part of sea-level rise from other drivers, e.g. changes in surface melting, the simulations

in PISM were performed with a constant surface mass balance, which corresponds to the monthly averages over the period

1958-1967 (Noël et al., 2018). The warming signal, a temperature anomaly which corresponds to the RCP 8.5 scenario from

the CMIP5 project (Dufresne et al., 2013), applies to the ice surface temperature and diffuses into the ice sheet thus making

the ice softer, without changes in the climatic mass balance.175

The subsequently observed mass loss is driven by changes in ice flow only, and is depicted in Figure 6 in units of global

sea-level rise. High parameters for the activation energies (Qw = 200 kJ/mol andQc = 85 kJ/mol) lead to increased mass losses

compared to standard parameters (Qw = 139 kJ/mol and Qc = 60 kJ/mol) and even more compared to low activation energies

(Qw = 120 kJ/mol and Qc = 42 kJ/mol): The higher activation energies yield an additional mass loss of up to 250% in the year

8

42,120

60,120

85,120

42,139

60,139

85,139

42,200

60,200

85,200

Activation energies Qc Qw (kJ/mol)

−20

0

20

40

60

80

Rel

ativ

ech

ange

inav

erag

eve

loci

ty(%

)

Standard

2

3

4

2

3

4

Flow exponent n

Figure 4. The effect of the flow exponent n on averaged velocities of the Greenland Ice Sheet as diagnosed with PISM under present day

condition for n= 2 (hatched bars), n= 3 (solid bars), and n= 4 (crosshatched bars).

2100, which is a more than a threefold increase compared to standard parameters. The lower activation energies reduce mass180

loss by approximately 50%, i.e. the mass loss is halved compared to standard parameters. Overall, comparing high and low

activation energies, this results in a difference by a factor of six for the mass loss in the year 2100. While the relative difference

in mass loss decreases over time as the warming increases, the absolute difference continues to rise.

Comparing the thickness to the control simulation (without temperature anomaly) shows that for high Q the main ice losses

are in the central part of the ice-sheet (see Figure 7) By increasing the softness of the ice, high activation energies lead to higher185

velocities, higher basal melt rates and higher discharge rates than low activation energies under otherwise equal conditions.

5 Discussion and conclusion

Here, we compile an overview over the parameter uncertainties associated with Glen’s flow law, a law widely used to describe

the deformation of ice in numerical ice-sheet models. Although the parameters are assumed to be well constrained by many

ice-sheet modelers, we found that the flow exponent n might vary between 1 and 4.5, and the activation energies Q might190

vary between 42 and 84 kJ/mol for cold ice and between 44 and 193 kJ/mol for warm ice. Despite these uncertainties, many

ice-sheet simulations neglect this uncertainty by using a set of standardised parameters instead. Therefore we argue that it

would be worthwhile for future ice-sheet simulations to incorporate the vast uncertainties from the flow-law parameters.

The overview in Figure 1 compiles values for flow parameters which have been reported under diverse conditions, such as

cold and warm conditions, measurements in the lab or in the field, and under tertiary creep. While these values are not directly195

comparable, we explore the impact of the upper and lower limits on flow driven ice losses in ice-sheet modeling.

9

Qc, Qw (kJ/mol)

42, 120

85, 200

42, 12

060, 12

085, 12

042, 13

960, 13

985, 13

942, 20

060, 20

085, 20

0

Activation energies Qc, Qw (kJ/mol)

0

200

400

600

800

1000

Rel

ativ

ech

ange

inar

eave

loci

ty(%

)

Standard

Ice stream area

Average velocity

a b

100

101

102

103

Surfacevelocity

(m/yr)

Figure 5. Influence of activation energy on modeled surface velocities for present-day Greenland under the assumption of high basal

drag and SIA flow only. a) The area with fast-flowing ice (surface velocities larger than 100 m/a), as simulated with PISM in the shallow ice

approximation without sliding for present-day climatic boundary conditions.Blue shading: Qc=42 kJ/mol and Qw=120 kJ/mol. Red outline:

Qc=85 kJ/mol and Qw=200 kJ/mol. For comparison, observed surface velocities are given by grey-shading (Rignot and Mouginot, 2012). b)

Relative change in surface velocities (hatched bars) and the area of fast-flowing ice (solid bars) of the Greenland Ice Sheet due to different

flow parameters, as simulated with PISM with shallow ice approximation only. Here the flow exponent is fixed to n= 3.

In order to explore the impacts of those uncertainties on the results of large-scale modeling we perform simulations of

the Greenland Ice Sheet with the Parallel Ice Sheet Model PISM with perturbed parameters. In these simulations the chosen

flow parameters are supposed to cover the full range consistent with literature values. For the flow exponent n, the lowest

values of 1 were discarded. There are clear indications that the ice flow in ice-sheets and glaciers is not Newtonian globally,200

although a linear flow plays a role locally, at low stresses (Pettit et al., 2011). A lower bound of n= 2 was chosen instead. For

Qw, 120 kJ/mol was chosen as the lower bound, although values of down to 44 kJ/mol were reported in literature, in order to

maintain the difference between warm and cold ice. The temperature threshold, which distinguishes warm ice from cold ice

was not varied. However, these upper and lower bounds are within two standard deviations from the mean for each of the flow

parameters.205

A discussion on how these uncertainties in flow parameters might affect our understanding of ice dynamics with respect to

shear-heating, grounding line flux or basal sliding is summarised in (Zeitz et al., 2020), together with a first exploration of the

impacts on an idealized flow-line ice sheet.

