On the Spatial and Temporal Characterization of Motion Induced Sea Ice Internal Stress
Jennifer K. Hutchings*
* International Arctic Research Center, University of Alaska Fairbanks, Fairbanks,
jenny@iarc.uaf.edu
AK, USA
Cathleen Geiger**, Andrew Roberts* Jacqueline Richter-Menge#, and Bruce Elder#
** University of Delaware, DE, USA
# Cold Regions Research and Engineering Laboratory, Hanover, NH, USA
features (Kwok 2003), is of order 100km.
ABSTRACT
In April 2007 an array of buoys was deployed in the Beaufort Sea with
one aim (among others) of examining the relationship between internal
ice stress and ice pack strain-rate or deformation. Here we present
preliminary analysis of stress data from this experiment. This analysis
is discussed in the context of strain-rate analysis that has been
performed previously. In order to identify ice motion induced stress
from stress measurements recorded at a point in the ice pack, we first
need to remove the thermal stress signal from the measurement time
series. We introduce a conceptual model of thermal stresses to support
a method of extracting ice motion induced stress from stress buoy data.
The model will require independent verification, which we outline,
however is useful for understanding our results. In this paper we focus
on spectral and scaling analysis of ice motion induced stresses, and
compare these to similar analysis of sea ice strain-rate. By comparing
spectral properties of stress and divergence we estimate that dynamic
stress events (such as ridge building) may be felt at a stress sensor up to
45km from the site of deformation. Ice motion induced stresses
demonstrate fractal scaling properties, and are anti-persistent. This
echoes similar results that have been identified for sea ice strain rate
across spatial scales from 10 to 1000 km. Ice motion induced stress and
sea ice strain rate can not be described by Gaussian statistics, and have
“fat tailed” probability distribution functions. These findings provide
insight into how to model risk of large deformation, with large ice
motion induced stress, events impacting any given place in the Arctic
ice pack.
KEY WORDS: sea ice, Beaufort Sea, deformation,
stress measurement, scaling relationship
INTRODUCTION
The Arctic ice pack undergoes continuous deformation, which is
manifest as fracturing, resulting in active leads (opening) and ridges
(closing). Active leads and ridges are linear features that can be
organized into systems that extend 100s of kilometers across the Arctic
Basin. The spacing between active leads or ridges is typically around
10km, whereas the spacing between lead systems, or linear kinematic
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Sea ice deformation is localized (Marsan et al. 2004), occurring at leads
and ridges, and has fractal scaling properties both in space (Marsan et
al. 2004; Rampal et al. 2008; Stern and Lindsay 2009) and time
(Hutchings et al. 2010). Sea ice strain-rate can be characterized as a
pink noise process (Hutchings et al. 2010a), which is controlled by the
balance between synoptic scale surface forcing on the ice pack (winds
and ocean currents) and localized dissipation at leads and ridges. At
local scales, O(10 km), strain-rate is dominated by dissipative
processes, and time series of strain-rate is close to white noise. As the
spatial scale increases, to the atmospheric synoptic scale, O(100km),
strain-rate time series become increasingly red. It is an emergent
property of the wind forcing on the ice pack that results in the
organization of sea ice deformation into systems that are coherent
across the atmospheric synoptic scale and regional scales (Hutchings et
al. 2010a). Over sub-synoptic scales leads and ridges do not deform
coherently, and are randomly distributed. Such emergent properties of
lead organization on synoptic, and greater, scales, and the knowledge
that active lead/ridge spacing is controlled by the surface forcing and
ice strength to be around 10km (Hutchings et al. 2005), provides a
physical explanation for previous observations (Overland et al. 1995)
that sea ice deformation appears to exhibit a set of hierarchical scaling
properties.
For those working in pack ice, leads and ridges present both hazards to
structures and navigation opportunities. Understanding how stress is
transmitted through the ice pack, and how this relates to the local and
regional scale deformation field, could be invaluable for logistics
planning in the Arctic. In particular we are interested in identifying the
likelihood of an energetic deformation event being felt at any particular
location. Our previous strain-rate analysis is tantalizing in that we
might expect ice-motion induced internal ice stress to follow similar
scaling laws, in both space and time. However, to investigate this
possibility, we need to have a more thorough understanding of how to
measure dynamic stresses at a point in the ice pack and how internal ice
stress due to dynamic (wind and tide for example) loading is
transmitted through the ice pack.
In this paper we provide a hypothesis for the relationship between
thermal and dynamic stresses experienced at a point in the ice pack.
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Data supporting this hypothesis is discussed, however further fieldwork
is required to identify if the hypothesis is correct. From previous work
(Richter-Menge and Elder 1998) we have confidence that a measure of
dynamic stress can be extracted from in-situ point stress measurements.
