Chapter 4.The weather: nothing but turbulence… and don’t ...



Chapter 4.The weather: nothing but turbulence… and don’t mind the gap 4.1 “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, 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” soon established the quantitative basis for interpreting radar precipitation scans; the famous “Z-R” relation (reflectivity- rain rate). 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 scales.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 ones. 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 outset. 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 directions ADDIN EN.CITE <EndNote><Cite><Author>Taylor</Author><Year>1935</Year><RecNum>1116</RecNum><DisplayText><style face="superscript">5</style></DisplayText><record><rec-number>1116</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1256921881">1116</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Taylor, G. I.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Statistical theory of turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Proc. Roy. Soc.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">421-478</style></pages><volume><style face="normal" font="default" size="12">I-IV, A151</style></volume><dates><year><style face="normal" font="default" size="12">1935</style></year></dates><urls></urls></record></Cite></EndNote>5. 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 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 direction. 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 important. 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 stirring. 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 cascade idea. 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 the 1930’s. During this time, the Soviet mathematician and physicist Andrei Kolmogorov (1903-1987) was axiomatizing probability theory ADDIN EN.CITE <EndNote><Cite><Author>Kolmogorov</Author><Year>1933</Year><RecNum>303</RecNum><DisplayText><style face="superscript">8</style></DisplayText><record><rec-number>303</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">303</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kolmogorov, A.N.</author></authors></contributors><titles><secondary-title>Grundebegrisse der Wahrscheinlichkeitrechnung</secondary-title></titles><dates><year>1933</year></dates><urls></urls></record></Cite></EndNote>8 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 obtains the Kolmogorov law ADDIN EN.CITE <EndNote><Cite><Author>Kolmogorov</Author><Year>1941</Year><RecNum>305</RecNum><DisplayText><style face="superscript">9</style></DisplayText><record><rec-number>305</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">305</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Kolmogorov, A. N.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Local structure of turbulence in an incompressible liquid for very large Reynolds numbers. (English translation: Proc. Roy. Soc. A434, 9-17, 1991)</style></title><secondary-title><style face="normal" font="default" size="12">Proc. Acad. Sci. URSS., Geochem. Sect.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">299-303</style></pages><volume><style face="normal" font="default" size="12">30</style></volume><dates><year><style face="normal" font="default" size="12">1941</style></year></dates><urls></urls></record></Cite></EndNote>9 that relates the turbulent velocity fluctuations across a structure to its scale:(Velocity Fluctuations) = (Energy Rate Density)1/3 x(Scale) 1/3Sometimes, 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, Obhukhov ADDIN EN.CITE <EndNote><Cite><Author>Obukhov</Author><Year>1941</Year><RecNum>1250</RecNum><DisplayText><style face="superscript">10</style></DisplayText><record><rec-number>1250</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1290742390">1250</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Obukhov, A.M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">On the distribution of energy in the spectrum of turbulent flow</style></title><secondary-title><style face="normal" font="default" size="12">Dokl. Akad. Nauk SSSR</style></secondary-title></titles><pages><style face="normal" font="default" size="12">22-24</style></pages><volume><style face="normal" font="default" size="12">32</style></volume><number><style face="normal" font="default" size="12">1</style></number><dates><year><style face="normal" font="default" size="12">1941</style></year></dates><urls></urls></record></Cite></EndNote>10 - 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 Onsager ADDIN EN.CITE <EndNote><Cite><Author>Onsager</Author><Year>1945</Year><RecNum>1246</RecNum><DisplayText><style face="superscript">11</style></DisplayText><record><rec-number>1246</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1290721945">1246</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Onsager, L.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The distribution of energy in turbulence (abstract only)</style></title><secondary-title><style face="normal" font="default" size="12">Phys. Rev.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">286</style></pages><volume><style face="normal" font="default" size="12">68</style></volume><dates><year><style face="normal" font="default" size="12">1945</style></year></dates><urls></urls></record></Cite></EndNote>11 (1945), who was the first to explicitly link the law to a cascade of energy flux from large to small scales. 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 Heisenberg ADDIN EN.CITE <EndNote><Cite><Author>Heisenberg</Author><Year>1948</Year><RecNum>1762</RecNum><DisplayText><style face="superscript">13</style></DisplayText><record><rec-number>1762</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1492048989">1762</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Heisenberg, W.</author></authors></contributors><titles><title>On the theory of statistical and isotrpic turbulence</title><secondary-title>Proc. of the Roy. Soc. A</secondary-title></titles><periodical><full-title>Proc. of the Roy. Soc. A</full-title></periodical><pages>402-406</pages><volume>195</volume><dates><year>1948</year></dates><urls></urls></record></Cite></EndNote>13 (1948) and von Weizacker ADDIN EN.CITE <EndNote><Cite><Author>von Weizacker</Author><Year>1948</Year><RecNum>1247</RecNum><DisplayText><style face="superscript">14</style></DisplayText><record><rec-number>1247</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1290722113">1247</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">von Weizacker, C. F.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Das spektrum der turbulenz bei grossen Reynolds&apos;schen zahlen</style></title><secondary-title><style face="normal" font="default" size="12">Z. Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">614</style></pages><volume><style face="normal" font="default" size="12">124</style></volume><dates><year><style face="normal" font="default" size="12">1948</style></year></dates><urls></urls></record></Cite></EndNote>14 (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) ADDIN EN.CITE <EndNote><Cite><Author>Corrsin</Author><Year>1951</Year><RecNum>121</RecNum><DisplayText><style face="superscript">15</style></DisplayText><record><rec-number>121</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">121</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>S. Corrsin</author></authors></contributors><titles><title>On the spectrum of Isotropic Temperature Fluctuations in an isotropic Turbulence</title><secondary-title>Journal of Applied Physics</secondary-title></titles><pages>469-473</pages><volume>22</volume><dates><year>1951</year></dates><urls></urls></record></Cite></EndNote>15-Obhukov (1949) ADDIN EN.CITE <EndNote><Cite><Author>Obukhov</Author><Year>1949</Year><RecNum>487</RecNum><DisplayText><style face="superscript">16</style></DisplayText><record><rec-number>487</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">487</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>A. Obukhov</author></authors></contributors><titles><title>Structure of the temperature field in a turbulent flow</title><secondary-title>Izv. Akad. Nauk. SSSR. Ser. Geogr. I Geofiz</secondary-title></titles><pages>55-69</pages><volume>13</volume><dates><year>1949</year></dates><urls></urls></record></Cite></EndNote>16 law”:(Temperature fluctuations)= (Turbulent fluxes) x(Scale) 1/3and again with the Bolgiano (1959) ADDIN EN.CITE <EndNote><Cite><Author>Bolgiano</Author><Year>1959</Year><RecNum>841</RecNum><DisplayText><style face="superscript">17</style></DisplayText><record><rec-number>841</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">841</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Bolgiano, R.</author></authors></contributors><titles><title>Turbulent spectra in a stably stratified atmosphere</title><secondary-title>J. Geophys. Res.</secondary-title></titles><periodical><full-title>J. Geophys. Res.</full-title></periodical><pages>2226</pages><volume>64</volume><dates><year>1959</year></dates><urls></urls></record></Cite></EndNote>17 – Obhukov (1959) ADDIN EN.CITE <EndNote><Cite><Author>Obukhov</Author><Year>1959</Year><RecNum>842</RecNum><DisplayText><style face="superscript">18</style></DisplayText><record><rec-number>842</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">842</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Obukhov, A.