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# What is glacier sliding?

Robert LAW1,2 David CHANDLER3,2 Andreas BORN1,2<EMAIL_ADDRESS>1Department
of Earth Science, University of Bergen, Bergen, Norway
2Bjerknes Centre for Climate Research, Bergen, Norway
3NORCE Norwegian Research Centre, Bergen, Norway

###### Abstract

Glacier and ice-sheet motion is fundamental to glaciology. However, there is
still no clear consensus for the optimal way to describe glacier and ice-sheet
sliding. Typically, sliding is parameterised using a traction coefficient
nominally linked to a given theory describing one or a limited set of sliding
processes. However, this approach precludes the possibility of multiple
simultaneous and spatio-temporally varying sliding modes with inaccuracies
resulting in model error propagation as the system evolves away from the
conditions under which it was optimised for. Here, revisiting early
theoretical work,we describe glacier sliding as a scale- and setting-dependent
‘inner flow’ that arises from multiple overlapping sub-processes, bridging
divides between hard and soft beds, rough and smooth beds, and stick-slip and
continuous sliding as well as providing a consistent definition for form drag.
The corresponding ‘outer flow’ then accounts for ‘normal’ ice deformation. We
propose that the significance of ‘Iken’s bound’ is then reduced if form drag
dominates over subglacial cavitation in a given region, possibly explaining
the persistent functionality of Weertman-type sliding in process-agnostic
sliding studies over rough topography. Last, reviewing observation-based
studies we suggest that a simple ‘unified’ sliding relationship controlled by
a single tunable coefficient may not be a realistic prospect, but that a
Weertman-type relationship with careful consideration of the power value is
presently the ‘least bad’ option.

##  1  Background

_The question then occurs, is the viscosity real or apparent?_ John Tyndall on
glacier motion in 1857.

Glacier sliding is a phenomenologically distinct problem within earth
sciences, separated from simpler plane-on-plane sliding problems between two
elastic solids by multiple confounding factors. The relative softness of ice
allows for both viscous deformation under a non-Newtonian power law _and_
discrete slip at the ice-bed interface, which may occur in a stick-slip or
continuous manner. A ‘soft’ sediment phase provides a further medium to convey
deformation in many instances (and is shaped in turn by glacial processes),
while ‘hard’ underlying bedrock can constitute a highly topographically
irregular base ranging from <1 cm asperities to dramatic fjords, necessitating
ice deformation around rigid obstacles over length scales covering 5 orders of
magnitude. A subglacial hydrological system comprised largely of melted ice
may significantly alter the effective pressure of the ice upon its substrate,
and influence the roughness and material characteristics of the base itself.
Debris entrained within basal ice interacts with the subglacial environment
and influences both basal ice properties and the frictional properties of the
glacier sole. These considerations present a wide parameter space for basal
boundary conditions, which may also exhibit considerable spatial and temporal
variations. The net outcome is significant difficulty in the derivation of a
‘unified’ or ‘generalised’ sliding relationship, usually expressed in the form

$\tau_{b}=f(u_{b})\>\>\text{or}\>\>\tau_{b}=f(u_{b},N)$ (1)

where $\tau_{b}$ is basal traction, $u_{b}$ is ice velocity tangential to the
ice-bed interface, $N=P_{i}-P_{w}$ is effective pressure, $P_{w}$ is
subglacial water pressure, and $P_{i}$ is the ice overburden pressure.

Determining an appropriate sliding relationship is central to producing
tractable, physically-based projections of the contribution of ice sheets and
glaciers to sea-level rise and fresh-water fluxes over the coming centuries.
Extant sliding relationships (Fig. 1, Table. 1) influence ice-sheet model
output at catchment to entire ice sheet scales, with the distinction between
bounded traction (plastic, regularised-Coulomb) and unbounded traction (all
other relationships) being particularly consequential (Gillet-Chaulet and
others,, 2012; Parizek and others,, 2013; Ritz and others,, 2015; Tsai and
others,, 2015; Bons and others,, 2018; Kyrke-Smith and others,, 2018; Brondex
and others,, 2019; Åkesson and others,, 2021; Lippert and others,, 2024;
Trevers and others,, 2024), yet around five111Some sliding relationships can
be considered a simplified versions of one another. For example Weertman-type
and pseudo-plastic can be manipulated into linear or plastic relationships,
the $N$ in Budd sliding is effectively subsumed into the $C$ of Weertman-type
sliding during inversion if it is assumed that subglacial water pressure does
not vary in time, making this number subjective. (Section 3) sliding
relationships implicitly or explicitly capturing different sliding processes
are used in the Ice Sheet Model Intercomparison Project (ISMIP) models
(Goelzer and others,, 2020; Seroussi and others,, 2020), which guide worldwide
policy on sea level rise mitigation. This state of affairs is not immediately
straightforward to resolve, as the inverse methods that tune for $C$, the
‘traction coefficient’, will produce reasonable basal stress states whichever
equation in Table 1 is chosen as the global stress balance condition must be
met (Joughin and others,, 2009; Arthern and Gudmundsson,, 2010). There is
therefore no simple approach to discounting sliding relationships given solely
a snapshot of glacier or ice sheet velocity and geometry. Errors associated
with an inappropriate choice of basal sliding relationship subsequently
increase as a function of model run time (Aschwanden and others,, 2021).

Figure 1: Existing sliding parameterisations and how their development relates to soft- and hard-bed theory. Traction and velocity values are plausible, but for illustrative purposes only. Sliding relationship |  Equation
---|---
Linear† |  $\tau_{b}=Cu_{b}$ or $\tau_{b}=NCu_{b}$
Plastic† |  $\tau_{b}=C$ or $\tau_{b}=CN$
Weertman(-type)† |  $\tau_{b}=Cu_{b}^{\frac{2}{1+n}}$ or $\tau_{b}=Cu_{b}^{1/m}$
Pseudo-plastic† |  $\tau_{b}=\tau_{c}\frac{u^{1/m}}{u_{t}^{1/m}}$
Budd† |  $\tau_{b}=C\left({N}^{q}u_{b}\right)^{1/m}$ or $\tau_{b}=CNu_{b}^{1/m}$
Regularised-Coulomb |  $\tau_{b}=C\left(\frac{u_{b}}{u_{b}+u_{t}}\right)^{1/m}$
Table 1: Selected sliding relationships applied in numerical glacier and ice-
sheet models, expressed in one-dimensional form. † Denotes that the sliding
relationship is used in Goelzer and others, (2020) or Seroussi and others,
(2020) ISMIP experiments. In all equations $\tau_{b}$ is basal traction and
$u_{b}$ is the basal velocity tangential to the bed. Where present, $n$ is the
exponent used in Glen’s flow law (usually 3) and is only used where an
explicit link to Glen’s flow law is made in the paper proposing the sliding
relationship, $m$ is an exponent often related to Glen’s flow law, but not
always explicitly, $N$ is the effective pressure, $C$ is the traction
coefficient which may be adapted from its use in the original paper for
simplified intercomparison (for example we replace $1/C$ with $C$ in the Budd
relationship), $\tau_{c}$ is the yield stress in the pseudo-plastic
relationship, $q$ is a fitting parameter in the Budd relationship, and $u_{t}$
is the threshold velocity in the pseudo-plastic and regularised-Coulomb
relationships. The rate-weakening 2D cavities relationship in Fig. 1 is not
included, as no numerical ice-sheet or glacier models include rate-weakening
behaviour above a given velocity threshold. Depending on convention, the RHS
may be negative in the original paper to indicate a traction force opposite to
velocity direction. In dimensions higher than 1 the sliding relationship is
vectorised. Notes: Linear: Adapted from Nye, (1969) and Kamb, (1970). As
applied in e.g. Morlighem and others, (2013). The version with $N$ is used in
some Goelzer and others, (2020) experiments. Weertman: Equation featuring $n$
is adapted from Weertman, (1957) where $n$ is set as 3. $m$ is often used in
models instead of $n$ where $m$ is usually between 3 and 4. Pseudo-plastic:
Effectively the same as Weertman but included for frequent use in PISM
applications e.g. Aschwanden and others, (2016), the pseudo-plastic
relationship can be varied between plastic ($m=\infty$), linear ($m=1$), or
Weertman-type ($2\leq m\leq 4$) behaviour. Regularised-Coulomb: The simplest
form excluding $N$, adapted from (Joughin and others,, 2019). Other
formulations exist such as
$\tau_{b}=Cu_{b}N\left(\frac{{u_{b}}^{-n+1}}{u_{b}+A_{s}C^{n}{N}^{n}}\right)^{1/n}$
(Helanow and others,, 2021) where $A_{s}$ is a parameter specific to this
formulation. The differences between regularised-Coulomb relationships are not
negligible (Appendix A). Plastic: As used in e.g. Bougamont and others, (2014)
and the PISM default (Winkelmann and others,, 2011). Budd: As developed in
Budd and others, (1984) where $q=1$ and $m=3$ and used in Budd and Jacka,
(1989) with $q=2$ and $m=1$. The alternative and simplified Budd formulation
is as used in Choi and others, (2022). The proposal of Tsai and others, (2021)
also bears similarities with Budd sliding.

In numerical ice-sheet models, these sliding relationships form a critical
thread between the processes occurring at the sub-grid-scale and the numerical
solution defined at the grid scale (where ‘grid’ refers to discretised model
grid cells or mesh elements) 222This was not always the case, with purely
analytical sliding relationships predating the first numerical models by
decades (Weertman,, 1957; Rasmussen and Campbell,, 1973; Jenssen,, 1977) which
may explain why sub-grid-scale processes were not initially a concern.. In
discretised continuum mechanics models, the distinction between sub-grid-scale
and grid-scale is usually made using a ‘representative volume element’ that is
large enough to be representative of the bulk behaviour of micro-scale
processes within the constitutive material it represents. For ice deformation,
this is straightforward – an ice volume of 1 m3, much below the resolution of
even the most detailed numerical models, will contain sufficient variability
in crystal orientation, size, and impurities as to have effectively the same
response to an applied stress field as a different 1 m3 of the same material.
Lab-based experiments using small volumes of ice (e.g. Glen,, 1952) can
therefore be reasonably applied to much larger-scale models, even if separate
issues such as the importance of tertiary creep remain (Adams and others,,
2021).

