tc 9 675 2015

tc 9 675 2015
The Cryosphere, 9, 675–690, 2015
www.the-cryosphere.net/9/675/2015/
doi:10.5194/tc-9-675-2015
© Author(s) 2015. CC Attribution 3.0 License.
Thermal structure and basal sliding parametrisation at Pine Island
Glacier – a 3-D full-Stokes model study
N. Wilkens1,2 , J. Behrens3 , T. Kleiner2 , D. Rippin4 , M. Rückamp2 , and A. Humbert2,5
1 Institute
for Geophysics, University of Hamburg, Germany
Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany
3 Numerical Methods in Geosciences, University of Hamburg, Germany
4 Environment Department, University of York, Heslington, UK
5 Department of Geosciences, University of Bremen, Germany
2 Alfred
Correspondence to: N. Wilkens (nina.wilkens@zmaw.de)
Received: 21 August 2014 – Published in The Cryosphere Discuss.: 16 September 2014
Revised: 16 March 2015 – Accepted: 17 March 2015 – Published: 13 April 2015
Abstract. Pine Island Glacier is one of the fastest changing
glaciers of the Antarctic Ice Sheet and therefore of scientific
interest. The glacier holds enough ice to raise the global sea
level significantly ( ∼ 0.5 m) when fully melted. The question addressed by numerous modelling studies of the glacier
focuses on whether the observed changes are a start of an
uncontrolled and accelerating retreat. The movement of the
glacier is, in the fast-flowing areas, dominated by basal motion. In modelling studies the parametrisation of the basal
motion is therefore crucial. Inversion methods are commonly
applied to reproduce the complex surface flow structure of
Pine Island Glacier by using information of the observed
surface velocity field to constrain, among other things, basal
sliding. We introduce two different approaches of combining
a physical parameter, the basal roughness, with basal sliding
parametrisations. This way basal sliding is again connected
closer to its original formulation. We show that the basal
roughness is an important and helpful parameter to consider
and that many features of the flow structure can be reproduced with these approaches.
1
Introduction
In the past decades the fastest changes in ice flow velocity,
ice thickness and grounding line retreat in the Antarctic Ice
Sheet have been observed in the region of Pine Island Glacier
(PIG), Amundsen Sea Embayment, West Antarctica (Rignot, 2008, 1998; Wingham et al., 2009; Joughin et al., 2010;
Park et al., 2013; Helm et al., 2014). Additionally, the currently observed mass loss from the Antarctic Ice Sheet is also
concentrated in the area around PIG (Horwath and Dietrich,
2009; Shepherd et al., 2012). Thus PIG shows an increased
contribution to global sea level rise (Mouginot et al., 2014).
The bed lies below sea level in large areas, making it part
of a so-called marine ice sheet. In combination with a retrograde bed, which slopes down from the ocean towards the
centre of the glacier, this setting was postulated to be intrinsically unstable by the so-called “Marine Ice Sheet Instability”
hypothesis (Hughes, 1973). This hypothesis is still up for debate (Vaughan, 2008; Gudmundsson et al., 2012), while the
trigger for the changes is thought to be enhanced ocean melting of the ice shelf (Dutrieux et al., 2014).
The dynamics of PIG are crucial to its future behaviour
and therefore to its contribution to sea level rise. Due to the
fast changes observed at PIG, a variety of modelling studies have been conducted. These studies address questions focusing on the sensitivity to changes in external conditions
(ice shelf buttressing, basal conditions) (e.g. Schmeltz et al.,
2002) and on the contribution to future sea level rise (e.g.
Joughin et al., 2010). The overarching question is whether
the system will stabilise again in the near future or whether
retreat may even accelerate (e.g. Katz and Worster, 2010;
Gladstone et al., 2012; Favier et al., 2014; Seroussi et al.,
2014).
Ice flow models simulate glacier ice flow which is due to
a combination of internal deformation and basal motion. Depending on the subglacial setting, basal motion can dominate
Published by Copernicus Publications on behalf of the European Geosciences Union.
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
the overall motion of a glacier, which is also the case for large
areas of PIG. Therefore, the basal sliding behaviour might be
the crucial process to cause a further retreat or halt of the system. Gudmundsson et al. (2012) show that stable grounding
line positions can be found on a retrograde bed using models
with two horizontal dimensions. The basal sliding behaviour
could be a similarly important process as the lateral buttressing.
On the one hand, the parametrisation of basal motion in
ice flow models is important for the overall dynamics of a
glacier. On the other hand, the difficulty of observing basal
properties renders the parametrisation one of the most challenging parts of ice flow modelling. In the absence of information on basal properties like bed type, structure and availability of liquid water, control methods are applied to simulate a complex glacier flow pattern, such as that present
at PIG (e.g. MacAyeal, 1992; Joughin et al., 2009, 2010;
Morlighem et al., 2010; Seroussi et al., 2014; Favier et al.,
2014). These methods use the measured surface velocity field
to invert for basal properties or effective viscosity and to adjust basal sliding parameters. Depending on the focus of the
study, these approaches can provide important insights into
glacier dynamics.
The prognostic studies on PIG all use control methods to
constrain basal sliding. Thus they define a spatially varying
basal sliding parameter for the present flow state and keep
it constant during the prognostic simulations. This way the
basal sliding is somehow decoupled from the rest of the system. As the observable surface velocity field is a superposition of basal motion (sliding and bed deformation) and ice
deformation, an inversion for basal sliding may introduce
errors in the basal sliding that compensate for errors in the
deformational part of the ice velocity. This is especially important for the strong temperature dependence of the flow
rate factor used for the ice viscosity. In addition, the spatial
distribution of a basal sliding parameter represents not only
the flow state at a specific time of observation, but also the
assumed thermal state in the model at the same time. One
could imagine that an area with significant sliding at the time
of the inversion could freeze at later time steps but is allowed
to slide at all times.
As the inversion for basal sliding parameters is not sufficient for the physical understanding of basal motion, we focus on basal sliding parametrisations that consider measured
basal roughness distributions. This accessible bed information could in further steps be combined with, for example, a
sufficient realistic and time-dependent hydrological model to
consider changing basal conditions for the sliding behaviour
of the glacier.
