In section 2.4 the turbulent surface fluxes have been identified as the quantities which should be provided to the vertical scheme as a lower boundary condition. To compute them, the LSS will need the atmospheric conditions at the previous time-step. Depending on the numerical scheme of the LSS it may also require other information from the atmosphere so that an equation can be written for the atmospheric conditions of the next time-step.
In the present section we will discuss the possibilities opened to LSSs for solving the surface energy balance and describe the methods currently in use. A knowledge about the methods used for solving the vertical turbulent diffusion is needed to appreciate the different couplings. In Appendix A the standard ``explicit coefficient, implicit temperature'' [Kalnay and Kanamitsu, 1988] method is presented for reference. In the following discussion only a local diffusion scheme is considered but it can be shown that the same equations can be written for a non-local scheme.
The numerical schemes used for solving equation 1 are
discussed using its discretized formulation which integrates from
to 
. The fluxes which make up the
coupling to the vertical diffusion are then written as follows :
Where i and j are time indices.
It is only in these fluxes that the lowest level prognostic variables
of the GCM come into play in the energy balance equation. To simplify
the discussion we have chosen to use a `` 
-formulation''
[Mahfouf and Noilhan, 1991] in the equations for the latent heat flux presented here.
The reasoning can easily be extended to an `` 
-formulation'' or
bulk aerodynamic formulas which use resistances to control
evaporation. Some LSSs use a threshold formulation for the latent heat
flux, that is below a given flux or amount of available moisture the
method for determining evaporation changes. We will point out in which
numerical schemes this formulation cannot easily be implemented.
The choice of the time step at which the atmospheric variables in the sensible and latent heat fluxes are taken determines the type of coupling. It will also dictate the time in the GCM at which the surface energy balance has to be solved and the approximation that need to be made. In the following we will use a simple nomenclature for describing these schemes :
The discussion in this section can also be applied to the coupling between the surface energy balance and the ground heat flux which is the result of a diffusion equation of the same type as the one used for the atmospheric PBL. As pointed out earlier, the ground heat flux is internal to the land-surface scheme and is thus irrelevant to the coupling with the atmosphere.
To simplify the equations in this section, we will not consider the
surface temperature dependence of the emitted radiation in the net
radiative flux ( 
). It is sufficient to say that the surface
temperature at time-step t can be used, or with the help of a Taylor
expansion and the value at t+1, an approximation of the emitted
radiation at t+1 can be obtained. This topic is discussed in detail
in section 3.