Institute for Research in Electronics and Applied Physics

Mikhail I. Sitnov
Associate Research Scientist

Mailing Address:
Institute for Research in
Electronics and Applied Physics
A.V. Williams Bldg., Rm. 3337
University of Maryland
College Park, Maryland
20742-3511

Phone: (301) 405-1632
Email: sitnov at umd.edu
FAX: (301) 405-1678


Research • CV • Publications

Mikhail Sitnov is an Associate Research Scientist in the Institute for Research in Electronics and Applied Physics at the University of Maryland, College Park. He received his PhD in physics from Moscow State University, Russia, in 1986. His current research interests are: (1) thin current sheets, (2) onset of magnetic reconnection in collisionless plasmas, (3) finite element analysis and particle simulations, (4) nonlinear time series analysis of magnetospheric data, including global indices of activity and magnetospheric magnetic field, and (5) phase transition physics with applications to magnetospheric activity.


The Second Workshop on Thin Current Sheets

Selected Research Topics


Thin Current Sheets

Curved magnetic fields in plasmas, which are of particular interest in studies of magnetic reconnection and dynamo, are provided by the localized electric currents flowing in plasmas. The more is the magnetic field line curvature the more must be the current concentration. The most entangled magnetic fields must be therefore maintained by singular current formations, current lines and current sheets. How singular may be these singular formations? It is well known in particular that when the current density in a current sheet reaches some critical maximum (or correspondingly, its thickness reaches a minimum), the sheet is disrupted leading to magnetic reconnection, which provides the release of the magnetic energy and its transformation into the kinetic energy of plasma particles.

A longstanding belief, that the minimum current sheet thickness is much in excess of any micro scales such as the Larmor radii of electrons and ions, was based largely on the assumption of particle collisions in plasmas. Collisions make the medium resistive and thus provide the reconnection in the current sheet far before its thickness reaches micro scales. However, in many interesting plasma objects the mean-free path exceeds the characteristic scales of the system. In particular, solar flares and onsets of magnetic reconnection in the tail of Earth's magnetosphere, occur in practically collisionless plasmas. The current sheet thickness prior to the reconnection onset in such plasmas may be as small as the thermal ion gyroradius in the field outside the sheet. The ion orbits in these thin sheets may strongly deviate from the classical Larmor gyration and more resemble the figure of eight. As a result, thin collisionless current sheets have a number of unusual properties. For example, a thin current sheet (TCS) can be embedded within a thicker plasma sheet. Also, the current density in TCS can be dominated by electrons, in contrast to the classical theory, which predicts domination of ions because of their higher temperature.

One of the most recent findings is the explicit demonstration that the TCS may have a bifurcated structure with two current density peaks separated by a current depression region at the sheet center. These results have been obtained using four-spacecraft CLUSTER observations in the tail of Earth's magnetosphere, which allowed, for the first time, the unambiguous separation of spatial and temporal variability. CLUSTER revealed in particular that in some cases the profile of the magnetic field across the sheet has characteristic inlection points (Figure 1, left panels), which correspond to the off-center peaks of the current density dJ/dz ~ d2B/dz2 = 0. It turns out that this bifurcation effect can be explained neither by the classical current sheet theory [Harris, 1962] nor by other conventional mechanisms. For example, it might be explained as a statistical effect of TCS flapping motions, which were indeed observed together with the bifurcation effect (Figure 1, right panels). However, the bifurcation has been detected in individual TCS crossings (Figure 1, left panel). Another explanation might be reconnection effects related either to the Hall-MHD physics or to slow shocks. However, CLUSTER detected neither the dawn-dusk magnetic field consistent with the Hall-MHD nor the plasma flows typical for slow shocks.

Figure 1. Bifurcated current sheets observed by CLUSTER mission. Left panels show the magnetic field profile with characteristic inflection points [Sergeev et al. GRL, 30, 1327, 2003; More information]. Right panels show bifurcated sheet flapping motions inferred from observations [Runov et al., GRL, 30, 1036, 2003; More information].

Recently we proposed a generalization of the Harris current sheet theory, which explains the effects of embedding and bifurcation. It also explicitly takes into account the distortion of the Larmor orbit in thin current sheets. In fact, the main new element of the theory is the special invariant of the particle motion in thin current sheets [Speiser, 1965; Sonnerup, 1971], similar to the magnetic moment for conventional Larmor gyration. The new invariant implies some plasma anisotropy, which is indeed very efficient in providing the current bifurcation [Cowley, 1978]. The results of the new theory are shown on the left panels in Figure 2.

