TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-12-3735-2018Comparison of four calving laws to model Greenland outlet glaciersCalving laws in GreenlandChoiYoungminyoungmc3@uci.eduhttps://orcid.org/0000-0002-2656-5371MorlighemMathieuhttps://orcid.org/0000-0001-5219-1310WoodMichaelBondzioJohannes H.Department of Earth System Science, University of California, Irvine, 3218 Croul Hall, Irvine, CA 92697-3100, USAYoungmin Choi (youngmc3@uci.edu)29November201812123735374625June201823July20186November20188November2018This 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/12/3735/2018/tc-12-3735-2018.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/12/3735/2018/tc-12-3735-2018.pdf
Calving is an important mechanism that controls the dynamics of marine terminating glaciers of
Greenland. Iceberg calving at the terminus affects the entire stress regime of outlet glaciers,
which may lead to further retreat and ice flow acceleration. It is therefore critical to
accurately parameterize calving in ice sheet models in order to improve the projections of ice
sheet change over the coming decades and reduce the uncertainty in their contribution to
sea-level rise. Several calving laws have been proposed, but most of them have been applied only to a
specific region and have not been tested on other glaciers, while some others have only been
implemented in 1-D flowline or vertical flowband models. Here, we test and compare
several calving laws recently proposed in the literature using the Ice Sheet System Model (ISSM).
We test these calving laws on nine tidewater glaciers of Greenland. We compare the modeled ice
front evolution to the observed retreat from Landsat data collected over the past 10 years, and
assess which calving law has better predictive abilities for each glacier. Overall, the von Mises
tensile stress calving law is more satisfactory than other laws for simulating observed ice front
retreat, but new parameterizations that better capture the different modes of calving should be
developed. Although the final positions of ice fronts are different for forecast simulations with
different calving laws, our results confirm that ice front retreat highly depends on bed
topography, irrespective of the calving law employed. This study also confirms that calving
dynamics needs to be 3-D or in plan view in ice sheet models to account for complex bed topography
and narrow fjords along the coast of Greenland.
Introduction
Mass loss from marine terminating glaciers along coastal Greenland is a
significant contributor to global sea-level rise. Calving is one of the
important processes that control the dynamics, and therefore the discharge,
of these glaciers e.g.,. Ice
front retreat by enhanced calving reduces basal and lateral resistive
stresses, resulting in upstream thinning and acceleration, which may lead to
a strong positive feedback on glacier dynamics
e.g.. Recent observations have shown that
many outlet glaciers along the coast of Greenland are currently experiencing
significant ice front retreat e.g.,. It is
therefore important to accurately parameterize calving in ice sheet models in
order to capture these changes and their effect on upstream flow and,
consequently, improve the projections for future global sea level.
The first attempts to model calving dynamics focused on empirical
relationships between frontal ablation rate and external variables such as
water depth or terminus height . Later
studies included ice properties
and dynamics to specify calving front position. In these studies, the ice
front position is based on a height-above-buoyancy criterion (HAB), with
which numerical models were able to reproduce more complex observed behaviors
of Arctic glaciers. This criterion, however, was not suitable for glaciers
with floating ice shelves and failed to reproduce seasonal cycles in ice
front migration . introduced a
crevasse-depth criterion (CD), which defines the calving front position as being where the
surface crevasses reach the waterline. modified this criterion by including basal crevasses and their
propagation for determining calving front position in a flowline model. This
model successfully reproduced observed changes of several glaciers
and simulated future changes of main outlet
glaciers of Greenland . proposed to
define the calving rate as proportional to the product of along- and
across-flow strain rates (eigencalving, EC) for Antarctic glaciers. This
calving law showed encouraging results for some large Antarctic ice shelves,
such as Larsen, Ronne, and Ross, but this parameterization has not been
applied to Greenland glaciers, which terminate in long and narrow fjords.
proposed a calving parameterization based on von Mises
tensile stress (VM) to model Store glacier, Greenland. This law only relies
on tensile stresses and frontal velocity, and does not include all of the
processes that may yield to calving (such as damage, hydro-fracture, or
bending), but it has shown encouraging results on some Greenland glaciers
. Recently, several studies have developed new
approaches based on a continuum damage model
or linear elastic fracture mechanics (LEFM) , and
combined damage and fracture mechanics to model calving
dynamics in Greenland. These studies investigated fracture formation and
propagation involved in calving, but have only focused so far on individual
calving events in small-scale cases, and it is not clear how to extend these
studies to 3-D large-scale models of Greenland.
