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Chapter 4. Weather, macroweather, the climate and beyond

4.1 The weather: nothing but turbulence… and don’t mind the gap

“This afternoon, the sky will start to clear, with cloud shreds, runners and thin bars followed by flocks”. If Jean-Bapiste Lamarck (1744 –1829) had had his way, this might have been an uplifting early morning weather forecast announcing the coming of a sunny day. Unfortunately for poetry, several month’s after Lamarck proposed this first cloud classification, in 1803, the “namer of clouds” Luke Howard (1772 –1864) introduced his own staid Latin nomenclature that is still with us today, including “cumulus”, “stratus”, and “cirrus”a.

For a long time, human scale observation of clouds was the primary source of scientific knowledge of atmospheric morphologies and dynamics. This didn’t change until the appearance of the first weather maps fifty years later based on a meagre collections of ground station measurements. This marked the beginning of the map-based field of “synoptic” (“map scale”) meteorology. Under the leadership of Wilhelm Bjerknes (1862-1951) it spawned the Norwegian school of meteorology that notably focused on sharp gradients, “fronts”. This was the situation when in the mid 1920’s - Richardson proposed his scaling 4/3 diffusion law. The resolution of these “synoptic scale” maps, was so low that features smaller than a thousand kilometers or so could not be discerned. Between these and the human “microscales”, virtually nothing was known. Richardson’s claim that a single scaling law might hold from thousands of kilometers down to millimeters therefore didn’t seem so daring: not only was it compatible with the equations that he had elaborated and that had no characteristic scales, but there were no scalebound paradigms to contradict it.

By the late 40’s and 50’s the development of radar finally opened a window onto the intermediate scales. During the war, the first radars had picked up precipitation as annoying noise that regularly ruined their signals. In 1943, in an attempt to better understand the problem, the Canadian Army Operational Research Group initiated “project stormy weather”. After the war, the team - headed by John Stuart Marshall – set up the “Stormy Weather Group” at McGill, which – thanks to the “Marshal-Palmer relation b” soon

a Luke not only had a more scientific sounding jargon, but was soon given PR in the form of a poem by Goethe; Lamarck’s names didn’t stand a chance.b This is still the name used by meteorologists for the humble exponential distribution of rain drop densities as functions of drop radii. In 1948 Marshall’s graduate student Palmer had use chemically coated blotting paper to relate the size of a drop to the diameter of a “blot”. Marshall and Palmer had used small many pieces of blotting paper placed in the bottom of small jars, in order to establish the relative number of small and large drops, information needed to interpreted the radar backscatter. But they had assumed that they were uniformly distributed in space whereas real turbulence distributed them in a hierarchical (cascade) like manner. Forty years later, using a huge (128x128cm) piece of blotting paper, a student and I recalibrated the blotting paper in the same McGill staircase, only this time showing that the spatial distribution of drops was a scaling fractal set:1 Marshall, J. S. & Palmer, W. M. The distribution of raindrops with size. Journal of Meteorology 5, 165-166 (1948); 2 Lovejoy, S. & Schertzer, D. Fractals, rain drops and resolution dependence of rain measurements. Journal of Applied Meteorology 29, 1167-1170 (1990).

A decade later, tens of thousands of drops in a volume 10m3 were analyzed using stereophotography. This confirmed that the homogeneity assumption is only valid up to about 40- 50 cm, not up to kilometers as is still routinely assumed:3 Desaulniers-Soucy., N., Lovejoy, S. & Schertzer, D. The continuum limit in rain and the HYDROP

experiment,. Atmos. Resear. 59-60,, 163-197 (2001); 4 Lovejoy, S. & Schertzer, D. Turbulence, rain

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established the quantitative basis for interpreting radar precipitation scans; the famous “Z-R” relation (reflectivity- rain rate)c. Beyond this quantification of precipitation, the key advance of the radar was the ability to image the first weather patterns in the range 1- 100 kilometers in size: the discovery of structures and motions in the middle (“meso”) scales between the human micro and the synoptic map scales.

At the same time that this atmospheric window was opened, statistical theories of turbulence - the path pioneered by Richardson – were advancing rapidly. The idea of turbulence theory was to derive high level statistical laws governing the behaviour of strongly nonlinear flows such as those in the atmosphere where the nonlinear terms we typically a thousand billion times larger than the linear onesd. In order to make progress, three important simplifications were made. First, only incompressible fluids were considered. Since gravity only affected flows with density variations, at the very outset, this had the effect of eliminating the main real world source of anisotropy and stratification. Second, boundaries, walls - and for the atmosphere, the earth’s surface and north-south temperature gradients - are also sources of anisotropy, so that the additional assumption of statistical isotropy was made: that the flow itself was on average the same in all directions5. Third, although at any instant in time, the actual turbulent flow would be highly variable from one place to another, it was assumed that on average, the turbulence was everywhere the same: that was statistically homogeneous.

It is important to take a moment to examine the notions of homogeneity and isotropy more closely. In common parlance, soothing that is homogeneous means that it is spatially uniform, that it is the same everywhere, constant. Similarly, something isotropic is the same in all directions, it is spherically symmetric. If the atmosphere was literally - in this deterministic sense - both homogeneous and isotropic, then the wind, temperature, pressure and other atmospheric parameters would have identical values everywhere (and hence also in all directions), such an atmosphere would be totally unrealistic.

The notion of a turbulence that is statistically isotropic and statistically homogeneous is much more subtle than this, it has to do with the same symmetries - translational and rotational invariance – but over a statistical average. A statistical average is neither a spatial nor a temporal average, it is rather an average over a statistical ensemble. To understand an ensemble one must imagine re-enacting (almost) exactly the same experiment a large number of times under identical conditions. For each experiment, the details of the resulting turbulent flow would be different because infinitesimally small differences would be amplified by the strongly nonlinear character of the flow (the “butterfly effect”, ch. ?). A statistical average would then be obtained by averaging the flow over this huge (in principle infinite) ensemble of experiments.

