TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-12-3229-2018Grounding-line flux formula applied as a flux condition in numerical simulations fails for buttressed Antarctic ice streamsGrounding-line flux formula fails for buttressed Antarctic ice streamsReeseRonjahttps://orcid.org/0000-0001-7625-040XWinkelmannRicardaricarda.winkelmann@pik-potsdam.dehttps://orcid.org/0000-0003-1248-3217GudmundssonG. Hilmarhttps://orcid.org/0000-0003-4236-5369Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, 14412 Potsdam, GermanyUniversity of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24–25, 14476 Potsdam, GermanyDepartment of Geography and Environmental Sciences, Northumbria University, Newcastle upon Tyne, NE1 8ST, UKRicarda Winkelmann (ricarda.winkelmann@pik-potsdam.de)9October201812103229324230December201715January201816August201826August2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://tc.copernicus.org/articles/12/3229/2018/tc-12-3229-2018.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/12/3229/2018/tc-12-3229-2018.pdf
Currently, several large-scale ice-flow models impose a condition on ice flux
across grounding lines using an analytically motivated parameterisation of
grounding-line flux. It has been suggested that employing this analytical
expression alleviates the need for highly resolved computational domains
around grounding lines of marine ice sheets. While the analytical flux
formula is expected to be accurate in an unbuttressed flow-line setting, its
validity has hitherto not been assessed for complex and realistic geometries
such as those of the Antarctic Ice Sheet. Here the accuracy of this
analytical flux formula is tested against an optimised ice flow model that
uses a highly resolved computational mesh around the Antarctic grounding
lines. We find that when applied to the Antarctic Ice Sheet the analytical
expression provides inaccurate estimates of ice fluxes for almost all
grounding lines. Furthermore, in many instances direct application of the
analytical formula gives rise to unphysical complex-valued ice fluxes. We
conclude that grounding lines of the Antarctic Ice Sheet are, in general, too
highly buttressed for the analytical parameterisation to be of practical
value for the calculation of grounding-line fluxes.
Introduction
Estimating the future impact of the Antarctic Ice Sheet (AIS) on global sea
levels invariably involves calculating changes in ice fluxes across grounding
lines, as well as determining the migration of the grounding lines
themselves. Accurately describing grounding-line dynamics can therefore be
considered an essential prerequisite for any numerical ice-flow simulation of
marine ice sheets such as the AIS. Accordingly, over the last decades,
considerable efforts have focused on ensuring that large-scale ice-flow
models are capable of correctly capturing the dynamical behaviour of grounding
lines e.g..
As part of these efforts, several model intercomparison experiments have
been conducted to assess different approaches within the ice-sheet modelling
community regarding the numerical modelling of marine-type ice sheets
.
Although it is still a subject of active research, one of the outcomes of these
intercomparison experiments has been to stress the need for a sufficiently
fine resolution of the computational domain around grounding lines. Within
the context of the shallow ice-stream computational models – a
commonly used flow approximation for describing the flow of ice streams and
ice shelves e.g. – it has, for example,
been suggested that for many applications a horizontal resolution of around
1 km or less is suitable . However, for
large-scale ice flow models using uniform grids, employing such a high
resolution globally for large ice sheets such as the AIS can be
computationally prohibitively expensive. As a way of resolving this issue,
and to allow for an accurate description of grounding-line dynamics without
resorting to high spatial resolution, in a number of numerical modelling
studies a “flux condition” is imposed at the grounding line, whereby the
grounding-line flux is prescribed using an analytical expression
e.g..
In other instances, the grounding-line migration rate is prescribed directly
e.g. without buttressing parameterisation,.
The analytical flux expression most often used is based on a theoretical
study by and was derived under the assumption
that the ice shelf provides no buttressing to the ice at the grounding
line. The absence of buttressing implies that the (vertically integrated)
horizontal stresses at the grounding line are not affected by the presence of
the ice shelf, and were the ice shelf to be removed and replaced by ocean
water, the state of stress (in a vertically integrated sense) would remain
unaffected e.g.. However, in general, and
this is certainly the case for the AIS today
e.g., ice
shelves do provide some buttressing. To account for this, numerical models
use a modified analytical expression of ice flux based on
involving an additional buttressing parameter (θ) describing the
modification in axial stress due to the mechanical impact of the ice shelf on
the stress state at the grounding line. The buttressing parameter (θ)
needs to be calculated by the numerical ice flow model and then inserted
into the analytical flux expression. The resulting flux is then used by the
corresponding numerical model as a flux condition along all grounding lines.
