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I Observations
II Assimilation
III Dynamics
IV Physics
V Ensemble
VI Technical
VII Waves

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.

3.10.1 Moisture adjustment and first time-step treatment

The parametrizations of cloud and convection show a sensitivity to the initial profiles. Therefore, at the initial time-step of the model "first guess" predictors are generated by a two step iteration of the parametrizations of cloud and convection consistent with the provisional dynamic fields as described above.

The effective profiles of temperature and humidity (including all contributions from the departure as well as the arrival point) are computed after all physical processes have been accounted for. A final moist adjustment is performed on these effective profiles and any amount of surplus humidity is added to the rainfall or snow fluxes in the next time-step. Note, that after this adjustment the temperature profile may still be altered as a result of the semi-implicit solution procedure.