An approximate analytic solution to the coupled problems of coronal heating and solar-wind acceleration

Between the base of the solar corona at $r=r_\textrm {b}$ and the Alfvén critical point at $r=r_\textrm {A}$, where $r$ is heliocentric distance, the solar-wind density decreases by a factor $ \mathop > \limits_∼ 10^5$, but the plasma temperature varies by a factor of only a few. In this paper, I show that such quasi-isothermal evolution out to $r=r_\textrm {A}$ is a generic property of outflows powered by reflection-driven Alfvén-wave (AW) turbulence, in which outward-propagating AWs partially reflect, and counter-propagating AWs interact to produce a cascade of fluctuation energy to small scales, which leads to turbulent heating. Approximating the sub-Alfvénic region as isothermal, I first present a brief, simplified calculation showing that in a solar or stellar wind powered by AW turbulence with minimal conductive losses, $\dot {M} ∼eq P_\textrm {AW}(r_\textrm {b})/v_\textrm {esc}^2$, $U_{\infty } ∼eq v_\textrm {esc}$, and $T∼eq m_\textrm {p} v_\textrm {esc}^2/[8 k_\textrm {B} \ln (v_\textrm {esc}/δ v_\textrm {b})]$, where $\dot {M}$ is the mass outflow rate, $U_{\infty }$ is the asymptotic wind speed, $T$ is the coronal temperature, $v_\textrm {esc}$ is the escape velocity of the Sun, $δ v_\textrm {b}$ is the fluctuating velocity at $r_\textrm {b}$, $P_\textrm {AW}$ is the power carried by outward-propagating AWs, $k_\textrm {B}$ is the Boltzmann constant, and $m_\textrm {p}$ is the proton mass. I then develop a more detailed model of the transition region, corona, and solar wind that accounts for the heat flux $q_\textrm {b}$ from the coronal base into the transition region and momentum deposition by AWs. I solve analytically for $q_\textrm {b}$ by balancing conductive heating against internal-energy losses from radiation, $p \textrm {d} V$ work, and advection within the transition region. The density at $r_\textrm {b}$ is determined by balancing turbulent heating and radiative cooling at $r_\textrm {b}$. I solve the equations of the model analytically in two different parameter regimes. In one of these regimes, the leading-order analytic solution reproduces the results of the aforementioned simplified calculation of $\dot {M}$, $U_\infty$, and $T$. Analytic and numerical solutions to the model equations match a number of observations.