In order to clarify the link between ocean salt content (OSC) conservation and the freshwater flux formulation in ocean general circulation models (OGCMs), a varying level thickness, nonlinear free surface version of the OPA is presented. Linear/nonlinear free surface equations are solved using an original approach based on an explicit damping of fast external gravity waves. The method leaves both the potential vorticity equation and the equilibrium state unchanged. Its numerics and cost are quite similar to those of implicit schemes. When nonlinearities are kept, a variable first level thickness is required. Its discretization is determined by volume and energy constraints. The OSC conservation depends on the surface kinematic equation used. Four formulations are presented: (1) virtual salt flux (fixed ocean volume and no volume flux), (2) natural (fixed ocean volume and volume flux), (3) linear free surface (fixed volume and volume flux computed from a linear free surface equation), and (4) assumption free (variable volume computed from a nonlinear free surface equation). Their impact is illustrated in 25 year low-resolution global OGCM simulations. In all cases the first-order ocean response is quite similar, as the concentration-dilution effect always exists. Formulations 4 and 2 ensure a strict conservation of the OSC. Nevertheless, the difference in formulation 3 is not strong enough to play a significant role: the conservation is almost perfect. Only formulation 1 neglects fresh water-driven surface pumping. This mainly modifies the sea surface salinity of the ocean basin where river runoffs are strong. No significant difference is found between the other formulations as a large time step dampens high-frequency free surface motion. The best compromise for climate is the linear free surface formulation. It allows a nearly exact OSC conservation, introduces the fresh water-driven pumping, and runs faster than all the other formulations.