Regulation of the electron heat flux

6.8 Regulation of the electron heat flux

Figure 22:

Left: Correlation of the total electron heat flux with the halo particle flux from $2$ to $2.3 AU$ . $VH$ is the halo drift speed in the Sun’s (inertial) frame of reference. Similar correlations are seen over the whole distance range from $1$ to $5 AU$ . Right: The electron heat flux and the local magnetic field magnitude show a clear and lasting correlation (after Scime et al., 1994

Significant progress, on the basis of Ulysses electron measurements (Scime et al., 1994 ), was made in the understanding of the electron heat flux regulation. The empirical heat conduction law suggested long time ago by Feldman et al. (1975 ) gives for the heat flux value, $q∥e$ , the following empirical scaling relation with the parallel and perpendicular core and halo temperatures:

which shows that the heat flux is carried mainly by the halo electrons (including the then still unresolved strahl), and essentially scales like the halo particle flux times the halo thermal energy. This dependence has experimentally been verified, demonstrating that solar wind heat conduction has nothing to do with the local temperature gradient but with the thermal energy convected by halo electrons. Empirically, the heat flux scales with radial distance, $R$ in units of $AU$ , from the Sun according to the formula: $q∥ e = 12.7 R −3.1μ Wm − 2$ , which describes the average observations in full agreement with Equation (62

). This scaling with distance gives values of $q∥ e$ being much larger than predicted by the collisional Spitzer–Härm theory that was discussed in previous sections.

The heat flux regulation mechanism enters formula (62 ) through the zero-current condition for the combined core and halo drifts, which reads $nH △VH + nC △VC = 0$ , whereby quasineutrality of courserequires that $nH + nC = ne$ , and through the fact that $VH ∼ VA$ , which is in proportion to the magnetic field magnitude $B$ . One also finds the relation $T ≈ 7 T H C$ , which was however predicted by collisional transport theory Scudder and Olbert (1979a ,b ) and is not easily explained by wave effects. From the work of Maksimovic et al. (2005 ) it appears that the strahl electron density is comparable to the halo density at heliospheric distances below $0.5 AU$ . Therefore, in a future study it should be investigated if the strahl electrons play a role in the zero current condition closer to the Sun.

That the halo drift speed was observationally found to be closely tied to the Alfvén velocity, supports a regulation of the heat flux by Whistler-mode waves. This correlation is shown in Figure 22 after Scime et al. (1994 ), where on the left the halo flux density and total heat flux density are displayed, and on the right the halo flux density and $B$ . Both plots illustrate the similar course versus time of the compared quantities, thus confirming that electron heat conduction in the distant solar wind is due to convection of differential heat (between halo and core electrons) at the halo speed, and that it follows closely the Alfvén speed.

The results in Figure 22 indicate that the halo electrons (a strahl could not be resolved in those measurements) carry the electron heat flux which is observed to vary with $B$ or $VA$ . This points to the importance of waves regulating the halo drift. Gary et al. (1994 ) studied in detail the possible whistler regulation of the electron heat flux, within the electron model VDF of two drifting anisotropic bi-Maxwellians. They considered models of local and global scalings of the heat flux with plasma parameters. The global model yields a heat flux at the threshold of the whistler instability, which scales with distance in the same way as the average observed heat flux from Ulysses and provides an upper bound. A closure relation was also suggested. Gary and Li (2000 ) provide further parametric studies for the instability over a wide range of the electron plasma beta.

Figure 23:

The normalised VDFs are plotted as a function of $v⊥ ∕Vc$ , for different radial distances ( $0.3– 0.41$ , $0.48 –0.53$ , $0.7 –0.75$ , $1$ , $1.35– 1.50 AU$ , in black, red, blue, green, and magenta, respectively). As one can see, the normalised core component remains unchanged at all radial distances but the relative number of halo electrons, as compared to the ones of the core, increases with radial distance (after Maksimovic et al., 2005

Figure 24:

The dimensionless heat flux at threshold of the whistler instability versus radial distance in units of $1 AU$ . Open squares (triangles) correspond to $γm∕Ωp = 0.1(0.01)$ . The continuous line gives $qe∕qmax = 1.78 R0.16$ , respectively the broken one $qe∕qmax = 1.68 R0.19$ . Obviously, the measured heat flux is, on average and everywhere, clearly below the threshold for the whistler instability, which provides an upper bound on $qe$ (after Gary et al. (1994 )).

The “maximal” heat flux is globally given as a function of radial distance by the empirical relation $−2 qmax = 22.85 R −3.19μ Wm$ after Scime et al. (1994 ), whereby the local values can substantially fluctuate about this mean value. This maximum is defined by the flux that is carried by all electrons, when having the core thermal energy, and that is convected at the core thermal speed, i.e., $3 qmax = 3 ∕2nemeV c$ , where the core thermal speed is given by $∘ --------- Vc = kBTC∕me$ . The dimensionless heat flux in units of $qmax$ at the threshold of the whistler instability is for two growth rates plotted versus radial distance in units of $1 AU$ in Figure 24 . Apparently, the observed value of $qe$ stays well below the threshold, suggesting the heat flux may be controlled and regulated by the whistler mode instability. However, Scime et al. (2001 ) then concluded from a large statistical study that the whistler heat flux instability does on average not provide the observed constraint on the measured $qe$ . For its latitudinal variation see Scime et al. (1995 ).

