Chapter 1 Introduction
5.3 Canopy Interface Model integration in WRF
5.3.2 WRF-CIM coupling strategy
CIM computes highly resolved vertical profiles of meteorological variables, but it doesnt include horizontal fluxes like a mesoscale model such as WRF (see Eq.5.1).
In such a context, it is possible to force CIM with WRF in a one-way nesting but it will not be valuable to correct the values calculated by WRF using CIM values as it could have been proposed in a traditional two-way nesting.
Thus two methodologies are tested : the first one is based on a coupling us- ing fixed top boundary conditions as done by Muller [2007] ; the second is a new proposition to add an additional term in CIM’s calculation in order to account for the processes described by the flux divergence term in Eq. (5.1).
Coupling by fixing top boundary condition - Method FT
CIM can calculate vertical profiles using prescribed top boundary conditions and description of the surface obstacles in each grid (geometry and surface tempera- ture). In an offline mode, the boundary conditions may be fixed at the top with a constant value, while when coupled with a meso-scale model, this value is interpo- lated from the meso-scale model at each time step. At the initialization time step, the meso-scale values are interpolated on each of CIM vertical level and used to initialize the computation of the surface fluxes done by the BEP-BEM system. At other time steps, CIM high resolution vertical profiles (wind speed, temperature and humidity) are given to BEP-BEM which then proceeds to a potentially more detailed estimation of compute sources/sinks. The sources and sinks are then given back to CIM to compute new vertical profiles, and to the meso-scale model (the surface fluxes are in this way aggregated at each of the meso-scale vertical levels and represent the FNs terms in the Eq.5.1 from Sect. 5.2).
This coupling may be enough when the mixing boundary layer is well developed but could be limited in stable conditions when the exchanges between air layers are low. Indeed, in such cases the horizontal fluxes cannot be neglected anymore as compared to the vertical fluxes and the method will not conserve the coherence between the two models from a fluxes point of view.
Coupling by fixing fluxes - Method FF 5-9
Figure 5.1: WRF scheme with the implementation of CIM(all in blue corresponds to WRF, in red variables corresponding to CIM and the fluxes are represented in green)
We hence propose in this section a methodology to keep the coherence between the models and take into account the horizontal transport in CIM as well as a new forcing at the top of CIM using fluxes. To develop this new methodology, an analysis of the fluxes budget is done over the vertical column of CIM and for a corresponding volume from the meso-scale model. Figure5.2gives a representation of the fluxes considered in both CIM and the meso-scale model. The following statements may be noted to ensure the coherence between the models and a balance of the fluxes:
• The mean value of each variables calculated on the CIM column should be the same as the one computed by the meso-scale model (both models proposing an estimation of the same real profiles);
• Bottom surface fluxes (i.e. surface fluxes calculated to take into account the effects of buildings at each level of the column) are computed once for forcing both the meso-scale model and CIM; the values should hence be equal
5.3 Canopy Interface Model integration in WRF
Figure 5.2: Representation of fluxes calculated on the vertical column in CIM (right) before correction and in the corresponding volume in WRF (left)
in both models (FBOT T OMM =FBOT T OMC =FBOT T OM);
• Far enough from the surface the flux at the top of both columns should be equal as it would be less influenced by the surface effects. In this case, a constant flux layer is considered and it is assumed that the flux at the top is equal to the bottom fluxes (FT OPM =FT OPC =FT OP).
Based on the above statements, CIM’s profiles may be corrected after each time step using an estimation of the horizontal fluxes. The formulation is done to allow a computation of these values that are not knowna priori in order to ensure a coherence between the models. Equation 5.9points out the consequences of this condition on the new CIM profiles.
For i < n
NiCt+1 = NiC∗+ ∆FHi
For i=n
NnCt+1 = NnC∗+ ∆FHi− FT OP
(5.9)
where N is one of the variables calculated by CIM (wind speed, potential temperature or humidity),tis the time step considered,iis an index corresponding to the center of a grid cell in CIM, NiCt+1 is the updated vertical value of CIM, NiC∗ is an “initial” value and ∆FHi the horizontal fluxes to be added. A different equation is proposed for the top most level of CIM since the objective is to not
5-11
force the model with a value of wind, temperature and humidity but with a flux value at the top, FT OP, that ensures the balance of both models. For each of the other levels, this flux may be computed at the cell faces. However the flux value on the top surface of the CIM column cannot be determined and has to be fixed.
Thus, NiC∗ represents NiCt including all fluxes except the horizontal ones and the top one.
To ensure coherence between the models using these formulation, we can write that the mean value of the variables calculated by CIM have to be equal to the meso-scale value:
NiM t+1 =NiCt+1 =NiC∗+ ∆FHi− FT OP
n (5.10)
whereNiM t+1 is the mean meso-scale value interpolated from the meso-scale model over the n levels present in CIM’s column and where n is the number of levels in the urban grid. As a first assumption, the horizontal fluxes, can be assumed constant over CIM’s column (equal to their mean) and it can then be written using Equation 5.10 as:
∆FHi = ∆FHi =NiM t+1−NiC∗+FT OP
n (5.11)
This then leads with Eqs.5.9to the Eqs.5.12, which give the new formulations used in CIM.
Fori < n
NiCt+1 = NiC∗+ NiM t+1− NiC∗+ FT OP n Fori=n
NnCt+1 = NnC∗+ NiCt+1− NiC∗+ FT OP
n − FT OP
(5.12)
When this correction is made, the results from CIM and the meso-scale mod- els should be coherent. It is proposed here to fix FT OP equal to FBOT T OM, in accordance with the statement formulated earlier.