Jennifer Hutchings, Petra Heil and Andrew Roberts

Scaling properties of sea ice deformation during
winter and summer
Jennifer Hutchings, Petra Heil and Andrew Roberts
Time series and spectral analysis of divergence and shear for sub-arrays
within the ISPOL and SEDNA arrays revealed that deformation did not
vary smoothly over length scales of 10 to 40 km. This indicates variability of
internal ice stress on the scale of 10 km. Across the ISPOL array divergence
varied, and did not show a distinct coherence length scale. The SEDNA
array displayed low values for coherence between divergence at 20 and 140 km
scales, with the coherence reducing as the season progressed towards summer.
The ISPOL and SEDNA arrays were split into sets of sub-arrays with varying
length scales. We find that variance of divergence decreases as the length scale
increases. The mean divergence for each length scale set follows a log-linear
scaling relationship with length scale. This is an independent verification
of a previous result of Marsan et al. (2004) (referred to as M04 from here
on). This scaling is indicative of a fractal process. Deformation occurs at
linear features (cracks, leads and ridges) in the ice pack, that are distributed
with scales that range from meter to hundreds of kilometers in length. The
magnitude of deformation at these linear features varies by two orders of
magnitude across scales. We demonstrate that the deformation at all these
scales is important in the mass balance of sea ice. Which has important
implications for the design of sea ice deformation monitoring systems.
Figure 9: Time series of divergence
(top) and maximum shear strain rate
(bottom) for the 20 km hexagon (green)
and 140 km hexagon (black) buoy arrays shown in figure 2.
We consider the time period March 24 to April 14 2007, when all the GPS
stations illustrated in figure 2 were reporting. This is late winter, when the
ice pack is close to it’s maximum volume and we expect ice pack deformation
to be correlated over larger distances than a looser late summer ice pack.
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Figure 12: Time series of divergence
(top), lead area (estimated from divergence), volume of lead ice grown, volume of new ice ridge and total volume
of ice grown in leads (ridged plus level,
bottom), for 140 km hexagon buoy array shown in figure 2. Ice growth rates
are taken from Maykut & Unterstiener
(1971).
Two ways of looking at spatial scaling
Figure 4: There were two distinct regions in the ISPOL array. Region B (see
figure 1c) was dominated by shear along
the continental shelf break. The rest of
the array diverged in concert with the
tides. See Hutchings et al. (2009) for
more details.
Can small scale deformation be predicted given larger scale measurements?
Three ways of looking at spatial scaling
Is there a decorrelation length scale?
Figure 5: Net divergence of the ISPOL
array was highly heterogeneous on the
10km scale, with no discernible pattern.
Net shear, on the other hand, decreased
uniformally across the continental shelf
break.
ISPOL: Austral Summer, Weddell Sea
Figure 1:
The
ISPOL
array
was deployed in
a region of over
90% ice cover,
straddling
the
continental shelf
and slope. The
array contained
a
shear
zone
(region
A)
as
well as on the
shelf that were
predominantly
tidally
driven
(B
and
C).
The equilateral
triangle
design
can be split into
sets
of
strain
arrays of varying
length scale from
50 km
(yellow)
to 10 km (blue).
Buoy array divergence was used to
force a model of lead ice growth
and ridging.
This provides an
estimate of the impact of leads
within the buoy array on ice
mass balance. An example of the
estimate of ice grown in leads is
show in figure ?? for the largest
SEDNA hexagon.
During the
study period, basal ice growth
is estimated to be about 7.5 cm.
Lead ice growth as a fraction of
the total buoy array area varied
across the SEDNA region.
Figure 3: Divergence and Maximum
Shear Strain rate estimated from drift of
buoys around ISPOL array perimeter.
We investigate sea ice deformation observed with ice drifting buoy arrays
during two field campaigns. Ice Station POLarstern [ISPOL], deployed in
the western Weddell Sea during November 2004 to January 2005, included
a study of small-scale (sub-synoptic) variability in sea ice velocity and
deformation using an array of 24 buoys. Upon deployment the ISPOL buoy
array measured 70 km in both zonal and meridional extent, and consisted
of sub-arrays that resolved sea ice deformation on scales from 10 to 70 km.
The Sea Ice Experiment: Dynamic Nature of the Arctic (SEDNA) used two
nested arrays of six buoys each as a backbone for the experiment, that were
deployed in late March 2007. The two arrays were circular with diameter
140 km and 20 km.
ISPOL and SEDNA provide insight into the scaling
properties of sea ice deformation over scales of 10 to 200 km during early
astral summer and late boreal winter.