Here we present results of the thermomechanical 3-D fullStokes model COMice (implemented in the COMmercial finite element SOLver COMSOL Multiphysics© , cf. Wilkens,
2014) applied diagnostically to PIG. Initially we conduct a
diagnostic inversion for a basal sliding parameter, as done
in previous studies, to generate a reference simulation and
The Cryosphere, 9, 675–690, 2015
analyse the thermal structure of the glacier. In subsequent
experiments we introduce two methods of connecting basal
roughness measures to the parametrisation of basal sliding
and therefore constrain basal sliding with more physically
justified assumptions. Additionally, we couple the sliding
behaviour to the basal temperature, adding another physically based constraint. The first method matches a singleparameter basal roughness measure for PIG, as presented in
Rippin et al. (2011), onto a basal sliding parameter. The second method is based on ideas from Li et al. (2010), with a
two-parameter basal roughness measure which we apply to
connect basal roughness to basal sliding.
2
2.1
The numerical flow model
Governing equations
The governing equations for the thermomechanical ice flow
model COMice are the fluid dynamical balance equations together with a formulation for the non-Newtonian rheology of
ice. The balance equations are set up for mass, momentum
and energy and solved for the velocity vector u, pressure p
and temperature T .
The mass balance equation is given in case of incompressibility as
div u = 0.
(1)
The momentum balance equation is the Stokes equation,
given by
div σ = −ρi g,
(2)
with the Cauchy stress tensor σ , the density of ice ρi and the
acceleration of gravity g = (0, 0, −g)T . The stress tensor σ
is split into a velocity dependent part τ , the deviatoric stress,
and a pressure dependent part pI with the identity matrix I,
such that σ = τ − pI. For incompressible materials only the
deviatoric stress τ can result in strains and is thus related to
the velocity field u via the strain-rate tensor ε̇ and the effective viscosity µ, such that τ = 2µε̇. The strain-rate tensor ε̇
is given in components as
1 ∂ui ∂uj
+
,
(3)
ε̇ij =
2 ∂xj
∂xi
in relation to Cartesian basis vectors. The effective viscosity
µ is described with use of Glen’s flow law (Glen, 1955; Nye,
1957), such that
µ(T 0 , ε̇e ) =
−1/n (1−n)/n
1
A(T 0 )
ε̇e
,
2
(4)
with the rate factor A(T 0 ), the stress exponent n and the effective strain rate
r
1
ε̇e =
tr(ε̇2 ),
(5)
2
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which is the second invariant of the strain-rate tensor ε̇. The
homologous temperature T 0 is the temperature relative to the
pressure melting point Tpmp , defined as
T 0 = T + βc p,
(7)
with the melting point at low pressures T0 . The rate factor
A(T 0 ) parametrises the influence of the temperature and the
pressure onto the viscosity µ and is described by A(T 0 ) =
0
A0 e−Q/RT (Greve and Blatter, 2009), with a pre-exponential
constant A0 , the activation energy for creep Q and the gas
constant R.
The energy balance equation is given as
∂T
+ u grad T = div(κ(T ) grad T ) + ψ, (8)
ρcp (T )
∂t
with thermal conductivity κ(T ), specific heat capacity cp (T )
and an internal heat source term ψ = 4µε̇e2 which connects
mechanical and thermal energy.
2.2
Boundary conditions
The balance equations are defined under the assumption that
the thermodynamic fields are sufficiently smooth and thus
continuously differentiable, which is only the case for the
inner parts of the glacier. The outer boundaries need specifically formulated boundary conditions. The vertical boundaries are the upper surface zs and the base zb of the glacier.
The lateral boundaries are given by the ice divide, an inflow
area and the calving front. The grounding line indicates the
change of the basal boundary conditions from grounded to
floating ice.
2.2.1
Stokes flow
The upper surface can be seen to be traction free by assuming that wind stress and atmospheric pressure are negligible
compared to the typical stresses in the ice sheet, such that
σ · n = 0.
Since the model is applied in a diagnostic manner and
therefore the geometry is fixed, only the ice base needs a
kinematic boundary condition to prevent the flow to point
into the ground and is given as u·n = 0, with the unit normal
vector n pointing outwards from the surface. This Dirichlet
condition is applied to the entire ice base, including grounded
and floating parts, and also implies that no basal melting is
considered.
At the base of the floating ice shelf shear stress induced by
circulating sea water can be neglected (Weis et al., 1999) and
the only stress onto the ice is exerted by hydrostatic water
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pressure. As the ice shelf floats it is assumed to fulfil the
floating condition and the stress applied equals the stress of
the displaced water column (Greve and Blatter, 2009), such
that
(6)
with the Clausius–Clapeyron constant βc . The pressure melting point Tpmp is described for typical pressures in ice sheets
(p.50 MPa) by a linear relation, such that
Tpmp = T0 − βc p,
677
σ · n = −psw n,
(9)
with the water pressure psw defined as
0
for z ≥ zsl
psw =
ρsw g (zsl − z) for z < zsl ,
(10)
with the density of sea water ρsw and the mean sea level zsl .
Thus, the boundary conditions at the ice shelf base read
u · n = 0,
(σ · n) · t x = 0 and
(σ · n) · t y = 0.
(11)
with the unit tangential vectors t x in the xz plane and t y in
the yz plane.
For the boundary condition of the grounded ice, it is assumed that the stress vector σ · n is continuous across the interface such that σ · n = σ lith · n, with the Cauchy stress tensor of the lithosphere σ lith . Since this tensor is not known,
the condition has to be approximated. This is done with
a sliding law that connects the basal sliding velocity ub =
(u·t x , u·t y )T to the basal drag τ b = ((σ ·n)·t x , (σ ·n)·t y )T .
So-called “Weertman-type sliding laws” are commonly
applied in ice flow modelling studies, of which the basis
was established by Weertman (1957). He developed a mathematical description for the mechanisms that influence basal
sliding. One focus lay hereby on connecting small-scale
processes with larger-scale sliding effects. Nye (1969) and
Kamb (1970) worked on related problems and they all found
that the basal sliding velocity ub varies with some power of
the basal shear stress τ b , depending on the dominant mechanism. Additionally they find that the sensitively of basal sliding velocity ub depends on the roughness of the bed.
The processes considered by Weertman (1957), Nye
(1969) and Kamb (1970) are only relevant for sliding over
hard bedrock, where an upper limit for sliding velocities is
found (ub < 20 m a−1 ; Cuffey and Paterson, 2010). For faster
sliding velocities, weak deformable substrate or water-filled
cavities have to be present. Water-filled cavities reduce the
contact between the ice and the bedrock, therefore effectively reducing the roughness of the bed; their effect can be
considered via the effective pressure N b = −Nb · n (Bindschadler, 1983). Fast sliding velocities can only occur when
the glacier base is at pressure melting point, but some sliding can also be present below these temperatures (Fowler,
1986). This mechanism can be reflected by a temperature
function f (T ), which regulates sub-melt sliding. Considering the above-stated thoughts leads to a sliding law of the
form
−q
ub = Cb |τ b |p−1 Nb
f (T ) τ b =
1
τ b,
β2
(12)
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
whereby Cb is originally seen as a roughness parameter and
p and q are basal sliding exponents. When written in a linearised form, all effects influencing the basal sliding velocity
ub , other than the linear relation to the basal shear stress τ b ,
are summarised in a basal sliding parameter β 2 .
