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Pet lenko A.V. Kopytenko Yu.A. (SPbF IZMIRAN)

About possibility of Pi2 field fine structure interpretation in terms of anomalous diffusion theory. Pet lenko A.V. Kopytenko Yu.A. (SPbF IZMIRAN). 1 Problem.

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Pet lenko A.V. Kopytenko Yu.A. (SPbF IZMIRAN)

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  1. About possibility of Pi2 field fine structure interpretation in terms of anomalous diffusion theory Petlenko A.V. Kopytenko Yu.A. (SPbF IZMIRAN)

  2. 1 Problem • We shall study Pi2 pulsations field fine structure as follows from Uozumi T, Yumoto K., Kawano H., et al., Propagation characteristics of Pi2 magnetic pulsations observed at ground high latitudes. J, Geophys. Res., 2004, 109, A0, 8203 To do this we shall : • analyze the field of high latitude stations (of BEAR network) only • compare time series of Pi2 magnetic pulsations field distributions with distributions of auroral luminosity based on “Polar” UV TV data. This part of the work has been performed in active collaboration with research workers of Institute of Earth Physics Martines V. and Pilipenko V. • find positions of local ionosphere sources (LIS) of magnetic pulsations by using gradient technique • investigate coherency of these sources in the narrow band of frequencies (6-8)10-3 Hz by phase functions detecting instead of correlation data analysis

  3. φ 75 70 65 60 BEAR magnetic stations network 10 15 20 25 30 λ

  4. 200 nT 20 nT 00 06 12 18 24 UT 18:00 18:30 19:00 19:30 20:00 Z-component magnetic field variations 1998.07.09 and 2 events of Pi2 generation 150s filtered by narrowband FIR Bea And Abi Hop Hom Iva Kev Kil Kir Lah Lon Lov Mas Muo Nor Nya Sor Tro

  5. Joined distributions of 150 nm auroral luminosity (more dark for more intensive local brightness) and X (upper), Z (lower panels) components of magnetic pulsations field (isolines). Positive X,Z values (red), negative ones (blue), 0-isolines (black dot lines). Boarder (light-green dots) of “BEAR” stations (circles).

  6. 0-isolines of X component are close to boarder of auroral luminosity region and local • brightness maxima at this region are close to ones of Z component distributions • during short time (~20 s) few local brightness maxima disappear or change their • position (arrows) that allows to speak about fast local brightness redistribution.

  7. -15 5 nT -5 15 18:59:10 18:59:40 18:59:20 18:59:50 19:00:00 18:59:30 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 30 nT +5 nT -4 nT (X2+Y2)1/2 Z +2.5 nT +1.2 nT +2.5 nT -1.5 nT -3.0 nT -3.0 nT More regular sequence of Pi2 magnetic field distributions study shows that

  8. -15 5 nT -5 15 18:59:10 18:59:40 18:59:20 18:59:50 18:59:30 19:00:00 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 30 nT +5 nT -4 nT (X2+Y2)1/2 Z +2.5 nT +1.2 nT +2.5 nT -1.5 nT -3.0 nT -3.0 nT • Horizontal and vertical field components represent the same field peculiarities. Vectors formed by X, Y components every time moment are tangent to Z isolines

  9. -15 5 nT -5 15 18:59:10 18:59:40 18:59:20 18:59:50 18:59:30 19:00:00 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 30 nT +5 nT -4 nT (X2+Y2)1/2 Z +2.5 nT +1.2 nT +2.5 nT -1.5 nT -3.0 nT -3.0 nT • Two zones with few Z component extremes of the same sign could be marked • These zones seems to be changed by their positions periodically

  10. -15 5 nT -5 15 18:59:10 18:59:40 18:59:20 18:59:50 18:59:30 19:00:00 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 30 nT +5 nT -4 nT (X2+Y2)1/2 Z +2.5 nT +1.2 nT +2.5 nT -1.5 nT -3.0 nT -3.0 nT • During the most part of Pi2 semi-period intensities of local Z component extremes are redistributed. During the least one extremes change their positions rapidly.

