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
.
Additionally, the currently observed mass loss from the Antarctic Ice Sheet
is also concentrated in the area around PIG .
Thus PIG shows an increased contribution to global sea level rise
.
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 . This
hypothesis is still up for debate ,
while the trigger for the changes is thought to be enhanced ocean
melting of the ice shelf .
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. and on the
contribution to future sea level rise e.g.. The
overarching question is whether the system will stabilise again in the near
future or whether retreat may even accelerate
e.g..
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 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.
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.. 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 full-Stokes model
COMice (implemented in the COMmercial finite element SOLver COMSOL
Multiphysics©, cf. ) 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 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 single-parameter basal roughness measure for PIG, as
presented in , onto a basal sliding parameter. The
second method is based on ideas from , with a two-parameter
basal roughness measure which we apply to connect basal roughness to
basal sliding.
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
divu=0.
The momentum balance equation is the Stokes equation, given by
divσ=-ρig,
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
ε˙ and the effective viscosity μ, such
that τ=2με˙. The
strain-rate tensor ε˙ is given in components as
ε˙ij=12∂ui∂xj+∂uj∂xi,
in relation to Cartesian basis vectors. The effective viscosity μ is
described with use of Glen's flow law , such
that
μ(T′,ε˙e)=12A(T′)-1/nε˙e(1-n)/n,
with the rate factor A(T′), the stress exponent n and the effective strain rate
ε˙e=12tr(ε˙2),
which is the second invariant of the strain-rate tensor
ε˙. The homologous temperature T′ is the
temperature relative to the pressure melting point Tpmp,
defined as
T′=T+βcp,
with the Clausius–Clapeyron constant βc. The pressure melting
point Tpmp is described for typical pressures in ice sheets
(p≲50MPa) by a linear relation, such that
Tpmp=T0-βcp,
with the melting point at low pressures T0. The rate factor A(T′)
parametrises the influence of the temperature and the pressure onto the
viscosity µ and is described by A(T′)=A0e-Q/RT′
, with a pre-exponential constant A0, the
activation energy for creep Q and the gas constant R.
The energy balance equation is given as
ρcp(T)∂T∂t+ugradT=div(κ(T)gradT)+ψ,
with thermal conductivity κ(T), specific heat capacity
cp(T) and an internal heat source term
ψ=4με˙e2 which connects mechanical and
thermal energy.
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.
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 and the only
stress onto the ice is exerted by hydrostatic water 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
, such that
σ⋅n=-pswn,
with the water pressure psw defined as
psw=0forz≥zslρswg(zsl-z)forz<zsl,
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)⋅tx=0and(σ⋅n)⋅ty=0.
with the unit tangential vectors tx in the xz plane and
ty 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=σlith⋅n, with the Cauchy stress
tensor of the lithosphere σlith. 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⋅tx,u⋅ty)T to the basal
drag τb=((σ⋅n)⋅tx,(σ⋅n)⋅ty)T.
So-called “Weertman-type sliding laws” are commonly applied in ice flow
modelling studies, of which the basis was established by
. 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. and
worked on related problems and they all found that the basal
sliding velocity ub varies with some power of the
basal shear stress τb, 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 , and
are only relevant for sliding over hard bedrock, where an
upper limit for sliding velocities is found (ub<20 m a-1; ). 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
Nb=-Nb⋅n .
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
. 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
ub=Cb|τb|p-1Nb-qf(T)τb=1β2τb,
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 τb,
are summarised in a basal sliding parameter β2.
The overburden pressure of the ice is reduced in marine parts by the uplifting water pressure , such that
Nb=ρigHforzb≥zslρigH+ρswg(zb-zsl)forzb<zsl,
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.
), we stick with the strong assumption stated above, as water
is likely present below all fast-flowing parts of PIG
which coincide with the marine regions. Equation () 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 in the Supplement.
The temperature function f(T) is taken, as suggested by , as an exponential function such that
f(T)=eν(T-Tpmp),
with a sub-melt sliding parameter ν.
The dynamic boundary condition at the base is implemented with
a tangential part (Eq. ) and a normal part
(Eq. ) such that
u⋅n=0,(σ⋅n)⋅tx=β2u⋅txand(σ⋅n)⋅ty=β2u⋅ty.
The stress boundary condition is a Robin boundary condition as it depends on the velocity and the
velocity gradients. With β2=0 for the floating part, this boundary condition equals Eq. .
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)⋅tx=0and(σ⋅n)⋅ty=0,
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. () and Eq. (), 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
. A no-slip condition is assigned to the outer wall of the ice rises, as they are implemented as holes in the
geometry.
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
gradT⋅n=qgeo+qfricκ(T),
with the geothermal flux qgeo and the friction heating due to basal
sliding qfric=ub⋅τb . 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:
T=Tpmp if grounded,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)gradT)⋅n=0. Lastly,
temperatures at the inflow boundary are prescribed by a linear profile
Tlin=Tpmp-Tszs-zb(zs-z)+Ts.
