Structure of the barotropic mode in layered ocean models

We present a near-exact analytic solution of the vertical modes in a hydrostatic ocean model in Lagrangian coordinates that is particularly well suited to studying the structure of vertical modes in layered models. In particular, we do not make the common approximation of incompressibility that is usually invoked in vertical mode analysis. This allows us to perform a study of the vertical mode errors in this approximation as well as in the common Boussinesq approximation. We find that the barotropic mode is not very sensitive to model assumptions and is vertically uniform to within about 0.03% for the case of the Levitus (1994) global mean data. The baroclinic modes, on the other hand, deviate substantially from the exact case for the incompressible case, but not for Boussinesq approximation. Most ocean models rely on an approximate mode splitting to separate the barotropic from the baroclinic modes. We find that the approximation of the velocity barotropic mode by a mass-weighted vertical average, as is usually done in layer models, is extremely good and considerably better than the corresponding approximation by the depth-average in z-coordinate Boussinesq models. Layer models have the additional task of approximating the barotropic layer thickness distribution. We find that the usual assumption that layer thickness varies in proportion to the nominal layer thickness distribution is viable but not as good as might be desired because the proportionality “constant” varies with depth by about 2%.