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VII Waves

2.1 Eulerian reformulation of the continuous equations

Following Ritchie (1988,1991), the first step in developing a semi-Lagrangian version of the ECMWF spectral model was to convert the existing Eulerian -D (vorticity-divergence) model to a - formulation, where and are the wind images defined by , ( and are the components of the horizontal wind in spherical coordinates, and is latitude). In this section we describe the Eulerian - model.

First we set out the continuous equations in coordinates, where is longitude and is the hybrid vertical coordinate introduced by Simmons and Burridge (1981); thus is a monotonic function of the pressure , and also depends on the surface pressure in such a way that

and .

The momentum equations are

(2.1)

,

(2.2)

where is the radius of the earth, is the -coordinate vertical velocity (), is geopotential, is the gas constant for dry air, and is the virtual temperature defined by

where is temperature, is specific humidity and is the gas constant for water vapour. and represent the contributions of the parametrized physical processes, while and are the horizontal diffusion terms.

The thermodynamic equation is

(2.3)

where ( is the specific heat of dry air at constant pressure), is the -coordinate vertical velocity (), and ( is the specific heat of water vapour at constant pressure).

The moisture equation is

(2.4)

In (2.2) and (2.3), and represent the contributions of the parametrized physical processes, while and are the horizontal diffusion terms.

The continuity equation is

(2.5)

where is the horizontal gradient operator in spherical coordinates and is the horizontal wind.

The geopotential which appears in (2.1) and (2.2) is defined by the hydrostatic equation

(2.6)

while the vertical velocity in (2.3) is given by

(2.7)

Expressions for the rate of change of surface pressure, and for the vertical velocity , are obtained by integrating (2.5), using the boundary conditions at and at

(2.8)

(2.9)

Since we use rather than as the surface pressure variable, it is convenient to rewrite (2.8) as

(2.10)