Introduction
Many ocean-terminating glaciers in the Arctic are currently undergoing
rapid retreat, thinning and strong acceleration in flow. These dynamic
mass losses contribute to about half of the Greenland ice sheet's
contribution to sea level rise and are
expected to further increase in the future . The
mechanism of iceberg calving is thereby at the heart of these rapid
dynamic changes of ocean-terminating glaciers. However, the
understanding of the involved processes and the capability of
predictive flow models to represent calving are limited
.
Tidewater glacier evolution is the result of an interplay between mass
flux from upstream and the rate and size of calving events
. Both processes are strongly influenced by the
geometry of the glacier surface, the glacier bed and the bathymetry of
the proglacial fjord as well as external
forcings such as submarine melt due to heat advection by ocean
currents
or changes in ice mélange .
Iceberg calving is a dynamical process of material failure which
occurs when the local stress field in the vicinity of the calving
front exceeds the fracture strength of ice, driving the formation and
propagation of cracks and eventually leading to the detachment of
a block of ice from the glacier front. The local geometry and water
level at the terminus determine the stress field and thereby the
fracture processes and the geometry evolution. Further, buoyancy
forces of submerged ice and erosion from subaqueous melt are expected
to enhance near-terminus stress intensity and hence calving rates,
while a reclining terminus should reduce extensional stresses.
Model parameters, notations, units and values for constant parameters.
Parameter
Notation
Value
Units
Fluidity parameter
A
75
MPa-3a-1
Effective damage rate
B̃
65
MPa-ra-1
Bed slipperiness
C
mMPa-1a-1
Initial damage
D0
0.2
Critical damage
Dc
0.7
Gravitational acceleration
g
9.81
ms-2
Sediment layer thickness
hs
10
m
Ice thickness
H
m
Water level height
Hw
m
Glen exponent
n
3
Damage law exponent
r
0.43
Velocity vector
u
ma-1
Basal velocity
ub
ma-1
Calving rate
u¯c
md-1
Reference velocity
uref
∼[1.7×10-6m-3a-1]H4
ma-1
Hayhurst parameter 1
α
0.21
Hayhurst parameter 2
β
0.63
Hayhurst stress
χH
MPa
Strain rate tensor
ε˙
a-1
Effective strain rate
εe˙
a-1
Effective viscosity
η
MPaa
Sediment layer viscosity
ηs
MPaa
Finite strain rate parameter
κε
5.98×10-6
a-1
Ice density
ρi
917
kgm-3
Seawater density
ρw
1028
kgm-3
Freshwater density
ρw
1000
kgm-3
Cauchy stress tensor
σ
MPa
Von Mises stress
σe
MPa
Maximum principal stress
σ1
MPa
Mean stress
σm
MPa
Deviatoric stress tensor
σ′
MPa
Reference stress
σref
ρigH∼[0.009MPam-1]H
MPa
Damage threshold stress
σth
0.17
MPa
Basal shear stress
τb
MPa
Relative water level
ω
Relative water level at flotation
ωf
0.89
Several empirical and semi-empirical parametrizations of the calving
rate for different terminus geometries have been proposed. A simple
empirical relationship of linearly increasing calving rate with water
depth, based on observations of tidewater glaciers in Alaska, has been
established, used and extended for different regions
. This approach only depends on the local
water depth at the terminus only and is not process-based, and it is
therefore independent of glacier geometry and dynamics
. In contrast, the flotation calving criterion,
proposed by and modified by
, determines the position of the terminus by
calving away all ice that is close to flotation. In this approach the
calving rate is an emergent quantity resulting from ice flow dynamics.
introduced a physics-based approach
by setting the terminus position at the location where crevasses
penetrate below the water level. The crevasse depth is computed using
the theory, which relies on the equilibrium between
longitudinal stretching and overburden stress of the ice. This
dynamic approach for calving allowed for successful reproduction of
calving front variations of ocean-terminating glaciers in Greenland
and Antarctica . Although the crevasse depth model can be
calibrated to observations , it lacks validation
with field observations and is based on a snapshot of the stress
balance, neglecting the pre-existence of cracks and their effect on
the stress state of the glacier . A recent, more
sophisticated approach by predicts calving
positions based on the maximum principal stress distribution and
accounts for the effect of water pressure in the submerged parts of
the glacier front by combination of a continuum flow model with
a discrete element model to simulate calving events.
