7.3 Kinetic model of coronal electrons

The observations shown in Figure 2View Image reveal pronounced deviations from a Maxwellian, a fact that reflects the local as well as global kinetic nature of solar wind electrons. To model these features certainly requires a kinetic approach. We discussed in Subsection 7.1 the model results for the solar wind. Only Tam and Chang (2001) investigated the kinetic effects of electron cyclotron-resonant heating on the evolution and acceleration of the fast solar wind. Their model followed the evolution of the electron VDFs along a non-uniform magnetic field line under the influence of resonant wave heating, Coulomb collisions, and the electric field consistent with the VDFs themselves.

As discussed in previous sections, there is a distinct tail in the observed electron VDFs in the solar wind. This skewness along the magnetic field due to suprathermal electrons may contribute substantially to the ambipolar electric field, which was also discussed in Subsection 3.3 on exospheric models. To include wave effects as well, a three-dimensional (in velocity space) kinetic model has been developed by Vocks and Mann (2003Jump To The Next Citation Point). In this model, it was shown that a suprathermal tail in the electron VDF can directly originate from coronal plasma processes. Their model describes the kinetics of coronal electrons, including their Coulomb collisions and diffusion in a field of outward propagating whistler waves.

The coronal source of these waves is unclear. However, an electron temperature with T ∕T > 1 ⊥ e ∥ e is known to drive whistler waves unstable, which then may lead to enhanced wave-electron scattering. For example, Nishimura et al. (2002) have shown this by particle-in-cell simulation, and thus estimated the maximum scattering rate. They found a numerical constraint which reads:

T⊥ e Se -----− 1 = --αe, (70 ) T∥ e β ∥ e
where Se = 0.7 and αe = 0.55 are fitting parameters derived from the simulation, and 2 β ∥ e = 8πnekBT ∥ e∕B0. Equation (70View Equation) defines a theoretical bound for the measured electron temperature anisotropy. The effective temperature isotropization rate &tidle;νe is found to be high, of the order of a tenth of the maximal instability growth rate, and thus &tidle;ν ∕ | Ω | e e may vary between 0.01 and 0.1.

Quasi-linear theory according to Equation (50View Equation) describes the wave-electron interaction as pitch-angle diffusion in the reference frame of the whistler waves. They are assumed to be generated below the coronal base and to propagate antiSunward through the corona. For high whistler-wave phase speeds, the resonant interaction causes electrons to be accelerated from relatively small Sunward velocities parallel to the background magnetic field to high speeds perpendicular to the magnetic field. As Vocks and Mann (2003Jump To The Next Citation Point) showed, suprathermal coronal electrons can in this way be generated by wave-particle interactions, a result that is illustrated in Figure 28View Image.

View Image

Figure 28: Contours of gyrotropic model VDF of coronal electrons after Vocks and Mann (2003Jump To The Next Citation Point). Left: Contours (at 1.014R ⊙) displaying a perpendicular temperature anisotropy and resonant plateaus (indicated by dotted circular lines) on the Sunward left side. Right: Contours (at 6.5R⊙) showing a sizable skewness (non-classical heat flux) along the field away from the Sun on the right.

In the paper by Vocks and Mann (2003) a simplified Boltzmann-type equation is numerically solved, which specifically can be written as

( ) ( ) ( ) ( ) ∂f- + v∥∂f-+ g∥ − qE-∥ -∂f-+ v⊥-∂A- v ⊥ ∂f-− v ∥ ∂f- = ∂f- + ∂f- . (71 ) ∂t ∂s m ∂v ∥ 2A ∂s ∂v ∥ ∂v⊥ ∂t w ∂t c
In this kinetic equation, g∥ is the gravitational acceleration along the field. The collision terms describe binary collisions and electron pitch-angle scattering by waves. They produce resonant shells on the Sunward side, and subsequent focusing that leads to an anti-Sunward strahl. The electron model VDFs reveals deviations from a Maxwellian, in accord with theoretical expectations and qualitatively similar than the observed electron VDFs shown in Figure 1View Image.

The kinetic model enables one to study electron acceleration and the evolution of the electron VDF from the coronal base into interplanetary space. It includes not only the resonant interaction with whistler waves but also Coulomb collisions and the mirror force that electrons experience in the opening magnetic structure of a coronal funnel. Moreover, wave absorption of the electrons is accounted for to guarantee energy conservation. Apparently, whistler waves can generate suprathermal electrons. Towards interplanetary space, the mirror force focuses the electrons into a narrow field-aligned strahl.

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