Classification of probability densities on the basis of Pearson’s curves with application to coronal heating simulations

Podladchikova, O.; Lefebvre, Bertrand; Krasnoselskikh, V.; Podladchikov, Vladimir N.

doi:10.5194/npg-10-323-2003

Cited by 14 publications

(18 citation statements)

References 26 publications

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“…4) that could be characterized by a broken power law (see also Fig. 12 in Aschwanden et al 2000b), by an exponential rollover (Kadanoff et al 1989;Lu et al 1993;Charbonneau et al 2001), or by Pearson distributions (Podladchikova et al 2002). Deviations from pure power laws can also be seen in our synthesized frequency distributions (Figs.…”

Section: Discussion Of Methodical Uncertaintiesmentioning

confidence: 79%

Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

Aschwanden

Parnell

2002

ApJ

174

154

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We derive universal scaling laws for the physical parameters of flarelike processes in a low-plasma, quantified in terms of spatial length scales l, area A, volume V, electron density n e , electron temperature T e , total emission measure M, and thermal energy E. The relations are specified as functions of two independent input parameters, the power index a of the length distribution, NðlÞ / l Àa , and the fractal Haussdorff dimension D between length scales l and flare areas, AðlÞ / l D . For values that are consistent with the data, i.e., a ¼ 2:5 AE 0:2 and D ¼ 1:5 AE 0:2, and assuming the RTV scaling law, we predict an energy distribution NðEÞ / E À with a power-law coefficient of ¼ 1:54 AE 0:11. As an observational test, we perform statistics of nanoflares in a quiet-Sun region covering a comprehensive temperature range of T e % 1 4 MK. We detected nanoflare events in extreme-ultraviolet (EUV) with the 171 and 195 Å filters from the Transition Region and Coronal Explorer (TRACE), as well as in soft X-rays with the AlMg filter from the Yohkoh soft X-ray telescope (SXT), in a cospatial field of view and cotemporal time interval. The obtained frequency distributions of thermal energies of nanoflares detected in each wave band separately were found to have powerlaw slopes of % 1:86 AE 0:07 at 171 Å (T e % 0:7 1:1 MK), % 1:81 AE 0:10 at 195 Å (T e % 1:0 1:5 MK), and % 1:57 AE 0:15 in the AlMg filter (T e % 1:8 4:0 MK), consistent with earlier studies in each wavelength. We synthesize the temperature-biased frequency distributions from each wavelength and find a corrected powerlaw slope of % 1:54 AE 0:03, consistent with our theoretical prediction derived from first principles. This analysis, supported by numerical simulations, clearly demonstrates that previously determined distributions of nanoflares detected in EUV bands produced a too steep power-law distribution of energies with slopes of % 2:0 2:3 mainly because of this temperature bias. The temperature-synthesized distributions of thermal nanoflare energies are also found to be more consistent with distributions of nonthermal flare energies determined in hard X-rays ( % 1:4 1:6) and with theoretical avalanche models ( % 1:4 1:5).

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Section: Discussion Of Methodical Uncertaintiesmentioning

confidence: 79%

Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

Aschwanden

Parnell

2002

ApJ

174

154

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“…Preliminary results concerned with the influence of the type of magnetic field energy dissipation on statistical properties of the total radiation energy flux were presented by Podladchikova et al (1999). It was shown that the dissipated energy has approximately normal distribution when the source of the magnetic field is random and the dissipation is provided by anomalous resistivity.…”

Section: Resultsmentioning

confidence: 99%

Quiet Sun coronal heating: A statistical model

Krasnoselskikh

Podladchikova

Lefebvre

et al. 2002

A&A

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Abstract. Recent observations of give strong support to Parker's hypothesis (1988) that small scale dissipative events make the main contribution to quiet Sun coronal heating. They also showed that these small scale events are associated not only with the magnetic network, but also with the cell interiors . Taking into account in addition the results of the analysis performed by Priest with co-authors (2000) who demonstrated that the heating is quasi-homogeneous along the arcs we come to the conclusion that the sources driving these dissipative events are also small scale sources. Typically they are of the order of or smaller than the linear scale of the events observed, that is < 2000 km. To describe statistical properties of quiet Sun corona heating by microflares, nanoflares, and even smaller events, we consider a cellular automata model subject to uniform small scale driving and dissipation. The model consists of two elements, the magnetic field source supposed to be associated with the small scale hydrodynamic turbulence convected from the photosphere and local dissipation of small scale currents. The dissipation is assumed to be provided by either anomalous resistivity, when the current density exceeds a certain threshold value, or by the magnetic reconnection. The main problem considered is how the statistical characteristics of dissipated energy flow depend upon characteristics of the magnetic field source and on physical mechanism responsible for the magnetic field dissipation. As the threshold value of current is increased, we observe the transition from Gaussian statistics to power-law type. In addition, we find that the dissipation provided by reconnection results in stronger deviations from Gaussian distribution.

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“…While the laborious fitting procedure by the method of moments, made somewhat easier by ad hoc routines and tables [23], still finds some applications [24], Pearson's distributions are nowadays seldom resorted to, owing to later introduction of more user friendly families.…”

Section: Empirical Distributionsmentioning

confidence: 99%

Uncertainty inherent in empirical fitting of distributions to experimental data

Barbato

Genta

Levi

2013

Int. J. Metrol. Qual. Eng.

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Abstract. Treatment of experimental data often entails fitting frequency functions, in order to draw inferences on the population underlying the sample at hand, and/or identify plausible mechanistic models. Several families of functions are currently resorted to, providing a broad range of forms; an overview is given in the light of historical developments, and some issues in identification and fitting procedure are considered. But for the case of fairly large, well behaved data sets, empirical identification of underlying distribution among a number of plausible candidates may turn out to be somehow arbitrary, entailing a substantial uncertainty component. A pragmatic approach to estimation of an approximate confidence region is proposed, based upon identification of a representative subset of distributions marginally compatible at a given level with the data at hand. A comprehensive confidence region is defined by the envelope of the subset of distributions considered, and indications are given to allow first order estimation of uncertainty component inherent in empirical distribution fitting.

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Classification of probability densities on the basis of Pearson’s curves with application to coronal heating simulations

Cited by 14 publications

References 26 publications

Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis

Quiet Sun coronal heating: A statistical model

Uncertainty inherent in empirical fitting of distributions to experimental data

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