Introduction
According to recent studies ,
most of the total volume of ice (∼ 60 %) calved from
the Antarctic continent is transported into the Southern Ocean by
large icebergs (i.e. > 18 km in length). However, their basal melting,
that is of the order of 320 km3 yr-1, accounts for
less than 20 % of their mass loss, and the majority of ice loss (1500 km3 r-1 ∼ 80 %)
is achieved through breaking into smaller icebergs .
Large icebergs actually act as a reservoir to transport ice away from
the Antarctic coastline into the ocean interior, while fragmentation
can be viewed as a diffusive process. It generates plumes of small
icebergs that melt far more efficiently than larger ones that have a
geographical distribution that constrains the input into the ocean.
Global ocean models that include iceberg components
show that basal ice-shelf and iceberg melting have different effects
on the ocean circulation. Numerical model runs with and without icebergs
show that the inclusion of icebergs in a fully coupled general circulation
model (GCM) results in significant changes in the modelled ocean circulation
and sea-ice conditions around Antarctica .
The transport of ice away from the coast by icebergs and the associated
freshwater flux cause these changes . Although
the results of these modelling studies are not always in agreement
in terms of ocean circulation or sea ice extent they all highlight
the important role that icebergs play in the climate system, and they
also show that models that do not include an iceberg component are
effectively introducing systematic biases .
However, despite these modelling efforts, the current generation of
iceberg models are not yet able to represent the full range of iceberg
sizes observed in nature from growlers (≤ 10 m) to
“giant” tabular icebergs
(≥ 10 km).
The iceberg size distribution has also strong impact on both circulation
and sea ice as shown by . Furthermore, all current
iceberg models fail in accounting for the size transfer of ice induced
by fragmentation, as in these models small icebergs cannot stem from
the breaking of bigger ones.
The two main decay processes of icebergs, melting and fragmentation,
are still quite poorly documented and not fully represented in numerical
models. Although iceberg melting has been widely studied
,
very few validations of melting laws have been published ,
especially for large icebergs. Large uncertainties still remain for
the melting laws to be used in numerical models.
The calving of icebergs from glaciers and ice shelves has been quite
well studied (e.g. )
and empirical calving laws have been proposed .
However, very few studies have been dedicated to the breaking of icebergs. Analysing
the decay of Greenland icebergs, proposed three
distinct fragmentation mechanisms. Firstly, flexural breakups by swell
induced vibrations, in the frequency range of the iceberg bobbing on
water, that could cause fatigue and fracture at weak spots .
Secondly, two mechanisms resulting from wave erosion at the waterline,
calving of ice overhangs and buoyant footloose mechanism .
, using satellite images, ICESat altimeter and
field measurements analysed the evolution of two Antarctic icebergs
and identified three styles of calving during the drift: “rift calving”,
which corresponds to the calving of large daughter icebergs by fracturing
along preexisting flaws, “edge wasting”, the calving of numerous
small narrow icebergs and “rapid disintegration”, which is characterized
by the rapid calving of numerous icebergs.
The pieces calved from icebergs drift away from their parent under
the action of wind and ocean currents as a function of size, shape,
and draft . This dispersion can create large plumes
of icebergs that can represent a significant contribution to the freshwater
flux over vast oceanic regions where no large icebergs are observed
. The size distribution of the calved
pieces is needed to analyse and understand the transfer of ice between
the different iceberg scales and thus to estimate the freshwater flux.
It is also important for modelling purposes. ,
using aerial images and in situ measurements, estimated the size distribution
of small bergy bits (< 20 m in length) calved from deteriorating
Greenland icebergs. However, until now, no study has been published
on the size distribution of icebergs calved from large Southern Ocean
icebergs.
Recent progress in satellite altimeter data analysis allows us to
estimate the small (< 3 km in length) iceberg distribution and volume
as well as the freeboard elevation profile and volume of large icebergs
. The location, area, and volume of small
icebergs from 1992 to 2018 is
contained in a database distributed
by CERSAT, as well as monthly fields of probability of presence, mean
area and volume of ice . It is now possible
to estimate the thickness variations and thus the melting of large
icebergs. A crude estimate of the large iceberg area is also available
from the National Ice Center but it is not precise enough to allow
analysis of the area lost by fragmentation. A more precise area analysis
can be conducted by analysing satellite images such as those for the
Moderate Resolution Imaging Spectroradiometer (MODIS) onboard the
Aqua and Terra satellites .
Two large icebergs, B17a and C19a, which have drifted for more than
one year in open water (see Fig. 1) away from other large icebergs
and which have been very well sampled by altimeters and MODIS, have
been selected to study the melting and fragmentation of large Southern
Ocean tabular icebergs. Their freeboard evolution, and thus thickness,
is estimated from satellite altimeter data, while their area and shape
have been estimated from the analysis of MODIS images. The icebergs
area and thickness evolution is then used to test the validity of
the melting models used in iceberg numerical modelling and to analyse
the fragmentation process. The two icebergs were also chosen because
they have very different characteristics. While C19a was one of the
largest iceberg on record (> 1000 km2) and drifted for more
than 2 years in the South Pacific, B17a was relatively small (200 km2)
and drifted in the Weddell Sea. The large plumes of small icebergs
generated by the decay of both large icebergs can be detected by altimeters
and MODIS images. The ALTIBERG database and selected MODIS images
can be used to analyse the size distribution of fragments.
