Nonextensive Dynamics of Drifting Sea Ice

Cycles of ice pack fragmentation in the Arctic Ocean are caused by the irregular dri dynamics. In February 2004, the Russian ice-research camp North Pole 32 established on a �oe in the Arctic Ocean ceased its working activity and was abandoned aer a catastrophic icequake. In this communication, the data collected during the last month of the �eld observations were used for calculating the changes in the kinetic energy of the ice �oe.e energy distribution functions corresponding to periods of different dri intensity were analyzed using the Tsallis statistics, which allow one to assess a degree of deviation of an open dynamic system, such as the driing ice, from its equilibrium state. e obtained results evidenced that the above-mentioned critical fragmentation has occurred in the period of substantially nonequilibrium dynamics of the system of ice �oes. e determination of the state of the pack (in the sense of its equilibrium/nonequilibrium) could provide some useful information on forthcoming icequakes.


Introduction
From the viewpoint of conventional mechanics, the Arctic sea ice cover (ASIC) is the consolidated, mobile, deformable system.Prevalently shearing deformations result in regular pattern of fragmented pack in accordance with the Mohr's mechanism of semibrittle failure of solids.
At the same time, the size distribution of sea ice �oes does not exhibit the random (Poissonian-like) statistics and follows the power law typical for scaling (fractal) structures [1,2].e conventional mechanics cannot explain this phenomenon.
e scaling is a manifestation of long-range correlations between separated events in a statistical system.e correlation radius is determined by the spatial decay of the event effect.e decaying is fast (exponential) in equilibrium systems but slow (power law) in nonequilibrium ones.erefore, the fractal structures are formed only under nonequilibrium conditions in statistical systems driven by external forcing.
In recent years, a new statistical concept called the nonextensive statistical mechanics (NESM) was developed by Tsallis [3,4] for thermodynamic description of the behavior of multiscale systems revealing �uctuations around their equilibrium state.e NESM was successively applied for assessing the degree of deviation of natural dynamic systems from their equilibrium state prior to large-scale hazards, such as earthquakes [5,6] and �oods [7].In the ASIC, cycles of pack fragmentation and signi�cant icequakes occur due to irregularity in the ice dri, which causes critical deformations in the ice cover.According to the NESM, the formation of fractal structures, such as fragmented ice pack, is a result of nonequilibrium dynamics of open systems.In this work, the sea ice dri dynamics during a few bursts of signi�cant fragmentations, which could be regarded as natural hazards, were analyzed in terms of the NESM.
e data on the local dri dynamics were collected in the Russian ice research camp North Pole 32 (NP 32) that was established on ice pack and dried in 2003-2004 in the region north from Franz Josef Land from 88 ∘ N to 84 ∘ N [8].In this work, the time series of the kinetic energy variation of an individual ice �oe were analyzed using the Tsallis statistics in order to obtain a thermodynamic characteristic of the system of driing ice prior to and during large-scale sea ice fragmentations.

Geometry and Size
A distinguishing feature of the Arctic sea-ice cover (ASIC) is a rectilinear lead pattern characterized by the constant angle ∼30 ∘ between directions of intersecting leads (Figure 1).is regular structure consists of parallelogram-like �gures ranging from 1 to ∼10 2 km.Marco and ompson [9] explained this phenomenon on the base of the Mohr's theory of failure, which suggests the following relation between the applied shear stress at failure, , and the normal stress,   , on the failure plane (or line in 2D-case): where  is the material cohesion strength, and  is the angle of internal friction,      ∘ ); here  is the intersection angle.e resolution of (1) relatively the intersection angle with parameters characteristic for sea ice gives one    ∘ , that is the value observed in many satellite images (Figure 1).e parallelogram-like pattern appears only at sufficiently rapid strain rates when the semibrittle failure prevails.At relatively slow strain rates, the ductile behavior takes place, which results in formation of polygonal (diamond-like) or fully disordered structures due to cohesion �ow at low yield stress [9].
At the same time, there are some features in the lead pattern, which cannot be explained in the framework of classical mechanics.Rothrock and orndike [10] and Matsushita [11] studied the size distribution of sea ice �oes and revealed a scale invariant relation between the number of �oes and their size in the form where    � ) is the number of �oes larger than  � ;  is the dimensional characteristics, such as area, , or length, ;  is the constant.e power law dependence means the selfsimilarity of the ensemble of ice �oes because the power law function (2) is the unique solution of the scaling relation where  is the constant (scaling factor).In general, the power law behavior is regarded as a representation of the scale invariance of an object or process in space, time, or energy (in dependence of the measured scaling parameter).e scaling (power law) number-size distributions were found for lead patterns formed both by ice �oes of regular geometry, such as parallelogram-like and diamond-like �oes, and by "smoothed" (oval) �oes typical for frost-free seasons.
Another example of the invariance of the �oe geometry was found by Chmel et al. [2] who had shown that the �oe area and �oe perimeter are interconnected through the following relation: e spatial invariance of this kind is characterized by the parameter  called the "fractal dimension", which, as distinct from the integer Euclidian dimension, could take a fractional value.In the "two-dimensional" case,  could take any value from 1 to 2. Highly fragmented pack with a vast of one dimensional features, such as leads and cracks is characterized by  close to 1, while the fractal dimension of consolidated, frozen ice �elds should be closer to 2.
Figures 2 and 3 show examples of the power law dependences ( 2) and ( 4) calculated for lead-and-crack pattern in the ice cover shown in Figure 1.
e fragmentation of the Arctic ice pack is caused by nonuniform sea ice dri, which is accompanied with substantial stress redistributions.To understand the driving force of self-organizing in fractal structures one should consider the thermodynamic conditions of the dri dynamics.