There is not, however, an obvious representative volume, or ‘representative
surface’ element size for glacier sliding processes, with investigations into
glacier sliding covering a disparate range of scales extending over seven
orders of magnitude from millimetres to 10s of kilometres (Fig. 2). For
example, lab experiments for ice-sediment shearing behaviour may be conducted
at a scale of tens of centimetres (e.g. Iverson and others,, 1998; Zoet and
Iverson,, 2020), up-scaled to field settings covering an entire glacier where
subglacially entrained clasts, variable subglacial hydrology, or bedrock
obstacles further influence the relationship between traction and velocity
(e.g. Iverson and others,, 1995 Hedfors and others,, 2003, Gimbert and
others,, 2021), and ultimately applied to ice-sheet models where individual
grid cells can exceed 10s of kilometres (or have a highly variable size across
a given domain) and cover an uncertain and/or heterogeneous subglacial
landscape (e.g. Kyrke-Smith and others,, 2018; Holschuh and others,, 2020;
Paxman and others,, 2021). Nonetheless, routinely-used sliding
parameterisations in ice sheet models almost exclusively use theories derived
for one or a limited set of sliding processes at small scales (Goelzer and
others,, 2020; Seroussi and others,, 2020), without representing the full
range of scale-dependent processes.

Figure 2: Spatial scales for theories and parameterisations of basal sliding.
A solid line indicates coverage as explicitly defined in the associated paper
paper while a dashed line indicates probable situation dependent coverage and
an arrow indicates extension beyond the scale bar or an uncertain coverage
beyond the given limit. Reasoning behind the positioning of each spatial range
is provided in Appendix B.

Representation of sliding in glacier and ice-sheet models is further hindered
by a tendency to classify the bed either as ‘soft’ (less rigid than ice and
subject to sediment deformation and transport under glacier slip and
subglacial hydrology even at short time scales) or ‘hard’ (more rigid than ice
and only appreciably modified by sliding or hydrology at glacial-cycle time
scales). Glacier and ice-sheet models generally cannot represent settings that
may feature both soft- and hard-bed characteristics within a single grid cell
or across a model domain (Fig. 3). This ‘one or the other’ classification is
challenged by geological and geomorphological evidence of mixed soft- and
hard-bed regions in both recently deglaciated landscapes (e.g. Hogan and
others,, 2020; Garcia-Oteyza and others,, 2022) and in active subglacial
settings (Jordan and others,, 2023) but has received very little attention in
glaciological studies (Koellner and others,, 2019). A middle-ground between
hard and soft beds can also exist if the substrate is only deformable under
certain conditions, i.e. very thick ice cover with high subglacial water
pressure. Sub-grid-scale topography is furthermore known to significantly
influence ice flow (Kyrke-Smith and others,, 2018; Hoffman and others,, 2022;
Law and others,, 2023), and may even dominate grid-scale basal traction (Fig.
3b), but is also lacking from present sliding parameterisations.

Figure 3: Typically sized grid cells and mesh element overlain on previously
glaciated regions. Thwaites Glacier bathymetry is from Hogan and others,
(2020). Frafjord topography is from Kartverket.no.

In this paper we provide a clear framework for describing glacier sliding as
behaviour emerging from multiple scale- and setting-dependent processes
(summarised in Fig. 4) rather than as a single dominant process. We reconsider
the ‘inner-outer’ flow division (Section 2) first proposed by Fowler, (1977),
but with a more universal approach to the inclusion of numerous sliding sub-
processes (Section 3). We briefly cover standard soft- and hard-bed theory
(Sections 3.1 and 3.2); form drag, roughness, and temperate ice (Section 3.3);
stick-slip vs. continuous sliding (Section 3.4); the overlap between soft- and
hard-bed sliding (Section 3.5); the basal ice layer and ice rheology (Section
3.6); and how these sub-processes are modified by subglacial hydrology
(Section 4). We focus on annual timescales as these are the most relevant for
predictive and palaeo models, and largely avoid the transient influence of
e.g. supraglacial lake drainages (Das and others,, 2008), rainfall events
(Doyle and others,, 2015), or seasonal modulation (Bartholomew and others,,
2011; Sole and others,, 2013). In Section 5 we review process-agnostic studies
fitting sliding relationships to observational data. Last, we consider if a
truly unified sliding relationship is a realistic possibility, consider the
settings under which Iken’s bound can be considered to hold, and make
suggestions for the application of sliding relationships to models (Section
6).

Throughout this paper we use _sliding_ to refer to the net effect of processes
that fall within the ‘inner flow’ (covered below), and _slip_ to refer to
discrete slip at the ice-bed interface where a clear ice-bed interface can be
discerned. Unless otherwise specified, we refer to sliding situations where
the ice-bed interface is at the pressure melting point. We hope that this
paper is useful to those seeking an introduction to glacier sliding and
coverage of advancements over the last decade, but we do not attempt an
exhaustive review. Earlier reviews (Clarke,, 2005; Fowler,, 2010) and book
chapters (Benn and Evans,, 2010; Cuffey and Paterson,, 2010) provide more
comprehensive coverage. Our discussion is limited to continuum-scales in ice
and we do not cover crystal-scale deformation or molecular-level interaction –
Schulson and Duval, (2009) and Krim, (1996) provide more information on these
topics.

Figure 4: Schematics of glacier sliding sub-processes and controls discussed
in the text. Scale is intentionally omitted but scale generally increases left
to right and top to bottom through sub processes. Hydrology in k and l is
considered as a process modifier, as no ice mass is transported and these are
distinguished with blue outline boxes. a Form drag over sedimentary clasts. b
Shear of sediment. c Ice with clasts sliding over a flat hard bed. d Stick-
slip events. e Regelation. f Deformation within the basal boundary layer. g
Cavitation. h Spatially variable sliding rates over rough topography. i
Spatially variable ice deformation over rough topography. j Temperate layer
processes. k Hydrology and channelisation. l Subglacial lakes.

##  2  Inner-outer flows

Fowler, (1977), Fowler and Larson, (1978), and Fowler, (1981) (hereafter
Fowler1977) consider glacier sliding over a two-dimensional free-slip bumpy
bed analytically. Recognising that the smooth basal boundary used in larger-
scale analytical models is not a realistic representation of a rough glacier
bed, Fowler1977 separate ice motion into an ‘inner’ flow and an ‘outer’ flow
(reinterpreted in Fig. 5). The inner flow, which includes a component of basal
slip, is theorised to closely follow the _actual_ bed topography. The outer
flow accounts for internal deformation at the typical depth and length scale
of the entire glacier and is considered to be ‘sliding’ over a smoothed
representation of the actual bed (i.e. the _model_ bed). The outer and inner
flows are then joined by a sliding relationship under the condition that the
inner flow ‘feels’ the outer flow as a uniform shearing flow, and the outer
flow ‘feels’ the inner flow as a tangential stress at the smoothed bed
boundary.

Figure 5: Inner and outer flows. a Outer flow and inner flow with matching
surface (dash-dot line) which may be similar, but is not directly equivalent,
to the model bed surface. Depending on setting the inner flow may feature any
of the processes in Fig. 4. b The inner flow is collapsed to occur at the
model bed surface. The outer flow represents ice deformation that is not
significantly influenced by bed processes.

In our application, the inner flow accounts for the relevant subset of
processes outlined in Fig. 4 and Section 3 (as opposed to the simpler set of
processes considered in Fowler1977) and the outer flow comprises englacial ice
deformation with minimal influence from the processes in Fig. 4. In a
modelling sense, the outer flow represents all motion not captured by the
basal sliding component. A more complex example of multiple sub-processes
operating in an inner flow is given in Appendix C.

A distinction between an inner and outer flow begins to form an inclusive
description of glacier sliding, but a consistent definition for the position
of the inner-outer boundary does not immediately materialise – even if an
explicit inner-outer boundary is in keeping with its implicit use in previous
studies (e.g. Schoof and Clarke,, 2008; Zoet and Iverson,, 2020; Helanow and
others,, 2021). The inner flow does not neatly fit the idea of a boundary
layer as typically used in fluid dynamics to describe a region of turbulent
flow separation (Batchelor,, 1967 and discussed further in Section 3.3) and
shares only broad similarities with a bottom boundary layer as used in
oceanography models (Campin and Goosse,, 1999). A numerical approach also
means that the inner and outer flow can not be matched analytically as they
are in Fowler1977. However, perturbations in strain rates tend to decrease in
amplitude with increasing height above the bed, either exponentially (Balise
and Raymond,, 1985) or as a power law (Chandler and others,, 2006);
corresponding length scales for the decay of inner layer strain rate
perturbations could then form a basis for defining the inner-outer layer
boundary.

The height at which inner-flow processes become negligible will also vary as a
function of the spatial scale under consideration, heterogeneity in bed
properties, and the roughness characteristics of the bed topography. For
example, inner-flow processes will be important at a greater height above the
bed if considering an area of rough topography around a planar soft-bed
region, than if considering the planar soft-bed region in isolation. In most
situations however, a reasonable norm is already defined for setting the
position of the model bed surface. BedMachine (Morlighem and others,, 2017)
and BedMap (Fretwell and others,, 2013) products are used for the Greenland
and Antarctic ice sheets respectively while a more fragmented approach is
taken for mountain glaciers (e.g. Welty and others,, 2020; Grab and others,,
2021). These bed topography models are significantly smoother than the
roughness suggested by geostatistical analysis of radar flight lines (MacKie
and others,, 2021; Law and others,, 2023) and deglaciated forelands
immediately adjacent to the ice sheets, but function as convenient thresholds
for separating inner and outer flows. Crucially, we note that the imperfect
representation inherent to subglacial bed topography products means a degree
of ice deformation will unavoidably already be incorporated within existing
sliding parameterisations even if its inclusion has not been previously
formalised. The error associated with a collapsed inner flow is therefore not
introducing new errors, but rather providing a way to quantify and address
existing ones.

Using this description of the inner flow, the net basal velocity at the model
bed surface (m a-1) can then be treated as the sum of individual sliding
processes up to $\mathcal{P}_{n}$ with

$u_{b}=\mathcal{P}_{1}+\mathcal{P}_{2}+...+\mathcal{P}_{n}.$ (2)

and the net basal traction (Pa) as

$\tau_{b}=\hat{\mathcal{P}_{1}}+\hat{\mathcal{P}_{2}}+...+\hat{\mathcal{P}_{n}}$
(3)

where hats ( $\hat{}$ ) refer to the resistance opposing each sliding process
over the area considered. This is displayed schematically in Fig 6. Expanding
Eqs. 2 and 3 to cover an area $R$, which may represent a model cell (e.g. Fig.
3), gives

$u_{b}\rvert_{R}=\frac{1}{R}\iint_{R}\mathcal{P}_{1}+\mathcal{P}_{2}+...+\mathcal{P}_{n}\,\mathrm{d}R$
(4)

and

$\tau_{b}\rvert_{R}=\frac{1}{R}\iint_{R}\hat{\mathcal{P}_{1}}+\hat{\mathcal{P}_{2}}+...+\hat{\mathcal{P}_{n}}\,\mathrm{d}R\,.$
(5)

This framework facilitates quantification of the role of each sub-process in
the inner flow, though a full representation will be further complicated if
sliding sub-processes are in fact represented as functions of one another
(e.g. $\hat{\mathcal{P}_{1}}=f(\hat{\mathcal{P}_{2}},\hat{\mathcal{P}_{3}})$).