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280
Beaufort Sea
Weddell Sea
Abstract
Case Study of lead impact on ice mass balance
We consider the correlation of divergence time-series between a large buoy
arrays and smaller arrays embedded inside the larger array. The full range
of buoy arrays we can use to calculate divergence are illustrated in figure 2,
coloured blue, green, yellow and white. There are 6, non-overlapping, 10 km
scale triangles (blue) , contained by a 20 km hexagon (green). These are
contained within a 140 km hexagon (white). The 140 km hexagon also contains
6 70 km triangles (yellow). Table 1 contains the mean and standard deviation
of correlation between time series of total deformation for combinations of
small arrays embedded in a larger array.
Table 2: Note the correlation between large
arrays (70 or 140 km) and the small arrays
(10 km) they contain is low.
Is there a relationship between spatial scale and deformation magnitude?
The figures showing relationship between deformation rate (divergence plus
maximum shear strain rate) and length scale for SEDNA and ISPOL where
created using the same methodology as M04. M04 used RGPS ice deformation
data, covering the western Arctic, and have recently extended their analysis
to 7 winters (Stern & Lindsay 2009, SL09). Our analysis differs in that we
consider a time series of deformation rather than a spatial series.
Figure 6: We calculate the correlation
coefficient between time series of total deformation (divergence + maximum
shear strain rate) for pairs of buoys sub
arrays shown in figure 2B. This is plotted above against the distance between
sub-array centroids in each pair. All
possible permutations of sub-array pairs
are plotted.
Figure 13: Basal plus total lead ice
growth (per m2 ) within each buoy array identified in figure 2, plotted against
mean length scale of the buoy array.
Note that ice growth is higher and shows
more variability for the smallest set of
arrays (blue). This is consistent with localisation of ice deformation at the lead
scale (< 10 km). A log-linear relationship between ice growth and length scale
has H=-2.2.
Figure 10:
Divergence
demonstrates a log-linear
scaling relationship with
exponent H=-0.26.
The
relationship for maximum
shear strain rate is close to
zero.
Note the large variance in correlation coefficient that appears to increase as
distance decreases. We did not find any changes in these coherence results
when we considered cross-correlations between triangles only within or only
across the shear zone in the ISPOL array.
Summary and Initial Conclusions
Can small scale deformation be predicted given larger scale measurements?
In our investigations of correlation between divergence of smaller buoy
arrays contained in larger arrays we found no obvious trends with array size
or ratio of array sizes. What is apparent, however is that correlation
between the smallest arrays (10 − 20 km) and a larger (40 − 140 km) array
containing the smaller array is consistently below 0.5. This suggests low
predictability of deformation at 10 − 20 km scales given a large scale measurement.
We have not identified the reason behind this, however expect that it may
be related to the fact that lead systems (where the deformation occurs) have
widths of up to 10 km. A small 10 − 20 km array may or may not contain
deforming leads that reside within the larger array. This has important implications for processes that are affected by the local ice thickness distribution
and the presence of leads. Localised measurements of such processes should
include observations of ice deformation 10 − 20 km around the measurement
site.
Is there a decorrelation length scale for deformation?
We consider the correlation of divergence time-series between a large triangle
and smaller triangle embedded inside the larger triangle. The full range of triangles we can use to calculate divergence are illustrated in figure 1, coloured
blue, green, red, orange and yellow. There are 4, non-overlapping, 30km scale
triangles, each containing 6 20km triangles. The 40km triangle contains 11
20km triangles and 1 30km triangle ... and so on. Table 1 contains the mean
and standard deviation of correlation between time series of total deformation
for all possible combinations of small triangles embedded in a larger triangle.
Table 1: Note there is a
sweet spot in the ratio between large and small triangles at which point there
is maximum correlation between deformation time series.
Figure 2: APLIS 2007 was deployed on a multi-year ice floe within a pack
of mixed ice age. Two hexagons of GPS ice drifters defined a set of nested
strain arrays with approximate length scale 140 km (white), 70 km (yellow),
20 km (green) and 10 km (blue). Black vectors are ice velocity and red lines
are discontinuities in the velocity field, estimated as Thomas (2008).
1
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Is there a relationship between spatial scale and deformation magnitude?
The ISPOL buoy array was a collaborative effort between IARC, AAD,
AWI and FIMR. Many thanks to Christian Haas and Jouko Launiainen who
contributed buoys. Many thanks also to the captain and crew of the RV
Polarstern, Adrienne Tivy, Tony Worby, Sasha Willmes, and Carl Hoffman
who ensured the field campaign was a success. The IARC ISPOL buoy
deployments were funded by JAMSTEC. The AAD component was funded by
Australian Antarctic Science grants #742, #2559 and #2678; and supported
by the Australian Government’s Cooperative Research Centre Program
through the Antarctic Climate and Ecosystems Cooperative Research Centre.
