  We design and analyse an energy stable, structure preserving, well-balanced
and asymptotic preserving (AP) scheme for the barotropic Euler system with
gravity in the anelastic limit. The key to energy stability is the introduction
of appropriate velocity shifts in the convective fluxes of mass and momenta.
The semi-implicit in time and finite volume in space fully-discrete scheme
supports the positivity of density and yields the consistency with the weak
solutions of the Euler system upon mesh refinement. The numerical scheme admits
the discrete hydrostatic states as solutions and the stability of numerical
solutions in terms of the relative energy leads to well-balancing. The AP
property of the scheme, i.e. the boundedness of the mesh parameters with
respect to the Mach/Froude numbers and the scheme's asymptotic consistency with
the anelastic Euler system is rigorously shown on the basis of apriori energy
estimates. The numerical scheme is resolved in two steps: by solving a
non-linear elliptic problem for the density and a subsequent explicit
computation of the velocity. Results from several benchmark case studies are
presented to corroborate the proposed claims.
