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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 a. Unfortunately for poetry, in 1803, several month’s after Lamarck proposed this first cloud classification, 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”. 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.

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 for fifty years until the appearance of the first weather maps based on meagre collections of ground station measurements. This was 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 kilometric human “microscales”, virtually nothing was known. Richardson’s claim that a single scaling law might hold from thousands of kilometers down to millimeters didn’t seem so daring: not only was it compatible with the scale free equations that he had elaborated, 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 range. During the war, the first radars had picked up precipitation as annoying noise that regularly ruined the 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 “Marshall-Palmer relation b” soon

a English translations fail to do justice to Lamarck’s poetry; here are some of his cloud types in the original French: “brumeux”; “en voiles”; “lambeaux”, “boursouflés”, “attroupés”; “en balayures”; “en barres”; “pommels”; “coureurs”; “diablotins”; “groups”; “en montagnes”. b This is still the name used by meteorologists for the humble exponential distribution of raindrops as functions of drop size. In 1948 Marshall’s graduate student Palmer had used chemically coated blotting paper to relate the size of a drop to the diameter of a “blot”. Marshall and Palmer had used many such small pieces of blotting paper placed in the bottom of jars, in order to establish the relative number of small and large drops, information needed to interpret radar backscatter. But they had assumed that the drops were uniformly distributed in space whereas - thanks to turbulence - they were in fact distributed in a hierarchical (cascade) like manner. Forty years later, using a huge (128x128cm) piece of blotting paper, a student and I recalibrated similar paper in the same McGill staircase, only this time showing that the spatial distribution of drops was a 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 volumes 10m3 were analyzed using stereophotography. This confirmed that Marshall’s homogeneity assumption is only valid up to about 40- 50 cm, not up to

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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 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 scalesd.

As this window opened, the path pioneered by Richardson; statistical theories of turbulence, was rapidly advancing. 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 were typically a thousand billion times larger than the linear onese. In order to make progress, three important simplifications were made. First, only incompressible fluids were considered. Since gravity acts on density variations, this had the effect of eliminating the main real world source of anisotropy and stratification at the very outsetf. Second, boundaries, walls - and for the atmosphere, the earth’s surface and north-south temperature gradients - are also sources of anisotropy, so that an 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 it was statistically homogeneous.

It is important to take a moment to examine the notions of homogeneity and isotropy more closely. In common parlance, something that is homogeneous is spatially uniform, 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, a useless approximation.

The notion of a turbulence that is statistically homogeneous and statistically isotropic is much more subtle than this, it has to do with the same symmetries - translational and rotational invariance – but over statistical averages. 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 are amplified by the strongly nonlinear character of the flow (the “butterfly

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 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. d Beyond about 100 km, due to the curvature of the earth, ground based radars start to see over the top of the weather.eThe “Reynolds number”.f Later, various incompressible approximations were made such as the Boussinesq approximation, but the main theory was based on incompressibility. The atmosphere is highly compressible. Being much more dense near the surface than at altitude. However if the atmosphere was “barotropic” – with constant pressure and constant density surfaces parallel to each other, this wouldn’t matter, it is the “baroclinicity” that quantifies the deviations from incompressibility.

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effect”, ch. ?). Statistical averages 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 ensemble could easily be extremely inhomogeneous in space and could have a strong preferred directiong. 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 preferred direction. The problem with empirically testing this idea is that no one ever performs 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, Mars comes pretty close, see below!). Often, we have to somehow figure out what “typical” inhomogeneities and “typical” anisotropies might be expected on single realizations even of processes that are known to be statistically homogeneous and isotropic. This underlines the importance of multifractals generated by cascades: they can easily be constructed to be statistically homogeneous and isotropic but nevertheless they are far more wildly variable than anyone had imagined!

While these assumptions may sound academic, they are not 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 importanth. This emboldened theorists to apply these ideas to the atmosphere - even if over limited ranges of scales. Unfortunately for isotropy, gravity acts at all scales so that even if the boundaries only affect the nearby flow and even if the north-south temperature gradients are important only at the largest scales, the presence of gravity is sufficient to render the isotropic theories of the atmosphere academic.

We have considered statistical constancy in space and in direction, what about in time? A vigorous stirring of your coffee might lead to some approximation of statistical homogeneity and isotropy, but if the stirring stopped, then due to friction – viscosity - the motions would die down. 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 – on average - be dissipated as heat at the same rate at which it was input by the stirringi.

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 let the turbulence 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 have little whirls that feed on their velocity”, it was the basic

g 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.h 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).i I say on average, because typical experiments would be far from smooth with energy dissipation occurring very unevenly in both space and in time; even in “bursts”. This was the phenomenon of intermittency discussed earlier – the “spottiness” of turbulence, but the full significance of this was not understood until much later. For bursts, see:7 Schertzer, D. & Lovejoy, S. in State of the Planet, Frontiers and Challenges in Geophysics (eds R.S.J. Sparks & C.J. Hawkesworth) 317-334 (2004).

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cascade ideaj. Such a quasi-steady state means on average everything is the same at all times, it is an approximation to the temporal equivalent of statistical homogeneity: statistical “stationarity”.

