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.