Figure 6: Schematics of inner flow velocity and traction components for planar
soft-bed sliding. Sub-processes are sediment deformation ($\mathcal{P}_{1}$),
slip at the ice-till interface ($\mathcal{P}_{2}$), and viscous deformation
and regelation around clasts that are substantially larger than the
surrounding till matrix ($\mathcal{P}_{3}$). This reflects the transition zone
between viscous deformation and sediment shearing modes detailed in Zoet and
Iverson, (2020). An example featuring more processes is Appendix C.

##  3  Sliding processes

###  3.1  Sliding over ‘soft’ (sediment) beds

Early work on soft-bed sliding suggested that subglacial till behaves as a
mildly non-linear Bingham viscous fluid (such that resistance to deformation
increases with deformation rate), with distributed deformation extending
around 50 cm below the ice-bed interface (Boulton and Hindmarsh,, 1987). This
idea received some support from contemporaneous studies of subglacial
conditions (Blake,, 1992; Humphrey and others,, 1993) but most subsequent
studies – including laboratory, direct measurement, and analogous laboratory
experiments from different material science disciplines – find strain-
localisation close to the ice-bed interface, or manifest in multiple discrete
shear zones at multiple depths within the subglacial till, and a Coulomb-
plastic rheology (e.g. Biegel and others,, 1989; Kamb,, 1991; Hooke and
others,, 1997; Iverson and others,, 1997; Iverson and others,, 1998; Iverson
and Iverson,, 2001; Damsgaard and others,, 2013; Cuffey and Paterson,, 2010),
and we therefore do not discuss viscous sediment deformation further.

In laboratory experiments of glacial-till deformation beneath a rigid shear
ring, the till reaches its yield strength and then fails at a uniform rate
under steady conditions, with a near-constant or a modest decrease in basal
traction over a large (400 m a-1) shear-ring velocity increase (Iverson and
others,, 1998; Fischer and others,, 2001; Iverson,, 2010). Field observations
using strain-gauges emplaced in actively deforming subglacial sediment
repeatedly support Coulomb-plastic deformation (Iverson and others,, 1995;
Alley and Whillans,, 1984; Fischer and others,, 2001), with an inverse
relationship between pore-water pressure and sediment strength (Fischer and
others,, 2001). Numerical models indicate that the pore-water pressure-
strength relationship is further modulated by till permeability properties
(Damsgaard and others,, 2017, 2020), with till properties influenced in turn
by crushing and compression from overlying ice (Iverson,, 1999). Field
observations additionally emphasise the importance of ploughing of clasts
lodged at the ice-till interface, particularly where sediment pore pressure is
high, as the pressure exerted by the clast on the downflow sediment can
locally increase the water pressure and weaken the sediment (Iverson and
others,, 1994; Rousselot and Fischer,, 2005). In the case where sediment is
not weakened by an increase in pore pressure, a clast may be pushed downwards
and no longer be ploughed through the matrix (Clark and Hansel,, 1989).

More recently, Zoet and Iverson, (2020) used a ring of temperate ice $\sim$15
cm thick between the rigid shear ring and underlying sediment. In this
setting, rate-strengthening behaviour occurs at lower velocities (below around
50 m a-1 in sediment with clasts, Fig. 4a) as ice viscously deforms and
regelates around static clasts before the clasts begin to plough through the
finer-grained matrix at a stress limited by the till’s Coulomb strength (Fig.
4b). The net result of the Zoet and Iverson, (2020) experiment is a
regularised-Coulomb relationship (Fig. 1, Table 1) where the sliding interface
is defined up 15 cm above the ice-till interface. The relationship of Zoet and
Iverson, (2020) to form drag as proposed by Minchew and Joughin, (2020) is
covered in Section 3.3.

Bedforms, including drumlins, mega-scale glacial lineations, and ribbed
moraine (Stokes,, 2018) are ubiquitous in areas of continuous sediment and
likely represent a continuum of features formed by glacier dynamics and
subglacial hydrology (Ely and others,, 2023). These bedforms are generally
streamlined and have a low aspect ratio (height/length), but their
contribution to form drag has not been quantified to our knowledge.

###  3.2  Sliding over ‘hard’ (bedrock) beds

Hard-bed sliding studies can be divided into those considering slip across
planar surfaces of limited extent and those concerned with sliding over rough
or undulating beds at longer wavelengths. In studies considering planar
surfaces water pressure is typically kept spatially constant, while in studies
considering a rough or undulating bed subglacial water may be theorised as
either a microscopically thin and continuous layer, or as a pressurised system
of subglacial cavities occupying local bedrock depressions. Undulating beds
with a frictionless ice-bed interface were the first to attract attention in
sliding theory, but we begin with macroscopically flat (i.e. flat to the naked
eye) surfaces due to their simplicity. Roughness at length scales beyond 25 m
is covered in Section 3.3.

The slip of ice over a planar surface (Fig. 4c) is closer to the assumptions
of standard Coulomb behaviour (Coulomb,, 1785; Desplanques,, 2014), though
plane-normal velocity components, englacial clasts, and possible clast-scale
cavitation present complications (Hoffman and others,, 2022). Rather than a
sliding relationship of the form of Eq. 1, basal traction in these studies is
often reported as a static $\mu_{s}$ or transient $\mu_{t}$ coefficient of
friction, $\mu_{s}\>\textup{or}\>\mu_{t}=\frac{\tau_{b}}{N}$ where $\mu_{s}$
describes the force required to move the plane from stationary and $\mu_{t}$
describes the force required to keep the plane in motion. Shear-ring studies
using temperate ice and englacial sediment content of 0-20% over
macroscopically flat granite find a $\mu_{s}$ of 0.02-0.05 (Barnes and
others,, 1971; Zoet and others,, 2013; McCarthy and others,, 2017; Thompson,,
2020), lower than teflon-on-teflon with no lubrication (Fetfatsidis and
others,, 2013). In the above studies, sliding may be stick-slip or steady-
state (outlined further in Section 3.4) depending on water pressure and
particular material properties. Transient coefficients of friction between
0.05-0.08 were obtained from subglacial access to the ice-rock interface of
the Engabreen outlet glacier in Norway (Iverson and others,, 2003; Cohen and
others,, 2005). Unexpectedly high shear traction values of up to 0.5 MPa
accompany the low $\mu_{t}$ values at Engabreen and these are still not fully
explained, suggesting a possible gap in understanding – see Thompson and
others, (2020) for a more in depth discussion. The transient coefficient of
friction is much higher for cold ice (0.5 at -20oC) making sub-temperate slip
very low, but potentially non-negligible (Barnes and others,, 1971; McCarthy
and others,, 2017; Atkins,, 2013; Mantelli and others,, 2019; Section 3.7).

Where an ice block rests unconstrained on a plane, and water pressure beneath
the ice is close to atmospheric pressure, $\mu_{s}$ can reach 0.6 for a rough
(up to 0.25 cm roughness) pebbly surface or 0.2 for smooth concrete (Budd and
others,, 1979). These are not typical stress or hydrological conditions for
glacier beds (e.g. Hubbard and Nienow,, 1997; Woodard and others,, 2021) but
were used by Budd and others, (1979) to derive an empirical sliding
relationship (Table 1), with later modifications to exponents to produce a
better fit with field data at an ice-sheet scale (Budd and others,, 1984) that
are arguably large enough to mark a departure from the empirical
underpinnings. To summarise planar hard-bed sliding, it is reasonable to treat
the friction of ice on rock in the presence of pressurised water as low, but
it is not negligible, and may in certain configurations be an important local
control on basal traction. There is significant scope to improve understanding
in the processes responsible for ice-rock friction across a wide parameter
space (Thompson and others,, 2020).

Sliding over rough hard beds has also received much attention. This began with
the classical work of Weertman, (1957), who considered ice motion over
isolated cuboids (akin to the geometry in Fig. 4e) with no ice-bed separation
through two processes: regelation and enhanced creep (Fig. 4e). In regelation,
ice moves without deformation. Temperate ice approaching an obstacle is
subject to greater pressure, lowering its melting point. Consequently, some of
the ice melts at the upstream face, and the meltwater is driven to the
downstream face by the pressure gradient. Here, where the pressure is lower
and the melting point is higher, the meltwater refreezes. The higher
temperature at the downflow side of the obstacle results in heat flow back
through the obstacle, completing the cycle. Regelation is an ongoing process
in many subglacial settings (Kamb and Lachapelle,, 1964; Kamb,, 1970; Hallet
and others,, 1978; Hubbard and Sharp,, 1993; Iverson,, 2000; Cook and others,,
2011; Rempel and Meyer,, 2019), including in soft bed settings outside the
scope of the original Weertman, (1957) paper. However, the requirement for
return heat flow causes regelation rate to vary inversely with obstacle
wavelength, limiting the importance of regelation as a sliding process to
obstacles smaller than a few cm Weertman, (1957). Regelation is therefore
generally not viewed as a significant contributor to overall sliding rates
(MacAyeal,, 2019) and has received comparatively little attention since the
mid 1990s, though further contemporary study would not be unwelcome.

Enhanced creep is prompted by stress concentrations at bedrock obstacles,
leading to locally increased strain rates. Functionally, this means enhanced
creep can be considered as a synonym for form drag (Section 3.3). Conversely
to regelation, the contribution of enhanced creep to glacier sliding in
Weertman’s 1957 study increases with obstacle size. Focusing on the obstacle
sizes that produce the greatest rate of sliding when regelation and enhanced
creep act in unison (millimetre- to metre-scale) Weertman, (1957) reaches a
value of $m$ in a power law of $(n+1)/2=2$ with $C$ representing a combination
of bed-geometry characteristics. Technically, values of $m$ deviating from 2
when $n=3$ are therefore not physically based on the original Weertman theory
making ‘power-law’ or ‘Weertman-type’ sliding a more suitable descriptor.
Sliding over a sinusoidal two-dimensional bed with no ice-bed separation and a
linear ice rheology produces a linear sliding relationship ($m=1$ in Weertman-
type sliding) (Nye,, 1969), as does Weertman sliding with $n=1$.

