Introduction
Recent years have witnessed increasing interest of the sea ice research
community in topics related to the floe-size distribution (FSD). A number of
new studies are devoted to observational FSD data obtained from airborne and
satellite imagery of sea ice
e.g.,,
enhancing earlier observations among
others. Statistical fracture models
have been proposed attempting to explain the properties of probability
density functions (PDFs) obtained from those data
e.g.,. Substantial effort
has been made to develop parameterizations of FSD-related processes for
numerical sea ice models
.
Equations for the evolution of FSD in time, suitable for continuum sea ice
models, have been developed by ;
derived more general equations for joint floe-size and floe-thickness
distribution see also. This increasing interest
results from growing evidence that the FSD is a signature of dynamic and
thermodynamic processes acting on the ice cover
e.g.,
and, presumably even more importantly, that these processes themselves are
significantly affected by the floe-size distribution. In short, mutual
interactions between FSD and physics and dynamics of the upper ocean, lower
atmosphere, and sea ice itself have to be taken into account in order to
understand and predict short-term, synoptic, and long-term evolution of this
complex system.
In spite of substantial progress, many controversies regarding the
interpretation of the available FSD data – including the shape of these PDFs
– remain unsolved due to the lack of understanding of mechanisms that
contribute to the formation of FSD under different conditions. In a great
majority of studies, scale invariance of floe sizes is assumed a priori
and, accordingly, different versions of power-law PDFs are fitted to observational
data (tapered or truncated power laws, two power-law regimes separated by a
sudden change of slope, etc.). Deviations from power laws are often explained
with finite-size effects, i.e., limited spatial resolution and/or extent of
images used to determine the FSD, but they can also be produced by physical
processes affecting the FSD, e.g., lateral melting/freezing
. In many cases, no convincing arguments for assuming
power-law FSDs exist, except the fact that the floe sizes cover a wide range
of values. Typically, no alternative PDFs are considered, no measures of the
fit errors are provided, and no methods different than least-squares fitting
of a straight line to a log–log plot of a cumulative floe-size distribution
(CDF) is considered – in spite of the fact that this method has a number of
well-known shortcomings (see, e.g., , and , for a discussion of typical
problems with this approach, including the tendency to produce large
systematic errors in the estimated exponents, strong influence of binning on
the results, and difficulties with obtaining reliable error
estimates).
An example of the process leading to narrow FSDs, with preferred floe sizes,
is ice breaking by waves, which is one of the dominating ice fragmentation
mechanisms in the marginal ice zone (MIZ). It is still disputed whether the
size of ice floes formed in this process depends on wavelength as
assumed by many parameterizations, see or
rather on material properties and thickness of the ice as proposed
by, but wave-induced fracturing unquestionably imposes an
upper limit on the floe sizes: floes larger than this limit are broken by
tensile stresses related to flexural strain. In their recent numerical model
of ice breaking by waves, obtained narrow, unimodal
PDFs of floe sizes that they describe as “nearly normal”. Similar FSDs were
obtained with the coupled discrete-element–hydrodynamic model of
when it was run with random variations of ice thickness or
strength (unpublished results). In combination with other breaking
mechanisms, melting, etc., FSDs observed in MIZ may still be (and often are)
very wide, but one cannot expect to find scale invariance in the range of
large floe sizes. Accordingly, attempts to fit a power law to the tail of the
FSD from MIZ are unjustified, even if a straight line seems to provide a nice
fit to a graphic representation of that FSD. The data presented in this paper
provide a good illustration of this fact. The results show also the (quite
obvious, but often disregarded) fact that limiting the FSD analysis to
log–log plots of the respective CDFs provides a distorted and misleading
picture of the properties of the respective FSD.