Each member – “realization” - of such a statistically homogeneous and statistically isotropic flow could easily be extremely inhomogeneous in space and could have a strong preferred directione. However, the preferred locations of turbulent hotspots, or the preferred orientations of vortices would be different on each experiment, so that the average over all the experiments would be a constant everywhere and would display no

drops and the l **1/2 number density law . New J. of Physics 10, 075017(075032pp), doi:075010.071088/071367-072630/075010/075017/075017 (2008).c After Marshall retired in 1977, my phD supervisor Geoff Austin succeeded him as leader of the Stormy Weather Group and as director of McGill’s (later baptised) John S. Marshall radar observatory which was then attached to the physics department. When in 1980, I gave my first seminar on fractal models of rain in the McGill meteorology department, Marshall attended as a still somewhat active emeritus professor. dThe ratio is the “Reynolds number”.e Indeed, the breakthrough due to cascades and multifractals was precisely the realization that we should expect extreme variations from one realization of a turbulent process to another.

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preferred direction in space. The problem with empirically testing this idea is that no one ever does an infinite number of identical experiments; and when it comes to the weather and climate, there is only one planet earth (although, in many respects - statistically speaking - Mars comes pretty close, see below!). Often, we have to somehow figure out what typical inhomogeneities and typical anisotropies might be expected single realizations of processes that are statistically homogeneous and isotropic. In this regard, cascasdes and the multifractals that they generated, turned out to be far more wildly variable than anyone had imagined!

While these assumptions may sound academic, they might not be unreasonable approximations to appropriately stirred water in a tank - or even the coffee in your cup. Of course, in practical terms it is impossible to either stir your coffee in exactly the same way thoughout the cup (homogeneously) or to do so in a way that is the same in all directions (isotropically). However, there are reasonable arguments to the effect that if one was far enough from boundaries and at small enough scales, that these anisotropies and inhomogeneities would no longer be feltf, so that the theorists were emboldened to apply these ideas - even if over limited ranges of scales - to the atmosphere. Unfortunately though, gravity acts at all scales so that even if the boundaries and the solar forcing (e.g. the north-south temperature gradient) are not terribly serious sources of anisotropy, the presence of gravity was enough to render the isotropic theories academic.

We have considered statistically constancy in space and in direction, what about time? While a vigorous stirring of your coffee might lead to some approximation of statistical homogeneity and isotropy, if the stirring stopped, then due to friction – viscosity - the motions would all die down and eventually the coffee would be motionless. Therefore, an even simpler situation was usually considered; “quasi-steady” homogeneous and isotropic turbulence in which the fluid was stirred constantly so that the stirring energy would being dissipated as heat at – on average - the same rate at which it was input by the stirringg.

Since large structures (“eddies”) tended to be unstable and to break up into smaller ones, it was enough for the stirring to create large whirls and the turbulence would do the rest: create smaller and smaller structures until eventually dissipation took over. This hierarchical transfer of energy from large to small was what Richardson had referred to in his poem, “the big whirls having little whirls that feed on their velocity”, it was the basic cascade idea. Note that to obtain such a steady state required a constant input of energy so that he overall system was very far from thermodynamic equilibrium. Such a quasi-steady state means on average that everything is the same at all times, it was an approximation to the temporal equivalent of statistical homogeneity: statistical “stationarity”.

The paradigm of “isotropic, homogeneous turbulence” emerged by the end of theh 1930’s. During this time, the Soviet mathematician and physicist Andrei Kolmogorov (1903-1987) was axiomatizing probability theory7 thus laying the mathematical basis for the treatment of random processes. By the end of the 1930’s, Kolmogorov had begun to turn his attention to turbulence. His key breakthrough was the recognition that the key parameter controlling the flow of energy from the large scale stirring to the small scale dissipation was the energy rate density. Using this quantity one immediately obtains the

f For example, the idea of “return to isotropy” was interpreted in this way:6 Rotta, J. C. Statistische theorie nichtonogenr turbulenz. Z. Phys. 129, 547-572 (1951).g I say on average, because typical experiments would be far from smooth with energy dissipation occurring very unevenly in both space and in time. This was the phenomenon of intermittency discussed earlier – the “spottiness” of turbulence, but the full significance of this was not understood until much later. hNotably in the form of the Karman-Howarth equations (1938).

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Kolmogorov law8 that relates the turbulent velocity fluctuations across a structure to its scalei:

(Velocity Fluctuations) = (Energy Rate Density)1/2 x(Scale) 1/3

During the war scientific exchanges were limited and it appears that the Kolmogorov law was independently discovered no less than five times! One of them was at almost exactly the same time – by another Soviet, Obhukhov9 - but in the (equivalent) spectral domain (where it has the form k-5/3 where k is an inverse length). As a consequence, the law is also referred to as the “k-5/3 law” or the “Komolgorov-Obukov” law. The next to publish the law was Onsager10 (1945), and he was the first to explicitly link the law to a cascade of energy flux from large to small scalesj. But Onsager’s American publication was no more than a short abstract; it was no more visible than the earlier Soviet papers. This led the physicists Heisenberg12 (1948) and von Weizacker13 (1948) to their own rediscoveries. The pattern of independent Soviet and nearly concurrent western discoveries continued with the discovery of the closely analogous turbulent laws of turbulent mixing, the (also scaling) “Corrsin14-Obuhov15 law”:

(Temperature fluctuations)= (Turbulent fluxes) x(Scale) 1/3

and again (1959) with the Obhukov16 – Bolgiano17 law for buoyancy driven turbulence that we discussed in ch. 2:

(Velocity fluctuations)= (Turbulent fluxes) x(Scale) 3/5

By 1953, the theory of isotropic homogeneous turbulence had evolved to the point that it was already the subject of a landmark book18 by George Batchelor (1920-2000). But, the role of isotropy had been subtly changed. Whereas it had originally been introduced as a way of simplifying theoretical statistical treatments of turbulence, it had now taken on a life of its own. While the main application of the theory was to the atmosphere, Kolmogorov noted that the rather stringent “inertial rangek” assumptions that he had used to derive it – including the neglect of gravitational forces - would only be valid up to scales of several hundred meters, a conclusion amplified by Batchelor who speculated that the range was might only be between 100 m down to 0.2 cm.