Ice surface velocity (black contours) for study glaciers
(a) Upernavik Isstrøm, (b) Hayes, (c) Helheim,
(d) Sverdrup, (e) Kjer. The thick black line is the ice
edge.
While all of these parameterizations have been tested on idealized or single,
real-world geometries, most of them have not yet been tested on a wide range
of glaciers, and some of these laws have only been implemented in
1-D flowline or vertical flowband models . The
main objective of this study is to test and compare some of these calving
laws on nine different Greenland outlet glaciers using a 2-D plan-view ice
sheet model. Modeling ice front dynamics in a 2-D horizontal or 3-D model has
been shown to be crucial, as the complex 3-D shape of the bed
topography exerts an important control on the pattern of ice front retreat,
which cannot be parameterized in flowline or flowband models
e.g.. We do not include continuum damage
models and the LEFM approach in this study because these laws require individual calving events
to be modeled, whereas we focus here on laws that provide
an “average” calving rate, or a calving front position, without the need to
track individual calving events. While these approaches remain extremely
useful to derive new parameterizations, their implementation in large-scale
models is not yet possible due to the level of mesh refinement required to
track individual fractures.
We implement and test four different calving laws, namely the
height-above-buoyancy criterion HAB;, the crevasse-depth
calving law CD;, the eigencalving law
EC; and von Mises tensile stress calving law
VM;, and model calving front migration of nine
tidewater glaciers of Greenland for which we have a good description of the
bed topography . The glaciers of this study are three
branches of Upernavik Isstrøm (UI), Helheim glacier, three sectors of
Hayes glacier, Kjer, and Sverdrup glaciers (Fig. ). Each of these
four calving laws includes a calibration parameter that is manually tuned for
each glacier. These parameters are assumed to be constant for each glacier.
To calibrate this parameter, we first model the past 10 years (2007–2017)
using each calving law and compare the modeled retreat distance to the
observed retreat distance. Once a best set of parameters is found, we run the
model forward with the current ocean and atmospheric forcings held constant
to investigate the impact of the calving laws on forecast simulations. We
discuss the differences between results obtained with different calving laws
for the hindcast and forecast simulations and the implications thereof for
the application of the calving laws to real glacier cases.
Data and method
We use the Ice Sheet System Model ISSM; to implement
four calving laws and to model nine glaciers. Our model relies on a
Shelfy-Stream Approximation , which is
suitable for fast outlet glaciers of Greenland . The mesh
resolution varies from 100 m near the ice front to 1000 m inland, and the
simulations have different time steps that vary between 0.72 and 7.2 days
depending on the glacier in order to satisfy the Courant–Friedrichs–Lewy condition . We use the surface
elevation and bed topography data from BedMachine Greenland version 3
. The nominal date of this dataset is 2008, which is
close to our starting time of 2007. The surface mass balance (SMB) is from
the regional atmospheric model RACMO2.3 and is kept constant
during our simulations. We invert for the basal friction to initialize the
model, using ice surface velocity derived from satellite observations
acquired in a similar period (2008–2009) .
ISSM relies on the level set method to track the calving
front position. We define a level set function, φ, as being positive
where there is no ice (inactive) and negative where there is ice (active
region), and the calving front is implicitly defined as the zero contour of
φ. Here, we implement two types of calving laws: EC and VM provide a
calving rate, c, whereas HAB and CD provide a criterion that defines where
the ice front is located. These two types of law are implemented differently
within the level set framework of ISSM.
When a calving rate is provided, the level set is advected following the
velocity of the ice front (vfront), defined as a function of the ice velocity
vector, v, calving rate, c, and the melting rate at the calving
front, M˙:
vfront=v-c+M˙n,
where n is a unit normal vector that points outward from the ice.
EC defines c as proportional to strain rate along (ϵ∥)
and transversal (ϵ⟂) to horizontal flow
:
c=K⋅ϵ∥⋅ϵ˙⟂,
where K is a proportionality constant that captures the material properties
relevant for calving. K is the calibration parameter of this calving law.
