## 6. Dissipated wave energies and modulated electron distributionsIt 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 , 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). 3. The initial power spectra are of the following form: where are reference powers at , and where is the wave frequency measured in the solar rest frame. 4. The convective evolution of the power spectra with increasing solar distance is described by the following wave energy continuity equation: where 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 is the nonlinear wavepower growth rate due to nonlinear wave couplings. For the critical frequency one thus obtains from Eqs. (53) and (54): The nonlinear growth rate is expressed by Chashei and Shishov (1982a, 1982b) in the following form: a,f 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 yielding the critical frequency as: 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 Introducing from Eq. (54) one arrives at: 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 |