10

2000 2200 2400

0

5

10

15

20

25

Flo

w-d

rive

nse

ale

velrise

(cm

)

Activationenergies Q

low

std

high

2000 2100 2200

0

2

4

6

Flo

w-d

rive

nse

ale

velrise

(cm

)

2100 2200 2500

Year

0

100

200

Rel

ativ

ediff

eren

ce(%

)

c

b

a

Figure 6. Illustration of the influence of activation energy on flow-driven mass losses in Greenland. (a) and (b): Flow driven ice loss in

sea-level equivalent as simulated with PISM with a constant climatic mass balance and an increase in ice-surface temperature corresponding

to RCP8.5. For all simulations, surface warming leads to an acceleration in flow and thereby ice loss beginning in the first half of the

21st century. Red lines: Qw = 200 kJ/mol and Qc = 85 kJ/mol, orange lines: Qw = 139 kJ/mol and Qc = 60 kJ/mol (standard parameters),

blue lines: Qw = 120 kJ/mol and Qc = 42 kJ/mol. (c): Relative difference in flow-driven sea-level change for the low (blue) and high (red)

parameter choice for the activation energies Q compared to standard parameters for the years 2100, 2200, 2500.

Similarly to the simulations performed in an idealized flow-line setup, we find that high activation energies Q, which are

associated with softer ice, lead to increases in flow driven ice-loss in simulations of the Greenland Ice Sheet. For simulated210

velocities of the Greenland Ice Sheet diagnosed under present-day conditions the areas with fast flowing ice increase by more

than two-fold and the average velocity increases by 60% in simulations with high activation energies compared to those with

11

2000

low Q standard Q high Q

2100

2500

−200 −150 −100 −50 0 50 100 150 200

Thickness anomaly (m)

Figure 7. Illustration of the influence of activation energy on the thickness anomaly The simulations with low activation energies

(Qw = 120 kJ/mol and Qc = 42 kJ/mol) on the left, standard activation energies (Qw = 139 kJ/mol and Qc = 60 kJ/mol) in the middle and

high activation energies (Qw = 200 kJ/mol and Qc = 85 kJ/mol) on the right. The control simulation without warming serves as a reference.

Positive values (red) correspond to a higher thickness in the simulation and negative values (blue) correspond to a lower thickness in the

simulation.

12

standard values. Those changes lead to increased discharge and increased melt at the base of the ice. In transient simulations

with idealized warming conditions, where the temperature anomaly acts only on the ice without changing the melt rates at the

surface, we observe a more than threefold increase in flow-driven ice losses for high activation energies compared to standard215

values.

This manuscript provides a thorough investigation of the flow law parameters, but does not provide a sea-level projection. 1)

the temperature forcing is applied in an idealized way while the climatic mass balance remains unchanged and 2) we vary only

one set of parameters,Q and n, without calibrating other parameters which influence the velocities of the ice. Both idealizations

serve to disentangle flow related ice-losses from other contributions like increased melt or sliding. However, increased melt220

rates are a major contribution to realistic sea-level rise projections (see e.g. Goelzer et al. (2020)). The second idealization

allows for initial states which are different from observations, which can be seen e.g. the difference to observed velocities

(Figure 3). Tuning the parameters associated to other processes, e.g. the sliding at the bed of the ice sheet, would mitigate

the effect of the variation in flow parameters. For example Bons et al. (2018) have shown that a high flow exponent of n= 4

implies greater areas of frozen basal conditions than the standard value n= 3.225

In this study we assume globally constant flow parameters, which do not vary through space and time. While so far there

is no understanding, how the activation energies might vary (apart from the distinction between cold and warm ice, which

is included here), there is indication that the flow exponent might be related to the stress, as e.g. proposed by Goldsby and

Kohlstedt (2001). In this manuscript, however, we chose to focus on the classical formulation of the flow law, as shown in

Equation (1), since it is the most widely used formulation for large-scale ice-sheet modeling. The flow parameters are only230

one part of the uncertainties associated with the deformation of ice, among other poorly constrained processes like anisotropy,

impurities or water content within the ice.

In addition, uncertainties at the boundaries of the ice play an important role for sea-level rise projections, in particular

processes which determine the climatic mass balance, the interaction with the ocean and the processes at the base of the ice

sheet, like geothermal heat fluxes or bed conditions which allow for sliding. A lot of progress is continuously made in order to235

constrain those uncertainties.

Inverse modeling can help to constrain the flow parameters by optimizing for diagnostic velocities of e.g. the Greenland

Ice Sheet (Le clec’h et al., 2019; De Rydt et al., 2015). However, in order to reduce uncertainty in flow law parameters this

approach requires accurate information about the ice-sheet and bedrock topography, the sliding conditions at the base of the

ice and the temperature distribution within the ice since all of these factors influence the ice velocities.240

The flow of ice is at the heart of our understanding of ice dynamics, it is therefore important to include the associated

uncertainties in future studies and explore how they relate to other known or unknown uncertainties. As we have shown in this

study, the flow-driven ice losses in simulations of the Greenland Ice Sheet change up to threefold due to these uncertainties.