The present paper is organized as follows. Section 2 describes the
data used in the study, including the environmental parameters (such
as ocean temperature, current speed, etc.) necessary to estimate melting
and fragmentation. Section 3 presents the evolution of the two selected
icebergs. In Sect. 4, the two melting laws widely used in the literature,
forced convection and thermal turbulence exchange, are confronted
with the observed melting of B17a and C19a. The final section analyses
the fragmentation process and proposes a fragmentation law. It also
investigates the size distribution of pieces calved from large icebergs.
Trajectories of B17a (a) and C19a (b) icebergs. The black circle locates
the B17a grounding site. The colour scale represents the time along
the trajectory.
Data
Iceberg data
The National Ice Center (NIC) Southern Hemisphere Iceberg Database
contains the position and size (length and width), estimated by analysis
of visible or SAR images of icebergs larger than 10 nautical miles
(19 km) along at least one axis; it is updated weekly. Every iceberg
is tracked, and when imagery is available, information is updated
and posted. The Brigham Young University (BYU) Center for Remote Sensing
maintains an Antarctic Iceberg Tracking Database for icebergs larger
than 6 km in length . Using six different satellite
scatterometer instruments, they produced an iceberg tracking database
that includes icebergs identified in enhanced resolution scatterometer
backscatter. The initial position for each iceberg is located based
on a position reported by the NIC or by the sighting of a moving iceberg
in a time series of scatterometer images.
In 2007, demonstrated that any target emerging
from the sea surface (such as an iceberg) can produce a detectable
signature in high-resolution altimeter wave forms. Their method enables us to detect
icebergs in open ocean only, and to estimate their area. Due to
constraints on the method, only icebergs between 0.1 and
∼ 9 km2 can be detected. Nine satellite altimetry
missions have been processed to produce a 1992–2018 database of
small iceberg locations, area, volume, and mean backscatter .
The monthly mean probability of presence, area and volume of ice over
a regular polar (100 × 100 km2) or geographical (1∘ × 2∘)
grid are also available and are distributed on the CERSAT website.
Altimetry can also be used to measure the freeboard elevation profile
of large icebergs . Combining
iceberg tracks from NIC and the archives of three Ku band altimeters,
Jason-1, Jason-2, and Envisat, created a database
of daily position, freeboard profile, length, width, area, and volume
of all the NIC and BYU large icebergs covering the 2002–2012 period. For
example, B17a was sampled by 152 altimeter passes during its drift
and C19a by 258 passes (see Fig. ).
Sampling of B17a (a) and C19a (b) icebergs by MODIS (beige stars)
and altimeters (blue circles).
Visible images
The weekly estimates of iceberg lengths and widths provided by NIC
are manually estimated from satellite images and they are not accurate
enough to precisely compute the iceberg area and its evolution. A
careful re-analysis of the MODIS imagery from the Aqua and Terra satellites
was thus conducted to precisely estimate the C19a and B17a area until
their final detectable collapse. The images have been systematically
collocated with the two icebergs using the NIC/BYU track data. It
should be noted that in some areas of high iceberg concentration,
especially when B17a reaches the “iceberg alley”, NIC and BYU regularly
mistakenly followed another iceberg, or lost its track when it became
quite small. Here, more than 1500 images were collocated and selected.
The level 1B calibrated radiances from the two higher resolution (250 m)
channels (visible channels 1 and 2 at 645 and 860 nm frequencies, respectively)
were used to estimate the iceberg's characteristics. For each image
with good cloud clover and light conditions, a supervised shape analysis
was performed. Firstly, a threshold depending on the image light conditions
is estimated and used to compute a binary image. The connected components
of the binary image are then determined using standard Matlab© image
processing tools and finally the iceberg's properties, centroid
position, major and minor axis lengths, and area are estimated. On
a number of occasions, the iceberg's surface was obscured by clouds,
but visual estimation was possible because the image contrast was
sufficient to discern edges through clouds. For these instances, the
iceberg's edge and shape were manually estimated. The final analysis
is based on 286 valid images for B17a, and 503 for C19a. The locations
of the MODIS images for B17a and C19a are given in Fig.
while four examples of iceberg area estimates are given in Fig. .
The comparison of area for consecutive images shows
that the area precision is around 2–3 %.
Environmental data
Several environmental parameters along the icebergs trajectories are
also used in this study. Due to the lack of a better alternative,
the sea surface temperature (SST) is used as a proxy for the water
temperature. The difference between the SST and the temperature at
the base of the iceberg will introduce an error in the melt rate computation,
as shown by . Using results from an Ocean General
Circulation Model, they also compared the mean SST and the average
temperature over the first 150 m from the surface showing that the
mean difference is less than 0.5 ∘C for most of the Southern
Ocean. The level-4 satellite analysis product ODYSSEA, distributed
by the Group for High-Resolution Sea Surface Temperature (GHRSST)
has been used. It is generated by merging infrared and microwave sensors
and using optimal interpolation to produce daily cloud-free SST fields
at 10 km resolution over the globe. The sea ice concentration data
are from the CERSAT level-3 daily concentration product, available
on a 12.5 km polar stereographic grid from the SSM/I radiometer observations.
The wave height and wave peak frequencies come from the global Wave
Watch3 hindcast products from the IOWAGA project
(http://wwz.ifremer.fr/iowaga/, last access: September 2017).
The AVISO Maps of Absolute Dynamic Topography & absolute geostrophic
velocities (MADT) provides a daily multi-mission absolute geostrophic
current on a 0.25∘ regular grid that is used to estimate
the current velocities at the iceberg locations.
Example of B17a (a, b) and C19a (c, d) area estimate using
MODIS
images. The blue lines represent the iceberg perimeter and the red and
green crosses represent the NIC and MODIS iceberg's positions, respectively.