In VM, c is assumed to be proportional to the tensile von Mises stress,
σ̃, which only accounts for the tensile component of the
stress in the horizontal plane:
c=‖v‖σ̃σmax,
with
σ̃=3Bε˙̃e1/n,
where σmax is a stress threshold that is calibrated, B
is the ice viscosity parameter, n=3 is Glen's exponent, and
ε˙̃e is the effective tensile strain rate
defined as
ε˙̃e2=12max0,ε˙12+max0,ε˙22,
where ε˙1 and ε˙2 are the two eigenvalues
of the 2-D horizontal strain rate tensor . To prevent
unrealistic calving rates caused by an abrupt increase in velocity upstream
from the ice front, we limit the maximum calving rate to 3 km yr-1.
For HAB and CD, we proceed in two steps at each time iteration as they do not
provide explicit calving rates, c. First, the ice front is advected
following Eq. (), assuming that c=0 and using the
appropriate melt rate, M˙, which simulates an advance or a retreat of
the calving front without any calving event. The calving front position is
then determined by examining where the condition of each law is met. The
level set, φ, is explicitly set to +1 (no ice) or -1 (ice) on each
vertex of our finite element mesh depending on that condition.
For HAB, the ice front thickness in excess of floatation cannot be less than
the fixed height-above-buoyancy threshold, HO:
HO=1+qρwρiDw,
where ρw and ρi are the densities of sea water and ice,
respectively, and Dw is the water depth at the ice front, here represented
by the bed depth below sea level. The fraction q∈0,1 of
the floatation thickness at the terminus is our calibration parameter.
For CD, the calving front is defined as where the surface crevasses reach the
waterline or surface and basal crevasses join through the full glacier
thickness. The depth of surface (ds) or basal (db) crevasses is
estimated from the force balance between tensile stress in the along-flow
direction or any direction, water pressure in the crevasse, and the
lithostatic pressure:
ds=σρig+ρwρidwdb=ρiρp-ρiσρig-Hab,
where g is the gravitational acceleration, Hab is the height
above floatation, and dw is the water height in the crevasse, which allows
the crevasse to penetrate deeper . The water depth in
the crevasse (dw) is the calibration parameter of this calving law. In
this study, we use two different estimations for the crevasse opening stress,
σ. First, we use the stress only in the ice-flow direction to estimate
σ in which changes in direction are taken into account
. The other estimation for σ is the largest principal
component of deviatoric stress tensor to account for tensile stress in any
direction . Here we use the term “CD1” (flow
direction) and “CD2” (all directions), respectively, to refer to these two
estimations for σ.
Chosen calibration parameters. The values in brackets are the range
of calibration parameters that produce a qualitatively similar ice front
retreat pattern to the chosen calibration parameter.
We use the frontal melt parameterization from to estimate
M˙ in Eq. (). The frontal melt rate, M˙,
depends on subglacial discharge, qsg, and ocean thermal forcing,
TF, defined as the difference in temperature between the potential
temperature of the ocean and the freezing point of seawater, as
M˙=Ahqsgα+bTFβ,
where h is the water depth, A=3×10-4 m-α dayα-1∘C-β,
α=0.39, b=0.15 m day-1∘C-β, and β=1.18. We use ocean temperature from the
Estimating the Circulation and Climate of the Ocean, Phase 2 (ECCO2) project
. To estimate subglacial discharge, we integrate the
RACMO2.3 runoff field over the drainage basin, assuming that surface runoff is
the dominant source of subglacial fresh water in summer .
(a) The observed ice front positions between 2007 and 2017 and
(b–f) modeled ice front positions obtained with different calving
laws between 2007 and 2100 overlaid on the bed topography of Upernavik
Isstrøm. The white lines are the flowlines used to calculate retreat or
advance distance of the ice front.