  11. Bφ Bφ r1 ψ r0 X1 d Y1 X0 Y0 αi,βi,τi,φi–decrement, amplitude logarithm, appearance time and phase of pulse response, ωk–frequency of filtration Use gradient technique to check local Z extremes positions coincide with LIS ones. Short impulse life-time of these sources is supposed.

  12. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 Z nT -45 -30 -15 0 15 30 45 Results of few ionosphere sources effective centers location during 3 sequent Pi2 semi-periods are presented at following Z-component distributions. 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ

  13. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 Results of few ionosphere sources effective centers location during 3 sequent Pi2 semi-periods are presented at following Z-component distributions. 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ • the sign of local Z extremes changes periodically two times for Pi2 period

  14. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 • LIS positions could be replaced during only one period of Pi2 that supports the  • hypothesis about their short life-time 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ

  15. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 • LIS positions could be replaced during only one period of Pi2 that supports the • hypothesis about their short life-time 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ • for the most part of Pi2 semi-period LIS intensities are redistributed between different sources the velocity of these redistributionsdoesn’t depend on Z sign

  16. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 • Spatial-temporal characteristics of these redistributions are close to those of • local auroral brightness 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ ~100 km

  17. Sor Tro Mas And Tro Kil And Abi Abi 18:56:20 φ φ φ φ φ φ 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 • Spatial-temporal characteristics of these redistributions are close to those of • local auroral brightness 18:57:20 18:58:40 18:56:00 Tro Sor Mas Mas And Kil Abi Muo Ull 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:20 18:57:40 Sor Tro Tro Kev Kev Iva Kil Mas Iva Iva Kil Kil Abi Abi Sod Sal Sod Sod 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ And we’re to check different LIS coupled oscillations character.

  18. Phase functions Фi(t) of NB-filtered horizontal field components for gradient stations could be represented by xi(t) = Ai(t) sin(ωkt+Фi(t)) and expressed in terms of αi,βi,τi,φi

  19. Фi(t) are sectionally smooth functions. At sections where Фi(t) and Фj(t) of the same field components at different stations are continuously differentiable they’re collinear

  20. For ФZ of field being detected at different stations that means the field oscillations at those points couldn’t be considered as independent or self-supported ones. On the contrary we’re to interpret them as coupled oscillations with an almost constant phase shift or a partial coherent related waveforms.

  21. Periodical differentiability brokening of phase functions are corresponded to field reconstructions being associated with Pi2’s periodic Z-component sign changes.

  22. 18:45:30 18:45:50 φ φ φ φ φ φ 19:00:00 75 75 75 75 75 75 70 70 70 70 70 70 65 65 65 65 65 65 60 60 60 60 60 60 One can see these fast non-stationary reconstructions at following field distributions 18:45:40 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ 18:59:40 18:59:50 15 20 25 30 λ 15 20 25 30 λ 15 20 25 30 λ

  23. j ||1– j ||1+ s21– s31– s11– s11+ s21+ j ||2+ s21– j ||2– s31– s22+ s22+ s32+ s32+ s12+ s12+ s11– s12– s12– s22– s22– s11+ s21+ Whereas its stationary behavior could be represented by following schematic which demonstrates that time (out Pi2 semi-period) or spatial (out of zone of Z component the same sing) field averaging leads to high-latitude Pi2 classic model

  24. 2 problems appear: • description of stationary LIS intensities redistributions and fast magnetic field reconstructions from the single position • explanation of LIS fields partial coherence To resolve these problems: • we shall use the model of vertical magnetic dipole random walk at the ionosphere plane (2) modify this model by walk randomization of its motion path and by changing its “point” size to more realistic one. It is well known Brockmann D., Sokolov I.M. Levy flights in external force fields: from models to equations. Chemical Phys. V.284. P.409-421 2002 that this subordinate walk path randomization leads to dipole motion which yields to Levy statistic.

