  We study the thermal structure of the widely adopted two-dimensional
advection dominated accretion flow (ADAF) of Narayan & Yi (1995a). The critical
radius for a given mass accretion rate, outside of which the optically thin hot
solutions do not exist in the equatorial plane, agrees with one-dimensional
study. However, we find that, even within the critical radius, there always
exists a conical region of the flow, around the pole, which cannot maintain the
assumed high electron temperature, regardless of the mass accretion rate, in
the absence of radiative heating. This could lead to torus-like advection
inflow shape since, in general, the ions too will cool down. We also find that
Compton preheating is generally important and, if the radiative efficiency,
defined as the luminosity output divided by the mass accretion rate times the
velocity of light squared, is above sim 4x10^-3, the polar region of the flow
is preheated above the virial temperature by Compton heating and it may result
in time-dependent behaviour or outflow while accretion continues in the
equatorial plane. Thus, under most relevant circumstances, ADAF solutions may
be expected to be accompanied by polar outflow winds. While preheating
instabilities exist in ADAF, as for spherical flows, the former are to some
extent protected by their characteristically higher densities and higher
cooling rates, which reduce their susceptibility to Compton driven overheating.
