Governing equations

The present simplified GCM is the Kühlungsborn 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 η as follows. Pressure p is represented as a function of η and surface pressure ps:

(A1)

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

(A2)

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

(A3)

Here, p00:=1013 mb corresponds to the mass of the atmosphere in case of zero orography. The prognostic equations for horizontal vorticity ξ‚ horizontal divergence D, temperature T, and surface pressure ps may be written as

(A4)
(A5)
(A6)
(A7)
(A8)

The prognostic equations (A4)-(A8) are completed by (Al) and (A3), and expressions for geopotential Ф, vertical velocity ή, and pressure velocity ω:

(A9)
(A10)
(A11)

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

(A12)

with Фs denoting the orography (Fig.1).

Figure 1 : Model topographie Фs/g. Contours are from 0.5 to 4 km in intervals of 0.5 km. The Himalayas and the Rockies reach maximum elevations of about 4.4 and 1.8 km, respectively.

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

(A13)

with respect to the kinematic boundary conditions

(A14)

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