Melting and fragmentation of B17a and C19a
B17a
Iceberg B17a originates from the breaking of giant tabular B17 near
Cape Hudson in 2002. It then drifted for 10 years along the continental
slope within the “coastal current”, until it reached the Weddell
Sea in summer 2012 (see Fig. 1a). It travelled within sea ice at
a speed ranging from 2 to 12 cm s-1, coherent with previous
observational studies . It crossed the Weddell
Sea while drifting within sea ice and reached the open water in April 2014.
It was then caught in the western branch of the Weddell Gyre
and drifted north in the Scotia Sea until it grounded, in October 2014,
near South Georgia, a common grounding spot for icebergs. It
remained there for almost 6 months until it finally left its trap
in March 2015 and drifted back northward until its final demise in
early June 2015. B17a was a “medium size” big iceberg, with primary
dimensions of 35 × 14 km2 and an estimated freeboard of
52 m, resulting in an original volume of 113 km3 and a corresponding
mass of ∼ 103 Gt. Before 2014, B17a freeboard and area
remained almost constant while it drifted within sea ice. After March
2014, B17a started to drift in open water and to melt and break. During
its drift in open water, from March 2014 to June 2015, B17a was sampled
by 200 MODIS images and 41 altimeter passes. Figure a
presents the satellite freeboard and area measurements as well as
the daily interpolated values. The standard deviation of freeboard
estimate computed from the freeboard elevation profiles is ±3 m.
The standard deviation of the iceberg area has been estimated by
analysing the area difference between images taken the same day. It
is of the order of 3–4 %. During this drift in the Weddell Sea, it
experienced different basal melting regimes: firstly, when it left
the peninsula slope current, with negative SSTs and low drift speeds
(see Fig. b and d), it was subject to an average melt
rate of 5.7 m month-1, then drifted more rapidly within the
Scotia Sea and experienced a mean thickness decrease of 15 m month-1,
and finally it melted at a rate close to 20 m month-1 as it
accelerated its drift before its grounding. As for fragmentation,
the area loss was limited (40 km2 in 250 days, i.e. less than
10 %) but then accelerated as B17a became trapped (80 km2
in 70 days). The area loss slowed down for the second half of the
grounding, only to increase dramatically once B17a was released and
before it collapsed a few days later. This could be related to an
embrittlement of the iceberg structure, potentially under the influence
of unbalanced buoyancy forces while grounded .
The cumulative total volume loss, basal melting, breaking are presented
in Fig. e. These terms are computed from the mean thickness
and area as follows: the basal melting volume loss M at day i
is the sum of the products of iceberg surface, S (in m2),
by the daily variation of thickness, dT
M(i)=∑k=1iS(k)dT(k),
and the breaking loss B (in m3) is the sum of the products
of thickness, T, by the daily variation of surface, dS
B(i)=∑k=1idS(k)T(k).
For large icebergs, the sidewall erosion and melting, which is of the
order of some metres per day, can be considered negligible compared
to breaking. As B17a started to drift in open water, its mass varied
slowly at first, mainly through melting. Between January 2014 and
March 2015, basal melting accounted for more than 60 % of the total
volume loss, whereas fragmentation was responsible for 30 % of the
loss. However, after November 2014 breaking became dominant as the
iceberg started to break up more rapidly.
(a) B17a area (in km2) and freeboard (in m). The black and blue
lines represent the interpolated daily area and freeboard and the
black circles and blue crosses the MODIS area and altimeter freeboard
estimates. (b) ODYSSEA sea surface temperature (in ∘C).
(c) Significant wave height in m (blue line) and peak frequency in
Hz (green line). (d) AVISO geostrophic current (black arrows),
current velocity (blue line), and iceberg velocity (dashed black line).
(e) Total volume loss (dotted line), volume loss by melting (dashed
line), and by fragmentation (solid line).
C19a
Our second iceberg of interest is the giant C19a which was one of
the fragments resulting from the splitting of C19, the second largest
tabular iceberg on record. C19a was born offshore of Cape Adare (170∘ E)
in 2003 and was originally oblong and narrow, around 165 km long
and 32 km wide with an estimated freeboard of ∼ 40 m, i.e. a
volume of about 1000 km3 and a mass of 900 Gt. It
drifted mainly in a northeasterly direction for almost 4 years, most
of the time in sea ice, until it first entered the open ocean in summer 2005
(see Fig. ). It was temporarily re-trapped by the floes
in winter 2006 and eventually left the ice coverage permanently in
late spring 2007. It then drifted within the Antarctic Circumpolar
Current and eventually close to the Polar Front and its warm waters
until its final demise in April 2009 in the Bellingshausen Sea. Before
November 2007, C19a experienced very little change except a very mild
melting (not presented in the figure). Its volume was 880 km3
(∼ 790 Gt) in December 2007 when it finally entered
the open sea. During its final drift, from December 2007 to March 2009,
C19a was sampled by 317 MODIS images and 69 altimeter passes
(see Fig. b). The C19a area and freeboard are presented
in Fig. as well as SST, sea state, and volume loss. While
the volume loss was mainly due to melting before this date, breaking
dominated afterwards. Basal melting only explains 25 % of the total
volume decrease (see Fig. e). B17 thickness loss was
almost 5 times faster than that of C19, the latter experiencing mean
basal melt rates ranging from 1 to 3 m month-1
in most of its drift (and as much as 13 m month-1 in its last
month, which was characterized by very high water temperatures). As
for fragmentation, its main volume loss mechanism (75 %), its area
loss was first mild while it progressed in colder waters (around 2.6 km2 day-1),
and started to increase as soon as it entered positive temperature
waters, with an average loss of 9.5 km2 day-1 and with
dramatic shrinkages of 340 km and 370 km2, lost in 10 days that correspond to large fragmentation events.
