Diffusion processes in the transition layer of the Earth's magnetosphere

Turbulence has a different nature in the interplanetary magnetic field and in the transition region, thus it requires a different type of analysis. The "Cluster 2" satellite mission provides magnetic measurements with a temporal resolution of 22.5 Hz. We analysed the evolution of the proba...

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Datum:	2015
Hauptverfasser:	Prokhorenkov, A.S., Kozak, L.V., Lui, A.T.Y., Gala, I.V.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Головна астрономічна обсерваторія НАН України 2015
Schriftenreihe:	Advances in Astronomy and Space Physics
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Zitieren:	Diffusion processes in the transition layer of the Earth's magnetosphere / A.S. Prokhorenkov, L.V. Kozak, A.T.Y. Lui, I.V. Gala // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 2. — С. 99-103. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1199322017-06-11T03:02:51Z Diffusion processes in the transition layer of the Earth's magnetosphere Prokhorenkov, A.S. Kozak, L.V. Lui, A.T.Y. Gala, I.V. Turbulence has a different nature in the interplanetary magnetic field and in the transition region, thus it requires a different type of analysis. The "Cluster 2" satellite mission provides magnetic measurements with a temporal resolution of 22.5 Hz. We analysed the evolution of the probability density function over time, as well as that of the structural function. From the analysis we can conclude that for small time scales, the fluctuation distribution differs significantly from the Gaussian. Furthermore, we see that in the foresho k region, the fluctuation be comes almost Gaussian. Using the extended self-similarity structure function we compare the experimental data with the Kolmogorov K41 model. Calculated diffusion coeficients have a good agreement with the analysis of the probability density function and this can prove the existence of superdiffusion processes in the transition region of the Earth's magnetosphere. 2015 Article Diffusion processes in the transition layer of the Earth's magnetosphere / A.S. Prokhorenkov, L.V. Kozak, A.T.Y. Lui, I.V. Gala // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 2. — С. 99-103. — Бібліогр.: 23 назв. — англ. 2227-1481 DOI:10.17721/2227-1481.5.99-103 http://dspace.nbuv.gov.ua/handle/123456789/119932 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Turbulence has a different nature in the interplanetary magnetic field and in the transition region, thus it requires a different type of analysis. The "Cluster 2" satellite mission provides magnetic measurements with a temporal resolution of 22.5 Hz. We analysed the evolution of the probability density function over time, as well as that of the structural function. From the analysis we can conclude that for small time scales, the fluctuation distribution differs significantly from the Gaussian. Furthermore, we see that in the foresho k region, the fluctuation be comes almost Gaussian. Using the extended self-similarity structure function we compare the experimental data with the Kolmogorov K41 model. Calculated diffusion coeficients have a good agreement with the analysis of the probability density function and this can prove the existence of superdiffusion processes in the transition region of the Earth's magnetosphere.
format	Article
author	Prokhorenkov, A.S. Kozak, L.V. Lui, A.T.Y. Gala, I.V.
spellingShingle	Prokhorenkov, A.S. Kozak, L.V. Lui, A.T.Y. Gala, I.V. Diffusion processes in the transition layer of the Earth's magnetosphere Advances in Astronomy and Space Physics
author_facet	Prokhorenkov, A.S. Kozak, L.V. Lui, A.T.Y. Gala, I.V.
author_sort	Prokhorenkov, A.S.
title	Diffusion processes in the transition layer of the Earth's magnetosphere
title_short	Diffusion processes in the transition layer of the Earth's magnetosphere
title_full	Diffusion processes in the transition layer of the Earth's magnetosphere
title_fullStr	Diffusion processes in the transition layer of the Earth's magnetosphere
title_full_unstemmed	Diffusion processes in the transition layer of the Earth's magnetosphere
title_sort	diffusion processes in the transition layer of the earth's magnetosphere
publisher	Головна астрономічна обсерваторія НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/119932
citation_txt	Diffusion processes in the transition layer of the Earth's magnetosphere / A.S. Prokhorenkov, L.V. Kozak, A.T.Y. Lui, I.V. Gala // Advances in Astronomy and Space Physics. — 2015. — Т. 5., вип. 2. — С. 99-103. — Бібліогр.: 23 назв. — англ.
