TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-497-2016Modelling calving front dynamics using a level-set method: application to Jakobshavn Isbræ, West GreenlandBondzioJohannes H.jbondzio@uci.eduSeroussiHélènehttps://orcid.org/0000-0001-9201-1644MorlighemMathieuhttps://orcid.org/0000-0001-5219-1310KleinerThomashttps://orcid.org/0000-0001-7825-5765RückampMartinhttps://orcid.org/0000-0003-2512-7238HumbertAngelikahttps://orcid.org/0000-0002-0244-8760LarourEric Y.Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research, Bremerhaven,
GermanyJet Propulsion Laboratory – California Institute of Technology, Pasadena, CA, USADepartment of Earth System Science, University of California Irvine, Irvine, CA, USAFaculty 05: Geosciences, University of Bremen, Bremen, GermanyJohannes H. Bondzio (jbondzio@uci.edu)3March201610249751014September201515October201513February201616February2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/497/2016/tc-10-497-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/497/2016/tc-10-497-2016.pdf
Calving is a major mechanism of ice discharge of the Antarctic and Greenland
ice sheets, and a change in calving front position
affects the entire stress regime of marine terminating glaciers. The
representation of calving front dynamics in a 2-D or 3-D ice sheet
model remains non-trivial. Here, we present the theoretical and technical
framework for a level-set method, an implicit boundary tracking scheme, which
we implement into the Ice Sheet System Model (ISSM). This scheme allows us to
study the dynamic response of a drainage basin to user-defined calving rates.
We apply the method to Jakobshavn Isbræ, a major marine terminating
outlet glacier of the West Greenland Ice Sheet. The model robustly reproduces
the high sensitivity of the glacier to calving, and we find that enhanced
calving triggers significant acceleration of the ice stream. Upstream
acceleration is sustained through a combination of mechanisms. However, both
lateral stress and ice influx stabilize the ice stream. This study provides
new insights into the ongoing changes occurring at Jakobshavn Isbræ and
emphasizes that the incorporation of moving boundaries and dynamic lateral
effects, not captured in flow-line models, is key for realistic model
projections of sea level rise on centennial timescales.
Introduction
Calving of icebergs is a major mean of ice discharge for marine terminating glaciers around the
world. It accounts for about half of the ice discharge of the Greenland and Antarctic ice sheets
. This process causes calving front retreat, which leads to
reduced basal and lateral resistive stress and results in upstream flow acceleration.
In order to assess the impact of calving on the dynamics of outlet glaciers using an ice sheet
model, we need to include a dynamically evolving calving front.
This requires tracking the calving front position and adjusting the boundary
conditions accordingly.
Addressing these issues is rather straightforward for 1-D flow-line or 2-D flow-band models
, where the calving front is tracked along the flow line. However,
this type of model lacks the consistent representation of lateral momentum transfer
and lateral ice influx from tributaries for example, which have to be parameterized instead. This
parameterization may neglect feedback effects important for simulations on decadal to centennial timescales, e.g. catchment area entrainment .
It is therefore critical to include a front tracking scheme in 2-D horizontal and 3-D models,
which has been addressed by only a few ice sheet models e.g.. Various
approaches to model the evolution of the shape of ice exist. Explicit methods track the position of
a set of points, which represent the calving front. They require a complex technical framework to
allow for geometric operations like folding and intersection of the continuum boundary, tracking
singularities in curvature, and determining the position of a point in space relative to the
modelled continuum.
Alternatively, the level-set method LSM; represents the
continuum boundary implicitly by a contour, or “level set”, of a so-called
“level-set function” (LSF). It easily handles topological changes of the
modelled continuum, like splitting and merging. The LSM is based on a partial
differential equation similar to the mass transport equation solved by ice
sheet models. This makes the method straightforward to implement and allows
for the application to continental-scale ice sheet simulations. Although the
method does not necessarily conserve volume accurately, it is well
established in continuum fluid mechanics . A LSM
has been applied to represent the ice surface in 2-D flow-band models
but not to model real ice sheets yet.
Understanding calving dynamics remains challenging because of the diversity of factors involved in
calving events. Bathymetry, tides, and storm swell, as well as sea ice cover, ice mélange, and temperatures of
both sea water and air are possible factors influencing calving rates. However, their effect, their
respective share, and their interplay seem to vary from glacier to glacier and are not well
understood . Therefore, no universal calving rate parameterization exists
to date , and we rely here on user-defined calving rates.
However, incorporating calving rate parameterization in the LSM should be straightforward.
Jakobshavn Isbræ is a major marine terminating glacier in West Greenland, which drains about
6.5 % of the Greenland Ice Sheet . It is characterized by two branches, which
today terminate into a 30 km long ice-choked fjord (Figs. and
).