The paradigm of “isotropic, homogeneous turbulence” emerged by the end of thek 1930’s. During this time, the Soviet mathematician and physicist Andrei Kolmogorov (1903-1987) was axiomatizing probability theory8 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. The 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 (ch. 1). Using this quantity one immediately obtainsl the Kolmogorov law9 that relates the turbulent velocity fluctuations across a structure to its scalem:

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

Sometimes, scientific ideas are so ripe that they are “in the air”, the Kolmogorov law is a classic example: 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, Obhukhov10 - but in the (equivalent) spectral domain (where it has the form k-5/3 where k is an inverse length, a wavenumber, the spatial equivalent of a frequency). As a consequence, the law is also referred to as the “5/3 law” or the “Komolgorov-Obukov” law. During the war scientific exchanges were limited so that the next was several years later by Onsager11

(1945), who was the first to explicitly link the law to a cascade of energy flux from large to small scalesn. But Onsager’s American publication was no more than a short abstract; it was no more visible than the earlier Soviet papers had been. This led the physicists Heisenberg13 (1948) and von Weizacker14 (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) “Corrsin (1951)15-Obhukov (1949)16 law”:

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

and again with the Bolgiano (1959)17 – Obhukov (1959)18 law for buoyancy driven turbulence that we discussed in ch. 2:

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

j Note that to obtain such a steady state required a constant input of energy so that the overall system was very far from thermodynamic equilibrium.kNotably in the form of the Karman-Howarth equations (1938).l The basic result follows from dimensional analysis, although various arguments and models – such as cascades – are increasingly convincing.mTled Richardson’s 4/3 law of turbulent diffusion, proposed by Richardson largely on empirical grounds. n Many years later, in an important update on his law (taking intermittency into account) Kolmogorov explained that – although they were not explicitly mentioned - during the period 1939-1941 both he and Obhukov had been inspired by Richardson’s cascades:12 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).

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By 1953, the theory of isotropic homogeneous turbulence had evolved to the point that it was already the subject of a landmark book19 “The theory of homogeneous turbulence” by George Batchelor (1920-2000). By then, the role of isotropy had subtly changed. Whereas it had originally been introduced as a way of simplifying theoretical treatments of turbulence, it now had a life of its own. While the main application of the theory was to the atmosphere, Kolmogorov noted that the rather stringent “inertial range o” 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 might only be between 100 m and 0.2 cm.

Writing twenty years later in the influential book “Weather forecasting as a problem in physics20”, A. S. Monin (1921-2007) reproduced (with embellishments) Richardson’s original 4/3 law figure (fig. 2.8), commenting – in accord with Richardson - that it “is valid for nearly the entire spectrum of scales of atmospheric motion from millimeters to thousands of kilometers”. Yet on the opposite page he claims that the Kolmogorov law should hold only to about 600 meters! In a later publication21, 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”.

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. This drastic limitation was not due to any evidence nor to the discovery of any scale breaking mechanism. Rather, the restriction and its implied scale break were hypothesized because atmospheric stratification: contradicted isotropy. Isotropy had come to dominate the theory and the reason for its introduction had been forgotten: theoretical simplicity! Rather than finding the best theory to fit reality, the theorists were trying to find realities to fit their theories.

The full irony had to wait over fifty years to be savoured, for the analysis of the drop sonde data discussed in ch. 1 involving 238 sondes. It turned out that scientists had been so confident in atmospheric isotropy that they hadn’t bothered to check the Kolomogorov law in the vertical direction - and when they finally did so at the end of the 1960’s (see ch. 2) - the contrary Bolgiano-Obhukov results had simply been ignored rather than repeated p. The change (difference) of the horizontal wind was calculated over layers of increasing thickness (fig. 4.1), first only for the near surface region (bottom of the plot), and then from the surface to higher and higher altitudes (the upper curves offset for clarity); the set of points shows the result using all the layers between the ground and 12.8 km (roughly the tropopause). For all altitude ranges, one obtains nearly perfect straight lines indicating scaling over its internal layers covering layers with thicknesses ranging from 5 m to nearly 10 km. Even at small scales, the Kolmogorov law (the line with the 1/3 slope at the bottom of the figure, in red) is completely unrealistic, with the real data being very close to the Bolgiano-Obhukov law (slope 3/5).

Although the slopes in the figure increase a little at higher altitudes even the theories predicting a slope of 1 can be rejected (see the line marked “GW” for gravity waves”). This slope is predicted by both gravity wave theories22 as well as by Charney’s quasi-geostrophic turbulence theory which is thus also seen to be quite unrealistic. The conclusion from the

o So-called because the law was valid only when the inertial terms in the equations dominated the dissipation/friction terms.p Actually, their confidence in isotropy was so strong that often, they didn’t even bother checking it in the horizontal: they measured the spectrum at a fixed location and then converted the result to space as described below.

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data is unequivocal: the original (isotropic) Kolmogorov law simply does not hold anywhere in the atmosphere (unless it is hiding at scales below 5 m!). Kolmogorov and Batchelor’s speculation that the Kolmogorov law would hold up to hundreds of meters was doubly wrong: in reality it holds much less in the vertical… but – as we will see below – much more in the horizontal: up to planetary scales.

As a corollary - but also represents a fundamental discovery – the data showed that the atmosphere was divided into a fractal hierarchy of stable and unstable layers (see box 4.1). Rather than the traditional low resolution view that the atmosphere was generally unstable near the surface and then stable at higher altitudes, the dropsondes showed that within each apparently stable layer that here were unstable sub-layers and within the unstable sub layers there were stable sub-sub layers etc. This discovery proved to be difficult for various theories (in particular of the propagation of gravity waves) that assumed that wide stable layers existedq.

Fig. 4.1 The average mean absolute difference in the horizontal wind from 238 drop sondes over the Pacific Ocean taken in 2004. The data were analyzed over regions from the surface to higher and higher altitudes (the different lines from bottom to top, separated by a factor of 10 for clarity). Layers of thickness z increasing from 5m to the thicknesses spanning the region were estimated, and lines fit corresponding to power laws with the exponents as indicated. At the bottom reference lines with slopes 1/3 (Kolmogorov, K), 3/5 (Bolgiano-Obhukov, BO), and 1 (Gravity waves, GW and quasi-geostrophic turbulence) are shown for reference. Reproduced from25.