Implementation
The thermomechanically coupled 3-D full-Stokes model COMice is implemented
in the COMmercial finite element SOLver COMSOL
Multiphysics© (cf. for
implementation details). The model has been successfully applied in the
diagnostic tests in the MISMIP 3-D model intercomparison project
and the ISMIP-HOM experiments .
The ice flow model solves for the velocity vector u, the
pressure p and the temperature T. The unstabilised Stokes equation
(Eq. ) 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. () is discretized with linear elements. To avoid numerical
instabilities due to strong temperature advection, and thus to ensure that
the element Péclet number is always <1, we use consistent stabilisation
methods provided by COMSOL. Eq. () is solved using a Galerkin least-square formulation in streamline direction and
crosswind diffusion 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 εe˙ (Eq. )
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 . The scalar values for
all parameters used throughout this study are listed in
Table .
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. ), together with the Dirichlet
condition for the basal temperature at the base (Eq. ),
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
.
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.
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 . This allows for reduced working
memory usage. For the remaining linear systems of equations, the direct solver
Pardiso ( 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.
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
and the ice thickness data of . The
grounding line position used is given by a combination of the positions
in the MODIS Mosaic Of Antarctica
corresponding to the years 2003/2004, the position in ,
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. . 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 . Please note that
ice rises are not indicated in Fig. .
Ice flow velocity
The observed surface velocity is taken from , shown in
Fig. , and used to validate the reference
simulation. The numbering of the 1–10 tributaries
feeding the central stream is based on . The
numbering used in , and
is the same for the even numbers but shifted by 1 for
the odd numbers, as they missed tributary 1 from the numbering by
. We extended the numbering from
to the tributaries 11–14, which are entering the ice shelf. Finally,
Fig. indicates the locations of the different types of
lateral boundaries (ice divide, inflow and calving front).
Parameter values.
Parameter
Value
Unit
Description
ρi
918
kg m-3
Ice density
ρsw
1028
kg m-3
Seawater density
g
9.81
m s-2
Acceleration of gravity
n
3
Stress exponent
R
8.314
J mol-1 K-1
Gas constant
Q
60 for T′≤263.15 K
kJ mol-1
Activation energy for creep
139 for T′>263.15 K
A0
3.985×10-13 for T′≤263.15 K
s-1 Pa-3
Pre-exponential constant
1.916×103 for T′>263.15 K
T0
273.15
K
Melting point for low pressures
βc
9.8×10-8
K Pa-1
Clausius–Clapeyron constant
Tsw
271.15
K
Freezing temperature of seawater
ν
0.1
Sub-melt sliding parameter
κ(T)
9.828e(-5.7× 10-3T[K-1])
W m-1K-1
Thermal conductivity
cp(T)
152.5+7.122T[K-1]
J kg-1K-1
Specific heat capacity
zsl
0
m
Sea level
spy
31 536 000
s a-1
Seconds per year
RADARSAT Antarctic Mapping Project (RAMP) mosaic with the
observed surface velocities from and the model
domain of Pine Island Glacier, with the different lateral boundaries,
the grounding line and the numbered tributaries indicated.
Temperature
The surface temperature used here is on a 5 km grid compiled by
(ALBMAP v1), based on the temperature data described
in . We use the geothermal flux qgeo from
Purucker (2012, updated version of ), because a
variety of sensitivity tests showed that other data sets lead to too-high velocities in regions with no or little basal sliding.
Roughness
In this study we use two different measurements of basal roughness
beneath the PIG. The first one is the single-parameter roughness measure
as presented in (cf. Fig. 4b therein) and represents
the methodology often employed to define subglacial roughness
.
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 ,
and more recently . They
introduce a second parameter which effectively provides a further
measure of the frequency or wavelength of roughness
obstacles . 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 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 from a RES data set
generated in austral summer 2004/05 . It is the same
data set the model geometry is based on
(Sect. ), 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. ). Both data
sets are then gridded with 1 km spacing.
By applying the work of , 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 . provide guidance on how
to interpret these two parameters in terms of their different basal
topographies, along with their geomorphic implications. The
interpretation from is based on ideas by
which give an interpretation for the
single-parameter roughness. 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 . 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.. 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 ,
to be able to apply the data in a sliding relation.
The received fields
of ξ and η for PIG are shown in Fig. . According to
, 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 . Nonetheless we continue here referring to
η as the roughness frequency for consistency with
. The suspected low-amplitude, low-frequency roughness is
not necessarily found in the central trunk area, as can be seen in
Fig. . Instead it seems to be more dominated by low-amplitude,
high-frequency roughness which can, following , 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
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.
