# Governing equations

The present simplified GCM is the KÃ¼hlungsborn Mechanistic general Circulation Model (KMCM) coded by the author. It is based on the primitive equations (Becker, 2003, section 2.4). These governing equations are transformed using a terrain-following vertical hybrid coordinate Î· as follows. Pressure p is represented as a function of Î· and surface pressure p_{s}:

The coefficients a and b must guarantee monotonic growth of p with Î·, as well as

The flexibility of (A1) is used to let surfaces of constant Î· correspond to Ïƒ-levels near the ground and to pressure levels at high altitudes. To achieve such behaviour we define

Here, p_{00}:=1013 mb corresponds to the mass of the atmosphere in case of zero orography. The prognostic equations for horizontal vorticity Î¾â€š horizontal divergence D, temperature T, and surface pressure p_{s }may be written as

The prognostic equations (A4)-(A8) are completed by (Al) and (A3), and expressions for geopotential Ð¤, vertical velocity Î®, and pressure velocity Ï‰:

Equation (A9) follows from vertical integration of the hydrostatic approximation

Vertical velocity, pressure velocity, and surface pressure tendency follow from integrations of the continuity equation

with respect to the kinematic boundary conditions