146
Joseph Pedlosky
Generally, the determination of these relationships involves the inversion of a rather
complex functional relationship and it is rare that a simple form emerges. However,
as I stared at the relation f /h 2 = Q(h) it suddenly occurred to me that at the latitude
of the outcrop h 2 was in fact equal to h since h 1 , by definition, vanishes there. Further,
if the outcrop line is a line of constant latitude where f is a constant, say f 2 , the
potential vorticity of every fluid column in layer 2 along the outcrop line would just
be f 2 /h and that relation would be preserved for all streamlines issuing from the
outcrop line so that the function Q was simply f 2 /h.
4 I remember being startled by
the utter simplicity of the result. It hardly seemed possible to me that it could be so
simple. Assuming it was correct, it allowed each layer thickness to be simply related
to the total depth, h, and the Sverdrup relation then gave a direct result for the field
of h and so h 1 and h 2 over the gyre. It was a matter of 20 minutes to write up the
result in a clean copy that I brought down to Hank’s office. As luck would have it,
he was not there so I left the clean copy on his desk and went back to my office. I
was still in a state of some excitement when Hank called me and thanked me for the
notes but said my result was clearly wrong. According to their numerical calculations
it appeared to him that the thermocline depth shallowed exponentially as the equator
was approached and my solution became shallow only algebraically. Well, I could
not see the error in the calculation but it was so simple that I confess to having had a
residue of doubt that such a simple result would solve a problem that had bedeviled
us for so long. So I just decided to go home, sleep on it that night, and check the result
again in the morning.
I did not have to wait. Later that same evening, in the first of what would be a
series of after-hours telephone conversations, Hank called and said that he had plotted
the analytical solution over the numerical solution he and Jim had obtained and they
matched perfectly. Could we talk about it in the morning? Well, one can just imagine
my state of mind at that moment.
The next day Hank and I spoke about the analytical solution. Jim had grasped
it immediately and had explained it to Hank and I sensed that Hank was both excited
and a little disappointed that the numerical work was no longer necessary. Indeed,
he wondered whether the analytical solution was intrinsically limited to the twomoving-layer model and that is what led to dealing with the three-layer model that
later formed the basis of the Ventilated Thermocline paper (Luyten et al., 1983). In
retrospect, Hank and I agreed that the development of the three-layer model was
probably a poor decision. The basic idea of the model is already captured in the
two-layer version and three layers added complexity with a rather small return in
understanding. However, with the model in hand we were able to think about what
aspects of the boundary conditions and forcing determined the structure.
4 Ironically, the case of zonal distribution of temperature, although the easiest in this case, was not possible
for the similarity solutions.
Joseph Pedlosky
Generally, the determination of these relationships involves the inversion of a rather
complex functional relationship and it is rare that a simple form emerges. However,
as I stared at the relation f /h 2 = Q(h) it suddenly occurred to me that at the latitude
of the outcrop h 2 was in fact equal to h since h 1 , by definition, vanishes there. Further,
if the outcrop line is a line of constant latitude where f is a constant, say f 2 , the
potential vorticity of every fluid column in layer 2 along the outcrop line would just
be f 2 /h and that relation would be preserved for all streamlines issuing from the
outcrop line so that the function Q was simply f 2 /h.
4 I remember being startled by
the utter simplicity of the result. It hardly seemed possible to me that it could be so
simple. Assuming it was correct, it allowed each layer thickness to be simply related
to the total depth, h, and the Sverdrup relation then gave a direct result for the field
of h and so h 1 and h 2 over the gyre. It was a matter of 20 minutes to write up the
result in a clean copy that I brought down to Hank’s office. As luck would have it,
he was not there so I left the clean copy on his desk and went back to my office. I
was still in a state of some excitement when Hank called me and thanked me for the
notes but said my result was clearly wrong. According to their numerical calculations
it appeared to him that the thermocline depth shallowed exponentially as the equator
was approached and my solution became shallow only algebraically. Well, I could
not see the error in the calculation but it was so simple that I confess to having had a
residue of doubt that such a simple result would solve a problem that had bedeviled
us for so long. So I just decided to go home, sleep on it that night, and check the result
again in the morning.
I did not have to wait. Later that same evening, in the first of what would be a
series of after-hours telephone conversations, Hank called and said that he had plotted
the analytical solution over the numerical solution he and Jim had obtained and they
matched perfectly. Could we talk about it in the morning? Well, one can just imagine
my state of mind at that moment.
The next day Hank and I spoke about the analytical solution. Jim had grasped
it immediately and had explained it to Hank and I sensed that Hank was both excited
and a little disappointed that the numerical work was no longer necessary. Indeed,
he wondered whether the analytical solution was intrinsically limited to the twomoving-layer model and that is what led to dealing with the three-layer model that
later formed the basis of the Ventilated Thermocline paper (Luyten et al., 1983). In
retrospect, Hank and I agreed that the development of the three-layer model was
probably a poor decision. The basic idea of the model is already captured in the
two-layer version and three layers added complexity with a rather small return in
understanding. However, with the model in hand we were able to think about what
aspects of the boundary conditions and forcing determined the structure.
4 Ironically, the case of zonal distribution of temperature, although the easiest in this case, was not possible
for the similarity solutions.