Our hypothesis suggests that this measure is actually the maximum
shear stress due to ice motion. The relationship between sea ice strainrate, measured over a set of spatial scales between 6km and 140km, and
dynamic stress measured at a point is investigated. Finally we use our
dynamic stress estimates to identify temporal scaling relations for icemotion induced ice stress experienced at a point.
components, σ 1 − σ 2 , measured by a stress buoy, provides
information about dynamic stresses contained in the stress buoy data
without an apparent thermal signature. However they did not provide a
physical explanation supporting this assertion.
IN-SITU MEASUREMENTS
Figure 2: Cartesian coordinates used for tensor analysis of planar
stress fields detected by a stress gauge in an ice floe located at the
origin of xi .
Figure 1: Map of SEDNA buoys on April 7th 05Z. GPS drifter (squares)
and stress-buoy (triangle) positions are plotted. The right panel is a
zoom into the inner (10km radius) buoy array. The stress buoy
deployed on multi-year ice, with weather station and IMB within 100m
is identified as the bold grey triangle.
In-situ stress (Cox and Johnson 1983) and strain-rate (Hutchings et al.
2010a) measurements were collected during the Sea Ice Experiment:
Dynamic Nature of the Arctic (SEDNA), March to June 2007
(Hutchings et al. 2008; Hutchings 2009). The locations of buoys
deployed during SEDNA that are discussed in this paper are shown in
figure 1. Pack ice strain-rate was estimated from the drift of two
hexagonal buoy arrays, with approximately 10km and 70km radii (as
Hutchings et al. 2010b).
Ice stress was estimated using biaxial stress sensors (Cox and Johnson
1983) deployed at 5 locations at 1m depth in the ice in early April
2007. The stress buoys also recorded ice temperature at the stress
sensor depth and air temperature. Four of the stress buoys were
deployed on first year ice pans that were 1.5m thick, three on the
perimeter of the 10km radius strain-rate buoy array, and one close to
the ice camp at the center of this array. The last stress buoy was
deployed in multi-year ice, 2.83m thick, near an ice mass balance buoy
(Richter-Menge et al. 2006) and weather station, close to the ice camp.
Identifying the dynamic component of stress
The stress buoys measure internal ice stress experienced at a point in a
floe parallel to the sea surface. This gives principal stresses σ 1 and σ 2
illustrated in Figure 2 (Cox and Johnson, 1983). An example of these
hourly-sampled time series is shown in Figure 3, top panel. They
represent a combination of dynamic and thermally induced stresses.
The thermal component results from the ice sheet changing temperature
at the upper surface relative to the ocean temperature surrounding the
floe, and the dynamic component is a response to convergence and
divergence of adjacent floes. The thermal stresses are driven by
changes in the ice surface temperature (Lewis 1993), and hence have
power at seasonal, synoptic and diurnal frequencies. Richter-Menge
and Elder (1998) identified that the difference between principle stress
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Figure 3: Time series of stress measured by a stress buoy on multi-year
ice close to the ice camp. The top panel shows the principle stress
components: compressive stress, σ 1 (black) and maximum shear stress,
σ 2 (grey). The middle panel shows σ 1 − σ 2 , which can be thought of
as the maximum shear stress due to ice motion. The bottom panel
shows near surface air temperature at the buoy (black), and ice surface
temperature from nearby IMB data (grey). Note that the ice stresses
are reduced after mid-May (day 115) due to increasing surface
temperature (see lower panel) and the commencement of spring breakup (Hutchings et al. 2010).
Such a physical explanation can be found with a tensor analysis of the
planar stress field in thermodynamically dependent solid. Planar stress
parallel to the ocean surface in a floating sheet of sea ice is given by:
σ =




σ 11 − P
σ 12
0
σ 21
σ 22 − P
0
0
0
σ 33 − P




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for the stress tensor σ ij in Cartesian coordinates, xi , oriented with x 3
perpendicular to the mean sea surface, given i, j ∈ {1, 2, 3} (Hunter,
1983). Here we apply P = P(Δθ , ρ ) as an augmented sub-floe scale
hydrostatic pressure resulting from a temperature differential within the
material, Δθ , and the ice density, ρ . It is different from the
∗
commonly used P describing the multi-floe compressive strength
following Rothrock (1975). The principal stresses along each of the
three axes in Figure 3 are:
σ1 =
1
2
(σ 11 + σ 22 ) − P +
1
4
(σ 11 − σ 22 ) + σ 12
σ2 =
1
2
(σ 11 + σ 22 ) − P −
1
4
(σ 11 − σ 22 ) + σ 12
σ3 =
1
2
(σ 11 + σ 22 ) − P
2
2
2
2
result of correlation between Tair and wind stress on the ice. This is
possibly the main reason for regions of statistically significant
coherence between Tair − Tice and σ 1 − σ 2 in Figure 4 (top panel).
