Introduction
Accelerated discharge of ice into the oceans from land ice is a major
contributor to sea level rise, and constitutes the largest source of
uncertainty in sea level predictions for the twenty-first century and beyond
. To a large degree, this uncertainty reflects the limited
understanding of processes impacting calving from tidewater glaciers and ice
shelves, and associated feedbacks with glacier dynamics. In particular,
calving occurs by the propagation of fractures, which are not explicitly
represented in the continuum models used to simulate ice flow and glacier
evolution.
Recently, it has been suggested that ocean warming could play an important
role in determining glacier calving rate and acceleration, by impacting
submarine melt rates .
proposed two mechanisms responsible for the increase of submarine melt rates
at the ice–ocean interface in Greenland: a warmer and thicker layer of
Atlantic water in the fjords and an increase in subglacial discharge mainly
during summer and autumn. Buoyant meltwater plumes entrain warm ocean water
and are thought to enhance melt undercutting
at the ice cliff triggering collapse of the ice above.
investigated controls on seasonal variations in calving
rates and showed that calving variations at Kronebreen, the glacier this
study focuses on, are strongly correlated with sub-surface ocean temperature
changes linked to melt undercutting of the calving front. However, direct
measurements of oceanic properties, ice dynamics, frontal geometries and mean
volumetric frontal ablation rates are still too scarce to quantify the
relationship between ocean processes, subglacial discharge and ice dynamics
and one must rely on modelling. Complex coupled process models can help to
gain a better understanding of the physics taking place at tidewater glacier
fronts.
In previous modelling work
,
the dynamics of ice masses have been simulated using continuum models, in
which the continuum space is discretised and includes processes of mass and
energy balance. In addition to the lack of process understanding, continuum
models cannot explicitly model fracture but must use simple
parameterisations such as damage variables or phenomenological calving
criteria.
These problems can be circumvented using discrete particle models, which
represent ice as assemblages of particles linked by breakable elastic bonds.
Ice is considered as a granular material and each particle obeys Newton's
equations of motion. Above a certain stress threshold, the bond is broken,
which allows the ice to fracture. showed that
complex crevasse patterns and calving processes observed in nature can be
modelled using a particle model, the Helsinki Discrete Element Model (HiDEM).
used a similar particle model and suggested that glacier
geometry provides the first-order control on calving regime. However, the
drawback of these models is that, due to their high computer resource demand,
they can only be applied to a few minutes of physical time.
A compromise should be found by coupling a continuum model, such as
Elmer/Ice, to a discrete model, such as HiDEM, to successively describe the
ice as a fluid and as a brittle solid. Sliding velocities and ice geometry
calculated with the fluid dynamic model are used by the discrete particle
model to compute a new calving front position. The effect of subglacial
drainage mixing with the ocean during the melt season is taken into account
by using a plume model that estimates melt rates at the front according to
pro-glacial observed ocean temperatures, subglacial discharge derived from
surface runoff and ice front height.
In this paper, we use both the capabilities of the continuum model Elmer/Ice
and the discrete element model HiDEM. We harness the ability of HiDEM to
model fracture and calving events, while retaining the long-term ice flow
solutions of a continuum approach. The aim is to investigate the influence of
basal sliding velocity, geometry and undercutting at the calving front on
calving rate and location. We determine the undercutting with a high-resolution plume model calculating melt rates from subglacial discharge. The
simple hydrology model that calculates the subglacial discharge is based on
surface runoff that is assumed to be transferred directly to the bed and
routed along the surface of calculated hydrological potential. We illustrate
the approach using data from Kronebreen, a fast-flowing outlet glacier in
western Spitsbergen, Svalbard (topography, meteorological and oceanographic
data, as well as horizontal surface velocity and front positions from 2013),
to assess the feasibility of modelling calving front retreat (rate and
position).
Methods
Observed geometry, surface velocities and front positions
The bed topography, zb, is derived from profiles of airborne and
ground-based common-offset ice-penetrating radar surveys distributed over
Kronebreen from 2009, 2010 and 2014 . The initial
surface topography includes different available surface digital elevation models (DEMs) and is described
in .
Ice surface velocities were derived from feature tracking of TerraSAR-X image
pairs in slant range using correlation windows of
200×200 pixels at every 20 pixels, and subsequently
ortho-rectified to a pixel size of 40 m using a DEM . Images were acquired roughly every
11 days for the period May–October 2013. Uncertainties in surface
velocity are estimated to be ∼ 0.4 md-1 and comprise a
co-registration error (±0.2 pixels) and errors arising from
unavoidable smoothing of the velocity field over the feature-tracking window.
Ice-front positions were manually digitised from the same images used for
feature tracking after they had been orthorectified to a pixel size of 2 m
using a surface DEM .
Offline coupling approach
We use surface velocity and frontal position data described above to test the
effects of sliding and undercutting on calving using different models in a
global approach. This one-way offline coupling approach is divided into three
parts using six models (see Fig. ): inversion for sliding and
computation of geometry evolution (with Elmer/Ice), determining undercutting
(with the energy balance model, subglacial hydrology model, plume model and
undercutting model) and computing calving (with HiDEM). In this paper, we use
the output of five different models as input for the discrete particle model,
HiDEM, in order to compare the modelled calving front to observations for
different configurations of sliding, geometry and undercutting.
Model scheme presenting the calculation of the sliding and geometry
(Elmer/Ice) as well as the undercutting at the subglacial discharge as input
to the glacier calving from the HiDEM.
Observation times of velocity acquisitions, ti,
associated dates and time interval between two observations (Δti).
