  We investigate the atmosphere, ocean, and crust of neutron stars accreting at
rates sufficiently high (typically in excess of the local Eddington limit) to
stabilize the burning of accreted hydrogen and helium. For hydrogen-rich
accretion at global rates in excess of 10^-8 solar masses per year (typical of
a few neutron stars), we discuss the thermal state of the deep ocean and crust
and their coupling to the neutron star core, which is heated by conduction
(from the crust) and cooled by neutrino emission. We estimate the Ohmic
diffusion time in the hot, deep crust and find that it is noticeably shortened
(to less than 10^8 yr) from the values characteristic of the colder crusts in
slowly accreting neutron stars. We speculate on the implications of these
calculations for magnetic field evolution in the bright accreting X-ray
sources. We also explore the consequences of rapid compression at local
accretion rates exceeding ten times the Eddington rate. This rapid accretion
heats the atmosphere/ocean to temperatures of order 10^9 K at relatively low
densities; for stars accreting pure helium, this causes unstable ignition of
the ashes (mostly carbon) resulting from stable helium burning. This unstable
burning can re-occur on timescales as short as hours to days, and might be the
cause of some flares on helium accreting pulsars, in particular 4U~1626--67.
Such rapid local accretion rates are common on accreting X-ray pulsars, where
the magnetic field focuses the accretion flow onto a small fraction of the
stellar area.
