A parameterisation of non-linear dynamical interactions with unresolved scales is needed in most atmospheric models to ensure realistic fluxes of energy and enstrophy near and at the truncation limit. In this paper, a minimisation of tendency errors in low to medium resolution versions of the ARPEGE/IFS and the ECHAM4 spectral general circulation models is sought in order to obtain spectral empirical interaction functions (EIFs) to be used in the formulation of horizontal diffusion. The tendency errors are calculated relative to high resolution adiabatic versions of the models themselves, meaning that the EIFs reflect all non-linear adiabatic processes. Different EIFs are obtained for vorticity, divergence and temperature. The most striking feature is that the vorticity and temperature EIFs have non-negligible negative values for low wave numbers in large parts of the troposphere. This implies that these waves are enhanced in amplitude due to non-linear interactions with unresolved scales. For low resolution models the generation/dissipation of kinetic energy due to the interactions is not well parameterised by either the standard horizontal diffusion in the models, or the EIFs computed here. When the EIFs are used in the formulation of horizontal diffusion it is seen that the kinetic energy spectrum is closer to observations than in the original model versions. Furthermore, the large scale systematic model errors are reduced in a medium resolution simulation, while less clear improvement is seen in a simulation at low resolution. It is argued that the technique used in this paper is quite general and may be used to develop a more realistic parameterisation of unresolved dynamical scale interactions. The method should, with some modifications, also be applicable in semi-Lagranxgian spectral models and in grid paint models.