Figure 2. Left panels show profiles of embedded (top) and bifurcated (bottom) current sheets obtained using a new equilibrium theory [Sitnov et al., 2003]. b, f, n and j denote the dimensionless magnetic field, electrostatic potential, plasma and current density, respectively; z is the distance from the sheet central plane in units of the thermal ion gyroradius in the field outside the sheet. Dashed lines show the corresponding profiles of the Harris current sheet. Dash-dotted and dash-triple dotted lines show the contributions to the current density of ion and electron species. Right panels show flapping motions of the bifurcated current sheets [Sitnov et al., 2004] reproduced using the new equilibrium theory and the explicit PIC code P3D [Zeiler et al., 2002]. y and z denote the coordinates along the equilibrium current and across the sheet plane, respectively, normalized by the ion inertial length d.

The effect of the current sheet embedding shown in the four top panels on the left appears in case of cigar ion distributions outside the sheet when the ion temperature along the magnetic field exceeds their temperature across the field. It can be best seen from the comparison of plasma and current density profiles. The current sheet bifurcation requires an opposite anisotropy with the larger temperature across the magnetic field. It is shown on the four bottom panels and can be best seen on the current density plot. On the other hand, the corresponding magnetic field profile is most relevant for comparison with observations shown in Figure 1 (left panel). Another interesting feature of the new theory, which is consistent with observations (not shown; for details see [Sergeev et al., 2003]), is the formation of a plateau in the profile of the plasma density between the current density peaks.

However, the TCS flapping motions (Figure 1, right panels) cannot be explained by the quilibrium theory. To explore the evolution of the bifurcated sheet we performed a set of 2D particle simulations using the massively parallelized code P3D [Zeiler et al., 2002] and starting from the equilibria of the type of those shown on the lower left panels in Figure 2. To load the complicated strongly non-Maxwellian ion distributions we used the rejection method [e.g., Press et al., 1999] in 3D space, including one spatial coordinate and two coordinates in velocity space. The right panels in Figure 2 show some results of these simulations. In particular, the left column of color plates shows the development of the lower-hybrid drift instability at the outer edges of the split current. The LHDI results in the transition to another quasi-equilibrium state with a considerable contribution of electrons to the bifurcated current. At later time the instabilities of both parities are detected inside the sheet. They have much larger wavelengths and end up with the kink-type flapping motions of the bifurcated sheet as a whole (right column of color panels in Figure 2). Overall, the comparison of Figures 1 and 2 reveals the amazing consistency between recent CLUSTER observations and the newly developed current sheet theory.

This research is supported by the National Aeronautics and Space Administration, the National Science Foundation and the U.S. Department of Energy.



Reconnection Onset


Recent studies of the collisionless magnetic reconnection process are largely limited to the magnetic field configurations with the X-line pattern being already formed. Thus, they leave open the problem of the reconnection onset, which requires the stability study of configurations where no magnetic field line is reconnected yet. These configurations can be divided into two basic classes, namely, the sheared magnetic field, which is typical for fusion devices and Earth's magnetopause, and the stretched magnetic field, which is a distinctive feature of Earth's magnetotail. While the onset of reconnection in the sheared magnetic field is understood relatively well, its mechanism in tail-like magnetic field configurations has been a big puzzle for more than a quater of century preventing proper modeling of many interesting space phenomena.

The instability responsible for the onset of reconnection, namely the tearing instability, was first considered for the collisionless plasma with applications to the magnetotail by Coppi et al. [1966] in the simplest geometry with antiparallel magnetic field lines. The instability was described as a growth of the negative energy wave (due to the mutual attraction of parallel current filaments) with the dissipation provided by the Landau resonance of the nonmagnetized electrons near the neutral plane, where the magnetic field has a null. In contrast, the more realistic current sheets with strongly stretched magnetic field (without null points) were proved to be stable under very general conditions [Lembege and Pellat, 1982]. The stabilization effect arises due to the drift motion of electrons in the crossed fields, the equilibrium magnetic field component normal to the sheet plane and the tearing electric field component directed along the equilibrium current. Drifting electrons drag ions along the direction of the main (antiparallel) magnetic field components, and the energy necessary for dragging exceeds the free energy available for the tearing mode due to the mutual attraction of parallel current filaments. This effect looks quite universal and difficult to cirumvent. In fact, a number of attempts to destabilize the tearing mode either by the external diffusion or by dynamical chaos in electron orbits have failed.