The overburden pressure of the ice is reduced in marine
parts by the uplifting water pressure (Huybrechts, 1992),
such that
ρi gH
for zb ≥ zsl
Nb =
(13)
ρi gH + ρsw g(zb − zsl ) for zb < zsl ,
with the unit normal vector n pointing outwards from the
surface and lying in the xy plane.
The boundary condition at the vertical calving front is
given by Eq. (9) and Eq. (10), where the normal vector is
pointing towards the ocean and lies in the xy plane. For the
inflow region a Dirichlet condition prescribes an inflow velocity field calculated analytically with the shallow ice approximation (Hutter, 1983; Morland, 1984). A no-slip condition is assigned to the outer wall of the ice rises, as they are
implemented as holes in the geometry.
2.2.2
where H is the ice thickness. The assumptions made above
imply that the base is perfectly connected to the ocean at any
location in the domain that is below sea level. This assumption is plausible near the grounding line but becomes highly
speculative towards the marine regions further inland. An additional hydrological model would be needed to realistically
simulate the effective basal pressure but is beyond the scope
of this study. Even though more sophisticated parametrisations for the effective pressure exist (e.g. Leguy et al., 2014),
we stick with the strong assumption stated above, as water
is likely present below all fast-flowing parts of PIG (Smith
et al., 2013) which coincide with the marine regions. Equation (13) further implies that the overburden pressure is cryostatic and neglects the dynamic contribution from the Stokes
solution to the pressure. In general, the dynamic part is small
compared to the cryostatic part, even if the bed is rough or
sliding changes over short distances (Pattyn et al., 2008, in
the Supplement).
The temperature function f (T ) is taken, as suggested by
Budd and Jenssen (1987), as an exponential function such
that
f (T ) = eν(T −Tpmp ) ,
(14)
with a sub-melt sliding parameter ν.
The dynamic boundary condition at the base is implemented with a tangential part (Eq. 12) and a normal part
(Eq. 13) such that
u · n = 0,
(σ · n) · t x = β 2 u · t x
(σ · n) · t y = β 2 u · t y .
and
(15)
The stress boundary condition is a Robin boundary condition
as it depends on the velocity and the velocity gradients. With
β 2 = 0 for the floating part, this boundary condition equals
Eq. 11.
Ice divides can be seen as mirror points where the direction of the driving stress and flow on one side of the divide
opposes that of the other side. No flow across the ice divide
is allowed, the tangential stresses vanish and therefore the
boundary condition for ice divides is given by
u · n = 0,
(σ · n) · t x = 0
(σ · n) · t y = 0,
and
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(16)
Temperature
The boundary conditions for the upper surface is given by
Dirichlet conditions in prescribing the average annual surface
temperature Ts (x, y, t). At the base of the grounded ice two
cases are to be distinguished. For a cold base, that is a basal
temperature below the pressure melting point, the boundary
condition has to be formulated as a Neumann condition and
the temperature gradient is prescribed as
grad T · n =
qgeo + qfric
,
κ(T )
(17)
with the geothermal flux qgeo and the friction heating due to
basal sliding qfric = ub · τ b (Pattyn, 2003). If the basal temperature reaches the pressure melting point in the grounded
part or the freezing temperature of seawater Tsw in the floating part, it has to be switched to a Dirichlet condition:
(
Tpmp if grounded,
T =
(18)
Tsw
if floating.
The boundary condition for the ice divide and the calving
front are based on the assumption that there is no temperature
gradient across the surface. It can thus be written in the form
of a thermal insulation (κ(T ) grad T )·n = 0. Lastly, temperatures at the inflow boundary are prescribed by a linear profile
T −Ts
Tlin = pmp
zs −zb (zs − z) + Ts .
2.3
Implementation
The thermomechanically coupled 3-D full-Stokes model
COMice is implemented in the COMmercial finite element
SOLver COMSOL Multiphysics© (cf. Wilkens (2014) for
implementation details). The model has been successfully
applied in the diagnostic tests in the MISMIP 3-D model intercomparison project (Pattyn et al., 2013) and the ISMIPHOM experiments (Pattyn et al., 2008). The ice flow model
solves for the velocity vector u, the pressure p and the temperature T . The unstabilised Stokes equation (Eq. 2) is subject to the Babuska–Brezzi condition, which states that the
basis functions for p have to be of lower order than for u.
Therefore, we use linear elements for p and quadratic elements for u (P1+P2). The energy balance equation Eq. (8)
is discretized with linear elements. To avoid numerical instabilities due to strong temperature advection, and thus to
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
ensure that the element Péclet number is always < 1, we
use consistent stabilisation methods provided by COMSOL.
Eq. (8) is solved using a Galerkin least-square formulation
(Codina, 1998) in streamline direction and crosswind diffusion (Hauke and Hughes, 1998) orthogonal to the streamline
direction. The chosen stabilisation methods add less numerical diffusion the closer the numerical solution comes to the
exact solution.
To the effective strain rate ε˙e (Eq. 5) a small value of
10−30 s−1 is added to keep the term non-zero. Model experiments have shown that this does not affect the overall results
(Pattyn, 2003; Cornford et al., 2012). The scalar values for all
parameters used throughout this study are listed in Table 1.
All Dirichlet boundary conditions are implemented as a
weak constraint, which means that constraints are enforced
in a local average sense. This gives a smoother result than
the standard method in COMSOL where constraints are enforced pointwise at node points in the mesh. The Neumann
condition (Eq. 17), together with the Dirichlet condition for
the basal temperature at the base (Eq. 18), is implemented
in a way that a switch between these two types is avoided.
This is done because a jump from Tpmp to Tsw in the area
of the grounding line leads to non-convergence of the flow
model. Therefore, a heat flux is prescribed as long as T <
(Tb,max − 0.01). The expression Tb,max prescribes a spatially
variable field that defines the maximal basal temperature allowed for a region (Tpmp for grounded areas, Tsw for floating areas). If T ≥ (Tb,max − 0.01), the heat flux is gradually
reduced and becomes zero when T = (Tb,max + 0.01). This
procedure ensures that the basal heat flux can not increase Tb
above Tb,max + 0.01. The smoothing of the step function ensures numerical stability, which was not found with a sharp
step. The implementation is similar as in Aschwanden and
Blatter (2009).