series	Advances in Astronomy and Space Physics
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first_indexed	2025-07-08T16:55:56Z
last_indexed	2025-07-08T16:55:56Z
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fulltext	Di�usion pro esses in the transition layer of the Earth's magnetosphere A. S. Prokhorenkov 1∗ , L.V.Kozak 1 , A.T.Y. Lui 2 , I. V.Gala 1 Advan es in Astronomy and Spa e Physi s, 5, 99-103 (2015) A.S. Prokhorenkov, L.V.Kozak, A.T.Y.Lui, I. V.Gala, 2015 1 Taras Shev henko National University of Kyiv, Glushkova ave. 4, 03127, Kyiv, Ukraine 2 Johns Hopkins University Applied Physi s Laboratory, Laurel MD, USA Turbulen e has a di�erent nature in the interplanetary magneti �eld and in the transition region, thus it requires a di�erent type of analysis. The �Cluster 2� satellite mission provides magneti measurements with a tem- poral resolution of 22.5 Hz. We analysed the evolution of the probability density fun tion over time, as well as that of the stru tural fun tion. From the analysis we an on lude that for small time s ales, the �u tuation distribution di�ers signi� antly from the Gaussian. Furthermore, we see that in the foresho k region, the �u tuation be omes almost Gaussian. Using the extended self-similarity stru ture fun tion we ompare the experimental data with the Kolmogorov K41 model. Cal ulated di�usion oe� ients have a good agreement with the analysis of the probability density fun tion and this an prove the existen e of superdi�usion pro esses in the transition region of the Earth's magnetosphere. Key words: solar wind-magnetosphere intera tions nonlinear phenomena; transport pro esses introdu tion A new type of instability arises in inhomogeneous plasma, as ompared with a lassi al des ription. In the ase of evolution of these instabilities, the plasma an be ome a turbulent medium. In addition, the transverse di�usion of plasma in the magneti layer an play a signi� ant role in areas where magneti re onne tions are ine�e tive. Thus, di�usion in the absen e of a tual parti le ollisions an be ensured as a wave-parti le intera tion (anomalous resistive di�usion). Sin e gradients of density, temperature, magneti �eld and �ow are observed in the transi- tion region of the magnetosphere, a large number of instabilities for anomalous di�usion evolves. A number of ma ro-instabilities may also ontribute to the di�usion. The Kelvin-Helmholtz instability and eddy turbulen e are the most important instabilities to evolve the inhomogeneous plasma hara teristi s. Both me hanisms an promote mixing of the plasma on a large s ale, and an reate the ne essary on- ditions for mi ro-instabilities. Chaoti ele tri and magneti �elds that o ur, lead to anomalous trans- port pro esses that are orders of magnitude higher than that of lassi al pro esses. Ma ros opi �ows of parti les, momentum, and energy are de�ned not only by the mean �eld and pro�les, but also by the �u tuation spe trum [17℄. The notions and on epts of anomalous dynami properties, su h as long-range spatial or temporal orrelations manifested in power-laws, stret hed ex- ponentials, or non-Gaussian probability distribution fun tions (PDFs), have been predi ted and observed in numerous systems from various dis iplines in lud- ing physi s, hemistry, engineering biology, meteo- rology, astrophysi s, and others. The standard tools to des ribe anomalous dynami s are ontinuous time random walks [7, 8, 20℄ and fra tional dynami al equations [15, 16, 21℄. The anomalous features usually stret h over the