However, it is the large difference between the diurnal coherences in
the top and bottom panels in Figure 4 that supports our physical
explanation of σ 1 − σ 2 being independent of Δθ in contrast to
σ1 + σ 2 .
assuming the material is incompressible, isotropic and isostatic. As a
result, twice the maximum shear is given by:
2
2
σ 1 − σ 2 = (σ 11 − σ 22 ) + 4 σ 12
which is independent of P and therefore independent of any thermal
gradient in the locale of the stress gauges. However, the first stress
invariant I σ = σ 1 + σ 2 + σ 3 depends on P and hence the influence
of Δθ cannot be removed from the measured compressive stress
without corrections using the heat equation.
The isotropic assumption used here is probably sufficient at the floe
scale (e.g. Schulson and Duval, 2009), and assumes that thermal strain
associated with P is also isotropic. Whilst our chosen coordinate
system is convenient for understanding the influence of Δθ , the
underlying physics remains independent of the orientation of xi . It is
also important to understand that we assume incompressibility of a floe
in three dimensions, not in the traditional two-dimensional coordinates
used to represent sea ice mechanics in Earth System Models.
Notwithstanding, ice compression, tilting, or an imbalance of buoyancy
and gravity forces on a stress-measured floe will introduce some error
into our measurements. If any of these occur, the aforementioned
equation for σ 1 − σ 2 would be inaccurate. However, a multi-scale time
series analysis of our observations in Figure 4 suggests this seldom
happens.
Figure 4 provides the wavelet cross-coherence of Tair − Tice against
invariants σ 1 − σ 2 (top panel) and 2σ 3
=
σ 1 + σ 2 (bottom panel) from
the multi-year floe in Figure 1. Tair is the surface air temperature on the
measured floe, and Tice is the temperature in the floe near the stress
gauge, so that Tair − Tice serves as a proxy for Δθ . The top panel in
Figure 4 indicates minimal coherence between shear stress and P ,
while the bottom panel connotes strong coherence between
compressive stress and P within the measured floe. While coherence
does not mean causation, Figure 4 supports the conclusions of RichterMenge and Elder (1998) and the outcomes of our tensor analysis:
Shear stress is independent of temperature differentials in the ice but
compressive stress is not. Note, however, that in our coherence tests,
we have implicitly assumed a linear relation between P and Δθ
which negates complexities of brine-channel interactions in the
material. Also, there will necessarily be a degree of correlation
between synoptically driven Tair and the internal stress in sea ice as a
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Figure 4: Wavelet cross-coherence between Tair − Tice and the stress
invariants σ 1 − σ 2 (top panel) and 2σ 3 (bottom panel) for the multiyear-floe stress sensor denoted with a bold grey triangle in Figure 1.
Dashed shading indicates results influenced by edge effects, and black
contours encircle regions with greater than 95% confidence. Crosscoherence has been calculated using a sixth derivative-of-Gaussian
wavelet with 0.025 octaves per scale following the methods outlined by
Grinsted et al. (2004).
Richter-Menge and Elder (1998) assert that σ 1 − σ 2 is an estimate of
ice motion induced stresses. They based their assertion on regression
analysis against air temperature and found that σ 1 − σ 2 is uncorrelated
to temperature, whereas σ2 was. We performed similar correlation
analysis between the SEDNA multi-year stress data and temperatures
measured by a close by IMB (not shown), and found both σ1 and σ2 are
not well correlated to air temperature, varying from 0.4 to 0.001 across
the five stress buoys. However, σ 1 − σ 2 was found to be less
correlated to ice surface temperature, reducing by an order of
magnitude in general. The lack of correlation between stress
components and ice temperature may in part be due to the transition to
warmer spring temperatures, where thermal stresses become reduced,
and hence correlation is lost in the later part of the time series. It is also
likely to be related to the dependence of P on the temperature
difference from a background state ( Δθ ), rather than an absolute
temperature. Correlation between stress components and air
temperature is higher in the first half of each stress buoy record. An
observation that supports Richter-Menge and Elder (1998) is that σ1
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Figure 5: Stress components measured at buoy on first year ice close to ice camp. Left panel shows these in principle stress space; top right shows
the time series of σ1 (black solid line) and σ2 (grey dashed line); and bottom right shows σ 1 − σ 2 . We identify points with σ 1 − σ 2 >40KPa with
grey crosses.
and σ2 have significant diurnal peaks in power spectra of their time
series, whereas σ 1 − σ 2 does not. Diurnal cycling is a signature of
thermal stresses, following diurnal cycling in the ice surface
temperature. Dynamic stresses may be influenced by tides at this time
period, however in the Beaufort Sea these tides are exceptionally small
(Kowalik and Proshuntinsky 1994). Considering a time series of σ1, σ2
and σ 1 − σ 2 from a single buoy (figure 5), we can see that observed
stress values with high shear stress correspond to times when stress
magnitude was large and σ1 or σ2 loses its diurnal fluctuation character.