Code availability. TEXT

13

Data availability. TEXT245

Code and data availability. TEXT

Sample availability. TEXT

Video supplement. TEXT

Appendix A: Glen’s flow law

A1 Measuring the flow exponent n250

In laboratory experiments, the flow exponent n and the activation energy Q are obtained either under constant stress or with a

constant strain rate. A measurement, typically in the secondary creep regime, is extracted from the resulting time series (Figure

A1). The flow exponent n is measured with constant temperature under a different stresses (or a different strain rate) (Figure

A2, left panel).

In the primary creep regime the strain rate is a fast changing function of strain and can not serve to establish a material law,255

thus the data point must be taken from the secondary or tertiary creep regime. In this regime, dynamic recrystallization does

not occur yet and the flow law is independent of material properties (Cuffey and Paterson, 2010; Greve and Blatter, 2009).

Secondary creep is reached only after long times, at low stresses and low temperatures the time to reach secondary creep can

take up to 3 years (Jacka, 1984). Measurements conducted under primary instead of secondary creep tend to systematically

reduce the apparent value of the flow exponent n. Ice masses in glaciers or in ice-sheets are usually highly deformed and in260

tertiary creep, thus efforts to model those glaciers and ice-sheets need experiments which measure the flow parameters in this

flow regime. Recent developments allow to achieve extremely high strains and thus tertiary creep in a laboratory setting (Qi

et al., 2017; Cyprych et al., 2016; Treverrow et al., 2012).

A2 Measuring activation energy Q

Activation energyQ is measured by stress-versus-strain experiment (Figure A1) under constant stress (or strain rate) at different265

temperatures. Each experiment is then one data point and plotting the logarithm of the strain rate against the inverse pressure

adjusted temperature in the so called Arrhenius plot allows to determine Q from the slopes below and above -10C(Figure A2

b). Due to pre-melt processes (e.g. water between ice grains and thus facilitated grain boundary sliding) close to the pressure

melting point the ice appears to soften stronger with increasing temperatures. This is typically expressed by assuming a higher

activation energy at T′

>-10C.270

14

Time

Str

ain

Str

ess

Constant strain rate

Constant stress

˙min

˙ss

τmax

τss

a

b

prim

ary

seco

ndar

y

tertiary

Figure A1. Schematic representation of commonly used stress-strain experiments. Shown are the three creep regimes, from primary

(green), to secondary (yellow) to tertiary (blue) creep. In ice-sheets and glaciers, the ice is typically in the tertiary creep regime. Two types

of experiments are performed in the laboratory, either applying a constant stress (upper panel) or a constant strain rate (lower panel). These

experiments are used to derive the flow parameters n and Q, as described in the text

Author contributions. R.W. and A.L conceived the study. M.Z., A.L. and R.W. designed the research and contributed to the analysis. M.Z.

carried out the literature review, the simulations, and the analysis. M.Z., A.L., and R.W. wrote the manuscript.

Competing interests. The authors declare no competing interests.

Acknowledgements. M.Z. and R.W. are supported by the Leibniz Association (project DOMINOES), Development of PISM is supported by

NASA grant NNX17AG65G and NSF grants PLR-1603799 and PLR-1644277. The authors gratefully acknowledge the European Regional275

Development Fund (ERDF), the German Federal Ministry of Education and Research and the Land Brandenburg for supporting this project

by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research. We thank G Hilmar

Gudmundsson, David J. Prior and Thomas Kleiner for insightful discussions.

15

nLo

g st

rain

rate

Log stress

Single experimentLinear fit

a

Q

Log

stra

in ra

te

Inverse temperature

warm cold

Single experimentLinear fit

b

Figure A2. Derivation of flow parameters n and Q from stress-vs.-strain experiments. a) The flow exponent n is derived from a linear

fit on the log-log scale of stress-vs-strain-rate pairs, measured from experiments under fixed temperature conditions and varied stress. b)

Arrhenius-plot for deriving the activation energy Q: strain-rate for a given stress vs. inverse temperature allows to determine the parameters

Qw (for T >−10C) and Qc (for T <−10C) with two linear fits on a semi-log scale.

References

Aschwanden, A., Fahnestock, M. A., and Truffer, M.: Complex Greenland outlet glacier flow captured, Nature Communications, 7, 10 524,280

https://doi.org/10.1038/ncomms10524, http://dx.doi.org/10.1038/ncomms10524, 2016.

Barnes, P., Tabor, D., and Walker, J. C. F.: The friction and creep of polycrystalline ice, Proceedings of the Royal Society A: Mathematical,

Physical and Engineering Sciences, 324, 127–155, 1971.