(a) C19a area (in km2) and freeboard (in m). The black and blue
lines represent the interpolated daily area and freeboard, and the
black circles and blue crosses the MODIS area and altimeter freeboard
estimates. (b) ODYSSEA sea surface temperature (in ∘C).
(c) Significant wave height in m (blue line) and peak frequency in
Hz (green line). (d) AVISO geostrophic current (black arrows),
current velocity (blue line) and iceberg velocity (dashed black line).
(e) Total volume loss (dotted line), volume loss by melting (dashed
line), and by fragmentation (solid line).
Melting models
Apart from fragmentation, the basal melting of icebergs accounts for
the largest part of the total mass loss .
Although firn densification (see Appendix for an estimate
of the associated freeboard change) and surface melting can also contribute,
it is the main cause of thickness decrease. It can mainly be attributed
to the turbulent heat transfer arising from the difference of speed
between the iceberg and the surrounding water. Two main approaches
have been used to compute the melting rate and to model the evolution
of iceberg and the freshwater flux (see for example ).
The first one is based on the forced convection formulation proposed
by , while the second one uses the thermodynamic
formulation of and the turbulent exchange velocity
at the ice-ocean boundary. The first model has been exclusively used
to compute iceberg basal melt rate while the second model has been
primarily developed and used to estimate ice shelf melting. The B17a
and C19a data sets allow us to confront these two formulations with
melting measurements for two icebergs of different shapes and sizes
and under different environmental conditions and to test their validity
for large icebergs.
Forced convection of Weeks and Campbell
The forced convection approach of is
based on the fluid mechanics formulation of the heat-transfer coefficient
for a fully turbulent flow of fluid over a flat plate. The basal convective
melt rate Mb is a function of both temperature and velocity
differences between the iceberg and the ocean. It is expressed (in
m day-1) as follows :
Mb=C|Vw-Vi|0.8Tw-TiL0.2,
with Vw being the current speed (at the base
of the iceberg), Vi the iceberg speed, Ti
and Tw the iceberg and water temperature, L the iceberg's
length (longer axis) and C=0.58 K-1 m0.4 s0.8 day-1.
This expression has been widely used in numerical models .
As water temperature at keel depth is not available, the sea surface
temperature (SST) is used as a proxy. The SST for each iceberg is
presented in Figs. b and b. The first unknown
quantity in Eq. (), the iceberg's temperature Ti, can be
at the time of calving as low as -20 ∘C .
Icebergs can sometimes drift for several years. During its travel
the iceberg's surface temperature will depend on the ablation rate.
When ablation is limited, i.e. in cold waters, the ice can theoretically
warm up to 0 ∘C, while in warmer waters the rapid disappearance
of the outer layers tends to leave colder ice near the surface. The
surface ice temperature could thus theoretically vary from -20
to 0 ∘C but is commonly taken at -4 ∘C
.
The mean daily iceberg speed can be easily estimated from the iceberg
track. Numerical ocean circulation models are not precise enough to
provide realistic current speed in this region. The comparison of
iceberg velocities and AVISO geostrophic currents presented in Figs. d
and d shows that the iceberg velocity is
sometimes significantly larger than the AVISO velocities. They are
thus not reliable enough to compute the melt rate. Vw is thus
treated as unknown.
The basal melt is computed using Eq. () for Vw
from 0 to 3 m s-1 by 0.01 steps and Ti from -20 to 2 ∘C
by 0.1 ∘C steps. The positive temperatures are used to test
the model's convergence. The uncertainties in the different parameters
and measurements are too large for a direct comparison of the modelled
and measured daily melt rate. However, it is possible to test the
model validity by comparing the bulk melting rate, i.e. the modelled
and measured cumulative loss of thickness, Σi=1nMb(ti).
As current velocities and iceberg temperatures are not constant during
the iceberg's drift, the modelled thickness loss is fitted to the
measured loss for each time step ti over a ±20-day period
by selecting the Vw(ti) and Ti(ti) that minimize
the distance between model and observations. When no SST is available,
i.e. when the iceberg is within sea ice for a short period, Tw
is fixed to the sea water freezing temperature. The model allows us
to reproduce the thickness variations extremely well, with correlations
larger than 0.999 for both B17a and C19a (see Figs. a
and a) and mean differences of thickness loss of 3.1 and
0.5 m, respectively, and maximum differences less than 8 and 1.5 m.
However, the current velocity inferred from the model, presented in
Figs. b and b, reaches very high and unrealistic
values (>2 m s-1). Compared to the altimeter geostrophic
currents from AVISO, the current speed can be overestimated by more
than a factor of 10.
The second model parameter Ti (see Figs. c and
c) varies between -20 and -0.6 ∘C
with a -10.9 ± 7.1 ∘C mean for B17a. For C19a, it is between
-9 and 1 ∘C with a -10.6 ± 5.8 ∘C
mean, although the model sometimes fails to converge to realistic
iceberg temperature, i.e. for Ti<0 ∘C. It happens when
the measured melting is weak and SST is positive (for example from
January to May 2007, Figs. c and b). The
model can reproduce this inhibition by decreasing the water and ice temperature
difference up to zero, resulting in an artificial increase of the
iceberg temperature to positive values. For B17a, the model always
converges, and the lower temperatures (-20 ∘C) are observed
during extremely rapid melting period or during the grounding period.