***By the mid 1950’s, established empirically based synoptic scale meteorology had

already relegated the “microscales” to mere turbulence, but this had been done mostly for practical 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. The emerging synoptic, meso

q It was argued that the results of wave theories could be reinterpreted in terms of strongly nonlinear but scaling dynamics:23 Lovejoy, S., Schertzer, D., Lilley, M., Strawbridge, K. B. & Radkevitch, A. Scaling turbulent atmospheric stratification, Part I: turbulence and waves. Quart. J. Roy. Meteor. Soc. 134, 277-300, doi:DOI: 10.1002/qj.201 (2008).24 Pinel, J. & Lovejoy, S. Atmospheric waves as scaling, turbulent phenomena. Atmos. Chem. Phys. 14, 3195–3210, doi:doi:10.5194/acp-14-3195-2014 (2014).

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and micro scale regimes were thus not theoretically ordained, they were 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 ones.

That the larger stratified scales might be different type of turbulence was already suggested by the discovery by Fjorthoft (1953) that completely flat, two dimensional turbulence was fundamentally different from three dimensional isotropic turbulencer. 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.

This was the situation when Panofsky and Van der Hoven began their famous measurements of the wind spectrum that they published between 1955 and 195728,29. At this point, wind data at sub-second scales for durations of minutes had already confirmed the Kolmogorov laws, 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 mesoscale between 1 and 100 km, such 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 the spectrum from this analysis was plotted on the same graph as a one minute spectrum that had been 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. 4.3). 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 simply ignoring these scales; later, it justified “parameterising” them. The gap idea was so popular that Van der Hoven’s spectrum was reworked and republished many times notably in meteorological textbooks, throughout the 1970’s. Soon, the actual data points were replaced with smooth artists’ impressions (see e.g. fig 4.3) thus inadvertently hiding the fact that his spectrum was actually a composite

r This was not quite the discovery of Kraichnan’s law of two dimensional turbulence:27 Kraichnan, R. H. Inertial ranges in two-dimensional turbulence. Physics of Fluids 10, 1417-1423 (1967).s The confirmation was indirect since the measurements were in time, not in space.

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taken under four different sets of conditions; even today, his paper is still frequently and approvingly cited.

Fig. 4.2: The famous “meso-scale gap” between the right-most bump and the “synoptic maximum” (left-most bump”), adapted from Van der Hoven 1957 in30. The ellipses show the rough ranges of the four experiments which were combined to give the composite spectrum (the actual data points had already been replotted from the original). The vertical line that corresponds to about 6 minutes and was often superposed to indicate the limit of 3D isotropic turbulence.

Yet within ten years, the gap was strongly criticized31,32,33 with critics pointing out that it was essentially based on a single high frequency bulge (fig. 4.3, near scales of 100 s) due a single experiment data taken “under near hurricane” conditions. By the end of the 1970’s, satellites were routinely imaging interesting mesoscale featurest. Practionners of the nascent field of mesoscale meteorology such as Atkinson34 were sceptical of any supposedly barren “gap” that could relegate their entire field to a mere footnoteu.

Although for many the gap was too convenient to kill, the mesoscale itself underwent a transformation. The new developments were of two dimensional isotropic turbulence by Robert Kraichnan (1928-2000) in 196735, 27, and in 1971 its extension to quasi-geostrophic turbulencev by Jules Charney (1917-1981)36. 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.

While in some quarters, the gap lived on, the development of two dimensional turbulence changed the focus: rather than searching for the gap, the new goal was to search for signs of large scale isotropic 2D turbulence. Such a discovery promised to transform the meso-scale from a barren gap into the site of a 3D-2D “dimensional transition”37. After

t 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.u Atkinson approvingly quotes the early critic Robinson who noted “… I find it unconvincing the argument that disturbances on scales between the cyclone and the thunderstorm do no exist because we do not see them on synoptic charts”:31 Robinson, G. D. Some current projects for global meteorological observation and experiment, . Quart. J. Roy. Meteor. Soc., 93, 409–418 (1967).v This was effectively a derivation of Kraichnan’s pure two dimensional turbulence starting from a series of nontrivial approximations to the governing equations of the atmosphere.

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Kraichnan published his paper, the 2D isotropic turbulence idea was so seductive that claims of 2D turbulence sprang up almost immediately. Whereas 3D isotropic turbulence followed the k-5/3 law, 2D turbulence was expected to have a k-3 regimew, 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 theoryx.

The first experiment devoted to testing the new 2D/3D model was the EOLE experimenty (1974). It used the dispersion of 480 constant density balloons39 (at about 12km altitudez dispersed over the southern hemisphere; this is was the same as some of Richardson’s methods used to obtain fig. 2.8; it was effectively an updated versionaa. 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 of the former. But contrary to Richardson, and several confirmations of Richardson’s law in the 1950’s and 1960’s (including in the ocean) – before the 3D/2D theory - the original analysis of the EOLE analysis by Morel and Larchevesque39 concluded that the turbulence in the 100 km -1000 km range did not follow his law, but rather those predicted by two dimensional turbulence.

Yet even in the mid 1970’s, internal discrepancies in the EOLE analysis had been notedbb. More importantly, the EOLE conclusions soon contradicted those of the GASP cc (1983) and later MOZAICdd (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 al40 and it was concluded that the original EOLE conclusions were not founded, that on the contrary, the data vindicated Richardson over the range 200 - 2000 km.

But the saga was still not over. Strangely, in spite of supporting Richardson at the largest scales, the Lacorte et al reanalysis contradicted him at the smallest EOLE scales -from 200 km down to the smallest available scales (50 km) - over which they claimed to have validated the original 2D turbulence interpretation! Nearly 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, 2000 km, thus (finally!) vindicating Richardson nearly 90 years later!ee.

***Science is a quintessentially human activity. At each epoch, it depends on the

available technology, the reigning scientific theories, on the key scientific problems: on its historical level of development. Yet it also depends on society’s attitude and on its

w It also generally had a k-5/3 regime - only this was at very large scales - a fact that was often conveniently forgotten.x Some authors admitted to “eyeballing” their spectra over a mere factor of 2 in scales to back up such claims:38 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). y After the Greek wind God.z More precisely, close to the 200 mb pressure level.aa 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.bb Between the relative diffusivity and the velocity structure function results.cc GASP = Global Atmospheric Sampling Program.dd MOZAIC = Measurements of OZone and water vapour by in-service AIrbus airCraft experiment.ee See appendix 2A of:30 Lovejoy, S. & Schertzer, D. The Weather and Climate: Emergent Laws and Multifractal Cascades. (Cambridge University Press, 2013).