While Richardson had been blissfully ignorant of isotropic theory and had dared to propose that his scaling law would hold over the whole range of atmospheric scales, now - the nearly equivalent - Kolmogorov law was claimed to be limited to a tiny range. But the proposed limitation was not due to any evidence nor from the discovery of any scale breaking mechanism. Rather, the limitation to small scales and the implied scale break was hypothesized because of the breakdown in the isotropy condition – atmospheric stratification. Isotropic theory had taken such a life of its own that it had already been forgotten that it had only been introduced on the grounds of theoretical expediency!

iTled Richardson’s 4/3 law of turbulent diffusion that Richardson had proposed largely on empirical grounds. j Many years later, in an important update on his law (taking intermittency into account) Kolmogorov explained that during the period 1939-1941 both he and Obukov had been inspired by Richardson’s cascades:11 Kolmogorov, A. N. A refinement of previous hypotheses concerning the local structure of turbulence in viscous incompressible fluid at high Raynolds number. Journal of Fluid Mechanics 83, 349 (1962).k So called because the law was valid only when the inertial terms in the equations dominated the dissipation/friction terms.

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The final important early turbulence development was the discovery by Fjorthoft (1953) that completely flat, two dimensional turbulence was fundamentally different from three dimensional isotropic turbulencel. Although Fjorthoft was cautious in interpreting his results in terms of real atmospheric flows, the seed had been planted for the isotropic 2D-3D model that followed a decade or so later.

By the mid 1950’s, thanks to radar, the meso-scale had come to the fore. Established empirically based synoptic scale meteorology had already relegated the “microscales” to mere turbulence, but this had been done mostly for pragmatic reasons. Similarly, the new mesoscale was pragmatically viewed as the connection between the two, while simultaneously promising a better understanding of thunderstorms and other previously inaccessible meteorological phenomena. But the emerging synoptic, meso and micro scale regimes were not theoretically ordained, they were essentially practical distinctions awaiting theoretical clarification. Yet, the theorists, were loath to drop their isotropy assumptions, and were happy to find convenient justifications for dividing up the range of scales into small scale isotropic three dimensional turbulence and something stratified - albeit not yet clearly discerned - at the larger scales. It was already tempting to knit all this together and to identify the microscales with 3D isotropic turbulence and the weather with the larger stratified scales.

This was the situation when Panofsky and Van der Hoven began their famous measurements of the spectrum of the wind that they published in a series of papers between 1955 and 195720,21. At this point, wind data at sub-second scales for durations of minutes had already confirmed the Kolmogorov law, but data were lacking at the longer time scales. Given the lack of computers, the researchers averaged their data at 10 s intervals using “eye averages” and collected data in this way for an hour or so. The spectrum was then laboriously calculated by hand. Finally, knowing the average wind speed allowed the scientists to make a rough conversion from time to space. For example if the average wind over a minute was 10 m/s, then the variability at one minute was interpreted as information about the variability at spatial scales of 600m. To investigate the key mesoscale between 1 and 100 km, data were needed spanning periods of minutes to several hours.

The new element was the use of lower resolution series that could be eye averaged at 5 minute resolutions and that lasted several days. When this analysis – was plotted on the same graph as a one minute spectrum - taken from a completely different experiment under different conditions – Panofsky and van der Hoven discovered that there was a dearth of variability precisely in the range of about 1- 100km, centered on time scales of 1 hour, corresponding to 10 km. In the authors’ words: “The spectral gap suggests a rather convenient separation of mean and turbulent flow in the atmosphere: flow averaged over periods of about an hour… is to be regarded as ' mean ' motion, deviations from such a mean as ' turbulence’.” The meso-scale gap was born.

However, the first (1955) paper was based on a single location and on only two experiments and their figures were not very convincing. This led van der Hoven to perform another series of four experiments that produced what later became the iconic mesoscale gap spectrum (fig. 4.1). The gap cleanly and “conveniently” separated the synoptic weather scales from the small scale turbulence. With the development of the first computer weather models whose resolutions – even today – don’t include the microscales, this gap became even more seductive. At first it justified ignoring these scales, later, it justified

l This was not quite the discovery of Kraichnan’s law of two dimensional turbulence:19 Kraichnan, R. H. Inertial ranges in two-dimensional turbulence. Physics of Fluids 10, 1417-1423 (1967).

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“parameterising” them. The gap idea was so popular that van der Hoven’s spectrum was reworked and republished many times notably in meteorological textbooks, even through the 1970’s. Soon, the actual data points were replaced with smooth artists’ impressions (see e.g. fig 4.1) thus inadvertently hiding the fact that his spectrum was actually a composite taken under four different sets of conditions; even today, his paper is still frequently and approvingly cited.

Fig. 4.1

Yet within ten years, the mesoscale gap was strongly criticized22,23,24 with critics pointing out that it was essentially based on a single high frequency bulge (fig. 4.1, near scales of 100 s) due an experiment data taken “under near hurricane” conditions. By the end of the 1970’s, satellites were routinely imaging interesting mesoscale featuresm. Practionners of the nascent field of mesoscale meteorology such as Atkinson25 were highly sceptical of any supposedly barren “gap” that would have reduced their entire field to a mere meteorological footnote.

m Due to the curvature of the earth ground based weather radars start looking above the weather at distances of around 100 km; satellites gave a much more satisfactory range of scales.

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For many however, the gap was too convenient to kill. Indeed, the development of two dimensional isotropic turbulence by Robert Kraichnan in 196726, 19, and especially its extension to quasi-geostrophic turbulencen by Jules Charney (1971)27, one of the most respected meteorologists of his generation, revived the mesoscale albeit in a slightly different form. While modern data had filled in the gap with lots of structures and variability, these isotropic turbulence theories supported a new interpretation of the mesoscale as a regime transitional between isotropic 3D and isotropic 2D (quasi-geostrophic) turbulence, supposedly near the atmospheric thickness, of about 10 km.