Recently the reconnection onset problem was reconsidered [Sitnov et al., 2002]. The stability problem was solved using the finite element technique and the drift-kinetic description of the electron species with additional averaging over the bounce motion of trapped electrons. The tearing mode was found unstable for large ion-to-electron temperature ratios typical for the tail current sheet of Earth's magnetosphere (Ti/Te >> 1) when this sheet is sufficiently long, so that the electrons leaving it may be treated as transient particles. Comparison of the theory with earlier fluid modeling showed that the onset of reconnection is controlled by the Hall effect and a purely kinetic effect arising from different responses of the trapped and transient electrons.

Figure 3. Schematic picture of the X- and Y- line reconnection processes in the tail of Earth's amgnetosphere emerging from the stability analysis [Sitnov et al., 2002].

The effect of transient electrons results in a peculiar feature of the reconnection onset on tail-like systems. Based on general properties of wave perturbations in a current sheet, one would expect the formation of magnetic islands (plasmoids) at the distance Lon from the tail footing (the near-Earth end of the tail in the case of the magnetosphere) comparable to the length Lp of the plasmoid itself. However, the new onset theory predicts Lon >> Lp (Figure 3). It also predicts the formation of a thin current sheet in the region earthward of Lon. The matter is that the theory of current sheet perturbations [Syrovatskii, 1971; Kulsrud and Hahm, 1982; Schindler and Birn, 1993] assumes the formation of either X-lines with the change of magnetic topology or Y-lines (discontinuities in fluid models or thin current sheets in kinetics). Therefore, far from the tail footing the formation of the X-line is possible, whereas at smaller distances (closer to the Earth in the case of the magnetosphere) only thin current sheets can be formed.

Figure 4. Left panel: Length of plasmoids versus the XGSM coordinate (The X GSM axis coincide with the direction to the Sun with its origin at the center of the Earth) [Ieda et al., JGR, 103, 4453, 1998]. Right panel: Evolution of the current density around the onset (a) for the earthward flow events and (b) for the tailward flow events and their averages (c) [Asano et al., JGR, 109, A05213, 2004].

Surprizingly, these theoretical predictions are fully consistent with recent observations from the GEOTAIL mission , which provided large statistics of plasmoids and current sheets in the tail of the magnetosphere. In particular, Ieda et al. [1998] showed that while the minimum distance, at which GEOTAIL observed the formation of plasmoids, is 24 RE (Earth radii), the characteristic size of the plasmoid itself in that region is only 4 RE (Figure 4, left panel). On the other hand, Asano et al. [2004] compared the current sheet thinning properties for earthward and tailward flow events that correspond to regions earthward and tailward of the near-Earth neutral line, respectively. They found a significant increase of the current density for earthward flow events prior to the substorm onset and no such increase for tailward flow events (Figure 4, right panel), suggesting that prior to the reconnection onset the TCS forms only earthward of the neutral line.

This research is supported by the National Aeronautics and Space Administration, the National Science Foundation the U.S. Department of Energy.



Phase Transitions in Magnetosphere


Earth's magnetosphere is a huge cavity created by the magnetic field of our planet in the flow of the plasma coming from the Sun (solar wind). Part of the solar wind energy penetrates this cavity due mainly to the reconnection at the outer edge of the magnetosphere (magnetopause), accumulates in the magnetosphere, and is then suddenly released. The most strongly pronounced phenomena associated with these storage and release processes are called magnetospheric substorms. The magnetosphere is usually far from equilibrium because of the persistent external driving by the turbulent solar wind as well as its own inherently unstable plasmas.

At first sight, such a complex system as the magnetosphere requires very sophisticated first-principle modeling tools. An example may be global MHD simulations. However, the use of first-principle models has many limitations. They often require too much computer resources and yet remain imprecise, for instance, in determining the timing of the substorm onset. One can propose another approach to complex system modeling. It is a more pragmatic approach without recourse to the tools of classical physics. Let us consider our system, the magnetosphere, as a black box with some input and output (Figure 5) and let us try to create a model of system's dynamics directly from data, using the techniques of signal processing, nonlinear dynamics and statistics. This is an empirical or data-derived approach. It often provides the optimum level of resolution and strongly complements the first-principle models being more robust and efficient in many applications.