For near-vertical calving fronts, the main driver for calving is the
horizontal deviatoric stress σxx′ in the vicinity of the
laterally confined calving front. Its magnitude can be estimated from
the difference of vertically integrated hydrostatic pressure within
the ice and of ocean water at the calving front
p. 353. The resulting extensional stress
within the ice depends on the ice thickness H and the water depth
Hw at the calving front:
σxx′≃ρigH41-ρwρiω2,
where ρi,ρw and ω=Hw/H are the ice density, water density and relative
water depth (Table ). This equation illustrates
the square dependence of the horizontal extensional stress on relative
water level at the terminus. However, it should be noted that this
vertically integrated stress is not representative for the stress
state near the surface of the terminus, and such a “depth-averaged”
longitudinal stress may be inaccurate as bending stresses are
neglected.
Using the above longitudinal stress at the front, the maximum height
for which a grounded glacier with a dry calving front can sustain
a stable vertical front is approximately 110 m when crevasse
depth is computed according to the theory and
221 m when the ice is considered as undamaged and without
crevasses . However, the presence of water
along the calving front influences this maximum stable height, as an
increase of water depth for a constant ice thickness reduces the
stresses and hence tends to increase the stability of the glacier
front. Thus, a thicker glacier must terminate in deeper water in
order for its calving front not to exceed a certain stress limit and
to remain stable .
Calving termini can also be over-steepened by melt undercutting, which
leads to higher stress intensities and may
facilitate calving . Ice flow model results
suggest that an increase of water depth leads
to a higher rate of over-steepening development at the calving front
and thus an increase of calving activity. However, model results seem
to indicate that melt undercutting does not significantly affect
calving rates , while other studies
suggest that calving rates are strongly related to melt undercutting
for some arctic glaciers
. Conversely,
a calving front inclined towards the inland is expected to be more
stable than a vertical cliff.
The state of stress near the calving front is determined by ice
geometry and water depth and controls the intensity and location of
material degradation processes. Material creep and fracture processes
in turn change the geometry of the glacier front. Observations and
theoretical considerations indicate a tendency of increasing relative
water level with increasing thickness . This
implies that thick glaciers approach flotation at their front but for
shallow water depth the bounds on stress, and hence cliff geometry,
are less well constrained.
The relationship between water depth, stress state, front geometry and
related calving type is well illustrated at the example of Eqip
Sermia, a medium-sized ocean-terminating outlet glacier on the West
Greenland coast. Figure
shows that this glacier is characterized by two distinct calving front
lobes with contrasting geometries: the grounded northern lobe exhibits
a 200 m high inclined calving face with slope angles exceeding
45∘ while the southern lobe features a vertical ice cliff
of ∼50 m freeboard with a water depth of ∼100 m . These substantially different
geometries lead to distinct velocity and stress regimes in the
proximity of the calving front which also determine the type of
calving. The high, grounded, inclined northern cliff collapses at
timescales of weeks, releasing large ice masses of up to
106m3 and generating 50 m tsunami waves
. In contrast, the southern part of the front
calves smaller volumes of ice at intervals of several hours.
Calving front of Eqip Sermia glacier in July 2016. The
boxes in the picture describe the geometrical properties of the two
distinct parts of the calving front.
Motivated partly by the case of contrasting calving front geometries
at Eqip Sermia, the aim of this study is to better understand the
detailed flow and stress regimes in the vicinity of the calving front of
tidewater glaciers, including those that are far from flotation. Using
a numerical model that solves the full equations for ice flow, we
investigate the sensitivity to variations in front thickness and
slope, the water depth and the strength of the coupling to the bed
which results from sliding processes. We perform these model
experiments on idealized geometries of grounded glacier termini and
succeed to explicitly express the results as function of relative
water depth.
