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IFS documentation Front Page
Table of contents
Chapter 1. Introduction
Chapter 2. Basic equations and discretization
Chapter 3. Semi-Lagrangian formulation
Chapter 4. Computational details
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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) |
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
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) |
(
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) |
and 
represent the contributions of the parametrized physical processes,
while 
and 
are the horizontal diffusion terms. The continuity equation is
|
(2.5) |
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) |
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) |
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08.04.2002 |
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