The southern branch exhibits high flow velocities, which are confined to a narrow, deep trough of
about 5 km width. The trough retrogradely slopes inland to a maximum depth of about 1700 m below
sea level and discharges most of the ice of the drainage basin
.
We refer to the fast-flowing area as “ice stream” and to the surrounding slow-moving ice as “ice
sheet”.
Those areas are separated by pronounced shear margins on either side of the ice stream.
Observations have shown that the fast flowing areas of Jakobshavn Isbræ
exhibit a weak bed with a basal layer of temperate, soft ice .
Basal sliding and shear in this layer cause most of these areas' horizontal motion. A large fraction of
the ice stream's momentum is transferred to the adjacent ice sheet by lateral stress. It is
thus well justified to use the 2-D shelfy-stream approximation
SSA, to simulate this glacier.
Until the late 1990s, Jakobshavn Isbræ had a substantial floating ice tongue, which extended well into
the fjord and was fed by both branches.
The calving front position remained fairly constant from 1962 to the 1990s , and the glacier exhibited
negligible seasonal variations in flow speed .
In the 1990s, the glacier started a phase of acceleration, thinning, and retreat that followed the breakup of its ice tongue.
Seasonal variations in calving front position and flow velocity increased sharply
.
Today, the glacier is one of the fastest ice streams in the world.
It is still far from equilibrium and is a major contributor to global sea level rise .
Observations suggest that the calving front position is a major control on the ice stream dynamics .
Various hypotheses have been proposed to explain the mechanisms behind this
change. All identify the breakup of the floating ice tongue as the initial
trigger of this dramatic chain of events, but different mechanisms have been
proposed to explain the sustained acceleration, thinning, and retreat of the
glacier. On the one hand, studies by
and propose loss of buttressing and
changes in basal conditions as the main cause behind the ongoing
acceleration. On the other hand, argue that weakening of the lateral
shear margins has significantly amplified the upstream acceleration. Several
modelling studies of the glacier, which use 1-D flow-line and
2-D flow-band models, project unstable retreat of the glacier along
its southern trough for up to 60 km inland within the next century
. Other modelling studies argue that
this type of ice stream is stable as long as it is fed by the surrounding ice
sheet . However, numerical 2-D plan-view modelling efforts
of Jakobshavn Isbræ so far lacked the representation of a dynamically
evolving calving front. Hence, the hypotheses could not be tested in a
satisfactory manner.
We present here a LSM-based framework to model the dynamic evolution of a calving front. This method is a step
towards better physical representation of calving front dynamics in 2-D and 3-D ice sheet models. We
describe the implementation of the method into the Ice Sheet System Model
ISSM,, a parallel, state-of-the-art ice sheet model, and apply it here to
Jakobshavn Isbræ in order to model its dynamic response to perturbations in calving rate.
TheoryIce flow model
We employ the SSA on both floating and grounded ice.
It neglects all vertical shearing but includes membrane stresses.
The ice viscosity, μ, follows Glen's flow law :
2μ=Bε˙e1-nn.
Here, n=3 is Glen's flow law coefficient, B the ice viscosity parameter, and ε˙e the effective strain rate.
We apply a Neumann stress boundary condition at the ice–air and ice–water interface, corresponding
to zero air pressure and hydrostatic water pressure respectively.
A linear friction law links basal shear stress, σb, to basal sliding velocity,
vb, on grounded ice:
σb=-α2Nvb,
where α denotes the basal friction parameter.
We calculate the effective basal pressure, N, assuming that sea water pressure applies everywhere at
the glacier base, which is a crude approximation far from the grounding line.
The ice thickness, H, evolves over time according to the mass transport equation:
∂H∂t=-∇⋅Hv+as+ab.
Here, v is the depth-averaged horizontal ice velocity, and as and ab are the surface and basal mass balance respectively.
We determine the grounding line position using hydrostatic equilibrium and treat it with a
sub-element parameterization .
We refer the reader to for details on the treatment of these equations in ISSM.
Let Ω be a computational domain in 2-D or 3-D space and φ a
real, differentiable function on Ω×R+, called LSF.
For any c∈R, we define the contour, or “c level set”, of
φ as φ(x,t)=c. Taking its material derivative yields
the “level-set equation” (LSE):
∂φ∂t+w⋅∇φ=0.
This Hamilton–Jacobi type partial differential equation describes how
level sets move with the local
value of the velocity w, which is called level-set velocity. We need
to provide an initial condition
φ0(x)=φ(x,t=0) to solve the LSE.