We determine each calibration parameter (Table ) by
simulating the ice front change between 2007 and 2017 and comparing the modeled
pattern of retreat to observed retreat. We manually adjust these parameters
for each calving law and for each basin to qualitatively best capture the
observed variations in ice front position. In order to compare modeled ice
front dynamics with observations, we estimate the retreat distance along five
flowlines across the calving front of each glacier so that we are able to
account for potential asymmetric ice front retreats. We only calculate the
retreat distance between 2007 and 2017 and choose the parameters that can
produce similar retreat distance for each flowline. We do not take into
account the timing of the retreat or advance between 2007 and 2017 when
choosing calibration parameters (Figs. S5–S7 in the Supplement). Based on
our calibrated models, we run the models forward until 2100 to investigate
and compare the influence of different calving parameterizations on future
ice front changes. For better comparison, we keep other factors (e.g., SMB,
basal friction) constant in our runs. We also keep our ocean thermal forcing
(Eq. ) the same as the last year of the hindcast
simulation (2016–2017) until the end of our forecast simulations. The
simulations are therefore divided into two time intervals: the hindcast
period (2007–2017) that we use to calibrate the tuning parameters of the
different calving laws and the forecast time period (2018–2100).
Results
The observed and modeled ice front evolutions in our simulations are shown in
Figs. –. The modeled retreat distances along five
flowlines are compared to observed retreat distances in Fig. . We
first notice that, in all cases, the calving laws that model a calving rate
(EC and VM) have a smoother calving front than other laws. This results from
the numerical implementation of these laws in which it is only required that
the advection equation of the calving front is solved, and a
local post-processing step is not relied upon that may yield a more irregular shape of the
calving front.
Same as Fig. 2 but for Hayes glacier.
If we look at individual glaciers, Fig. a shows the observed
pattern of retreat between 2007 and 2017 for the three branches of UI. The
northern and southern branches have been rather stable over the past 10 years, but the central branch has retreated by 2.6 to 4 km.
Figure b shows the pattern of modeled ice front position between 2007 and 2017
(warm colors) and 2017 and 2100 (cold colors) using HAB. We observe that the ice
front in the central branch jumps upstream by about 2–3.5 km at the beginning
of the simulation and slows down as the bed elevation increases. The ice
front starts retreating again after 2017 and stops when it reaches higher
ground about 5 km upstream. The modeled northern and southern branches are
stable until 2017 and the northern branch retreats significantly to another
ridge upstream between 2017 and 2100. The modeled ice front using EC does not
match the observed pattern of ice front retreat well (Fig. c). This
approach causes the calving front to be either remarkably stable or creates
an ice front with a strongly irregular shape. Figure d and
e present the modeled ice front evolution using the CD1 and CD2,
respectively. Both models have similar ice front retreat patterns between
2007 and 2017, and they both overestimate the retreat of the central branch
compared to observations (Fig. ). In the forecast simulations, the
central branch retreats more when only the flow-direction stress is
considered (Fig. d). However, in both cases, the ice front stops
retreating at the same location on a pronounced ridge. The model that relies
on VM shows a gradual terminus retreat and stabilizes at the end of 2017
(Fig. f). After 2017, the retreat behavior is similar to the one
with the height-above-buoyancy law. We observe that HAB and VM reproduce the
observed changes reasonably well, although they do not capture the exact
timing of the 2007–2017 retreat (Fig. b and f).
The second region of interest is Hayes glacier. Currently, the three branches
of this system rest on a topographic ridge, ∼300 m below sea level,
which is likely responsible for the observed stability in the position of the
ice front over the past 10 years (Fig. a). The ice front of the
northern glacier, however, retreated by up to 3 km from 2007 to
2014 and readvanced in 2016 and 2017. In this region, HAB produces a stable
ice front for the northern (Hayes) and the southern sector (Hayes N), but the
central sector (Hayes NN) retreats more than the observations by 0.5–0.7 km
(Fig. b). After 2017, Hayes NN and Hayes N only retreat by a few
kilometers and stabilize there until the end of the simulation. The model using EC shows
very little change between 2007 and 2100 (Fig. c). As in the
previous region, both the CD1 and CD2 show very similar results (Fig. d and e). In the hindcast simulation (2007–2017), both
models overestimate the retreat at the western part of the northern branch
(Hayes). After 2017, Hayes and Hayes NN retreat quickly by 2.2–6 km into an
overdeepening in the bed topography. The final positions of the ice front
derived from two crevasse-depth laws are 5 km upstream of their initial
position on higher ridges further upstream. Figure f shows the
modeled ice front evolution using VM. This model reproduces the stable ice
front positions for two sectors (Hayes and Hayes N) but tends to overestimate
the retreat for Hayes NN. Although, for the forecast simulation, VM results
in more retreat than obtained with other laws for Hayes, the ice front ends
up resting on the same ridges as the ones based on the crevasse-depth laws.