Calculated roughness parameters at Pine Island Glacier, given
by the roughness amplitude ξ (a) and the roughness frequency
η (b).
Spatial distribution of the basal sliding parameter β2.
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.. 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
. 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. used for our reference
simulation starts by assuming the linearised form of Eq. ():
thus τb=β2ub, where
β2 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 sliding
velocity ub can be approximated by subtracting the
surface velocity due to internal deformation us,nosl
from the measured surface velocity field uobs
(, Fig. ). The basal drag from the
simulation where no basal sliding is allowed,
τb,nosl, is taken as a good first representation of
the real basal drag distribution, τb. With this, the
field of the basal sliding parameter β2 is defined as
β2=|τb,nosl|(|uobs|-|us,nosl|)-1,
as shown in Fig. . This is a significantly different approach
from in e.g. , and
, who minimise a cost function.
The basal sliding parameter β2 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. ), 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=263.15K. The resulting surface velocity field
us,nosl,1 and basal drag
τb,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 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=263.15K) → 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=1m∑i=1m(||us,sim|i-|us,ref|i|)2,
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 (“slow”) and the
entire model region (“all”) (detailed description in
). 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. b and c)
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. ), 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 ,
Sect. , 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. 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
Cb=(|uobs|-|us,nosl|)Nbq|τb,nosl|p.
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], such that
Cb(ξ)=(Cb,max-Cb,min)⋅(ξ-ξmin)(ξmin-ξmax)+Cb,max.
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
.
Experiment 3: parametrisation with two-parameter roughness
The aim of this experiment is to test the idea of 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
of describing the sliding mechanisms of regelation
and enhanced creep, such that
ub=CWτbl2a2(1+n)2,
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. ) and n=3 is the stress exponent.
state that the two-parameter roughness measures ξ and
η, representing the amplitude and frequency of the roughness
(cf. Sect. ), 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=c1ξ and
l=c2η, where c1 and c2 are proportionality factors.
Entering this into Weertmans original formulation (Eq. ) and
additionally including a temperature function f(T) as introduced
above leads to
ub=CLf(T)τbηξ(n+1)2,
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
CL=(|uobs|-|us,nosl|)|τb,nosl|-(n+1)2ξη(n+1)2.
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. ) is varied in this range.
For all simulations conducted for Experiment 3, only Eq. () is
solved for due to time constraints (cf. ). 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.
Results
Experiment 1
The resulting surface velocity field from the reference simulation is shown
in Fig. . The general pattern of the surface velocity field
is well reproduced in the reference simulation compared to the observed
surface velocity field (Fig. ). 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. ). 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. . For slower velocities the
angle offset is bigger, coinciding with a higher measurement error for
slower velocities.
Surface velocity field from the reference simulation
|us,ref| with the numbered tributaries.
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
Δα between the vectors of the surface velocity field
uobs and the reference surface velocity field
us,ref is shown as the colour code.
The simulation shows that large areas under PIG are temperate
(Fig. ). In general the overall flow pattern is reflected
in the basal temperature structure, with fast-flowing areas being
underlain by a temperate base. Figure shows the
homologous temperature Tref′ at three vertical slices, of
which the locations are indicated in Fig. . 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. a). A similar picture is found in the next slice,
which is located further downstream towards the ice shelf
(Fig. b). Here, additionally a cold core can be seen,
located in the fast-flowing central stream.
The next slice partly crosses the ice shelf (Fig. c).
It can be well observed that a cold core is entering the ice shelf. In
the vicinity of tributary 11 (cf. Fig. ) a small
temperate layer is found.
The basal homologous temperature from the reference
simulation Tb,ref′, with tributary locations in black and
the location of the vertical slices (a), (b) and (c) in
Fig. in grey.
The internal homologous temperature from the reference
simulation Tref′ at three vertical slices (a), (b) and
(c)
(horizontal locations indicated in Fig. ).
(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.
Experiment 2
We show in Fig. a 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. b shows the
surface 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. c. 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 tested yet due to cpu time
constraints (for a detailed description of the solution time of the
simulations refer to ), 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. ) 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. b). 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. ). 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. ). 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 .
Experiment 3
The RMSus deviations between the reference
simulation and all conducted parameter combinations in Experiment 3 show
a somewhat regular pattern (Fig. a). 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 decreasing and then
increasing with increasing CL, with a minimum
RMSus value of 271 m a-1 for
CL=1 Pa-2 m a-1.
Basal homologous temperature of Simulation 2.
(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.
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. b 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.
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 β2 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
β2, as τb,nosl is not vanishing near
the grounding line as would be expected (cf. and
).
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.. 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
and .
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 is located
at or near the base. To form a significant basal temperate layer,
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. and c), 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 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 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 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 , 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
( 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 be beneficial to separate the hydrology component from
the sliding relation. However, this is beyond the scope of this work.