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Electron Heating and Acceleration while Magnetoshere Substorm due to Varying Phase Velocity

Karas` V.I., Potapenko I.F*.
NSC “Kharkov Institute of Physics &Technology” of NAS of Ukraine,61108, Kharkov,1Akademicheskaya Str.
*Keldysh Institute of Applied Mathematics of RAS, 125047, Moscow, Miusskaya sq.4
Abstract. In the paper some properties of chorus radiation while magnetosphere substorm are discussed. The
influence of the hydro magnetic waves on the electron distribution function is studied by numerical simulations. A
quasi-linear 2D in velocity space operator models the electron damping of plasma eigenmodes. The dynamic of
process is estimated under condition of varying in time of phase velocity and hence of phase resonance on the base
of chorus radiation while substorms. This allows us to explain acceleration and heating of energetic electrons that
double up energy during the stage of substorm
Keywords: plasma-wave interaction, electron heating and acceleration, magnetosphere, Alfvén maser.
PACS numbers: 52.65 Ff; 52.55.Dy
The effects of particle precipitation in the Earth’s aurora zone are discussed in numerous publications (for example,
[1,2]). Studies have shown that the electron precipitation related to substorms can be induced by wave-particle
interactions around the magnetospheric equatorial plane. Those waves can be generated in the Earth’s magnetosphere due
to the maser-effect [3]. A significant number of observational data on electron precipitation has been correlated to chorus
[4]. In this paper we address the following problem. We consider that the turbulence is composed of hydro magnetic
waves that are assumed to be propagation along the ambient magnetic field. The wave power absorption mechanism due
to Landau damping is considered in the framework of the standard quasi-linear theory of wave-particle interaction. For
simplicity we use the local approximation in which the velocity space is connected with the given force line of the
magnetic field. Thus the magnetized plasma is assumed to be space homogeneous and that charge neutrality is provided.
The hydro magnetic wave level is not too high, so the weak turbulence theory can be applied. Starting with the initial
Maxwellian distribution we describe the evolution of the electron distribution function with following equation

density and energy, respectively. The phase resonance region and the values of the diffusion coefficient, which are the
parameters of the problem, define the electron scattering into the loss cone, i.e. energy and the particle flux, due to
waves. Any external particle sources usually are not taken into account and the plasma dynamics is studied over the
plasma decay. Therefore, we deal with quasi-stationary state problem. Under the wave influence the electron
distribution function tends to the form of a `plateau’ with respect to the parallel velocity in the resonance region. The
anisotropy of the distribution function over pitch angles depends on time and after some relaxation period the electron
function takes on a quasi-stationary form. The particles diffused toward high parallel velocities would enter the loss
cone and would escape from the trap at once. Thus, the waves induce precipitation in two ways: due to a distortion of
the electron distribution over pitch angles and due to the plasma heating.
Magnetosphere is considered being Alfvén maser and the characteristic time of the electron losses out of the
magnetic trap with the mirror ratio R is chosen equal to T = R » 10 C . Then in the above diffusion equation the loss
term is d × f , where -1
C d = T if v v ³ R / ^ and d = 0, otherwise. The dynamic of the electron precipitation
process is estimated under the condition that the phase velocity of the whistler waves in not constant in time. Chorus
radiation while magnetosphere substorm (see for example [4]) consists in successive discrete positively inclined
elements, dw / dt > 0 , that follow consequently with frequency 1-10 kHz. Micro precipitation of electrons with energy
more than 20 keV is closely connected with chorus. From the observation data of chorus dynamic while magnetosphere
substorm we take typical parameters of the process. The velocity is normalized on phase velocity and the characteristic
time unit is 1 sec.
We present the results of simulations for the following parameters. We start with the initial Maxwellian distribution and
present the results of numerical simulations of the electron distribution function and rf- enhanced energy. We give two
examples of simulation results: for the diffusion coefficient D=10-2 and D=10-3. The phase resonant region moves over
parallel velocity with time following data obtained from observation. Characteristic time period of one pulsation is
subdivided on two unequal periods: during time period 0.9 1 Dt = the resonant region is maintained stable with the
phase velocity equals = 1.5 ph v and the width to D = 0.5 ph v . Then during the period 0.1 2 Dt = corresponding to
chorus precipitation the resonant region it is extending until = 2.5 ph v . The wave packet does not change its phase
velocity width. Such a process is successively repeated during about 0.5-1 hour. While relatively short initial stage the
Maxwellian adopts the loss cone form, then the quasi-stationary state is established. In the figures 1 and 2 the averaged
energy of precipitated electrons and of the electrons that are trapped are shown as a function of time for two values of
the diffusion coefficient D. The established value of precipitated electron energy does not differ much for different
diffusion coefficients. The variance can be seen in the initial stage, see fig.1-3. Obviously the time relaxation of the
system to quasi-state is shorter for larger diffusion coefficient. The dissipated wave power and the electron velocity can
be enhanced for the wave phase velocity that increasing in time. The quasi- linear operator with moving phase resonant
region rakes up electrons from the domain with higher density to the higher energetic region. That is why dissipated
wave power is larger in comparison with the case when the phase velocity region is constant. The diffusion operator
forms the distribution function plateau within the region of its action. Figure 4 demonstrates the electron distribution
function in the steady state for D=0.01. It should be noted that, to form plateau for relatively small diffusion coefficients
D = 0.001-0.01 within “changing ” in time phase resonant region there is necessary to pass over hundreds seconds.
Explanations of field-aligned particle precipitation by means of Landau damping with varying phase velocity in time is
able to provide sufficient increase in electron energy of chorus while substorm. This allows us to explain acceleration
and heating of energetic electrons that double up energy on the stage of substorm. In this preliminary study, the
observational data could be interpreted in terms of the phenomena observed.

1. R. G Lundin,. A. Gustafsson , I. Eriksson, and G. Marklund, J. Geophys. Res., 95, 5905, (1990).
2. E. Ungstrup , A. Bahnsen, H.K. Wong, M. André, and L. Matson, J. Geophys. Res., 95, 5973, (1990).
3. V.Yu. Trakhtengerts, Eur. Space Agency Spec. Publ., ESA-195, 67, (1983).
4. T.G.Rosenberg, J.C.Siren, D.L.Matthews, et al., J.Geophys. Res., 86, 5819-5832 (1981).
5. I.F.Potapenko, C.A.Azevedo, Computer Physics Communications, 121-122, 274-277 (1999).

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