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willingness to allocate resources. In order to understand the trajectory of atmospheric science following the heady nonlinear 1980’s decade, we need to briefly discuss the changing fortunes of fundamental science in the scientifically advanced countries. Although the following account is tainted by my own situation in Canada, its Friedmanite economic policies are fairly representative of this group.

The post World War Two élan of scientific optimism was vividly expressed in the title of Vannevar Bush’s report: “Science, the endless frontier”ff. It was the beginning of the era of “Big Science”, i.e. of science being directly harnessed by big business albeit in the form of a partnership between publically and privately funded efforts. It was recognized that investment in inappropriate scientific concepts or unrealistic models could squander huge sums of money; that no matter how urgent a problem might be, its solution required a balance between fundamental and applied research. But fundamental research was expensive and accountants were unable to determine the corresponding rates of return on investment. Corporations were happy to let academic or other publically funded institutionsgg pickup the bill. By the 1990’s, high costs and risks had created a situation where only a handful of giant corporations still carried out any fundamental research. Marking the end of an epoch, in 1996, even the famous Bell labs were sold offhh, while others were downsized or refocused toward more practical mattersii. Today, the fundamental research required for technological advance is virtually entirely publically funded.

This evolution came at a price. Until then, fundamental sector scientists had been given free reign to investigate the areas of greatest scientific significance: it was “curiosity driven”. Now, all research required an economic justification and curiosity and kindred terms became the kiss of death. Businesses lobbied governments for direct control over public research, over both its priorities and the management of its funds. Violating its very nature as a long-term enterprise, funding of fundamental science was retargeted towards short-term corporate gain with public research agencies reoriented accordingly. To reassure the public, officials mouthed the new mantra of “excellence” according to which special interests were also excellent and where doing more with less was particularly excellent. At the same time, official R&D figures were doped by including tax shelters for businesses claiming to invest in high techjj, effectively hiding the pirating of public funds from close scrutiny.

Concomitant with the industrial focus, was a growing disinterest – even by scientists -in fundamental issues including those that the nonlinear revolution had promised to solve. In atmospheric science, resources were tightly focused on the development of numerical weather models (Global Circulation Models, GCMs)kk. The only justification for funding became the promise of improving GCM outputs. In the past, it had been possible to obtain

ff The title of a report in July 1945 by Vannevar Bush, Director of the Office of Scientific Research and Development, US.gg In the US, especially the military.hh 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.ii In my case, Canada’s branch plant economy had never enjoyed much corporate research. jj Specific Canadian examples that stick in my memory include a case where hundreds of millions of dollars in tax relief was given to banks who had “invested” in science and technology simply by upgrading their office equipment. Another example also from the mid 1990’s was of tax relief given to a companying for research into new flavours of beer. This type of accounting trick allowed the government to boast of stable levels of support for science.kk In Canada for example, the staff of the federal weather office, the “Atmospheric Environment Service” (now the Meteorological Services of Canada, a part of Environment Canada) was cut by 50% during the 1990’s and this included the elimination of a small fund used to support academic research.

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support for an applied science project and – thanks to deliberately loose controls – scientists would regularly siphon off some of the funds to (illicitly) do “real science”. But in the brave new world of excellence, sponsors required rigorous accountability. Not only were research priorities imposed from without, but every dollar had to be spent exactly as specified in increasingly detailed submissions often written years earlier ll. University accounting departments administering the grants played a police role protecting the sponsors’ money from being irresponsibly spent in advancing science rather than following the sponsors’ dictates. Academic scientists were gradually being transformed into cheap labour.

But technology continued to advance and rapidly increasing computer sizes and speeds, improved algorithms and mushrooming quantities of remotely sensed data ensured that the 1990’s were a golden age for atmospheric science. Fundamental GCM issues were also being resolved. In particular, advances in data assimilation opened the doors to the widespread “ingestion” of satellite and other disparate and hitherto under exploited sources of datamm so that by the decade’s end, weather forecasts had significantly improved. For atmospheric science, GCMs increasingly appeared to be the only way of the future. In GCMs, scale and scaling might have intruded in the choice of grid size but in practice, this was determined by computer technology. The main choice - the ratio of the number of vertical to horizontal pixels – was made by empirical experience rather than by using theory. The fact that it ended up following the 23/9D model (fig. 1.8) was by empirical necessity rather than by explicit theorynn. Scale and scaling seemed abstract and unnecessaryoo.

The divorce between practical atmospheric science and turbulence theory was hardly new. Atmospheric science had always suffered from the gulf between the idealized smooth, calm models concocted by theoreticianspp and real worldqq wild irregularity. In the 1970’s, a popular adage was “No one believes a theory except the person who invented it. Everyone believes the data except the person who took them”. In the new ambiance, cynicism about theory was especially strong. When discussing atmospheric scaling in the early 1990’s a prominent colleague commented: “if no progress is made for long enough, the problem is considered solved and we move on.” It was thus not surprising that when the budgetary screws were turned, this ambient negativism hit the nonlinear revolution and conventional turbulence theory alike: interest was scant and funding even more so. Theory of any kind was increasingly seen as superfluous – it was either irrelevant or a luxury that could no longer be afforded. Any and all atmospheric questions were answered using the now standard tool: GCMs.