Fig. 4.2: The spectrum of temperature (T), humidity (h) and log potential temperature (log) averaged over 24 legs of aircraft flight over the Pacific Ocean at 200 mb. Each leg had a resolution of 280m and had 4000 points (1120 km). A reference line corresponding to k-2 spectrum is shown in red. The meso-scale (1 – 100 km) is shown between the dashed blue lines.

While in some quarters, the gap lived on, the development of two dimensional turbulence changed the focus from the gap to finding evidence for large scale isotropic 2D turbulence. Such a discovery could potentially transform the meso-scale from a barren gap into the site of a 3D-2D “dimensional transition”28. By 1967, when Robert Kraichan (1928-2008) published his paper on 2D isotropic turbulence, the idea was so seductive that empirical claims of 2D turbulence started to appear almost immediately. Whereas 3D isotropic turbulence followed the k-5/3 law, 2D turbulence was expected to have a k-3

regimeo, so that anything resembling a k-3 regime was considered to be a “smoking gun” for the purported 2D behaviour. Even tiny ranges with only two or three data points that were vaguely aligned with the right slope were soon interpreted as confirmation of the theoryp.

n This was effectively a derivation of Kraichnan’s pure two dimensional turbulence starting from a series of nontrivial approximations to the governing fluid equations.o It also generally had a k-5/3 regime - only this was at very large scales - a fact that was often conveniently forgotten.p Some authors admitted to “eyeballing” their spectra over a factor of 2 in scales to back up such claims:29 Julian, P. R., Washington, W., M., Hembree, L. & Ridley, C. On the spectral distribuiton of large-scale atmospheric energy. J. Atmos. Sci., 376-387 (1970).

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The first major experiment devoted to testing this 2D/3D model was the EOLE experiment (1974). It used the dispersion of 480 constant density balloons30 (at about 12km altitudeq dispersed over the southern hemisphere; this is was the same as some of Richardson’s methods used to obtain fig. 1.?; it was effectively an updated versionr. Since Kolmogorov’s 5/3 law was essentially the same a Richardson’s 4/3 law, the dispersion of the balloons was an indirect test. But contrary to Richardson, the original (1974) analysis of the EOLE analysis by Morel and Larchevesque30 concluded that the turbulence in the 100 km -1000 km range did not follow his law, but rather those predicted by two dimensional turbulence.

But the matter didn’t rest there. Even in the mid 1970’s, internal discrepancies in the EOLE analysis had been noteds. More importantly, the EOLE conclusions soon contradicted those of the GASP (1983) and later MOZAIC (1999) analyses that found Kolmogorov turbulence out to hundreds of kilometres. Two decades later, the original (and still unique) EOLE data set was reanalyzed by Lacorte et al31 and it was concluded that the original EOLE conclusions were not founded and that on the contrary the data vindicated Richardson over the range of about 200 - 2000 km range.

But the saga was still not over. Strangely, in spite of supporting Richardson at the larger scales, the Lacorte et al reanalysis contradicted him at the smallest EOLE scales from 200 km down to the smallest available (50 km), over which they claimed to have validated the original 2D turbulence interpretation! A decade later, this conclusion prompted a re-revisit that found an error in this smaller scale analysis thus eliminating any evidence for two dimensional turbulence up to the largest scale covered by EOLE, 1000 km, thus (finally!) vindicating Richardsont.

***Over the 1980’s and 1990’s evidence for wide range scaling in other atmospheric

fields slowly accumulated, notably by the study of radar rain reflectivities39,41, and satellite cloud radiances40-45 (see e.g. fig. 4.2). But these analyses were generally only up to several hundred kilometers in scale and crucially, they didn’t involve the wind field that could not be reliably sensed by remote means. For the wind field, the only alternative to aircraft data was the analysis of the outputs of numerical models, but at the time, these didn’t have a wide enough range of scales to be able to settle the issue eitheru.

The 1990’s were a transitional period for science. Up until the end of the 1980’s, there had been a general recognition that high quality fundamental research was necessary for high quality applied research. Only a handful of giant corporations had significant

q More precisely, close to the 200 mb pressure level.r The balloons stayed (nearly) on isopycnals (i.e. surfaces of constant density), not on isobars (surfaces of constant pressure), the key difference being that while the latter are gradually sloping, the former are highly variable with large scale average slopes diminishing at larger and larger scales.s Between the relative diffusivity and the velocity structure function results.t See appendix 2A of:32 Lovejoy, S. & Schertzer, D. The Weather and Climate: Emergent Laws and Multifractal Cascades. (Cambridge University Press, 2013).u A partial exception was a paper by Strauss and Ditlevsen in 1999 that analyzed reanalyses. A reanalysis is a complex data – model hybrid that effectively fills in holes in the data (and partially corrects for errors), by constraining the system using the equations of the atmosphere as embodied in a numerical model. Although these authors strongly criticized the reigning 2D picture, rather than simply analyzing the scaling directly, they instead analyzed complex theoretically inspired constructs. 46 Strauss, D. M. & Ditlevsen, P. Two-dimensional turbulence properties of the ECMWF reanalyses. Tellus 51A, 749-772 (1999).

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fundamental research programs; most research and development relied on academic or other publically funded institutions for their fundamental side (in the US, especially the military). But this slowly changed, some corporations such as Bell labs - sold off – their research departmentsv, others lobbied governments to allow them greater control over priorities. In the past, scientists in the fundamental sector had generally been given free reign to investigate the areas of greatest scientific significance. Now, funding became increasingly restricted to areas of short term corporate significance. It was at this time of tighter budgets and increasing restrictions accompanied by the new administrative mantra of “excellence”. This was surely reassuring to anyone worried about the impacts of cutbacks and corporate oversight.

