Regulation of the proton core temperature anisotropy

6.4 Regulation of the proton core temperature anisotropy

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

We now return to the observed characteristics of the proton VDFs in the solar wind. Their core parts often show a distinct thermal anisotropy (illustrated before in Figure 3 ), which was first discovered in the early days of space observations near 1 AU (Feldman et al., 1974 ). Subsequently, the Helios observations also indicated that the proton core temperature ratio, $T ∕T ⊥c ∥c$ , was relatively high and found to range from about 2 to 3 in the high-speed streams measured in situ near 0.3 AU (Marsch et al., 1982c ). An example is shown in Figure 16 . Schwartz et al. (1981 ) provided an early explanation of the coupling between the core anisotropy and the beam component of the protons. More recently, similar pronounced anisotropies were also found in the solar wind proton data, including the entire velocity distribution, from the Ulysses SWOOPS instrument (Gary et al., 2002 ) and near 1 AU from the Advanced Composition Explorer (ACE) satellite (Gary et al., 2001c ) and the WIND spacecraft (Kasper et al., 2003 ). When the beam in the proton VFD is sufficiently tenuous, a simple two-parameter description determined by the parameters anisotropy, $A = T⊥ ∕T∥ − 1$ , and proton plasma beta, $βp$ , usually provides a good characterization of the instability consequences of the thermal anisotropy.

Following the basic rationale of collisionless kinetic plasma theory, according to which enhanced wave-particle interactions induced by a kinetic instability would limit and regulate possible free energy sources, some early simulations were carried out by Gary et al. (1997 , 1998 ). They predicted that anisotropy-beta relations which represent proton anisotropy constraints should exist. Subsequently, solar wind observations (the last three citations in the first paragraph) presented experimental confirmation of these predictions. Gary et al. (2000b ) further investigated the proton cyclotron anisotropy instability and derived scattering rates, or temperature anisotropy relaxation rates, from numerical hybrid simulations. The threshold anisotropy estimated from theory can be written as:

where $Sp$ (of order unity) and $αp$ ( $≈ 0.4$ ) are fitting parameters derived from various growth rates, and $β∥p = 8πnpkBT ∥p∕B20$ . Equation (57

) completely determines the properties of the instability in space plasma conditions. Computer simulations allow one to infer scaling relations that describe the non-linear saturation of this instability. Defining the variable $0.4 xp = β ∥p (T⊥p∕T ∥p − 1)$ , permits one to write the maximum proton scattering rate, $ν&tidle;p$ , as follows:

which shows that scattering in its non-linear phase sensitively depends on the proton anisotropy and effectively ceases when isotropy is reached. Further hybrid simulations were carried out by Gary et al. (2003

) on the proton and alpha-particle anisotropies. They show a clear tendency of the fluctuations to reduce the temperature anisotropies (consistent with the linear constraint of Equation (57

)), and also reduce the initial differential speed that was assumed to be at a sizable fraction of the Alfvén speed. The numerical results were found to be consistent with Ulysses observations.

Kasper et al. (2002 ) also evaluated the total proton thermal anisotropy and investigated the firehose instability, which may arise in the solar wind. They demonstrated, with a large data set of more than seven years, that the observed limit to the proton temperature anisotropy for $T ∥∕T ⊥ > 0$ was in agreement with constraints posed by theory and simulations on the firehose instability (Gary et al., 1997 , 1998 , 2000a ). This constraint is analogous to the anisotropy limit obtained by theoretical and computational methods for the electromagnetic cyclotron instability driven by $T⊥ ∕T∥ > 1$ (see Gary et al., 2001b and references therein). Matteini et al. (2005 ) demonstrated that the proton fire hose instability, which can develop when $T∥ > T⊥$ , is able to counteract and limit the growth of the anisotropy as naturally caused by adiabatic expansion.

Concerning other recent theoretical interpretations of the measured anisotropies, Araneda et al. (2002 ) studied the proton core temperature effects on the relative drift and anisotropy evolution of the ion beam instability in the fast solar wind, and Gary and Saito (2003 ) presented theoretical evidence obtained through numerical simulations for the regulation of the core anisotropy in association with plateau formation through pitch-angle diffusion of protons in cyclotron resonance.

