%0 Journal Article
%T Numerical solution of conservative form of two-dimensional compressible and non-hydrostatic equations of the atmosphere using second-order MacCormack method
%J Journal of the Earth and Space Physics
%I Institute of Geophysics, University of Tehran
%Z 2538-371X
%A Ghader, Sarmad
%A Ali-Akbari Bidokhti, Abbas-Ali
%A Falahat, Saeed
%D 2011
%\ 07/23/2011
%V 37
%N 2
%P 171-191
%! Numerical solution of conservative form of two-dimensional compressible and non-hydrostatic equations of the atmosphere using second-order MacCormack method
%K Atmosphere
%K Compressible
%K Finite difference
%K MacCormack Scheme
%K Non-hydrostatic
%R
%X 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.
%U https://jesphys.ut.ac.ir/article_23095_98626d26e9dc900d7784f80a7e833613.pdf