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Wang S, Lu S, Lu Q, Wang R, Ren J, Gao X, Guo J. Origin of reconnecting current sheets in shocked turbulent plasma. SCIENCE ADVANCES 2024; 10:eado4639. [PMID: 39141737 PMCID: PMC11323938 DOI: 10.1126/sciadv.ado4639] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2024] [Accepted: 07/10/2024] [Indexed: 08/16/2024]
Abstract
Magnetic reconnection, the rearrangement of magnetic field topologies, is a fundamental plasma process throughout the universe, which converts magnetic energy to plasma kinetic energy and results in particle energization. A current sheet is a prerequisite for the occurrence of magnetic reconnection. It has been well documented that reconnecting current sheets are prevalent in turbulent plasmas. However, how these current sheets are formed remains unclear. Among natural plasmas, the region downstream of the Earth's bow shock, the magnetosheath, is one of the most turbulent. Here, we show that the reconnecting current sheets in the turbulent magnetosheath originate from the waves in the region upstream of the shock. Once excited, the upstream waves are transmitted across the shock, compressed, and then transformed into current sheets in the downstream region. Magnetic reconnection subsequently occurs in these downstream current sheets. This process can be generalized to various shocked plasmas in astrophysical and laboratorial environments where turbulent magnetic reconnection should be common.
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Affiliation(s)
- Shimou Wang
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - San Lu
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - Quanming Lu
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - Rongsheng Wang
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - Junyi Ren
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - Xinliang Gao
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
| | - Jin Guo
- Deep Space Exploration Laboratory/School of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui, China
- CAS Center for Excellence in Comparative Planetology/CAS Key Lab of Geospace Environment, University of Science and Technology of China, Hefei, Anhui, China
- Collaborative Innovation Center of Astronautical Science and Technology, Harbin, Heilongjiang, China
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Oka M, Birn J, Egedal J, Guo F, Ergun RE, Turner DL, Khotyaintsev Y, Hwang KJ, Cohen IJ, Drake JF. Particle Acceleration by Magnetic Reconnection in Geospace. SPACE SCIENCE REVIEWS 2023; 219:75. [PMID: 37969745 PMCID: PMC10630319 DOI: 10.1007/s11214-023-01011-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/04/2023] [Accepted: 10/05/2023] [Indexed: 11/17/2023]
Abstract
Particles are accelerated to very high, non-thermal energies during explosive energy-release phenomena in space, solar, and astrophysical plasma environments. While it has been established that magnetic reconnection plays an important role in the dynamics of Earth's magnetosphere, it remains unclear how magnetic reconnection can further explain particle acceleration to non-thermal energies. Here we review recent progress in our understanding of particle acceleration by magnetic reconnection in Earth's magnetosphere. With improved resolutions, recent spacecraft missions have enabled detailed studies of particle acceleration at various structures such as the diffusion region, separatrix, jets, magnetic islands (flux ropes), and dipolarization front. With the guiding-center approximation of particle motion, many studies have discussed the relative importance of the parallel electric field as well as the Fermi and betatron effects. However, in order to fully understand the particle acceleration mechanism and further compare with particle acceleration in solar and astrophysical plasma environments, there is a need for further investigation of, for example, energy partition and the precise role of turbulence.