2.3.1
Mesh
To maximise the resolution while minimising the amount of
elements, we use an unstructured finite element mesh. The
upper surface is meshed first with triangles. The horizontal edge lengths are 5–500 m at the grounding line and the
calving front, 50–1000 m at the inflow area and 100–2000 m
at the rest of the outer boundary. The resulting 2-D surface
mesh is extruded through the glacier geometry with a total of
12 vertical layers everywhere. The thickness of the vertical
layers varies only with ice thickness. The spacing between
the layers is refined towards the base. The ratio of the lowest
to the upper most layer thickness is 0.01, leading to a thickness of the lowest layer of about 5 m for a total ice thickness
of 3000 m. The final mesh consists of ∼ 3.5 × 105 prism elements, which results in ∼ 5 × 106 degrees of freedom (DOF)
when solved for all variables.
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2.3.2
679
Solver
For solving the nonlinear system, a direct segregated solver
is used which conducts a quasi-Newton iteration. It solves
consecutively: first for the velocity u and the pressure p and
thereafter for the temperature T (COMSOL, 2012). This allows for reduced working memory usage. For the remaining
linear systems of equations, the direct solver Pardiso (COMSOL (2012) and http://www.pardiso-project.org/, last access:
9 December 2014) is applied. While uncommon for such
large numbers of DOF, it proved to be computationally viable and robust.
3
3.1
Data
Geometry
The geometry of the model was built with a consistent set
of surface elevation, ice thickness and bed topography on a
1 km grid, created by A. Le Brocq and kindly provided by
her for this work. The data set represents the thickness distribution of PIG for the year 2005 and earlier. The Le Brocq
data are based on the surface elevation data of Bamber et al.
(2009) and the ice thickness data of Vaughan et al. (2006).
The grounding line position used is given by a combination of the positions in the MODIS Mosaic Of Antarctica
(Bohlander and Scambos, 2007) corresponding to the years
2003/2004, the position in Rignot (1998), corresponding to
1996 and the positions that give the smoothest ice thickness joined between grounded and floating ice, assuming the
floatation condition. The model domain and grounding line
are indicated in Fig. 1. The location of the ice rises pinning
the ice shelf at present are detected on TerraSAR-X images
from 2011, with assistance of interferograms from Rignot
(2002). Please note that ice rises are not indicated in Fig. 1.
3.2
Ice flow velocity
The observed surface velocity is taken from Rignot et al.
(2011), shown in Fig. 1, and used to validate the reference
simulation. The numbering of the 1–10 tributaries feeding
the central stream is based on Stenoien and Bentley (2000).
The numbering used in Vaughan et al. (2006), Karlsson et al.
(2009) and Rippin et al. (2011) is the same for the even numbers but shifted by 1 for the odd numbers, as they missed
tributary 1 from the numbering by Stenoien and Bentley
(2000). We extended the numbering from Stenoien and Bentley (2000) to the tributaries 11–14, which are entering the
ice shelf. Finally, Fig. 1 indicates the locations of the different types of lateral boundaries (ice divide, inflow and calving
front).
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
Figure 1. RADARSAT Antarctic Mapping Project (RAMP) mosaic
with the observed surface velocities from Rignot et al. (2011) and
the model domain of Pine Island Glacier, with the different lateral
boundaries, the grounding line and the numbered tributaries indicated.
3.3
Temperature
The surface temperature used here is on a 5 km grid compiled by Le Brocq et al. (2010) (ALBMAP v1), based on
the temperature data described in Comiso (2000). We use the
geothermal flux qgeo from Purucker (2012, updated version
of Fox Maule et al., 2005), because a variety of sensitivity
tests showed that other data sets lead to too-high velocities in
regions with no or little basal sliding.
3.4
Roughness
In this study we use two different measurements of basal
roughness beneath the PIG. The first one is the singleparameter roughness measure as presented in Rippin et al.
(2011) (cf. Fig. 4b therein) and represents the methodology often employed to define subglacial roughness (Hubbard
et al., 2000; Taylor et al., 2004; Siegert et al., 2004, 2005;
Bingham and Siegert, 2007; Bingham et al., 2007; Rippin
et al., 2011). This usual approach effectively provides a measure of bed obstacle amplitude or vertical roughness. The
second measure, which we calculate for this study, follows
the work of Li et al. (2010), Wright et al. (2012) and more
recently Rippin et al. (2014). They introduce a second parameter which effectively provides a further measure of the frequency or wavelength of roughness obstacles (Rippin et al.,
2014). Both roughness measures are based on fast Fourier
transforms (FFTs). FFT can be used to transform any surface into a sum of several periodically undulated surfaces.
In a number of recent glaciological studies, basal topography data are transformed into a single-parameter roughness
measure (ξ ) which is defined as the integral of the spectrum
The Cryosphere, 9, 675–690, 2015
within a specified wavelength interval. This method represents the amplitude of the undulations, but information about
the frequency is lost. For PIG the single-parameter roughness measure ξ was calculated by Rippin et al. (2011) from a
RES data set generated in austral summer 2004/05 (Vaughan
et al., 2006). It is the same data set the model geometry is
based on (Sect. 3), although the roughness measure includes
higher-resolution information as the derivation is based on
along-track sample spacing of the order of 30 m (cf. Rippin
et al., 2011). Both data sets are then gridded with 1 km spacing.
By applying the work of Li et al. (2010), we introduce a
second parameter so that as well as being able to represent
the amplitude (ξ ), we are also able to explore the frequency
(η) of the undulations. This measure is calculated as the total
roughness divided by the bed slope roughness (Rippin et al.,
2014). Li et al. (2010) provide guidance on how to interpret these two parameters in terms of their different basal
topographies, along with their geomorphic implications. The
interpretation from Li et al. (2010) is based on ideas by Bingham and Siegert (2009) which give an interpretation for the
single-parameter roughness. Rippin et al. (2014) extended
the interpretation for the two-parameter roughness measure.
The implications for PIG will be discussed below.
Because of the statistical meanings of ξ and η, they can be
used as representatives for the vertical and horizontal length
scales present at the base. To do so the integration interval for
{ξ, η} should be in the metre-scale waveband (Li et al., 2010).
The two-parameter roughness measure for PIG was calculated for this study. The spatial resolution of the underlain
data for PIG is 34 m. A moving window is calculated with
N = 5 (2N = 32), which is the minimum for N that should
be used (e.g. Taylor et al., 2004). With a spatial resolution of
34 m this leads to a moving window length of 1088 m, which
is in the metre-scale waveband required by Li et al. (2010),
to be able to apply the data in a sliding relation.
