Introduction
175
A principal mathematical feature of this early theoretical attempt to deal
with the thermocline as a thermal boundary layer was the technical difficulty of
the governing nonlinear partial differential equation. The vertical diffusion of
density in the theory is balanced by both lateral and vertical advection of
density. Both of the latter are of the same order as a consequence of estimates
connected with the continuity equation linking the vertical and horizontal
velocities. The great mathematical difficulty of including both diffusion and
three-dimensional advection led to the search for structurally special solutions
of the governing equations of so-called similarity type. That is, it was assumed
(see, for example, Robinson and Stommel1959) that the basic vertical structure
of the thermocline is the same everywhere except insofar as the vertical scale is
stretched differently at different horizontal locations. Similarity solutions of
this type are common in fluid dynamics and are rigorously valid only when
there are no significant external length scales to provide a natural scale against
which the phenomenon can be measured . This is certainly not the case for the
thermocline where the basin scale, the length scale of the wind, and the length
scale associated with the surface density field are all important for the physics.
It was also hoped that although the similarity solutions were unable to satisfy
arbitrary and realistic surface boundary conditions, they would nonetheless be
physically revealing. The complexity of the mathematics, however, limited the
solutions to a fairly narrow class, and, most disappointingly of all , the physical
content of the solutions remained unsatisfactorily sparse. The solutions all had
structures that were frozen by the requirement of the similarity form and
proved to be largely independent, for example, of the degree of vertical mixing
that was assumed. Thus it was impossible to ask the solutions to reveal
parametrically interesting structural variations with the degree of mixing.
Worst of all, even though the mixing might be assumed small, the
mathematical requirement that the similarity form of the solution be consistent
with each term in the governing partial differential equation, regardless of its
size, meant that dissipation played a determining role in selecting the form of
the solution everywhere. This is not physically realistic. The search for
complete similarity solutions was technically so challenging that the physical
connection with the thermocline problem became increasingly tenuous.
An interesting alternative approach was taken by Welander (1959, 1971) in
which he proposed a purely adiabatic, advective model for the thermocline
more in line with the Iselin and Montgomery ideas discussed above. In
Welander's theory the potential vorticity, q, is the key dynamic variable. In a
nondissipative, steady flow it is constant on both density, p, and Bernoulli
function, B, surfaces (Pedlosky 1987b; see also below). Thus in general
q = Q(p,B) where Q is an arbitrary function of its arguments. Welander chose
certain forms for Q and found interesting solutions but was unable to describe
a relationship between the function Q and the prescribed external boundary
conditions for the problem specified in a forward manner. That is, given the
surface density field and the Ekman pumping and conditions of adiabatic flow
it should be possible to determine the flow and the potential vorticity, i.e.,
175
A principal mathematical feature of this early theoretical attempt to deal
with the thermocline as a thermal boundary layer was the technical difficulty of
the governing nonlinear partial differential equation. The vertical diffusion of
density in the theory is balanced by both lateral and vertical advection of
density. Both of the latter are of the same order as a consequence of estimates
connected with the continuity equation linking the vertical and horizontal
velocities. The great mathematical difficulty of including both diffusion and
three-dimensional advection led to the search for structurally special solutions
of the governing equations of so-called similarity type. That is, it was assumed
(see, for example, Robinson and Stommel1959) that the basic vertical structure
of the thermocline is the same everywhere except insofar as the vertical scale is
stretched differently at different horizontal locations. Similarity solutions of
this type are common in fluid dynamics and are rigorously valid only when
there are no significant external length scales to provide a natural scale against
which the phenomenon can be measured . This is certainly not the case for the
thermocline where the basin scale, the length scale of the wind, and the length
scale associated with the surface density field are all important for the physics.
It was also hoped that although the similarity solutions were unable to satisfy
arbitrary and realistic surface boundary conditions, they would nonetheless be
physically revealing. The complexity of the mathematics, however, limited the
solutions to a fairly narrow class, and, most disappointingly of all , the physical
content of the solutions remained unsatisfactorily sparse. The solutions all had
structures that were frozen by the requirement of the similarity form and
proved to be largely independent, for example, of the degree of vertical mixing
that was assumed. Thus it was impossible to ask the solutions to reveal
parametrically interesting structural variations with the degree of mixing.
Worst of all, even though the mixing might be assumed small, the
mathematical requirement that the similarity form of the solution be consistent
with each term in the governing partial differential equation, regardless of its
size, meant that dissipation played a determining role in selecting the form of
the solution everywhere. This is not physically realistic. The search for
complete similarity solutions was technically so challenging that the physical
connection with the thermocline problem became increasingly tenuous.
An interesting alternative approach was taken by Welander (1959, 1971) in
which he proposed a purely adiabatic, advective model for the thermocline
more in line with the Iselin and Montgomery ideas discussed above. In
Welander's theory the potential vorticity, q, is the key dynamic variable. In a
nondissipative, steady flow it is constant on both density, p, and Bernoulli
function, B, surfaces (Pedlosky 1987b; see also below). Thus in general
q = Q(p,B) where Q is an arbitrary function of its arguments. Welander chose
certain forms for Q and found interesting solutions but was unable to describe
a relationship between the function Q and the prescribed external boundary
conditions for the problem specified in a forward manner. That is, given the
surface density field and the Ekman pumping and conditions of adiabatic flow
it should be possible to determine the flow and the potential vorticity, i.e.,