entire data window, but there exist examples when they develop after an initial period of sampling (�- nite size/time e�e ts), or they may be transient, i. e., eventually the nature of anomalous pro ess turns into normal transport or relaxation dynami s. The most fundamental de�nition of anomaly is the devi- ation of the mean squared displa ement 〈(∆r)2〉 = 〈(r − 〈r〉)2〉 ∼ Dt∝ from the `normal' linear depen- den e 〈(∆r)2〉 ∼ Dt over time. Here D is the gen- eralized di�usion onstant. The anomalous di�usion exponent α 6= 1 determines whether the pro ess will be ategorized as subdi�usive (dispersive, slow, et .) if 0 < α < 1, or superdi�usive (enhan ed, fast) if 1 < α. Usually, the domain 1 < α ≤ 2 is on- sidered, α = 2 being the ballisti limit des ribed by the wave equation, or its forward and ba kward modes [14℄. Pro esses with α > 2 are known, su h as the Ri hardson pair di�usion in fully developed turbulen e [18℄. Generally, one an not reate a losed system of equations des ribing the anomalous transport pro esses; the results are mostly limited to semi- quantitative assessments. The weak turbulen e, ∗ andrew.prokhorenkov�gmail. om 99 Advan es in Astronomy and Spa e Physi s A. S. Prokhorenkov, L.V.Kozak, A.T. Y. Lui, I. V.Gala with quasi-linear approximation, is a ase when the anomalous transport pro esses an be analyti ally des ribed. Therefore, it is ne essary to determine the statis- ti al properties of the environmental parameter �u - tuations from the experiment whi h are related to the s ale invarian e, and get estimates for the type of di�usion pro esses. This will qualitatively and quantitatively des ribe transport pro esses in di�er- ent regions of the magnetosheath. methods to determine the type of diffusion pro esses evolution of the flu tuations PDF Probability distribution of �u tuation amplitudes of a random pro ess obeys the Gaussian law (the so- alled normal law) [3, 10℄. The dependen e of the maximum probability density fun tion in the analy- sis of data series X(t) at di�erent time s ales an be approximated by a power law: Pτ (0) ∼ τ−s , where τ is the s ale shift over time, and s is a param- eter that hara terizes the homogeneity or hetero- geneity of the studied pro ess (for Gaussian distri- bution s ∼ 0.5 and the presen e of heterogeneity s > 0.5 [3, 6℄. From the behaviour of the PDF the spatial or temporal s ales where the PDF losses the Gaussian properties, an be de�ned. Evaluation of parti le biases an be arried out using the �Levy �ight� (Levy �ights) [2, 22℄: 〈X2(τ)〉 ∝ τ2s, where s is the maximum exponent of magneti �eld �u tuations PDF Pτ (0). Statisti al averages properties. For the turbulent �eld X(t), the stru tural fun - tion (statisti al average) of order q is de�ned as a statisti al average over the ensemble of relations [9℄: δτX = X(t + τ) − X(t), Sq(t) = 〈\|δtX\|q〉 ∼ tζ(q), where the exponent ζ(q) � des ribes the type of pro ess and turbulent di�usion properties of the medium. The linear dependen e of ζ(q) indi ates homogeneity of turbulent pro esses (parti ularly, for the Kolmogorov model � ζ(q) = q/3, and Iroshnikov-Krai hnan model ζ(q) = q/4) [9, 12, 13, 23℄. For turbulen e intermitten e of high order stru - ture fun tions there is a nonlinear dependen e of ζ(q) on q. This re�e ts the deviations from the Gaussian law [5℄. In addition, the stru ture fun tions of high orders allow one to hara terize the properties of het- erogeneity on small s ale pro ess. Analysis of experimental data has a great impor- tan e to the presen e of extended