Barthel, A., Agosta, C., Little, C. M., Hattermann, T., Jourdain, N. C., Goelzer, H., Nowicki, S., Seroussi, H., Straneo, F., and Brace-

girdle, T. J.: CMIP5 model selection for ISMIP6 ice sheet model forcing: Greenland and Antarctica, The Cryosphere, 14, 855–879,285

https://doi.org/10.5194/tc-14-855-2020, https://tc.copernicus.org/articles/14/855/2020/, 2020.

Blinov, K. and Dmitriev, D.: Otsenka reologicheskih parametrov Ida po rezultatam mnogoletnih nabludeniy v skvazhinah na st. Vostok v

Antarktide [Rheological ice par- ameters estimations on the basis of measurements in bore-holes at Vostok Station, Antarctica], Antarctica,

26, 95–106, 1987.

Bons, P. D., Kleiner, T., Llorens, M.-G., Prior, D. J., Sachau, T., Weikusat, I., and Jansen, D.: Greenland Ice Sheet: Higher290

Nonlinearity of Ice Flow Significantly Reduces Estimated Basal Motion, Geophysical Research Letters, 45, 6542–6548,

https://doi.org/10.1029/2018GL078356, http://doi.wiley.com/10.1029/2018GL078356, 2018.

Bowden, F. P. and Tabor, D.: The friction and lubrication of solids. Vol. 2, OUP, 1964.

Box, J. E.: Greenland ice sheet mass balance reconstruction. Part II: Surface mass balance (1840-2010), Journal of Climate, 26, 6974–6989,

https://doi.org/10.1175/JCLI-D-12-00518.1, 2013.295

16

Bromer, D. J. and Kingery, W. D.: Flow of polycrystalline ice at low stresses and small strains, Journal of Applied Physics, 39, 1688–1691,

https://doi.org/10.1063/1.1656416, 1968.

Budd, W. F. and Jacka, T. H.: A review of ice rheology for ice sheet modelling, Cold Regions Science and Technology, 16, 107–144, 1989.

Bueler, E., Brown, J., and Lingle, C. S.: Exact solutions to the thermocoupled shallow ice approximation: effective tools for verification,

Journal Of Glaciology, 53, 499–516, https://doi.org/10.3189/002214307783258396, 2007.300

Butkovich, T. R. and J.k, L.: The Flow Law For Ice, http://hydrologie.org/redbooks/a047/04734.pdf, 1957.

Butkovich, T. R. and Landauer, J. K.: Creep of Ice at Low Stresses, https://doi.org/10.1038/232222b0, 1960.

Cole, D. M.: On the physical basis for the creep of ice: The high temperature regime, Journal of Glaciology, 66, 401–414,

https://doi.org/10.1017/jog.2020.15, 2020.

Cole, D. M. and Durell, G. D.: A dislocation-based analysis of strain history effects in ice, Philosophical Magazine A, 81, 1849–1872,305

https://doi.org/10.1080/01418610108216640, https://doi.org/10.1080/01418610108216640, 2001.

Craw, L., Qi, C., Prior, D. J., Goldsby, D. L., and Kim, D. D. D.: Mechanics and microstructure of deformed natural anisotropic

ice, Journal of Structural Geology, 115, 152–166, https://doi.org/10.1016/j.jsg.2018.07.014, https://linkinghub.elsevier.com/retrieve/pii/

S0191814118300646https://doi.org/10.1016/j.jsg.2018.07.014, 2018.

Cuffey, K. M. and Paterson, W. S. B.: The Physics of glaciers, Elsevier Inc., 4 edn., http://linkinghub.elsevier.com/retrieve/pii/310

0016718571900868, 2010.

Cyprych, D., Piazolo, S., Wilson, C. J., Luzin, V., and Prior, D. J.: Rheology, microstructure and crystallographic preferred orientation of

matrix containing a dispersed second phase: Insight from experimentally deformed ice, Earth and Planetary Science Letters, 449, 272–281,

https://doi.org/10.1016/j.epsl.2016.06.010, http://dx.doi.org/10.1016/j.epsl.2016.06.010, 2016.

De Rydt, J., Gudmundsson, G. H., Rott, H., and Bamber, J. L.: Modeling the instantaneous response of glaciers after the collapse of the315

Larsen B Ice Shelf, Geophysical Research Letters, 42, 5355–5363, https://doi.org/10.1002/2015GL064355, 2015.

Doake, C. S. M. and Wolff, E. W.: Flow law for ice in polar ice sheets, Nature, 314, 255–257, https://doi.org/10.1038/314255a0, http:

//www.nature.com/articles/314255a0, 1985.

Dufresne, J.-L., Foujols, M.-A., Denvil, S., Caubel, A., Marti, O., Aumont, O., Balkanski, Y., Bekki, S., Bellenger, H., Benshila, R., et al.:

Climate change projections using the IPSL-CM5 Earth System Model: from CMIP3 to CMIP5, Climate dynamics, 40, 2123–2165, 2013.320

Duval, P.: Fluage de la glace polycristalline, Rheologica Acta, 13, 562–566, https://doi.org/10.1007/BF01521756, 1974.

Duval, P.: Lois du fluage transitoire ou permanent de la glace polycristalline pour divers états de contrainte, Annals of Geophysics, 32,

335–350, 1976.