Concomitant with the new focus on industrial needs, was a disinterest in the fundamental issues that the nonlinear revolution had promised to resolve. In Canada for example, the staff of the federal weather branch, the Atmospheric Environment Service was cut by 50% during the 1990’s and this included the elimination of their small fund to support academic research. The research resources were tightly focused on the development of numerical weather models (Global Circulation Models, GCMs); increasingly the only justification for funds in atmospheric science was the improvement of GCMs, including improved data inputs. Whereas in the past, it had been possible to obtain funding for an applied project and then to siphon off some of the funds for related, more fundamental ones, sponsors were now requiring detailed accountability. Every dollar had to be spent exactly as preapproved and university accounting departments administering the grants were increasingly transformed into policing agencies. Their main job was no longer to help scientists, but to protect the sponsors by ensuring that research dollars were spent according to elaborate rules.

This was the beginning of a golden age for atmospheric science; fundamental issues for GCMs were being resolved and by the end of the 1990’s advances in data assimilation opened doors to widespread “ingestion” of satellite and other sources of remotely sensed data, and thus to significant improvements in weather forecasts (helped of course by rapidly increasing computer sizes and speeds and of algorithmic efficiency). In the 1970’s, a popular geoscience adage was “No one believes a theory except the person who invented it. Everyone believes the data except the person who took them”. This expressed the divorce between the idealized theoretical atmospheric models and messy real world data; in my view, the fact that the theories made many conventional smoothness and regularity assumptions that were totally unrealistic and that the scaling approach would overcome. But the ambient cynicism about atmospheric theories was hitting the nonlinear revolution and turbulence theorists alike. Funding was scarce – and interest scant, even for the theoretically dominant 2D/3D paradigm. Theory was increasingly seen as being unnecessary, irrelevant: virtually any and all atmospheric questions were answered with the help of the numerical models. The problem was the models themselves were massive constructs built by teams spanning generations of scientists. These were increasingly seen as big “black boxes” that did not deliver understanding. Overall, the field was being gradually transformed into a purely applied science, with new areas – such as the climate – being driven by applications and technology: climate change and computers.

***

v In 1996, the parent company, AT&T sold Bell labs to Lucent Technologies, 1996, in 2006, Lucent was acquired by Alcatel and in 2015, by Nokia.

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Even if one believed the original interpretation of the EOLE experiment in terms of 2D turbulence, by its nature, it measured the dispersion of balloons, the not wind speeds that were needed for a direct test of the 2D-3D model. It was therefore only with the first large scale aircraft campaigns in the 1980’s that the theory was seriously tested - and this turned out to be the beginning of another multidecadal saga. The first and still most famous experiment was GASPw whose data were analyzed by Nastrom and Gage 198333. On this basis, it was claimed that the empirical spectrum did indeed show a transition from the Kolmogorov 3D isotropic turbulence, to two dimensional isotropic turbulencex.

The most glaring problem with the GASP results was that the apparent 2D-3D transition scale was typically at several hundred kilometers34,35,36,37. At such distances, 3D isotropic turbulence would imply that clouds and other structures extended well into outer space! Simply calling the phenomenon “squeezed 3D isotropic turbulence”38, or “escaped” 3D turbulence34 explained nothing.

were generally loath to In addition, the crucial vertical structure started to be clarified, first with the help of

direct vertical sections from lidar (fig.. 1.7) and later CloudSat (fig. 3.?), but also by dropsondes. …

Explain dropsondes… show fractal stable layers… lack of K law in vertical!!

The picture started to change in the late 2000’s with the beginning of widespread availability of global scale atmospheric data sets, notably massive satellite data sets and these invariably showed excellent wide range scaling, although again, not of the hard to measure wind field (see fig 4.2). Also at this time, the numerical models were getting big enough to analyse, and again, wide range scaling was found (refs, fig. 4.2 for reanalyses). By 2008, it seemed that the only empirical evidence that the wide range scaling hypothesis could not explain, was the spectra of aircraft wind measurements.

In order to get to the bottom of this, with the help of Daniel Schertzer and Adrian Tucky, we reanalysed47 the published wind spectra and showed that a key point had been overlooked: the spectra did not transition between k-5/3 and k-3 but rather between k-5/3 and k-2.4. In the 2009 paper “Reinterpreting aircraft measurements in anisotropic scaling turbulence47” we proposed a simple explanation: that the low wavenumber (k-2.4) part of the spectrum was not simply a poorly discerned k-3 signature of isotropic two dimensional turbulence, but rather the spectrum of the wind in the vertical rather than the horizontal direction! If the turbulence was never isotropic but rather anisotropic with different exponents in the horizontal and vertical directions, then everything could be easily explained by aircraft following gently sloping isobars (rather than isoheights), see fig. ? (aircraft trajectories).

w GASP= Global Atmospheric Sampling Program.x Interestingly, Nastrom and Gage themselves interpreted their results as support for yet another theory based on waves.y Tuck was real pioneer in aircraft measurements. At the time, he was head of the atmospheric chemistry group at NOAA, boulder. He had been responsible for the Antarctic aircraft campaign in the late 1980’s that conclusively established the existence of the ozone hole and the link to CFC’s.

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But even this didn’t satisfy the die-hard 2D-3D theorists, notably Eric Lindborg. He incited experimentalist colleagues at NCAR (Frehlic and Sharman ref.) to use the big AMDAR aircraft data base to

In order to prove this interpretation to be correct and to end the debate (notably with

48, and 49), it proved necessary to determine the joint (horizontal- vertical) velocity structure function (here, the root mean square wind differences over lags in vertical sections). This was finally done with the help of wind data from 14,500 aircraft trajectories that allowed its first direct determination. The technical difficulty was to distinguish isoheights from isobars and this required accurate (GPS) altitude measurements that had not been previously available. The horizontal wind turned out to be scaling and anisotropic with in-between “elliptical dimension” 2.56±0.02, close to the theoretical value 23/9 50.