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Affiliation(s)
- Mitsuo Oka
- Space Sciences Laboratory, University of California Berkeley, 7 Gauss Way, Berkeley, 94720 CA USA
| | - Joachim Birn
- Center for Space Plasma Physics, Space Science Institute, 4765 Walnut Street, Boulder, 80301 CO USA
- Los Alamos National Laboratory, Los Alamos, 87545 NM USA
| | - Jan Egedal
- Department of Physics, University of Wisconsin-Madison, 1150 University Avenue, Madison, 53706 WI USA
| | - Fan Guo
- Los Alamos National Laboratory, Los Alamos, 87545 NM USA
| | - Robert E. Ergun
- Laboratory for Atmospheric and Space Physics, University of Colorado, 1234 Innovation Drive, Boulder, 80303 CO USA
- Department of Astrophysical and Planetary Sciences, University of Colorado, 2000 Colorado Avenue, Boulder, 80309 CO USA
| | - Drew L. Turner
- The Johns Hopkins Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, 20723 MD USA
| | | | - Kyoung-Joo Hwang
- Southwest Research Institute, 6220 Culebra Road, San Antonio, 78238 TX USA
| | - Ian J. Cohen
- The Johns Hopkins Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, 20723 MD USA
| | - James F. Drake
- Department of Physics, The Institute for Physical Science and Technology and The Joint Space Science Institute, University of Maryland, College Park, 20742 MD USA
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3
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The Transport and Evolution of MHD Turbulence throughout the Heliosphere: Models and Observations. FLUIDS 2021. [DOI: 10.3390/fluids6100368] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
A detailed study of solar wind turbulence throughout the heliosphere in both the upwind and downwind directions is presented. We use an incompressible magnetohydrodynamic (MHD) turbulence model that includes the effects of electrons, the separation of turbulence energy into proton and electron heating, the electron heat flux, and Coulomb collisions between protons and electrons. We derive expressions for the turbulence cascade rate corresponding to the energy in forward and backward propagating modes, the fluctuating kinetic and magnetic energy, the normalized cross-helicity, and the normalized residual energy, and calculate the turbulence cascade rate from 0.17 to 75 au in the upwind and downwind directions. Finally, we use the turbulence transport models to derive cosmic ray (CR) parallel and perpendicular mean free paths (mfps) in the upwind and downwind heliocentric directions. We find that turbulence in the upwind and downwind directions is different, in part because of the asymmetric distribution of new born pickup ions in the two directions, which results in the CR mfps being different in the two directions. This is important for models that describe the modulation of cosmic rays by the solar wind.
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Li JH, Yang F, Zhou XZ, Zong QG, Artemyev AV, Rankin R, Shi Q, Yao S, Liu H, He J, Pu Z, Xiao C, Liu J, Pollock C, Le G, Burch JL. Self-consistent kinetic model of nested electron- and ion-scale magnetic cavities in space plasmas. Nat Commun 2020; 11:5616. [PMID: 33154395 PMCID: PMC7644639 DOI: 10.1038/s41467-020-19442-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2020] [Accepted: 10/15/2020] [Indexed: 12/05/2022] Open
Abstract
NASA’s Magnetospheric Multi-Scale (MMS) mission is designed to explore the proton- and electron-gyroscale kinetics of plasma turbulence where the bulk of particle acceleration and heating takes place. Understanding the nature of cross-scale structures ubiquitous as magnetic cavities is important to assess the energy partition, cascade and conversion in the plasma universe. Here, we present theoretical insight into magnetic cavities by deriving a self-consistent, kinetic theory of these coherent structures. By taking advantage of the multipoint measurements from the MMS constellation, we demonstrate that our kinetic model can utilize magnetic cavity observations by one MMS spacecraft to predict measurements from a second/third spacecraft. The methodology of “observe and predict” validates the theory we have derived, and confirms that nested magnetic cavities are self-organized plasma structures supported by trapped proton and electron populations in analogous to the classical theta-pinches in laboratory plasmas. Magnetic cavities play important roles in the energy cascade, conversion and dissipation in turbulent plasmas. Here, the authors show a theoretical insight into magnetic cavities by deriving a self-consistent, kinetic theory of these coherent structures.