Unfortunately, GCMs are massive constructs built by teams of scientists spanning generations. They were already “black boxes”, and even when they answered questions, they did not deliver understanding. Atmospheric science was being gradually transformed

ll Since the outcome of research was intrinsically unpredictable, the best strategy for scientists was to somehow get ahead of the game and ask for funding for projects that they had already performed, thus guaranteeing that they could fulfill the precise terms of the grant.mm The development of revolutionary new data assimilation techniques: (spatial) “3D var”, and later, (space-time) “4D var”.nn Recall that the necessary theory – the 23/9 D model – had been around since the mid 1980’s.oo By the end of the 1990’s the flow at scales below those resolved by the grids (corresponding to 100 km or so) started to be parameterized stochastically – using random numbers (“stochastic parametrization”). But even this was done in relatively ad hoc way, without regard to cascade theory.pp The first (potentially realistic), strongly non smooth, violently variable models came out of only one of the strands of turbulence - the scaling strand, multifractals.qq In my view, this was because the theories made many conventional smoothness and regularity assumptions that were totally unrealistic and that the scaling approach would overcome.

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from an effort at comprehending the atmosphere into one of numerically imitating it; i.e. into a purely applied field. New areas – such as the climate – were being totally driven by applications and technology: climate change and computers. In this brave new world, few felt the need or had the resources to solve the basic scientific problems.

***So it was that the excitement engendered by the nonlinear revolution in the 1980’s,

slowly faded and with it, the general scientific interest in geosystem scales and scaling. Yet the revolution had succeeded in establishing a beach-head, a community of like minded scientists organized first in the European Geophysical Society’srr (EGS) Nonlinear Processes divisionss (1989), around the Nonlinear Processes in Geophysics journal (1994), and a little later, in the American Geophysical Union’s Nonlinear Geophysics focus group (1997). Following a 2009 workshop on Geocomplexitytt, a dozen scientists published a kind of nonlinear manifesto entitled “Nonlinear Geophysics: Why we need it” 41. It proclaimed that “the disciplines coalescing in the NG movement are united by the fact that many disparate phenomena show similar behaviours when seen in a proper nonlinear prism. This hints at some fundamental laws of self-organization and emergence that describe the real nature instead of linear, reductive paradigms that at best capture only small perturbations to a solved state of problem…”.

It was largely in nonlinear geophysicsuu that evidence for wide range atmospheric scaling slowly accumulated, notably by the study of radar rain reflectivities42,41, and satellite cloud radiances43-48 (see e.g. fig. 4.4). But these analyses were generally restricted to scales smaller than a thousand kilometers 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 and at the time, these didn’t have a wide enough range of scales to be able to settle the issue eithervv.

rr Since 2002, following a merger with the European Union of Geodesy, it became the European Geosciences Union.ss This precocious development was greatly helped by EGS’s visionary director Arne Richter (1941-2015).tt Organized by Q. Cheng and me at York University, Toronto, Canada.uu At the European Geosciences Union, Nonlinear Processes division.vv A partial exception was a paper by Strauss and Ditlevsen in 1999 that considered 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 directly analyzing the scaling of model outputs, they instead analyzed more complex theoretically inspired constructs so that the conclusions were less clear. 49 Strauss, D. M. & Ditlevsen, P. Two-dimensional turbulence properties of the ECMWF reanalyses. Tellus 51A, 749-772 (1999).

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Fig. 4.4: Spectra from three different satellites from largely cloudy regions. Meteosat, geostationary, 8 km resolution, Landsat, at 83 m resolution and the number 1- 5 from the NOAA-9 satellite with channel 1 in the visible, channel 5 the thermal infra red and 2-4 in the in between wavelengths. Scaling, power laws are straight lines on the log-log plot; the range 1 – 100 km is indicated “mesoscale” and it shows no signs of a break in the scaling. Overall, the figure covers the range of 166m to 5000 km. Reproduced from50.

The next major advance appeared in the late 2000’s with the beginning of widespread availabilityww of truly global scale atmospheric data sets, notably massive satellite archives and these invariably showed excellent wide range scaling, although again, not for the hard-to-measure wind field (see fig 4.5). Also at this time, the numerical models were getting big enough to analyse, and once again, wide range scaling was found60 (see the reanalyses, fig. 4.5 lower right). Whether or not meteorologists liked scaling (or were even aware of it!), their models respected the symmetry extremely wellxx! By 2008, it seemed that the only evidence that apparently contradicted the wide range scaling hypothesis, were aircraft wind spectra.

ww The key advance was not so much the actual existence of large scale data sets - this was not so new - it was rather due to the rapidly increasing internet speed and the online availability. xx But this didn’t prevent the continued development of all kinds of scalebound models attempting to better understand the model outputs!

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Fig. 4.5: Upper left: Spectra from over 1000 orbits the Tropical Rainfall Measurement Mission (TRMM); of five channels visible through thermal IR wavelengths displaying the very accurate scaling down to scales of the order of the sensor resolution (≈ 10 km).Upper left: Spectra from five other (microwave) channels from the same satellite. The data are at lower resolution and the latter depends on the wavelength, again the scaling is accurate up to the resolution. Lower Left: 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. Lower right: Zonal Spectra of reanalyses from the European Centre for Medium Range Weather Forecasting (ECMWF), once daily for the year 2008 over the band ±45o latitude. These figures are adapted from ones in the review: 30.

Fig. 4.6a: The GASP spectrum of long-haul flights (more than 4800 km) with reference lines corresponding to the horizontal and vertical behaviour with slope indicated. The rough position of the scale break is shown, it is near 1000 km, much larger than any possible 2D-3D transition scale. Adapted from Gage and Nastrom61 (in particular, the lines with slope -2.4 were added).

Fig. 4.6b: Trajectories of an aircraft following lines of constant pressure to with 0.1% near the 200 mb pressure level (height in meters). Data were taken every second (≈ 280 m). The trajectories are far from constant in altitude as can been seen in the blow of one of them below. Adapted from62.