At the same time as the empirical situation was being clarified, the theoretical debate between the 3D-2D isotropic model or the anisotropic scaling alternative (the 23/9D model, notably with 51) was also being resolved. In order to theoretically justify wide range scaling, appeal must be made to the laws governing atmospheric dynamics. Since the 1950’s it has been known that the simpler equations of incompressible hydrodynamics (where gravity is irrelevant), are invariant under isotropic scale changes 52, 53, symbolically:

Incompressible fluid laws-> Isotropic blowup (factor -> H(Incompressible fluid laws)

This isotropic invariance of the laws is the theoretical basis of the 3D and 2D turbulence theories that we mentioned above and the scaling symmetries of the fluid equations are at the root of the many “similarity” laws of classical fluids mechanics. But what about the stratification induced by gravity? While Charney was able to extend Kraichnan’s 2-D turbulence to include some limited vertical structure - “quasi-geostrophic turbulence” – the latter predicted a horizontal wind spectrum of k-3 in both the horizontal and vertical directions, a possibility that was excluded by a large experiment involving hundreds of accurate dropsondes 54.

6.1.1 The isotropy assumption, historical overview

We have argued that atmospheric scaling holds over a wide range in the horizontal,

this is clearly not possible if the turbulence is isotropic since it would imply the

existence of roundish structures hundreds or even thousands of kilometres thick.

Such wide range scaling is only possible because of the stratification. Following a

brief historical overview, in this chapter we discuss the nature of the stratification.

Initially motivated by mathematical convenience, starting with 5, the paradigm of

isotropic and scaling turbulence was developed initially for laboratory applications,

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but following 8, three dimensional isotropic turbulence was progressively applied to

the atmosphere. However, there are several features (including gravity, the Coriolis

force and stratification) that bring into question the simultaneous relevance of both

isotropy and scaling. In particular since the atmosphere is strongly stratified, a

model with a single wide range of scaling which is both isotropic and scaling is not

possible so that theorists had to immediately choose between the two symmetries:

isotropy or scale invariance. Following the development of models of two

dimensional isotropic turbulence (26 and 19 but especially quasi geostrophic

turbulence 27 which can be seen as quasi-2D turbulence (see section 2.6), the

mainstream choice was to first make the convenient assumption of isotropy and to

drop wide range scale invariance; this could be called the “isotropy primary”

paradigm. In ch. 2 we saw how - starting at the end of the 1970’s, this has lead to a

series of increasingly complex isotropic 2D/isotropic 3D models of atmospheric

dynamics and we noted that justifications for these approaches have focused almost

exclusively on the horizontal statistics of the horizontal wind in both numerical

models and analyses and from aircraft campaigns, especially the highly cited GASP

33, 55,56 and MOZAIC 57 experiments. Since understanding the anisotropy clearly

requires comparisons between horizontal and vertical statistics/structures it is not

surprising that this focus has had deleterious consequences.

Over the same thirty year period that 2D/3D isotropic models were being

elaborated, evidence slowly accumulated in favour of the opposite theoretical

choice: to drop the isotropy assumption but to retain wide range scaling. This

change of paradigm from isotropy primary to scaling primary was explicitly

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proposed by 28, and 39 who considered strongly anisotropic scaling so that vertical

sections of structures become increasingly stratified at larger and larger scales

albeit in a power law manner. Similarly, although not turbulent, anisotropic wave

spectra were proposed in the ocean by 58 and an anisotropic buoyancy driven wave

spectrum was proposed in the atmosphere by (59. Related anisotropic wave

approaches may be found in 60,61, 62, 63, 64 and 65. These authors used anisotropic

scaling models not so much for theoretical reasons, but rather because the data

could not be explained without it. In addition, many experiments found non-

standard vertical scaling exponents thus implicitly supporting this position, see the

review (and many additional references) in 66. Below we shall see that state-of-the-

art lidar vertical sections of passive scalars or satellite vertical radar sections of

clouds (section 6.5) give direct evidence for the corresponding scaling (power law)

stratification of structures. These analyses show that the standard bearer for

isotropic models - 3D isotropic Kolmogorov turbulence - apparently doesn’t exist in

the atmosphere at any scale - at least down to 5 m in scale - or at any altitude level

within the troposphere 67, section 6.1.5. In ch. 1 and 4 we used large quantities of

high quality satellite data to directly demonstrate the wide range horizontal scaling

of the atmospheric forcing (long and short wave radiances) and showed that

reanalyses and atmospheric models display nearly perfect scaling cascade

structures over the entire available horizontal ranges. This shows also that the

source/sink free “inertial ranges” used in the classical models are at best academic

idealizations and at worst unphysical: atmospheric dynamics has multiple energy

sources that do not prevent it from being scaling.

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At a theoretical level, quasi-geostrophic and similar systems of equations purporting

to approximate the synoptic scale structure of the atmosphere are justified by

“scale” analysis of various terms in the dynamical equations in which “typical” large

scale values of atmospheric variables (usually their fluctuations) are assumed and

terms eliminated if they are deemed too small. However, in section 2.3.2 we showed

that by replacing such scale analyses – valid at most over narrow ranges of scales –

by scaling analysis, focusing on horizontal and vertical exponents valid over wide

ranges, that the equations were symmetric with respect to anisotropic scale

changes. However, just as the classical isotropic scaling analysis of the Navier-

Stokes equations is insufficient to determine the Kolmogorov exponent 1/3 – for

this we also need dimensional analysis on the scale by scale conserved energy flux –

so here, the value of the new vertical exponents depend on a new turbulent flux that

is the subject of this section.

6.1.2 Empirical status of the isotropy assumption

Kolmorogov’s theory of isotropic 3D turbulence is based on the key assumptions:

that there exists an “inertial” range where the turbulence is isotropic and depends

only on the energy flux across scales (or wave number in Fourier space) yielding

the k-5/3 regime for the energy spectrum. In the atmosphere, the main reason for

supposing the existence of any isotropic ranges is to do with anisotropic boundary

conditions, i.e. it ignores gravity which acts at all scales throughout the flow.