Weertman-type sliding, usually with $m=3$, is the most commonly used sliding
relationship in the latest round of ISMIP experiments (with linear sliding
also frequently used) (Goelzer and others,, 2020; Seroussi and others,, 2020),
yet the theoretical underpinnings behind these relationships do not hold up to
scrutiny – including by Weertman’s own reappraisal (Weertman,, 1979).
Weertman’s original theory features a non-linear rheology following Glen,
(1955), but it is apparent that no glacier beds actually have the morphology
used by Weertman, (1957), and even changes to a different unrealistic bed
geometry produces a materially different relationship (Lliboutry,, 1979).
Weertman sliding is further challenged by ubiquitous evidence and theoretical
support for subglacial cavitation, or regions of bounded ponded water at high
pressure between the glacier sole and underlying hard or soft bed, which form
when the pressure on the downstream face of an obstacle falls below a critical
level (Fig. 4g and e.g. Walder and Hallet,, 1979; Kamb,, 1987; Hooke,, 1989;
Helanow and others,, 2021). Weertman sliding also does not provide an
explanation for increased sliding rates when subglacial water pressure
increases. The persistent utility of Weertman-type sliding in heuristic
studies, though generally with $m\geq 3$, despite these major problems is
covered in Sections 5 and 6.4.

Subglacial cavities (Fig. 4g) act to reduce contact between the glacier sole
and the ice bed. In the two-dimensional formulations of Lliboutry, (1968),
Iken, (1981), Schoof, (2005), and Gagliardini and others, (2007), amongst
others, where the bed is comprised of frictionless sinusoids, this produces an
upper bound for basal traction, followed by rate-weakening behaviour (Fig. 1).
Put mathematically, ’Iken’s bound’ as coined by Schoof, (2005) is given as

$\tau_{b}\leq N\textup{tan}(\theta)$ (6)

where $\theta$ is the maximum up-slope angle between the bed and the mean flow
direction of the ice (Iken,, 1981; Schoof,, 2005). Roldan-Blasco and others,
(2022) suggest that in the simplified two-dimensional framework of compound
sinusoids non-negligible friction at the ice-bed interface will not
significantly alter the form of the sliding relationship or the existence of
Iken’s bound. However, the geometry used in these studies can also be
considered unrealistic, with the absence of a third dimension preventing
lateral escape of subglacial water that would act to reduce the cavity size
(Gimbert and others,, 2021). Helanow and others, (2019) and Helanow and
others, (2021) address this and show that realistic three-dimensional bed
topographies up to 25 m still result in bounded basal traction, but that
significant rate-weakening behaviour is not expected (the regularised-Coulomb
relationship in Fig. 1). The validity of the assumptions underlying Eq. 6 when
considering inner flows over larger areas and rough topography is covered in
Section 6.3.

###  3.3  Roughness, form drag, and temperate ice

Glaciated and deglaciated landscapes are characterised by roughness not only
at scales of centimetres to metres (as featured in the previous section), but
also by roughness at scales of tens to thousands of metres. This includes
cnoc-and-lochan landscapes, drumlins, incised fjords, and other erosional or
depositional features that bridge a very subjective boundary between
‘roughness’ and ‘topography’ (Figs. 4h-j). Radar flight lines reveal extensive
and variably rough beds beneath the Greenland and Antarctic ice sheets (e.g.
Bingham and Siegert,, 2009; Rippin,, 2013; Munevar Garcia and others,, 2023),
often quantified through a Fast Fourier Transform or variogram (e.g. MacKie
and others,, 2021) which pares back the geomorphological uniqueness of a
landscape but enables regional intercomparison. At present, only roughness up
to a length scale of 25 m is explicitly incorporated in a process-based
sliding relationship (Helanow and others,, 2021), but it is clear from
borehole observations (Ryser and others,, 2014; Doyle and others,, 2018; Maier
and others,, 2019; Law and others,, 2021) and simulations of ice motion over
rough terrain at multi-kilometre scale (Hoffman and others,, 2022; Law and
others,, 2023; Liu and others,, 2023) that roughness at larger scales
significantly influences patterns of ice motion. Deformation rates are highly
variable in proximity to the bed (Fig. 4i) with basal slip ranging between
5%-95% of total surface velocity (Fig. 4h) and much higher slip rates over
topographic highs (Law and others,, 2023). Existing soft- and hard-bed sliding
theories and parameterisations do not incorporate these processes – yet the
limited studies that have investigated its influence on bed properties suggest
a notable control (Wilkens and others,, 2015; Gillet-Chaulet and others,,
2016; Falcini and others,, 2018).

In polythermal glacier and ice-sheet settings, a layer of lower-viscosity
basal temperate ice (Fig. 4j) may furthermore modulate the complex motion
patterns caused by topographic roughness (Krabbendam,, 2016; Law and others,,
2023), with greater thicknesses of temperate ice found in topographic troughs
than peaks, and the formation of a shear band at the top of the temperate zone
(Law and others,, 2023; Liu and others,, 2023). The rheology of temperate ice,
and in particular temperate ice with an impurity content, is very poorly
constrained and possibly even linear viscous (Lliboutry,, 1971; Adams and
others,, 2021), making future studies into temperate ice rheology an urgent
objective.

The influence of roughness on sliding can be theorised through form drag.
Kyrke-Smith and others, (2018) consider the influence of increasing the
spatial resolution of a numerical model from 5 km to 0.5 km (and therefore
also increasing the fidelity of the underlying topography) on the basal
traction obtained from an inversion procedure. Intuitively, Kyrke-Smith and
others, (2018) find that as spatial resolution is decreased, basal traction
increases. Form drag is then described as the area-averaged basal traction
with high topographic fidelity subtracted from the area-averaged basal
traction with low topographic fidelity, i.e. the basal traction arising from
topographic obstacles not explicitly represented by the basal boundary
position. This clearly aligns with the idea that form drag can be theorised as
part of the inner-flow which collapses to the modelled bed surface (Section
2). At a much smaller scale, Minchew and Joughin, (2020) comment on Zoet and
Iverson, (2020) and invoke form drag as the viscous resistance that arises as
ice deforms over entrained clast ‘micro-topography’ in laboratory soft-bed
sliding experiments before the sediment reaches its yield strength described
previously in Section 3.1.

Form drag may therefore be a very useful concept in descriptions of glacier
sliding, but its use in glaciology differs fundamentally from form drag as
originally described for aerodynamic applications. The differences are
sufficiently large that it may be more appropriate to use a different phrase
altogether but we maintain the the phrase ‘form drag’ for continuity with
previous studies. In aerodynamic applications, form drag refers to the force
acting on a solid body moving through a viscous and locally-turbulent fluid
that is opposite and parallel to its velocity and which is not accounted for
by induced drag, where the induced drag is a force resulting from turbulent
vorticies connected to the lift on the body (Batchelor,, 1967, sections 5.11,
7.8). Form drag has also been used in atmospheric and ocean dynamics, where it
typically refers to the entire drag force resulting from turbulence around an
obstacle or obstacles fixed to the surface, rather than a division of it (e.g.
Arya,, 1973; Renfrew and others,, 2019; Jagannathan and others,, 2023).
Difficulties are still encountered when applying the concept of form drag to
settings where there is a fixed continuous surface however, even if turbulence
is present (MacCready and others,, 2003).

Under the inner-outer flow framework described in Section 2 we fully define
form drag as the viscous creep contribution to the inner flow not accounted
for by slip processes occurring directly at the ice-bed interface or within
subglacial sediment. In this manner the resistance from form drag is

$\hat{\mathcal{F}}=\tau_{b}-\hat{\mathcal{P}_{s_{1}}}-\hat{\mathcal{P}_{s_{1}}}-...-\hat{\mathcal{P}_{s_{n}}}$
(7)

where $\hat{\mathcal{P}_{s}}$ terms are traction provided by sediment or slip
processes and which can be expanded to an area $R$ in the same manner as Eq.
5. The contribution of form drag to sliding velocity then follows Eqs. 2 and
4. We consider temperate ice processes (this Section) and basal ice layer
processes (Section 3.6) within the inner flow to be components of form drag
such that

$\hat{\mathcal{F}}=\hat{\mathcal{P}_{d_{1}}}+\hat{\mathcal{P}_{d_{2}}}+...+\hat{\mathcal{P}_{d_{n}}}$
(8)

where $\hat{\mathcal{P}_{d}}$ terms refer to sliding sub-processes occurring
within the ice itself within the inner flow. Eq. 8 can also be expanded to an
area $R$ or to describe the sliding velocity component in the established way.

###  3.4  Stick-slip and continuous sliding

In most cases, glacier and ice-sheet models neglect acceleration in the
Navier-Stokes equations and treat ice as inelastic. These generally reasonable
assumptions at typical modelling time-scales require that basal traction and
basal velocity are connected through a continuous relationship in the form of
Eq. 1 and that rapid (e.g. order of seconds) changes in velocity are not
accounted for. However, these requirements are sometimes inconsistent with
field observations of sliding, where near-instantaneous stick-slip behaviour
(Fig. 4d) is frequently recorded by seismometers and geophones as basal
icequakes (e.g. Neave and Savage,, 1970; Walter and others,, 2013) and also by
direct observation at the glacier bed (Theakstone,, 1967; Hubbard,, 2002).
Icequakes are also generated englacially by processes such as extensional
faulting near the surface, but these phenomena are not covered here.

Basal icequakes occur due to a rapid release of elastically stored energy,
resulting in ice motion that is transiently far higher than the long-term
average (Winberry and others,, 2009; Podolskiy and Walter,, 2016). Moment
magnitudes have been recorded from negative in alpine settings (Helmstetter
and others,, 2015) to magnitude seven at Whillans Ice Stream in West
Antarctica (Wiens and others,, 2008). For a basal icequake to occur, the local
frictional shear strength of the bed must be lower when the sliding interface
is in motion than when it is static, i.e. $\mu_{t}$ must be lower than
$\mu_{s}$ (Bahr and Rundle,, 1996; Rice and others,, 2001). Stick-slip is
usually associated with diurnal or tidal variations in subglacial water
pressure allowing a clear build-up period for elastic energy (Bahr and
Rundle,, 1996; Bindschadler and others,, 2003; Walter and others,, 2008;
Stevens and others,, 2024) but basal icequakes are also recorded outside of
such cycles (Hubbard,, 2002). Stick-slip behaviour is usually described
through a rate-and-state framework (Rice and others,, 2001) – as often applied
in geological-fault settings (Gomberg and others,, 2000; Appendix D) – where a
state variable, $\psi$, is introduced to account for the strength of the fault
(Rice and others,, 2001; van den Ende and others,, 2018), usually based on the
slip displacement (Ruina,, 1983) or its time-dependent evolution (Dieterich,,
1979) giving

$\tau_{b}=f(u_{b},N,\psi)\>.$ (9)

In contrast to Eq. 1, the expectation is that $u_{b}$ in Eq. 9 can vary
rapidly in both time and space. A rate-and-state friction relationship can
still account for the aseismic steady slip of a fault (van den Ende and
others,, 2018) but is not well-suited for models entirely excluding rapid
velocity changes and elastic behaviour by design.

