A non-hydrostatic semi-Lagrangian scheme on convective dynamics

A non-hydrostatic, semi-Lagrangian transport scheme is proposed. The scheme is able to solve the fully-compressible Euler equations with a minimum of filtering and approximations, allowing high-speed waves such as acoustic waves. Acoustic waves put severe limitations on the maximum time step that can be taken ensuring stability by not violating the Courant-Friedrich-Lewy criterion. Therefore, a time-split technique is incorporated such that the pressure type waves are treated with an Eulerian forward-backward method, and the advective part is treated with a semi- Lagrangian method. The scheme is able to reliably simulate convective thermals in both neutrally and stably stratified environments, including features as gravity wave oscillations and Kelvin-Helmholtz instability. A review of Eulerian and semi-Lagrangian methods is given along with a discussion of previous methods used in compressible models. Qualitative comparisons to earlier non-hydrostatic model studies are made.