Specifically, the unimportance of boundary conditions is consequence of the fact

that structures at a given scale are mostly coupled with structures at neighbouring

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scales, so that the effects of large scale boundary conditions are progressively

“forgotten” at small scales. Classically this tendency to “return to isotropy” 6 has

been modelled using second order statistical closure techniques; however even

within this framework, as soon as buoyancy forces are included, their effect is found

to be relatively large 68, just as in laboratory flows it is found that even small

buoyancy forces readily destroy isotropy 69. Even recent theoretical developments

70,71 assume a priori that fluctuation statistics for points separated by r follow the

form:

11\* MERGEFORMAT ()

where is the angle of r with the vertical, the length of the separation vector

. To select only functions of this form is to implicitly assume that the angular

variation in the anisotropy is the same at each scale, in the terminology developed

below, this mild anisotropy could be called “trivial”. The main originality of Arad et

al., 1998 was the proposal that there exists a hierarchy of such isotropic terms each

with different H’s and ’s. Since the theory ignores buoyancy, when it was checked

in the atmosphere, the data were restricted to the horizontal 72! Indeed, virtually all

empirical surface layer atmospheric tests of isotropy (i.e. those with the best quality

data) simply assume eq. 1 with H = 1/3 and test the anisotropy at unique scales. It is

even common to study the spatial anisotropy of scalars without using any spatial

data whatsoever! For example, a common technique involves using time series of

temporal gradients and converting time to space with “Taylor’s hypothesis” of

frozen turbulence. Finally the “anisotropy” is estimated by using a third order

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statistical moment (the skewness) to determine the forward/backward trivial

anisotropy 73! Even in the analysis of laboratory (Rayleigh-Bénard) convection

where there is a debate about whether H = 1/3 or 3/5 (respectively energy and

buoyancy variance flux dominance, see below), isotropy is still assumed and one still

uses data from time series at single points 74,75.

6.1.3 The Obukhov and Bolgiano buoyancy subrange 3D

isotropic turbulence

Kolmogorov theory was mostly used to understand laboratory hydrodynamic

turbulence which is mechanically driven and can be made approximately isotropic

(unstratified) by the use of either passive or active grids. In this case, we have seen

that fluctuations in v for points separated by r can be determined essentially via

dimensional analysis using (ch. 2); the latter choice being justified since it is a

scale by scale conserved turbulent flux. The atmosphere however is fundamentally

driven by solar energy fluxes which create buoyancy inhomogeneities; in addition to

energy fluxes, buoyancy is thus also fundamental. In order to understand

atmospheric dynamics we must therefore determine which additional dimensional

quantities are introduced by gravity/ buoyancy. As discussed in 76, this is necessary

for a more complete dimensional analysis.

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Figure 6.11: The stability of the atmosphere as determined by a drop sonde using

the stability criterion Ri>1/4 where the Richardson number (Ri) is estimated using

increasingly thick layers: 5, 20, 80, 320 m thick (black, red, blue, cyan respectively). The

figure shows atmospheric columns, the left one from the ocean to 11520 m (just below

the aircraft), the right is a blow up from 8000-9000 m. The left of each column indicates

dynamically unstable conditions (Ri<1/4) whereas the right hand side, indicates

dynamically stable conditions (Ri>1/4). The figure reveals a Cantor set-like (fractal)

structure of unstable regions. Reproduced from 77.

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Reproduced from L+S2013 fig. 8.15b. form IR satellites over the Pacific

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Fig. 1a: Top row: The zonal spectra of Earth (top left) and Mars (top right) as functions of the nondimensional wave numbers for the pressure (p, purple), meridional wind (v, green), zonal wind (u, blue), and temperature (T, red) lines. The data for Earth were taken at 69% atmospheric pressure for 2006 between latitudes ±45∘. The data for Mars were taken at 83% atmospheric pressure for Martian Year 24 to 26 between latitudes ±45∘. The reference lines (top left, Earth) have absolute slopes, from top to bottom: 3.00, 2.40, 2.40, and 2.75 (for p, v, u, and T, respectively). Top right (Mars) have reference lines with absolute slopes, from top to bottom: 3.00, 2.05, 2.35, and 2.35 (for p, v, u, and T, respectively). The spectra have been rescaled to add a vertical offset for clarity and wavenumber k = 1 corresponds to the half circumference of the respective planets.

Bottom row: The same as top row except for the meridional spectra of Earth (left) and Mars (right). The reference lines (left, Earth) have absolute slopes, from top to bottom: 3.00, 2.75, 2.75, and 2.40 (for p, v, u, and T, respectively). The reference lines (right, Mars) have absolute slopes, from top to bottom: 3.00, 2.40, 2.80, and 2.80 (for p, v, u, and T, respectively). The spectra have been rescaled to add a vertical offset for clarity. Adapted from 78.

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Fig. 1b: The three known weather - macroweather transitions: air over the Earth (black and upper purple), the Sea Surface Temperature (SST, ocean) at 5 o resolution (lower blue) and air over Mars (Green and orange). The air over earth curve is from 30 years of daily data from a French station (Macon, black) and from air temps for last 100 years (5ox5o resolution NOAA NCDC), the spectrum of monthly averaged SST is from the same data base (blue, bottom). The Mars spectra are from Viking lander data (orange) as well as MACDA Mars reanalysis data (Green) based on thermal infrared retrievals from the Thermal Emission Spectrometer (TES) for the Mars Global Surveyor satellite. The strong green and orange “spikes” at the right are the Martian diurnal cycle and its harmonics. Adapted from 79.

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Fig. 2: The weather-macroweather transition scale w estimated directly from break points in the spectra for the temperature (red) and precipitation (green) as a function of latitude with the longitudinal variations determining the dashed one standard deviation limits. The data are from the 138 year long Twentieth Century reanalyses (20CR, 80), the τw estimates were made by performing bilinear log-log regressions on spectra from 180 day long segments averaged over 280 segments per grid point. The blue curve is the theoretical w obtained by estimating the distribution of from the ECMWF reanalyses for the year 2006 (using w =-1/3L2/3 where L = half earth circumference), it agrees very well with the temperature w. w is particularly high near the equator since the winds tend to be lower, hence lower . Similarly, w is particularly low for precipitation since it is usually associated with high turbulence (high ). Reproduced from 32.

