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
[51],
and in the rigid-lid formulation by Holland and Lin
[30].
More recent formulations that allow the layers to surface have
been done by Bleck and Boudra
[2]
and Oberhuber
[49].