5/6/23 Chapter 1 1

Words: 11242 (without references), 8774, no figs.

The Weather and Climate: Emergent Laws and Multifractal cCasades

1. Introduction

1.1 The new synthesis

1.1.1 Two (irreconcilable?) approaches to understanding the atmosphere

In the last twenty years there has been a quiet revolution in atmospheric modelling. It’s not just that computers and numerical algorithms have continued their development, but rather that the very goal of the modelling has profoundly changed. Whereas twenty years ago, the goal was to determine the (supposedly) unique state of the atmosphere, today with the advent in Ensemble Forecasting Systems (EFS), the aim is to determine the possible future atmospheric states including their relative probabilities of occurrence: this new goal is stochastic. A stochastic process is a set of random variables indexed by time (Kolmogorov 1933), and this definition includes that of deterministic processes as a special case.

At present, the EFS are really hybrids in the sense that they operate by first generating an initial ensemble of atmospheric states compatible with the observations and then they use conventional deterministic forecasting techniques to advance each member in time to produce a distribution of future states. Once the leap was taken to go beyond the forecasting of a unique state to forecasting an ensemble, the next step was to make the subgrid parametrisations themselves stochastic (e.g. (Buizza et al., 1999), (Palmer, 2001)), (Palmer and Williams, 2010)). This is an attempt to take into account the variability of different subgrid circulations. The artificial deterministic / stochastic nature of these hybrids suggests that the development or pure stochastic forecasts would be advantageous, a possibility we explore in ch. 9.

Interestingly, the tension between determinism and stochasticity has been around pretty much since the beginning, although for most of the (still brief) history of atmospheric science, the deterministic approaches have been in the ascendancy and the stochastic ones left in the wings. To see this, let us recall the important developments. Drawing on the classical (deterministic) laws of fluid mechanics, Bjerknes (1904) and (Richardson, 1922) extended these to the atmosphere in the familiar form of a closed set of nonlinear partial differential governing equations. From a mathematical point of view, their deterministic character is evident from the absence of probability spaces; from a conceptual point of view, it is associated with classical Newtonian thinking. In physics, Newtonian determinism began to disappear with the advent of statistical mechanics (starting with the “Maxwellian” distribution of molecular velocities (Maxwell, 1890)) - which showed that physical theories could indeed be stochastic. The break with determinism was consecrated with the development of quantum mechanics -

DRAFT

1

1

2

3

4

5

6

789

10111213141516171819202122232425262728293031323334353637383940

2

5/6/23 Chapter 1 2

which is a fundamental yet stochastic theory where the key physical variable - the wavefunction - determines probabilities.

At roughly the same time as the basis of modern deterministic numerical weather prediction was being laid, an alternative stochastic “turbulent” approach was being developed by G.I. Taylor, L. F. Richardson, A. N. Kolmogorov and others. Just as in statistical mechanics where huge numbers of degrees of freedom exist but where only certain “emergent” macroscopic qualities (temperature, pressure etc.) are of interest, in the corresponding turbulent systems the new theories sought to discover new emergent statistical turbulence laws.

The first of these emergent turbulent laws was the Richardson “4/3 law” of atmospheric diffusion: n(L) ≈ KL4/3 where n(L) is the effective viscosity at scale L and K is a constant to which we return (Richardson, 1926), see fig. 1.1. This law is famous not only as the precursor of the (Kolmogorov, 1941) law of 3D isotropic homogeneous turbulence (the “5/3” law for the spectrum - or if expressed for the fluctuation Dv(L), the “1/3” law: Dv(L) = e1/3L1/3 where Dv is the velocity fluctuation and e is the energy flux) but it is also celebrated thanks to the ingenious way that Richardson experimentally confirmed his theory with the help of balloons and later even with parsnips and thistledown (Richardson and Stommel, 1948)! While this attention is all well deserved, the law was perhaps even more remarkable for something else: that Richardson had the audacity to conceive that a unique scaling (power) law - i.e. a law without characteristic length scales - could operate over the range from millimeters to thousands of kilometers i.e. over essentially the entire meteorologically significant range. In accord with this, Richardson believed that the corresponding diffusing particles had “Weierstrass function like” (i.e. fractal) trajectories. Nor was the 4/3 law an isolated result. In the very same pioneering book “Weather Prediction by Numerical Process” (Richardson, 1922) in which he wrote down essentially the modern equations of the atmosphere (Lynch, 2006) and even attempted a manual integration, he slyly inserted:

“Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity – in the molecular sense”.

Thanks to this now iconic poem, Richardson is often considered the grandfather of the modern cascade theories that we discuss at length in this book.

Fig. 1 Here

Had Richardson been encumbered by later notions of the meso-scale - or of isotropic turbulence in either two or three dimensions - he might never have discovered his law. Already - fifteen years after he proposed it - (Kolmogorov, 1941) humbly claimed only a relatively small range of validity of the stringent “inertial range” assumptions of statistical isotropy and homogeneity which he believed were required for the operation of his eponymous law (which was also discovered independently by (Obukhov, 1941b), (Obukhov, 1941a), (Onsager, 1945), Heisenberg (1947) and (von Weizacker, 1948)); and this even though it has strong common roots with Richardson’s law. Indeed, it implies that Richardson’s proportionality constant depends on the energy

DRAFT

3

41424344454647484950515253545556575859606162636465666768697071727374757677787980818283848586

4

5/6/23 Chapter 1 3

flux e: n(L) = LDu(L) = e1/3L4/3; in this sense Kolmogorov’s contribution was to find K = e1/3. Echoing Kolmogorov’s reservations, (Batchelor, 1953) speculated that the Kolmogorov law should only hold in the atmosphere over the range 100 m to 0.2 cm! Even in Monin’s influential book “Weather Forecasting as a problem in Physics” (Monin, 1972), the contradiction between the small and wide ranges of validity of the Kolmogorov and Richardson 4/3 laws is pushed surprisingly far since on the one hand Monin confines the range of validity of the Dv(L) = e1/3L1/3 law to “micrometerological oscillations… up to ≈ 600 m in extent”, while on the other hand publishing (on the opposite page!) a reworked copy of Richardson’s figure demonstrating the validity of the n(L) ≈ L4/3 up to thousands of kilometers (fig. 1.1). For the latter, he comments that it “is valid for nearly the entire spectrum of scales of atmospheric motion from millimeters to thousands of kilometers”, in accord with Richardson. In (Monin and Yaglom, 1975), the contradiction is noted with the following mysterious explanation: “…in the high frequency region one finds unexpectedly, that relationships similar to those valid in the inertial subrange of the microturbulence spectrum are again valid…”. In ch. 6 we argue on the basis of modern reanalyses and other data that the law Dv(L) = e1/3L1/3 does indeed hold up to near planetary scales in the horizontal but paradoxically, that even at scales as small as 5 m, it does not hold in the vertical (and hence 3D isotropic turbulence does not seem to hold anywhere in the atmosphere)! By proposing a theory of anisotropic but scaling turbulence, we attempt to explain how it is possible that Kolmogorov was simultaneously both so much more accurate (the horizontal) and yet so much less accurate (the vertical) than anyone expected. This was achieved with the help of a generalized notion of scaling (Schertzer and Lovejoy, 1985b), (Schertzer and Lovejoy, 1985a), which ironically led to an effective “in between” dimension of atmospheric turbulence D =23/9=2.55….