Figure 5. Representation of the magnetosphere in a data-derived approach. Input parameters are represented here by the solar wind speed v, southward component of the interplanetary magnetic field Bs, and dynamical pressure Pdyn, while the output is represented by the auroral indices AL and AE reflecting the substorm activity, and the disturbance storm-time index Dst.

The magnetosphere is an open system driven by the turbulent solar wind and exhibits complex behavior with global and multiscale characteristics. Consequently, its data-based modeling involves various linear and nonlinear filter techniques. Its complexity was interpreted as a result of the tubulent solar wind driving, dynamical chaos effects as well as the self-organized criticality. Recently a new class of models, which take into account both the organized nonlinear component of the magnetospheric dynamics and their scale-invariant features as well as their dependence on the solar wind input, has been elaborated [Sitnov et al., 2000, 2001; Ukhorskiy et al., 2004a,b]. Sitnov et al. [2000, 2001] applied the technique of the singular spectrum analysis (SSA) [Broomhead and King, 1986] to the combined set of input and output substorm data [Bargatze et al., 1985] and revealed that the magnetospheric dynamics share a number of important properties with non-equilibrium phase transitions. It was shown in particular that, after some averaging in phase space, the substorm dynamics looks as a counter-clockwise motion of a point on a two-level surface in the 3D space formed by two functions of the input vBs and output AL (Figure 6, left panel). It was also shown that, in contrast to classical SOC models and in full analogy with classical phase transition physics, some scale-free properties of the magnetospheric activity are connected to the corresponding scale-free properties of the solar wind input, and the connection exponent is similar to the well-known exponent b (Figure 6, right panel), relating the density and temperature fluctuations near the critical point of water-steam transitions [Sitnov et al., 2001].
Figure 6. First and second order phase transition in the magnetosphere. Left panels: Average substrom dynamics of the magnetosphere inferred from the singular spectrum analysis of AL-vBs for the whole year 1998 [Sitnov et al., 2004]. Variables X1, X2 (color coded) and X3 are obtained by projecting the time-delayed input-output vector X(t)=(It,...It-(M-1),Ot,...Ot-(M-1)), It=vBs(t), Ot=AL(t) onto three principal SSA eigenvectors V1, V2 and V3 (j=1,...,2M) shown on the yellow-blue panels. Right panel: Scale invariant relation between solar wind (P1~X1) and magnetospheric (P2~X2) parameters [Sitnov et al., 2001].

The phase transition analogy made several important contributions to our understanding of modeling solar wind-magnetosphere interaction. First, it showed that global coherent and multiscale phenomena may co-exist like first- and second order transitions in the water-steam system do. Second, it revealed the importance of averaging in the phase space necessary to reveal the "mean-field" dynamical features. Third, it confirmed that the mutiscale features of the magnetospheric dynamics depend on the solar wind conditions. Therefore, the probabilistic description of those multiscale features must have the form of conditional probabilities. The utilization of these new concepts have resulted in creating a new generation of tools to model and forecast the magnetospheric activity, which combine the deterministic and probabilistic forecasts [Ukhorskiy et al., 2004a; Ukhorskiy et al., 2004b].

This work is supported by the National Science Foundation.


Publications since 1997
  1. Sitnov, M. I., M. Swisdak, P. N. Guzdar, and A. Runov, Structure and dynamics of a new class of thin current sheets, J. Geophys. Res., v.111, A08204, doi:10.1029/2005JA011517, 2006. Full text (PDF, 906 KB)

  2. Sitnov, M. I., P. N. Guzdar, and M. Swisdak, On the formation of a plasma bubble, Geophys. Res. Lett., v.32, L16103, doi:10.1029/2005GL023585, 2005. Full text (PDF, 702 KB)

  3. Tsyganenko, N. A., and M. I. Sitnov, Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms, J. Geophys. Res., v.110, A03208, doi:10.1029/2004JA010798, 2005.