Previous numerical model intercomparison experiments
have shown that in the unbuttressed case there is, in general, good
agreement between the analytically and numerically calculated ice fluxes for
steady-state conditions. For one particular synthetic model
set-up, also found, in places, good agreement between
analytically and numerically calculated ice fluxes for buttressed ice. The
question now arises as to how accurately the analytical expression predicts
grounding-line ice fluxes for realistic geometries such as that of the
present-day AIS. More specifically, if one were to apply sufficiently high
resolution around all Antarctic grounding lines, would fluxes calculated
directly by such a high-resolution numerical model agree with the predictions
of the analytical flux formula? Answering this question is the subject of
this study. Here we assess the accuracy and the general applicability of the
analytical flux formula for calculating ice fluxes across grounding lines of
present-day Antarctica. We do this by comparing predicted analytical fluxes
with independently numerically calculated ice fluxes using the community
ice-flow model Úa . The ice flow model is applied
to the whole continent, using high spatial resolution around all grounding lines of
a few hundreds of metres.
The paper is structured as follows: first, we describe our numerical ice flow
model Úa, and the model initialization procedure in
Sect. . We then give a brief overview over the flux formula
derived by and discuss several different approaches to
quantifying ice-shelf buttressing. The following Sect.
on the comparison between numerically calculated grounding-line ice fluxes
and those by the flux formula forms the main part of the paper. This is
followed by a discussion of the results and final conclusions in
Sects. and .
Model description
We diagnose the fluxes at the grounding line with the finite-element ice-flow
model Úa . The flow model Úa has been used to
calculate the ice-flow for various geometries involving ice-shelf buttressing
e.g.,
and results obtained by the model submitted to a number of model
intercomparison experiments MISMIP, and
MISMIP3d,. The model employs an unstructured
grid and hence allows for resolving the grounding-line zone locally with high
resolution. Simultaneous inversion for the ice rate factor (A) and the basal
slipperiness (C) can be done either over nodal or over element values, and using
either Bayesian- or Tikhonov-type regularisation. The gradient of the resulting objective function is calculated using the adjoint method.
Here we use Úa to solve the shallow ice-stream equations
e.g. in a diagnostic mode using a
Weertman-type sliding law (see Eq. ) and Glen's flow law
(see Eq. ). In the glaciological literature the shallow
ice-stream equations are also referred to as the shallow shelf approximation or shelfy stream
approximation and often abbreviated as SSA. In the two horizontal dimensional
situation (2HD) the SSA momentum equations are
∇xy⋅(hR)-τbh=ρigh∇xys+12gh2∇xyρi,
where
∇xy=∂x,∂y,
and R
is the tensor of resistive stresses given by
Eq. (), h is the ice thickness, s is the ice surface elevation,
ρi is the vertically averaged ice density, and τbh is
the horizontal part of the bed-tangential basal traction τb.
Where the ice is floating τbh=0.
In the SSA the flotation criterion has the form h<hf with
hf=(S-B)ρw/ρi,
where S is the ocean surface, B the bedrock, and ρw is the ocean
density. The flotation criterion in Úa is evaluated at each integration
point of the elements of the finite element mesh and the basal drag term is
evaluated accordingly through a standard finite-element procedure involving
element-wise integration.
Methodology
Using the ice flow model Úa, we calculate ice velocities for the entire
Antarctic Ice Sheet, including all ice shelves. The SSA equations are solved
throughout the computational domain. Stress boundary conditions (i.e. Neumann
boundary conditions) are applied at the margins of the computational
domain. Since the modelling domain covers the whole of the AIS, no inflow or
outflow boundary conditions (i.e. Dirichlet boundary conditions) need to be
applied at any sections of the boundary.
Two different computational meshes were generated and the sensitivity of the
results was evaluated using linear (3-node), quadratic (6-node) and cubic
(10-node) triangular elements. All results presented here were obtained using
a very high-resolution mesh generated with the finite-element mesh generator
Gmsh with 1 360 894 triangular linear
elements and 689 042 nodes. Within 5 km distance to the grounding
line, the mesh was refined such that element sizes decrease towards the
grounding line to a maximum size of 250 m directly at the grounding
line. Overall, the elements have a maximal size of 179 307 m in the
interior of the continent and a minimal size of 56 m along the
grounding line. The mean element size is 1596 m and the median is
480 m. A regional example of the mesh is given in
Fig. S1 in the Supplement. The robustness of the results was also tested based on
the mesh used in , as discussed in Appendix .
Ice thickness and bed geometry input is based on the Bedmap2 estimates
. Vertically averaged ice densities were
calculated using firn thickness fields from RACMO2
and assuming a constant ice density of
910 kg m-3 and a firn density of 500 kg m-3.
Resulting densities range from 770 to 910 kg m-3 and the horizontal gradients in vertically averaged
densities are hence small; see Fig. S2. In a few places the
bathymetry around the grounding lines was vertically modified to improve its
alignment with , with vertical adjustments of
maximally 50 m being allowed.
