  This study proposes an algorithm for modeling compressible flows in spherical
shells in nearly incompressible and weakly compressible regimes based on an
implicit direction splitting approach. The method retains theoretically
expected convergence rates and remains stable for extremely small values of the
characteristic Mach number. The staggered spatial discretization on the MAC
stencil, commonly used in numerical methods for incompressible Navier-Stokes
equations, was found to be convenient for the discretization of the
compressible Navier-Stokes equations written in the non-conservative form in
terms of the primitive variables. This approach helped to avoid the
high-frequency oscillations without any artificial stabilization terms.
Nonlinear Picard iterations with the splitting error reduction were also
implemented to allow one to obtain a solution of the fully nonlinear system of
equations. These results, alongside excellent parallel performance, prove the
viability of the direction splitting approach in large-scale high-resolution
high-performance simulations of atmospheric and oceanic flows.