Facing colossal mathematical difficulties, turbulence theorists starting with (Taylor, 1935) concentrated their attentions on the simplest turbulence paradigm: turbulence that is statistically isotropic, first in 3D, and then - following (Fjortoft, 1953) and (Kraichnan, 1967) - on the special isotropic 2D case. While Charney did extend Kraichnan’s 2D theory to the atmosphere in his seminal paper “Geostrophic Turbulence” (Charney, 1971), meteorologists had already begun focusing on numerical modelling. By the end of the 1970’s, there had thus developed a wide divergence between on the one hand, the turbulence community with its focus on statistical closures and statistical models of intermittency (especially cascades) and on the other hand the meteorology community with its focus on practical forecasting and which treated turbulence primarily as a subgrid parametrization problem.

1.1.2 Which chaos for the geophysics, for atmospheric science?

The divergence between statistical and deterministic approaches was brought into sharp relief thanks to advances in the study of nonlinear systems with few degrees of freedom. The new science of “deterministic chaos” can be traced back to the pioneering paper “Deterministic nonperiodic Flow” (Lorenz, 1963) (and has antecedents in (Poincaré, 1892)). Lorenz’s 1963 paper caused excitement by showing that three degrees of freedom were sufficient to generate chaotic (random-like) behaviour in a purely deterministic system. At the time, it was widely believed (following (Landau,

DRAFT

5

87888990919293949596979899

100101102103104105106107108109110111112113114115116117118119120121122

123124125126127128129130131

6

5/6/23 Chapter 1 4

1944)) that on the contrary, random-like behaviour was a consequence of very large number of degrees of freedom so that as the nonlinearity increased (e.g. the Reynold’s number), a fluid became fully turbulent only after successively going through a very large (even infinite) number of instabilities. By showing that as few as three degrees of freedom were necessary for chaotic behaviour, Lorenz’s paper opened the door to the possibility that turbulence could have a relatively low dimensional “strange attractor” so that effectively only a few degrees of freedom might matter. However, Lorenz’s observation did not immediately lead to practical applications because theorists can readily invent nonlinear models and at the same time it appeared that each model would require its own in-depth study in order to understand its behaviour. The problem of apparent lack of commonality in different nonlinear systems is the now familiar problem of “universality” which Fischer, Kadanoff and Wilson were only then successfully understanding and exploiting in the physics of critical phenomena; we shall revisit universality later in this book (ch. 3). It is therefore not surprising that the turning point for deterministic chaos was precisely the discovery of “metric” (i.e. quantitative) “universality” by (Grossman and Thomae, 1977) and (Feigenbaum, 1978): the famous Feigenbaum constant in period doubling maps. Soon, with the help of theorems such as the extension of the Whitney embedding theorem (Whitney, 1936) and the practical “Grassberger-Procaccia algorithm” (Grassberger and Procaccia, 1983b), (Grassberger and Procaccia, 1983a) all manner of time series were subjected to nonlinear analysis in the hope of “reconstructing the attractor” and of determining its dimension, which was interpreted as an upper bound on the number of degrees of freedom needed to reproduce the system’s behaviour. In fact - as argued by (Schertzer et al., 2002) - the mathematics do not support such a statement: they showed that indeed a stochastic cascade process may yield a finite correlation dimension, whereas the process itself has an infinite dimension! They therefore raised the question “which chaos?”. Other developments in the 1980’s helped to transform the “deterministic chaos revolution” into a more general “nonlinear revolution”. Of particular importance for this book, was the idea that many geosystems were fractal (scale invariant) (Mandelbrot, 1977), (Mandelbrot, 1983) and later, that they commonly displayed “Self-Organized Criticality” (SOC), (Bak et al., 1987), (Bak, 1996) implying that many real world systems could be “avalanche-like”. Indeed, SOC is so extreme that even “typical” structures are determined by extreme events (see ch. 5 for the connection between SOC and turbulent cascades).

The success of the apparently opposed paradigms of deterministic chaos and (stochastic) fractal systems thus sharply posed the question “which chaos for atmospheric science: deterministic or stochastic?” The question was not the philosophical one of whether or not the world is deterministic or stochastic, but rather whether deterministic or stochastic models are the most fruitful: which is the closest to reality (Lovejoy and Schertzer, 1998). The answer to this question essentially depends on the number of degrees of freedom that are important: since stochastic systems are usually defined on infinite dimensional probability spaces they are good approximations to systems with large numbers of degrees of freedom. As applied to the atmosphere, the classical estimate of that number is essentially the number of dissipation scale fluid elements in the atmosphere, roughly 1027 -1030 (see ch. 2 for this estimate). However, at any given moment clearly many of these degrees of freedom are inactive, indeed we shall see that multifractals (via the codimension function c(g), ch.5) provide a precise

DRAFT

7

132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177

8

5/6/23 Chapter 1 5

estimate of the fraction of those at any given level of activity and at any space-time scale.

1.2 The golden age resolution revolution and paradox: an up to date empirical tour of atmospheric variability

1.2.1 The basic form of the emergent laws and spectral analysis

Without further mathematical or physical restrictions, the high number of degrees of freedom paradigm of stochastic chaos is too general to be practical. But with the help of a scale invariant symmetry such that in some generalized sense, it repeats scale after scale, it becomes tractable and even seductive. It turns out that the equations of the atmosphere are indeed formally scale invariant (ch. 2), and even fields for which no theoretically “clean” equations exist (such as for precipitation) still apparently respect such scale symmetries. However, even if the equations respect a scaling symmetry, the solutions (i.e. the real atmospheric motions) would not be scaling were it not for the scale invariance of the relevant boundary conditions.

We have briefly mentioned the Kolmogorov law as being an example of an emergent law. Indeed, all the emergent laws discussed in this book are of the form:

Fluctuations ≈ turbulent flux( )× scale( )H

The Kolmogorov law mentioned in the previous section is recovered as a special case if the velocity differences Dv across a fluid structure of a given scale (L) is used for the fluctuations and we take the scaling exponent H = 1/3 and the turbulent flux = e1/3 (a power of a turbulent flux is still a turbulent flux). The book is structured around a series of generalizations of this basic equation. For example, rather than considering smooth or weakly varying (for example quasi Gaussian) fluxes, we show in ch. 3, 5 how to treat wildly variable fluxes that are the results of multiplicative (and multifractal) cascades (this involves interpreting the equality in eq. 1 in the sense of random variables). Then in ch. 6, 7 we generalize the notion of “scale” to include strongly anisotropy - needed in particular for handling atmospheric stratification (“Generalized Scale Invariance”). In chs. 8, 9 this is further generalized from anisotropic space to anisotropic space-time (including causality). Finally in ch. 10 we show how that the long time behaviours of space-time cascades involve “dimensional transitions” and low frequency weather fluctuations with H<0. According to equation 1, since the mean of the turbulent flux is independent of scale this “low frequency weather” regime is characterized by mean fluctuations that decrease with scale. This contrasts with the higher frequency “weather” regime in which typically H>0 so that on the contrary, mean weather fluctuations increase with scale.

We now proceed to give an empirical tour of some the fields relevant either directly or indirectly to atmospheric dynamics. This overview is not exhaustive and it partly reflects the availability of relevant analyses and partly the significance of the fields in question. Our aim is to exploit the current “golden age” of geophysical observations so as to demonstrate as simply as possible the ubiquity of wide range scaling- even up to planetary scales - and hence the fundamental relevance of scaling

DRAFT

9

178179

180181

182183184185186187188189190191192193194

195

196197198199200201202203204205206207208209210211212213214215216217218219

10

5/6/23 Chapter 1 6

symmetries for understanding the atmosphere. However, before setting out to empirically test eq. 1 on atmospheric fields, a word about fluctuations. Often, the definition of a fluctuation as simply a difference is adequate (strictly speaking when 0<H<1), but sometimes other definitions are needed. Indeed, there has arisen an entire field - wavelets - centred essentially around systematic ways of defining and handling fluctuations. For most of the following, thinking of fluctuations as differences is adequate but some mathematical formalism is developed in section 5.6 and as a practical matter, differences are not adequate in ch. 10 where we treat low frequency weather which has H<0 and requires other definitions of fluctuations (we recommend the simple Haar fluctuation, but others are possible).