The HiDEM model is run for observational times t0, t4, t6 and
t11 indicated.
ti
Δti
Date
Comment
t0
2 Jun 2013
Before the onset of
the melting season
t1
11 d
13 Jun 2013
First melt
t2
11 d
24 Jun 2013
t3
11 d
5 Jul 2013
t4
26 d
31 Jul 2013
Period of high
surface runoff
t5
11 d
11 Aug 2013
t6
11 d
22 Aug 2013
Minimum
basal friction
t7
11 d
2 Sep 2013
t8
11 d
13 Sep 2013
t9
11 d
24 Sep 2013
t10
11 d
5 Oct 2013
t11
11 d
16 Oct 2013
After the last melt
We set t0 at the velocity acquisition just before the first melt and the
following observational times are set at each observation of surface
velocity. The exact dates are summarised in Table .
First, we infer the sliding velocity at each observational time from surface
velocities using an adjoint inverse method implemented in Elmer/Ice with an
updated geometry from observations. At each iteration, i, corresponding to
an observed front position, Fiobs, the front and the surface are
dynamically evolved during the observation time interval (roughly
11 days) with Elmer/Ice with a time step of 1 day. By the end
of the observation interval, the front has advanced to a new position,
Fi+1elmer. Here we use i+1 because this is the position the front
would have at ti+1 in the absence of calving. Second, given subglacial
drainage inferred from modelled surface runoff, a plume model calculates melt
rates based on the subglacial discharge for each iteration, which are
subsequently applied to the front geometry at subglacial discharge locations.
At each iteration, the front geometry takes into account the undercutting
modelled at the former iteration. Finally, the sliding velocity, geometry and
undercutting (when applicable) are taken as input to the calving particle
model HiDEM for each iteration and a new front, Fi+1hidem, is
computed for four iterations, i={0,4,6,11}, which represent interesting
cases (see comments in Table ). More details about each
aspect of the model process are given in the following sections.
We call this approach an offline coupling because inputs to the HiDEM are
output results from Elmer/Ice and undercutting model but not vice versa. In
Elmer/Ice, we use the observed frontal positions. A completely coupled
physical model would use the output of HiDEM, the modelled front position, as
input to the ice flow model Elmer/Ice and the undercutting model. It would
also calculate the basal friction from a sliding law rather than an
inversion. In principle, such an implementation is possible using the same
model components as this study.
Sliding and frontal advance with continuum model Elmer/Ice
At the base of the glacier, we use a linear relation for sliding of the form
τb+βvb=0,
with τb the basal shear stress and vb the basal
velocity. The basal friction coefficient, β, is optimised at each
observational time to best reproduce observed velocity distribution at the
surface of the glacier as described in . This is done by
using a self-adjoint algorithm of the Stokes equations for an inversion
e.g. and
implemented in Elmer/Ice . The inversion is performed
using the method of Lagrange multipliers to minimise a cost function
including the observed horizontal surface velocities and a Tikhonov
regularisation. We use an unstructured mesh, with spatial repartition of
elements based on the mean observed surface velocities in the horizontal
plane (roughly 30 m resolution close to the front). Vertically, the
2-D mesh is extruded with 10 levels (roughly 10 m resolution close to
the front). More details on the Elmer/Ice modelling (viscosity, ice
temperature, iterations, etc.) are given in .
Front position and surface elevation changes with Elmer/Ice during
Δt=ti+1-ti.
After each inversion, the temporal evolution of the glacier is mathematically
described by the kinematic boundary condition defined at the surface,
∂zs∂t+vx(zs)∂zs∂x+vy(zs)∂zs∂y-vz(zs)=a˙s(ti),
which describes the evolution of the free surface elevation, z=zs, for a
given net accumulation, a˙s(ti), calculated using a coupled
modelling approach after , described in the next section.
We use a time step of 1 day during the interval of time between two
acquisitions. Equation () is solved alongside the Stokes equation,
coupled to the latter by the velocities. The basal sliding velocity is not
evolved and stays equal to the result of the inversion. When the front is
advanced, the mesh is stretched to match the new front position. No new
element or node is created and the basal sliding coefficients are
extrapolated towards the new front. The new surface is in fact only used as
an input for the next iteration. There is no interpolation of the basal
sliding coefficients between two observational dates.
We assume that the front is vertical above the water line so that the
observed front position (at the surface of the glacier) is the same at sea
level. We call Fiobs(z=0), the front position observed at time ti
with z=0 at the sea level and Fi+1elmer(z=0), the advanced modelled
front position after Δt=ti+1-ti (see Fig. ).
The front is advanced by imposing a Lagrangian scheme over a distance equal
to the ice velocity multiplied by the time step. We do not account for the
submarine melting during the advance because we only have observations at the
beginning and the end of each time span. Instead, we lump frontal melting by
applying an undercutting after the advance, as explained hereafter.
Surface runoff and subglacial discharge model
The surface mass balance, a˙s, and runoff are simulated with a
coupled energy balance–snow modelling approach . The
coupled model solves the surface energy balance to estimate the surface
temperature and melt rates. The subsurface routine simulates density,
temperature and water content changes in snow and firn while accounting for
meltwater percolation, refreezing and storage. The model is forced with
3-hourly meteorological time series of temperature, precipitation, cloud
cover and relative humidity from the Ny-Ålesund weather station
(eKlima.no; Norwegian Meteorological Institute). Elevation lapse rates for
temperature are calculated using output from the Weather Research and
Forecast (WRF) model , while the precipitation lapse
rate is taken from ; zero lapse rates are assumed for
cloud cover and relative humidity. Surface runoff is modelled on a
100 m × 100 m grid.
The temporal subglacial discharge at the calving front is estimated from
integration of daily surface runoff assumed to be directly transferred down
to the glacier bed. Assuming the basal water pressure at over burden, the
flow path of the meltwater towards the glacier front is determined from the
hydraulic potential surface defined as
ϕ=ρig(zs-zb)+ρwgzb,
with g the gravitational acceleration. The grid is the same as that used
for surface runoff. The flow path along the hydraulic potential surface is
determined by D-infinity flow method where the flow direction from a grid
cell is defined as the steepest triangular facets created from the
eight neighbouring grid cells . The flow from the centre grid
cell is distributed proportionally to the two cells that define the steepest
facet. The flow is accumulated as the meltwater is routed along the
calculated hydraulic potential surface towards the front and outlet points at
the front are determined by identifying flow rates higher than
1 m3s-1. The hydraulic potential surface is filled before flow
accumulation is calculated to avoid sinks.
