Self-Similar Theory of Thermal Conduction and Application to the Solar Wind

We propose a self-similar kinetic theory of thermal conductivity in a magnetized plasma, and discuss its application to the solar wind. We study a collisional kinetic equation in a spatially expanding magnetic flux tube, assuming that the magnetic field strength, the plasma density, and the plasma temperature decline as power laws of distance along the tube. We demonstrate that the electron kinetic equation has a family of scale-invariant solutions for a particular relation among the magnetic-, density-, and temperature-scaling exponents. These solutions describe the heat flux as a function of the temperature Knudsen number $\gamma$, which we require to be constant along the flux tube. We observe that self-similarity may be realized in the solar wind; for the Helios data $0.3–1\mathrm{AU}$ we find that the scaling exponents for density, temperature, and heat flux are close to those dictated by scale invariance. We find steady-state solutions of the self-similar kinetic equation numerically, and show that these solutions accurately reproduce the electron strahl population seen in the solar wind, as well as the measured heat flux.

Figure 1

2D histogram of $\gamma$ versus heliocentric distance $r$, derived from fits to Helios eVDF data. Each column is normalized by its peak, to bring out the functional dependence. We observe $\gamma (r)\approx \text{const}$, required in order to apply the self-similar theory.

Figure 2

Perpendicular and parallel cuts of $F^{*}$ from the Helios measurements (lines) and Langevin simulations (dots). The Helios data are averaged into logarithmically spaced bins ${10}^{-13/8}<\gamma <{10}^{-7/8}$. The numerical solutions are found with $\gamma$ chosen at the logarithmic center of a bin.

Figure 3

2D histogram of $q/q_0$ versus heliocentric distance ${\lambda }_{mfp}/L_T$, derived from fits to Helios eVDF data, correcting for the Parker spiral angle and assuming ${\alpha }_T=-2/5$. Each column is normalized by its peak, to bring out the functional dependence. The Spitzer-Härm prediction is shown as a line. Results of our numerical solutions are shown as dots.