In the following scaling overview, it will therefore be convenient to use the Fourier (spectral) domain version of eq. 1 which avoids these technical issues. In Fourier space, eq. 1 reads:

Varianceobservables

wavenumber⎛⎝⎜

⎞⎠⎟ =

Varianceflux

wavenumber⎛⎝⎜

⎞⎠⎟ wavenumber( )−2H

Consider a random field f(r) where r is a position vector. Its “Variance/wavenumber” or “spectral density” E(k) is the total contribution to the variance of the process due to structures of with wavenumber between k and k + dk (i.e. due to structures of size 2p/l

where l is the corresponding spatial scale; k is the modulus of the wavevector k = k we postpone a more formal definition to chapter 2 (including appendix 2A)). The spectral density thus satisfies:

f r( )2 = E(k)dk0

∞

∫

where f r( )2

is the total variance (assumed to be independent of position; the angular brackets “< . >” indicate statistical averaging). .

In the following examples, we demonstrate the ubiquity of power law spectra:E k( )≈k−b

If we now consider the real space (isotropic) reduction in scale by factor l we obtain: r → l−1r corresponding to a “blow up” in wavenumbers: k → lk ; power law E(k) (eq. 4) maintain their form under this transformation: E → l−bE so that E is “scaling” and the (absolute) “spectral slope” b is “scale invariant”. If empirically we find E of the form eq. 4, we take this as evidence for the scaling of the field f. For the moment, we consider only scaling and scale invariance under such conventional isotropic scale changes; in ch. 6 we extend this to anisotropic scale changes.

1.2.2 Atmospheric data in a Golden AgeAs little as 25 years ago, few atmospheric data sets spanned more than two

orders of magnitude in scale; yet they were challenging even to visualize. Global models had even lower resolutions, yet required heroic computer efforts. The

DRAFT

11

220221222223224225226227228229230231232

233

234235236

237238239

240

241242243

244

245

246247248249250251

252253254255

12

5/6/23 Chapter 1 7

atmosphere was seen through a low resolution lens. Today, in situ and remote data routinely span scale ratios of 103- 104 in space and /or time scales and operational models are not far behind. We are now beginning to perceive the true complexity of atmospheric fields which span ratios of over 1010 in spatial scales (the planet scale to the dissipation scale). One of the difficulties in establishing the statistical properties of atmospheric fields is that it is impossible to estimate spatial fields without making important assumptions about their statistical properties. We now survey the main data types indicating some of their limitations and briefly discuss the various relevant data sources.

In situ networks: In situ measurements have the advantage of directly measuring the quantities of greatest interest: the variables of state: pressure, temperature, wind, humidity etc., however, at the outset, these fields are rarely sampled on uniform grids, more typically they are sampled on sparse fractal networks (see fig. 3.6a, for an example). In addition, standard geostatistical techniques such as Kriging require various regularity and uniformity assumptions which are unlikely to be satisfied by the data (as we see in section 5.4, the latter are more accurately densities of measures which are singular with respect to the usual Lebesgue measures). This means that the results will depend in power law ways on their resolutions.

At first sight, an in situ measurement might appear to be a “point” measurement, however this is misleading since while their spatial extents may be tiny compared to the analysis grids, what is relevant is rather their space-time resolutions, and in practice, this is never point-like – nontrivial amounts of either spatial or temporal averaging are required. The main exceptions would be measurements simultaneously near 10 kHz in time and at 0.1 -1 mm in space which would allow one to approach the typical viscous dissipation (and hence true homogeneity) space and time scales.

In situ measurements, aircraft, sondes: In situ measurement techniques such as aircraft (horizontal) or sondes (vertical) have other problems, some of which we detail in later chapters. Aircraft data are particularly important. In many cases they provide our only direct measurements of the horizontal statistics. Unfortunately aircraft don’t fly in perfectly flat straight trajectories; due to the very turbulence that they attempt to measure, the trajectories turn out to be more nearly fractal and – this turns out to be even more important - their average slopes with respect to the vertical are typically non negligible. If one assumes that the turbulence is isotropic (or at least has the same statistical exponents in the horizontal as in the vertical), then this issue is of little importance: if one measures a scaling exponent, then by the isotropy assumption it is unique so that the exponent estimate is assumed to be correct. However, it turns out that if the turbulence is strongly anisotropic, with different exponents in the horizontal and vertical directions then – as we show in ch. 7 - the interpretation of the measurements is fraught with difficulties and one will generally observe a break in the spectrum/ scaling. For the smaller scales the statistics are dominated by the horizontal fluctuations while at the larger scales, they are dominated by the vertical fluctuations. In ch. 2 we see that naïve use of isotropy assumptions have commonly led researchers to misinterpret this spurious transition from horizontal to vertical scaling as a signature of a real physical transition from an isotropic 3D turbulence regime at small scales to an isotropic 2D

DRAFT

13

256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291292293294295296297298299300301

14

5/6/23 Chapter 1 8

turbulence regime at large scales.

Remote sensing: One way of overcoming the problems of in situ sampling is to use remotely sensed radiances. There is a long history of using radiances in “inversion algorithms” to attempt to use them to directly estimate atmospheric parameters (Rodgers, 1976). However to be useful in numerical weather models, the data extracted from the inversions must generally be of high accuracy. This is because models typically require gradients of wind, temperature, of humidity etc. and taking the gradients greatly amplifies errors. The fundamental problem is that classical inversion techniques aim to estimate the traditional numerical model inputs (variables of state) and they rely on unrealistic subsensor resolution homogeneity assumptions to relate these parameters to the measured radiances. Since the heterogeneity is generally very strong (scaling, multifractal) there are systematic power law dependencies on the resolution of the measurements (a consequence of the cascades structure, section 5.3). Therefore, new resolution independent algorithms are needed (Lovejoy et al., 2001).

Reanalyses: Having recognized that in situ measurements have frequent “holes”, and that the inversion of remote measurements is error-prone, one can attempt to combine all the available data as well as the theoretical constraints implied by the governing atmospheric equations to obtain an “optimum estimate” of the state of the atmosphere; these are the meteorological “reanalyses”. Reanalyses are effectively attempts to provide the most accurate set of fields consistent with the data and with the numerical dynamical models, themselves believed to embody the relevant physical laws. The data are integrated in space with help of a variational algorithm either at regular intervals (“3D var”); or - in the more sophisticated “4D var” - both in space and time (see e.g. (Kalnay, 2003)). In these frameworks, remotely sensed data can also be used, but in a forward rather than an inverse model: one simply calculates theoretically the radiances from the guess fields of the traditional atmospheric variables. Once all the guess fields are calculated at the observation times and places, then the two are combined by weighting each guess and measurement pair according to pre-established uncertainties. While these sophisticated data assimilation techniques are elegant, one should not forget that they are predicated on various smoothness and regularity assumptions which are in fact not satisfied because of the very singular scaling effects discussed in this book. These resolution effects introduce non negligible uncertainties and possible biases on the reanalyzed fields.