Both these observations provide confidence that the hypothesis of equal
thermal stress components ( P applies equally in all three principle
directions) may be reasonable. Should this hypothesis be correct, we
can state that σ 1 − σ 2 is a measure of the maximum shear stress due to
ice motion.
Without information about the magnitude of the thermal stresses, it is
not possible to estimate the compressive stress due to ice motion. In
this paper we only consider the maximum shear stress, assumed to be
0.5( σ 1 − σ 2 ). In future we believe it will be possible to estimate the
compressive stress given a model of thermal stress (Lewis 1993) or
function relating ice surface temperature to thermal stresses. These
proposed methods will have to be developed through further fieldwork.
Comparison of stress and strain – rate
At first glance you may not see any similar patterns between stress data
and strain-rate data (see figure 6). Correlation between strain-rate and
stress is small (less than 0.2). However, on closer inspection there are
physical characteristics of the time series that suggest information
about ice motion induced stresses is present.
Events when ice motion induced stress dominates over thermal stress
can be identified from in the stress buoy time series. During these
events σ1, the compressive stress measured by the buoy, loses any
diurnal cycling character; and σ 1 − σ 2 is amplified. Figure 6 reveals
several deformation events that are identifiable in the stress-buoy data.
For example, consider the event between days 107 and 111. This event
was characterized by regional scale shear, and a stress response to this
shear is evident in all stress buoy time series. What is notable regarding
this event (and others prior to day 115) is that though during this event
dynamic stress, σ 1 − σ 2 , loading and release is felt by all or some
buoys, the magnitude of the dynamic stress is not consistent between
buoys. On the other hand, the strain-rate (divergence and maximum
shear strain rate) is well correlated across larger spatial scales, 10km to
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200km, as measured with GPS drifters (Hutchings et al. 2010a).
To understand this difference, consider the nature of the measurements
being made. The strain-rate is an area integrated measurement, where
stress is measured at a point. For strain-rate we can expect correlation
between measurements over different spatial scales to break down if
the area of the small array becomes smaller than the active lead/ridge
spacing, and Hutchings et al. (2010a) show reducing correlation
(increasing white noise character of the strain rate) as spatial scale
decreases. In the late winter connected ice pack the deformation across
leads, in an average sense, is highly correlated. In our data we see a
reduction in coherence and correlation of divergence of the 10km and
70km arrays only after the commencement of break-up around May
15th (Hutchings et al. 2010a). A discussion of how stress measurements
are characterized is given in the following subsection.
Towards a conceptual model of internal sea ice stress at a point
Dynamic stress is transmitted through the ice pack, over the distances
that the pack is connected. In a diverging ice pack, with opening leads,
the compressive stress due to ice motion may fall to zero. On the other
hand, in a closing ice pack this compressive stress is not experienced
uniformly over the pack.
As a simple example, consider ice blown towards a coast. In response
to compressive stresses the ice pack fractures into a set of active cracks
and ridges running parallel to the coast. Numerical models of this can
be tuned to reproduce similar fracture spacing as observed in nature
(Hutchings et al. 2005), though these models assume a heterogeneous
ice strength field with uniform mean properties across the model
domain. Stress is concentrated, and maximum, at the fractures, where
the ice pack experiences brittle failure.
To understand the stress observed at a point in the ice pack we need to
understand how the stress field across an ice floe is related to the
confining stresses on that floe. Under compression this confining stress
is related to the stress experienced at the ridges on the floe perimeter.
Note, here we define an ice floe as a region of ice experiencing rigid
motion. The size of the floe is related to the wind stress leading to
failure (Hutchings et al. 2005): during more energetic failure events ice
floes are smaller. Richter-Menge and Elder (1998) found that stress
measured by a stress buoy decreases with distance from the perimeter
of the ice floe. We do not expect distance to floe edge and ice motion
induced stress at a point are linearly related, however there is likely to
be a relationship that could be determined through numerical modeling.
The SEDNA data set is particularly well suited to this as four buoys
were deployed on uniform ice of thickness 1.5m, which should display
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homogenous strength.