</author></authors></contributors><titles><title>Effect of archimedean forces on the structure of the temperature field in a turbulent flow</title><secondary-title>Dokl. Akad. Nauk SSSR</secondary-title></titles><pages>1246</pages><volume>125</volume><dates><year>1959</year></dates><urls></urls></record></Cite></EndNote>18 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 book ADDIN EN.CITE <EndNote><Cite><Author>Batchelor</Author><Year>1953</Year><RecNum>45</RecNum><DisplayText><style face="superscript">19</style></DisplayText><record><rec-number>45</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">45</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Batchelor, G. K.</author></authors></contributors><titles><title>The theory of homogeneous turbulence</title></titles><dates><year>1953</year></dates><publisher>Cambridge University Press</publisher><urls></urls></record></Cite></EndNote>19 “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” 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 physics ADDIN EN.CITE <EndNote><Cite><Author>Monin</Author><Year>1972</Year><RecNum>471</RecNum><DisplayText><style face="superscript">20</style></DisplayText><record><rec-number>471</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">471</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Monin, A. S.</author></authors></contributors><titles><title>Weather forecasting as a problem in physics</title></titles><dates><year>1972</year></dates><pub-location>Boston Ma</pub-location><publisher>MIT press</publisher><urls></urls></record></Cite></EndNote>20”, 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 publication ADDIN EN.CITE <EndNote><Cite><Author>Monin</Author><Year>1975</Year><RecNum>472</RecNum><DisplayText><style face="superscript">21</style></DisplayText><record><rec-number>472</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">472</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Monin, A. S.</style></author><author><style face="normal" font="default" size="12">Yaglom, A. M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Statistical Fluid Mechanics</style></title></titles><dates><year><style face="normal" font="default" size="12">1975</style></year></dates><pub-location><style face="normal" font="default" size="12">Boston MA</style></pub-location><publisher><style face="normal" font="default" size="12">MIT press</style></publisher><urls></urls></record></Cite></EndNote>21, 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. 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 theories ADDIN EN.CITE <EndNote><Cite><Author>Dewan</Author><Year>1997</Year><RecNum>863</RecNum><DisplayText><style face="superscript">22</style></DisplayText><record><rec-number>863</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">863</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Dewan, E.</author></authors></contributors><titles><title>Saturated-cascade similtude theory of gravity wave sepctra</title><secondary-title>J. Geophys. Res. </secondary-title></titles><periodical><full-title>J. Geophys. Res.</full-title></periodical><pages>29799-29817</pages><volume>102</volume><dates><year>1997</year></dates><urls></urls></record></Cite></EndNote>22 as well as by Charney’s quasi-geostrophic turbulence theory which is thus also seen to be quite unrealistic. The conclusion from the 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 existed. 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 from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2007</Year><RecNum>928</RecNum><DisplayText><style face="superscript">25</style></DisplayText><record><rec-number>928</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">928</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F.</style></author><author><style face="normal" font="default" size="12">Hovde, S. J.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Is isotropic turbulence relevant in the atmosphere?</style></title><secondary-title><style face="normal" font="default" size="12">Geophys. Res. Lett.</style></secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages><style face="normal" font="default" size="12">L14802</style></pages><volume><style face="normal" font="default" size="12">doi:10.1029/2007GL029359</style></volume><dates><year><style face="normal" font="default" size="12">2007</style></year></dates><urls></urls></record></Cite></EndNote>25. ***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 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 turbulence. 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 1957 ADDIN EN.CITE <EndNote><Cite><Author>Panofsky</Author><Year>1955</Year><RecNum>754</RecNum><DisplayText><style face="superscript">27</style></DisplayText><record><rec-number>754</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">754</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Panofsky, H. A.</author><author>Van der Hoven, I.</author></authors></contributors><titles><title>Spectra and cross-spectra of velocity components in the mesometeorlogical range</title><secondary-title>Quarterly J. of the Royal Meteorol. Soc.</secondary-title></titles><pages>603-606</pages><volume>81</volume><dates><year>1955</year></dates><urls></urls></record></Cite></EndNote>27, ADDIN EN.CITE <EndNote><Cite><Author>Van der Hoven</Author><Year>1957</Year><RecNum>693</RecNum><DisplayText><style face="superscript">28</style></DisplayText><record><rec-number>693</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">693</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Van der Hoven, I.</author></authors></contributors><titles><title>Power spectrum of horizontal wind speed in the frequency range from 0.0007 to 900 cycles per hour</title><secondary-title>Journal of Meteorology</secondary-title></titles><pages>160-164</pages><volume>14</volume><dates><year>1957</year></dates><urls></urls></record></Cite></EndNote>28. 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 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 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 in ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2013</Year><RecNum>1219</RecNum><DisplayText><style face="superscript">29</style></DisplayText><record><rec-number>1219</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1287684944">1219</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Weather and Climate: Emergent Laws and Multifractal Cascades</style></title></titles><pages><style face="normal" font="default" size="12">496</style></pages><dates><year><style face="normal" font="default" size="12">2013</style></year></dates><pub-location><style face="normal" font="default" size="12">Cambridge</style></pub-location><publisher><style face="normal" font="default" size="12">Cambridge University Press</style></publisher><urls></urls></record></Cite></EndNote>29. 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 criticized ADDIN EN.CITE <EndNote><Cite><Author>Robinson</Author><Year>1967</Year><RecNum>1186</RecNum><DisplayText><style face="superscript">30</style></DisplayText><record><rec-number>1186</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1282327361">1186</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Robinson, G.D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Some current projects for global meteorological observation and experiment, </style></title><secondary-title><style face="normal" font="default" size="12">Quart. J. Roy. Meteor. Soc., </style></secondary-title></titles><pages><style face="normal" font="default" size="12">409–418</style></pages><volume><style face="normal" font="default" size="12">93</style></volume><dates><year><style face="normal" font="default" size="12">1967</style></year></dates><urls></urls></record></Cite></EndNote>30, ADDIN EN.CITE <EndNote><Cite><Author>Vinnichenko</Author><Year>1969</Year><RecNum>699</RecNum><DisplayText><style face="superscript">31,32</style></DisplayText><record><rec-number>699</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">699</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>N. K. Vinnichenko</author></authors></contributors><titles><title>The kinetic energy spectrum in the free atmosphere for 1 second to 5 years</title><secondary-title>Tellus</secondary-title></titles><pages>158</pages><volume>22</volume><dates><year>1969</year></dates><urls></urls></record></Cite><Cite><Author>Goldman</Author><Year>1968</Year><RecNum>756</RecNum><record><rec-number>756</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">756</key></foreign-keys><ref-type name="Report">27</ref-type><contributors><authors><author>Goldman, J. L.