properties of self- similarity (ESS), for a power law of the stru ture fun tions Sq ∼ S ζ(q)/ζ(p) p and allows one to hara - terize �u tuations in the turbulent �ow for a large range of Reynolds numbers [1℄. Using the extended self-similarity property, the good a ura y values of ζ(q) and di�usion properties of the plasma an be evaluated. To interpret the nonlinear dependen e ζ(q) the log-Poisson model is used, for whi h ζ(q) is given by [4, 11, 19℄: ζ(q) = (1−∆) q 3 + ∆ 1− β [ 1− β q 3 ] , where β and ∆ are parameters that hara terize the intermediate and singularity pro esses, respe tively. As a result of ESS-analysis for log-Poisson s al- ing parameters β and ∆ are used to determine the hara teristi s of turbulent plasma transport. The generalized di�usion oe� ient depends on the ζ(q) as [22℄: D ∝ tR, where R = ∆(1/β − l). This relation is used to evaluate transport prop- erties in a heterogeneous environment. In general, the parameter R is de�ned by the fra tal properties of the medium. Thus, there is a relation between the exponent that hara terizes the evolution of PDF, and the pa- rameter R. Displa ement law for the parti les is given by the relation: 〈δX2(τ)〉 ∝ Dτ ∝ τ2s ∝ τα, with order of 2S = 1 + R [2, 22℄. Where, normal di�usion orresponds to α = 1, and the onve tive (ballisti ) motion is hara terized by the α = 2. satellites used for measurement We used the data from �Cluster 2� spa e mission. We analysed 5 events of satellites C1 passing the magnetosheath from 2004 to 2010, with a temporal resolution of 22.5Hz. The examples of analysed mag- neti �eld �u tuations are shown in Fig. 1. In two events out of �ve (2009/05/11 and 2010/03/31), the satellite rossed the magnetosheath, moving from the magnetopause region to the interplanetary medium. In other ases it was moving from the solar plasma wind medium. Fig. 1: Example of analysed magneti �eld. In di�erent regions of magnetosheath there are di�erent levels of magneti �eld �u tuations ob- served: � in foresho k region, the varian e of variations normalized with the urrent mean value is δB/B = 0.1− 0.25; � after rossing the sho k wave in postsho k re- gion, �u tuations in rease several times, om- pared to the foresho k δB/B = 0.5 − 0.6; 100 Advan es in Astronomy and Spa e Physi s A. S. Prokhorenkov, L.V.Kozak, A.T. Y. Lui, I. V.Gala � in deep magnetosheath region, �u tuations de- rease to a value of δB/B ∼ 0.15 − 0.2. Even with almost omplete absen e of magneti �eld �u tuations in the solar wind, there are high �u tuations in the magnetosheath region. results To study the probability density fun tion of �u - tuations of the magneti �eld B(t) we hose the o�set time τ , whi h is a multiple of dis rete measurements τ min = 0.0445s, and we analysed the statisti al prop- erties of the absolute value of the magneti �eld vari- ations dB = B(t+ τ)−B(t) in the foresho k region, postsho k region, and magnetosheath, on di�erent time s ales. Fig. 2: Maximal value for magneti �eld �u tuations PDF P (0) in logarithmi s ale for 3rd Mar h 2004. Ex- perimental data approximated with a line: P ∼ t−s : 1 � foresho k, 2 � magnetosheath, 3 � postsho k. Val- ues of s presented in Table 1. Good temporal resolution of the measurement allows us to observe the dependen e of the maxi- mum value of the PDF of magneti �eld �u tuations Pτ (0) on a small s ale (less than 1 se ond), whi h is very important be ause we an analyse on a s ale smaller than the ion y lotron frequen y. Logarith- mi s ale plots for measurements on Mar h 27, 2010 and Mar