In this work, we present the results of two groups of ice breakup tests
performed in 2015 and 2016 in the Large Ice Model Basin (LIMB) of the Hamburg
Ship Model Basin (Hamburgische Schiffbau-Versuchsanstalt, HSVA). The tests
belong to the first experiments specifically devoted to observing ice
breaking by waves under controlled, laboratory conditions. The data collected
are used to analyze the FSDs resulting from breaking of initially continuous
ice sheets by regular waves with prescribed characteristics. We present
floe-size data obtained from digital images of the broken ice sheets, from
five test runs. The PDFs of floe sizes are wide (up to 5 orders of magnitude
of floe surface area) and have nontrivial shapes, excellently illustrating
typical problems with interpretation of FSD data. We show that the method of
presenting the data – in terms of PDFs of binned data, CDFs of unbinned
data, and so on – may influence data interpretation by accentuating certain
aspects and eliminating others. We fit the observed PDFs with a function that
is a weighted sum of two probability distributions, a tapered power law and a
Gaussian; we discuss theoretical arguments underlying this choice of PDF and
interpret the obtained values of the fitted parameters.
The paper is structured as follows: Sect. provides a
description of the research facility and of the two groups of experiments
(Sect. ), as well as image processing methods used and the
collected floe-size data (Sect. ). In Sect. ,
after a short analysis of floe shapes and orientation, a theoretical
probability distribution function that combines a tapered power law with a
normal distribution is proposed and fitted to the experimental data.
Section provides a discussion of the results in view of
theoretical research on fragmentation of brittle materials and finishes with
conclusions.
Instrument setup during test group A (a) and
B (b): single pressure sensors are marked in red, a double pressure
sensor in violet, ultrasound sensors in blue, and Qualisys markers in green;
dashed black lines show fields of view of sideward-looking GoPro cameras
(GoProS and GoProB for Silver and Black models, respectively), and
dashed blue lines show fields of view of the cameras mounted on the
ceiling.
Results
Floe shapes and orientation
Histograms of the absolute value of floe orientation
|θf| (a) and eccentricity ef
(b) in the five analyzed tests, for floes with
s>5×10-3 m2. Bin widths equal 5∘ and 0.04,
respectively; bar heights are normalized so that their total area in each
panel equals one.
Visual inspection of Fig. and Supplement
Figs. S3–S5 shows that the floe shapes are far from regular. Most floes are
polygonal and elongated, and they tend to be longer in the across-tank
direction than in the along-tank direction. As can be seen from
Fig. , the histograms of floes' orientation and
eccentricity are similar in all five cases analyzed, with only a small
fraction of floes oriented at |θf|<45∘, i.e., with
their longer axis closer to the x axis than to the y axis, and there are
almost no floes with eccentricity ef<0.5. Moreover, importantly
for the further analysis, over the whole range of values of s and
bf (6 and 3 orders of magnitude, respectively) there is a strong
linear relationship between bf2 and s
(Fig. ). Thus, bf can be regarded as a
meaningful measure of the linear floe size l in the wave propagation
direction, and due to the fact that bf2∝s, we may expect
ps(s) and pl(bf) to be related through relationships
described in the previous section.
Scatter plot of the floe minor axis bf (m) vs.
floe surface area s (m2). Data from all five tests. The slope of the
black line equals 2.
Floe-size distributions
Histograms of bf (a–e) and s (f–j)
from all five tests analyzed. Bin width equals 0.05 m and 0.05 m2,
respectively.
As already mentioned, the floe surface areas cover roughly 5 orders of
magnitude, from ∼5×10-4 m2 to over 10 m2.
