Figure 3: Vertical discretization in ocean models.
About the same time that the z-level models were developed, several investigators have proceeded to exploit the fact that the ocean is shallow and hydrostatic, and that its vertical structure has a semi-permanent tendency. A modal decomposition by empirical orthogonal functions or by linear dynamic modes has shown that the first three or four modes can usually capture 95 percent of the energy. Thus by vertically integrating the equations of motion in z between isopycnal (i.e. equal density) surfaces (employing the hydrostatic relation (1)) they have derived a system where the layer-averaged velocities and layer-thicknesses are the dependent variables. Since the isopycnal surfaces move with the fluid, this representation is in fact quasi-Lagrangian. The first model using this approach in the free surface formulation was built by O'Brien , and in the rigid-lid formulation by Holland and Lin . More recent formulations that allow the layers to surface have been done by Bleck and Boudra  and Oberhuber .