Based on these model results, we derive a novel parametrization of
calving rate that is calibrated with observations from Arctic
tidewater glaciers. This parametrization only requires the relative
water level and is based on a fit to the modeled stress field at the
surface and an isotropic damage evolution relation.
Methods
Ice flow model and rheology
We used the finite-element library libMesh to
implement the Stokes equations for continuum momentum and mass
conservation:
div(σ)+ρig=0,div(u)=0,
where σ is the Cauchy stress tensor, ρi
the ice density, g the gravitational force vector and
u the velocity vector. As we assume ice to be incompressible
and isotropic, the Cauchy stress tensor can be decomposed into an
isotropic and a deviatoric part σ′:
σ=σ′+σmI,
where σm=13tr(σ)=13σii is the isotropic mean stress and I the
identity matrix. The ice rheology is described as viscous power-law
fluid (Glen's flow law), linking the deviatoric stress tensor
σ′ to the strain rate tensor
ε˙:
σ′=2ηε˙.
The effective shear viscosity η is defined as
η=12A-1n(εe˙+κε)1-nn,
where ε˙e=(12ε˙ijε˙ij)12
is the effective strain rate, A the fluidity parameter, n=3 the
power-law exponent and κε is a finite strain rate
parameter included to avoid infinite viscosity at low stresses
p. 56.
The model domain was discretized with second-order nine-node quadrangle
elements with Galerkin weighting. Model variables are approximated
with a second-order approximation for the velocities u and w and
a first-order approximation for the mean stress σm
(forming a LBB-stable set). The accuracy of the solution was improved
with adaptive mesh refinement near the calving front. The Stokes
equations with the nonlinear rheology were solved with the PETSc
nonlinear solver SNES to a relative accuracy of 10-4
.
Model geometry and scaling
We used a two-dimensional version of the model to conduct the
geometrical tests, as illustrated in Fig. . The
geometry is defined in a Cartesian coordinate system with horizontal
axis x and vertical axis z with origin at sea level at the calving
front (where x=0). The ice moves from right to left. The idealized
glacier geometry used in all model experiments consists of a block of
ice resting on a flat bed with a characteristic length L=2000m and a characteristic ice thickness H=200m. The domain was discretized with 20 elements in the
vertical and 200 elements in the horizontal which, after mesh
refinement, led to a spatial resolution of 2.5m in the
terminus area.
All numerical results are scalable with reference values for ice
thickness Href and overburden stress σref and
are therefore independent of the geometrical extent. This validity of
the scaling was tested by running the model for different ice
thicknesses, which recovered identical flow and stress results. The
velocity scale uref was chosen as the vertical surface
velocity caused by uniaxial confined compression in pure shear of an
ice block under its own weight p. 377:
Href=H,σref=ρigH∼[0.009MPam-1]H,uref=AHσrefn8(n+1)∼[1.7×10-6m-3a-1]H4.
The coordinates and the water depth at the calving front
Hw are scaled by the ice thickness Href:
x^=xHref,z^=zHref,ω=HwHref.
All stress and velocity components are scaled according to
σ^=σσref,u^=uuref.
Geometry of the idealized grounded glacier. α is the slope
angle of the calving front above the vertical cliff.
Boundary conditions
The upper surface of the glacier was described as a traction-free
surface boundary. Basal motion was parametrized with a slipperiness
coefficient C, which relates the basal velocity ub with
basal shear stress τb
:
ub=Cτb.