Figures and show the number-weighted
floe-size and floe-area data from the five test runs analyzed. In
Fig. , histograms of binned bf and s values
are shown in linear coordinates (with constant bin spacing);
Fig. shows exceedance probabilities for unbinned bf
and s values in logarithmic coordinates. Obviously, the histograms
correspond to probability density functions pl and ps, and the plots in
Fig. to Pl and Ps. However, although they are just
different ways of presenting the same data, it is clear that they underline
certain aspects of those data and tend to obscure others. In most studies in
which FSD is discussed, only log–log plots of CDFs are used, similar to
those in Fig. e.g.,. The
shape of the curves in Fig. suggests – again similarly to
data from many studies, including those cited above – the existence of two
“regions”, for small and large floes, with a sudden change of slope between
them. Qualitatively similar shapes of Pl(l) obtained from satellite and
airborne floe-size data have been interpreted by the above authors as two
power-law regimes. Obviously, all cumulative distributions from our tests
could be fitted with two straight lines just as well. However, there are at
least three important arguments against this choice. First, the regime of
large floes covers no more than 1 order of magnitude in the case of
bf (and, consequently, less than 2 orders of magnitude of s),
which is not sufficient to speak about power-law dependence. Secondly, the
histograms in Fig. clearly show that in the range of
medium-sized floes, roughly between 0.2 and 1.0 m in size, power law is not
a good candidate distribution. In tests A 2030 and 2060, the
histograms have an especially clear maximum at bf∼0.4 m; in the remaining
three tests, no pronounced maximum exists, but nevertheless a kind of
“plateau” can be observed, with values higher than a power law would imply.
Thirdly, there are well-established theoretical arguments against the
concept of two power laws that are relevant to the present setting: some have
been mentioned in the introduction, others will be discussed in
Sect. at the end of this paper.
Based on the data from our experiments, as well as insights from available
research on fragmentation of brittle materials (see further
Sect. ), we consider the following function as a candidate
for probability distribution that approximates the empirical floe-size
distributions shown in Figs. and :
pl(l)=εpPL(l)+(1-ε)pG(l),
where
pPL(l)=1β1-αΓ(1-α,lm/β)l-αe-l/β,pG(l)=12πσ211-erflm-μσ2e-(l-μ)2/2σ2,
where α, β, μ, σ, and ε are adjustable
parameters, Γ(u,x)=∫x∞tu-1e-tdt is the upper
incomplete gamma function,
erf(x)=2π∫0xe-t2dt is the
error function, and lm denotes the lowest value of l for which
the distributions are valid. The scaling factors in Eqs. () and
() ensure that ∫lm∞pPL(l)dl=1 and ∫lm∞pG(l)dl=1.
As can be seen from Eqs. ()–(), pl is a weighted
sum of two functions: a tapered power law and a normal distribution, the
relative contribution of each component dependent on the value of
ε∈[0,1]. The power-law component has a slope α, and the
value of β determines the onset of the exponential tail at large floe
sizes. The second, Gaussian component of pl is significant within a
limited region around l=μ, with σ describing the width of that
region. The exceedance probabilities PPL(l) and
PG(l), corresponding to pPL(l) and pG(l),
are
PPL(l)=Γ(1-α,l/β)/Γ(1-α,lm/β),PG(l)=1-erfl-μ2σ/1-erflm-μ2σ,
and the total exceedance probability Pl(l) is given by Pl(l)=εPPL(l)+(1-ε)PG(l). A detailed discussion of
the properties of equations ()–() and justification
for their choice to represent the observed FSDs are provided in
Sect. .
The distribution given by Eqs. ()–() has five
adjustable parameters, which makes fitting it to the data a nontrivial task,
mainly due to problems with multiple local minima in the parameter space.
Moreover, specific features of the PDFs analyzed here, described briefly
above, make it difficult to choose a suitable approach. Methods that perform
satisfactorily in terms of fitting the tails of the PDFs tend to fail in the
region in the middle, and methods that successfully fit the middle parts of
the PDFs fail to reproduce the tails. Nonlinear least-squares fitting of
Pl(l) to the observed exceedance probabilities (those shown in
Fig. ) belongs to the first category. This is not surprising
as even in tests 2030 and 2060, in which the maxima at floe sizes of
∼0.4 m are most pronounced, can hardly any signature of these maxima be
seen in cumulative distributions. Another widely used fitting method, the
maximum-likelihood estimation (MLE), captures the middle regions of the PDFs,
but produces tails that very strongly deviate from the observed ones.
Moreover, our tests showed that both these methods are very sensitive to the
value of lm (MLE is known to encounter problems with truncated
distributions), as well as to the initial guess of the parameters. When run
many times with different initial conditions, both algorithms converged to
very different local minima, characterized with almost identical
goodness-of-fit measures, which made the choice of the “best” fit a matter
of subjective preference.