For the entire Antarctic set-up we inverted for basal slipperiness C (see
Eq. ) and ice softness fields A (see Eq. ) to
match observed 2015–2016 velocities derived from Landsat 8 imagery
. The stress exponent of Glen's flow law was set to
n=3 and we repeated the inversion for a whole sequence of sliding law
exponents m=1, 2, 3, 4, 5, 7, 9, 11. We inverted for A and C over the
computational nodes using Tikhonov-type regularisation. The inversion
procedure minimizes the function
J(u,p)=I(u)+R(p),
with respect to p, where p stands for model parameters to be determined
(i.e. A and C), u are modelled surface velocities, I is the data misfit
function, and R is the regularisation term. The misfit function I has the form
I=12A∫vmodelled-vobserved2/e2dA,
where A=∫dA is the total area, vmodelled and
vobserved are modelled and observed velocities, respectively,
and e are the data errors. The regularisation function R has the form
R=12A∫γs2∇log10(p)-log10(p^)2+γa2log10(p)-log10(p^)2dA=12A∫γs2∇log10(p/p^)2+γa2log10(p/p^)2dA,
where γa and γs are regularisation parameters, and
p^ are the a priori values for model parameters. Inversions were done for
a wide range of γs and γa and optimal values were determined from
an L-curve analysis. In the results shown here, we use γa=1 and
γs=10000 m. However, our results are insensitive to the
particular values chosen.
For γs=10000 m, γa=1 and the sliding exponent
m=3, the corresponding basal slipperiness C and the ice rate factor A
distributions are shown in Figs. S3 and S4. The
average difference between modelled and observed ice speed is 29 m per
year with a median of 13 m per year and a root mean square error of
103 m per year. The measured and modelled velocity fields for the
region of Institute Ice Stream are displayed in the upper panels of
Fig. . They agree
well in this area, as the residual histogram for this region shows in the
lower-left panel, but also for the entire continent; see Fig. S5. As a consequence of our
inverse methodology, modelled ice velocities are in close agreement with measurements.
From the modelled stresses obtained with our ice-flow model we calculate the
buttressing parameter θ as defined in Sect. . We do
this for each of the three different definitions for θ (see
Eqs. , , and ). We then
calculate the analytical fluxes predicted by the flux formula,
i.e. Eq. (). Note that we refer to these
fluxes as analytical fluxes, although their calculation involves the use of
our numerical ice-flow model for estimating the buttressing number θ.
We also calculate modelled grounding-line fluxes from modelled ice
velocities. Since our modelled velocities are in good agreement with
observed velocities, these modelled grounding-flux estimates will be in an
equally good agreement with fluxes estimated directly from observed
velocities. The analytical and the modelled flux estimates are then compared
and analysed.
Observed a; and
modelled (b) ice speed in the region of Institute Ice Stream. The
inset displays the location of the plotted area in Antarctica. Grounding
lines are shown as black lines and streamlines are displayed in blue.
Panel (c) shows a normalised bivariate histogram of the velocity residuals
which are the differences between modelled and observed velocities within this
area, that is, Δu=umodelled-uobserved and
Δv=vmodelled-vobserved, and u and v are the
horizontal components of the surface velocity vector, respectively.
Panel (d) shows an ice-speed profile along the central line of Institute
Ice Stream that is indicated in green
in (a).
When calculating grounding-line fluxes we interpolate nodal quantities of the
computational mesh onto the (calculated) grounding line. The grounding line
does not, as such, enter the numerical calculations made by our numerical ice
flow model. As described in Sect. , it is the flotation mask – evaluated
at the integration points – that determines the impact of the
basal drag term. However, in a post-processing step we determine the
positions of the grounding lines from the flotation mask. Our approximation
of the grounding line is a piecewise linear curve, with each linear segment
representing the grounding line within a given computational element (see
Figs. S1 and ). We then interpolate
nodal values onto the central point of each linear segment. The same
procedure is employed when calculating both analytical and modelled fluxes.
Ice-shelf buttressing and grounding-line ice flux
In , an expression for the grounding-line flux (q)
of marine ice sheets is derived. While the analysis is primarily
focused on a flow-line configuration where ice-shelf buttressing plays no
role, also estimates how the flux might be affected by a
reduction θ in axial stress at the grounding line due to ice-shelf
buttressing. The resulting analytical flux expression is
q(x)=θnmm+1ρih1+m(n+3)m+114nAρign+11-ρi/ρwnC1/mmm+1,
where q is the ice flux across the grounding line, h is the ice thickness,
ρi the ice density, ρw the density of ocean water and g the
gravitational acceleration (please note that in the related Eq. 17 of
for the flux q there is a typo in the exponent of
the basal slipperiness C). For grounded ice, the tangential component of
the basal traction (τb) is related to the basal velocity (vb)
through the Weertman-type sliding law
τb=C-1/m|vb|1/m-1vb,
where C is the basal slipperiness, and m is the stress exponent, while
deviatoric stresses τij and strain rates ϵ˙ij in ice
flow are linked via Glen's flow law
ϵ˙ij=Aτn-1τij,
with τ=τijτij/2 the second invariant of the
deviatoric stress tensor, exponent n (often set to 3) and rate factor A.