</author></authors></contributors><titles><title>The power spectrum in the atmosphere below macroscale</title></titles><dates><year>1968</year></dates><pub-location>Houston Texas</pub-location><publisher>Institue of Desert Research, University of St. Thomas</publisher><urls></urls></record></Cite></EndNote>31,32 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 features. Practionners of the nascent field of mesoscale meteorology such as Atkinson ADDIN EN.CITE <EndNote><Cite><Author>Atkinson</Author><Year>1981</Year><RecNum>1757</RecNum><DisplayText><style face="superscript">33</style></DisplayText><record><rec-number>1757</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1490735458">1757</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Atkinson, B. W. </author></authors></contributors><titles><title>Meso-scale atmospheric circulations</title></titles><pages>469</pages><dates><year>1981</year></dates><pub-location>London</pub-location><publisher>Academic Press</publisher><urls></urls></record></Cite></EndNote>33 were sceptical of any supposedly barren “gap” that could relegate their entire field to a mere footnote. 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 1967 ADDIN EN.CITE <EndNote><Cite><Author>Fjortoft</Author><Year>1953</Year><RecNum>194</RecNum><DisplayText><style face="superscript">34</style></DisplayText><record><rec-number>194</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">194</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Fjortoft, R.</author></authors></contributors><titles><title>On the changes in the spectral distribution of kinetic energy in two dimensional, nondivergent flow</title><secondary-title>Tellus</secondary-title></titles><pages>168-176</pages><volume>7</volume><dates><year>1953</year></dates><urls></urls></record></Cite></EndNote>34, ADDIN EN.CITE <EndNote><Cite><Author>Kraichnan</Author><Year>1967</Year><RecNum>313</RecNum><DisplayText><style face="superscript">26</style></DisplayText><record><rec-number>313</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">313</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kraichnan, R. H.</author></authors></contributors><titles><title>Inertial ranges in two-dimensional turbulence</title><secondary-title>Physics of Fluids</secondary-title></titles><pages>1417-1423</pages><volume>10</volume><dates><year>1967</year></dates><urls></urls></record></Cite></EndNote>26, and in 1971 its extension to quasi-geostrophic turbulence by Jules Charney (1917-1981) ADDIN EN.CITE <EndNote><Cite><Author>Charney</Author><Year>1971</Year><RecNum>834</RecNum><DisplayText><style face="superscript">35</style></DisplayText><record><rec-number>834</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">834</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Charney, J.G.</author></authors></contributors><titles><title>Geostrophic Turbulence</title><secondary-title>J. Atmos. Sci</secondary-title></titles><pages>1087</pages><volume>28</volume><dates><year>1971</year></dates><urls></urls></record></Cite></EndNote>35. 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” ADDIN EN.CITE <EndNote><Cite><Author>Schertzer</Author><Year>1985</Year><RecNum>586</RecNum><DisplayText><style face="superscript">36</style></DisplayText><record><rec-number>586</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">586</key></foreign-keys><ref-type name="Book Section">5</ref-type><contributors><authors><author>Schertzer, D.</author><author>Lovejoy, S.</author></authors><secondary-authors><author>L. J. S. Bradbury et al.</author></secondary-authors></contributors><titles><title>The dimension and intermittency of atmospheric dynamics</title><secondary-title>Turbulent Shear Flow</secondary-title></titles><pages>7-33</pages><dates><year>1985</year></dates><publisher>Springer-Verlag</publisher><urls></urls></record></Cite></EndNote>36. After 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 regime, 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 theory.The first experiment devoted to testing the new 2D/3D model was the EOLE experiment (1974). It used the dispersion of 480 constant density balloons ADDIN EN.CITE <EndNote><Cite><Author>Morel</Author><Year>1974</Year><RecNum>1175</RecNum><DisplayText><style face="superscript">38</style></DisplayText><record><rec-number>1175</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1279899322">1175</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Morel, P.</style></author><author><style face="normal" font="default" size="12">Larchevêque, M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Relative dispersion of constant level balloons in the 200 mb general circulation</style></title><secondary-title><style face="normal" font="default" size="12">J. of the Atmos. Sci.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">2189-2196</style></pages><volume><style face="normal" font="default" size="12">31</style></volume><dates><year><style face="normal" font="default" size="12">1974</style></year></dates><urls></urls></record></Cite></EndNote>38 (at about 12km altitude 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 version. 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 Larchevesque ADDIN EN.CITE <EndNote><Cite><Author>Morel</Author><Year>1974</Year><RecNum>1175</RecNum><DisplayText><style face="superscript">38</style></DisplayText><record><rec-number>1175</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1279899322">1175</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Morel, P.</style></author><author><style face="normal" font="default" size="12">Larchevêque, M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Relative dispersion of constant level balloons in the 200 mb general circulation</style></title><secondary-title><style face="normal" font="default" size="12">J. of the Atmos. Sci.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">2189-2196</style></pages><volume><style face="normal" font="default" size="12">31</style></volume><dates><year><style face="normal" font="default" size="12">1974</style></year></dates><urls></urls></record></Cite></EndNote>38 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 noted. 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 al ADDIN EN.CITE <EndNote><Cite><Author>Lacorta</Author><Year>2004</Year><RecNum>1176</RecNum><DisplayText><style face="superscript">39</style></DisplayText><record><rec-number>1176</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1279899689">1176</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lacorta, G.</style></author><author><style face="normal" font="default" size="12">Aurell, E.</style></author><author><style face="normal" font="default" size="12">Legras, B.</style></author><author><style face="normal" font="default" size="12">Vulpiani, A.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Evidence for a k^-5/3 spectrum from the EOLE Lagrangian balloons in the lower stratosphere</style></title><secondary-title><style face="normal" font="default" size="12">J. of the Atmos. Sci.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">2936-2942</style></pages><volume><style face="normal" font="default" size="12">61</style></volume><dates><year><style face="normal" font="default" size="12">2004</style></year></dates><urls></urls></record></Cite></EndNote>39 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!. ***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 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”. 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 institutions 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 off, while others were downsized or refocused toward more practical matters. 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 tech, 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). The only justification for funding became the promise of improving GCM outputs. In the past, it had been possible to obtain 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. 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 data 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 theory. Scale and scaling seemed abstract and unnecessary.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 theoreticians and real world 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 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’s (EGS) Nonlinear Processes division (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 Geocomplexity, a dozen scientists published a kind of nonlinear manifesto entitled “Nonlinear Geophysics: Why we need it” PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Mb3Zlam95PC9BdXRob3I+PFllYXI+MjAwOTwvWWVhcj48

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ADDIN EN.CITE.DATA 40. 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 geophysics that evidence for wide range atmospheric scaling slowly accumulated, notably by the study of radar rain reflectivities ADDIN EN.CITE <EndNote><Cite><Author>Schertzer</Author><Year>1987</Year><RecNum>589</RecNum><DisplayText><style face="superscript">41</style></DisplayText><record><rec-number>589</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">589</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Schertzer, D.</author><author>Lovejoy, S.</author></authors></contributors><titles><title>Physical modeling and Analysis of Rain and Clouds by Anisotropic Scaling of Multiplicative Processes</title><secondary-title>Journal of Geophysical Research</secondary-title></titles><periodical><full-title>Journal of Geophysical Research</full-title></periodical><pages>9693-9714</pages><volume>92</volume><dates><year>1987</year></dates><urls></urls></record></Cite></EndNote>41,41, and satellite cloud radiancesPEVuZE5vdGU+PENpdGU+PEF1dGhvcj5HYWJyaWVsPC9BdXRob3I+PFllYXI+MTk4ODwvWWVhcj48

