176
Theory of the Ventilated Thermocline
Q, from first principles. Without such a forward method solutions of the
adiabatic problem as well as the similarity solutions of the dissipative problem
remained as isolated examples of shrewd analysis whose relationship with the
underlying physical problem was obscure.
The Rhines and Young theory discussed in Chapter 3, limited though it is
to a quasi-geostrophic theory for the thermocline velocity field, nevertheless
restored the physics of the problem to center stage. Shortly after the
appearance of the Rhines and Young theory, Luyten, Pedlosky, and Stommel
(1983), stimulated by that work, presented a theory for the thermocline which
included the physics of outcropping which was lacking in the Rhines and
Young theory. The Luyten et al. theory is meant to explain both the vertical
and the horizontal variation in the density field for layers which outcrop in the
subtropical gyre in terms of the surface density field. The two theories together
form a complementary picture of the thermocline circulation, and their union
and the subsequent work of others have led to a rich theoretical picture of the
midocean thermocline structure. The purpose of this chapter is to describe
some of the key aspects of these developments.
As in earlier chapters we assume that the Sverdrup theory is valid in the
interior, and that the solutions which we find for the interior structure of the
thermocline circulation are consistent with closure of the circulation through a
western boundary current. The reader should keep in mind that neither of these
two assumptions has been clearly demonstrated.
Before taking up the thermocline theories themselves we review some
general dynamical fundamentals. This is useful for placing our approximations
in context and allows us later to discuss extensions of the theory to the
equatorial regions where the geostrophic approximation fails.
4.2 Formulation of the Model
We concentrate here on layer models of the thermocline and defer to Section
4.11 the formulation and solution of the continuously stratified problem.
Consider the situation depicted in Fig. 4.2.1. A layer of constant density Pn is
bounded above by the interface z = Zn where Zn is a function of longitude, l/J,
latitude (), and time, t. The horizontal velocity in each layer is independent of z.
This is a modeling decision which becomes rigorously required in the limit
when the motion is geostrophic. The thickness of the layer is hn ( cp, 8, t ). The
local value of the Coriolis parameter is f = 2Q sin e. At each interface a crossisopycnal velocity W* is allowed. As the fluid crosses the interface, its density
changes instantaneously to adapt to the density of the layer that it enters. As
discussed in Chapter 3, this is equivalent to the specification of a local heating
(or cooling) to accomplish the density change.
If we consider a small cylindrical control volume in which the area in the
horizontal plane is fixed with time such that fluid flows are unimpeded across
Theory of the Ventilated Thermocline
Q, from first principles. Without such a forward method solutions of the
adiabatic problem as well as the similarity solutions of the dissipative problem
remained as isolated examples of shrewd analysis whose relationship with the
underlying physical problem was obscure.
The Rhines and Young theory discussed in Chapter 3, limited though it is
to a quasi-geostrophic theory for the thermocline velocity field, nevertheless
restored the physics of the problem to center stage. Shortly after the
appearance of the Rhines and Young theory, Luyten, Pedlosky, and Stommel
(1983), stimulated by that work, presented a theory for the thermocline which
included the physics of outcropping which was lacking in the Rhines and
Young theory. The Luyten et al. theory is meant to explain both the vertical
and the horizontal variation in the density field for layers which outcrop in the
subtropical gyre in terms of the surface density field. The two theories together
form a complementary picture of the thermocline circulation, and their union
and the subsequent work of others have led to a rich theoretical picture of the
midocean thermocline structure. The purpose of this chapter is to describe
some of the key aspects of these developments.
As in earlier chapters we assume that the Sverdrup theory is valid in the
interior, and that the solutions which we find for the interior structure of the
thermocline circulation are consistent with closure of the circulation through a
western boundary current. The reader should keep in mind that neither of these
two assumptions has been clearly demonstrated.
Before taking up the thermocline theories themselves we review some
general dynamical fundamentals. This is useful for placing our approximations
in context and allows us later to discuss extensions of the theory to the
equatorial regions where the geostrophic approximation fails.
4.2 Formulation of the Model
We concentrate here on layer models of the thermocline and defer to Section
4.11 the formulation and solution of the continuously stratified problem.
Consider the situation depicted in Fig. 4.2.1. A layer of constant density Pn is
bounded above by the interface z = Zn where Zn is a function of longitude, l/J,
latitude (), and time, t. The horizontal velocity in each layer is independent of z.
This is a modeling decision which becomes rigorously required in the limit
when the motion is geostrophic. The thickness of the layer is hn ( cp, 8, t ). The
local value of the Coriolis parameter is f = 2Q sin e. At each interface a crossisopycnal velocity W* is allowed. As the fluid crosses the interface, its density
changes instantaneously to adapt to the density of the layer that it enters. As
discussed in Chapter 3, this is equivalent to the specification of a local heating
(or cooling) to accomplish the density change.
If we consider a small cylindrical control volume in which the area in the
horizontal plane is fixed with time such that fluid flows are unimpeded across