Here τij denote the components of the deviatoric stress tensor
and ϵ˙ij the components of the strain rate tensor.
Buttressing ratio θ1 along the grounding lines of
Filchner–Ronne Ice Shelf (a) and Ross Ice Shelf (b).
Insets indicate the ice shelves' locations in Antarctica.
Regions where the grounding line is over-buttressed, that is,
θ≤0, are displayed in black. Modelled speed is plotted in grey
ranging up to 1500 m a-1. Grounding line and ice front locations are
indicated in black. IS denotes ice streams; IR denotes ice rises or
rumples.
As mentioned above, θ is a scalar quantity that describes the deviation
in deviatoric axial stress at the grounding line from the unbuttressed
situation. For an unbuttressed grounding line in one horizontal dimension
(i.e. no variations in any quantities transverse to the flow direction) and
assuming that the x axis of the coordinate system is aligned with the flow,
we have τxx=τf where
τf=ρig41-ρiρwh,
which can be derived from the stress boundary condition at the calving front
(see Appendix ). In the buttressed case, τxx is, however,
no longer necessarily equal to τf, and θ is defined as
θ1HD=τxx1HDτf.
Here we have used the superscript 1HD to indicate that this definition of θ
is only unambiguous in the one horizontal dimensional situation (1HD).
In the more general two horizontal dimensional situation (2HD), where
the flow direction is not necessarily aligned with the (horizontal) normal to
the grounding line, several different definitions of θ are possible,
and in the literature at least three different definitions of θ have
been suggested. In the following we denote these by θ1, θ2,
and θ3, with
θ1=n1⋅Rn12τf,
where n1 is the normal to the grounding line, pointing
horizontally outwards from the grounded ice into the ice shelf, and
θ2=n1⋅τn1τf,
and
θ3=n2⋅τn2τf,
where n2 is the direction of ice flow at the grounding line and
τ=τxxτxyτxyτyy,
is the (horizontal) deviatoric stress tensor, and
R=2τxx+τyyτxyτxyτxx+2τyy,
is the tensor of resistive stresses. In the 1HD unbuttressed case where
n1=n2, τxx=ρigh(1-ρi/ρw)/4,
and τyy=τxy=0, all these three
definitions of θ result in θ1=θ2=θ3=1. The first
definition (i.e. θ1) has, for example, been used by
to diagnose buttressing at the grounding line of an
idealised set-up. The second definition has, for example, been used by ,
, and as a flux condition, and the
third one has been used by to diagnose “flow buttressing” within
Antarctic ice shelves. Note, however, that see
Sect. 2.3, (, see Sect. 4.4),
(, see Supplementary Eq. 2), and (; see
Eq. 20) appear to use a different expression for τf, with
τf=ρigh(1-ρi/ρw)/2, in which case θ=1/2 in the
unbuttressed case and θ in Eq. ()
must be replaced by 2θ.
Buttressing ratio and differences in grounding-line flux for
Institute Ice Stream draining into the Filchner–Ronne Ice Shelf (location
shown in inset). (a) Buttressing values θ1 are displayed
along the grounding line and principle deviatoric stresses are shown with
compression in red and extension in blue. The length of the vectors indicate
the magnitude of each principle stress. (b) Differences between
analytical and modelled fluxes and observed ice velocities ranging up to
500 m a-1. Analytical fluxes are set to 0
where θ1<0. Grounding-line positions are indicated in
black.
Comparison of fluxes calculated with Úa (blue) and analytical
fluxes (black) along the grounding lines of four major ice streams draining
into the Filchner–Ronne Ice Shelf. Locations where the flux formula provides
unphysical results are marked in grey. Plotted grounding-line segments are
located as displayed in the inset with western margins indicated by a yellow
dot.
The definition of θ1 is motivated by the form of the boundary
condition at the calving front in the shallow ice-stream approximation (see
Appendix ). For θ1=1 the normal traction at the grounding
line equals that of a calving front. In the general 2HD situation, this same
interpretation does not hold for the definitions of θ2 and θ3.