This suggests that the ice motion induced stress estimated with a stress
buoy need to be scaled to estimate the local failure stress of the ice
pack around the stress buoy. Which is an important point when you
consider that sea ice models typically estimate ice stress over relatively
large (10-100 km) spatial scales. At scales greater than the fracture
spacing, during compression of the ice pack, it is reasonable to expect
the model ice stress should represent the failure stress of the ice pack in
the region of the model grid cell. This is also an important point when
comparing stress and strain-rate observations. Strain-rate is estimated
over a relatively large area (100-1000 km2), and if we wish to relate
stress to strain-rate we need a measure of the failure stress within that
area. Determination of the scaling factors for stress is not trivial, but
will be aided by knowledge of the distance between stress buoy and
active leads and ridges. Also, some estimate of the ice strength spatial
variability will be needed. This could be estimated given ice thickness
and type observations.
properties is debatable. Stress measured at a point is reacting to thermal
and dynamic stress on the ice pack over an unknown length scale.
Thermal stress is linearly related to ice surface temperature, and hence
we can expect it to have a red noise character with a single diurnal peak
(as ice surface temperature is controlled by weather and radiative
heating of the ice surface). If σ 1 − σ 2 is truly a measure of dynamic
stress, then we can assume that it is not “contaminated” with red noise
from thermal stresses. To first order dynamic stress is proportional to
divergence. By interpolation of the gradients of divergence spectra over
four length scales (determined by Hutchings et al. 2010a), we can
estimate that a divergence spectrum with a gradient of 1.7 corresponds
to a length scale of about 80 to 90km. This suggests that dynamic ice
stress measured by a stress buoy is influenced by dynamic events up to
45km away. This length scale is smaller than the atmospheric synoptic
scale, yet larger than the typical active lead/ridge spacing.
TEMPORAL SCALING OF INTERNAL ICE STRESS AT A POINT
Given the above limitations on our understanding of what is measured
by a stress buoy, there is still useful information provided by these
buoys. It is reasonable to assume that σ 1 − σ 2 is some measure of the
magnitude of ice motion induced shear stress that is experienced at a
point in the ice pack. Hence stress buoys provide time series that can be
analyzed to determine probabilities of experiencing large stress events
at particular locations. The SEDNA data set is interesting in that buoys
were deployed in first year ice, which is more vulnerable to failure. We
expect this data can provide some information about the risk of large
stress events occurring at a given location in first year ice embedded
within the perennial ice pack.
We estimate spectral power density of the measured stress components
(σ1 and σ2) and σ 1 − σ 2 with a fast Fourier transform method (Jenkins
and Watts 1969), using hourly linearly interpolated σ1 and σ2 data, with
a Hann window that spans the 50 day time series. Spectra of the
observed stress components, σ1 and σ2 (figure 7), indicate a significant
peak at one cycle per day. This peak is most likely related to diurnal
cycling of thermal stresses. Note that for σ 1 − σ 2 (figure 7) the diurnal
peak is not present. The spectra for stress components and σ 1 − σ 2
have pink noise properties, meaning they are log-log linear with a
gradient between 1 (white noise) and 2 (red noise). Wind forcing has a
red noise spectra, so the pink character of the dynamic stress ( σ 1 − σ 2 )
spectra indicates that the ice pack dynamics acts to dissipate energy,
resulting in a cascade of power from low to high frequencies.
Hutchings et al. (2010a) performed spectral analysis of SEDNA strain
rate over four spatial scales (roughly 10km, 20km, 70km and 140km).
They found that divergence of the ice pack displayed red or pink noise
character, depending on spatial scale. The gradient reduces from 2 (red
noise) at the largest scale (140km) to 0.6 (pink noise) at the 20km scale.
It was identified that at the smallest scales, where individual leads or
ridges are sampled, that strain-rate was closer to white noise than at
larger scales. It was found that coherence of ice deformation was an
emergent property over large spatial and temporal scales, and due to
this emergent coherence with the wind forcing large scale strain-rate
has a red noise character.
Whether internal ice stress can be treated as having similar scaling
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Figure 6: Time series of σ 1 − σ 2 for five stress buoys (top panel: three
stress buoys on the perimeter of 10km radius array; second panel:
stress buoys near ice camp, black is the buoy on multi-year ice), and
time series of maximum shear strain rate (third panel) and divergence
(bottom panel) of 10km (black) and 70km (grey) buoy arrays.
This estimate of the length scale over which dynamic stress is felt
depends on our assumption that thermal stress components are equal in
all directions. If this is not the case, the gradient of the σ 1 − σ 2 spectra
will be larger than that of a purely dynamic stress spectra, as σ 1 − σ 2
will retain some thermal stress component that would, most likely, have
red noise character. In which case we can expect the length scale over
which dynamic stress is transmitted is less than 45km. We believe the
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utility of stress buoy data could benefit from further work in
understanding thermal stresses and their spectral signature. However, at
present the data is useful for estimating upper limits on dynamic shear
stresses felt at a point in the ice pack.