Plume model and submarine melt rates
A high-resolution plume model is used here to simulate the behaviour of
subglacial discharge at the terminus of Kronebreen. The model is based upon
the fluid dynamics code Fluidity , which solves the
Navier–Stokes equations on a fully unstructured three-dimensional finite
element mesh. The model formulation builds upon the work of
, with the addition of a large eddy simulation (LES)
turbulence model and the use of the synthetic eddy
method (SEM) at the inlet .
The geometry of the model is adapted to Kronebreen by setting the water depth
to 100 m and initialising the model with ambient temperature and salinity
profiles collected from ringed seals instrumented with GPS-equipped
conductivity, temperature and depth satellite relay data loggers
(GPS-CTD-SRDLs) . These data were collected
between 14 August and 20 September 2012 from a region between 1 and 5 km away from the glacier terminus and are taken as
representative of the ambient conditions in the fjord during summer. Melt
rates are calculated on the terminus using a three-equation melt
parameterisation described by and
and implemented in Fluidity by . Velocities driven by
ocean circulation are typically around 2–3 orders of magnitude smaller than
plume velocities and therefore neglected.
The model is spun up for 1000 model seconds until the turbulent kinetic
energy in the region of the plume reaches a steady state and thereafter run
for 10 min of steady-state model time. Melt rates are extracted from the
duration of the steady-state period and then time-averaged and interpolated onto
a uniform 1 m × 1 m grid covering a 400 m wide section of the glacier
terminus.
The high computational cost of the model means that it cannot be run
continuously over the study period, nor can the full range of discharges and
oceanographic properties be tested. Instead, representative cases Md using
the ambient ocean properties described above and discharges d of 1, 10, 50
and 100 m3s-1 were tested and the melt rate profiles for
intermediate discharges were linearly interpolated from these cases.
Undercutting model
We assume a vertically aligned surface front at the beginning of the melt
season. We know the position of the front, F0obs(z=0), for the time
span of each satellite image. The front is spatially digitised with 10 m
spacing in the horizontal space and 1 m spacing in the vertical space. We
use the combination of observed front, advanced front from Elmer/Ice and melt
rates from the plume model to estimate the daily amount of undercutting. At
each iteration, i, the sum of the daily undercutting during the observation
interval is subtracted from the front.
Three cases of undercutting i+1 at ti+1 (black line)
depending on former undercutting i at ti (grey line) at z relative to
Fiobs(z=0) (black line with circles) in plan view (left) and side view
(right). The red star represents the discharge location. On the side view,
the dashed line represents the simplified undercut geometry where the ice
foot has been removed, which is given as input to the HiDEM. (a)
Fiobs(z=0) is behind Fi+1elmer(z=0) and in front of
Fielmer(z). The undercutting from Fielmer(z) is translated to
Fi+1elmer(z) (grey line) and the new undercutting is superposed (red
line). (b) Fiobs(z=0) is in front of Fielmer(z). The
remnant from Fielmer(z) (which is behind Fiobs(z=0)) is
translated to Fi+1elmer(z) (grey line) and the new undercutting is
superposed (red line). (c) Fiobs(z=0) is behind
Fielmer(z). The undercutting from Fielmer(z) is ignored and the
undercutting created at ti+1 is the only one (red line).
When the first discharge occurs, the melt rate calculated with the plume
model in 2-D is summed for the period of time between t0 and t1 and
projected to the advanced front F1elmer(z=0) (advanced from
F0obs(z=0)) at the location of the subglacial outlets and ice is
removed normal to the front. This yields a new position of the front at depth
z below sea level called F1elmer(z). At the second iteration,
t2, we know where the front would be if there had not been any calving
between t1 and t2: F2elmer(z=0), which is the advanced front
from the observed position at t1, F1obs(z=0). Therefore we can transfer the
whole undercutting from previous iteration to F2elmer(z) if
F1obs(z=0) is situated in front of F1elmer(z) (see
Fig. b–c). Otherwise, the undercutting would have
been fully or partly calved away (see Fig. b–c).
We then apply the new undercutting on this new geometry given the melt rates
between t1 and t2.
At time ti, the modelled front position at depth z (advanced by
Elmer/Ice from the observed front position at ti-1) is
Fielmer(z) and the observed front position is Fiobs(z=0). We
advance this observed front with Elmer/Ice during Δt=ti+1-ti to
obtain the front position Fi+1elmer(z=0) at ti+1. We want to
determine Fi+1elmer(z) and depth z given the melt rate calculated
between ti and ti+1 and the state of the undercutting from the
previous front Fielmer(z) updated by the observed front
Fiobs(z=0). Three different cases, depending on the relative position
of the observed and modelled fronts at depth z, are then possible as shown
in Fig. :
if the new observed position Fiobs(z=0) is behind Fielmer(z=0) and
in front of Fielmer(z), the melted undercutting is kept and advances in the flow
direction the same distance as the surface modelled front Fi+1elmer(z=0)
(see Fig. a);
if the new observed position Fiobs(z=0) is in front of Fielmer(z),
the undercutting is displaced to the next modelled front Fi+1elmer(z=0)
(see Fig. b);
if the new observed position Fiobs(z=0) is behind Fielmer(z), the
front starts from a vertical profile again (see Fig. c).
The melt summed up between ti and ti+1 is then applied to
Fielmer(z) to obtain Fi+1elmer(z) and so on. Frontal melt
above the surface has not been taken into account so that the effect of
submerged ice feet is not described.