Nonextensive Dynamics
e thermodynamic state determines the principal trends in the behavior of every statistical system.e most signi�cant perturbations occur when the system deviates from its equilibrium state.Under nonequilibrium conditions, individual "events" in the system are not independent but affect each other at distances much longer than the size of an excited locality.A substantial effect of local event on adjacent sites disturbs the entropic additivity of the process, which in the simplest case of two independent equilibrium subsystems  and  can be expressed as: where  is the entropy.e thermodynamic description of processes in nonequilibrium statistical systems was developed by Tsallis [3,4].e main idea forwarded in his approach was to introduce into additive expression (5) a cross-member that would take into account some interactions between subsystems through a "parameter of nonextensivity" : Here  is the Boltzmann constant.Passing over details of the Tsallis' consideration (see [3,4] for details), we note that the classical statistical mechanics responds the limit    ((6) transforms into (5)); for    the formalism imposes a limited number of events in equilibrium system; �nally, the value    indicates the nonequilibrium state of the dynamic system-this is the case of non-extensive dynamics, which is incompatible with the entropic additivity.
e value of the parameter  serves as a measure of deviation of the statistical system from its equilibrium state.e stronger deviation, the higher .Accordingly, the higher , the larger correlation radius of interactions, and the cooperative effects are more pronounced.
Abe and Suzuki [12,13] used the non-extensive paradigm in their analysis of the distance and waiting time distributions between successive earthquakes and derived a universal expression for the probability of events in nonequilibrium system: where (   �  is the probability of being the value  higher than a threshold  � ,  0 is the mean value of  in the series.Equation ( 7) was used by Abe and Suzuki for �tting the parameters  and  0 to known distances [12] and waiting time [13] distributions between earthquakes.e expression for the cumulative energy distribution (   �  versus  � can written in the following form: where (   �  is the number of events satisfying the inequality    � , and  is the total number of events.Some opportunities given by the applying of ( 8) in studies of sea ice dri dynamics were demonstrated recently by Chmel et al. [14].
In this work we used (8) for analyzing the time series of changes in dri speed of an individual ice �oe.In this context, an "event" should be regarded as a change in the kinetic energy of the �oe;  0 is the mean value of  in a selected time interval.

Energy Exchange
e energy exchange between a driing ice �oe and its environment can be found from changes in the speed of the �oe caused by impact or shear interactions with neighboring �oes.e kinetic energy change, , is directly proportional to the speed change squared of the ice �oe,    2 (where −  �� 2  |,  � and  �� are the ice �oe speeds measured in two subsequent time intervals).erefore, the distribution function, (   � , is equivalent to the distribution of the values  2 .
e time series of changes in the kinetic energy of the ice �oe were recorded in the Russian ice research camp NP 32.
In February 2004, a series of icequakes caused the intensive ice cover fragmentation in the region of the NP 32 activity.e GPS monitoring of the camp position showed a series of local pack displacements with the most intensive shocks on 18 February.In addition, multiple fragmentations of the ice �oe, on which the NP 32 dried, were detected as beginning from the day of the main lead propagation and during subsequent days of February.As a result, the scienti�c observations were ceased on 26 February; on 2 March 2004 the research station NP 32 was abandoned in connection with multiple breakage of the ice �oe.e time series of the �oe kinetic energy changes were now used for assessing the disequilibrium of the pack state prior to, during, and aer the occurred icequake.e values  were found from the changes in the geopositioning data determined by a GPS transducer.e changes in the ice �oe speed were measured in 10 minutes intervals.e measurement procedure and the accuracy of the determination of changes in the value of the ice �oe speed using GPS transducer are given in [15].A time series of kinetic energy changes (proportional to the measured Δ 2 values) in the time period of dri from 1 February to 26 February 2004 is depicted in Figure 4(a).e whole period of observations was divided in four "time windows" in accordance the varied intensity of �oe displacements in different periods of time.Figure 5(a) shows the energy distributions    � ) versus  � in each time window.e measured distributions were approximated by the function (8) using the procedure of best �tting applied to the parameters  0 and .e calculated analytical curves are shown in Figure 5(a) by lines.e found best-�tted values of the mean value of the energy change and the parameter of nonextensivity are shown as histograms in Figures 4(b) and 4(c), respectively.One can see that the parameter  0 (Figure 4(b)) reproduces adequately the intensity of ice �oe mobility (Figure 4(a)) in different time intervals.In a similar way, the -value is minimal in the window A that is prior to the beginning of the sea ice fragmentation.Its closeness to unity evidences the almost equilibrium state of the driing sea ice.In the subsequent interval � characteri�ed by the moderate ice �oe mobility, the -value increased from 1.11 to 1.19 thus indicating an increased deviation of the ice pack from the equilibrium state.e highest value of the parameter  (1.31) was achieved in window C when the largest changes in the �oe kinetic energy were recorded.At the �nal stage of observations (window �), the value  returned to its initial value.
e variation of the -value re�ects trends in the behavior of ice pack in�uenced by the external forcing.A deviation of the parameter  from unity means the nonequilibrium state of driing sea ice.Under this condition, the fragmentation of the pack would tend to form fractal structures.