If θ1>1 the ice shelf can be considered to be “pulling”
the ice at the grounding line, while θ1<1 implies that the ice shelf
causes a reduction in normal traction at the grounding line; i.e. the ice
shelf “holds the ice back”. Note that, for all these three different
definitions, it is possible for θ to become negative. If, however, a
negative θ value is inserted into
Eq. (), the resulting value for the flux q
is a negative or even a complex number for most combinations of n and m – a
clear indication that the analytical flux formula fails in such
situations. Only the specific combinations of n and m, such that nm/(m+1)=2k
for k∈N (for instance the combination n=3 and m=2),
“fix” the flux to a positive real number; however they introduce a
non-substantiated dependency between the flow law and the sliding law.
Furthermore, for these combinations and θ<0, enhanced
buttressing inconsistently yields an increase in ice flux. Physically, θ1<0
corresponds to a situation where the traction vector at the grounding line
points in upstream direction. One possible situation giving rise to
θ1<0 would be τxx<0 while τyy=0, with x being the
flow direction and the grounding line aligned with the y axis. In this
case, the ice at the grounding line experiences compression in the along-flow
direction and, hence, longitudinal strain rates are negative and ice
velocities become smaller as the grounding line is approached from upstream
direction. Another situation giving rise to θ1<0 is that of equal
transversal compression and vertical extension of the ice column at the
grounding line; i.e. τyy=-τzz<0 while τxx=0.
Results
From the numerically modelled stress field we calculate the buttressing
parameter θ1 (given by Eq. ) for all grounding lines
of the Antarctic set-up described in Sect. . While here we
focus on the buttressing parameter θ1, our findings are
independent of the exact definition of θ, the choice of the sliding
law exponent m, the mesh and the details of the inverse methodology
applied (see Appendix ).
We find that the grounding lines of the Filchner–Ronne and Ross ice shelves are,
in general, highly buttressed with buttressing values significantly different
from unity (see Fig. ). Typically, θ1≤0.4,
and in many cases θ1<0. Among the ice streams of these two
biggest ice shelves of the AIS, the dormant Kamb Ice Stream is the
least buttressed one, with θ1≈0.4. Over all other ice streams
θ values are even smaller. Negative θ values are also found
over grounding-line segments located between active ice streams, for example
along the grounding line running between the Rutford and Institute ice streams.
An example of an ice stream where θ1<0 over most of its grounding
line is the Institute Ice Stream (see Figs.
and ). Inspection of the velocity field in the
vicinity of the grounding line of that ice stream reveals that ice flow
velocities decrease with distance as the grounding line is approached from
upstream direction (see also Fig. d). Consequently, both along-flow strain rates and along-flow deviatoric
stresses are negative (compressive). This general feature of ice flow around
the grounding line of Institute Ice Stream implies that its grounding line is
“over-buttressed” with the traction vector at the grounding line pointing in
an inland direction. Hence, independently of our numerical simulation of the
stress field, it is clear that for this ice stream a negative value for θ is obtained.
As discussed in Sect. the analytical flux formula
(Eq. ) is clearly not applicable in
situations where θ becomes negative. As θ is found to become
negative over large sections of the grounding lines of many the ice streams
of the two largest Antarctic ice shelves, i.e. Ross and Filchner–Ronne ice
shelves, it follows that the formula cannot be used to calculate
grounding-line ice fluxes over significant parts of the AIS.
We furthermore compare analytical and numerically modelled grounding-line ice
fluxes in all regions where θ1≥0, i.e. where the application of
the analytical flux formula (Eq. )
results in real-valued ice fluxes. In particular, we compare both the flux
values pointwise along all grounding lines
(Fig. ) and the total cumulative fluxes over
grounding lines of ice streams and ice shelves
(Table ). When comparing cumulative
analytical fluxes, we are forced to assume values for those sections of
grounding lines for which θ is negative (and q complex). There we
assume q=0, which is equivalent to setting θ=0.
In general, we find significant differences between analytically calculated
and numerically modelled flux values. Analytical fluxes are much lower than
modelled in many locations of the Filchner–Ronne Ice Shelf, especially along the
grounding lines of the Rutford, Institute and Moeller ice streams
(Fig. ). However, cumulative analytical fluxes
over all grounding lines of the Filchner–Ronne Ice Shelf are about 30 %
larger than modelled for θ1, and this difference is considerably
larger for θ2 and θ3 (Table ). A similar disagreement between
analytical and modelled fluxes is found for the Siple Coast ice streams such
as Bindschadler and MacAyeal ice streams, and for Byrd Glacier
(Fig. b). For Ross Ice Shelf the overall
difference is only 5 %, but, given the fact that θ1 is negative over
significant sections of its grounding line (where we set the analytical flux
values to zero), this agreement appears somewhat fortuitous.
