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DYNAMICS
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Chapter 1. Introduction
Chapter 2. Basic equations and discretization
Chapter 3. Semi-Lagrangian formulation
Chapter 4. Computational details
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3.8 Modified semi-Lagrangian equations
3.8.1 Momentum equations
The momentum equations are treated in vector form (Eq. (3.14)). Following Rochas (1990) and Temperton (1997), the Coriolis terms can be incorporated in the semi-Lagrangian advection. Thus, the advected variable becomes
where 
is the earth's rotation and 
is the radial position vector, while the Coriolis terms are
dropped from the right-hand side. As described by Temperton (1997), this
reformulation is beneficial provided that the spherical geometry is treated
accurately in determining the departure point and in rotating the vectors
to account for the change in the orientation of the coordinate system as
the particle follows the trajectory.The discretization of the momentum equations in the notation of Eq. (3.1) is then:
|
(3.31) |
|
(3.32) |
|
(3.33) |
where
is the gas constant for dry air, 
is a reference temperature, 
is geopotential and 
is the linearized hydrostatic integration matrix defined in
Eq. (2.32) of Ritchie (1995).In component form,
is just 
where 
is the earth's radius and 
is latitude. Since the latitude of the departure point is known,
the term 
in the advected variable 
is computed analytically rather than interpolated. An alternative
semi-implicit treatment of the Coriolis terms has also been developed (Temperton
1997).3.8.2 Continuity equation
Modelling flow over mountains with a semi-Lagrangian integration scheme can lead to problems in the form of a spurious resonant response to steady orographic forcing. The mechanism was clarified by Rivest et al. (1994). Strictly speaking, the problem has little to do with the semi-Lagrangian scheme itself; rather, it is a result of the long time steps permitted by the scheme, such that the Courant number becomes greater than 1. Recently, Ritchie and Tanguay (1996) proposed a modification to the semi-Lagrangian scheme which alleviates the problem. It turned out that their suggestion was easy to implement in the ECMWF model, and had additional benefits besides improving the forecast of flow over orography.
Although Ritchie and Tanguay start by introducing a change of variables in the semi-implicit time discretization, this is not necessary and a slightly different derivation is presented here. The continuity equation is written in the form
|
(3.34) |
where
represents right-hand-side terms. The total
derivative on the left-hand side is discretized in a semi-Lagrangian fashion,
and the final form of the discretized equation involves a vertical summation.Now split
into two parts:
|
(3.35) |
where the time-independent part
depends on the underlying orography 
:
|
(3.36) |
and
is a reference temperature. This choice gives
|
(3.37) |
so that
is (to within an additive constant) the value
of 
appropriate for an isothermal state at rest with underlying
orography.Using (3.35) and (3.36),
|
(3.38) |
The second term on the right-hand side is computed in an Eulerian manner and transferred to the right-hand side of the continuity equation (3.34), which becomes
|
(3.39) |
The new advected variable is much smoother than the original, since the influence of the underlying orography has been subtracted out; hence, the semi-Lagrangian advection is presumably more accurate.
3.8.3 Thermodynamic equation
As mentioned above, the semi-Lagrangian treatment of the continuity equation is improved by changing the advected variable to a smoother quantity which is essentially independent of the underlying orography. A similar modification has been implemented in the thermodynamic equation, borrowing an idea from the treatment of horizontal diffusion. To approximate horizontal diffusion on pressure surfaces, thereby avoiding spurious warming over mountain tops in sigma or hybrid vertical coordinates, the diffused quantity is
, with
|
(3.40) |
where the subscript `ref' denotes a reference value which is a function only of model level. For the purposes of the semi-Lagrangian advection
is replaced by a time-independent value as in Eq.
(3.36) above, to define a "temperature"
which depends only on the model level and the underlying orography:
|
(3.41) |
The semi-Lagrangian advection is now applied to the quantity
, while a compensating expression
|
(3.42) |
appears on the right-hand side of the equation and is computed in an Eulerian fashion (note that this time it includes a vertical advection term).
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08.04.2002 |
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