numerical solution of conservative form of two-dimensional compressible and non-hydrostatic equations of the atmosphere using second-order maccormack method

this work reports the results of the application of the second-order maccormack method for numerical solution of‎ the conservative form of two-dimensional non-hydrostatic and fully compressible navier-stokes equations governing an inviscid and adiabatic atmosphere‎. various aspects of the computational approach such as discretization of the governing equations for the interior and boundary points‎, ‎the details of implementation of boundary conditions for different boundary types, i.e., ‎rigid and open boundaries‎, ‎time step‎, ‎grid resolution and dissipation are presented‎. in addition, it is shown that application of the second-order maccormack scheme to spatial discretization of the source term in the vertical momentum equation of two-dimensional non-hydrostatic and fully compressible navier-stokes equations ‎needs special treatment‎. ‎in other words‎, ‎the spatial discretization of this source term should be consistent with the hydrostatic equation and must not degrade its balance‎. ‎the details of the procedure to reach the discretized version of the vertical momentum equation are also presented. ‎several well known test cases including evolution of a warm bubble in a neutral atmosphere (in domains with rigid and open boundary conditions), evolution of a cold bubble in a neutral atmosphere (density current benchmark proposed by straka et al. (1993)) and a gravity current, ‎are used for numerical experiments. qualitative and quantitative comparisons indicate the validity of the results and show that the results of the second-order maccormack scheme are in good agreement with the published results for the evolution of the warm bubble and the reference solution presented by straka et al.