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 = rb and the Alfvén critical point at r = rA, where r is heliocentric distance, the solar-wind density decreases by a factor ≳ 105, 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 = rA 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, Ṁ ≃ PAW(rb)/v2esc, U∞ ≃ vesc, and T ≃ mpv2esc/[8kB ln(vesc/δvb)], where Ṁ is the mass outflow rate, U∞ is the asymptotic wind speed, T is the coronal temperature, vesc is the escape velocity of the Sun, δvb is the fluctuating velocity at rb, PAW is the power carried by outward-propagating AWs, kB is the Boltzmann constant, and mp 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 qb from the coronal base into the transition region and momentum deposition by AWs. I solve analytically for qb by balancing conductive heating against internal-energy losses from radiation, p dV work, and advection within the transition region. The density at rb is determined by balancing turbulent heating and radiative cooling at rb. 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 Ṁ, U∞, and T. Analytic and numerical solutions to the model equations match a number of observations.