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Fig. 3: The zonal, meridional and temporal spectra of 1386 images (~ two months of data, September and October 2007) of radiances fields measured by a thermal infrared channel (10.3-11.3 μm) on the geostationary satellite MTSAT over south-west Pacific at resolutions 30 km and 1 hr over latitudes 40°S – 30°N and longitudes 80°E – 200°E. With the exception of the (small) diurnal peak (and harmonics), the rescaled spectra are nearly identical and are also nearly perfectly scaling (the black line shows exact power law scaling after taking into account the finite image geometry. Reproduced from 81.

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Fig. 5c: Haar fluctuation analysis of globally, annually averaged outputs of past Millenium simulations over the pre-industrial period (1500-1900) using the NASA GISS E2R model with various forcing reconstructions. Also shown (thick, black) are the fluctuations of the pre-industrial multiproxies showing that they have stronger multi centennial variability. Finally, (bottom, thin black), are the results of the control run (no forcings), showing that macroweather (slope<0) continues to millennial scales. Reproduced from 82.

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Fig. 5d: Haar fluctuation analysis of Climate Research Unit (CRU, HadCRUtemp3 temperature fluctuations), and globally, annually averaged outputs of past Millenium simulations over the same period (1880-2008) using the NASA GISS E2R model with various forcing reconstructions (dashed). Also shown are the fluctuations of the pre-industrial multiproxies showing the much smaller centennial and millennial scale variability that holds in the pre-industrial epoch. Reproduced from 82.

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Fig. 6: Variation of τw (bottom) and τc (top) as a function of latitude as estimated from the 138 year long 20CR reanalyses, 700mb temperature field (the c estimates are only valid in the anthropocene). The bottom red and thick blue curves for w are from fig. 2; also shown at the bottom is the effective external scale (eff) of the temperature cascade estimated from the European Centre for Medium-Range Weather Forecasts interim reanalysis for 2006 (thin blue). The top τc curves were estimated by bilinear log-log fits on the Haar structure functions applied to the same 20CR temperature data. The macroweather regime is the regime between the top and bottom curves.

.

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Fig. 1a: The trajectory (green) of a scientific aircraft (Gulfstream 4) following a constant pressure level (to within 0.1%), data at 1 s resolution (≈280 m). The red shows the variation in the longitudinal component of the horizontal velocity (in m/s, deviations from 24.5 m/s), and the blue is the transverse component (in m/s, deviations from 1.2 m/s). Moveing from upper left to right, top to bottom, we show three blowups by factors 8 in scale. Green shows the deviations of z from the 12 700m of the altitude of the aircraft, (in m) but divided by 8, 4, 2, 1 respectively (the overall trajectory thus changes altitude by over 160 m overall). This is for flight leg 15, but was typical. Reproduced from 83.

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Fig. 1b: The figures corresponding to fig. 1a, but for the change in altitude (in m, red) and the turbulence energy flux (blue; the units are 4x10-3 W/Kg). All except the upper left are multiplied by 40 to make them visible. The energy flux is estimated from the cube of the absolute wind gradient at the smallest scale.

Readers of the blog “23/9 D atmospheric motions: an unwitting constraint on Numerical Weather Models” will recall that larger and larger atmospheric structures become flatter and flatter at larger and larger scales, but that they do so in a scaling (power law) way. Contrary to the postulates of the classical 3D/2D model of isotropic turbulence, there is no drastic scale transition in the atmosphere’s statistics. However, since the famous Global Atmospheric Sampling Program (GASP) experiment (fig. 2) there have been repeated reports of drastic transitions in aircraft statistics (spectra) of horizontal wind typically at scales of several hundred kilometers. We are now in a position to resolve the apparent contradiction between scaling 23/9D dynamics and observations with broken scaling. At some critical scale – that depends on the aircraft characteristics as well as the turbulent state of the atmosphere - the aircraft “wanders” sufficiently off level so that the wind it measures changes more due to the level change than to the horizontal displacement of the aircraft. It turns out that this effect can

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easily explain the observations. Rather than a transition from characteristic isotropic 3D to isotropic 2D behavior (spectra with transitions from k-5/3 to k-3 where k is a wavenumber, an inverse distance), instead, one has a transition from k-5/3 (small scales) to k-2.4

at larger scales (fig. 2), the latter being the typical exponent found in the vertical direction (for example by dropsondes, 54).

Since the 1980’s, the wide range scaling of the atmosphere in the both the horizontal and the vertical was increasingly documented; many examples are shown in WC, ch. 1. By around 2010, the only remaining empirical support the 3D/2D model was the interpretation of fig. 2 (and others like it) in terms of a “dimensional transition” from 3D to 2D. These interpretations were already implausible since a re-examination of the literature had shown that the large scales were closer to k-2.4 than k-3, as expected due to the “wandering” aircraft trajectories. Finally, just last year, with the help of ≈14500 commercial aircraft flights with high accuracy GPS altitude measurements, it was possible for the first to determine the typical variability in the wind in vertical sections (fig. 3), and this was almost exactly the predicted 23/9=2.555… value: the measured “elliptical dimension” being ≈2.57. It is hard to see how the 3D/2D model can survive this finding.

So next time you buckle up, celebrate the fact that the turbulence you feel is still stimulating scientific progress!

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Fig. 2: Spectrum of the long haul aircraft flights from the GASP experiment. The lines with slopes -5/3 and -3 shows the predictions of the 3D/2D model; the -5/3 and -2.4 slope the prediction of the 23/9D model with aircraft “wandering” in the vertical. Reproduced from WC, fig. 2.10, adapted from 56.

Fig. 3: This shows typical variations in the transverse (top) and longitudinal (bottom) components of the wind; black are the measurements, purple are the theoretical contours for a 23/9D atmosphere. Reproduced from WC fig. 2.15a and 50.

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