Duval, P. and Gac, H. L.: Mechanical behaviour of antarctic ice, Annals of Glaciology, 3, 92–95, 1982.

Fletcher, N.: The chemical physics of ice. Cambridge, Cambridge University Press, Camebridge, 1970.325

Glen, J. W.: Experiments on the Deformation of Ice, Journal of Glaciology, 2, 111–114, https://doi.org/10.1017/S0022143000034067, https:

//www.cambridge.org/core/product/identifier/S0022143000034067/type/journal_article, 1952.

Glen, J. W.: Rate of Flow of Polycrystalline Ice, Nature, 172, 721–722, https://doi.org/10.1038/172721a0, 1953.

Glen, J. W.: The Creep of Polycrystalline Ice, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sci-

ences, 228, 519–538, https://doi.org/10.1098/rspa.1955.0066, http://linkinghub.elsevier.com/retrieve/pii/0165232X85900527http://rspa.330

royalsocietypublishing.org/cgi/doi/10.1098/rspa.1955.0066, 1955.

Glen, J. W.: The mechanical properties of ice I. The plastic properties of ice, Advances in Physics, 7, 254–265,

https://doi.org/10.1080/00018735800101257, 1958.

17

Goelzer, H., Nowicki, S., Payne, A., Larour, E., Seroussi, H., Lipscomb, W. H., Gregory, J., Abe-Ouchi, A., Shepherd, A., Simon, E., Agosta,

C., Alexander, P., Aschwanden, A., Barthel, A., Calov, R., Chambers, C., Choi, Y., Cuzzone, J. K., Dumas, C., Edwards, T., Felikson,335

D., Fettweis, X., Golledge, N. R., Greve, R., Humbert, A., Huybrechts, P., Le Clec’H, S., Lee, V., Leguy, G., Little, C., Lowry, D. P.,

Morlighem, M., Nias, I., Quiquet, A., Rückamp, M., Schlegel, N.-J. J., Slater, D. A., Smith, R. S., Straneo, F., Tarasov, L., van de Wal,

R., and Van Den Broeke, M.: The future sea-level contribution of the Greenland ice sheet: a multi-model ensemble study of ISMIP6, The

Cryosphere, 14, 3071–3096, https://doi.org/10.5194/tc-14-3071-2020, https://tc.copernicus.org/articles/14/3071/2020/, 2020.

Goldsby, D. L. and Kohlstedt, D. L.: Grain boundary sliding in fine-grained ice I, Scripta Materialia, 37, 1399–1406,340

https://doi.org/10.1016/S1359-6462(97)00246-7, 1997.

Goldsby, D. L. and Kohlstedt, D. L.: Superplastic deformation of ice: Experimental observations, Journal of Geophysical Research, 106,

11 017–11 030, https://doi.org/10.1029/2000JB900336, http://doi.wiley.com/10.1029/2000JB900336, 2001.

Gow, A. J.: Results of measurements in the 309 meter bore hole at byrd station, antarctica, Journal of Glaciology, 4, 1963.

Greve, R. and Blatter, H.: Dynamics of Ice Sheets and Glaciers, Advances in Geophysical and Environmental Mechanics and Math-345

ematics, Springer Berlin Heidelberg, Berlin, Heidelberg, https://doi.org/10.1007/978-3-642-03415-2, http://link.springer.com/10.1007/

978-3-642-03415-2, 2009.

Hansen, B. and Landauer, J. K.: Some results of ice cap drill hole measurements, IASH Publ., 47, 313–317, 1958.

Hobbs, P.: Ice Physics, Clarendon Press, Oxford, 1974.

Holdsworth, G. and Bull, C.: The flow law of cold ice: investigations on Meserve Glacier, Antarctica, International Symposium on Antarctic350

Glaciologic Exploration, 86, 204–216, https://doi.org/(Symposium at Hanover 1968 — Antarctic Glaciological Exploration (ISAGE)),

1970.

Jacka, T. H.: The time and strain required for development of minimum strain rates in ice, Cold Regions Science and Technology, 8, 261–268,

1984.

Jellinek, H. H. G. and Brill, R.: Viscoelastic Properties of Ice, Journal of Applied Physics, 27, 1198–1209, https://doi.org/10.1063/1.1722231,355

http://aip.scitation.org/doi/10.1063/1.1722231, 1956.

Kuiper, E.-J. N., de Bresser, J. H. P., Drury, M. R., Eichler, J., Pennock, G. M., and Weikusat, I.: Using a composite flow law to model

deformation in the NEEM deep ice core, Greenland – Part 2: The role of grain size and premelting on ice deformation at high homologous

temperature, The Cryosphere, 14, 2449–2467, https://doi.org/10.5194/tc-14-2449-2020, https://tc.copernicus.org/articles/14/2449/2020/,

2020a.360

Kuiper, E.-J. N., Weikusat, I., de Bresser, J. H. P., Jansen, D., Pennock, G. M., and Drury, M. R.: Using a composite flow law to model

deformation in the NEEM deep ice core, Greenland – Part 1: The role of grain size and grain size distribution on deformation of the

upper 2207 m, The Cryosphere, 14, 2429–2448, https://doi.org/10.5194/tc-14-2429-2020, https://tc.copernicus.org/articles/14/2429/2020/,

2020b.

