International audienceThis paper focuses on the energy conservation properties of a hydrostatic, Boussinesq, coastal ocean model using a classic finite difference method. It is shown that the leapfrog time-stepping scheme, combined with the sigma-coordinate formalism and the motions of the free surface, prevents the momentum advection from exactly conserving energy. Because of the leapfrog scheme, the discrete form of the kinetic energy depends on the product of velocities at odd and even time steps and thus appears to be possibly negative when high-frequency modes develop. Besides, the study of the energy balance clarifies the numerical choices made for the computation of mixing processes. The time-splitting technique used to reduce the computation costs associated to the resolution of surface waves leads to the well-known external and internal mode equations. We show that these equations do not conserve energy if the coupling of these two modes is forward in time. Even if non-linear terms are negligible, this shortcoming can be significant regarding the pressure gradient term ‘frozen' over a baroclinic time step. An alternative energy-conserving time-splitting technique is proposed in this paper. Discussion and conclusions are conducted in the light of a set of numerical experiments dedicated to surface and internal gravity waves

This paper reviews the usual open boundary conditions (OBCs) for coastal ocean models and proposes a complete set of open boundaries based on stability criteria, on mass and energy conservation arguments, and on the ability to enforce external information. This set includes a radiation condition for barotropic variables, an equation of wave propagation for baroclinic velocities, and an advection equation for tracers. Considerations on mass and energy conservation properties suggest a suitable numerical treatment of the barotropic scheme, which is different from what is commonly used. Restoring terms, when classically added in the Sommerfeld OBCs, are not consistent with external fields. It is shown that this can be avoided if proper numerical schemes are used or if OBCs are applied on differences between the model and forcing rather than on absolute variables. Finally, this paper shows that simplistic advection-type methods for temperature and salinity should not be used in sigma coordinate models because this introduces errors in the computation of the horizontal pressure gradient.

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