Quantifying the contribution of stick-slip basal sliding to glacier motion is
challenging due to large uncertainties in fault dimensions, slip displacement,
and stress calculated from seismic data (Abercrombie,, 2015). Some estimates
suggest stick-slip accounts for all basal sliding over an annual period
(Helmstetter and others,, 2015), while in other cases, coseismic and aseismic
regions coexist (Barcheck and others,, 2020; Kufner and others,, 2021) –
analogous to coexisting seismogenic and creeping faults in actively deforming
geological faults (e.g. Azzaro and others,, 2020).

Despite the growing interest in stick-slip motion in glaciers (e.g. Podolskiy
and Walter,, 2016; Zoet and others,, 2020) there are limited connections
between rate-and-state theory (of form Eq. 9) and continuous basal traction
theory relationships (of form Eq. 1). We are only aware of the statistical
mechanical approach of Bahr and Rundle, (1995), where the glacier sole is
represented as numerous elastically interconnected blocks with stochastically
varying roughness parameters and the similar but simplified 1D approach of
(Köpfli and others,, 2022). Bahr and Rundle’s approach, intended for thin,
steep, valley glaciers, finds that the average total force on an individual
block (including elastic forces from adjacent blocks) is proportional to the
area-average basal traction, $\tau_{b}$. If this holds for greater ice
thicknesses and larger scales then stick-slip behaviour alone will not
invalidate continuous basal traction relationships over a sufficiently large
grid cell, but much further work is required to integrate this theory with
recent developments in understanding the stick-slip behaviour of glaciers, and
how this can relate to the spatial scales of numerical model grid cell
resolution.

###  3.5  Overlap between soft- and hard-bed sliding

Most glacier and ice sheet modelling studies use basal traction relationships
that implicitly assume the bed is either soft or hard across the entire domain
with the ostensibly hard-bed Weertman-type sliding relationship (or
equivalently, pseudo-plastic sliding with $2\geq m\geq 4$) being the modal
choice in ISMIP6 experiments. In some situations a hard-soft division is
appropriate, for example most studies are in agreement that extensive,
relatively planar sediment lies beneath the ice streams draining into the
Ronne and Ross ice shelves in Antarctica (Tulaczyk and others,, 2000; Vaughan
and others,, 2003; Peters and others,, 2007), while geographically limited
areas of Canadian continental shield may qualify as ‘true’, relatively planar,
sediment-free hard beds (Slaymaker and Kovanen,, 2017). Many individual valley
glaciers can also be reasonably categorised as predominantly soft- (e.g.
Murray and Porter,, 2001) or hard-bedded (e.g. Hubbard,, 2002) (Section 6.1).
However, taking Isunnguata Sermia in west Greenland as one well-studied
example, borehole data mostly from topographic highs are used to suggest hard-
bed conditions are dominant (Harper and others,, 2017; Maier and others,,
2019) while nearby seismic surveys are used to suggest the opposite (Booth and
others,, 2012; Dow and others,, 2013; Kulessa and others,, 2017). Meanwhile,
ice-marginal studies in west Greenland (Klint and others,, 2013) find a
complex mix of bedrock, sediment-filled depressions, and extensive sandurs
filling valley bottoms in front of the major land-terminating outlets (Grocott
and McCaffrey,, 2017). In the case of the seismic survey of Kulessa and
others, (2017) the recorded seismograms also allow for hard-bed conditions at
topographic highs which would be in agreement with Harper and others, (2017).
Similarly, recent work at Thwaites Glacier in Antarctica further suggests the
coexistence of soft- and hard-bed regions over distances less than 10 km
(Jordan and others,, 2023), supported by distinct along-flow variations in
roughness (Holschuh and others,, 2020). While soft- or hard-bed categories can
be a reasonable distinction for individual glaciers, all available evidence
indicates that a more appropriate default for present and palaeo ice-sheets is
a mixed continuum between the two end-members.

Some recent attention has been focused on the interaction between soft- and
hard-beds, with Koellner and others, (2019) considering 2D flow over
sinusoidal beds with wavelengths from 8-60 km and setting the value of $m$ in
the Weertman-type sliding relationship (Table 1) to 8 to approximate
regularised-Coulomb sliding in sinusoid troughs, while keeping it at 3 over
the sinusoid peaks. This demonstrates that a hybrid $m=3\leftrightarrow 8$
Weertman rheology results in a modelled glacier response between $m=3$ and
$m=8$ end-members. Given sliding rates at the modelled ice-bed interface can
vary significantly over much shorter distances (Law and others,, 2023) than
considered in Koellner and others, (2019), further work is needed to
effectively isolate the influence of realistic variations in soft- and hard-
bed sliding, and to assess whether hard-bed highs and soft-bed depressions are
a reasonable model for the geologic composition of the Greenland and Antarctic
ice sheets. Joughin and others, (2010) also apply a plastic sliding
relationship to inferred soft-bedded regions and Weertman-type sliding with
$m=3$ to inferred hard-bedded regions Joughin and others, (2009) at Pine
Island Glacier, Antarctica, finding good agreement in this mixed model over an
8-year period (Section 5).

The recent suggestion that similar forms of the empirically derived sliding
relationships for hard-bed cavitation (Schoof,, 2005; Gagliardini and others,,
2007) and soft-bed sediment deformation present a ‘universal glacier slip law’
(Minchew and Joughin,, 2020), excluding the other sliding sub-processes
discussed in this paper merits reconsideration. It is challenging to reconcile
the relationships of Zoet and Iverson, (2020) and Helanow and others, (2021)
without requiring an unrealistically low friction angle in Zoet and Iverson,
(2020) (Appendix A), indicating that the set-up of Zoet and Iverson, (2020)
produces a stronger bed than Helanow and others, (2021) under equivalent
effective pressure and velocities. Despite the general conception that soft
beds are weaker than their hard-bed equivalents (e.g. Koellner and others,,
2019; Joughin and others,, 2009), the strength difference may be explained by
(i) a natural requirement for lower subglacial water pressure in hard beds (by
around 4%), which is supported by available evidence (e.g. Engelhardt and
others,, 1990; Doyle and others,, 2018), (ii) non-negligible ice-rock
interface friction, which is not included in Helanow and others, (2021), or
(iii) unaccounted for form-drag occurring in most hard-bed settings that leads
to a view of a stronger bed. There is some ground to cover yet to reconcile
these differences, which may still not necessarily constitute a truly unified
sliding relationship incorporating all sliding sub-processes (Section 6.4).

Last, Tsai and others, (2021) take an agnostic approach to the soft- and hard-
bed division, focusing instead on the region of the bed where the hydrological
system is expected to be active (where $\tau_{b}$ is locally set to 0) and
inactive (where $\tau_{b}$ is locally determined using Weertman-type sliding).
Under this formulation Tsai and others, (2021) find a relationship similar to
Budd and others, (1984) sliding (their Eq. 6) which matches well with annual
variations in velocity and moulin discharge at a land-terminating sector of
the Greenland Ice Sheet. A simplification of this relationship may be well-
applicable in large-scale models (Section 6.4).

###  3.6  The basal ice layer and ice rheology

A basal ice band of a distinctly different nature to clean meteoric ice,
referred to here as the basal ice layer (Figs. 4f, 7), but which may also be
called the frozen fringe (Meyer and others,, 2018), is frequently a feature of
the lowermost section of glaciers and ice sheets. The basal ice layer is
characterised by entrained debris and the diagenetic modification (or
metamorphism) of meteoric ice by hydrologic processes, melting, refreezing,
and intense strain (see Hubbard and Sharp,, 1989; Knight,, 1997; Souchez and
others,, 2000; and Hubbard and others,, 2009 for comprehensive reviews). A
basal ice layer is observed in deep ice-divide Greenlandic and Antarctic ice
sheet ice cores (e.g. Gow and others,, 1979; Gow and Meese,, 1996; Souchez and
others,, 1998; Tison and others,, 1998; Souchez and others,, 2002),
Greenlandic and Antarctic ice sheet margins (e.g. Swinzow,, 1962; Tison and
others,, 1993), and across many alpine and ice-cap settings (e.g. Hubbard and
Sharp,, 1995; Lawson and Kulla,, 1978). Proposed formation mechanisms for the
basal ice layer are basal freeze on entraining sediment through regelation (at
a scale of up to around one meter) (e.g. Weertman,, 1961; Hubbard and Sharp,,
1993), a pressure-driven heat pump of temperate ice (up to a scale of several
meters) (Robin,, 1976), or movement of subglacial meltwater along hydro-
potential pathways to a region of the glacier bed below the pressure melting
point (e.g. Bell and others,, 2011). Basal melting associated with frictional
heat generated through slip is expected to decrease the thickness of the basal
ice layer and impede its growth (Hubbard and Sharp,, 1989; Knight,, 1997).
However, large basal ice layer thicknesses (up to 30 m) are recorded at the
western terminus of the GrIS (Fig. 6, Knight and others,, 2002) and it is not
conclusive if these sequences formed from freeze on in the immediate vicinity
of the margin, or survived passage through up to 100 km of thawed bed
conditions and slip-driven basal melt (MacGregor and others,, 2016). Recent
modelling and radar mapping of Antarctica suggests thick (>100 m) sequences of
sediment-laden basal ice may persist for tens of kilometres at least beyond a
transition from frozen to thawed bed settings (Franke and others,, 2024) but
further work is required to fully explore the parameter space of these
processes. Further, spatially and temporally varying basal ice characteristics
ranging from debris-rich ice to frozen sediment, will blur the distinction
between soft- and hard-bedded regions.

Figure 7: The basal ice layer at Russel Glacier, Greenland. From Knight and
others, (2002) showing characteristic stratigraphy.

Shear-ring laboratory experiments for the deformation of a 1-2 cm thick ice-
sediment melange between thawed sediments and sediment (Hansen and others,,
2024) suggest similar regularised-Coulomb behaviour to shear-ring experiments
of clean ice over sediment (Zoet and Iverson,, 2020). Meyer and others, (2018)
also construct an analytical model arguing that ice infiltration into
previously thawed sediments limits bed strength (in effect, plastic sliding)
in pervasively soft-bedded regions. Beyond these studies however, we are not
aware of experiments considering basal ice sequences exceeding a few cm in
thickness or the large diversity of facies found in the basal ice layer and
their interaction (Hubbard and others,, 2009).

