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Joseph Pedlosky
diffusion coefficient, κ, needed by the model. The authors note that an extremely large
value would turn the thermocline into an unrealistic, linear top-to-bottom profile of
temperature while a very small value of κ, less than 1 cm
2 /sec, would lead to an
advective model of the thermocline which, using the similarity form introduced by
the authors, is claimed to provide the wrong form of the latitude dependence of
the thermocline. This I believe is an unfortunate consequence of ignoring the zonal
advective term.
Welander’s approach, on the other hand, strikes the contemporary reader as
being remarkably prescient. He points out that “the importance of diffusion processes
in large-scale ocean dynamics has not yet been proved” and he suggests ignoring
diffusion altogether. Doing so, he obtains a purely advective model described by a
nonlinear partial differential equation for a variable related to the vertical integral
of the pressure anomaly, itself related to an integral of the density, his M function.
The difficult, resulting equation has a simple solution in which the density field exponentially approaches its uniform abyssal value with a scale height that is directly
proportional to the Coriolis parameter. We recognize today that this solution is one in
which the planetary potential vorticity (the Coriolis parameter multiplied by the vertical temperature gradient) is constant on density surfaces so the adiabatic advection
of density and potential vorticity are automatically satisfied. Welander extended his
advective model in a series of later papers (Welander, 1971a,b) to include more complex dependence of the potential vorticity on Bernoulli function
3 as well as density.
My own personal view is that Welander had a deeper insight, shown especially in the
second of these two 1971 papers, than almost all his contemporaries. He anticipated
a thermocline in which diffusion was important only on the boundary of an adiabatic
region separating the ideal-fluid thermocline from the abyss and gave clear scaling
arguments and a cogent physical discussion of the way in which the model’s parts
should fit together. Again, what was lacking was a theoretical tool to provide a basis
for determining the functional relationship between potential vorticity, density, and
Bernoulli function. Rather than pursuing this line further, Welander and Robinson
collaborated on a model in which the advective/diffusive balance was added back to
Welander’s formulation. This leads to a very complex nonlinear partial differential
equation. Only special similarity solutions of this kind of model were ever found, i.e.,
models whose functional behavior with depth is the same everywhere except insofar
as the vertical coordinate is stretched as a function, in the general case, of latitude
and longitude. Veronis (1969) gives a clear review of this theoretical approach and
its results. Many were the authors who approached the problem in this form in an attempt to obtain a physical understanding of the mechanics that shape the thermocline.
Perhaps the most distinguished of these is the formulation and solution of Needler
(1967) but his solutions, again of similarity type, were mostly of the constant potential
vorticity form and shared with all the other models of this type the inability to deal
with a simple forward problem. Namely, if the surface density (or temperature ) field
3 The planetary scale Bernoulli function p + ρgz is also called the Montgomery stream function.
Joseph Pedlosky
diffusion coefficient, κ, needed by the model. The authors note that an extremely large
value would turn the thermocline into an unrealistic, linear top-to-bottom profile of
temperature while a very small value of κ, less than 1 cm
2 /sec, would lead to an
advective model of the thermocline which, using the similarity form introduced by
the authors, is claimed to provide the wrong form of the latitude dependence of
the thermocline. This I believe is an unfortunate consequence of ignoring the zonal
advective term.
Welander’s approach, on the other hand, strikes the contemporary reader as
being remarkably prescient. He points out that “the importance of diffusion processes
in large-scale ocean dynamics has not yet been proved” and he suggests ignoring
diffusion altogether. Doing so, he obtains a purely advective model described by a
nonlinear partial differential equation for a variable related to the vertical integral
of the pressure anomaly, itself related to an integral of the density, his M function.
The difficult, resulting equation has a simple solution in which the density field exponentially approaches its uniform abyssal value with a scale height that is directly
proportional to the Coriolis parameter. We recognize today that this solution is one in
which the planetary potential vorticity (the Coriolis parameter multiplied by the vertical temperature gradient) is constant on density surfaces so the adiabatic advection
of density and potential vorticity are automatically satisfied. Welander extended his
advective model in a series of later papers (Welander, 1971a,b) to include more complex dependence of the potential vorticity on Bernoulli function
3 as well as density.
My own personal view is that Welander had a deeper insight, shown especially in the
second of these two 1971 papers, than almost all his contemporaries. He anticipated
a thermocline in which diffusion was important only on the boundary of an adiabatic
region separating the ideal-fluid thermocline from the abyss and gave clear scaling
arguments and a cogent physical discussion of the way in which the model’s parts
should fit together. Again, what was lacking was a theoretical tool to provide a basis
for determining the functional relationship between potential vorticity, density, and
Bernoulli function. Rather than pursuing this line further, Welander and Robinson
collaborated on a model in which the advective/diffusive balance was added back to
Welander’s formulation. This leads to a very complex nonlinear partial differential
equation. Only special similarity solutions of this kind of model were ever found, i.e.,
models whose functional behavior with depth is the same everywhere except insofar
as the vertical coordinate is stretched as a function, in the general case, of latitude
and longitude. Veronis (1969) gives a clear review of this theoretical approach and
its results. Many were the authors who approached the problem in this form in an attempt to obtain a physical understanding of the mechanics that shape the thermocline.
Perhaps the most distinguished of these is the formulation and solution of Needler
(1967) but his solutions, again of similarity type, were mostly of the constant potential
vorticity form and shared with all the other models of this type the inability to deal
with a simple forward problem. Namely, if the surface density (or temperature ) field
3 The planetary scale Bernoulli function p + ρgz is also called the Montgomery stream function.