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The climate is not what you expect
complexity and emergent scale invariant laws
Table of Contents
Preface / I describe how the book came to be written. It includes some personal details about how - during my PhD research - of Mandelbrot’s book on fractals, and the subsequent pursuit of scale invariant ideas throughout my career, on to the (2014) statistical testing (and rejecting) of the Giant natural fluctuation hypothesis for explaining global warming.1. Zooming through scales by the billion
1.1 What is weather? What is climate? Why do we need more categories?
1.2 From milliseconds to the age of the earth: a voyage through scales in time
Box: Paleotemperatures
Box: removing annual, daily cycles
1.3 From millimeters to the size of the planet
Box: Leonardo da Vinci
1.4 Complexity, emergent laws, scale invariance and the unfinished nonlinear revolution
Box: Complexity or scale invariant simplicity?
Box: A metric for geo and cosmic complexity
1.5 Overview of the book / This is the introductory chapter. In section 1.1 I explain what is meant by the terms in the title: the negation of the popular saying “the climate is what you expect, the weather is what you get”, the notion of “emergent laws”, and the theme of the book, “scale invariance”. To paraphrase the founder of numerical weather prediction, Lewis Richardson: I argue that “while at first sight strange, scale invariance grows upon acquaintance”.
The reader will learn that although weather has always been a metaphor for the unpredictable that nevertheless – within “Predictability limits” it can be predicted, and how scale invariance can help thanks to the long memory it implies. The weather can also be a metaphor for the extreme, and we discuss the corresponding “black swan events” and the effect of global warming. I will discuss common misconceptions about time and space scales in atmospheric science particularly in the climate. Reading this book will allow the reader to clarify basic ideas about the atmosphere such as what is “weather” and “climate”?, what is “scale” (i.e. what is “big” and “small”?). A more complete list of what the reader will learn is at the end of the table of contents.
In sections 1.2, 1.3, the reader is taken through a sequence of plots of temperature and proxy temperature series spanning time scales from milliseconds to hundreds of millions of years, this chapter gives a visual idea of the incredible range of time scales over which the atmosphere is variable. Similarly, satellite and other images visually demonstrate the same idea in space: structures within structures within structure. This introductory chapter interrogates the reader: how can we possibly understand, theorize and model such behaviour?
This is followed (section 1.4) by a general discussion (with much historical flavour) of the nonlinear revolution, complexity, emergent (high level) laws: scale invariance manifested by fractal structures, multifractal fields (these are described in more detail later).
This is followed by an overview of the book and its structure.
2. New worlds to scale invariance: van Leeuwenhoek to Mandelbrot and beyond
2.1 A new world in a drop of water: scale bound thinking
Box: Antoni van Leeuwenhoek
2.2 Scale invariance: Big whirls have little whirls and little whirls have lesser whirls
Box: Jean Perrin: the coast of Brittany
Box: Lewis Fry Richardson: cascades
Box: Benoit Mandelbrot: fractals
Box: Edwin Hurst: long range memory
2.3 Scale invariance and the phenomenological fallacy
2.4 Scale invariant versus scale bound classifications
Box: Henry Stommel: space-time diagrammes / The two extreme opposite approaches for dealing with systems with structures over huge ranges of scale are the “scale bound” and the “self-similar scale invariant” approaches associated with van Leuwenhoek (17th C) and Mandelbrot.
In the former, every factor of ten or so of “zooming” leads to something totally different, in the latter, on average zooming changes nothing. Then (section 2.3), we generalize Mandelbrot’s scale invariance to include processes that are the same from big to small, but as we move from one scale to another require involve squashing and/or rotation in order to remain invariant. The result is that structures are different at different scales even though they are produced by the same scale invariant mechanism. This demonstrates the common “phenomenological fallacy” whereby differences in structure/appearance from one scale to another are used to hypothesize the dominance of qualitatively different processes at different scales. Scale invariant systems generally have structures with different appearances even though the underlying mechanisms are the same at all scales.
3. Testing scale invariance: fluctuations as a microscope
3.1 Fluctuations
3.2 The fluctuation exponent H and the H model
Box: Spectra and the missing quadrillion / This chapter describes fluctuation analysis that allows us to quantitatively distinguish and characterize scale bound and scale invariant approaches. To do this we need to define fluctuations that are then evaluated at large and small scales and then compared statistically. We introduce the only equation of the book:
(Fluctuation) = (Scale)H
In this prototypical expression of scale invariance, the fluctuations are said to be “scaling” while the exponent H is “scale invariant”. We demonstrate this with some simple geometric fractal constructions (the “H model”).