This could reflect the decrease of ice surface temperature during
rapid ablation events or an underestimation of the melt rate.
Thickness loss (in m) for B17a (a). Measured thickness loss (black
line); modelled loss using forced convection (dashed blue line) and
turbulent exchange (solid blue line). (b) Iceberg velocity (dotted
black line). Modelled velocity using forced convection (solid blue
line) and using turbulent exchange (dotted blue line). AVISO Geostrophic
current velocity (solid black line). (c) Modelled iceberg temperature
using forced convection (dashed line) and using thermal exchange (solid
line).
Thickness loss (in m) for C19a (a). Measured thickness loss (black
line); modelled loss using forced convection (dashed blue line) and
turbulent exchange (solid blue line). (b) Iceberg velocity (dotted
black line). Modelled velocity using forced convection (solid blue
line) and using turbulent exchange (dotted blue line). AVISO Geostrophic
current velocity (solid black line). (c) Modelled iceberg temperature
using forced convection (dashed line) and using thermal exchange (solid
line).
Thermal turbulent exchange of Hellmer and Olbers
The second melt rate formulation is based on thermodynamics, and on
heat and mass conservation equations. It assumes heat balance at the
iceberg–water interface and was originally formulated for estimating
ice shelf melting . The turbulent
heat exchange is thus consumed by melting and the conductive heat
flow through the ice:
ρwCpwγT(Tb-Tw)=ρiLMb-ρiCpiΔTMb.
Thus,
Mb=ρwCwγTρiTb-TwLH-CpiΔT,
where Mb is the melt rate (in m s-1), LH=3.34×105 J kg-1
is the fusion latent heat, and Cpw=4180 J kg-1 K-1
and Cpi=2000 J kg-1 K-1 are the heat
capacity of seawater and ice, respectively. Tb=-0.0057 Sw+0.0939–7.64 × 10-4Pw
is the freezing temperature at the base of the iceberg, Sw (in
g kg-1) and Pw (in 104 Pa) are the
salinity (here fixed at the averaged value of 35 g kg-1)
and pressure at the bottom of the iceberg, ΔT=Ti-Tb
represents the temperature gradient within the ice at the iceberg
base . γT is the thermal turbulent
velocity that can be expressed as follows :
γT=u*2.12log(u*lν-1)+12.5Pr2/3-9,
where Pr=13.1 is the molecular Prandtl number of sea water,
l=1 m the mixing length scale, ν=1.83×10-6 m2 s-1
is the water viscosity, and u* the friction velocity. The latter,
which is defined in terms of the shear stress at the ice-ocean boundary,
depends on a dimensionless drag coefficient, or momentum exchange
coefficient, CD=0.0015 and the current velocity in the boundary
layer, u≃Vw-Vi, by u*2=CDu2.
modelled the evolution of a large iceberg (A38b)
using this formulation for melting. They calibrated their model using
IceSat elevation measurements and found γT ranging from
0.4×10-4 to 1.8×10-4 m s-1
close to the 1×10-4 m s-1 proposed by .
, who estimated the Southern Ocean freshwater flux
by combining the NIC iceberg data base and a model of iceberg thermodynamics
also based on this formulation, considered a unique and much larger
γT of 6×10-4 m s-1.
The basal melt is thus computed using Eq. () for γT
from 0.1×10-5 to 10×10-4 m s-1
by 0.1×10-5 steps and Ti from -20 to 2 ∘C
by 0.1 ∘C steps. As for forced convection, the model is fitted
for each time step over a ±20 day period to estimate γT(ti)
and Ti(ti). The current speed is then estimated using Eq. ().
This model also reproduces the thickness variations extremely well, with a correlation better than 0.999 for both B17a and C19a (see Figs. , a).
The mean differences of thickness are
3.7 and 0.3 m for B17a and C19a respectively, and the maximum differences
are 14.1 and 0.8 m. The modelled current velocity (Figs. b
and b) is always smaller than the forced convection velocity
except for B17a during the three months (September to November 2014)
of very rapid drift and melting. Although it is still significantly
larger than the AVISO velocities, especially for B17a, the values
are more compatible with the ocean dynamics in the region .
For B17a, γT varies from 0.41×10-4 to 10×10-4 m s-1
with a (2.9±2.8)×10-4 m s-1 mean. If the
period of very rapid melting (September to November 2014), during
which γT increases up to 10×10-4, is not considered,
γT varies only up to 2.5×10-4 m s-1
with a (1.6±0.92)×10-4 m s-1 mean. These
values are comparable to those presented by
for A38b whose size was similar to that of B17a. For C19a, γT
has significantly lower values ranging from 0.3×10-5 to
1.6×10-4 m s-1 with (0.34±0.37)×10-4 m s-1 mean.
These values, which correspond to the lower γT found by
, might reflect a different turbulent behaviour
for very large icebergs that can more significantly modify their environment,
especially the ocean circulation .
The mean iceberg temperature is -10.8±5.0 ∘C for B17a
and -10.6±5.8 ∘C for C19a. It oscillates quite rapidly
and certainly more erratically than in reality.
