List of Figures

View Image Figure 1:
Top: Electron velocity distribution function in the solar wind as measured by the plasma instrument on the Helios spacecraft at 1 AU. Note the distinct bulge along the magnetic field, which is the so-called strahl, a suprathermal population carrying the heat flux together with the halo, the hotter isotropic component which is slightly displaced with respect to the maximum of the core part (indicated in red) (after Pilipp et al., 1987b). Below: Radial decline (increase) of the number of strahl (halo) electrons with heliocentric distance from the Sun according to the Helios, WIND and Ulysses measurements (after Maksimovic et al., 2005).
View Image Figure 2:
Electron velocity distribution functions as energy spectra (top) and velocity space contours (bottom) for fast (left), intermediate (middle) and slow (right) solar wind. Isodensity contours are in steps by a factor of 10. Note the core-halo structure and the strahl of suprathermal electrons in fast solar wind (after Pilipp et al., 1987a).
View Image Figure 3:
Proton velocity distribution functions in the fast solar wind as measured by Helios at 0.5 AU (top left), 0.54 AU (top right), 0.4 AU (bottom left) and 0.3 AU (bottom right). Note in the lower VDFs a distinct temperature anisotropy in the core and the strong beam (after Marsch et al., 1982c).
View Image Figure 4:
Top: The proton magnetic moment is observed to increase with heliocentric distance and indicates through its non-conservation proton heating. Bottom: Selected velocity distribution functions measured in high-speed wind. The solid isodensity contours correspond to 20% steps of the maximum, and the last broken contour is at 0.1%. Note the large temperature anisotropy in the core and the tails along the magnetic field direction (after Marsch, 1991a).
View Image Figure 5:
Velocity distribution functions of helium (top), oxygen (intermediate) and neon (bottom curves) ions as measured by the ion mass spectrometer on the WIND spacecraft for various solar wind speeds. Note the extended power-law tails in the VDFs which are fitted well by kappa functions, in particular for helium (after Gloeckler et al., 2001).
View Image Figure 6:
Left: Electrostatic interplanetary potential from the exobase (at 2R ⊙) out to 215R ⊙ in an exospheric model with kappa VDFs of the solar wind consisting of protons and electrons, with base temperature, Te(r0) = Tp (r0) = 106 K and κe = 2.5. Right: Module of the ratio of outward directed electric force, qE (r), and inward directed gravity, m g(r) p, acting on a proton. This ratio in the solar wind is plotted versus radial distance (after Pierrard et al., 2004).
View Image Figure 7:
Left: Different examples of κ-functions (after Maksimovic et al., 1997a), all normalised to unity at v = 0. Obviously, in the limit κ → ∞, these functions transform into a Maxwellian or Gaussian (solid line). Right: Measured electron VDF (after Feldman et al., 1975) in the solar wind (diamonds). The dashed lines correspond to the classical model VDF, being composed of two Maxwellians: a core with − 3 nc = 30.8 cm and 5 Tc = 1.6 × 10 K, and a halo with − 3 nh = 2.2 cm and Th = 8.9 × 105 K. The full line represents the κ-VDF model fit with n = 33.9 cm −3, T κ = 1.9 × 105 K and κ = 4.
View Image Figure 8:
Left: Electron temperature profile in the corona in an exospheric model with kappa function tails of the VDF. The kappa values are indicated at the different lines. The dashed vertical line corresponds to 1 AU. Only κ larger than about 5 is compatible with empirical constraints. Right: Contours of the terminal (at 1 AU) solar wind speed in km s–1 for an electron VDF composed of core and halo Maxwellians. The contours are shown as a function of the relative density, α0 = nh(r0)∕nc(r0), and the temperature ratio, τ0 = Th(r0)∕Tc(r0), of the electron core and halo at the exobase, being located on the solar surface, r0 = R ⊙ (after Zouganelis et al., 2004).
View Image Figure 9:
Electron heat flux in transition region as calculated for a kappa VDF at the chromospheric lower boundary. The horizontal dash-dot line (left frame) and dashed line (right) are the Spitzer and Härm (1953) predictions drawn for reference. Left: The squares show the predicted normalised heat flux from the kappa-function model equation. The vertical dashed line shows the value of κ for which the electron heat flux vector changes sign. Positive values of 𝜃 correspond to heat flowing up the temperature gradient (antiSunward) for strong tails (κ < 10). Negative values of 𝜃 = q∕qsat correspond to heat flowing down the temperature (Sunward) gradient (after Dorelli and Scudder, 1999). Right: Similar plot, but for a full numerical solution obtained by solving the kinetic problem according to Fokker–Planck diffusion. Note that here non-classical conduction occurs only for strong suprathermal tails, κ < 5, assumed to prevail at the lower boundary (after Landi and Pantellini, 2001).