Larour, E. Y., Seroussi, H., Morlighem, M., and Rignot, E.: Continental scale, high order, high spatial resolution, ice sheet modeling using365

the Ice Sheet System Model (ISSM), Journal of Geophysical Research: Earth Surface, 117, https://doi.org/10.1029/2011JF002140, http:

//doi.wiley.com/10.1029/2011JF002140, 2012.

Le clec’h, S., Charbit, S., Quiquet, A., Fettweis, X., Dumas, C., Kageyama, M., Wyard, C., and Ritz, C.: Assessment of the Greenland ice

sheet–atmosphere feedbacks for the next century with a regional atmospheric model coupled to an ice sheet model, The Cryosphere, 13,

373–395, https://doi.org/10.5194/tc-13-373-2019, https://www.the-cryosphere.net/13/373/2019/, 2019.370

18

Levermann, A. and Winkelmann, R.: A simple equation for the melt elevation feedback of ice sheets, Cryosphere, 10, 1799–1807,

https://doi.org/10.5194/tc-10-1799-2016, 2016.

Ma, Y., Gagliardini, O., Ritz, C., Gillet-Chaulet, F., Durand, G., and Montagnat, M.: Enhancement factors for grounded ice and ice

shelves inferred from an anisotropic ice-flow model, Journal of Glaciology, 56, 805–812, https://doi.org/10.3189/002214310794457209,

http://www.agu.org/pubs/crossref/2011/2011GL048892.shtmlhttps://www.cambridge.org/core/product/identifier/S0022143000214068/375

type/journal_article, 2010.

Martin, P. J. and Sanderson, T. J. O.: Morphology and Dynamics of Ice Rises, Journal of Glaciology, 25, 33–46,

https://doi.org/10.3189/S0022143000010261, https://www.cambridge.org/core/product/identifier/S0022143000010261/type/

journal_article, 1980.

Mellor, M.: Creep tests on antarctic glacier ice, Nature, 184, 1959.380

Mellor, M. and Smith, J. S.: Creep of Snow and Ice, US Army Cold Regions Research and Engineering Laboratory, pp. 843–855, 1967.

Mellor, M. and Testa, R.: Effect of temperature on the creep of ice, Journal of Glaciology, 8, 131–145, 1969.

Montagnat, M., Castelnau, O., Bons, P. D., Faria, S. H., Gagliardini, O., Gillet-Chaulet, F., Grennerat, F., Griera, A., Lebensohn, R. A.,

Moulinec, H., Roessiger, J., and Suquet, P.: Multiscale modeling of ice deformation behavior, Journal of Structural Geology, 61, 78–108,

https://doi.org/10.1016/j.jsg.2013.05.002, http://dx.doi.org/10.1016/j.jsg.2013.05.002, 2014.385

Morlighem, M., Williams, C. N., Rignot, E., An, L., Arndt, J. E., Bamber, J. L., Catania, G. A., Chauché, N., Dowdeswell, J. A., Dorschel,

B., Fenty, I., Hogan, K. A., Howat, I. M., Hubbard, A., Jakobsson, M., Jordan, T. M., Kjeldsen, K. K., Millan, R., Mayer, L. A., Moug-

inot, J., Noël, B. P., O’Cofaigh, C., Palmer, S., Rysgaard, S., Seroussi, H., Siegert, M. J., Slabon, P., Straneo, F., van den Broeke,

M. R., Weinrebe, W., Wood, M., and Zinglersen, K. B.: BedMachine v3 : Complete Bed Topography and Ocean Bathymetry Map-

ping of Greenland From Multibeam Echo Sounding Combined With Mass Conservation, Geophysical Research Letters, 44, 51–61,390

https://doi.org/10.1002/2017GL074954, 2017.

Muguruma, J.: Effects of surface condition on the mechanical properties of ice crystals, J. Phys. D: Appl. Phys., 2, 1517, 1969.

Naruse, R., Okuhira, F., Ohmae, H., Kawada, K., and Nakawo, M.: Closure Rate of a 700 m Deep Bore Hole at Mizuho Station, East

Antarctica, Annals of Glaciology, 11, 100–103, https://doi.org/10.3189/S0260305500006406, https://www.cambridge.org/core/product/

identifier/S0260305500006406/type/journal_article, 1988.395

Noël, B., Van De Berg, W. J., Van Wessem, J. M., Van Meijgaard, E., Van As, D., Lenaerts, J. T. M., Lhermitte, S., Munneke, P. K., Smeets,

C. J., Van Ulft, L. H., Van De Wal, R. S., and Van Den Broeke, M. R.: Modelling the climate and surface mass balance of polar ice sheets

using RACMO2 - Part 1: Greenland (1958-2016), Cryosphere, 12, 811–831, https://doi.org/10.5194/tc-12-811-2018, 2018.

Nye, J. F.: The Flow Law of Ice from Measurements in Glacier Tunnels, Laboratory Experiments and the Jungfrau-

firn Borehole Experiment, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 219,400

477–489, https://doi.org/10.1098/rspa.1953.0161, http://rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1974.0120http:

//rspa.royalsocietypublishing.org/cgi/doi/10.1098/rspa.1953.0161, 1953.