We use φ to partition Ω into three disjoint subdomains:
the ice domain, Ωi(t), its complement, Ωc(t),
and their common boundary, Γ(t),
φ(x,t)<0⇔x∈Ωi(t)φ(x,t)=0⇔x∈Γ(t)φ(x,t)>0⇔x∈Ωc(t).
We omit the time dependence of these sets in the remainder of this article.
By construction, Γ, the 0 level set of φ, separates Ωi and
Ωc.
We extend n, the outward-pointing unit normal of Γ, onto Ω using the LSF by
n=∇φ|∇φ|.
For details on the level-set method and its applications, we refer to and
.
Schematic of the numerical ice margin. The red dashed lines denote different contour
lines (level sets) of the LSF φ. The thick red line marks the 0 level set,
Γ, and
the yellow line the numerical calving front Γh. Dark blue triangles are
ice-free elements, white ones are ice-filled, and the light blue ones are the front elements.
The three vectors show an example of the level-set velocity w=v+c,c=-c⟂n at a finite element node.
The boundary position of an ice sheet evolves with the sum of the ice
velocity v along n and an ablation rate, a⟂:
w⋅n=v⋅n-a⟂.
It follows that the ice boundary is stationary if and only
if a⟂=v⋅n, i.e. the ablation rate matches the ice velocity perpendicular to the ice
boundary.
The ablation rate needs to be prescribed, either based on observations or through a parameterization.
Note that no limitations have been made so far with respect to the dimension of the problem in this section.
Accordingly, the method could be applied to model the evolution of the glacier thickness and
lateral extent simultaneously .
However, in this 2-D plan-view model study, we use the LSM to model only the horizontal extent of the ice sheet.
Its vertical extent is described by the mass transport Eq. ().
We model lateral ablation as the sum of a melting rate, m⟂, and a
calving rate, c⟂: a⟂=c⟂+m⟂. For simplicity, we
assume in the remainder of the article that lateral ablation occurs in the
form of calving exclusively, i.e. m⟂=0. Calving itself is assumed to
be a quasi-continuous process, consisting of frequent, but small, calving
events. With Eqs. () and
(), the LSE (Eq. 4) becomes
∂φ∂t+v⋅∇φ=c⟂|∇φ|,
which is also known as “kinematic calving front condition” KCFC;
. Both the calving rate and ice velocity need to be
provided on the entire 2-D computational domain Ω in order to solve
the KCFC. An example of a calving rate field will be given in
Sect. and is shown in
Fig. . The KCFC implies that all level
sets of φ, including the
calving front Γ, move at a given time with the local sum of the
horizontal ice velocity and calving rate along the normal n
(Fig. ). We define the calving flux,
Qcf, as the ice flux crossing the calving front:
Qcf=∫Γc⟂(r)H(r)dr.
The ice velocity components and the ice thickness are only defined on Ωi and need to be
extended onto Ωc (see also Sect. ). Any scalar field,
S, is extrapolated onto Ωc
by solving
n⋅∇S=0,
while keeping S fixed on Ωi.
This type of extrapolation has the tendency to preserve |∇φ|=O(1) when
we use the extrapolated ice velocity field to solve the KCFC .
Implementation
ISSM relies on the finite element method (FEM) to solve partial differential equations.
It applies a continuous Galerkin FEM using triangular (2-D) and prismatic (3-D) Lagrange finite
elements and uses anisotropic mesh refinement to limit the number of degrees of freedom while maximizing spatial
resolution in regions of interest.
We discretize the KCFC (Eq. ) and extrapolation equation
(Eq. ) using linear finite elements on the same mesh as the one used to model the ice dynamics.
We stabilize both equations with artificial diffusion , which after thorough testing proved to be the most robust stabilization scheme.
We integrate over time using a semi-implicit time-stepping scheme.
We solve the KCFC and the field equations for ice flow modelling in a decoupled fashion.
The KCFC is solved first with input data from the previous time step.
We then update the numerical ice domain using the new LSF as described below and update boundary
conditions accordingly.
Finally, we solve the momentum balance and the mass transport equation on the updated ice domain.
The 0 level set of φ,
Γ, does not in general coincide with the finite element mesh edges due
to its implicit representation. It intersects a number of elements (“front
elements”) with a hyperplane, which divides them into an ice-filled and an
ice-free part (Fig. ). This has various
implications on the numerical level. When assembling the system stiffness
matrices for ice flow modelling, exclusive integration over the ice-filled
part of the element would be required. The stress boundary condition at the
calving front would have to be applied at the intersecting hyperplane.
Currently, ISSM is not capable of resolving those submesh-scale processes.