For other ice shelves, cumulative fluxes are generally underestimated by the
flux formula. Analytical fluxes for Pine Island Glacier and Thwaites Glacier,
for example, deviate by -33 % and -52 % from the modelled fluxes,
respectively. For George VI ice shelf, cumulative analytical fluxes are
several times smaller than modelled ones (Table ).
The analytical flux formula tends to strongly overestimate fluxes over
grounding lines where ice flow is approximately tangential to the grounding
line. The failure of the flux formula to correctly predict fluxes in such
circumstances is not surprising, as the underlying assumptions of the formula
are clearly not met in these situations. Nevertheless, this demonstrates the
inherent conceptual difficulties in applying the formula to the Antarctic Ice Sheet.
Moreover, the analytical formula produces much higher spatial variability in
fluxes than the numerically modelled ones. This can be clearly seen in
Fig. , where analytical and modelled ice fluxes
are plotted along the grounding lines of Rutford, Institute, Foundation and
Recovery ice streams. Here, the grey background indicates sections of the
respective grounding lines where the flux formula yields unphysical results.
Variability in fluxes calculated with Úa occurs when ice flow is nearly
aligned with the grounding line. We calculate fluxes within each triangular
element using the normal of the piecewise-linear grounding-line curve, which
may vary among individual line segments.
Ice flux integrated along the grounding lines of Antarctic ice
shelves. QÚa denotes the modelled ice flux with Úa, Q1
was derived from the analytical flux formula based on θ1,
Q2 based on θ2 and Q3 based on θ3,
respectively. The last column shows the deviation of the analytical flux Q1
from the modelled QÚa.
Difference between the analytical and the modelled fluxes along the
grounding lines of Filchner–Ronne Ice Shelf (a) and Ross Ice
Shelf (b). Analytical fluxes are calculated based on θ1
defined in Eq. (). In locations where the formula yields
unphysical results, fluxes are set to zero. Grey arrows show the modelled ice
flow. IS denotes ice streams; IR denotes ice rises or rumples. Grounding line
and ice front locations are indicated in
black.
We test the sensitivity of our analytical flux calculations to different
degrees of regularisation (γs and γa), different values of
the sliding law stress exponent (m) and relaxation of the ice geometry, for
which our findings are summarised in Appendix .
Numerically modelled fluxes are, as expected, mostly independent of the value
of the sliding law stress exponent m. This can be considered to be a
consequence of the inversion procedure, which ensures that modelled velocity
fields agree closely with measured data, independently of the value of m.
On the other hand, analytically calculated flux values are highly sensitive
to the value of m (see Fig. ). For example, cumulative
analytical fluxes for Filchner–Ronne Ice Shelf increase by about a factor of
5 as m is changed from 1 to 7, while numerically modelled fluxes change
by less than 10 %. Numerically modelled fluxes are also insensitive to the
exact degree of regularisation applied, whereas analytically calculated flux
values change significantly (Fig. ). The
dependency of the analytically calculated fluxes on the amount of
regularisation used in the numerical model is due to the impact
regularisation has on modelled stresses and, therefore, on the value of θ.
We also compare analytical fluxes as calculated using the three different
definitions (Eqs. , and ) for θ.
While overall spatial variability of θ is similar for these
three definitions, with all definitions giving rise to extended areas of
negative θ values, the cumulative flux for the alternative
definitions θ2 and θ3 are generally higher than for θ1 (see also
Fig. ). Extended negative θ values and large
flux deviations are consistently found after a short relaxation period (see
Figs. S6 and S7).
Discussion
The analytical grounding-line flux formula
Eq. () was derived for a flow-line
configuration , and there is no reason to doubt its validity
in that particular case. When applied to a flow-line configuration, many
current ice-flow models employing the shallow ice-stream approximation (SSA)
with Weertman-type sliding law, have demonstrated excellent agreement
between modelled and analytical grounding-line fluxes .
Ice fluxes and grounding-line positions calculated with the ice flow model
Úa also agree closely with those predicted by
Eq. (), where such an agreement is to be
expected. The inclusion of the buttressing parameter θ was used by
to illustrate the potential impacts of ice-shelf
buttressing on ice flux, provided its effects were sufficiently small
to not invalidate the basic assumption of a flow-line setting too strongly.
However, we find that most of the grounding lines of the AIS are highly
buttressed, with θ significantly different from unity. It seems likely
that at least part of the reason that the analytical flux formula fails
relates to the high degree of buttressing that we find to be characteristic
for most Antarctic ice streams.
When applied to the current geometry and the current flow field of the AIS,
the flux formula predicts either unphysical or highly inaccurate flux values
when compared to modelled ones. While we have done the comparison with
numerically modelled fluxes, a comparison with observed fluxes – calculated
from measured surface velocities, observed grounding-line positions, and
measured ice thicknesses – would not alter our conclusions as, due to our
inversion procedure, observed and modelled surface velocities are in good agreement.