This boundary condition was implemented as a two-element layer with
constant viscosity ηs=hs/C, which was added at the
bottom of the model domain representing the glacier. At the lower
boundary of this “sediment layer”, a Dirichlet boundary condition
with zero velocity (u=v=0) was imposed. A layer thickness of hs=10m was chosen, although tests with varying hs showed no
significant differences. This simple approach allowed us to capture the
physical processes that are relevant to this study. In the case of
vanishing basal motion the two-element layer was not used for the computation, and
Dirichlet boundary conditions (u=v=0) were imposed directly at the
bottom of the model domain representing the glacier.
At the calving front a normal stress boundary condition was imposed
below the water level, while the surface above water was kept
stress-free. The stress boundary condition thus reads
σnn=min(ρwgz,0)σnti=0(i=1,2),
where σnn and σnti are the normal and
tangential tractions applied on the calving front (σnn is
negative, i.e., compressive since z<0 below water) and ρw is water
density (Table ).
At the upstream boundary of the glacier domain velocities were fixed
to zero. Additional modeling experiments showed that different values
for this upstream boundary condition do not affect the results of the
analysis.
Sensitivity analysis strategy
The stress state and flow field near the calving front is analyzed in
three suites of numerical experiments that investigate the effect of
variations in relative water level ω, the slope of the calving
front and basal motion.
The water level sensitivity experiments were performed for relative
water levels ω=0,0.25,0.5,0.75,0.85 and
ωf=ρiρw,
where the last value is the relative water level at flotation. The
calving front for this experiment was vertical and the bottom
boundary without sliding (i.e., zero velocity Dirichlet boundary
condition). All these experiments were undertaken with both the
density of ocean water (ρw=1028 kgm-3)
and freshwater (ρw=1000 kgm-3).
The calving front slope sensitivity experiments were performed on
a geometry with the upper part of the calving front reclining at
various angles. The lower 25 % of the calving front height
was set vertical, and the upper part inclined at angles from
90, 75, 60 and
45∘, until it reached the maximum surface height (see
Fig. for illustration). This particular geometrical
setup was chosen to represent a simplified geometry of Eqip Sermia,
which has a 50 m high vertical cliff at the bottom with
a 45∘ inclined slope up to the top at 200 m. For
this experiment, the relative water level was set to ω=0 and
the sliding velocity was set to zero.
The bed slipperiness sensitivity experiments were performed on a block
geometry with a vertical calving front and a relative water level
ω=0.5. The basal slipperiness coefficient C was varied from
0 to 1000 mMPa-1a-1 with
333 mMPa-1a-1 increments. A slipperiness of
1000 mMPa-1a-1 corresponds to a sliding speed of
300 ma-1 for a typical tidewater outlet glacier in
Greenland with a driving stress of 0.3 MPa.
Stress invariant combinations
Any criterion for fracture propagation or damage evolution should be
independent of the choice of coordinate system and can therefore be
expressed as a function of the invariants and eigenvalues of the
stress tensor. proposed a linear combination of
three stress invariants to describe the creep rupture of ductile and
brittle materials under multi-axial states of stress. The invariants
chosen were maximum principal stress σ1, first stress
invariant I1=σm=13σii and the
von Mises stress
J2=σe=(32σij′σij′)12
to form the stress combination
χH=ασ1+βσe+γσm,
where the weights α, β and γ fulfill the conditions
0≤α,β,γ≤1,α+β+γ=1.
The Hayhurst stress χH has been used as a criterion for the
initiation and evolution of damage in several glaciological studies
.
Sensitivity experiment results for varying water depth. (a)
Scaled Hayhurst stress distribution. (b) Scaled horizontal velocity
distributions. (c) Scaled Hayhurst stress along the surface. (d)
Scaled horizontal velocity magnitude along the surface. In panels
(a, b), the subplots show increasing water depth from the
bottom to the top (water level at z^=0). Solid and dashed
lines in panels (c, d) correspond to experiments with sea- and freshwater densities, respectively.
Scaled velocities along the vertical face of the calving
front (solid lines) for different relative water levels. Horizontal
line markers show the relative water level for each curve.