View Image Figure 10:
Two-dimensional contours of gyrotropic model VDFs of protons at various solar distances in the solar wind frame (with speed components c ∥ and c ⊥). The VDF is normalised to unity, and contours (from outside to maximum) correspond to fractions of 0.01, 0.03, 0.05, 0.2, 0.4, 0.6, and 0.8 of the maximum. Note the unphysical negative values (−0.005) indicated by the dotted contour. The positive parallel speed points away from the Sun along the local magnetic field direction (after Li, 1999).
View Image Figure 11:
Comparison of a measured proton VDF in the solar wind (1-D cut along the field of the normalised VDF indicated by crosses) with various models. Left frames: Bi-Maxwellian (continuous line) background and expansions in c∥,⊥ up to order 3 (dashed) and 4 (dotted line) in the top panel, and up to order 5 (dashed) and 7 (dotted line) in the bottom panel. Right fames: Same format but now for a skewed weight function as zeroth-order solution, which better interpolates the beam (after Leblanc and Hubert, 1997).
View Image Figure 12:
Left: The proton cyclotron dissipation wave number for parallel Alfvén-cyclotron waves as a function of βp. The solid line represents kdlp, and the dashed line represents kdrp, where the thermal proton gyroradius, r = v ∕ Ω p p p, and the proton inertial length, l = c∕ω = V ∕Ω p p A p, are used for normalization, and kd is the dissipation wave vector. Right: The damping rate divided by the real frequency, γ∕ω, of the oblique Alfvén-cyclotron waves as a function of the perpendicular wave number for three values of the electron βe as labelled. Here k∥ = kd∕3 and − 0.01 ≤ γ∕Ωp ≤ 0. Note the increasing damping, leading to oblique wave dissipation, with growing electron temperature, i.e. with increasing β e (after Gary and Borovsky, 2004).
View Image Figure 13:
The dispersion relation of kinetic Alfvén waves for βp = 0.01 versus k∥lp. The solid and dashed lines represent the real frequency, and the dotted chains represent the corresponding damping rates after Gary and Nishimura (2004). Left: Dispersion results for parallel and very oblique propagation. Right: Three dispersion curves relating to different temperature ratios.
View Image Figure 14:
Left: Comparison of measured proton velocity contours with the quasilinear plateau. The horizontal axis gives v∥ and the vertical v⊥ (in units of km s–1). The measured contours correspond to fractions of 0.8, 0.6, 0.4, 0.2 of the maximum. The dotted lines (circular arcs) delineate the theoretical contours shaped by diffusion Marsch and Tu (after 2001b). Right: Another comparison of the cyclotron diffusion plateau in velocity space with proton observations. The solid lines on the left hand (right hand) side of the vertical axis represent the theoretical contours (for zero pitch-angle gradient) formed by the cyclotron resonance of protons with outward propagating (inward) left hand circularly polarised cyclotron waves. The dotted contours are measured and correspond to fractions 0.8, 0.6, 0.4, 0.2, 0.1, of the maximum of the VDF, respectively. The dispersion relation of a cold plasma with protons and electrons was used to calculate the phase speed. The Alfvén speed is 184 km s–1 Tu and Marsch (after 2002).
View Image Figure 15:
Numerical simulation of the proton velocity distribution fp(v∥,vy) at tΩp = 56. The phase speeds, ω ∕k ∥, are for five left hand polarised wave modes indicated on the right hand side by the five dots at the locations with v⊥ = 0 and v∥ > 0. The five dots on the left axis represent the corresponding cyclotron resonant velocities. The related solutions of the plateau Equation (56View Equation) are indicated by the five heavy solid lines. Note that the diffusion can even render the ions cross the v∥ = 0 line (after Gary and Saito, 2003).
View Image Figure 16:
A highly anisotropic proton VDF with a large core-temperature anisotropy as measured by Helios 2 in fast solar wind near 0.3 AU. The data were taken in the year 1976 on day 107, during the time span from 01:24:38 to 01:25:18. The proton fluid speed is 729 km s–1. The contour lines correspond to fractions 0.8, 0.6, 0.4, 0.2 of the maximum located at the central point (continuous lines), and to fractions 0.1, 0.032, (dashed lines) and 0.01, 0.0032, 0.001 (dotted lines). The velocity plane is determined by the unit vectors in the direction of the proton fluid velocity (x-axis) and the magnetic field (straight solid line) and centered at the maximum. Apparently, the symmetry axis is well defined by the magnetic field direction that represents the axis of gyrotropy. The distribution also reveals a hot tail travelling along the field (after Tu and Marsch, 2002).
View Image Figure 17:
Comparison of the measured proton core temperature ratio, A + 1 = T ⊥c∕T∥c, with theoretical predictions. This empirical ratio is plotted versus the plasma beta based on the core VDF only. A least-squares fit to the binned data with variance bars is also given, together with other lines indicating the A-β-relations derived from various sources in the literature (after Tu and Marsch, 2002).