The Hurst exponent (H) of a data set provides an estimate of the fractal
dimension of that data and information about statistical independence
of the data, whether there is persistence, or lag correlation, in the time
series. It can be estimated from a power spectra of the data time series,
as β=2H+1 where β is the gradient of the power spectra (PSD 
over the period, normalized by the standard deviation of σ 1 − σ 2 over
the period. In figure 8 we plot the natural logarithm of the rescaled
mean against the natural logarithm of time period length T. The Hurst
exponent is estimated as the gradient of a linear least squares fit,
weighted by the standard error on the mean rescaled range. Results
from five stress buoys are consistent and we estimate H to be 0.22 +/0.01. Our estimates of H suggest the fractal dimension of ice motion
induced maximum shear stress is around 1.7 to 1.8.
1/fβ ). From figure 7 we can estimate that for σ 1 − σ 2 , H=-0.3. Another
way of estimating H is with the rescaled range method (Hurst et al.
1965). In this method each time series of σ 1 − σ 2 is split into periods
of uniform length of time T. For each period the rescaled range of
σ 1 − σ 2 is calculated as
Figure 8: Rescaled range method to estimate Hurst exponent. The
logarithm of rescaled range (R/S where R is the range between
minimum and maximum values of σ 1 − σ 2 , and S is standard deviation
of σ 1 − σ 2 ), plotted against logarithm of time period (T). Grey crosses
are mean values of ln(R/S) for each stress buoy time series. The solid
black lines are least square fit to crosses for each buoy. Note that we
did not de-trend the σ 1 − σ 2 time series, and the curve in the date
points probably reflects a trend in the time series towards smaller
stresses as spring progresses.
As H is less than 0.5 for σ 1 − σ 2 , this indicates that dynamic stress is
anti-persistent, and has short-term memory. In other word increasing
ice stress will, in general, be followed by decreased ice stress, and there
is little long-term memory in the ice pack for dynamic stress events.
The fact that H is not equal to 0.5 means that dynamic stress, estimated
at a point in the ice pack, is not randomly distributed in time. Hence our
spectral and Hurst exponent analysis provides insight into the statistical
properties of σ 1 − σ 2 , which may be used if one wishes to model ice
motion induced maximum shear stress time series.
PROBABILITY DISTRIBUTION FUNCTIONS OF
ICE MOTION INDUCED STRAIN-RATE AND
STRESS
Figure 7: Power spectral density (PSD) of stress components, σ1 (top),
σ2 (middle) and σ 1 − σ 2 (bottom). Grey lines are single buoy spectra.
Solid black lines are mean spectra over all buoys. The 99% confidence
interval (99% CI), at PSD=1, is shown for the mean spectra. The
dashed line is a log-log linear least square fit to the mean spectra.
the difference between the maximum and minimum value of σ 1 − σ 2
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Variables that have fractal scaling properties are also known to have
probability distribution functions (PDFs) that display power law shape.
These probability distributions have “fat tails” that are not well
described with the Gaussian distribution function. Figure 9 shows PDFs
for divergence, σ2 and σ 1 − σ 2 , estimated from the SEDNA buoy
array data set. The divergence PDF, for both small and large buoy
arrays, has a shape that can be defined by a power law. This indicates
that the tails of the divergence PDF are thicker than tails in a Gaussian
distribution. The PDFs of σ2 vary across the five stress buoys, though
all have shapes that are not dissimilar from Gaussian. The σ 1 − σ 2
PDF is a closer fit to a power law.
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Thermal stresses are driven by ice surface temperature, and therefore
we expect them to have similar statistical properties as surface
temperature. During the SEDNA experiment, the ice surface
temperature (figure 2), are close to Gaussian distributed. Hence it is not
unreasonable to assume that thermal stresses should be normally
distributed. The fact that the stress components measured by stress
buoys, σ1 and σ2, are reasonably approximated by Gaussian
distributions indicates that stress measured at the buoy is
predominantly thermally induced. Ice motion induced stresses,
σ 1 − σ 2 , are due to wind driven changes in strain rate experienced by
the ice pack, and the strength of the ice pack. The fact that σ 1 − σ 2 has
a power law shaped PDF, provides further evidence that this is a
measure of dynamics stresses, as sea ice strain rate also has a power
law shaped PDF.
-3, which is the average gradient of least squares fit to each of the five
stress buoys.
The fact that the PDFs of the separate stress components measured by
each stress buoy, σ1 and σ2, have a different characteristic shape to the
PDF of σ 1 − σ 2 is remarkable. This indicates that the information
controlling the shape of the σ1 and σ2 distributions has been removed
through the σ 1 − σ 2 operation. This provides some anecdotal support
for Richter-Menge and Elder’s (1998) postulate that σ 1 − σ 2 is a
measure of the ice motion induced ice stress.