Jun
0
2007
Wavelet software was provided by C. Torrence and G. Compo, and is available
at URL: http://paos.colorado.edu/research/wavelets/. ESA is thanked for
c
providing Envisat ASAR data (!ESA
(2004)). RADARSat imagery was provided by the Canadian Space Agency, and analysis performed by M. Thomas,
C. Khambhatmettu and C. Geiger.‘
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Period (days)
0.5
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May
Jun
As M04 we find an inverse relationship between deformation magnitude and
length scale, with fractal scaling properties. The relationship we find is
similar to SL09, however this is surprising.
Our time step is 1 hour, and based on work by Rampal et al. (2008) we
could expect such a short time step should lead to higher exponents than we
observed. This needs to be investigated further, and may be due to differing
error characteristics (the ISPOL and SEDNA buoys have position, velocity
and deformation accuracy an order of magnitude higher than both ARGOS
and RGPS)
Is ice mass dependent on lead scale deformation?
Yes! High spatial variability, and localisation of the deformation at leads
and ridges means that for accurate estimates of new ice growth in leads, the
deformation at the lead scale must be resolved.
2007
Small buoy array divergence wavelet power (s−2)
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May
Jun
2007
Figure 8: We find a loglinear scaling relationship
with
exponent
H=-0.21,
similar to M04.
Our
relationship for variance
does not match M04’s
results however.
Is there a relationship between spatial scale and deformation magnitude?
0.2
May
Period (days)
SEDNA is funded by the National Science Foundation, ARC0612527. It
is a collaborative effort between IARC, CRREL, U. Delaware and various
European collaborators.
The U.S. Navy’s Arctic Submarine Laboratory
provided access to APLIS07, supporting camp construction. Many thanks
to Fred Karig and the APL team who ran APLIS07. Many thanks to Pat
McKeown, ’Andy’ Anderson and Randy Ray who deployed the SEDNA strain
array. The International Arctic Buoy Program archives the SEDNA buoy
data presented here.
decorrelation length scale is larger than the ISPOL array.
0.6
32
The colour codes used for buoy arrays of varying size magnitudes in figures 1 and 2 is
used throughout this poster to identify data from particular buoy arrays.
The
4
16
Figure 7: Distribution of total strain rate for sub-arrays
of length scale set ranging
from 10 km (blue) to 50 km
(yellow). Right panel is the
cumulative probability density function.
Figure 14: Least squares fit (red) indicates a decorrelation length scale of
380 km. Note the large variability in
correlation between buoy arrays close
to each other.
This variability reduces as the arrays move further apart.
Close arrays can have higher correlations, and we estimate the decorrelation
length scale of the most correlated arrays
to be 250 km (blue line).
Coherence between large and small buoy array divergence
Large buoy array divergence wavelet power (s−2)
Acknowledgments
The ISPOL experiment was designed to investigate this question.
Is there temporal evolution in coherence between large
and small arrays?
Period (days)
SEDNA: Boreal Winter, Beaufort Sea
Can small scale deformation be predicted given larger scale measurements?
SL09 indicate that we might expect higher exponent values in the southern
Beaufort, where deformation rate is greater, than central Arctic. Hence our
divergence estimate appears to be reasonable. Shear strain rate needs to be
looked at a little more carefully.
Figure 11: Wavelet spectra of divergence time series (fig. 9) for small, 20km hexagon
(bottom right) and large, 140km hexagon (bottom left). The coherence between these two
is shown at top. 95% significance levels are solid line, and the cone of influence is shown
with bold solid line.
Note reducing coherence, at longer time scales (8-32 days) with time. This
is consistent with an ice pack becoming looser and less able to support large
scale coherent motion during Spring.
References
Hutchings, J.K., P. Heil, A.Steer, W.D. Hibler III, Small-scale spatial variability of sea ice deformation in the western Weddell Sea during early summer.
Submitted to J. Phys. Oceanogr.
Hutchings, J. K. and I. G. Rigor (2008), Mechanisms explaining anomalous
ice conditions in the Beaufort Sea during 2006 and 2007. Submitted to J.
Geophys. Res.
Marsan, D., H. Stern, R. Lindsay and J. Weiss (2004), Scale dependence and
localization of the deformation of Arctic sea ice, Phys. Rev. Lett, 93(17),178.
Maykut, G. A., and N. Untersteiner (1971), Some results from a timedependent thermodynamic model of sea ice, J. Geophys. Res..
Rampal, P., J. Weiss, D. Marsan, R. Lindsay and H. Stern (2008), Scaling
properties of sea ice deformation from buoy dispersion analysis, J. Geophys.
Res. (113), doi:10.1029/2007JC004143.
Stern, H. L. and R. W. Lindsay (2009), Spatial scaling of Arctic sea ice deformation, J. Geophys. Res. (114), doi:10.1029/2009JC005380.
Thomas, M. V. (2008), Analysis of Large Magnitude Discontinuous Non-rigid
Motion, Ph.D. Dissertation, University of Delaware.