The received fields of ξ and η for PIG are shown in Fig. 2.
According to Li et al. (2010), different basal properties and
related geomorphic implications can be distinguished from
patterns of ξ and η. A marine setting with intensive deposition and fast and warm ice flow, as proposed for the central
part of PIG, is characterised by low values of ξ and high values of η, thus low-amplitude, low-frequency roughness. Here
it has to be noted that the second parameter η should be more
accurately seen as representing the wavelength of roughness,
rather than the frequency, as high values correspond to low
frequencies (Rippin et al., 2014). Nonetheless we continue
here referring to η as the roughness frequency for consistency with Li et al. (2010). The suspected low-amplitude,
low-frequency roughness is not necessarily found in the central trunk area, as can be seen in Fig. 2. Instead it seems to
be more dominated by low-amplitude, high-frequency roughness which can, following Li et al. (2010), be interpreted as a
continental setting after intensive erosion, also with fast and
warm ice flow. Still, this interpretation can not be seen as a
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
681
Table 1. Parameter values.
Parameter
Value
Unit
Description
ρi
ρsw
g
n
R
Q
kg m−3
kg m−3
m s−2
T0
βc
918
1028
9.81
3
8.314
60 for T 0 ≤ 263.15 K
139 for T 0 > 263.15 K
3.985 × 10−13 for T 0 ≤ 263.15 K
1.916 × 103 for T 0 > 263.15 K
273.15
9.8 × 10−8
Tsw
ν
271.15
0.1
K
κ(T )
cp (T )
zsl
spy
−3
−1
9.828 e(−5.7 × 10 T [K ])
152.5 + 7.122 T [K−1 ]
0
31 536 000
W m−1 K−1
J kg−1 K−1
m
s a−1
Ice density
Seawater density
Acceleration of gravity
Stress exponent
Gas constant
Activation energy for creep
(Paterson, 1994)
Pre-exponential constant
(Paterson, 1994)
Melting point for low pressures
Clausius–Clapeyron constant
(Hooke, 2005)
Freezing temperature of seawater
Sub-melt sliding parameter
(Budd and Jenssen, 1987)
Thermal conductivity
Specific heat capacity
Sea level
Seconds per year
A0
J mol−1 K−1
kJ mol−1
s−1 Pa−3
K
K Pa−1
contradiction to the suspicion of the presence of marine sediments. It is important to state that absolute values of roughness cannot be derived from these calculations. It is rather
the pattern relating to relative roughness values that is significant.
ing velocity ub can be approximated by subtracting the surface velocity due to internal deformation us,nosl from the
measured surface velocity field uobs (Rignot et al., 2011,
Fig. 1). The basal drag from the simulation where no basal
sliding is allowed, τ b,nosl , is taken as a good first representation of the real basal drag distribution, τ b . With this, the field
of the basal sliding parameter β 2 is defined as
4
β 2 = |τ b,nosl | (|uobs | − |us,nosl |)−1 ,
4.1
Experiment description
Experiment 1: reference simulation
The main difficulty is to capture the distinct surface flow
pattern by making appropriate assumptions about the basal
sliding behaviour. Many ice modelling studies use a constant
set of basal sliding parameters to reproduce somewhat realistic surface velocity fields (e.g. Rückamp, 2011; Kleiner
and Humbert, 2014). This approach can not be adopted for
PIG, as it leads to a shut down of parts of the fast-flowing
main trunk due to very low basal shear stresses in that region
(Joughin et al., 2009; Morlighem et al., 2010). Instead, for
our reference simulation, an inversion for basal parameters
is conducted. This approach will lead to a realistic reproduction of the surface flow velocity field and lets us analyse the
thermal structure of the glacier.
The inversion method (cf. Schmeltz et al., 2002) used for
our reference simulation starts by assuming the linearised
form of Eq. (12): thus τ b = β 2 ub , where β 2 is the basal
sliding parameter to be inferred. Additionally, a simulation
is conducted where the glacier base is not allowed to slide.
Therefore the resulting surface velocity field us,nosl can be
seen to be solely due to internal deformation. The basal slidwww.the-cryosphere.net/9/675/2015/
(19)
as shown in Fig. 3. This is a significantly different approach
from in e.g. Joughin et al. (2009), Morlighem et al. (2010)
and Seroussi et al. (2014), who minimise a cost function.
The basal sliding parameter β 2 is subsequently applied in
the forward model in the linear sliding law. Since the amount
of internal deformation in the ice crucially depends on the
ice temperature (Eq. 4), it is important to consider a realistic
temperature distribution within the ice. At this point it is important to note that the model is applied in a diagnostic manner and therefore the received temperature distribution is a
steady state one for a fixed geometry with constant boundary
conditions which might differ from the actual transient field.
Nonetheless, the received field is likely to show a better approximation to reality than simply assuming a certain distribution. To consider a realistic temperature distribution within
the ice, we conduct the above-described procedure in an iterative manner. We first conduct a “no sliding” simulation
nosl,1 with a constant temperature of T = 263.15 K. The resulting surface velocity field us,nosl,1 and basal drag τ b,nosl,1
lead to a basal sliding parameter β12 . This basal sliding parameter enters the next simulation step, in which basal sliding sl,1 is accounted for and the temperature field is solved
The Cryosphere, 9, 675–690, 2015
682
N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
Figure 2. Calculated roughness parameters at Pine Island Glacier, given by the roughness amplitude ξ (a) and the roughness frequency η (b).
(“slow”) and the entire model region (“all”) (detailed description in Wilkens, 2014). The regions of all tributaries (1–
14), the central stream (CS) and the shelf area (shelf) are
combined to the region “fast”, while the remainder is the region “slow” (cf. Fig. 8b and c)
4.2
Figure 3. Spatial distribution of the basal sliding parameter β 2 .
for. The temperature distribution enters the next “no sliding”
simulation nosl,2 as a constant field. Again a basal sliding
parameter is found, entering the next simulation step sl,2,
which is our final reference simulation ref. Thus the procedure is stopped after two iterations and listed in a schematic
manner as nosl,1(T = 263.15 K) → β12 → sl,1(T solved) →
nosl,2(T from sl,1) → β22 → sl,2/ref(T solved).