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ADDIN EN.CITE.DATA 42-47 (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 either. 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 from ADDIN EN.CITE <EndNote><Cite><Author>Tessier</Author><Year>1993</Year><RecNum>672</RecNum><DisplayText><style face="superscript">49</style></DisplayText><record><rec-number>672</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">672</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Tessier, Y.</author><author>Lovejoy, S.</author><author>Schertzer, D.</author></authors></contributors><titles><title>Universal Multifractals: theory and observations for rain and clouds</title><secondary-title>Journal of Applied Meteorology</secondary-title></titles><pages>223-250</pages><volume>32</volume><number>2</number><dates><year>1993</year></dates><urls></urls></record></Cite></EndNote>49.The next major advance appeared in the late 2000’s with the beginning of widespread availability 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 found ADDIN EN.CITE <EndNote><Cite><Author>Stolle</Author><Year>2009</Year><RecNum>1050</RecNum><DisplayText><style face="superscript">50</style></DisplayText><record><rec-number>1050</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1050</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Stolle, J.</style></author><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The stochastic cascade structure of deterministic numerical models of the atmosphere</style></title><secondary-title><style face="normal" font="default" size="12">Nonlin. Proc. in Geophys.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">1–15</style></pages><volume><style face="normal" font="default" size="12">16</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>50 (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 well! By 2008, it seemed that the only evidence that apparently contradicted the wide range scaling hypothesis, were aircraft wind spectra.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: ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2013</Year><RecNum>1219</RecNum><DisplayText><style face="superscript">29</style></DisplayText><record><rec-number>1219</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1287684944">1219</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Weather and Climate: Emergent Laws and Multifractal Cascades</style></title></titles><pages><style face="normal" font="default" size="12">496</style></pages><dates><year><style face="normal" font="default" size="12">2013</style></year></dates><pub-location><style face="normal" font="default" size="12">Cambridge</style></pub-location><publisher><style face="normal" font="default" size="12">Cambridge University Press</style></publisher><urls></urls></record></Cite></EndNote>29.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 Nastrom ADDIN EN.CITE <EndNote><Cite><Author>Gage</Author><Year>1986</Year><RecNum>869</RecNum><DisplayText><style face="superscript">51</style></DisplayText><record><rec-number>869</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">869</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Gage, K.S.</author><author>Nastrom, G.D.</author></authors></contributors><titles><title>Theoretical Interpretation of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft during GASP</title><secondary-title>J. of the Atmos. Sci.</secondary-title></titles><pages>729-740</pages><volume>43</volume><dates><year>1986</year></dates><urls></urls></record></Cite></EndNote>51 (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 from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1067</RecNum><DisplayText><style face="superscript">52</style></DisplayText><record><rec-number>1067</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1067</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. and Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">1-19</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>52.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 from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1245</RecNum><DisplayText><style face="superscript">53</style></DisplayText><record><rec-number>1245</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1290615077">1245</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. Phys. Discuss., </style></secondary-title></titles><pages><style face="normal" font="default" size="12">3871-3920</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>53.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 from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1067</RecNum><DisplayText><style face="superscript">52</style></DisplayText><record><rec-number>1067</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1067</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. and Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">1-19</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>52.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 from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1067</RecNum><DisplayText><style face="superscript">52</style></DisplayText><record><rec-number>1067</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1067</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. and Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">1-19</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>52.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 1983-1986PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5OYXN0cm9tPC9BdXRob3I+PFllYXI+MTk4MzwvWWVhcj48

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ADDIN EN.CITE.DATA 51,54-56. 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. 4.6a. The most glaring problem with the GASP results was that the apparent 2D-3D transition scale was typically at several hundred kilometers ADDIN EN.CITE <EndNote><Cite><Author>Lilly</Author><Year>1989</Year><RecNum>1077</RecNum><DisplayText><style face="superscript">57</style></DisplayText><record><rec-number>1077</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1232043726">1077</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lilly, D. K. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Two-dimensional turbulence generated by energy sources at two scales</style></title><secondary-title><style face="normal" font="default" size="12">J. Atmos. Sci. </style></secondary-title></titles><periodical><full-title>J. Atmos. Sci.</full-title></periodical><pages><style face="normal" font="default" size="12">2026–2030</style></pages><volume><style face="normal" font="default" size="12">46</style></volume><dates><year><style face="normal" font="default" size="12">1989</style></year></dates><urls></urls></record></Cite></EndNote>57, ADDIN EN.CITE <EndNote><Cite><Author>Bacmeister</Author><Year>1996</Year><RecNum>1065</RecNum><DisplayText><style face="superscript">58</style></DisplayText><record><rec-number>1065</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1065</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Bacmeister, J. T.</author><author>Eckermann, S. D.</author><author>Newman, P. A.</author><author>Lait, L.</author><author>Chan, K. R.</author><author>Loewenstein, M.</author><author>Proffitt, M. H.</author><author>Gary, B. L.</author></authors></contributors><titles><title>Stratospheric horizontal wavenumber spectra of winds, potnetial temperature, and atmospheric tracers observed by high-altitude aircraft</title><secondary-title>J. Geophy. Res.</secondary-title></titles><pages>9441-9470</pages><volume>101</volume><dates><year>1996</year></dates><urls></urls></record></Cite></EndNote>58, ADDIN EN.CITE <EndNote><Cite><Author>Gao</Author><Year>1998</Year><RecNum>1064</RecNum><DisplayText><style face="superscript">59</style></DisplayText><record><rec-number>1064</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1064</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Gao, X.</author><author>Meriwether, J. 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K. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Two-dimensional turbulence generated by energy sources at two scales</style></title><secondary-title><style face="normal" font="default" size="12">J. Atmos. Sci. </style></secondary-title></titles><periodical><full-title>J. Atmos. Sci.</full-title></periodical><pages><style face="normal" font="default" size="12">2026–2030</style></pages><volume><style face="normal" font="default" size="12">46</style></volume><dates><year><style face="normal" font="default" size="12">1989</style></year></dates><urls></urls></record></Cite></EndNote>57 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 theory ADDIN EN.CITE <EndNote><Cite><Author>Lindborg</Author><Year>2001</Year><RecNum>790</RecNum><DisplayText><style face="superscript">62,63</style></DisplayText><record><rec-number>790</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">790</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Lindborg, E.</author><author>Cho, J.