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Affiliation(s)
- Jing-Huan Li
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China
| | - Fan Yang
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China
| | - Xu-Zhi Zhou
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China.
| | - Qiu-Gang Zong
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China.
| | - Anton V Artemyev
- Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, 90095, USA.,Space Research Institute, Russian Academy of Sciences, Moscow, 117997, Russia
| | - Robert Rankin
- Department of Physics, University of Alberta, Edmonton, AB, T6G2G7, Canada
| | - Quanqi Shi
- Institute of Space Sciences, Shandong University, Weihai, 264209, China
| | - Shutao Yao
- Institute of Space Sciences, Shandong University, Weihai, 264209, China
| | - Han Liu
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China
| | - Jiansen He
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China
| | - Zuyin Pu
- School of Earth and Space Sciences, Peking University, 100871, Beijing, China
| | - Chijie Xiao
- School of Physics, Peking University, Beijing, 100871, China
| | - Ji Liu
- National Space Science Center, Chinese Academy of Sciences, Beijing, 100190, China
| | | | - Guan Le
- NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA
| | - James L Burch
- Southwest Research Institute, San Antonio, TX, 78238, USA
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Abstract
Small-scale magnetic flux ropes in the solar wind have been studied for decades via both simulation and observation. Statistical analysis utilizing various in situ spacecraft measurements is the main observational approach, which helps investigate the generation and evolution of these small-scale structures. In this study, we extend the automated detection of small-scale flux ropes based on the Grad-Shafranov reconstruction to the complete data set of in situ measurements of the Ulysses spacecraft. We first discuss the temporal variation of the bulk properties of 22,719 flux ropes found through our approach, namely, the average magnetic field and plasma parameters, etc., as functions of the heliographic latitudes and heliocentric radial distances. We then categorize all identified events into three groups based on event distributions in different latitudes separated by 30°, at different radial distances, and under different solar activities. With the detailed statistical analysis, we conclude the following: (1) the properties of flux ropes, such as the duration, scale size, etc., follow power-law distributions, but with different slope indices, especially for distributions at different radial distances. (2) They are also affected by the solar wind speed, which has different distributions under different solar activities, manifested as a latitudinal effect. (3) The main difference in flux rope properties between the low and high latitudes is attributed to possible Alfvénic structures or waves and to flux ropes with relatively high Alfvénicity. (4) Flux ropes with longer durations and larger scale sizes occur more often at larger radial distances. (5) With a stricter Walén slope threshold, more events are excluded at higher latitudes, which further reduces the latitudinal effects on flux rope properties. The entire database is published online at http://www.fluxrope.info.
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Vlahos L, Anastasiadis A, Papaioannou A, Kouloumvakos A, Isliker H. Sources of solar energetic particles. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20180095. [PMID: 31079581 PMCID: PMC6527952 DOI: 10.1098/rsta.2018.0095] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/15/2019] [Indexed: 06/09/2023]
Abstract
Solar energetic particles are an integral part of the physical processes related with space weather. We present a review for the acceleration mechanisms related to the explosive phenomena (flares and/or coronal mass ejections, CMEs) inside the solar corona. For more than 40 years, the main two-dimensional cartoon representing our understanding of the explosive phenomena inside the solar corona remained almost unchanged. The acceleration mechanisms related to solar flares and CMEs also remained unchanged and were part of the same cartoon. In this review, we revise the standard cartoon and present evidence from recent global magnetohydrodynamic simulations that support the argument that explosive phenomena will lead to the spontaneous formation of current sheets in different parts of the erupting magnetic structure. The evolution of the large-scale current sheets and their fragmentation will lead to strong turbulence and turbulent reconnection during solar flares and turbulent shocks. In other words, the acceleration mechanism in flares and CME-driven shocks may be the same, and their difference will be the overall magnetic topology, the ambient plasma parameters, and the duration of the unstable driver. This article is part of the theme issue 'Solar eruptions and their space weather impact'.
