  Anelastic convection at high Rayleigh number in a plane parallel layer with
no slip boundaries is considered. Energy and entropy balance equations are
derived, and they are used to develop scaling laws for the heat transport and
the Reynolds number. The appearance of an entropy structure consisting of a
well-mixed uniform interior, bounded by thin layers with entropy jumps across
them, makes it possible to derive explicit forms for these scaling laws. These
are given in terms of the Rayleigh number, the Prandtl number, and the bottom
to top temperature ratio, which measures how stratified the layer is. The top
and bottom boundary layers are examined and they are found to be very
different, unlike in the Boussinesq case. Elucidating the structure of these
boundary layers plays a crucial part in determining the scaling laws. Physical
arguments governing these boundary layers are presented, concentrating on the
case in which the boundary layers are thin even when the stratification is
large, the incompressible boundary layer case. Different scaling laws are
found, depending on whether the viscous dissipation is primarily in the
boundary layers or in the bulk. The cases of both high and low Prandtl number
are considered. Numerical simulations of no-slip anelastic convection up to a
Rayleigh number of $10^7$ have been performed and our theoretical predictions
are compared with the numerical results.