Therefore, we either fully activate or deactivate a mesh element at every time step.
Only active elements are considered for the numerical discretization of the respective field
equations.
We activate an element if at least one of its vertices is in Ωi or Γ,
and the element is then considered to be entirely filled with ice.
We flag the element as ice free if it lies entirely inside Ωc, and it is
deactivated.
As a consequence, the numerical calving front, Γh, runs along mesh edges and is updated
in a discontinuous manner (Fig. ).
We apply the stress boundary condition along Γh for numerical consistency.
Calving front normals on Γ and Γh may differ significantly in direction.
However, stress components tangential to n cancel out along Γh, so that the
integrated stress exerted at the calving front is close to the one applied along Γ.
For all further calculations where a normal is involved, like extrapolation, the normal to the LSF (Eq. ) is used.
The numerical calving front is by definition further downstream than Γ.
This may lead to slightly higher resistive lateral stress at the calving front, the magnitude of which depends on the excess ice area
of the intersected front element and the front geometry.
We use a fine mesh resolution in the vicinity of the calving front to limit this effect.
We extrapolate the calving front thickness onto the ice-free domain using
equation (Eq. ).
This yields realistic ice thickness and ice thickness gradients across the front elements
that would otherwise lead to overestimated driving stress and underestimated water pressure at the
ice–ocean interface.
If not corrected, those two effects unrealistically increase ice velocities at
the calving front, which then feed back into the mass transport and LSM schemes.
Calving rate field c0⟂ in the region of fast flow, derived from modelled ice velocities at the end of the relaxation run.
The red line indicates the 0 level set of the initial LSF used for geometry relaxation and as start position for the calving front during the experiments.
The turquoise line marks the grounding line.
Purple contours indicate zero bedrock elevation.
Black lines are the “along-trough” (a) and “across-trough” (b) profiles used in Fig. .
The finite element mesh is displayed in grey.
We present two experiments for validation of the LSM implemented here in the Appendix.
The first experiment shows that the ice margin is advected with the prescribed level-set velocity w.
The linear representation of the LSF on an unstructured mesh causes a small error
in the exact level-set position, which depends on element size and cancels out over time.
The second test shows that errors in volume conservation introduced by the LSM decrease with finer
mesh resolution and are below 0.2 % after 100 years for a mesh resolution of 1 km.
In the application to Jakobshavn Isbræ, we use a front element size of 0.5 km.
The potential volume loss inherent to this implementation of the LSM is thus far below current uncertainties of model input data.
Inclusion of the LSM requires additional computational effort for the extrapolation of field variables, to
solve the KCFC and for extra iterations of the momentum balance solver, since the stress boundary
conditions at the calving front change frequently.
Its amount depends on the flow approximation and especially on whether the model setup is close to a
stable configuration or not.
Using SSA, the additional computational cost reaches up to 25 %, of
which 11 % is caused by the solution of the KCFC.
Data and model setupJakobshavn Isbræ model setup
We use Jakobshavn Isbræ's drainage basin from to generate a 2-D horizontal finite element
mesh with element size varying from 500 m in the fjord and areas of fast flow to 10 km inland
(Fig. ).
We choose this high mesh resolution to minimize calving front discretization errors, and to resolve the fjord and the deep trough accurately in the
model.
The resulting mesh has about 10 000 vertices and 19 000 elements.
Due to high flow velocities, the Courant–Friedrich–Lewy condition CFL;
dictates a time step on the order of days for the solution of the momentum balance equation, the
mass transport equation, and the KCFC.
We use bed topography from , derived using a mass conservation approach .
The ice surface elevation is taken from GIMP , and ice thickness is the difference between ice surface and ice base elevation.
Bathymetry of the ice-choked fjord of Jakobshavn Isbræ is difficult to measure and currently poorly known.
As a first-order estimate, we apply a parabolic profile of 800 m depth along the ice fjord, fitted via spline interpolation to known topography data.
We rely on for the surface mass balance and, as a first approximation, use their
surface temperatures to initialize our ice viscosity parameter, B, based on the table from .
Surface temperatures range from ∼-8∘C near the coast to ∼-28∘C near the divide.
Basal mass balance is set to 0 and no thermal model is run.
All these forcings are kept constant over time.
We infer a basal friction coefficient, α, in Eq. () using an adjoint-based
inversion of surface velocities from 2009 .
In regions like the fjord, where there is no ice today, we apply an area-averaged
value of α=30 a1/2m-1/2.
At the margins of the computational domain we prescribe zero horizontal ice velocities in order to
prevent mass flux across this boundary.
The friction parameter is kept fixed over time for all model simulations.