4. Scale invariant regimes: weather, macroweather, climate, macroclimate and megaclimate
4.1 Scale invariance, scaling and atmospheric dynamics
Box: Stochastic versus deterministic chaos
Box: Statistics versus deterministic mechanism
Box: “Fractals: where’s the physics?”
4.2 The weather is a scale invariant, turbulent cascade
Box: Andrei Kolmogorov: turbulent laws
Box: World record wind
Box: How wet is the coast of Brittany?
Box: Numerical weather and climate models are scale invariant
4.3 Expect Macroweather: fluctuations decreasing with scale
Box: A Martian family goes for a picnic
4.4 The climate: don’t expect it
Box: Solar, volcanic climate forcings are scaling
Box: How accurately do we know the temperature of the Earth?
4.5 Macroclimate and the ice ages: scaling or cycles?
Box: Svante Arrhenius: doubling CO2
Box: Milutin Milankovitch, orbital forcing
4.6 Megaclimate: long term instability and the end of Gaia
Box: James Lovelock / Having discussed the tool needed to quantitatively investigate the variability (fluctuation analysis), we apply it to the time domain to the data and proxy data discussed and displayed in ch. 1. This shows that between milliseconds and hundreds of millions of years, that are five different regimes, each defined by the way fluctuations change as we zoom from long to short time intervals.
In this chapter we consider each scale invariant regime separately, discussing topics such as turbulence and cascades, including the pedagogical additive-multiplicative H-a model, the existence of stable atmospheric layers exist, the dimension of atmospheric motions, the difference between earth and mars, ice ages, and the Gaia hypothesis.
5. What is scale?
5.1: Scale as an emergent turbulent property: Generalized Scale Invariance
Box: Distance as a emergent property in General Relativity
Box: Is isotropic turbulence relevant in the atmosphere?
5.2 What is the dimension of atmospheric motions?
Box: Numerical Weather models: 23/9 dimensional?
5.3 Aircraft measurements are not what they seem / In this chapter we go beyond scale invariance in time to discuss scale invariance in space (and even space-time). It turns out that to do so we must treat the notion of scale not as something pre-ordained by the observer, but as an emergent property that is determined by the complex nonlinear dynamics: “Generalized Scale Invariance”. This generalization of the notion of scale is necessary to account for the stratification and rotation of structures as we zoom into the atmosphere. Generalized Scale Invariance is somewhat analogous to General relativity in which the distribution of mass and energy determines the notion of distance.
This allows us to account for vertical stratification and space-space relations. We can understand convection and other processes classically explained by scale bound models as being on the contrary manifestations of scaling process. It also turns out that aircraft data are not what they seem; they need to be reinterpreted.
6. Scale Invariance and extremes
6.1 White, Grey and Black Swan events
6.2 The multifractal butterfly effect
Box: Per Bak: Sandpiles, Self-organized Criticality / Scale invariant processes have dynamics that repeat scale after scale from large to small. It turns out that this builds up stronger and stronger variability as we move to from large to small scales so that atmospheric variability has extreme fluctuations (corresponding to extreme storms, winds, heat waves etc.). These extremes are so much stronger than the conventional weak “bell curve” extremes that they have been termed “grey” or “black swan” type events. We discuss the scaling of extremes and how they are generated by a multifractal (scale invariant) version of the “butterfly effect” and its relation of black swans to epistemological uncertainty.
7. Scale invariance and giant natural fluctuations: climate closure
7.1 Why the warming can’t be natural
Box: “A mephitic ectoplasmic emanation of the forces of darkness”
7.2 The $100,000 Giant Natural climate Fluctuation and Anthropogenic warming
/ Scale invariance in time was used to define the macroweather regime; in the pre-industrial epoch it ranges over time scales from weeks to centuries. In the previous chapter we also discussed the link between scale invariance and extremes. In this chapter, we bring the two together to explain how we can statistically test the hypothesis that the industrial epoch warming is simply a giant natural fluctuation. We give various anecdotes about the author’s dealings with climate sceptics including the ongoing $100,000 climate contest.