Tu and Marsch (2002 ) analysed proton VDFs in the solar wind with respect to the dependence of the temperature anisotropy on the plasma beta and established an empirical relationship and theoretical explanation of their result on the basis of resonant diffusion of the protons by dispersive cyclotron waves. Marsch et al. (2004 ) provided solid statistical evidence on the relation between the anisotropy and the proton plasma beta, parameters that are believed to play a key role in the wave regulation of the shape of the core VDF. They found a clear linear correlation between $T⊥ ∕T∥$ and the plasma beta $β$ , and used these data to make a least-squares-fit analysis and to compare the resulting empirical fit with theoretical predictions.

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

These results are shown in Figure 17 . The black isolated dots represent the mean values of the observed Helios data points, with the data binned in various $β$ intervals. The whole $β$ range extends from $log(0.06)$ to $log(1.0)$ and is divided into 36 bins. The vertical bars give the corresponding standard deviations, respectively. The thick solid line shows the result of a least-squares fit to the data points. The light-dotted line shows the temperature ratio resulting from quasi-linear diffusion caused by dispersive cyclotron waves, which obey the cold plasma dispersion relation Tu and Marsch (2002 ). The dotted-dashed line shows the function $A = 0.65β− 0.40$ , which is the anisotropy instability threshold as inferred from numerical simulations for a limiting growth rate $γ ∕Ωp = 0.01$ (Gary et al., 2001c ).

The least-squares fit presented in Figure 17 involves a large number of data points: 25439. The fit gives the functional relation $A = eaβb − 1$ , with the coefficients $a = 1.505 × 10−1 ± 4.358 × 10−3$ and $b = − 5.533 × 10−1 ± 2.809 × 10−3$ . This correlation with a coefficient of 0.78 indicates that in high-speed wind there exists an empirical relation between the proton core anisotropy and plasma beta determined by the proton core parallel temperature.

Large heavy-ion thermal anisotropies were also detected in the solar corona. The Ultraviolet Coronagraph and Spectrograph (UVCS) on SOHO measured the O vi line widths and inferred that very high temperature anisotropies of the O⁵⁺ ions must exist in the Sun’s polar corona hole (Kohl et al., 1998 ). According to these remote-sensing observations, $T ∕T ⊥o ∥o$ , may become higher than 100. However, Ofman et al. (2001 ) have shown by numerical simulations that the ion-cyclotron instability constrains the anisotropy of the O⁵⁺ ions that can be sustained in the corona. Maximum growth is obtained for parallel wave propagation. Using the linear dispersion relation after Gary and Lee (1994 ), the exact numerical solution of the dispersion equation allows one to derive for any heavy ion species $i$ a threshold condition involving the parallel plasma beta, $β∥i$ , such that the limiting anisotropy for maximal growth rate reads as follows:

where $Si$ is of order unity, and for the exponent one obtains $αi ≈ 0.4$ . Hybrid numerical simulations were performed by Gary et al. (2001c ), which carried the instability to its non-linear saturation. The simulation results are in general agreement with the instability threshold scaling of Equation (59

), however the best description of the non-linear stage yields higher $S i$ values, with $1 < Si ≤ 10$ . There were limitations in this work in so far as no relative flows between protons and heavy ions were considered. The effective wave-ion scattering rate turned out to be about of the order the ion gyrofrequency and rather independent of the plasma beta, with $&tidle;νi ≈ 0.3Ωi$ .

It is now widely believed that the observed large ion temperature anisotropies indicate the physical mechanism by which the solar corona and solar wind are heated (see the review by Hollweg and Isenberg, 2002 ). Recently, Li and Habbal (2005 ) have also carried out hybrid simulation of the ion-cyclotron resonance in the solar wind and studied the evolution of velocity distribution functions. However, the coronal and interplanetary origin of the strong perpendicular ion heating is still not well understood.