Log–log plots of the exceedance probability Pl(l) (a)
and Ps(s) (b) for unbinned data from all five tests, for floes
larger than 5×10-4 m2.
Results of the linear least-squares fit of predicted and observed
CDFs for bf data from test group A: histograms of bf
with fitted pPL, pG, and pl (a, c, e) and
observed exceedance probabilities with fitted Pl (b, d, f). The
insets show P–P plots of the fitted vs. observed
CDFs.
Due to these problems, we tested a third approach, in which predicted
cumulative probabilities are linear-least-squares-fitted to the empirical ones
– an idea based on the fact that for a perfect fit, the CDFs should lie on a
straight 1:1 line on a P–P plot (see insets in Figs. b,
d, f and b, d). More precisely, the goal is to find the values
of the coefficients ε, α, β, μ and σ that
minimize a metric D defined as a weighted sum of the squared distances to
the 1:1 line. The weights w are expressed in terms of the empirical
probabilities as w=1/Pl(1-Pl); i.e., they are lowest in the
center and highest at the extremes in order to compensate for the variance of
the fitted probabilities, which is lowest in the tails and highest near the
median. For our data, this procedure produced stable results and meaningful
values of the coefficients, even though their ranges of validity had not been
specified beforehand. By “meaningful values” we mean values fulfilling a
few basic criteria, for example that ε>0 (i.e., the contribution
of both components is nonnegative) and μ>0; the two other methods often
converged to ε<0, μ≈0 or α<0. Moreover, when
tested on artificially generated data from purely Gaussian and purely
power-law distributions, this method consistently produced values of
ε below 0.03 and above 0.97, respectively; the two methods
mentioned earlier failed this test.
As in Fig. , but for test group B.
The values of the parameters obtained with this method are provided in
Table . Figures and show the
results in terms of both PDFs and CDFs for tests from test group A and B,
respectively.
In order to obtain a measure of standard errors of the estimates, Monte Carlo
simulations were used. For each fitted model, N=100 data sets with random
numbers drawn from that model were generated, the parameters were estimated
by applying the procedure described above, and the standard deviation of
these parameter values was used as a standard error, given in
Table . Two-sample Kolmogorov–Smirnov tests were performed
pairwise between the observed data and those generated with the fitted
models. The percentage of cases in which the test rejected the null
hypothesis that the two samples were from the same distribution (at the 5 %
significance level) varied between 2 and 3 % for tests 2020 and 2060,
22 and 25 % for tests 1450 and 1510, and 35 % for test 2030. Additionally,
the metric D was calculated for each generated model, and a p value was
computed defined as the percentage of cases in which D was smaller than
that obtained for the original data . The
lowest p value was obtained for test 1450 (p=0.07); the highest one for
test 1510 (p=0.9); all other p values were close to 0.3–0.35. Thus, with
the exception of test 1450 (see further), all other data can be regarded as
drawn from distribution ().
The results show that in both test groups A and B, as fragmentation
progresses, the power-law parts of the FSDs evolve towards lower values of
α and lower values of β: the slope of the PDFs in the range of
small values of l decreases, and the cut-off shifts towards smaller floe
sizes, which is reasonable as fewer and fewer large floes survive without
breaking. The two trends together produce larger and larger differences
between the slopes of the large and small floe regions in CDF plots, giving
the impression of a “regime shift”. The Gaussian part of the PDFs is
relatively stable, with a slight tendency for the value of μ to shift to
the left, again as a result of breaking. The values of ε in both
tests decrease in time, indicating decreasing (increasing) contribution of
pPL (pG). Notably, in test B 1450 the predicted
contribution of pG equals ∼ 3 %, and Monte Carlo
simulations produced very scattered results – note large error estimates in
Table , especially for μ and σ. The tapered power
law alone seems to be a more appropriate model to explain the data (last row in
Table ). Generally, the tests in test group A were conducted for much
longer than those of test group B (see Table ). Test group B represents
early stages of fragmentation caused by relatively long waves; accordingly,
the pl(l) in these tests are wider than those from test group A, and the change
of slope between the region of small and large floe sizes is less pronounced.