The bed topography, the new geometry (surface elevation, front position with or without
undercutting) and the basal friction are then interpolated onto a
10 m × 10 m
grid to feed the HiDEM and a new front, Fi+1hidem, is modelled after calving for the
four selected iterations (i={0,4,6,11}).
Calving with the first-principles ice fracture model HiDEM
The fracture dynamics model is described in detail in
. This first-principles model is constructed
by stacking blocks connected by elastic and breakable beams representing
discrete volumes of ice. For computational efficiency, we use a block size of
10 m.
At the beginning of a fracture simulation, the ice has no internal stresses
and contains a few randomly distributed broken beams, representing small
pre-existing cracks in the ice. The dynamics of the ice are computed using a
discrete version of Newton's equation of motion, iteration of time steps, and
inelastic potentials for the interactions of individual blocks and
beams. As the ice deforms under its own weight, stresses on the beams
increase, and if stress reaches a failure threshold the beam breaks and the
ice blocks become disconnected but continue to interact as long as they are
in contact. In this way cracks in the ice are formed. For computational
reasons, we initialise the glacier using a dense packed face-centred cubic
(fcc) lattice of spherical blocks of equal size. This introduces a weak
directional bias in the elastic and fracture properties of the ice. The
symmetry of the underlying fcc lattice is, however, easily broken by the
propagating cracks. The ground under the ice or at the seafloor is assumed
to be elastic with a linear friction law that varies spatially
(Eq. ).
The time step is limited by the travel time of sound waves through a single
block and is thereby set to 10-4 s. If the stress in the ice
exceeds a fracture threshold, crevasses will form and ice may calve off the
glacier. The duration of a typical calving event at Kronebreen is a few tens
of seconds followed by a new semi-equilibrium when the ice comes to a rest. The
model runs for ∼100 s, which takes 2 days of computing time. As
HiDEM cannot be triggered too often because of computational limitations, we
simulate ice flow with Elmer/Ice and compute calving with HiDEM thereafter
for the selected iterations. Calving events will then appear as fewer but
bigger events compared to observations. If the time step is changed, the
overall rate of change stays roughly within ±50 % .
The basal friction coefficients, β, at the front of Kronebreen are in
the order of 108–1012 kgm-2s-1
,
and in order to avoid instabilities building up, a cut-off value, above which
particles are assumed to be stuck to the bed substrate, is fixed at
β=1012 kgm-2s-1.
HiDEM reads a file with surface and bed coordinates on a grid and a file with
surface and basal ice (to take into account the undercutting) coordinates.
For simulations with an undercutting at a discharge location and in order to
avoid complication in the HiDEM (position of the basal ice), we remove
particles below the maximum melt (no ice foot as shown by the dashed line in
Fig. ). In the ocean, the basal friction
coefficient is extrapolated downstream of the front and taken equal to the
mean of the values further up from the terminus in case the ice advances. An
ice block is calved when all bonds are broken from the glacier even though it
does not separate from the front.
There is a clear separation of timescales between the velocities of
sliding (∼mday-1) and calving ice (∼ms-1).
This gives us the opportunity to rescale friction so that we can more
effectively simulate calving: even if we scale down friction by, for example,
2
orders of magnitude and increase sliding accordingly to
∼100 md-1, there is still negligible sliding during
calving events which last tens of seconds or perhaps a minute. However, a
rescaling speeds up the frequency of calving, and we can thus “speed up”,
within reason, the few minutes of HiDEM simulation to effectively model
calving which would otherwise take tens of hours or days and thus be
practically impossible to simulate with HiDEM. By applying scaling, the
calving events modelled during the simulation of HiDEM (a few minutes)
correspond to the sum of calving events that would happen during the timescale of sliding. The scaling factor that we use is the same for the whole
domain and for all simulations. We use a friction scaling factor for β
equal to 10-2 (or sliding velocity scaled up by 102), and simulations
run until calving stops and a new quasi-static equilibrium is reached.
In a fully coupled model, the altered ice geometry after calving could then
be re-implemented in the flow model, acting as the initial state for a
continued prognostic simulation with the continuum model. Here, this
back-coupling is replaced by prescribing the next observed configuration.
Frontal ablation calculation
The mean volumetric frontal ablation rate (or mean volumetric frontal calving rate) at the
ice front at time ti, a˙c(ti), is the difference between the ice velocity at the
front, vw(ti), and the rate of change of the frontal position, ∂L/∂t,
integrated over the terminus domain Σw as defined in .
This yields
a˙c(ti)=∫Σwvw(ti)-∂L∂tdΣw,
with
∫Σw∂L∂tdA=ΔA(ti)ti-ti-1∫z∈Σwdz
and ΔA(ti), the area change at the terminus over the interval of time between two
observations ti-ti-1. We want to compare the ablation rates from Fielmer for
observed and modelled cases. For the observed case, the mean volumetric ablation rate is
calculated between the advanced front Fielmer and the observed front Fiobs.
For the modelled case, during one simulation with HiDEM, several calving events are
triggered. Volumetric calving rate is then inferred from the difference between the initial,
Fielmer, and final position, Fihidem, of the front, after calving has stopped.
The total subaqueous melt rate, a˙m, at the front of the glacier is omitted in this balance.
Calving scenario simulations
We investigate the effect of three different parameters on calving activity:
the geometry, gi, corresponding to the frontal position and topography;
the sliding velocity, mainly influenced by the basal friction parameter, βi; and the undercutting, ui, at the subglacial discharge
locations for four distinct times ti=t0,t4,t6,t11
(see Table ). The different configurations are referred as
C(gi,βj,ui). If ui=0, there is no undercutting and hence a vertical
ice front at the subglacial discharge location. At t=0, the melt season has
not started yet, so there is no modelled undercutting. At t=11, the melt
season is finished and there is no modelled undercutting. If j≠i, the
geometry, gi, is taken at ti and the basal friction, βj, at
tj to assess the roles of geometry and basal sliding velocity. We
investigate basal friction at t0 and t6 since the former has maximum
friction and the latter minimum friction of the studied cases. The
configurations studied in this paper are summarised in
Table .
