We show that, due to the high optical depth of the intergalactic medium to
Lyman-alpha photons before the Epoch of Reionization, the Lyman-alpha
scattering rate responsible for the Wouthuysen-Field effect from an isolated
source will be negligible unless (1) there is sufficient time for the
scattering photons to establish a steady state, or (2) the scattering gas is
undergoing internal expansion or has a peculiar motion of tens to hundreds of
km/s away from the source. We present steady-state solutions in the radiative
diffusion approximation for the radiation field trapped in a clump of gas and
show that this may result in an enhancement, by a factor of up to 10^6, of the
strength of the Wouthuysen-Field effect over that obtained from the
free-streaming limit. Solutions to the time-dependent diffusion equation,
however, suggest that the timescales required to reach such a steady state will
generally exceed the source lifetimes. In the presence of internal expansion, a
steady state may be established as photons are redshifted into the red wing,
and significant enhancement in the scattering rate may again be produced.
Alternatively, a substantial scattering rate may arise in systems with a
peculiar motion away from the source that redshifts the received radiation into
the resonance line centre. As a consequence, at epochs z<30, when collisional
decoupling is small except in dense regions, and prior to the establishment of
any large-scale diffuse radiation field of resonance line photons, the 21cm
signature from the Intergalactic Medium produced by the Wouthuysen-Field effect
will in general trace the peculiar velocity field of the gas in addition to its
density structure.Comment: 11 pages, 7 figures. Accepted for publication in MNRA