Calving front positions for experiments A, B1, B2, and C4 at the start of each
year, plotted over basal topography (grey).
Inconsistencies in model input data cause sharp readjustments of the glacier
state at the beginning of each simulation, which would make it difficult to
distinguish between such effects and those of the applied forcing
. Therefore, we relax the model prior to the experiments
using a fixed, piecewise linear LSF φ0, whose 0 level
set corresponds to the mean annual
calving front position of 2009 (Fig. ). Since the
glacier in this configuration is far from steady state, model relaxation
causes considerable thinning across the glacier's catchment area. In order
not to deviate too much from present day's geometric setting we choose a
100-year relaxation time period. Note that the grounding line retreats during
the relaxation due to dynamic thinning, so that the glacier forms a new
floating ice tongue. This ice tongue extends about 15km to a local
topographic maximum in the southern trough and 3km into the
northern one (Fig. ). The relaxed geometry
constitutes the initial state for our experiments. Due to this deviation in
geometry, providing quantitative insights into Jakobshavn Isbræ is beyond
the scope of this study. However, the main characteristics of the ice stream
(e.g. its large drainage basin and the narrow outlet channel) are preserved,
so that the results presented in this paper qualitatively represent the
behaviour of Jakobshavn Isbræ.
Description of experiments
We set
c0⟂=q|v0|
as a basic calving rate estimate, where v0 is the velocity field at the end of the geometry relaxation run, extended
onto Ωc (Fig. ).
The continuous function q is equal to 1 in areas where the bed lies below -300 m,
and it
linearly drops to 0 in areas of positive bed elevation.
It prevents calving occurring in areas with a glacier bed above sea level, as suggested by observations of tidewater glaciers .
The choice of the calving rate estimate is motivated by the small observed angle between v0 and
n at the calving front (v0≈|v0|n).
Then w0⋅n=v0⋅n-c0⟂≈0, so that we
can expect this calving rate estimate to yield a stationary calving front, if applied to a geometry that is in steady state.
We scale c0⟂ over time with a scaling function, s, which allows for the
representation of seasonal cycles, and a perturbation function, p, to modify the
calving rate for some period of time.
The applied calving rate is then c⟂(x,t)=c0⟂(x)s(t)p(t).
We perform three suites of experiments in order to analyse the impact of the
calving rate on the glacier's dynamics.
The calving front is now allowed to freely evolve in response to c⟂.
All experiments run for 120 years.
In experiment A, we keep the calving rate constant over time, i.e. we set
both s(t)=p(t)=1. Hence, c⟂(x,t)=c0⟂(x). This experiment, although not physically motivated,
is used to evaluate whether a stable calving front position can be reached
using the LSM and for comparison to the experiments described below.
In experiment suites B and C, we represent the seasonal cycle by scaling c0⟂ by
s(t)=max(0,πsin2π(t/L-ϕ0)), with a phase shift
ϕ0=4/12 and a period L=1 a.
We perturb the calving rate during a limited duration, Δt, with a perturbation
strength
p0≥0:p(t)=p0,ift0<t<t0+Δt,and1,else.
We start the perturbation at t0=20a for all experiments. In
experiment suite B, we perform five experiments with Δt=1a, while varying p0 from 0 to 4 by increments of 1. In
experiment suite C, we keep p0=2 fixed and set Δt as 2, 4, and 8
years. We use the notation B <p0> and C <Δt> to identify single
experiments, e.g. B1 for experiment B with perturbation strength p0=1,
which represents the case of unperturbed periodic calving. B1 is used as the
control run to which the other experiments are compared.
Table lists all the experiments performed here.
Figure shows calving front positions for experiments A, B1, B2, and C4.
Under constant calving rate forcing, the calving front remains at a stable position after minor
readjustments in the first decade of the simulation.
In experiment A, the calving front undergoes gradual retreat over time due to the slowdown of the
glacier caused by its ongoing thinning.
When we perturb the calving rate, the calving front migrates, and higher calving rates lead to larger retreats.
The retreat is highest in areas of fast flow and strongly decreases towards the ice stream margins.
This yields the characteristic concave shape of a retreating calving front.
The modelled calving front positions and their shape are in good agreement with observations (Fig. ).
The retreat rate during continued phases of calving decreases to 0, so that the calving front
reaches a new stable position 9 km upstream of its initial position (Fig. d).
In experiments B and C, the calving front returns to a similar position as in the unperturbed
experiment B1 within 10 to 20 years after the perturbation stops.
Section profiles of ice geometry (a) and ice velocity (b) along-trough, as well as ice
geometry (c) and effective strain rates (d) across-trough for experiment
C4 at the end of the calving season in October each year. Positions of the
lines are given in Fig. .