In contrast to these statistical results, Dum et al. (1980 ) found individual measured electron VDFs to be at the margin of the whistler instability. We recall the quasilinear diffusion scenario that was discussed previously for the ions interacting with cyclotron waves. A similar interaction takes place between electrons and the right hand polarised whistler mode waves, which pitch-angle scatter the electrons to the effect that their thermal energy perpendicular to the field will increase. Since the fluctuation level of whistlers (Gurnett, 1991 ) is usually low in the solar wind, the net resulting heating may be weak. However, by elastic scattering the influence of waves on the shape of the halo and the broadening of the strahl may be substantial.

That the halo shape varies with radial distance was recently demonstrated with Helios, WIND and Ulysses observations by Maksimovic et al. (2005 ), who used a mixed model for the electron VDF, taking the fact into account that the VDF at high speeds varies more like a power law rather than a Maxwellian. The simplified model VDF (neglecting the strahl), which would correspond to a pure energy distribution as obtained by pitch-angle averaging, is assumed to be composed of two Maxwellians (Feldman et al., 1975 ; Pilipp et al., 1987a ,b ; McComas et al., 1992 ), one for the core and one for the halo (see the previous Figure 2 ). However, Maksimovic et al. (1997a ,b ) modelled the Ulysses VDF as a generalised Lorentzian or kappa function, as in our Equation (6 ). After a careful analysis of their data, Maksimovic et al. (2005 ) used as the best fit an anisotropic sum of a bi-Maxwellian for the core and a bi-kappa for the halo.

Some of their results are shown in Figure 23 , where the VDF is plotted versus $v⊥$ . Obviously, the normalised core component remains unchanged at all radial distances, and thus the relative importance of the halo component is increasing with radial distance. This enhancement would qualitatively be consistent with enhanced pitch-angle scattering in a background whistler mode field. Also, scattering or mirroring by meso-scale field fluctuations might be a possible cause.

Figure 25:

WIND electron data from 50 consecutive days after Salem et al. (2003

). Left: Scatter plot of 11-minute averages of the electron temperature anisotropy as a function of the inverse of the collisional age, $1∕Ae$ . Right: Scatter plot of 11-minute averages of the normalised heat flux $Qn = Qe ∕q0$ in the solar wind as a function of the ratio between the electron mean free path, $Lfp = Ve∕ νee$ , and the scale of the temperature gradient $− 1 LT = (dTe∕dR ) ∼ R$ . The classical Spitzer–Härm value of $Qn$ (given by the straight line indicated SH) represents an upper limit to the observations.

Certainly, Coulomb collision would be insufficient to modify the VDF to the observed degree. However, they seem to matter according to WIND observations made at $1 AU$ by Salem et al. (2003 ), who could show that the electron temperature anisotropy, $Te∥∕Te⊥$ , which seems to depend mainly on the solar wind speed, $Vsw$ , and electron density, $ne$ , and heliocentric distance, $R$ , actually depends on the number of Coulomb collisions through what they called the electron collisional age, $Ae ∼ νeeR ∕Vsw$ . It is the number of transverse collisions, at a rate $νee$ , suffered by a thermal electron during the expansion time of the wind over the density-gradient scale. The age $Ae$ also depends on the spatial coordinates, like $R$ , and it may change considerably at stream-interface crossings. Salem et al. (2003 ) demonstrated that $T ∕T e∥ e⊥$ was strongly correlated with $ν ee$ , and they also found that the normalised heat flux $Qn = Qe∕q0$ displays an upper bound inversely proportional to the collisional age, a result being in favour of an overall regulation of the heat flux by Coulomb collisions. Here the free streaming heat flux for the entire VDF is defined as $q0 = 3∕2nekBTeVe$ , where the mean thermal speed is given by $∘ ---------- Ve = 2kBTe ∕me$ . The classical Spitzer–Härm value of $Q n$ , which was given in Equation (29 ) and linearly increases with the free path, represents an upper limit to the observations. Apparently, other than collisional friction keeps $Qn$ constrained. The results of the correlation analysis are shown in the two frames of Figure 25 .

In summarizing this section, one must conclude that the electron heat flux appears to be regulated by a variety of processes, in which local wave-particle interactions as well as collisions act in combination with global ballistic effects (on the almost collision-free suprathermal electrons), so as to produce and regulate together the observed features like core-halo structure, heat-flux tail, and skewness and thermal anisotropy in the thermal range of the VDFs. Note that all electrons are locally coupled to the ions by the quasineutrality condition, violations of which lead immediately (on the fast scale of the ion or electron plasma period, which is to say within a few milliseconds at $1 AU$ ) to strong electrostatic couplings, which tend to equilibrate ion and electron charge densities.

Finally, a note of caution is in order concerning the measured electron VDF. The Helios electron measurements (Rosenbauer et al., 1977 ) first showed the presence of a narrow strahl in the electron velocity distribution. A more recent paper Gosling et al. (2001 ) states that the suprathermal electrons consist of two separable components: a relatively isotropic halo and a narrow strahl which carries the heat flux. Therefore, future electron measurements at higher pitch-angle resolution will hopefully allow us to better understand the still open problem of solar wind heat conduction.