Four further rheological factors present significant complications for ice
deformation in the vicinity of the bed and are hence captured at least
partially within the inner flow. First, pre-Holocene ice with a higher dust
concentration and typically smaller grain size deposited during the last
glacial period deforms at a rate around 2.5 times that of Holocene ice under
the same stress and temperature conditions (Paterson,, 1991). Such ice is
widely present in the strata of the Greenland and Antarctic Ice Sheets
(Macgregor and others,, 2015; Winter and others,, 2019; Ashmore and others,,
2020) but uncommon in glaciers and ice caps where it is limited to very slow
moving areas (Thompson and others,, 1997), though valley glaciers generally
feature much higher rates of englacial debris impurities (e.g. Goodsell and
others,, 2005). Second, ice at depth has a highly anisotropic rheology due to
the development of a strong crystallographic preferred orientation under
consistent uni-directional deformation (Lile,, 1978; Baker,, 1981; Duval,,
1981; Wilson and Sim,, 2002). Such mechanical anisotropy presents major
challenges in implementation, bench-marking, and interpretation within
numerical modelling (Martín and others,, 2004; Gillet-Chaulet and others,,
2005) but is nonetheless likely of significant importance within viscous
deformation of ice in the inner flow where strain rates are both high and
spatially complex (Law and others,, 2023). To our knowledge, this aspect of
glacier sliding is not covered by any existing glaciological literature.
Third, folds close to the bed may arise from mechanical differences in ice
layers (Whillans and Jezek,, 1987; Bons and others,, 2016; Zhang and others,,
2023) with the possibility to significantly influence the bulk rheological
behaviour of the inner flow. Fourth, while $n=3$ is the default in glacier and
ice-sheet models (in fact already rounded down from the 3.2 reported in Glen,,
1955), there are numerous field-observation based studies suggesting that
$n=4$, representing a dislocation creep regime, is more appropriate in higher
stress settings (away from ice-sheet ice divides, for example) (Bons and
others,, 2018; Millstein and others,, 2022; Ranganathan and Minchew,, 2024)
and perhaps also at areas dominated by pure shear (Gillet-Chaulet and others,,
2011). Considering a smooth BedMachine lower boundary in northern Greenland,
$n=4$ dramatically decreases the area over which basal sliding is expected to
contribute significantly to total surface displacement (Bons and others,,
2018). The influence of $n=4$ on form drag and sliding parameterisations, and
its interaction with rough topography is to our knowledge entirely unexplored.

Where present, these rheological complications will already be implicitly and
unavoidably accounted for via inversion procedures under parameterisations
where they are not explicitly accounted for. The potential heterogeneity of
their interactions creates significant complexity that future work should seek
to disentangle.

###  3.7  Frozen-thawed transitions and sub-temperate sliding

In some simpler theories both sliding and slip are considered to be negligible
for frozen beds (Lliboutry,, 1966; Cuffey and Paterson,, 2010, 7.1.1), but
most models either simply neglect a temperature control on sliding (including
all models inverting for basal traction with present-day velocity fields in
ISMIP6 experiments) or take a heuristic approach to scaling basal traction
with temperature to avoid a stress singularity at the frozen thawed boundary
(e.g. Tarasov and Peltier,, 2007; Mantelli and others,, 2019) giving

$\tau_{b}=f(u_{b},N,T_{h})$ (10)

as an adaptation of Eq. 7 where $T_{h}=T_{b}-T_{pmp}$ is the homologous
temperature calculated from the pressure melting point, $T_{pmp}$, and ice
temperature at the base, $T_{b}$. This corroborates with field studies finding
slow (<0.2 m a-1) slip rates even with temperatures some degrees below
freezing for both soft- and hard-bed settings (Hallet and others,, 1986;
Echelmeyer and Zhongxiang,, 1987; Cuffey and others,, 1999; Fitzsimons and
others,, 1999). The mechanical process or processes responsible for sub-
temperate sliding in these instances remains uncertain (Cuffey and Paterson,,
2010, 7.2.8) but may be related to a very thin (nm) premelt layer at the ice
bed (Shreve,, 1984), the inclusion of solid friction (Fowler,, 1986), or
sediment deformation despite freezing conditions Waller, (2001). Separately,
form drag will still be expected under freezing bed conditions, though at a
slower rate than under equivalent stress conditions in colder ice. This
processes is framed by Weertman, (1967) as the continuation of enhanced creep
despite the cessation of regelation.Expanding sub-temperate sliding out to a
grid cell under consideration also allows for the situation of both freezing
and thawed areas within the same grid cell (Fig. 7), which may be a common
situation in frozen-thawed transition zones (e.g. Oswald and Gogineni,, 2012;
Dawson and others,, 2024). Generally, a pattern of greater basal traction in
interior ice-sheet regions where a frozen bed is considered more likely is
obtained in ice-sheet wide inversions (Larour and others,, 2012; Morlighem and
others,, 2013).

Figure 8: Subtemperate sliding. Triangle represents area $R$ under
consideration. Solid black line represents frozen-thawed transition at the bed
and cooler colours represent decreasing temperature of frozen portion of bed.
Sub-temperate sliding is not sufficiently well-studied to provide a definite
spatial range, but speculatively this may range from 100 m - 10 km. Adapted
from Wilch and Hughes, (2000).

##  4  Subglacial Hydrology

Subglacial hydrology and basal sliding are intimately linked in many glacial
settings (see reviews by Irvine-Fynn and others,, 2011; Ashmore and Bingham,,
2014, and Nienow and others,, 2017). Importantly, water pressure influences
sliding velocity via the effective pressure $N$ and sliding velocity can in
turn influences $N$ as changes in cavitation influence the hydraulic
conductivity and connectivity of the drainage system (e.g. Schoof,, 2010). The
subglacial hydrological system is also important for transporting and
evacuating glacial melt water, sediment, and nutrients (Swift and others,,
2005; Death and others,, 2014; Hawkings and others,, 2015; Hofstede and
others,, 2023) – each of which can be a product of sliding, via frictional
heat dissipation or subglacial erosion.

In most numerical models, subglacial hydrology is either absent or is
parameterised in a basic fashion through the effective pressure $N$ term in
the sliding relationship (Table. 1, Fig. 9) – with at least three materially
different ways of including $N$ in Eq. 1 (Table 1). While in the physical
sense the meaning of $N$ as subglacial water pressure subtracted from the
overburden pressure at a point in time and space is very clear, neither its
meaning nor ways of quantifying a representative $N$ at a given scale are
straight forward in a modelling sense. Subglacial hydrological environments
are very spatially diverse even at scales of tens of metres, with various
terms (e.g. ‘channelised; distributed; partly-connected’) referring to various
end members in a continuum of characteristics (e.g. Hubbard and Nienow,, 1997;
Hoffman and others,, 2016; Rada and Schoof,, 2018). Borehole observations
evidence this complexity, with widely varying $N$ time series observed in
adjacent boreholes (Doyle and others,, 2018; Rada and Schoof,, 2018; Doyle and
others,, 2021) capturing distinctly separate components of subglacial
hydrology, consistent with theoretical studies finding low-pressure conduits
adjacent to higher-pressure regions with reduced hydrological connectivity
(Engelhardt and Kamb,, 1997; Hewitt and others,, 2018). Moreover, subglacial
drainage systems are observed to evolve diurnally and seasonally in response
to supraglacial melt water inputs (Hubbard and Nienow,, 1997), even beneath
ice $\sim$1 km thick (Chandler and others,, 2013; Doyle and others,, 2021).
This evolution in turn leads to diurnal and seasonal variability in sliding
(Willis,, 1995; Zwally and others,, 2002; Bartholomew and others,, 2011;
Nienow and others,, 2017). Such temporal variability is not captured by ice
sheet or glacier models employing temporally constant parameters in the
friction law.

While the influence of $N$ is fairly well understood for specific sliding sub-
processes in isolation – for example, cavitation over a relatively flat
bedrock area (Helanow and others,, 2021), and sediment shearing in a
laboratory setting (e.g. Iverson and others,, 1998) or through a numerical
continuum model (e.g. Damsgaard and others,, 2020) – the influence of $N$ over
a large area encompassing a diverse set of sliding sub-processes is not well
constrained. Additionally, it is not comprehensively established how $N$
should be incorporated within Eq. LABEL:eq:1, whether through a linear
multiplication (e.g. Zoet and Iverson,, 2020; Choi and others,, 2022), raised
to a power (Budd and others,, 1984; Budd and Jacka,, 1989), or included in a
more complicated manner (Helanow and others,, 2021) – a situation that may be
more uncertain if the influence of rough topography is also incorporated.

Despite the importance of subglacial hydrology as a control on sliding at sub-
annual scales, its overall influence on long-term glacier and ice sheet
evolution remains debated, particularly with respect to future changes of the
Greenland Ice Sheet (e.g. Tedstone and others,, 2015; Doyle and others,, 2015;
Beckmann and Winkelmann,, 2023), and Antarctic ice streams (Bougamont and
others,, 2015), with major changes in glacier and ice-sheet velocity fields
typically linked to geometry changes and ocean forcing rather than intrinsic
changes in the hydrological system whether or not substantial surface melt
occurs (e.g. Vincent and others,, 2012; Habermann and others,, 2013; Moon and
others,, 2014; Cook and others,, 2016; Seroussi and others,, 2017; Catania and
others,, 2018; Dehecq and others,, 2018; Gimbert and others,, 2021; Frank and
others,, 2022) with the exception of surging glaciers (Sevestre and Benn,,
2015). However, large-scale observations do indicate that major changes in the
hydrological system – such as a switch from marine-terminating to land-
terminating outlet glaciers, or an advancement of the ablation zone – may
influence large-scale geometry and velocity patterns over a multi-annual
timescale (Maier and others,, 2022). It may therefore be reasonable to suggest
that diurnal and seasonal hydrology variations are of secondary importance to
glacier geometry and ocean forcing at the annual and greater timescales
important for future predictions. The exception to this is that the subglacial
water pressure of marine-terminating outlet glaciers grounded below sea level
is likely significantly controlled by their proximity to the ocean, with
inferred very low basal traction values and very high subglacial water
pressure very close to overburden (e.g. Stearns and Van Der Veen,, 2018;
Minchew and others,, 2019).

Figure 9: Subglacial hydrology as observed in the field or represented in an
ice flow model element.  a Cavities (pale blue), channels (dark blue lines)
and striation flow markers from fieldwork at Blackfoot Glacier (Walder and
Hallet,, 1979). b simplification to a single mesh element.

