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.