On deformation of electron holes in phase space

epl draft On deformation of electron holes in phase space R. A. Treumann,1,2 C. H. Jaroschek3 and R. Pottelette4 1 Department of Geophysics, Munich University, Theresienstr. 41, D-80333 Munich, Germany 2 Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 3 Department of Earth and Planetary Sciences, Tokyo University, Tokyo, Japan 4 CETP/CNRS St. Maur des Foss´es, Cedex, France PACS 94.30.Aa – Auroral phenomena PACS 94.20.wj – Wave-particle interactions PACS 94.05.Dd – Radiation processes Abstract. - This Letter shows that for particularly shaped background particle distributions momentum exchange between phase space holes and the distribution causes acceleration of the holes along the magnetic field. In the particular case of a non-symmetric ring distribution (ring with loss cone) this acceleration is nonuniform in phase space being weaker at larger perpendicular velocities thus causing deformation of the hole in phase space. Introduction. – In configuration space, phase space holes appear as localised intense electrostatic fields E∥h = −∇∥φh with broadband spectral signature parallel to the ambient magnetic field. In velocity space they form nar- row regions of lacking particles of one signature kept alive for a limited time by the electrostatic field. Ion holes are local deficiencies of ions while electron holes are local defi- ciencies of electrons. Thus the former correspond to weak negative, the latter to positive space charges Qi,e. In this Letter we deal with electron holes which can be excited by beam or current instabilities parallel to the ambient mag- netic field B, like the two-stream instability which works for electron drifts vd > ve, larger than the electron ther- mal velocity ve [3, 4]. At lower drifts this instability is replaced by a modified version [14] which is a form of the modified two-stream instability [13, 16, 17, 24]. Their the- ory has been given by Bernstein, Greene and Kruskal [2], Schamel [30–32], Dupree [8, 9] and Turikov [38]. Simula- tions by Newman et al. [22, 23], Muschietti et al. [20, 21] and others have shown that electron holes are the natu- ral nonlinear state of these instabilities, being Debye scale entities along B in configuration space, and of short exten- sion in the parallel velocity component v∥in velocity space. They contain a dilute component of trapped electrons of density Nt of low energy mv2 t /2 ≤|eφh|. In configuration space they are oblate in the direction perpendicular to the magnetic field (pancakes). Their behaviour in v⊥has not yet been investigated in detail. It is, however, reasonable to assume that the holes are either gyro-limited, being of transverse spatial extension up to the thermal gyrora- dius ∆h⊥∼rc = ve/ωc or inertia limited ∆h⊥∼c/ωp. Their life time is determined by the stability of the holes with respect to the generation of whistlers, trapped par- ticle instabilities, particle trapping, heating and diffusion and the corresponding generation of dissipation (see, e.g., Newman et al. [23]). One might believe that these micro- scopic entities are of minor importance for the behaviour of the plasma. However, in collisionless plasmas they form an important dynamical source of dissipation. They heat and accelerate electrons, cause beam cooling, and are sus- pected to provide a substantial part of the dissipation that is needed in collisionless shocks and in reconnection. In collisionless shocks they might contribute to the emission of radiation causing the badly understood type II bursts. Some time ago we proposed [25] that phase space holes contribute to electron cyclotron maser emission [35] gen- erating auroral kilometric radiation in the upward current source region where the holes have been identified [26] subsequently, forming what we called ‘elementary radi- ation sources’. For this to work the holes must become deformed in phase space in order to attain a perpendicu- lar phase space gradient ∂F(v∥, v⊥)/∂v⊥on the electron distribution function, which is required by the cyclotron maser mechanism [35]. A qualitative discussion of how this can be achieved has rcently been provided [36]. Momen- tum exchange between the background electron distribu- tion and the hole has been made responsible for defor- mation of the phase space shape of the hole, with the p-1 arXiv:0810.4642v1 [physics.space-ph] 25 Oct 2008 R. A. Treumann, C. H. Jaroschek, and R. Pottelette dynamics of the hole depending sensitively on the shape of the background electron distribution. In this Letter we present a more quantitative mechanism which is developed for electron holes. However, in a similar way it should also work for ion holes in the presence of, say, ion conics, which have been found in multitude under auroral conditions. Mechanism. – Under auroral upward current con- ditions the bulk distribution is kind of a non-symmetric (downgoing) ring distribution with loss cone (due to the presence of the absorbing ionosphere) as shown in Fig- ure 1. We assu

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