Different configurations, C, characteristics and
periods.
Configuration
Characteristics
Applied to
Geometry at ti
C(gi,βi,0)
Sliding at ti
i∈[0,4,6,11]
Vertical front
Geometry at ti
C(gi,βi,ui)
Sliding at ti
i∈[4,6]
Undercutting at discharge
Geometry at ti
C(gi,βj,0)
Sliding at tj
(i,j)∈[(0,6),
Vertical front
(6,0)]
Basal friction coefficient obtained from inverse modelling and
observed frontal position for (a) t0: 2 June 2013, (b) t4: 31 July 2013,
(c) t6: 22 August 2013 and, (d) t11: 16 October 2013.
Results
Basal friction coefficients
(a) Subglacial flow following the hydraulic potential
surface (in m3 d-1) in logarithmic scale on 22 July 2013.
(b) Daily discharge for the northern and southern discharge (ND and
SD respectively) during the melting season (data gaps correspond to no discharge).
The basal friction coefficient, β, for the four runs presented above,
is shown in Fig. . At t0, before the melt season, the basal
friction is high and roughly homogeneous over the first kilometre. At t4,
when the surface runoff is the highest, the pattern is similar but with a
large offset. The lowest friction is reached at t6, particularly at the
front and in the southern part of the glacier. The highest friction is
reached at t11 a kilometre from the front. Close to the front position,
however, the friction is still high.
Subglacial discharge and submarine melt rates
The hydrological model predicts that there are two main subglacial channels
with discharge exceeding 1 m3s-1 of water (see
Fig. a). This is in accordance with satellite and
time-lapse camera images showing upwelling at these locations
. Modelled surface
melt and discharge at the northern outlet – in short northern discharge (ND)
– starts 6 June and ends 1 October, while the discharge at the southern
outlet (SD) starts 21 June and ends 22 August. Fluxes at ND clearly exceed
those at SD as shown in Fig. b and
Table .
Total volume of subglacial discharge modelled per period
of calving front recording.
Start date
End date
Days
Volume (m3)
ND
SD
2 Jun (t0)
13 Jun (t1)
11
1.27 × 105
13 Jun (t1)
24 Jun (t2)
11
8.73 × 106
4.94 × 105
24 Jun (t2)
5 Jul (t3)
11
6.24 × 107
2.05 × 106
5 Jul (t3)
31 Jul (t4)
26
1.10 × 108
3.54 × 106
31 Jul (t4)
11 Aug (t5)
11
6.2 × 107
1.36 × 106
11 Aug (t5)
22 Aug (t6)
11
4.69 × 107
1.04 × 106
22 Aug (t6)
2 Sep (t7)
11
3.91 × 107
2.03 × 105
2 Sep (t7)
13 Sep (t8)
11
1.18 × 107
0
13 Sep (t8)
24 Sep (t9)
11
6.20 × 106
0
24 Sep (t9)
5 Oct (t10)
11
8.04 × 105
0
24 Sep (t10)
5 Oct (t11)
11
0
0
The melt rate profiles calculated by the plume model for four different
volumes of subglacial discharge are shown in Fig. .
Melt rates, Md, from the plume model given a discharge, d, of
(a) 1 m3s-1, (b) 10 m3s-1,
(c) 50 m3s-1 and (d) 100 m3s-1.
At a discharge of 1 m3s-1, melt rates are low
(< 2.5 md-1), with the maximum melt rate occurring at depth
and negligible melt rates close to the water line. At 10 m3s-1,
the melt profile reaches the surface and has highest melt rates
(∼ 3.5 md-1) along the plume column. With
50 and 100 m3s-1 discharge, the highest
melt rates are attained at the ocean surface on the sides of the plume column
(∼ 5 and ∼ 6 md-1 respectively).
In general, low discharges drive maximum melt within the plume and at depth,
while higher discharges drive stronger surface gravity currents and
therefore give higher melt rates at the surface.
(a) Plan view of the observed frontal position of Kronebreen
at six different dates, defined by different colours, corresponding to the satellite
data acquisition dates during the melt season in 2013 (up to 22 August).
At ti, the observed front, Fiobs, is represented by a dashed line and
the advanced front, Fielmer(z=0), by a thin line. The discharge location
is defined by a star. Enlargement at (b) the northern discharge (ND)
area at z=-3 m and at (c) the southern discharge (SD) area at z=-42 m with
the advanced front at depth z where undercutting has been applied, Fielmer(z),
represented by a thick line. Vertical section (d) at the northern discharge (ND)
location and at (e) the southern discharge (SD) location. The stars in (d, e) indicate the plan view elevation z from (b, c). Horizontal lines in (d, e)
represent the sea level for each iteration.
Undercutting
The modelled frontal position is summarised in Fig. in plan view
and vertical view at the discharge locations. In most cases for the ND
location, where the discharge is the highest, the melt profile
(Fig. d) creates an undercut profile concentrated right near the
waterline. found similar results when modelling melt rates
at shallow grounding lines (100–250 m) given 250 m3s-1
discharge. It is interesting to see that the observed front after calving,
Fiobs (dashed line in Fig. a–b), generally falls behind
the undercut front before calving, Fielmer(z) (thick line in
Fig. b).