1.2.3 The horizontal scaling of atmospheric FieldsWe start our tour by considering global scale satellite radiances since they are

quite straightforward to interpret. Fig. 1.2 shows the “along track” 1-D spectra from the VIRS (Visible Infra Red Sounder) instrument of the TRMM (Tropical Rainfall Measurement Mission) at wavelengths of 0.630, 1.60, 3.75, 10.8, 12.0 mm i.e. for visible, near infra red, and (the last two) thermal infra red. Each channel was recorded at a nominal resolution of 2.2 km and was scanned over a “swath” 780 km wide and ≈1000 orbits were used in the analysis. The scaling apparently continues from the largest scales (20,000 km), to the smallest available. At scales below about 10 km, there is a more rapid falloff but this is likely to be an artefact of the instrument whose sensitivity starts to drop off at scales a little larger than the nominal resolution. The scaling

DRAFT

15

302

303304305306307308309310311312313314315

316317318319320321322323324325326327328329330331332333334

335336337338339340341342343344345

16

5/6/23 Chapter 1 9

observed in the visible channel (1) and the thermal IR channels (4, 5) are particularly significant since they are representative respectively of the energy containing short and long wave radiation fields which dominate the earth’s energy budget. One sees that thanks to the effects of cloud modulation, the radiances are very accurately scaling. This result is incompatible with classical turbulence cascade models which assume well-defined energy flux sources and sinks with a source and sink - free “inertial” range in between. Also of interest is the fact that the spectral slope b is close (but a little lower) than the value

€

b =5 /3 expected for passive scalars in the classical Corrsin-Obukhov theory discussed in ch. 2. This result is consistent with theoretical studies of radiative transfer through passive scalar clouds ((Watson et al., 2009), (Lovejoy et al., 2009b)). Although we cannot directly interpret the radiance spectra in terms of the wind, humidity or other atmospheric fields, they are strongly nonlinearly coupled to these fields so that the scaling of the radiances are prima facie evidence for the scaling of the variables of state. To put it the other way around: if the dynamics was such that it predominantly produced structures at a characteristic scale L, then it is hard to see how this scale would not be clearly visible in the associated cloud radiances.

Fig. 1.2, 1.3 Here

To bolster this interpretation, we can also consider the corresponding images at microwave channels (corresponding to black body thermal emission with wavelengths in the range 0.351 to 3.0 cm), fig. 1.3. In order to extend these results to smaller scales, we can use either finer resolution satellites such MODIS, SPOT or LANDSAT, or we can turn to ground based photography, see fig. 1.4 a, b. Again, we see no evidence for a scale break. Interestingly, the average exponent b ≈ 2 indicates that the downward radiances captured here (with near uniform background sky) are smoother (larger b) than for the upward radiances analysed in fig. 1.2, 1.3 (the variability falls off more rapidly with wavenumber since b is larger).

Fig. 1.4a, b Here

The remotely sensed data analyzed above give strong direct evidence of the wide range scaling of the radiances and hence indirectly for the usual meteorological variables of state. For more direct analyses, we therefore turn our attention to reanalyses. Fig. 1.5a shows representative reanalyses taken from the European Medium Range Weather Forecasting Centre (ECMWF) “interim” reanalysis products, the zonal and meridional wind, the geopotential height, the specific humidity, the temperature, vertical wind. The ECMWF interim reanalyses are the successor products to the ECMWF 40 year reanalysis (ERA40) and are publicly available at 1.5o resolution in the horizontal and at 37 constant pressure surfaces (every 25 mb in the lower atmosphere). At the time of writing, the fields were available every 6 hours from 1989 - present. The data in fig. 1.5a were taken from the 700 mb level. The 700 mb level was chosen since it is near the data-rich surface level, but suffers little from the extrapolations necessary to obtain global 1000 mb fields (which is especially especially problematic in mountainous regions); it gives a better representation of the “free” atmosphere (section 4.2.2 for more information and analyses and see (Berrisford et al., 2009) for complete reanalysis

DRAFT

17

346347348349350351352353354355356357358359360361362363

364365366367368369370371372373374375

376377378379380381382383384385386387388389390

18

5/6/23 Chapter 1 10

details).

The data analyzed were daily data for the year 2006 with only the band between ±45o latitude used (with a cylindrical projection). The reason for this choice was two -fold: first, this region is fairly data-rich compared to the more extreme latitudes (although there are more grid points for the latter), second it allows us to conveniently compare the statistics in the east-west and north-south directions in order to study the statistical anisotropies between the two. In addition, the east-west direction was similarly broken up into 2 sections, one from 0-180o and the other from 180-0o

longitude. For technical reasons discussed in ch. 6, the spectrum was estimated by performing integrals around ellipses with aspect ratios 2:1 (EW:NS) the wavenumber scale in fig. 1.5b indicates the east-west scale, a full discussion of the anisotropy is postponed to ch. 6.

From the figure we can see that the scaling is convincing (with generally only small deviations at the largest scales ≥ 5000 km), although for the geopotential, the deviations begin nearer to 2500 km. In spite of this generally excellent scaling, the values of the exponents are not “classical” in the sense that they do not correspond to the values predicted by any accepted turbulence theory. An exception is the value b ≈ 1.6 for the humidity which is only a bit bigger than the Corrsin-Obukhov passive scalar value 5/3 (minus intermittency corrections which for this are of the order of 0.15; see ch. 3), although in any case classical (isotropic) turbulence theory would certainly not be expected to apply at theses scales. We could also mention that classically, the atmosphere is “thin” at these scales (since the horizontal resolution ≈ 166 km is much greater than the exponential “scale height” ≈10 km), hence according to the classical isotropic 3D/2D theory one would expect 2D isotropic turbulence to apply. This leads to the predictions b = 3 for the horizontal wind field (a downscale enstrophy cascade) and b = 5/3 (an upscale energy cascade; see ch. 2). In comparison, we see that the actual value for the zonal wind (b = 2.35) is in between the two. In ch. 6 we argue that this is an artefact of using gradually sloping isobars (rather than isoheights) in a strongly anisotropic (stratified) turbulence. These spectra already caution us that in spite of the intentions of their creators, the reanalyses should not be mistaken for real world fields. Indeed, it is only by comparing the reanalysis statistics (especially the scaling exponents) with those from other (e.g. aircraft) sources, that they can be validated through scale by scale statistical intercomparisons.

Fig. 1.5a,b, 1.6a,b,c here

Satellite imagery and reanalyses are the only sources of gridded global scale fields and we have mentioned some of the limitations of each. We therefore now turn our attention to in situ aircraft data; first consider the 12 m resolution data from an experimental campaign over the sea of China (fig. 1.6 a, b). We see that the scaling for both the temperature and horizontal wind is excellent. In both cases, the value b ≈ 1.7 (near the Kolmogorov value 5/3) is reasonable, although in the case of the temperature, we have added reference slopes with b = 1.9 which seems closer to those of the more recent data analyzed in fig. 1.6c over the larger range 560 m to 1140 km. Once again, the scaling is excellent. We have deliberately postponed discussion of the larger scale wind field to ch. 2 and 6 since somewhere between ≈ 30 and 200 km (i.e. a bit beyond

DRAFT

19

391

392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423

424

425426427428429430431432433434

20

5/6/23 Chapter 1 11

the range of fig. 1.6b) it displays what is apparently a spurious break due to the aircraft flying on isobars at nontrivial slopes.