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Fig. 4.6c: The trajectory (black) of a scientific aircraft following a constant pressure level (to within 0.1%), data at 1 s resolution (≈280 m). The purple shows the variation in the longitudinal component of the horizontal velocity (in m/s, deviations from 24.5 m/s), and the orange is the transverse component (in m/s, deviations from 1.2 m/s). Moving from upper left to right, top to bottom, we show three blowups (over the region indicated by the parentheses) by factors 8 in scale. Black shows the deviations of altitude (z) from the 12700m 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. It can be seen that the changes in horizontal wind cannot be dissociated from the variability of the aircraft altitude. Reproduced from63.

Fig. 4.7a: Spectra from the Pacific Winter Storm experiment; the average over 24 legs, each 280 m in resolution of length 1120 km at 200 mb (about 12 km altitude), longitudinal and transverse components of the horizontal wind are shown along with reference lines indicating the horizontal (Kolmogorov) exponent 5/3 and the vertical exponent 2.4 (close the Bolgiano-Obukov value 11/5). Adapted from62.

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Fig. 4.7b: Same as fig 4.7a except for temperature (T), humidity (h) and log potential temperature (log). A reference line corresponding to k-2 spectrum is shown in red. The mesoscale (1 – 100 km) is shown between the dashed blue lines. Adapted from62.

Even if one accepted the original interpretation of the EOLE experiment in terms of 2D turbulence, it measured the dispersion of balloons, not the 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 a parallel multidecadal saga. The first and still most famous experiment was GASP whose data were analyzed by Nastrom and Gage in a series of papers from 1983-198661,64-66. The dominant interpretation was that the empirical spectrum did indeed show a transition from the Kolmogorov 3D isotropic turbulence, to two dimensional isotropic turbulence, see e.g. fig. 4.6ayy.

The most glaring problem with the GASP results was that the apparent 2D-3D transition scale was typically at several hundred kilometers67,68,69,70. 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”71, or “escaped” 3D turbulence67 explained nothing. In 1999, an update of the GASP experiment MOZAIC found essentially the same results, that were strongly interpreted as support for the 2D-3D theory72,73.

In order to get to the bottom of this, with the help of Daniel Schertzer and Adrian Tuckzz, we reanalysed62 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 (see e.g. fig. 4.6a, 4.7a). In the 2009 paper “Reinterpreting aircraft measurements in anisotropic scaling turbulence62” 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 a theory was needed to correctly interpret aircraft measurementsaaa and the new theory easily

yy Interestingly, Nastrom and Gage themselves interpreted their results as support for yet another theory based on waves.zz Tuck was 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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explained the results by aircraft following gently sloping isobars (rather than isoheights), fig. 4.6a,b.

Fig. 4.8: 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 74.

But even this didn’t satisfy the die-hard 2D-3D theorists, notably Eric Lindborg. He incited experimentalist colleagues at the National Center for Atmospheric Research (NCAR) Frehlic and Sharman75 to use the big Aircraft Meteorological Data Relay (AMDAR) data base to disprove our hypothesis by attempting to empirically demonstrate the statistical equivalence of wind data at constant heights and constant pressures (isoheights versus isobars). But the AMDAR technology didn’t include GPS altitude determinations that were needed to distinguish accurately enough between the two. In order to prove that our explanation was correct and to close the debate76, 75, it was necessary to determine the joint (horizontal- vertical) velocity structure functionbbb. This was finally done with the help of wind data from 14,500 aircraft trajectories that allowed its first direct determination (fig. 4.8). The aircraft used in our study had Tropical Airborne Meteorological Reporting (TAMDAR) technology with metric scale GPS altitude measurements. The results of this massive study74 showed that the horizontal wind was scaling with an anisotropic in-between “elliptical dimension” 2.56±0.02, close to the theoretical value 23/9 discussed in ch. 2 and 3.

***The have summarized a series of attempts to shoehorn the atmosphere into a series

of scale frameworks, followed by long struggles to empirically disprove them. First, the meso-scale gap that would have allowed the

We have seen that scalebound

With the reinterpretation of aircraft measurements, , Sixty years after the meso-scale gap, forty years after the EOLE experiment and thrity years after the GASP aircraft measurements had brought Richardson’s wide range scaling into question, the only thing preventing closure was the development of theory showing that the observed anisotropic

aaa Until then the turbulence was assumed isotropic so that a theory of how the aircraft was affected by the turbulence and how this might bias the results hadn’t seemed necessary. Isotropy implied that there was a unique exponent for each field so that if aircraft measured scaling spectra that the exponents would be unbiased.bbb This means that we estimated the typical change in the horizontal wind for arbitrary displacements in vertical cross-sections.

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scaling was indeed compatible with the equations. As discussed in ch. 3, this was indeed done at nearly at the same time so that the theoretical debate between the 3D-2D isotropic model or the anisotropic scaling alternative (the 23/9D model, notably with 77) was finally nearing closure.

2.6.5 The continuing difficulties with the classical model and inferences from numerical simulations

In order to improve on these speculative mechanisms needed to combine isotropic 2D and isotropic 3D turublence, efforts have been made to reproduce “realistic” k-3 to k-5/3 transitions in numerical models. This is not a trivial question because of the possibility of “three dimensionalization” of two dimensional flows discussed in 51, i.e. the likelyhood that three dimensional turbulence can destabilize an otherwise 2D flow. For the moment, the results are at best equivocal. For example, most numerical weather models do not display the transition 52, while others may display it although over very small ranges e.g. the 53, WRF (regional) model spectra which are in fact very close to k-2.4, see fig. 2.20. To date, the most convincing k-3 to k-5/3

transitions in numerical models have been produced using the SKYHI model on the earth simulator in 54 and 55, yet as pointed out in 56 they have the poorest fit to GASP observations precisely over the range ≈ 400 – 3000 km which their (painstakingly crafted) k-5/3 to k-3 transition is supposed to explain. In other words, this model “success” may make them less rather than more realistic! In addition, 57 has shown that at least in the case of the quasi-geostrophic simulation by 58, high wavenumber k-

5/3 regimes are in reality spurious consequences of energy build-up due to unresolved high wavenumbers with respect to an incorrectly “tuned” hyperviscosity.