Discussion
Classical statistical mechanics are constructed on the implication of the independence of individual events involved in the mechanical process; each event produces an additive contribution.e additivity (extensivity) of this kind is the basic property of closed systems.
e mechanical behavior of open, nonequilibrium thermodynamic systems, such as the ASIC, is affected by the energy exchange between its components.It is the open character of the ASIC that causes nonlinear, synergic effects, such as the fractal geometry of lead-and-crack pattern.e �ow of the external energy through the system increases the correlation radius of the event cross impact.e effect of local perturbations decays as the power law with covering much more signi�cant neighborhood than that in the case of exponential decay inherent in the system under equilibrium condition.e principle of independence of individual events becomes disturbed, and the system exhibits a trend to selforganization.
Intuitively, one could expect that the scaling would be more pronounced under more nonequilibrium conditions.However, during a long time aer the discovery of fractals by Mandelbrot no quantitative parameters were suggested for assessing the deviation of a dynamic system from its equilibrium state.Meanwhile, this problem grows out of abstract discussions because of its signi�cance for forecasting the behavior of open natural systems.Apparently, the occurrence of natural hazards depends on the actual state of the given system.It appears that it is for this reason �rst attempts to estimate in numerical parameters the role of nonequilibrium state were made in seismology [5,6,16].
In a similar way, the parameter of nonextensivity could serve as a characteristics of the state of driing sea ice, which would provide information on potential changes in the pack motion and connectedness.
e example given in this communication demonstrates the dri dynamics of an individual ice �oe during a substantial ice pack fragmentation, which had catastrophic subsequences for the research camp established on the pack.e terminal stage of the �oe fracturing was preceded by the steady rise of the degree of nonequilibrium.A similar scenario one could expect for any object of marine engineering, which undergoes irregular impacts from the driing ice.A response of the sea ice fragmentation on �uctuations in the thermodynamic state of the ice pack (equilibrium/nonequilibrium) contain a certain potential for forecasting icequakes in the Arctic region.

Conclusion
Multilevel fractal structures, such as the crack-and-lead pattern of fragmented sea ice, are formed in open dynamic systems driven by the external forcing.e statistical analysis of the dri dynamics in the Arctic Ocean in the period of time covering a large-scale ice pack fragmentation allowed us to assess the �uctuations in the thermodynamic state of the mobile sea ice.It was revealed that the most signi�cant sea ice fragmentation characterized by the strongest �uctuations in the ice �oe kinetic energy occurred in highly nonequilibrium state of the pack.e deviation of the driing pack from its equilibrium state could be regarded as a precursor of forthcoming icequakes.

F 1 :
AVHRR image taken from NOAA satellite on 6 December 2003 in the region around 82 ∘ N, 5 ∘ E.

F 2 :)F 3 :
Cumulative distribution of �oe sizes calculated by processing the satellite image shown in Figure1.Straight line is the best �tted power law dependence(2).Slope = 1.77 ± 0.05 D = 1.13 ± 0Area versus perimeter dependences for ice-�oes fragments resolved in the satellite image shown in Figure1.e straight line �ts the power law dependence (4).

F 5 :
Experimental (signs) and calculated (curves) energy distributions    � ) versus  � in time windows A to D (see Figure4) in the periods of time covering the icequakes in February 2004.e lines �t(8).