1.2.4 The atmosphere in the verticalIn spite of the fact that gravity acts strongly at all scales, the classical theories of

atmospheric turbulence have all been quasi-isotropic in either two or three dimensions. While a few models tentatively predict possible transitions in the horizontal (for example between k-5/3 and k-3 spectra for a 3D to 2D transition for the wind), in contrast, in the vertical any 3D/2D “dimensional transition” would be even more drastic (Schertzer and Lovejoy, 1985b). This is also true for passive scalars: if a passive scalar variance flux is injected at wavenumbers ki, then we expect E(k) ≈ k+1 if k<ki and k-5/3 for k > ki; see e.g. (Lesieur, 1987). It is therefore significant that wind spectra from relatively low resolution (50 m) radiosondes (fig. 1.7a taken as part of the same experiment as the horizontal data analysed in fig. 1.6a, b) are scaling over nearly the entire troposphere (up to 13.3 km). The slope is near (but a little larger) than that predicted by Bolgiano and Obukhov (11/5), a fact we discuss in ch. 6. In any case, the empirical vertical spectral slope value bv ≈ 2.4 is greater than the horizontal spectral slope value bh ≈ 5/3 (fig. 1.6b). Although it is not obvious, this implies in fact that the atmosphere is more and more stratified at larger and larger scales. Let us mention that the analysis of another radiosonde dataset - (Schertzer and Lovejoy, 1985b), see fig. 5.19a) - convinced the authors that the dimensional transition does not occur along the vertical and hence would not occur along the horizontal either; that this was evidence for the existence of a new type of scaling. This differential stratification can be observed directly by eye in fig. 1.7b, c which are vertical cross-sections of state-of-the-art lidar aerosol backscatter fields with resolutions down to 3 m in the vertical. Starting at the low resolution image (fig. 1.7b), we see that structures are generally highly stratified. However, zooming in closer (fig. 1.7c) we can already make out waves and other vertically (rather than horizontally) oriented structures. In fig. 1.7d we confirm - by direct spectral analysis - that the fields are scaling in both the horizontal and in the vertical directions and that the exponents are indeed different in both directions: the critical exponent ratio (bh-1)/(bv-1) = Hz is quite near the theoretical value 5/9 discussed in ch. 6.

Fig. 1.7 a,b,c,d here

1.2.5 The smallest scales

Conventional turbulence theory has primarily been applied at small scales and there are a great many published spectra showing that they are scaling with various exponents over various ranges. Indeed, classical turbulence theory predicts that viscosity becomes dominant (and breaks the scaling) when the turbulent viscosity from Richardson’s 4/3 law equals the molecular viscosity (equivalently, when the turbulent Reynolds number is unity), i.e. at scales L ≈ (n3/e)1/4 where n ≈ 10-5 m2/s is the kinematic viscosity of air, and e ≈ 10-3 m2/s3 is the typical energy flux to smaller scales (see discussion in ch. 8); this leads to the estimate L ≈ 0.1-1 mm. Also classically, the number of degrees of freedom is roughly the number of these 0.1-1 mm sized cubes contained in the troposphere, therefore a number somewhere around 1027- 1030 (for an

DRAFT

21

435436

437438439440441442443444445446447448449450451452453454455456457458459460461462463464465

466

467

468469470471472473474475476477

22

5/6/23 Chapter 1 12

atmosphere of 104 km of horizontal and 10 km vertical extent.Up until now rather than survey the abundant literature on small scale scaling of

turbulence, we have deliberately concentrated on the far less numerous (and more controversial) large scale analyses showing the little known fact that scaling applies not only at the smallest but also to the largest scales. However, an interesting nonclassical exception to this is the case of rain where the interdrop distances even in fairly heavy rain is of the order of 10 cm and hence much larger than the turbulent dissipation scales. In addition, at large enough scales, rain clearly follows the wind field (except for a superposed mean drop fall speed), so that it is important to determine the scale where the turbulence and rain drops effectively decouple. Unfortunately up until now the study of rain and turbulence have been almost entirely divorced from each other so that it is only very recently with the help of stereo-photography of individual drops, that this question can finally be answered. Fig. 1.8a shows a representative 3D “drop reconstruction” in which the positions and sizes of roughly 20,000 drops in a 2x2x2 m volume were determined (for clarity only the largest 10% are shown). 90% of the drops larger than 0.2 mm in diameter were identified and the positional accuracy is of the order of ±4 cm (depth) and ±2 cm (left to right). The drop liquid water volumes were binned to this accuracy (i.e. on 4 cm cubes) and the 3D isotropic spectrum estimated (using spherical shells in Fourier space, eq. 5 in 3D). The result is shown in fig. 1.8b for five storms (a total of 18 reconstructions from sets of three images (“triplets”) taken a slightly different angles in 3D). We see a clear transition from the white noise spectrum E(k) ≈ k2 at small scales corresponding to the usual “homogeneous” assumption (Poisson drop statistics), to the form E(k) ≈ k-5/3 at scales larger than 30-50 cm (depending somewhat on the rain), which is the spectrum predicted for passive scalars in Corrsin-Obukhov theory. Interestingly, the introduction of cascade for the rainfall (Schertzer and Lovejoy, 1987) was argued on the coupled cascades of dynamics and passive scalar. As detailed in (Lovejoy and Schertzer, 2008) the transition occurs where the mean turbulent Stokes number is of order unity; this is effectively the scale at which the turbulence and drops decouple due to the drop inertia. Down to this homogeneous “patch” scale, rain is a thoroughly turbulent field.

Fig. 1.8a,b here

1.2.6 Temporal scaling, weather, low frequency weather and the climate

If the wind field is scaling in space, then atmospheric fields are likely to be scaling in time – at least up to scales of the order of the lifetime of the largest eddies/structures in the wind field. This is because physically, the wind transports the fields and dimensionally a velocity is all that is needed to convert spatial fluctuations into temporal ones. A typical example of temporal scaling is shown in fig. 1.9 a; the mean of hourly temperature spectra from four years and four stations from the Northwestern US from the US Climatological Reference Network. We can already note two key features: the division of the spectrum into two scaling ranges with transition roughly frequency a ww= (7 days)-1, and the very sharp diurnal (and harmonic) “spikes” roughly three orders of magnitude above an otherwise scaling “background”.

Fig. 1.9 b gives more justification for the existence of a straightforward space-time relation. It is based on two months of hourly 30 km resolution thermal IR data over

DRAFT

23

478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507

508

509

510511512513514515516517518519520521

24

5/6/23 Chapter 1 13

the west Pacific from 30o S to 40o N from the geostationary MTSAT satellite. One can see that if the time scales are converted to space using a fixed speed of ≈900 km/day then the 1-D spatial (zonal, meridional) and temporal spectra are nearly identical. Although at the largest scales (corresponding to ≈ 5,000 km), the spectrum is slightly curved, the curvature for both space and time are virtually identical so that even over the full range, time and space are statistically connected by this constant speed. In addition, much of this small curvature can be explained by the spectral anisotropy and the finite range of wave numbers empirically available. In ch. 8 and 9 we discuss this in more detail and argue that the same data show evidence that atmospheric waves also display emergent scaling laws.

But what about the transition and the behaviour at frequencies <ww? As we discuss in detail in chs. 8 and 10, this transition scale is roughly the lifetime of planetary sized structures and the break is a “dimensional transition”, whose mechanism is fairly obvious. The high frequencies where both spatial and temporal interactions are important are statistically quite different from the lower frequencies where (almost) only temporal interactions are important. In the former case, this means interactions between neighbouring structures of all sizes and at their various stages of development whereas in the latter case, only interactions between very large structures at various stages in their development are important. This break - which is universally observed (see ch. 8 and 10 for many examples) – provides an objective basis for determining the low frequency limit of the weather regime.

Fig. 1.9a, b here

What about the frequencies below wc? It turns out that this lower frequency regime is at least roughly scaling down to the beginning of a third new very low frequency wc ≈ (10yr)-1 -(100yr)-1; see fig. 1.9c for a instrumental/paleotemperature composite spectrum of the “three scaling regime” model, the figure is a modern update of that originally proposed in (Lovejoy and Schertzer, 1986).

