The aim of this method is to keep the atmospheric profiles of temperature and humidity and the surface conditions synchronous. This scheme has been used in most GCMs when they were coupled to the simple bucket model [Manabe, 1969]. Only a few models have kept this method when they moved to more complex land surface schemes, among them are the LMD-GCM coupled to SECHIBA [Ducoudré et al., 1993] and the UKMO-GCM [Warrilow et al., 1986]. Newly developed schemes have also adopted it; the ECMWF model (CY 48) [Viterbo and Beljaars, 1995] and ISBA coupled to ARPEGE [Mahfouf et al., 1995].
In order to implement this method, information on the dependence of the
atmospheric conditions on the surface forcing is needed. As shown in
appendix A the atmospheric variables can be given as functions of the
surface conditions (see eq. 16) within the PBL. This
information can be provided to the LSS by passing through the
interface the coefficients 
and 
. this introduces a
much stronger coupling between the two systems. It means that the
closure of equation 1 is transformed from a Dirichlet to a mixed
boundary condition problem.
Applying this to the discretized version of equation 1 a fully implicit equation for the energy balance is obtained:
As the saturated water-vapor mixing ratio is a nonlinear function of surface temperature, we need to replace it by its truncated expansion:
Thus equation 8 can be solved and 
computed. During these steps we have implicitly computed the sensible
and latent heat flux which need to be diagnosed from the new values of
temperature and moisture at the surface and the first level. These
fluxes can then be used to do the back-substitution which solves the
vertical diffusion equations.
Besides its mathematical elegance and low computational cost, this method has the advantage of displaying numerical stability. In the case of constant transfer coefficients it can be shown to be unconditionally stable [Davies, 1983] else it is linearly stable [Kalnay and Kanamitsu, 1988]. It conserves energy, as at any point in time the surface fluxes are coherent to the profiles of temperature and moisture in the atmosphere. The disadvantage of this method is that the solution of equation 8 may become very difficult as land surface schemes become more complex and include sub-grid scale variability. If a threshold method is used in the land surface scheme then equation 8 will have to be supplemented by a predictor-corrector method, which will make it computationally more expensive. This need arises from the fact that in equation 8 the fluxes are computed implicitly and thus no conditions can be applied while it is being solved.
In SECHIBA the coupling of the thermodynamical soil model to the surface energy balance is also accomplished with an implicit method leading to a closed set of equations which describe the diffusion from the top of the PBL to the bottom of the soil. In a similar way the equations for soil water transfer in the ECMWF scheme are solved with an implicit method from the surface throughout the entire soil column depth.