In contrast, tests 2030 and 2060 from test group B represent ice at advanced
stages of breaking by short waves, in which a dominating floe size can be
clearly seen in pl(l) data. Note that the Gaussian component of these PDFs
contributes to the sudden change of slope in log–log CDF plots.
Note also that in the tests from test group A, the floes described by the Gaussian
component of FSDs represent the “dominant” or “significant” floes in the
sense that they cover the largest fraction of the total surface area, i.e.,
the area-weighted floe-size distributions have very peaked maxima at
bf∼ 0.5 m (see Supplement Fig. S6a–c). In fact, this is
also the range of values estimated by a human looking at an image of the ice
like that in Fig. and Supplement Figs. S3 and S4.
These maps definitely do not look “fractal”. Analogous area-weighted PDFs
from test group B, in which the power-law component is dominant, have a very
different shape, with larger floes occupying a larger fraction of the total
surface (Supplement Fig. S6d, e).
Results of least-squares fit of Eq. () to observed FSD
data.
Run no.
ε
α
β
μ
σ
A 2020
0.821 ± 0.119
1.039 ± 0.250
0.736 ± 0.271
0.574 ± 0.039
0.160 ± 0.060
A 2030
0.685 ± 0.039
0.590 ± 0.084
0.298 ± 0.039
0.431 ± 0.012
0.111 ± 0.014
A 2060
0.610 ± 0.068
0.245 ± 0.115
0.204 ± 0.037
0.463 ± 0.022
0.154 ± 0.021
B 1450
0.968 ± 0.042
1.136 ± 0.115
2.408 ± 0.676
1.117 ± 3.280
0.055 ± 1.265
B 1510
0.695 ± 0.030
0.513 ± 0.169
0.155 ± 0.035
0.924 ± 0.053
0.391 ± 0.037
B 1450
ε=1
1.123 ± 0.065
2.743 ± 1.776
—
—
The error estimates are standard deviations obtained with
Monte Carlo simulations (see text). The last row shows least-squares fit of data from
test 1450 to a tapered power law (ε=1).
Discussion and conclusions
One of the conclusions of this study is that even in a simple laboratory
configuration, under controlled conditions, the interpretation of the
obtained floe-size distributions is far from trivial. With uniform ice,
regular waves, and an approximately one-dimensional setting, one could expect a
straightforward relationship between the wave forcing and ice mechanical
properties on the one hand, and the resulting floe sizes on the other hand.
However, this is not the case, and one of the main reasons for this is that
laboratory-grown ice is softer, weaker, and thinner than
real-world sea ice. Consequently, a number of processes contribute to
breaking and overall wear out of the ice, wave-induced flexural stress being
only one of them. Our video material clearly shows strong over-wash of
the upper ice surface, floe–floe collisions, grinding of small ice fragments
between larger ice floes, and “erosion” of the ice, producing significant
amounts of slush filled spaces between ice floes at later stages of the
experiments, especially those from test group A, in which individual runs
were much longer than in test group B (Table ). In runs with
steeper waves (e.g., 2030 and 2050 in test group A), several cases of floe
rafting were observed as well. Importantly, the effects of these processes
are visible already shortly after the formation of the first cracks; i.e., it
is not possible to identify a phase of ice breaking due to flexural stresses,
followed by a later phase of breaking induced by the remaining processes:
they all contribute to ice fragmentation simultaneously. Consequently,
although it may seem to be a paradox, we do not observe any regular breaking
patterns similar to that repeatedly reported from the field.