The strongest indication that the analytical flux formula fails when applied
to the Antarctic Ice Sheet is arguably the fact that it predicts non-real
valued fluxes over significant parts of Antarctic grounding lines. This
happens whenever θ becomes negative. Although for specific
combinations of n and m (such as n=3 and m=2) the resulting exponents
in the flux formula are even numbers – in which case the analytical fluxes
are always real positive numbers – the flux values are still unphysical
(see Sect. ). As we point out above, even a cursory
inspection of the velocity field of the AIS suffices to show that θ is
negative for a number of grounding lines (e.g. the Institute Ice Stream
grounding line). Hence, the occurrence of negative θ values is not
simply a feature of our particular numerical approach, but a general aspect
of the current ice-flow regime of the AIS.
As analytical ice fluxes are strongly dependent on ice thickness (h) at the
grounding line, they depend somewhat on the specifications of the numerical
model: the exact location of the grounding line is influenced by the mesh
resolution used by the model. The resulting error is an example of a
discretization error that becomes smaller as the mesh is refined. Other
numerical models using a different computational mesh may locate the
grounding line differently and hence calculate different ice flux values. We
tested the dependency of our modelled ice fluxes to grid resolution by using
several different meshes – an example of two such meshes is given in
Fig. – and found none of our main conclusions to
be affected by differences in mesh resolution.
As measured by the buttressing parameter θ1, almost all grounding
lines of the AIS can be considered to be strongly buttressed with, in most
cases, θ<0.4. Hence, theoretical concepts based on the assumption of
none, or insignificant, ice-shelf buttressing may not apply to present-day
Antarctica. One such theoretical prediction of considerable relevance for the
possible future of the AIS relates to the stability of its grounding lines.
In the absence of ice-shelf buttressing, grounding-line stability is
predicted to be related to local bed slope
. However, in
the presence of ice-shelf buttressing no such simple conclusions can be drawn
e.g.
Possibly, rather than being dominated by local bed slope, the stability
regime of the Antarctic Ice Sheet is to a leading order dependent on the
properties of the ice shelves downstream of its grounding lines
(e.g. geometry and structural integrity), as also supported by
and . Further work is needed to address the
question of the stability of Antarctica's grounding lines.
Conclusions
In our study, we compare grounding-line ice fluxes obtained by an ice-sheet
model with fluxes predicted by an analytical flux formula based on
. The formula includes a parameter (θ) to
account for ice-shelf buttressing, and the resulting flux is sometimes
applied as a grounding-line flux condition in numerical simulations. We find
that the formula results in unphysical and grossly inaccurate grounding-line
fluxes for most of the AIS. We furthermore find that almost all Antarctic
grounding lines are highly buttressed, suggesting that the underlying
assumptions of the analytical flux formula are not met for the current
configuration of the Antarctic Ice Sheet.
The data and code that support the findings of this
study are available from the corresponding author upon request.
Vertically integrated stress boundary condition at a free calving front
A derivation of the boundary condition at the calving front for the momentum
equations in 2HD can be found in and
, for example. At the calving face it holds that
∫bsσ⋅ncfdz=-∫bSpw⋅ncfdz,
where ncf=(nx, ny, 0) is the normal of the calving front pointing outwards,
s is the ice surface, S is the sea-level and pw is hydrostatic pressure in
the ocean pw=ρwg(S-z). The balance in x direction reads
∫bsσxxnx+σxynydz=∫bS-ρwg(S-z)nxdz=-ρi2g2ρwh2nx.
We can rewrite σxx=2τxx+τyy+σzz (since
σxx=τxx+p, σzz=τzz+p and τxx+τyy=-τzz).
Under the assumptions of the cryostatic stress approximation, σzz=-ρig(s-z).
The vertically integrated horizontal stress balance equals
∫bsσxxnx+σxynydz=2hτxxnx+hτyynx+hτxyny-ρig2h2nx,
since τxx, τyy, nx and ny do not vary vertically.
Inserting this in Eq. () yields the following:
2τxx+τyynx+τxyny=ρig21-ρiρwhnx.
A similar expression is obtained for the y direction. This can be abbreviated as
R⋅n=ρig21-ρiρwhn.
Following we obtain the normal buttressing value,
which compares the RHS and LHS of the equation above in the direction of the
normal n at the grounding line:
θ=n⋅Rnρig21-ρiρwh=n⋅Rn2τf.
In the case of a laterally uniform unconfined ice shelf with τyy=0
and τxy=0, this reduces to τxx/τf.
