Changes in steric sea level are caused when changes in the density of the water column imply an expansion or contraction of the column. It is essentially produced through surface heating/cooling and to a lesser extent through non-linear effects of the equation of state (cabbeling, thermobaricity...). Non-Boussinesq models contain all ocean effects within the ocean acting on the sea level. In particular, they include the steric effect. In contrast, Boussinesq models, such as NEMO, conserve volume, rather than mass, and so do not properly represent expansion or contraction. The steric effect is therefore not explicitely represented. This approximation does not represent a serious error with respect to the flow field calculated by the model [Greatbatch, 1994], but extra attention is required when investigating sea level, as steric changes are an important contribution to local changes in sea level on seasonal and climatic time scales. This is especially true for investigation into sea level rise due to global warming.
Fortunately, the steric contribution to the sea level consists of a spatially uniform component that can be diagnosed by considering the mass budget of the world ocean [Greatbatch, 1994]. In order to better understand how global mean sea level evolves and thus how the steric sea level can be diagnosed, we compare, in the following, the non-Boussinesq and Boussinesq cases.
Let denote the total mass of liquid seawater ( ), the total volume of seawater ( ), the total surface of the ocean ( ), the global mean seawater (in situ) density ( ), and the global mean sea level ( ).
A non-Boussinesq fluid conserves mass. It satisfies the following relations:
In a Boussinesq fluid, is replaced by in all the equation except when appears multiplied by the gravity ( in the hydrostatic balance of the primitive Equations). In particular, the mass conservation equation, (11.4), degenerates into the incompressibility equation:
Nevertheless, following [Greatbatch, 1994], the steric effect on the volume can be diagnosed by considering the mass budget of the ocean. The apparent changes in , mass of the ocean, which are not induced by surface mass flux must be compensated by a spatially uniform change in the mean sea level due to expansion/contraction of the ocean [Greatbatch, 1994]. In others words, the Boussinesq mass, , can be related to , the total mass of the ocean seen by the Boussinesq model, via the steric contribution to the sea level, , a spatially uniform variable, as follows:
The above formulation of the steric height of a Boussinesq ocean requires four remarks. First, one can be tempted to define as the initial value of , set , so that the initial steric height is zero. We do not recommend that. Indeed, in this case depends on the initial state of the ocean. Since has a direct effect on the dynamics of the ocean (it appears in the pressure gradient term of the momentum equation) it is definitively not a good idea when inter-comparing experiments. We better recommend to fixe once for all to . This value is a sensible choice for the reference density used in a Boussinesq ocean climate model since, with the exception of only a small percentage of the ocean, density in the World Ocean varies by no more than 2 from this value (Gill , page 47).
Second, we have assumed here that the total ocean surface, , does not change when the sea level is changing as it is the case in all global ocean GCMs (wetting and drying of grid point is not allowed).
Third, the discretisation of (11.10) depends on the type of free surface which is considered. In the non linear free surface case, key_ vvl defined, it is given by
The fourth and last remark concerns the effective sea level and the presence of sea-ice. In the real ocean, sea ice (and snow above it) depresses the liquid seawater through its mass loading. This depression is a result of the mass of sea ice/snow system acting on the liquid ocean. There is, however, no dynamical effect associated with these depressions in the liquid ocean sea level, so that there are no associated ocean currents. Hence, the dynamically relevant sea level is the effective sea level, the sea level as if sea ice (and snow) were converted to liquid seawater [Campin et al., 2008]. However, in the current version of NEMO the sea-ice is levitating above the ocean without mass exchanges between ice and ocean. Therefore the model effective sea level is always given by , whether or not there is sea ice present.
In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to changes in ocean density arising just from changes in temperature. It is given by:
Both steric and thermosteric sea level are computed in diaar5.F90 which needs the key_ diaar5 defined to be called.
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17