##  5  Heuristic studies

A separate approach to investigating glacier sliding is to use large-scale
remotely sensed or field data to heuristically (i.e. through process-agnostic
iterative experimentation and adjustment) determine the form of the sliding
relationship, which may then be compared to theoretical expectations (Section
3). A time series with significant variation in surface geometry is best
suited to the decadal time scales over which mass loss is most often
considered (Gillet-Chaulet and others,, 2016; Gilbert and others,, 2023), but
several other variations have been exploited (Table 2). Considered in their
entirety, these studies indicate that variation in the form of the sliding
relationship across variable glacier settings cannot be captured by a single
tunable coefficient, $C$, obtained through a standard inversion procedure. For
example, Gimbert and others, (2021) and Gilbert and others, (2023) find
$m=3.1\pm 0.3$ in Weertman-type sliding for Argentière Glacier in the French
Alps using a combination of surface velocity changes, length changes, and
direct sliding measurements while Gillet-Chaulet and others, (2016) find
$m\leq 5$ and up to 50 well reproduces surface velocities at Pine Island
Glacier, Antarctica. Maier and others, (2021) meanwhile indicate that a single
$m$ value, also for Weertman-type sliding, cannot be applied to the entire
Greenland Ice Sheet with, for example, an $m$ of 8 in northeast Greenland
contrasting an $m$ of 4 in central southwest Greenland. Note that in standard
applications using Weertman-type sliding, $C$ is the subject of the inversion
and varies in space, while $m$ is kept constant.

Setting |  Variation |  Study |  $m$
---|---|---|---
Argentière, French Alps – valley glacier‡ |  Length (velocity) over 100 (15) years to $\sim$ 2020 |  Gilbert and others, (2023) |  3
Hofsjökull, Iceland – ice cap§ |  Summer-winter velocity comparison |  Minchew and others, (2016) |  $\infty$/plastic
Greenland Ice Sheet – ice sheet |  Single velocity field averaged 2005-2015 |  Maier and others, (2021) |  3-10
Northwest Greenland – outlet glaciers |  Forward modelling of mass loss 2007-2018 |  Choi and others, (2022) |  6 in modified Budd
Northeast Greenland Ice Stream – ice stream |  Forward modelling of mass change 2006-2021 |  Khan and others, (2022) |  Regularised-Coulomb
Western Greenland – land-terminating outlet glacier |  Annual velocity and hydrology variations 2006-2013 |  Tsai and others, (2021) |  4 in modified Budd
Pine Island Glacier, Antarctica – outlet glacier |  Geometry 2003-2008 |  Joughin and others, (2010) |  mix of 3 and $\infty$/plastic
Pine Island Glacier, Antarctica – outlet glacier |  Velocity in 1996 and 2007-2010 |  Gillet-Chaulet and others, (2016) |  $\geq$5
Pine Island Glacier, Antarctica – outlet glacier |  Forward modelling of velocity field and geometry 2002-2017 |  Jouvet and Huss, (2019) |  Regularised-Coulomb
Rutford Ice Stream, Antarctica – ice stream |  Displacement over fortnightly tidal cycles |  Gudmundsson, (2011) |  3
High Mountain Asia – valley glaciers |  Velocity fields 2000-2017 |  Dehecq and others, (2018) |  4

Table 2: Heuristic studies on sliding parameters. ‡ indicates largely hard
bed, §

indicates largely soft bed. Absence of a superscript indicates ambiguous or
mixed conditions. Notes: Gudmundsson, (2011) did not test $m$ values above 3.
Dehecq and others, (2018) combine ice deformation and sliding in a 1D model.
Choi and others, (2022) also include the influence of $N$, defined as ice
pressure above hydrostatic equilibrium with a sheet perfectly connected to the
ocean, finding best fitting with $N$ included.

Such studies offer a testing ground for proving the efficacy of a given
sliding relationship, but it is important to note that a match to a given
theory does not on its own demonstrate that the physical processes that build
it up are actually occurring in the study location. This is straightforward in
the case of $m\leq 5$ from Gillet-Chaulet and others, (2016) at Pine Island
Glacier. No existing theory postulates this relationship (Section 3) so the
utility of $m\leq 5$ in Weertman-type sliding comes either from the fact that
it emulates regularised-Coulomb sliding that does have a large body of theory
behind it, or it represents hitherto unappreciated processes not presently
incorporated into a process-based sliding relationship. Meanwhile, it is
difficult to interpret the findings of Minchew and others, (2016) from a
setting known to have significant till coverage Björnsson and others, (2003)
as anything other than plastic deformation of till. In contrast, Gimbert and
others, (2021) and Gilbert and others, (2023) suggest that their findings of
$m\sim 3$ reflects only the ice deformation component of the original
Weertman, (1957) paper without the regelation component where $m=1$,
neglecting the fact that the original Weertman theory is somewhat dependent
upon a combination of both regelation and ice deformation, and making no
allowance for the slip component actively observed beneath Argentière (Vincent
and Moreau,, 2016; Gimbert and others,, 2021). Maier and others, (2021) also
invoke Weertman-type hard-bed physics as justification for lower values of
$m$, despite the fundamental issues underlying the theory (Section 3.2) and
the fact that no theory building upon Weertman, (1957), or the original study,
provide a physical justification for $n=4$ or above.

Taken collectively, heuristic studies emphasise a heterogeneity between
settings that is not easily reconciled with a single sliding relationship
controlled by a single tunable parameter. However, they also present perhaps
the most definitive method of accurately determining the optimum sliding
relationship for a given modelling setting. We suggest that development of a
standardised methodology that can be efficiently applied to varying settings
utilizing increasingly long datasets will be of great use in transparently
tailoring sliding relationships for given modelling applications.

##  6  Discussion

###  6.1  General differences between sliding in glaciers and ice-sheets

By convention, the sliding of ice sheets is treated in the same manner as the
sliding of valley glaciers from which foundational sliding theories are
typically derived (e.g. Hodge,, 1974; Engelhardt and others,, 1978; Fischer
and Clarke,, 1997; Willis,, 1995; Bouchayer and others,, 2024). However,
differences intrinsic to the two settings, and choices made in model
development, may significantly influence the combination of sliding sub-
processes occurring and those represented within a given grid cell. Valley
glaciers (i.e. those occupying a limited geographical area, not draining a
large accumulation area ice field, and comprising only a handful of
tributaries) have several features distinguishing them from both ice-sheet
interiors and ice-sheet outlet glaciers. First, valley glaciers generally
occupy a sufficiently small area such that, as a rule, major geological
boundaries are not crossed beneath glacier, making a distinction between soft-
and hard-bed sliding more reasonable (and indeed consequential, with surging
glaciers generally occurring over inferred soft beds; Cuffey and Paterson,,
2010, Section 12.2). Second, valley glaciers tend to be much steeper – though
the distribution of driving stresses is similar (Meyer and others,, 2018) –
with underlying topography in the mountainous regions where valley glaciers
are present typically rough, compared to the more mixed geomorphology that
underlies present and palaeo ice sheets. Third, valley glaciers generally do
not reach the high velocities that characterize the large outlet glaciers
important for dynamic changes in ice sheets with velocities only exceeding 100
m a-1 for large glaciers comprising an extensive accumulation area and with
several tributaries (Laumann and Wold,, 1992; Jouvet and Huss,, 2019). Fourth,
if valley glaciers do terminate in lakes or the ocean then the glacier
immediately inland tends to be much steeper than ice-sheet outlet glacier
counterparts meaning the hydrological influence of the water body also
decreases rapidly inland.

The net effect of these factors, and the generally much greater resolution in
modern models of valley glaciers (cf. Aschwanden and others,, 2019; Gilbert
and others,, 2020) is that that the processes represented by a single grid
area, $R$, can differ substantially between glaciers and ice sheets (Fig. 2).
This is born out in the limited evidence from heuristic studies (Section 5)
where all valley glaciers investigated in this manner are found to be better
represented by unbounded sliding relationships (Table 2). In fact, we are not
aware of valley glaciers being represented by a regularised-Coulomb
relationship in any instance. Care is needed therefore, in applying the
findings of sliding studies conducted at valley glaciers (Gimbert and others,,
2021; Gilbert and others,, 2023) to ice sheets, and vice versa (Tsai and
others,, 2021), in the same manner that care is needed in applying laboratory
and numerical modelling studies to glacier and ice sheet settings, even though
the individual sub-processes will remain valid in both settings.

###  6.2  Threshold velocities for regularised-Coulomb sliding

In Weertman-type sliding two parameters ($C$ and $m$) control the
relationship, but even in the simplest regularised-Coulomb relationship at
least one further parameter is required ($u_{t}$ in our usage in Table 1) to
set the transition point to bounded traction. Laboratory experiments over
sediment indicate a transition to bounded traction occurs at around 50 m a-1
(Zoet and Iverson,, 2020) in a controlled setting, while the transition for
cavitation over hard beds at small scales (<25 m) is more variable, and
possibly as low as 5 m a-1 (Helanow and others,, 2021). Given that uniformly
smooth soft beds of consistent material properties, or hard beds of consistent
geometry, are somewhat unlikely, it becomes complicated to suggest a
reasonable physically based $u_{t}$ value that can be universally applied and
we therefore recommend avoiding arbitrary assumptions on its value. Joughin
and others, (2019) set a threshold velocity of 300 m a-1 for Pine Island
Glacier, Antarctica, without physical justification, which sets a clear
spatial boundary for the region of bounded traction. As bounded traction
significantly increases the sensitivity of outlet glaciers to climate forcing
(Ritz and others,, 2015; Tsai and others,, 2015; Brondex and others,, 2019),
further work determining the factors controlling the threshold velocity
(Trevers and others,, 2024) is necessary, though we note that the increased
complexity inherent to a regularised-Coulomb sliding may make it less
versatile when applied universally (Section 6.5). We further note that in
marine-terminating settings it may be difficult to distinguish whether the
$u_{t}$ value used results in a good match with observations because of an
intrinsic switch in processes beyond the threshold velocity, or due to the
increasing influence of the ocean on subglacial hydrology, with ice velocity
simply acting as a proxy for proximity to the coast.

###  6.3  Revisiting Iken’s bound

The assertion by Lliboutry, (1968), Iken, (1981), Schoof, (2005), Gagliardini
and others, (2007) and others that sliding relationships of the form Eq. 1
must be bounded following Eq. 6 is based on two-dimensional topography
comprised of compound sinusoids (a ‘washboard topography’, Fig. 10a). In these
studies an upper size for subglacial cavitation is not theorised and so
cavitation is implicitly assumed to hold at all scales. However, in realistic
three-dimensional topography there are many more pathways for pressurised
water to escape from cavities (Gimbert and others,, 2021; Helanow and others,,
2021) which may naturally taper the number of large cavities as they grade
into relatively scarce subglacial lakes (Bowling and others,, 2019). If large
topographic obstacles are present, but large-scale cavities are not
concomitant, form drag may come to surpass ‘classical’ cavitation as the
dominant sliding process in a given grid cell area (Fig. 10). At present, no
studies outline the size-distribution of cavities that may occur with
realistic three-dimensional bed geometries, but cavitation studies generally
do not insinuate cavities larger than 15 m (e.g. Vivian and Bocquet,, 1973;
Walder and Hallet,, 1979; Helanow and others,, 2021). Establishing this
distribution and its interplay with form drag and grid cell area is an urgent
area for future research. We suggest that the unsuitability of Iken’s bound in
regions of large topographic obstacles may be the main reason for the
continued efficacy of a Weertman-type sliding relationship in ice-sheet
settings (e.g. Maier and others,, 2021) despite the questionable underlying
assumptions (Section 3.2), if form drag around obstacles is unmitigated by
large cavities bounding the potential traction of the area.