A different approach that defines θ would be based on this vertically
integrated stress boundary condition in 1HD with θ1HD=τxx/τf.
In 1HD the normal at the grounding line is equal to the
flow direction. In 2HD, this is not necessarily true. Thus, to generalize the
longitudinal direction in the 1HD buttressing ratio, a choice needs to be
made. The longitudinal direction can either be generalized as the normal at
the grounding line (θ2) or as the flow direction (θ3).
Consistent results using different model parameters
We test the robustness of our findings with respect to the mesh, the sliding
law stress exponent m, the definition of the buttressing parameter θ,
the regularisation parameter γs and relaxation of the ice
geometry. In a second pan-Antarctic set-up, based on a different mesh with quadratic base functions (instead of linear elements; see
Fig. ), we find a similar pattern of
θ1≤0,
which yields similar flux differences as exemplified in
Fig. for the Filchner–Ronne Ice Shelf. In this case,
inversion was done for element-based basal slipperiness and ice softness
(instead of inverting on a nodal basis) using a Bayesian methodology (instead
of Tikhonov regularisation) and the MEASURES velocity data set
( instead of Landsat 8, ).
This set-up is further described in . In this
set-up, Bedmap2 bathymetry is not adjusted around the grounding line. This
indicates that the exact location of the grounding line does not affect our findings.
For the Antarctic-wide set-up described in Sect. , we
test the choice of the stress exponent m in the sliding law. Different
choices of m=1, 3, 7 yield good agreement in modelled fluxes but large
disagreement between analytical fluxes; see Fig. .
Comparing shelf-wide integrated fluxes for major Antarctic ice shelves shows
that the definitions θ2 and θ3 of the buttressing
parameter also yield large deviations from the modelled fluxes; see
Fig. . Similarly, we find that the choice of the
regularisation parameter γs does not influence the results
significantly; see Fig. . Extended areas of
negative θ values and large flux deviations are consistently found
after a short relaxation run; see Figs. S6 and S7. Our findings are hence
independent of the details of numerical modelling choices.
Close-up of the Bindschadler grounding line for two different meshes.
(a) Elements and nodes of the mesh presented in the main text. The
mesh was refined, especially around the grounding line, and linear 3-node
elements were employed. (b) An alternative mesh with 6-node elements
with quadratic base functions. The grounding-line position is indicated in
both meshes in orange.
Difference between formula-derived and modelled fluxes along the
grounding lines of Filchner–Ronne Ice Shelf. In contrast to
Fig. a different mesh was employed (exemplified
in the right panel in Fig. ), the data assimilation
was conducted using Bayesian inversion and based on the MEASURES velocity
data set . The analysis was done using quadratic
elements. This Antarctic-wide set-up is described in more detail in
.
Comparison of fluxes calculated with Úa (x axis) and with the
analytical flux formula (y axis, using θ1), integrated along the
grounding lines of the ice shelves indicated in the legend. Symbols indicate
the different sliding law exponents m=1, 3, 7. All other
parameters agree with the reference run (indicated by a circle). The dashed
line shows where fluxes calculated with Úa and predicted by the formula
would agree.
Comparison of fluxes calculated with Úa (x axis) and with the
analytical flux formula (y axis), integrated along the grounding lines of
the ice shelves indicated in the legend. Symbols indicate the different
definitions of θ as described in Sect. . All other
parameters agree with the reference run (indicated by a circle). The dashed
line shows where fluxes calculated with Úa and predicted by the formula
would agree.
Comparison of fluxes calculated with Úa (x axis) and with the
analytical flux formula (y axis, using θ1), integrated along the
grounding lines of the ice shelves indicated in the legend. Symbols indicate
the different regularisation parameters γs. All other
parameters agree with the reference run (indicated by a circle). The dashed
line shows where fluxes calculated with Úa and predicted by the formula
would agree.
The supplement related to this article is available online at: https://doi.org/10.5194/tc-12-3229-2018-supplement.
All authors designed the study. RR carried out the analysis
based on an Antarctic set-up and using the ice-flow model Úa, both
developed by GHG. RR prepared the manuscript with contributions from GHG and RW.
The authors declare that they have no conflict of
interest.
Acknowledgements
We would like to thank Christian Schoof, the anonymous reviewer and the
editor Olivier Gagliardini for their helpful comments on the manuscript.
Ronja Reese was supported by the COMNAP Antarctic Research
Fellowship 2016–2017, the German Academic National Foundation, the
postgraduate scholarship programme of the state of Brandenburg and the
Evangelisches Studienwerk Villigst.
The article processing charges for this open-access publication
were covered by the Potsdam Institute for Climate Impact Research (PIK).
Edited by: Olivier Gagliardini
Reviewed by: Christian Schoof and one anonymous referee
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