Although we are used to the idea that the climate is simply the long term statistics of the weather, this figure shows that this idea is both vague and misleading. To start with, as shown graphically in fig. 1.9 d, the intermediate regime ww< w < wc has statistics which are very close to those predicted by simply extending the weather scale models to low frequencies. This includes the stochastic Fractionally Integrated Flux (FIF) model developed below (ch. 5) which predicts a realistic dimensional transition as well as standard GCM’s when these are run without special anthropogenic, solar, orbital or other “climate forcings”: i.e. in “control runs”. This regime is therefore no more than “low frequency weather”, without any new internal dynamical element, nor any new forcing mechanism. Although the three scaling regime picture seems quite realistic, the transition frequency wc varies from place to place and even from epoch to epoch with the Greenland Holocene series are exceptional in having particularly small values corresponding to very stable (weakly variable) conditions; see ch. 10.

On the contrary, the lowest frequencies w<wc corresponding to multidecadal, multicentennial, multimillennial variability and correspond to our usual ideas about “climate”. At these really long time scales – in addition to various “climate forcings”, all kinds of complex oceanic, cryogenic and other internal mechanisms become

DRAFT

25

522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559560561562563564565566567

26

5/6/23 Chapter 1 14

important – and these may also be expected to be scale invariant: in effect the synergy between nonlinearly interacting parts of the “climate system” result in the emergence of a unique scaling regime; in this case between about 10 – 100 yrs and 100 kyrs (see fig. 1.9 c). At the really low frequencies below this, the spectrum flattens out; this is the pseudo-periodicity of the interglacials. Note that the spectral spikes corresponding to the Milankovitch (orbital) forcing mechanism are at fairly narrow bands near the precessional (≈ (19 kyr)-1, (23 kyr)-1), and obliquity frequencies (41 kyr)-1 and the “wobbling” of the eccentricity at around (100 kyrs)-1 are apparently quite weak: almost all of the variability is due to the scaling “background”, presumably to internal nonlinear (and apparently scaling) variability.

Fig. 1.9c, d here

1.2.7 The scaling of the atmospheric boundary conditionsIn ch. 2 we shall see that the basic equations of the atmosphere are scaling, so

that solutions can potentially also be scaling. However, for this to be true, the boundary conditions must be scaling. We will therefore now take a quick tour of some of these.

One of the lower boundary condition is the topography, it is of its prime importance for surface hydrology and oceanography, and therefore the interactions hydrosphere-atmosphere. The issue of scaling in topography has an even longer history than it does in atmospheric science, going back nearly 100 years to when (Perrin, 1913) eloquently argued that the coast of Brittany was nondifferentiable. Later, (Steinhaus, 1954) expounded on the nonintegrability of the river Vistula, and (Richardson, 1961) quantified both aspects using scaling exponents and (Mandelbrot, 1967) interpreted the exponents in terms of fractal dimensions. Indeed, scaling in the earth’s surface is so prevalent that there are entire scientific specializations such as river hydrology and geomorphology which abound in scaling laws of all types (for a review see (Rodriguez-Iturbe and Rinaldo, 1997), (Tchiguirinskaia et al., 2000) for a comparison of multifractal and fractal analysis of basins), and which virtually require the topography to be scaling.

The first spectrum of the topography was the very low resolution one computed in (Venig-Meinesz, 1951) who already noted that it was nearly a power law with b ≈ 2. After this pioneering work, (Balmino et al., 1973) made similar analyzes on more modern data and confirmed Vening Meinesz’s results. (Bell, 1975) followed, combining various data (including those of abyssal hills) to produce a composite power spectrum that was scaling over approximately 4 orders of magnitude in scale (also with b ≈ 2). More recent spectral studies of bathymetry over scale ranges from 0.1 km to 1000 km can be found in (Berkson and Matthews, 1983) (b ≈ 1.6−1.8), (Fox and Hayes, 1985) (b ≈ 2.5), (Gilbert, 1989) (b ≈ 2.1−2.3) and (Balmino, 1993) (b ≈ 2). Attempts were even made (Sayles and Thomas, 1978) to generalize this to many natural and artificial surfaces: the resulting spectrum exhibited scaling

DRAFT

27

568569570571572573574575576577578579

580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611

28

5/6/23 Chapter 1 15

over 8 orders of magnitude with b ≈2; see however the critique by (Berry and Hannay, 1978) and (Gagnon et al., 2006).

In fig. 1.10a we show a grey scale rendition of the modern ETOPO5 data set which is the earth’s topography (including bathymetry) at 5 minutes of arc, roughly 10 km, and in fig. 1.10b, we show the corresponding spectrum along with those of other higher resolution but regional Digital Elevation Models (DEM’s). These include GTOPO30 (the continental US at ≈1 km) as well as two other DEMS; the US at 90 m resolution and part of Saxony in Germany at 50 cm resolution. Overall, the spectrum follows a scaling form with b ≈ 2.1 down to at least ≈40 m in scale. The remarkable thing about the spectra are that the only obvious breaks are near the small scale end of each data set. In Gagnon et al 2006, it is theoretically shown that starting from the arrows (which are always near the high wavenumber end of the scaling part), that the data are corrupted by an inadequate dynamical range. For example, the DEM at 90 m spatial resolution only had a 1 m altitude resolution implying that huge swathes of the country had nominally zero gradients and hence overly smooth spectra. We shall see later that such power law spectra imply fractal isoheight contours and that the topography itself (the altitude as a function of horizontal position) is multifractal since each isoheight contour has a different fractal dimension.

The ocean surface is particularly important for the exchanges with the atmosphere; fig. 1.11 shows a particularly striking wide range scaling result: a swath over 200 km long at 7 m resolution over the St. Lawrence estuary in 8 different narrow visible wavelength channels the from the airborne MIES sensor. The use of different channels allows one to determine “ocean colour”, which itself can be used as a proxy for phytoplankton concentration. For example the channel 4th and 8th from the top in the figure exhibit nearly perfect scaling over the entire range; these are the channels which are insensitive to the presence of chlorophyll, they give us an indication that over the corresponding range, that ocean turbulence itself is scaling. In comparison, other channels show a break in the neighbourhood of ≈200 m in scale, these are sensitive to phytoplankton. The latter are “active scalars” undergoing both exponential growth phases (“blooms”) as well as being victim to grazing by zooplankton; in (Lovejoy et al., 2000) a turbulence theory is developed to explain the break with a zooplankton grazing mechanism. Other important ocean surface fields that have been found to be scaling over various ranges include the Sea Surface Temperature field (SST). The scaling of ocean currents and SST is discussed at length in section 8.1.4.

Finally, many surface fields are scaling over wide ranges, particularly as revealed by remote sensing. Fig. 1.12 shows 6 MODIS channels at 250 m resolution over Spain (a 512X512 pixel “scene”). The scaling is again excellent except for the single lowest wavenumber which is probably an artifact of the contrast enhancement algorithm which is was applied to each image before analysis. These channels are used to yield vegetation and surface moisture indices by dividing channel pair differences by their means, so that the scaling is evidence that both vegetation and soil moisture is also scaling. In-situ measurements show that not only underground flows are also scaling but that the hydraulic conductivity is extremely variable and strongly anisotropic (Tchiguirinskaia, 2002).