Discussion
The two classical parameterizations of iceberg basal melting have
been tested against observations. Both models can reproduce the
iceberg thickness variations well by fitting the iceberg temperature and
the current velocity. Nevertheless, the two melting strategies fail
on several occasions in reproducing the observed melt rates, namely
when thickness variations are important.For instance, the forced convection
approach of requires very large current
velocities and/or very high iceberg–ocean temperature difference to
reproduce the measured melt rate. The large overestimation of current
speed and temperature differences indicates that this model tends
to underestimate the melt rate. If realistic velocities and temperatures
were used, the melt rate could be underestimated by a factor of 2
to 4. This formulation is mainly a bulk parameterization based on
heat transfer over a flat plate. It was proposed in the 1970s to analyse
the melting of small icebergs and relies on typical mean values of
water viscosity, Prandtl number, thermal conductivity, and ice density.
These approximations might not be valid, especially for very large
tabular icebergs, and can not take into account the impact of the iceberg
on its environment. The velocity and temperature differences for the
second formulation usually assume values that are more compatible with
the ocean flow properties in the region. This parameterization was
developed for numerical models and represents the conservation of
heat at the iceberg surface. It depends on both the ocean–ice and
the ice surface–ice interior temperature gradients, although the ocean–ice
gradient is dominant. Compared to the forced convection, for similar
temperature and velocity gradients, the Hellmer and Olbers formulation
leads to melt rates that are 2 to 4 times more efficient. Thus, although
the current velocity can reach quite high values, this melt rate formulation
is certainly better suited to reproduce the bulk melting of icebergs
than forced convection. The comparison of the Mb values computed
using the two formulations for identical environmental parameters
which shows a factor 5 difference between the forced convection and
thermal turbulence for B17a (L=35 km) and 6–8 for C19a (L=150 km),
confirms the underestimation of the melting by the forced convection
approach.
As a consequence, our study brings out some of the limitations of
the classical modelling strategies of iceberg basal melting. To make
sure the second strategy is able to reproduce realistic melt rates,
especially for large icebergs, we need to extend our study to more
iceberg cases, namely to be able to have a broader view on the variability
range of the γT parameter.
Fragmentation
As said earlier, fragmentation is the least known and documented decay
mechanism of icebergs. It has been suggested that swell induced vibrations
in the frequency range of the iceberg bobbing on water could cause
fatigue and fracture at weak spots . Small
initial cracks within the iceberg are likely to propagate in each
oscillation until they become unstable resulting in the iceberg fracture
. suggested from model simulations
that increasing ocean temperatures along the iceberg drift and enhanced
melting cause a rapid ablation of the warmer basal ice layers, while
the iceberg core temperature remains relatively constant and cold.
The resulting large temperature gradients at the boundaries could
be important for possible fracture mechanics during the final decay
of iceberg.
Fragmentation law
Like the calving of icebergs from glacier or ice shelves ,
fragmentation is a stochastic process that makes individual events
impossible to forecast. However, the probability that an iceberg will
calve during a given interval of time can be described by a probability
distribution. This probability distribution depends on environmental
conditions that can stimulate or inhibit the fracturing mechanism
. If the environmental parameters conditioning
the probability of fracture can be determined, it would thus be possible
to propose at least bulk fracturing laws that could be used in numerical
models. The correlation between the relative volume loss, i.e. the
adimensional loss, dV/V (which was filtered using a 20-day Gaussian
window) and different environmental parameters (namely SST, current
speed, difference of iceberg and current velocities, wave height,
wave peak frequency and wave energy at the bobbing period) has thus
been analysed in detail. The highest correlation is obtained for SST,
with similar values for both icebergs, namely 0.63 for B17a and 0.64
for C19a. It is high enough to be statistically significant and to
show that SST (or the temperature difference) is certainly one of
the main drivers of the fracturing process. SST is followed by the
iceberg velocity which has a low correlation of 0.30 for B17a and
0.28 for C19a showing a potential second order impact. The correlation
for all the other parameters, in particular for the sea state parameters,
is below 0.15. Figure , which presents the 20 day-Gaussian
filtered relative surface loss as function of SST, iceberg velocity
and wave height, confirms the strong impact of the temperature. The
logarithm of the loss clearly increases almost linearly with temperature.
The regression gives similar slopes of 1.06 ± 0.04 for B17a and
0.8 ± 0.04 for C19a. There also exists a slight increase of loss
with iceberg velocity. However, the regression slopes are very different
for B17a (1.8±0.8) and C19a (6.3±0.8). The significant wave
height has no impact on the loss.
The cumulative sums of the relative volume loss for the two icebergs,
presented in Fig. , exhibit very similar behaviour, suggesting
that a general fracturing law might exist.
We investigate this matter step by step, by progressively including
the dependence to environmental parameters in a simple model of bulk
volume loss. Firstly, only the temperature difference between the ocean
and the iceberg is considered in the model
Mfr=αexp(β(Tw-Ti)),
where Mfr is the relative volume loss by fragmentation and α,β
are model coefficients. In a first step, the daily volume loss is
computed and compared to the observed loss. The model best fit presented
in Fig. (black line) gives similar results for B17a
and C19a: α= 1.9×10-5 and 2.7×10-5, β=1.3 and 0.91, Ti=-3.4 and -3.7 ∘C, respectively. Although
the correlation between model and measurement is high (0.96 and 0.98,
respectively), the model does not reproduce the final decay
of the iceberg very well.
A possible second order contribution of the iceberg velocity is thus
taken into account by introducing a second correction term in the
model in the form:
Mfr=αexp(β(Tw-Ti))(1+exp(γVi)).
The model is first fitted by setting the β coefficient to the
value found using the simple model. The best fit of the model is presented
as a blue line in Fig. . The fitting parameters have
quite similar values for the two icebergs, α=5×10-6 for
both, γ=5.3 and 6.2 and Ti=-3.3 and -4 ∘C, respectively.