The reference simulation serves as a validation parameter
for the subsequent experiments. As a quantitative measure
the root-mean-square (RMS) deviation RMSus (unit: m a−1 )
between the simulated surface velocity field us,sim and the
reference surface velocity field us,ref is given by
RMSus
v
u
m
u1 X
(||us,sim |i − |us,ref |i |)2 ,
=t
m i=1
(20)
where m is the number of discrete values on a regular grid
with 1 km spacing. The comparison is done in three distinct
regions: fast flow velocities (“fast”), slower flow velocities
The Cryosphere, 9, 675–690, 2015
Experiment 2: parametrisation with
single-parameter roughness
The approach in the reference simulation is dissatisfying
when aiming to constrain basal sliding with physical parameters at the base of the glacier. Therefore, we introduce in
this experiment a parametrisation of basal sliding that considers the basal roughness below the glacier in the formulation
of the commonly used Weertman-type sliding law (Eq. 12),
with the aim of reproducing the surface velocity field of PIG.
Instead of inverting for one spatially varying parameter, we
now connect the basal sliding parameter Cb to the measured
single-parameter roughness measure ξ (Rippin et al., 2011),
Sect. 3.4, as it is closest to the original physical meaning of
Cb .
The absolute values of the roughness measure ξ are dependent on parameters chosen for its derivation (e.g. wavelength interval in Rippin et al., 2011). At the same time the
sliding parameter Cb depends not only on mechanical properties, such as basal roughness, but also thermal properties.
Therefore, the roughness measure ξ can not directly be used
as the sliding parameter. To use the roughness information,
we select a range for the sliding parameter Cb , obtained via
the approximation
q
Cb =
(|uobs | − |us,nosl |) Nb
.
|τ b,nosl |p
(21)
To obtain a roughness parameter that depends on the
roughness measure ξ , the resulting range [Cb,min , Cb,max ] is
mapped onto the range of the roughness measure [ξmin , ξmax ],
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
such that
Cb (ξ ) =
(Cb,max − Cb,min ) · (ξ − ξmin )
+ Cb,max .
(ξmin − ξmax )
(22)
The lowest roughness correlates with the highest basal sliding and therefore the highest values of Cb (ξ ). In the following we will refer to the basal sliding parameter Cb (ξ ) as Cξ
when it is related to the basal roughness measure ξ .
In total we conduct 15 simulations for Experiment 2,
where each parameter combination represents a potential
subglacial setting. In all simulations the coefficient p = 1 is
kept constant, while q is varied to investigate the effect of the
effective pressure onto the sliding velocities (q ∈ {0, 1, 2}).
This results in varying minimum and maximum values for Cξ
for each parameter combination. To account for potential outliers in the Cξ distribution, we have subsequently narrowed
the range of Cξ . This results in varying minimum and maximum values for Cξ for each parameter combination. Hence,
simulations with an identifier 1–5 are conducted with q = 0
and five different ranges of Cξ , simulations 6–10 with q = 1
and five different ranges of Cξ and simulations 11–15 with
q = 2 and five different ranges of Cξ . The ranges of Cξ can
be found in Table 5.2 in Wilkens (2014).
4.3
Experiment 3: parametrisation with
two-parameter roughness
The aim of this experiment is to test the idea of Li et al.
(2010) for its applicability to PIG. This approach also relates
the basal roughness to the basal sliding velocity. This time
the roughness is represented by a two-parameter roughness
measure for the amplitude ξ and frequency η of the undulations. The approach is based on Weertmans original formulation (Weertman, 1957) of describing the sliding mechanisms
of regelation and enhanced creep, such that
ub = CW
l2
τb 2
a
(1+n)
2
,
(23)
where CW is a parameter defined by thermal and mechanical properties of the ice, l is the obstacle spacing, a is the
obstacle size (cf. Weertman, 1957) and n = 3 is the stress exponent.
Li et al. (2010) state that the two-parameter roughness
measures ξ and η, representing the amplitude and frequency
of the roughness (cf. Sect. 3.4), can be used as a proxy for the
vertical and horizontal length scales present√at the base due to
√
their statistical meanings, such that a = c1 ξ and l = c2 η,
where c1 and c2 are proportionality factors.
Entering this into Weertmans original formulation (Eq. 23)
and additionally including a temperature function f (T ) as
introduced above leads to
(n+1)
2
η
ub = CL f (T ) τ b
,
ξ
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(24)
683
with the constant CL = CW (c2 /c1 )1+n . As the proportionality factors c1 and c2 are not further defined, we take CL as a
single parameter to adjust.
The upper and lower bounds for CL are obtained via
− (n+1)
2
CL = (|uobs | − |us,nosl |) |τ b,nosl |
(n+1)
2
ξ
.
η
(25)
The vast majority of the values lie within CL = [3×10−2 ; 3×
102 ] Pa−2 m a−1 . We conduct 18 simulations for this experiment, whereby the value of CL (Eq. 24) is varied in this
range.
For all simulations conducted for Experiment 3, only
Eq. (2) is solved for due to time constraints (cf. Wilkens,
2014). The temperature distribution within the ice is taken
from the reference simulation. Use of the temperature field
from the reference simulation gives the opportunity to connect the sliding behaviour to the basal temperature, thus only
allowing ice to slide where T is close to Tpmp .
5
5.1
Results
Experiment 1
The resulting surface velocity field from the reference simulation is shown in Fig. 4. The general pattern of the surface
velocity field is well reproduced in the reference simulation
compared to the observed surface velocity field (Fig. 1). The
tributaries are all in the right location and the velocity magnitudes agree in most areas well. The highest differences between |us,ref | and |uobs | are found in the ice shelf, where the
simulated velocities are up to 1 km a−1 smaller than the observed ones.
When solely looking at the velocity magnitudes we again
find that for higher velocities the simulated velocity field is
lower than the observed field (Fig. 5). The spread around the
diagonal for lower velocities appears bigger, which is mainly
due to the logarithmic axes chosen. For higher flow velocities the direction of flow of the simulated field agrees well
with the direction of the observed field. This is shown as a
colour code for the angle offset between the velocity vectors
in Fig. 5. For slower velocities the angle offset is bigger, coinciding with a higher measurement error for slower velocities.
The simulation shows that large areas under PIG are temperate (Fig. 6). In general the overall flow pattern is reflected
in the basal temperature structure, with fast-flowing areas being underlain by a temperate base. Figure 7 shows the ho0 at three vertical slices, of which
mologous temperature Tref
the locations are indicated in Fig. 6. The slice located furthest away from the ice shelf shows that the base is mainly
temperate, while the inner ice body (away from the base) is
predominantly cold (Fig. 7a). A similar picture is found in
the next slice, which is located further downstream towards
the ice shelf (Fig. 7b). Here, additionally a cold core can be
seen, located in the fast-flowing central stream.
