Turbulent Heating between 0.2 and 1 au: A Numerical Study

Abstract

The heating of the solar wind is a key to understand its dynamics and acceleration process. The observed radial decrease of proton temperature in the solar wind is slow compared to the adiabatic prediction and it is thought to be caused by turbulent dissipation. To generate the observed 1/R decrease, the dissipation rate has to reach a specific level which varies in turn with temperature, wind speed, and heliocentric distance. We want to prove that MHD turbulent simulations can lead to the 1/R profile. We consider here the slow solar wind, characterized by a quasi-2D spectral anisotropy. We use the EBM (expanding box model) equations, which incorporate into 3D MHD equations the expansion due to the mean radial wind, allowing to follow the plasma evolution between 0.2 and 1 AU. We vary the initial parameters which are: Mach number, expansion parameter, plasma beta, and properties of the energy spectrum as the spectral range and slope. Assuming turbulence starts at 0.2 AU with a Mach number equal to unity, with a 3D spectrum mainly perpendicular to the mean field, we find radial temperature profiles close to 1/R in average. This is done at the price of limiting the initial spectral extent, corresponding to the small number of modes in the inertial range available, due to the modest Reynolds number reachable with high Mach numbers.

Figures (12)

• Figure 1: Initial and final domains of simulation (and plasma volume as well) in the ecliptic plane. Thin lines: direction of mean magnetic field. For all runs, the aspect ratio of the domain varies from 1/5 to unity. In the figure, the mean magnetic field angle with the radial varies varies from $\tan^{-1}(1/5) \simeq11.3^0$ to $\tan^{-1}(1)=\pi/4$.
• Figure 2: Run A. Evolution of basic quantities vs heliocentric distance $R/R_0$. (a) Velocity amplitude $u_{rms}$ (solid line), compressible velocity $u^c_{rms}$ (dotted) and magnetic field fluctuation $b_{rms}/\bar{\rho}^{1/2}$ (dashed). (b) Visco-resistive dissipation $Q_\nu$ (solid line) decomposed as the sum of $\mu |\tilde{\nabla} \times u|^2$ (dotted), $\eta \tilde{j}^2$ (dashed) and $\frac{4}{3}\nu |\tilde{\nabla} .u|^2$ (dotted-dashed). (c) Dissipation rates per unit mass: $dE/dt$ (solid line), expansion-driven damping $Q_{exp}$ (dotted), visco-resistive dissipation $Q_\nu$ (dashed), nonlinear loss during cascade $Q_{NL}$ (dotted-dashed, thick when increasing the decay rate, thin when decreasing the decay rate). (d) Temperature compensated by $1/R$ decrease. Distance is normalized by the initial distance $R_0$ = 0.2 AU.
• Figure 3: Heating ratio $Q_{\nu}/Q_c$ and temperature profiles, all with $M=1,\epsilon=0.2$ but varying small-scale initial excitation. Runs A (thick solid line), B (dotted), C (dashed), E (dashed-dotted). (a) Heating ratio $Q_{\nu}/Q_{c}$ vs heliospheric distance $R$. (b) Average temperature (normalized by its initial value) compensated by $R_0/R$. Distance is normalized by the initial distance $R_0=0.2$ AU. The thin solid straight line in panel (b) corresponds to $T/T_0=(R/R_0)^{-4/3}$.
• Figure 4: Heating ratio $Q_{\nu}/Q_c$ and temperature profiles, with same $\epsilon=0.2$ but varying initial Mach number. Runs C (M=1, solid thick line) and D (M=0.77, dotted). Same caption as in fig. \ref{['fig1']}.
• Figure 5: Heating ratio $Q_{\nu}/Q_c$ and temperature profiles, with same $M=0.77$ but varying expansion parameter. Runs F ($\epsilon=0.12$, solid thick line) and K ($\epsilon=0.2$, dotted); same caption as in fig. \ref{['fig1']}.