Predictive Chaos

A common argument in the current debate about global climate is that the Earth climate system is chaotic and so cannot be computationally modeled and predicted. The argument is used to question alarmistic predictions of climate change, but also open to unrestricted alarmism in the sense that anything can happen from virtually nothing. 

Basic examples of chaotic systems are (i) coin tossing and (ii) turbulent fluid motion. In both cases pointwise prediction in space and time is impossible, but mean-values can be predicted with high precision. In coin tossing with a perfect coin the quotient of number of heads and tails quickly approach 1 as the number of tosses increases. The drag and lift as mean-values in space and time of the forces on an airplane wing subject to turbulent flow of air, can be accurately computationally predicted by solving the Navier-Stokes equations. This is the subject of the monumental treatise Computational Turbulent Incompressible Flow. See also previous posts under label chaotic system.  

How is it possible that mean-values can be accurately computed but point-values not? The reason is that turbulent flow is (like the weather) fluctuating with alternating ups and downs (like the motion of a tossed coin before landing). Therefore mean-values can be predicted by computational simulation of the fluctuations without asking for point-wise accuracy. 

We understand that chaotic motion as fluctuating turbulent motion in a certain sense is more predictive than non-fluctuating motion like that of a pen left in upright position.

While the weather as pointwise chaotic motion is not predictable over more than a week, climate as mean value of weather changes only slowly and so can be predictive, more or less. For example, we can well expect to enter into a new ice age within hundreds of years as an effect of major factors like the orbit of the Earth. 

Stock markets appear chaotic, yet can be predicted over time using models including major relevant factors. 

In any case, reference to chaotic systems as being unpredictable can be misleading and so has to be qualified.  

PS This post connects to the previous post on the unfortunate present formulation of the Navier-Stokes Clay Millennium Problem forgetting the completely foundational aspect of well-posedness in the sense of small output effects of small perturbations, which is a property of turbulent solutions of Navier-Stokes equations: Drag and lift of a wing remain the same under small perturbations of incoming flow and geometry. The present formulation appears to confuse smoothness with well-posedness.

Corruption of Modern Physics 9: Misuse of Chaotic Systems

Waterfall as partially predictable chaotic system

The idea that global climate is a chaotic system and as such cannot be predicted, because of sensitivity to small perturbations, is often presented as an expression of deep insight into mathematical modeling. But it may hide a common misunderstanding of the nature of a chaotic system. The basic example of a chaotic system is turbulent flow. The nature of turbulent flow is to be unpredictable pointwise in space and time (because of sensitivity) while being predictable in a mean-value sense (because of insensitivity). 

This is developed in detail in Computational Turbulent Incompressible Flow and Computational Thermodynamics recommended for download. The combination of mean-value predictability and pointwise unpredictability is expressed by the fact that the drag of a car as total resistance to motion through air, is computable/predictable while the pressure at specific points on the car body cannot be computed/predicted. The reason mean-values are predictable is the fluctuating nature of turbulence with high pressure followed by low pressure forming stable mean-values. This is the reason nature can function as a more ot less ordered system even if being a chaotic system, which can be seen as a form of order in chaos.  

The Earth climate system can be described as a turbulent thermodynamic ocean-atmosphere system which forms weather local in space and time and global climate as mean-values over space and time. Experience shows that local weather acts as a chaotic system which is unpredictable pointwise in time over more than a week. The question is then to what extent climate as mean-value weather is computable/predictable? 

If all the equations (Navier-Stokes equations and more) modeling the Earth system were known, we would be able to compute/predict for example global mean temperature year 2100 or the onset of the next Ice Age, because of the fluctuating nature of turbulent flow. But we do not know all the parameters entering in the equations nor the initial conditions. Therefore such computation/prediction for now is impossible, but not because the system in principle is chaotic, rather because present climate models contains unknowns. 

This means that with better climate models it could be possible to predict e.g. the onset of the next Ice Age, or on shorter time scales the Winther weather over Europe depending on jet streams and La Nina and more. Even without climate model we can predict the global mean temperature 2023 to be about the same as 2022. 

In short, global ocean-atmosphere weather system is a chaotic system which as global climate is computable/predictable to a certain degree. Work on better climate models is not meaningless.

What is remarkable is the stability of Earth climate without runaway global warming yet with global cooling into repeated Ice Ages,   

Climate vs Chaos and Turbulence

Both climate alarmists and skeptics like to suggest deep understanding by expressing that global climate is a non-linear chaotic system and as such is unpredictable (as discussed by Kip Hansen in a recent sequence of posts):

• The climate system is a coupled non-linear chaotic system, and therefore the long-term prediction of future climate states is not possible. (IPCC TAR WG1, Working Group I: The Scientific Basis)
• It is my belief that most climate variability and even climate change could simply be the result of chaos in the climate system. (Roy Spencer)

But to simply say that a chaotic system is unpredictable is not the entire story. It is true that point values in space/time of a chaotic system are unpredictable, due to strong pointwise sensitivity to pointwise perturbations, but mean values of a chaotic system typically are predictable.

It is certainly impossible to predict the daily temperature of a specific city within one degree one year ahead, but meaningful monthly temperatures are routinely reported in tourist guides.

The book Turbulent Incompressible Fluid Flow presents the following analysis of turbulence as prime example of chaos:

1. Point values are unpredictable due to local exponential instability.
2. Mean values are predictable due to cancellation of instability effects.

It may thus well be possible (with a high degree of certainty) to predict that the global mean temperature will be the same 100 years from now, within a degree up or down.

For the Lorenz system, as a key example of a chaotic system, it is impossible to predict in which lobe a trajectory will be long ahead in time, but the total time spent in each lobe is observed to become nearly equal over long time. About the weather in Scandinavia, we know for sure that it will be variable with alternating low and high pressures, with sunshine following rain and vice versa as a result of the dynamics.