8. Using scale invariance for prediction: exploiting long range memory
8.1 Weather forecasting and the butterfly effect: deterministic predictability limits
Box: Edward Lorenz: Texas tornadoes and Brazilian butterflies
8.2 Macroweather forecasting and stochastic predictability limits: The Stochastic Seasonal and Interannual Prediction System (StocSIPS)
8.3 The future of weather and climate forecasting / Scale invariance implies long range (in space and in time) interactions and correlations. It turns out that this can be used for macroweather forecasting (months to decades).
How accurate can we forecast? In the weather regime, usual, deterministic forecasts are limited to about 10 days by the butterfly effect (“sensitive dependence on initial conditions”). In the macroweather regime, we make statistical forecasts and these are instead limited by “stochastic” (statistical) limits to predictability. We discuss the new scaling based StocSIPS prediction system for macroweather forecasting (months to decades) and compare it to the standard numerical model (Global Circulation Model, GCM) approach, showing why and by how much the new StocSIPS method is better while simultaneously being both simpler and about a million times faster.
9. Earth, water, fire, air
9.1 Scale invariance in the hydrosphere
9.2 Scale invariance in volcanoes
9.3 Scale invariance in the solid earth / Having explained scale invariance in space and in time, and the link to the extremes, in this final chapter we give numerous examples showing how the same ideas can be applied in the hydrosphere (precipitation, river flows, floods) and to volcanoes and volcanic processes and phenomena more to the traditional solid-earth geophysics including topography, mantle convection, geogravity, geomagnetism. This brings out a new aspect of the unity of the geosciences: the fact that geofields generally exhibit wide range scale invariance, including the scaling of their extremes.
What the reader will learn:
a) That science is an interlocking hierarchy of theories and how while both low high level theories can be correct, the high level theories are usually more useful. The high level theories – here based on scale invariance – are “emergent” with respect to the low level theories - here those of continuum mechanics and thermodynamics. Whereas the latter are deterministic, the former are statistical (“stochastic”).
b) What is scale invariance including the main scale invariant objects: fractal sets and multifractal fields.
c) That structures such as clouds, “weather systems”, eddies can be of very different size yet still be produced by fundamentally the same scale invariant mechanism. Classifying, analyzing and modelling the atmosphere on the basis of appearance “phenomenology” is not justified: the “phenomenological fallacy”.
d) Readers will learn what is weather, what is the climate and why we need more categories.
e) How weather and climate forecasting is done today, how scale invariance can improve it.
f) Why the industrial epoch warming can’t be natural.
g) How scale invariance can help understand other areas of geoscience including the hydrosphere, the lithosphere, volcanoes.
Synopsis
This book describes in layman’s terms a new paradigm for understanding the atmosphere from millimeters to the size of the planet and from milliseconds to the age of the earth. Whereas the popular expression states that “the climate is what you expect, the weather is what you get”, in this book, we take the reader by the hand and explain that there is a third regime –macroweather – in between the weather and climate so that on the contrary, the climate is not what you expect: expect macroweather.
In order to understand this new view, the book takes the reader on a journey through scales in both space and in time. It describes why the traditional “scale bound” (“powers of 10”) approach - inherited from van Leeowenhoek in the 17th century – is not adequate for understanding the atmosphere’s astonishing variability. In its place, the book describes the new paradigm of scale invariance associated with fractal structures and multifractal processes. In its simplest form championed by Mandelbrot – “self-similarity” - it describes systems that are the opposite of scale bound: under “zooming” they just reveal just more of the same: they are “scale invariant”.
As the book progresses, more nuanced ideas of scale invariance are described wherein one must zoom and possibly squash and/or rotate in order to obtain the same. This is the more general case needed to deal with stratification and rotation both in the atmosphere and in many other geosystems including the rocks (lithosphere). It reveals the “phenomenological fallacy” whereby the quite different appearances of small and large structures are used to justify the elaboration of separate theories and models: in scale invariant processes, a unique mechanism repeats scale after scale yet the large and small may easily have quite different appearances. The scale invariance paradigm emerged in the 1970’s and 80’s as part of the nonlinear “revolution”; the book gives some of this history. Indeed, the book will have many roughly one page “boxes” that are intended to be asides on key historical characters and concepts. In addition, there will be footnotes for readers who want to dig deeper.