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684
N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
Figure 4. Surface velocity field from the reference simulation
|us,ref | with the numbered tributaries.
Figure 5. Observed surface velocity field |uobs | versus reference
surface velocity field |us,ref |. The logarithmic scales exaggerate the
spread around the low speeds. The angle offset 1α between the
vectors of the surface velocity field uobs and the reference surface
velocity field us,ref is shown as the colour code.
The next slice partly crosses the ice shelf (Fig. 7c). It can
be well observed that a cold core is entering the ice shelf. In
the vicinity of tributary 11 (cf. Fig. 1) a small temperate layer
is found.
5.2
Experiment 2
We show in Fig. 8a the RMSus deviations between the reference simulation and Experiment 2 for all 15 conducted parameter combinations. It can be seen, that the “fast” regions
differ most for all parameter combinations tested here. Additionally, for the entire region “all” there seems to be no single parameter combination that minimises the RMSus value.
Although the RMS is relatively high, some of the complex
surface flow features could be reproduced with our approach,
which can only be seen by looking at the qualitative structure
of the resulting surface flow fields. Fig. 8b shows the surface
The Cryosphere, 9, 675–690, 2015
Figure 6. The basal homologous temperature from the reference
0 , with tributary locations in black and the location
simulation Tb,ref
of the vertical slices (a), (b) and (c) in Fig. 7 in grey.
Figure 7. The internal homologous temperature from the reference
0 at three vertical slices (a), (b) and (c) (horizontal
simulation Tref
locations indicated in Fig. 6).
velocity field with q = 0, which means the effect of the effective pressure is cancelled out (simulation identifier 2). The
location of tributary 7 (and slightly 11) and the central stream
are well reproduced.
Although the central stream is in general well reproduced,
the inflow into the ice shelf is characterized by a drop of flow
velocities which does not coincide with the observed velocities. In the simulations where the effective pressure is considered with q = 2 (simulation identifiers 11–15), a much better
representation of the central stream at the inflow into the ice
shelf across the grounding line is found, as can be seen for
example in the surface flow field from Simulation 11 shown
in Fig. 8c. The influence of the effective pressure Nb is thus
emphasised. At the same time, this method does not lead to
a full reproduction of the surface flow structure. This suggests that other processes not considered here may also be
important for the basal sliding behaviour. A possibility, not
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
685
Figure 8. (a) RMS error to the surface velocity field of the reference simulation versus the simulation number; (b) surface velocity field of
Simulation 2 with q = 0; (c) surface velocity field of Simulation 11 with q = 2.
tested yet due to cpu time constraints (for a detailed description of the solution time of the simulations refer to Wilkens,
2014), is the effect of the basal stress exponent p. Increasing
it would possibly regulate to some extent the high velocities
in some areas due to low basal stresses.
The basal homologous temperature from Simulation 2
(Fig. 9) shows a very clear structure of the temperate base
below the tributaries, even though they are not clearly visible
in the flow field (cf. Fig. 8b). The temperature-driven separation between tributaries 2 and 4 and tributaries 7 and 9 is even
more visible than in the reference simulation (cf. Fig. 6). The
structure of the basal homologous temperature of all other
simulations looks very similar to that of Simulation 2, but
the total area fraction of ice at pressure melting point, as well
as the separation between the tributaries, varies.
Another interesting feature found in the structure of the
basal temperature from Simulation 2 is the advection of
warmer ice into the shelf. This feature can be attributed to the
implementation of the thermal basal boundary condition in
the shelf (cf. Sect. 2.3). While the heat flux is not allowed to
raise the temperature above 271.15 K, it does not hinder the
advection of warmer ice from the grounded areas. The structure of the bands of warmer ice agree well with melt channels
below the ice shelf, as found by Vaughan et al. (2012).
5.3
Experiment 3
The RMSus deviations between the reference simulation and
all conducted parameter combinations in Experiment 3 show
a somewhat regular pattern (Fig. 10a). For the slower-flowing
areas, the RMSus value increases with increasing CL . For
the faster-flowing areas, the RMSus value first slightly decreases with increasing CL and, after reaching a minimum
of RMSus = 500 m a−1 for CL = 1.58 Pa−2 m a−1 , increases
with increasing CL . Since we conduct simulations with discrete values for CL , the value of RMSus = 500 m a−1 represents the minimum value for the simulations conducted here
and not an absolute minimum. The RMSus value for the entire region “all” shows a similar behaviour of first decreaswww.the-cryosphere.net/9/675/2015/
Figure 9. Basal homologous temperature of Simulation 2.
ing and then increasing with increasing CL , with a minimum
RMSus value of 271 m a−1 for CL = 1 Pa−2 m a−1 .
Although RMS values reveal a slight minimum, the surface velocity field of PIG is not reproduced with all its features. For higher CL values that reproduce the velocities in
the central stream in a better manner, the velocities in the
slower-flowing area around tributaries 3, 5, 7 and 9, located
to the south of the main stream, are simulated much too high.
A striking feature of all simulations is that the central
stream is partitioned into a faster-flowing upper part and a
slower-flowing lower part in the vicinity of the ice shelf. We
show an example for a CL value with a low RMS value and a
CL value with a high RMS value (Fig. 10b and c). However,
when looking at the structure of the surface flow fields it is
apparent that some features of the observed surface flow field
are reproduced.
Additionally, the area around tributary 14 behaves slightly
different to most other tributaries. It speeds up much faster
for much lower values of CL . This is related to the low roughness measures ξ and η in that region.
The Cryosphere, 9, 675–690, 2015
686
N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
Figure 10. (a) RMS error to the surface velocity field of the reference simulation versus CL value; (b) surface velocity field with CL =
1 Pa−2 m a−1 ; (c) surface velocity field with CL = 31.56 Pa−2 m a−1 .
6
Discussion
We have shown that the complex surface flow structure of
PIG could be well reproduced with our simplified approach
of an inversion for a basal sliding parameter β 2 in the reference simulation. Although the simulated flow pattern agrees
well with observations, some differences in the magnitude of
the surface flow velocities to observations were found. These
differences are highest in the ice shelf and might be partly
related to a slower inflow from the grounded areas where the
difference is about 1 km a−1 . This might be due to the position of the grounding line in our model which is further
downstream than the location in 2009 to which the observed
surface velocity field belongs (2007–2009). As we run the
flow model in a diagnostic manner without a relaxation of
the geometry, the simulated ice flow could not be consistent
with the geometry. Alternatively, it might be caused by the
simplified method of inferring β 2 , as τ b,nosl is not vanishing
near the grounding line as would be expected (cf. Joughin
et al., 2010 and Morlighem et al., 2010).