  4. Sitnov, M.I., A. Y. Ukhorskiy, A. S. Sharma, and R. S. Weigel, Roles of chaos, self-organized criticality and phase transitions in magnetospheric physics, in: "Multiscale Coupling of Sun-Earth Processes", Ed. by A. T. Y. Lui, Y. Kamide, and G. Consolini, Elsevier, B. V., pp.195-215, 2005.

  5. Sitnov, M. I., M. Swisdak, J. F. Drake, P. N. Guzdar, and B. N. Rogers, A model of the bifurcated current sheet 2: Flapping motions, Geophys. Res. Lett., v.31, L09805, doi:10.1029/2004GL019473, 2004. Full text (PDF, 314 KB)

  6. Ukhorskiy, A. Y., Sitnov, M. I., Sharma, A. S., Anderson, B. J., Ohtani, S., Lui, A. T. Y., Data-derived forecasting model for relativistic electron intensity at geosynchronous orbit, Geophys. Res. Lett., v.31, L09806, doi:10.1029/2004GL019616, 2004. Full text (PDF, 314 KB)

  7. Ukhorskiy, A. Y., Sitnov, M. I., Sharma, A. S., Papadopoulos, and K., Global and multi-scale features of solar wind-magnetosphere coupling: From modeling to forecasting, Geophys. Res. Lett., v.31, L08802, doi:10.1029/2003GL018932, 2004. Full text (PDF, 314 KB)

  8. Sitnov, M. I., Lui, A. T. Y., Guzdar, P. N., Yoon, P. H., Current-driven instabilities in forced current sheets, J. Geophys. Res., v.109, A3, A03205, doi:10.1029/2003JA010123, 2004. Full text (PDF, 314 KB)

  9. Sitnov, M. I., Guzdar, P. N., and Swisdak, M., A model of the bifurcated current sheet, Geophys. Res. Lett., v.30(13), 712, doi:10.1029/2003GL017218, 2003. Full text (PDF, 314 KB)

  10. Ukhorskiy, A. Y., M. I. Sitnov, A. S. Sharma, and K. Papadopoulos, Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling, Ann. Geophys., 21, 1913, 2003. Full text (PDF, 314 KB)

  11. Shao,X., M. I. Sitnov, A. S. Sharma, K. Papadopoulos, C. C. Goodrich, P. N. Guzdar, G. M. Milikh, M. J. Wiltberger, and J. G. Lyon, Phase Transition-Like Behavior of Magnetospheric Substorms: Global MHD Simulation Results, J. Geophys. Res. 108 (A1), 1037, doi: 10.1029/2001JA009237, 2003.

  12. Sharma, A. S., A. Y. Ukhorskiy, M. I. Sitnov, and J. A. Valdivia, Modeling the magnetosphere using time series data, in: "Disturbances in Geospace: The storm-substorm relationship", Ed. by A. S. Sharma, Y. Kamide and G. S. Lakhina, Geophys. Monogr. 142, pp.231-241, 2003.

  13. Sitnov, M. I., A. S. Sharma, P. N. Guzdar, and P. H. Yoon, Reconnection onset in the tail of Earth's magnetosphere, J. Geophys. Res., v.107, A9, SMP-20, 2002. Full text (PDF, 314 KB)

  14. Sitnov, M. I., A. S. Sharma, A. T. Y. Lui, and P. H. Yoon, The Significance of Tail Instabilities in Triggering Substorm Onset, Proc. 6-th International Conference on Substorms, Seattle, 25-29 March, pp.231-238, 2002. Full text (PDF, 168 KB)

  15. Ukhorskiy, A. Y., Sitnov, M. I., A. S. Sharma, and K. Papadopoulos, Modeling and Forecasting of the Multi-Scale Features of Magnetospheric Dynamics during Substorms, Proc. 6-th International Conference on Substorms, Seattle, 25-29 March, pp. 496-501, 2002. Full text (PDF, 209 KB)

  16. Ukhorskiy, A. Y., M. I. Sitnov, A. S. Sharma, and K. Papadopoulos, Global and multi-scale aspects of magnetospheric dynamics in local-linear filters, J. Geophys. Res., 107(A11), 10.1029/2001JA009160, 2002.

  17. Yoon, P. H., A. T. Y. Lui, and M. I. Sitnov, Generalized lower-hybrid drift instabilities in current-sheet equilibrium, Phys. Plasm., v.9, p.1526, 2002.