CONCLUSIONS AND RECOMMENDATIONS
In this paper we presented a conceptual model of how to estimate ice
motion induced maximum shear stress from stress buoy time series
data. We argue that the difference between the major and minor
principle stress estimates ( σ 1 − σ 2 ) from a stress buoy provides a
measure of the maximum shear stress due to ice motion. Analysis of the
probability distribution functions of divergence, σ2 and σ 1 − σ 2 ; cross
wavelet coherence between surface temperature and 2σ3 or σ 1 − σ 2 ;
and spectra features of these two support the concept of Richter-Menge
and Elder (1998) that σ 1 − σ 2 is a measure of ice motion induced ice
stress. Spectral and Hurst exponent analysis of this indicates certain
properties of dynamic ice stress that have not been previously
identified.
First, the distance over which the internal ice stress at a point is related
to dynamic events (active leads and ridges) in the vicinity of a stress
buoy is estimated to be less than 45km. Essentially the stress buoy only
feels stress from dynamic events that are local (with 45km range) of the
buoy. Ice pack strain-rate (deformation) is correlated and coherent over
much longer scales, reflecting a degree of long-range connectivity in
the ice pack. The fact that ice motion induced stress at a point is
influenced by more local deformation indicates a heterogeneous ice
stress field. Our scaling analysis of stress and strain-rate paints a
picture of an ice pack that experiences an extensive area (greater than
4000km2) of reaction to changing surface forcing, resulting in
deformation, within which the pack has a complicated distribution of
stress.
Figure 9: Probability distribution function (PDF) of divergence (top),
maximum shear stress (σ2, middle) and ice motion induced stress,
σ 1 − σ 2 (bottom). Divergence of 20km (black line) and 70km (grey
line) GPS buoy arrays is shown. Least squares fit to the PDF tail, in
log-log space, indicates a gradient of -1.5 (bold grey line). σ2, and
σ 1 − σ 2 , PDFs are plotted for all buoys. The thin grey line in the
middle and bottom panels is the PDF of stress buoy on multi-year ice.
Fitted Gaussian is shown as a bold grey line in the middle panel. The
PDFs of compressive stress (σ1) are not shown, however are similar in
shape to those of σ2. In the bottom panel the bold line has a gradient of
Paper No. ICETECH10-XYZ-R0
Hutchings
Second, ice motion induced maximum shear stress is anti-persistent,
and hence is not a random walk process. Stress loading on the ice pack
results in later stress release or reduction. This is consistent with the
brittle nature of the ice pack that failure occurs on stress loading, such
that once a maximum load is attained ice stress cannot increase and
may only decrease. We expect that thermal stresses, on the other hand,
are probably a red noise process (with Hurst exponent of 0.5), that
could be modeled well as a random walk process. However this
hypothesis requires testing.
Stress buoy time series data does not provide the magnitude of the
maximum loads due to ice motion, unless the sensor is deployed at the
exact location of failure. However the time series data can provide
information about the timing of these maximum stress events within
about 45km of the stress buoy, and the stress that is felt at a particular
point in the ice pack during these events. Dynamic events may be
identified with increasing magnitude of σ 1 − σ 2 , and loss of diurnal
cycling in the measured stress components.
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One feature of fractal processes is that they have distributions with
heavy tails, and we find that ice motion induced ice stress has such
properties. A power law describes the probability distribution function
of dynamic ice stress reasonably well. One important point that follows
is that ice motion induced internal ice stress should not be modeled as a
random walk process. The possibility of experiencing a destructive
stress event would be seriously under-estimated if dynamic ice stress is
assumed to follow a Gaussian distribution.
In order to test our hypotheses regarding the nature of thermal stresses
in sea ice, and to test our method of how to dissociate dynamic stresses
from thermal stresses, further fieldwork is required. Time series data
from a stress buoy deployed in a location that is not influenced by ice
motion induced stresses would allow us to test the hypotheses that (1)
thermal stresses are isotropic, (2) have red noise spectral properties and
(3) has a Hurst exponent of 0.5. Such a location could be a lagoon, for
example, Eilson lagoon near Barrow Alaska. Numerical modeling
should also be able to shed some light on the results presented in this
paper. In particular models could help us understand the length scales
over which dynamic stress are felt throughout the ice pack. Idealized
simulations of sea ice internal stress could test our assumption that to
first order ice motion induced stress is proportional to the divergence of
the ice pack, and therefore should have similar spatial scaling
properties.