However, the main cause seems to be that we did not account for the highly rifted shear margins. These shear margins have been shown to be rheologically softer than undamaged ice (e.g. Humbert et al., 2009). In reality the shear margins partly uncouple the fast-flowing central part from the
surrounding ice. In our study we neglect the effect of shear
margins and treat them as rheologically equal to undamaged ice. This leads to an overestimation of the flow outside
the central stream and an underestimation within the central
stream in the main trunk. The softening due to shear margins can be included in different ways, as for example done
in Joughin et al. (2010) and Favier et al. (2014).
From the simulated temperature distribution we found that
the base of the glacier is predominantly temperate with the
absence of a significant temperate layer; the rest of the inner ice body is mainly cold. This finding is consistent with
the general definition of an Antarctic glacier where, due to
cold conditions at the surface, the cold-temperate transition
surface (Blatter and Hutter, 1991) is located at or near the
base. To form a significant basal temperate layer, Blatter and
The Cryosphere, 9, 675–690, 2015
Hutter (1991) state that strain heating is the necessary mechanism. This also agrees well with our results, as the flow of
PIG is dominated by basal sliding, and therefore strain heating due to internal deformation is small. Only an area around
tributary 11 (cf. Figs. 6 and 7c), where strain heating is much
higher, shows the existence of a somewhat larger temperate
layer at the base. Unfortunately there are no measured (deep)
ice core temperatures available at PIG to which our results
could be compared. Nonetheless, our findings of a temperate base below some parts of PIG are supported by findings
from Smith et al. (2013) of the existence of water below the
glacier.
As the first new parametrisation for basal sliding we tested
the applicability of including actual measured roughness data
in the commonly used Weertman-type sliding law. The new
parameter Cξ was applied in the basal sliding law and we
were able to reproduce many of the tributaries, although not
the full complexity of the flow structure. For instance, the
central stream is in large areas underlain by a very smooth
bed, indicated by a low roughness measure ξ , which becomes
rougher towards the grounding line.
We have shown that the influence of the effective pressure
onto basal sliding must be large close to the grounding line
to keep the flow velocities high. As part of the overburden
pressure is supported by the basal water of the marine setting
of the glacier, the effective pressure is low and basal motion
is therefore facilitated.
The locations of the fast-flowing tributaries and the central
stream are well indicated by a temperate base. The structure
is visible even more clearly than for the reference simulation.
This supports the idea that the location of some tributaries is
influenced by basal temperatures.
A full reproduction of the surface flow structure is not
achieved with the single-parameter roughness measure. This
suggests that other processes, not considered here, are also
important for the basal sliding behaviour. In addition, the
basal stress exponent p would, to some extent, perhaps regulate the high velocities in some areas due to low basal
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N. Wilkens et al.: Thermal structure and basal sliding parametrisation at Pine Island Glacier
stresses. The effect of p on the flow field has not been investigated here but should be considered in the future.
For the second new parametrisation for basal sliding we
test the applicability of a theory developed by Li et al. (2010)
to the region of PIG that connects a two-parameter roughness
measure {ξ, η} to the basal sliding law. This approach additionally accounts for the frequency of roughness but neglects
the effective pressure. The results of the surface flow field
show that the central stream in all the simulations from this
experiment is partitioned into a faster-flowing upper part and
a slower-flowing lower part in the vicinity of the ice shelf.
No single value for CL could be found that reproduces the
surface velocity field of PIG with all its features. To account
for the frequency roughness does not lead to an overall better
representation of the flow compared to the single-parameter
approach. We expect that if the effective pressure at the base
is considered in the sliding formulation of Li et al. (2010),
the results would significantly improve as the reduced effective pressure at the grounding line in the marine setting of
PIG would favour higher sliding velocities.
Despite the inability to completely reproduce the surface
flow field of PIG with the methods using the roughness measure, this approach represents some important flow features,
like the location of the fast-flowing central stream and some
of the numerous tributaries.
To derive basal properties and to adjust basal sliding parameters, pure inversion methods use the observed velocity
field and minimise the misfit between observation and model
results. They require no or only very little information about
the bedrock properties (e.g. bed type, temperature or availability of basal water) and result in a very good representation of the flow for the time of observation. The derived basal
sliding parameters then contain not only processes related to
sliding but all assumptions and approximations of the applied
flow model, as the surface flow is a superposition of ice deformation and basal motion. Inversion methods are therefore
not sufficient to gain the knowledge about the processes at
the base and their complex interplay.
In comparison to the inversion methods our approach relates sliding to the physical parameter of the subglacial bed
roughness. Although the measured bed roughness is only
valid for a certain period as the subglacial environment
changes over time, we do not expect the main features to
change in the near future. One important process for prognostic simulations over longer timescales could be basal erosion (Smith et al. (2012) report relatively high erosion rates
at PIG of 0.6 m a−1 ±0.3 m a−1 ), but this is beyond the scope
of this work.
In this study the effective pressure at the base is only influenced by the height above buoyancy and affects only the
areas below sea level. This is a strong restriction of the model
as the sliding formulation is not connected to the very diverse hydrology at the base of the glacier. A sufficiently complex/realistic hydrology model would be a great benefit for
this and for every other ice flow model. In that case, it might
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687
be beneficial to separate the hydrology component from the
sliding relation. However, this is beyond the scope of this
work.
7
Conclusions
The overall motion of the fast-flowing parts of PIG are dominated by basal motion. The parametrisation of basal motion
is therefore crucial for simulating the flow of PIG. Especially
when running prognostic simulations of the glacier and aiming at analysing the stability of the system, parametrisation
of basal motion is important. We introduce two different approaches for connecting a basal sliding formulation to an actually measurable subglacial parameter, the basal roughness
measure. Our results show that the roughness measure is a
very useful parameter to be considered for parametrisation
of basal motion at PIG because important features of the flow
field could be reproduced. Nonetheless the full complexity of
the problem was not captured. Our approach is a step towards
a more physically based parametrisation for basal sliding,
which is very important for realistic simulations of glacier
dynamics.
Acknowledgements. This work was supported through the Cluster
of Excellence “CliSAP” (EXC177), University of Hamburg,
funded by the German Science Foundation (DFG). We would like
to thank Anne Le Brocq for providing the compiled data set for
the geometry of Pine Island Glacier. We thank Stephen Cornford,
an anonymous referee and the editor, Olivier Gagliardini, for their
very helpful suggestions which improved the manuscript.
Edited by: O. Gagliardini
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