Let us take a closer look at the components of equation () in the
context of what is known about fragmentation of sea ice and other brittle
materials. The equation postulates that the observed floe-size distributions
are a result of two (groups of) processes, one leading to scale invariance of
floe sizes, with some tapering effects present at large floe sizes, and the
other producing a preferred floe size, with some random scatter around the
mean value. A similar general approach, in which the probability distribution
of fragment sizes is expressed as a sum of two (or more) terms, is well known
in studies on fracture of brittle materials. Multimodal distributions
observed in some fragmentation experiments are often fitted with bilinear
Poisson distributions, with individual components attributed to distinct
fracture mechanisms significant at distinct spatial scales see,
e.g.,. One interesting example, relevant in the present
context, is the breaking of slender, elongated rods made of a brittle material,
such as in the experiments of , in which rods made of
dry pasta, glass, steel, and so on, impacted axially, undergo a dynamic
buckling instability and break. The resulting fragment-length distributions
are non-monotonic; i.e., they exhibit maxima corresponding to the dominating
wavelengths of the perturbation developing in the material shortly before the
onset of breaking see Fig. 5 in. As the authors
note, the effects of fragmentation in this case are not purely random, but
“include the imprint of the deterministic buckling process leading to
breakup”. referred to this mode of fragmentation as
“patterned breaking” and proposed a one-dimensional mathematical model of
this fragmentation mechanism in which the probability density of breaking is
a prescribed function of location. The model successfully predicted the
observed distributions of fragment sizes. Crucially, although stress maxima
corresponding to the locations of maximum curvature of the rod are regularly
distributed along its length, the observed fragment-size distributions are
very wide, due to a number of competing effects acting in parallel, including
flexural waves associated with stress release after individual breaking
events, pre-existing flaws in the material, or the so-called delayed-fracture
phenomenon . In a different context,
analyzed calving rates of observed and simulated
grounded tidewater glaciers and floating ice shelves. They showed that
fragment-size distributions obtained from their data can be described as a
sum of two components, one representing the largest fragments and dependent
on the large-scale pattern of parent cracks, and the other resulting from
crack propagation and grinding within individual fracture zones.
used the same approach in their discrete-element model
of glacier ice and analyzed how model parameters influenced the relative
contribution of the two components to the resulting fragment-size
distributions.
Analogously to the studies mentioned above, it seems reasonable to represent
the floe-size data with a function given by Eq. (), with one
component describing the “patterned breaking” due to wave-induced flexural
stress, acting at a clearly defined spatial scale, and the other component
representing the remaining fracture mechanisms, producing floes with sizes
spanning a few orders of magnitude. As has been mentioned in the
introduction, the recent numerical studies on ice breaking by waves suggest
that the Gaussian distribution pG(l) is a suitable candidate for
the first component. For an ice sheet floating on the water “foundation”
and subject to flexural deformation, the location of the maximum bending
stress relative to the ice edge – and thus the most probable breaking
location – can be estimated from
xm=π2Ehice33kw(1-ν2)1/4,
where kw is the foundation (in this case water) modulus and
ν is the Poisson's ratio see, e.g.,. For
kw=104 N m-3, ν=0.3 and the values of E and
hice measured in our experiments (see Sect.
and ), we obtain xm=0.48 m for test group A and
xm=0.62 m (based on the average ice thickness) for test group B.
Remarkably, this is very close to the values of μ obtained during the
fitting process (Table ), especially for the first group of
tests in which, as we describe in Sect. , breaking
progressed gradually from the ice edge, so that the assumptions
underlying () should be valid. This is in agreement with
and with the recent results by showing that
the floe size resulting from breaking by waves depends not on the incoming
wavelength, but rather on the mechanical properties of the ice itself.
The second component of Eq. () is more problematic, as its
suitable form depends on the character of the fragmentation process.