The inclusion of velocity clearly improves the modelling of the final
decay and increases the correlation to more than 0.99.
The possibility of a general law has been further investigated by
testing the model with a common β of 1 for both icebergs. The
best fit is presented as green lines. The best fit is only slightly
degraded (correlation about 0.992). The γ and Ti fitting
parameters slightly vary and are of the same order of magnitude for
the two icebergs. Only the α parameter strongly differs for
B17a (3×10-5) and C19a (5×10-6). This can result
from the fact that the variability of iceberg temperature is not taken
into account. Indeed, a change of Ti of ΔT introduces
a change of α of exp(-βΔT).
Fragmentation models and parameters. The bold characters represent
fitted parameters while the regular characters represent the fixed
values.
Iceberg
B17a
C19
Model/parameters
α
β
Ti
γ
α
β
Ti
γ
1 – αexp(β(Tw-Ti))
1.9×10-5
1.3
-3
2.7×10-5
0.91
-3.7
2 – αexp(β(Tw-Ti))(1+exp(γVi)
5.0×10-6
1.3
-3.3
5.3
5.0×10-6
0.91
-4
6.2
3 – αexp(β(Tw-Ti))(1+exp(γVi)
3.0×10-6
1
-3
6
5.0×10-6
1
-3.2
7.2
4 – αexp(β(Tw-Ti))(1+exp(γVi)
1.0×10-6
1
Piecewise
6.5
1.0×10-6
1
Piecewise
6.5
A final model is tested in the same way as the melting law. The α,
β, and γ parameters are fixed at 1×10-6, 1
and 6.5 respectively, and the model is fitted at each time step over
a ±20 day period to determine the best fit Ti. The model
fit the data with correlation higher than 0.998. The iceberg temperature
varies by less than 2 ∘C and has a mean of -3.7±0.6 ∘C
for B17a and -2.9±0.6 ∘C for C19a (see Fig. ).
Table summarizes the different models and fitted parameters
for the two icebergs.
Other model formulations including wave height, iceberg
speed, and wave energy at the bobbing period were tested but brought
no improvement.
(a) Relative volume loss dV/V as a function of SST. The colour
represents the significant wave height in m. (b) dV/V as a function
of the iceberg velocity. The colour represents the SST in ∘C.
(c) dV/V as a function of significant wave height. The circles
correspond to C19a and the triangles to B17a. The red lines represent
the regression lines. The ordinate scale is logarithmic.
Cumulative relative volume loss, ∑ dV/V, measured (solid
blue line), modelled depending on temperature difference only (dashed
black line), on temperature difference and iceberg velocity (dashed
blue line), on temperature difference and iceberg velocity with β=1
(dotted black line), and full model fitted piecewise (solid black line);
(a) B17a, (b) C19a. The ordinate scale is logarithmic.
Fitted iceberg temperature for B17a (a) and C19a (b).
Transfer of volume and distribution of sizes of fragments
The fragmentation of both icebergs generates large plumes of smaller
icebergs that drift on their own path and disperse the ice over large
regions of the ocean. The knowledge of the size distribution of the
calved pieces is as important as the fragmentation law for modelling
purposes as the fragment size will condition their drift and melting
and ultimately the freshwater flux. The fragment size distribution
is analysed using both the ALTIBERG small icebergs database and the
analysis of three clear MODIS images that present large plumes of
pieces calved from C19a and B17a. Figure a and c present
the small icebergs detected by altimeters in the vicinity (same day
and 400 km in space) of B17a and C19a. To restrict as much as possible
a potential influence of icebergs not calved from the one considered,
the analysis of the iceberg size is restricted to the period when
C19a drifted thousands of kilometres away from any large iceberg.
During this period more than 2400 icebergs were detected. The corresponding
size distribution is presented in Fig. .
The small iceberg detection algorithm used to analyse the MODIS images
is similar to those used to estimate the large iceberg area. Firstly,
the cloudy pixels are eliminated by using the difference between channel
1 and 2 radiances. The image is then binarized using a radiance threshold.
A shape analysis is then applied to the binary images to detect and
characterize the icebergs. The results are then manually validated.
Figure presents an example of such a detection for C19a.
The full resolution images are available in the Supplement
(Figs. S1–S4). The analysis detected 1057, 817, 1228 and 337
icebergs for the four images respectively. The size distributions
for the four images and for the overall mean are given also in Fig. .
The six distributions are remarkably similar between
0.1 and 5 km2. The tail of the distributions (i.e. for area
larger than 7 km2) is not statistically significant because
too few icebergs larger than 5–6 km2 were detected.
The slopes of the distributions have thus been estimated by linear
regression for areas between 0.1 and 5 km2. The values for the
four images are -1.49 ± 0.13, 1.63 ± 0.15, -1.41 ± 0.15,
-1.44 ± 0.24 respectively and 1.53 ± 0.12 for the overall mean
distribution. The slope of the ALTIBERG iceberg distribution is -1.52 ± 0.07.
These values are all close to the -3/2 slope previously presented
by for icebergs from 0.1 to 10 000 km2.
A -3/2 slope has been shown both experimentally and theoretically
to be representative of brittle fragmentation .
This size distribution represents a statistical view of the fragmentation
process over a period of time that can correspond to several days
or weeks. Indeed, it is impossible to determine from satellite image
analysis or altimeter detection the exact calving time of each fragment,
and it is thus impossible to estimate the exact distribution of the
calved pieces at their time of calving. In the same way as fragmentation
is characterized by a probability distribution, the size of the fragment
will also be characterized by a probability distribution. The size
distribution represents the integration over a period of time of this
probability distribution. It can be used to model the transfer of
volume calved from the large iceberg into small pieces.
