My comment on Kip Hansen’s post “Chaos and Weather” at WattsUpWithThat, July 25, 2020:
Your last figure is a nice diagram of Hopf bifurcations leading up to the transition to chaos. Generally it is at the border of transition to chaos, where the system is still low-dimensional, where the interesting emergent pattern phenomena occur; rather than in high dimensional turbulent “full blown chaos”.
Reading some research in chemical engineering by Matthias Bertram and others, I came to realise that one scenario highly relevant to the climate system is the opposite of the Hopf diagram; that is, rather than the progress of an initially linear system over the threshold into chaos, the transition of an already chaotic turbulent system from high down to lower dimensionality, nearer to the Hopf boundary regime where pattern formation can occur. In other words, the reduction rather than the increase in chaotic behaviour and dimensionality.
How can this occur? Bertram and others give some interesting examples from chemical engineering model systems on thin films, including the platinum-catalysed oxidation of CO and the Belousov-Zhabotinsky reaction. Bertram’s goal as an engineer was to control chaotic processes, and he shows two ways to do this: adding feedback, and adding periodic external forcing. To quote:
Spontaneous pattern formation and spatiotemporal chaos (turbulence) are common features of spatially extended nonlinear systems maintained far from equilibrium. The aim of this work is to control and engineer such phenomena. As an example, the catalytic oxidation of carbon monoxide on a platinum (110) single crystal surface is considered. In order to control turbulence and to manipulate pattern formation in this reaction, two different control methods, global delayed feedback and periodic forcing, are employed.
In a nutshell what they do is reduce the dimensionality of the chaotic system by either of these two factors, delayed feedback or periodic forcing. In this way they reduce the “chaoticness” of the system bringing it to the borderline chaos region where interesting and – for them – useful pattern and oscillation emerge.
This made me think of oceanic systems where feedbacks are linked to oscillation. For instance ENSO. You have the Bjerknes feedback whereby Peruvian oceanic upwelling (linked to the Humboldt current) interacts with the trade winds to create intermittent positive feedback which reinforces both the upwelling and the trade winds. (The cold upwelling sets up a sea surface temperature gradient which impels the trade winds).
Another longer term oscillation is the Atlantic Meridional Overturning Circulation (AMOC) which oscillates in strength, giving rise to the AMO – Atlantic Multidecadal Oscillation. Here again there is an intermittent positive feedback. The Gulf Stream increases the salinity of sea surface water in the far North Atlantic and the Norwegian Sea. As this water cools it becomes very dense, causing substantial downwelling all the way to the ocean floor resulting in the “deep water formation” that drives the ocean circulation system. This down welled water flows back south along the ocean floor, completing the 3D loop of the AMOC and reinforcing the Gulf Stream.
In both these cases, ENSO and AMOC, it’s fair to say that the ocean circulation system in 3D is chaotic and turbulent. A glance at the NullSchool ocean circulation animations will confirm this:
Applying the paradigm of Matthias Bertram to this, we can suggest that the presence of feedback in these oceanic systems – the Bjerknes feedback with ENSO and the salinity feedback with AMOC – is “reducing the dimensionality” of the turbulent chaotic circulation systems and causing quasi-regular oscillations to arise.
And as Bertram also found with the BZ reaction, periodic forcing can also bring about emergent oscillation in a chaotic ocean system. ENSO is known to be phase-locked to the annual cycle such that El Niños typically happen at Christmas (this their name).
Tziperman, Cane and Zebiak have shown how ENSO can be modelled as a delayed oscillator periodically forced by the annual cycle:
Warmists try to write off chaos as just noise in the evolving climate system. They may be right if they are talking about only high dimensional turbulence. However they miss the fact that both feedbacks and external periodic forcing (from annual, solar and other astrophysical sources) can reduce the dimensionally of climate subsystems with the result of emerging pattern and oscillation. This can be on many timescales up to century and millennial.
This process, the reduction of dimensionality of chaotic climate systems by feedback or periodic forcing, provides a paradigm to understand how observed fluctuations and oscillations occur in the climate where a direct proximal cause seems elusive. ENSO, AMO and PDO are some examples. Here a real case can be made for chaotic-nonlinear dynamics being the cause of much more substantive climate change than just short term noise. It makes such a model the null hypothesis for much natural ocean driven climate change that is observed over many time scales.