</author></authors></contributors><titles><title>Horizontal velocity structure functions in the upper troposphere and lower stratosphere ii. Theoretical considerations</title><secondary-title>J. Geophys. Res.</secondary-title></titles><periodical><full-title>J. Geophys. 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Res.</full-title></periodical><pages>10223-10232</pages><volume> 106</volume><dates><year>2001</year></dates><urls></urls></record></Cite></EndNote>62,63.In order to get to the bottom of this, with the help of Daniel Schertzer and Adrian Tuck, we reanalysed ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1067</RecNum><DisplayText><style face="superscript">52</style></DisplayText><record><rec-number>1067</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1067</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. and Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">1-19</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>52 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 turbulence ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2009</Year><RecNum>1067</RecNum><DisplayText><style face="superscript">52</style></DisplayText><record><rec-number>1067</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">1067</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Hovde, S. J. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Reinterpreting aircraft measurements in anisotropic scaling turbulence</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. and Phys. </style></secondary-title></titles><pages><style face="normal" font="default" size="12">1-19</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>52” 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 measurements and the new theory easily 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 ADDIN EN.CITE <EndNote><Cite><Author>Pinel</Author><Year>2012</Year><RecNum>1410</RecNum><DisplayText><style face="superscript">64</style></DisplayText><record><rec-number>1410</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1330306742">1410</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Pinel, J., </style></author><author><style face="normal" font="default" size="12">Lovejoy, S. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12"> Joint horizontal - vertical anisotropic scaling, isobaric and isoheight wind statistics from aircraft data</style></title><secondary-title><style face="normal" font="default" size="12">Geophys. Res. Lett.</style></secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages><style face="normal" font="default" size="12">L11803</style></pages><volume><style face="normal" font="default" size="12">39</style></volume><dates><year><style face="normal" font="default" size="12">2012</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">10.1029/2012GL051698</style></electronic-resource-num></record></Cite></EndNote>64.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 Sharman ADDIN EN.CITE <EndNote><Cite><Author>Frehlich</Author><Year>2010</Year><RecNum>1171</RecNum><DisplayText><style face="superscript">65</style></DisplayText><record><rec-number>1171</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1278008196">1171</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Frehlich, R. G. </style></author><author><style face="normal" font="default" size="12">Sharman, R. D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Equivalence of velocity statistics at constant pressure or constant Altitude </style></title><secondary-title><style face="normal" font="default" size="12">GRL </style></secondary-title></titles><pages><style face="normal" font="default" size="12">L08801, doi:10.1029/2010GL042912, 2010</style></pages><volume><style face="normal" font="default" size="12">37</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls></record></Cite></EndNote>65 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 debate ADDIN EN.CITE <EndNote><Cite><Author>Lindborg</Author><Year>2009</Year><RecNum>1117</RecNum><DisplayText><style face="superscript">66</style></DisplayText><record><rec-number>1117</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1256926158">1117</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lindborg, E.</style></author><author><style face="normal" font="default" size="12">Tung, K.K.</style></author><author><style face="normal" font="default" size="12">Nastrom, G.D.</style></author><author><style face="normal" font="default" size="12">Cho, J.Y.N.</style></author><author><style face="normal" font="default" size="12">Gage, K.S.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Comment on &quot;Reinterpreting aircraft measurments in anisotropic scaling turbulence&quot; by Lovejoy et al 2009</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. Phys. Discuss.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">22331-22336</style></pages><volume><style face="normal" font="default" size="12">9</style></volume><dates><year><style face="normal" font="default" size="12">2009</style></year></dates><urls></urls></record></Cite></EndNote>66, ADDIN EN.CITE <EndNote><Cite><Author>Frehlich</Author><Year>2010</Year><RecNum>1171</RecNum><DisplayText><style face="superscript">65</style></DisplayText><record><rec-number>1171</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1278008196">1171</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Frehlich, R. G. </style></author><author><style face="normal" font="default" size="12">Sharman, R. D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Equivalence of velocity statistics at constant pressure or constant Altitude </style></title><secondary-title><style face="normal" font="default" size="12">GRL </style></secondary-title></titles><pages><style face="normal" font="default" size="12">L08801, doi:10.1029/2010GL042912, 2010</style></pages><volume><style face="normal" font="default" size="12">37</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls></record></Cite></EndNote>65, it was necessary to determine the joint (horizontal- vertical) velocity structure function. 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 study ADDIN EN.CITE <EndNote><Cite><Author>Pinel</Author><Year>2012</Year><RecNum>1410</RecNum><DisplayText><style face="superscript">64</style></DisplayText><record><rec-number>1410</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1330306742">1410</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Pinel, J., </style></author><author><style face="normal" font="default" size="12">Lovejoy, S. </style></author><author><style face="normal" font="default" size="12">Schertzer, D. </style></author><author><style face="normal" font="default" size="12">Tuck, A. F. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12"> Joint horizontal - vertical anisotropic scaling, isobaric and isoheight wind statistics from aircraft data</style></title><secondary-title><style face="normal" font="default" size="12">Geophys. Res. Lett.</style></secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages><style face="normal" font="default" size="12">L11803</style></pages><volume><style face="normal" font="default" size="12">39</style></volume><dates><year><style face="normal" font="default" size="12">2012</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">10.1029/2012GL051698</style></electronic-resource-num></record></Cite></EndNote>64 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. ***We have summarized a series of attempts to shoehorn the atmosphere into a series of theoretically inspired scalebound frameworks, followed by long struggles against them to exonerate Richardson’s wide range scaling. First, there was the meso-scale gap. Had it existed, it would have allowed the small scales to be explained by isotropic three dimensional turbulence theory, it would have justified Van der Hoven’s “convenient” division of the atmosphere into large scale weather and small scale turbulence. Then there came the 3D/2D dimensional transition theory that is still with us today. Although the gap had never enjoyed much empirical support, it was marginalized not so much because it contradicted the data, but rather because of the theoretical convenience of isotropic theory. Although initial evidence supporting the theory was eagerly embraced (EOLE, GASP), when decades later more careful analyses eventually showed that the data supported symmetries of scale rather than direction, isotropy had become far too entrenched to drop. Instead of considering the (anisotropic) scaling of the equations themselves or at least the scaling of model-data hybrids - the reanalyses (fig. ?) - attention shifted to the numerical models: the GCMs were called upon for support. The idea was to show that the GCMs reproduced “realistic” 2D-3D transitions (detected in spectra as transitions from k-3 to k-5/3 behaviour). Indeed, the failure of GCMs to reproduce such transitions is taken as a sign of model failure, rather than of model success ADDIN EN.CITE <EndNote><Cite><Author>Berner</Author><Year>2009</Year><RecNum>1764</RecNum><DisplayText><style face="superscript">68</style></DisplayText><record><rec-number>1764</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1494603660">1764</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Berner, J. </author><author>Shutts, G. J. </author><author>Leutbecher, M. </author><author>Palmer, T. N. </author></authors></contributors><titles><title>A Spectral Stochastic Kinetic Energy Backscatter Scheme And Its Impact On Flow-Dependent Predictability In The ECMWF Ensemble Prediction System</title><secondary-title>J. Atmos. Sci., </secondary-title></titles><periodical><full-title>J. Atmos. Sci.,</full-title></periodical><pages>603-626</pages><volume>66</volume><dates><year>2009</year></dates><urls></urls><electronic-resource-num>10.1175/2008JAS2677.1</electronic-resource-num></record></Cite></EndNote>68! Once again, the trouble is that most numerical weather models are scaling and do not display the transition ADDIN EN.CITE <EndNote><Cite><Author>Palmer</Author><Year>2001</Year><RecNum>1112</RecNum><DisplayText><style face="superscript">69</style></DisplayText><record><rec-number>1112</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1255446698">1112</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Palmer, T. N.