Sensitivity experiment results for varying calving front
slope. Panels (a–d) are the same as Fig. . In panels (a, b), the subplots show decreasing
calving front slopes from the bottom to the top. In panel (c), the local
minimum of stress close to the calving front is located where the
front reaches its maximal height. In panel (d), vertical lines on the
curves for inclined fronts mark the distance at which the maximal
surface height is reached.
To investigate the full spectrum of possible stress states that lead
to the initiation of damage, we investigated linear combinations of
five stress invariants: σ1,σe,σm and
additionally the third invariant of the stress tensor
I3=det(σ) and third invariant of the
deviatoric stress tensor J3=det(σ′). This
extended linear combination reads
χ=ασ1+βσe+γσm+ϕI3+μJ3
with weights α,β,γ,ϕ and μ that fulfill the
conditions
0≤α,β,γ,ϕ,μ≤1,α+β+γ+ϕ+μ=1.
We performed a sensitivity analysis based on the five stress
invariants of Eq. () by systematically varying the weights
with 0.1 increments (Eq. ).
Sensitivity experiment results for varying bed slipperiness
C. Panels (a–d) are the same as Fig. .
In panels (a, b), the subplots show increasing bed slipperiness
from the bottom to the top. Units for bed slipperiness C are
mMPaa-1.
Stress parametrization and calving relation
The similarity of stress distribution curves along the glacier surface
for varying relative water levels (Fig. c) allows
for an explicit parametrization of the stresses. With some simple
assumptions on a damage evolution law, a calving rate parametrization
can be derived that is expressed as a function of total ice thickness
and relative water level. Specifically, we assume that surface
crevasses open under the extensional stress σ1. The Hayhurst
stress would be a similarly suited stress measure for the extensional
stress state under small compressive load at the glacier surface. The
above stress state analysis showed that the three stress intensity
measures σ1, σe and χH along the
glacier surface are very similar, as demonstrated in
Fig. .
Modeled (dashed lines) and corresponding parametrized (solid
lines) maximum extensive stresses σ^1 at the surface
for different water depths. The dotted lines show the horizontal
deviatoric stresses at the calving front for all water depths
based on Eq. ().
Stress parametrization
The distribution along the glacier surface of scaled maximum principal
stress σ^1s is shown for different relative water
levels in Fig. (this is approximately the
tensile stress along the surface, whereas Fig. c
shows Hayhurst stress). The similarity in shape of these stress
curves allows for an approximate representation by a function that
depends on the relative water level ω alone:
σ^1s(x^)=a(ω)x^^exp(-x^^),
where x^^ is a stretched and shifted version of
the scaled (by ice thickness) horizontal coordinate x^.
This stretching function is somewhat cumbersome and is given in
Appendix . The extensional stress reaches the maximum
at x^^=1 (setting the derivative of
Eq. to 0) with magnitude σ^1,m=a(ω)exp(-1) and can be approximated
by
σ^1,m(ω)=0.4-0.45ω-0.0652≃0.41-ρwρiω-0.0652,and therefore
a(ω)=1.087-1.223ω-0.0652.
It is interesting to note that the maximum extensional stress at the
surface has a similar form as the mean deviatoric stress in
Eq. () but is ∼60% higher.
The scaled horizontal position of the stress maximum can be
approximated by
x^m=0.671-ω2.8.
Analytical calving relation
Using the parametrizations of magnitude and position of the maximum
extensional stress at the surface (Eqs.
and ) the calving rate can be estimated under
simple assumptions on crevasse formation.
One major assumption is that a large crevasse forms at the location of
the maximum tensile surface stress where the ice is weakened until
failure. Such crevassing seems realistic as both observations and
model results show the formation of huge crevasses. When failure of
the surface ice is complete, we assume that all ice in front of the
crevasse is removed and a new calving front forms at the location of
the crevasse. Here, we do not consider explicitly which processes are
responsible for downward propagation of the crevasse. Several
processes could be considered, such as bottom crevassing,
hydro-fracturing by ponding water in surface crevasses, rapid elastic
crevasse propagation , ice break-off in multiple
steps (e.g., a surface slump, followed by subaqueous buoyant calving)
or continued material fatigue due to tidal forcing. The proposed
calving relation relies on the major assumption that processes
responsible for ice break-off act on faster timescales than the
formation of the surface crevasse and, therefore, that the calving
process is uniquely determined by the time to failure at the surface
stress maximum. Thus, the average calving rate u¯c
can be calculated as the distance of the stress maximum divided by the
time to failure Tf. In dimensional coordinates this is
u¯c=x^mHrefTf.