Figure shows ice velocity, geometry, and strain rates for experiment C4 along two lines, which
go along and across the southern trough respectively (Fig. ).
During the first 20 years prior to the perturbation, the ice thickness in the floating part decreases
by about 100 m (Fig. a).
As the calving front retreats during the perturbation, the ice velocity increases (Fig. b), and
the ice thickness adjusts accordingly (Fig. a).
The ice thinning leads to a fast retreat of the grounding line in the regions of locally retrograde bed and
temporarily stabilizes over local along-trough topographic maxima, referred to as “local highs”.
The southern trough has many local highs, which act as pinning points
and are critical for flow dynamics, in agreement with earlier results from .
The acceleration of the ice stream extends tens of kilometres upstream to areas of grounded ice (Fig. b).
Thinning and acceleration are strongest over the ice stream and spread out to the surrounding ice sheet in a dampened fashion.
These thinning and acceleration patterns increase surface and velocity gradients, especially in the
shear margins (Fig. c), where the effective strain rates gradually increase up to a factor of 4 in
experiment C4 (Fig. d), which corresponds to a drop in viscosity of about
60 %
(Eq. ).
This substantially weakens the mechanical coupling between the ice stream and the surrounding ice
sheet.
Calving front and grounding line positions along-trough (left column),
calving front thickness, ice velocity, and effective strain rate relative to their initial value
along-trough (right column) over time for experiments A, B1, B2, and C4.
Perturbation intervals are marked in grey.
Relative values for ice velocity have been shifted up by 0.5 for better
visibility (red y axis).
Figure shows the intra-annual variability of ice properties at the calving front and
grounding line for experiments A, B1, B2, and C4.
All shown variables reflect the characteristics of the applied calving rate forcing.
The constant calving rate applied in experiment A leads to a steady configuration (Fig. a and b).
For an unperturbed periodic calving rate forcing (Fig. c and d), the calving
front position oscillates around a constant annual mean value by ±3 km, while the grounding line
position remains unchanged at kilometre 29.
Ice velocities and thickness at the calving front act in phase with the calving front position,
while the response of strain rates at the grounding line is delayed by about a month.
The ice velocity varies over a year by ±20 %, which corresponds to about ±2 km a-1,
the ice thickness by ±13 %, or ±100 m,
and effective strain rates at the grounding line by ±7 %, or ±0.1 a-1.
The response to a calving rate perturbation scales with p0 and Δt.
When the calving rate doubles (B2, C4), the calving front retreats initially at an average rate of 4.5 km a-1.
The calving front stabilizes 9 km upstream for longer perturbations (Fig. g).
The intra-annual variability of the calving front position doubles to ±6.5 km.
The grounding line position is hardly affected by small calving rate perturbations, but large
perturbations trigger fast retreats of several kilometres, which in turn cause drastic, but
short-lived, flow accelerations (Fig. g and
h).
The annual average ice velocity increases by 10 %, and its intra-annual variability doubles to ±38 % (Fig. h).
The mean calving front thickness decreases by 30 % towards the end of the perturbation of experiment
C4 and experiences large variations up to ±75 %.
This high thickness variability is due to the front retreating into areas of thick ice in summer followed
by stretching and thinning during calving front advance in winter.
For small perturbations, variations of effective strain rates at the grounding line quadruple to ±25 %
(Fig. f).
Once the calving rate perturbation stops, all variables display remarkable reversibility.
When calving is temporarily turned off (experiment B0, not shown here), the response
of the glacier is reversed: the calving front advances, creating a convex ice tongue.
Meanwhile, the ice stream decelerates, thickens, and the grounding line advances.
After the perturbation, the glacier retreats into a state slightly thicker and faster than
the one of experiment B1.
(a) Absolute difference in ice volume for the different simulations with respect to experiment B1.
The non-oscillating ice volume profile of experiment A causes its difference to experiment B1 to oscillate.
(b) The volume differences from experiments B and C divided by Δt(1-p0), the measure of the time-integrated calving rate perturbation.
Figure a shows the evolution of the ice volume with respect to experiment
B1, the control run.
The glacier in experiment B1 continues to lose volume at an average rate of -22.8km3a-1
due to the ongoing geometry relaxation.
In experiment A, Jakobshavn Isbræ loses an additional 0.4km3a-1,
which corresponds to the gradual retreat of the calving front. Enhanced calving
causes additional volume loss proportional to Δt(1-p0), the
measure of the time-integrated calving rate perturbation
(Fig. b). If the calving rate is doubled, the
additional volume loss reaches -35.7 km3a-1 in the
first year but decreases with time, as the calving front thins and retreats
into areas of lower calving rates. Those numbers agree well with recent ice
discharge observations . Over the first decade after the
perturbation, all modelled glaciers recover 40 to 60 % of the volume
deviation to the control run.
