It is evident that the truncated Maxwellians introduced in Sects. 2 and 3 can only be considered as appropriate to describe the effective kinetics of the solar wind electrons in parametrized form, if these functions can be physically motivated. Without collisional or "quasi-collisional" influences on the electrons at their evaporation from the lower corona by no means truncated Maxwellians could be good approximations since the hemispherical pitch angle isotropy would be violated. The classical "Liouville'an" distribution function resulting in case of collisionless evaporation is strongly pitch-angle dependent both in the antisunward and in the sunward part of the distribution with no particles populated in the elliptic branch of the velocity space (see Fahr & Shizgal 1983). To nevertheless explain the transport of electrons into these branches, and to better approach a truncated Maxwellian, either wave-induced pitch-angle diffusion and energy diffusion processes of electrons have to occur or the electron distribution functions have to be revealed as unstable with respect to driving waves by themselves.
The latter process has been discussed by Scime et al. (1994) and Gary et al. (1994). These authors point to the possibility of a heat flux instability with respect to whistler wave excitation. Representing the electron distribution function as given by two anisotropic Maxwellians (i.e. core and halo) with a relative drift Gary et al.(1994) can calculate positive whistler wave growth rates pointing to the fact that the electron heat flow may be instability-limited to a value of the order of , with being the thermal velocity of the electron core and being the local Alfven velocity. However, as already noticed by Dum et al. (1980) these growth rates are highly sensitive to specific features of the distribution function. Thus no clearcut result can be obtained with respect to the effectiveness of this wave growth with respect to reshaping the distribution function.
We therefore look into an alternative relaxation mechanism with explicit influences on the shaping of the electron distribution function. Here we think of quasilinear interactions of the electrons with preexisting whistler wave turbulence. Connected with such turbulences specific Fokker Planck diffusion coefficients can be evaluated which describe wave-induced electron diffusion processes in velocity space. The process operating with the highest rate, higher than the expansion rate, is pitch-angle scattering of electrons by resonant whistler waves (see e.g. Denskat et al. 1983). This process is appropriately described by the so-called pitch-angle diffusion coefficient given by (see Schlickeiser et al. 1991; Achatz et al. 1993):
where , v and µ are gyrofrequency, velocity and pitch-angle cosine of the electron, and where and are whistler waves with resonant wavenumbers k propagating parallel or antiparallel to . Both for negative and positive values of µ the pitch-angle diffusion process operates quite efficient and rapidly tends to isotropize the distribution function, whereas due to a resonance gap in the cyclotron interaction of electrons with whistler waves (e.g. see Dusenbery & Hollweg 1996; Schlickeiser et al. 1991) around pitch-angles with the pitch-angle diffusion between the two hemisphere and is strongly impeded. This quite naturally justifies the assumption of truncated Maxwellians since these are µ-isotropic in both hemispheres with the µ-anisotropy limited to . Besides in the principles of this effect we are also interested in its quantitative strength which is connected with the level of whistler wave turbulence.
To clarify this point, we consider processes producing whistler turbulence. We consider the interaction of high frequency Alfvén and fast magnetosonic waves with electrons starting from the following assumptions:
1. The low frequency turbulence is described as a mixture of Alfvén (a) and fast magnetosonic (f) waves.
2. The source of the turbulent energy is due to pumping of wave energy from the largest to the smallest wavelengths (i.e from the lower to the higher frequencies, the whistler modes, where a part of the spectral energy flux is resonantly absorbed by solar wind electrons).
where are reference powers at , and where is the wave frequency measured in the solar rest frame.
where U is the solar wind velocity and is the group velocity of the waves in the solar wind reference frame. The source term on the right hand side describes the wavepower gain at frequency due to divergence in -space of the cascading wave energy flow in a saturated turbulence field. This term does not contain linear (L), but only nonlinear (NL) contributions and essentially allows to separate the frequency space into two regions using a critical frequency according to the following rule roughly valid here:
Here the nonlinear terms are due to couplings between waves of the Alfvén (a) and of the fast magnetosonic (f) types. The above terms can be estimated by simple expressions if a radial symmetry of the problem with constant solar wind velocity U can be assumed, yielding:
where the numerical factors describing the efficiency of mode-couplings are shown to be of the order of 0.1. Using Eqs. (56) and (47) and approximating by U (i.e. super-Alfvénic solar wind flow), we then arrive with Eq. (55) at:
Associated with the approximate expression for the linear wave power sources one can derive the following expression for the total energy generation which cascades up to the nonlinear regime from the critical frequency
The above expression for denotes the total spectral energy flux in modes "a" and "f" respectively, integrated over the frequency range of the inertial range where is valid. Thus the active heating sources resulting from energy dissipation in the two modes are given by:
where the wave energy is dissipated to the electrons in the whistler frequency domain, mostly at the highest frequency end, i.e. at , where is the electron cyclotron frequency.
© European Southern Observatory (ESO) 2000
Online publication: December 5, 2000