Assuming further that crevasse formation can be described by isotropic
damage formation with damage variable D, the stress in the damaged
material is σ̃=(1-D)-1σ
. The isotropic damage evolution
relationship employed here is
dDdt=Bσ0-σthr(1-D)k+r,
where B is the rate factor for damage evolution, r and k are
constants, σ0 is the stress in the work zone and
σth a stress threshold for damage creation.
Integrating this relation over time, the time to failure, i.e., the
time required for damage to evolve from an initial value D0 to
a critical value Dc, reads
Tf=(1-D0)k+r+1-(1-Dc)k+r+1(k+r+1)B(σ0-σth)r.
We further assume that the stress in the work zone is the maximum
tensile stress σ0=σ1,m. Inserting the
parametrizations for maximum tensile stress and stress maximum
position (Eqs. and
) in the above relation yields
u¯c=x^mHTf=0.67(k+1+r)B(1-D0)k+r+1-(1-Dc)k+r+1×1-ω2.8(σ1,m-σth)rH.
The term in square brackets is constant, and after renaming it the
effective damage rate B̃ the expression reads
u¯c=B̃1-ω2.8×0.4-0.45ω-0.0652ρigH-σthrH
with parameter values B̃=65MPa-ra-1 and
σth=0.17MPa, which were determined from
a calibration with data discussed below in Sect. 5.3. The parameter
value r=0.43 is chosen according to .
Discussion
Sensitivity analyses
The stress intensity and therefore ice deformation rates are
decreasing as the relative water level increases due to the pressure
exerted by the water at the calving front. This feature is already
captured by the depth-integrated extensional stress at the front
(Eq. ) and, in more detail, in the parametrized
maximum extensional stress (Eq. ),
illustrated in Fig. . In both cases the
square dependence of the horizontal stress on relative water level
controls fracture or damaging processes, the magnitude and rate of
which depend linearly on the stress intensity.
In addition, the detailed modeling shows that the stress peak at the
glacier surface moves upstream for lowering relative water level
(Figs. and ),
implying that crevasses are likely to open in greater distance from
the calving front and leading to detachment of larger masses during
calving.
A higher relative water level results in a more stable calving front
as emphasized by, which seems to be in
contrast with the often-used relations which predict that calving
rates increase with water depth
. In nature,
however, glaciers terminating in deeper waters are also thicker and
calve at higher rates as they experience higher absolute (unscaled)
stresses. Furthermore, submarine frontal melting is likely to lead to
higher calving rates by over-steepening of the front
, although the melt undercutting
effect on calving rates seems to be limited .
Using freshwater instead of seawater at the calving front yields
slightly higher stresses and velocities
(Fig. c, d). This difference can be explained by
the reduced back pressure applied by freshwater on the calving front,
which results from a lower water density.
The model results demonstrate that reclining calving fronts lead to
lower velocities and stresses and thereby implicitly confirm that
inclined calving fronts should reach larger stable heights than
vertical cliffs, as observed for example at Eqip Sermia (200 m
high at 45∘). This sensitivity analysis on front slope
may, together with observational data on non-vertical calving fronts,
provide constraints on parameters of ice resistance to failure.
Further, the presence of extrusion flow along the reclining calving
face of an idealized glacier was demonstrated. Such a velocity
pattern has been observed and measured on an inclined slope at the
northern front of Eqip Sermia but is rarely
discussed in modeling studies
.