Fig. 1.10 a,b, 1.11, 1.12 hereThe phenomenological fallacy

DRAFT

29

612613614615616617618619620621622623624625626627628629630631632633634635636637638639640641642643644645646647648649650651652653654655656657

30

5/6/23 Chapter 1 16

We have presented a series of striking wide range scaling spectra covering most of the meteorologically significant fields as well as for several important atmospheric boundary conditions. In this “tour” of the scaling we have exclusively used a common statistical analysis technique (the power spectrum). The conclusion that scaling is a fundamental symmetry principle of wide applicability is hard to escape, although it is still greeted with deep skepticism by some. Part of the difficulty probably stems from the feeling that “the real world can’t be so simple”, or “that wide range scaling must imply that the morphologies of all clouds or all landscapes are basically the same at all scales and this is absurd” etc. Such reactions illustrate the “phenomenological fallacy” (Lovejoy and Schertzer, 2007), which arises when phenomenological approaches are only based on morphologies, rather than underlying dynamics. This fallacy has two aspects. First, form and mechanism are confounded so that different morphologies are taken as prima facie evidence for the existence of different dynamical mechanisms. Second, scaling is reduced to its special isotropic “self-similar” special case in which small and large scales are statistically related by an isotropic “zoom”/ “blowup”. In fact, as we explore later in this book - we see that scaling is a much more general symmetry: it suffices for small and large scales to be related in a way that doesn’t introduce a characteristic scale and the relation between scales can involve differential squashing, rotation etc. so that small and large scales can share the same dynamical mechanism yet nevertheless have quite different appearances.

In order to illustrate how morphologies can change with scale when the scaling is anisotropic, consider fig. 1.13. This is a multifractal simulation of a rough surface (with the parameters estimated for the topography), its anisotropy is in fact rather simple in the framework of the generalised scale invariance (GSI) that we will discuss in various chapters. More precisely, it is an example of linear GSI (with a diagonal generator) or “self-affine” scaling. The technical complexity with respect to self-similarity is that the exponents are different in orthogonal directions, which are the eigenspaces of the generator, so that structures are systematically “squashed” (stratified) at larger and larger scales. The underlying epistemological difficulty, which was not so simple to overcome and which still puzzles phenomenologists, corresponds to a deeply change in the underlying symmetries. The top of the figure illustrates the morphology at a “geologists scale” as indicated by the traditional lens cap reference. If this were the only data available, one might invoke a mechanism capable of producing strong left-right striations. However, if one only had the bottom image available (at a scale 64 times larger), then the explanation (even “model”) of this would probably be rather different. In actual fact, we know by construction that there is a unique mechanism responsible for the morphology over the entire range.

Figure 1.14 gives another example of the phenomenological fallacy this time with the help of multifractal simulations of clouds. Again (roughly) the observed cascade parameters were used yet each with a vertical “sphero-scale” (this is the scale where structures have roundish vertical cross-sections) decreasing by factors of 4 corresponding to zooming out at random locations. One can see from the vertical cross-section (bottom row) that the degree of vertical stratification increases from left to right. These passive scalar cloud simulations (liquid water density bottom two rows, single scattering radiative transfer, top row) show that by zooming out (left to right) diverse morphologies appear. Although a phenomenologist might be tempted to introduce more

DRAFT

31

658659660661662663664665666667668669670671672673674675676677678679680681682683684685686687688689690691692693694695696697698699700701702703

32

5/6/23 Chapter 1 17

than one mechanism to explain the morphologies at different scales in the figure we are simply seeing the consequence of single underlying mechanism repeating scale after scale. The phenomenological fallacy can undermine many classical ideas. For example, in Lovejoy and Schertzer (2009) it is argued that the classical two-scale theories of convection are incompatible with the data which are scaling, that the division into qualitatively distinct small and large regimes is unwarranted.

Fig. 1.13, 1.14 here1.4 References:

P. Bak, How nature works, Copernicus, New York (1996).P. Bak, C. Tang and K. Weiessenfeld, Self-Organized Criticality: An explanation of 1/f

noise, Physical Review Letter 59(1987), pp. 381-384.G. Balmino, The spectra of the topography of the Earth, Venus and Mars, Geophys. Res.

Lett. 20(11)(1993), pp. 1063–1066.G. Balmino, K. Lambeck and W. Kaula, A spherical harmonic analysis of the Earth’s

topography, J. Geophys. Res. 78(2)(1973), pp. 478–481.G.K. Batchelor, The theory of homogeneous turbulence, Cambridge University Press

(1953).T.H. Bell, Statistical features of sea floor topography, Deep Sea Research 22(1975), pp.

883-891.J.M. Berkson and J.E. Matthews, Statistical properties of seafloor roughness. In: N.G.

Pace, Editor, Acoustics and the Sea-Bed, Bath University Press, Bath, England (1983), pp. 215–223.

P. Berrisford et al., The ERA interim archive. ERA Report SEries, Shinfield Park, Reading (2009).

M.V. Berry and J.H. Hannay, Topography of random surfaces, Nature 273(1978), p. 573.

R. Buizza, M. Miller and T.N. Palmer, Stochastic representation of model uncertainties in the ECMWF Ensemble Prediction System, Q. J. Roy. Meteor. Soc. 125(1999), pp. 2887–2908.

J.G. Charney, Geostrophic Turbulence, J. Atmos. Sci 28(1971), p. 1087.Y. Chigirinskaya, D. Schertzer, S. Lovejoy, A. Lazarev and A. Ordanovich, Unified

multifractal atmospheric dynamics tested in the tropics Part 1: horizontal scaling and self organized criticality, Nonlinear Processes in Geophysics 1(1994), pp. 105-114.

M.J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19(1978), p. 25.

R. Fjortoft, On the changes in the spectral distribution of kinetic energy in two dimensional, nondivergent flow, Tellus 7(1953), pp. 168-176.

C.G. Fox and D.E. Hayes, Quantitative methods for analyzing the roughness of the seafloor, Rev. Geophys. 23(1985), pp. 1–48.

J. Gagnon, S., S. Lovejoy and D. Schertzer, Multifractal earth topography, Nonlin. Proc. Geophys. 13(2006), pp. 541-570.

L.E. Gilbert, Are Topographic data sets fractal?, Pageoph 131(1989), pp. 241-254.P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50(1983a), pp. 346-349.

DRAFT

33

704705706707708709710711712713714715716717718719720721722723724725726727728729730731732733734735736737738739740741742743744745746747748749

34

5/6/23 Chapter 1 18

P. Grassberger and I. Procaccia, Measuring the strangeness of Strange atractors, Physica 9D(1983b), pp. 189-208.

S. Grossman and S. Thomae, Invariant distributions and stationary correlation functions of one-dimensional discrete processes, Zeitschrift für Naturforschung a 32(1977), pp. 1353-1363.

E. Kalnay, Atmospheric Modelling, Data Assimilation and Predictability, Cambridge University Press, Cambridge (2003).

A.N. Kolmogorov, Local structure of turbulence in an incompressible liquid for very large Reynolds numbers. (English translation: Proc. Roy. Soc. A434, 9-17, 1991), Proc. Acad. Sci. URSS., Geochem. Sect. 30(1941), pp. 299-303.

R.H. Kraichnan, Inertial ranges in two-dimensional turbulence, Physics of Fluids 10(1967), pp. 1417-1423.

L. Landau, C. R. (Dokl.) Acad. Sci. USSR 44(1944), p. 311.A. Lazarev, D. Schertzer, S. Lovejoy and Y. Chigirinskaya, Unified multifractal

atmospheric dynamics tested in the tropics: part II, vertical scaling and Generalized Scale Invariance, Nonlinear Processes in Geophysics 1(1994), pp. 115-123.

M. Lesieur, Turbulence in fluids, Martinus Nijhoff Publishers, Dordrecht (1987).M. Lilley, S. Lovejoy, K. Strawbridge and D. Schertzer, 23/9 dimensional anisotropic

scaling of passive admixtures using lidar aerosol data, Phys. Rev. E 70(2004), pp. 036307-036301-036307.