ACKNOWLEDGEMENTS
This research was funded by the U.S. National Science Foundation
(grants NSF ARC 0612527, 0612105, and 0612402). The U.S. Navy’s
Arctic Submarine Laboratory provided access to the Applied Physics
Laboratory Ice Station (APL) in 2007 (APLIS07). Many thanks to Fred
Karig and the APL team who provided logistical support for the
SEDNA field campaign. Pat McKeown, “Andy” Anderson and Randy
Ray are thanked for deploying GPS ice drifters. Wavelet coherence
software provided by A. Grinsted has been adapted for the purpose of
this study.
Hutchings, J. K., P. Heil and W. D. Hibler III, On modeling linear
kinematic features in sea ice, Mon. Wea. Rev., 2005.
Hutchings, J. K., P. Heil, A. Steer and B. W. Hibler III, Sub-synoptic
scale spatial variability of sea ice deformation in the western Weddell
Sea during early summer., submitted to J. Phys. Oceanogr. 2010b.
Hutchings, J.K., A. Roberts, C. Geiger, J. Richter-Menge, Spatial and
temporal characterization of sea ice deformation, submitted to Ann.
Glaciol., 2010a.
Jenkins, W. M., and D. G. Watts, Spectral Analysis and its applications,
Holden-Day, Cambridge, 525pp, 1969.
Kowalik, Z., and A.~Proshutinsky, The Arctic Ocean tides, in The
Polar Oceans and Their Role in Shaping the Global Environment,
edited by O.~M.Johannessen, R.~D. Muench, and J.~E. Overland,
Geophysical Monograph, 85, pp. 137--158, AGU, Washington, D. C,
1994.
Kwok, R., RGPS Arctic Ocean sea ice deformation from SAR ice
motion: linear kinematic features Winter 1996-1997, Winter 1997
1998, Summer 1998, Tech. rep., Polar Remote Sensing Group, Jet
Propulsion Laboratory, 2003.
Lewis, J. K., A model for thermally-induced stresses in multiyear sea
ice, Cold Reg. Sci. Technol. 2, 1,337-348, 1993.
Marsan, D., H. Stern, R. Lindsay and J. Weiss. Scale dependence and
localization of the deformation o f Arctic sea ice, Phys. Rev. Lett.,
93(17), 178501, 2004.
Overland, J. E., B. A. Walter, T. B. Curtin and P. Turet, Hierarchy and
sea ice mechanics: A case study from the Beaufort Sea, J. Geophys.
Res, 1995.
REFERENCES
Rothrock, D. A., The energetics of the plastic deformation of pack ice
by ridging, J. Geophys. Res., 80(33), 4514-4519, 1975.
Cox, G.F.N. and J.B. Johnson. Stress measurements in ice. USA Cold
Regions Research and Engineering Laboratory, CRREL Rep., 83–23,
1983.
Rampal, P., J. Weiss, D. Marsan, R. Linday and H. Stern. Scaling
properties of sea ice deformation from buoy dispersion analysis, J.
Geophys. Res, 2008.
Grinsted, A., J. C. Moore, and S. Jevrejeva, Application of the cross
wavelet transform and wavelet coherence to geophysical time series.
Nonlinear Processes in Geophysics, 11, 561–566, 2004.
Richter-Menge, J.A. and B.C. Elder. Characteristics of pack ice in the
Alaskan Beaufort Sea. Journal of Geophysical Research 103(C10):
21,817–21,829, 1998.
Hunter, S. C., Mechanics of continuous media, Ellis Horwood, 640 pp,
1983.
Richter-Menge, J.A., D.K. Perovich, B.C. Elder, K. Claffey, I. Rigor,
M. Ortmeyer, Ice mass balance buoys: A tool for measuring and
attributing changes in the thickness of the Arctic sea ice cover, Annals
of Glaciology, 44, 205-210, 2006.
Hurst, H.E., Black, R.P. and Simaika, Y.M., Long-Term Storage: An
Experimental Study, Constable, London, 1965.
Hutchings, J. K., The Sea Ice Experiment: Dynamic Nature of the
Arctic (SEDNA), Applied Physics Laboratory Ice Station (APLIS) 2007,
Field Report, Tech. rep., International Arctic Research Center,
University of Alaska Fairbanks, 2009.
Schulson, E. M., and P. Duval, 2009: Creep and fracture of ice.
Cambridge University Press, 401 pp.
Stern, H. and R. Lindsay, Spatial scaling of Arctic sea ice deformation,
J. Geophys. Res, 2009.
Hutchings, J. K., C. Geiger, A. Roberts, J. Richter-Menge, M. Doble,
R. Forsberg, K. Giles, C. Haas, S.M. Hendricks, C. Khambhamettu,
S.W. Laxon, T. Martin, M. Pruis, M. Thomas, P. Wadhams and J.
Zwally, Exploring the role of ice dynamics in the sea ice mass balance,
EOS, 2008.
Paper No. ICETECH10-XYZ-R0
Hutchings
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