  A three-dimensional Monte Carlo Ly-alpha radiative transfer (RT) code, named
LaRT, is developed to study the Ly-alpha RT and the Wouthuysen-Field (WF)
effect. Using the code, we calculate the line profile of Ly-alpha radiation
within the multiphase interstellar medium (ISM), with a particular emphasis on
gas at low densities. We show that the WF effect is in action: the central
portion of the line profile tends to approach a small slice of the Planck
function with a color temperature equal to the kinetic temperature of the gas,
even in a system with an optical thickness as low as tau_0~100-500. We also
investigate the effects of the turbulent motion of the ISM on the emergent
Ly-alpha spectrum and color temperature. The turbulent motion broadens, as
generally expected, the emergent spectrum, but the color temperature is not
affected by the turbulent motion in typical astrophysical environments. We
utilize two multiphase ISM models, appropriate for the vicinity of the Sun, to
calculate the 21-cm spin temperature of neutral hydrogen, including excitation
via the Ly-alpha resonant scattering. The first ISM model is a simple clumpy
model, while the second is a self-consistent magnetohydrodynamics simulation
model using the TIGRESS framework. Ly-alpha photons originating from both H II
regions and the collisionally cooling gas are taken into account. We find that
the Ly-alpha radiation field is, in general, likely to be strong enough to
bring the 21-cm spin temperature of the warm neutral medium close to the
kinetic temperature. The escape fraction of Ly-alpha in our ISM models is
estimated to be ~7-20%.
