A note on the representation of steric sea level in models that conserve volume rather than mass

This note discusses the representation of steric sea level in ocean circulation models. Changes in steric sea level are caused when changes in the density of the water column imply an expansion or contraction of the column. Models usually make the Boussinesq approximation and conserve volume, rather than mass, and so do not properly represent expansion or contraction. This means that although expansion/contraction is included in the equation of state, it is not accounted for by the model dynamics. In this note, we examine the equation governing the time evolution of the sea level displacement. It is shown that requiring conservation of mass, rather than volume, introduces a new term to this equation. A simple example is used to show the relationship of the new term to the surface buoyancy flux. The equilibrium response to the new term has two parts. One part consists of the Goldsbrough and Stommel gyres, for which, in the ocean interior, vortex stretching due to the local expansion/contraction of the water column is balanced by changes in planetary vorticity. The other part corresponds to the “inverse barometer.” The effect is to adjust sea level by a globally uniform but timevarying factor, determined by the net expansion/contraction of the global ocean. Since this correction is globally uniform, it has no dynamical significance. Both the Goldsbrough/Stommel gyres and the inverse barometer solution are missing from models as currently formulated. This does not represent a serious error. However, if comparison is made with observations of sea level, model-calculated sea level should be adjusted by a globally uniform, time-varying factor, determined by the net expansion/contraction of the global ocean. This would be important for assessing the likely rise in sea level in response to global warming.