E.N. Lorenz, Deterministic Nonperiodic flow, J. Atmos. Sci. 20(1963), pp. 130-141.S. Lovejoy et al., Universal Multfractals and Ocean patchiness Phytoplankton, physical

fields and coastal heterogeneity, J. Plankton Res. 23(2000), pp. 117-141.S. Lovejoy and D. Schertzer, Scale invariance in climatological temperatures and the

spectral plateau, Annales Geophysicae 4B(1986), pp. 401-410.S. Lovejoy and D. Schertzer, Stochastic chaos and multifractal geophysics. In: F.M.

Guindani and G. Salvadori, Editors, Chaos, Fractals and models 96, Italian University Press (1998).

S. Lovejoy and D. Schertzer, Scale, scaling and multifractals in geophysics: twenty years on. In: J.E. A.A. Tsonis, Editor, Nonlinear dynamics in geophysics, Elsevier (2007).

S. Lovejoy and D. Schertzer, Turbulence, rain drops and the l **1/2 number density law, New J. of Physics 10(2008), pp. 075017(075032pp), doi:075010.071088/071367-072630/075010/075017/075017.

S. Lovejoy and D. Schertzer, Space-time cascades and the scaling of ECMWF reanalyses: fluxes and fields, J. Geophys. Res. 116(2011).

S. Lovejoy, D. Schertzer, M. Lilley, K.B. Strawbridge and A. Radkevitch, Scaling turbulent atmospheric stratification, Part I: turbulence and waves, Quart. J. Roy. Meteor. Soc.(2008), p. DOI: 10.1002/qj.1201.

S. Lovejoy, D. Schertzer and Y. Tessier, Multifractals and Resolution independent remote sensing algorithms: the example of ocean colour., Inter. J. of Remote Sensing 22(2001), pp. 1191-1234.

S. Lovejoy , A. Tarquis, H. Gaonac'h and D. Schertzer, Single and multiscale remote sensing techniques, multifractals and MODIS derived vegetation and soil moisture, Vadose Zone J. 7(2007), pp. 533-546.

DRAFT

35

750751752753754755756757758759760761762763764765766767768769770771772773774775776777778779780781782783784785786787788789790791792793794795

36

5/6/23 Chapter 1 19

S. Lovejoy, A.F. Tuck and D. Schertzer, The Horizontal cascade structure of atmospheric fields determined from aircraft data, J. Geophys. Res. 115(2010), p. D13105.

S. Lovejoy, A.F. Tuck, D. Schertzer and S.J. Hovde, Reinterpreting aircraft measurements in anisotropic scaling turbulence, Atmos. Chem. and Phys. 9(2009a), pp. 1-19.

S. Lovejoy, B. Watson, Y. Grosdidier and D. Schertzer, Scattering in Thick Multifractal Clouds, Part II: Multiple Scattering, 388(2009b), pp. 3711-3727.

P. Lynch, The emergence of numerical weather prediction: Richardson's Dream, Cambridge University Press, Cambridge (2006) 279 pp.

B.B. Mandelbrot, How long is the coastline of Britain? Statistical self-similarity and fractional dimension, Science 155(1967), pp. 636-638.

B.B. Mandelbrot, Fractals, form, chance and dimension, Freeman, San Francisco (1977).

B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco (1983).J.C. Maxwell, Molecules. In: W.D. Niven, Editor, The scientific papers of James Clerk

Maxwell, Cambridge (1890), p. 373.G. Members, Climate instability during the last interglacial period recorded in the GRIP

ice core, Nature 364(1993), pp. 203-207.A.S. Monin, Weather forecasting as a problem in physics, MIT press, Boston Ma

(1972).A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics, MIT press, Boston MA

(1975).A.M. Obukhov, On the distribution of energy in the spectrum of turbulent flow, Dokl.

Akad. Nauk SSSR 32(1941a), pp. 22-24.A.M. Obukhov, Spectral energy distribution in a turbulent flow, Akad. Nauk SSSR Ser.

Georg. Geofiz. 5(1941b), pp. 453-466.L. Onsager, The distribution of energy in turbulence (abstract only), Phys. Rev.

68(1945), p. 286.T. Palmer, Predicting uncertainty in numerical weather forecasts. In: R.P. Pierce, Editor,

Meteorology at the Millenium, Royal Meteorological Society (2001), pp. 3-13.T.N. Palmer and P. Williams, Editors, Stochastic physics and Climate models,

Cambridge University PRess, Cambridge (2010) 480 pp.J. Perrin, Les Atomes, NRF-Gallimard, Paris (1913).J. Pinel, The space-time cascade structure of the atmosphere. Physics, McGill, Montreal

(2012), p. (in preparation).H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste, vol. 1, Gautier-Villars,

Paris (1892).L.F. Richardson, Weather prediction by numerical process, Cambridge University Press

republished by Dover, 1965 (1922).L.F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph, Proc.

Roy. Soc. A110(1926), pp. 709-737.L.F. Richardson, The problem of contiguity: an appendix of statistics of deadly quarrels,

General Systems Yearbook 6(1961), pp. 139-187.L.F. Richardson and H. Stommel, Note on eddy diffusivity in the sea, J. Met. 5(1948),

pp. 238-240.

DRAFT

37

796797798799800801802803804805806807808809810811812813814815816817818819820821822823824825826827828829830831832833834835836837838839840841

38

5/6/23 Chapter 1 20

C.D. Rodgers, Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation, Rev. of Geophys. 14(1976), pp. 609-624.

I. Rodriguez-Iturbe and A. Rinaldo, Fractal River Basins, Cambridge University Press, Cambridge (1997) 547 pp.

D. Sachs, S. Lovejoy and D. Schertzer, The multifractal scaling of cloud radiances from 1m to 1km, Fractals 10(2002), pp. 253-265.

R.S. Sayles and T.R. Thomas, Surface topography as a nonstationary random process, Nature 271(1978), pp. 431–434.

D. Schertzer and S. Lovejoy, Generalised scale invariance in turbulent phenomena, Physico-Chemical Hydrodynamics Journal 6(1985a), pp. 623-635.

D. Schertzer and S. Lovejoy, The dimension and intermittency of atmospheric dynamics. In: B. Launder, Editor, Turbulent Shear Flow 4, Springer-Verlag (1985b), pp. 7-33.

D. Schertzer and S. Lovejoy, Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling of Multiplicative Processes, Journal of Geophysical Research 92(1987), pp. 9693-9714.

D. Schertzer et al., Which chaos in the rainfall-runoff process?, Hydrol. Sci. 47(2002), pp. 139-148.

H. Steinhaus, Length, Shape and Area, Colloquium Mathematicum III(1954), pp. 1-13.G.I. Taylor, Statistical theory of turbulence, Proc. Roy. Soc. I-IV, A151(1935), pp. 421-

478.I. Tchiguirinskaia, Scale invariance and stratification: the unified multifractal model of

hydraulic conductivity, Fractals 10 (3)(2002), pp. 329-334.I. Tchiguirinskaia, S. Lu, F.J. Molz, T.M. Williams and L. D., Multifractal versus

monofractal analysis of wetland topography, SERRA 14, 1(2000), pp. 8-32.F.A. Venig-Meinesz, A remarkable feature of the Earth's topography, Proc. K. Ned.

Akad. Wet. Ser. B Phys. Sci. 54(1951), pp. 212-228.C.F. von Weizacker, Das spektrum der turbulenz bei grossen Reynolds'schen zahlen, Z.

Phys. 124(1948), p. 614.B.P. Watson, S. Lovejoy, Y. Grosdidier and D. Schertzer, Multiple Scattering in Thick

Multifractal Clouds Part I: Overview and Single Scattering Scattering in Thick Multifractal Clouds, Physica A 388(2009).

H. Whitney, Differentiable Manifolds, Ann. Math. 37(1936), p. 645.

DRAFT

39

842843844845846847848849850851852853854855856857858859860861862863864865866867868869870871872873874875876877

40