246
Theory of the Ventilated Thermocline
4.11 Continuous Models of the Ventilated Thermocline
The passage from a layer model of the ventilated thermocline to a continuous
model is both important and technically difficult. If for no other reason, it is
important to be able to demonstrate that the layer models are physically
relevant in the sense that with increased resolution the layer solutions smoothly
tend to the solutions for a continuous fluid. As opposed to the continuous
quasi-geostrophic model of Chapter 3, the very nonlinear nature of the
problem on the planetary scale, in which the isopycnals suffer large vertical
excursions over the gyre, makes the mathematical problem a great challenge.
This is, after all, why the layer models with their few vertical degrees of
freedom are attractive.
Huang (1988, 1989a,b, 1991) was the first to take up this problem and in a
series of remarkably fruitful papers described a method of dealing with the
continuous problem. Important contributions have also been made by
Williams (1989). As our discussion in Section 4.10 suggests, the problem must
ultimately be dealt with numerically. However, the analytical formulation of
the problem by Huang and Williams is of great conceptual value in itself. With
the advantage of hindsight we see that the mathematical character of the
problem in its final form is a straightforward extension of the layer models as
the number oflayers becomes infinite. This is a very important result. It verifies
that the layer models smoothly tend to the solution for a continuous fluid.
Since the layer models are the exact representation of conceptually realizable, if
highly idealized, physical systems, the fact that they are also the finite difference
approximations of a continuous model invests even the simplest layer model
with a high degree of physical significance. It implies that a low-order layer
model is simultaneously an exact representation of a real but crude physical
system as well as a crude finite-difference representation, in the vertical, of the
exact continuous model. As the number of layers increases both the physical
and mathematical representations continuously improve.
The relationship between the layer models and the continuous representation of the solution is best seen in a coordinate system in which the vertical
coordinate used is the density itself. This is after all what we have already used
in the layer models where all fields are represented as function of latitude,
longitude, and the layer index n. The depths of the layer interfaces, z,., or their
thicknesses, h,., become the dependent variables. Using density as a vertical
coordinate is simply a way of maintaining this representation in the continuous
limit.
As long as there is a 1:1 relation between depth and density, the density may
be used as a vertical coordinate. This is certainly the case as long as the ocean is
stably stratified, i.e., as long as 8pj8z < 0. Thus any variable, u, say, can then be
represented as a function of time, horizontal position, and density, i.e.,
u = u(t, cp, (), p ).
(4.11.1)
Theory of the Ventilated Thermocline
4.11 Continuous Models of the Ventilated Thermocline
The passage from a layer model of the ventilated thermocline to a continuous
model is both important and technically difficult. If for no other reason, it is
important to be able to demonstrate that the layer models are physically
relevant in the sense that with increased resolution the layer solutions smoothly
tend to the solutions for a continuous fluid. As opposed to the continuous
quasi-geostrophic model of Chapter 3, the very nonlinear nature of the
problem on the planetary scale, in which the isopycnals suffer large vertical
excursions over the gyre, makes the mathematical problem a great challenge.
This is, after all, why the layer models with their few vertical degrees of
freedom are attractive.
Huang (1988, 1989a,b, 1991) was the first to take up this problem and in a
series of remarkably fruitful papers described a method of dealing with the
continuous problem. Important contributions have also been made by
Williams (1989). As our discussion in Section 4.10 suggests, the problem must
ultimately be dealt with numerically. However, the analytical formulation of
the problem by Huang and Williams is of great conceptual value in itself. With
the advantage of hindsight we see that the mathematical character of the
problem in its final form is a straightforward extension of the layer models as
the number oflayers becomes infinite. This is a very important result. It verifies
that the layer models smoothly tend to the solution for a continuous fluid.
Since the layer models are the exact representation of conceptually realizable, if
highly idealized, physical systems, the fact that they are also the finite difference
approximations of a continuous model invests even the simplest layer model
with a high degree of physical significance. It implies that a low-order layer
model is simultaneously an exact representation of a real but crude physical
system as well as a crude finite-difference representation, in the vertical, of the
exact continuous model. As the number of layers increases both the physical
and mathematical representations continuously improve.
The relationship between the layer models and the continuous representation of the solution is best seen in a coordinate system in which the vertical
coordinate used is the density itself. This is after all what we have already used
in the layer models where all fields are represented as function of latitude,
longitude, and the layer index n. The depths of the layer interfaces, z,., or their
thicknesses, h,., become the dependent variables. Using density as a vertical
coordinate is simply a way of maintaining this representation in the continuous
limit.
As long as there is a 1:1 relation between depth and density, the density may
be used as a vertical coordinate. This is certainly the case as long as the ocean is
stably stratified, i.e., as long as 8pj8z < 0. Thus any variable, u, say, can then be
represented as a function of time, horizontal position, and density, i.e.,
u = u(t, cp, (), p ).
(4.11.1)