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">A nonlinear dynamical perspective on model error: a proposal for non-local stochastic-dynamic paramterisation in weather and climate prediction models</style></title><secondary-title><style face="normal" font="default" size="12">Quart. J. Roy. Meteor. 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The likely explanation is that the models generally use an approximation called the “hydrostatic approximation”, so that their intrinsic horizontal direction is in actual fact an isobar, not an isoheight, for example the popular ADDIN EN.CITE <EndNote><Cite><Author>Skamarock</Author><Year>2004</Year><RecNum>1110</RecNum><DisplayText><style face="superscript">70</style></DisplayText><record><rec-number>1110</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1253801033">1110</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Skamarock, W.C.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Evaluating Mesoscale NWP models using kinetic energy spectra</style></title><secondary-title><style face="normal" font="default" size="12">Mon. Weath. 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O.</style></author><author><style face="normal" font="default" size="12">Ohfuchi, W.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Mesoscale spectrum of atmopsheric motions investigated in a very fine resolution global general ciruculation model</style></title><secondary-title><style face="normal" font="default" size="12">J. Geophys. Res.</style></secondary-title></titles><periodical><full-title>J. Geophys. Res.</full-title></periodical><pages><style face="normal" font="default" size="12">D18110, doi:10.1029/2008JD009785</style></pages><volume><style face="normal" font="default" size="12">113</style></volume><dates><year><style face="normal" font="default" size="12">2008</style></year></dates><urls></urls></record></Cite></EndNote>72 or alternatively - in at least one prominent case of a quasi-geostrophic simulation ADDIN EN.CITE <EndNote><Cite><Author>Tung</Author><Year>2003</Year><RecNum>1261</RecNum><DisplayText><style face="superscript">73</style></DisplayText><record><rec-number>1261</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1291651265">1261</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Tung, K. K. </style></author><author><style face="normal" font="default" size="12">Orlando, W. W.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The k^-3 and k^-5/3 energy spectrum of atmospheric turbulence: Quasigeostrophic two-level model simulation</style></title><secondary-title><style face="normal" font="default" size="12">J. Atmos. Sci.</style></secondary-title></titles><periodical><full-title>J. Atmos. Sci.</full-title></periodical><pages><style face="normal" font="default" size="12">824–835</style></pages><volume><style face="normal" font="default" size="12">60</style></volume><dates><year><style face="normal" font="default" size="12">2003</style></year></dates><urls></urls></record></Cite></EndNote>73 - it has been shown ADDIN EN.CITE <EndNote><Cite><Author>Smith</Author><Year>2004</Year><RecNum>1143</RecNum><DisplayText><style face="superscript">74</style></DisplayText><record><rec-number>1143</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1264108824">1143</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Smith, K. S. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Comment on: &quot;The k-3 and k-5/3 energy spectrum of atmospheric&#xD;turbulence: Quasigeostrophic two-level model simulation&quot;</style></title><secondary-title><style face="normal" font="default" size="12">J. Atmos. Sci</style></secondary-title></titles><pages><style face="normal" font="default" size="12">937-941</style></pages><volume><style face="normal" font="default" size="12">61</style></volume><dates><year><style face="normal" font="default" size="12">2004</style></year></dates><urls></urls></record></Cite></EndNote>74 that the carefully crafted high wavenumber k-5/3 regime is spurious. Sixty years after the meso-scale gap, forty years after the EOLE experiment and thirty years after GASP, Richardson’s wide range scaling has finally been empirically vindicated. The last obstacle preventing closure of the scaling versus scalebound debate was a theoretical demonstration that the observed anisotropic scaling is compatible with the equations. As discussed in ch. 3, this was indeed done at nearly at the same time ADDIN EN.CITE <EndNote><Cite><Author>Schertzer</Author><Year>2012</Year><RecNum>1407</RecNum><DisplayText><style face="superscript">75</style></DisplayText><record><rec-number>1407</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1328132018">1407</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Schertzer, D.</style></author><author><style face="normal" font="default" size="12">Tchiguirinskaia, I.</style></author><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Chem. Phys.</style></secondary-title></titles><periodical><full-title>Atmos. Chem. Phys.</full-title></periodical><pages><style face="normal" font="default" size="12">327-336</style></pages><volume><style face="normal" font="default" size="12">12</style></volume><dates><year><style face="normal" font="default" size="12">2012</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">10.5194/acp-12-327-2012</style></electronic-resource-num></record></Cite></EndNote>75 so that the theoretical debate between the 3D/2D isotropic model and the anisotropic 23/9D scaling alternative was finally nearing closure. Fig. 4.9: 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 ADDIN EN.CITE <EndNote><Cite><Author>Skamarock</Author><Year>2004</Year><RecNum>1110</RecNum><DisplayText><style face="superscript">70</style></DisplayText><record><rec-number>1110</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1253801033">1110</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Skamarock, W.C.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Evaluating Mesoscale NWP models using kinetic energy spectra</style></title><secondary-title><style face="normal" font="default" size="12">Mon. Weath. Rev.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">3020</style></pages><volume><style face="normal" font="default" size="12">132</style></volume><dates><year><style face="normal" font="default" size="12">2004</style></year></dates><urls></urls></record></Cite></EndNote>70. 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 down pointing arrows, upper left). Except for the extremes, the spectra again follow the isobaric predictions k-2.4 (red) very well over most of the range. Reproduced from ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2010</Year><RecNum>1105</RecNum><DisplayText><style face="superscript">76</style></DisplayText><record><rec-number>1105</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1242154116">1105</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Tuck, A. F.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Horizontal cascade structure of atmospheric fields determined from aircraft data</style></title><secondary-title><style face="normal" font="default" size="12">J. Geophys. Res.</style></secondary-title></titles><periodical><full-title>J. Geophys. Res.</full-title></periodical><pages><style face="normal" font="default" size="12">D13105</style></pages><volume><style face="normal" font="default" size="12">115</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">doi:10.1029/2009JD013353</style></electronic-resource-num></record></Cite></EndNote>76.***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” ADDIN EN.CITE <EndNote><Cite><Author>Kolesnikov</Author><Year>1965</Year><RecNum>302</RecNum><DisplayText><style face="superscript">77</style></DisplayText><record><rec-number>302</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="0">302</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>V. N. Kolesnikov</author><author>A. S. Monin</author></authors></contributors><titles><title>Spectra of meteorological field fluctuations</title><secondary-title>Izvestiya, Atmospheric and Oceanic Physics</secondary-title></titles><pages>653-669</pages><volume>1</volume><dates><year>1965</year></dates><urls></urls></record></Cite></EndNote>77, it was eventually theorized as due to the unstable nature of atmospheric layers. 