  25. () 18:57:40 76 + 9 нТл + 8 нТл 72 68 64 - 4 нТл - 4 нТл 60 () 18:57:30 76 + 8 нТл 72 68 - 3 нТл 64 60 () 76 72 68 64 60 18:57:50 150 с 140 с 130 с 120 с Considering shapes transformations for zones where the same signs of Z-component are obtained (for different field’s spectral component) one can see that 15 20 25 30 ( ) 15 20 25 30 ( ) 15 20 25 30 ( ) they look like a forms being spilled over themselves. Hence we can imagine every field distribution at these zones as integration of different Hall currents of the same directivity or (in virtue of their oscillations couples) as field integral of vertical magnetic dipole random walks at ionosphere plane.

  26. 18:57:50 (N) (E) 72 70 20 25 Our model of dipole motion will differ from Brockman-Sokolov one in details: • we shall not permit the walk path to exceed the bounds of zone where the field of Hall currents has the same sign • we shall control on the walk path self-intersections density to be proportional to intensity of magnetic field Z component

  27. 18:57:50 (N) (E) 72 70 20 25 • we shall not permit the walk path to exceed the bounds of zone where the field of Hall currents has the same sign • we shall control on the walk path self-intersections density to be proportional to intensity of magnetic field Z component Within the limits of walk path covering • mobile dipole field approximate the measured field distribution with high (~2%) precision • turn-points of Levy trajectory (flight) being subordinate to the simple walk path are very close to determined LIS positions It’s evident that for stable LIS positions at different instants Levy flights turn-points remain stable. More interesting is when their intensifications-depressions occur

  28. To answer this question we are to remember that probability distribution function (PDF) of symmetric Levy process being subordinated to ordinary Brownian motion relates to corresponding PDF in the form: where is an extreme one-sided Levy distribution of index α and τ is the process operational time. In the case of discreet process τ is estimated in random steps units ; operational time is given by and p(x,t) could be written by Cauchy distribution: That assumes t is scaled as v·t to have x dimension and v has a sense of diffusion velocity. And we can propose field distribution model as a sum of mobile LIS fields contributions; everyone LIS moves along Levy trajectory. By assuming LIS as a system of concentric currents within disk diameter ~100 km one can obtain good (~2%) approximation for field distribution within a zone where these currents have the same directivity.This model is suitable not only for stationary sources but also for non- stationary ones; it is enough to prevent latest to have less sizes and more high velocities (to be them more torrent).

  29. Another remarkable property of stable infinitely dividual (partible) Levy distributions is their “orthogonality” as follows from Вершик А.М., ЦилевичН.В. Фоковские факторизации и разложения пространств L2над общими процессами Леви. УМН Т. 58. Вып. 3(351) C.3-50. 2003. This property allows considering Pi2 generation as two relatively independent processes. One of them is responsible for Pi2 field align currents (FAC) forming. The other one possibly supplies particles to FAC environments. These particles can produce local FAC (and corresponding LIS) intensifications as could be seen in following schematic which explains origin of LIS field partial coherency. DD DS DD DD DE DE DE

  30. Hence LIS fields’ intensifications-depressions could be explained by the process of normal diffusion type whereas their stabilizations-dissipations are possibly clarified in terms of anomalous diffusion process. That is in good accordancewith conception on Pi2 generation at magnetosphere tail suggestedby Зеленый Л.М., Милованов А.В. Фрактальная топология и странная кинетика: от теории перколяции к проблемам космической электродинамики. УФН Т.174. № 8. С.809-852. 2004 Another remarkable property of stable infinitely dividual (partible) Levy distributions is their “orthogonality” as follows from Вершик А.М., ЦилевичН.В. Фоковские факторизации и разложения пространств L2над общими процессами Леви. УМН Т. 58. Вып. 3(351) C.3-50. 2003. This property allows considering Pi2 generation as two relatively independent processes. One of them is responsible for Pi2 field align currents (FAC) forming. The other one possibly supplies particles to FAC environments. These particles can produce local FAC (and corresponding LIS) intensifications as could be seen in following schematic which explains origin of LIS field partial coherency. DD DS DD DD DE DE DE

  31. Conclusions: • Partial coherency of Pi2 Local Ionosphere Sources (LIS) is found • This LIS coherency could be explained in terms of anomalous diffusion process • That allows to consider measured Pi2 field in terms of ionosphere Hall currents field dynamic

  32. Thank You for Attention

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