Same as Fig. 2 but for Helheim glacier.
Same as Fig. 2 but for Sverdrup glacier.
Figure a shows the observed ice front pattern for Helheim glacier.
Since 2007, this glacier has shown a stable ice front evolution, retreating
or advancing only by a few kilometers over the past 10 years . All
calving parameterizations, except for EC, result in a stable or a little
advanced ice front pattern (Fig. b–f), and only the VM model
reproduces the observed retreat distance from 2007 to 2017
for this region reasonably well (Fig. g), although it never readvances. The other
calving laws do not capture the observed retreat distance or the shape of ice
front properly with our ocean parameterization. In the forecast simulations,
all model results show an advance or stable pattern of ice front evolutions
at the end of 2100. The model with EC results in a significantly different
shape of ice front compared to other models (Fig. c).
Same as Fig. 2 but for Kjer glacier.
From 2007 to 2014, the mean terminus position of Sverdrup glacier
(Fig. a) has been around a small ridge ∼300 m high. In 2014,
the glacier was dislodged from its sill and the glacier started to retreat.
The models with HAB and EC show that the ice front jumps to a similar
location to the 2017 observed ice front (Fig. b and c). The glacier
does not retreat much after 2017 in these two models. The two CDs tend to
produce more retreat than other parameterizations after the ice front is
dislodged from the ridge (Fig. d and e). The ice front
retreat, after 2017, starts slowing down near another ridge 9 km upstream and
the glacier stabilizes there until 2100. Only VM captures the timing of the
retreat reasonably right (Fig. f). After 2017, the forecast
simulation shows that ice front retreats up to 4.5 km before slowing down at
the second ridge upstream. The ice front then retreats past this ridge
quickly and keeps retreating until it reaches a bed above sea level further
upstream, where the retreat stops.
Modeled retreat distances (with respect to the calving front initial
position in 2007) for different calving laws compared to observed retreat
distance for nine study glaciers. The retreat distances between 2007 and 2017
from each calving law are shown as bar solid colors. The hatched bars are the
retreated distances in 2100 for each calving law. Shaded areas represent the
range of 500 m from the 2017 observed retreat, and the modeled retreats that
fall into this range are shown with the red edge.
The ice front of Kjer glacier retreated continuously between
2007 and 2017 (Fig. a). All calving parameterizations, except for EC,
simulate the observed retreat well (Figs. b–f and i).
The forecast simulations, however, show different retreat patterns. HAB shows
relatively less retreat than other models (Fig. b). The calving
front slows down and stabilizes at the location where the direction of the trough
changes. The calving front from two crevasse-depth parameterizations retreats
past this pinning point and stops retreating at the next pinning point where
the small ridge is located (Fig. d and e). In the model
with VM, the retreat rate slows down near this ridge as well. The ice front,
however, keeps retreating beyond this ridge and stabilizes on another ridge
further upstream (Fig. f).
Discussion
Our results show that different calving laws produce different patterns of
ice front retreat in both timing and magnitude, despite equal climatic
forcing. In the hindcast simulations, we calculate the modeled retreat
distance from 2007 to 2017 for a total of 45 flowlines from our study
glaciers to investigate which calving law, with the best tuning parameter,
better captures the observed ice front changes (Fig. ). We find
that overall, VM captures the observed retreat better than other calving
laws. For 67 % (30 out of 45) of these flowlines, VM reproduces the retreat
distance within 500 m from the observations, which we assume to be a
reasonable range based on the seasonal variability of ice fronts, error in
observations, and model resolution . With HAB,
the modeled retreat distance is within 500 m of the observed retreat
distance for 53 % of the flowlines, while CD1 and CD2 capture the retreat for
51 % and 40 % of the the flowlines. EC reproduces only 31 % of the retreat
that falls into the 500 m range.
