The semi-Lagrangian dynamical core adopts the same hybrid vertical coordinate () as the Eulerian core defined by

where is pressure, is surface pressure, and is a specified constant reference pressure. The coefficients and specify the actual coordinate used. As mentioned by Simmons and Burridge [1981] and implemented by Simmons and Strüfing [1981] and Simmons and Strüfing [1983], the coefficients and are defined only at the discrete model levels. This has implications in the continuity equation development which follows.

In the system the hydrostatic equation is approximated in a general way by

where k is the vertical grid index running from 1 at the top of the model to at the first model level above the surface, is the geopotential at level , is the surface geopotential, is the virtual temperature, and R is the gas constant. The matrix , referred to as the hydrostatic matrix, represents the discrete approximation to the hydrostatic integral and is left unspecified for now. It depends on pressure, which varies from horizontal point to point.