  We discuss a hydrodynamical model for the dispersal of protoplanetary discs
around young, low mass (<1.5 M_sun) stars by photoevaporation from the central
object's energetic radiation, which considers the far-ultraviolet as well as
the X-ray component of the radiation field. We present analytical scaling
relations and derive estimates for the total mass-loss rates, as well as
discussing the existence of similarity solutions for flows from primordial
discs and discs with inner holes. Furthermore, we perform numerical
calculations, which span a wide range of parameter space and allow us to
provide accurate scalings of the mass-loss rates with the physical parameters
of the systems (X-ray and FUV luminosity, stellar mass, disc mass, disc
temperature and inner hole radius).
  The model suggest that the X-ray component dominates the photoevaporative
mass-loss rates from the inner disc. The mass-loss rates have values in the
range from 10e-11 to 10e-7 M_sun/yr and scale linearly with X-ray luminosity,
with only a weak dependence on the other parameters explored. However, in the
case of high FUV to X-ray (L_FUV/L_X>100) luminosity ratios, the FUV constricts
the X-ray flow and may dominate the mass-loss.
  Simulations of low mass discs with inner holes demonstrate a further stage of
disc clearing, which we call `thermal sweeping'. This process occurs when the
mid-plane pressure drops to sufficiently low values. At this stage a bound,
warm, X-ray heated region becomes sufficiently large and unstable, such that
the remaining disc material is cleared on approximately dynamical time-scales.
This process significantly reduces the time taken to clear the outer regions of
the disc, resulting in an expected transition disc population that will be
dominated by accreting objects, as indicated by recent observations.
