Conservative Scheme for the Compressible Nonhydrostatic Models with the Horizontally Explicit and Vertically Implicit Time Integration Scheme

A new dynamical scheme with the conservative forms of the equations of density, momentum, and internal energy is proposed for the nonhydrostatic models. With this scheme, the conservations of the mass and the total energy are satisfied within round-off errors. In particular, methods for the integration of energy are discussed in detail, and three of the approaches are compared; one is in the form of the pressure equation, the second is the integration of internal energy with corrections on the transformation terms, and the third is use of the sum of internal energy and kinetic energy as a prognostic variable. This scheme is incorporated into a nonhydrostatic model with the horizontally explicit and vertically implicit time integration scheme for sound waves, and various numerical experiments for the dry atmosphere are performed. The numerical results show that the conservative properties are well satisfied.