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

Abstract

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_\sim 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, $\dot {M} \simeq P_\textrm {AW}(r_\textrm {b})/v_\textrm {esc}^2$ , $U_{\infty } \simeq v_\textrm {esc}$ , and $T\simeq m_\textrm {p} v_\textrm {esc}^2/[8 k_\textrm {B} \ln (v_\textrm {esc}/\delta v_\textrm {b})]$ , 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, $\delta v_\textrm {b}$ 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.