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Affiliation(s)
- Loukas Vlahos
- Department of Physics, Aristotle University, Thessaloniki 54124, Greece
| | - Anastasios Anastasiadis
- Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, Penteli 15236, Greece
| | - Athanasios Papaioannou
- Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, Penteli 15236, Greece
| | | | - Heinz Isliker
- Department of Physics, Aristotle University, Thessaloniki 54124, Greece
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Pavlos EG, Malandraki OE, Khabarova OV, Karakatsanis LP, Pavlos GP, Livadiotis G. Non-Extensive Statistical Analysis of Energetic Particle Flux Enhancements Caused by the Interplanetary Coronal Mass Ejection-Heliospheric Current Sheet Interaction. ENTROPY 2019; 21:e21070648. [PMID: 33267362 PMCID: PMC7515141 DOI: 10.3390/e21070648] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/18/2019] [Revised: 06/19/2019] [Accepted: 06/27/2019] [Indexed: 11/17/2022]
Abstract
In this study we use theoretical concepts and computational-diagnostic tools of Tsallis non-extensive statistical theory (Tsallis q-triplet: qsen, qrel, qstat), complemented by other known tools of nonlinear dynamics such as Correlation Dimension and surrogate data, Hurst exponent, Flatness coefficient, and p-modeling of multifractality, in order to describe and understand Small-scale Magnetic Islands (SMIs) structures observed in Solar Wind (SW) with a typical size of ~0.01–0.001 AU at 1 AU. Specifically, we analyze ~0.5 MeV energetic ion time-intensity and magnetic field profiles observed by the STEREO A spacecraft during a rare, widely discussed event. Our analysis clearly reveals the non-extensive character of SW space plasmas during the periods of SMIs events, as well as significant physical complex phenomena in accordance with nonlinear dynamics and complexity theory. As our analysis also shows, a non-equilibrium phase transition parallel with self-organization processes, including the reduction of dimensionality and development of long-range correlations in connection with anomalous diffusion and fractional acceleration processes can be observed during SMIs events.
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Affiliation(s)
- Evgenios G. Pavlos
- IAASARS, National Observatory of Athens, 15236 Penteli, Greece
- Research Team of Chaos and Complexity, Department of Environmental Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
- Correspondence: (E.G.P); (L.P.K.); Tel.: +30-2108109114 (E.G.P.); +30-2541079114 (L.P.K.)
| | | | - Olga V. Khabarova
- Heliophysical Laboratory, Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation RAS (IZMIRAN), Troitsk, Moscow 108840, Russia
| | - Leonidas P. Karakatsanis
- Research Team of Chaos and Complexity, Department of Environmental Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
- Correspondence: (E.G.P); (L.P.K.); Tel.: +30-2108109114 (E.G.P.); +30-2541079114 (L.P.K.)
| | - George P. Pavlos
- Research Team of Chaos and Complexity, Department of Environmental Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
- Department of Electrical Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
| | - George Livadiotis
- Space Science & Engineering, Southwest Research Institute, San Antonio, TX 78238, USA
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Particle Acceleration at 5 au Associated with Turbulence and Small-scale Magnetic Flux Ropes. ACTA ACUST UNITED AC 2019. [DOI: 10.3847/1538-4357/aafcb2] [Citation(s) in RCA: 41] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Automated Detection of Small-scale Magnetic Flux Ropes in the Solar Wind: First Results from the
Wind
Spacecraft Measurements. ACTA ACUST UNITED AC 2018. [DOI: 10.3847/1538-4365/aae57d] [Citation(s) in RCA: 41] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Chen Y, Hu Q, Roux JL, Zheng J. Observational Analysis of Small-scale Magnetic Flux Ropes from Ulysses In-situ Measurements. ACTA ACUST UNITED AC 2018. [DOI: 10.1088/1742-6596/1100/1/012006] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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11
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Self-consistent Energetic Particle Acceleration by Contracting and Reconnecting Small-scale Flux Ropes: The Governing Equations. ACTA ACUST UNITED AC 2018. [DOI: 10.3847/1538-4357/aad8b3] [Citation(s) in RCA: 44] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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12