The frontal submerged undercutting driven by the plume differs in shape from
one location to another. In the first 50 m below the surface, the
undercutting at the SD is not as abrupt as at the ND and is also smaller
(Fig. c–e). Where the discharge is the highest, the melt rate
peaks just below the waterline and stretches laterally from the vertical
centreline of the plume. The lateral extent of melting is much lower at
depth. At the SD, melting is strongest at depth due to lower discharge rates
and less vigorous buoyant ascent of the plume. One should keep in mind that
our modelling approach neglects the change of the front during the period of
interest between two observations of frontal positions (11 days for most
cases). In reality, calving would occur more often during that period,
causing
such large undercuttings, like the modelled ones, to not be possible. This
simplification has consequences for the next step when the particle model
handles the calving of icebergs due to front imbalance.
Observed mean volumetric calving rates and modelled calving
The observed mean volumetric calving rate averaged over the entire calving
front volume of ice, a˙cobs, is the difference between the frontal
velocity, vwobs(ti), and the rate of position change, ∂Lobs/∂t, integrated over the terminus domain. These quantities
and the total modelled ice mass melted by the plume normalised per day (when
an undercutting is prescribed) are given in Table .
Observed and modelled mean volumetric calving rates, a˙c, in
m3d-1, are presented as the integrated tangential (ice flow
direction) ice front velocity a˙c,v (dark grey), the integrated
rate of change of the frontal position, a˙c,L (light grey), and the
total subaqueous melt rate, a˙m (red), if an undercutting is prescribed
for each configuration. The mean distance differences between the modelled
and the observed front positions, L‾, are shown on the right. A
negative value corresponds to underprediction of calving position (modelled
in front of observed).
Observed mean volumetric calving rate,
a˙cobs=a˙c,vobs-a˙c,Lobs, in
105m3d-1, as the difference between the tangential (ice flow
direction) ice velocity at the front and the rate of change of the frontal
position integrated over the terminus domain, and estimated subaqueous melt
rate, a˙m, in 105m3d-1.
t0
t4
t6
t11
a˙c,vobs
2.63
3.68
4.31
2.56
a˙cobs
a˙c,Lobs
-5.30
-4.28
-22.63
-22.43
Total
7.93
7.97
26.94
24.99
SD
0
0.08
0.14
0
a˙m
ND
0
0.86
1.25
0
Total
0
0.94
1.39
0
Ratio a˙m/a˙c
0 %
11.8 %
5.2 %
0 %
To assess the performance of the offline coupling, we evaluate the mean
volumetric calving rate averaged over the entire calving front volume of ice
(see Eq. ), and the mean absolute distance between the
modelled and the observed front, |L|‾. These are presented in
Fig. for each configuration as well as the observed mean
volumetric calving rate. Figure shows the different front
positions after the HiDEM simulation for each configuration of the studied
time. Figure shows strain rates modelled by HiDEM that
resemble an observed crevasse patterns (yellow lines representing crevasses).
Basal velocity, advanced front before calving modelled with
Elmer/Ice, Fielmer, at ti in plain black; observed front after
calving, Fiobs, in dashed black; and calving front modelled with HiDEM,
Fihidem, given the different configurations summarised in
Table for (a) i=0, (b) i=4,
(c) i=6, and (d) i=11. Discharge locations (for
i=4,6) are marked with a red star.
At t0, before the melt started, the front has retreated at a rate of
7.93 ×105 m3 d-1 with a frontal ice flux of
2.63 ×105 m3 d-1, mostly in the middle part with a
calved area of 5.1×104 m2. The HiDEM produces a slightly
higher mean volumetric calving rate, 9.76 ×105 m3 d-1,
with a vertical ice front configuration (red line C(g0,β0,0) in
Fig. a) at a mean distance of 32 m from the observed
front. However, calving is concentrated south of SD in a zone of high ice
velocity and high strain rates as modelled by HiDEM (see
Fig. ).
Strain rates modelled with HiDEM for each configuration. Yellow
colouring shows the crevasse pattern and is denser close to the front where the
difference between each configuration for the four selected iterations can be
observed.
With peak surface runoff, at t4, the observed mean volumetric calving rate
equals 7.97 ×105 m3 d-1, similar to t0 but with
higher ice velocities (3.68 ×105 m3 d-1). Observed
retreat at and north of ND is significant but is not reproduced by the
configuration with a vertical ice front (red line C(g4,β4,0) in
Fig. b). Instead the front is retreating south of SD in the
same fashion as for t0. The mean volumetric calving rate
(6.82 ×105 m3 d-1) is therefore close to the observed
value, but the mean distance between the observed and the modelled front is
close to 60 m (see Fig. ). For the undercutting
configuration (blue line C(g4,β4,u4) in Fig. b), the
mean volumetric calving rate is also overestimated at the same location but
the observed retreat around ND is matched by the HiDEM. The mass removed by
undercutting represents 11.8 % of the total observed mean volumetric
calving rate (see Table ) and is therefore
non-negligible. At the SD, the observed front is advancing (see
Fig. b) and regardless of the applied modelled front
configuration (with or without undercutting), a similar slight retreat is
modelled. In this case, the undercutting has no influence on the calving.
Vertical front configuration at t6 (red line C(g6,β6,0) in
Fig. c), during a period of accelerated glacier flow, results
in slower modelled mean volumetric calving rate (16.26 ×105 m3 d-1) than observed (26.94 ×105 m3 d-1)
and no front position change at both SD and ND, leading to a mean distance to
the observed front close to 60 m. With the undercut configuration
(blue line C(g6,β6,u6) in Fig. b), modelled mean
volumetric calving rate (23.60 ×105 m3 d-1) is similar
to observation and the front positions at discharge locations are reproduced
even though the undercutting only represents 5.2 % of the observed
mean volumetric calving rate. The modelled front is still intensively
breaking up south of SD, but, at that date, it matches the observed retreat.