Figure 10: A spatial bound to Iken’s bound? a Cavitation (blue) occurring
where topography is sufficiently small adapted from Schoof, (2005). b
Cavitation can occur at a sufficiently small scale (yellow highlight) but will
not occur in larger wavelength depressions. z is height from baseline and x is
distance along flow. The thick black line is the glacier bed. $R$ represents a
possible region in consideration for the sliding relationship. This figure is
drawn in two dimensions for comparison with previous studies, but the same
principles apply in three dimensions with realistic topography.

###  6.4  Does a unified basal sliding relationship exist? What would it look
like?

A defining objective of much research into sliding is to uncover a ‘universal’
sliding relationship which can provide an accurate description of sliding
processes across all settings, scales, and time periods with the frequently
implicit assumption that this is controlled by a single tunable coefficient
(henceforth single coefficient) and/or calculated subglacial water pressure
(e.g. Minchew and Joughin,, 2020). Lamentably, we suggest that a simple
universal sliding relationship defined by a single coefficient may not be a
reasonable prospect, whether based on physical processes or heuristic
modelling. The complex nature of myriad sliding sub-processes and their scale
and setting dependence (Section 3) present no intrinsic physical reason why
they should neatly fall within a single-coefficient relationship and it is not
possible to control the shape of the available range of sliding relationships
(Sections 3, 5) through a single parameter (Fig. 11). It follows that we do
not hold that similarities in the sliding relationships for small-scale
cavitation (Helanow and others,, 2021) and ring-shear sediment sliding (Zoet
and Iverson,, 2020) indicate that a unified sliding relationship has been
obtained (Minchew and Joughin,, 2020). Instead, the studies reviewed here
point instead to a compound relationship that bears closer resemblance to
descriptions of ice strain comprising several deformation mechanisms (e.g. Eq.
3 of Goldsby and Kohlstedt,, 2001). As a purely illustrative example,
$\tau_{b}$ could, at a given setting and scale, be well represented by

$\tau_{b}=f_{w}(u_{b})+f_{rC}(u_{b})$ (11)

where $f_{w}$ and $f_{rC}$ are Weertman-type and regularised-Coulomb sliding
respectively, each controlled by distinct $m$ and $C$ values.

Figure 11: The issue of unification. As for Fig. 1 but where a is the maximum
traction for a bounded relationship, b is the threshold velocity for a bounded
relationship, and c is the slope of an unbounded relationship or a switch to a
bounded relationship.

However, as standard inversion procedures are ill-suited to determining
multiple parameters simultaneously (Morlighem and others,, 2010) and as a
compound sliding relationship (Eq. 11) would complicate implementation, the
choice between a compound friction law (e.g. Eq. 3) and single-term equation
from Table 1 is a compromise between a more realistic representation of
friction and less practical numerical implementation. The path to a ‘unified’,
or in more realistic terms ‘optimum’, sliding relationship could therefore
come by disregarding process-based models for traction and the development of
an efficient, transparent, and easily applicable method for determining the
best parameters at a more setting-specific scale. Nonetheless, while this
approach could address several pressing issues in glacier and ice-sheet
modelling, it is still important to grasp the physical underpinnings to
produce tractable predictions of sea level rise and to asses how a change in
physical conditions can influence sliding. If a process-based sliding
relationship is sought, we suggest de-emphasizing a single process in favor of
the interplay between multiple sub-processes. Additionally, an optimum sliding
relationship should conform to the following:

  * •

$\bullet$ A clear breakdown of which sliding sub-processes are considered, and
the conditions under which a given sub-process becomes dominant for overall
traction.

  * •

$\bullet$ Explicit inclusion of form drag.

  * •

$\bullet$ An assessment of scale dependence and the spatial range over which
the relationship can be considered valid.

  * •

$\bullet$ An indication of the position of the inner-outer flow boundary, and
if this will vary based on the conditions of the bed. This is not
straightforward and may be difficult to determine, but transparent inclusion
is preferable to omission.

A very broad range of methodologies have been used to investigate sliding
(Section 3), but it may be that an empirical numerical-modelling approach
(e.g. Helanow and others,, 2021), building on the results of more granular
studies of sub-processes, is best suited to addressing the necessarily large
parameter space inherent to this question over a variety of scales.

###  6.5  Implications for glacier and ice-sheet modellers

####  6.5.1  Which basal sliding relationship to use?

No single sliding relationship accounts for the full breadth of processes or
observations that comprise glacier and ice-sheet sliding (Sections 3, 5),
making it challenging to recommend one single-tunable-coefficient sliding
relationship. Pending future investigations, we make the following suggestions
based solely on heuristic studies (Section 5), whose empirically-validated
nature makes them more suitable for application to glacier and ice-sheet
models.

We suggest that for ice sheets Weertman-type sliding (or equivalently, pseudo-
plastic sliding), with a setting-dependent $m$, presents the least-bad option.
Given information from the limited number of studies available (Table 2) lower
$m$ values ($\leq 4$) are then better suited for rougher settings inferred to
be predominantly hard-bedded and higher $m$ values ($\geq 5$) are better
suited for regions where topography is extensively smooth and the setting is
inferred to be largely soft-bedded. Therefore, while $C$ is the main object of
the inversion procedure, $m$ may be assigned based on additional constraints,
such as topographic roughness. Where applied, high $m$ values approximating
regularised-Coulomb-type sliding should feature careful justification for
their use, particularly given the influence of bounded traction (and
implicitly unbounded sliding) on ice-sheet stability (e.g. Parizek and
others,, 2013; Ritz and others,, 2015; Lippert and others,, 2024). The
particular mapping of parameters in the pseudo-plastic relationship to easily
understandable outcomes in $u_{b},\tau_{b}$ space is also preferable to a link
to quantities such as till friction angle, which implies a greater physical
meaning in the sliding relationship than may actually be the case, though the
outcome is the same. Except for where a clear need is identified, regularised-
Coulomb sliding may be less optimum due to its dependence on at least three
tuning parameters, and its inability to represent unbounded traction (unless
unrealistically high $u_{t}$ values are used).

If hydrology is to be incorporated, we recommend a simplified implementation
of Budd sliding as used in Choi and others, (2022) and set as the default for
ISSM (Larour and others,, 2012), which is functionally the same as Weertman-
type sliding but with $N$ included as a multiplication factor. It is possible
that this represents a more physically based alternative to regularised-
Coulomb sliding at marine terminating glaciers (Section 6.2). A simplified
hydrological model as used in Choi and others, (2022) based on elevation above
sea level may then be reasonable over annual timescales (Section 4), though we
stress that this is a pressing area for future research – as is exact way in
which $N$ should be incorporated within Eq. LABEL:eq:1.

Separately, it seems reasonable to suggest that Weertman-type sliding with
$3\leq m\leq 4$ is a a fairly appropriate choice for most if not all valley
glaciers, and particularly for studies covering a very large number of valley
glaciers in a simple manner (Dehecq and others,, 2018; Cook and others,,
2023), with no contradictory evidence that we are aware of.

####  6.5.2  A note on terminology

We hope that this paper provides a clear description of the inner flow sub-
processes that a sliding relationship will account for under ideal
circumstances, and may also ease some of the angst experienced by modelling
practitioners (ourselves included) in labelling bed properties. When we invert
for basal properties we are not solely inverting for ‘roughness’ (e.g. Berends
and others,, 2023), ‘till friction angle’ (e.g. Sutter and others,, 2021), or
‘friction’ at the ice-bed interface (e.g. Sergienko and Hindmarsh,, 2013) – we
are likely inverting for all of these processes, and several more,
simultaneously. We use ‘ideal circumstances’ here because the traction field
is often one of the few tunable parameters in an ice-sheet model and so will
in practice also account for so-called ‘compensating errors’ arising from
incorrectly prescribed englacial temperature, ice viscosity, subglacial
topography, and surface and basal mass balance (Berends and others,, 2023). We
propose ‘basal traction’ as a least-bad descriptor here as its more abstract
nature does not feature the same physical-process misunderstanding that comes
with terms such as basal friction which appears to imply slip at the ice-bed
interface rather than sliding as defined here. We further suggest that
‘sliding relationship’ is more appropriate than ‘sliding law’. Physical laws
may hold up well at atomic or astronomical scales, but the overlapping nature
of phenomena in Earth Sciences applications makes a single ‘law’ unrealistic.
Similarly, ‘Weertman-type’ and ‘regularised-Coulomb-type’, or ‘unbounded’ and
‘bounded’ are preferable to ‘Weertman’ or ‘regularised-Coulomb’ which
emphasise a physical basis that may not exist. Last, while we use ‘glacier
sliding’ throughout this paper to remain in keeping with its common usage, but
make a distinction with ‘slip’ to refer strictly to ice-bed displacement,
‘glacier basal motion’ (e.g. Waller,, 2001; Hubbard,, 2002; Law and others,,
2023) is a more precise term.

##  7  Conclusions

Glacier sliding is not a straightforward topic. Here, we have done our best to
integrate existing understanding into a relatively straightforward framework.
This is centred upon an inner flow (Fig. 5) comprised of potentially numerous
sliding sub-processes (Fig. 4), which in ideal circumstances are accurately
captured by the sliding relationship used in numerical models and which allows
for the co-existence of soft- and hard-bed sliding, stick-slip and continuous
sliding, and form drag. However, while this framework provides a pathway for
incorporating these processes into an ‘optimum’ process-based sliding
relationship, the existing range of heuristic studies (Table 2) likely
indicates that a simple ‘unified’ sliding relationship controlled by a single
coefficient is not a realistic outcome. Nonetheless, Weertman-type sliding, or
simplified Budd sliding where hydrology is included, with an $m$ determined
based on setting, preferably with the use of sensitivity studies, is a
reasonable working assumption based on available observational evidence.

Despite obvious obstacles there are clear priorities for future work.
Principally, determining a simple universally deployable methodology for
determining $m$ across a wide range of settings and formally reconciling
process-based studies with heuristic studies to understand the physical
reasoning for the efficacy of the sliding relationships outlined in Section 5.
Overall, we hope that the framework presented here allows for a transparent
approach to understanding glacier sliding, applying glacier sliding
relationships to numerical models, and for directing future research.

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