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An Unusual Energetic Particle Flux Enhancement Associated with Solar Wind Magnetic Island Dynamics. ACTA ACUST UNITED AC 2018. [DOI: 10.3847/2041-8213/aaddf6] [Citation(s) in RCA: 55] [Impact Index Per Article: 9.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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13
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Kilpua E, Koskinen HEJ, Pulkkinen TI. Coronal mass ejections and their sheath regions in interplanetary space. LIVING REVIEWS IN SOLAR PHYSICS 2017; 14:5. [PMID: 31997985 PMCID: PMC6956910 DOI: 10.1007/s41116-017-0009-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2017] [Accepted: 10/03/2017] [Indexed: 06/09/2023]
Abstract
Interplanetary coronal mass ejections (ICMEs) are large-scale heliospheric transients that originate from the Sun. When an ICME is sufficiently faster than the preceding solar wind, a shock wave develops ahead of the ICME. The turbulent region between the shock and the ICME is called the sheath region. ICMEs and their sheaths and shocks are all interesting structures from the fundamental plasma physics viewpoint. They are also key drivers of space weather disturbances in the heliosphere and planetary environments. ICME-driven shock waves can accelerate charged particles to high energies. Sheaths and ICMEs drive practically all intense geospace storms at the Earth, and they can also affect dramatically the planetary radiation environments and atmospheres. This review focuses on the current understanding of observational signatures and properties of ICMEs and the associated sheath regions based on five decades of studies. In addition, we discuss modelling of ICMEs and many fundamental outstanding questions on their origin, evolution and effects, largely due to the limitations of single spacecraft observations of these macro-scale structures. We also present current understanding of space weather consequences of these large-scale solar wind structures, including effects at the other Solar System planets and exoplanets. We specially emphasize the different origin, properties and consequences of the sheaths and ICMEs.
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Affiliation(s)
- Emilia Kilpua
- Department of Physics, University of Helsinki, Helsinki, Finland
| | - Hannu E. J. Koskinen
- Department of Physics, University of Helsinki, Helsinki, Finland
- Finnish Meteorological Institute, Espoo, Finland
| | - Tuija I. Pulkkinen
- Department of Electronics and Nanoengineering, Aalto University, Espoo, Finland
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14
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The Origin of Compressible Magnetic Turbulence in the Very Local Interstellar Medium. ACTA ACUST UNITED AC 2017. [DOI: 10.3847/1538-4357/aa7685] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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15
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Servidio S, Haynes CT, Matthaeus WH, Burgess D, Carbone V, Veltri P. Explosive Particle Dispersion in Plasma Turbulence. PHYSICAL REVIEW LETTERS 2016; 117:095101. [PMID: 27610862 DOI: 10.1103/physrevlett.117.095101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Indexed: 06/06/2023]
Abstract
Particle dynamics are investigated in plasma turbulence, using self-consistent kinetic simulations, in two dimensions. In the steady state, the trajectories of single protons and proton pairs are studied, at different values of plasma β (ratio between kinetic and magnetic pressure). For single-particle displacements, results are consistent with fluids and magnetic field line dynamics, where particles undergo normal diffusion for very long times, with higher β's being more diffusive. In an intermediate time range, with separations lying in the inertial range, particles experience an explosive dispersion in time, consistent with the Richardson prediction. These results, obtained for the first time with a self-consistent kinetic model, are relevant for astrophysical and laboratory plasmas, where turbulence is crucial for heating, mixing, and acceleration processes.
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Affiliation(s)
- S Servidio
- Dipartimento di Fisica, Università della Calabria, I-87036 Cosenza, Italy
| | - C T Haynes
- School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
| | - W H Matthaeus
- Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA
| | - D Burgess
- School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
| | - V Carbone
- Dipartimento di Fisica, Università della Calabria, I-87036 Cosenza, Italy
| | - P Veltri
- Dipartimento di Fisica, Università della Calabria, I-87036 Cosenza, Italy
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