h 20, 2006 are shown in Fig. 2, where the experimental points were approximated by a straight line Pτ ∼ τ−s . The values for the exponents of the events are presented in Table 1. From the obtained values, we an on lude that for small s ales in the transition region of the Earth's magnetosphere, the distribution signi� antly devi- ates from the Gaussian distribution. The largest deviations are observed for the post- sho k region. Furthermore, in foresho k region we have an almost Gaussian distribution for magneti �eld �u tuations. The generalized di�usion oe�- ient R is shown in Table 1. R > 0 indi ates the existen e of superdi�usion. The extended self-similarity stru ture fun tions of high orders are spe i�ed by the equation: Sq(t) = 〈\|B(t+τ)−B(t)\|q〉 ∼ τ ζ(q), where 〈. . .〉 � statisti al averaging of experimental data over time. Fig. 3: Relation of stru tural fun tion on time s ale for a magnetosheath region (03/03/2004). Stru tural s aling fun tion normalized to s aling for the third moment, ζ(q)/ζ(3) an be obtained from the slope of a logarithmi s ale. This will allow us to ompare experimental data to the Kolmogorov model of turbulen e (K41) for whi h ζ(3) = 3/3 = 1. In Fig. 3 power law Sq(τ) ∼ τ ζ(q) (i. e. self-similarity) is observed only in a limited range of time s ales. This interval orresponds to the inertial range, whi h is onsidered a lassi model of developed isotropi turbulen e (K41, et .) [5℄. In the transition region of the magnetosphere this interval is observed on s ales less than 1 se ond (this will allow us to ompare the results obtained by di�erent approa hes � from the analysis of maximum height of magneti �eld �u - tuation probability density fun tion and statisti al average). Cal ulated results for ζ(q)/ζ(3) of di�erent orders q in the analysis of small-s ale turbulen e and om- parison with Kolmogorov model for the events on 3rd Mar h 2004 and 3rd April 2006 are shown in Fig. 4 and 5. The most signi� ant deviation from the Kol- mogorov model is observed in the postsho k region. Determined by the ESS analysis parameters β and ∆ and al ulated values R = 0.3÷0.98 are presented in Table 2. The obtained values are in good agreement with the data presented in Table 2. 101 Advan es in Astronomy and Spa e Physi s A. S. Prokhorenkov, L.V.Kozak, A.T. Y. Lui, I. V.Gala Thus, using two independent te hniques, we proved the existen e of superdi�usion pro esses in the transition region of the Earth's magnetosphere. Fig. 4: Relation of exponential value of stru tural fun - tion of q-order to the 3-rd order stru tural fun tion for 3rd Mar h 2004. K41 � values al ulated for Kol- mogorovmodels; FSH� foresho k region, MSH�mag- netosheath, PSH � postsho k. Fig. 5: Relation of exponential value of stru tural fun - tion of q-order to the 3-rd order stru tural fun tion for 3rd April 2006. K41 � values al ulated for Kol- mogorovmodels; FSH� foresho k region, MSH�mag- netosheath, PSH � postsho k. on lusions The relative variations of the magneti �eld in magnetosheath ex eed the values in the solar wind by fa tor of 2�5. However, the onverse proposition is in orre t � not all variations of parameters in the magnetosheath are generated by �u tuations of the solar wind or the interplanetary magneti �eld. The largest deviation from a Gaussian pro ess is observed in the postsho k region, and the distribu- tion losest to a Gaussian one is in the foresho k region. The value of the generalized di�usion oe� ient in reases with time-s aling. Thus, the analysis of the evolution of