At the end of the melt season at t11, when subglacial discharge has
ended, the observed front retreats at a rate of 24.99 ×105 m3 d-1 despite a frontal basal friction higher than at the last
studied iteration resulting in an averaged frontal velocity of
2.56 ×105 m3 d-1. But, as shown in Fig. ,
the sliding velocity is higher (lower basal friction, β11) close to
the front than further upglacier. Large calving events occur at both former
discharge locations where the bed elevation is lower than anywhere else. The
calving front modelled by HiDEM (red line C(g11,β11,0) in
Fig. d) manages to reproduce this behaviour but overestimates
the retreat for the region in between, where the pattern of high strain rate
is also denser (see Fig. ).
Two configurations vary the friction coefficient, β, to assess the role
of sliding in the calving process. If the basal friction is set according to
t6 and the geometry to t0 (orange line C(g0,β6,0) in
Fig. a), the mean volumetric calving rate exceeds
observations by more than a factor of 2 (16.40 ×105 m3 d-1), similar to C(g6,β6,0), yet with matching spatial
frontal patterns as C(g0,β0,0) as well as strain rate distribution
with elevated rates close to the calved zones. If the geometry of t6 is
simulated with the basal friction of t0 (orange line C(g6,β0,0) in
Fig. c), it is striking to notice again that the calved zones
are similar to the vertical front configuration at t6 but the mean
volumetric calving rate is similar to the observed one at t0. High strain
rates are less pronounced than with the basal friction of t6 but
concentrated at the same locations.
Discussion
Plume model and undercutting
Our plume model uses a fixed, planar ice front to calculate submarine melt
rates rather than a time-evolving geometry. This assumption is supported by
, who showed that the shape of the submerged ice front
does not have a significant feedback effect on plume dynamics or submarine
melt rates. However, the same study suggests that the total ablation driven
by submarine melting will increase due to the greater surface area available
for melting. To take this effect into account in our undercutting model,
submarine melt rates are horizontally projected onto the undercut front
modelled at the previous iteration.
By using ambient temperature and salinity profiles that do not vary in time,
we neglect the inter- and intra-annual variability in Kongsfjorden. This
variability can affect the calculated melt rate in two ways: (i) the
three-equation melt parameterisation explicitly includes the temperature and
salinity at the ice face, and (ii) the ambient stratification affects the
vertical velocity and neutral buoyancy height of the plume. The direct effect
of changes in temperature and salinity on the melt equations are well tested.
Past studies using uniform ambient temperature and salinity conditions have
found a linear relationship between increases in ambient fjord temperatures
and melt rates, with the slope of the relationship dependent upon the
discharge volume . Salinity, on
the other hand, has been shown to have a negligible effect on melt rates
. However, with a non-uniform ambient temperature and
salinity, the effects of changes in the stratification on the plume vertical
velocity and neutral buoyancy are much more complex. The stratification in
Kongsfjorden is a multi-layer system, with little or no direct relationship
between changes in different layers . Therefore, testing
cases by uniformly increasing or decreasing the salinity would not be
informative for understanding the true effects of inter- and intra-annual
variability. The high computational expense of the plume model used here
means that it is not yet feasible to run the model on the timescales
necessary to understand this variability, nor to run sufficient
representative profiles to provide a useful understanding of the response.
Previous work has suggested that intra-annual changes in the ambient
stratification are small enough that plumes are relatively insensitive to
these changes and that plume models forced with
variations in runoff and a constant ambient stratification can qualitatively
reproduce observations . For these reasons, we highlight
this as a limitation of the current implementation and suggest that this
should be addressed in future investigations of plume behaviour. A model
based upon one-dimensional plume theory e.g. would be less computationally expensive and may
allow some of these limitations to be addressed. However, such a model would
not capture the strong surface currents driven by the plume which are
important for the terminus morphology studied here.
For ND (Fig. b and d), the undercutting is in line with the
observed front to a certain extent, particularly for t4. However, for SD,
apart from t3, no apparent correlation between modelled undercutting and
observed front location seems to exist. However, Fig. shows
that modelling calving with undercutting at SD and ND for t4 and t6
gives a good fit to observation. The difference in agreement with the
observed front position and the modelled calving could possibly be explained
by the uncertainty in discharge or the different character of the plume at
high and low discharge. The low dependence of calving front position on
modelled undercutting in situations of low discharge seems to have no major
influence on the performance of the calving model. At Kronebreen, the high
discharge relative to the shallow depth of the terminus drives strong gravity
currents at the surface as water is rapidly exported horizontally away from
the plume. The melt rates driven by these gravity currents are significant, as
shown in Fig. , and in some cases dominate over the melt rates
driven by the plume at depth. The difference between low and high discharges
is therefore slightly counterintuitive. At low discharges, when maximum melt
rates occur at depth, the terminus is more undercut but in a narrower area;
meanwhile, at higher discharges, strong undercutting occurs but over a much
wider area of the terminus. This suggests that calving behaviour may be very
different in these two situations.
Calving model
Because the imposed undercuttings are the product of melt during the whole
interval between observations, the model results should be treated with
caution. compared HiDEM calving for specified undercuttings
of different sizes and showed that calving magnitude increases with
undercutting size. For small undercuttings, calving simply removes part of
the overhang, but for large undercuttings calving removes all of the overhang
plus additional ice. The mechanisms are different in each case: low-magnitude
calving for small undercuttings occurs through collapse of part of the
unsupported overhang, whereas high-magnitude calving for large undercuttings
involves forward rotation of the whole front around a pivot point located at
the base of the undercut cliff. The long time-step intervals (11 or 18 days)
between the starting geometry and the HiDEM simulation in the present study
might therefore bias the results towards higher calving events. Testing this
possibility is beyond the scope of the present paper, but it remains an
important goal for future research. Despite this caveat, our results compare
well with observations and yield valuable insights into the calving process.