Discussion
The applied calving rate determines the behaviour of the calving front and the ice stream.
In our simulations, larger perturbation strengths p0 lead to faster calving front retreats.
In the case of long perturbations (experiments C4 and C8), the calving front reaches a new stable
position.
A stable calving front position requires the calving rate to be larger than the ice velocity
if the calving front advances; similarly, if the calving front retreats, the calving rate needs to be lower than the ice velocity.
Several mechanisms determine how the model responds to the calving rate forcing.
First, a calving rate increase leads to a retreat of the calving front position, ice stream
acceleration and dynamic thinning in the vicinity of the terminus.
Second, this dynamic thinning increases surface slopes and therefore the driving stress. As the
glacier locally speeds up, the ice thinning propagates upstream.
The ice stream thins much faster than the surrounding ice sheet, which steepens the surface across the shear margins.
Lateral inflow of ice into the ice stream hence increases until it balances the calving flux.
This limits the thinning of the ice stream.
Thinning of the ice stream in turn leads to grounding line retreat and reduction in basal effective pressure, both of which reduce basal drag significantly in the vicinity of the grounding line.
We showed that grounding line retreat leads to short-lived but drastic increases in ice flux.
This mechanism is qualitatively the same as the one described in and .
Several pinning points along the retrograde trough of the southern branch, as well as the
lateral stress transfer and mass influx, prevent the modelled ice stream from
being prone to the marine ice sheet instability , a hypothesis which
states that grounding line positions are unstable on retrograde slopes.
This corroborates earlier results by , who presented examples of
stable grounding line positions on retrograde beds.
However, due to large uncertainties in the input data, and since some physical processes are
not represented in our experiments, evaluation of this question for Jakobshavn Isbræ is beyond the scope of this study.
A third mechanism is related to the calving front lengthening during its retreat (e.g.
Fig. ).
The lengthening causes tributaries of the main ice stream to calve directly into the fjord, thereby
increasing the calving flux Qcf (Eq. ) and thinning of the
terminus vicinity.
Finally, the ice stream accelerates faster than the surrounding ice sheet, which increases strain
rates at the shear margins. This reduces the ice viscosity in these areas, which mechanically
decouples the ice stream from the ice sheet, allowing the ice stream to accelerate further. This
positive feedback confines the initial thinning to the ice stream and is controlled by the rate at which
ice enters the ice stream. This mechanism is essential for enabling ice stream acceleration
tens of kilometres upstream of the grounding line, since large fractions of the ice stream's driving stress are
balanced by lateral stress. This corroborates force balance arguments produced earlier by
.
In experiments A and B1, we apply the same annual mean calving rate.
However, due to the lack of seasonal cycle in calving rate the mechanical coupling between the ice
stream and ice sheet is higher in experiment A.
The ice stream velocity is therefore lower, causing gradual calving front retreat and additional ice
volume loss.
This illustrates that volume change estimates from models with and without seasonal cycles of
calving may differ.
Our results suggest that including both a dynamically evolving calving front as well as seasonal
cycles are critical for accurate projections of future contributions of ice sheets to
global sea level rise on decadal to centennial timescales.
Response mechanisms not covered here will likely include feedbacks with damage mechanics and
thermodynamics due to the increased strain rates. During longer perturbations, ice surface lowering
will probably affect the surface mass balance and the drainage basin outline.
The reversibility of the calving front configuration after the calving rate perturbation is a robust
feature across all experiments.
The short duration of the perturbation, the prescribed calving rates, and the geometry of the
glacier are responsible for this behaviour.
The volume change in all experiments never exceeds 0.1 % of the initial glacier volume in the
experiments shown here.
Once the perturbation stops, the surrounding ice sheet continues to replenish the ice stream,
which allows for its quick recovery.
The modelled glacier response to enhanced calving is in good qualitative agreement with
observations, which corroborates that calving is a major control on this glacier.
The similar shape of the modelled and observed calving front suggests that calving rates are indeed
proportional to its flow speed during the glacier's current retreat.
However, the reversibility of the modelled calving front position is in contrast to Jakobshavn
Isbræ's actual behaviour.
Sustained high calving rates are therefore necessary to explain the continued retreat of the
glacier, as our results suggest that the glacier would have readvanced otherwise.
Accurate model input data, representation of all relevant physical processes, and incorporation of a
suitable calving rate parameterization will be necessary for quantitative analysis of this dynamic
ice stream.
