The problem of data assimilation in the specification of open boundary conditions for limited area models is addressed in this paper. Optimization approaches are detailed, which are based on combining available data on an open boundary with the physics of the hydrodynamical model. In our case the physics is in terms of the flux of energy through the open boundary. These optimized boundary conditions, for both barotropic and baroclinic situations, interpreted physically as special linearizations of the Bernoulli equation for each normal mode. Because of the complexity of decomposing variables into normal modes for open boundaries with varying bathymetry, we present two alternative approaches. The first is a simplification of the optimized baroclinic boundary condition based on normal modes. The second makes use of empirical orthogonal functions instead of normal modes. The results of testing and comparisons of these approaches are presented for coupling coarse- and fine-resolution models. In this case our approach is in assimilating values and variables from a large-scale model (along the open boundaries of a limited area model). In the proposed coupling schemes the energy fluxes are estimated either from coarse or from fine-grid model results. With the progress of oceanographic observing systems we would like to explore ways of combining model outputs with the oceanographic measurements in order to estimate energy fluxes used in optimized open boundary conditions. ¿ 1999 American Geophysical Union