Fig. 2.20: This shows sample spectra from WRF forecasts of zonal wind averaged over the isobaric surfaces covering roughly the range 3-9 km in altitude, adapted from 53. Although they claimed that this shows a “clear k-3 regime” for the solid (oceanic) spectrum it only spans a range of factor 2 to 3 in scale, and this at the relatively unreliable extreme low wavenumbers (between the downpointing arrows, upper left).

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Except for the extremes, the spectra again follow the isobaric predictions k-2.4 (red) very well over most of the range. Reproduced from 59.

***

We have outlined in considerable detail the history and developments surrounding the issue of wide range horizontal scaling. Not only is it important in its own right, but - due to the wind - it implies temporal scaling, at least up to scales of about ten days (the size of the planet divided by the typical large scale wind). Fig. 4.5 (lower left) shows that indeed, up to scales of about 5000 km and 7 days in time, that the spatial and temporal spectra are essentially indistinguishable from each other and also scaling.

In order to understand this, and in particular to work out the fundamental time scale at which the weather regime breaks down and makes a transition to the lower frequency macroweather regime, we can again go back to [Van der Hoven, 1957]. Aside from the ill- starred spectral gap, his spectrum also showed a more robust feature: a drastic change in atmospheric statistics at time scales of several days (fig. 4.3, the “bump” on the left). At first ascribed to “migratory pressure systems”, later termed the “synoptic maximum” 78, it was eventually theorized as due to the unstable nature of atmospheric layers ccc. However, neither its presence in all the atmospheric fields nor its true origin and fundamental implications could be explained until - as explained above - the turbulent laws were extended to planetary scales.

The key feature of anisotropic scaling is that the vertical is controlled by the buoyancy force variance flux (the Bolgiano-Obukov law) and the horizontal dynamics by the energy flux to smaller scales (in units of W/Kg, also known as the “energy rate density”). This is the same dimensional quantity upon which the Kolmogorov law is based, although here the law only holds in the horizontal, not the vertical. The classical lifetime size relation is then obtained by using dimensional analysis:

(lifetime) = (energy rate density)-1/3 (scale) 2/3

This is the space-time relation for structures such as storms; it would be obtained if one followed the storm as it propagated, it is the “Lagrangian” space-time relation (80,30, see the straight line in fig. 2.4). Rather than identifying a structure and following it, it is usually easier simply take a series of snapshots – such as those taken by geostationary satellite in fig. 4.5 lower left - and to compare typical sizes in space with typical sized in time (durations); the “Eulerian” – fixed frame - space-time relation. Using the same Infra Red imagery in fig. 4.5 one finds that the space-time - is linear ddd, fig. 4.9 shows the relationship that this spectrum implies; an average wind of about 900 km/day (≈11m/s).

ccc Specifically, due to Baroclinic instabilities. These occur when the pressure and density gradients in a stratified fluid are strongly misaligned. One of the many problems with this explanation is that it only works in midlatitudes whereas the transition time scale is not much different in equatorial regions, see fig. 4.10. 79 Vallis, G. in Stochastic Physics and Climate Modelliing (ed P. Williams T. Palmer) 1-34 (Cambridge University Press, 2010).ddd For some theoretical explanation see:81 Radkevitch, A., Lovejoy, S., Strawbridge, K. B., Schertzer, D. & Lilley, M. Scaling turbulent atmospheric stratification, Part III: empIrical study of Space-time stratification of passive scalars using lidar data. Quart. J. Roy. Meteor. Soc., DOI: 10.1002/qj.1203 (2008).

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Fig. 4.9: The Eulerian (fixed frame) space-time diagram obtained from the satellite pictures analyzed in fig. 4.5, lower left, reproduced from 82. The slopes of the reference lines correspond to averages winds of 900 m/s, i.e. about 11 m/s.

If the above explanation for the space-time relation is correct, then it depends on the energy rate density. This fundamental quantity ought to have a fundamental explanation in terms of the main force driving the atmosphere: the sun. Indeed, starting with the solar power delivered to the earth, it turns out to be nearly trivial to estimate it. The idea is that - thanks to the turbulent dynamics (the weather!) – the average power is more or less uniformly redistributed over the atmosphere. One can estimate trgy rate density simply by dividing the total atmospheric masseee - about 10 tons per square meter – by the average solar power (about 240 Watts per square meter) that is transformed into mechanical energy (about 4%), i.e. about 10 Watts per square meter, yielding an energy rate density of 0.001 Watts per Kilogram. Combining this with the earth scale; 20,000km (the largest great circle distance) this value implies that the lifetime of planetary structures ≈ 5- 10 days. Fig. 4.10 shows that this simple theory even explains the latitudinal variations: as the winds (and hence the rate density) are lower near the equator, the transition time scale is a little longer, exactly as the theory predicts.

According to our derivation, structures that are larger and larger live longer and longer, but eventually, due to the finite size of the earth, this must break down. The scale that we just estimated is the breakdown scale, the lifetime of the largest possible structures. At the same time – since structures can only be reliably forecast over their lifetimes – this is also the overall deterministic predictability limitfff.

If we go beyond this, then we are considering several lifetimes of structures, we should not be surprised to find that the fluctuations over consecutive lifetimes tend to cancel; this is the macroweather regime discussed in ch. ? From the point of view of turbulent laws, the transition from weather to macroweather is a “dimensional transition” since at longer time scales, the spatial degrees of freedom are essentially “quenched” so that

eee If one takes only the lower- troposphere- this may be a little more accurate, but the troposphere has about 80% of the overall mass, so that this distinction isn’t very important.fff This is actually not as trivial as it may seem. One of the consequences of the 2D-3D model – and one of its attractive features - would have been a potentially much longer predictability limit of 2D turbulence.