EC was designed to model calving of large-scale floating shelves by including
strain rates along and across ice flow . Our results
show that it does not work well in the case of Greenland fjords because
these glaciers flow along narrow and almost parallel valleys. The transversal
strain rate, ϵ⟂, is small and noisy in these valleys, leading
to a significantly different pattern of ice front changes with either a
remarkably stable (e.g., Fig. c) or some complex shape of the
modeled ice front (e.g., Figs. c, c). The forecast
simulations with this calving law also show different retreat patterns
compared to other calving laws. While this calving law may be appropriate in
the case of unconfined ice shelves, we do not recommend using this calving
law for Greenland glaciers.
The two crevasse-depth calving laws are very similar in terms of the ice
front retreat patterns they produce. For the regions of fast flow, the
maximum principal strain rate is almost the same as the along-flow strain
rate, which leads to a similar amount of stress for opening crevasses. We
note that for almost all of the glaciers that match the observed retreat, the
model is very sensitive to the water depth in crevasses, the calibration
parameter, for both laws (Table ). Even a 1 m
increase in water depth significantly changes the calving rate, and thus the
entire glacier dynamics. This behavior has been noticed in other modeling
studies . Only one glacier (Hayes N) allows the water depth to be changed by up to ∼18 m and still reproduces the observed ice
front pattern. One reason why CDs do not capture the rate of retreat well in
the hindcast simulations might also be this high sensitivity to water depth
in crevasses. Models relying on this law should be taken with caution because
it is hard to constrain the water depth in crevasses. The water depth in
crevasses is certainly different from one year to another, and can be
significantly affected by changes in surface melting and the hydrology of glacier
surfaces for the forecast simulations e.g.,.
applied a CD calving law with a 3-D full Stokes model and were
able to reproduce the seasonal calving variability of Store Glacier without
any tuning of water depth in crevasses. For our study glaciers, however,
tuning the water depth was necessary to reproduce the observed ice front
changes (Figs. S1–S4). This either shows that this calving law works well
for Store but not for other glaciers without the tuning process, or that full
3-D stresses are required to model calving. Further studies need to
investigate the stresses from different models and their relationships with
water depth in crevasses.
The model results with HAB indicate that this calving law reproduces the
final position of observed calving front well for some glaciers, but does not
capture continuous retreat patterns and the timing of retreat between 2007
and 2017. The ice front generally tends to jump to its final position. This
may be due to the fact that we keep the height above floatation fraction
(q) constant during our simulations. This constant fraction value also
explains a relatively limited retreat compared to other calving laws for the
forecast simulations. The sensitivity of the model to the parameter q is
different for every glacier (Table ). The glaciers
with an ice front that is in shallow water (e.g., Hayes N, Kjer) are less
sensitive to the choice of q than the ones with a deeper ice front. A wide
range of grounding conditions in the study glaciers also explains the wide
range of the parameter q between different glaciers. Because determining
q is empirical and buoyancy conditions may change through time, this
calving law becomes less reliable than other physics-based calving laws for
the forecast simulations. Another disadvantage of this law is that it does
not allow for the formation of a floating extension, and cannot be applied to
ice shelves.
All calving laws implemented in this study rely on parameter tuning for each
glacier in order to match observations. However, this tuning process makes it
difficult to apply any of these calving laws to glaciers for which we have no
observations of ice front change, and it is not clear whether these
parameters should be held constant in future simulations or whether they may
change. In particular, when the parameters span a wide range between
different glaciers, as in HAB or EC, it is hard to constrain these parameters
for forecast simulations. Model simulations with these calving laws should be
taken with caution.
Our results for forecast simulations suggest that ice front retreat strongly
depends on the bed topography. Although different calving laws do not always
have the same final positions, the extent of glacier retreat shows a similar
pattern: topographic ridges slow down or/and stop the retreat, and retrograde
slopes accelerate the retreat, which has been shown in several studies
e.g.,. Whether the glaciers continue to
retreat beyond these ridges depends on the calving law used and may also
depend on the choice of tuning parameters. For the forecast simulations, it
is not clear whether the tuning coefficients of the calving laws should be
kept constant, as we did here. Some parameters potentially vary depending on
future changes in external climate forcings or ice properties. These changes
may affect the final locations where glaciers eventually stabilize. However,
the bed topography still plays a crucial role in determining stable positions
of ice fronts and the general pattern of retreat before the glaciers
stabilize.