Basal sliding leads to increased stresses at the surface throughout
the computational domain. Thus, basal sliding may cause an onset of
ice damaging and crevasse opening in a greater distance from the
calving front (Fig. c). The velocity patterns in
Fig. b show that the influence of bed slipperiness is
only apparent in the proximity of the calving front, even for high
sliding coefficients. Moreover, stress distributions are almost
identical for all bed slipperiness experiments, which implies that
basal sliding has a negligible effect on the stability of the calving
front. Basal sliding adds a constant velocity at the bottom of the
domain rather than affecting the velocity gradients. This result does
not include any spatial variation in bed slipperiness, which would
likely be caused by including a water-pressure-dependent sliding
relation. Effective pressure (the difference between ice normal
stress at the bottom and water pressure) typically decreases towards
the calving front for real glaciers with sloping surface and may
cause additional sliding towards the front, an effect that is not
considered in this modeling effort.
For a sloping glacier surface the location and magnitude of the stress
maximum in the vicinity of the calving front remain almost identical, as
shown in Fig. . Similar results are obtained for
a reverse bed slope with a flat surface, with a smaller influence on
stresses and velocities than for the reclining
surface. However, the effect on stresses and velocities upstream of the calving
front is not visible for the reverse bed slope with a flat
surface. This indicates that, for a glacier with a reclining surface
slope, ice can potentially start damaging and forming crevasses at the
surface far upstream from the calving front.
Calving relation
The proposed calving rate parametrization
(Eq. ) is simple and only requires two
geometrical quantities: frontal ice thickness H and water depth
Hw. The assumptions about the failure process are
lumped into three parameters – B̃, σth and
r –
which can be determined by data calibration (Sect. 5.3). The
parametrization exhibits many similarities with established calving
relations but is formulated in terms of two quantities that are
calculated by any ice flow model. It therefore is a drop-in
replacement for other calving relations used in glacier models of
different complexity.
The calving rate parametrization (Eq. )
has some interesting properties, which are illustrated in
Figs. and .
Holding constant the relative water depth, the absolute water depth or
the ice thickness results in different calving laws:
for constant relative water level ω the calving rate
grows roughly like u¯c∝H1+r (black and gray lines
in Fig. );
for constant absolute water depth Hw=ωH a fit shows
that roughly u¯c∝H1.25 (red and orange lines in
Fig. );
for constant ice thickness the calving rate decreases with
increasing relative water level (Fig. )
roughly likeu¯c∝1-ω2.81-1.3ω2r≃1-ω2.81-ρwρiω2r.
The predicted calving rate for a given water depth depends on the
thickness of the glacier, which is the result of the mass fluxes in the
terminus area. Thus, calving rates depend on the surface evolution
and hence the upstream dynamics of the glacier. The semi-empirical
calving rate parametrization is therefore, in the sense of inclusion
of upstream dynamics, similar to the position based calving models
. The formulation as a calving rate also makes this
parametrization relatively easy to use in larger-scale fixed grid
models.
Calving rates predicted by the parametrization in relation to
ice thickness. Calving rates increase with increasing total ice
thickness for a given water depth Hw=ωH (red and orange
lines), relative water level ω (black and gray lines) or
freeboard H-Hw (blue lines). Note that the gray line refers to
a front at flotation.
Calving rates predicted by the parametrization as a function
of relative water level. Calving rate decreases under increase of
the relative water level ω for constant total ice thickness
H.
Calving rates (md-1) predicted by the parametrization are shown as
contours in dependence of H and ω. The hatched region
indicates the states excluded by the maximum calving front criterion
. The gray area indicates states where the
stress threshold σth precludes calving. Blue dots
with numbers indicate calving rates determined from measurements,
shown in Table .
Comparison of measured calving rates with predictions from
the calving parametrization. The glacier names are abbreviated
according to Table .
