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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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3.10 Numerical coupling of the physical parameterizations to the "dynamical" equations (SLAVEPP)
Due to the diffusive nature of the mostly parabolic equations in the physics the contributions of the physical parameterizations are computed separately from the "dynamical" equations. The coupling of these two parts can use the SLAVEPP (Semi-Lagrangian Averaging of Physical Parameterizations) method which is described and discussed in detail by Wedi (1999).
In equation (3.14)-(3.16) the contribution of the physical parameterizations are denoted as
indicating an evaluation of the parameterizations
at the arrival point only. In the two time level scheme as described in
section 3.9 this is replaced by
a partly second order accurate coupling of the parameterizations in time
and space, which is achieved by evaluating part of the "physics" at the
arrival point and the remainder at the departure point of the semi-Lagrangian
trajectory. Due to the different nature of the parameterized processes the
contributions of radiation, convection and cloud parameterization are averaged
"along" the semi-Lagrangian trajectory while the contributions of vertical
diffusion and parameterized gravity waves are taken at the arrival point
only. Equation (3.43) becomes then
|
(3.53) |
Part of the implicit calculations of the physical parameterizations use the following tendency:
 ,
|
(3.54) |
with equation (3.43) modified to yield
|
(3.55) |
The "~" denotes that only provisional values of the dynamic fields are available because semi-implicit correction terms are still to be computed (see section 3.6). Therefore
is used for the linear terms. Equation
(3.54) describes local tendencies,
which are computed subtracting the new provisional explicit values 
of the dynamic fields (at the arrival point) from their values
at the previous time step. The parameterizations at the time
step 
are computed at the arrival point as
shown in the following equation:
|
(3.56) |
where the "first guess" predictor
of the model variables at the arrival point at time step 
is computed from the tendency of the
"dynamics", the tendency of the parameterizations of radiation, convection
and clouds at the previous time-step t and the tendency of vertical
diffusion and gravity waves at 
:
 .
|
(3.57) |
denotes an explicit interaction of the parameterizations
of cloud and convection.The parameter 
has been introduced in order to achieve a better balance
between the physical parameterizations when the "first guess" predictor
is computed.Next Section
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08.04.2002 |
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