The models of Bryan
[4],
Madala and Piacsek
[37]
and Killworth et al.
[36]
have all used fixed
levels in the vertical 
-direction, with a variable spacing of the
depth levels to resolve the rapid changes in the surface layers.
On the other hand, Blumberg and Mellor
[3]
and Haidvogel et al.
[25]
have introduced a stretched coordinate usually referred to as
sigma (for its previous usage in meteorology), defined as
where 
is the fluid depth. This leads to boundary surfaces that
can be put at 
and an automatic accommodation of the
depth contours. In addition Haidvogel et al.
[25]
have introduced a
semi-spectral representation of the vertical dimension (sigma
layout) in terms of Chebyshev polynomials (collocation method).
Both of these have employed the free-surface formulation. See
Figure 2
and Figure 3.
the sea surface deflection.
View Figure
is the
free surface deflection.
View Figure
About the same time that the -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 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].