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. There is an irony here. Ironically a factor that contributed to its 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 in the 23/9D picture, 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) 2/3This 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 ( ADDIN EN.CITE <EndNote><Cite><Author>Pinel</Author><Year>2014</Year><RecNum>1462</RecNum><DisplayText><style face="superscript">79</style></DisplayText><record><rec-number>1462</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1345485338">1462</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Pinel, J.</author><author>Lovejoy, S.</author><author>Schertzer, D.</author></authors></contributors><titles><title>The horizontal space-time scaling and cascade structure of the atmosphere and satellite radiances</title><secondary-title>Atmos. Resear.</secondary-title></titles><periodical><full-title>Atmos. Resear.</full-title></periodical><pages>95–114</pages><volume>140–141</volume><dates><year>2014</year></dates><urls></urls><electronic-resource-num>10.1016/j.atmosres.2013.11.022</electronic-resource-num></record></Cite></EndNote>79, ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2013</Year><RecNum>1219</RecNum><DisplayText><style face="superscript">29</style></DisplayText><record><rec-number>1219</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1287684944">1219</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Weather and Climate: Emergent Laws and Multifractal Cascades</style></title></titles><pages><style face="normal" font="default" size="12">496</style></pages><dates><year><style face="normal" font="default" size="12">2013</style></year></dates><pub-location><style face="normal" font="default" size="12">Cambridge</style></pub-location><publisher><style face="normal" font="default" size="12">Cambridge University Press</style></publisher><urls></urls></record></Cite></EndNote>29, it explains the slope of the straight line in fig. 2.4). Rather than identifying a structure and following it, it is usually easier to 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 this Eulerian space-time relation - is linear, fig. 4.9 shows the relationship that this spectrum implies; an average wind of about 900 km/day (≈11m/s).Fig. 4.9: The Eulerian (fixed frame) space-time diagram obtained from the satellite pictures analyzed in fig. 4.5, lower left, reproduced from ADDIN EN.CITE <EndNote><Cite><Author>Pinel</Author><Year>2012</Year><RecNum>1085</RecNum><DisplayText><style face="superscript">81</style></DisplayText><record><rec-number>1085</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1233172533">1085</key></foreign-keys><ref-type name="Thesis">32</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Pinel, J.</style></author></authors><tertiary-authors><author><style face="normal" font="default" size="12">S. Lovejoy</style></author></tertiary-authors></contributors><titles><title><style face="normal" font="default" size="12">The space-time cascade structure of the atmosphere</style></title><secondary-title><style face="normal" font="default" size="12">Physics</style></secondary-title></titles><pages><style face="normal" font="default" size="12">(in preparation)</style></pages><volume><style face="normal" font="default" size="12">phD</style></volume><dates><year><style face="normal" font="default" size="12">2012</style></year></dates><pub-location><style face="normal" font="default" size="12">Montreal</style></pub-location><publisher><style face="normal" font="default" size="12">McGill</style></publisher><urls></urls></record></Cite></EndNote>81. 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 the energy rate density is a fundamental quantity and we should expect that it is directly linked to the main force driving the atmosphere: the sun. Think of the atmosphere as a giant machine – a “heat engine” - for converting solar energy into mechanical energy, wind. Almost all the solar energy is delivered to atmosphere by solar heating of the surface, and then - thanks to the turbulent dynamics (the weather!) – the average power is more or less uniformly redistributed over the atmosphere.The energy rate density turns out to be easy to estimate. Start with the average solar power delivered to the earth: about 240 Watts per square meter. Then, divide it by the total atmospheric mass - about 10 tons per square meter - yielding 0.024 W/Kg. This is the average amount of energy per unit time per unit mass that is received from the sun, almost all of it is at the “short” (visible) wavelengths and is delivered to the surface of the earth. From the surface, warm air rises distributing the energy throughout the troposphere, and north south winds distribute the energy from the equator to the higher latitudes. But most of this energy ends up being re-emitted to outer space at “long” (infra red) wavelengths with only a small fraction doing work, i.e. being transformed into wind - and this is the part that we seek. This fraction is the thermodynamic efficiency of the atmospheric heat engine.To get an estimate of the efficiency, we can follow Pauluis ADDIN EN.CITE <EndNote><Cite><Author>Pauluis</Author><Year>2011</Year><RecNum>1766</RecNum><DisplayText><style face="superscript">82</style></DisplayText><record><rec-number>1766</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1494670946">1766</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Pauluis, O.</author></authors></contributors><titles><title>Water Vapor And Mechanical Work: A Comparison Of Carnot And Steam Cycles</title><secondary-title>J. Atmos. Sci.</secondary-title></titles><periodical><full-title>J. Atmos. Sci.</full-title></periodical><pages>91-102 </pages><volume>68</volume><dates><year>2011</year></dates><urls></urls><electronic-resource-num>10.1175/2010JAS3530.1</electronic-resource-num></record></Cite></EndNote>82 who modelled the atmosphere as a heat engine operating between 12 and 27 oC (285 to 300 K). The theoretical efficiency of a (maximally efficient) Carnot cycle operating between these temperatures is 5% (=(300-285)/300) but Pauluis also considers a “steam cycle” that treats water vapour more realistically and whose efficiency varies over the range 1 - 5% (low to high humidity). Using the intermediate value 4% yields an estimate of 0.024x0.04≈0.001 W/Kg. This is essentially the same as the direct estimate of the energy rate density based on global wind data ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2010</Year><RecNum>1094</RecNum><DisplayText><style face="superscript">84</style></DisplayText><record><rec-number>1094</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1235165959">1094</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S., </style></author><author><style face="normal" font="default" size="12">Schertzer,D. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Towards a new synthesis for atmospheric dynamics: space-time cascades</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Res.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">pp. 1-52</style></pages><volume><style face="normal" font="default" size="12">96</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">doi: 10.1016/j.atmosres.2010.01.004</style></electronic-resource-num></record></Cite></EndNote>84. Although the amount was somewhat variable with latitude and altitude, the global average value was indeed close to 0.001 W/Kg.The thing about the thermodynamic analysis is that it only depends on the succession of thermodynamic states of a “packet” of air, it doesn’t require assumptions about the detailed dynamics. Since the energy flux density is the basic horizontal scale invariant quantity, this analysis reveals the physical mechanism behind the thermodynamic cycle: turbulent cascades. If we analyse the data at any scale (above the millimetric dissipation scale), we expect to get – on average - the same value. Alternatively, by using turbulence theory to estimate the energy flux density from wind gradients, we were able to estimate the thermodynamic efficiency. ***Now that we have obtained an estimate of 0.001 Watts per Kilogram, we can use the formula: (lifetime) = (energy rate density)-1/3 (scale)2/3 to estimate the lifetime of the largest structures which is also our estimate of the weather-macroweather transition time scale. Using the largest great circle distance (20,000km), we obtain a lifetime of ≈ 5- 10 days. Fig. 4.10 shows that this simple theory even explains the latitudinal variations: since 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 limit.If we go beyond this, then we are effectively 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 the system’s dimension is effectively reduced from (time+space) to (time). Both turbulent cascade models and GCMs control runs (i.e. with constant external forcings) reproduce the transition and also produce realistic low frequency variability ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2013</Year><RecNum>1219</RecNum><DisplayText><style face="superscript">29</style></DisplayText><record><rec-number>1219</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1287684944">1219</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Weather and Climate: Emergent Laws and Multifractal Cascades</style></title></titles><pages><style face="normal" font="default" size="12">496</style></pages><dates><year><style face="normal" font="default" size="12">2013</style></year></dates><pub-location><style face="normal" font="default" size="12">Cambridge</style></pub-location><publisher><style face="normal" font="default" size="12">Cambridge University Press</style></publisher><urls></urls></record></Cite></EndNote>29. The fact that these weather models reproduce this justifies the term “macroweather”. Surprisingly, the energy rate density based explanation of this basic feature of the atmosphere was not discovered until 2010 ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2010</Year><RecNum>1094</RecNum><DisplayText><style face="superscript">84</style></DisplayText><record><rec-number>1094</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1235165959">1094</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S., </style></author><author><style face="normal" font="default" size="12">Schertzer,D. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Towards a new synthesis for atmospheric dynamics: space-time cascades</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Res.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">pp. 1-52</style></pages><volume><style face="normal" font="default" size="12">96</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">doi: 10.1016/j.atmosres.2010.01.004</style></electronic-resource-num></record></Cite></EndNote>84, 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. To test this, we used data from thousands of ocean “drifters” and empirically estimated the near surface energy rate density. We found that it was about one hundred thousand times smaller than in the atmosphere: 10 billionths of a Watt per square meter. This value yields an ocean weather- ocean macroweather transition time of about 1- 2 years which is also observed, see fig. 4.11 ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2010</Year><RecNum>1094</RecNum><DisplayText><style face="superscript">84</style></DisplayText><record><rec-number>1094</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1235165959">1094</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S., </style></author><author><style face="normal" font="default" size="12">Schertzer,D. </style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">Towards a new synthesis for atmospheric dynamics: space-time cascades</style></title><secondary-title><style face="normal" font="default" size="12">Atmos. Res.</style></secondary-title></titles><pages><style face="normal" font="default" size="12">pp. 1-52</style></pages><volume><style face="normal" font="default" size="12">96</style></volume><dates><year><style face="normal" font="default" size="12">2010</style></year></dates><urls></urls><electronic-resource-num><style face="normal" font="default" size="12">doi: 10.1016/j.atmosres.2010.01.004</style></electronic-resource-num></record></Cite></EndNote>84. While this estimate of the ocean surface energy rate density corresponds gives a good estimate of the observed ocean weather-ocean macroweather transition of the ocean surface temperature, the ocean is fundamentally different from the atmosphere inasmuch as the energy rate density is very far from being uniform. As we moive from the surface to lower depths, it decreases very rapidly: by factors of thousands or more in the 100 m and perhaps as much as a billion over the first kilometer or so. Since the transition time scale varies as the inverse cube root, this corresponds to deep currents with millennial scale lifetimes.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, PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5Db21wbzwvQXV0aG9yPjxZZWFyPjIwMTE8L1llYXI+PFJl

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ADDIN EN.CITE.DATA 85), 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 ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2013</Year><RecNum>1219</RecNum><DisplayText><style face="superscript">29</style></DisplayText><record><rec-number>1219</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1287684944">1219</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author><style face="normal" font="default" size="12">Lovejoy, S.</style></author><author><style face="normal" font="default" size="12">Schertzer, D.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="12">The Weather and Climate: Emergent Laws and Multifractal Cascades</style></title></titles><pages><style face="normal" font="default" size="12">496</style></pages><dates><year><style face="normal" font="default" size="12">2013</style></year></dates><pub-location><style face="normal" font="default" size="12">Cambridge</style></pub-location><publisher><style face="normal" font="default" size="12">Cambridge University Press</style></publisher><urls></urls></record></Cite></EndNote>29.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 ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2014</Year><RecNum>1563</RecNum><DisplayText><style face="superscript">86</style></DisplayText><record><rec-number>1563</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1406554008">1563</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Lovejoy, S.</author><author>Muller, J.P.</author><author>Boisvert, J.P.</author></authors></contributors><titles><title>On Mars too, expect macroweather</title><secondary-title>Geophys. Res. Lett.</secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages>7694-7700</pages><volume>41</volume><dates><year>2014</year></dates><urls></urls><electronic-resource-num>doi: 10.1002/2014GL061861</electronic-resource-num></record></Cite></EndNote>86.*** “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 dynamics of the two atmospheres 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 affect 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 value). The ratio yields an expected 0.04 Watts per Kilogram on Mars, 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 sols. 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) years. 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 macroweather ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2014</Year><RecNum>1563</RecNum><DisplayText><style face="superscript">86</style></DisplayText><record><rec-number>1563</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1406554008">1563</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Lovejoy, S.</author><author>Muller, J.P.</author><author>Boisvert, J.P.</author></authors></contributors><titles><title>On Mars too, expect macroweather</title><secondary-title>Geophys. Res. Lett.</secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages>7694-7700</pages><volume>41</volume><dates><year>2014</year></dates><urls></urls><electronic-resource-num>doi: 10.1002/2014GL061861</electronic-resource-num></record></Cite></EndNote>86.The paper was published in one of planet Earth’s leading Geoscience journals ADDIN EN.CITE <EndNote><Cite><Author>Lovejoy</Author><Year>2014</Year><RecNum>1563</RecNum><DisplayText><style face="superscript">86</style></DisplayText><record><rec-number>1563</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1406554008">1563</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Lovejoy, S.</author><author>Muller, J.P.</author><author>Boisvert, J.P.</author></authors></contributors><titles><title>On Mars too, expect macroweather</title><secondary-title>Geophys. Res. Lett.</secondary-title></titles><periodical><full-title>Geophys. Res. Lett.</full-title></periodical><pages>7694-7700</pages><volume>41</volume><dates><year>2014</year></dates><urls></urls><electronic-resource-num>doi: 10.1002/2014GL061861</electronic-resource-num></record></Cite></EndNote>86 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. 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! 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 ADDIN EN.CITE <EndNote><Cite><Author>Chen</Author><Year>2016</Year><RecNum>1733</RecNum><DisplayText><style face="superscript">87</style></DisplayText><record><rec-number>1733</rec-number><foreign-keys><key app="EN" db-id="rvavvvrwivsszmeezw95tf26fe0w95wfrstx" timestamp="1478805450">1733</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Chen, W., </author><author>Lovejoy, S. </author><author>Muller, J. P. </author></authors></contributors><titles><title>Mars’ atmosphere: the sister planet, our statistical twin</title><secondary-title>J. Geophys. Res. Atmos.</secondary-title></titles><periodical><full-title>J. Geophys. Res. Atmos.</full-title></periodical><volume>121</volume><dates><year>2016</year></dates><urls></urls><electronic-resource-num>10.1002/2016JD025211</electronic-resource-num></record></Cite></EndNote>87.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). References ADDIN EN.REFLIST 1Marshall, J. S. & Palmer, W. M. The distribution of raindrops with size. Journal of Meteorology 5, 165-166 (1948).2Lovejoy, S. & Schertzer, D. 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