Conclusions
In this study, we present the theoretical framework for coupling a LSM
to ice dynamics and implement it into ISSM. The LSM proves to be a robust method for modelling the dynamic evolution of a calving
front. We apply this technique to Jakobshavn Isbræ using prescribed
calving rates, and we find that the glacier is highly sensitive to this forcing, which agrees well with
observations.
Calving rate perturbations strongly affect the ice stream through several linked mechanisms.
First, changes in calving rate cause calving front migration and alter the ice discharge.
Second, the resulting thickness change at the calving front spreads out to the surrounding ice sheet.
Third, thinning-induced grounding line retreat causes further ice stream acceleration and creates a positive
feedback.
Finally, shear margin weakening caused by the ice stream acceleration decreases lateral drag resisting ice flow.
This positive feedback mechanism sustains significant acceleration of the ice stream tens of
kilometres upstream of the grounding line.
The surrounding ice sheet is barely affected by short periods of enhanced calving.
It stabilizes the ice stream and allows for quick reversibility of the calving front position
through lateral ice influx and stress transfer once the calving rates are set back to their initial
values.
Since the calving front position and dynamic lateral effects are critical to simulate and understand the behaviour of
marine terminating glaciers, the inclusion of moving boundaries in 2-D plan-view and 3-D models is key for
realistic sea level rise projections on centennial timescales.
This method is a step towards better physical representation of calving front dynamics in ice sheet models.
Validation of the level-set method
We present two simple test setups to validate the LSM.
The first is designed to show the accurate advection and shape preservation properties of the method.
The second setup aims to give an estimate for the volume change introduced by the LSM for different
mesh resolutions.
Advection
Let Ω be a 50 km square with the initial LSF as
φ0(x)=‖x-x0‖2-R,
where x0=(25,25) km and R=12.5 km, so that the initial 0
level set describes a circle in the
middle of the domain. We prescribe a constant velocity
v=cos(π/4),sin(π/4) km a-1 everywhere.
We advect φ0 over 10 years with time steps of 0.1 a and keep track
of the 0 level set.
Figure shows the 0-level-set position at the
beginning of every year. The LSM preserves the initial circular shape and can
be used to model both advance and retreat of a calving front. We measure the
advection speed of the 0 level set
along the diagonal marked in white in Fig. .
Figure shows the standard deviation of the
numerical error relative to the prescribed advection speed taken over time
for different element sizes. The numerical error is due to the linear
interpolation of the curved shape, which causes variations of the level-set
velocity around the prescribed value. The standard deviation of the error
linearly decreases with mesh resolution and drops below 1 % for elements
sizes below 0.5 km. We therefore choose a mesh resolution of 0.5 km in the
vicinity of the calving front in our simulations.
Volume conservation
Let Ω be a 200×20 km2 rectangle with an initial LSF given by
φ0(x)=1,0⋅x-100km.
The initial lateral extent is thus a 100×20 km2 rectangle.
The geometry corresponds to the ice shelf ramp presented in .
The ice thickness linearly decreases from 400 m at the grounding line (x=0 km) to 200 m at the
calving front (x=100 km).
We apply zero surface accumulation and basal melt, as well as zero grounding line velocity and free
slip boundary conditions at y=0 km and y=20 km.
The ice sheet spreads under its own weight for 100 years.
Figure shows the evolution of the ice volume for different element sizes.
All simulations show volume loss due to the free flux boundary condition at the numerical ice
front, which is not entirely balanced by the volume added through the ice thickness extrapolation.
The volume loss decreases with element size and is below 0.2 % of the initial ice volume after 100
years for an element size of 1 km.
This volume loss is far below current uncertainties of other model input data.
Zero-level-set positions at the start of every year, plotted over φ0, which is
in grey scale.
An example of the mesh with element size 1 km is marked in black.
The white diagonal marks the line along which the velocity of the 0 level set is tracked.
Standard deviation of the relative numerical error in advection velocity of the 0 level set
depending on element size.
Evolution of the relative ice volume change for different element sizes.
The red line shows volume conservation.
Acknowledgements
A. Humbert acknowledges support of the DLR proposal HYD2059 which provides
TerraSAR-X data for the project HGF-Alliance Remote Sensing and Earth System
Dynamics. H. Seroussi, M. Morlighem, and E. Y. Larour are supported by grants from the National Aeronautics and Space
Administration, Cryospheric Sciences, and Modeling, Analysis and Prediction
programs. The authors thank the referees G. Jouvet and J. Bassis as well as
the editor O. Gagliardini for their helpful and insightful comments. Edited by: O. Gagliardini
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