The results for Helheim glacier are very similar for all calving laws, and
none of them captures the pattern of ice front migration perfectly. In the
forecast simulations, the modeled ice front slightly advances until 2100 for
all calving laws. This ice front advance is mostly caused by the ocean
thermal forcing data used in the forecast simulations. The thermal forcing
has been slightly decreasing after 2012 and a relatively cold water is
applied to our forecast simulations, which leads to a similar advance of ice
front for all calving law simulations. However, according to the bed
topography of this region, this glacier might potentially retreat upstream if
the ocean temperature increases, which may trigger more frequent calving
events.
Ocean forcing is one of the limitations of this study: the frontal melt rate
is simply parameterized. The ocean parameterization does not take into
account ocean circulation within the fjords, which could cause localized melt
higher or lower than the parameterization. We need to account for these ocean
processes that may affect melt rate and could potentially vary the retreat
rate of ice front. We also assume that the calving front remains vertical and the
melt is applied uniformly along the calving face . Future
studies should include more detailed ocean physics and coupling to better
calibrate our calving laws and improve results.
Based on our results, we recommend using the von Mises stress calving law
(VM) for modeling centennial changes in Greenland tidewater glaciers within a
2-D plan-view or 3-D model. This calving law captures the observed pattern
of retreat and rate of retreat better than other calving laws, and does allow
for the formation of a floating extension. VM does not, however, necessarily
capture specific modes of calving as it is only based on horizontal tensile
stresses, which may be a reason why it does not always capture the pattern of
ice front migration perfectly. The strong correlation between calving rate
and ice velocity produces reasonable calving rates (Fig. S8) but whether
these relationships hold for forecast simulations needs further
investigation. Another disadvantage of this law is that it strongly depends
on the stress threshold, σmax, that needs to be calibrated. Some
modeled glaciers (e.g., Helheim, Sverdrup, Kjer) are very sensitive to
σmax, in which case a ∼50 kPa change significantly affects
the calving dynamics of these glaciers (Table ). As
a result, the modeled ice front dynamics is dependent on this one single
value that we keep constant through time and uniform in space, which adds
uncertainty to model projections. It is therefore critical to further
validate the stress threshold and improve this law by accounting for other
modes of calving or to develop new parameterizations. Current research based
on discrete element models e.g, or on damage mechanics
may help the community derive these new parameterizations.
Conclusions
We test and compare four calving laws by modeling nine tidewater glaciers of
Greenland with a 2-D plan-view ice sheet model. We implement the
height-above-buoyancy criterion, eigencalving law, crevasse-depth calving
laws, and von Mises stress calving parameterization in order to investigate
how these different calving laws simulate observed front positions and affect
forecast simulations. Our simulations show that the von Mises stress calving
law reproduced observations better than other calving laws although it may
not capture all the physics involved in calving events. Other calving laws do
not capture the pattern or pace of observed retreat as well as the VM. In
forecast simulations, the pattern of ice front retreat is somewhat similar
for most calving laws because of the strong control of the bed topography on
ice front dynamics. Based on our results, we recommend using the tensile von
Mises stress calving law, but new parameterizations should be derived in
order to better capture and understand the complex processes involved in
calving dynamics. It is not clear, however, whether these recommendations
would apply to Antarctic ice shelves. These ice shelves calve large tabular
icebergs that may be governed by different physics.
The data used in this study are freely available at the
National Snow and Ice Data Center, or upon request to the authors. The ISSM is
open source and is available at http://issm.jpl.nasa.gov (Version 4.13,
released on 10 May 2018).
The supplement related to this article is available online at: https://doi.org/10.5194/tc-12-3735-2018-supplement.
YC and MM designed the experiments. YC conducted the numerical
modeling simulations with help from MM and JHB. MW provided the data related to
ocean melt rates. YC wrote the first version of the manuscript with inputs
from MM, JHB, and MW.
The authors declare that they have no conflict of interest.
Acknowledgements
We would like to thank the editor, Olivier Gagliardini, for his constructive comments after the
initial submission of this manuscript. This work was performed at the University of California
Irvine under a contract with the National Science Foundation's ARCSS program (no. 1504230), the
National Aeronautics and Space Administration, Cryospheric Sciences Program (no. NNX15AD55G), and
the NASA Earth and Space Science Fellowship Program (no. 80NSSC17K0409).
Edited by: Olivier Gagliardini
Reviewed by: Doug Benn and Jeremy Bassis
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