Fragment-size distributions in the form of a power law with an exponential
cut-off, as given by pPL(l), have been reported in numerous
studies of fragmentation in both two and three dimensions including
those bycited above, and models explaining the
emergence of power-law fragment size distributions have a sound theoretical
basis see, e.g.,. In
these models, the power-law regime of fragment sizes results from
branching and merging of cracks produced around major parent cracks, and as
the energy available for new crack production is limited, the width of the
fracture zone and thus the fragment size is limited as well, producing the
exponential cut-off in the observed probability distributions. Thus, the
cut-off results from the nature of the process itself. Another source of
cut-off is the finite-size effects that obviously are significant or even
dominating in many configurations. Undoubtedly, in a laboratory experiment
the floe sizes are subject to a global constraint ∑isi=Stot, where Stot denotes the surface area of the
ice sheet. The influence of global constraints of this kind on the tails of
power-law PDFs is discussed in . Together with waves acting
as a floe-size limiting factor, this eliminates the possibility of obtaining
FSDs with heavy, power-law tails. As it is well documented that the gamma
distribution is found in critical phenomena in the presence of finite-size
effects, this function seems suitable for representing FSD data. Notably,
use a very similar functional form – a power
law with an exponential cut-off – to describe the observed FSD data from
four different regions. More importantly, they analyze two different
statistical models of fragmentation, both of which are shown to produce power
laws with exponential cut-offs. Notably as well, used the
Weibull distribution (i.e., a PDF in the form of a product of a power-law
term and an exponential term) to fit their observational FSD data.
Importantly, branching models of fragmentation predict that the exponent of
the power law is universal and depends only on the dimension D in which the
process takes place: αD=(2D-1)/D
. In two dimensions, relevant for sea
ice breaking at scales larger than ice thickness, this value relates to PDF
of surface areas, ps(s): αs=α/2=3/2. Values of αs>3/2
are expected in situations when fragmentation due to crack propagation is
accompanied by further grinding of the material under combined compressive
and shear deformation e.g.,. Scale invariance in
these models and observations suggests that fragmentation takes place as a
self-organized process, as opposed to random breaking that results in
exponential fragment-size distributions e.g.,, i.e.,
αs=0.
In the experiments described here, α was close to 1 during initial
tests (2020 in test group A and 1450 in test group B) and decreased to values
as low as 0.24 in test 2060. This suggests, reasonably, that the random
breaking model is more appropriate in this case. The video material collected
during the experiments shows that individual cracks seem to form
independently of each other, with a simple linear form, i.e., without
secondary, rapidly forming side branches. To the contrary, formation of
individual cracks is relatively stretched in time – it begins at the lower
side of the ice sheet and may take a few wave periods until the two new ice
floes detach from each other. This behavior is very different from processes
that are described by the branching models, in which crack formation is rapid
and their dynamic instability is the source of branching and the resulting
scale invariance of fragment sizes. It must be also remembered that the ice
floes in our experiments were allowed to drift towards the open water area in
front of the wavemaker, so that the conditions were very far from those
favorable for grinding. High values of α are rather expected under
confined conditions dominated by compressive, not tensile, deformation.
Finally, it is worth noting that the processes that lead to breaking of the
ice influence not only the sizes, but also the shape of the ice floes. As has
been noted in Sect. (see also
Fig. and Supplement Figs. S3–S5), the floes
obtained in the experiments described here were polygonal, with relatively
straight edges and sharp angles. Similar (or even more regular, rectangular)
floe shapes have been observed in sea ice broken by waves
e.g.,. They are very
different from approximately circular floes often observed in satellite
images. In the literature, floe shapes have attracted much less attention than the
floe-size distribution but see and little is
known about factors influencing their evolution, but it is tempting to
speculate that in an initially intact ice sheet broken by waves, angular
floes are formed that subsequently gradually evolve towards more rounded
shapes (and wider size distributions) in a process of grinding, known to
produce rounded grains in other granular materials.
In general, the results presented here, obtained under controlled laboratory
conditions, illustrate how difficult the interpretation of real-world
floe-size data is when the ice floes are a product of many cycles of
breaking, freezing, melting, and so on. In most cases, only snapshots of the
ice cover are available, without information on its history and the forcing
acting on it. Nevertheless, we believe that more insight could be gained from
the existing FSD data sets. It would be worthwhile to reexamine the published
floe-size data without commonly made a priori assumptions regarding the
form of the PDFs and to test alternative floe-size distribution models.