Calibration of the parametrization
The calving rate parametrization (Eq. )
contains three empirical parameters: B̃=65MPa-ra-1 and σth=0.17MPa,
which were obtained by calibration with data, and r=0.43, which is
taken from . Calving rates thus obtained are
not very sensitive to the exact choice of parameter values, which are
within the range of previous studies
.
Values of calving front height, water depth and
calving rate for different glaciers.
Abbr
Glacier
Hs
Hw
uc
Source
(m)
(m)
(m d-1)
Bow
Bowdoin 2015
25
220
1.5
, Guillaume Jouvet, personal communication (2017)
Col
Columbia 1983
53
213
6.7
Fig. 4
Col
Columbia 1988
29
243
10.6
Fig. 4
Col
Columbia 1994
61
280
15.4
Fig. 4
Col
Columbia 1998
103
253
29.7
Fig. 4
Col
Columbia 2000
122
260
24.8
Fig. 4
Eqi
Eqip Sermia 2015
50
80
8.2
Hel
Helheim 2015
80
615
25.0
, Fig. 2
Hum
Humboldt N 2015
25
250
1.2
Hum
Humboldt S 2015
30
125
0.2
JI
Jakobshavn Isbræ 2008
100
800
35.6
Kng
Kangilgata 1962/63
40
350
4.5
Lil
Lille 1962/63
25
230
1.5
Moe
Moench 2006
50
0
0.1
RI
Rink Isbræ1962/63
70
560
13.0
Sto
Store 1962/63
65
500
17.3
Sto
Store 2015
60
500
16.0
Yak
Yakutat 2015
30
325
0.4
To calibrate these parameter choices, data on calving rate, ice
thickness and water depth for a wide variety of tidewater glaciers in
the Arctic were collected. The data set covers the full range of water
levels (relative and absolute), velocities and ice thicknesses that
are found in Arctic tidewater glaciers. Unfortunately, many studies
report only width-averaged data on calving front geometry and calving
rate, which are not suitable for our proposed relation which relies on
local stresses on a flowline. Only a limited set of point data on
calving front geometries are available from the published literature
from which total ice cliff thickness, water depth and calving rate can
be obtained. For the calibration, we used the values shown in
Table from diverse data sources.
Contours of calving rates calculated with
Eq. () are shown in
Fig. together with the maximum
theoretical calving front height predicted by
. Figure
plots the same calving rate data against results from the
parametrization. While a sizable spread of the data is visible,
especially for low calving rates, the general agreement shows that the
parametrization is well suited to estimate calving rates for this set
of tidewater glaciers in the Arctic.
Note that the derivation of the parametrization is independent of the
specific geometry or location of a tidewater glacier and thus the
calibration is expected to be “global” and valid for any tidewater
glacier.
Conclusions
This study improves our knowledge on the influence of geometry and
water depth on the stress and flow regimes in the vicinity of the
calving front and proposes a novel calving rate parametrization.
The magnitude of the stresses and flow speeds near a grounded vertical
calving front are dominantly dependent on water depth and increase
with decreasing water depth. Thus, the presence of water at the
calving front has a strong stabilizing effect. Importantly, the
extensional stress at the surface can be parametrized as a function of
relative water level only. Further, we find that grounded tidewater
glaciers with reclining calving faces have the potential to reach
larger maximum stable heights than those with vertical calving
fronts. Spatially uniform variations in basal sliding likely have
a weaker effect than water depth and calving front slope on the
stability, as the magnitude and location of the stress maximum show
a small sensitivity to variations in bed slipperiness.
A simple calving rate parametrization was derived that was calibrated
with calving rate data of a set of tidewater glaciers in the Arctic.
This approach can be used to compute calving rates for grounded
tidewater glaciers with relatively simple geometries when front
thickness and water depth are known. The application of this
parametrization in flow models of different complexity should be
straightforward.
The present study lays the foundation for future, more detailed,
studies of the calving process on more realistic geometries. Detailed
analyses including time evolution, further processes such as frontal
melt and water-filled crevasses, and data validation will be necessary
for the implementation of improved calving parametrizations.