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the system’s dimension is effectively reduced from (time+space) to (time). Both turbulence models and GCMs control runs (i.e. with constant external forcings) reproduce the transition and also produce realistic low frequency variability. The fact that these weather models reproduce this justifies the term “macroweather”.

Amazingly, this basic feature of the atmosphere was not discovered until 201083, presumably because no one had dared to suggest that the Kolmogorov law could possibly be relevant at planetary scales! This result is so fundamental that we need to validate it as thoroughly as possible. For example, the ocean is also a turbulent fluid; in many respects it is similar to the atmosphere, only it is stirred not by the sun but by the wind - so that it too should have an “ocean-weather” – “ocean- macroweather” transition determined by its own typical energy rate density using data from thousands of ocean “drifters”. To test this, we simply used the empirically estimated near surfaceggg energy rate density finding that it was about one hundred thousand times smaller than the atmosphere: 10 billionths of a Watt per square meter. This value yields a transition time of about 1- 2 years which is also observed, see fig. 4.1183.

We have discussed the fact that these turbulent scaling laws are expected to hold “universally” i.e. under fairly general conditions in roughly similar atmospheres. But for the most convincing test of the scaling theory, we need another planet!

Fig. 4.10: 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, 84), the w estimates were made by performingτ 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 30.

ggg In the atmosphere, the average energy rate density is not very different throughout so that a single value is already a good approximation. However, in the ocean, it decreases very rapidly with depth implying – as is well known - that deep currents may live for thousands of years.

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Fig. 4.11: The three known weather - macroweather transitions: air over the Earth (black and upper purple), the Sea Surface Temperature (SST, ocean) at 5o 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 85.

*** “Good afternoon Martians, I bring good tidings from planet Earth: we’re twins!” So

began my presentation to a room packed with Mars specialists at a session at the European Geosciences Union, April 2016. Mars may be our sister planet, but when it comes to its atmosphere, up until now, scientists had focused on the differences, not the similarities. My introduction caught the audience off guard since I had inverted the usual procedure. I had not gone to Mars to better understand the red planet, but rather the blue one.

The most important differences between the atmospheric dynamics of the two polanets are the strong control of Martian atmospheric temperature by dust, the larger role of topography, the stronger diurnal and annual cycles, the larger role of atmospheric tides. But these differences mostly affected the forces driving the system and the nature of the boundaries: if the turbulence approach is correct, at small enough scales and far enough from the boundaries, then we expect to find the same statistics, the same scaling: the behavior was expected to be independent of the details: it should be “universal”.

Using the theory based on the energy rate density, we can easily calculate the Martian weather – Martian macroweather transition time scales. All we need are its distance from the sun – and hence solar insolation (about 55% of ours), and the surface pressure - and hence the mass per square meter (about 1.5% of our valuehhh). The ratio yields an expected 0.04 Watts per Kilogram on Marsiii, therefore – taking this and the small planetary size into account (about half the earth size), we predicted a Martian weather - Martian macroweather transition time scale of about 1.8 solsjjj.

hhh The surface atmospheric pressure is about 200 times smaller than on earth. The mass/area estimate takes into account the differing surface gravity.iii The larger value is due to the much smaller atmospheric mass.jjj A sol is a Martian day, about 25 earth hours.

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The problem was then to find data to test the prediction. There had been many Mars landers carrying meteorological instruments, but it turned out that the first and oldest – the two Viking landers (1976, 1977) were still the best for the purpose with hourly temperature and wind data spanning (nearly continuously) three (earth) yearskkk. The spectra obtained from these landers showed that the theory result is indeed well respected (fig. 4.11). Since the spectrum implies that fluctuations longer than 1.8 sols have nearly identical behavior as those on earth, on Mars too, we should expect macroweather85.

The paper was published in one of planet Earth’s leading Geoscience journals85 yet, the Martians appeared to be far more interested interested than the Earthlings. A few days after the Geophysical Research Letters paper had been published, the story was picked up by the amateur Astronomy Magazine, and was prominently featured on its website. This just happened to be at the same time that the comet lander Philae discovered organic molecules on comet 67P/Churyumov–Gerasimenko; the two stories on the site were right next to each other. Whereas Martian macroweather garnered 3360 likes, poor Philae had only 324!

But not only should the wind field be universal, so should the other atmospheric fields, and this should include the spatial scaling as well. It turned out that full reconstructions of the Martian atmosphere existed for nearly 3 Martian years lll. These had been calculated using an Earth weather model adapted to Mars, and then using an orbiting Infra red satellite (“Mars express”?) to estimate the temperature profile, “reanalyses”. Fig. 4.12 and 4.13 show that Earth and Mars do indeed have virtually the same statistics of pressure, wind, temperature in the east-west and north south directions. Even the turbulent intermittencies (fig. 4.13) turned out to be the same!

kkk Perhaps surprisingly, the more recent landers such as Pathfinder had much higher frequency response (seconds), but the records were much shorter; the Viking data turned out to be optimal.lll About 8 Earth years, after this time, the satellite went out of operation. The reanalysis had a resolution of 5 o

and 6 hours in time.

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Fig. 4.12: A comparison of spectra from terrestrial and Martian reanalyses (left are right columns respectively) showing the universality of the scaling behaviour; the top row shows the zonal (east-west), the bottom row the meridional (north-south) spectra. Adapted from 86.

Fig. 4.13: A comparison of trace moments (M = <q> ) terrestrial (left) and Martian (right) for

moments q =0.2, 0.4, 0.6, …1.8, 2 and = (half planet circumference)/ (spatial resolution), the same reanalyses as in fig. 1.2.3a. The graphs (top to bottom) are for the surface pressure, north-south wind, east-westwind, temperature (the latter three at about 70% surface pressure). The Martian trace moments should be compared to the terrestrial ones on the left of the thin black dashed line (the points to the right are at scales not represented in the lower resolution Martian reanalysis).

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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 87.

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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 87.

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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.

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

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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 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, 88).

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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