Model equations

In a generic pressure-based terrain following vertical coordinate η, the governing equations of the atmosphere under the hydrostatic assumption can be derived following Kasahara (1974). This equation set, also known as the primitive equations, reads

　

Here v is the  horizontal velocity vector (on the sphere);  k is the  unit vector in the radial outward direction;  f  is the Coriolis parameter; is the vertical component of the relative vorticity; is the kinetic energy per unit mass.  p and T denote pressure and temperature, respectively. is geopotential, R is the gas constant and Cp  the specific heat capacity of dry air under constant pressure. Note that the momentum equation is written in the vector invariant form.

　

The hybrid vertical coordinate

The generalized p-sigma hybrid vertical coordinate used in ICOHDC is the same as in ECHAM5 and many other global hydrostatic models. It was first introduced by Simmons and Strüfing (1981) for the grid-point forecast model of the ECMWF. The vertical domain of the model atmosphere is divided into NLEV layers. Pressure values of the interfaces between layers are given by

　

and are parameters independent of the horizontal location.  Near the earth’s surface equals zero and the interfaces are pure sigma levels; near the model top equals zero and the interfaces coincide with isobaric surfaces. The levels in between provide a smooth transition. Pressure of the "full levels" on which the horizontal wind and temperature are predicted is defined as