the magneti �u tuations PDF � the exponent �eld dependen e on the generalized di�usion oe� ient of the s ale (R) in the transi- tion regions of the Earth's magnetosphere: foresho k region, postsho k and magnetosheath � is in the range of 0.32�0.92. The highest value is observed for the postsho k region. The analysis of the statisti al properties of points determined the parameter R to be in the range of 0.3�0.98. Likewise, the highest val- ues are observed for the postsho k region in this ase as well. It an be noted that di�erent approa hes to the analysis of turbulent pro esses give similar re- sults and indi ate the presen e of superdi�usion pro- esses in the transition region of the Earth's magne- tosphere. This fa t must be taken into a ount when onstru ting quantitative models of transport. a knowledgement The work is done in the frame of omplex program of NAS of Ukraine on spa e resear h for 2012-1016, the grant Az. 90 312 from the Volkswagen Founda- tion, and within the framework of the edu ational program No.2201250 �Edu ation, Training of Stu- dents, PhD Students, S ienti� and Sedagogi al Sta� Abroad� laun hed by the Ministry of Edu ation and S ien e of Ukraine. referen es [1℄ Benzi R., Ciliberto S., Tripi ione R. et al. 1993, Phys. Rev. E, 48, R29 [2℄ Che hkinA.V., Goren�oR. & Sokolov I.M. 2002, Phys. Rev. E, 66, 046129 [3℄ Consolini G., Kretzs hmarM., LuiA.T. Y., ZimbardoG. & Ma ekW.M. 2005, J. Geophys. Res., 110, A07202 [4℄ DubrulleB. 1994, Phys. Rev. Lett., 73, 959 [5℄ FreakP.G. 1999, `Turbulen e: models and approa hes', Perm National Te hni al University Press, Perm [6℄ FryshW. 1998, `Turbulen e: Kolmogorov heritage', Pha- sis, Mos ow [7℄ HughesB. D. 1995, `Random Walks and Random Envi- ronments, Volume 1: Random Walks', Oxford University Press, Oxford [8℄ Klafter J., ShlesingerM. F. & ZumofenG. 1996, Physi s Today, 49, 33 [9℄ KolmogorovA.N. 1941, Doklady Akademiia Nauk SSSR, 30, 301 [10℄ KozakL. V. 2010, Spa e S ien e and Te hnology, 16, 1, 28 [11℄ Kozak L. V., Pilipenko V. A., Chugunova O. M. & KozakP.N. 2011, Cosmi Resear h, 49, 194 [12℄ Krai hnanR.H. 1959, J. Fluid Me hani s, 5, 497 [13℄ Krai hnanR.H. 1970, J. Fluid Me hani s, 41, 189 102 Advan es in Astronomy and Spa e Physi s A. S. Prokhorenkov, L.V.Kozak, A.T. Y. Lui, I. V.Gala Table 1: Generalized di�usion oe� ient al ulated from the evolution of magneti �eld �u tuation PDF. Date Region s R = 2s− 1 2009/03/23 Postsho k 0.78 ±0.085 0.56 Magnetosheath 0.69 ±0.092 0.38 2006/04/03 Foresho k 0.66 ±0.076 0.32 Postsho k 0.96 ±0.098 0.92 Magnetosheath 0.89 ±0.088 0.78 2009/05/01 Magnetosheath 0.9 ±0.099 0.81 2004/03/03 Foresho k 0.76 ±0.083 0.52 Postsho k 0.96 ±0.087 0.92 Magnetosheath 0.66 ±0.054 0.32 2010/03/31 Magnetosheath 0.68 ±0.069 0.36 Table 2: Generalized di�usion oe� ient al ulated from ESS analysis. Date Region β ∆ R = ∆( 1 β − 1) 2004/03/03 Foresho k 0.26 ±0.012 0.2 ±0.01 0.58 Postsho k 0.45 ±0.017 0.8 ±0.011 0.98 Magnetosheath 0.66 ±0.009 0.6 ±0.009 0.3 2006/04/03 Foresho k 0.67 ±0.008 0.69 ±0.007 0.35 Postsho k 0.71 ±0.013 2.2 ±0.01 0.91 Magnetosheath 0.45 ±0.01 0.68 ±0.007 0.82 2009/03/23 Postsho k 0.5 ±0.012 0.6 ±0.013 0.6 Magnetosheath 0.67 ±0.008 0.69 ±0.007 0.35 2009/05/01 Magnetosheath 0.5 ±0.011 0.4 ±0.012 0.81 2010/03/31 Magnetosheath 0.62 ±0.009 0.51 ±0.008 0.31 [14℄ LandauL.D., Lifshitz E.M. & Pitaevskii L. 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Diffusion processes in the transition layer of the Earth's magnetosphere

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