Firstly, the HiDEM results show that undercutting associated with meltwater
plumes is an essential factor for calving during the melt season (t4 and
t6). Surface melt leads to the formation of a subglacial drainage system
that ultimately releases the water into the ocean from discharge points at
the front of the glacier. Simulations without frontal undercutting at these
subglacial discharge locations do not agree well with observed frontal
positions and mean volumetric calving rates. In contrast, simulations with
frontal undercutting reproduce the retreat reasonably well at these
locations, particularly where the discharge is high such as at ND. The
largest discrepancy between modelled and observed calving is in the region
south of SD at t4. Here, the model predicts calving of a large block,
whereas the observed front underwent little change. This largely reflects the
rules used for calving in HiDEM: any block that is completely detached from
the main ice body is considered as calved, even if only separated by a narrow
crack from the rest of the glacier and still sitting at its original
position. This is the case for the large “calved” region south of SD at
t4, where the block may have been completely detached but remained
grounded and in situ. If this were to occur in nature, it would not register
as a calving event on satellite images. The discrepancy between model results
and observations at this locality therefore may be more apparent than real.
Secondly, the model results replicate the observed high calving rates at
t11, after the end of the melt season, when there is no undercutting. At
this time, the observed mean volumetric calving rate is
24.99×105 m3d-1, which compares well with the HiDEM
rate of 28.50×105 m3d-1. These values are much higher
than those at the start of the melt season, when there is also zero
undercutting. This contrast can be attributed to the high strain rates in the
vicinity of the ice front at t11, which would encourage opening of
tensile fractures (Fig. ). In turn, the high strain rates
result from low basal friction (Fig. d), likely reflecting
stored water at the glacier bed after the end of the melt season. It is
possible that geometric factors also play a role in the high calving rates at
t11, because the mean ice front height is greater at that time than at
t0, reflecting sustained calving retreat during the summer months, which
would have increased longitudinal stress gradients at the front
. This interpretation is supported by experiments C(g0,β6,0) and C(g6,β0,0), in which the basal friction values are
transposed for non-undercut ice geometries at t0 and t6. Imposing low
friction (β6) at t0 produces mean volumetric calving rates similar
to (but smaller than) those observed at t6, whereas imposing high basal
friction (β0) at t6 produces low volumetric calving rates similar
to those observed at t0. The influence of basal friction on calving rates
is consistent with the results of , who found that a
strong correlation exists between frontal ablation rates and ice velocity at
Kronebreen when velocity is high. Low basal friction is associated with both
high near-terminus strain rates and high velocities, facilitating fracturing
and high rates of ice delivery to the front. Our experiments do not include
varying fjord water temperature, so we cannot corroborate the strong
correlation between frontal ablation and fjord temperature observed by
. However, our results are consistent with their finding
that melt undercutting is a primary control on calving rates, with an
additional role played by ice dynamics at times of high velocity.
Conclusions
In this study, we use the abilities of different models to represent
different glacier processes at Kronebreen, Svalbard, with a focus on calving
during the melt season of 2013. Observations of surface velocity, front
position, topography, bathymetry and ocean properties were used to provide
data for model inputs and validation.
The long-term fluid-like behaviour of ice is best represented using the
continuum ice flow model Elmer/Ice, which computes basal velocities by
inverting observed surface velocities and evolves the geometry, including the
front position. During the melt season, a subglacial hydrology system is
created and the water is eventually evacuated at the front of the glacier. We
used a simple hydrology model based on surface runoff directly transmitted to
the bed and routing the basal water along the deepest gradient of the
hydraulic potential. Two subglacial discharge locations have been identified
by this approach: the northern one evacuates water with a high rate
(∼ 10–100 m3s-1) and the southern one with a low rate
(∼ 1–3 m3s-1). This fresh water is subsequently mixed with
ocean water. Rising meltwater plumes entrain warm fjord water and melt the
subaqueous ice creating undercuttings at the subglacial discharge location.
We modelled the plume with a simplified 2-D geometry using a high-resolution
plume model based upon the fluid dynamics code Fluidity adapted to the front
height and the ocean properties of Kronebreen. Melt rates depend on the
discharge rate and the shape of the plume differs greatly with its magnitude.
Higher discharges tend to let the plume rise to the surface close to which
melt rates are the highest, while low discharges concentrate the melt at lower
elevations. The melt rates are then projected to the actual frontal geometry
taking into account the subaqueous ice-front shape of the former time step. It
is interesting to note that modelled undercuttings for high subglacial
discharges are spatially close to the observed calving front, whereas such a
correspondence is not evident for small discharges. The elastic–brittle
behaviour of the ice, such as crevasse formation and calving processes, is
modelled using a discrete particle model, HiDEM. Two factors impacting
glacier calving are studied here using HiDEM: (i) melt undercutting associated
with buoyant plumes and (ii) basal friction, which influences strain rates
and velocity near the terminus. The performance of the calving model is
evaluated quantitatively by comparing observed and modelled mean volumetric:
calving rate and qualitatively by comparing calved regions. Results show that
modelled calving rates are smaller than observed values during the melt
season in the absence of melt undercutting, and that there is a closer match
with observations if undercutting is included. Additionally, there is good
agreement between modelled and observed calving before (t0) and after
(t11) the melt season, when there is no undercutting. Both modelled and
observed calving rates are much greater after the melt season than before,
which we attribute to lower basal friction and higher strain rates in the
near-terminus region at t11. The influence of basal friction on calving
rates is corroborated by model experiments that transposed early- and
late-season friction values, which had a large effect on modelled calving.
These results are consistent with the conclusions of ,
that melt undercutting is the primary control on calving at Kronebreen at the
seasonal scale, whereas dynamic factors are important at times of high
velocity (i.e. low basal friction).
In this paper, we have shown that offline coupling of ice-flow, surface melt,
basal drainage, plume-melting, and ice-fracture models can provide a good
match to observations and yield improved understanding of the controls on
calving processes. Full model coupling, including forward modelling of ice
flow using a physical sliding law, would allow the scope of this work to be
extended farther, including prediction of glacier response to atmospheric and
oceanic forcing.