View Image Figure 18:
Left: This panel shows 2-D contours of a typical proton beam VDF, based on a 3-D interpolation of the data obtained by Helios 2 at 09:08:33 on day 70 in 1976 after Tu et al. (2004). The horizontal axis gives the velocity component parallel (v∥) and the vertical perpendicular (vx) to the magnetic field. The solid, dashed and dotted curves, respectively, show the contours of the VDF relative to its maximum value which is located at v∥ = 0 and vx = 0. The respective values of the contours from the center correspond to 0.8, 0.6, 0.4, 0.2, 0.1, 0.032, 0.01, 0.0032, and 0.001 of the maximum. Right: This panel shows a 1-D reduced VDF, ΔF (v∥), obtained by integration along the vertical vx-direction indicated in the left panel. The VDF is normalised to its maximum value. The horizontal axis gives the velocity component v ∥ in km s–1.
View Image Figure 19:
Left: The normalised beam drift speed is plotted versus the plasma beta after Tu et al. (2004). Each cross point represents a single Helios plasma measurement of the proton beam drift speed plotted against the core plasma beta, β ∥c. The dash-dot line shows the result of a linear least-squares fit to the logarithm of the observed data points. The dotted line and the dashed line show the threshold of the Alfvén I instability after Daughton and Gary (1998) with a constant ratio of the proton beam density, nb, to the electron density, ne. The values of their ratio, nb ∕ne, are 0.05 (upper line) and 0.2 (lower line). The maximum instability growth rate at the threshold is γ ∕Ω = 0.01 m p. The diamonds show the few data points for which the VDFs are found to be unstable. Right: The normalised proton-proton relative drift speed is shown versus the relative beam density. Individual data points as measured by Ulysses are given after Goldstein et al. (2000). The four lines represent threshold conditions for two proton-proton instabilities as shown in Daughton and Gary (1998). The upper and lower solid lines represent the thresholds of the magnetosonic instability at β = 0.2 ∥c and 1.0, respectively, whereas the upper and lower dashed lines display the thresholds of the Alfvén instability at β∥c = 1.0 and 0.2, respectively.
View Image Figure 20:
Power spectrum arranged according to the dispersion relation of transverse magnetic field fluctuations propagating along the field, for the case of zero background-wave amplitude, A = 0, on the upper (a) and large amplitude, A = 0.3, on the lower panel (b). The bottom gray codes correspond to the logarithms of the wave power. A strong beam is present, but the original beam instability (a) is highly suppressed in the presence of the ambient wave (b) (after Araneda and Gomberoff, 2004).
View Image Figure 21:
Left: Time evolution of the average differential streaming speed Upp for a relative beam density nb = 0.25. Initially, the main and beam protons are isotropic and have the same temperatures. Curves for five values of the starting speed are plotted, each for the case without waves (dotted lines) and with initial waves (solid lines). Only the fast initial beams show significant deceleration in the presence of waves. Right: Time evolution (extending to 3000 cyclotron periods) of Upp for a strong beam with nb = 0.5 and starting speed 1.57 VA. The case without waves is given by the dotted line, and the cases with waves are indicated by solid lines for different relative wave amplitudes, A. All cases show speeds saturating well below VA. The maximum deceleration rate (for A = 0.0, 0.1, 0.25, and 0.5) is 1.13, 1.31, 2.04, and 2.53, respectively, in units of 10−3VA Ωp (after Kaghashvili et al., 2004).
View Image 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).
View Image 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).
View Image 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)).
View Image 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.
View Image Figure 26:
Model electron VDFs at a distance of 1, 7.5, and 14.8R⊙ from the Sun (from bottom to top). In the left column we see isocontours, and in the right column cuts through the VDF (logarithmically displayed) along the magnetic field (continuous line) and perpendicular to it (dashed line). The dotted line is the equivalent Maxwellian. In the top panels the escape speed of 11,000 km s–1 is indicated by dotted vertical lines. Note the occurrence of a pronounced skewness, equivalent to the electron strahl, and the evolution of a slight temperature anisotropy, with Te∥ > Te⊥, for increasing solar distance (after Lie-Svendsen et al., 1997).
View Image Figure 27:
Two-dimensional gyrotropic model VDF of the heavy coronal ion O5+ at 0.44R ⊙ (left) and 0.73R ⊙ (right). The left gyrotropic VDF shows plateau formation leading to marginal stability at the Sunward side. Note on the left the contours with a large perpendicular temperature anisotropy, and on the right the skewness developing along the magnetic field with increasing distance (after Vocks and Marsch, 2002).
View Image Figure 28:
Contours of gyrotropic model VDF of coronal electrons after Vocks and Mann (2003). 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.