The transfer of volume from the large icebergs to smaller pieces can
also be estimated using the small iceberg area data from the ALTIBERG
database. The sum of the areas of the detected fragments is presented
in Fig. b and d as well as the large iceberg surface
loss by fragmentation. The difference between the two curves can result
from: (1) an underestimation of the number of small icebergs, (2) the
total area of pieces larger than ∼ 8 km2 not
detected by altimeters. While (1) is difficult to estimate, (2) can be
computed, assuming that the piece distribution follows a power law.
Appendix presents the details of the computation. For both
icebergs, as long as the surface loss is limited, the number of calved
pieces is small and the probability for a fragment to be too large
to be detected by altimeter is also small. The total surface of the
detected small icebergs represents thus almost all the parent iceberg
surface loss. As the degradation increases so does the surface loss.
The number of calved pieces as well as the probability of larger pieces
calving become significantly larger resulting in a larger proportion
of the surface loss due to pieces larger than 8 km2 (thus not
detected). The overall proportion of the surface loss due to small
icebergs is about 50 %, which is in good agreement with the power law model
of Appendix .
Time–longitude trajectory of B17a (a) and C19a (c) (red line) and
coincident small icebergs detected in its vicinity. The colour represents
the area of the iceberg in log scale. Surface loss by breaking (black
lines) and surface of the detected small icebergs (dashed line) for
B17a (b) and C19a (d).
Example of fragment detection using a MODIS image (C19a 5 February 2009).
The contour of the detected icebergs are represented with red lines.
Probability density function of the fragment size detected on MODIS
images (dark blue line C19a 5 February 2009, orange line C19a 15 August 2008,
yellow line 21 August 2008, violet line B17a 2 March 2015, green line all
images), and detected by altimeter in the vicinity of C19a (turquoise
line). The dashed straight lines represent the power law fit to the
data. Dates are listed as MM.DD.YYYY.
Conclusions
The evolution of the dimensions and shape of two large Antarctic icebergs
was estimated by analysing MODIS visible images and altimeter measurements.
These two giant icebergs, named B17a and C19a, were worthy of interest
because they have drifted in the open ocean for more than a year,
are relatively remote from other big icebergs, and were frequently
sampled by our sensors (altimeters and MODIS). Furthermore, the two
of them exhibited very different features, in terms of size and shape
as well as in their drift characteristics. We thus expect their joint
study to be an opportunity to obtain a more comprehensive insight
into the two main processes involved in the decay of icebergs, melting,
and fragmentation.
Basal melting is the main cause of an iceberg's thickness decrease.
The two main formulations employed to represent the melting of iceberg
in numerical models have been confronted to the evolution of the iceberg
thickness. The two melting models, which differ in their formulation
depend primarily on the same two quantities: the iceberg–water differential
velocity and their temperature difference. The classical bulk parameterization
of the forced convection is shown to strongly underestimate the melt
rate, while the forced convection approach, based on the conservation
of heat, appears better suited to reproduce the iceberg thickness
variations.
The main decay process of icebergs, fragmentation, involves complex
mechanisms and is still poorly documented. Due to the stochastic nature
of fragmentation, an individual calving event cannot be forecast.
Yet, fragmentation can still be studied in terms of a probability
distribution of a calving. We carried out a sensitivity study to identify
which environmental parameters that likely favour fracturing. We thus
analysed the correlation between the relative volume loss of an iceberg
and some environmental parameters. The highest correlations are found
firstly for the ocean temperature and secondly for the iceberg velocity,
for both B17a and C19a. All other parameters (namely the wave-related
quantities) show no significant link with the volume loss. We then
formulated two bulk volume loss models: firstly one that depends only
on ocean temperature, and secondly one that takes into account the
influence of both identified key parameters. The two formulations
are fitted to our relative volume loss measurements, and the best
fitting parameters are estimated. Using iceberg velocity along with
ocean temperature clearly reproduces the volume loss variations better,
especially the quicker ones seen near the final decays of both bergs.
Moreover, if the variability of the iceberg temperature is taken into
account, the model coefficients are in this case quite similar for
the two icebergs.
Finally, we have estimated the size distribution of the fragments
calved from B17a and C19a, using MODIS images and altimetry data.
For both icebergs and both methods, the slope of the distribution
is close to -3/2, consistent with our previous altimetry-based global
study and typical of brittle fragmentation processes.
While giant icebergs are not included in the current generation of
iceberg models, they transport most of the ice volume in the Southern
Ocean. Furthermore, the impact of icebergs on the ocean in global
circulation models strongly depends on their size distribution .
As a consequence, it is believed that the current modelling strategies
suffer from a “small iceberg bias”. To include large icebergs
in models would require us to ascertain that the previous modelling
strategies are still valid for large icebergs. We also ought to gain
more knowledge on how these bigger bergs constrain a size transfer
to produce medium to small pieces via fragmentation. Eventually, these
smaller pieces are those that account for the effective fresh water
flux in the ocean. Our study showed that a classical modelling strategy
is able to reproduce the basal melting of large icebergs, provided
that relevant parameters are chosen. It has also demonstrated that
a simple bulk model with appropriate environmental parameters can
be used to account for the effect of the fragmentation of large icebergs,
and highlighted the consequent size distribution of the pieces. These
results could prove valuable for including a more realistic representation
of large icebergs in models. Our analyses could be extended to the
cases of more large icebergs, namely to validate our bulk modelling
approaches on a more global scale.