Recent studies of turbulence driven solar winds indicate that fast winds are obtained only at the price of unrealistic bottom boundary conditions: too large wave amplitudes and small frequencies. In these works, the incompressible turbulent dissipation is modelled with a large-scale von Karman-Howart-Kolmogorov-like phenomenological expression (Q 0 K41 ). An evaluation of the phenomenology is thus necessary to understand if unrealistic boundary conditions result from physical or model limitations. To assess the validity of the Kolmogorov-like expression, Q 0 K41 , one needs to compare it to exact heating, which requires describing the cascade in detail. This has been done in the case of homogeneous MHD turbulence, including expansion, but not in the critical accelerating region. To assess the standard incompressible turbulent heating in the accelerating region, we use a reduced MHD model (multishell model) in which the perpendicular turbulent cascade is described by a shell model, allowing to reach a Reynolds number of 10 6 . We first consider the homogeneous and expanding cases, and find that primitive MHD and multishell equations give remarkably similar results. We thus feel free to use the multishell model in the accelerating region. The results indicate that the large-scale phenomenology is inaccurate and it overestimates the heating by a factor at least 20, thus invalidating earlier studies of winds driven by incompressible turbulence. We conclude that realistic 1D wind models cannot be based solely on incompressible turbulence, but probably need an addition of compressible turbulence and shocks to increase the wave reflection and thus the heating.

We derive two new forms of the Kármán–Howarth–Monin (KHM) equation for decaying compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results of a weakly compressible, 2D, moderate-Reynolds-number Hall MHD simulation and compare them with an isotropic spectral transfer (ST) equation. The KHM and ST equations are automatically satisfied during the whole simulation owing to the periodic boundary conditions and have complementary cumulative behavior. They are used here to analyze the onset of turbulence and its properties when it is fully developed. These approaches give equivalent results characterizing the decay of the kinetic + magnetic energy at large scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. The Hall cascade appears when the MHD one brings the energy close to the ion inertial range and is related to the formation of reconnecting current sheets. At later times, the pressure dilatation energy exchange rate oscillates around zero, with no net effect on the cross-scale energy transfer when averaged over a period of its oscillations. A reduced 1D analysis suggests that all three methods may be useful to estimate the energy cascade rate from in situ observations.

The heating of the solar wind is key to understanding its dynamics and acceleration process. The observed radial decrease of the 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 that 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 expanding box model equations, which incorporate into 3D MHD equations the expansion due to the mean radial wind, allowing us to follow the plasma evolution between 0.2 and 1 au. We vary the initial parameters: Mach number, expansion parameter, plasma β, 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 on 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.