Nye, J. F.: The distribution of stress and velocity in glaciers and ice-sheets, Proceedings of the Royal Society of London. Series A. Mathemat-

ical and Physical Sciences, 239, 113–133, https://doi.org/10.1098/rspa.1957.0026, http://www.royalsocietypublishing.org/doi/10.1098/

rspa.1957.0026, 1957.405

Paterson, W. S. B.: Observarions on Athabaska Glacier and their relation to the theory of glacier flow, Ph.D. thesis, The University of British

Columbia, https://doi.org/10.14288/1.0085872, 1962.

19

Paterson, W. S. B.: Secondary and Tertiary Creep of Glacier Ice as Measured by Borehole Closure Rates, Reviews of Geophysics and Space

Physics, 15, 47–55, https://doi.org/10.1029/RG015i001p00047, http://doi.wiley.com/10.1029/RG015i001p00047, 1977.

Paterson, W. S. B. and Savage, J. C.: Measurements on Athabasca Glacier relating to the flow law of ice, Journal of Geophysical Research,410

68, 4537–4543, https://doi.org/10.1029/jz068i015p04537, 1963.

Pattyn, F.: Sea-level response to melting of Antarctic ice shelves on multi-centennial timescales with the fast Elementary Thermomechanical

Ice Sheet model (f.ETISh v1.0), Cryosphere, 11, 1851–1878, https://doi.org/10.5194/tc-11-1851-2017, 2017.

Pattyn, F. and Morlighem, M.: The uncertain future of the Antarctic Ice Sheet, Science, 367, 1331–1335, 2020.

Pettit, E. C. and Waddington, E. D.: Ice flow at low deviatoric stress, Journal of Glaciology, 49, 359–369,415

https://doi.org/10.3189/172756503781830584, 2003.

Pettit, E. C., Waddington, E. D., Harrison, W. D., Thorsteinsson, T., Elsberg, D., Morack, J., and Zumberge, M. A.: The crossover stress,

anisotropy and the ice flow law at Siple Dome, West Antarctica, Journal of Glaciology, 57, 39–52, 2011.

Pfeffer, W. T., Harper, J. T., and O’Neel, S.: Kinematic Constraints on Glacier Contributions to 21st-Century Sea-Level Rise, Science, 321,

1340–1343, https://doi.org/10.1126/science.1159099, http://www.sciencemag.org/cgi/doi/10.1126/science.1159099, 2008.420

Pörtner, H.-O., Roberts, D., Masson-Delmotte, V., Zhai, P., Tignor, M., Poloczanska, E., Mintenbeck, K., Nicolai, M., Okem, A., Petzold, J.,

Rama, B., and Weyer, N.: The IPCC Special Report on the Ocean and Cryosphere, Tech. rep., 2019.

Pritchard, H., Ligtenberg, S. R. M., Fricker, H. A., Vaughan, D. G., Van Den Broeke, M. R., and Padman, L.: Antarctic ice-sheet loss driven

by basal melting of ice shelves, Nature, 484, 502–505, https://doi.org/10.1038/nature10968, 2012.

Qi, C., Goldsby, D. L., and Prior, D. J.: The down-stress transition from cluster to cone fabrics in experimentally deformed ice, Earth425

and Planetary Science Letters, 471, 136–147, https://doi.org/10.1016/j.epsl.2017.05.008, http://dx.doi.org/10.1016/j.epsl.2017.05.008http:

//linkinghub.elsevier.com/retrieve/pii/S0012821X17302625, 2017.

Quiquet, A., Dumas, C., Ritz, C., Peyaud, V., and Roche, D. M.: The GRISLI ice sheet model (version 2.0): Calibration and validation for

multi-millennial changes of the Antarctic ice sheet, Geoscientific Model Development, 11, 5003–5025, https://doi.org/10.5194/gmd-11-

5003-2018, 2018.430

Raraty, L. E. and Tabor, D.: The adhesion and strength properties of ice, Proceedings of the Royal Society of London. Series A. Mathematical

and Physical Sciences, 245, 184–201, https://doi.org/10.1098/rspa.1958.0076, 1958.

Raymond, C. F.: Inversion of flow measurements for stress and rheological parameters in a valley glacier, Journal of Glaciology, 12, 1973.

Rignot, E. and Mouginot, J.: Ice flow in Greenland for the International Polar Year 2008-2009, Geophysical Research Letters, 39, 1–7,

https://doi.org/10.1029/2012GL051634, 2012.435

Robinson, A., Calov, R., and Ganopolski, A.: Multistability and critical thresholds of the Greenland ice sheet, Nature Climate Change,

2, 429–432, https://doi.org/10.1038/nclimate1449, http://dx.doi.org/10.1038/nclimate1449%5Cnhttp://www.nature.com/doifinder/10.

1038/nclimate1449http://dx.doi.org/10.1038/nclimate1449, 2012.

