Formation and dynamics of coherent structures involving phase-space vortices in plasmas

Bengt Eliasson, Padma Shukla

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109 Citations (Scopus)


We present a comprehensive review of recent theoretical and numerical studies of the formation and dynamics of electron and ion holes in a collisionless plasma. The electron hole is characterized by a localized positive potential in which a population of electrons is trapped, and a depletion of the electron number density, while the ion hole is associated with localized negative potential in which a population of ions is trapped and a depletion of both the ion and electron densities occur. We present conditions for the existence of quasi-stationary electron and ion holes in unmagnetized and magnetized electron–ion plasmas, as well as in an unmagnetized pair-ion plasma. An interesting aspect is the dynamic interactions between ion and electron holes and the surrounding plasma. The dynamics is investigated by means of numerical simulations in which analytical solutions of quasi-stationary electron and ion holes are used as initial conditions. Our Vlasov simulations of two colliding ion holes reveal the acceleration of electrons and a modification of the initially Maxwellian electron distribution function by the ion hole potentials, which work as barriers for the electrons. A secondary effect is the excitation of high-frequency plasma waves by the streams of electrons. On the other hand, we find that electron holes show an interesting dynamics in the presence of mobile ions. Since electron holes are associated with a positive electrostatic potential, they repel positively charged ions. Numerical simulations of electron holes in a plasma with mobile ions exhibit that the electron holes are, in general, attracted by ion density maxima and repelled by ion density minima. Therefore, standing electron holes can be accelerated by the self-created ion cavities. In a pair plasma, both the temperatures and masses of the positively and negatively charged particles are assumed equal, and therefore one must treat both species with a kinetic theory on an equal footing. Accordingly, a phase-space hole in one of the species is associated with reflected particles in the opposite species. Here standing solitary large-amplitude holes are not allowed, but they must have a propagation speed close to the thermal speed of the particles. We discuss extensions of the existing non-relativistic theories for the electron and ion holes in two directions. First, we describe a fully nonlinear kinetic theory for relativistic electron holes (REHs) in an unmagnetized plasma, and show that the REHs have amplitudes and widths much larger than their non-relativistic counterparts. Second, we extend the weakly nonlinear theories for the electron and ion holes to include the effects of the external magnetic field and the plasma inhomogeneity. The presence of the external magnetic field gives rise to the electron and ion hole structures which have differential scale sizes along and across the magnetic field direction. Consideration of the density inhomogeneity in a magnetized plasma with non-isothermal electron distribution function provides the possibility of bipolar electrostatic pulses that move with the electron diamagnetic drift across the density inhomogeneity and the magnetic field directions. In a plasma with sheared magnetic field we can have fast reconnection involving phase-space vortices in the lower-hybrid frequency range. We further present theoretical and numerical investigations of the nonlinear interaction between ion and electron holes and large amplitude high-frequency electrostatic and electromagnetic waves in plasmas. Since the ion hole is associated with a depletion of both the electron and ion densities, it can work as a resonance cavity for trapped Langmuir waves whose frequencies are below the electron plasma frequency of the unperturbed plasma. Large-amplitude Langmuir waves can modify the ion hole due to the ponderomotive force which pushes the electrons away from the center of the hole. This process can be modeled by a coupled nonlinear Schrödinger and Poisson system of equations, which exhibit a discrete set of trapped Langmuir eigenmodes in the ion hole in plasmas without and with high-Z charged impurities. In a kinetic description, however, the trapped Langmuir waves are Landau damped due to the relatively small length scale of the ion hole. Intense electromagnetic waves (IEMWs) can also be trapped by the REHs in an unmagnetized plasma. We study the localization of arbitrary large amplitudes IEMWs by incorporating nonlinearities associated with relativistic electron mass increase, relativistic ponderomotive force of the IEMWs, and non-isothermal electron distributions. We find that the combined effects of these three nonlinearities produce extremely large amplitude REHs which traps localized IEMWs. Such a nonlinear structure can accelerate electrons to very high energies. The relevance of our investigation to numerous localized structures observed in laboratory and space plasmas is discussed. We also propose to conduct new experiments that can confirm our new theoretical and simulation models dealing with kinetic nonlinear structures in electron–ion and electron–positron plasmas. Finally, we present our views of possible extensions of theoretical models and computer simulations that have been reviewed in here.
Original languageEnglish
Pages (from-to)225–290
Number of pages66
JournalPhysics Reports
Issue number6
Publication statusPublished - Jan 2006


  • electron holes
  • ion holes
  • phase-space vortices
  • electromagnetic waves

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