  18. Sitnov, M. I., A. S. Sharma, K. Papadopoulos, D. Vassiliadis, Modeling substorm dynamics of the magnetosphere: From self-organization and self-organized criticality to nonequilibrium phase transitions, Phys. Rev. E, v.65, p.016116, 2001. Full text (PDF, 410 KB)

  19. Sharma, A. S., M. I. Sitnov, and K. Papadopoulos, Substorms as nonequilibrium transitions of the magnetosphere, J. Atm. Sol.-Terr. Phys., v. 63, p.1399, 2001.

  20. Sitnov, M. I., A. S. Sharma, K. Papadopoulos, D. Vassiliadis, J. A. Valdivia, A. J. Klimas, D. N. Baker, Phase transition-like behavior of the magnetosphere during substorms,  J. Geophys. Res., v. 105, p. 12,955, 2000. Full text (GZIP PDF, 6.67 MB)

  21. Sitnov, M. I., H. V. Malova, L. M. Zelenyi, and A. S. Sharma, Thin current sheet embedded within a thicker plasma sheet: Self-consistent kinetic theory, J. Geophys. Res, v. 105, p. 13,029, 2000. Full text (GZIP PDF, 3.07 MB)

  22. Sitnov, M. I. and  A. S. Sharma, Magnetotail thin current sheet models and their role in substorm physics, Proc. 5-th Int. Conf. on Substorms, St. Petersburg, Russia, 16-20 May 2000, ESA SP-443, p.113, 2000.

  23. Sitnov, M. I., A. S. Sharma, L. M. Zelenyi, H. V. Malova, Distinctive features of forced current sheets: Electrostatic effects, Proc. 5-th Int. Conf. on Substorms, St. Petersburg, Russia, 16-20 May 2000, ESA SP-443, p.197, 2000.

  24. Sharma, A. S., V. A. Sergeev, M. I. Sitnov, and K. Papadopoulos, Global and multi-scale features of substorms inferred from ground-based and multi-spacecraft data, Proc. 5-th Int. Conf. on Substorms, St. Petersburg, Russia, 16-20 May 2000, ESA SP-443, p.193, 2000.

  25. Malova, H. V., M. I. Sitnov, L. M. Zelenyi, A. S. Sharma, Self-consistent model of 1D current sheet:the role of drift, magnetization and diamagnetic currents, in: "Magnetospheric Current Systems", Geophys. Monogr. 118, AGU, p.313, 2000.

  26. Zelenyi, L. M., M. I. Sitnov, H. V. Malova, and A. S. Sharma, Thin and superthin ion current sheets. Quasiadiabatic and nonadiabatic models, Nonlin. Proc. in Geophys., v.7, p.127, 2000.

  27. Sitnov, M. I., H. V.Malova, and A. S. Sharma, Linear stability of a tearing mode in a quasi-neutral current sheet, Plasma Phys. Rep., v. 25, p. 227, 1999.

  28. Sitnov, M. I., and A. T. Y. Lui, Cross-field current instability as a catalyst of the explosive reconnection in the geomagnetotail, J. Geophys. Res, v. 104, p. 6941, 1999.

  29. Kropotkin, A. P., M. I. Sitnov, and H. V. Malova, Self-consistent structure of thin anisotropic current sheet, Izvestiya AN, ser. Phys., v.63, no.8, p. 10 (in Russian), 1999.

  30. Sitnov, M. I., and A. S. Sharma, Role of transient electrons and microinstabilities in the tearing instability of the geomagnetotail current sheet and the general scenario of the substorm as a catastrophe, In: SUBSTORMS-4, Edited by S. Kokubun and Y. Kamide, Terra Scientific Publ. Co./Kluwer Academic Publishers, p. 539, 1998.

  31. Sitnov, M. I., H. V.Malova, and A. S. Sharma, Role of the temperature ratio in the linear stability of the quasi-neutral sheet tearing mode, Geophys. Res. Lett., v. 25, p. 269, 1998

  32. Kropotkin, A.P., H.V.Malova, M.I.Sitnov. Self-consistent structure of a thin anisotropic current sheet, J. Geophys. Res., v. 102, p. 22,099, 1997.

  33. Sitnov, M.I., Malova H.V., Lui A.T.Y. Quasineutral sheet tearing instability induced by electron preferential acceleration from stochasticity, J. Geophys. Res., v. 102, p. 163, 1997.

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