Saruya, T., Nakajima, K., Takata, M., Homma, T., Azuma, N., and Goto-Azuma, K.: Effects of microparticles on deformation and microstruc-

tural evolution of fine-grained ice, Journal of Glaciology, 65, 531–541, 2019.440

Schaefer, J. M., Finkel, R. C., Balco, G., Alley, R. B., Caffee, M. W., Briner, J. P., Young, N. E., Gow, A. J., and Schwartz, R.: Greenland

was nearly ice-free for extended periods during the Pleistocene, Nature, 540, 252–255, https://doi.org/10.1038/nature20146, http://www.

nature.com/doifinder/10.1038/nature20146, 2016.

Seroussi, H., Nowicki, S. M. J., Payne, A. J., Goelzer, H., Lipscomb, W. H., Abe-Ouchi, A., Agosta, C., Albrecht, T., Asay-Davis, X.,

Barthel, A., Calov, R., Cullather, R. I., Dumas, C., Gladstone, R., Golledge, N. R., Gregory, J. M., Greve, R., Hattermann, T., Hoffman,445

20

M. J., Humbert, A., Huybrechts, P., Jourdain, N. C., Kleiner, T., Larour, E. Y., Leguy, G. R., Lowry, D. P., Little, C. M., Morlighem, M.,

Pattyn, F., Pelle, T., Price, S. F., Quiquet, A., Reese, R., Schlegel, N. J., Shepherd, A., Simon, E., Smith, R. S., Straneo, F., Sun, S., Trusel,

L. D., Van Breedam, J., van de Wal, R. S. W., Winkelmann, R., Zhao, C., Zhang, T., and Zwinger, T.: ISMIP6 Antarctica: a multi-model

ensemble of the Antarctic ice sheet evolution over the 21st century, The Cryosphere Discussions, pp. 1–54, https://doi.org/10.5194/tc-

2019-324, https://www.the-cryosphere-discuss.net/tc-2019-324/, 2020.450

Shepherd, A., Ivins, E. R., A, G., Barletta, V. R., Bentley, M. J., Bettadpur, S., Briggs, K. H., Bromwich, D. H., Forsberg, R., Galin, N.,

Horwath, M., Jacobs, S., Joughin, I., King, M. A., Lenaerts, J. T. M., Li, J., Ligtenberg, S. R. M., Luckman, A., Luthcke, S. B., McMillan,

M., Meister, R., Milne, G., Mouginot, J., Muir, A., Nicolas, J. P., Paden, J. D., Payne, A. J., Pritchard, H., Rignot, E., Rott, H., Sorensen,

L. S., Scambos, T. A., Scheuchl, B., Schrama, E. J. O., Smith, B., Sundal, A. V., van Angelen, J. H., Van De Berg, W. J., van den Broeke,

M. R., Vaughan, D. G., Velicogna, I., Wahr, J., Whitehouse, P. L., Wingham, D. J., Yi, D., Young, D., and Zwally, H. J.: A Reconciled455

Estimate of Ice-Sheet Mass Balance, Science, 338, 1183–1189, https://doi.org/10.1126/science.1228102, http://www.sciencemag.org/cgi/

doi/10.1126/science.1228102, 2012.

Song, M., Cole, D. M., and Baker, I.: An investigation of the effects of particles on creep of polycrystalline ice, Scripta materialia, 55, 91–94,

2006.

Steinemann, S.: Experimentelle Untersuchungen zur Plastizität von Eis, Ph.D. thesis, 1958.460

Talalay, P. and Hooke, R. L.: Closure of deep boreholes in ice sheets: a discussion, Annals of Glaciology, 47, 125–133,

https://doi.org/10.3189/172756407786857794, https://www.cambridge.org/core/product/identifier/S0260305500253457/type/

journal_article, 2007.

the PISM authors: PISM, a Parallel Ice Sheet Model, http://www.pism-docs.org, 2018.

Thomas, R. H.: The creep of ice shelves: interpretation of observed behaviour, Journal of Glaciology, 12, 55–70, 1973.465

Treverrow, A., Budd, W. F., Jacka, T. H., and Warner, R. C.: The tertiary creep of polycrystalline ice: experimental evidence for stress-

dependent levels of strain-rate enhancement, Journal of Glaciology, 58, 301–314, https://doi.org/10.3189/2012JoG11J149, 2012.

Weertman, J.: Dislocation Climb Theory of Steady-State Creep, Transactions of the ASM, 61, 681–694, 1968.

Weertman, J.: Creep of ice, in: Physics and Chemistry of Ice, edited by Whalley, E., Jones, S. J., and Gold, L. W., pp. 320–337, Royal Society

of Canada, Ottawa, 1973.470

Winkelmann, R., Martin, M. A., Haseloff, M., Albrecht, T., Bueler, E., Khroulev, C., and Levermann, A.: The Potsdam Parallel Ice

Sheet Model (PISM-PIK) – Part 1: Model description, The Cryosphere, 5, 715–726, https://doi.org/10.5194/tc-5-715-2011, https:

//www.the-cryosphere.net/5/715/2011/, 2011.

Zeitz, M., Levermann, A., and Winkelmann, R.: Sensitivity of ice loss to uncertainty in flow law parameters in an idealized one-dimensional

geometry, Cryosphere, 14, 3537–3550, https://doi.org/10.5194/tc-14-3537-2020, 2020.475

21